A simpler formula for the singular values of a certain Hankel operator

This paper deals with the problem of computing the singular values and vectors of a Hankel operator with symbol m^*W where m ε H^∞ is arbitrary inner and W ε H^∞ is rational. A simplified version of the formula given in [6] is obtained for computing the singular values of the Hankel operator. This result is applied to the (one-block) H^∞ optimal control of SISO stable infinite dimensional plants and rational weights. Using this new formula a simple expression is derived for the H^∞ optimal controller whose structure was observed in [9].     

On Stieltjes functions and Hankel operators

Let G be a Stieltjes function which is analytic in the open right half plane. It is shown that G is in H^∞(RHP) if and only if the Hankel operator H_G on H²(RHP) with symbol G is nuclear. If G is in H^∞(RHP) it is shown that the non-tangential limit of G at s = 0 equals twice the nuclear norm of H_G.     

Simultaneous H control of a finite collection of linear plants with a single nonlinear digital controller

This paper considers the problem of simultaneous H^∞ control of a finite collection of linear time-invariant systems via a nonlinear digital output feedback controller. The main result is given in terms of the existence of suitable solutions to Riccati algebraic equations and a dynamic programming equation. Our main result shows that if the simultaneous H^∞ control problem for k linear time-invariant plants of orders n₁,n₂,…,n_k can be solved, then this problem can be solved via a nonlinear time-invariant controller of order nn₁+n₂++n_k.     

Nonlinear H∞-control and estimation of optimal H∞-gain

The structure of nonlinear H_∞-controller and the estimation of optimal H_∞-gain are investigated in this paper. The essential problem boils down to the existence of semipositive solution to the Hamilton-Jacobi-Isaacs inequality. Further, the solvability of this first-order fully nonlinear differential inequality is discussed. We look for one kind of special semipositive radial solution to the Hamilton-Jacobi-Isaacs inequality. An explicit estimation of optimal H_∞-gain and explicit formulas of semipositive radial solutions to the Hamilton-Jacobi-Isaacs inequality are obtained. The results are quite simple and intuitive. They even shed a new insight on the linear H_∞-theory.     

L approximation of a parabolic system regarded as a delay system

We present an approach to the problem of finding an L^∞ approximant of the infinite-dimensional system describing the diffusion of heat in a wall. We show that this system can be regarded as a delay system with Laplace variable √s. We are able to get results, established in Zwart et al. (1988), about partial fraction expansion for delay systems, achieved by some adjustment to the specificities of our particular case. The determination of an L^∞ approximant is realised in two steps, using the optimal Hankel-norm approximation.     

Optimal H∞ model reduction via linear matrix inequalities: continuous- and discrete-time cases

Necessary and sufficient conditions are derived for the existence of a solution to the continuous-time and discrete-time H_∞ model reduction problems. These conditions are expressed in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint set. In addition, an explicit parametrization of all reduced-order models that correspond to a feasible solution is provided in terms of a contractive matrix. These results follow from the recent solution of the H_∞ control design problem using LMIs. Particularly simple conditions and a simple parametrization of all solutions are obtained for the zeroth-order H_∞ approximation problem, and the convexity of this problem is demonstrated. Computational issues are discussed and an iterative procedure is proposed to solve the H_∞ model reduction problem using alternating projections, although global convergence of the algorithm is not guaranteed.     

Optimal measurement trajectories for distributed parameter systems

The paper deals with the optimal sensor location problem associated with minimax filtering for linear distributed parameter systems under the moving sensors. The problem of optimal choice of a measurement trajectory is treated here as an H^∞-optimal control one under phase constraints with the objective to minimize a given ‘weak’ performance index under ‘worst’ possible disturbances. Existence of a solution to such a problem is established for the case of quadratic constraints on the disturbances and necessary conditions for optimally are derived on the basis of constructing a sequence of suboptimal solutions associated, in turn, with a sequence of finite-dimensional maximum principles.     

Minimax control of switching systems under sampling

We consider a general class of systems subject to two types of uncertainty: A continuous deterministic uncertainty that affects the system dynamics, and a discrete stochastic uncertainty that leads to jumps in the system structure at random times, with the latter described by a continuous-time finite state Markov chain. When only sampled values of the system state is available to the controller, along with perfect measurements on the state of the Markov chain, we obtain a characterization of minimax controllers, which involves the solutions of two finite sets of coupled PDEs, and a finite dimensional compensator. For the linear-quadratic case, a complete characterization is given in terms of coupled generalized Riccati equations, which also provides the solution to a particular H^∞ optimal control problem with randomly switching system structure and sampled state measurements.     

Computational complexity of a problem arising in fixed order output feedback design

This paper is concerned with a matrix inequality problem which arises in fixed order output feedback control design. This problem involves finding two symmetric and positive definitive matrices X and Y such that each satisfies a linear matrix inequality and that XY=I. It is well-known that many control problems such as fixed order output feedback stabilization, H_∞ control, guaranteed H₂ control, and mixed H₂/H_∞ control can all be converted into the matrix inequality problem above, including static output feedback problems as a special case. We show, however, that this matrix inequality problem is NP-hard.     

Uniformly optimal control of linear time-invariant plants: Nonlinear time-varying controllers

It is shown that in the problems of uniformly (or H^∞−) optimal control of linear time-invariant plants, arbitrary nonlinear, time-varying controllers offer no advantage over linear, time-invariant controllers.     

Characterization of controllers in simultaneous stabilization

This paper gives a convenient parametrization for the class of all stabilizing controllers for two or more plants. The result represent a generalization of the Youla parametrization of the class of all stabilizing controllers in terms of an arbitrary stable proper transfer function. Although, as expected, the additional constraints for simultaneous stabilization are not readily incorporated into H^∞, H² optimization procedures as in the standard case, there is immediate application of the theory for reduction of controllers which simultaneously stabilize two or more plants.     

Simultaneous H2/H∞ optimal control The state feedback case

A simultaneous H₂/H_∞ control problem is considered. This problem seeks to minimize the H₂ norm of a closed-loop transfer matrix while simultaneously satisfying a prescribed H_∞ norm bound on some other closed-loop transfer matrix by utilizing dynamic state feedback controllers. Such a problem was formulated earlier by Rotea and Khargonekar (Automatica, 27, 307–316, 1991) who considered only so called regular problems. Here, for a class of singular problems, necessary and sufficient conditions are established so that the posed simultaneous H₂/H_∞ problem is solvable by using state feedback controllers. The class of singular problems considered have a left invertible transfer function matrix from the control input to the controlled output which is used for the H₂ norm performance measure. This class of problems subsumes the class of regular problems.     

Lipschitz continuity of H interpolation

Optimal H^∞ interpolants may be infinitely sensitive to data. However, δ-suboptimal interpolants of the AAK central (maximal entropy) type are shown to satisfy a Lipschitz condition with respect to data.     

Optimum H designs under sampled state measurements

We present the complete solution to the H^∞-optimal control problem when only sampled values of the state are available. For linear time-varying systems the optimum controller is characterized in terms of the solution of a particular generalized Riccati-differential equation, with the optimum performance determined by the conjugate point conditions associated with a family of generalized Riccati differential equations. For the infinite-horizon time-invariant problem, however, the optimum controller is characterized in terms of the solution of a particular generalized algebraic Riccati equation, and the performance is determined in terms of the conjugate-point conditions of a single generalized Riccati equation, defined on the longest sampling interval. If the distribution of the sampling times is also taken as part of the general design, uniform sampling turns out to be optimal for the infinite horizon case, while for the finite horizon problem a nonuniform sampling generally leads to a better performance.     

Foundations of a theory of synchronous systems

A semantic algebra construction is introduced to model the stepwise behavior of synchronous systems in an arbitrary pointed algebraic theory T. The theory T is extended to a feedback theory F^∞T in which the bottom morphism is the designated point of T. The feedback theory F^∞T is obtained as the inverse limit of the theories n-res T that describe the stepwise behavior of systems in T restricted to the first n clock cycles. It is shown that in F^∞T, iteration satisfies the functorial dagger condition. Some suggestions are made about how to generalize the construction to handle infinite systems.     

The singular minimum entropy H∞ control problem

In this paper we search for controllers which minimize an entropy function of the closed loop transfer matrix under the constraint of internal stability and under the constraint that the closed loop transfer matrix has H_∞ norm less than some a priori given bound γ. We find an explicit expression for the infimum. Moreover, we give a characterization when the infimum is attained (contrary to the regular case, for the singular minimum entropy H_∞ control problem the infimum is not always attained).     

Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues

In many calculations, spectral discretization in space is coupled with a standard ordinary differential equation formula in time. To analyze the stability of such a combination, one would like simply to test whether the eigenvalues of the spatial discretization operator (appropriately scaled by the time step k) lie in the stability region for the o.d.e. formula, but it is well known that this kind of analysis is in general invalid. In the present paper we rehabilitate the use of stability regions by proving that a discrete linear multistep ‘method of lines’ approximation to a partial differential equation is Lax-stable, within a small algebraic factor, if and only if all of the -pseudo-eigenvalues of the spatial discretization operator lie within O() of the stability region as → 0. An -pseudo-eigenvalue of a matrix A is any number that is an eigenvalue of some matrix A + E with E ; our arguments make use of resolvents and are closely related to the Kreiss matrix theorem. As an application of our general result, we show that an explicit N-point Chebyshev collocation approximation of u_t = −xu_x on [−1, 1] is Lax-stable if and only if the time step satisfies k = O(N⁻²), although eigenvalue analysis would suggest a much weaker restriction of the form k CN⁻¹.     

H∞-control by state feedback: An iterative algorithm and characterization of high-gain occurence

We present an iterative algorithm to compute the achievable H_∞-norm by state feedbak for a standard regular problem and give a characterization when high-gain feedback is necessary to approach the optimal value.

If there is no need for high-gain components to approximate the optimal value, we can reduce the problem to the computation of the H_∞-norm of a certain stable transfer matrix.     

Robust H∞ control design for systems with structured parameter uncertainty

In a recent paper a unification of the H₂ (LQG) and H_∞ control-design problems was obtained in terms of modified algebraic Riccati equations. In the present paper these results are extended to guarantee robust H₂ and H_∞ performance in the presence of structured real-valued parameter variiations (ΔA, ΔB, ΔC) in the state space model. For design flexibility the paper considers two distinct types of uncertainty bounds for both full- and reduced-order dynamic compensation. An important special case of these results generates H₂/H_∞ controller designs with guaranteed gain margins.     

Duality theory for MIMO robust disturbance rejection

Banach space duality theory is used to characterize the solutions of a nonstandard H^∞ optimization problem which is shown to be allpass in general and unique in the single-input single-output (SISO) ease. The theory leads to a numerical solution of duality and convex optimization, which is applied to an example. For a limiting case of sharp cutoff filters, an explicit solution of the optimal robust disturbance attenuation problem (ORDAP) resembling the two arc theorem of complex analysis is derived