-like control for nonlinear stochastic systems

In this paper we develop a H_∞-type theory, from the dissipation point of view, for a large class of time-continuous stochastic nonlinear systems. In particular, we introduce the notion of stochastic dissipative systems analogously to the familiar notion of dissipation associated with deterministic systems and utilize it as a basis for the development of our theory. Having discussed certain properties of stochastic dissipative systems, we consider time-varying nonlinear systems for which we establish a connection between what is called the L₂-gain property and the solution to a certain Hamilton–Jacobi inequality (HJI), that may be viewed as a bounded real lemma for stochastic nonlinear systems. The time-invariant case with infinite horizon is also considered, where for this case we synthesize a worst case-based stabilizing controller. Stability in this case is taken to be in the mean-square sense. In the stationary case, the problem of robust state feedback control is considered in the case of norm-bounded uncertainties. A solution is then derived in terms of linear matrix inequalities.     

Dynamic output feedback control synthesis for stochastic time-delay systems

This article deals with the dynamic output feedback control synthesis problem for Itô-type stochastic time-delay systems. Our aim is to design a full order dynamic output feedback controller to achieve the desired control objectives. We will formulate the controller design problem as an H _∞ optimisation problem in the mean-square sense. The main contributions of this article are as follows: (i) for stochastic systems, the design of a controller with multiple objectives can be addressed without employing a unique Lyapunov function; (ii) using an inequality technique and Finsler Lemma, we provide convex controller synthesis conditions described by linear matrix inequalities (LMIs). Some examples are presented to show the effectiveness of the developed theoretical results.     

An improved delay‐dependent bounded real lemma (BRL) and H∞ controller synthesis for linear neutral systems

In this paper, a novel delay‐dependent bounded real criterion and an improved sufficient condition are derived for the design of an H_∞ state‐feedback controller for linear neutral time‐delay systems. On the basis of an augmented Lyapunov‐Krasovskii functional, a new bounded real lemma is introduced in terms of a convex linear matrix inequality (LMI) condition that can be solved using interior point algorithms. The bounded real lemma is extended to obtain a sufficient condition for the existence of a delay‐dependent H_∞ memoryless state‐feedback controller. Neither any model transformation nor bounding of any of the cross terms are utilized while deriving the bounded real lemma. Moreover, the use of any free slack matrix variable approach is avoided to a certain extent in order not to increase the complexity of the synthesis problem. A cone complementary nonlinear minimization algorithm is employed to achieve a feasible solution set for the synthesis conditions. Finally, seven numerical examples are given to illustrate the effectiveness of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Dissipative control for a class of uncertain nonlinear control systems

A general dissipative controller is proposed to achieve robust tracking control performance for a class of uncertain single‐input single‐output (SISO) nonlinear systems. The feedback linearization technique is employed to transform the nonlinear system into an assignable inner linear system with a differential control input so that the relationship of the external (input) power and the stored energy of system can be shown clearly. Then, a dissipative controller with an assignable attenuation level is proposed to make the system energy dynamics fit a required dissipative inequality. The unstable factors of the system can then be attenuated accordingly. The system stability is guaranteed even if the system has permanent unavoidable uncertainties. The proposed design can be achieved without the use of traditional means, i.e. optimal control, which requires solution of a Hamilton inequality (or Riccati equation). The Lyapunov stable condition is assured in our approach when the system uncertainties belong to L _∞. Moreover, due to the compatibility of the proposed controller, the controller can be embedded into the designs of other controllers. In those designs, knowledge of the system functions is not required. Basically, the proposed dissipative controller is independent of the system functions. The use of the bounds of the system function is considered to prove the system stability only. Two simulations are given to illustrate the effectiveness of the proposed controller. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Singular H∞ control in nonlinear systems: positive semidefinite storage functions and separation principle

A framework is developed for the general nonlinear H_∞ output feedback control problem, in which two major restrictions are relaxed, i.e., the non-singular penalty in H_∞ cost and the positive definite solution of Hamilton–Jacobi inequality at present state space nonlinear H_∞ control literatures. As illustrated in an example, positive semidefinite solution simplifies the structure of the H_∞ controller. Based on this framework, some sufficient conditions are derived. While specialized to linear systems, the controller reduces to the so-called central controller. © 1997 by John Wiley & Sons, Ltd.     

A family of H∞ controllers for dissipative Hamiltonian systems

Using structure properties of dissipative Hamiltonian systems, this paper investigates the parameterization problem of H_infty controllers for such systems. A family of H_∞ controllers with full information is first obtained by interconnecting an H_∞ controller with a generalized zero‐energy‐gradient (ZEG) detectable, free generalized Hamiltonian system. Then, a family of H_∞ controllers with partial information is presented in terms of the solution to an inequality only in 2n independent variables (twice as many as the one used to characterize the state feedback) and without imposing an additional cascade condition. Both of the parameterization methods avoid solving Hamilton–Jacobi–Issacs equations (or inequalities), and thus the proposed controllers are relatively simple in form and easy in operation. Numerical experiments show the effectiveness and feasibility of the proposed methods. Copyright © 2011 John Wiley & Sons, Ltd.     

连续T-S模糊系统的严格二次型耗散控制

研究一类连续Ｔ-Ｓ模糊系统的严格二次型耗散控制问题,给出了保证系统严格二次型耗散稳定的状态反馈控制器的设计方法．控制器可通过求解一组线性矩阵不等式获得,所得结果不仅提供了解决犎∞ 控制与正实控制的统一框架,而且提供了一种更灵活、保守性更小的控制器设计方法．最后通过仿真说明了所提出方法的有效性、可行性和优越性．     

A variational inequality for measurement feedback almost-dissipative control

In this paper, an optimal measurement feedback control problem that yields an almost- (or practically) dissipative closed loop system is considered. That is, the aim is to consider optimal control problems which, when solved, yield a closed loop system which almost satisfies a dissipation inequality. The main idea is that by weakening the required dissipation inequality, a broader class of open loop systems and controllers are admissible, leading to broader application. In obtaining the main results of this paper, dynamic programming is applied to the optimal control problem of interest to derive a variational inequality that generalizes the information state based partial differential equation associated with measurement feedback nonlinear dissipative control. This variational inequality can in principle be used to derive the optimal controller. In the special case of certainty equivalence, an explicit solution of the variational inequality exists and is a functional of the solution of the corresponding optimal state feedback almost-dissipative control problem.     

General quadratic performance analysis and synthesis of differential algebraic equation (DAE) systems

In this paper, control of linear differential-algebraic-equation systems, subject to general quadratic constraints, is considered. This setup, especially, includes the H_∞ control problem and the design for strict passivity. Based on linear matrix inequality (LMI) analysis conditions, LMI synthesis conditions for the existence of linear output feedback controllers are derived by means of a linearizing change of variables. This approach is constructive: a procedure for the determination of controller parameterizations is given on the basis of the solution of the LMI synthesis conditions. A discussion of the possible applications of the presented results concludes the paper.     

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.     

Improved LMI conditions for gain scheduling and related control problems

The gain scheduling problem considered in this paper concerns a linear system whose state-space equations depend rationally on real, time-varying parameters, which are measured in real time. A stabilizing, parameter-dependent controller is sought, such that a given ℒ︁₂-gain bound for the closed-loop system is ensured. Sufficient linear matrix inequality (LMI) conditions are known, that guarantee the existence of such ‘gain-scheduled’ controllers. This paper improves these results in two directions. First, we show how to exploit the realness of the parameters using a ‘skew-symmetric scaling’ technique. Moreover, we show how to apply this technique in a time-varying and/or nonlinear setting. We first devise a general result pertaining to control synthesis of interconnection of dissipative operators, and apply it to the gain-scheduling problem. Owing to its generality, this result can be applied to other problems such as anti-windup control, nonlinear control and model reduction. © 1998 John Wiley & Sons, Ltd.     

Reliable dissipative control for uncertain time‐delayed stochastic systems with Markovian jump switching and multiplicative noise

This paper considers the robust reliable dissipative control problem for a class of hybrid systems, which includes stochastics, Markovian jumping, state time delay, parameter uncertainty, possible actuator failure, multiplicative noises and impulsive effects. We propose a linear feedback memoryless controller and impulsive controller such that the hybrid system is stochastically stable and strictly (Q, S, R) dissipative, which include H_∞ performance as a special case, for all the admissible uncertainties and actuator failures occurring among a prescribed subset of actuators. Based on Itô's differential formula and Lyapunov stability theory, sufficient conditions are obtained in terms of linear matrix inequalities. A numerical example is constructed to show the effectiveness of the controller designed in this paper. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Sufficient conditions for the synthesis of H∞ fixed‐order controllers

This brief note deals with the synthesis of H_∞ fixed‐order controllers for linear systems. It is well known that this problem can be formulated as a bilinear matrix inequality optimization problem which is non convex and NP hard to solve. In this paper sufficient conditions are provided which allow to convert the controller design into a linear matrix inequality feasibility problem. A numerical example on a practical control problem shows the application of the proposed technique. Copyright © 2000 John Wiley & Sons, Ltd.     

H ∞ output tracking control for delta operator systems with insensitivity to controller coefficient variations

The problem of designing insensitive H _∞ output tracking controller for discrete-time systems is studied in a unified framework by using the delta operator approach. A type of coefficient sensitivity performance measure is first developed tomeasure the sensitivity of the transfer function with respect to controller coefficient variations, and the corresponding controller design problem is naturally reformulated as a mixed-objective H _∞ control problem. It is worth mentioning that a novel bounded real lemma (BRL), i.e. Theorem 1 in this article, for delta operator systems is obtained, which is adapted to deal with the multi-objective optimisation problem or polytopic uncertain systems in a potentially less conservative framework. Then, based on the novel lemma, an linear matrix inequality (LMI)-based design method is proposed for achieving the output tracking purpose. Finally, a numerical example based on the linearised model of F-18 aircraft is given to illustrate the effectiveness of the proposed method.     

Dissipative control and filtering of discrete-time singular systems

In this paper, the problems of dissipative control and filtering of discrete-time singular systems are investigated. Based on parametrising the solutions of the constraint set, a necessary and sufficient condition is established in terms of strict linear matrix inequality which makes the condition more tractable. By using the system augmentation approach, a static output feedback controller design method is proposed to guarantee that the closed-loop system is admissible and strictly (Q, S, R) dissipative. Then, the result is applied to tackle the reduced-order filtering problem. The effectiveness of the obtained results in this paper is illustrated by numerical examples.     

不确定时滞系统的鲁棒耗散控制

耗散不确定性是已经被广泛讨论的范数有界的不确定性和正实不确定性的广义化,因此,从二次型供给率的角度,将耗散不确定性引入线性时变时滞系统,研究了含有此类不确定性的线性时滞系统的鲁棒耗散控制问题,给出了线性矩阵不等式（LMI）形式的充分条件。而且,考虑了不确定时滞系统在状态反馈控制器作用下的闭环系统的耗散控制问题,同样给出了LMI形式的充分条件,并通过线性矩阵不等式的可行解构造出鲁棒耗散的状态反馈控制器。最后。通过实例证明了定理的可行性。     

Adaptive control of uncertain port-controlled Hamiltonian systems subject to actuator saturation

This paper deals with the problem of adaptive control of uncertain nonlinear port-controlled Hamiltonian systems subject to actuator saturation, and proposes a number of results on the control design. Firstly, the adaptive stabilization problem is studied, and a control design method is developed by using both the dissipative Hamiltonian structural and saturated actuator properties. Secondly, for the case that there are both parametric uncertainties and external disturbances in the AS systems, an adaptive H _∞ control design approach is presented. Finally, study of an example of power systems with simulations shows that the adaptive controller proposed in this paper is effective.     

Feedback control of nonlinear differential algebraic systems using Hamiltonian function method

Many complex dynamic systems, such as power systems, robotic systems, etc. can be modeled as the following nonlinear differential algebraic systems (NDAS) [1-4](1) where the vector 1 x1 ∈ X 1 ? Rn represents the state variable, and x2 ∈X2? Rn2 is the al…     

Robust H2/H∞ control for time‐delay systems with polytopic uncertainty

In this paper, new separated H_∞ and H₂ performance criteria are derived for a class of time‐delay systems. When used in robust performance analysis and synthesis for real polytopic uncertainty and in multiobjective controller synthesis, they can partially rule out the technical restriction of using a single Lyapunov function, and therefore, lead to potentially less conservative linear matrix inequality (LMI) characterizations. Based on the criteria, robust multiobjective H₂/H_∞ controller is designed for time‐delay systems with polytopic uncertainty. All the conditions are given in terms of LMIs. Numerical examples are given to illustrate the proposed method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A Simultaneous Mixed LQR/H∞ Control Approach to the Design of Reliable Active Suspension Controllers

This paper is concerned with the synthesis of reliable controllers for quarter‐car active suspension systems. By a simultaneous mixed LQR/H_∞ control approach, a static output feedback controller is derived for guaranteeing good suspension performance under possible sensor fault or suspension component breakdown. The considered simultaneous mixed LQR/H_∞ control problem is a nonconvex optimization problem; therefore, the linear matrix inequality approach is not applicable. Based on the barrier method, we solve an auxiliary minimization problem to get an approximate solution for the simultaneous mixed LQR/H_∞ control problem. Necessary conditions for the local optimum of the auxiliary minimization problem are derived. Moreover, a three‐stage solution algorithm is developed for solving the auxiliary minimization problem. The simulation shows that the obtained static output feedback suspension controllers can improve suspension performance in nominal mode and all considered failure modes.