Two-degree-of-freedom H∞-optimization ofmultivariable feedback systems

A solution to the two-degree-of-freedom H_∞ -minimization problem that arises in the design of multivariable optimal continuous-time stochastic control systems is derived. A decoupling approach that enables a partially independent design of the prefilter and the feedback controller and yields a simple solution to the optimization problem is applied. This solution is obtained by transforming the optimization problem into two standard form (four-block) problems     

Correction of the FI result in H∞ controland parameterization of H∞ state feedbackcontrollers

The authors correct the parameterization of the H_∞ controller of the full-information (FI) problem derived by J.C. Doyle et al. (1989). Then they parameterize the H_m0 state feedback controller and explain how dynamical free parameters implied in it are related to constant feedback gains different from the central solution F_∞     

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     

H∞ control for nonlinear systems withoutput feedback

The basic question of nonlinear H^∞ control theory is to decide, for a given two-port system, when feedback that makes the full system dissipative and internally stable exists. This problem can also be viewed as a question about circuits, and, after translation, also has a game-theoretic statement. Several necessary conditions for solutions to exist are presented, and sufficient conditions for a certain construction to lead to a solution are given     

H∞-control by state-feedback and fastalgorithms for the computation of optimalH∞-norms

The suboptimality of some parameter for H_∞ -optimization by dynamic state-feedback is characterized in terms of the solvability of Riccati inequalities. This is done without restricting the finite zero structure of the plant. If there are no system zeros on the imaginary axis, the H_∞-problem can be treated in a complete and satisfactory way. Explicit characterizations optimum to be achieved are provided, and a closed formula for the optimal value is derived in terms of the H_∞-norm of some fixed transfer matrix. If the optimum is not attained, any sequence of controllers of bounded size which is constructed to approach the infimal norm must necessarily be high-gain. A globally and quadratically convergent algorithm to compute the optimal value is proposed. This algorithm is generalized to the H_∞-optimization problem by measurement feedback     

H∞-optimal control with state-feedback

A H_∞-optimal control problem in which the measured outputs are the states of the plant is considered. The main result shows that the infimum of the norm of the closed-loop transfer function using linear static state-feedback equals the infimum of the norm of the closed-loop transfer function over all stabilizing dynamic (even, nonlinear time-varying) state-feedback controllers     

Controller reduction by H∞-balancedtruncation

H_∞-balanced truncation may be used to obtain reduced-order plants or controllers. The plant (possibly unstable) is compensated using a particular robustly stabilizing controller. The two Riccati equations involved are then used to define a set of closed-loop input-output invariants called the H_∞-characteristic values. That part of the plant or controller corresponding to small H_∞-characteristic values is discarded to give a reduced-order plant or controller. By exploiting an intimate connection with coprime factorization, a simple a priori test is derived for the ability of such a reduced-order controller to stabilize the full-order plant. The performance of the resulting closed-loop may also be bounded a priori, i.e. in terms of the prespecified level of robustness and the discarded H_∞-characteristic values     

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     

On the parameterization H∞ controllers

The author clarifies some of the results of J.C. Doyle et al. (ibid., vol.34, no.8, p.831-47, Aug. 1989) and gives some new interpretations. In particular, the author parameterizes all suboptimal H_∞ controllers for the full information (FI) and state feedback control problems and indicates when this FI H_∞ control problem can or cannot be given a differential game saddle point interpretation     

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     

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     

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     

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     

H∞-minimum error state estimation oflinear stationary processes

A state estimator is derived which minimizes the H_∞-norm of the estimation error power spectrum matrix. Two approaches are presented. The first achieves the optimal estimator in the frequency domain by finding the filter transfer function matrix that leads to an equalizing solution. The second approach establishes a duality between the problem of H_∞-filtering and the problem of unconstrained input H_∞-optimal regulation. Using this duality, previously published results for the latter regulation problem are applied which lead to an optimal filter that possess the structure of the corresponding Kalman filter. The two approaches usually lead to different results. They are compared by a simple example which also demonstrates a clear advantage of the H_∞-estimate over the conventional l ₂-estimate     

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     

A two-sided interpolation approach to H∞optimization problems

A solution to the two-sided interpolation problem which arises in H_∞ optimization theory is obtained. This solution is found in closed form, explicitly in terms of the required interpolation directions. It is simple to obtain and it does not require the application of the relatively complicated matrix Pick-Nevanlinna theory. The solution obtained is of minimum order; due to its simplicity, the order reduction, which occurs at the minimum value of the H_∞-norm, is clearly explained     

Relative-error H∞ identification fromautocorrelation data-a stochastic realization method

A variant of the balanced stochastic truncation (BST) method for approximated realization of power spectrum matrices is shown to form the basis for an identification procedure that is well-suited to the task of determining relative-error-bounded approximate plant models for use in control design from input-output cross correlation data. Central to the theory is a novel L_∞-norm bound on the relative-error between an exact realization of the data and BST approximate realization     

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     

Frequency response algorithms for H∞optimization with time domain constraints

A very broad framework for control system design is considered that encompasses frequency-response methodologies for H_∞ optimization that solve various aspects of the control design problem and that are less well known that state-space methods. The focus is on linear programming, Lawson's algorithm, and Trefethen's algorithm. A modified Lawson's algorithm is proposed and related to Trefethan's method. The modified algorithm is shown to be significantly faster than linear programming and Lawson's algorithm. It is also shown how to extend the modified Lawson's algorithm so as to handle time-domain constraints in addition to frequency-domain specifications, which distinguishes it from other H_∞ optimization methods. Some steps are taken toward dealing with time domain constraints within an H_∞ optimization framework     

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