PID controller optimization for H∞ control by linear programming

The constraints on the PID gains, which are derived from the H_∞ norm performance index by discretization of the frequency, are convex or concave depending on frequencies. This problem is a non‐convex problem, and a new method of approximating these constraints as adequate linear inequalities is proposed. Then, the optimal solution can be efficiently and successfully searched for by applying linear programming iteratively. This method is compared with methods based on barrier function and linear matrix inequality. Copyright © 2000 John Wiley & Sons, Ltd.     

Parametric uncertainty bounds for guaranteed H∞-norm specifications

We derive bounds on the parameters of systems represented by a transfer functions such that a system satisfying these bounds will satisfy a bound on the H_∞-norm of the uncertainty in a given nominal system. We develop the bound and give examples for illustration. Copyright © 1999 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∞ 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.     

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.     

Optimal H∞ scaling for sensitivity optimization of detection filters

The paper deals with the sensitivity optimization of detection filters in linear time‐varying (LTV) systems which are subject to multiple simultaneous faults and disturbances. The robust fault detection filter design problem as a scaled H_∞ filtering problem is considered. The effect of two different input scaling approaches to the optimization process is investigated. The objective is to provide the smallest scaled L₂ gain of the unknown input of the system that is guaranteed to be less than a prespecified level, i.e., to produce a filter with optimal disturbance suppression capability in such a way that sufficient sensitivity to failure modes should still be maintained. It is shown how to obtain bounds on the scaled L₂ gain by transforming the standard H_∞ filtering problem into a convex feasibility problem, specifically, a structured, linear matrix inequality (LMI). Numerical examples demonstrating the effect of the scaled optimization with respect to conventional H_∞ filtering is presented. Copyright © 2002 John Wiley & Sons, Ltd.     

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∞-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.     

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.     

Network‐based robust H∞ control of continuous‐time systems with uncertainty

This paper studies the problem of robust H_∞ control for continuous‐time networked control systems (NCSs). A new type of Lyapunov functionals is exploited to derive sufficient conditions for guaranteeing the robust exponential stability and H_∞ performance of the considered system. It is shown that the new result is less conservative than the existing corresponding ones. Meanwhile, a method of eliminating redundant variables to reduce computational complexity is given, which is also applied to design state feedback H_∞ controllers, and the design condition is given in terms of solutions to a set of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the proposed methods. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

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.     

Multiple robust H∞ controller design using the nonsmooth optimization method

This paper proposes a systematic technique to design multiple robust H_∞ controllers. The proposed technique achieves a desired robust performance objective, which is impossible to achieve with a single robust controller, by dividing the uncertainty set into several subsets and by designing a robust controller to each subset. To achieve this goal with a small number of divisions of the uncertainty set, an optimization problem is formulated. Since the cost function of this optimization problem is not a smooth function, a numerical nonsmooth optimization algorithm is proposed to solve this problem. This method avoids the use of Lyapunov variables, and therefore it leads to a moderate size optimization problem. A numerical example shows that the proposed multiple robust control method can improve the closed‐loop performance when a single robust controller cannot achieve satisfactory performance. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust H2/H∞ control for time‐delay systems with polytopic uncertainty

In this paper, new separated H_∞ and H₂ performance criteria are derived for a class of time‐delay systems. When used in robust performance analysis and synthesis for real polytopic uncertainty and in multiobjective controller synthesis, they can partially rule out the technical restriction of using a single Lyapunov function, and therefore, lead to potentially less conservative linear matrix inequality (LMI) characterizations. Based on the criteria, robust multiobjective H₂/H_∞ controller is designed for time‐delay systems with polytopic uncertainty. All the conditions are given in terms of LMIs. Numerical examples are given to illustrate the proposed method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

H∞-control with measurement-feedback for Pritchard-Salamon systems

In this paper we extend the finite-dimensional results for the H_∞-control problem with measurement-feedback to a large class of infinite-dimensional systems, allowing for a certain type of unboundedness in the input and output operators (the Pritchard-Salamon class). The main result of the paper relates the solvability to the suboptimal H_∞-control problem to the existence of stabilizing solutions to certain operator Riccati equations. Furthermore, a characterization of all suboptimal controllers is given.     

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.     

Unconstrained H∞ predictive control with H∞ prediction: single‐input–single‐output case

Does the replacement of the quadratic (H₂) predictor by the worst‐case (H_∞, or cumulative minimax) predictor robustify the predictive control laws? The present work provides a partial answer to this question, positive for the examples considered, representative of three broad classes of systems. The H_∞ prediction is demonstrated to be a powerful and convenient tool for frequency shaping of the gain of the closed‐loop complementary sensitivity function, capable of robustifying the closed loop for systems with different stability properties. The H_∞‐optimal k‐step ahead predictor is derived for an unstable single‐input–single‐ output CARMA model. A BIBO unstable filter for the disturbance rejection is obtained using the internal model principle and included into the closed loop, and the H_∞ predictor is applied to the combination of this filter with the plant. The sum over a finite horizon of the current and the predicted tracking error and control signal power spectral densities (PSDs) is decomposed into two parts, one induced by the worst‐case predicted disturbance and the other—by the known future reference input. A two degrees of freedom algorithm, referred to as the multi‐step closed‐loop polynomial H_∞ predictive control law, is obtained that minimizes the peaks of the PSD of the first part and the integral on the unit circle of the PSD of the second. It is demonstrated on several systems that H_∞ prediction introduces a very intuitive tuning knob in the form of the prediction horizon capable of setting a trade‐off between the steady‐state disturbance rejection perfor mance in terms of the output error variance and the closed‐loop robustness, however the efficacy of the knob strongly depends on the stability properties of the system and its inverse. The trade‐off becomes less pronounced or completely disappears when the H_∞ predictor is replaced by the quadratic one. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

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.     

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.