Uniformly optimal control of linear time-varying systems

An optimal control problem is formulated in the context of linear, discrete-time, time-varying systems. The cost is the supremum, over all exogenous inputs in a weighted ball, of the sum of the weighted energies of the plant's input and output. The controller is required to be causal and to achieve internal stability. Existence of an optical controller is proved and a formula for the minimum cost is derived.     

Simultaneous H control of a finite collection of linear plants with a single nonlinear digital controller

This paper considers the problem of simultaneous H^∞ control of a finite collection of linear time-invariant systems via a nonlinear digital output feedback controller. The main result is given in terms of the existence of suitable solutions to Riccati algebraic equations and a dynamic programming equation. Our main result shows that if the simultaneous H^∞ control problem for k linear time-invariant plants of orders n₁,n₂,…,n_k can be solved, then this problem can be solved via a nonlinear time-invariant controller of order nn₁+n₂++n_k.     

Finite-dimensional optimal controllers for nonlinear plants

Optimal risk sensitive feedback controllers are now available for very general stochastic nonlinear plants and performance indices. They consist of nonlinear static feedback of so called information states from an information state filter. In general, these filters are linear, but infinite dimensional, and the information state feedback gains are derived from (doubly) infinite dimensional dynamic programming. The challenge is to achieve optimal finite dimensional controllers using finite dimensional calculations for practical implementation.This paper derives risk sensitive optimality results for finite-dimensional controllers. The controllers can be conveniently derived for ‘linearized’ (approximate) models (applied to nonlinear stochastic systems). Performance indices for which the controllers are optimal for the nonlinear plants are revealed. That is, inverse risk-sensitive optimal control results for nonlinear stochastic systems with finite dimensional linear controllers are generated. It is instructive to see from these results that as the nonlinear plants approach linearity, the risk sensitive finite dimensional controllers designed using linearized plant models and risk sensitive indices with quadratic cost kernels, are optimal for a risk sensitive cost index which approaches one with a quadratic cost kernel. Also even far from plant linearity, as the linearized model noise variance becomes suitably large, the index optimized is dominated by terms which can have an interesting and practical interpretation.Limiting versions of the results as the noise variances approach zero apply in a purely deterministic nonlinear H_∞ setting. Risk neutral and continuous-time results are summarized.More general indices than risk sensitive indices are introduced with the view to giving useful inverse optimal control results in non-Gaussian noise environments.     

Nonlinear H∞-control and estimation of optimal H∞-gain

The structure of nonlinear H_∞-controller and the estimation of optimal H_∞-gain are investigated in this paper. The essential problem boils down to the existence of semipositive solution to the Hamilton-Jacobi-Isaacs inequality. Further, the solvability of this first-order fully nonlinear differential inequality is discussed. We look for one kind of special semipositive radial solution to the Hamilton-Jacobi-Isaacs inequality. An explicit estimation of optimal H_∞-gain and explicit formulas of semipositive radial solutions to the Hamilton-Jacobi-Isaacs inequality are obtained. The results are quite simple and intuitive. They even shed a new insight on the linear H_∞-theory.     

Simultaneous H2/H∞ optimal control The state feedback case

A simultaneous H₂/H_∞ control problem is considered. This problem seeks to minimize the H₂ norm of a closed-loop transfer matrix while simultaneously satisfying a prescribed H_∞ norm bound on some other closed-loop transfer matrix by utilizing dynamic state feedback controllers. Such a problem was formulated earlier by Rotea and Khargonekar (Automatica, 27, 307–316, 1991) who considered only so called regular problems. Here, for a class of singular problems, necessary and sufficient conditions are established so that the posed simultaneous H₂/H_∞ problem is solvable by using state feedback controllers. The class of singular problems considered have a left invertible transfer function matrix from the control input to the controlled output which is used for the H₂ norm performance measure. This class of problems subsumes the class of regular problems.     

H∞ control via output feedback for linear systems with time-varying delays in state and control input

In this paper, an observer-based H_∞ controller for systems with time-varying delays in state and control input is proposed. Using the solution of proposed modified algebraic Riccati equation, we can construct a controller which depends on the maximum value of the time derivative of time-varying delay and guarantees a prescribed H_∞ norm bound of the closed-loop transfer matrix from the disturbance to the controlled output. A given example illustrates the availability of the proposed design method.     

Robust memoryless H∞ controller design for linear time-delay systems with norm-bounded time-varying uncertainty

We consider the robust H_∞ control problem for linear time-delay systems with norm-bounded time-varying uncertainty. Based on the Riccati-equation approach, we present a method for designing a robust memoryless linear time-invariant state feedback control law, which stabilizes the system and reduces the effect of the disturbance input on the controlled output to a prescribed level for all admissible uncertainties.     

On the robust stabilizability of uncertain linear time-invariant plants using nonlinear time-varying controllers

We consider general families of linear time-invariant plants described by a nominal plant model with additive or multiplicative unstructured modeling uncertainty. We show that such families of plants are robustly (uniformly) stabilizable by nonlinear time-varying controllers, if and only if they are robustly stabilizable by linear time-invariant ones. Some auxiliary results on Hankel and L_∞-norm approximation are also obtained.     

Robust dissipative control for linear systems with dissipative uncertainty and nonlinear perturbation

This paper focuses on the problem of dissipative control for linear systems which are subjected to dissipative uncertainty and matched nonlinear perturbation. Specifically, quadratic dissipative uncertainty is considered, which contains norm-bounded uncertainty, positive real uncertainty and uncertainty satisfying integral quadratic constraints (IQCs) as special cases. We develop a linear matrix inequality (LMI) approach for designing a robust nonlinear state feedback controller such that the closed-loop system is quadratic dissipative for all admissible uncertainties. Furthermore, under some condition on the dissipative uncertainty, we show that the controller also guarantees the asymptotic stability of the closed-loop system. As special cases, robust H_∞ control and robust passive control problems for systems with nonlinear perturbation and norm-bounded uncertainty (respectively, generalized positive real uncertainty) are solved using the LMI approach.     

Direct computation of infimum in discrete-time H∞-optimization using measurement feedback

A direct and non-iterative method for the computation of the infimum for a class of discrete-time H_∞ optimal control problem is considered in this paper. The problem formulation is fairly general and does not place any restrictions on any direct feedthrough terms of the given systems. The method is applicable to systems where (i) the transfer function from the disturbance input to the measurement output is free of unit circle invariant zeros and left invertible, and (ii) the transfer function from the control input to the controlled output of the given system is free of the unit circle invariant zeros and right invertible.     

On the order of nonlinear time-invariant stabilizing controllers

Here we consider the problem of stabilizing a finite-dimensional, stabilizable and detectable, linear time-invariant (LTI) plant in the sense that the plant state goes to zero while the controller state converges. With r the dimension of the plant output, we show that there exists a nonlinear time-invariant controller of order r+3 which stabilizes all such systems.     

Finite-time control with H-infinity constraints of linear time-invariant and time-varying systems

This paper presents finite-time control methods with H-infinity constraints for linear time-invariant (LTI) and time-varying (LTV) systems. The basic idea of the proposed approaches is to construct controllers for the LTI and LTV in such a way that a constant quadratic Lyapunov function and a time-varying quadratic Lyapunov function can be used to establish the finite-time stability and the H-infinity performance of the resulting closed-loop systems. It is shown that the control laws can be obtained by solving a set of linear matrix inequalities (LMIs) and Differential Riccati Inequalities (DRIs) that are numerically feasible with commercially available software. Finally, the results are illustrated by application to the design of guidance law for a class of terminal guidance system.     

Nonlinear controllers for solar thermal plants: A comparative study

Solar plants have nonlinear dynamics which must be taken into account when a control system is applied to them. The main purpose of the control systems is to maintain the outlet temperature in a desired reference value and, at the same time, attenuate the undesirable transients caused by the disturbances. Linear controllers, like PID ones, are not able to obtain good performance over the whole operation range of these kind of plants. To overcome these limitations two nonlinear controllers, a nonlinear model-based predictive controller and a distributed sliding mode controller, are applied to a solar plant in this work. The performance of these controllers is tested through experimental and simulation results, which show the tracking and disturbance rejection capabilities of the proposed controllers.     

A partial parameterization of nonlinear output feedback controllers for saturated linear systems

This paper addresses the stabilization problem of linear systems subject to input saturation. The major purpose is to introduce nonlinearity into control laws so as to expand the design freedom for performance enhancement. To this end, a new approach is developed which involves a partial differential matrix inequality (PDMI). A class of stabilizing feedback laws is explicitly obtained by solving this PDMI analytically and the feedback laws are parameterized by a nonlinear function. Further, it is revealed that any linear observer can be used to realize the output feedback stabilization. Numerical examples, including the seek control of a hard disk drive, show that the introduced nonlinearity does contribute to the improvement of system performance. The application to integral control is also discussed.     

Optimal H∞ model reduction via linear matrix inequalities: continuous- and discrete-time cases

Necessary and sufficient conditions are derived for the existence of a solution to the continuous-time and discrete-time H_∞ model reduction problems. These conditions are expressed in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint set. In addition, an explicit parametrization of all reduced-order models that correspond to a feasible solution is provided in terms of a contractive matrix. These results follow from the recent solution of the H_∞ control design problem using LMIs. Particularly simple conditions and a simple parametrization of all solutions are obtained for the zeroth-order H_∞ approximation problem, and the convexity of this problem is demonstrated. Computational issues are discussed and an iterative procedure is proposed to solve the H_∞ model reduction problem using alternating projections, although global convergence of the algorithm is not guaranteed.     

Reduced order LQG controllers for linear time varying plants

The problem of reduced order LQG optimization is addressed in a finite horizon, linear time-varying (LTV) system setting. First-order necessary conditions for local optimality in the parameter space is provided in terms of four coupled matrix differential equations. This result provides a transparent generalization of the optimal projection equations of Hyland and Bernstein for the optimal, steady-state compensation of linear time-invariant (LTI) plants.     

Linear parameter-varying versus linear time-invariant control design for a pressurized water reactor

The applicability of employing a parameter-dependent control to a nuclear pressurized water reactor is investigated and is compared to that of using an ℋ︁_∞ control. A linear time-invariant controller cannot maintain performance over the entire operating range. The parameter-dependent synthesis technique produces a controller which achieves specified performance against the worst-case time variation of a measurable parameter which enters the plant in a linear fractional manner. The plant can thus have widely varying dynamics over the operating range. The controllers designed perform well over the entire operating range. Copyright © 1999 John Wiley & Sons, Ltd.     

Nonlinear optimal control as quantum mechanical eigenvalue problems

A novel approach for approximating the nonlinear optimal feedback control of a system with a terminal cost is proposed. To lessen the difficulty due to nonlinearity, we try to treat the system in a framework of linear theories. For this, we assume a quantum mechanical linear wave associated with the system. Since the control system is constrained by state equations, we handle the system according to quantum mechanics of constrained dynamics. A Hamiltonian is represented as a linear operator acting on a function that describes behavior of waves. Subsequently, nonlinear feedback is calculated without any time integration in the backward direction. Using eigenvalues and eigenfunctions of the linear Hamiltonian operator, an optimal feedback law is given as a combination of analytic functions of time and state variables. We take as an example a system described by two scalar variables for state and control input. Simulation studies on the system by the eigenvalue analysis show that the proposed method reduces calculation time to nearly a tenth that of a numerical calculation of a Hamilton-Jacobi equation by a finite difference method.     

Non-fragile output feedback H∞ vehicle suspension control using genetic algorithm

This paper presents an approach to design static output feedback and non-fragile static output feedback H_∞ controllers for active vehicle suspensions by using linear matrix inequalities and genetic algorithms. A quarter-car model with active suspension system is considered in this paper. By suitably formulating the minimization problem of the sprung mass acceleration, suspension deflection and tyre deflection, a static output feedback H_∞ controller and a non-fragile static output feedback H_∞ controller are obtained. The controller gain is naturally constrained in the design process. The approach is validated by numerical simulation which shows that the designed static output feedback H_∞ controller can achieve good active suspension performance in spite of its simplicity, and the non-fragile static output feedback H_∞ controller has significantly improved the non-fragility characteristics over controller gain variations.