Computing polytopic controlled invariant sets which are maximal inside a prescribed region often ... more Computing polytopic controlled invariant sets which are maximal inside a prescribed region often yields sets which have a really complex representation. Since the associated control has a complexity which drastically increases with that of the region, this turns out to be a major problem in the implementation of the theory of controlled invariant regions. In this paper, we consider the problem of reducing the complexity of these regions and/or the complexity of the associated compensator. We propose two heuristic techniques based on spectral properties of some relevant matrices and on vertex-elimination methods. The paper presents several preliminary results which are interesting on their own such as dynamic augmentation and the properties of complex (as opposed to real) polytopic invariant regions.
We face the problem of determining a tracking domain of attraction, say the set of initial states... more We face the problem of determining a tracking domain of attraction, say the set of initial states starting from which it is possible to track reference signals in given class, for discrete-time systems with control and state constraints. We show that the tracking domain of attraction is exactly equal to the domain of attraction, say the set of states which can be brought to the origin by a proper feedback law. For constant reference signals we establish a connection between the convergence speed of the stabilization problem and tracking convergence which turns out to be independent of the reference signal. We also show that the tracking controller can be inferred from the stabilizing (possibly nonlinear) controller associated with the domain of attraction. We refer the reader to the full version [17], where the continuous-time case, proofs and extensions are presented.
The bidiagonal-Frobenius form for single-input systems is presented. The new form is a compromise... more The bidiagonal-Frobenius form for single-input systems is presented. The new form is a compromise between the bidiagonal canonical form introduced by Policastro (1980) and the classical Frobenius one. It can be employed in the state-feedback pole-assignment problem, with the advantage that it requires neither the computation of the system eigenvalues as the bidiagonal form does, nor the characteristic polynomial coefficient evaluation as necessary using the Frobenius form. The final result is obtained using only the desired eigenvalues, and the coefficients of the feedback vector are given directly from the bidiagonal-Frobenius form without any computation.
Journal of Optimization Theory and Applications, Sep 1, 1993
The problem is considered of finding a control strategy for a linear discrete-time periodic syste... more The problem is considered of finding a control strategy for a linear discrete-time periodic system with state and control bounds in the presence of unknown disturbances that are only known to belong to a given compact set. This kind of problem arises in practice in resource distribution systems where the demand has typically a periodic behavior, but cannot be estimated a priori without an uncertainty margin. An infinite-horizon keeping problem is formulated, which consists in confining the state within its constraint set using the allowable control, whatever the allowed disturbances may be. To face this problem, the concepts of periodically invariant set and sequence are introduced. They are used to formulate a solution strategy that solves the keeping problem. For the case of polyhedral state, control, and disturbance constraints, a computationally feasible procedure is proposed. In particular, it is shown that periodically invariant sequences may be computed off-line, and then they may be used to synthesize on-line a control strategy. Finally, an optimization criterion for the control law is discussed.
IEEE Transactions on Automatic Control, Jun 1, 2000
We consider production-distribution systems with buffer and capacity constraints in the presence ... more We consider production-distribution systems with buffer and capacity constraints in the presence of uncertain inputs. We face the problem of finding a control which drives the buffer levels to a prescribed value. We consider a continuous-time model and show that some important properties which fail in general in the discrete-time case. In particular we show that the case in which the controller knows the external input is “equivalent” to the case in which such information is not available. Furthermore, a decentralized strategy can be determined. We also show that in the case of network failures the controller does not need to know the current network configuration in order to guarantee the tracking goal as long as a certain necessary and sufficient condition is satisfied
7.1 Motivations and Preliminaries 7-1 Motivating Example: DC Electric Motor with Uncertain Parame... more 7.1 Motivations and Preliminaries 7-1 Motivating Example: DC Electric Motor with Uncertain Parameters 7.2 Description of the Uncertainty Structures ........7-3 7.3 Uncertainty Structure Preservation with Feedback 7-4 7.4 Overbounding with Affine Uncertainty: The Issue of Conservatism 7-5 7.5 Robustness Analysis for Affine Plants 7-7 Value Set Construction for Affine Plants • The DC-Electric Motor Example Revisited 7.6 Robustness Analysis for Affine Polynomials ..7-10 Value Set Construction for Affine Polynomials • Example of Value Set Generation • Interval Polynomials: Kharitonov’s Theorem and Value Set Geometry • From Robust Stability to Robust Performance • Algebraic Criteria for Robust Stability • Further Extensions: The Spectral Set 7.7 Multiaffine Uncertainty Structures 7-14 7.8 General Uncertainty Structures and Controller Synthesis 7-16 References 7-17 Roberto Tempo IEIIT−CNR Politecnico di Torino
Constraints are encountered practically in every real control problem. It is a tradition (althoug... more Constraints are encountered practically in every real control problem. It is a tradition (although questionable) that in many textbooks constraints are mentioned but, with several exceptions, the design of control systems which take into account constraints is frequently disregarded. In this chapter is is shown how set-theoretic methods are appropriate to control design when constraints are present.
ABSTRACT We consider the problem of stabilizing a dynamic system by means of bounded controls. We... more ABSTRACT We consider the problem of stabilizing a dynamic system by means of bounded controls. We show that the largest domain of attraction can be arbitrarily closely approximated by a “smooth” domain of attraction for which we provide an analytic expression. Such an expression allows for the determination of a (non-linear) control law in explicit form.
In this paper we consider the problem of controlling a multi-inventory system in the presence of ... more In this paper we consider the problem of controlling a multi-inventory system in the presence of uncertain demand. The demand is unknown but bounded in an assigned compact set. The control input is assumed to be also constrained in a compact set. We consider an integral cost function of the bu er levels and we face the problem of minimizing the worst-case cost. We show that the optimal cost of a suitable auxiliary problem with no uncertainties is always an upper bound for the original problem. In the special case of minimum-time control, this upper bound is tight, namely its optimal cost is equal to the worst-case cost for the original system. Furthermore, the result is constructive, since the optimal control law can be explicitly computed.
Lyapunov functions are crucial in the present book aims, given the strict relation between Lyapun... more Lyapunov functions are crucial in the present book aims, given the strict relation between Lyapunov functions and invariant sets. In this chapter, basic notions of Lyapunov and Lyapunov-like functions will be presented. Before introducing the main concept, a brief presentation of the class of dynamic models will be considered and some preliminary mathematical notions are given.
Journal of Dynamic Systems Measurement and Control-transactions of The Asme, Feb 1, 2021
Motivated by two engineering applications, we address nonlinear bounded steady-state optimal cont... more Motivated by two engineering applications, we address nonlinear bounded steady-state optimal control of linear dynamical systems undergoing steady-state bandlimited periodic oscillations. The optimisation can be cast as a minimisation problem by expressing the state and the input as finite Fourier series expansions, and using the expansions coefficients as parameters to be optimised. With this parametrisation we address linear quadratic (LQ) problems involving periodic bandlimited dynamics by using quadratic minimisation with parametric time-dependent constraints. We hence investigate the implications of a discretisation of linear continuous time constraints and propose an algorithm that provides a feasible sub-optimal solution whose cost is arbitrarily close to the optimal cost for the original constrained steady-state problem. Finally, we discuss practical case studies that can be effectively tackled with the proposed framework, including optimal energy harvesting from pulsating mechanical energy sources, and optimal control of DC/AC power converters.
In this paper we show that (under some input matrix rank conditions) there exists a single compen... more In this paper we show that (under some input matrix rank conditions) there exists a single compensator which achieves simultaneously the performances of r ≤ n (the system order) given static state feedback (local) compensators. The compensator, whose order is r(n − 1), is then capable of matching the r (possibly different) optimality criteria defined for each input-output pair. An explicit and easy construction procedure (we refer to such procedure as "compensator blending") is provided. We also consider the dual version of the problem, precisely, we show how to achieve simultaneous optimality by blending a set of given filters.
In this chapter, we survey a few decades of robustness investigation for uncertain systems. We ai... more In this chapter, we survey a few decades of robustness investigation for uncertain systems. We aim at embracing most of the robustness literature, starting from the Lyapunov approach of the ’70s, which involved both quadratic and non-quadratic Lyapunov functions, until recent developments on polynomial techniques for robustness. We consider both time-varying and time-invariant uncertainties, in an inclusive way: important techniques are presented, such as the Lur’e systems framework, qualitative feedback theory, parametric robustness analysis, linear matrix inequalities, parameter-dependent Lyapunov functions, H-infinity, small-gain theorems, non-quadratic Lyapunov functions and Lyapunov–Metzler inequalities. The chapter proposes a critical view on all these techniques, highlighting both advantages and limitations. Illustrative examples and applications are proposed. Technicalities are kept to the least possible level to render the chapter accessible to a broad, possibly interdisciplinary, audience. The chapter is written with a historic view. Nonetheless, future perspectives are emphasized, and several open problems and future research directions are pointed out. The chapter is inspired by the spirit, attitude and fairness of our great friend Roberto Tempo and is written following his invaluable teaching.
Computing polytopic controlled invariant sets which are maximal inside a prescribed region often ... more Computing polytopic controlled invariant sets which are maximal inside a prescribed region often yields sets which have a really complex representation. Since the associated control has a complexity which drastically increases with that of the region, this turns out to be a major problem in the implementation of the theory of controlled invariant regions. In this paper, we consider the problem of reducing the complexity of these regions and/or the complexity of the associated compensator. We propose two heuristic techniques based on spectral properties of some relevant matrices and on vertex-elimination methods. The paper presents several preliminary results which are interesting on their own such as dynamic augmentation and the properties of complex (as opposed to real) polytopic invariant regions.
We face the problem of determining a tracking domain of attraction, say the set of initial states... more We face the problem of determining a tracking domain of attraction, say the set of initial states starting from which it is possible to track reference signals in given class, for discrete-time systems with control and state constraints. We show that the tracking domain of attraction is exactly equal to the domain of attraction, say the set of states which can be brought to the origin by a proper feedback law. For constant reference signals we establish a connection between the convergence speed of the stabilization problem and tracking convergence which turns out to be independent of the reference signal. We also show that the tracking controller can be inferred from the stabilizing (possibly nonlinear) controller associated with the domain of attraction. We refer the reader to the full version [17], where the continuous-time case, proofs and extensions are presented.
The bidiagonal-Frobenius form for single-input systems is presented. The new form is a compromise... more The bidiagonal-Frobenius form for single-input systems is presented. The new form is a compromise between the bidiagonal canonical form introduced by Policastro (1980) and the classical Frobenius one. It can be employed in the state-feedback pole-assignment problem, with the advantage that it requires neither the computation of the system eigenvalues as the bidiagonal form does, nor the characteristic polynomial coefficient evaluation as necessary using the Frobenius form. The final result is obtained using only the desired eigenvalues, and the coefficients of the feedback vector are given directly from the bidiagonal-Frobenius form without any computation.
Journal of Optimization Theory and Applications, Sep 1, 1993
The problem is considered of finding a control strategy for a linear discrete-time periodic syste... more The problem is considered of finding a control strategy for a linear discrete-time periodic system with state and control bounds in the presence of unknown disturbances that are only known to belong to a given compact set. This kind of problem arises in practice in resource distribution systems where the demand has typically a periodic behavior, but cannot be estimated a priori without an uncertainty margin. An infinite-horizon keeping problem is formulated, which consists in confining the state within its constraint set using the allowable control, whatever the allowed disturbances may be. To face this problem, the concepts of periodically invariant set and sequence are introduced. They are used to formulate a solution strategy that solves the keeping problem. For the case of polyhedral state, control, and disturbance constraints, a computationally feasible procedure is proposed. In particular, it is shown that periodically invariant sequences may be computed off-line, and then they may be used to synthesize on-line a control strategy. Finally, an optimization criterion for the control law is discussed.
IEEE Transactions on Automatic Control, Jun 1, 2000
We consider production-distribution systems with buffer and capacity constraints in the presence ... more We consider production-distribution systems with buffer and capacity constraints in the presence of uncertain inputs. We face the problem of finding a control which drives the buffer levels to a prescribed value. We consider a continuous-time model and show that some important properties which fail in general in the discrete-time case. In particular we show that the case in which the controller knows the external input is “equivalent” to the case in which such information is not available. Furthermore, a decentralized strategy can be determined. We also show that in the case of network failures the controller does not need to know the current network configuration in order to guarantee the tracking goal as long as a certain necessary and sufficient condition is satisfied
7.1 Motivations and Preliminaries 7-1 Motivating Example: DC Electric Motor with Uncertain Parame... more 7.1 Motivations and Preliminaries 7-1 Motivating Example: DC Electric Motor with Uncertain Parameters 7.2 Description of the Uncertainty Structures ........7-3 7.3 Uncertainty Structure Preservation with Feedback 7-4 7.4 Overbounding with Affine Uncertainty: The Issue of Conservatism 7-5 7.5 Robustness Analysis for Affine Plants 7-7 Value Set Construction for Affine Plants • The DC-Electric Motor Example Revisited 7.6 Robustness Analysis for Affine Polynomials ..7-10 Value Set Construction for Affine Polynomials • Example of Value Set Generation • Interval Polynomials: Kharitonov’s Theorem and Value Set Geometry • From Robust Stability to Robust Performance • Algebraic Criteria for Robust Stability • Further Extensions: The Spectral Set 7.7 Multiaffine Uncertainty Structures 7-14 7.8 General Uncertainty Structures and Controller Synthesis 7-16 References 7-17 Roberto Tempo IEIIT−CNR Politecnico di Torino
Constraints are encountered practically in every real control problem. It is a tradition (althoug... more Constraints are encountered practically in every real control problem. It is a tradition (although questionable) that in many textbooks constraints are mentioned but, with several exceptions, the design of control systems which take into account constraints is frequently disregarded. In this chapter is is shown how set-theoretic methods are appropriate to control design when constraints are present.
ABSTRACT We consider the problem of stabilizing a dynamic system by means of bounded controls. We... more ABSTRACT We consider the problem of stabilizing a dynamic system by means of bounded controls. We show that the largest domain of attraction can be arbitrarily closely approximated by a “smooth” domain of attraction for which we provide an analytic expression. Such an expression allows for the determination of a (non-linear) control law in explicit form.
In this paper we consider the problem of controlling a multi-inventory system in the presence of ... more In this paper we consider the problem of controlling a multi-inventory system in the presence of uncertain demand. The demand is unknown but bounded in an assigned compact set. The control input is assumed to be also constrained in a compact set. We consider an integral cost function of the bu er levels and we face the problem of minimizing the worst-case cost. We show that the optimal cost of a suitable auxiliary problem with no uncertainties is always an upper bound for the original problem. In the special case of minimum-time control, this upper bound is tight, namely its optimal cost is equal to the worst-case cost for the original system. Furthermore, the result is constructive, since the optimal control law can be explicitly computed.
Lyapunov functions are crucial in the present book aims, given the strict relation between Lyapun... more Lyapunov functions are crucial in the present book aims, given the strict relation between Lyapunov functions and invariant sets. In this chapter, basic notions of Lyapunov and Lyapunov-like functions will be presented. Before introducing the main concept, a brief presentation of the class of dynamic models will be considered and some preliminary mathematical notions are given.
Journal of Dynamic Systems Measurement and Control-transactions of The Asme, Feb 1, 2021
Motivated by two engineering applications, we address nonlinear bounded steady-state optimal cont... more Motivated by two engineering applications, we address nonlinear bounded steady-state optimal control of linear dynamical systems undergoing steady-state bandlimited periodic oscillations. The optimisation can be cast as a minimisation problem by expressing the state and the input as finite Fourier series expansions, and using the expansions coefficients as parameters to be optimised. With this parametrisation we address linear quadratic (LQ) problems involving periodic bandlimited dynamics by using quadratic minimisation with parametric time-dependent constraints. We hence investigate the implications of a discretisation of linear continuous time constraints and propose an algorithm that provides a feasible sub-optimal solution whose cost is arbitrarily close to the optimal cost for the original constrained steady-state problem. Finally, we discuss practical case studies that can be effectively tackled with the proposed framework, including optimal energy harvesting from pulsating mechanical energy sources, and optimal control of DC/AC power converters.
In this paper we show that (under some input matrix rank conditions) there exists a single compen... more In this paper we show that (under some input matrix rank conditions) there exists a single compensator which achieves simultaneously the performances of r ≤ n (the system order) given static state feedback (local) compensators. The compensator, whose order is r(n − 1), is then capable of matching the r (possibly different) optimality criteria defined for each input-output pair. An explicit and easy construction procedure (we refer to such procedure as "compensator blending") is provided. We also consider the dual version of the problem, precisely, we show how to achieve simultaneous optimality by blending a set of given filters.
In this chapter, we survey a few decades of robustness investigation for uncertain systems. We ai... more In this chapter, we survey a few decades of robustness investigation for uncertain systems. We aim at embracing most of the robustness literature, starting from the Lyapunov approach of the ’70s, which involved both quadratic and non-quadratic Lyapunov functions, until recent developments on polynomial techniques for robustness. We consider both time-varying and time-invariant uncertainties, in an inclusive way: important techniques are presented, such as the Lur’e systems framework, qualitative feedback theory, parametric robustness analysis, linear matrix inequalities, parameter-dependent Lyapunov functions, H-infinity, small-gain theorems, non-quadratic Lyapunov functions and Lyapunov–Metzler inequalities. The chapter proposes a critical view on all these techniques, highlighting both advantages and limitations. Illustrative examples and applications are proposed. Technicalities are kept to the least possible level to render the chapter accessible to a broad, possibly interdisciplinary, audience. The chapter is written with a historic view. Nonetheless, future perspectives are emphasized, and several open problems and future research directions are pointed out. The chapter is inspired by the spirit, attitude and fairness of our great friend Roberto Tempo and is written following his invaluable teaching.
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