Non-fragile robust H ∞ control for uncertain spacecraft rendezvous system with pole and input constraints

This article investigates the robust H _∞ control problem for a class of spacecraft rendezvous systems with parameter uncertainties and subject to input constraint, pole constraint and external and controller perturbations. Based on the Lyapunov theory, a sufficient condition for the existence of the non-fragile robust state-feedback controller is given in terms of linear matrix inequalities (LMIs). Then, proper non-fragile controller design can be cast as a convex optimisation problem subject to LMI constraints. With the obtained controller, the spacecraft rendezvous mission can be accomplished. An illustrative example is provided to show the effectiveness of the proposed control design method.     

Robust H2 guaranteed cost fuzzy control for uncertain discrete‐time fuzzy systems via poly‐quadratic Lyapunov functions

In this paper, new approaches regarding H₂ guaranteed cost stability analysis and controller synthesis problems for a class of discrete‐time fuzzy systems with uncertainties are investigated. The state‐space Takagi‐Sugeno fuzzy model with norm‐bounded parameter uncertainties is adopted. Based on poly‐quadratic Lyapunov functions, sufficient conditions for the existence of the robust H₂ fuzzy controller can be obtained in terms of linear matrix inequalities (LMIs). Furthermore, a convex optimization problem with LMI constraints is formulated to design a suboptimal fuzzy controller which minimizes the upper bound on the quadratic cost function. The effectiveness of the proposed design approach is illustrated by two examples. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust H ∞ control for a class of uncertain mechanical systems

In this article, the problem of H _∞ control is investigated for a class of mechanical systems with input delay and parameter uncertainties which appear in all the mass, damping and stiffness matrices. Two approaches, norm-bounded and linear fractional transformation (LFT) uncertainty formulations, are considered. By using a new Lyapunov–Krasovskii functional approach, combined with the advanced techniques for achieving delay dependence, improved robust H _∞ state-feedback controller design methods are developed. The existence condition for admissible controllers is formulated in the form of linear matrix inequalities (LMIs), and the controller design is cast into a convex optimisation problem subject to LMI constraints. If the optimisation problem is solvable, a desired controller can be readily constructed. The result for the norm-bounded uncertainty case improves the existing ones in terms of design conservatism, and that for the LFT uncertainty case represents the first attempt in this direction. An illustrative example is provided to show the effectiveness and advantage of the proposed controller design methodologies.     

Robust H∞ fuzzy observer‐based tracking control design for a class of nonlinear stochastic Markovian jump systems

This paper investigates the stochastic H_∞ tracking control problem for a class of nonlinear stochastic Markovian jump systems. The attention is focused on the design of a fuzzy observer‐based fuzzy controller such that an H_∞ model reference tracking performance is guaranteed for admissible disturbances. A sufficient condition is established to guarantee the existence of the desired robust controller, which is given in terms of a set of coupled matrix inequalities. Moreover, a novel decoupled method is proposed to transform the sufficient condition into some linear matrix inequality (LMI) form such that observer gains and control gains can be simultaneously obtained by solving a set of LMIs. Finally, a simulation example is presented to illustrate the effectiveness of the proposed design method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

LMI optimization approach to robust H∞ observer design and static output feedback stabilization for discrete‐time nonlinear uncertain systems

A new approach for the design of robust H_∞ observers for a class of Lipschitz nonlinear systems with time‐varying uncertainties is proposed based on linear matrix inequalities (LMIs). The admissible Lipschitz constant of the system and the disturbance attenuation level are maximized simultaneously through convex multiobjective optimization. The resulting H_∞ observer guarantees asymptotic stability of the estimation error dynamics and is robust against nonlinear additive uncertainty and time‐varying parametric uncertainties. Explicit norm‐wise and element‐wise bounds on the tolerable nonlinear uncertainty are derived. Also, a new method for the robust output feedback stabilization with H_∞ performance for a class of uncertain nonlinear systems is proposed. Our solution is based on a noniterative LMI optimization and is less restrictive than the existing solutions. The bounds on the nonlinear uncertainty and multiobjective optimization obtained for the observer are also applicable to the proposed static output feedback stabilizing controller. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust H∞ dynamic output feedback control for spacecraft rendezvous with poles and input constraint

This paper is concerned with the output feedback control problem for spacecraft rendezvous subject to target angular velocity uncertainty and controller uncertainty, external disturbance and input constraint. A general full-order dynamic output feedback (DOF) controller is proposed. As a stepping-stone, the H_∞ performance requirement, poles and input constraint are analysed separately via linear matrix inequalities (LMIs). Then, with the obtained results, the controller design problem is cast into a convex problem subject to a set of LMI constraints through a critical change of controller variables. Furthermore, when the system states are all available, a reduced sufficient condition of the non-fragile state feedback controller is given. Compared with existing results, the designed controller has overcome the disadvantage of strictly proper DOF controller, where the initial value of the control input is zero. Besides, the constraint on poles placement is relaxed. A numerical simulation is performed to verify the effectiveness of the proposed method.     

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.     

Dynamic output-feedback <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control for active half-vehicle suspension systems with time-varying input delay

This paper addresses the new output-feedback H _∞ control problem for active half-vehicle suspension systems with time-varying input delay. By introducing multi-objective synthesis, a new dynamic output-feedback H _∞ controller is designed such that the closed-loop suspension system is asymptotically stable with guaranteed robust performance in the H _∞ sense. The proposed controller is formulated in terms of linear matrix inequality (LMI) based on the auxiliary function-based integral inequality method and the reciprocally convex approach. A new delay-dependent sufficient condition for the desired controller offers a wider range of control input delay. Numerical examples are provided to validate the effectiveness of the proposed design method.     

A parametric insensitive H2 control design approach

H₂ and H_∞ control design methodologies are known to be efficient to deal with multivariable control problems. However, most of them do not take explicitly the parametric uncertainties into account. This paper proposes a low parametric sensitivity H₂ control design method as an alternative to µ‐synthesis or robust H₂ control design. In addition to the standard H₂ criterion, the H₂ norm of the parametric sensitivity function is introduced in order to improve the robustness of the resulting controller. Unfortunately, this problem is a difficult one. Its equivalence to structured feedback H₂ control problem will be shown. The underlying BMI will be solved by making use of an iterative LMI procedure. Two examples will illustrate the interest of the approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Implementation of FIR control for H ∞ output feedback stabilisation of linear systems

In this article, finite impulse response (FIR) control is addressed for H _∞ output feedback stabilisation of linear systems. The problem we deal with is the construction of an output feedback controller with a certain FIR structure such that the resulting closed-loop system is asymptotically stable and a prescribed H _∞ norm bound constraint is guaranteed. Some solvability conditions are suggested in this article. Sufficient conditions are derived to obtain a suboptimal solution of the H _∞ FIR control problem via convex optimisation. Also, an equivalent condition for the existence of H _∞ FIR control is presented in the set of linear matrix inequalities (LMIs) and a reciprocal matrices equality constraint. An effective computational algorithm involving LMIs is suggested to solve a concave minimisation problem characterising a local optimal solution of the H _∞ FIR control problem. Numerical examples demonstrate the validity of the proposed H _∞ FIR control and the numerical efficiency of the proposed algorithm for FIR control.     

Robust discrete-time control design with time-varying parametric uncertainties

A novel multiplier approach for robust controller design in discrete-time systems with real, time-varying parametric uncertainty is presented. An important feature of our approach is that bounds on the rate of variation of the uncertain parameters are assumed and, unlike in most related approaches, dynamic multipliers are obtained that utilize this information. A convex minimization procedure formulated as an LMI problem is presented to obtain multipliers that satisfy the robustness conditions derived. Such conditions are transformed to an equivalent scaled H_∞ norm condition and a μ/k_m-synthesis approach is proposed to design robust controllers. Copyright © 1999 John Wiley & Sons, Ltd.     

Parameter dependent H∞ control by finite dimensional LMI optimization: application to trade‐off dependent control

In this paper, we consider the design of an H_∞ trade‐off dependent controller, that is, a controller such that, for a given Linear Time‐Invariant plant, a set of performance trade‐offs parameterized by a scalar θ is satisfied. The controller state space matrices are explicit functions of θ. This new problem is a special case of the design of a parameter dependent controller for a parameter dependent plant, which has many application in Automatic Control. This last design problem can be naturally formulated as a convex but infinite dimensional optimization problem involving parameter dependent Linear Matrix Inequality (LMI) constraints. In this paper, we propose finite dimensional (parameter independent) LMI constraints which are equivalent to the parameter dependent LMI constraints. The parameter dependent controller design is then formulated as a convex finite dimensional LMI optimization problem. The obtained result is then applied to the trade‐off dependent controller design. A numerical example emphasizes the strong interest of our finite dimensional optimization problem with respect to the trade‐off dependent control application. Copyright © 2005 John Wiley & Sons, Ltd.     

Robust Reliable H∞ Control for Discrete‐Time Systems with Actuator Delays

This article focuses on the robust state feedback reliable H_∞ control problem for discrete‐time systems. Discrete‐time systems with time‐varying delayed control input are formulated. Based on the Lyapunov–Krasovskii method and linear matrix inequality (LMI) approach, delay‐dependent sufficient conditions are developed for synthesizing the state feedback controller for an uncertain discrete‐time system. The parameter uncertainty is assumed to be norm bounded. A design scheme for the state feedback reliable H_∞ controller is proposed in terms of LMIs, which can guarantee the global asymptotic stability and the minimum disturbance attenuation level. Finally, numerical examples are provided to illustrate the effectiveness and reduced conservatism of the proposed methods.     

Robust H ∞ reliable control of time delay nonlinear systems via Takagi–Sugeno fuzzy models

This article is focused on reliable fuzzy H _∞ controller design for a class of Takagi–Sugeno (T–S) fuzzy systems with state delay, actuator failures, disturbance input and norm bounded uncertainties. In the design, the H _∞ performance of the closed-loop system is optimised during normal operation (without failures) while the system satisfies a prescribed H _∞ performance level in the case of actuator failures. Two methods are presented in this study. In the first method, delay-dependent conditions are derived based on a single Lyapunov–Krasovskii function. This method improves delay-independent results existing in the literature. Next, to further reduce the conservatism, we use a parameter-dependent Lyapunov–Krasovskii function. The new sufficient conditions for the existence of the suboptimal robust reliable controller are shown in terms of linear matrix inequalities (LMIs), which can be solved by using LMI optimisation techniques. A simulation example shows the effectiveness of the proposed methods.     

Robustness design of a dynamic output-feedback decentralized controller using <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> synthesis and LMI paradigm

This paper proposes developing a H _∞ dynamic output-feedback decentralized control design method for nonlinear interconnected systems subject to time-varying parameters and external disturbances. The designed controller is formulated as an optimization problem subject to linear matrix inequalities (LMIs) for the concurrent computation of the decentralized observation and control gains, and for the external disturbance mitigation by means of a H _∞ performance criterion minimization. The propounded optimization problem, designed in LMI conditions, is expeditiously resolved by a one-step procedure to override the conservatism generated by using many step-based procedures often used in the analysis and synthesis of interconnected systems. The effectiveness of the developed control scheme is demonstrated through simulation results of multimachine power systems.     

A linear matrix inequality approach to H∞ control

The continuous- and discrete-time H_∞ control problems are solved via elementary manipulations on linear matrix inequalities (LMI). Two interesting new features emerge through this approach: solvability conditions valid for both regular and singular problems, and an LMI-based parametrization of all H_∞-suboptimal controllers, including reduced-order controllers. The solvability conditions involve Riccati inequalities rather than the usual indefinite Riccati equations. Alternatively, these conditions can be expressed as a system of three LMIs. Efficient convex optimization techniques are available to solve this system. Moreover, its solutions parametrize the set of H_∞ controllers and bear important connections with the controller order and the closed-loop Lyapunov functions. Thanks to such connections, the LMI-based characterization of H_∞ controllers opens new perspectives for the refinement of H_∞ design. Applications to cancellation-free design and controller order reduction are discussed and illustrated by examples.     

New iterative linear matrix inequality based procedure for H2 and H∞ state feedback control of continuous‐time polytopic systems

This article provides new linear matrix inequality (LMI) sufficient conditions for a generalized robust state feedback control synthesis problem for linear continuous‐time polytopic systems. This generalized problem includes the robust stability, H₂ ‐norm, and H_∞ ‐norm problems as special cases. Using a novel general separation result, which separates the state feedback gain from the Lyapunov matrix but with the state feedback gain synthesized from the slack variable, then allows the formulation of LMI sufficient conditions for the generalized problem. Compared to existing parameterized LMI based conditions, where auxiliary scalar parameters are introduced in order to include the quadratic stability conditions (ie, assuming a constant Lyapunov matrix) as a special case, the proposed new conditions are true LMIs and contain as a particular case the optimal quadratic stability solution. Utilizing any initial solution derived by the quadratic or some existing methods as a starting solution, we propose an algorithm based on an iterative procedure, which is recursively feasible in each update, to compute a sequence of nonincreasing upper bounds for the H₂ ‐norm and H_∞ ‐norm. In addition, if no feasible initial solution can be found for some uncertain systems using any existing methods, another algorithm is presented that offers the possibility of obtaining a robust stabilizing gain. Numerical examples from the literature demonstrate that our algorithms can provide less conservative results than existing methods, and they can also find feasible solutions where all other methods fail.     

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     

Computer algebra tailored to matrix inequalities in control

A major advance in linear systems theory over the last decade has been a formalism for converting systems problems to matrix inequalities. In this tutorial paper we describe computer algebra algorithms, methodology, and implementation which allows users to convert many systems problems to linear matrix inequalities (LMIs). We shall focus on computer algebra methodology which can assist the user in producing LMIs for control design. We provide a step-by-step computer derivation of LMI formulas for the design of linear time-invariant dynamic controllers that achieve a prespecified performance measured by the H _∞ norm of a certain closed loop transfer function.     

Computationally Efficient Algorithm For Frequency‐Weighted Optimal H∞ Model Reduction

In this paper, a frequency‐weighted optimal H_∞ model reduction problem for linear time‐invariant (LTI) systems is considered. The objective of this class of model reduction problems is to minimize H_∞ norm of the frequency‐weighted truncation error between a given LTI system and its lower order approximation. A necessary and sufficient solvability condition is derived in terms of LMIs with one extra coupling rank constraint, which generally leads to a non‐convex feasibility problem. Moreover, it has been shown that the reduced‐order model is stable when both stable input and output weights are included, and its state‐space data are given explicitly by the solution of the feasibility problem. An efficient model reduction scheme based on cone complementarity algorithm (CCA) is proposed to solve the non‐convex conditions involving rank constraint.