Robust nonlinear ship course‐keeping control by H∞ I/O linearization and μ‐synthesis

In this paper, the H_∞ input/output (I/O) linearization formulation is applied to design an inner‐loop nonlinear controller for a nonlinear ship course‐keeping control problem. Due to the ship motion dynamics are non‐minimum phase, it is impossible to use the ordinary feedback I/O linearization to resolve. Hence, the technique of H_∞ I/O linearization is proposed to obtain a nonlinear H_∞ controller such that the compensated nonlinear system approximates the linear reference model in I/O behaviour. Then a μ‐synthesis method is employed to design an outer‐loop robust controller to address tracking, regulation, and robustness issues. The time responses of the tracking signals for the closed‐loop system reveal that the overall robust nonlinear controller is able to provide robust stability and robust performance for the plant uncertainties and state measurement errors. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

A new static output feedback approach to the suboptimal mixed H2/H∞ problem

In this paper we propose a new approach to solve the static output feedback suboptimal mixed H₂/H_∞ control problem using a state fixed‐structure feedback design. We formulate the static output feedback problem as a constrained static state feedback problem and obtain three coupled design equations: one Riccati equation, one Lyapunov equation, and a gain equation. We will prove the equivalence of the proposed solution to the existing solution. A very simple iterative algorithm is then presented to solve the design equations for the stabilizing output feedback gain that minimizes an upper bound of H₂ norm while satisfying the H_∞ disturbance attenuation requirement. A unique feature of the new approach is that it admits the Kalman gain as an initial stabilizing gain to start the above iterative solution procedure, which is computationally attractive and advantageous compared to the direct approach, as the latter has to deal with the difficult algorithm initialization problem. Some illustrative numerical examples are given to demonstrate the effectiveness of the approach. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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 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.     

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.     

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.     

A generalized robust H ∞ filtering for singular Markovian jump systems and its applications

In this paper, a generalized robust H _∞ filtering method is proposed for a class of singular Markovian jump systems, whose generality is mainly embodied that the desired filter could bear perturbances in terms of uncertainties on its parameter matrices. Firstly, an LMI condition of robust mode‐dependent filter is developed. Based on the given result, a new approach to mode‐independent H _∞ filter is presented, which establishes a direct connection between mode‐dependent and mode‐independent filters. Secondly, when the transition rate matrix is with elementwise bounded uncertainties or partially unknown, sufficient conditions of such robust mode‐dependent and mode‐independent filters are all developed within LMI frameworks. Finally, a numerical example is used to demonstrate the effectiveness of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust nonlinear H ∞ control based on the incremental gain

The incremental gain is proposed as an alternative to the usual gain for designing nonlinear H _∞ controllers. Considering a class of plants with Lipschitz nonlinearities and using linear matrix inequalities, a state feedback controller is designed such that the closed‐loop system is exponentially stable in the absence of disturbance inputs and has incremental gain less than or equal to a minimized number in the presence of disturbances as well as model uncertainties. Moreover, a norm‐wise robustness analysis of the proposed technique against nonlinear uncertainties has been accomplished. Our result is verified through stabilization of both certain and uncertain systems in an incremental sense and also input tracking of a chaotic plant. Copyright © 2014 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     

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.     

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.