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     

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     

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     

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     

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     

Game theory approach to optimal linear state estimation and itsrelation to the minimum H∞-normestimation

A possible game theory approach to optimal state estimation is presented. It is found that in a certain differential game, the minimizer's policy is identical to the one obtained by optimal estimation in the minimum H_∞-norm sense. This interpretation of H_∞-optimal state estimation provides better insight into the mechanism of H_∞ -optimal filtering, especially in the case where the exogenous signals are not energy bounded     

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     

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     

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∞-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     

Robust stability and performance via fixed-order dynamiccompensation: the discrete-time case

A discrete-time feedback-control-design problem involving parametric uncertainty is considered. A quadratic bound suggested by recent work on discrete-time state-space H_∞ theory is utilized in conjunction with the guaranteed cost approach to guarantee robust stability with a robust performance bound. The principal result involves sufficient conditions for characterizing robust full- and reduced-order controllers with a worst case H ₁ performance bound     

The robust optimal control of uncertain systems-state space method

The robust optimal control of linear uncertain systems with L ₂-bounded uncertainties is addressed. The optimal state feedback control law in the so-called worst disturbance case is derived. The LQL method, minimax state estimation-maximin state feedback closed loop control scheme (LQL means linear quadratic L₂-bounded uncertainties), is proposed. In a sense, the LQL method uses the idea of the linear-quadratic-Gaussian (LQG) method to treat H_∞-optimizing design problems     

Robust stabilization of uncertain linear systems: quadraticstabilizability and H∞ control theory

The problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time-domain results on the quadratic stabilization of uncertain systems and the frequency-domain results on H^∞ optimization. A complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given. Relations between these robust stabilization problems and H^∞ control theory are explored. It is also shown that in a number of cases, if a robust stabilization problem can be solved via Lyapunov methods, then it can be also be solved via H ^∞ control theory-based methods     

Robust stability under structured and unstructured perturbations

The problem of robust stability for linear time-invariant single-output control systems subject to both structured (parametric) and unstructured (H_∞) perturbations is studied. A generalization of the small gain theorem which yields necessary and sufficient conditions for robust stability of a linear time-invariant dynamic system under perturbations of mixed type is presented. The solution involves calculating the H_∞ -norm of a finite number of extremal plants. The problem of calculating the exact structured and unstructured stability margins is then constructively solved. A feedback control system containing a linear time-invariant plant which is subject to both structured and unstructured perturbations is considered. The case where the system to be controlled is interval is treated, and a nonconservative, easily verifiable necessary and sufficient condition for robust stability is given. The solution is based on the extremal of a finite number of line segments in the plant parameter property of a finite number of line segments in the plant parameter space along which the points closest to instability are encountered     

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     

Nondefinite least squares and its relation toH∞-minimum error state estimation

The problem of recursive nondefinite least-squares state estimation of continuous-time stationary processes is solved, by applying Pontryagin's maximum principle. A comparison of the derived solution to the result that is obtained for the H_∞ -minimum error estimation suggests a new interpretation for the H_∞-optimal estimation mechanism. According to this interpretation, the estimator tries to optimally estimate the required combination of the states, in the l₂-norm sense, against the worst disturbance signal that stems from a fictitious measurement of this combination     

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     

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     

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     

Comments on Stein's postulated high-frequency behavior ofH∞ optimal controllers

G. Stein (26th IEEE Conf. Decision Control, Los Angeles, CA, Dec. 1987) showed that H_∞ controller designs often give very unrealistic high-frequency behavior. The polynomial systems approach to H_∞ is used by the commenter to demonstrate that the high frequency gain of H_∞ controllers can be made to fall off at any desired rate provided improper noise and weighting models are chosen