LMI-based robust control of uncertain discrete-time piecewise affine systems

The main contribution of this paper is to present stability synthesis results for discrete-time piecewise affine (PWA) systems with polytopic time-varying uncertainties and for discrete-time PWA systems with norm-bounded uncertainties respectively. The basic idea of the proposed approaches is to construct piecewise-quadratic (PWQ) Lyapunov functions to guarantee the stability of the closed-loop systems. The partition information of the PWA systems is taken into account and each polytopic operating region is outer approximated by an ellipsoid, then sufficient conditions for the robust stabilization are derived and expressed as a set of linear matrix inequalities (LMIs). Two examples are given to illustrate the proposed theoretical results.     

Robust observer-based control for uncertain discrete-time piecewise affine systems

The main contribution of this paper is to present a novel robust observer-based controller design method for discrete-time piecewise affine systems with norm-bounded uncertainties.The key ideas are to ...     

Improved Adaptive Controller Synthesis of Piecewise‐Affine Systems

This paper considers the adaptive control problem for piecewise affine systems (PWS), a novel synthesis framework is presented based on the piecewise quadratic Lyapunov function (PQLF) instead of the common quadratic Lyapunov function to achieve the less conservatism. First, by designing the projection‐type piecewise adaptive law, the problem of the adaptive control of PWS can be reduced to the control problem of augmented piecewise systems. Then, we construct the piecewise affine control law for augmented piecewise systems in such a way that the PQLF can be employed to establish the stability and performance. In particular, the Reciprocal Projection Lemma is employed to formulate the synthesis condition as linear matrix inequalities (LMIs), which enables the proposed PQLF approach to be numerically solvable. Finally, an engineering example is shown to illustrate the synthesis results.     

A sum of squares approach to backstepping controller synthesis for piecewise affine and polynomial systems

This paper develops a backstepping controller synthesis methodology for piecewise polynomial (PWP) systems in strict form. The main contribution of the paper is to formulate sufficient conditions for controller design for PWP systems in strict form as a sum of squares feasibility problem under the assumption that an initial control Lyapunov function exists to start the iterative backstepping procedure. This problem can then be translated into a convex SDP problem and solved by available software packages. The controller synthesis problem for PWP systems in strict feedback form is divided into two cases. The first case consists of the construction of a sum of squares polynomial control Lyapunov function for PWP systems with discontinuous vector fields. The second case addresses the construction of a PWP control Lyapunov function for PWP systems with continuous vector fields. One major advantage of the proposed method is the fact that it can handle systems with discontinuous vector fields and sliding modes. The new synthesis method is applied to several numerical examples. One of these examples offers the first convex optimization solution to piecewise affine (PWA) control of a benchmark circuit system addressed before in the literature using non‐convex PWA control solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Input-to-state stability and interconnections of discontinuous dynamical systems

In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting piecewise smooth ISS Lyapunov functions. The main motivation for investigating piecewise smooth ISS Lyapunov functions is the success of piecewise smooth Lyapunov functions in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov’s solution concept, that is appropriate for ‘open’ systems so as to allow interconnections of DDS. It is proven that the existence of a piecewise smooth ISS Lyapunov function for a DDS implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two DDS that both admit a piecewise smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to construct piecewise quadratic ISS Lyapunov functions for piecewise linear systems (including sliding motions) via linear matrix inequalities.     

A duality-based convex optimization approach to L2 -gain control of piecewise affine slab differential inclusions

This paper introduces the dual parameter set of a piecewise affine (PWA) system. This is a key concept to enable a convex formulation of PWA controller synthesis for PWA slab differential inclusions using a new convex relaxation. Another important contribution of the paper is to present PWA L₂-gain analysis and synthesis results for PWA systems whose output is also a PWA function of the state (as opposed to a piecewise-linear (PWL) function). Unlike other results existing in the literature, the sufficient LMI conditions in this paper are valid for synthesis, even when the PWA systems include sliding modes. A numerical example with sliding modes illustrates the new approach.     

H_∞ controller synthesis of piecewise discrete time linear systems

This paper presents an H∞ controller design method for pieccwise discrete time linear systems based on a piecewise quadratic Lyapunov function. It is shown that the resulting closed loop system is globally stable with guaranteed H∞ perfomiance and the controller can be obtained by solving a set of bilinear lnatrLx inequalities. It has been shown that piecewise quadratic Lyapunov functions are less conservative than the global qnadnmc Lyapunov functions. A simulation example is also given to illustrate the advantage of the proposed approach.     

DC programming and DCA for solving Brugnano–Casulli piecewise linear systems

Piecewise linear optimization is one of the most frequently used optimization models in practice, such as transportation, finance and supply-chain management. In this paper, we investigate a particular piecewise linear optimization that is optimizing the norm of piecewise linear functions (NPLF). Specifically, we are interested in solving a class of Brugnano–Casulli piecewise linear systems (PLS), which can be reformulated as an NPLF problem. Speaking generally, the NPLF is considered as an optimization problem with a nonsmooth, nonconvex objective function. A new and efficient optimization approach based on DC (Difference of Convex functions) programming and DCA (DC Algorithms) is developed. With a suitable DC formulation, we design a DCA scheme, named ℓ₁-DCA, for the problem of optimizing the ℓ₁-norm of NPLF. Thanks to particular properties of the problem, we prove that under some conditions, our proposed algorithm converges to an exact solution after a finite number of iterations. In addition, when a nonglobal solution is found, a numerical procedure is introduced to find a feasible point having a smaller objective value and to restart ℓ₁-DCA at this point. Several numerical experiments illustrate these interesting convergence properties. Moreover, we also present an application to the free-surface hydrodynamic problem, where the correct numerical modeling often requires to have the solution of special PLS, with the aim of showing the efficiency of the proposed method.     

Non-zenoness of piecewise affine dynamical systems and affine complementarity systems with inputs

In the context of continuous piecewise affine dynamical systems and affine complementarity systems with inputs, we study the existence of Zeno behavior, i.e., infinite number of mode transitions in a finite-length time interval, in this paper. The main result reveals that continuous piecewise affine dynamical systems with piecewise real-analytic inputs do not exhibit Zeno behavior. Applied the achieved result to affine complementarity systems with inputs, we also obtained a similar conclusion. A direct benefit of the main result is that one can apply smooth ordinary differential equations theory in a local manner for the analysis of continuous piecewise affine dynamical systems with inputs.     

An efficient algorithm for optimal control of PWA systems with polyhedral performance indices

We present an algorithm for the computation of explicit optimal control laws for piecewise affine (PWA) systems with polyhedral performance indices. The algorithm is based on dynamic programming (DP) and represents an extension of ideas initially proposed in Kerrigan and Mayne [(2003). Optimal control of constrained, piecewise affine systems with bounded disturbances. In Proceedings of the 41st IEEE conference on decision and control, Las Vegas, Nevada, USA, December], and Baoti? et al. [(2003). A new algorithm for constrained finite time optimal control of hybrid systems with a linear performance index. In Proceedings of European control conference, Cambridge, UK, September]. Specifically, we show how to exploit the underlying geometric structure of the optimization problem in order to significantly improve the efficiency of the off-line computations. An extensive case study is provided, which clearly indicates that the algorithm proposed in this paper may be preferable to other schemes published in the literature.     

Stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems

This paper provides several stability tests for piecewise linear systems and proposes a method of stabilization for bimodal systems. In particular, we derive an explicit and exact stability test for planar systems, which is given in terms of coefficients of transfer functions of subsystems. Restricting attention to the bimodal and planar case, we show simple stability tests. In addition, we drive a necessary stability condition and a sufficient stability condition for higher-order and bimodal systems. They are given in terms of the eigenvalue loci and the observability of subsystems. All the stability tests provided in this paper are computationally tractable, and our results are applied to the stabilizability problem. We confirm the exactness and effectiveness of our approach by illustrative examples.     

The analysis of global input-to-state stability for piecewise affine systems with time-delay

In this paper, global input-to-state stability (ISS) for discrete-time piecewise affine systems with time-delay are considered. Piecewise quadratic ISS-Lyapunov functions are adopted. Both Lyapunov-Razumikhin and Lyapunov-Krasovskii methods are used. The theorems of Lyapunov-Razumikhin type and Lyapunov-Krasovskii type for piecewise affine systems with time-delay are shown, respectively.     

Popov–Belevitch–Hautus type controllability tests for linear complementarity systems

It is well-known that checking certain controllability properties of very simple piecewise linear systems are undecidable problems. This paper deals with the controllability problem of a class of piecewise linear systems, known as linear complementarity systems. By exploiting the underlying structure and employing the results on the controllability of the so-called conewise linear systems, we present a set of inequality-type conditions as necessary and sufficient conditions for controllability of linear complementarity systems. The presented conditions are of Popov–Belevitch–Hautus type in nature.     

Synthesis of optimal controllers for piecewise affine systems with sampled-data switching

This paper discusses the optimal control problem of the continuous-time piecewise affine (PWA) systems with sampled-data switching, where the switching action is executed based upon a condition on the state at each sampling time. First, an algebraic characterization for the problem to be feasible is derived. Next, an optimal continuous-time controller is derived for a general class of PWA systems with sampled-data switching, for which the optimal control problem is feasible but whose subsystems in some modes may be uncontrollable in the usual sense. Finally, as an application of the proposed approach, the high-speed and energy-saving control problem of the CPU processing is formulated, and the validity of the proposed methods is shown by numerical simulations.     

Output feedback controller design for uncertain piecewise linear systems

This paper proposes output feedback controller design methods for uncertain piecewise linear systems based on piecewise quadratic Lyapunov function. The α-stability of closed-loop systems is also considered. It is shown that the output feedback controller design procedure of uncertain piecewise linear systems with α-stability constraint can be cast as solving a set of bilinear matrix inequalities （BMIs）. The BMIs problem in this paper can be solved iteratively as a set of two convex optimization problems involving linear matrix inequalities （LMIs） which can be solved numerically efficiently. A numerical example shows the effectiveness of the proposed methods.     

Stabilization of Fractional‐Order Linear Systems with State and Input Delay

This paper describes a variable structure control for fractional‐order systems with delay in both the input and state variables. The proposed method includes a fractional‐order state predictor to eliminate the input delay. The resulting state‐delay system is controlled through a sliding mode approach where the controller uses a sliding surface defined by fractional order integral. Then, the proposed control law ensures that the state trajectories reach the sliding surface in finite time. Based on recent results of Lyapunov stability theory for fractional‐order systems, the stability of the closed loop is studied. Finally, an illustrative example is given to show the interest of the proposed approach.     

An improved LMI-based approach for stability of piecewise affine time-delay systems with uncertainty

The stability problem for uncertain piecewise affine (PWA) time-delay systems is investigated in this article. It is assumed that there exists a known constant time delay in the system and the uncertainly is norm-bounded. Sufficient conditions for the stability of nominal systems and the stability of systems subject to uncertainty are derived using the Lyapunov–Krasovskii functional with a triple integration term. This approach handles switching based on the delayed states (in addition to the states) for a PWA time-delay system, considers structured as well as unstructured uncertainty and reduces the conservativeness of previous approaches. The effectiveness of the proposed approach is demonstrated by comparing with the existing methods through numerical examples.     

Relaxed delay-dependent stabilization criteria for discrete-time fuzzy systems based on a switching fuzzy model and piecewise Lyapunov functions

This paper presents delay-dependent stability analysis and controller synthesis methods for discrete-time Takagi-Sugeno （T-S） fuzzy systems with time delays. The T-S fuzzy system is transformed to an equivalent switching fuzzy system. Consequently, the delay-dependent stabilization criteria are derived for the switching fuzzy system based on the piecewise Lyapunov function. The proposed conditions are given in terms of linear matrix inequalities （LMIs）. The interactions among the fuzzy subsystems are considered in each subregion, and accordingly the proposed conditions are less conservative than the previous results. Since only a set of LMIs is involved, the controller design is quite simple and numerically tractable. Finally, a design example is given to show the validity of the proposed method.     

Polynomial-time probabilistic observability analysis of sampled-data piecewise affine systems

This paper proposes a polynomial-time probabilistic approach to solve the observability problem of sampled-data piecewise affine systems. First, an algebraic characterization for the system to be observable is derived. Next, based on the characterization, we propose a randomized algorithm that can determine if the system is observable in a probabilistic sense or the system is not observable in a deterministic sense. Finally, it is shown with some examples, for which it is hopeless to check the observability in a deterministic way, that the proposed algorithm is very useful.     

Stability and stabilization of periodic piecewise positive systems: A time segmentation approach

This paper is concerned with the stability analysis and stabilization of periodic piecewise positive systems. By constructing a time-scheduled copositive Lyapunov function with a time segmentation approach, an equivalent stability condition, determined via linear programming, for periodic piecewise positive systems is established. Based on the asymptotic stability condition, the spectral radius characterization of the state transition matrix is proposed. The relation between the spectral radius of the state transition matrix and the convergent rate of the system is also revealed. An iterative algorithm is developed to stabilize the system by decreasing the spectral radius of the state transition matrix. Finally, numerical examples are given to illustrate the results.