202 zyxw zyxz zyxwvutsrqpon IEEE TRAMACTTONS ON AUTOH&flC COXTROL. VOL. AC-29. NO. 3. M4RCH 1984 Robust Redesign of Adaptive Control PETROS A. IOANNOU AND PETAR V. KOKOTOVIC. FELLOW. Absiruct-Effects ofunmodeledhighfrequencgdynamicsonstability and performance of adaptive control schemes are analped. In the regulation problemglobal stabiity propertiesarenolongerguaranteed,but a region of attraction exists for exact adaptive regulation. The dependence of the region of attraction on unmodeled parasiticsis examined first. Then the general case ofmodel reference adaptivecontrol is consideredinwhich parasitics can destroy stabilie andboundednessproperties. A modified adaptive law is proposed guaranteeing the existence of a region of attraction from whichall signals converge to a residual set which contains the equilibrium for exact tracking.The size of this set dependsondesign parameten,thefrequencyrange of parasitics, andthe reference input signal Characteristics. IEEE zy the frequency content of the plant input. The lack of this mechanism has caused the loss of robustness reported in [12]. [13j. The two main results of this paper are: first, an estimate of the region of attraction for adaptive regulation, and second. a modification of the adaptive laws to guarantee boundedness in the case of tracking. The frequency content and magnitude of the reference input signal. the speed ratio p of slow versus fast phenomena, the adaptive gain, and initial conditions are shown to have crucial effects on the stability of the adaptive control schemes. These results are first analytical conditions for robustness of direct adaptivc control with respect to high frequency dynamics. They are obtained for a continuous-time SISO adaptive control scheme 113. The same methodology can be extended to more complicated continuous and discrete-time adaptive control problems. The paper is organized in two main sections. The first section containsa simple motivating scalar example which illustrates the salient features of the general methodology developed in the second section. zyxwvutsrqpo INTRODUCTION LOBAL stability of adaptive control systems, an open problem for almost two decades, was recentlysolved for both continuous and discrete SISO (single-input single-output) systems [1]-[5]. However, there still remains a significant gap between the available theoretical methodologies and the potential applications of such adaptive schemes. Global stability properties are guaranteed under the “matching assumption” that the model order is not lower than the order of the unknown plant. Since this restrictive assumption is likely to be violated in applications, it is important to determine the robustness of adaptive schemes with respect to such modelin, errors. Several attempts have been made to formulate and anal5ze reduced-order adaptive systems.Specific results such as error bounds have been obtained for adaptive observers and identifiers [6]-[lo]. In [ l l ] local stability has been proved for a reduced-order indirect adaptive regulator. Efforts on reduced-order direct adaptive control [12], [13] concentrated on simple examples where it wasshown by “linearization” [12] or demonstrated by simulations [13] that unmodeled parasitics can lead to an unstable closed-loop system. Analysis [6]. [9] of the effects of high frequency plantinputson the performance of identifiers and adaptive observers with parasitics has determined that the inputs should be restricted to dominantly rich inputs. As a design concept, the dominant richness requires that in the presence of parasitics the richness condition be satisfied outside the parasitic range. It excludes wideband inputs such as noise and square waves as undesirable. The situation in adaptive control is more difficult because the plant input is generated by adaptive feedback which incorporates the u r h o w n plant with parasitics. The schemes proposed thus far do not contain a mechanism to restrict G 0 zyxw zy zyxwvutsr zyxw 17. 1983. Paper Manuscript received May 7. 1982; revisedMarch recommended by G. C. Goodwin. Past Chairman of the Adaptive. Learning Systems. Pattern Recognition Committee. This work was supported in part by the U.S. Am Force under Grant AFOSR 78-3633.the Joint ServicesElectronicsProgram underContract N00014-79-C-0414. Ford Motor Company. and by the U.S. Department of Energy under Contract DE-AC01-81RA50568 with Dvnamic Svstems. Urbana. IL 61801. -. P. A. Ioannou~iswith thi Univeriity of’ Southem CalifoAa. Los Angeles, CA 90089-0781. P. K. Kokotokic is with the Coordinated Science Laboratory. University of Illinois, Urbana, IL 61801. THE SCALAR REDUCED-ORDER ADAPTIVE CONTROL PROBLEM We start with a simple example of reduced-order adaptive control in which the output yp of a second-order plant -i.P = a P -~P + 2 z - u . =-Z S u, (1.1) (1.2) with unknown constant parameters u p and p, is required to track the state rg of a first-order model jn,= - a,,,yn1+ r a,, > 0 (1.3) where u is the control input and r = r ( r ) is a reference input. a uniformly bounded function of time. This example serves as a motivation for and an introduction to the general methodology to be developed in the next section. As in our earlier work [6], the model-plant mismatch is due to some “parasitic” time constants which appear as multiples of a singular perturbation parameter p and introduce the “parasitic” state 7. In (1.1).(1.2) the parasitic state is defined as 9 = z - u resulting into the following representation: j p = a,Fp + 2 q +u (1.4) (1.5) p*=-q-pU where the “dominant” part (1.4) and “parasitic” part (1.5) of the plant appear explicitly. Ifwe apply to the plant with parasitics (1.4), (1.5) the same adaptive law which we would have applied to the plant without parasitics: that is, if we use the control + (14 y > 0, (1.7) u=- K(C)J> r(t) ~~~ and the adaptive law 0018-9286/84/0300-0202%01.00 01984 IEEE K = yevp 203 IOANNOU AND KOKOTOVIC: ROBUST REDESIGN OF ADAPTIVE CONTROL and the objective is to drive yp to zero despite the presence of parasitics while assuring that all the signals in the closed-loop system (1.15)-(1.18) remain bounded. It is important to note that the open-loop system (1.15):(1.16) might not be stabilizable by constant gain output feedback for a given value of p . If this is the case, then there is no hope that the adaptive controller (1.17)? (1.18) will stabilize the equilibrium of (1.15), (1.16). The following lemma characterizes parasitics for which a linear output stabilizing feedback law exists. Lemma 2: There exists a p1 > 0 and a constant KO such that for all p E (0, pl] the system (1.15), (1.16) with the feedback law we obtain zyxw zyxwvutsrqpon zyxwvutsrq zyxwvutsr zyxwvu zyxwvutsrq zyxwvutsr zyxwv zyxwvuts where zyxwvutsr e=yP-yfll. K*=a,+a,. (1.11) The existing theory of adaptive control [14],[15] guarantees stability properties for the case without parasitics p = 0, when (1.8)-(1.10) reduce to t=-a,,,~-(K(t)-~*)(~+y,,,) x = y2( 2 + y,,,). (1.19) u - KOyp is an asymptotically stable closed-loop system. Furthermore, (1.20) (1.12) (1.13) and 1 Lemma I : For any bounded initial conditions i?(O)? X ( 0 ) the solution F ( t ) , K ( t ) of (1.12), (1.13) is bounded and lim,+=F(t) = 0, limrzXF( t ) = K, where constant K , is in general a function of i?(O), K(0). Furthermore, .if r ( t ) is sufficiently rich, then K ( t ) = K *, independent of F(0).K(0). lim, The above example illustrates some of the robustness questions to be answered, in this paper. Given that the adaptive system without parasitics, in this case (1.12), (1.13), possesses properties such as in Lemma 1, how will these properties be altered by the parasitics, that is, what are the stability properties of (1.8)-(1.10)? Which modification of the adaptive law would help to preserve some of the desirable properties? The perturbation parameter p provides us with a means to answer such questions in a semiquantitative way using the orders of magnitude O(p’), noting that for p small, the quantity O( p ” ) is small when v > 0 and large when v 0. The smallness of p implies that the parasitics are fast and that neglecting them, p = 0, we concentrate on the slow. that is. the “dominant,” part of the plant. As we shall see, a first property to be lost due to parasitics is global stability. In the case of regulation, that is. when y,,, = 0, r ( t ) = 0, the boundedness of the solutions e( t ) , K( t ) and the convergence of e ( r ) to zero as t + ce is preserved, but is not global. It possesses a domain of attraction whose size we describe by estimating the orders of magnitudes of the axes of an ellipsoid D(p). In the tracking problem, when r( r ) 0 the adaptive system with parasitics such as (1.8)-(1.10) may not converge to or may not evenpossess an equilibrium. A practical goal is then to guarantee some boundedness properties. Weshow that a redesign, which may sacrifice some propeoies of the ideal system without parasitics, results in the convergence from any point in D(p) to a uniformly bounded residual set D,(p). Thedesign objective is then to make D(p) as large as possible and D,(p) as small as possible. Let us illustrate this discussion by analqzing the regulation problem and the tracking problem for the example (1.1)-(1.3). + - P - up > K, > u p . (1.21) We now establish the stability properties of the adaptive control system (1.15)-(1.18) for p < p , . Theorem 1: There exists p* < p , and positive numbers a < 1/2, cl, c2 such that for each p s ( O , p * ] any solutionyp(r),q(r):K(t) of (1.15)-(1.18) starting from the set D(p) = { ~ b , qK: , lypl+ IKl< c,p-=, 191 < c 2 p - u - 1 / 21 (1.22) is bounded and y ( t )+ 0: q ( t ) + 0, K ( t ) ---* constant as t + m. ProoJ Let > u p be a finite constant and consider the function k, Observe that for each p > 0, c > 0, a > 0 the equality V(yp,q,K)=cp-*“ (1.24) defines a closed surface . Y ( p , a , c) in R 3 . The time derivative of V(y,,q, K ) along the solution of (1.15)-(1.18) is A . Regulation In the regulation problem expressions (1.8)-(1.10) become r ( t ) = O , y,,,,(t)=O, e ( t ) = y p ( t ) (1.14) ,$ = ap-”p+ u + 2 9 (1.15) pQ=-q-pU (1.16) u = - K(t)yp (1.17) K = yy; (1.18) where /3 = K , - a p > 0. Inside 9 ( p t a , c), IynI,1K1 can grow up to O ( P - ~ ) . whereas 191 can grow up to O(pLil/Z-a).Therefore, there exist constants 204 zyxwvutsrqponm zyxwvutsr IEEETRANSACTIONS Oh' AUTOMATICCONTROL,VOL. p l , p 2 , p 3 such that insideY(p,a,c) AC-29, NO. 3, MARCH 1984 1.5 zyxw zyx zyxwvuts zyxwvutsrq zyxwvutsrqpo 1.0 - 82(+ - 2p2p1-a - (1.26a) 4p) for some positive constants 6,, 6,. Choosing a < 1/2 we can find a p* > 0 such that for any P E (0,P*l > 6 , p 1 - ~ a+ a2p2(1-la) , 1 > 2p2p1-" +4p. 2 -0.5 1 1 , , , 1 , , , , 1 , , , , 1 1 , , , 1 - ' - - m ~ - ~ zyxwvu 0 1 2 3 4 Time,t (sed 5 6 (a) Therefore, ~ O ; p , ~ , K ) ~ O . f o r e a c h ~ ~ ( O , p * ] a n d a l l ~ , , ~ . K enclosed in 9( p , a, c), and V = 0 only at the equilibrium J, = 0, 7 = 0, K = constant. Moreover, there exist positive constants cl, c2 20 2'5s$ \ such that the set D(p) given by (1.22) is enclosed by the surface :! Y ( p , a, c) and any solution of (1.15)-(1.18) starting from D(p) v 15; \ remains inside Y ( p , a, c). Furthermore, inside Y ( p , a , c), - I V (y,, q , K ) is a nonincreasing function of time which is bounded 1.03 * ii from below, and hence converges to a finite value V,. Since V ( q , q , K ) is bounded, V ( y , ( r ) , q ( t ) , K ( t ) ) is uniformly conj i 0.5i y p , q, K enclosed in Y ( p , a, c) and t > 0. Theretinuous for 1 fore, lim,+zV(y,(t), q ( t ) , K(rN = 0, i.e., q ( t ) 0, q ( t ) -,0. ; \ and K ( t )+ constant as t + w . 0.0 , , ,Ll 1 , ' , , , , , , , I , I , , , I 0 1 2 3 4 5 6 Remark I: It can also be shown that increasing adaptive gain Tlme,t (sed y for a fixed p reduces the size of the domain D(p) and the stability properties of Theorem 1 can no longer be guaranteed if (bl Y 3 0(1/F). Fig. 1. Adaptive regulation for p = 0.05. ~ ~ = ( 1, 0 p(0) ) = 1. K(0) = 3, Remark 2: As p -,0, domain D(p) becomes the whole space and y = 5. Response ma. stable. R3; that is, the adaptive regulation problem (1.15)-(1.18) is well posed with respect t o parasitics. Remark 3: Theorem 1 is more than a local result because it : shows that given any bounded initial condition y (0).~ ( 0 )K(0). there always exist p* such that for each p E (0,p*? the solution of (1.15)-(1.18)isboundedand~~(r)-,O,~(t)+O.K(t)-,constant I\ + I , , , ast-m. Remark 4: Since Theorem 1 is only a sufficient condition it is of interest to examine whether the stability properties of Lemma 1 are indeed lost for initial conditions outside the set (1.22). From Lemma 2 and the fact that K ( r ) is nondecreasing it can be seen that instability occurs if K(0) > (l/p)- a,. As an illustration of the stability properties established by Theorem 1 simulation results for (1.15)-(1.18) with a p = 4 and different values of p, y and initial conditions are plotted In Figs. 1-4. In addition to y,(t), V ( y p ,17, K ) = V with K , .= 7 is plotted against time to show whether all the signals in the closed loop remain bounded. In, Fig. 1 where p = 0.05, y = 5 . ~ ~ (= 1.0. 0 ) q(O)=l.O, K(0) = 3, the objective of the regulator is acheved since y, 0 and Vis bounded. Increasingy,(O) from 1 to 2.4 and keeping all the other conditions the same as in Fig. 1, the regulator fails its objective and y, -,oc as s h o m in Fig. 2. With the same initial conditions as in Fig. 1, but with p = 0.07 instead of p = 0.05. y, -, 03 as indicated in Fig. 3. Fig. 4 shows the effect of increasing the adaptive gain y. With the same initial conditions as in Fig. 1, but with y = 30 instead of y = 5. regulation fails and - zyxwvuts Y, --*a. B. Tracking Returning now to the tracking problem we note that for a general r ( t ) + 0 the system(1.8)-(1,10)need not possess an equilibrium. The best we can expect to achieve in this case is to guarantee that the solutions startingin D(p) remain bounded and converge to a residual set Do(p). Fig. 2. Adaptive regulation for p = 0.05, ~ ~ (= 02.4.) q(0) = 1, K(O) = 3, and y = 5. Instability due to largerxD(0). To prove such a result we modify the adaptive law (1.10) as k=-aK+ye(e+yn2) (1.27) where is a positive design parameter. In viewof (1.27) the equations describing the stability properties of the tracking problem in the presence of parasitics are: (J k=-a,e-(~(t)-~*)(e+~,,,)+2q ,++, = - ?I + [ ye( e + yn2 1'+ 2 ~ +q Kr - 4 k=-aK+ye(e+yn,). (1.28) K ( K - a, - .)(e + y m ) . (1.29) (1.30) zyxwvutsrqpo zyxwvutsrqpo z zy zyxwvutsrqponmlkjihgfedc zyxwvutsrqpon zyxwvutsrqpo zyxwvutsrqpo zyxwvutsrq 205 IOANNOU AND KOKOTOVIC: ROBUST REDESIGN OF ADAPTI\T CONTROI =-a,,,e'-aK(K-K*)-q2+p(q+2e) Y ~[ye(e+ynz)'-K(K-ap-u)(e+y,,,)+2Kq+Kr YP 50+ 1 Equation (1.36) can be rewritten as o i a P ( e , q , K ) = - ~ ~ [4~ - p ( 2 - y e ~ + 4 y y ~ , e + 2 y y ~ - 2 K ' A I----- (1.36) -2an,e-i-2(K-K*)(e+ym)+4q]. 1 ' ' ' ' 0.0 ID 05 2.0 1.5 Tlme,t ! ' ' ' ' I 2.5 30 +2Kap-4am-4K+4K*+2Ko) (sed Fig. 3. Adaptive regulation for 1.1 = 0.07, ~ ~ (= I0, ~) ( 0=I. ) K(O) = 3. and y = 5. Instabilitylarger due to 1.1. -p2(4K+8+ye'+2yyme+yy,i -K2+Kap-2a,,,-2K+2K*+oK I') - - [1p . - 2 p e ( 4 K + 8 + y e 2 + 2 y y m e + ~ y y ~ 4 -K2+Kap-2a,-2K+2K*+oK)]' zyxwvutsrqponmlk 1 - -100 1, 4 [ 71 - 2 ~ ~ ( a p . ~-n~z y 2 + m -2ym + u ~ m ) ] - - [41q - 2 p ( 2 K * y m - i ) ] * - q ' ( 8 -1p ( 4 + 2 K ) ) 1 - 1 2 5 . ~ y r - - y r ~ ~ - ~ - - r - - - - . ~ I 0.0 0.2 0.4 C16 0.8 LO 12 Tlme,t (sed Fig. 4. Adaptive regulation for 1.1 = 0.05, y (0) =1, q(0)= 1, K(0) = 3, and y = 30. Instability due & larger y. Theorem 2: Let the reference input r ( t ) satisfy Ir(t)l<r,, li.(t>l<rz Vt>O (1.31) where r,, r, are given positive constants. Then there exist positive constants ti, p*, r3,a < 1/2, c1 to c 3 , and c4 >, 1 such that for each p E (O,p*] every solution of (1.28)-(1.30) starting at r = 0 from the set D(p) = { e , K , qlei+ : IKI < clp-", (1.37) PI, 1711 < c~"~-'.''} (1.32) enters the residual set h i d e Y ( p , a, c,), lKl can grow up to O(p-") and 1711 can grow up to O(p-" -"). Hence, there exists positive constants 11,5,. 5; such that inside Y (p , a,c,) le1 < 1K1< S Z P - ~ , Iql< 5 3 ~ - ~ ' ~ - ~ (1.38) . For e, 7 1 , K inside Y ( p , a,c,), (1.37) can be simplified to for some positive constants PI, p2, and we can see thatfor each p > 0, co > 0, a > 0 the equality V ( eK, )q , = coppZa (1.35) defiies a closed surfaceY(p, a, cp) in R 3 space. The time derivative of V( e , 7, K ) along the solution of (1.28)-(1.30) is ,, > , c3p2(1-a) /I3. Choose (1.40) where cj = 2yP,, and a < 1/2. Then there exists a p* such that for each p E (0, p * ] 206 zyxwvutsrqponm zyxwvut AC-29.NO. 3, 1984 zyxwvutsr zyxwvutsrq zyxwvuts zyxwvutsr zyxwvutsr zyxwvuts IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. MARCH 7 (1.41) Hence, for each p E (0, p * ] and all e, q , K inside Y ( p , a, co) i + p 2 I + - 12K*yn,- i j 2 . a:,) (1.42) Since r, i. are uniformly bounded the set Do@) is uniformly bounded and is enclosed by Y ( p ,a, co). Outside D,(p) and inside Y ( p , a , co), k(e , q , K ) < 0, and therefore V (e, q , K ) decreases. Hence, there exists positive constants c,, c2 such that the set D(p) is enclosed by Y ( p , a , co) and any solution e( t ) . K( t ) . q ( t ) which starts in D(p) remains inside Y ( p , a , c,). Furthermore, there exist constants c4 > 1 and t1 such that every solution of(1.28)-(1.30) starting at t = 0 from D(p)/DO(p) mill enter Do(p) at r = r, and remain in D,(p) thereafter. Similarly, any solution starting at t = 0 from Do(p ) will remain in D,(p) for all t > 0. Remark 5: The set Do(p) depends on r, and rz, i.e., the bounds for the magnitude and frequency content of the reference input signal. For, a given p an increase in Irl or 1i.l can no longer guarantee that V(e, q , K ) < 0 everywhere in D(p)/D,(p). For this reason, ow formulation excludes high frequency or high amplitude reference' input signals such as square or random waveforms. the traditional favorites of the adaptive control literature. Remark 6: It can also be shown that increasing the adaptive gain y for given p , r , and i reduces the size of the domain D(p). For y > O(l/p) the stability properties of Theorem 2 can no longer be guaranteed. Remark 7: The analysis in the proof of Theorem 2 shows that the useof u is essential in obtaining sufficient conditions for boundedness in the presence of parasitics. However, in the absence of parasitics ( p = 01, o > o causes an output error of o(&). This is a tradeoff between boundedness of all signals in the presence of parasitics and the loss of exact convergence of the output error to zero in the absence of parasitics. The sizeof u reflects our ignorance about p. For high frequency parasitics p is small, and therefore u can be small. Remark 8: Conditions (1.20),(1.21) of Lemma 2 specify the relation between the range of parasitics and the open-loop and desired closed-loop poles. From (1.21) we cansee that for a desired closed-loop pole a,, the gain K ( t ) has to converge to K * = a,, + a,, which has to satisfy "ri zyxwvuts zyxwvu 1 - a p> K * > a p . P (1.43) From (1.43) it is clear that a,, should satisfy 1 -Zap > a,, > 0 - P (1.44) indicating that the desired closed-loop system has to beslower than the fast subsystem. Therefore, for stability the poles of the reference model have to be chosen away from the parasitic range. It is of interest to examine whether for initial conditions outside the set D(p) we can loose boundedness. Simulation results with a p = 4, a,, = 3, and y = 5 are summarized in Figs. 5-12. Plots of the output error and the function V ( e ,q . K ) = V versus time are obtained for different initial conditions, p, u, and reference input characteristics. In Fig. 5 the output error e and function V are plotted for p = 0.01. e(0)=l. q(0)=1, K(0) = 3. o = 0.06, and r( t ) = 3sin2t. The output error decreases and remains close to zero. The function V is strictly decreasing for LO I'"'I-1 0.0 0.0 2.5 5.0 7.5 no.15.0 Tlme,t (sed 12.5 (b) Fig. 5. Tracking for p = 0.01. e(0)= 1, q(0) = 1. K ( 0 ) = 3. r( t ) = 3 sin 2 t. Bounded response. (I = 0.06 and, V > 0.05, but V changes sign in the region V < 0.05 as shown in Fig. 5(b). Keeping the same conditions as in Fig. 5, but increasing p from 0.01 to 0.05 we can still achieve similar results as shown in Fig. 6. However, in this case the steady-state error and V are larger and V changes sign for V < 0.4. Increasing the value of p from 0.05 to 0.08 the output error becomes unbounded for all u 2 0 as indicated in Fig. 7. The effects of the input characteristics are summarized in Figs. 8-10. In Fig. 8, p = 0.05, e(0)= 1, q(0) = 1, K(0) = 3, u = 0.06. and r( t ) = 3sin10t results into an unbounded output error due to the increase of the frequency of r ( t ) from 2 to 10. The same instability result has been observed for u = 0.0,0.02. However, for u = 0.08 the output error became bounded as shown in Fig. 9 illustrating the beneficial effects of u when parasitics are present. The effect of the amplitude of the reference input r( t ) is shown in Fig. 10. With p = 0.05, u = 0 or 0.06 and the same initial conditions as before, but with r( r) = 15sin2t, the output error goes unbounded. Fig. 11 shows the effect of initial conditions on boundedness. By increasing e(0) from 1 to 2.5 and keeping p = 0.05, q(0)=1, K(0) = 3, and r( t ) = 3 sin2r the output error becomes unbounded for all u > 0. In Fig. 12 we show the loss of exact convergence of the output error to zero in the absence of parasitics ( p = 0) due to the design parameter u. ADAPTIVE CONTROL OF A SISO PLANTIN THE PRESENCE OF PARASITICS . zyxwv We now consider the general problem of adaptive control of a SISO time-invariant plant of order n + m where n is the order of the dominantpart of the plant and rn is the order of the parasitics. The plant is assumed to possessslow and fast parts zyxwvutsrqp zyxwvutsrqp zyxwvutsrqponm zyx I zyxwvutsr zyxwvutsrqponmlkjih 207 IOANNOL! AND KOKOTOWC: ROBUST REDESIGN OF ADAPTIVE C O m R O L L5 -l -1 __ __ - -. . __- - -__ - - -- - - e - - - -- - 200 ___ --- -- - -a5- zyxwvutsr - l . O ~ y ~ - r ~ - - - ~ - r - -TO - 2 4 IO 8 6 I 12 -203 O - P r 7 , Q2 0.1 0.0 Tm,t (sed (a) -. - 0.4 0.3‘ (sed T1me.t 0.5 Fig. 8. Tracking for p = 0.05, e(0) = 1, q(0) =1, K(0) = 3, (I = 0, or 0.06 and r ( t ) = 3sin10t. Response unbounded due to high reference input frequency. e o zyxwvutsrqpon zyxwvutsrqpo zyxwvut 1 m IO 0.0 0 2 4 6 12 8 Time,t (sed -1 (b) Fig. 6 . Tracking for p = 0.05, e(0)= 1, q(0) = 1, K ( 0 ) = 3, u = 0.06, and r ( t ) = 3sin2t. Bounded response, error and V larger due to larger p. -2 ao 1 1 1 1 , 1 25 1 1 1 1 1 5.0 I I I I I I 1 Tm,t 2000 : no 7.5 I I , I 125 , , , 1- (sed Fig. 9. Tracking for p = 0.05, e(0) =1, q(0)=1, K(0) = 3, u = 0.08, and r( t ) = 3 sinlot. Input frequency high, but e is bounded due to larger u. ~ 1500 - e IOOO- 500- e 200 0 - v a2 0.0 , I OS a4 Time,t -? I a8 (sed Fig. 7. Tracking for p = 0.08, e(0) = 1, q(0) = 1, K(0) = 3, u 2 0.0, and r( t ) = 3 sin2t. Response unbounded due to large p. OD a2 ~4 and is represented in the explicit singular perturbation form k = A 1 , x + A 1 2 Z + b,u pi y = Az,x + A,?” + b2u, LO IZ (sed Fig. 10. Tracking for p = 0.05, e(0) =1, q(0) =1, K(0) = 3, u = 0, or 0.06 and r( t ) = 15sin2t. Response unbounded due to large reference input amplitude. (2.1) ReA(A,,) < 0 (2.2) (2.3) = c;x where x, z are n and m vectors, respectively, and u , y are the scalar input andoutput of the plant, respectively. State z is formed of a “fast transient” and a “quasi-steady state” defined as the solution of (2.2) with p i = 0. This motivates the definition of the fast parasitic state as q =z aa ~6 Tirne,t + A,’(A2,x + b,u). (2.4) The two important restrictions on the unknown plantare Re A(A2,) 0 and y = c r x . The first restriction, which prohibits unstable or oscillatory parasitic modes, is natural and cannot be removed. The second restriction allows the parasitics to be only “weakly observable,” that is observable through the slow part of the plant. For plants with strongly observable parasitics y = clx + c2z, the static output feedback is nonrobust, that is, it can destabilize a stable plant [16]. In this case a dynamic compensator must be used, containing a low-pass filter. As shown in [9], the original plant with a first-order filter can be represented as an 208 zyxwvutsrqponm zyxwvutsrqponz IEEE TRAWSACTIONS ON AUTOMATIC CONTROL, VOL. AC-29. NO. 6oo- I 3, p+=Ar2q+p(A1x+Azu+A3q+A4k) MARCH 1984 (2.7) (2.8) I'= c;x. The output y of the system (2.6)-(2.8) is required to track the output;,'I of an nth order reference model m i = AnrXnt + bnrr (2-9) zyxwvutsrqponmlkjihgf zyxwvutsrqp zyxwvutsr zyxwvutsr ynt= (2.10) CLXm where r( t ) is a uniformly bounded reference input signal. Without loss of generality let us assume that the transfer function W n t ( s ) -2000 8 . 0 0 . 1 , , , I I ' I I ! I I I I I L O L ! 5 2 ~ 2 TIW,~ (sed , , 5 I I I 5 3 D ~ Fig. 11. Tracking,forp = 0.05, e(0) = 2.5. ~ ( 0=)1. K ( 0 ) = 3. w > 0. and r ( t ) = 3an2t. Response unbounded due to large e ( 0 ) . is strictly positive real. The reduced-order plant obtained by setting p = 0 in (2.6)-(2.8) is assumed to satisfy the following conditions. i) The triple ( A o , b o co) , is completely controllable and completely observable. ii) In the transfer function W o ( s )= c:(sI- A o ) - ' b 0 = K Np ( sm) (2.12) Hurwitz polynomial of degree n - 1 and D ( s ) is a monic polynomial of degree n. For ease of exposition we assume that K p = K , = 1. The controller structure has the same form as that used in [l] for the parasitic-free plant, that is for p = 0 in (2.6)-(2.8). In this controller the plant input u and measured output y are used to generate a (2 n - 2) dimensional auxiliary vector u as X ( s) is a monk -1 2.5 0.0 zyxwvutsrqpon 7.5 50 Time) 10.0 12.5 15.0 (set) 8 , = Av, w1= c'( B, v + gu (2.13) t ) v1 = Av, + gy W, = d o ( t ) y + dT(t)v2 1.0- (2.14) where A is an ( n - l ) ~ ( n - l ) stable matrix and ( A , g ) is a controllable pair. The plant input is given by 0.5- (2.15) u=r+era 0.0 1 - I I , 2.5 loa 7.5 Tlme. t Eo (sed (b) Fig. 12. Tracking for p = 0. e(0)= 1. ~ ( 0=)1, K(0) = 3. w = 0.08. and r ( r ) = 3sin2r. Loss of u.a.s. in the parasitic freecase. p = 0. due to w > 0. wherewT=[~~,v~,~]and8(t)=[cT(t),d~(t),do(t)lTiSa( 1) dimensional adjustable parameter vector. It has been shown in [l] that a constant vector 8* exists such that for O ( t ) = 8* the transfer function of the parasitic-free plant (2.12) with controller (2.13)-(2.15) matches that of the model (2.11). Ifwe apply to the plant with parasitics (2.6)-(2.8) the controller described by (2.13)-(2.15) we obtain the following set of equations for the overall feedback system zyxwvutsrqp zyxwvutsr 0.05.0 augmented plant in the form (2.1)-(2.3) with weakly observable parasitics. Defining [tl) c2 = A , =A l l - A12A,'A21, [ I: :][rl]+ gc, $r A 0 u, [:](era + [*,I. zyxwvutsr A , = A$A,,AO A, bO=bl-A,,A,'b2, A? = AglAzlbo, = A G I A Z I A ,I Z , = AG'bz (2.16) (2.5) and substituting (2.4) into (Z.l), (2.2) we obtain a representation of (2.1)-(2.3) with the dominant part (2.6) and the parasitic part (2.7) appearing explicitly i= A o x + bou + A12q (2.6) pQ = A,2q i p ( A l x + A28'o -k A,d'a + A , r + A,q + A,B'cb + A d ? ) (2.18) y =.x:. Introducing 8 * , Y' (2.17) ', a,: t , : ] and = [X zyxwvutsrqpon zyxwvutsrqpo zyxwvutsrq 209 IOANNOU AND KOKOTOVIC: ROBUST REDESIGN OF ADMTIVE CONTROL where u is a design scalar parameter. The resulting adaptive control system with parasitics is described by zyxwvutsrqp zyxwvutsrq zyxwvutsrqp zyxwvutsrq (2.19) we rewrite (2.16), (2.17) in a form convenient for our stability analysis Y= A , Y + b,((e + r)+&2p - (2.32) (2.20) p3=~22p+p(&~+~2eTW+~2r+~3p + A,dTw + A4B% + A , i ) - ~ , ~ e + x , , , ~ ~ r ~ e + x , , ~ e , + ~ , ~ ~ ~ ~ , e (2.21) where 4, = [AT, 0 0IT and Kl = [AT 0 O]? An advantage of this form is that for e( t ) = e* in the parasitic-free case (2.20) becomes a nonminimal representation of the reference model ... Ole d-rel(e+x,,). e,= [I O e=- (2.33) (2.34) Theorem 3: Let the reference input r( t) satisfy zyxw for some given positive constants r l , r2. Then there exist positive constants p*, u, (Y < 1/2, c1 to c,, y1 and t , such that for each p E (0, p * ] every solution of (2.31)-(2.34) starting at ( = 0 from the set , The equations for the error e = Y - x can be expressed as t.=A,e+b,(e-e*)'(P+x,,)+&~p (2.23) p 3 = ~ ~ ~ p 1 p [ [ ( e + x , , ) + ~ ~ e ~ ~ , + ~ ~ , , )enters the residual set + A 2 r + A 3 p + A , P ( P + x,,) + A407f(e, e , s ) + A4i] e,=kTe=[l O ... Ole (2.24) (2.25) where where X, to X, are positive constants, at t D,(p) for all t > t,. Furthermore, =t, and remains in The -corresponding regulation result follows as a corollary. Corollary I: Assume r( t ) = 0, x,, = 0. Then there exists a p* such that for each p E (0, p * ] and u = 0 in (2.34) any solution e ( ? )= Y ( t ) ,p ( t ) , e( t ) of (2.31)-(2.34) which starts from D(p) given by (2.36) is bounded, l i m , + x ~ [ Y ( = r )0, ~ ~lim,+xllp(()]l = 0, lirn,+,llO(t)ll= 0, and l i m , + J p ( r ) - O * ~ ~ r - ~=constant. Proof of Theorem 3: Choose the function e'" = zyxwv x - x,, e(" = u1 - u l n , , e(2)= u2 - van,. (2.28) We now need to design an adaptive law for updating the parameter vector e ( t ) . For the parasitic-free case [l]the adaptive law zyxwvutsr [ zyxw (2.39) P, p - P;'( e'PA,,A;')'] guarantees that the output error goes to zero as t 4c ~ jand the signals in the closed loop remain bounded for any uniformly where P = P T > 0 satisfies bounded reference input r ( t ) . As demonstrated in the first secATP+PA,=-qqT-~L tion for the scalar tracking problem, the best we can expect in the presence of parasitics is to guarantee that the solutions starting in Pb, = h a domain D(p) remain bounded and converge to a residual bounded set D,(p). To achieve this we modify the adaptive law for some vector q, matrix L = LT> 0, and scalar (2.29) as P, = PT> O satisfies (2.40) (2.41) 4 0, and 210 zyxwvutsrqpon zyxwvutsrqp zyxwvu zyxw zyxwvutsr zyxwvutsrq zyxwvuts zyxwvutsr IEEE TRANSACTIONS ON AUTOhL4TTC CONTROL. VOL. AC-29. NO. Equations (2.40), (2.41) follow from the fact that hT(sZ - A,)-'b, is strictly positive real [l] and (2.42) follows from the assumption that Re h ( A), < 0. Observe that for each p > 0, p o > 0, and a > 0 the equality v(e,q.8)=pop-2a 3. XIARCH 1984 for some positive constants 6, to 6, which depend on r , , r, and the norms of matrices. For a < 1/2 there exists a p* such that for each p E (0, p * ] (2.43) (2.48) defines a closed surface S(p,a , p o ) in R S ' 1 + m .- 3The time derivative of V ( e ,q , 8 ) along the solution of (2.31)-(2.34) is V(.,v,e) 1 = - -eT(qq'+ 2 rL)e- o(e- e*)'r-le - 1 ?qTQIq +p[9-pll(e~p~,~,')~]Tp~[~,(e;x,,,) + A28T( 2 + x , , ) + A , r + A39 T -~A48'(e+x,,)-A,(P+x,,)T(e+x,,)e, +A,0Tf(8,e,q,r)+A4t -p- A~,P(A,.~+~=(B-~*)~(~~X,,,)+A,~~)I. 22t7-7~ (2.44) Let E h,=-minh(L), 2 1 h,=minA(P). A , = ~ ~ ~ ~ A ( Qx 4 J= ,l p ll. (2.45) Then (2.45) is simplified to Choose u 3 c3p2("-"' where c3 = (4/X,)6,. Then there exist constants c1 to c2 such that the set D,(p) is enclosed by [[P(p).and D(p) isenclosed by S ( p , a . p o ) . Furthermore. V ( e ,77.6) < 0 everywhere inside S(p. a , p , ) except possibly in D,(p). The constants cl, c2 are chosen such that every solution of (2.31)-(2.34) starting from D ( p ) remains in S ( p , a , p,). Since in D(p)/[[P,(p). I/( e, q , 6 ) is strictly decreasing there exist constants c4 2 1 and t , such that any solution starting at t = 0 from D ( p ) / ~ , ( p ) will enter Do(p) at t = r , and remain in [[P,,(p) thereafter. Any solution starting from D,(p) at t = 0 cannot escape and remains in D o ( p ) for all t > 0. Pruuf of Curulhy I: The proof of Corollary 1 follows directly from the proof of Theorem 3 by noting that when r(r) = 0, .xn, = 0, and u = 0, the set D,(p) reduces to the origin e = 0, q = 0, i.e.? in (2.38) y1 =.O. Therefore, V(e,q,8)< 0 everywhere inside S ( p , a , p o ) and V( e. 17.8) = 0 at the origin e = 0, 17 = 0. Hence, any solution e(t) = Y ( t ) ,q ( t ) ,O ( t ) which starts from D(p) is bounded. Furthermore, V ( e( t ) , q( t ) , 8( t ) ) is uniformly continuous, and therefore lim, ,V( e( t ) , q( t ) . 8( t ) ) = 0. Hence, lim,+,ll~(t)ll=0. and Lm,,,ll8(r)ll=O. limt,,llY(t)ll=O, Since inside S ( p , a , p , ) , V(e, q, 8) is nonincreasing and bounded below, it reaches a limit k ' ,which is a finite constant. Therefore, lirn,,,(6(r)-e*)'r-'(6(r)-6*) = 27/,. Le., limr-xlle(t)8*llr-1 =constant. In Theorem 3 and Corollary 1, it is assumed that p* < p1where p1is defined in the following lemma. Lemma 3: There exists a p1> 0 such that constant output feedback u = B&J stabilizes (2.16)-(2.18) for all p E (0.pl]. The proof of Lemma 3 is more complicated than that of Lemma 2 and can be found in [17] where an explicit expression has been obtained for p,. Remark 9: The dependence of constants 6, to 8, and y, on r l . r, shows that for a given p a reference input signal with high magnitude or high frequencies can no longer guarantee that V ( e ,17.8) < 0 everywhere in D(p)/D,(p). Such a reference signal introduces frequencies in the input control signal which are in the parasitic range. Thus, the control signal is no longer dominantly rich [6] and, hence, it excites the parasitics considerably and leads to instability. This explains the instability phenomena observed by other authors in simulations such as [18] where a square wave was used as a reference input signal. Remark IO: For ease of exposition we have assumed that r( t ) has bounded derivative everywhere on [O.m). This assumption can be relaxed byallowing a finite number of jump points or comer points for r ( t ) on the t-axis d h o u t any significant changes on the results of Theorems 2 and 3. Similar comments as in Remark 8 apply here also. zyx zyxwvutsr where a, to a,,PI to & and y1 are positive constants determined from r,, r, and the norms of the system and the reference model matrices. Constant y1 depends on rlr r, and y1 = 0 when r, = 0. r, = 0.For all e, 8, and q enclosed in S(p,a , pol. (2.46) becomes CONCLUSION We have analyzed reduced-order adaptive control schemes in which reference models can match the dominant part of the plant, while the model-plant mismatch is caused by the neglected high frequency parasitic modes. In presence of parasitics the global stability properties of the parasitic-free schemes can be lost. However. we have shown that in the regulation problem a region of attraction exists for exact adaptive regulation. This region is a function of the adaptive gains and the speed ratio p , and as p -+ 0, it becomes the wholespace. Thus the adaptive regulation problem is well posed with respect to parasitics. In the case of tracking we proposed a modified adaptive law. The modified scheme guarantees the existence of a region of attrac- D zyxwvutsrqpon zyxwvutsrqp zyxwvutsr KOICOTOVlC: ROBUST REDESIGN OF ADAPTIVE CONTROL IOANNOU tion from which all signals converge to a residual set which contains the equilibrium for exact tracking. The dependence of the size of this set on design parameters indicates that a tradeoff can be made sacrificing some of the ideal parasitic-free properties, in order to achieve robustness in presence of parasitics. The crucial effects of the frequency range of parasitics, the adaptive gains, and the reference input signal characteristics on the stability properties of adaptive control schemes explain the undesirable phenomena observed in [12], [13]. The results of this paper are obtainedfor a continuous-time SISO adaptive control scheme where the transfer function of the dominant part of the plant has a relative degree of one. The same methodology can be extended to more complicated continuous and discrete-time adaptive control problems. REFERENCES ~ [16] H. K. Khalil, “On the robustness of output feedback control methods to modehng errors,” IEEE Trans. Aurotmt. Contr., 1701. AC-26, Apr. 1981. [17] P. A. Ioannou, “Robustness of absolute stability,” Znr. J . Confr., vol. 34, no. 5, pp. 1027-1033, 1981. [18] C. Rohrs, L. Valavani, and M. Athans, “Convergence studm of adaptive control algorithms, Part I: Analysis,” in Proc. 19th IEEE Conf. on Decision and Conrr., Albuquerque, NM, Dec. 1980. Petros .A. Ioannou was born in Cyprus, on February 3, 1953. He received the B.Sc. degree with First Class Honors from University College. London, England. in 1978 and the M.S. and Ph.D. degrees from the University of Illinois, Urbana, in 1980 and 1983, respectively. In the summer of 1979 he worked with the Ford Motor Company on the control problem of the car engine. From 1979 to 1982 he was a Research Assistant at the Coordinated Science Laboratory, University of Illinois. During his graduate studies, he also worked on research projects from the Ford Motor Company, General Electric, and Dynamic Systems. In 1982 he joined the Department of Electrical Engineering-Systems, University of Southern California, Los Angeles, as an Assistant Professor. He currently teaches and conducts research in the areas of adaptive systems. large scale systems, and singular perturbations. Dr.Ioannou is a member of the American Society of Mechanical Engineers, EtaKappaNu,and Tau Beta Pi. He has held a Commonwealth Scholarship from The Association of Common\vealth Universities, London. England, and was awarded the Goldsmid Prize and the A.P. Head Prize from University College. zyxw zyxwvutsrqp zyxwvuts [l] K. S. Narendra. Y. H. Lin, and L. S. Valavani, “Stable adaptive controller desi Part 11: Proof of stability.” IEEE Trans. Automar. . Conrr.. vol. A F h , pp. 440-448, June 1980. [2]A. Feuer and S. Morse, “Adaptive control of single-input singleoutput linear systems,” IEEE Trans. Automar. Contr., vol. AC-23, pp. 557-570, Aug. 1978. [3] G. Kreisselmeier, “Adaptive control via adaptive observation and asymptotic feedback matrix synthesis.” IEEE Trans. Automar. Conrrl., vol. AC-25, Aug. 1980. . [4] G. C . Goodwin, P. S. Ramadge, and P. E. Caines. “Discrete time multivariable adaptive control.” IEEE Trans. Automar. Conrr., vol. AC-25. June 1980. ~[SI I. D. Landau and H. M. Silveira, “A stability theorem with applications to adaptive control,” IEEE Trans. Auromat. Contr.. vol. AC-24. pp. 305-312. Apr. 1979. [6] P.A. Ioannou and P. V. Kokotovic. “An asymptotic error analysis of identifiers and adaptive observers in the presence of parasitiis,” IEEE Trans. Auromat. Contr., vol. AC-27, Aug. 1982. [7] B. D. 0. Anderson and C. R.Johnson, Jr.. “On reduced order adaptive output error identification and adaptive IIR filtering,” IEEE Trans. Automat. Conrr., vol. A5;27. Aug. 1982. [SI P. A. Ioannou and C. R. Johnson, Jr., Reduced-order performance of parallel and series-parallel identifiers of two-time-scale systems.” in Proc. Workshop on Applications of .Adaptice SJst. Theoy, Yale Univ., New Haven, CT, May 1981. [9] P. V. Kokotovk and P.A. Ioannou. “Robustness redesign of continuous-time adaptive schemes,” in Proc. 2Orh IEEE Conf. on Decision and Conrr.. San Diego, CA, 1981, pp. 522-527. [lo] M. S. Balas and C. R. Johnson. Jr..“Adaptive identification and control of large scale or distributed parameter systems using reduced order models.” in Proc. Workshop on Applicutioru of Aduprice S U I . Theor3., Yale University. New Haven, CT, May 1981. [ll]G. Kreisselmeier. “On adaptive state regulation:” IEEE Trans. ,4utomut. Contr.. vol. AC-27, Feb. 1982. [12] C. E. Rohrs, L. Valavani, M. Athans. and G. Stein, “Analytical verification of undesirable properties of direct model reference adaptive control algorithms,” in Proc. 20th IEEE Conf. on Decision und Contr.. San Diego. CA. Dec. 1981. [13] C. R.Johnson. Jr.. and M.J.Balas. “Reduced-order adaptive controller studies.” in Proc. Joinr Automar. Conrr. Conf.. June 1980. [14] I. D. Landau, “A survey of model reference adaptive techniquesTheory and applications,” Automatica, vol. 10. no. 4, pp. 353-379. [15] B. B. Peterson and K. S. Narendra,“Boundederror adaptive control,” part I, Dep. Eng. AppI. Sci.. Yale Univ., New Haven, CT. Rep. 8005, Dec. 1980. ~1~~~~~ 211 zyxwvuts zyxwvutsrq zyxwvutsrqponm Petar V. Kokotovic (SM’74-F’80) received graduate degrees in 1962 under Prof. Mitrovic in Belgrade, Yugoslavia, and in 1965 under Prof. Feldbaum and Prof. Tsypkin in Moscow, USSR. He has been active in control theory and its applications for more than 20 years. From 1959 to 1966 he was writh the Pupin Research Institute. Belgrade, Yugoslavia. Since1966he has been with the Department of Electrical Engineering and theCoordinated Science Laboratoq? University of Illinois, Urbana, where he teaches control theory and conducts research in modeling. sensitivity analysis, optimal and adaptive control, singular perturbations. and large scale systems. He has held visiting appointments with ETH. Zurich, Switzerland. INRIA, France, and has lectured at other institutions in the U.S. and Europe. His consulting activities include General Electric Company and Ford Motor Company. Prof. Kokotovic has served on the IDCand AdCom of the IEEE Control Systems Society and on committees of the International Federation of Automatic Control. He is an Associate Editor of $vrretm and Control Lerters.