Brief paperA new class of Lyapunov functions for the constrained stabilization of linear systems☆
Introduction
The state-feedback stabilization of constrained uncertain linear systems, covering saturations of the control inputs, state constraints and model uncertainties, is equivalent to the design of a robust Control Lyapunov Function (CLF). Since the particular choice of the candidate CLF also provides an estimation of the controlled invariant set (Blanchini, 1999), the exact solution consists in providing the largest controlled invariant region of the state space, according to both state and control constraints (Balestrino, Caiti, & Grammatico, 2011). In general, nontrivial classes of candidate CLFs are required to shape the maximal controlled invariant set. For instance, Polyhedral CLFs (PCLFs) are a universal class of functions for the stabilizability of uncertain linear systems (Blanchini, 1995), or equivalently Linear Differential Inclusions (LDIs). PCLFs can be smoothed with standard norms (Blanchini & Miani, 1999) in order to obtain an everywhere differentiable smoothed PCLF that can be used together with nonlinear gradient-based continuous controllers (Petersen & Barmish, 1987). Recently, the class of Truncated Ellipsoids (TEs) (O’Dell and Misawa, 2002, Thibodeau et al., 2009) has been proposed as candidate LFs and CLFs for constrained uncertain linear systems to provide a good approximation of the maximal controlled invariant region with a reduced number of parameters (O’Dell & Misawa, 2002). In Thibodeau et al. (2009) a linear state-feedback control is designed by solving a Bilinear Matrix Inequality (BMI), maximizing the volume of the estimated controlled invariant set.
The main contribution of this paper is the definition of a novel composition rule for merging two different CLFs, allowing the design of a non-homothetic smooth CLF with the following properties: (a) the external level set exactly shapes the maximal controlled invariant set; and (b) the inner sublevel sets can be made arbitrarily close to any given choice of smooth ones. This properties allow us to define a stabilizing nonlinear gradient-based control law that is continuous everywhere inside the maximal controlled invariant set. The results of Balestrino et al., 2010, Balestrino, Caiti et al., 2011, Balestrino, Crisostomi et al., 2011, where a basic composition rule is introduced, are extended to the class of constrained uncertain linear systems by deriving the more general class of so-called Control Lyapunov R-Functions (CLRFs). Moreover, a Linear Matrix Inequality (LMI) feasibility test for the candidate CLRF is here proposed. As in Chesi and Hung, 2008, Hu and Blanchini, 2010, the synthesis condition is obtained via BMIs. CLRFs can smooth both PCLFs and TEs in a non-homothetic way and they can be made everywhere differentiable.
The novel smoothing technique follows from the framework of R-functions, referred to in the next section. In Sections 3 Stability analysis of nonlinear systems via Lyapunov R-functions, 4 Constrained stabilization of uncertain linear systems via control Lyapunov R-functions the main results are provided. Section 5 heuristically addresses the constrained LQ control problem as an application. All the proofs are in the Appendix.
denotes the identity matrix. denotes . denotes . denotes the convex hull. The closed -level set of a continuous function , i.e. , is denoted by . A convex and compact set s.t. is called 0-symmetric (Blanchini, 1995). A set is a controlled invariant set if there exists an admissible control to keep (Hu & Blanchini, 2010). If is the largest controlled invariant set associated to the CLF , then the level set is called the external level set, while the level sets such that are addressed as internal level sets.
Section snippets
A novel composition rule for R-functions
The framework of R-functions has been first proposed in the setting of state-feedback stabilization in Balestrino et al., 2010, Balestrino, Crisostomi et al., 2011. Here a novel composition rule is defined.
Definition 1 A function is an R-function if there exists a Boolean function , where , such that where is the standard Heaviside step function.
Stability analysis of nonlinear systems via Lyapunov R-functions
In this section, we consider the stability analysis of nonlinear dynamical systems , .
Given two differentiable LFs , , respectively in , , the candidate LF in is derived as follows: Now if , then, according to Lemma 1: , i.e. and hence . We consider the interior of to avoid the lack of
Problem statement and discussion
Consider the constrained stabilization of an uncertain linear system where , , via a continuous state-feedback control such that asymptotically converges to the origin, in accordance with the state and control input constraints. The constraints are assumed to be convex and 0-symmetric. In particular .
A polyhedral approximation (with arbitrary precision) of the maximal controllable set for system (7) can
Application to constrained linear quadratic optimal control
Designing the shape of the candidate CLRF, via the novel composition rule, suggests the application to the constrained LQ optimal control problem. In fact, while the external set can be designed in accordance to the shape of the maximal controllable set, the inner sublevel sets can be (independently) made arbitrarily close to the locally-optimal quadratic ones.
Consider the constrained (nominal) linear system , , with standard quadratic performance index
Conclusion and future work
The constrained stabilization of linear uncertain systems is addressed via the class of control Lyapunov R-functions, that are differentiable and non-homothetic. The novel composition rule looks useful for constrained linear quadratic control problems because the proposed Lyapunov function can be designed with both a large domain of attraction and inner sublevel sets close to the optimal ones. Investigations for theoretical bounds of sub-optimality and for the best tuning of the free trade-off
Acknowledgments
The authors thank Prof. Franco Blanchini for useful discussions and the anonymous reviewers for improving the quality of the paper.
Aldo Balestrino was born in Mercato San Severino, Italy, in 1941. He received the “Laurea” degree in electronic engineering from the University of Naples, Naples, Italy, in 1968. From 1980 to 1983, he was a Full Professor of Systems Theory at the University of Naples and since 1984, he has been a Professor of Automatic Control at the University of Pisa, Pisa, Italy, where he has been Head of the Department of Electrical Systems and Automation and the Coordinator of the “Doctorate” program in
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Aldo Balestrino was born in Mercato San Severino, Italy, in 1941. He received the “Laurea” degree in electronic engineering from the University of Naples, Naples, Italy, in 1968. From 1980 to 1983, he was a Full Professor of Systems Theory at the University of Naples and since 1984, he has been a Professor of Automatic Control at the University of Pisa, Pisa, Italy, where he has been Head of the Department of Electrical Systems and Automation and the Coordinator of the “Doctorate” program in Automation and Industrial Robotics since 1989 and 2001, respectively. His research interests include systems theory, control engineering, neural networks, and electrical drives. Since 1994, he has been the National Coordinator of several research projects supported by the “Ministero dell’Istruzione, dell’Università e della Ricerca”, the Italian Ministry of Research.
Andrea Caiti was born in Naples, Italy, in 1963. He received the “Laurea” degree (“cum laude”) in electronic engineering from the University of Genova, Genoa, Italy, in 1988. After working as a Staff Scientist at the North Atlantic Treaty Organization Supreme Allied Commander Atlantic Undersea Research Center, La Spezia, Italy, he was with the Department of Electrical Systems and Automation, University of Pisa, in 1996, as an Assistant Professor. In 1998, he was an Associate Professor at the “Dipartimento di Ingegneria dell’Informazione”, University of Siena, Siena, Italy. Since 2001, he has been with the University of Pisa, where he currently is a Full Professor of Systems Theory and Automatic Control. He is the Director of the Interuniversity Center of Integrated Systems for the Marine Environment. He is a member of the IEEE Control Systems and IEEE Oceanic Engineering Societies.
Sergio Grammatico was born in Marsala, Italy, in 1987. He received the B.Sc. and M.Sc. degrees (both “cum laude”) in computer engineering and automation engineering, respectively, from the University of Pisa, Pisa, Italy, in 2008 and 2009, respectively. He also received a M.Sc. degree (“cum laude”) in engineering from the Sant’Anna School of Advanced Studies, Pisa, Italy, in 2011. He is currently working towards his Ph.D. in automation, robotics and bioengineering at the University of Pisa. His research interests include systems theory, robust Lyapunov-based and model-predictive control.
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The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Zongli Lin under the direction of Editor Andrew R. Teel.
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