H∞ control and filtering with initial uncertainty for infinite dimensional systems

The H_∞-control problem with a non-zero initial condition for infinite dimensional systems is considered The initial conditions are assumed to be in some subspace. First the H_∞ problem with full information is considered and necessary and sufficient conditions for the norm of an input-output operator to be less than a given number are obtained, The characterization of all admissible controllers is also given. This result is then used to solve the general H^∞ control problem and the filtering problem with initial uncertainty. The filtering problem on finite horizon involves the estimate of the state at final time. The set of all suboptimal filters is given both on finite and infinite horizons.     

Observer-based H ∞-control for discrete-time T–S fuzzy systems

This article further studies the observer-based H _∞-control problem for discrete-time Takagi–Sugeno (T–S) fuzzy systems. By using fuzzy Lyapunov functions and introducing slack variables, a sufficient condition, which can guarantee observer-based H _∞-control performance for T–S fuzzy systems, is proposed in terms of a set of bilinear matrix inequalities. Moreover, in the so-called two-step procedure, results of the first step are allowed to select in order to reduce the conservatism of previous approaches. In comparison with the existing literature, the proposed approach not only provides more relaxed H _∞-control conditions but also ensures better H _∞-control performance. Finally, the validity and applicability of the proposed approach are successfully demonstrated through two numerical examples.     

Performance and robustness preservation in MIMO systems when applying SPR Substitutions†

We are concerned in this article with controlled linear time-invariant multi input multi output systems in continuous-time. We study here the preservation under strict positive real substitutions (of zero relative degree) of coprime factorisations as well as stabilising properties in parametrised controllers. We also tackle here both the preservation of well-posedness (of the tracking feedback control scheme) and the preservation of H _∞-optimality properties in some classes of closed-loop systems affected by multiplicative unstructured uncertainty, including the so-called suboptimal H _∞-control problem by stable controllers (i.e. preservation of both strong stabilisation and H _∞-boundedness).     

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.     

Multiplicative Stochastic Systems with Multiple External Disturbances

With the methods of H₂/H_∞-control theory, in the presence of noise we solve the optimization problem for a multiplicative stochastic system with several external disturbances (the multiperturbation problem) and vector Wiener processes with arbitrary intensity matrices. We obtain matrix differential equations of Riccati type, reducing the original optimization problem to solving these equations.     

On H∞-control of discrete-time nonlinear systems

Following recent works on continuous-time nonlinear H_∞-control, where connections with game theory and passivity conditions have been set, the present paper studies the corresponding problem for discrete-time systems. The paper describes sufficient conditions for the existence and the construction of a feedback law which imposes a prescribed level of disturbance attenuation with internal stability. Both cases of state feedback and measurement feedback are considered.     

H∞ estimation for uncertain systems

This paper deals with the problem of H_∞ estimation for linear systems with a certain type of time-varying norm-bounded parameter uncertainty in both the state and output matrices. We address the problem of designing an asymptotically stable estimator that guarantees a prescribed level of H_∞ noise attenuation for all admissible parameter uncertainties. Both an interpolation theory approach and a Riccati equation approach are proposed to solve the estimation problem, with each method having its own advantages. The first approach seems more numerically attractive whilst the second one provides a simple structure for the estimator with its solution given in terms of two algebraic Riccati equations and a parameterization of a class of suitable H_∞ estimators. The Riccati equation approach also pinpoints the ‘worst-case’ uncertainty.     

Finite horizon H∞ with parameter variations

In this paper we consider the finite horizon H_∞ performance robustness problem with parameter variations. Assuming a nominal model, an iterative procedure is given to synthesize a suboptimal H_∞ controller, which yields the required performance even under parameter variations. As a byproduct, an expression for the variation of performance due to parameter variations is given for this specific controller. An example which illustrates the methodology is worked out under parameter uncertainties.     

State feedback H ∞ control of commensurate fractional-order systems

This article focuses on the state feedback H _∞ control problem for commensurate fractional-order systems with a prescribed H _∞ performance. For linear time-invariant fractional-order systems, a sufficient condition to guarantee stability with H _∞ performance is firstly presented. Then, by introducing a new flexible real matrix variable, the feedback gain is decoupled with complex matrix variables and further parametrised by the new flexible matrix. Moreover, iterative linear matrix inequality algorithms with initial optimisation are developed to solve the state feedback H _∞ suboptimal control problem for fractional-order systems. Finally, illustrative examples are given to show the effectiveness of the proposed approaches.     

Product of nonnegative operators and infinite-dimensional H∞ Riccati equations

In this paper we collect some useful properties of the product of nonnegative operators in a Hilbert space. We then apply them to the standard H_∞-control problem for infinite-dimensional time-varying systems and give necessary and sufficient conditions for the existence of a suboptimal controller by three conditions involving two independent Riccati equations with a coupling inequality.     

Explicit constructions of global stabilization and nonlinear H∞ control laws for a class of nonminimum phase nonlinear multivariable systems

This paper investigates a global stabilization problem and a nonlinear H_∞ control problem for a class of nonminimum phase nonlinear multivariable systems. To avoid the complicated recursive design procedure, an asymptotic time‐scale and eigenstructure assignment method is adopted to construct the control laws for the stabilization problem and the nonlinear H_∞ control problem. A sufficient solvability condition is established onthe unstable zero dynamics of the system for global stabilization problem and nonlinear H_∞ control problem, respectively. Moreover, based on the sufficient solvability condition, an upper bound of the achievable L₂‐gain is estimated for the nonlinear H_∞ control problem. Copyright © 2007 John Wiley & Sons, Ltd.     

An optimization algorithm for checking feasibility of robust H ∞-control problem for linear time-varying uncertain systems

An algorithm for checking feasibility of the robust H _∞-control problem for systems with time-varying norm bounded uncertainty is suggested. This algorithm is an iterative procedure on each step of which an optimization problem for a linear function under convex constraints determined by LMIs is solved. The effectiveness of the proposed algorithm is demonstrated on the numerical example of a parametrically disturbed pendulum.     

Nonstationary H∞ dynamic output feedback control for discrete‐time Markov jump linear systems with actuator and sensor saturations

This paper is concerned with the nonstationary dynamic output feedback control problem for a class of discrete‐time Markov jump linear systems (MJLSs) under simultaneous consideration of actuator and sensor saturations. The so‐called nonstationary controllers dominated by two other different piecewise‐stationary Markov chains are introduced, making the designed controllers not only mode‐dependent but also dependent on other variations associated with the mode transitions in the original MJLSs. The sufficient conditions formulated in terms of bilinear matrix inequalities for the H_∞ control problem are established such that the resulting closed‐loop system is stochastically stable and achieves a prescribed H_∞ noise attenuation level. A suboptimal algorithm with line search is exploited to solve for the associated controller gains. Effectiveness of the developed theoretical results is verified via a numerical example. Copyright © 2015 John Wiley & Sons, Ltd.     

Distributed consensus observer-based H∞ control for linear systems with sensor and actuator networks

This paper proposed a distributed consensus observer (DCO) based H_∞ control method for a class of linear time-invariant (LTI) continuous systems with a sensor and actuator network (SAN). The communication topology of the SAN under consideration is represented by a directed graph, in which the sensor nodes are not able to acquire all the control inputs applied to the target system from the actuator nodes. To overcome this difficulty, a set of novel DCOs embedded in the sensor nodes and a set of DCO-based controllers embedded in the actuator nodes are initially constructed to estimate and control the state of the target system in a fully distributed way, respectively. The constructed DCOs take full advantage of their consensus property and replace the unavailable control inputs with the approximate ones computed on the basis of the state estimates of the underlying sensor node and its neighboring sensor nodes. Subsequently, a design method of DCO-based H_∞ control is proposed in terms of bilinear matrix inequality (BMI) to ensure that the closed-loop system is exponentially stable while satisfying a prescribed overall H_∞ performance of disturbance attenuation. Moreover, in order to make attenuation level as small as possible, a suboptimal H_∞ control design problem is formulated as a BMI optimization problem, and a modified path-following method is provided for solving this problem by using the existing linear matrix inequality (LMI) optimization techniques. Finally, simulation results demonstrate the effectiveness of the proposed method.     

Polynomial solutions to H∞ problems

The paper presents a polynomial solution to the standard H_∞-optimal control problem. Based on two polynomial J-spectral factorization problems, a parameterization of all suboptimal compensators is obtained. A bound on the McMillan degree of suboptimal compensators is derived and an algorithm is formulated that may be used to solve polynomial J-spectral factorization problems.     

The H∞ control problem using static output feedback

In this paper we shall consider the H_∞ control problem using static output feedback. The approach uses some recent results from linear algebra. The main result shows that the H_∞ control problem is solvable by a static output feedback controller if and only if there exists a positive definite matrix satisfying two certain quadratic matrix inequalities. A parametrization of all static output feedback H_∞ controllers is given.     

Singular nonlinear H∞ optimal control problem

The theory of nonlinear H_∞ of optimal control for affine nonlinear systems is extended to the more general context of singular H_∞ optimal control of nonlinear systems using ideas from the linear H_∞ theory. Our approach yields under certain assumptions a necessary and sufficient condition for solvability of the state feedback singular H_∞ control problem. The resulting state feedback is then used to construct a dynamic compensator solving the nonlinear output feedback H_∞ control problem by applying the certainty equivalence principle.     

LMI-based H ∞-optimal control with transients

In this article, the worst-case norm of the regulated output over all exogenous signals and initial states as a performance measure of the system is characterised in terms of linear matrix inequalities (LMIs). Optimal time-invariant state- and output-feedback controllers are synthesised as minimising this performance measure. The essential role in this synthesis plays a weighting matrix reflecting the relative importance of the uncertainty in the initial state contrary to the uncertainty in the exogenous signal. H _∞-optimal control with transients is shown to be actually a trade-off between H _∞-control, being optimal under unknown exogenous disturbances and zero initial state, and γ-control, being optimal under zero exogenous signal and unknown initial conditions, if and only if the weighting matrix satisfies a fundamental inequality. If this inequality is met, the performance measure is achieved and the explicit formulae for the worst-case disturbance and initial state are provided. If this inequality fails, the performance measure coincides with the H _∞-norm and the trade-off gets broken.     

Model reduction of discrete time systems through linear matrix inequalities*

In this paper, model reduction of discrete time linear systems is revisited. The main objective is to define a convex programming problem expressed in terms of linear matrix inequalities which provides a good suboptimal solution to the model reduction problem. The quality of the proposed suboptimal solution is assessed through comparisons, performed via simulation, with the already classical balanced truncation method. The results lead to the conclusion that the proposed method performs equivalently when the H ₂ norm of the reduction error is considered and significantly better when the H _∞ norm is considered instead.     

H∞ generalized minimum sensitivity control design

This paper presents a two-stage approach to the H_∞ generalized sensitivity minimization problem (H_∞-GSM) for SISO, continuous-time plants. Besides some possible advantages in the control law evaluation with respect to alternative polynomial methods, the proposed approach provides a direct link with the solution of an underlying generalized minimum variance (H₂-GMV) problem and allows one to identify the class of the joint H₂/H_∞-GMV (equalizing H₂-GMV) optimal controllers.