Robust H∞ control for linear systems withnorm-bounded time-varying uncertainty

Robust H_∞ control design for linear systems with uncertainty in both the state and input matrices is treated. A state feedback control design which stabilizes the plant and guarantees an H_∞-norm bound constraint on disturbance attenuation for all admissible uncertainties is presented. The robust H_∞ control problem is solved via the notion of quadratic stabilization with an H_∞ -norm bound. Necessary and sufficient conditions for quadratic stabilization with an H_∞-norm bound are derived. The results can be regarded as extensions of existing results on H_∞ control and robust stabilization of uncertain linear systems     

Direct state-space solution to the discrete-timeH∞ one-block problem

A general state-space representation is used to allow a complete formulation of the H_∞ optimization problem without any invertibility condition on the system matrix, unlike existing solutions. A straightforward approach is used to solve the one-block H_∞ optimization problem. The parameterization of all solutions to the discrete-time H_∞ suboptimal one-block problem is first given in transfer function form in terms of a set of functions in H _∞ that satisfy a norm bound. The parameterization of all solutions is also given as a linear fractional representation     

H∞ control and filtering for sampled-datasystems

H_∞ control and filtering problems for sampled-data systems are studied. Necessary and sufficient conditions are obtained for the existence of controllers and filters that satisfy a specified H_∞ performance bound. When these conditions hold, explicit formulas for a controller and a filter satisfying the H_∞ performance bound are also given     

H2-optimal sampled-data control

The H₂-optimal control of continuous-time linear time-invariant systems by sampled-data controllers is discussed. Two different solutions, state space and operator theoretic, are given. In both cases, the H₂ sampled-data problem is shown to be equivalent to a certain discrete-time H₂ problem. Other topics discussed include input-output stability of sampled-data systems, performance recovery in digital implementation of analog controllers, and sampled-data control of systems with the possibility of multiple-time delays     

Mixed H2/H∞ control:a convex optimization approach

The problem of finding an internally stabilizing controller that minimizes a mixed H₂/H_∞ performance measure subject to an inequality constraint on the H_∞ norm of another closed-loop transfer function is considered. This problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint. Both the state-feedback and output-feedback problems are considered. It is shown that in the state-feedback case one can come arbitrarily close to the optimal (even over full information controllers) mixed H₂/H_∞ performance measure using constant gain state feedback. Moreover, the state-feedback problem can be converted into a convex optimization problem over a bounded subset of (n×n and n ×q, where n and q are, respectively, the state and input dimensions) real matrices. Using the central H_∞ estimator, it is shown that the output feedback problem can be reduced to a state-feedback problem. In this case, the dimension of the resulting controller does not exceed the dimension of the generalized plant     

The application of scheduled H∞controllers to a VSTOL aircraft

The authors apply H_∞-designed controllers to a generic VSTOL (vertical and short takeoff and landing) aircraft model GVAM. The design study motivates the use of H _∞ techniques, and addresses some of the implementation issues which arise for multivariable and H_∞-designed controllers. An approach for gain scheduling H_∞ controllers on the basis of the normalized comprime factor robust stabilization problem formulation used for the H_∞ design is developed. It utilizes the observer structure unique to this particular robustness optimization. A weighting selection procedure, has been developed for the associated loop-shaping technique used to specify performance. Multivariable controllers pose additional problems in the event of actuator saturations, and a desaturation scheme which accounts for this is applied to the GVAM. A comprehensive control law was developed and evaluated using the Royal Aerospace Establishment piloted simulation facility     

Linear and nonlinear algorithms for identification in H∞ with error bounds

A linear algorithm and a nonlinear algorithm for the problem of system identification in H_∞ posed by Helmicki et al. (1990) for discrete-time systems are presented. The authors derive some error bounds for the linear algorithm which indicate that it is not robustly convergent. However, the worst-case identification error is shown to grow as log(n), where n is the model order. A robustly convergent nonlinear algorithm is derived, and bounds on the worst-case identification error (in the H_∞ norm) are obtained     

LQG control with an H∞ performance bound:a Riccati equation approach

An LQG (linear quadratic Gaussian) control-design problem involving a constraint on H^∞ disturbance attenuation is considered. The H^∞ performance constraint is embedded within the optimization process by replacing the covariance Lyapunov equation by a Riccati equation whose solution leads to an upper bound on L₂ performance. In contrast to the pair of separated Riccati equations of standard LQG theory, the H^∞-constrained gains are given by a coupled system of three modified Riccati equations. The coupling illustrates the breakdown of the separation principle for the H^∞ -constrained problem. Both full- and reduced-order design problems are considered with an H^∞ attenuation constraint involving both state and control variables. An algorithm is developed for the full-order design problem and illustrative numerical results are given     

The robust H2 control problem: a worst-casedesign

The worst-case effect of a disturbance system on the H ₂ norm of the system is analyzed. An explicit expression is given for the worst-case H₂ norm when the disturbance system is allowed to vary over all nonlinear, time-varying and possibly noncausal systems with bounded L₂-induced operator norm. An upper bound for this measure, which is equal to the worst-case H₂ norm if the exogeneous input is scalar, is defined. Some further analysis of this upper bound is done, and a method to design controllers which minimize this upper bound over all robustly stabilizing controllers is given. The latter is done by relating this upper bound to a parameterized version of the auxiliary cost function studied in the literature     

H∞ analysis and synthesis ofdiscrete-time systems with time-varying uncertainty

The problems of H_∞ analysis and synthesis of discrete-time systems with block-diagonal real time-varying uncertainty are considered. It is shown that these problems can be converted into scaled H_∞ analysis and synthesis problems. The problems of quadratic stability analysis and quadratic stabilization of these types of systems are dealt with as a special case. The results on synthesis are established for general linear dynamic output feedback control     

Minimum H∞-norm regulation of lineardiscrete-time systems and its relation to linear quadratic discretegames

A solution is derived to the H_∞-optimization problem that arises in multivariable discrete-time regulation when the controller has full access to the state vector. The solution method is based on the close relations that exist between linear quadratic differential game theory and H_∞-optimization. The existing theory of discrete-time quadratic games is readily applied in order to derive the solution to a finite-time horizon version of the H_∞ -optimization problem. The solution of the infinite-time horizon H_∞-optimization problem is obtained by formally taking the limit of the number of stages to infinity     

H∞ optimal control as a weightedWiener-Hopf problem

It is shown that H_∞ optimization is equivalent to weighted H₂ optimization in the sense that the solution of the latter problem also solves the former. The weighting rational matrix that achieves this equivalence is explicitly computed in terms of a state-space realization. The authors do not suggest transforming H_∞ optimization problems to H₂ optimization problems as a computational approach. Rather, their results reveal an interesting connection between H_∞ and H₂ optimization problems which is expected to offer additional insight. For example, H₂ optimal controllers are known to have an optimal observer-full state feedback structure. The result obtained shows that the minimum entropy solution of H_∞ optimal control problems can be obtained as an H₂ optimal solution. Therefore, it can be expected that the corresponding H_∞ optimal controller has an optimal observer-full state feedback structure     

Worst-case/deterministic identification in H∞: the continuous-time case

Results obtained by the authors (1991) worst-case/deterministic H_∞ identification of discrete-time plants are extended to continuous-time plants. The problem involves identification of the transfer function of a stable strictly proper continuous-time plant from a finite number of noisy point samples of the plant frequency response. The assumed information consists of a lower bound on the relative stability of the plant, an upper bound on a certain gain associated with the plant, an upper bound on the roll-off rate of the plant, and an upper bound on the noise level. Concrete plans of identification algorithms are provided for this problem. Explicit worst-case/deterministic error bounds for each algorithm establish that they are robustly convergent and (essentially) asymptotically optimal. Additionally, these bounds provide an a priori computable H_∞ uncertainty specification, corresponding to the resulting identified plant transfer function, as an explicit function of the plant and noise prior information and the data cardinality     

Plant and controller reduction problems for closed-loop performance

Model reduction problems which consider preserving closed-loop performance (H₂, H_∞, and μ) in the presence of reduction error are developed. These are formulated as weighted multiplicative error problems (for plant reduction) and weighted additive error problems (for controller reduction), with the weight function incorporating explicitly such control information as the desired sensitivity operator bound, the setpoint/disturbance spectrum, and the plant uncertainties. These problems are efficiently solved using the frequency-weighted balanced realization technique. The benefits of these reduction problems are illustrated with examples taken from the control of a binary distillation column     

L2-gain analysis of nonlinear systems andnonlinear state-feedback H∞ control

Previously obtained results on L2-gain analysis of smooth nonlinear systems are unified and extended using an approach based on Hamilton-Jacobi equations and inequalities, and their relation to invariant manifolds of an associated Hamiltonian vector field. On the basis of these results a nonlinear analog is obtained of the simplest part of a state-space approach to linear H_∞ control, namely the state feedback H_∞ optimal control problem. Furthermore, the relation with H_∞ control of the linearized system is dealt with     

An H∞ approach to feedback design withtwo objective functions

The problem of tightly bounding and shaping the frequency responses of two objective functions T_i(s)( i=1,2) associated with a closed-loop system is considered. It is proposed that an effective way of doing this is to minimize (or bound) the function max {∥T₁(s)∥ _∞, ∥T₂(s)∥_∞} subject to internal stability of the closed-loop system. The problem is formulated as an H^∞ control problem, and an iterative solution is given     

Necessary and sufficient conditions for mixedH2 and H∞ optimal control

It is shown that D.S. Bernstein and W.M. Hadad's (ibid., vol.34, no.3, p.293, 1989) necessary condition for full-order mixed H ₂ and H_∞ optimal control is also sufficient, and that J.C. Doyle et al.'s (Proc. Amer. Control Conf., p.2065, 1989) sufficient condition for full-order mixed H₂ and H_∞ optimal control is also necessary. They are duals of one another     

Some new properties of the structured singular value

J.C. Doyle et al. (1982) have shown that a necessary and sufficient condition for robust stability or robust performance in the H^∞-frame work may be formulated as a bound on the structured singular value (μ) of a specific matrix M which includes information on the system model, the controller, the model uncertainty, and the performance specifications. Often it is desirable to express the robust stability and performance conditions as norm bounds on transfer matrices (T) which are of direct interest to the engineer, e.g. sensitivity or complementary sensitivity. The present paper shows how to derive bounds on σ(T) from bounds on μ(M)     

Further results on the optimal rejection of persistent boundeddisturbances

The problem of designing controllers that optimally reject persistent disturbances is studied. The focus is on the case where the plant to be controlled has either zeros or poles on the stability boundary, i.e. the unit circle in the discrete-time case and the extended jω axis in the continuous-time case. For the discrete-time case, the problem of minimizing a cost functional of the form ∥f-rg∥₁, where the transform g˜ of g has some unit circle zeros is studied. A previously published dual problem formulation is extended, and it is shown that an optimal controller need not exist. The construction of a sequence of suboptimal controllers whose performance approaches the unattainable infimum of the cost function is studied. It is shown that two results which hold in the case of H_∞ optimization do not hold in the presented situation. Specifically, the introduction of unit circle zeros can increase the value of the infimum, even when every unit circle zero of g˜ is also a zero of f˜, and a sequence of controllers constructed in an obvious fashion fails to be an optimizing sequence. Similar results are obtained for the continuous-time case