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.     

Singular H∞ control for nonlinear systems

This paper presents a solution to the singular H_∞ control problem via state feedback for a class of nonlinear systems. It is shown that the problem of almost disturbance decoupling with stability plays a fundamental role in the solution of the considered problem. We also point out when the singular problem can be reduced to a regular one or solved via standard H_∞ technique. We must stress that the solution of the singular problem is obtained without making any approximation of it by means of regular problems. © 1997 John Wiley & Sons, Ltd.     

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.     

Approximate solutions to a weighted mixed-sensitivity H∞-control design for irrational transfer matrices

Approximate solutions to a weighted mixed-sensitivity H^∞-control problem for an irrational transfer matrix are obtained by solving the same problem for a reduced-order (rational) transfer matrix. Upper and lower bounds are given in terms of the solution to the reduced-order problem and the approximation error. A delay example and a pde example are discussed.     

H∞-type control for discrete-time stochastic systems

In this paper we consider discrete-time, linear stochastic systems with random state and input matrices which are subjected to stochastic disturbances and controlled by dynamic output feedback. The aim is to develop an H^∞-type theory for such systems. For this class of systems a stochastic bounded real lemma is derived which provides the basis for a linear matrix inequality (LMI) approach similar to, but more general than the one presented in Reference 1 for stochastic differential systems. Necessary and sufficient conditions are derived for the existence of a stabilizing controller which reduces the norm of the closed-loop perturbation operator to a level below a given threshold γ. These conditions take the form of coupled nonlinear matrix inequalities. In the absence of the stochastic terms they get reduced to the linear matrix inequalities of deterministic H^∞-theory for discrete time systems. Copyright © 1999 John Wiley & Sons, Ltd.     

H∞ balancing for nonlinear systems

In previously obtained balancing methods for nonlinear systems a past and a future energy function are used to bring the nonlinear system in balanced form. By considering a different pair of past and future energy functions that are related to the H_∞ control problem for nonlinear systems we define H_∞ balancing. Furthermore, we investigate the monotonicity of the Hamilton-Jacobi-Bellman equations that appear in this balancing method. The method is used as a tool for model reduction.     

Robust H∞ nonlinear control via measurement feedback

This paper deals with the problem of finite-time-horizon robust H_∞ control via measurement feedback, for affine nonlinear systems with nonlinear time-varying parameter uncertainty. The problem addressed is the design of a control law, which processes the measured output and guarantees a prescribed level of closed-loop disturbance attenuation. Conditions for the existence of such a controller are obtained by solving an auxiliary control problem for a related system which is obtained from the original one by converting the parameter uncertainty into exogenous bounded energy signals. This approach allows us to apply the recently developed H_∞ nonlinear control techniques to solve the robust control problem. The problem is investigated in both the continuous- and discrete-time cases. The results are demonstrated by a simple example. © 1997 by John Wiley & Sons, Ltd.     

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.     

H∞ estimation for discrete-time linear uncertain systems

This paper is concerned with the problem of H_∞ estimation for linear discrete-time systems with time-varying norm-bounded parameter uncertainty in both the state and output matrices. We design an estimator such that the estimation error dynamics is quadratically stable and the induced operator norm of the mapping from noise to estimation error is kept within a prescribed bound for all admissible uncertainties. A Riccati equation approach is proposed to solve the estimation problem and it is shown that the solution is related to two algebraic Riccati equations.     

H∞ nonlinear filtering

This paper investigates the problem of H_∞ estimation of nonlinear processes. An estimator, which may be nonlinear, is looked for so that a given bound on the ratio between the energy of the estimation error and the energy of the oxogeneous inputs to the estimated process is achieved. Conditions for the existence of such an estimator and formulas for its derivation are obtained using both the game theory approach and the theory of dissipative systems. The results of the paper extend the recent results on H_∞ nonlinear control. They are demonstrated by a simple example of a linear system with a nonlinear measurement rule and compared with corresponding results that are obtained by the extended Kalman filter.     

H∞ Control with Transients for Singular Systems

In this paper, the problem of a generalized type of H_∞ control is investigated for continuous‐time singular systems, which treats a mixed attenuation of exogenous inputs and initial conditions. First, a performance measure that is essentially the worst‐case norm of the regulated outputs over all exogenous inputs and initial conditions is introduced. A necessary and sufficient condition is obtained to ensure the singular system to be admissible and the performance measure to be less than a prescribed scalar. Based on the criterion a sufficient condition for the existence of a state‐feedback controller is established in terms of linear matrix inequalities. Moreover, the relationship between the performance measure and the standard H_∞ norm of the system is provided. Two numerical examples are given to demonstrate the properties of the obtained results.     

Fixed-order H∞ compensator design

A method is presented for the construction of fixed-order compensators to provide H_∞ norm constraint for linear control systems with exogenous disturbances. The method is based on the celebrated bounded-real lemma that predicates the H_∞ norm constraint via a Riccati inequality. The synthesis of fixed-order controllers whose dimensions are less than the order of a given plant, is demonstrated by a set of sufficient conditions along with a numerical algorithm.     

Reduced order H∞ filter design for discrete time-variant systems

This paper addresses the reduced-order H_∞ filter design for linear time varying discrete-time systems in which all measurements are not noise-free. The proposed filter has a Luenberger observer structure and its order is n - p, where n and p are the order of signal systems and the number of measurements, respectively. Several bounded real lemmas are developed and they are the only tools used to establish the reduced-order filter design in this paper. It is shown that the full order a priori and a posteriori filters can be obtained by properly choosing the filter order and parameters based on the reduced-order filter design results. The resultant filters are the same as the known full-order optimal H_∞ filters for time-invariant systems. © 1997 John Wiley & Sons, Ltd.     

The discrete-time H∞ control problem with measurement feedback

This paper is concerned with the discrete-time H_∞ control problem with measurement feedback. We extend previous results by having weaker assumptions on the system parameters. We also show explicitly the structure of H_∞ controllers. Finally, we show that it is in certain cases possible, without loss of performance, to reduce the dynamical order of the controllers.     

H∞-optimal controller discretization

A redesign method for discretizing a continuous-time controller is proposed. The resulting hybrid control system, for example with continuous-time plant and discrete-time controller, is stable, and performance including the system's inter-sampling behaviour can be optimized by approximating some chosen reference transfer function of the continuous-time control system. In order to obtain a tractable problem, the continuous-time part of the hybrid system and the reference transfer function are approximated by a discrete-time system with arbitrary fast sampling. After lifting the resulting periodic system, the approximation problem can be formulated as a standard H_∞-problem which is solved using standard software for H_∞-controller design.     

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.     

H∞ loop transfer recovery synthesis of discrete-time systems

This paper addresses an H_∞ disturbance attenuation problem of discrete-time systems, and presents a new discrete-time H_∞ loop transfer recovery (LTR) synthesis method. The method involves two design steps. First, an H_∞ state-feedback controller is designed to satisfy the design specifications, such as stability robustness and disturbance attenuation. Subsequently, the properties of the H_∞ state-feedback are reobtained by designing full/minimal order observers. A numerical examples is included to illustrate the presented method. © 1998 John Wiley & Sons, Ltd.     

Robust H∞ Control Problem For General Nonlinear Systems With Uncertainty

In this paper, both state and output feedback robust H^∞ control problems for general nonlinear systems with norm‐bound uncertainty are considered. Sufficient conditions for the existence of robust output feedback H^∞ controller are provided. State space formulas for robust H^∞ output controller are provided.     

H∞ controller design for boilers

This paper examines the ability of the H_∞ design methodology to provide a solution to a gas- or oil-fired boiler control problem, and addresses the nontrivial application issues of the H_∞ design. The H_∞ methodology is applied to an experimentally verified heating-cogeneration boiler model which exhibits nonlinearities, instability, time delays, non-minimum phase behaviour, and coloured noise disturbances with sensor noise in the frequency range of the significant plant dynamics. The design shows that, to satisfy performance criteria, a high order controller is needed. The paper also demonstrates a trade-off between the reduction of controller order and the loss of controller performance.