Guaranteed H∞ robustness bounds for Wiener filtering and prediction

The paper deals with special classes of H_∞ estimation problems, where the signal to be estimated coincides with the uncorrupted measured output. Explicit bounds on the difference between nominal and actual H_∞ performance are obtained by means of elementary algebraic manipulations. These bounds are new in continuous‐time filtering and discrete‐time one‐step ahead prediction. As for discrete‐time filtering, the paper provides new proofs that are alternative to existing derivations based on the Krein spaces formalism. In particular, some remarkable H_∞ robustness properties of Kalman filters and predictors are highlighted. The usefulness of these results for improving the estimator design under a mixed H₂/H_∞ viewpoint is also discussed. The dualization of the analysis allows one to evaluate guaranteed H_∞ robustness bounds for state‐feedback regulators of systems affected by actuator disturbances. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

H∞ control for a class of cascade switched nonlinear systems

In this paper, we investigate the H_∞ control problem for a class of cascade switched nonlinear systems consisting of two nonlinear parts which are also switched systems using the multiple Lyapunov function method. Firstly, we design the state feedback controller and the switching law, which guarantees that the corresponding closed‐loop system is globally asymptotically stable and has a prescribed H_∞ performance level. This method is suitable for a case where none of the switched subsystems is asymptotically stable. Then, as an application, we study the hybrid H_∞ control problem for a class of nonlinear cascade systems. Finally, an example is given to illustrate the feasibility of our results. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

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.     

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.     

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.     

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.     

Plant zero structure and further order reduction of a singular H∞ controller

A new class of reduced‐order controllers is obtained for the H_∞ problem. The reduced‐order controller does not compromise the performance attained by the full‐order controller. Algorithms for deriving reduced‐order H_∞ controllers are presented in both continuous and discrete time. The reduction in order is related to unstable transmission zeros of the subsystem from disturbance inputs to measurement outputs. In the case where the subsystem has no infinite zeros, the resulting order of the H_∞ controller is lower than that of the existing reduced‐order H_∞ controller designs which are based on reduced‐order observer design. Furthermore, the mechanism of the controller order reduction is analysed on the basis of the two‐Riccati equation approach. The structure of the reduced‐order H_∞ controller is investigated. Copyright © 2002 John Wiley & Sons, Ltd.     

H∞ control with weighting functions having purely imaginary poles—implementing internal models to H∞ controllers

The present paper considers a variant of the standard H_∞ control problem which allows one to use weighting functions having jω poles. Using the solution, one can design H_∞ controllers having prescribed jω poles such as internal models. To solve the problem, the authors propose a new requirement of closed-loop stability, called essential stability, and alternative standing assumptions imposed on the generalized plant. To write the results in the form of the so-called 2-ARE solution, the authors introduce the notion of quasi-stabilizing solutions to the algebraic Riccati equations arising in H_∞ control. The solution involves the same Riccati equations and parametrization of the controllers given by Glover and Doyle except the stability requirement on the solutions to the Riccati equations.     

Structured H∞‐control of infinite‐dimensional systems

We develop a novel frequency‐based H_∞‐control method for a large class of infinite‐dimensional linear time‐invariant systems in transfer function form. A major benefit of our approach is that reduction or identification techniques are not needed, which avoids typical distortions. Our method allows to exploit both state‐space or transfer function models and input/output frequency response data when only such are available. We aim for the design of practically useful H_∞‐controllers of any convenient structure and size. We use a nonsmooth trust‐region bundle method to compute arbitrarily structured locally optimal H_∞‐controllers for a frequency‐sampled approximation of the underlying infinite‐dimensional H_∞‐problem in such a way that (i) exponential stability in closed loop is guaranteed and that (ii) the optimal H_∞‐value of the approximation differs from the true infinite‐dimensional value only by a prior user‐specified tolerance. We demonstrate the versatility and practicality of our method on a variety of infinite‐dimensional H_∞‐synthesis problems, including distributed and boundary control of partial differential equations, control of dead‐time and delay systems, and using a rich testing set.     

Robust H∞ control of uncertain singular systems based on equivalent‐input‐disturbance approach

In this paper, a robust H_∞ control problem is considered for an uncertain singular system. An active disturbance rejection method called equivalent input disturbance (EID) is used to reduce the influence of exogenous disturbances and uncertainties on the system. At the first, there exists an EID, which can produces the same effect on the system as disturbances and uncertainties do in the control channel according to the EID concept. Then, an EID estimator is constructed to estimate the influence of EID on the system. Finally, based on Lyapunov stability theory, a static output feedback‐based robust H_∞ controller combined with EID estimate is designed, guaranteeing that closed‐loop system is admissible (regular, impulse‐free, and stable) with a prescribed H_∞ performance level. Compared with traditional H_∞ control method, H_∞ control based on EID method improve the control performance of the system. A numerical example demonstrates the validity of the method.     

State feedback H∞ control for quantized discrete‐time systems

This paper is concerned with the quantized state feedback H_∞ control problem for discrete‐time linear time‐invariant systems. The quantizer considered here is dynamic and composed of an adjustable “zoom” parameter and a static quantizer. Static quantizer ranges are with practical significance and fully considered here. A quantized H_∞ controller design strategy is proposed with taking quantizer errors into account, where an iterative linear matrix inequality (LMI) based optimization algorithm is developed to minimize static quantizer ranges with meeting H_∞ performance requirement for quantized closed‐loop systems. An example is presented to illustrate the effectiveness of the proposed method. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Delay‐dependent adaptive reliable H∞ control of linear time‐varying delay systems

This paper considers the problem of delay‐dependent adaptive reliable H_∞ controller design against actuator faults for linear time‐varying delay systems. Based on the online estimation of eventual faults, the parameters of adaptive reliable H_∞ controller are updating automatically to compensate the fault effects on the system. A new delay‐dependent reliable H_∞ controller is established using a linear matrix inequality technique and an adaptive method, which guarantees the stability and adaptive H_∞ performance of closed‐loop systems in normal and faulty cases. A numerical example and its simulation results illustrate the effectiveness of the proposed method. Copyright © 2008 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.     

Further results on H∞ filtering for discrete‐time systems with state delay

This article studies the problem of H_∞ filtering for linear discrete‐time systems with state delay. Via delay partitioning idea, two new H_∞ filter design methods are proposed with much less conservatism than most existing results. The improvement lies in constructing two new Lyapunov–Krasovskii functionals by partitioning the known constant lower bound of delay into several segments equally. Using free‐weighting matrix and Jensen inequality methods, two new delay‐dependent bound real lemmas (BRLs) are obtained, which depend on both the delay and the partitioning number. Based on the obtained BRLs, new H_∞ filter design approaches are proposed in terms of linear matrix inequalities. The results are immediately extended to multiple time delay case and polytopic uncertain case, respectively. Three numerical examples are presented to illustrate the effectiveness and advantage of the proposed approaches. Copyright © 2010 John Wiley & Sons, Ltd.     

Three‐dimensional nonlinear H∞ guidance law

This paper proposes a novel three‐dimensional missile guidance law design based on nonlinear H_∞ control. The complete nonlinear kinematics of pursuit–evasion motion is considered in the three‐dimensional spherical co‐ordinates system; neither linearization nor small angle assumption is made here. The nonlinear H_∞ guidance law is expressed in a simple form by solving the associated Hamilton–Jacobi partial differential inequality analytically. Unlike adaptive guidance laws, the implement of the proposed robust H_∞ guidance law does not require the information of target acceleration, while ensuring acceptable interceptive performance for arbitrary target with finite acceleration. The resulting pursuit–evasion trajectories for both the H_∞‐guided missile and the worst‐case target are determined in closed form, and the performance robustness against variations in target acceleration, in engagement condition, and in control loop gain, is verified by numerical simulations. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

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.     

Quadratic stabilizability and H∞ control of linear discrete‐time stochastic uncertain systems

This paper considers quadratic stabilizability and H_∞ feedback control for stochastic discrete‐time uncertain systems with state‐ and control‐dependent noise. Specifically, the uncertain parameters considered are norm‐bounded and external disturbance is an l²‐square summable stochastic process. Firstly, both quadratic stability and quadratic stabilization criteria are presented in the form of linear matrix inequalities (LMIs). Then we design the robust H_∞ state and output feedback H_∞ controllers such that the system with admissible uncertainties is not only quadratically internally stable but also robust H_∞ controllable. Sufficient conditions for the existence of the desired robust H_∞ controllers are obtained via LMIs. Finally, some examples are supplied to illustrate the effectiveness of our results.     

Finite‐Time H∞ Filtering for Nonlinear Continuous‐Time Singular Semi‐Markov Jump Systems

The stochastic finite‐time H_∞ filtering issue for a class of nonlinear continuous‐time singular semi‐Markov jump systems is discussed in this paper. Firstly, sufficient conditions on singular stochastic H_∞ finite‐time boundedness for the filtering error system are established. The existence of a unique solution for the corresponding system is also ensured. Secondly, based on the bounds of the time‐varying transition rate, without imposing constraints on slack variables, a novel approach to finite‐time H_∞ filter design is proposed in the forms of strict LMIs, which guarantees the filtering error system is singular stochastic H_∞ finite‐time bounded and of a unique solution. Compared with the existing ones, the presented results reveal less conservativeness. Finally, one numerical example is exploited to testify the advantage of the proposed design technique.