Brief paperConvex liftings-based robust control design☆
Introduction
Originated in the seminal work (Lyapunov, 1907), Lyapunov stability stands as a fundamental concept in control theory (Loría & Panteley, 2006). In stability analysis, a Lyapunov function is usually of use to prove closed-loop stability, see Kalman and Bertram (1960), Brayton and Tong (1979) and Molchanov and Pyatnitskiy (1989). On the other hand, in control design, control Lyapunov functions are usually employed to design stabilizing/robust controllers, see among others (Khalil, 2002, Zubov and Boron, 1964). Accordingly, whenever such control Lyapunov functions are used in optimization based strategies, these should be chosen such that the recursive feasibility and closed-loop stability are all fulfilled. Different classes of control Lyapunov functions have been proposed in control theory (Michel et al., 1984, Polanski, 1995). In the context of linear quadratic control, infinite/finite quadratic cost functions usually serve as control Lyapunov functions, as shown in Anderson and Moore (2007), Chmielewski and Manousiouthakis (1996) and Sznaier and Damborg (1987). In particular, in linear model predictive control (MPC), such a control Lyapunov function has been used to design robust controllers to cope with polytopic uncertainties, leading to a linear matrix inequality problem, see Kothare, Balakrishnan, and Morari (1996). Polyhedral control Lyapunov functions have also been exploited in several studies, e.g., Bitsoris (1988b), Bitsoris and Vassilaki (1995), Blanchini, 1994, Blanchini, 1995, Gutman and Cwikel (1987), Lazar (2010) and Vassilaki, Hennet, and Bitsoris (1988), since they lead to simple design procedures, i.e., composed of linear constraints. Convex piecewise affine control Lyapunov function for piecewise affine systems has also been considered in Baotic, Christophersen, and Morari (2006) and solved using dynamic programming, which may be impractical if disturbances and uncertainties are considered.
It is worth emphasizing that the robust control design proposed in Kothare et al. (1996) requires at each sampling time solving a linear matrix inequality (LMI) problem, the online evaluation thus becomes computationally demanding. Some improvements of this method are presented in Cuzzola, Geromel, and Morari (2002) and Wan and Kothare (2003). An effort to simplify this complexity has been proposed in Kouvaritakis, Rossiter, and Schuurmans (2000). However, this method can only guarantee the positive invariance of the initially ellipsoidal feasible set instead of asymptotic stability of the origin. Also, although the number of LMIs is decreased, however, solving online an LMI problem is still expensive in comparison to strict real-time requirements. Some extensions of the latter method have been proposed to reduce complexity, e.g., Khan and Rossiter (2012). Note however that making use of degree of freedom is nothing other than solving a finite horizon MPC problem. Also, in the context of MPC, the optimal cost function usually serves as a Lyapunov function, therefore minimizing a nominal cost function as in this reference is meaningless, and robust stability is thus guaranteed by the constraint set. Further, the pre-imposition on the structure of controllers leads to conservativeness and possible loss of recursive feasibility. An alternative robust MPC scheme has been presented in Mayne, Seron, and Raković (2005) to take bounded additive disturbances into account. However, polytopic uncertainties considerably increase its computational complexity with respect to the prediction horizon. As an extension of this method, parameterized tube MPC has recently been proposed in Rakovic, Kouvaritakis, Cannon, Panos, and Findeisen (2012) to cope with bounded additive disturbances. Although implicit controller is computed based on its decomposed elements, the number of decision variables is of order , with to be the number of vertices of the given disturbance set and to be the prediction horizon. As a consequence, accounting for polytopic uncertainty makes the online computation much more demanding, as the number of decision is of order , with to be the number of vertices of the given polytopic uncertainty set. Further, dealing with tube cost function in this case becomes more complicated.
This paper proposes a method which only requires resolution of a linear programming problem at each sampling instant. Moreover, unlike the method in Blanchini (1994), which guarantees robust stability in the sense of Lyapunov (input-to-state stability), this paper proves a more flexible result by guaranteeing that the state converges to a given robust positively invariant set (minimal/maximal robust positively invariant set) as time tends to infinity. Note that such a constructed convex lifting is not a control Lyapunov function, which represents a relaxation and a supplementary degree of freedom with respect to the method in Blanchini (1994). Finally, to our best knowledge, convex liftings have never been used in control design and can be a valuable tool, offering additional flexibility for the existing constrained control methods.
Section snippets
Notation and definitions
Throughout this paper, denote the set of nonnegative integers, the set of positive integers, the set of real numbers and the set of nonnegative numbers, respectively. For ease of presentation, with a given , by , we denote the index set: . Also, we use to denote the set defined as: .
A polyhedron is the intersection of finitely many closed halfspaces. A polytope is a bounded polyhedron. If is an arbitrary polytope, then by , we denote the set of
Problem settings
In this paper, we consider a discrete-time linear system: where denote the state, control variable and additive disturbance at time . The state-space matrices are time-varying and assumed to belong to an uncertainty matrix polytope denoted by and defined below: The state, control variables and disturbances are subject to constraints: where , and are polytopes containing
Robust positively invariant sets
Positively invariant sets have been studied over several decades. Due to their relevance in control theory, they turn out to be useful in many control related studies, e.g., Bitsoris, 1988a, Bitsoris, 1988b, Bitsoris and Vassilaki (1995), Blanchini and Miani (2007) and Kerrigan (2001). The definition of a robust positively invariant set for system (1) is recalled below. Definition 4.1 Given an admissible control law , a set is called robust positively invariant with respect to (1) if
Numerical example
To illustrate the proposed procedure, consider Example 1 in Kothare et al. (1996) where an angular antenna positioning system is modeled by the following equation: where and the uncertain parameter ranges in interval . The state and control variables are subject to the following constraints: . Unconstrained controller is chosen as follows: . Accordingly, the maximal robust positively invariant set associated
Conclusions
This paper presented a new method to design robust control law for constrained linear systems affected by bounded additive disturbances and polytopic uncertainties. This method was based on convex liftings. It was shown to guarantee the recursive feasibility and robust stability as well. The benefit of the proposed method was also shown via a numerical example relative to several MPC methods.
Acknowledgments
The authors would like to thank the Associate Editor and the anonymous reviewers for constructive comments to improve this manuscript. This work has partially been supported by the Linz Center of Mechatronics (LCM) in the framework of the Austrian COMET-K2 program. The research by the second author has benefited from the financial support of the European Unions 7th Framework Programme under EC-GA No. 607957 TEMPO—Training in Embedded Model Predictive Control and Optimization. Michal Kvasnica
Ngoc Anh Nguyen received a double Dip.Ing. degree in Electrical Engineering at Hanoi University of Science and Technology, Vietnam and Grenoble Institute of Technology, France, both in 2012. He is currently doing a postdoc at the Johannes Kepler University Linz, Austria. Prior to that, he obtained his Ph.D. degree at Laboratory of Signals and Systems, CentraleSupelec, University Paris Saclay, France in December 2015. His main research interests lie in computational geometry, optimization based
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Ngoc Anh Nguyen received a double Dip.Ing. degree in Electrical Engineering at Hanoi University of Science and Technology, Vietnam and Grenoble Institute of Technology, France, both in 2012. He is currently doing a postdoc at the Johannes Kepler University Linz, Austria. Prior to that, he obtained his Ph.D. degree at Laboratory of Signals and Systems, CentraleSupelec, University Paris Saclay, France in December 2015. His main research interests lie in computational geometry, optimization based control and set theoretic methods.
Sorin Olaru is a Professor in CentraleSupélec, member of the CNRS Laboratory of Signals and Systems and of the INRIA team DISCO, all these institutions being part of the Paris-Saclay University in France. His research interests are encompassing the optimization-based control design, set-theoretic characterization of constrained dynamical systems as well as the numerical methods in control. He is currently involved in research projects related to embedded predictive control, fault tolerant control and time-delay systems.
Pedro Rodríguez-Ayerbe received the technical engineering Diploma in electronics from Mondragon University, Arrasate, Spain, in 1993, and the Engineering degree in electrical engineering from SUPELEC, Gif sur Yvette, France, in 1996. In 2003, he received the Ph.D. degree in automatic control from SUPELEC and the Université Paris Sud, Orsay, France. He is currently an Associate Professor in CentraleSupélec, member of the CNRS Laboratory of Signals and Systems being part of the Paris-Saclay University in France. His research interests include constrained predictive control and robust control theory.
Michal Kvasnica was born in 1977. He received his diploma in chemical engineering from the Slovak University of Technology in Bratislava, Slovakia and the Ph.D. in electrical engineering from the Swiss Federal Institute of Technology in Zurich, Switzerland. Since 2011 he is an Associate Professor at the Slovak University of Technology in Bratislava. His research interests are in model predictive control, modeling of hybrid systems, and development of software tools for control. He is the co-author and developer of the MPT Toolbox for explicit model predictive control.
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The material in this paper was partially presented at the 8th IFAC Symposium on Robust Control Design, July 8–11, 2015, Bratislava, Slovakia. This paper was recommended for publication in revised form by Associate Editor Akira Kojima under the direction of Editor Ian R. Petersen.
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