Exponentially incremental dissipativity for nonlinear stochastic switched systems

ABSTRACT

In this paper, we investigate the exponentially incremental dissipativity for nonlinear stochastic switched systems by using the designed state-dependent switching law and multiple Lyapunov functions approach. Specifically, using incremental supply rate as well as a state dissipation inequality in expectation, a stochastic version of exponentially incremental dissipativity is presented. The sufficient conditions for nonlinear stochastic switched systems to be exponentially incrementally dissipative are given by the designed state-dependent switching law. Furthermore, the extended Kalman–Yakubovich–Popov conditions are derived by using two times continuously differentiable storage functions. Moreover, the incremental stability conditions in probability for nonlinear stochastic switched systems are derived based on exponentially incremental dissipativity. The exponentially incremental dissipativity is preserved for the feedback-interconnected nonlinear stochastic switched systems with the composite state-dependent switching law; meanwhile, the incremental stability in probability is preserved under some certain conditions. A numerical example is given to illustrate the validity of our results.     

Robust H ∞ control via a stable decentralized nonlinear output feedback controller

This paper presents a new method to construct a decentralized nonlinear robust H ^∞ controller for a class of large‐scale nonlinear uncertain systems. The admissible uncertainties and nonlinearities in the system satisfy integral quadratic constraints and global Lipschitz conditions, respectively. The decentralized controller, which is required to be stable, is capable of exploiting known nonlinearities and interconnections between subsystems without treating them as uncertainties. Instead, additional uncertainties are introduced because of the discrepancies between nondecentralized and decentralized nonlinear output feedback controllers. The H ^∞ control objective is to achieve an absolutely stable closed‐loop system with a specified disturbance attenuation level. A solution to this control problem involves stabilizing solutions to algebraic Riccati equations parametrized by scaling constants corresponding to the uncertainties and nonlinearities. This formulation is nonconvex; hence, an evolutionary optimization method is applied to solve the control problem considered. Copyright © 2012 John Wiley & Sons, Ltd.     

On designing network‐based H ∞ controllers for stochastic systems

This paper is concerned with network‐based H _∞ stabilization for stochastic systems, where network‐induced delays, packet dropouts, and packet disorders are taken into account simultaneously. The packet disorders arising from both the sampler‐to‐controller channel and the controller‐to‐actuator channel are considered by introducing a logic controller and a logic zero‐order hold. The network‐induced delays and packet dropouts are modeled as a constant delay plus a non‐differentiable time‐varying delay in the input. By employing Lyapunov–Krasovskii functional approach, we establish results that parallel well‐known bounded real Lemmas. More specifically, these results provide conditions to bound the H _∞ level of the system, which means the worst case energy of the output of the system when subjected to a unitary norm deterministic disturbance signal. On the basis of these results, suitable network‐based H _∞ controllers are designed by using cone complementary linearization method. An air vehicle system is finally taken as an example to show the effectiveness of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

H ∞ control for discrete‐time nonlinear Markov jump systems with multiplicative noise and sector constraint

This paper addresses the finite horizon H _∞ control problem for a class of discrete‐time nonlinear Markov jump systems with multiplicative noise and nonlinear feedback device. The system nonlinearity occurs in a random way specified by a Bernoulli process, whereas the actuator and sensor nonlinearities are restricted to a sector region. Both the state and the dynamic output feedback H _∞ controllers are devised in terms of difference LMIs. The proposed approach not only allows the resulting system to achieve a prescribed disturbance attenuation level, but also enables the output of actuator/sensor to meet the designated sector condition. Moreover, it is also shown that our approach is well‐adapted for dealing with the discrete‐time Markov jump systems with saturated actuator and sensor. Finally, a backward iterative algorithm is provided to solve the obtained difference LMIs and a numerical example is presented to verify the efficiency of the theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

H ∞ observer design for uncertain nonlinear discrete‐time systems with time‐delay: LMI optimization approach

We present a robust H _∞ observer for a class of nonlinear discrete‐time systems. The class under study includes an unknown time‐varying delay limited by upper and lower bounds, as well as time‐varying parametric uncertainties. We design a nonlinear H _∞ observer, by using the upper and lower bounds of the delay, that guarantees asymptotic stability of the estimation error dynamics and is also robust against time‐varying parametric uncertainties. The described problem is converted to a standard optimization problem, which can be solved in terms of linear matrix inequalities (LMIs). Then, we expand the problem to a multi‐objective optimization problem in which the maximum admissible Lipschitz constant and the minimum disturbance attenuation level are the problem objectives. Finally, the proposed observer is illustrated with two examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Disturbance attenuation by output feedback for linear systems subject to actuator saturation

In this paper, we study the problem of disturbance attenuation by output feedback for linear systems subject to actuator saturation. A nonlinear output feedback, expressed in the form of a quasi‐linear parameter‐varying system with state‐dependent scheduling parameter, is constructed that leads to the attenuation of the effect of the disturbance on the output of the system. The level of disturbance attenuation is measured in terms of the restricted ℒ︁₂ gain and the restricted ℒ︁₂–ℒ︁_∞ gain over a class of bounded disturbances. Copyright © 2008 John Wiley & Sons, Ltd.     

H ∞ preview tracking control of retarded state‐multiplicative stochastic systems

The problem of infinite‐horizon H _∞ state‐feedback tracking control for linear continuous time‐invariant retarded systems with stochastic parameter uncertainties is investigated. Two tracking patterns are considered depending on the nature of the reference signal; that is, whether it is measured online or previewed in a fixed time‐interval ahead. The stochastic uncertainties appear in the dynamics matrices for both the retarded and the non‐retarded states of the system. The delayed system is transformed via the input–output approach, to an uncertain norm‐bounded system. A new method that efficiently yields a min–max strategy to the solution of each of the aforementioned two cases is suggested where, given a specific reference signal, the controller plays against nature, which chooses the maximizing energy‐bounded disturbance. The theoretical results are demonstrated by two examples that show the impact of the delay length and the preview length on the system performance. Copyright © 2013 John Wiley & Sons, Ltd.     

Observer‐based H ∞ control for networked systems with bounded random delays and consecutive packet dropouts

This paper is concerned with the H _∞ control problem for a class of systems with bounded random delays and consecutive packet dropouts that exist in both sensor‐to‐controller channel and controller‐to‐actuator channel during data transmission. A new model is developed to describe possible random delays and packet dropouts by two groups of Bernoulli distributed stochastic variables. To avoid the state augmentation, a full‐order observer‐based feedback controller is designed via LMI approach. Based on the Lyapunov theory, a sufficient condition is provided to guarantee the closed‐loop networked system to be asymptotically mean‐square stable and achieve the prescribed H _∞ disturbance‐rejection‐attenuation level. The simulation examples illustrate the effectiveness of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

On dissipativity, passivity and dynamic operability of nonlinear processes

Passivity and dissipativity have long being important tools in nonlinear systems analysis and control design. In this article, connections between dissipativity and the dynamic operability of nonlinear processes are explored. In particular, we study nonlinear processes that are dissipative with respect to a supply rate quadratic in the inputs and outputs. These type of dissipative systems include processes that can be unstable and can have unstable zero dynamics. Two methods to quantify operability are proposed. The first method analyses the speed of response of the closed-loop with no exogenous signals. The second method studies the output disturbance attenuation problem. The proposed framework reveals important insights into those nonlinear process characteristics that can limit dynamic operability.     

H∞ fuzzy control design of discrete‐time nonlinear active fault‐tolerant control systems

This paper is concerned with the problem of H_∞ fuzzy controller synthesis for a class of discrete‐time nonlinear active fault‐tolerant control systems (AFTCSs) in a stochastic setting. The Takagi and Sugeno (T–S) fuzzy model is employed to exactly represent a nonlinear AFTCS. For this AFTCS, two random processes with Markovian transition characteristics are introduced to model the failure process of system components and the fault detection and isolation (FDI) decision process used to reconfigure the control law, respectively. The random behavior of the FDI process is conditioned on the state of the failure process. A non‐parallel distributed compensation (non‐PDC) scheme is adopted for the design of the fault‐tolerant control laws. The resulting closed‐loop fuzzy system is the one with two Markovian jump parameters. Based on a stochastic fuzzy Lyapunov function (FLF), sufficient conditions for the stochastic stability and H_∞ disturbance attenuation of the closed‐loop fuzzy system are first derived. A linear matrix inequality (LMI) approach to the fuzzy control design is then developed. Moreover, a suboptimal fault‐tolerant H_∞ fuzzy controller is given in the sense of minimizing the level of disturbance attenuation. Finally, a simulation example is presented to illustrate the effectiveness of the proposed design method. Copyright © 2008 John Wiley & Sons, Ltd.     

H disturbance attenuation for state delayed systems

In this note, the differential game and dissipation inequality are applied to the disturbance attenuation or H^∞-control for linear systems with delayed state. Firstly, a simple sufficient condition on the existence of a γ-suboptimal H^∞ state feedback controller is given, which is independent of delay, and an observer-based dynamic output feedback solution is presented in terms of Riccati inequalities (or Riccati equations). Secondly, a sufficient condition on the existence of a delay-dependent state feedback is presented and the criterion is presented by a matrix inequality which can be solved by numerical methods.     

Exponential H ∞ filtering for stochastic Markovian jump systems with time delays

This paper investigates the problem of exponential H _∞ filtering for stochastic systems with time delays and Markovian jumping parameters. On the basis of Lyapunov–Krasovskii functional theory and generalized Finsler lemma, a delay‐dependent bounded real lemma is established without using any model transformations, bounding techniques for cross terms, or additional free matrix variables. The obtained bounded real lemma guarantees that the filtering error system is both mean‐square exponentially stable and almost surely exponentially stable with a prescribed H _∞ noise attenuation level. Then an exponential H _∞ filter is designed for stochastic retarded Markovian jump systems in terms of a set of LMIs. Meanwhile, the mathematical equivalence of the proposed method to one recent method is presented, but our proposed method is more computationally efficient with fewer matrix variables than that recent method. The validity of the method is verified by a numerical example.Copyright © 2012 John Wiley & Sons, Ltd.     

A dissipative dynamical systems approach to stability analysis of time delay systems

In this paper the concepts of dissipativity and the exponential dissipativity are used to provide sufficient conditions for guaranteeing asymptotic stability of a time delay dynamical system. Specifically, representing a time delay dynamical system as a negative feedback interconnection of a finite‐dimensional linear dynamical system and an infinite‐dimensional time delay operator, we show that the time delay operator is dissipative with respect to a quadratic supply rate and with a storage functional involving an integral term identical to the integral term appearing in standard Lyapunov–Krasovskii functionals. Finally, using stability of feedback interconnection results for dissipative systems, we develop sufficient conditions for asymptotic stability of time delay dynamical systems. The overall approach provides a dissipativity theoretic interpretation of Lyapunov–Krasovskii functionals for asymptotically stable dynamical systems with arbitrary time delay. Copyright © 2004 John Wiley & Sons, Ltd.     

Robust delay‐dependent ℋ︁∞ control of time‐delay systems with state and input delays

This paper studies the design problem of robust delay‐dependent ??_∞ controller for a class of time‐delay control systems with time‐varying state and input delays, which are assumed to be noncoincident. The system is subject to norm‐bounded uncertainties and ??₂ disturbances. Based on the selection of an augmented form of Lyapunov–Krasovskii (L‐K) functional, first a Bounded Real Lemma (BRL) is obtained in terms of linear matrix inequalities (LMIs) such that the nominal, unforced time‐delay system is guaranteed to be globally asymptotically stable with minimum allowable disturbance attenuation level. Extending BRL, sufficient delay‐dependent criteria are developed for a stabilizing ??_∞ controller synthesis involving a matrix inequality for which a nonlinear optimization algorithm with LMIs is proposed to get feasible solution to the problem. Moreover, for the case of existence of norm‐bounded uncertainties, both the BRL and ??_∞ stabilization criteria are easily extended by employing a well‐known bounding technique. A plenty of numerical examples are given to illustrate the application of the proposed methodology of this note. The achieved numerical results on the maximum allowable delay bound and minimum allowable disturbance attenuation level are exhibited to be less conservative in comparison to those of existing methods in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

A generalized approach to stabilization of linear interconnected time‐delay systems

The objective of this paper is to propose a generalized approach to stabilization of systems which are composed of linear time‐delay subsystems coupled by linear time‐varying interconnections. The proposed algorithms, which are formulated within the convex optimization framework, provide decentralized solutions to the problem of delay‐dependent asymptotic stability with strict dissipativity. It is established that the new methodology can reproduce earlier results on passivity, positive realness and disturbance attenuation. Then a decentralized structure of dissipative state‐feedback controllers is designed to render the closed‐loop interconnected system delay‐dependent asymptotically stable with strict dissipativity. Numerical examples are presented to illustrate the applicability of the design method. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Finite-time H∞ control for discrete-time switched singular time-delay systems subject to actuator saturation via static output feedback

This study employs the multiple Lyapunov-like function method and the average dwell-time concept of switching signal to investigate the finite-time H_∞ static output-feedback (SOF) control problem for a class of discrete-time switched singular time-delay systems subject to actuator saturation. First, sufficient conditions are presented to guarantee the discrete-time switched singular time-delay system regular, causal and finite-time boundedness. Meanwhile, sufficient conditions are presented to ensure the H_∞ disturbance attenuation level, and the design method of H_∞ SOF controller is developed by solving matrix inequalities optimisation problem without any decompositions of system matrices and equivalent transformation. Finally, the effectiveness and merit of the theoretical results are shown through some numerical examples and several vivid illustrations.     

Stability of systems with the Duhem hysteresis operator: The dissipativity approach

In this paper, we discuss the dissipativity property of the counterclockwise Duhem operator. Sufficient conditions on the functions which define the Duhem operator are given such that the Duhem operator has counterclockwise input–output dynamics. In particular, an explicit construction of the storage functions satisfying the counterclockwise dissipation inequality is given. The constructed storage function is also related to the underlying anhysteresis function which is commonly used to describe hysteresis in magnetic materials. The results can thus facilitate analysis of systems with the counterclockwise Duhem operator via the dissipativity approach.     

On nonlinear control of Euler-Lagrange systems: Disturbance attenuation properties

In this brief note we analyse the (local) disturbance attenuation properties of some asymptotically stabilizing nonlinear controllers for Euler-Lagrange systems reported in the literature. Our objective in this study is twofold: first, to compare the performance of these schemes from a perspective different from stabilizability; second, to quantify the basic tradeoff between robust stability and robust performance for these designs. We consider passivity-based and feedback linearization schemes developed for the control of DC-to-DC converters and rigid robots. For the DC-to-DC problem we show that for both controllers there exists a lower bound to the achievable attenuation level, i.e. a lower bound to the ₂-gain of the closed loop operator from disturbance to regulated output, which is independent of the design parameters. Also, for the passivity based scheme we obtain an upper bound for the disturbance attenuation, which is insured provided we sacrifice the convergence rate. For rigid robots we show that both approaches yield arbitrarily good disturbance attenuation without compromising the convergence rate.     

H∞ control of networked control systems based on Markov jump unified model

This paper focuses on a new H_∞ controller design issue for networked control systems with external disturbance as well as random time delays and packet dropouts in forward and feedback channels, which are modeled by multiple Markov chains in a unified style. The output feedback controller is designed to stabilize the networked control system and also achieves the prescribed H_∞ disturbance attenuation level. The addressed controller design problem is transformed into a nonlinear minimization problem with LMI constraints. An illustrative example is provided to show the effectiveness of the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd.     

A generalized robust H ∞ filtering for singular Markovian jump systems and its applications

In this paper, a generalized robust H _∞ filtering method is proposed for a class of singular Markovian jump systems, whose generality is mainly embodied that the desired filter could bear perturbances in terms of uncertainties on its parameter matrices. Firstly, an LMI condition of robust mode‐dependent filter is developed. Based on the given result, a new approach to mode‐independent H _∞ filter is presented, which establishes a direct connection between mode‐dependent and mode‐independent filters. Secondly, when the transition rate matrix is with elementwise bounded uncertainties or partially unknown, sufficient conditions of such robust mode‐dependent and mode‐independent filters are all developed within LMI frameworks. Finally, a numerical example is used to demonstrate the effectiveness of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.