Robust H∞ performance using lifted polynomial parameter-dependent Lyapunov functions

This paper investigates the robust H _∞ performance of time-invariant linear uncertain systems where the uncertainty is in polytopic domains. Robust H _∞ is checked by constructing a quadratic parameter-dependent Lyapunov function. The matrix associated with this quadratic Lyapunov function is a polynomial function of the uncertain parameters, expressed as a particular polynomial matrix involving κ powers of the dynamic matrix of the system and one symmetric matrix to be determined. The degree of this polynomial matrix function is arbitrary. Finsler's Lemma is used to lift the obtained stability conditions into a larger space in which sufficient stability tests can be developed in the form of linear matrix inequalities. As κ increases, less conservative H _∞ evaluations are obtained. Both continuous and discrete-time systems are investigated. Numerical examples illustrate the method and compare the present results with similar works in the literature.     

Reliable observer-based H∞ control of uncertain state-delayed systems

This paper is concerned with the reliable H_∞ control design problem for linear state-delayed system using observed-based output feedback. It proposes a reliable control design scheme for the case of possibly a simultaneous presence of actuator failures and sensor failures. Modified algebraic Riccati inequalities are developed to solve the problem addressed. Based on this approach, observer-based feedback control laws are designed that guarantee closed-loop asymptotic stability and reduction of the effect of an augmented disturbance input on the controlled output of a prescribed level, not only when the system is operating properly, but also under actuator and sensor failures. A numerical example is presented to demonstrate the applicability and effectiveness of the proposed approach.     

Gain-scheduled H∞ control via parameter-dependent Lyapunov functions

Synthesising a gain-scheduled output feedback H_∞ controller via parameter-dependent Lyapunov functions for linear parameter-varying (LPV) plant models involves solving an infinite number of linear matrix inequalities (LMIs). In practice, for affine LPV models, a finite number of LMIs can be achieved using convexifying techniques. This paper proposes an alternative approach to achieve a finite number of LMIs. By simple manipulations on the bounded real lemma inequality, a symmetric matrix polytope inequality can be formed. Hence, the LMIs need only to be evaluated at all vertices of such a symmetric matrix polytope. In addition, a construction technique of the intermediate controller variables is also proposed as an affine matrix-valued function in the polytopic coordinates of the scheduled parameters. Computational results on a numerical example using the approach were compared with those from a multi-convexity approach in order to demonstrate the impacts of the approach on parameter-dependent Lyapunov-based stability and performance analysis. Furthermore, numerical simulation results show the effectiveness of these proposed techniques.     

Gain-scheduled H ∞ filtering of parameter-varying discrete-time systems via parameter-dependent Lyapunov functions

This paper is concerned with the problem of gain-scheduled H _∞ filter design for a class of parameter-varying discrete-time systems. A new LMI-based design approach is proposed by using parameter-dependent Lyapunov functions. Recommended by Editorial Board member Huanshui Zhang under the direction of Editor Jae Weon Choi. This work was supported in part by the National Natural Science Foundation of P. R. China under Grants 60874058, by 973 program No 2009CB320600, but also the National Natural Science Foundation of Province of Zhejiang under Grants Y107056, and in part by a Research Grant from the Australian Research Council. Shaosheng Zhou received the B.S. degree in Applied Mathematics and the M.Sc. and Ph.D. degrees in Electrical Engineering, in January 1992, July 1996 and October 2001, from Qufu Normal University and Southeast University. His research interests include nonlinear control and stochastic systems. Baoyong Zhang received the B.S. and M.Sc. degrees in Applied Mathematics, in July 2003 and July 2006, all from Qufu Normal University. His research interests include and nonlinear systems, robust control and filtering. Wei Xing Zheng received the B.Sc. degree in Applied Mathematics and the M.Sc. and Ph.D. degrees in Electrical Engineering, in January 1982, July 1984 and February 1989, respectively, all from the Southeast University, Nanjing, China. His research interests include signal processing and system identification.     

Robust control of electrically-stimulated muscle using polynomial H∞ design

The work is concerned with the design and experimental evaluation of robust feedback systems for the control of ankle moments generated by the electrical stimulation of the human calf muscle. This is an important part of the problem of designing feedback controllers for stabilising the upright posture of people with spinal cord injuries while they stand. Robust controllers are designed using the polynomial approach to mixed sensitivity H_∞ feedback design. The approach was found to give a convenient and transparent way to design for performance. Moreover, the method gives a useful indicator of the robustness properties of a given design. This is highly useful for experimental controller “tuning”.     

H ∞ predictive control of networked control systems

This article is concerned with the problem of H _∞ predictive control of networked control system with random network delay. A new control scheme termed networked predictive control is proposed. This scheme mainly consists of the control prediction generator and network delay compensator. While designing the predictor, the control input to the actuator may be different due to networked induced time-delay and data dropout, and two cases are considered depending on the way that the observer obtains the plant control input u _t . The necessary and sufficient conditions are given for the closed-loop networked predictive control system to be stochastically stable for different u _t and random network delays in controller to actuator channel (CAC) and sensor to controller channel (SCC). A simulation study shows the effectiveness of the proposed scheme.     

Fault tolerant H ∞ control for a class of polynomial non-linear discrete-time systems

This paper studies the fault tolerant control (FTC) problem for a class of polynomial nonlinear discrete-time systems with guaranteed H _∞ performance in the presence of actuator faults. The concerned fault is considered as a multi-model of the typical aberration in actuators’ effectiveness. A quadratic-like polynomial Lyapunov function is presented for the H _∞ specification. The main contribution of this paper is that the effect of the nonlinear terms appear in FTC analysis is described as an index in order to transform the controller design into a semi-definite programming (SDP). A numerical example is given to verify the applicability of this new approach for the nonlinear FTC synthesis.     

H∞ state-feedback control of time-delay systems using reciprocally convex approach

In this paper, state-feedback H_∞ control of systems with time-varying delay is considered. The time-delay is assumed to be a time-varying continuous function which is bounded below and above by positive constants. The reciprocally convex approach based on linear matrix inequality (LMI) optimization is adapted to design the H_∞ state feedback control law. For the illustrative examples the physical model of liquid monopropellant rocket motor with a pressure feeding system and the problem of controlling the yaw angles of a satellite system with delays are considered along with their numerical simulations to show effectiveness of derived results. Furthermore few other numerical examples are also given to illustrate the advantages of the derived results.     

Robust H ∞ control of uncertain nonlinear switched systems using constructive method

This paper focuses on the problem of robust H _?? control of nonlinear switched systems with parameter uncertainty via the multiple Lyapunov functions (MLFs) approach. The uncertain parameters are assumed to be in a known compact set and are allowed to enter the system nonlinearly. Based on the explicit construction of Lyapunov functions, which avoids solving the Hamilton-Jacobi-Isaacs (HJI) inequalities, sufficient conditions for the solvability of the robust H _?? control problem of cascade nonlinear switched systems are derived under some switching signal. Then, the result is extended to solve the robust H _?? control problem of nonlinear switched systems in strict feedback form. Finally, the effectiveness of the proposed results is illustrated through a simulation example.     

Passivity and H∞ control of switched discrete-time nonlinear systems using linearisation

Local passivity and H_∞ control of switched discrete-time nonlinear systems are studied using the linearisation technique in this paper. We first establish LMI-based sufficient conditions under which theconsidered system is locally strictly QSR-dissipative. Then, two special cases of QSR-dissipativity, local passivity and l₂ gain, are investigated. In view of the derived conditions being all convex in linearised system matrices, local feedback passification and H_∞ control problems of switched discrete-time nonlinear systems are solved. The efficiency of the proposed method is verified through numerical examples.     

Discrete-time H∞ control of linear parameter-varying systems

We introduce new conditions for the H_∞ synthesis of discrete-time linear parameter-varying systems in the form of linear matrix inequalities. A distinctive feature of the proposed conditions is the ability to handle variation in both the dynamics and the input matrices without resorting to dynamic augmentation or iterative procedures. We show that this new condition contains the existing poly-quadratic H_∞ synthesis result as a particular case. We also derive a corollary which shows improvement even in the stronger case of quadratic H_∞ synthesis. Additionally, we show that, surprisingly, a dynamic gain-scheduled quadratic H_∞ controller can result in inferior performance compared to a static robust controller. Numerical examples illustrate our results.     

Conjugation and H ∞ control of discrete-time systems

We establish conjugation notion in discrete-time systems, first introduced into the H ^∞ control theory of continuous-time systems by Kimura (1989). In discrete-time systems, conjugation is a very elementary operation on rational transfer functions that replaces some of their poles by their reflections with respect to the unit circle. With the aid of J-lossless conjugation conjugation by a J.lossless system), it is shown that the parametrization of sub-optimal solutions of H ^∞ model-matching problems is reduced to a Lyapunov-type equation. The parametrization of all solutions is given in an extremely simple way. It is further proved that the J-lossless conjugation of the H ^∞ model-matching problem is a natural state-space representation of classical interpolation in discrete-time systems.     

Homotopy solution of polynomial equations in H∞ optimal control

A method is described to solve the ‘standard’ H_∞ optimal control problem polynomially. The polynomial equations for equalizing solutions are of a homotopy type, so that the equations may be transformed accordingly to a set of non-linear ordinary differential equations.     

H ∞ control of networked control systems with state quantisation

This article addresses the problem of controller design for networked control systems over digital communication. The systems under consideration are stabilised via state feedback, where the effects of sampled signal, state quantisation, network-induced delay and packet dropout are considered. The proposed delay-dependent stability criteria are formulated in the form of a linear matrix inequality, which ensure asymptotic stability and a prescribed H _∞ performance level for networked control systems with admissible uncertainties. Maximum allowable delay bound of networked control systems is obtained by solving a convex optimisation problem. Furthermore, a numerical example is given to illustrate the effectiveness of the main result.     

Multiple Lyapunov functions approach to observer-based H ∞ control for switched systems

This article investigates the issue of H _∞ control for a class of continuous-time switched Lipschitz nonlinear systems. None of the individual subsystems is assumed to be stabilisable with H _∞ disturbance attenuation. Based on a generalised multiple Lyapunov functions (GMLFs) approach, which removes the nonincreasing requirement at switching points, a sufficient condition for the solvability of the H _∞ control problem under a state estimation-dependent switching law is presented. Observers, controllers and a switching law are simultaneously designed. As an extension, a sufficient condition for exponential stabilisability is also given.     

H ∞ control of networked control systems based on event-time-driven model

This article considers the H _∞ control problem for a class of networked control systems (NCSs) based on the event-time-driven model, under which the considered NCS can be changed into a class of switched delay systems including an unstable subsystem. The Lyapunov functional exponential estimation method is adopted to solve the problem of H _∞ control for such systems. The switching controller is designed to make the considered system exponentially stable with an H _∞ norm bound in terms of linear matrix inequalities. The obtained results are less conservative than existing ones. Finally, one example is given to illustrate the effectiveness and benefit of the proposed method.     

H ∞ sampled data control of systems with time-delays

Sampled-data H _∞ control of linear systems with constant state, control and measurement delays is considered. The sampling of the controlled input and of the measured output is not assumed to be uniform. The system is modelled as a continuous-time one, where the controlled input and the measurement output have piecewise-continuous delays. The input–output approach to stability and L ₂-gain analysis is applied to the resulting system. The discretised Lyapunov functional method is extended to the case of multiple delays, where the Lyapunov functional is complete in one of the delays (in the state) and is simple in the other delays (those in the input and in the output), which are constant. Solutions to the state-feedback and the output-feedback H _∞ control problems are derived in terms of linear matrix inequalities (LMIs).     

Finite-time H∞ control of periodic piecewise linear systems

In this paper, the finite-time stability, stabilisation, L₂-gain and H_∞ control problems for a class of continuous-time periodic piecewise linear systems are addressed. By employing a time-varying control scheme in which the time interval of each subsystem constitutes a number of basic time segments, the finite-time controllers can be developed with periodically time-varying control gains. Based on a piecewise time-varying Lyapunov-like function, a sufficient condition of finite-time stability and the relevant time-varying controller are proposed. Considering the finite-time boundedness of the closed-loop periodic system, the L₂-gain criterion with continuous time-varying Lyapunov-like matrix function is studied. A finite-time H_∞ controller is proposed based on the L₂-gain analysis. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed criteria.