A design procedure for the exact H∞ siso-robust performance problem

Presented in this paper is a loop-shaping solution to the H^∞ robust performance problem for a class of SISO systems. The key feature of this work is the explicit incorporation of phase information into the controller synthesis to facilitate loop-shaping over all frequencies. Consequently, a significant reduction in the required controller gain/bandwidth is achieved. It is also shown that the optimal solution, in the sense of controller gain/bandwidth minimization, to the robust performance problem is that which makes μ = 1 over all frequencies. This is different from the usual definition of optimality in the H^∞/μ literature. An example is included for completeness.     

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.     

H2/H∞ filtering theory and an aerospace application

In many filtering problems of practical interest, some of the noise signals satisfy the assumptions of H₂ (Kalman-Bucy) filtering, while others can be more accurately modelled as signals with bounded variance (hence more amenable to an H_∞ filtering approach). These problems may be addressed by considering a mixed H₂/H_∞ filtering problem. In this paper we present a novel theory which solves the mixed problem exactly and in a computationally efficient way. The applicability of the theory is illustrated by designing a filter to estimate the states of an aircraft flying through a downburst.     

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 first principles solution to the non-singular H∞ control problem

This paper presents an elementary solution to the non-singular H^∞ control problem. In this control problem, the underlying linear system satisfies a set of assumptions which ensures that the solution can be obtained by solving just two algebraic Riccati equations of the game type. This leads to the central solution to the H^∞ control problem. The solution presented in this paper uses only elementary ideas beginning with the Bounded Real Lemma.     

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.     

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.     

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.     

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.     

Mixed H2/H∞ filtering

In paper we consider the problem of finding a filter or estimator that minimizes a mixed H₂/H_∞ filtering cost on the transfer matrix from a given noise input to the filtering error subject to an H_∞ constraint on the transfer matrix from a second noise input to the filtering error. This problem can be interpreted and motivated in many different ways; for instance, as a problem of optimal filtering in the presence of noise with fixed and known spectral characteristics subject to a bound on the filtering error due to a second noise source whose spectral characteristics are unknown. It is shown that one can come arbitrarily close to the optimal mixed H₂/H_∞ filtering cost using a standard Kalman-Luenberger estimator. Moreover, the problem of finding suitable Kalman-Luenberger estimator gains can be converted into a convex optimization problem involving affine symmetric matrix inequalities.     

Application Of H2/H∞ Technique To Aircraft Landing Control

The paper describes the automatic control of an aircraft in the longitudinal plane during landing, taking into account sensor errors and disturbances. Aircraft auto‐landing is achieved by combining H₂ and H_∞ control techniques, thereby obtaining a robust H₂/H_∞ controller. It provides good precision tracking and robust stability with respect to the uncertainties caused by different disturbances and noise‐type signals. The weights of the H₂ and H_∞ control techniques within the robust H₂/H_∞ controller are adjusted so that the aircraft accurately tracks the desired trajectory during the two main stages of the landing process. The theoretical results are validated by numerical simulations for the landing of a Boeing; the results are very good (Federal Aviation Administration accuracy requirements for Category III are met) and prove the robustness of the new auto‐landing system even in the presence of disturbances and sensor errors.     

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∞-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∞ 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.     

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∞ control of phase‐locked loops with polytopic type uncertainties

A robust H_∞ control method is applied in the design of loop filters for phase‐locked loop (PLL) carrier phase tracking. The proposed method successfully copes with large S‐curve slope uncertainty and with a significant decision delay in the closed loop that may stem from the decoder and/or the equalizer there. The design problem is transformed into a state‐feedback control problem where phase and gain margins should be guaranteed in spite of the uncertainty. Of all the loop filters that achieve the required margins the one that minimizes an upper bound on the effect of the phase and the measurement noise signals is derived. Copyright © 2001 John Wiley & Sons, Ltd.     

A dynamic game approach to mixed H∞/H2 estimation

A deterministic, two-person, zero-sum, linear quadratic estimation game, which is closely related to the mixed H_∞/H₂ estimation problem, is introduced. The strategy of the estimator in this game must guarantee a prescribed H_∞-performance level γ. If the desired H_∞-performance level is not achieved by an H₂-optimal estimator, the H₂-performance of the estimator must be degraded. An optimal estimator, in the sense of this game, minimizes the ratio between the resulting H₂-performance degradation, and the performance degradation that would have been obtained if one had used the standard H_∞-central filter. Both the continuous and the discrete-time cases are treated.     

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.