Iterative algorithms for computing the feedback Nash equilibrium point for positive systems

The paper studies N-player linear quadratic differential games on an infinite time horizon with deterministic feedback information structure. It introduces two iterative methods (the Newton method as well as its accelerated modification) in order to compute the stabilising solution of a set of generalised algebraic Riccati equations. The latter is related to the Nash equilibrium point of the considered game model. Moreover, we derive the sufficient conditions for convergence of the proposed methods. Finally, we discuss two numerical examples so as to illustrate the performance of both of the algorithms.     

H2/H∞ control for stochastic systems with delay

This paper is concerned with the H₂/H_∞ control problem for stochastic linear systems with delay in state, control and external disturbance-dependent noise. A necessary and sufficient condition for the existence of a unique solution to the control problem is derived. The resulting solution is characterised by a kind of complex generalised forward–backward stochastic differential equations with stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the equivalent feedback solution via a new type of Riccati equations. To explain the theoretical results, we apply them to a population control problem.     

A structure-preserving algorithm for the minimum H ∞ norm computation of finite-time state feedback control problem

The algorithm being developed here is based on the generating function approach for finite-time H _∞ control and application of canonical transformation of linear Hamiltonian system. First, an equivalent finite-time H _∞ control law in terms of the third-type generating function is derived. Then, by using symplectic structure of the Hamiltonian system's state transition matrix, a group of matrix recursive formulae are deduced for the evaluation of the finite-time H _∞ control law. Combining with a matrix singularity testing procedure, this recursive algorithm verifies the existence condition of sub-optimal H _∞ controllers and gives the minimum H _∞ norm of finite-time control systems. Inherited from the canonical transformation, the matrix recursive formulae have a standard symplectic form; this structure-preserving property helps facilitate reliable and effective computation. Numerical results show the effectiveness of the proposed algorithm.     

Iterative algorithm to compute the maximal and stabilising solutions of a general class of discrete-time Riccati-type equations

In this article an iterative method to compute the maximal solution and the stabilising solution, respectively, of a wide class of discrete-time nonlinear equations on the linear space of symmetric matrices is proposed. The class of discrete-time nonlinear equations under consideration contains, as special cases, different types of discrete-time Riccati equations involved in various control problems for discrete-time stochastic systems. This article may be viewed as an addendum of the work of Dragan and Morozan (Dragan, V. and Morozan, T. (2009), ‘A Class of Discrete Time Generalized Riccati Equations’, Journal of Difference Equations and Applications, first published on 11 December 2009 (iFirst), doi: 10.1080/10236190802389381) where necessary and sufficient conditions for the existence of the maximal solution and stabilising solution of this kind of discrete-time nonlinear equations are given. The aim of this article is to provide a procedure for numerical computation of the maximal solution and the stabilising solution, respectively, simpler than the method based on the Newton–Kantorovich algorithm.     

Robust wedge-stability analysis and design of continuous constrained systems with stability radii

In this paper, based on stability radii, a sufficient condition is first proposed to ensure the robust wedge stability of continuous-time constrained systems with state feedback. The saturated actuator is reformulated as a conditioned linear actuator subject to structured uncertainties and then an auxiliary matrix is introduced so that the closed-loop system can be characterized into a scheme of stability radius approach. We also investigate the issue of maximizing complex stability radius subject to a wedge region by state feedback. Through iteratively solving a parametrized Riccati equation, a desired maximizing state feedback controller corresponding to the prescribed wedge subregion can be obtained. An example is given to illustrate the design algorithm and to reveal the feasibility of stability radius approach for continuous-time constrained systems.     

Robust consensus algorithm for multi-agent systems with exogenous disturbances under convergence conditions

In this paper, a robust consensus algorithm is developed and sufficient conditions for convergence to consensus are proposed for a multi-agent system (MAS) with exogenous disturbances subject to partial information. By utilizing H_∞ robust control, differential game theory and a design-based approach, the consensus problem of the MAS with exogenous bounded interference is resolved and the disturbances are restrained, simultaneously. Attention is focused on designing an H_∞ robust controller (the robust consensus algorithm) based on minimisation of our proposed rational and individual cost functions according to goals of the MAS. Furthermore, sufficient conditions for convergence of the robust consensus algorithm are given. An example is employed to demonstrate that our results are effective and more capable to restrain exogenous disturbances than the existing literature.     

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

In 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.     

Robust mixed performance switching control for uncertain discrete switched systems with time delay

In this paper, the problem of H₂ and H_∞ mixed switching control of uncertain discrete switched systems with time delay is considered via a switching signal design. New Lyapunov functional and Wirtinger inequality are used to guarantee the mixed performance for discrete switched time-delay system. The number of using LMI variables is less than our past proposed approach. Finally, a numerical example is illustrated to show the improvement of the developed results.     

Robust ℓ 1 performance analysis for linear systems with parametric uncertainties

In this contribution, a computational approach for analysing the robust ?_∞-gain (or the robust ?₁ performance) of uncertain linear systems is developed. In particular, the system's state-space matrices may have a rational dependence on structured parametric time-invariant or time-varying uncertainties. The computation is based on robust semi-definite programming and provides a trade-off between accuracy and computational effort. A novel matrix inequality condition to determine the star-norm of discrete-time systems is derived as an auxiliary result.     

状态饱和线性离散系统的H∞ 控制

研究一类具有状态饱和约束的离散线性系统的 ∞控制问题.通过引入一个无穷范数小于等于1的自由变量,将状态饱和约束下的离散线性系统状态变量约束在一个凸多面体内.在此基础上,给出了状态饱和离散线性系统的有界实引理,并研究了状态反馈控制律设计算法.所给出的结论表示为双线性矩阵不等式,可通过所提出的迭代线性矩阵不等式算法求解.最后通过数值例子验证了所提出算法的正确性和有效性.     

Distributed robust control of linear multi-agent systems with parameter uncertainties

This article considers the distributed robust control problems of uncertain linear multi-agent systems with undirected communication topologies. It is assumed that the agents have identical nominal dynamics while subject to different norm-bounded parameter uncertainties, leading to weakly heterogeneous multi-agent systems. Distributed controllers are designed for both continuous- and discrete-time multi-agent systems, based on the relative states of neighbouring agents and a subset of absolute states of the agents. It is shown for both the continuous- and discrete-time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the H _∞ control problems of a set of decoupled linear systems having the same dimensions as a single agent. A two-step algorithm is presented to construct the distributed controller for the continuous-time case, which does not involve any conservatism and meanwhile decouples the feedback gain design from the communication topology. Furthermore, a sufficient existence condition in terms of linear matrix inequalities is derived for the distributed discrete-time controller. Finally, the distributed robust H _∞ control problems of uncertain linear multi-agent systems subject to external disturbances are discussed.     

H ∞ control for non-linear stochastic systems: the output-feedback case

The H _∞ output-feedback control problem for non-linear stochastic systems is considered. A solution for a large class of non-linear stochastic systems is introduced (including non-linear diffusion systems as a subclass). This solution is based on a bounded real lemma for non-linear stochastic systems that was previously established via a stochastic dissipativity concept. The theory yields sufficient conditions for the closed-loop system to possess a prescribed L ₂-gain bound in terms of two Hamilton Jacobi inequalities: one that is associated with the state feedback part of the problem is n-dimensional (where n is the underlying system's state dimension) and the other inequality that stems from the estimation part is 2n-dimensional. Both stationary and non–stationary systems are considered. Stability of the closed-loop system is established, both in the mean-square and the in-probability senses. As the solution to the Hamilton Jacobi inequalities may, in general, lead to a non–realisable state estimator, a modification of the associated 2n-dimensional Hamilton Jacobi inequality is made in order to circumvent this realisation problem, while preserving the system's L ₂-gain bound. For time-invariant systems, the problem of robust output-feedback is considered in the case of norm-bounded uncertainties. A solution is then derived in terms of linear state-dependent matrix inequalities.     

Output-feedback control for sampled-data systems with variable sampling rate

In this paper, we propose design method of controller for sampled-data systems with variable sampling rate. First, we give design method for both H₂ and H_∞ controller. For H₂ control, performance of the system is introduced according to a standard sampled-data setting. A discrete-time H₂ control problem is employed for solving the original problem. Its solvability condition is then established as a parameter-dependent linear matrix inequality. A probabilistic approach is taken for coping with the parameter-dependency. H_∞ controller is designed by almost the same manner. Applying both results, we have design method for multi-objective control.     

H ∞ filtering for stochastic systems with time-varying delay

This article considers the problem of H _∞ filter design for stochastic systems with time-varying delay. The time delay is assumed to be of interval type. Attention is focused on the design of delay-dependent filters that guarantee the asymptotic stability in mean square and a prescribed noise attenuation level in an H _∞ sense for the filtering error dynamics. The delay-dependent H _∞ filter design scheme is proposed in terms of a linear matrix inequality. A numerical example is used to illustrate the effectiveness of the proposed approach.     

H ∞ output feedback stabilisation of linear discrete-time systems with impulses

This article addresses the issue of designing an H _∞ output feedback controller for linear discrete-time systems with impulses. First, a new concept of H _∞ output feedback stabilisation for general linear discrete-time systems with impulses is introduced. Then sufficient linear matrix inequality conditions for the stabilisation and H _∞ performance of general discrete systems with impulses are proposed. In addition, the result is applied to resolve typical output feedback control problems for systems with impulses, such as the decentralised H _∞ output feedback control and the simultaneous H _∞ output feedback control. Finally, a numerical simulation is also presented to illustrate the effectiveness of the proposed results.     

Decentralised H ∞ control for singular systems

In this article, considering the design problem of decentralised H _∞ controller of singular systems, the two cases of controllers via measurement feedback are designed: one is precise controller, and the other is additive controller gain variation. The design procedures of the two cases of controllers are presented in terms of the solutions to generalised algebraic Riccati inequalities. The designed controllers in each case guarantee that closed-loop singular systems are admissible and with H _∞-norm bound on disturbance attenuation. Finally, a numerical example to demonstrate the validity of the proposed approach is given.     

Optimal fuzzy control of exponential synchronisation via genetic algorithm

In this study, a novel approach via GA-based fuzzy control is proposed to realize the exponential optimal H^∞ synchronisation of MTDC systems. A robustness design of model-based fuzzy control is first presented to overcome the effect of modelling errors between the MTDC systems and T-S fuzzy models. Next, a delay-dependent exponential stability criterion is derived in terms of Lyapunov's direct method to guarantee that the trajectories of the slave system can approach those of the master system. Subsequently, the stability conditions of this criterion are reformulated into LMIs. According to the LMIs, a fuzzy controller is then synthesised to exponentially stabilise the error systems. Moreover, the capability of GA in random search for near-optimal solutions, the lower and upper bounds of the search space based on the feedback gains via LMI approach can be set so that the GA will seek better feedback gains of fuzzy controllers to speed up the synchronisation. Additionally, an IGA was proposed to overcome both the shortcomings of premature convergence of GA and local search. According to the IGA, a fuzzy controller is synthesised not only to realise the exponential synchronisation but also to achieve the optimal H^∞ performance by minimising the disturbance attenuation level.     

Robust H∞ filtering with error variance constraints for discrete time-varying systems with uncertainty

The robust H_∞ filtering problem with error variance constraints is considered for discrete time-varying systems subject to norm-bounded parameter uncertainties in both the state and the output matrices of the state-space model. Sufficient conditions for a finite-horizon filter to satisfy state estimation error variance constraints as well as prescribed H_∞ performance for all admissible perturbations are given in terms of two discrete Riccati difference equations, which are of a form suitable for recursive computation. The results are extended to cover the case of stationary filtering over an infinite-horizon for uncertain time-invariant systems.