Discrete-time Piecewise Linear Systems Research Articles

Parity Space-Based Fault Diagnosis in Piecewise Linear Systems*

This paper is concerned with the detection and isolation of faults in discrete-time piecewise linear systems, where parity space-based residuals are non-stationary and of varying dimensionality. A modification of the widely used cumulative sum control chart (CUSUM) is presented for fault detection under these conditions. Fault isolation is conducted using a sequential probability ratio test (SPRT). Simulated fault scenarios of an adaptive high-rise building, in which wind-induced vibrations are compensated by hydraulic actuators, are used to demonstrate the methods. Therein, the piecewise linear behavior of tension-only cross-bracing elements renders corresponding strain gauges temporarily unusable and affects the overall system dynamics. Standard linear methods are thus not applicable for the diagnosis of sensor and actuator faults, whereas the proposed approach is shown to perform well.

Chaos in Saw Map

We consider the dynamics of a scalar piecewise linear “saw map” with infinitely many linear segments. In particular, such maps are generated as a Poincaré map of simple two-dimensional discrete time piecewise linear systems involving a saturation function. Alternatively, these systems can be viewed as a feedback loop with the so-called stop hysteresis operator. We analyze chaotic sets and attractors of the “saw map” depending on its parameters.

Fault detection for discrete piecewise linear systems with infinite distributed time-delays

ABSTRACTIn this paper, the fault detection problem is investigated for a class of discrete-time piecewise linear systems with external disturbances and infinite distributed time-delays. As a modelling framework, piecewise linear system often arise when piecewise linear components are encountered, such as dead-zone, saturation, relays and hysteresis. The time-delays are assumed to be infinitely distributed in the discrete-time domain. The aim of this paper is to detect the possible faults and to estimate the system state. For this purpose, firstly, stability analysis is given based on a piecewise smooth Lyapunov function. Afterward, an appropriate approach of fault detection and filter design problem is provided to achieve a satisfactory balance between the disturbances attenuation level γ and the sensitivity to the fault for piecewise linear systems. As a consequence, a sufficient condition is obtained in terms of the linear matrix inequalities such that, for all admissible infinite distributed time-delays and external disturbances, the system is guaranteed to be asymptotically stable and the residual is guaranteed to satisfy H∞ filtering performance and fault detection performance. At last, a simulation example is provided to demonstrate the applicability and effectiveness of the fault detection filtering scheme proposed in this paper.

Distributed [formula omitted] filtering for piecewise discrete-time linear systems

This paper is concerned with the distributed H∞-consensus filtering problem for a class of piecewise discrete-time linear systems. Firstly, the modes and their transitions of augmented piecewise linear systems as well as distributed filters are formulated. Also, the structure of augmented distributed filter gains is presented in virtue of the adjacent matrix of sensor networks. Then, a set of sufficient conditions are provided for the distributed filter to ensure that its dynamics is global asymptotically stable with the H∞-consensus performance constraint. In addition, the distributed filter gains are obtained with the aid of the convex optimal method. At last, an illustrative simulation is presented to demonstrate the effectiveness and applicability of the proposed distributed filtering algorithm.

Stability analysis of discrete-time piecewise-linear systems: A generating function approach

This paper studies the exponential stability of a class of discrete-time piecewise-linear systems (DPLS). Some basic properties of the proposed DPLS are established, which enables the generating function approach to be used for the system stability analysis. By introducing the generating functions of DPLS and showing their properties, a sufficient and necessary condition for the exponential stability of DPLS is derived. Furthermore, the maximum exponential growth rate of system trajectories can be obtained exactly by computing the radii of convergence of the generating functions. The algorithm for computing the generating functions is developed and two examples are given to demonstrate the proposed approach.

Reliable H∞ filtering for discrete piecewise linear systems with infinite distributed delays

This paper is concerned with the reliable filtering problem for discrete-time piecewise linear systems subject to sensor failures and time delays. The considered sensor failures are depicted by bounded variables taking value on a certain interval. The time delays are assumed to be infinitely distributed in the discrete-time domain. The purpose of the addressed reliable filtering problem is to design a piecewise linear filter such that, for the admissible sensor failures and possible infinite distributed delays, the augmented dynamics is exponentially stable and the performance is guaranteed with a prescribed attenuation level . With the aid of the convex optimal method, the filter parameters are obtained in terms of the solution to a set of LMIs which can be solved by the Matlab Toolbox. At last, an illustrative simulation is presented to demonstrate the effectiveness and applicability of the proposed algorithms.

Robust observer-based fault estimation and accommodation of discrete-time piecewise linear systems

In this paper a new integrated observer-based fault estimation and accommodation strategy for discrete-time piecewise linear (PWL) systems subject to actuator faults is proposed. A robust estimator is designed to simultaneously estimate the state of the system and the actuator fault. Then, the estimate of fault is used to compensate for the effect of the fault. By using the estimate of fault and the states, a fault tolerant controller using a PWL state feedback is designed. The observer-based fault-tolerant controller is obtained by the interconnection of the estimator and the state feedback controller. We show that separate design of the state feedback and the estimator results in the stability of the overall closed-loop system. In addition, the input-to-state stability (ISS) gain for the closed-loop system is obtained and a procedure for minimizing it is given. All of the design conditions are formulated in terms of linear matrix inequalities (LMI) which can be solved efficiently. Also, performance of the estimator and the state feedback controller are minimized by solving convex optimization problems. The efficiency of the method is demonstrated by means of a numerical example.

Controllability of a class of bimodal discrete-time piecewise linear systems

In this paper we will provide full algebraic necessary and sufficient conditions for the controllability/reachability/null controllability of a class of bimodal discrete-time piecewise linear systems including several instances of interest that are not covered by existing works which focus primarily on the planar case. In particular, the class is characterized by a continuous right-hand side, a scalar input and a transfer function from the control input to the switching variable with at most two zeroes whereas the state can be of any dimension. To prove the main result, we will make use of geometric control theory for linear systems and a novel result on controllability for input-constrained linear systems with non-convex constraint sets.

Control of uncertain piecewise discrete-time linear systems via state and output feedback

Based on piecewise quadratic Lyapunov function techniques, this paper first provides a stability condition for the uncertain piecewise discrete-time linear system, which is a feasibility problem of linear matrix inequalities. Then the control methods for uncertain piecewise discrete-time linear systems via state feedback and output feedback controllers are proposed. It is shown that both the state feedback and output feedback controllers can be obtained by solving a set of bilinear matrix inequalities. A numerical example is given to illustrate the effectiveness of the developed methods

STATE ESTIMATION FOR A CLASS OF DISCRETE-TIME PIECEWISE-LINEAR SYSTEMS

In this paper, we investigate the problem of state estimation of a class of hybrid discrete-time piecewise-linear systems composed by linear discrete-time (LTI) subsystems with autonomous switching. The aim is to propose a method for the synthesis of a hybrid observer. The proposed approach consists in combining a discrete observer with a continuous observer. It is shown that under conditions related to minimum dwell time of each mode, the switching law can be expressed as a linear combination of the system input/output samples. The continuous observer is a piecewise-linear observer whose dynamics depend on the current active mode. Two estimation schemes are analyzed: on-line estimation and off-line estimation.

Reliable [formula omitted] control for discrete-time piecewise linear systems with infinite distributed delays

In this paper, the reliable H ∞ control problem is investigated for discrete-time piecewise linear systems with time delays and actuator failures. The time delays are assumed to be infinitely distributed in the discrete-time domain, and the possible failure of each actuator is described by a variable varying in a given interval. The aim of the addressed reliable H ∞ control problem is to design a controller such that, for the admissible infinite distributed delays and possible actuator failures, the closed-loop system is exponentially stable with a given disturbance attenuation level γ . The controller gain is characterized in terms of the solution to a linear matrix inequality that can be easily solved by using standard software packages. A simulation example is exploited in order to illustrate the effectiveness of the proposed design procedures.

L 2–l ∞ Filtering Design for Piecewise Discrete-Time Linear Systems

This paper is concerned with the l 2---l ? filtering problem for a class of piecewise discrete-time linear systems. Attention is focused on the design of a stable filter guaranteeing a prescribed noise attenuation level in the l 2---l ? sense. By using the piecewise Lyapunov function, a sufficient condition for the solvability of this problem is obtained in terms of linear matrix inequalities (LMIs). It has been shown that the l 2---l ? filtering problem can be solved as an LMI optimization problem. Two numerical examples are presented to demonstrate the validity of the proposed design method.

Static Output Feedback Control for Discrete-time Piecewise Linear Systems: an LMI Approach

This paper investigates the problem of static output feedback (SOF) control for discrete-time piecewise linear systems. Based on piecewise quadratic Lyapunov functions, new sufficient LMI conditions for the synthesis of SOF stabilization controllers are presented. Meanwhile, by using Finsler's lemma, a set of slack variables with special structure are introduced to reduce design conservatism. Compared to the existing methods, the proposed method has a good performance and can work successfully in situations where the existing methods fail. An extension of this method is also given in order to incorporate H ∞ performance. Three examples are given to illustrate the effectiveness of our method.

New results on robust H ∞ filtering design for discrete-time piecewise linear delay systems

This paper revisits the problem of robust H ∞ filtering design for a class of discrete-time piecewise linear state-delayed systems. The state delay is assumed to be time-varying and of an interval-like type, which means that both the lower and upper bounds of the time-varying delay are available. The parameter uncertainties are assumed to have a structured linear fractional form. Based on a novel delay-dependent piecewise Lyapunov–Krasovskii functional combined with Finsler's Lemma, a new delay-dependent sufficient condition for robust H ∞ performance analysis is first derived and then the filter synthesis is developed. It is shown that by using a new linearisation technique, a unified framework can be developed so that both the full-order and reduced-order filters can be obtained by solving a set of linear matrix inequalities (LMIs), which are numerically efficient with commercially available software. Finally, a numerical example is provided to illustrate the effectiveness and less conservatism of the proposed approach.

Output Tracking of Piecewise-Linear Systems via Error Feedback Regulator With Application to Synchronization of Nonlinear Chua's Circuit

This paper considers the output tracking problem of general piecewise discrete-time linear systems via error feedback scheme. A number of sufficient conditions are obtained based on piecewise-quadratic Lyapunov functions in the framework of output regulation theory. The resulting closed-loop system is guaranteed to be stable and the output tracking is achieved asymptotically. Moreover, the proposed piecewise-linear model based regulator has been successfully applied to the chaos synchronization of general nonlinear Chua's circuits. Simulation results are also given to illustrate the performance and advantages of the proposed approach.

Output regulation of discrete-time piecewise-linear systems with application to controlling chaos

This paper presents an approach to output regulation of discrete-time piecewise-linear systems. A sufficient condition to guarantee output regulation of such systems via state feedback has been obtained based on piecewise-quadratic Lyapunov functions. It is shown that the output regulation controller can be obtained by solving a set of linear-matrix inequalities. Application to controlling chaos is also given to illustrate the performance of the proposed approach.

Nonsynchronized state estimation of discrete time piecewise linear systems

This paper presents two approaches to H/sub /spl infin// state estimation of discrete-time piecewise linear systems based on a common quadratic Lyapunov function and a piecewise quadratic Lyapunov function, respectively. The key issue addressed in this paper is that the system states and their estimates may stay at different regions of the state space. It is shown that the resulting state estimation error systems is globally stable with l/sub 2/ performance, and the state estimator gains can be obtained by solving a set of linear matrix inequalities. Two simulation examples are finally given to illustrate the performance of the proposed approaches.

Null controllability of discrete-time planar bimodal piecewise linear systems

This paper addresses the null controllability problem for a class of discrete-time piecewise linear systems. Based on a general classification method, explicit necessary and sufficient conditions for the controllability of discrete-time planar bimodal piecewise linear systems are derived. The complexity of the controllability problem for piecewise linear systems can be clearly seen from the study of the present paper. We also briefly discuss the high-order and multi-modal cases.

Robust filtering design of piecewise discrete time linear systems

This work presents two filter design methods for piecewise discrete time linear systems based on a piecewise Lyapunov function. It is shown that the resulting filtering error system is globally stable with guaranteed H/sub /spl infin// or generalized H/sub 2/ performance, and the filter gains can be obtained by solving a set of linear matrix inequalities that is numerically feasible with commercially available software. Two simulation examples are also given to illustrate the performance of the proposed approaches.

H∞ FILTERING FOR DISCRETE-TIME PIECEWISE LINEAR SYSTEMS

This paper deals with the design of piecewise linear H∞ filters for discrete-time piecewise linear systems. Both the system and filter have structures that change according to their switching rules. The system switching rule is characterized in terms of a given partition of the system state-space. The filter switching rule is also characterized in terms of a suitable filter state-space partition which is determined, jointly with the filter matrices, in order to ensure a prescribed H∞ performance for the estimation error system. The filter design is based on a piecewise quadratic Lyapunov function and is given in terms of linear matrix inequalities (LMIs).