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∞-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∞ 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.     

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∞ 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.     

H∞ control via measurement feedback for affine nonlinear systems

This paper summarizes some recent results on the problem of disturbance attenuation via measurement feedback, with internal stability, for an affine nonlinear system. The solution of the problem is shown to be related to the existence of solutions of a pair of Hamilton-Jacobi inequalities in n independent variables, associated with state feedback and, respectively, output-injection design.     

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.     

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.     

Engineering design of a multirate nonlinear controller for robot manipulators

A new multirate nonlinear controller for robot manipulators is described. The algorithm stems from the computed torque method, but has several modifications on its structure, resulting in a more robust controller. The main philosophy behind those improvements resides on a sound engineering and management of the system resources. Consequently, the use of multirate sampling, memory tables, and predictive action arises naturally from that methodology. The resulting controller is easy to tune using standard industrial engineering practice; moreover, by alleviating the computational burden its implementation is feasible with present day microprocessors.     

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.     

Practical multi-constraint H∞ controller synthesis from time-domain data

This paper presents a practical algorithm for MIMO controller design with multiple H^∞ norm constraints. The plant is described by its sampled-data impulse response matrix, which could be determined directly from measurements; the controller design parameters are the tap weights of an FIR discretization of the Q-(or Youla-) parameter. We approximate the H^∞ norm by the maximum transfer matrix 2-norm on an even frequency sampling over θ = [0, π]. The control design algorithm uses the Newton method to minimize two cost functions sequentially: the first determines multiple H^∞ feasibility, and the second minimizes a generalized entropy. As a design example, we control the acoustic radiation from a (mathematically modelled) submerged spherical shell. The plant-model impulse response matrix has McMillan degree 8 800. We specify and synthesize a controller that simultaneously achieves ten H^∞ constraints: one on the radiation reduction, one for stability robustness, and eight on actuator authority.     

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.     

Robustness studies for H∞ feedback control in a structural acoustic model with periodic excitation

Results from an extensive study on the robustness of an H^∞ compensator for a 2-D structural acoustic model are presented. The effects of frequency uncertainties in an exogenous signal are studied for both the case where the signal is contained in the controller formulation and the case where it is excluded. Delays are inserted in the input and/or output signals and their effect on the controller performance is recorded. A comparison between the standard LQG/Kalman filter and the H^∞/Min-Max compensator reveals no significant differences in the overall controller performance. Modifications in the controller structure are studied to see whether loss of information (the tracking variable) that must be calculated a priori would result in performance degradation. This study provides valuable insight into the computational and implementational issues that arise when dealing with the control of large and complex systems that are governed by partial differential equations.     

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.     

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.     

Robust controller design for a class of nonlinear robot manipulators with actuator dynamics

A robust controller is proposed to achieve accurate tracking for a class of nonlinear uncertain robot manipulators with actuator dynamics. A dynamic model is derived that incorporates the manipulator dynamics with the actuator dynamics. The parameter uncertainty in the model is quantified using the linear parameterization technique. The proposed switching controller guarantees the global asymptotic stability. To rectify the chattering caused by the switching action of the controller, the controller is modified into a smoothed form by putting a boundary layer around the sliding surface. It is shown that the smoothing controller guarantees the uniform ultimate boundedness of the actual trajectory. Simulation results show the feasibility and excellent tracking performance of the controller.     

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.     

Robust filtering for a class of discrete-time uncertain nonlinear systems: An H∞ approach

This paper deals with the H_∞ filtering problem for a class of discrete-time nonlinear systems with or without real time-varying parameter uncertainty and unknown initial state. For the case when there is no parametric uncertainty in the system, we are concerned with designing a nonlinear H_∞ filter such that the induced l₂ norm of the mapping from the noise signal to the estimation error is within a specified bound. It is shown that this problem can be solved via one Riccati equation. We also consider the design of nonlinear filters which guarantee a prescribed H_∞ performance in the presence of parametric uncertainties. In this situation, a solution is obtained in terms of two Riccati equations.     

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.