Flatness, tangent systems and flat outputs Jaume Franch May 25, 1999 m UNIVERSITAT POLITÈCNICA £ CATALUNYA To Zael and Carlota /ß*V "' UWVERÄ1TAT POLITÈCNICA J ! CATÂ'-UMYA V Acknowledgements 1 During the years leading up to this thesis, I have had the good fortune to meet many people to whom I wish to express my gratitude. First of all, I would like to acknowledge my advisor, Enric Fossas. One day, believing that I could tackle a thesis, he introduced me to the world of control theory. He has maintained this belief throughout the years, even in the worse moments, when he encouraged me not to give up. Furthermore, he made it possible for me to attend conferences, where I was able to learn a lot. His continuing support, our infinite discussions on some topics, his patient revision of my work, have all been crucial in the realization of this thesis. I would like to thank all the members of the Department of Applied Mathematics and Telematics who have helped me at one time or another, especially those of the research group in differential geometry, dynamical systems and applications. I would like to single out Professor Miguel Carlos Muñoz, from whom I learned a great deal concerning differential geometry, especially in the subject he was lecturing on at that time at the Faculty of Mathematics and Statistics. I am also grateful to Xavier Gracia, who occasionally helped me with DMpjXproblems. I must not forget Antoni Palau, a student who in his final year made some Maple programs related to Chapter 4 under my supervision. I would also like to thank Jeff Palmer, my English teacher for four years (and I hope for many more), for his revision and correction of the english manuscript. I would also like to mention my parents. They provided me with a fine education, giving me the opportunity of studying mathematics, as well as ceaseless encouragement in my undergraduate studies and the preparation of this thesis. Last but not least, I wish to express my infinite gratitude to Zael, my wife. Without her support in good and bad moments, her patience when arriving home late because of work, or when she realized that often when she was talking to me I was thinking about a problem that had arised that day, and her unfailing love, this thesis would have not been possible. 'Partially supported by CICYT under Grant TAP94-0552-C03-02 and TAP97-0969-C03-01 Contents 1 Introduction 1.1 Linear systems 1.2 Static feedback linearization 1.3 Dynamic feedback linearization 1.4 Flatness in the differential algebraic setting 1.5 Contents and contributions 1 1 2 2 3 4 2 Linear systems 2.1 Linear control systems 2.2 Examples 2.3 Controllability and observability 2.4 Modes, poles and zeros 2.5 Examples 2.6 Final remarks 2.6.1 The matching condition in sliding control mode 2.6.2 The linear system interconnections 3 Linearization of nonlinear systems and 3.1 Different types of linearizations 3.2 Flatness and differential algebra 3.3 Linearization by prolongations 4 Linearization using differential algebra 4.1 Introduction 4.2 Single-input systems 4.3 Static feedback linearization of multi-input systems 4.4 Dynamic feedback linearization for multi-input systems 4.5 Software 4.6 Examples 4.7 An analogous procedure using field extensions vii 7 7 12 14 16 18 22 22 23 flatness 29 29 31 34 37 37 37 41 45 47 47 52 viii 5 Linearization by prolongations of 2-input systems 5.1 Prolongations of m inputs are not necessary 5.2 Main result 5.3 Examples 55 55 59 66 6 Improvement of the bounds for 3-input systems 6.1 Main Result 6.2 Where do the bounds 2r + 1 and In — 2 + r come from? 75 75 90 7 Linearization by prolongations of m-input systems 7.1 Main results 7.2 About the bounds 93 93 102 8 105 105 106 Conclusions and suggestions for further research 8.1 The differential algebraic approach 8.2 Linearization by prolongations and possible extensions A Introduction to differential algebra A.l Basics on differential algebra A.2 The Kahler differential 115 115 117 B Software package for Chapter 4 119 List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Elementary RLC-circuit worked in example 3 Mechanical system with springs and dampers corresponding to example 4 . . . . Not stabilizable example Reverse block diagram of 2.3 Ill-posed system Parallel interconnection Series interconnection Feedback interconnection Example illustrating possible changes of rank ix 20 21 23 24 24 25 25 26 27 LIST OF FIGURES Chapter 1 Introduction Feedback linearization of nonlinear control systems is a problem on which several scientists have been working during the last twenty years. Its importance lies in the fact that it enables us to transfer the properties of a linear system to a nonlinear one, as well as to propose a simple solution to one of the main problems of automática, which is stabilization around a given trajectory of the system. Feedback linearization is in general an open problem. There are solutions for specific cases, such as linearization by static feedback, the equivalence between static feedback and dynamic feedback linearization for single input systems, systems with m inputs and m + 1 state variables, some cases of systems without drift, For many years differential geometric tools have been used in order to solve the problem of feedback linearization. Notions such as Lie brackets and involutive fields or distributions are the most common tools in this context. But to solve partial differential equations is also needed. In the nineties, a new way of tackling the problem was proposed. This method, related to the notion of flatness, was introduced in the differential algebraic setting. This setting led to new concepts, and it has implied the introduction of new concepts for linear and nonlinear systems. It allows us to deal with a greater number of problems than the classic framework. Some years after the first works on the differential algebraic setting were done, two new versions of flatness appeared, one using the differential geometry of infinite jets, and the other in the exterior differential systems. 1.1 Linear systems In 1963, Kaiman [36] introduced a new method for describing linear control systems. In this first work the foundations for a good comprehension and a revision of the results known at that time can be found. Keywords introduced by Kaiman are state variables, controllability, observability, realization, minimal realization . . . . At the end of the sixties and the beginning of the seventies, the algebraic theory of linear control systems in an arbitrary field was developed . Roucheleau [55] and Roucheleau, Wyman and 1 2 CHAPTER 1. INTRODUCTION Kaiman [56] studied the realization problem over commutative rings. The situation in the non commutative case, treated in [60] by Sontag in 1976, is different because the Cayley-Hamilton fails to hold. Willem's papers [64] - [67] must be mentioned here, because the geometric concept of trajectories introduced therein plays a crucial role, and allows us to deal with many questions without distinguishing between inputs, outputs, states and other variables. Also important was the work of Brunovsky [5], who gave a classification of linear controllable systems. Since then, the researchers in this area have referred to the Brunovsky canonical form, both those who work in linear systems and those who try to linearize nonlinear control systems. At the beginning of the nineties, Michel Fliess suggested a new algebraic treatment for linear control systems ([10], [11], [15], [16], [20]). The cornerstone of his work resides in the fact that it enables us to put linear control theory in an algebraic setting which utilizes module theory in a more general manner than that commonly employed since Kaiman. According to Fliess, these papers sketch an attempt to rehabilitate Kalman's point of view in the new context of module theory. His work is based on a state variable representation, where the dynamics is strictly in the Kaiman form, but where the output map not only involves the state but also the control variables and their derivatives. This is the frame we wish to use in Chapter 2. 1.2 Static feedback linearization Since 1973, the problem of linearization of continuous nonlinear control systems has been extensively studied. Krener [41] found conditions for linearizing a system by means of state space diffeomorphisms. A particular type of state feedback transformation was first introduced by Brockett [4]. This was later generalized for single input systems by Su [61], who also related his results to the notion of relative degree. The problem for multi input systems was finally solved by Hunt, Su and Meyer [31] and Jakubczyk and Respondek [34]. Their works used mathematical tools such as Lie brackets and involutive distributions. In fact, they proved the equivalence between the static feedback linearization and the rank and involutivity of certain distributions. The Kronecker indices [51] were also a fundamental to this procedure. For non static feedback linearizable systems, some authors have considered partial linearizations ([42],[44]), as well as approximate feedbacks ([29], [32]). 1.3 Dynamic feedback linearization Partial feedback linearization is related to input-output decoupling. Necessary and sufficient conditions are available for this problem. For linear systems it is known that those conditions can be weakened if one allows for a dynamic compensator. This motivated the introduction of a nonlinear dynamic state feedback transformation, which is a generalization of the static state feedback transformation. In [6] and [7], the problem of dynamic feedback linearization was studied by Charlet, Levine and Marino. Approaching the problem from the differential geometric point of view, they showed that single input systems that are dynamically feedback 1.4. FLATNESS IN THE DIFFERENTIAL ALGEBRAIC SETTING 3 linearizable are also statically feedback linearizable, and two very special cases of dynamically feedback linearizable multi input systems are also given in [6]. In [7] they presented fairly general sufficient conditions for a system to be dynamic feedback linearizable by prolongations, as well as a necessary conditions. Unfortunately, as they also showed with examples, neither are the sufficient conditions necessary nor are the necessary conditions sufficient. Aranda-Bricaire, Moog and Pomet gave a different approach in [1] and [2]. They characterized the flat or linearizing outputs in their framework, the so-called infinitesimal Brunovsky form. Again, although their result establishes a sufficient condition for the existence of a dynamic feedback transformation that linearizes the system, this condition is not necessary in general. Sluis and Tilbury [59] gave an upper bound on the number of integrators needed to linearize a control system, but they proved only the sharpness of the bound for systems with two inputs. Their work was based on exterior differential systems. In the same framework, Rathinam and Sluis [53] obtained a test for dynamic feedback linearization by reduction to single input systems. 1.4 Flatness in the differential algebraic setting Differential algebra was established by Ritt [54], Kaplansky [39] and Kolchin [40]. What interests us most about this theory is the differential field extensions. Fliess was the first to introduce differential algebra into control theory for linear and nonlinear systems. One of the chief features of the utilization of differential algebra is the avoidance of explicit equations. This enables us to deal with a greater number of problems. Using the differential version of the theorem of the primitive element, Fliess proposed a generalized canonical form in [10]. This was followed by a series of papers as a result of his joint work with Levine, Martin and Rouchon. See, for instance, [13], [14], [17], [18], [19], [22], [45]. In these papers some concepts such as flatness and defect were introduced. One major property of flatness is the existence of what the authors called flat or linearizing outputs. The system is fiat, if and only if, without integrating any differential equation, the state and input variables can be directly expressed in terms of the flat outputs and a finite number of their derivatives. Flatness is best defined by not distinguishing between input, state, output and other variables. This standpoint matches Willems' approach in [64] well. He did not make distinctions among the different types of variables. Flatness might be seen as another nonlinear extension of Kalman's controllability. In fact, any flat nonlinear system is controllable. In addition, for linear systems, flatness is equivalent to controllability. A set of flat outputs is the nonlinear analogue of a basis of a free module. It must be emphasized that from trajectories of the flat outputs, trajectories for the states and the inputs are immediately deduced. The relationship between the nonlinear theory (using differential field extensions) and the linear theory, which utilizes modules, is given by what is called the Kahler differential [35]. This mathematical tool is used in this context to compute the associated tangent system to a nonlinear one. This tangent system is linear. Therefore, one strategy to obtain the flat outputs could be to compute the tangent system, and to find out an integrable basis of this tangent system. 4 CHAPTER 1. 1.5 INTRODUCTION Contents and contributions The aim of Chapter 2 is to present the state of the art on linear control systems within the framework of module algebraic theory. Fliess' papers are collected, although some proofs and examples are new. Among new proofs, we would like to emphasize the proof of proposition 1, which gives the equivalence between a linear control system in state-space representation and modules over a ring of differential operators. The proofs of section 2.4 are extensions of known proofs, including all the details required to make such proofs more clear. This chapter has been submitted as a survey to the journal Linear algebra and its applications. In Chapter 3 some background necessary for understanding the main results of this work is given. The different types of linearization are presented: namely, static feedback linearization, linearization by prolongations, dynamic feedback linearization, and flatness. Some known results in this field are stated, with appropriate references to locate the proofs. Chapter 4 deals with the problem of flatness in a nonlinear multi input (m inputs) system. In the framework of differential algebra, the tangent system is used in order to find out the mth fiat output when m — 1 flat outputs have been guessed. The quotient of modules is crucial in this procedure. The contributions in this Chapter include: 1. A new proof of the well known fact that linearization by static and dynamic feedback are equivalent for single-input systems. 2. A new algorithm to linearize single-input systems, as well as an algorithm to linearize multi-input systems by static feedback. 3. A theoretic procedure to linearize any multi-input systems, together with a software package to carry out the computations. Once the system is linearized, a condition to check whether or not the system can be linearized via prolongations is also derived. 4. An application of the procedure to a vertical take off and landing (VTOL) aircraft. Two new flat outputs have been obtained, and it is proven that these flat outputs can be found just by using prolongations. These results have been published in two conferences. In 1997 SAAEI [25], which refers to static feedback linearization, and in 1998 ACC [26], which is related to dynamic feedback linearization. Some parts of this Chapter, together with a part of Chapter 5, have been submitted to Automática. In Chapter 5, the problem of linearization by prolongations of systems with two inputs is studied. A bound on the number of integrators needed to linearize a control system is obtained, using the most elementary tools of differential geometry, such as Lie brackets and involutive distributions. An algorithm derived from this result is applied to some examples, some of which were thought until now to be not linearizable by prolongations. For instance, the VTOL and the planar ducted fan. A part of this Chapter will appear in 1999 SAAEI [28]. Chapter 7 generalizes the results of Chapter 5 to an arbitrary number of inputs, improving the existent bounds in the literature when the number of inputs is greater than or equal to four. It 1.5. CONTENTS AND CONTRIBUTIONS 5 also contains a new proof of the fact that, when linearization by prolongations are considered, not all the inputs must be prolonged. This Chapter has been submitted to Systems and Control Letters. In the case of three inputs, better results are given in Chapter 6. These results have appeared in 1999 IFAC [27]. This work ends with the conclusions and some suggestions for future research. 6 CHAPTER J. INTRODUCTION Chapter 2 Linear systems This Chapter is organized as follows: Section 2.1 is devoted to comparing two definitions of linear control systems in order to show their equivalence, and examples are given at the end. In Section 2.3, controllability and observability are presented in the module formalism. Modes, poles and zeros are treated in section 2.4. Some examples clarify the work. Finally, some applications to sliding control and linear systems interconnections are explained. 2.1 Linear control systems This section deals with two definitions of linear control systems, the classical one in statevariables and a new one using left modules ([10]). The equivalence between both definitions is shown and some examples are given. Definition 1 A linear control system in state-space representation is a system described by: X = A{t)X + B(t)U Y = C(t)X m where U = {u\,..., um) E R , A(t) € Mnxn, Y = (yi,...,yp)eRp B{t) £ MmXn, X = (x\,..., xn) S Rn and X are called state variables, U are input variables and Y are output variables. Definition 2 A linear control system using left modules is a left finitely-generated K[-^]-module A. ( K is supposed to be a field 1 ) Definition 3 A linear dynamic with input U — (u\, ...,u n ) is a linear control system A (that is to say a left finitely-generated K[-^]-module) which contains < U > and such that A/ < U > is a torsion module. An output Y = (yi, ...,y p ) is a finite set of elements of the system. X K — R or C for constant linear control systems, otherwise K is a field of meromorphic functions 7 8 CHAPTER 2. LINEAR SYSTEMS P r o p o s i t i o n 1 The above definitions are equivalent in the sense that if a system as in definition 1 is given, the left module of the definition 2 can be built, fulfilling the desired conditions. Conversely, if a system as in definition 2 is given, a realization as in 1 can be obtained. Proof. First we see that definition 1 implies definition 2. Let X = A(t)X + B{t)U (2.1) be a linear system as in definition 1. Consider the left ür[g¿]-module generated by X and U. That is to say M = K[^-]<X,U> On the other hand, let N be the submodule generated by the relations of 2.1. Consider A = M/N the quotient submodule. As M is finitely generated ( a finite number of X and U ), A will also be finitely generated. So, it remains to be shown that A/ < U > is torsion. Let z G M. Then, z = aiXi + ... + anxn + &iiii + ...bmum where ai,bi G K[^). Consider the natural projection over A, z = a\X\ + ... + anxn + b\Ui + ...bmum By construction, any element of A has this form. Making the quotient A/ < U > we get: z = a\X\ + ... + anxn If the torsion elements make up a sub module, it is only necessary to show that x~l is torsion Vi. In order to end the proof the following lemmas are stated (the proof will be performed later): L e m m a 1 Vx¿ 3P¿ G K{^\ such that PiXi G tf [4] < U > at This lemma states that Xi is torsion in A/ < U >. L e m m a 2 The torsion elements make up a submodule. Now, it must be proven that definition 3 implies definition 1. As will be seen in lemma 4, A — TOP, where T is a torsion submodule and F is a free submodule. Let be {x¿}"¿1 a set of generators of F and {ZJ}™^ a set of generators of T. Zj are torsion elements, so there exists Qj(j¡) such that Qi(4)*i=0 2.1. LINEAR CONTROL SYSTEMS 9 On the other hand, there exists a submodule U such that A/Î7 is torsion. That is to say: V z ¿ 3 P i ( | ) | Piij^xi EU An output y is an element of A. So For all ¿, j let the next integers be defined by: di = max{degree(Pi),degree(Ri) ej — max{degree(Qj),degree(Sj) + 1} + 1} Then, for any i we have a system of the form: •1 %i 2 3-j — di—1 fc=0 where a^ = a;¿ , the coeffiecients a¿ come from the equation Pj(^)x¿ G U or the (d¿ — degree(Pi))th derivative of this equation, if necessary, and u¿ = -F»(¡g) o r a l s 0 * n e (¿* ~~ degree{Pi))th if necessary. The same can be done for zy. ¿<? = z = j zl Z j ej-1 where Zj = zj ' and the coeffiecients bk come from the equation QJ{^¡)ZJ = O or the (ej — de#ree(Q¿))-th derivative of this equation, if necessary. With all these variables each output y can be written as a linear combination of x* and z!f (i = 1 , . . . , m , j = 1 , . . . , m i , k = 0 , . . . ,di — 1, /i = 0 , . . . ,ej — 1), which will be the state variables. So a system in the state-space form is obtained. Notice that the module generated by the new state-variables is A. And also ^ < ui,...,«ni > = A/C/ 10 CHAPTER 2. LINEAR SYSTEMS That is to say, the dynamic generated by the realization obtained is the dynamic A/U. Now we are going to prove the two lemmas previously stated. Proof of lemma 1: First case: A is a matrix with constant coeficients. Then, PA ( the characteristical polynomial of A ) accomplishes p 4?x'*Kù<u> Indeed, in A/ < U > the system is only X = AX The solution of this system is X(t) = eAtXo. If PA is applied to this system we obtain = PA(jt)eAtXo PA(^)X = PA(A)eAtXQ = 0 Where the Cayley-Hamilton theorem has been applied to the last equality. So, each component of X, labelled xt fulfills PA{Í¿)XÍ = °Since X(t) = eAtXo in the non constants coefficients case cannot be assured, this demonstration does not hold. The corresponding equation to x¿ must be derived, replacing the other variables by their corresponding equations. If this method is iterated until the nth derivative of X{, we obtain: / z i \ / 0 \ / l \ / 0 \ Xi \ *Sn) J Hi x\-\ H V<W + Xn h Xi -\ WW v<¿/ This is a (n+l)-vector which is a linear combination of n vectors. For this reason these vectors are linearly dependent. Hence their determinant is vanishing and this determinant yields a polynomial with indeterminate ^ such that when it is applied to x¿, it is zero. Obviously this can be done for any state-variable. Thus we can state that Vi 3Pi | PiXi = 0 in A/ < U >. In other words, x¿ is torsion in the quotient submodule. Example: Consider the system X = ' t i \ „ . ( os 1 t l X + \ 1 U • 2.1. LINEAR CONTROL SYSTEMS 11 Let A be < Xi,X2,U > < X\ — tX\ — X2, X2 — X\— tX2 — U > Following the above algorithm, torsion of x\ in A/ < U > will be proven. x\ fulfills the following equations (the same notation is used for x\ in A or in A/U): X\ = X\ X\ — tX\ + X2 Deriving this equation: X\ = x\ + tx\ + ¿2. And making the substitution for ¿1 and ¿2, it becomes (2.2) ¿1 = (£2 + 2)xi + 2tx2 The following system can be written So the following determinant is vanishing: Xl ¿1 xi t2 + 2 0 1 It That is ¿1 - 2t±i + (i 2 - 2)2:1 = 0 in A/U. So there exists a polynomial with inderminate ^ that voids xi in A/U. The same can be done for £2. Proof of lemma 2: The torsion elements must make up a submodule. Let T be the set of torsion elements. Two conditions must be proven: \/x,y eT=>x + y &T (1) VxeT,keK[~]^kxeT at (2) First of all a property of the ring of differential operators A = K[-^\ will be proven: Lemma 3 Va,òe A,a^0,b^0,3a',b' \0¿b'a = a'b 12 CHAPTER 2. LINEAR SYSTEMS Proof: Let be t—0 If there exist, j=0 n j m 7 "' = £<*(!)*. * = »(;!>' The equality o'6 = b'a must be verified; that is to say: (E4(|)')çw|)')-(E«i(5),)a:-(5)i) fc=0 j=0 ¡=0 t=0 Equaling term to term, a system with n+m + 1 homogeneous equations and n + m+2 unknowns is obtained. So it has a non trivial solution, and hence the existence of a' and b' fulfilling the required conditions can be deduced. H Once this fact has been proven, it is not difficult to prove lemma 2: As x G T, 3a G A \ ax = 0 Analogously, as y G T, 3b G A \ by = 0. Using the property just proved 3a', b' | a'b = b'a ^ 0. Then b'a(x + y) = b'ax + b'ay = b'ay = a'by = 0. S o i + y e T . On the other hand, a'kx = k'ax = 0 =ï kx G T. This fact finishes the proof of the lemma 2. • Now, another lemma will be stated and proven. This lemma will be useful in order to decompose the submodule A into a direct sum of a torsion submodule and a free submodule. L e m m a 4 A = T © F, where T is a torsion submodule and F is a free submodule ( that is to say, without torsion elements ) . Proof: Consider the canonical morphism: n : A —• A/r By definition this morphism is linear and exhaustive. It is clear that the kernel of this morphism is T. So it must must ensured that A/T is free. Let y ^ 0 be a torsion element of A/T. Such element is the image by the morphism II of an element y G A. As y is a torsion element 3pi G K[£\ | piy = 0 Thus p\y G T and, consequently, 3p2 S K[-^] | P2P\V = 0. This is the same as y is torsion or, in other words, y = 0, which contradicts our initial assumptions. Thus, there is no torsion element in A/T; that is to say A/T is a free module. F will be generated by the one element of each subset of inverse images of the generators of the free module A/T. • 2.2 Examples l. ¿ i = X2 ¿2 = U EXAMPLES 13 In this example, M = < Xi,X2,u >. N is generated by the above equations. So, < Xi,X2,U A= — = N < ±i — X2,X2~ > U> = < Xx > When the quotient A/ < u > is done, it can be seen that $pX\ = 0. Then an element of the ring of differential operators that cancels X\ is obtained. So A/ < u > is torsion because its only generator is torsion. ¿i = i i + u ¿2 = £3 ¿3 = u In this example, M =< xi,X2,x¡,u >. N is also generated by the above equations. So < Xi,X2,X¡,U > < £1,2:2 > < ¿1 — x\ +u,X2 — £3,2:3 — u > N <x\—x\+u> In A/ < u > the relations ( ^ — I)x\ = 0 and ^2:2 = 0 are satisfied. Thus, the quotient module is torsion again. X = AX + BU /0 1 0 1 0 0 1 0 1 0 A= \ (0 \ B = 0 1 0 1 0 ) 0 1 0 0 0 0 0 \o 0\ 0 0 0 0 1 0 0 0) Notice that ^2:3 = u\, $pX2 = u\ and ^3-2:1 = u\. The same happens to 2:4, 2:5 ,XQ and U2- Furthermore, 2:7, 2:8, X9 are torsion elements. Therefore, it can be written < X\, Xi, X7 > _ < X7 > < Xi,...,Xg,Ui,U2 > 3) 3) ' = < 2:1,2:4 > e < ¿1 — X2, ¿2 — 2:3, ¿3 — Ui, ... > <4 > <4 > So the system has a torsion submodule, generated by 2:7 and a free submodule generated by 2:1 and 2:4. Notwithstanding, A/ < u\,U2 > is a torsion submodule. 14 CHAPTER 2. LINEAR SYSTEMS 4. This is an example of a non-constant coefficient. The ring over which is defined the system is R{t)[-j¡i], where R(t) is the field of fractions of real polynomials. The system is rewritten in the following way: . _ < Xi,X2,U > < ¿ i — tX\ — X2, ¿2 ~ xl ~ tx2 — U > The relation xi = (— - tl)xi can be deduced from the first equation. And from the second equation u = (jt - tl)x2 - Ixx = ((jt)2 - 2tjt + (Í 2 - 2)I)X1 So it can be said that A=<zi > So, It is clear that, using the latest relations written, A/ < u > is a torsion module. 5. x = x + u (2) The derivatives of the input are not considered in the state-space representation. Let be y = x — u — ù. y is a generator of the module and the module is free. The equation in the variable y is: y=y+u 2.3 Controllability and observability In this section, simple characterizations of controllability and observability based on module theory techniques are given. See [10] again. The equivalence between control systems in statespace form and control systems in module theory will be used. T h e o r e m 1 A system A is controllable if and only if it is a free module. Proof. First assume that A is free. If the system is uncontrollable, as in [36], we have a Kaiman realization as follows: 2.3. CONTROLLABILITY AND OBSERVABILITY 15 where Ai is a square matrix and belongs to the uncontrollable part. Now lemmal can be applied. Therefore, the elements of X\ are torsion. So there exists a contradiction with the freeness of A. Now assume that the system is controllable. Let A = F © T be a descomposition in a direct sum of a free left module and a torsion left module. If A was not free, T ^ 0. Let be T =<xi,...,Xk > Thus Vx¿ 3Pj G K[^[ \ P{Xi = 0, because X{ are torsion elements. Thus a system with k equations of order nfc (polynomial degree) is obtained. This can be transformed into a system with order 1 equations where the state variables are xi¿, ...,xitJll, ...¡x^^, ...,Xk,nk This is an expression of the form: * i = AXXX Here there is a further contradiction because this is an uncontrollable Kaiman realization. The contradiction comes from the assumption of non-freeness of the module. • The above proof also shows an equivalence between the torsion submodule and the uncontrollable part of the Kaiman realization. It can also be seen that, if the system is controllable, each element of A is related, directly or indirectly, to the inputs: directly, if it can be expressed as a linear combination of the inputs; or indirectly, because each state-variable accomplishes a differential equation where there are inputs. Next the relation between observability and module theory is shown. In the classical theory, a system: X = AX + BU Y = CX is called observable if and only if rank < C\ {A1 + ^-)C\..., {A1 + ^-^C* at at where n is the dimension of the state-variable vector. >= n T h e o r e m 2 A system is observable if and only if A=<U,Y> That is to say, if and only if each variable of A can be written as a linear combination of inputs, outputs and their derivatives. Proof. Let Y = (yi,...,yp). There is no loss of generality in assuming that j / i , . . . , yp are linearly independent. First of all, we suppose that A = < U, Y > or, in other words, < U, Y >=< U, X >. This is also equivalent to <U,Y > _ <U,X> <U> ~ <U> CHAPTER 2. LINEAR 16 SYSTEMS So, the Kaiman realization is written in the quotient, where the variables are overlined: ~X = AX Y = CX Deriving k times: Consider the linear system obtained by gathering the former equation for A; = 0 , . . . , n — 1. Since we have assumed that j / i , . . . ,yp are linearly independent, the system has a unique solution if, and only if, rank{C\ {A1 + ~)C\..., {A* + ^T^C1) at at which is the classical observability condition. Therefore, rank(C\ (A' + j¿C\..., = n (A4 + | ) n - x C * ) = n if, and only if, X are written as a unique linear combination of Y and their derivatives; if, and only if, X are written as a unique linear combination of U, Y and their derivatives. Notice that, from this proof, an equivalence between the observability part of the Kaiman • realization and the submodule < U, Y > follows. 2.4 Modes, poles and zeros In this section we attempt to give an algebraic interpretation of the hidden modes, poles and zeros of the constant linear systems. It follows [11], although some proofs have been extended. Let us recall the Kaiman realizations in the uncontrollable and in the unobservable cases. In the uncontrollable case the Kaiman realization is: U)-(î5.)(sH^> where A\ is the uncontrollable part matrix. And in the unobservable case: (£)-(£ SOUKS)* -(f)(5) where A3 is the unobservable part matrix. 2.4. MODES, POLES AND ZEROS 17 Classically, the hidden modes were the eigenvalues of A\ (input-decoupling zeros ) and A$ (output-decoupling zeros). What is the interpretation of these matrix in the module theory? Let us begin with A\: As this is the uncontrollable part matrix, A is not free. So the module can be decomposed in a direct sum of its free part and its torsion part: A = F © T. Let us denote the linear mapping induced by the derivative ^ by T:T—>T This mapping is well defined because ^ is an element of the ring over which the module A has been defined, and T is a submodule of A. Recall also the equivalence between the uncontrollable part and the torsion submodule T. So A\ is the matrix of r, and therefore the input-decoupling zeros are the eigenvalues of the mapping T. Analogously, note that there is an equivalence between the observable part and < U, Y >. Consider the quotient submodule S = A/ < U, Y >. Obviously there is an equivalence between this quotient submodule and the unobservable part. Denote the linear mapping induced by J¿ by: a:S—>5 This mapping is again well-defined and its matrix will be A3. Therefore the output-decoupling zeros are the eigenvalues of this mapping. Now, an interpretation for poles is looked for. Let be A= A / r =<Xi,X2,U>/<Xl> <u> <u> which is torsion, because A/ < U > is also torsion. As in the above interpretations, denote the linear mapping induced by 4A by: dt Ô: A Again remember that poles are the eigenvalues of A3 ( the controllable part matrix ). Then, since a; G A, the equation x = A3X is satisfied, because when the quotient is done, the U part vanishes. Thus we obtain the proof that poles are the eigenvalues of ô. Lastly, let us look for an interpretation of zeros. Consider the greatest torsion submodule in < U,Y > and call it Ti. Let <U,Y> <Y,T1> be a quotient module. Notice that < U,Y > / < T\ > is free because the torsion part has been removed. If when the quotient by < Y > is done, a torsion submodule appears, then the CHAPTER 2. LINEAR 18 SYSTEMS quotient submodule J will be a dynamic where Y are now the inputs. More precisely, there exist two polynomial matrices P{-j¡¿) and Q ( ^ ) , in such a way that Now, it is straightforward procedure to obtain a realization of the system such as Z = ÄZ + BY U = CZ This is the inverse system, where U are the outputs and Y the inputs. We know that zeros are the poles of the inverse system. On the other hand denote the linear mapping induced by $ by: e: J J The poles of the inverse system are the eigenvalues of this mapping and, therefore, the zeros of the initial system are the eigenvalues of e. 2.5 Examples 1. Consider the linear control system described by X I -4 -4 0 -1 1 0 0 0 0 0 - 4 - 5 0 0 1 0 V o o o i 1 0 -10 Y -2 \ 0 -2 0 ( o 0 -1 0 x+ oy i \ 0 0 0 u V o o/ 0 0 0 1 0 0 X The module description of the system is _ < Xi,X2,X3,X4,X5,U\,U2 > < equations > = < X2,X5 > This is a free module and the system is therefore controllable. On the other hand it is clear that A/ < U > is torsion. The observability of the system can also be checked: x\ = j/i, *3 = Î/1 + V2, Z4 = (a* + 4 á + 4 / ) y i + (»i + VÚ - &uz> ^5 = - 1 / 2 $ ( i / i + y2) - 4(yi + 2/2) - 5x4 — «i and x-i = - l / 4 ( $ a ; i + 4a; 1 + £4 + X5 + ^2)- In short < X,U >—< Y, U >. So, the system is observable. Now, the derivatives of u\ and ui are written as functions of the outputs: à2 { ,d? dT^ = - d^ 2 Ad + A a¥ + d 5 nT„ dl+2I)^+y2) 2.5. EXAMPLES 19 d2 fd 3 „d2 Ad. ,d „, The submodule J such that J =< U, Y > / < U, Ti > is, in this example, the same as J =< U,Y > I < Y >. By the above equalities it can be affirmed that J is a torsion module. And the matrix of the mapping e is ( 0 0 A = 0 V 0 1 0 0 0 0 0 0 0 0 \ 0 1 0J Thus, it is clear that there is only one eigenvalue, which is zero. Moreover, it can be checked that this is the unique zero of the system. See [38] for more details. 2. Consider the linear control system described by an aircraft altitude dynamics ( [58] ). X = I -°4 1 -4 0 0 0 V 6 0 0\ 0 0 0 1 0 0 ) y=(0 x+ ( 3° \ 0 W/ 0 1 O) Now the module of the system is < equations > In order to find a generator for this module, the next procedure is shown: Write z = axi+bx2+cx2+dx4:. Then, the following condition must be imposed: derivatives up to order three cannot contain the control variable u. With this condition a generator of the system is z = 15xi + 2^2 + +2IZ3 + 6x4, because the following equalities are verified: ,(2) Xl 22 X3 = = 98 98 142 — 4i - *(2) 294 Uz - AzW - z& 294 So, z is a generator of A. Moreover, A = < z > is a free module or, in other words, the system is controllable. £4 20 CHAPTER 2. LINEAR SYSTEMS Figure 2.1: Elementary RLC-circuit worked in example 3 Let us recall that A = < u,y > is the observability condition. This is an easy computation and can be left to the reader. Looking for poles and zeros is equivalent to finding a relation like: 'dt' And in this case we have: d ,+4 d 3+4 d. 'dt' 2 4 d. +14/ d «s) (e» <¿' »»=(-(^- © '» Therefore, poles are the zeros of the polynomial p(x) = s + 4 i + 4a;2, that is to say, x = 0 and x = 2. On the other hand, zeros are the zeros of the polynomial q(x) =x'2+4x — lA and, therefore, are x = - 2 ± 3\/2. 4 3 3. Consider the circuit shown in figure 2.1. The equations are: X = -1/RiC 0 y = ( -l/Ri 0 -R2L X+ l)x + 1/RiC 1/L u l/RlU Using the same method as in the last example, a generator of the module is found: z = + X2- So A = < z >. This module is free in all cases except L = R1R2C. In this ~^cxi case, (Lgj + R%I)z = 0. That is to say, there are torsion elements in A, and therefore the system is not controllable. Let us consider the observability condition. It is necessary to check whether or not A is equal to < y,u >. The state variables are involved in the following equations: y — \/R\u = ~1/R\X\ + X2 2.5. EXAMPLES 21 —WN— —Am— K2 wal Ui mi T712 "2 c2 il il '1 «1 Vi V2 Figure 2.2: Mechanical system with springs and dampers corresponding to example 4 y - l/Riù + ( - ^ Ö - 1/L)U = ^QXÍ * R Lx îl 2 This system has a solution if, and only if, L ^ R1R2C. Therefore the system is observable if and only if L ^ RxR2C. In the case L = RxR2C there exist decoupling zeros. Remember that the input-decoupling zeros are the eigenvalues of the linear mapping induced by j¿ in the torsion submodule T. The generator of this submodule is z and it fulfills the following equation: 1 z + RXC z = 0 So, -g^ is the eigenvalue sought. The output-decoupling zeros are the eigenvalues of the linear mapping induced by ^ in S = <u,y> A generator of this submodule is x\. In this submodule the equation is xi = - R\C xx So, again, the output-decoupling zero is -j^ 4. Another example is drawn in figure 2.2 and modelled by the linear system: / X 0 0 fci + k2 7711 m2 0 0 12. mi 7712 1 0 C1+C2 m\ 7T12 0 1 \ / x+ C2 mi 7712 0 0 1/mi o / 0 0 0 \ i/m2 y 17 22 CHAPTER 2. LINEAR SYSTEMS Y = It can be shown that A = < x\, x2 >• This module is free and, for this reason, the system is controllable. It is also very easy to check the observability condition A = < ui, u2, j/i, y2 >• It is possible to find the following relation between the input and the output variables: If ,d_\ "\dt' _ mim2(^)4+(miC2+m2Ci+m2C2)(^:)3+(ciC2+A:2mi+A:im2+fc2m2)(^)2 raira2 (fc2ei+fcie2)(^)+fcifc2Jr mirU2 Then: rn2(&)2 + c2(£) + k2 c2{i) + k2 c2(£)+k2 m ^ ) 2 + (Cl + c2)(¿) + (h + k2) For this reason the zeros are the zeros of the determinant of the last matrix, and the poles are the zeros of p(—). at 2.6 Final remarks A formal Laplace tranform and the transfer function matrix are naturally defined in the module formalism in [20], where the relationship between left(right) coprime matrix decomposition and controllability (observability) is also studied. Most of the concepts and results in linear control systems have been presented within the framework of the new algebraic formalism introduced by- M. Fliess. In addition to the concision, clarity and stylishness of the concepts, this approach is specially appropriate for problems involving tracking of references and generation of signals. Moreover, this algebraic framework enables the classical results to be improved. On the other hand, the concision, clarity and stylishness make it easier to consider some phenomena which have sometimes been ignored in the control literature and seem difficult to explain in any classical framework, as will be seen in the next subsections: 2.6.1 T h e m a t c h i n g condition in sliding control m o d e Let Í X \y = AX + Bu = CX be a linear single-input system, and assume that we want y = 0 to be achieved as steady state; that is to say, a sliding regime on the sliding surface CX = 0. In classical references ( [62],[16] ) the existence of a sliding regime and the description of the ideal sliding dynamics is closely related to the equivalent control (ueq) , which is derived from y = CX = 0. A necessary 2.6. FINAL REMARKS 23 u . s-1 s+1 v . y 1 s-1 , Figure 2.3: Not stabilizable example condition for obtaining ueq from y = 0 is that the relative degree of y is 1. Otherwise ueq cannot be well defined. In the framework of module theory, the module over K[-jj¿] spanned by y is considered instead of the sliding surface equations. In this module an element (P-1 <P d (2.3) where n = reld°(y, u) — 1 ( relative degree minus one ) andp(z) = zn + a\zn~l +... + an-\z + an is a Hurwitz polynomial, can be chosen. Thus, equation 2.3 determines a well defined equivalent control ueq. The control policy «(v) = Í uHv) U[y) \u-(y) + if ¥B>0 if U B < 0 with u~(y) < ueq(y) < u (y) guarantees the achievement of equation 2.3. Finally y = 0 is the assymptotically stable equilibrium solution of equation 2.3. Hence y = 0 is obtained as steady state and the control objective is attained. 2.6.2 T h e linear s y s t e m interconnections Let us consider a motivating example. The system: ,(2) _ . , yv = u—u y— (2.4) whose transfer function is s-1 s2 - 1 s + 1 corresponds to the block diagram in figure 2.3, since ù — u = v-\-v,v = y — y. Let z = y + y — u. Then z — z = 0. So, z satisfies an unstable equation. This implies that system 2.4 is not stabilizable. The reverse block diagram is figure 2.4. It corresponds to u = w — w = y + y. That is to say, y+y = u Its transfer function is also ^ ; but this is input-output stable. Finally, let us consider the feedback system in figure 2.5. (2.5) 24 CHAPTER 2. LINEAR u . s-1 s+l w 1 SYSTEMS ,y Ä - 1 Figure 2.4: Reverse block diagram of 2.3 T{s) V S(s) Figure 2.5: Ill-posed system Its transfer function is x L . If TS = — 1, then the system is "ill-posed" in the sense of Willems [68]. It is difficult to explain these phenomena in any classic framework. There is no difference between systems 2.4 and 2.5 in the transfer function approach. But in the module framework we notice that system 2.4 is not free torsion because z is a torsion element. On the other hand, system 2.5 is free and y is a generator. These kinds of systems, called interconnections, have been examined by M. Fliess and H. Bourlès [21] via a standard algebraic tool, coproducts of modules ( [8],[43] ). They confirm Willem' standpoint [64]: "It is often misleading to distinguish between systems variables". Consider a family of modules {Ma, a E A}. Let E be a given module such that, for any a G A, there exists a morphism: ha : E —> Ma Define the submodule e of the cartesian product x a £ y i M a spanned by the elements of the form (..., 0 , . . . , hai ( e ) , . . . , 0 , . . . , -K2(e),..., 0,...) where e E E and ai ^ Q.%. The quotient module xaeAMa/E is called the coproduct ( or the fibered sum, or the amalgamated sum ) of the MQ's(referències). It is written Uaç.A,EMa. When the modules M a 's are viewed as linear systems, the above coproduct is called a system interconnection. These interconnections are defined without distinguishing between system variables. Some examples are studied below. Let D% be a dynamic with inputs ul = {u\,... ,u%m} and outputs yx = {y\,..., y^} for i = 1,2. 1. If m := m1 = m? and p := p 1 = p 2 , consider the parallel interconnection from figure 2.6. 2.6. FINAL REMARKS 25 D1 y1, D2 v\ u1 = u2t Figure 2.6: Parallel interconnection « \ D1 y 1 = u2 D2 v\ Figure 2.7: Series interconnection Consider the free module [5] = [Su..., Sm] of rank m, and the two canonical isomorphisms: <¿¿: [5] —» [u*] í« —• wls e is the submodule of D 1 x D2 spanned by the elements of the form {(^51(5S), — (p2(5s), s = 1 , . . . , m}. The interconnection is represented by Dl U u i =u 2 D2, which is defined, in practice, by the sets of equations of each module plus the equation u1 = u2. An output of this parallel interconnection will be any K-linear combination of the components of y1 and y2. 2. Assume that pl = m2 and y1 = u2. Consider the series interconnection 2.7. Consider the free module [ej = [ei,..., epi] of rank p1 and the canonical isomorphisms: ^: [e] —> [u|] es —> uzs The interconnection is D1 U y i =tí 2 D2, which is defined by the equations of each module plus the equation y 1 = u2. The first two cases of the motivating examples are examples of this type of interconnection. 3. Let D 3 be a third dynamic with input u 3 = {u\,..., u 3 ^} and output y 3 = {yf,..., y 3 3 }. Consider the feedback interconnection whose block diagram is figure 2.8. The input u3 = v U w is divided into two parts. Set y 3 = it1, y 1 = it2, y 2 = w. Therefore, the above block diagram corresponds to the coproduct Uy3:=uityi=u2iy2-W(D1,D2,D3). 26 CHAPTER 2. LINEAR SYSTEMS Figure 2.8: Feedback interconnection A frequent phenomena is the lack of controllability or observability; that is to say, interconnecting controllable (or observable) linear systems may give rise to an uncontrollable (or unobservable) one. This cannot be detected by transfer functions. Moreover, when K is a field of constants, the hidden modes corresponding to the lack of controllability (or observability) may exhibit positive real parts which imply unstability. Let us see the examples stated at the beginning of this subsection. As has been said, example 2.4 is not torsion free. In other words, it is not controllable. The corresponding input decoupling zero, which is 1, is unstable. In system 2.5, w cannot be expressed as a linear combination of u, y and a finite number of their derivatives. So, the system is unobservable. Consider the third example. Write T(s) = | f | j and 5(s) = 4 4 , a,b,c,d € R[s], abed ^ 0, a,b (resp. c, d) coprime. The system is governed by the equations: a(^)(«-w) bi )y (2.6) CÍ (2.7) Í Jt)y There are two possible situations: 1. If ac + bd 5¿ 0 (i.e. ST ^ - 1 ) , then (b(s)d(s) + a(s)c{s))y(s) = a(s)d{s)u{s). Therefore, y can be obtained from u; and v can also be obtained from y, and, consequently, from u. 2. If ac + bd = 0, then u must satisfy a ( ^ ) u = 0. S o u becomes a torsion element. The remaining variables y, v span a free module of rank 1. Here, the lack of controllability concerns the control variable. Another strange phenomena is the change of rank. Generally speaking the rank is the maximum number of independent channels, but may change in some interconnections. Let us consider an example: The system in figure 2.9 is governed by the equations: a{ )v - It d(^-)v K dr = = b ^y (2.8) c{ )y (2.9) ít 2.6. FINAL REMARKS 27 _+ -V. V T(s) V S(s) Figure 2.9: Example illustrating possible changes of rank There are also two possible situations: 1. If ac + bd / 0 (i.e. ST ^ —1) then (ac + bd)v = 0. This implies that v, and consequently y, are torsion. Therefore, the rank is zero. 2. If ac + bd = 0 (i.e. ST = —1) the module is free of rank 1. 28 CHAPTER 2. LINEAR SYSTEMS Chapter 3 Linearization of nonlinear systems and flatness This chapter serves as an introduction to different types of linearizations for nonlinear control systems: static feedback linearization, dynamic feedback linearization and linearization by prolongations, which is a particular case of dynamic feedback linearization. The concept of flatness will be also introduced, as well as the concept of flat outputs. The tools and concepts of the two different frameworks used throughout this thesis will be stated in this chapter. These frameworks are: differential geometry and differential algebra. 3.1 Different types of linearizations Definition 4 A nonlinear system m x = f{x) + ^gi(x)ui x G Rn i=i is said to be static feedback linearizable if it is possible to find a feedback u = a{z) + ß{z)v ueRm veRm z G Rn and a diffeomorphism z = 4>(x) such that the original sistem is transformad into a linear controllable system z = Az + Bv where A and B are matrices of appropiate size. The next theorem is a characterization of static fedback linearizability in the differential geometry framework. A proof can be found in ([34], [31], [51]). 29 30 CHAPTER 3. LINEARIZATION Theorem 3 Let OF NONLINEAR SYSTEMS AND FLATNESS m x = f(x) + Y^9i(x)ui 1=1 be a nonlinear system with m inputs. This system is static feedback linearizable if and only if the following distributions have constant rank and are involutive: D0= Di= and the rank of -D n -i <gu...,gm> <Di-1,adlfgi,...,adzfgm> n_ i = 1 ''"'' îS n In the case that the above system is static feedback linearizable, there exists a change of variables and a feedback such that the system is written in the following way Vi = Vi+x Vi = 1 , . . . , n i j¿ kj, j = 1 , . . . , m Vkj = Vj j = l,...,m kj are the so called Brunovsky indices [5],[51]. The definition of these indices is as follows: Define po = dirnDo Pi — dimDi — dimDi-i i> 1 Then, kj = #{pi >3,i> 0} A generalization of the static feedback is a dynamic feedback transformation. Definition 5 A nonlinear system x = f(x,u) x£Rn ueRm (3.1) is said to be dynamic feedback linearizable if there exists: 1. A regular dynamic compensator z = a(x,z,v) u = b(x,z,v) with z € Rg and v G Rm. input v and output u. (3.2) The regularity assumption implies the invertibility of 3.2 with 2. A diffeomorphism xp = $(x,z) n+q with tp G R , such that the original system 3.1 with the dynamic compensator after applying 3.3, becomes a constant linear controllable system: (3.3) 3.2, 3.2. FLATNESS AND DIFFERENTIAL ALGEBRA 31 This linear system may be written in Brunovsky canonical form ([5],[51]) by means of a static state feedback and a linear invertible change of coordinates: y¡ki)=Vi V¿ = l , . . . , m where {ki}1¡l.1 are the Kronecker indices. Therefore, setting ii - dh y — \ylj" i;(fcl-1) • iVl 7/ „( f c ™- 1 h ) • • • > y m ) • • • iVm I it is possible to write y = Tip, where T is an invertible matrix. This can be transforned, using the invertibility of the diffeomorphism 3.3, into That is to say, x and u can be expressed as real-analytic And from 3.2, u = b('9~1(T~ly),v). functions of the components of ( y i , . . . ,y m ) and their derivatives. The dynamic feedback 3.2 is called endogenous if, and only if, the converse holds; that is to say, if, and only if, ( y i , . . . , ym) can be expressed as real-analytic functions of x, u and a finite number of their derivatives. Definition 6 A dynamics 3.1 is called (differentially) flat if, and only if, is linearizable via dynamic endogenous feedback. The variables (yi,... ,y m ) are called flat or linearizing outputs. Therefore, a flat system is not only linearizable, but is also a system where x and u trajectories can be deduced immediately from ( y i , . . . , y m ) trajectories. In fact, this is the power of flatness. Once the flatness of a system is known, it does not imply that one intends to transform the system into a single linear one. When a system is flat, it is an indication that the nonlinear structure of the system is well characterized, and one can exploit that structure by designing control algorythms for motion planning, trajectory generation, and stabilization. Indeed, the flat outputs are the nonlinear analogue of a basis of the free module for linear controllable systems. Flatness was first introduced by Fliess and coworkers in [13],[14],[19],[22] using the formalism of differential algebra. In differential algebra, a system is viewed as a differential field generated by a set of variables (states and inputs). Recently, flatness has been defined in a more geometric context. One approach is to use exterior differential systems, and to regard a nonlinear control system as a Pfaffian system on an appropiate space (see, for instance [49] and referències therein). A somewhat different geometric point of view is to consider a Lie-Bäcklund framework as the underlying mathematical structure ([23],[24]). In this context, a system is a smooth vector field on a smooth manifold, possibly of infinite dimension. 3.2 Flatness and differential algebra For an introduction to differential algebra see [39],[40],[54]. 32 CHAPTER 3. LINEARIZATION OF NONLINEAR SYSTEMS AND FLATNESS Definition 7 Let k be a given differential field. A system is a finitely generated differential extension D/k. This corresponds to a finite number of quantities which are related by a finite number of algebraic differential equations over k. The differential order of the system D/k is the differential transcendence degree of the extension D/k. Let k < u > the differential field generated by k and a finite set u = ( i t i , . . . , um) of differential /c-indeterminates. Assume ui,...,um differentially fc-algebraically independent; that is to say, diff tr d°k < u> /k = m. A dynamics with input u is a finitely generated differentially algebraic extension D/k <u>. Note that the number of independent inputs is equal to the differential order of the system D/k as was proven in [63]. An output y = (j/i,... ,yp) is a finite set of differential quantities in D. According to theorem 6, there exists a finite trascendence basis x = (xi,..., xn) of D/k <u>. Therefore, since ¿¿,y¿ € D, Xi,yj are k < u >-algebraically dependent on x. That is to say, there exist Ai and Bj, polynomials over k, such that { Ai(ii, x,u,u,..., Bj{yj,x,u,ù,.. u(r*)) = 0 .,u^) Vi = 1 , . . . , n = 0 Vj = 1,... ,p Xi are called generalized states and n is the dimension of the dynamics D/k <u>. As was stated in the former section, linear systems are viewed as finitely generated modules over principal ideal rings. The relation between these two approaches (field extensions and modules) is established by what is called Kahler differential ([35]). See appendix A for an introduction to differential algebra and details on the Kahler differential. To a finitely generated differential field extension L/K, associate a mapping (the Kahler differential) dL/K '• L — > P L / K where CIL/K is a finitely generated left L[^]-module, such that: Va GL dL/Kft = i(dL/Ka) dL/Ka + dL/Kb Va,beL dL/K{a + b) = Va, 6 € L dL/i({ab) = bdi/^a + adi/^b VceK dL/Kc = 0 As was seen in the previous section, a module like this corresponds to a linear system. In this case, this system is called the tangent or variational system. The inputs of this tangent system are (dL/KUi,..., dL/Kum). Properties of the extension L/K can be translated into the linear module theoretic framework: • A set ijj = (tpi,... ,tpm) is a differential transcendence basis of L/K if, and only if, their respective Kahler differentials {di/Ripi,... ALIK^™) rnake up a maximal set of L[-^\linearly independent elements in ÍÍ¿/A"- I*1 other words, if, and only if, (dL/Ki(ji,... ,dL/K are a basis of ClL/K. Thus, diff tr d°L/K = rk SIL/K- 3.2. FLATNESS AND DIFFERENTIAL ALGEBRA 33 • The extension L/K is differentially algebraic if, and only if, the module fi¿/^ is torsion. And a set X = (x\,..., xn) is a transcendence basis of L/K (not differential) if, and only if, {di/ftX = CIL/K^I, • • • ) dL/Kxn) is a basis of ^LJK a s a £-vector space. • The extension L/K is algebraic if, and only if, ClLfK = {0}. The following definition states precisely what dynamic endogenous feedback means in this framework. Definition 8 Two systems D\jk andDi/k are said to be equivalent (by endogenous feedback) if, and only if, there exist two algebraic extensions (not differential algebraic) D\/D\ and D2/D2 and a differential k-automorphism between D\/k and D-i/k. In other words (identifying the systems with their respective images in the bigger fields), D\/k and .D2 A are equivalent if, and only if, any element ofD\ (respectively D<i) is algebraic overDi (respectively D\). Two dynamics D\/k < U > and A J A < ^ > are said to be equivalent if, and only if, their corresponding systems D\/k and D2¡k are equivalent. Proposition 2 Two equivalent systems have the same differential order (and, therefore, the same number of inputs). And the same happens to the dynamics. Proof: Let K be the differential field generated by D\ and Di- Since D\ and D2 are equivalent, K/D\ and K/D2 are algebraic extensions. So diff tr cPDi/k = diff tr d°K/k = diff tr d°D2/k m Remark: Consider two equivalent dynamics, D\/k < u > and Ü2¡k < v >. Let n\ and ri2 be the dimension of Di/k < u > and D2¡k < v > respectively. Write the generalized state variable representations of both dynamics: Al(ii,x,u,u,... ,u^r^) = 0 i = 1,... , n i =0 ¿ = l,...,n2 Cf{¿i,z,v,v,...,v^) On the other hand, since any element of D\ is algebraic over D2 and viceversa, ' p}(uj > « 1 t> l û 1 ... 1 « (i i ) ) qj{xi,z,v,v,...,vW)) pj(Vj,x,u,u,...,uKJ') k qf(zi,x,u,ù,...,u^) =0 =0 = 0 =0 3= i= j i= l,...,m l,...,r¡ l,...,m l,...,r? , ., where Pj,p"j, g},g^ are polynomials over k. 3.4 corresponds to two endogenous dynamic feedbacks because they do not use any variable trascendental over D\ and D2. 34 3.3 CHAPTER 3. LINEARIZATION OF NONLINEAR SYSTEMS AND FLATNESS Linearization by prolongations Definition 9 Let m i = f(x) + Y^9i{x)ui ¿=1 be a nonlinear system with m inputs. A prolongation of this system is x ul = ÛÏ-1 = /(*) + E£i$iOi)u? u- Vt = l. ,m Vi where u\, which corresponds to u\ , are new state variables Vi = l . . . m And the new inputs are V{. j = O . - . & i — 1. Definition 10 Let £ = f(x) + ^2gi(x)ui i=l be a nonlinear system with m inputs. This system is said to be linearizable by prolongations if there exists a prolongation of the original system which is static feedback linearizable. In fact, a system which is linearizable by prolongations is dynamic feedback linearizable. That is to say, a linearization by prolongations is a particular case of dynamic feedback linearization. Let us see the relationship between these two types of linearizations. Consider a dynamic feedback compensator, affine respect to the inputs: z = a°(x,z) + a1(x,z)v u = b°(x,z) + bl(x,z)v (3.5) with z G Rq and v € Rm. A dynamic feedback compensator is a prolongation if, and only if, z ki+l u = Zi where 1 < i < q. = Zi+i if i± kj, j = l , . . . , m Vj if i = kj, j = 1,... ,m 3.3. LINEARIZATION BY PROLONGATIONS 35 That is to say, / * i Z k!+l b°(x,z) = i+Er=Tlfci / b\x,z) 1V ' ' 1 if í = kj, j = 1, . . . , m 0 ,^-J ° flir, =0 if *^A¿« i = !»•••. m m The following lemma will be used in some proofs related to linearization by prolongations in chapters 4, 5 and 6. Lemma 5 Let 7j-, i G I a family of coordinate vector fields in R ; D1 = <a 1 ,... l a j ;{¿}0^ 1 > D 2 = where ai,..., < {-5^}kÍn+l+l > d_ _d_ ctj e< -^-,..., -57- > dyi ' " * * ' dyn and not depending on the variables Vn+l+l,- • • ,Vn+s Then D\ @ D% is involutive if, and only if, Di is involutive. Proof: It is straightforward. 36 CHAPTER 3. LINEARIZATION OF NONLINEAR SYSTEMS AND FLATNESS Chapter 4 Linearization using differential algebra 4.1 Introduction This chapter deals with the problem of linearization of nonlinear control systems; that is to say, with the problem of flatness. Using the methods of differential algebra, we explain the conditions which in this framework must be satisfied in order for a nonlinear control system to be linearizable. First of all, the tangent system is computed using the Kahler differential. Then, for a single-input system, we give a new proof of the fact that a single-input system is static feedback linearizable if, and only if, it is dynamic feedback linearizable. We also tackle the static feedback linearizability problem for a multi-input system, thereby obtaining the conditions that a system must fulfill in order to be transformed into a linear one in this context. The problem of dynamic feedback linearizability is solved by trying to guess m — 1 flat outputs and computing the last one. The quotient of modules appears to be a cornerstone in this procedure. Finally, without computing the tangent system, a procedure based also on guessing m — 1 flat outputs is be designed. The main tool here will be the intermediate differential field extensions. A helpful software package in Maple V is created in order to simplify the computations required. The listing of this program can be found in appendix B. 4.2 Single-input systems Consider the single-input system: (4.1) x = f(x,u) where the state x = (x\,... Thanks to the property ,xn) G Rn and the control u £ R. Assume that 4.1 is controllable. d(—a) = —(da) 37 Va G L 38 CHAPTER 4. LINEARIZATION USING DIFFERENTIAL ALGEBRA where d is the Kahler differential and L is the field extension corresponding to 4.1, the tangent system is: dx = -^-dx + -^-du (4.2) Therefore, the basis of the corresponding module Q contains just one element. Let w be such an element. In this context, a characterization of flatness for a single-input system is shown, and consequently the well known equivalence between dynamic and static feedback linearization for single input systems can be deduced (see [7], [2], [49] for other proofs of the same result): Proposition 3 : System 4-1 is static feedback linearizable if and only if the module f2 characterizing 4.2 is generated by an integrable one form w. Proof: The necessity is obvious. Sufficiency: Since w is assumed to be a generator: dxi = ql;{-)w Vi = l , . . . , n (4.3) where <?/( j¿) are elements of the ring K[-^], and l{ are the respective degrees. Therefore, 4.3 can be written as follows: U dxi = Y,b>{h) /i-0 (4-4) Note that w and its derivatives of any order are L-independents. If there exists a combination among them such as w^ = X)fc=o akW^ then, w is a torsion element, which contradicts the hypothesis of controllability (recall that the controllobality of a linear system is equivalent to the freeness of the module). On the other hand, dx\,... ,dxn are L-indepents too (there are no algebraic relations between the state variables). Since n independent elements can not be written as a combination of a set of I independent elements, with / < n, there exists U greater than or equal to n — 1. Let us take lj = max {li} > n — 1 l<j<n and substitute the expression of dx^ from 4.4 in the j t h equation of 4.2. Thus, in the left hand side we have dxj, which is a polynomial in the indeterminate ^ of degree lj + 1. And in the right hand side of the above equation we have dxj, which is a polynomial in the indeterminate ^ of degree smaller than or equal to lj. Therefore, 4.2. SINGLE-INPUT SYSTEMS 39 because, if it is zero, the above equation becomes ¿•H-(s).(¿* WW which implies that w is a torsion element, in contradiction with the controllability of the system. Now, since (£),*"• it is possible to isolate du: So, J du=pli+1(—)w at lj+1 where p (-^) is a polynomial in the indeterminate ^ of degree lj + 1 > n. Note that it is no possible that du=ps(4:)w dt with degree s < n, because if 4.6 holds, then, equalling 4.6 and 4.5: plj+l(-r)w K y dt' = du (4.5) (4.6) s (—)w =p F K dt' which implies that w i s a torsion element, in contradiction with the controllability hypothesis. Summarizing, the relative degree of w with respect to du is lj + 1 > n. But, it is well-known that the relative degree is always smaller than or equal to the dimension of the state. Therefore, the relative degree of w with respect to du is n. Now, using the integrability condition of w, a variable y (the flat output) such that dy = w is obtained. And since the Kahler differential commutes with the time derivative, this variable y, the flat output, fulfills the relative degree condition with respect to u . • Corollary 1 : Linearization by static and dynamic feedback are equivalent for single-input systems. Proof: As already stated in chapter 2, flatness (or linearization by dynamic feedback) is equivalent to the existence of an integrable basis of the tangent system. Prom the proposition this last fact is equivalent to linearization by static feedback. Hence, dynamic and static feedback linearization are equivalent for single-input systems. • 40 CHAPTER 4. LINEARIZATION USING DIFFERENTIAL ALGEBRA Now, imposing the former relative degree condition, w is computed as a solution of an homogenous linear system. Since w lies in fi the expression of w must be: w = aidx\ + ... + andxn := a • dX And an expression for w is obtained: ädx + a • (g-dx + g{¡du\ vj = = (ó + aË) • dx + a • Ëdu Therefore, the condition of the relative degree implies: -I;- 0 47 <-» Deriving 4.7, it is possible to compute another condition, useful in the following steps: du The value of w is calculated: -m- {à + a%)ILdu + The condition of the relative degree leads to: fà + a^V —= 0 dx) du V which thanks to equation 4.8, becomes: a.(Ël_±I)dl \dx dt J du =Q Here it is also possible to obtain another useful expression: •"•5£—!(59-- m Imposing once again the relative degree condition, now to to'3) : ( (2) o-dj_ ICI I dx dtdx (2L\2\ QL-o \dxj I du 4.3. STATIC FEEDBACK LINEARIZATION OF MULTI-INPUT SYSTEMS 41 Using 4.8 and 4.9, this last equation becomes: \dx dt J du Iterating this process, a system of equations is obtained: •(£-£)'§£- » - - This is an homogeneous system with n — 1 equations and n unknowns ( a\,... ,an ). So, the solution w will depend upon a certain function A. Now, a variable y such that dy — w must be obtained. The role of A is the role of an integrant factor. 4.3 Static feedback linearization of multi-input systems Once again the intention is to compute the basis of the module corresponding to the tangent system. As was stated in chapter 2, this basis contains the same number of elements as the number of the inputs. Let m be this number. And let w\,..., wm be the elements of this basis. And again, as in the single-input case, the conditions that must be imposed to find the basis are the relative degree conditions. The following lemmas will translate the relative degree conditions of a nonlinear system to its tangent system. Lemma 6 For all f, g vector fields, and for any w differential one form, the next equality is satisfied: Lf<w,g > = < LfW, g> + <w,[f,g}> (4.10) Lemma 7 The Lie derivative with respect to a vector field f commutes with the exterior derivative: Lf(dh) = dLf{h) (4.11) The proof of these two lemmas can be found in any elementary text on differential geometry. Lemma 8 For any C°° function y and any vector fields f and h, it follows the equality: < dLrfy,h >= ¿(-l) fc ( ¡ W * < d^ad)h > Proof: It will be proven by induction. For r = 1, and thanks to 4.11 < dLfy, h > = < Lfdy, h > Applying now 4.10, we have < Lfdy, h>=Lf <dy,h> -< dy, [/, h] > (412) 42 CHAPTER 4. LINEARIZATION USING DIFFERENTIAL ALGEBRA which is the desired equality for r = 1. Assuming the trueness of the statement for the case r, the case r + 1 will be proven. Applying 4.11 and 4.10: < dLrfy, h>-< < dLrf+1y, h >=< LfdLrfy, h>=Lf dLrfy, [/, h] > Using the induction hypothesis, the latter expression becomes h Í E í - W í W * < dV><4h >) - Í D - W Í W * < dy,adkfadfh >) which, in turn, is equal to ( £ ( - ! ) * (¿) L)+1~k < *V> ad)h A - ( ¿ V i ) * (¡) LT" < dV, «if + 1Ä >) Finally, the equality »hull H T L'W applied to the former expression, leads to \ fi/ \ 1/ A, , 'kj£\-l)k^liyrf+1-k<dy,adkfh>^ This is the desired expression for the case r + 1. Proposition 4 When a system x = f{x) + Yu9i{x)ui i=l is linearizable by static feedback, then —yi = Lrfyi Vr < k{ - 2 where yi and its derivatives up to orderfc¿— 1 are the coordinates of the change of variables that linearizes the nonlinear system; and ki are the Kronecker indices. Proof: Again it will be proven by induction. For r = 1 j —yj =< dyj,f{x) n + Yjgi{x)ui > = < dyjyf > 4.3. STATIC FEEDBACK LINEARIZATION OF MULTI-INPUT SYSTEMS 43 since the static feedback linearization conditions imply < dyj, cufygi > = 0 Vr < kj - 2 (4.13) Assuming that the statement is true up to r, the case r + 1 will be studied. Using the induction hypothesis we have ¿ (F+i d~F^ Vi = n J t ^ ^ =< fVJ>f( ) + J29i(x)ui x dLT > Applying now the equation 4.12, we get: ¿ ( - l ) f c ( T W * < dyj,adkf(f(x) + J29i(x)ui) > which thanks to 4.13 can be written Lrf<dyj,f>+Í2(-l)fcf fc=i l J L T k é u i < d W' a d )Si > W ¿=i Again, because of 4.13, ¿(-i)*K W * £ « i < «foi««*/* >= o jfc=i W ¿=i Therefore, ^ rr+1 Corollary 2 TTie Kahler differential of yj does noi depend on the inputs of the tangent system dui i = 1 , . . . ,m Proof: In the previous proposition it has been proven that ^FTTVJ does not depend on the input variables it¿ Vi = l , . . . , m . The commutativity between the time derivative and the Kahler differential was commented on in chapter 2. Putting them all together, the statement of the corollary is derived. So, the relative degree of Wj = dyj with respect to du must be fc¿, where ki are the Kronecker indices ([51]). Let x = f(x,u) xeRn ueRm a multi-input system. Its Kahler differential is dx = ^J-dx + ^-du ox ou (4.14) 44 CHAPTER 4. LINEARIZATION USING DIFFERENTIAL ALGEBRA The basis can be written Wi — a\dx\ + ... + azndxn = a1 • dx Vi = 1 , . . . , m Leading the relative degree condition to •'(i-l'Ts;- 0 w-".".*-2 « = !,...,» The solution of this system will be a basis of fi. Note that, if the Kronecker indices are such that k\ > k-2 > ..., then w\ depends upon, at least, n — (k\ — \)m parameters; IV2 depends at least upon n — fa — l)m parameters; and so on. These parameters act as integrant factors in order to find y¿ G L such that dyi = iu¿. Note that even though the original system was not affine in the inputs, this procedure can still be applied Example: Let us consider the following system: ¿1 ¿2 ¿3 = = = £2 + ^1^2 Z3U1U2 + X2U2 U2Z3 Its tangent system is: dx\ = uzdxi + dx2 + x\du2 < dx2 = U2dx2 + 2xzU\U2dxz + x\v,2du\ + {x2 + x1u\)dv,2 dxz = U2dx$ + xzdu2 The Kronecker indices are k\ = 2 y &2 = 1. Denoting Wi = a\dx\ + a\dx2 + a\dx^ ¿ = 1,2 Applying the conditions explained above, the next equations for w\ are obtained: a\ = 0 and a\x\ + 0,3X3 = 0 and no conditions for W2- Therefore u>i = \{xzdx\ — x\dxz) AG L and IU2 can b e choosen freely, taking into account that W2 has to be differentially independent with respect to 101. As A is an integrant factor for w\, it can be chosen in such a way that w\ is an exact one form. A possibility is to choose 2:1X3 4.4. DYNAMIC FEEDBACK LINEARIZATION 45 FOR MULTI-INPUT SYSTEMS Hence: and, for the sake of simplicity, w\ = —dx\ Xi dxz Xz W2 — dxi It can be checked that j/i = In — y2 = zs xz are such that Wi = dm, i = 1,2. In other words, the flat outputs of the system are yi, 1/2- 4.4 Dynamic feedback linearization for multi-input systems Let us consider a controllable multi-input system: (4.15) x = f(x,u) where x E Rn and u € R2. Recall that the tangent system is: dx = ^-dx + ?fdu (4.16) ox ou Let ÎÎ be the module characterization of 4.16. Suppose that a flat output can be guessed ( for example yi). In order to obtain the second flat output y2, a quotient of modules is made. Let us consider the quotient module: Ü/ < dyi > (4.17) This is a controllable single-input system, and by application of the algorythm explained in the former section a basis of this module can be found. Let W2 be this basis. Therefore, a basis of O, will be: m = dyi, w2 = w2 +p(-j^)dyi p( jj) must be chosen in such a way that W2 is a if-integrable one form. So a system of partial differential equations must be solved. This procedure can be generalized for systems with m inputs. In that case, m — 1 flat outputs must be guessed, but the algorythm to get the last one is the same. Once the flat outputs yi,...,ym have been obtained, it is possible to know if the original system is dynamic feedback linearizable by derivation of the inputs [7]. To get such a condition, let us define the parameters kji,rp,rij as follows: let Wj = dVj = Ajdx + Pfiif^dux + ...+ p ^ m (ft)dum (4.18) On the other hand, the system variables can be written using ui,..., dx = AÍ1 ( - ) c % + . . . + Rl£(-)dym wm: 46 CHAPTER 4. LINEARIZATION ¿"i = éH^)dyi USING DIFFERENTIAL + ••• + ALGEBRA q^(jt)dym are the maxiwhere R^ ( ^ ) , . . . , Rl™ (-^) are vectors with coefficients in K[(-^)] and h,...,lm 1 mum degrees of the indeterminate in each vector, ç f / ^ ) + . . . + ?,>£"( ¡g) are polynomials in the indeterminate (J¿) and fin,..., h{m are the degrees of the respective polynomials. Now Jb..._ J kJi ]l ' 1 —rel d°(dyj,dui) ifPji^O if pji = 0 rp := max{{Zj + kjp,Vj = 1 . . . m}, {/i¿,- + kjp,Vj = 1 . . . m, Vi = 1 . . . m}} rij := max{Zj, /i¿j, V¿ = 1 , . . . , m} It must be noticed that Z¿ := —oo (respectively fty = —oo) if Rj = 0 (respectively gy = 0). Corollary 3 Ï7ie system is dynamic feedback linearizable by prolongations if and only if the sets V and W have the same number of variables, where W = : dxn, dux,..., duY1',..., dum, •.., {dxi,..., du^} and V = {dyu...,dy^\...,dym,...,dy<£^} That is to say Proof: Note that m m p=i j-=i dx\,...,dxn,dui,...,duy',...,dum,...,du^m' can be written as linear functions of dyu...>dy^\...,dym,...,dy^ And conversely, dyi,...,dyini),...,dymi...Jdy^ are linear functions of dx\,..., dxn,dui,..., dui,..., dum,..., du^m' In other words, there exists a linear change of variables between the variables in V and W. This change of variables is, in fact, thanks to the integrability conditon, the jacobian of the change of variables between X\, . . . ,Xn, U l , . . . , U± ,Um,...,Um 4.5. SOFTWARE 47 and í/1 > • • • ) y i ) • • • i ym¡ • • • > í/m So, as the change of variables exists, the system x faW = f(x,u) = u¡j+1) V* = l,...,m V^O,...,^ is static feedback equivalent to JtVi)=Vi+l) Vi = l , . . . , m Vi = 0 , . . . , n < - 1 In other words, the original system x = f{x,u) is linearizable by prolongations. 4.5 Software In order to simplify the calculations needed in the method presented above, a software package has been developed. It consists in many Maple V functions that allow us to use a computer to perform the basic operations -such as the computation of the tangent system- and those that are harder. For example, given a single input system, a basis of the module fi can be determined and quotients in this module with arbitrary expressions can also be made. Given a two-input dynamic feedback linearizable system, it can be reduced to a single input one through a quotient, and, in this single input system, a basis can be computed. The integrability of the basis is automatically tested, and, if it holds, the integrals will be the flat outputs. 4.6 Examples 1. This example has been borrowed from [45]. Let: ¿1 = 1£l ¿2 = 112 ¿3 = UiU2 be a nonlinear system. The tangent system associated to it is a quotient Ä"[^]-module A defined by the generators {dx\,dx2, dx3,dui, duz] 48 CHAPTER 4. LINEARIZATION and the relations: USING DIFFERENTIAL dx\ = dx2 — dii2 dx¡ = U2du\ + u\du2 ALGEBRA du\ dx\ can be guessed as one of the two generators of the free module. Let us consider the quotient module ííi = <t¿? > . In the state space representation, Q,\ is given by the equations: dx2 = dv,2 dx$ = U\du2 Clearly, a basis of Í2i is given by UJï, = u\dx2 — dx%. Going back to 102 = U\dx2 — dxz + p(^)dxi; where p(jjj) € K[j^\. In order for W2 to be if-integrable, an appropriate choice for p ( ^ ) is d p{ di) d = X2 d-t Hence, an Ä"-integrable basis of Í3 is CJI CJ2 = = dxi U\dx2 — dx3 + X2du\ and the flat outputs are: Vi = y2 = xi U1X2 - £3 The relationship between the state variables of the tangent system and the basis of the tangent module is: dx\ = u)\ (2) , dX2 dxz = = • + (JL>2 X2W\ Ui X2U1U1 — UiX2W\ (2) : — U1W2 + UiU)2 du\ = (¿i du2 = involves third derivatives of wi and second derivatives of UJ2 4.6. EXAMPLES Therefore n = max{l,l,0,-oo,2,2} = 2 7-2 = max{—00, — 1, — 00, — 00,— oo,0} m max{2,1,3} = 3 = ri2 = max{l, —oo,2} = 2 Summarizing, the nonlinear system defined by xi = tin ¿2 = V2 «il = «12 UÍ2 = «1 is static feedback equivalent to the system 2/11 = 2/12 2/12 = 2/13 2/13 = 2/14 2/21 = 2/22 2/22 = 2/23 which is linear. 2. This example has been borrowed from [53]. X2X1 — x\¿2 = 2:3 X\X$ = X4 This system can be written down in the following way: x\ X2 ¿3 = = = xa+xiui 12 Itl II CHAPTER 4. LINEARIZATION USING DIFFERENTIAL ALGEBRA Its tangent system is: S3+zmi 1 \ X2 dx = 0 0 0 £1 0 0 (¿£+ 0 X2 1 0 | du ii Making quotient by dx\ — Adu2 ( that is to say, x\ — Xx^ in the original system ) we get: / CÍX _x 3 +«i«i Ul 0 = 12 0 0 0 0 V _L \ da;+ Applying the algorithm for single-input systems, the basis u>2 is obtained: / x2{u2\-xi) W2=ai( dxi+ —2\^r - u2X-xi +dxz) - ~2^r where ai is a function depending upon the variables x's and u's. Therefore: W2=W2+p(—) (dxi - \du2) at But this one form is not integrable. So, x\ — \x± in the original system cannot be a flat output. If A is zero ( that is to say, guessing x\ as a flat output ), the quotient is: dx2 dxz jca+nm X\X2 0 + Ü) X2 X\X2 0 dx2 dxz ¿Í2 Clearly, dx2 is a basis of this module. Therefore, the flat outputs are x\ and X2Comparing this method with the method used in [53], note that our algorithm reduces to: (a) Making a quotient of modules. 4.6. EXAMPLES 51 (b) Solving an homogeneous linear system. (c) Checking the integrability condition of just one form. Therefore, it seems simpler than the procedure appearing in [53] . 3. The following example is related to a vertical take off and landing aircraft (VTOL). This aircraft was assumed to be not linearizable by prolongations since [23], but it was known to be flat . In our framework we prove not only that is flat, but that it is also linearizable by prolongations. Here are the equations of the system: u\ sin 6— uze cos 6 uicos6+ U2£sin0 —1 u2 x= y= 0= Reducing the system to order 1 in order to be able to apply our algorithm: x\ = ¿2= ¿3 = £4 = X2 uisinrEô— U2ecosrc5 X4 —1+ í¿icos:E5+ uiesmxs ¿6 = U2 where x\ = x X2 — x X3 = y X4 = y x$ = 6 XQ —6 As in the preceding examples, its Kahler differential is computed: dx\ = dx2 = dxs = dxi= dx$ = dx§ = dx2 (ui cosís + U2£s'mx5)dxs+ sinxsdui — ecosa;5du2 dxi (— sin^ö + U2£Cosx5)dx5+ COSX5CÍU1+ esinx5<iu2 dxQ du2 Guessing y\ = i 5 as a flat output, the quotient is dx\ dx2 dxz dx4 = dx2 = sinx^dui = dxi = cos x^dui 52 CHAPTER 4. LINEARIZATION USING DIFFERENTIAL ALGEBRA A basis of this quotient module is __= ?^É^^lXdxi + Xdx2+ When A = XQ COS X5 ^ ) is the appropiate polynomial, y2 = /•2l2sini5+U2COSl5 2x1 sin 15 +U2 cos i s, ,\ 2 X\ y-* , ) + X2X6 cos 2igcosi5-U2sina:5 , _ Xs—B g . Z5 + . 1 X4XS S1I1Î5 Once the flat outputs have been obtained, we are able to decide whether or not this system is linearizable by prolongations. For this purpose the parameters previously defined are computed: ¿i=3 h =5 h\\ = 4 h\2 = 6 /121 = —00 = -4 Â12 = 0 &21 = - 0 0 kn /122 = 2 fc22 = -2 Therefore, ri =0 T2 = 4 ni = 4 ri2 = 6 and That is to say, n + 7-i+r 2 = 6 + 0 + 4 = 10 While «1 + ri2 = 10 Therefore, following the corollary, the VTOL is linearizable by prolongations. 4.7 An analogous procedure using field extensions The procedure explained in section 3.3 admits a nice counterpart using only field extensions. Let us consider a nonlinear control system x = f(x,u) xERn ueRm 4.7. AN ANALOGOUS PROCEDURE USING FIELD EXTENSIONS 53 As was stated in chapter 2, this is a differential field extension L/R, where L is the minimum field containing the variables x, u and where the equations of the system are satisfied. If m — 1 flat outputs yi,... ,ym are guessed, let us consider the intermediate differential field extension L/R < J / 1 , . . . , Vm-i > /R- Then, the extension L/R < y i , . . . , y m - i > is a single-input system. In this single-input system, the relative degree condition can be applied in order to find its flat output, which will be the remaining flat output yn of L/R. This procedure has a disadvantage with respect to the one explained in section 3.3. The procedure in section 3.3 uses quotient of modules, and because of this, some equations and m — 1 inputs are eliminated. In the field extension procedure, neither equations nor variables can be eliminated. Another difficulty is that there is no software package to apply the procedure, and neither is there a sistemàtic way to obtain the last flat output. Example: Let us consider again the VTOL. Let us also recall that the system equations can be written as follows: Xl = x2 = X2 ¿3 = x\ = X± -1+ ¿5 = X$ Ui s i n £5— U2E COS £5 U\ COS £ 5 + t o c s i n 25 Ul ¿6 = If yi = X5 is guessed as one of the flat outputs, the extension R < yi,V2 > /R < Vi > is represented by the following system Xl = U2E cos X5 ¿3 = X2 ui s i n X5 — X4 X4 = —I + U1COSX5+ U2£sina:5 ¿2 = which is a single-input system. Applying the usual static feedback linearizability conditions to this single-input system, the following distributions have to be involutive: / 0 \ sinzs £>,=< > 0 y cos 3:5 J is trivially involutive. D2=< ( °^ / sinx5 0 \ coss5 J Î \ — s i n x 5 "^ XQ cos 2:5 - COS X5 - 2 6 sin 25 J > 54 CHAPTER 4. LINEARIZATION USING DIFFERENTIAL ALGEBRA is involutive because the Lie bracket between its two elements is zero. —2XQ cos x$ ( Dz =< D2t \ —x\ sin £5 + U2 cos £5 > 2x6 sin £5 y — ¡Eg COS £5 — U2SÍnX5 J is involutive because the Lie brackets between its elements are zero. And, finally, D4 is also involutive because its dimension is 4. Therefore the system is static feedback linearizable and its flat output j/2 can be computed from the ortogonality between V7/2 and D3. The solution of this linear system leads to a system of partial differential equations. One solution of this system is —s J + X2£6 COS X5+ i 2x1 cos X5— U2 sinxs . £3—S 2 ^X4X6 smx 5 which coincides with the solution found using the quotient module algorithm. Chapter 5 Linearization by prolongations of 2-input systems This chapter deals with the problem of linearization by prolongations for 2-input systems. A necessary and sufficient condition for a two input system to be linearizable is derived. As an application, two flat outputs for the VTOL and a planar model of a ducted fan are obtained. These examples were thought to be not linearizable by prolongations in the existent literature ([59], [23]). Another example, which shows the sharpness of the bound obtained, is also given. 5.1 Prolongations of m inputs are not necessary The following proposition will be helpful for establishing conditions so that a two input system may be linearizable by means of prolongations. Thanks to this proposition, it suffices to prolong the system by derivatives of just one input. In other words, prolongations by derivatives of both inputs are not necessary. This proposition has been already proven in [59], but our proof uses a different approach. Proposition 5 If the system: x e Rm ¿ = f(x)+gi{x)ui+g2{x)u2 is linearizable by derivation ofu\ n\ times and u2 n2 times (with n\ > ri2 > I), then the system is linearizable by derivation of u\ n\ — 1 times and u2 ri2 — 1 times. Proof: Let E ^ and S ^ l J be the systems obtained by prolongation of u\ n\ times and u2 n2 times, and u\ n\ — 1 times and u2 n2 — 1 times respectively. The drifts associated to these systems are ni-l f 1 = ß f + giVn+x + 522/n+ni+i + ^2 yn+j+i-rT—; 55 ri2-l + ] C Vn+m+j+i-g- ß ; 56 CHAPTER 5. LINEARIZATION BY PROLONGATIONS OF 2-INPUT SYSTEMS for system S"?. And ni-2 f = f + 9lVn+l + 92yn+ni+l + J2 g n2-2 ; + JZ Vn+j+1^- ^»^dyn+j. g Vn+m+j+1 ^ j ^ m ^ ^ for system S„j_î- While the input fields are 01 and = »T vyn+n\ 2_ 32 = " uyn+ni+ri2 2_ ° dyn+ni-l dyn+rn+712-l for the systems T^\ and S ^ l J respectively. By hypothesis, the system E"J is static feedback linearizable. Therefore, the following distributions are involutive and constant rank: i _ d d üyn+ni ¿ w d 0yn-^.nijrn2 d C'?/n+ni+n2 uyn+ni It is a straightforward computation to check the following equalities d nl d d d dyn+rii oyn+ni—i Oyn+ni+n2 oyn+ni+n2—i d dyn+ni d d dyn+ni-Ti2 dyn+ni+n2 for all i < m — 1. And x 712 While __ d . . d uyn+ni—1 uyn+ni—l—i „a d dyn+ni+i . d d ^í/n+ni+n2—1-i ^yn+n\+n2—l for all i < U2 — 2. And 2 _ d d d d Cyn+ni-1 Cyn+ni-n2 Cyn+ni+n2-l Öyn+m+1 Therefore, for all i ^ ri2, D} = D>®< 9 d OVn+ni (Jyn+ni+Ti2 For i > n2, some computational lemmas are required. Lemma 9 n l _ d Oyn+ni wÄere d oyn+m-ri2-i d d j 0 yn+n 1 +n 2 Cj/n+m+l / , ni—1 7 = / + 5iyn+i + X) 2/"+i+i" 5.1. PROLONGATIONS OF M INPUTS ARE NOT NECESSARY 57 Proof: The proof will be done by induction. For i = 1, adfig2 = [A02] = [71,92} + 2M+m+l[S2,S2] = [7\ff2] Assuming the equality is true up to order i, adif11g2 = [f1,adiflg2] can be replaced, through application of the induction hypothesis, by [f\adLig2] which is equal to Î12-2 ß X > a d 7 ]92] ac¿t S2 + foVn+ni+l + Y) î/n+ni+j+lâ However, due to the involutivity of the distributions and the next lemma, the last summand belongs to D„2+i. Therefore, aéXlg2 can replace aétlg2 in D * 2 + i + 1 . L e m m a 10 odjii/2 -hifa+ 3/n+ifol, 52] where hi is a function depending on the variables (x,yn+i,... , y n +j_i) Proof: It is done by induction. When i = 1, adjig2 = [f,g2] + yn+i[ffi,S2] Assuming the trueness of the equality up to order i, actt 1 52 = [7,adLig2] = U^ig-^l + U,Vn+i\9u92]] = if,hi—} + [f+giyn+1 L + Y, d «i+ir- + yn+i+l[9ii92\ where hi+1 ~dx = ^'hidx^ + Vn+i\f>\9u92]] +Vn+lVn-H\gi,\gi,Sn]] 58 CHAPTER 5. LINEARIZATION BY PROLONGATIONS OF 2-INPUT SYSTEMS Lemma 11 x ni+i _ —^ d d d d o ' • • • i p, ' p, CJ/n+ni Cyn+m—n2—i uyn+ni+n2 öftere ni—2 1=1 , i•••' a Oyn+ni+i i Q-Q-f / o + 9lVn+\ + 2_s Vn+j+i: j=¡1 dyn+j Proof: The statement is clear for all i <n\ since adtlg2 = adL2g2 because of the former lemma. When i = ni, = [f.adjr1^] = a<ñ\g2 = [7\adlr1g2\ = adllg2 + 2/n+mta / Cj/n+TH-l ,a^_1p2] which, thanks to the former lemma, is equal to / acfög-2 + yn+m\9u92] Notice that 51,52 €E D\ C -D^ 1+n2 _ x . Therefore, due to the involutivity of this distribution, adH\g2 can be b replaced by adTLlg2 in -D n i + n 2 Lemma 12 D2 "2 = l d d d ' dyn+m-ri2-i-l dyn+m-l dyn+ni+n2-l d ¿ ¿tyn+m+l / Proof: This proof is equal to the one made for the first of these lemmas. Summarizing, Df®<<yyn+ni ,5 cl yn+ni+n2 >=A--i Since the former equality has also been proven in the case D?© < - - * - , — 2 CJ/n+ni c '2/n+ni+n2 Vi>n2 i^ri2, >=DU Vi Therefore, since the hypotheses of lemma 5 are satisfied, the static feedback linearizability conditions are the same for both systems. • Corollary 4 If a system is linearizable by prolongation of ui n\ times and u2 n2 times (with "1 > 1^2)) then the system is linearizable by prolongation of u\ n\ — n2 times. Proof: It is the result of applying the former proposition n 2 times. • Thus, in the following, prolongations by derivatives of just one input will be taken into consideration for two input systems. The same proof can be done for a system with m inputs, where only m — 1 have to be prolonged. 5.2. MAIN 5.2 59 RESULT Main result The following theorem, which establishes a necessary and sufficient condition for the existence of prolongations, also provides a finite algorithm to decide whether or not a system is linearizable by prolongations. Theorem 4 The system x E Rn x = f{x) + gi{x)u1+g2(x)u2 is linearizable by prolongations if and only if one of the following systems is static feedback linearizable: ( x = f(x)+gi(x)yn+i+ Vn+j - g2(x)w2 Vj = 1 , . . . , k - 1 Vn+j+i Vn+k = x yn+j = f(x)+g2(x)yn+i+ = Vn+j+l V)i or _ Sfc : Vn+k gi(x)wi = W2 where k = 1 , . . . , 2n — 3 and yn+j = u^~1) j = (or, respectively) yn+j — u2 Vj = 1 , . . . ,fc- 1 l,...,k j = 1, • • •, k are the new state variables and w\ = u\ w2 = u2 Í respectively w\ = u\ w2 = U2 ) are the new inputs. Proof: It will be proven that the static feedback linearizability conditions for the system: : £¡2n-3 * x Vn+j = f{x)+g1{x)yn+i+ = Vn+j+l 2/3n-3 = g2{x)w2 Vj = 1 , . . . , 2n - 4 Wi and the static feedback linearizability conditions for the system: s i : S = f(x)+gi(x)yn+i+ x Vn+j = Vn+j+i Vn+i = g2{x)w2 YJ = 1 , . . . , I -1 m with I > 2n — 3 are equivalent. So, this being proven, if a system is linearizable by prolongations adding I derivatives of u\ (I > In — 3), then it will also be linearizable by prolongations adding 60 CHAPTER 5. LINEARIZATION BY PROLONGATIONS OF 2-INPUT SYSTEMS only 2n — 3 derivatives of iti, and, obviously, the same fact occurs with U2- Therefore, a finite algorithm for checking the linearization by prolongations will be to check the static feedback linearizability conditions for S¿ and E¿, VA: = 1 , . . . , 2n — 3. As the same proof is valid for both types of prolongations (resulting from adding derivatives of u\ or U2), it will be proven just once, in this case for prolongations of ui. The details for the prolongations by derivation of U2 can be rewritten in the following proof by changing u\ by U2, and viceversa. So, let f2n~3{x,y) 2n-4 = f(x)+gi(x)yn+1 p. + ] £ Vn+j+i^ j=1 OVn+j be the drift associated with the system S2n-3> and let 9Ín-3(x,y) g22n-2(x,y) = -^— = 92{x) be the its control fields. I¡2n-3 1S static feedback linearizable if and only if the following distributions are involutive and constant rank: D T~Z = <T^3,---^pn-31^,92,...,adif2n_3g2> Denote T]k = adkpn-392 VA; > 0 The following lemmas clarify what {% k > 0} are. Lemma 13 rjk e S =< £ > Proof: It is proven by induction. For i = 1, 2n-4 ß m = [f + 9\Vn+i + X) vn+j+i-g-—-,92) = [f,92¡ + yn+i[gi,g2] e S Assuming T]Í E S, i <k, 2n-4 ß Vk+1 = [f +gWn+l + J2 yn+j+l-Q- ;,T}k] = 2n-4 f + yn+i9i + Y, yn+j+i-Q-—: I (%) - mif + Vn+igi)G S thanks to the induction hypothesis. Therefore, the maximum independent number of these functions is n. In other words Vn € < 7 ? 0 , •••>»7n-l > 5.2. MAIN RESULT 61 L e m m a 14 rjk depends only on the variables x,yn+i,-••,Vn+k- Proof: As in the previous lemma, Vi = [f,92]+Vn+i\gi,92] On the other hand, [f2n~3,Vk{x,yn+i,---,yn+k] Vk+i = where the induction hypothesis has been applied. Therefore, it is clear that £pí Vk+l = [f + Vn+l9l + 2 _ , Vn+j+l-g- d d -, r]k] + [Vn+k+l-Q- ,Vk] which depends only on the variables x,yn+i, • • • j Vn+k+i. Let us now compute ad3pn_3 g L e m m a 15 1. 1 Oyzn-Z Oyzn-z-j ad2ß-%1 Oyzn-3 = -gi Proof: 1. For j = 1, d . d d . d adpn-î-z = [yzn-3-z 1 ä J — — ) "5 J— ô ipn-3-z iy3n-3â 1 dyzn-z oy3n-2 dy3n-3 oyzn-2 Assuming the equality is true up to j , 1 ! OyZn-3 Oyzn-Z Oy3n-3-j where the induction hypothesis has been applied. This Lie bracket is equal to 2n—4 o 1 i a £ 1 i+1 [f+9iyn+i+ £ ^+i+i5r-»(- ) Är —l = (- ) ¿TÍ oyn+j dyzn-3-j 2. Prom the former proof we have d_ 4 a </ f22n-3 nc " ôy 3 n -3 a dyn+i Therefore, 7 ôy 3 „-3 oyn+i o sr- 2 —1 dy3n-3-j-i 62 CHAPTER 5. LINEARIZATION BY PROLONGATIONS OF 2-INPUT SYSTEMS It has been pointed out that 77n G< 770,... , 77n_i >• But n may not be is not the least integer satisfying such property. So, let us define r = min{k\r]k e< Vo, • • • ,Vk-i >} There are two possibilities for these distributions: 1. r = n 2. r < n 1. If r = n, the distribution D^[3 is equal to d d <í?o,···,%-i!^ >···in > Oy3n-3 OJ/2n-2 Thus, this distribution being involutive, D].n~3 are involutive for all k > n. The reason for this fact is that r^2n-3 d 9 oyzn-z oyzn-z-k and the Lie brackets [ Cy3n-3-i , Tfr] G 5 = < 770,. -., 77„_i > V* < k, \fj < n - 1 So, system I¡2n-3 is linearizable by static feedback (in the case r = n) if, and only if, [77j,77fc] €<r]o,...,î}k > Vj<k, V* = l , . . . , n - 2 Note that fe'p ] = 0 V -?<fc> dy3n-3-k because t)j depends upon the variables x,yn+i,... 3n - 3 - k > 2n - 1. Vfc = l , . . . , n - 2 ,yn+j and n + j < 2n — 2, while 2. If r < n, the conditions for the distributions of the system Y¡2n-3 to be involutive are • Vfc < r, d =<T ? 0 ,...,r7 j f c ,— ,...,> oyzn-z dyzn-z-k Conditions: [77^, 77*] G< 7/0,..., rjk >, V7 < fc. The reason is the same as in the former case. Df 2. MAIN RESULT • For k = r + 1,..., 2n — r — 4, have no conditions to check. Again, the reason is the dependence of r)j on x and the fact i Vn+li • • • i Vn+j n + j<n + r<n + r + l<3n — 3 — (2n — r — 4) • The following distributions to be studied are 7-)2n-3 _ ., n 2 " - 3 -s •^n-r-S+A: ~ ^ -ty2n-r-4î o > •••' Q -•" Cyn+r VUn+r-k with A = 0,..., r — 1. The involutivity conditions are h ,T7j] e<i7o,-..,í7r > Vr>i>r-fc oyn+T-k Particularizing for k = r — 1, £>2 2 "=4=< % , . . . , r ? r , ö ,···,or < <jyZn-Z OVn+1 • In order to establish the elements belonging to i^n-f+t' following lemma: Lemma 16 D%-Ui =< DlT-l+n «tjOi > w ^n z ^ 0, we Vi > 0 Proof: when i = 0, Thus, When i > 1, in fact we will prove, by induction, the following equality acf/2n_3 0i = ad/0i + •D2nïf-4 where ady5i + -D|£+3_4 means afygx + h heD%-?_4 64 CHAPTER 5. LINEARIZATION BY PROLONGATIONS OF 2-INPUT SYSTEMS For i = 1, [/ 2 n _ 3 ,yi] = [f,9i] + yn+i[gu9i] = [f,9i] For i > 1, • acTpn-sgi = [f + giUn+l + Vn+2-^ 9 h . . . + Z/3n-3"5 9 > adpn-sgi] Using the induction hypothesis, this expression becomes: fi [f + 9iyn+l ri , ad}"1 + • • • + VSn-3^ + yn+2TTi oy 3n _4 oyn+i + D^_s] = J by construction of the distributions -D^n+i^-s and £>^+f_4. Moreover, the elements d yi) o d oyn+l ac n 2r belong to D ¡^_i (^ ^ *' * tivityofI>^_4, nev >••• ' o Oyzn-A 2 belong to -D "-!)- Therefore, because of the involu- \giVn+l + yn+2JT, + • •• + y3n-3^¡of^Çl] Cf/n+1 Cí/3n-4 £ ^2nïf-4 The proof of 5.1 ends by using this fact and [/, ady_1yi] = ad^gi. • This concludes the study of system X¡2n-3Now, the system E2n-2 w * u be studied. Let 2n-3 2n 2 / " = / + yiyn+i + 53 a yn+j+l dy~ be the drift associated with the system E271-2, and let „2n-2 _ d ffl = ä Oy3n-2 „2n-2 _ „ #2 = 52 be its control fields. The distributions associated with this system are • \/k < r, n„_o K d Ó dyzn-2 oy3n-2-k So, the equality Df1-2 e ^ — = Df - 3 8 < ^ - > holds, and the hypotheses of lemma 5 are fulfilled. Therefore, the involutivity conditions are the same for I» 2 " - 2 and for D2kn~3, with k < r. 5.2. MAIN RESULT 65 For k = r,..., 2n — r — 4, 2n-2 d OyZn-2 + & Öy3n_3_fc satisfies the equality Z 2 @ < ^— = DfDU' k+l k dynn-2 3 > As the hypotheses of lemma 5 are also fulfilled, the involutivity conditions are the same for Dll~2 and for D2kn~z, with 2 n - r + 4 > f e > r . In fact, the former equality £2n-2 fc+l = jD2n-30 ~ ~k < _JL_ . ô y 3 n _2 -2 > is also valid for any k greater or equal than r. Thus, the static feedback linearizability conditions are the same for both systems when r <n. If r = n, the distribution D2^2 is equal to d < Vu,... ,---,^ ,T)n-i, d > oyzn-2 oy2n-i Thus, this distribution being involutive, D2kn~2 are involutive for all fe > n. The reason for this fact is that d _ <, -T 7 o , . . . , 7 7 n _ i , ^ - d Dtfn-2 = ,.-•,«> k and the Lie brackets [— rjn-i > -,rij] eS=<Tjo,..., Vi < fe, Vj < n - 1 So, system XÎ2n-2 is linearizable by static feedback (in the case r = n) if, and only if, [Vj, %] G< *7o, • • •, % > Vj < fe, Vfe = 1 , . . . , n - 2 m which are the same conditions as those for E2 n -3REMARKS 1. The proof required in order to show that X¡2n-3 is static feedback linearizable if, and only if, E; (Vi > 2n — 3) is static feedback linearizable, is the same using the fact ^ = ^Kii®<ÂTOyZn-2 s r ~ vyZn-Z+h > Vi r - where h = I — (2n — 3), and the lemma 5. This means that it is not necessary to add more than 2u — 3 derivatives of the input in order to check whether or not a system is linearizable by prolongations. In other words, it is sufficient to check the static feedback linearizability conditions of Ei,...,E2n-3 (and Ei,...,E2 n -3) in order to check the linearizability by prolongations of a certain system. 66 CHAPTER 5. LINEARIZATION BY PROLONGATIONS OF 2-INPUT SYSTEMS 2. Let us compare the static feedback linearizability conditions of I¡2n-3 anc ^ ^2n-4- In order to be involutive, the distribution 2n-4 D n-2 = < d ,Vo ) "lui • • • i ¡a Oyzn-A ;Vn-2 > o 0¡j2n-2 needs an extra condition (which is not necessary for the system S2 n -3): 1 '¿b^ ^or, in other words, h > Vn-2] € < 770,..., r/n_2 > oy2n-2 This is the reason why, in general, 2n — 4 or any smaller number cannot be a bound of the number of derivatives added to the original system. 5.3 Examples 1. As an application of the former theorem, a static feedback linearizable prolongation will be sought in the following system: X\ = ¿2 =' ¿3 = ¿4 = ¿5 = \, ¿6 = X2 Wising £4 —1 + x 6 Micosis — U2ecosxs + (5.2) U2£sin:E5 U2 This system comes from the Vertical Take Off and Landing (VTOL) aircraft model ([30], [23]), a model of a mechanical system with two inputs, whose evolution is restricted in the vertical plane. The original equations are: x = iii sino — p e c o s o y= uicosö + U2esm9 —1 (5.3) U2 Ö= The changes made in 5.3 to obtain 5.2 are x\ = x X2 = x x3 = y Xi = y x¡ = 6 XQ = 9 5.3. EXAMPLES 67 To apply the algorythm explained above, the input U2 is derivated up to 2n = 12 times. And the static feedback linearizability of the systems = = = = = = Vi 2/2 2/3 2/4 2/5 ye S f c :< k V2 -y7ecosy5 V4 -H-y7esmy5 ye Vi + uisiny 5 + v\ cos y5 y6+fc U2 is checked (VÄ; = 1,...,12). A straightforward computation shows that Si, S2, S3 does not satisfy the static feedback linearizability conditions. System S4. The distributions ( Dt=< V 0 \ siny 5 0 cosy 5 0 0 0 0 0 0 ) I siny° ^ (°\0 5 2*=< V 0 cosy 5 0 0 0 0 0 i 0 0 0 0 0 0 0 0 j ll / ) \ (°\0 1 0 0 0 0 0 0 0 > vW -siny5 \ y6 cos y5 - cos y5 -y 6 sin y5 0 ) 0 0 0 0 0 J f °0 \ 0 0 0 0 0 0 1 > 68 CHAPTER 5. LINEARIZATION BY PROLONGATIONS OF 2-INPUT SYSTEMS and 1D¡ = ) Vi ) ) -siny5 \ ye eos y5 - eos y5 -y6siny5 0 V 1 o ( o o o o f 0 \ 0 0 0 1 ) \o) i -2y 6 cosy 5 \ - y | s m y 5 + y7Cosy5 2y 6 siny 5 -y|cosy5 -y7smy5 0 o o o o ) / o o o o V o o o o o o o o o * / 0 \ V o f0\ o o o o j < 0 \ siny 5 0 cosy 5 0 o o o o ( ) 0 0 1 0 > ) \0 ) axe involutive because the Lie brackets between two of their elements are zero. The following distribution is ( D¡ =< D¡, 3y¡siny 5 -3y 7 cosy5 \ ( - y ¡ cos y5 - y7 siny 5 )y 6 - 2y6y7 siny 5 + y8 cos y5 3y|cosy 5 + 3y 7 siny 5 (yl siny 5 - y7 cos y5)y6 - 2y6y7 cos y5 - y 8 siny 5 0 0 0 0 o' 0 0 0 0 0 1 0 0 Voy 0 It is also involutive because ( S =<no = 0 \ siny 5 0 cosy 5 0 0 0 0 0 0 / i »7i = -siny5 \ y6 cos y5 - cos y5 -y 6 siny 5 0 0 0 0 0 0 i % V - 2 y 6 cos y5 - y | sin y5 + y7 cos y 5 2y 6 smy 5 - y | cos y 5 - y 7 sin y5 0 0 0 0 0 0 > 5.3. 69 EXAMPLES ( m 3y¡siny5-3y7cosy5 \ ( - y ¡ eos y5 - y 7 siny 5 )y 6 - 2y6y7 siny 5 + y8 eos y5 3y¡cosy 5 + 3y 7 siny 5 {yl siny 5 - y7 eos y5)y6 - 2y6y7 eos y5 - y8 siny 5 0 >= 0 0 0 0 0 {(ai,a2)ûi3 5 a4i0,0,0,0,0,0) where a¿ are functions of yi,...,yio} and all the Lie brackets between two elements of D§ belong to S. The next distribution is ( D\=<Dl 0 \ - e cos i/s 0 esiny 5 0 > 1 0 0 0 0 because V2 —ey 7 cos V5 \ V4 l + ey7siny5 ye ,V3 V7 L\ V8 V9 yio 0 J eS 70 CHAPTER 5. LINEARIZATION BY PROLONGATIONS OF 2-INPUT SYSTEMS and also the Lie brackets between two of its elements are in S. Finally e cos 2/5 ey6smy5 -e sin y5 ej/6 cos J/5 1 >=R 10 Di --<D*4) 0 0 0 0 0 which is obviously involutive. 2. Another interesting example, very similar to the VTOL, is exhibited by a planar model for the ducted fan ([46],[47] (veure ST)), given by: Xl X2 T£ ¿2 COS £5 ^•sinxö J m ¿3 ^ sinx5 + ^ cos ¿4 ¿5 £4 £4 x5-mg X6 T ¿6 5«! where m, J and r are constants. This system was thought to be not linearizable by prolongations ([59]). However, we are in fact able to prove that it is linearizable by prolongations. Take the prolongation coming from adding four derivatives of u\ and zero derivatives of U2Xl ¿2 ¿3 ¿4 ¿5 X2 §£ COS X5 Vi sin £5 X4 = ^ s i n x 5 + ^cosa;5 -mg XQ x XQ li ¿7 2>8 ¿8 ¿9 Xg Vi The distributions associated with this prolonged system are D0=< d dxiO ,Vo> a DI=<D0,—,T)I> 0x9 71 5.3. EXAMPLES D2=<Du-—,r]2> ox8 where sinxs 9 i cos 15 8 m 9i2 m 9x4 _ '0 sin i s 8 _ J6COSX5 9 _ cos xs 5 m on m ÔX2 m 9x3 '1 „ '2 2 • 2x6 cos X5 9 1 /X f i sinx5 m 9xi "> m _ xs sin xs 9 ni 9x4 rx7CQSX5-v 9 mJ ' 9x2 2 /X 6 cosx5 rxTSinx-n 9 ^ m "•" mJ ' 9i4 1 2x6 sin X5 9 w» 3x3 Remark that all the Lie brackets between their elements are zero. Therefore, are involutive. D3=<D2,¿-r,m> DQ,DI,D2 where 773 is such that d d d d <r/o,m,ïï2,%>=<^,^,^,^> Notice that the Lie brackets between the elements in D3 belong to < eiao, 771,772, % >• So, Dz is also involutive. Di =< D3, cos£5 d m 0x2 1 SÍTÍX5 d m 0x4 r d J 0x6 h — •»— > And, because of r [Vi, cos 25 d sinis d r d , "ô— + - 0 — + T T T - G < ^Vum^s m OT2 JTi öa;4 J OXQ > _, n _ w. V? = 0,1,2,3 w Di is also involutive. Finally, D5 = R . That is to say, the static feedback linearizability conditions are fulfilled for the prolonged system, or, in other words, the planar ducted fan is linearizable by prolongations. 3. This example shows the sharpness of the bound 2n — 3. Xi = X2 X2U1 X2 = X3 X\U\ Xk = Xk+l Xn-\ = Xn = fc = 3,.. .,n-2 U2 Xn-\U\ The system is clearly not static feedback linearizable since d d d d DQ =<x2^— + xih xn-i-—, > ax\ 0x2 oxn oxn-i 72 CHAPTER 5. LINEARIZATION BY PROLONGATIONS OF 2-INPUT SYSTEMS is not involutive. It is also easy to check that the system is not linearizable by prolongation of U2'- let ví •j d ^ d d be the drift of the system 3 prolonged with r derivations of U2, and let 9i=X2-z d d d i-xiha;n_i-— ¿tei ox2 oxn _ d oyn+r g2 = ~ be its corresponding input fields. Then, d d Di=< ä iñ »»70,»7i > Oyn+r OPn+T-l where d ' dx\ dx2 dxn and Vi = [f,9i¡ = x2-¿— + \x3 - XI)T, ¿7X2 OX\ h Vn+r OXn Therefore, + ( 2:r i [Vo, Vi] = -2XÍ-Q¿~ ~ x ^ ^ ^ ^ Hence, P i is not involutive. Finally, let us try with a r-prolongation of u\. Let - ^ d d d d £í d be the drift of such a prolongation, and d _ 9\ = -or. 92 dVn+r d dxn-i its corresponding input fields. The distributions associated with this system are: - d d D =< o a ' oyn+r > oxn-i which is involutive. 1 < i < n — 3, - d Vi —< -5 oyn+r d 1 •• •)ä óyn+T-i d )ö dxn-i d 1 Ö ax n _ 2 d •" 2/n+l-S oxn d 1 •••)ä axn-i-\ . .,,;_! 1" (,~1) d Vn+i-Z dxn > 5.3. 73 EXAMPLES are involutive if, and only if, n + r — i > n + i or, equivalently, if, and only if, r > 2i. Dn-2 =< Dn-3, -=. , (1 + Vn+l)^— + (-l)n~3î/2n-2^— > 1< t < n - 3 oyr+2 oxi dxn is involutive if, and only if, r + 2 > 2n — 2; that is to say, if, and only if, r > 2n — 4 (r > In — 3). On the other hand, the main result of this chapter states that r < In — 3. Therefore, since d -FT < ^ - > C D„_i ox there are no more conditions to check. Hence, the system 3 is linearizable by adding exactly 2n — 3 derivatives of the input tti. So, in general, the bound 2n — 3 cannot be improved. 74 CHAPTER 5. LINEARIZATION BY PROLONGATIONS OF 2-INPUT SYSTEMS Chapter 6 Improvement of t h e bounds for 3-input systems In Chapter 7, a bound in the number of integrators needed for the linearization of a nonlinear control system will be obtained. This bound is an improvement on the bounds existent in the literature ([59]) for systems with 4 or more inputs. However, for systems with 3 inputs the bound still needs to be better. This chapter, therefore, is devoted to the search for an inprovement on the bound for three-input systems obtained in Chapter 7. 6.1 Main Result The following theorem, which states a necessary and sufficient condition for the linearizability by prolongations of a three input system, also provides an algorithm for determining whether or not a 3-input system is linearizable by prolongations, and improves on the bounds that have appeared before in the literature. Theorem: The system S defined by 3 is linearizable by prolongations if, and only if, at least one of the following systems £J^ is static feedback linearizable. S^J is given by x = f{x) + ghl (x)yn+i + gh2 (x)yn+kl+i + gh3 (x)w3 Vn+j =Vn+j+l] j = Vn+kx Vn+ki+l Vn+ki+ki = Wl =Vn+ki+l+l] = W2 / = 1 , . • • , fo 75 l,...,ki-l _ 1 76 CHAPTER 6. IMPROVEMENT OF THE BOUNDS FOR 3-INPUT SYSTEMS k\ G { 1 , . . . , 2r + 1}, &2 € { 1 , . . . , 2n — 2 + r}. The new state variables ys and inputs Wj are related to the old ones by Vn+j = V% 1,. . . , & 1 3 = (f-1) i i .,¿2 = < 2 ) = «As W3 being {/ii,/i2,^3} = {1)2,3}. And, finally, r is defined in the following way: let consider the prolongation EJJ. Compute the Lie brackets T)i = adjgh3 where / is the drift associated with the prolonged system, and define r = min{i e {0, n - 1} | ifc+i € < rj0, • • •, m >} Proof: There is no loss of generality in assuming that hi = i, i = 1,2,3. The proof is based on the lemma 5 from Chapter 2, and on the justification of the following items: 1. The static feedback linearizability conditions for the system E2r+i + r and E£J (with r\ > 2r + 1 and r% > 2n — 2 + r) are the same. 2. The static feedback linearizability conditions for the system E | " _ 2 + r and Ej^ (with k\ < 2r + 1 and r% > In — 2 + r) are the same. 3. The static feedback linearizability conditions for the system E ^ + i and E ^ (with ki < 2n — 2 + r and r\ > 2r + 1) are the same. This being proven, and in order to check if a system is linearizable by prolongations, a finite algorithm can be applied: it is only necessary to check if any of the systems EJ^ (with k\ < 2 r + l and k% < 2n — 2 + r) is static feedback linearizable. Obviously, this fact must be checked for any permutation of the inputs. 1. First of all, let us study the static feedback linearizability conditions for the system V>2n-2+r ¿J 2r+l • Let f = f + QlVn+l + 022/n+2r+2+ + L,¿=lVn+j+155£7 + 2w=l 2/rx+2r+l+i+lei, n+2r+1+( be the drift of the system S2r+i + r {% must be understood as (j/i,... ,yn))- And let _ d 9l = ö oyn+2r+l _ d 92 = ö Cj/3n+3r-l __ 53 = 53 MAIN RESULT 77 be the vector fields associated with the new inputs. Using the conditions stated in Chapter 2 to check whether or not a system is static feedback linearizable, the following distributions of E J H T I ^ must be involutive and constant rank: d d _ >. n2r+l,2n-2+r _ _ . ^0 - ^ dyn+2r+i ' dy3n+3r-i ' 93 > n2r+l,2n-2+r n2r+l,2n-2+r _ for all t such that dim p2r+i,2n-2+r < 3 n+ 3 r _ fF n 2 r + l , 2 n - 2 + r i . L Let us examine these distributions in some detail. First of all, a computational lemma is stated and proven. L e m m a 17 (a) aJji—J1 \Oyn+2r+lJ )=(_i)'_i <J¡¡n+2r+l-i Vi<2r (b) ,oyn+2r+lJ (0) adl7(ir-?S )=(-l)<- \Oy3n+3r-lJ Vi<2n-3 + r 9y3n+3r-l-i (d) Proof: (a) This part will be proven by induction. The case i = 1: d d d d ] = [ n+2r ]= — ) = [/ ad- ( n T f dyn+2r ' dyn+2r+i dyn+2r \dyn+2r+iJ ' dyn+2r+i Assuming the equality is true up to i, the case i + 1 (with i + 1 < 2r): +I [7 °4 Gdb)= '°4( s Applying the induction hypothesis, this becomes n+2r+l/ 78 CHAPTER 6. IMPROVEMENT OF THE BOUNDS FOR 3-INPUT SYSTEMS (b) Using the former equality °4'+1 ( d b ) = [7 ^ ( d b ) 1 •u' (-1,2r Gwb;) 1 = (c) Again, it will be proven by induction. When i = 1 ad 7 (ô»3»+3r-J = W » öy3n+3r-J = [Mn+3r-l dy3n+3r-2 ' öy 8 »+3r-J = Ô1/3n+3r-2 Assuming the trueness of the statement up to i, the equality will be proven for i + 1 (withi + 1 < 2 n - 3 + r). 4•f + 1W2/3n+3r-l/ f ^ — ) = [ / ,' < ( f\dy3n+3r-it which becomes, by application of the induction hypothesis, [7,(_1)< U 3 JLJ ] =y[3n+3r 1i (_1)i ~- U J L J ' U 3 JLJ \9î/3n+3r-t-2/ (d) Using the former equality f \9y3n+3r-lJ ' S \dyzn-\-Zr-l ) n = [/,(-i)2n-3+rii3% 3 n + 3 r - l - 2 n + 3 - r >/)] = (-ir 1 i7.' Öt/n+d2r+2 = ("ir W+2T+2, ^ Oyn+2r+2 Remarks: (a) Denote rjk = adjÇ3 Vfc < r. Note that * e 5 = < ¿> ] = {-l)T92 ]= 6.1. MAIN RESULT 79 Proof: Vi = [7.53] = [(f + giVn+l + 92yn+2r+2),93] £ < ^ > Vk+l = [7i Vk] = 2r ^ 2n—3+r o [(/ + ffiyn+1 + 92Vn+2r+2) + T2 Vn+j+l^ + T] J/n+2r+l+2+lö jTx OVn+j ¡r[ Let us recall that, by the induction hypothesis, rji- 6 < ^ , Vk] Oyn+2r+l+l > . Therefore, ?7A:+1 = / ( % ) - % ( ( / + giVn+l + 92yn+2r+2)) which belongs to < J j > . • (b) % depends upon the variables ^i yn+l, • • • , yn+k, yn+2r+2, ••-, yn+2r+l+k Proof: Vi = [f,93¡ = [(/ + Slî/n+l + 52yn+2r+2),Ö3] = [/. 53] + yn+l [Si, 53] + yn+2r+2[32, 33] which depends on x, yn+\, yn+2r+i+iVk+l = [f, H(x,yn+i,. . . ,yn+k,yn+2r+2, • •• • ,yn+2r+l+k)-Q-] where the induction hypothesis has been applied. Therefore, it is clear that % + i = [/ + 3 i y n + l + 32!/n+2r+2 + £ lfe+J+1 g ^ . + g tfe+ír+l+l+l ^ ¡ ^ > ^ depends upon the variables x, yn+i, • • •, Vn+k+i, Vn+2r+2, • • •, Vn+2r+i+k+iLet us enumerate the conditions to be checked in order for the corresponding distribution to be involutive. Henceforth, when the involutivity of Dj is studied, we will assume the involutivity of Df, I < j — 1. (a) Distributions D f + ^ - ^ t k<r. d 'd|/n+2r+l' jD 2r+l,2n-2+r d = d ' %n+2r+l-fc'Öy3n+3r-l' d " dVZn+Zr-1-k Taking into account that r/¿ depends on the variables £ , y n + l , • • • 1 yn+k, yn+2r+2) • • • , yn+2r+l+fc note that n + k<n + 2r + l-k involutivity conditions are and n + 2r + 1 + k < 3n + 3r - 1 - k. So, the only [m,Vk] e<r?o,...,r/ fc > • 80 CHAPTER 6. IMPROVEMENT 2+r (b) Distributions D^k1,2n n2r+l,2n-2+r ^r+k _ - OF THE BOUNDS FOR 3-INPUT SYSTEMS , 1 < k < r. The condition r¡r+i €< rjo,...,T¡r > implies d d 'dyn+2r+l' d ' dyn+r+l-k' d dî/3n+3r-l ' dy3n+2r-l-k The additional involutivity conditions are: d ßVn+r+l-k » VT+1-J e<T)o,...,T]T > VI < j < k, because fy+i-j depends on, among other variables, y n + r + i _ ¿ (take j = k). Note also that 3n + 2r — 1 — k > n 4- 2r + 1 -j- k. So, there are no more conditions except those explained above, (c) The distribution D^i 1 ' 2 " -2 " 1 " 7- is spanned by d d < T?0>···>'7r>-3 oyn+2r+l d '•••>» 'Sliö Oyn+1 Oy3n+Zr-l d '•••'"ñ > Oyzn+r-2 If r = n — 1, then <7?o,...,r? r > = 5 Therefore, n2r+l,2n-2+r L· '2T+1 _ .. m — _ 9 " ^ '/Oi • • • ) '/r» o Cyn+2r+l is involutive since d d i • • • i "5 ) "5 Ctyn+l to,^]eSGD¡;t¡'2n-2+r Cy3n+3r-l d )— '> ! "ö oy3n+T-2 vi> n And, for the same reason, any distribution with subindex greater than 2r + l will also be involutive. Thus, it can be assumed that r < n — 2. This implies 3n + r - 2 > n + 3r + l (6.1) Proof: The above equality is equivalent to the following: 2n - 2 > 2r + 1 which, in turn, is equivalent to r < n — 3/2. The facts r < n — 2 and r, n integers imply r <n — 3/2. • 6.1 and the dépendance of r¡r (see the above remark (b)) on the variables £) J/n+li ••• i yn+r, 2/n+2r+2) • • • i î/n+3r+l 6.1. MAIN RESULT lead to the following involutivity conditions: [Vk,9l] € < % , - • • ,Vr,01 > V0</c<r since [m oyZn+Zr-l-j ]=0 V0<i<2r + 1 ]= 0 V0<i<2r + 1 ]= 0 V0<j<2r and [gi OySn+Zr-l-j [51,5 Oyn+2r+l-j In order to proceed, another computational lemma is required Lemma 18 D%ll*¡-2+r = D2r+l,2n-2+r - w d n2r+l,2n-2+r # ,_ njj„ 2n-2+r f = f + 92Vn+2r+2 + Y) Q î/n+2r+l+i+l Ô ¿Z¡ Proof: It will be proven by induction. For ,7 = 1, oyn+2r+l+l adjgi = [/,5i] = 2r o [/ + 012/n+l + 022/n+2r+2 + X I 2/n+J+l ¿fy 2fí—3+r +" X o ?/nH-2r+l+i+l ¿ f y n + 2 r + 1 + i ' Si] = [/.0l] +!/n+l[01,0l] +ï/n+2r+2[02i0l] = = [/ + 52yn+2r+2 + 2n-2+r o X JZ¡ Vn+2r+l+l+l Ö , 0l] = 0^/01 OVn+2r+l+l because [01,01] = 0. Assuming that the statement is true up to j , û4+15i = [/.Q4ffi] 82 CHAPTER 6. IMPROVEMENT span in D 2r +i+j+i OF THE BOUNDS FOR 3-INPUT SYSTEMS » by application of the induction hypothesis, the same as [/, adjf-gi] = + [/, adJf-gi] + yn+i[gi, adjf-gi] + yn+2r+2[g2, adJf-gi]+ E£r24T yn+2r+i+i+iät^^MJf-9i] However, due to the involutivity of the former distributions, [guaéf-9l]eDlZl%-i+r So, ad^gi can be replaced in I>2ríÍ+"+i 2+r by adjr+1gx. m So, let us define 5j = ad^gi. Remarks: i. Sj€S Proof: It will be performed by induction on j . Si = [f,gi] = [f,9l] + yn+2r+2[92,gi] since gi depends only on the x variables. í j + 1 = [/,í i ] = / ( A j ) - í i ( / ) The induction hypothesis implies that ôj G 5. Therefore 5j+i e S, because the only members in / depending on x are f{x) + Vn+2r+292 ii. 5j depends upon the variables x, yn+2r+2> • • • > yn+2r+i+jProof: As has been said, ¿1 = [ft Si] + yn+2r+2Í92, 9l] which proves the statement for j = 1. By application of the induction hypothesis and the preceding remark Sj+i = [/» ¿j] is equal to [f,H(x,yn+2r+2, [/, H—} + yn+2r+2[92, H~\ • • • ,!/n+2r+l+j)^] = + ^[yn+2T+l+l+l^~——, H—} 6.1. MAIN RESULT 83 which will depend on the same variables as Sj, plus the variable j/n+2r+i+j+i • This last variable comes from dyn+2r+l+j'Hd¿] [yn+2r+l+j+1 Define also h = min-fj G {0, n - r - 2}\5j+i € < 770,. • •, Vr, 50,..., 6j >} (d) For all 0 < j < h, the distributions n2r+l,2n-2+r „ n2r+l,2n-2+r <- 9 d %3n+r-2 dy 3n +r-2-¿ x +1+J must fulfill the following conditions in order to be involutive: &'^ J€<7?o,...,Tjr,ío,...,íj> V0</c<r V0<i<j-1 Proof: The definition of h implies j <h<n—r—2. This is equivalent to n — j > r+2. So, 3n + r - 2 - j > In + r - 2 + r + 2 = 2n + 2r. Note that r < n - 2. Therefore, 3n + r - 2 - j > 2 n + 2 r > n + 2r + r + 2 > n + 3r + l Observe that n + 3r + 1 is the maximum subindex of the variables y appearing in r¡r, while 3 n + r — 2—j is the minimum sub index of the coordinate charts in D 2r +i+¿ • So, the Lie brackets between rjk (k < r) and g-—-—— > are vanishing. On the other hand, 3n + r - 2 - 2j > n + r - 2 + 2(r + 2) = n + 3r + 2 > n + 3r + 1. Or, equivalently, 3n + r - 2 - j > n + 3r + l + i Note that n + 3r + 1 + j is the maximum subindex that appears in ôj. m Now, the static feedback linearizability conditions for TTr\ (with r\ > 2r + 1 and r<i > 2n - 2 4- r) will be studied. More precisely, the systems S | " + | + r , S^+ 1 1 + r and S ^ 2 1 + r will be detailed (that is to say, systems where an extra derivative of u\ and/or ui have been added). (a) System El?+ 1 1+r . Note that for all j <r D2r+l,2n+r-l J d = < ' ' J ' Öyn+2r+l' d d ' 9 y n + 2 r + l - i ' Öy3n+3r' _ 9 ' <9y3n+3r-j > 84 CHAPTER 6. IMPROVEMENT OF THE BOUNDS FOR 3-INPUT O—l "I O T Î - I - 7 * Comparing this distribution with D • p2r+l,2n-l+re < J 0 ' d SYSTEMS , the following equality can be written: > = ^+1,2^2+^ J Oy3n+3r-l~j d_ < > oy3n+3r When r < j <2r £,2r+l,2n+r-1 d dyn+2r+l ' = < J ' ^ ^ ' 9yn+2r+l-j ' 9j/3n+3r ' d ' 9j/3 n+ 3 r _j and the same equality as above is fulfilled: £)2r+l,2n-l+r0 < J Ö Oy3n+3r_i_j > = p2r+l,2n-2+r8 < J d_ Öy3„+3r > Furthermore, this equality is also true for 1r + 1 < j < 2r + 1 4- h. Moreover, in all three cases, the hypothesis of lemma 5 are satisfied. Thus, one may be sure that with the involutivity conditions are the same for ¿}J r +i,2n-i+r and D2r+1'2n~2+r, 0 < j < IT + 1 + h. When j>2r + l + h, £>2r+l,2n+r-l = d d ' dVn+2r+l' 'Ôyn+i' ' ->2r+l,2n-2+r «-r^»-^ ^ I ) ' d d dy3n+3r' ' dy3n+3r-.;-l Ô e < öy3„+3r > and the technical lemma 5 is also fulfilled. Thus, the conditions for both distributions to be involutive are the same. Hence, both systems satisfy the same static feedback • linearizability conditions. (b) System £¡?+ 2 2+r . Denoting (2r+l) y3n+3r = U{ . V3n+3r = U>1 which appears in the equation Vn+2r+l = y3n+3r the distributions associated with this system are n2r+2,2n-2+r D0 .„ 9 d =<Vo,-dl/3n+Zr' dV3n+3r-l So z ) 2r + 2,2n-2 + r e < _ 9 _ oyn+2r+l > = ^+1,2^2+^ < ö _ <?y3n+3r > 6.1. MAIN RESULT 85 If 1 < j < r, D2r+2'2n~2+r = d J d d d ' % 3 n + 3 r ' % n + 2 r + l ' 9yn+2r+2-j Then J Oyn+2r+l-j For r < j < 2r, D2r_^2'2n-2+r ' dVZn+Zr-1 ' 9 y 3 n + 3 r - l - j >= Df+1.*-a+·© < —^— > D2r+2'2n~2+r® < T-J. J d OVZn+Zr = d d d ' d¡j3n+Zr ' % n + 2 r + l d dyn+2r+l-j d ' 9î/3n+3r-l dyzn+Zr-2-j satisfies the equality £)2r+2,2n-2+r _ j-)2r+l,2n-2+r f f i J ·,+1 Moreover, jD|;f,2n-2+r - d d %3n+3r ' Öy3n+3r-2-j > = ^ 7/Q, . . . , T]r, , dyn+2r+i ' ' ' - ' Syn+i ' dy3n+3r > ¿0,. • • , í j - 2 r - l i öy3n+3r-i ' dy3n+3r-2-j also fulfills the above equality for 2r + 1 < j < 2r + 1 + h. = Finally, if ; > 2r + 1 + h, D2r+2'2n-2+r = < T?0, • • • , Vr, dy3n+3T ' Ôyn+2r+l ' " " * ' W¡^+\ ' ^ ° ' • • • ' í / l ' %5n+3r-l ' ÖWn+Sr-l-i = jD 2r+l,2n-2+re< Ö _ J > = > OJ/3n+3r Therefore, taking into account the lemma 5, both systems fulfill the same conditions. H (c) Finally, the relationship between system S | r + l + r a n d system E|"+2 + r *s t n e same as that one between system E2£jT2+r and system E2,"+11+r. Proof: The equality D2r+2,2n-l+re 3 < Ô dyzn+3r-l-j > = ^+2,2^2+^ 3 < d 9î/3n+3r+i . 86 CHAPTER 6. IMPROVEMENT OF THE BOUNDS FOR 3-INPUT SYSTEMS is satisfied for all 0 < j < 2r + 2 + h. And for j > 2r + 2 + h, p2r+2,2n+r-lp2r+2,2n-2+r» i+1 9 3 dyzn+3r+i In addition, the hypotheses of the lemma 5 are fulfilled. Therefore, the static feedback linearizability conditions are the same for both systems. To summariza, it is proven that the static feedback linearizability conditions for system ¿^2r+i+r and the system where one more derivative of ui and/or U2 have been added are the same. The same proof is also valid for addition of more derivatives of the inputs. 2. Systems E 2 ." _ 2 + r and E £ (with fa < 2r + 1 and r 2 > In - 2 + r) are compared. In fact, in order to clarify the proof, systems E ^ - 2 + r and E^" _ 1 + r are studied. Extra additions of U2 does not affect the proof. Using the same notations as before, let us define: hi = min{i 6 {0, r } | rji+i e< rjo, • • • » m > } The reason for this definition is that h\ could be smaller than r. There are two possibilities, namely hi < fa and hi>fa. Note that the drift of E^"~ 2 + r is f = , f + giVn+i + 92yn+ki+i+ v^i-1,, 8 i y^2n-3+r ., d while the input vector fields are 9 9l = ô "Vn+ki In+ki 9 92 = = "3 92 Ct/3*n+fc1+r_2 03 = 53 The distributions for the case hi < fa are: n fc 1 ,2n-2+r _ . n 9 " 9yn+ki ' 8 ' dyn+ki-j d ' 9y3n+ki+r-2 ' d ' %3n+/ci+r-2-;7 if J < Ai. On the other hand, nki,2n-l+r Í _ 9 ) • • • i J) dyn+kl ' d ' dyn+^-j d ' dyzn+ki+T-l ' ' d 9yzn+kl+T-\-j They satisfy the same involutivity conditions because rjj does not depend on the variables y 3 n + fc 1 + r _j_ 2 . Note that T]J depends on the variables X, Vn+U • • • ) Vn+j, 2/n+fci+l) • • • ! Vn+ki+j 6.1. MAIN RESULT 87 \/hi < j < ku D^'2n-2+r = d d Cj/n+fci d a OVn+ki-j d VZn+ki+r-2 OyZn+ki+r-2-j fci,2n— l + r and Dj a ^yzn+kl+r-\, ' dyn+h-j' dyn+ki' ' <9y3n+fci+r-l-¿ ' ' l ' Therefore, the equality jDfc1)2n-2+re Ö < > = ^1>2n-l+r0 Ö < J C?/3n+¿1+r_i > Oyzn+ki+r-2-j holds, and because of the lemma 5 they provide the same conditions. • Let us define h2 = min{z|5 i+ i e<r)Q)... and let Dkk^~2+r d ~ ,ôi >} = d = < ,r}hl,60,... d >···'o oyn+ki >» Oyn+l d . '•••'« >»70J • • • j ^ / i i j O o , • • . , ö i > Cy3n+r-2-i Oyzn+ki+r-2 be the following distributions to be studied, Vi < h2. Note that h2 < n — hi — 3, because when h2 = n — h\ — 2, then < VO, • • • » % i i ¿0) • • • i ¿71-/11-2 > = S and -Dfei+nZ^Ia *s trivially involutive, as well as all the distributions after it. Since j-)&i,2n— l + r ki+i U = < "5 - d »•••ió Oyn+ki d >« Oyn+l d oy^n+ki+r-l . d '•••''5 oyZn+r-1-i ,%···,%1)<'0r··,«i > Then *1+t fcl+Z %3n+fci+r-l %3n+r-2-i However, to assure that the hypotheses of lemma 5 are fulfilled, one must prove that 3n + r — 2 — i > n + k\ + i, which is the maximum subindex of the variables on which öi depends. It is enough to check this inequality when i = n — hi — 3, which is the highest possible value for i. Then, 2n + r + l + hi>2n + h-hi-3 must be proven or, equivalently, r — ki + 4 > — 2hi. This is obvious when fci < r. If &i > r, note that hi = r. In this case, the inequality to be proven becomes r —fci+ 4 > — 2r, also trivial. 88 CHAPTER 6. IMPROVEMENT OF THE BOUNDS FOR 3-INPUT SYSTEMS Finally,forall i > h + h2, £>^f n - 1 + r = =< d d d d dyn+ia' ' J dyn+i'dysn+kx+r-i' ' <9y3n+fc1+r_2-¿' ' s ' S *' ' ' " ' 2 •>fci,2n—2+r and D¡ '»+1 d =< -r-^—, ••• JÖ So, Oyn+ki d d ,^ Oyn+l d '•••'â ,Vo,···,i1hl,So,...,6fl2 > Oy3n+fci+r-2-t OySn+ki+r-2 T}fci,27i-l+r _ rjfci,2n-2+r 1+1 " W * %3n+Ä!+r-l which, thanks to lemma 5, assures that both distributions satisfy the same involutivity conditions. This ends the proof for the case hi < ki. Now, for the case hi > &i, • Vj < ki n fci,2n-2+r 3 _ 3, ' d dyn+ki'' d d ' dyn+ki-j'9y3n+ki+r-2' d ' %3n+fti+r—2-i and Dku2n-l+r 3 d = < ' ' J ^ '%n+A;i' d d ' 9yn+ki-i _ .( %3n+ifci+r-l' d > ' 0J/3n+Jfei+r-l-¿ • Vfci < j < hi (if /ii < &i + /i2- The case hi > ki + hi is treated analogously and is left to the reader), D- 1 ' n ~ r = ô ô x , . . . , -»- -< 7]0,...,rjj,— oy n+ fcj d * d ,d0,...,Új-A!ii^- ,•••> 07 o , y3 n+ fc 1+r _2-j öj/ 3n+ fc 1+r _2 oyn+i > and D*!. 2 »- 1 -^ = a _<%,...,y, dyn+ki,..., d ayn+i • V/i1<¿<fci+/i2,I>*1'2n-2+r = and D^'2n-l+r J --< r]o,...,r¡j,- d ,oo,..., o,.*, ^ Ô d öt/n+A:i oyn+i . ^ ^ , d . • • •. dy3n+kí+r_1_j d dy3î2+fc1+r_2 > d Cy3n+fcj+r-2-i = 9 Öyn+fcl !---!"5 ô 9yn+l x x jOOí'-'joj-Auñ d oysn+fci+r-l !••• i «: d <7y3n+fc 1 +r-l-j > 6.1. MAIN RESULT 89 Therefore, in any of the three cases, the following equality holds n fci,2n-2+r „ 9 _ _ ku2n-l+r 9 •* "VZn+kx+r-l dy3n+k1+r-2-j and the following inequalities must be fulfilled in order to satisfy the hypotheses of lemma 5 (just studying the maximum possible value for j , j = n + k\ — h\ — 3): 2n + r + hi + l>n + hi 2n + r + hi + l>n + ki+n — hi—3 or, equivalently, the trivial inequalities n + r + 1 > 0 r + 2hi+A> ki Finally, Vj >h + h2, nfci,2n-l+r _ 9 a _d ffci+r-l'"''' and -.fci,2n-2+r •p.Ki,¿,n—í-fT u i „ _ . - < „ ga d_ dg ' '···'7'/li'3j^7'···'5^7I T 0 ôo,-.. ,â/i 2 , a„, . . . _ . >• Ôy 3 n + fe 1 + r _2_jysn+ki+r-l ' ' •* •' i' o«w.fc.4.,_î_< > So, rjfci,2n-l+r _ r)fci,2n-2+r « J+1 d 3 9y3n+kl+r-i Therefore, both systems satisfy the same static feedback linearizability conditions. 3. The third case to compare is the static feedback linearizability conditions for the systems + r and rx > 2r +1). In fact, systems S ^ + 1 and E^-fi a n d S n ( w i t n 2r <k2 <2n-2 are compared (which still satisfies the restriction k2 + 1 < 2n — 2 + r). Denoting ^2r+2 yn+2r+l+fc2+l = u l (n+2r+l) (fc2) J/n+2r+l+fc 2 +2 = ^2 which appear in the equations yn+2r+l+k2+l (n+2r+2) = "l = ™1 . yn+2r+l+fc 2 +2 = u (fc2+l) 2 = w 2 Ifj<r,Df +1 '* 2 a J 'Öyn+2r+l' ' 9yn+2r+i-j ' 9y n +2r+l+fc 2 ' ' Öy n +2r+l+fc 2 -j 90 CHAPTER 6. IMPROVEMENT OF THE BOUNDS FOR 3-INPUT SYSTEMS and D2r+2M+i =< 3 a ^ §__ a ''°> • • • ' 'h ' ô y n + 2 r + i + f c 2 + i dyn+2r+l ' • • • ' 0y„+2r+2-> ' a a a 9yn+2r+3+k2 ' dVn+2r-i-l+k2 ' " ' ' > dyn+2r+2+k2-j then the following equality holds: D7r+2,k2+l Q d 3 £)2r+i,fc2 Q >= dyn+2r+l+lc2-j a Ôj/n+2r+l+fc 2 +l ' For r < j < 2r, Df+1M > 9yn-f.2r+l+k2+2 = d d Ct/n+2r+l n 2r+2,fc 2 +i ' a < 3 and d dyn+2r+l-j Oyn+2r+l-j _^ d d Oyn+2r+l+k2 0¡/n+2r+l+k2-j a a dyn+2r+3+k2 a s a a ' dyn+2r+i+k2 ' ' " ' Ôyn+2r+1+fc2_j > Therefore ^ ^ ¿ z + l _ D2r+l,k2Q J+ 3 ^ 9j/n+2r+H-fc2+l ' < d > 9yn+2r+l+k2+2 Note that this last equality is still valid when j > 2r. Moreover, all these distributions satisfy the hypotheses of lemma 5. Thus, one system is linearizable by static feedback if, and only if, the other one is also. The three different cases stated at the beginning of the proof having been proven, the proof is concluded. • 6.2 Where do the bounds 2r + 1 and 2n — 2 + r come from? If the bound 2r + 1 is relaxed to 2r, then the distribution D2r'2n'2+r n 2r,2n-2+r _ . „ U ~ ^ "0> ' • • ! '/r j «j T „ 9 d '•"•'a d ' a OVn+r Cy3 n+ 3 r _2 Oyn+2r of the system S ^ ~ 2 + r is d '"•''« Cy3n+2r-2 •* Therefore, a new involutivity condition appears, which is different from those appearing in n2r+l,2n-2+r. 9 [rir,g i ] E< rjo,... ,r¡r > 6.2. WHERE DO THE BOUNDS 2R + 1 AND 2N-2 + R COME FROM? 91 Therefore, it is not possible to relax that bound. In the same way, the purpose of the bound 2n — 2 + r is to avoid extra involutivity conditions among r]j (j < r), Si (i < h) and the other elements of £>2r+i+¿- Nevertheless, it remains to be seen whether or not this last bound can be improved. 92 CHAPTER 6. IMPROVEMENT OF THE BOUNDS FOR 3-INPUT SYSTEMS Chapter 7 Linearization by prolongations of m-input systems This chapter gives a bound on the number of integrators needed to linearize a control system with an arbitrary number of inputs. Although some work have been done in this direction in [59], our bound improves the existing results for systems with four or more inputs. The bound for two input systems is the same as the one that appeared in [59], and has already been studied in Chapter 5. The bound for three input systems is further improved in Chapter 6. Nevertheless, the sharpness of our bound remains an open question. 7.1 Main results First of all, we state and prove a proposition useful in the proof of the main result. Proposition 6 If a system with m inputs is linearizable by prolongation of U{ ki times (with ki> 1 for all i), then the system is also linearizable by prolongation of u¡ ki — 1 times. Proof: Note that there is no loss of generality in assuming that k\ < ki < ... < km. Let Sfc be the system obtained by prolongation of u¿ ki times (with ki > 1 for all i) and £ # be the system obtained by prolongation of t¿¿ ki — 1 times. And consider m m ¿=1 and fc¡—1 i = i i=i TTi m ¿=i °Vi ki~2 ¿=i i=i the drifts associated to S^ and Sfc' respectively. And let 93 p\ pi a Vl 94 CHAPTER 7. LINEARIZATION BY PROLONGATIONS 9i - OF M-INPUT SYSTEMS - the input vector fields of E^ and E^/. The main idea of this proof is to see the equalities Df = DJtL1®<{-^r,i (7.1) = l,...,m}> for all I > 1. Note that Dk and Dk are the distributions associated to system S*; and Ejf respectively. We will also check to see if lemma 5 can be applied. Some lemmas are needed for this proof. Lemma 19 1. ad/fc,i0i can replace adhkg\ in D%i+l, where m m ki—1 Q i=2 »=2 1=1 °yl 2. And, in general, a l d fk,jgj can replace adKkgj in D^.+l, where kj m m f ' = f+ £ rf* + £ i=j+l ki — 1 EvUign t=7+l /=1 Lemma 20 a dfk,jgj belongs to < J j > and depends on the variables x,yH\h = l,...,l; p=j + Lemma 21 adfkjgj can be replaced by adl in D kj+l- jk',jQj n l,...,m} y l 7.1. MAIN RESULTS Lemma 22 95 1. can replace ad1fk,g\ in D^i+l' w ere ^ t=2 ¿=2 Í = I °yi 2. And, in general, ad fk',j9j can replace ad1 k,gj in D%.+[) where 771 771 k{—2 r\ The proof of these lemmas is exactly the same as that carried out for lemmas 9,10,11 and 12 in Chapter 5. Thanks to these lemmas, the equality 7.1 holds, and lemma 5 can be applied. • Corollary 5 If a system with m inputs is linearizable by prolongation ofui n¿ times (withrii > 1 for alii), then the system is also linearizable by prolongation of Ui n¿ — mini<¿< m {nj} times. Proof: This is straightforward after the application of the former proposition mini<¿< m {ni} times. • Thanks to this corollary, we will only consider prolongations where one input is not prolonged. When a prolongation is considered, the inputs are ordered by the number of derivatives added to each one. So, the number of derivatives of u\ is zero, and the number of derivatives of IÍ¿ is less than or equal to the number of derivatives of u¿+i, for all i from one to m — 1. Our main result is established in the following theorem: T h e o r e m 5 Let S: m x = f{x) + Y/9i{x)ui xERn i=l be a m-input control system not static feedback linearizable, and such that the prolongation E¿ is static feedback linearizable, where E* is given by i yj = f{x)+g1(x)vi+ = y/ + 1 y\. = Ví+i Vcj+l = Vj vi = Vi YALÏ Vi9i (x) l-l,...,Cj,forcertainje{2,...,m} i — 2 , . . . , m, i 5¿ j l — 1 , . . . , ki — 1 i = 2,..., m 96 CHAPTER 7. LINEARIZATION BY PROLONGATIONS OF M-INPUT SYSTEMS k{ being fcj < c¿ = 2(i — l)n — (i — l)(z + 4)/2, i — 2 , . . . , m. The new state variables and inputs are related to the old ones by VÍ = 4 i _ 1 ) yj = uf'V V{ = u\ Vx = Ui l = l,...,Cj + l l = l,...,ki i = 2,... ,mi ^ j Then, Sfc/ is static feedback linearizable, where £&/ is given by x yj y{ = f{x) + gl{x)vl + Y,?=2y\9i{x) = y¡+i i= i,...,cj-i i = 2 , . . . , m, i < j I = 1 , . . . , ki — 1 = y\+i i = 2 , . . . ,m, i > j I = 1,... ,ki — 2 = Vî+i ylj = Vj ni = Vi Vki-i - Vi m i <j i > j Proof: Denote by fk, gf (resp. /*', gf) the drift and the input fields of Sfc (resp. S fc '). Then m m ¿=2 m ^ oy i=2,i,yi=i Si = 0 1 and fc;-l Si = 7dyí 0- yj 5 Í=I ß Uy í i=2,i¿jl=l ui ¡ „ i ? — n. ffï = Si ui O n™ — Si' = TO" Ui öyf duiCi+l m ¿=2 c; ui n™ — 9j 3 = * d 1=1 y í O dyí, The main idea of the proof is to see the equalities 1. 2. A + i = Df® < -j—, {---j-, i>j}> l> Ä,--i + r.-x 7.1. MAIN RESULTS 97 and to verify the hypotheses of lemma 5, where n = min{s|?4 +1 € < r\\, • • • i Vrx i • • • . 4 ~ \ • • •.Cii. and rf¡ = adlrkgi 4, • • •. vl >} i = 2 , . . . , m; l > 0 A series of lemmas lead us to the proof. Lemma 23 For 2 < i < m, 1. «44 b - r =(( -1}i ) S o - v¿<ft¿ U ü ~ W< 4 ( ¿ ) = i"1)?* fl Proof: 1. It is proven by induction. For / = 1, Assuming the trueness of the equality up to i, which becomes, by application of the induction hypothesis, V'(-1)lñCJ - [y k-<^Z? (-1) %^] 1 ; ôvUi-iJ ^(4)=íA^(¿ is equal, using the first part of the lemma, to [fk, ( - i ) f c < - x ^ — ] = [visu (-i) f c < _ 1 ¿] = (-i)to 98 CHAPTER Lemma 24 7. LINEARIZATION BY PROLONGATIONS OF M-INPUT SYSTEMS 1. ifs. e S = < — > dx Ví' = 2 , . . . , m ; s ¿ < r ¿ 2. T]ls only depends upon the variables (x, {yf, h = 2 , . . . , m ; l = 1 , . . . , s}). Proof: Both results are proven by induction. 1. For Si — 1, m k T)i = lf ,9i] = lf+EyÏ9h,9i} h=2 which clearly belongs to S. Assuming that the statement is true up to s, because rfs E S and because the part of fk depending on x is m f + T,Vi9h h=2 2. m foi = [/fc,<?i] = [/ + £ y i W i ] h=2 only depends on yf. H+i = [fk,r)Í} where it can be assured, thanks to the induction hypothesis and the previous lemma, that rfs € S and depends on the variables (x, {yf, h = 2 , . . . , m; l = 1 , . . . , s}). Therefore, m h m s o h=21=1 °yl tó+i = [/+Ey i9k + EEtf+iA.râ h-2 will depend only on the variables (x, {yf, h = 2 , . . . , m; í = 1 , . . . , s + 1}). Lemma 25 r¡[, l > 0, i = 2 , . . . , m, can be replaced by adjk.igi in all the distributions where they appear, where fc m m fcft—1 n / 'W + E ^ + E E ^ i À h=i h=i 1=1 Uy l • 7.1. MAIN RESULTS 99 Proof: Again, the proof uses induction. For I = 1 4 = {fk,9i] appears in the distribution D^.+l for the first time. Notice that Z)|. contains <?i,..., <?¿ and is involutive since the system is static feedback linearizable. Therefore, all the Lie brackets between them are in £>£.. So, m 4 = U + Y,yï9h,gi] h=2 can be replaced by m 4 = \f + ^VÎ9h,9i] h—i in -Djfc.+1 and all the distributions which follow. If the statement is assumed to be true up to I, r¡Í+i = lf\ví] can be replaced, by using the induction hypothesis, by m k l [f ,ad fk,igi\ m l + ^ = [f,ad fk,igi] + Y^vh^ad^gi) fcft—1 J2 tî/i+i«""*»0«*/*-1^ h=i i=i A=I o °y¡ And due to the involutivity of all the previous distributions, ¿ —1 ¿—1fcfc— 1 h=l h=l 1=1 o J2toïtoMfk,i9i] + E E [yï+irrK'adlf".i9i} 6 Dkki+( oy l Therefore, rfl+1 can be replaced by adfk,i9i in D%.+l+l and all the distributions which follow. • Thanks to this lemma, we are able to assume r¡¡ = adlfkiigi l>0, i = 2,..., m Now it is possible to write the distributions associated to both systems, Efc and Sfe . In order to do that, first we assume r¿ < k{+i —fc¿,Vi < j . Then: DQ=<VO dyij+i =9U-T-J—i{0-7-,* Wcj+i-i dy ki = 2 , . . . , m ; t ?¿ j } > ^ki-i 100 CHAPTER r^fc 7. LINEARIZATION i i _ - r,k ^2 9 , 9 % + 1 _ f c 2 + 1 dyk. 9 dyij+i +l kp+i 9 . àyk._k2+1 SYSTEMS .... n = < 4,...,7^,...,rff1,...,i$-_\,rf0,...,rjf, = OF M-INPUT ^ _i „i „2 _ „„ a .a a „a <7?o>···^r1,%-52,3^+i,···,a^+i_fc2,^,...,S2/,2_fc2+i, And, in general, if p < j , D\ and D BY PROLONGATIONS ^— Cj+1 ,--.,^r a "ITcj+i-kp-l = <Vh-.-,Tl}li...tTJr\...tTJ?;_\,T$,...,T#p,-Z- -^-L ^.+1'""'Vc.+1_fcpV 5 t = p + 1,...,m; i # i } > r . »» • = 2»• •• ••• .P}. {/$"• •dyl. ' •• •• •. •' ^ ôj/ï -"> . i - J . {flâ-i I f l ^ . •• •• •. • . STay¡¿._-—, fcp _, r p < l < kp+i - kp On the other hand, for Sfc', -^fc!,+/ = < % i - - • l ^ n i - • • >% >···>í7rp_i»7?0>··-i7^ > ¿ J 3 - > - " > ¿¡"J i {4;'···'aí'í = 2 '· , '' í j } ' { 4: , ;··'S4 < -* P - 1 ' i = í ' + 1 , ···' : '~ 1 } ' and nfc' _ "^Arl+Z — - _i „i „p-i ^ VOJ---Jï/ri)···>7/0 ^ 5 ^ ' • • • ' Wx'l = 2 „p-i „P „p _a_ > • • • ' 7 ' r p _ i ) '/O' • • • i '/r p > Ö T T J • • • > S v 7 ' • • ' , P ^ { d y i ^ * " ' Wi-kp-i'l a > = P + !'• • • ' i - 1 )' Therefore, £>* e < — ^ L , { , i > j } > = of© < — 3 — , {— r , i>j}> i< kj.i + rj_ 101 7.1. MAIN RESULTS It remains to be seen that lemma 5 can be applied. Indeed, it can be applied since rp < Cj - kj-x - ry_i Vp < j - 1 (7.2) In fact, it must also be checked rp<ki-l- kj-x - Tj-x Vp < j - 1; Vz > j (7.3) But if 7.2 holds, 7.3 also holds because ki — 1 > cy Vi > j since the inputs are ordered following the number of derivatives added to each one. To prove inequality 7.2, remark that Cj - kj.x > CJ - Cj-x = 2(i - l)n - (j - l)(i + 4)/2 - 2(j - 2) + (j - 2)(j + 3)/2 = In - j - 1 On the other hand r p < n — (p +• 1) for all p < j — 1. Then, r p + rj_i < n — (p + 1) + n — j < In — j — 2 < 2n — j — 1 < Cj — kj-x Vp < j — 1 which is 7.2. In conclusion, , i > j} >= Df® < - J — , { X , i>j}> Df® < —j—, {•—, l< kj-x + rj-x and lemma 5 can be applied. This implies that the involutivity conditions are the same for Df and Df for I < kj-x + rj-xNow, we are going to prove the equality fi Df+1 = Df® < —j—, fi {— r , i > j} > l> kj-x + rj-x also checking that lemma 5 can be applied. nfc - <r -n1 7 í TJ 1 í T ^ ^Vi+o-i+i - < ?o>···. ?r1>···! ro - 1 J~l d , , £ '•••» fri-i'¿tó. +1 '-"' <tó. +1 _ 4 . implies ni ., = n i ' . . ffi <r And, in general, using induction, A \ 2 = Afc+i+ < {[/V]> * e Afc+i} >=< T - j - , { X , i > j} > +Df + < {[/*, d], s G Df} >= =< >j}>+D ^:M" > k' 102 CHAPTER 7. LINEARIZATION BY PROLONGATIONS OF M-INPUT SYSTEMS Moreover, lemma 5 can be applied since r¡lh does not depend on y ¿ . + 1 , . . . , y™ . So, the involutivity conditions are the same for D* and £>f+1, for all I > kj-\ + rj_i. This, together with the fact that the involutivity conditions for £>f and D\ are the same for all / < fcj_i + rj_i, implies that Efc' is static feedback linearizable providing that Sfc is too. Remarks: 1. If for some i, r¿ > k{+\ — ki, it is not difficult to check that the above equalities between distributions of "Ef. and £&/ also hold. 2. Let Efc be a static feedback linearizable prolongation of S with k = (0, fo,..., km. Assume kis > Cis for some is. Then, Efc< is static feedback linearizable with k' = (0,k'2,... ,k'm), where ki if i < i k'1 = 1 ki — 1 if i > i\ The proof is exactly the same as the one previouly made. We have chosen to do the above proof in order to clarify the notation. 7.2 About the bounds The bounds c¿ = 2(i — 1) — (i — l)(i + 4)/2 have been chosen to enable us to apply lemma 5. Notice that the equality Cj — Cj_i = 2n — j — 1 has become fundamental for proving 7.2. This does not mean that these bounds are sharp. In order to prove the sharpness of these bounds, an example has to be constructed. This example must satisfy that the only static feedback linearizable prolongation was that with ki = c¿, for all i. To date, the authors have been unable to find such an example, except that mentioned in Chapter 5. In any case, the bounds obtained here improve the bounds existent in the literature when the number of inputs is greater than or equal to four. The two and three input cases have been treated in previous chapters. In [59], the bound for the number of integrators is 3m-2(2n-9-) + l while the bound obtained here is 2 ^ d = ¿ ^ 2(t - l)n ^ = m{m - l)n This equality is obvious for m = 1. Assuming the equality for the case m, we prove the case m + 1 using induction: TO+1 E2(,-l)n-' E î=l i -1»' i +4 " ' = f: 2 ( i -l)n-' i - 1 »i i + 4 » + 2mn- m ''" + 5 ' »=1 7.2. ABOUT THE BOUNDS 103 m3 + 6m 2 - 7 m . ,. = m[m — l)n . , .. = m(m + l)n „ \- 2mn O m(m + 5) — ¿i = (m + I) 3 + 6(m + I) 2 - 7(m + 1) m 3 + 9m 2 + 8m = m(m + l)n 6 6 Proposition 7 For m > 4, 9 , ,x N , 3 om-2/n 3 m 2 (2n - - ) + - > m(m - l)n m3 + 6 m 2 - 7 m Proof: Note that n > 5 because n > m > 4. Then 2n - 9/2 > n. Therefore, for m > 5, 3 m " 2 (2n - \ ) + \> 3 m _ 2 n + | > m(m - l ) n + | On the other hand, since m 3 + 6m2 — 7m > 0 (if m > 1), 3 m 3 + 6m 2 - 7m n 5>o> J Putting everything together, om-2/r, 9, 3 . ,, 3 m 2 (2n - - ) + - > m{m - l)n m3 + 6 m 2 - 7 m The case m = 4 reduces to prove 9 3 9(2n - - ) + - > 12n - 22 which is equivalent to see 6n > 17. This is true since n > 5. • Note that for the two inputs case, the bounds are equal, and also equal to 2n — 3, which is the bound obtained in Chapter 5. For the three inputs case, our general bound is worse than that in [59]. However, the different approach we adopted in Chapter 6 has improved the bound in [59]. Let us recall that the bound obtained in Chapter 6 is 2n — 2 + r + 2r + 1, while the bound in [59] is 6n - 12. 1. If r = n — 1, there is no need to consider derivations of U3 greater than 2r + 1. Therefore, the number of integrators is 2(2r + 1) = 4r + 2 = 4n - 2 which is smaller than 6n — 12 for all n > 5 (and equal if n = 5). 2. In the case r < n — 2, the maximum number of integrators is 2 n - 2 + r + 2r + l = 2n + 3 r - l < 2 n + 3 n - 6 - l = 5 n - 7 also smaller than 6n — 12 if n > 5 (and also equal for n = 5). 104 CHAPTER 7. LINEARIZATION BY PROLONGATIONS OF M-INPUT SYSTEMS Chapter 8 Conclusions and suggestions for further research In this dissertation we have presented different methods for linearizing nonlinear control systems or for studying differential flatness. These have been carried out using two different frameworks, namely: differential algebra and differential geometry. Since our approaches apply only to special classes of systems, the general problem remains open. 8.1 The differential algebraic approach There are two Chapters in which the differential algebra setting for control systems has been used. In Chapter 2, a survey on linear control systems from the module theory has been presented, while Chapter 4 deals with nonlinear control systems. By means of the tangent system, obtained by application of the Kahler differential, a procedure for finding the last flat output has been designed, providing that the first m — 1 flat outputs have been guessed. In this context, an easy new proof is given of the well known fact that dynamic and static feedback linearization are equivalent for single-input systems. This has been used to make a new algorithm to linearize single-input systems, working with the concept of relative degree. Another algorithm for linearizing multi-input systems by static feedback has been created from the translation of the meaning of relative degree into the differential algebraic framework. For not static feedback linearizable systems, a procedure for reducing to single-input systems has been carried out. This procedure is based on guessing the first m — 1 flat outputs and making a quotient of modules. Here, the results of Chapter two have been crucial. However, a way of obtaining the first flat output is still a problem. Sometimes, when working with a concrete problem, some variables with physical meaning (center of oscillation [23], center of mass, . . . ) can be flat outputs. In other problems, one can guess some flat outputs from the structure of the system (backstepping, variables not appearing in any equation, . . . ). Unfortunately, there is no general method which is good for all systems. One possibility could be to make quotients in the tangent system by the input variables, until a single-input system 105 106 CHAPTER 8. CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH is obtained. We have applied this procedure to some examples with good results, but we have not obtained any general solution for all systems. Another difficulty in such a procedure is to struggle with modules with torsion elements, which implies that there exists no basis that generates all the elements of the module. Another concept strongly related to dynamic feedback linearization and flatness is the concept of defect. The defect can measure how far a system is from being flat. Thus, a system with defect zero is, indeed, a flat system. Let us recall that a system is flat if, and only if, there exists an integrable basis of the module associate with the tangent system. One may suspect that the defect is the minimum number of not integrable elements of a basis, taking int« account that there are infinite different basis for a module. 8.2 Linearization by prolongations and possible extensions In Chapters 5, 6 and 7, a necessary and sufficient condition for a system to be linearizable by means of prolongations is given. This condition states that the involutivity of a finite number of distributions must be checked. The upper bounds on the number of derivatives of the controls added to the original system have been improved. For two-input systems, it has been shown that the bound is sharp, and the results have been applied to some systems that hitherto were thought to be not linearizable by prolongations. This procedure can be applied to other concrete examples. The first case that arises is a driftless system affine in the inputs. We have performed initial explorations in this direction for two-input systems. Let us write the equations of such a system: x € Rn E : x = gi(x)u\ + 52(^)^2 Let E r be a prolongation of E based on adding r derivatives of xii (recall that it is only necessary to add derivatives of just one input), the drift of the prolonged system becomes: r-l g f = 92{x)yn+\ + X ^ + Î + I  : — and the input vector fields are 9ri=9i{x) 92 = d dy,n+r where yn+i = u\ Remark that [/ r ,9l] = [52,5i] and, in general, (xPfrgi = a(Pg2g\ 8.2. LINEARIZATION BY PROLONGATIONS AND POSSIBLE EXTENSIONS 107 The conditions for S r to be static feedback linearizable are the involutivity of the distributions d d A = (dy +r ' ,gi,...,ad* dyn+r-i ' n gx) Vi = 0 , . . . , r - 1 and Dr = ( dyn+T ' " ' ' dyn+i ,gi,...,a(fg gi,g2) Since Dr must be involutive, a ^gtl9l = b2,a<Csi] € £ r while [f,ff 2 ] = 0 Therefore, DT+\ = Dr. Thus, it is not necessary to check the involutivity of more distributions. Furthermore, since the rank of DT must be n + r, then r + 2 > n. Summarizing, E is linearizable by prolongations if, and only if, (pi,...,a425i) are involutive for a l i i < r — 2 and the rank of {gi,...,a<fg~lgi,g2} is n. Or, exchanging <?i by g2 and viceversa, (S2,...,a4i22) are involutive for alH < r — 2 and the rank of {92,---,adrg~1g2,gi) is n. These good results encourage us to tackle systems with more inputs using this technique. As already stated in Chapter 3, a linearization by prolongation is a particular type of dynamic feedback linearization. Let us recall that a dynamic feedback linearization requires the existence of a dynamic compensator z = a°(x, z) + a 1 (x, z)v u = b°(x,z) + bl (x, z)v (8.1) with z G Rq and v € Rm. A dynamic feedback compensator is a prolongation if, and only if, \ /*i b°(x,z) ¿fci+i J i+£:=7lfc. / 108 CHAPTER S. CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH b1(x,z) = 0 di(x,z) 1 = y if i T¿ kj, j = if i = kj, j = l,...,m l,...,m I 1 if i = = ki, kj, ji = 1 , .l,...,m . . ,m In order to proceed with the research on dynamic feedback linearization, the authors suggest studying dynamic compensators in an increasing order of difficulty. Since the sharpness of the bounds obtained here is not clear, new work could well be done with the purpose of finding better bounds. Let us recall that the procedure used in the proofs has been the comparison between distributions of différents systems. However, to ensure the involutivity of a certain distribution of a given system, only some distributions of the other system have been considered. Taking into account all the distributions could lead to more restrictive bounds. Considering the great number of systems and distributions involved in the application of the procedure in Chapters 5, 6 and 7, a software package for carrying out all the computations should definitively be done. This software package must be programmed carefully, to avoid repetitions of the same calculations. Bibliography [1] E. Aranda-Bricaire, C.H. Moog, J.B. Pomet. "Infinitessimal Brunovsky form for nonlinear systems with applications to linearization" Proceedings of the Ext. Workshop Geometry Nonlinear Control Warsaw, Banach Center Publications, 1992. [2] E. Aranda-Bricaire, C. H. Moog, J. B. Pomet. 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A.l Basics on differential algebra Definition 11 An ordinary differential ring A is a commutative ring equipped with a single derivation ^ such that u WaeA, ... . ,, , A Va,beA, • A da . a= — e A at d, ,. da db d , ,. da, db -(ab) = -b + a- A constant c G A is an element such that c = 0. An ordinary field is an ordinary ring ring which is a field. A differential field extension L/K is given by two fields, L and K, such that the restriction to K of the derivation of L coincides with the derivation of K. Definition 12 An element x G L is said to be differentially algebraic over K if, and only if, it satisfies an algebraic differential equation with coefficients on K. The extension L/K is said to be differentially algebraic if, and only if, any element of L is differentially algebraic over K. 115 116 APPENDIX A. INTRODUCTION TO DIFFERENTIAL ALGEBRA Definition 13 An element x G L is said to be differentially K-trascendental if, and only if, it is not differentially algebraic over K. And the extension L/K is said to be differentially transcendental if, and only if, there exists at least one element x G L that is differentially Ktranscendental. Definition 14 A set {x¿ | i G 1} of elements in L is said to be differentially K-algebraically independent if, and only if, the set of derivatives of any order {xf \ i G I, j > 0} is Kalgebraically independent. Such an independent set which is maximal with respect to the inclusion is called a differential transcendence basis of the extension L/K. Two different transcendence basis of an extension L/K have the same number of elements. This cardinality is called the differential transcendence degree of L/K, and it is denoted diff tr d°L/K. The following theorem establishes a relation between differential algebraic extension and transcendental extensions. Theorem 6 For a finitely generated differential extension L/K, equivalent: the next two properties are 1. L/K is differentially algebraic. 2. The transcendence degree (not the differential transcendence degree) of the extension is finite. L/K Let K a given differential field. The ring of differential operators over K is denoted by K[-^], and it contains all the elements of the form A é ^0aidF This ring is commutative if, and only if, K is a field of constants. In the non-commutative case is always a principal ideal ring. Thus, the most important properties of the modules over commutative rings are fulfilled also by the left modules over K{j¿\. Let M be a left module over *[&• Definition 15 An element m € M is said to be a torsion element if, and only if, there exists p G K[^\ such that prh = 0. A torsion module is a module in which all the elements are torsion elements. The following proposition relates a torsion module with a vector space: Proposition 8 For a finitely generated left K[^]-module M, the next two properties are equivalent: 1. M is a torsion module. 2. The dimension of M as a K-vector space is finite. Definition 16 A finitely generated module over a principal ideal ring is free if, and only if, there does not exist any torsion element. A.2. A.2 THE KAHLER DIFFERENTIAL 117 The Kahler differential Let A be a differential ring and let B be a differential .A-algebra (in our case, A is a differential field and S is a differential field extension of A). Let p: B®AB —> B the canonical A-algebra homomorphism such that p{b®b') = 66'. Let I be the kernel of p. Then, (6 0 1 - 1 ( 8 ) 6 ) 6 / Since p is exhaustive and I an ideal of B, applying the theorem of isomorphism we have {B ®A B)/I = B Let us recall that I/I2 is a differential (B ®A 5)/7-module, because it is also a differential (B® v 4 5)-module. Therefore, J / 7 2 is a differential J3-module. Let us define now the differential B-module "a/A = 7 / / 2 and the application (Kahler differential): d = dB/A '• B —> &B/A defined by d(b) = (6 ® 1 - 1 ® 6) + 1 2 . Proposition 9 £IJB/A ^ a s a canonical structure of differential module over B such that, for any derivation S, S(d(b)) = d(ô(b)). Proof: The uniqueness of the differential structure is clear since the differential 5-module is generated by the elements d(b), for 6 € B. The existence of the differential structure comes from the above construction. And, for 6 G B, ö{d{b)) = í((6 (8) 1 - 1 <g> 6) +1 2 ) = (6(b) ® 1 - 1 ® 6{b)) + I2 = d{5{b)) • More details on the Kahler differential can be found on [35] and referències therein. 118 APPENDIX A. INTRODUCTION TO DIFFERENTIAL ALGEBRA Appendix B Software package for Chapter 4 ################################################## # # # MAPLE V PROGRAMS TO LINEARIZE CONTROL SYSTEMS # # USING THE KHLER DIFFERENTIAL # # # ################################################## ################################################################### # # # Function afegir_temps: it transforms expressions of the form # x_i, u_j, dx_i, du_j in x_i(t), u_j(t), dx_i(t), du_j(t) # # # # ################################################################### afegir_temps := # Ok! proc(expr) local r e s , i , j ; global F , x , u , d x , d u , n , m ; r e s := expr; for i t o n do r e s := subs({x[i] = x [ i ] ( t ) , d x [ i ] = dx[i] ( t ) } , e v a l ( r e s ) ) ; od; 119 APPENDIX B. SOFTWARE PACKAGE FOR CHAPTER 4 120 for j t o m do r e s := subs({u[j] = u [ j ] ( t ) , duCj] = du[j] ( t ) } , e v a l ( r e s ) ) ; od; eval(res); end; ################################################################### # # # Function treure_temps: it transforms expressions of the form # x_i(t), u_j(t), dx_i(t), du_j(t) in x_i, u_j, dx_i, du_j # # # # ################################################################### treure_temps := # Ok! proc(expr) local r e s . i , j ; global F , x , u , d x , d u , n , m ; r e s := expr; for i t o n do r e s := s u b s ( { x [ i ] ( t ) = x [ i ] , d x [ i ] ( t ) = d x [ i ] } , e v a l ( r e s ) ) ; od; for j t o m do r e s := subs({u[j] ( t ) = u [ j ] ,du[j] ( t ) = d u [ j ] } , e v a l ( r e s ) ) ; od; eval(res); end; #################################################### # # # Function d_dt: it computes temporal derivatives # # of expressions of the form x_i, dx_i, doing the # # substitutions given by the systems sys and lin. # # # #################################################### d_dt := # Ok! proc(expr,sys,lin) local res,i,j; global F,x,u,dx,du,n,m; res res res for := expr; := afegir_temps(res); := diff(res.t); i to n do res := subs({sys[i],lin[i]},eval(res)) ; od; for j to m do res := subs(diff(u[j](t),t)=Diff(u[j],t),eval(res)); od; res := treure_temps(res); eval(res); end; ############################################################ # # # Function Kahler: it computes the tangent system of a # # systema x'=F(x,u). Writing down this tangent system as # # dx'=Adx+Bdu, the outputs are the matrixes A and B, and # # the right hand side of the equations. # # # ############################################################ Kahler:= # Ok! proc 0 local A,B,lin,i,j; global F,x,dx,u,du,n,m; A:=matrix(n,n); B:=matrix(n,m); lin:=vector(n); i:=0; 122 APPENDIX B. SOFTWARE A:=jacobian(F,x); PACKAGE FOR CHAPTER 4 B:=jacobian(F,u); for i t o n do l i n [ i ] :=dotprod(row(A,i),dx)+dotprod(row(B,i),du); od; eval([evalm(A),evalm(B),eval(lin)]); end; ############################################################# # # # Function EqSys: it computes the equations of the original # # system in the form x'(t)=F(x(t),u(t)). # # # ############################################################# EqSys:= # Ok! procO global F,n,m,x,dx,u,du; local i,sortida; sortida :=vector(n); i:=l; for i to n do sortida[i]:= od; diff(x[i](t),t)=afegir_temps(F[i]); eval(sortida); end; ################################################################### # # # Function EqLin: it computes the equations of the tangent system # # in the form dx'(t)=A(x(t),u(t))dx(t)+B(x(t),u(t))du(t). # # The third output of the function Kahler must be given as a # # parameter. # # # ################################################################### EqLin:= # Ok! proc(kah3) global F,n,m,x,dx,u,du; local i,sortida; sortida:=vector(n); i:=l; sortida :=afegir_temps(eval(kah3)); for i to n do sortida[i]:=diff(dx[i](t),t)=sortida[i] ; od; eval(sortida) ; end; ############################################################ # # # Function equacions: it gives the result of the former # # two equations, EqSys i EqLin. # # # ############################################################ equacions:= # Ok! proc(kah3) global F,n,m,x,dx,u,du; eval([EqSys(),EqLin(kah3)]); end; ############################################################ # # # Function integrable: the result is 1 or 0, depending # # whether or not the 1-form given to the function is # # integrable (exact 1-form) or not. It is done using # # Schwarz's conditions of integrability: # # w=a_l dx_l+...+a_n dx_n integrable if, and only if, # # diff(a_i,x_j)==diff(a_j,dx_i), for all i,j # # # ############################################################ APPENDIX B. SOFTWARE PACKAGE FOR CHAPTER 4 124 i n t e g r a b l e s # Ok! proc(w) local i,j,sortida; global F,x,dx,u,du,n,m; sortida:=1; for i to n do for j to n do if diff (coeff (w,dx[i]) ,x[j])odiff (coeff (w,dx[j]) ,x[i]) then sortida:=0; fi; if sortida=0 then j:=n; fi; od; if sortida=0 then i:=n; fi; od; eval(sortida); end; ############################################################ # # # Function Singlelnput: it computes the coefficients of # # the basis w for single-input systems _ # # ' # ############################################################ Singlelnput:= # Ok! procO local sol,kah,eq; global F,n,m,x,dx,u,du; kah:=vector(3); eq:=vector(2); kah[l] :=matrix(n,n); kah[2]:=matrix(n,m); kah:=Kahler(); eq:=equacions(kah[3]); sol:=trobar_base(kah[l],kah[2] ,eq,n,l) ; eval(CevaKsol) ,eval(dotprod(sol,dx))]) ; end; ############################################################ # # # Function Singlelnput2: it computes the coefficients of # # the basis w for single-input systems coming from # # quotients done in dynamic feedback linearizable # # multiple-input systems. # # # ############################################################ Singlelnput2:= # Ok! proc(lin2,zeros) local i, j , compt, A ,B, eq, sol »variable,uns, auxA, auxxA, auxB, auxdx ; global F,n,m,x,dx,u,du; uns :=sum('zeros[i]','i'=1..n); auxA:=matrix(n,n); auxxA:=matrix(n,n-uns); auxB:=matrix(n,l); A:=matrix(n-uns,n-uns); B:=matrix(n-uns,l); eq:=vector(2); eq[l]:=vector(n); eq[2]:=vector(n); s o l : = v e c t o r ( n - u n s ) ; auxdx:=vector(n-uns); variable:=0; for i t o n do for j t o m do i f c o e f f ( l i n 2 [ i ] , d u [ j ] ) < > 0 then variable :=j; j:=m+i; fi; od; i f v a r i a b l e O O then i : = n + l ; f i ; od; eq:=equacions(lin2); auxA:=jacobian(lin2,dx); auxB:=iacobian(lin2,[du[variable]] ); 126 APPENDIX B. SOFTWARE PACKAGE FOR CHAPTER 4 compt:=l; for i t o n do i f z e r o s [ i ] = 0 then copy_vec_col(ii,col(auxA,i),auxxA,compt); compt:=compt+l; fi; od; compt:=1; for i to n do if zeros [i]=0 then copy_vec_row(n-uns,row(auxxA,i),A,compt); copy_vec_row(l,row(auxB,i),B,compt); compt:=compt+l; fi; od; sol:=trobar_base(A,B,eq,n-uns,2); compt:=1; for i to n do i f z e r o s [ i ] = 0 then auxdx[compt]:=dx[i] ; compt:=compt+l; fi; od; eval([eval(sol),eval(dotprod(sol,auxdx))]); end; ############################################################ # # # Function trobar_base: it is an auxiliar function for # # Singlelnput and Singlelnput2. It solves a linear system # # in order to find the coefficients of the basis w. # # # ############################################################ trobar_base:= # Ok! proc(A,col,eq,dim,flag) 127 local aux,M,i,o,a,oldm,oldn; global F,n,m,x,dx,u,du; oldm:=m; oldn:=n; if flag=2 then m:=l; n:=dim; fi; a:=vector(n) ; M:=matrix((dim-l)*m,n); o:=vector((dim-l)*m,t->0); aux:=matrix(n,l); aux:=col; M:=transpose(aux); for i from 2 to dim-1 do aux:=evalm(multiply(A,aux)-map(d_dt,aux,eq[l] ,eq[2])); M:=stack(M,transpose(aux)); od; m:=oldm; n:=oldn; eval(linsolve(M,o,'r',a)); end; ############################################################ # # # Function vec_col: It is an auxiliar function that # # transforms a vector in a column matrix. # # # ############################################################ vec_col:= # Ok! proc(dim.v) local i, M; global F,n,m,x,dx,u,du; M:=linalg[matrix] (dim,l) ; APPENDIX B. SOFTWARE PACKAGE FOR CHAPTER 4 128 for i t o dim do M[i,l] :=v[i] ; od; evalm(M); end; ############################################################ # # # Function col_vec: it is an auxiliar function that # # transforms a column matrix in a vector. # # # ############################################################ col_vec:= # Ok! proc(dim.B) local i, v; global F,n,m,x,dx,u,du; v:=linalg[vector](dim); for i to dim do v[i]:=B[i,l]; od; eval(v); end; ############################################################ # # # Function copy_vec_col: it is an auxiliar function that # # writes a vector 'vec' of dimension 'dim' in the column # # 'j' of the matrix 'Mat'. # # # ############################################################ copy_vec_col:= # Ok! proc(dim,vec,Mat,j) local i; global F,n,m,x,dx,u,du; for i t o dim do M a t [ i , j ] : = v e c [ i ] ; od; evalm(Mat); end; ############################################################ # # # Function copy_vec_row: it is an auxiliar function that # # writes a vector 'vec' of dimension 'dim' in the row 'i' # # of the matrix 'Mat' # # # ############################################################ copy_vec_row:= # Ok! proc(dim,vec,Mat,i) local j ; global F,n,m,x,dx,u,du; for j to dim do Mat[i,j]:=vec[j]; od; evalm(Mat) ; end; ############################################################ # # # Function Kronecker: it is an auxiliar function that # # computes the Kronecker indices for static feedback # # linearizable multi-input systems # # # ############################################################ Kronecker:= # Gk! proc(kah,eq) local D,d,j,k,rho,K,Aux; global F,n,m,x,dx,u,du; 130 APPENDIX B. SOFTWARE PACKAGE FOR CHAPTER 4 D:=matrix(n,m); Aux:=matrix(n,m); d : = v e c t o r ( n ) ; r h o : = v e c t o r ( n ) ; K:=vector(m); Aux :=evalm(kah[2] ); D:=evalm(Aux); d[l]:=rank(Aux); for k from 2 to n do Aux :=evalm(multiply(kah[l],Aux)-map(d_dt,Aux,eq[l],eq[2])); D :=augment(D,Aux); d[k]:=rank(D); od; rho[l]:=d[l]; for k from 2 to n do rho[k]:=d[k]-d[k-l]; od; for j from 1 to m do K[j]:=0; for k from 1 to n do if (rho[k]>=j) then K[j]:=K[j]+l; fi; od; od; eval(K); end; ############################################################ # # # Function MultiInputSFL: it computes the coefficients # # of the 1-forms w_l,... ,w_m that make a basis for # # static feedback linearizable multi-input systems. # # It is analagous to the function Singlelnput. # # # ############################################################ MultiInputSFL:= procO # Ok! local i,l,j,k,eq,kah,aux,aux2,K,W,a,o,M,formes; global Fjiijm.x.dx.Ujdu; kah:=vector(3); k a h [ l ] : = m a t r i x ( n , n ) ; kah[2]:=matrix(n,m) ; W:=matrix(n,m); K:=vector(m); formes:=vector(m); kah:=Kahler(); eq:=equacions(kah[3]); K:=Kronecker(kah,eq); for 1 from 1 to m do if (K[l] > 1) then a:=vector(n); M:=matrix((K[l]-l)*m,n) ; o:=vector((K[l]-l)*m,t->0); aux:=vector(n); aux2:=vector(n); aux:=vec_col(n,col(kah[2] ,1)) ; M :=transpose(aux); for i from 2 to (K[l]-1) do aux:=evalm(multiply(kah[l] ,aux)-map(d_dt,aux,eq[l] ,eq[2])); M:=stack(M,transpose(aux)); od; for j from 2 to m do aux: = 'aux'; aux:=vec_col(n,col(kah[2] ,j)); M:=stack(M,transpose(aux)); for i from 2 to (K[l]-1) do aux:=evalm(multiply(kah[l] ,aux)-map(d_dt,aux,eq[l] ,eq[2])) ; M:=stack(M,transpose(aux)); od; od; aux2:=linsolve(M,o,'r',a) ; copy_vec_col(n,aux2,W,l); else a:=vector(n); copy_vec_col(n,a,W,l); fi; APPENDIX B. SOFTWARE PACKAGE FOR CHAPTER 4 132 od; f o r j t o m do formes[j]:=dotprod(col(W,j),dx); od; eval([evalm(W),eval(formes)]); end; ############################################################ # # # Function quocient; it makes the quotient of a certain # # system by a general expression 'w'. # # # ############################################################ quocient:- # Ok! proc(w) global F,n,m,x,dx,u,du; local i,final,currenteq,expr,kah.sys,lin,lin2,k,var,zeros; f i n a l : = 0 ; expr:=w; kah:=Kahler(); sys:=EqSys(); lin:=EqLin(kah[3] ) ; l i n 2 : = k a h [ 3 ] ; zeros :=vector(n); for i t o n do z e r o s [ i ] : = 0 ; od; while f i n a i o l do i f coeff(expr,du[l])<>0 then currenteq:=du[1]=solve(expr,du[1] ) ; for k t o n do lin2[k]:=subs(currenteq,lin2[k] ); od; final:=1; e l i f coeff(expr,du[2])<>0 then currenteq:=du[2]=solve(expr,du[2] ) ; for k t o n do lin2[k]:=subs(currenteq,lin2[k]); 133 od; final:=1; eise var:=l; while coeff(expr,dx[var])=0 do var:=var+l; od; zeros [var]:=1; currenteq:=dx[var]=solve(expr,dx[var]); for k to n do lin2[k] :=subs(currenteq,lin2[k]); od; lin:=EqLin(lin2); expr:=d_dt(expr,sys,lin) ; expr:=subs(currenteq,expr); fi; od; eval([map(simplify,eval(lin2)) ,eval(zeros)]) ; end;