Reachability analysis of continuous-time piecewise affine systems

This paper proposes an algorithm for the characterization of reachable sets of states for continuous-time piecewise affine systems. Given a model of the system and a bounded set of possible initial states, the algorithm employs an LMI approach to compute both upper and lower bounds on reachable regions. Rather than performing computations in the state-space, this method uses impact maps to find the reachable sets on the switching surfaces of the system. This tool can then be used to deduce safety and performance results about the system.     

Well-posedness analysis of switch-driven piecewise affine systems

The well-posedness problem (existence and uniqueness of solutions) of a class of multi-modal piecewise affine systems is addressed, where binary-switches individually act under autonomous switching. First, a new transition rule on the discrete state, called the switch-based transition rule, is introduced and some relations with the mode-based transition rule are discussed. Next, a sufficient condition for such a multi-modal system to be well-posed for all external inputs is derived in terms of well-posedness of its subsystems of lower complexity "bimodal systems". Finally, an easily checkable condition for the bimodal system to be well-posed for all external inputs is given, which consequently allows us to algebraically determine well-posedness of the multi-modal systems in question.     

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.     

Classification and stabilizability analysis of bimodal piecewise affine systems

This paper presents a classification of bimodal piecewise affine systems from the viewpoint of well‐posedness. First, we address the feedback well‐posedness problem of a general class of bimodal piecewise affine systems, which is the problem of feedback equivalence to a well‐posed system. Next, based on this result, we classify all feedback well‐posed systems into four classes to address the control problem of piecewise affine systems in a systematic way. As its application, the stabilizability problem with well‐posedness is discussed for each class, and several remarks on stabilizability are given. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

On convergence properties of piecewise affine systems

In this paper convergence properties of piecewise affine (PWA) systems are studied. In general, a system is called convergent if all its solutions converge to some bounded globally asymptotically stable steady-state solution. The notions of exponential, uniform and quadratic convergence are introduced and studied. It is shown that for non-linear systems with discontinuous right-hand sides, quadratic convergence, i.e., convergence with a quadratic Lyapunov function, implies exponential convergence. For PWA systems with continuous right-hand sides it is shown that quadratic convergence is equivalent to the existence of a common quadratic Lyapunov function for the linear parts of the system dynamics in every mode. For discontinuous bimodal PWA systems it is proved that quadratic convergence is equivalent to the requirements that the system has some special structure and that certain passivity-like condition is satisfied. For a general multimodal PWA system these conditions become sufficient for quadratic convergence. An example illustrating the application of the obtained results to a mechanical system with a one-sided restoring characteristic, which is equivalent to an electric circuit with a switching capacitor, is provided. The obtained results facilitate bifurcation analysis of PWA systems excited by periodic inputs, substantiate numerical methods for computing the corresponding periodic responses and help in controller design for PWA systems.     

Optimal control of sampled-data piecewise affine systems

This paper discusses the optimal continuous-time control problem of a class of piecewise affine (PWA) systems, where the switching action of the discrete state is determined at each sampling time according to a condition on the continuous state. Such a system is called here the sampled-data PWA (SD-PWA) system. First, important remarks on the control design of this system via the continuous-(or discrete-)time PWA model are pointed out, which motivate us to use the SD-PWA model. Next, based on the good properties of the proposed model, an optimal continuous-time controller of the SD-PWA systems is proposed.     

Reference governor for constrained piecewise affine systems

We present a methodology for designing reference tracking controllers for constrained, discrete-time piecewise affine systems. The approach follows the idea of reference governor techniques where the desired set-point is filtered by a system called the “reference governor”. Based on the system current state, set-point, and prescribed constraints, the reference governor computes a new set-point for a low-level controller so that the state and input constraints are satisfied and convergence to the original set-point is guaranteed.In this note we show how to design a reference governor for constrained piecewise affine systems by using polyhedral invariant sets, reachable sets, multiparametric programming and dynamic programming techniques.     

Optimal complexity reduction of polyhedral piecewise affine systems

This paper focuses on the NP-hard problem of reducing the complexity of piecewise polyhedral systems (e.g. polyhedral piecewise affine (PWA) systems). The results are fourfold. Firstly, the paper presents two computationally attractive algorithms for optimal complexity reduction that, under the assumption that the system is defined over the cells of a hyperplane arrangement, derive an equivalent polyhedral piecewise system that is minimal in the number of polyhedra. The algorithms are based on the cells and the markings of the hyperplane arrangement. In particular, the first algorithm yields a set of disjoint (non overlapping) merged polyhedra by executing a branch and bound search on the markings of the cells. The second approach leads to non-disjoint (overlapping) polyhedra by formulating and solving an equivalent (and well-studied) logic minimization problem. Secondly, the results are extended to systems defined on general polyhedral partitions (and not on cells of hyperplane arrangements). Thirdly, the paper proposes a technique to further reduce the complexity of piecewise polyhedral systems if the introduction of an adjustable degree of error is acceptable. Fourthly, the paper shows that based on the notion of the hyperplane arrangement PWA state feedback control laws can be implemented efficiently. Three examples, including a challenging industrial problem, illustrate the algorithms and show their computational effectiveness in reducing the complexity by up to one order of magnitude.     

Fault recoverability analysis of nonlinear systems: A piecewise affine system approach

The recoverability reflects the capability of the system to tolerate the worst faults over a prescribed set under admissible energy constrains. In this paper, a piecewise affine (PWA) controller is designed for a PWA approximation of the nonlinear dynamics, which can stabilize the original nonlinear system under some conditions. By analyzing the upper bound of the piecewise linear quadratic control performance of the PWA system, a novel fault recoverability evaluation scheme is further proposed for the nonlinear system with respect to the given limit of admissible control performance. A longitudinal flight control example is taken to show the effectiveness of the proposed method.     

Trajectory tracking control of bimodal piecewise affine systems

This paper deals with a trajectory tracking problem for a class of bimodal piecewise affine systems, which is inherently difficult because of the discontinuous changes of their vector fields. First, we introduce an error variable and an error system as a generalization of the tracking error and its system. As an error variable, a function switched by the mode of a piecewise affine system is adopted to overcome the inherent difficulty in trajectory tracking control of piecewise affine systems. Next, we design a tracking controller which stabilizes the error system using a Lyapunov-like function, which can be applied to systems including state jumps. Furthermore, the feasibility condition of tracking for SISO piecewise linear systems is simplified. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.     

Analysis of discrete-time piecewise affine and hybrid systems

In this paper, we present various algorithms both for stability and performance analysis of discrete-time piece-wise affine (PWA) systems. For stability, different classes of Lyapunov functions are considered and it is shown how to compute them through linear matrix inequalities that take into account the switching structure of the systems. We also show that the continuity of the Lyapunov function is not required in discrete time. Moreover, the tradeoff between the degree of conservativeness and computational requirements is discussed. Finally, by using arguments from the dissipativity theory for nonlinear systems, we generalize our approach to analyze the l₂-gain of PWA systems.     

Identification of piecewise affine systems via mixed-integer programming

This paper addresses the problem of identification of hybrid dynamical systems, by focusing the attention on hinging hyperplanes and Wiener piecewise affine autoregressive exogenous models, in which the regressor space is partitioned into polyhedra with affine submodels for each polyhedron. In particular, we provide algorithms based on mixed-integer linear or quadratic programming which are guaranteed to converge to a global optimum. For the special case where the estimation data only seldom switches between the different submodels, we also suggest a way of trading off between optimality and complexity by using a change detection approach.     

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.     

Robust constrained control of piecewise affine systems through set‐based reachability computations

This article addresses finite‐horizon robust control of a piecewise affine system affected by uncertainty and characterized by different affine dynamics (modes) associated with a polyhedral partition of the state space. The goal is to design a static state‐feedback control law that maintains the state of the system within given—possibly time‐varying—sets, subject to actuation constraints. The proposed approach rests on two phases: a reference mode sequence with a sufficiently large robustness level is determined first, and then a tracking state‐feedback control law defined on the reach sets of the controlled system is designed to counteract uncertainty and maintain the reach sets within the reference sequence. If this is not possible and the reach sets split over different modes, then, further reference mode sequences and tracking controllers are computed. The designed state‐feedback control law is represented through a collection of controllers defined on precomputed reach sets of the closed‐loop control system. Performance of the approach is shown on some numerical examples.     

Lagrange stability and performance analysis of discrete-time piecewise affine systems with logic states

In this paper we consider discrete-time piecewise affine systems with boolean inputs, outputs and states and show that they can be represented in a logic canonical form where the logic variables influence the switching between different submodels but not the continuous-valued dynamics. We exploit this representation for studying Lagrange stability and developing performance analysis procedures based on linear matrix inequalities. Moreover, by using arguments from dissipativity theory for non-linear systems, we generalize our approach to solve the H _∞ analysis problem.     

Configuration selection for reconfigurable control of piecewise affine systems

In this paper, the problem of configuration selection, i.e. sensor/actuator placement for piecewise affine (PWA) systems subject to both sensor and actuator faults is considered. A method is proposed that provides a tool for the design phase to decide about the optimal placement of sensor/actuators where the reconfigurability of the system subject to sensor and actuator faults is also taken into account. Using a lattice of possible configurations (sensor/actuator placements), the reconfigurability of the system subject to faults for each configuration is evaluated and based on that one can draw conclusions about the reconfigurability of the system and the optimal configuration in the architecture design phase. A reconfigurable control must ensure stability of the reconfigured system and, if possible, a graceful degradation in the performance. Therefore, in the proposed reconfigurability analysis, we consider both stabilisability and performance of the system. The efficiency of the proposed method is demonstrated in several numerical examples.     

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.     

Observability and controllability of piecewise affine and hybrid systems

We prove, in a constructive way, the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics, automata, and propositional logic. By focusing our investigation on the former class, we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems. Instead, we propose practical numerical tests based on mixed-integer linear programming.     

Nonsynchronized state estimation of uncertain discrete-time piecewise affine systems

This paper investigates the problem of robust H-infinity state estimation for a class of uncertain discretetime piecewise affine systems where state space instead of measurable output space partitions are assumed so that the filter implementation may not be synchronized with plant state trajectory transitions. Based on a piecewise quadratic Lyapunov function combined with S-procedure and some matrix inequality convexifying techniques, two different approaches are developed to the robust filtering design for the underlying piecewise affine systems. It is shown that the filter gains can be obtained by solving a set of linear matrix inequalities (LMIs). Finally, a simulation example is provided to illustrate the effectiveness of the proposed approaches.