An alternative solution to the H∞-optimal control problem

In this paper we present an alternative solution to the problem min _{X ε H^n×n∞} |A + BXC|_∞ where A, B, rmand C are rational matrices in H^n×n_∞. The solution circumvents the need to extract the matrix inner factors of B and C, providing a multivariable extension of Sarason's H^∞-interpolation theory [1] to the case of matrix-valued B(s) and C(s). The result has application to the diagonally-scaled optimization problem int |D(A + BXC)D⁻¹|_∞, where the infimum is over D, X εH^n×n_∞, D diagonal.     

Nonlinear eigenvalue problems for quasilinear systems

The paper deals with the existence of positive solutions for the quasilinear system (Φ(u'))' + λh(t)f(u) = 0,0 < t < 1 with the boundary condition u(0) = u(1) = 0. The vector-valued function Φ is defined by Φ(u) = (q(t)(p(t)u₁), …, q(t)(p(t)u_n)), where u = (u₁, …, u_n), andcovers the two important cases (u) = u and (u) = up > 1, h(t) = diag[h₁(t), …, h_n(t)] and f(u) = (f¹(u), …, fⁿ (u)). Assume that fⁱ and h_i are nonnegative continuous. For u = (u₁, …, u_n), let

, f₀ = maxf¹₀, …, fⁿ₀ and f_∞ = maxf¹_∞, …, fⁿ_∞. We prove that the boundary value problem has a positive solution, for certain finite intervals of λ, if one of f₀ and f_∞ is large enough and the other one is small enough. Our methods employ fixed-point theorem in a cone.     

On the weighted sensitivity minimization problem for delay systems

We consider the H^∞-optimal sensitivity problem for delay systems. In particular, we consider computation of μ:= inf {|W-φq|_∞ : q ε H^∞(j )} where W(s) is any function in RH^∞(j ), and φ in H^∞(j ) is any inner function. We derive a new explicit solution in the pure delay case where φ = e^−sh, h > 0.     

On pure H reduced-order dynamic controllers: an erratum

There are at least two approaches advocated to obtain a pure H^∞ reduced-order dynamic controller for a given augmented plant. One approach is to eliminate completely the H² aspect from a standard H²/H^∞ setting. A second approach is to equate the H² aspect with the H^∞ aspect in that same setting. This paper invalidates the first approach but affirms the second approach and produces the correct equations resulting therefrom.     

Stabilization and H control of symmetric systems: an explicit solution

In this paper we address the H^∞ control analysis, the output feedback stabilization, and the output feedback H^∞ control synthesis problems for state-space symmetric systems. Using a particular solution of the Bounded Real Lemma for an open-loop symmetric system we obtain an explicit expression to compute the H^∞ norm of the system. For the output feedback stabilization problem we obtain an explicit parametrization of all asymptotically stabilizing control gains of state-space symmetric systems. For the H^∞ control synthesis problem we derive an explicit expression for the optimally achievable closed-loop H^∞ norm and the optimal control gains. Extension to robust and positive real control of such systems are also examined. These results are obtained from the linear matrix inequality formulations of the stabilization and the H^∞ control synthesis problems using simple matrix algebraic tools.     

H optimization problem with sign-indefinite quadratic form

This paper presents the solution to min-max control problem arising when the matrix C₁^TC₁ of the cost function in the standard H^∞ control problem (Doyle et al., 1989) is replaced by an arbitrary matrix Q 0. This difference is proved to be sufficient for results obtained in (Doyle et al., 1989) not to cover such the case. Their derivations essentially base on the cost function being H^∞ norm and can not be adjusted to deal with sign-indefinite quadratic form. With some sort of strict frequency condition assumed, state space technique is fruitful to obtain the necessary and sufficient conditions of the solvability of the problem. The solution is given by two Riccati equations and has some difference when compared to that of (Doyle et al., 1989).     

Stabilization of some fractional delay systems of neutral type

In this note, we give new stability tests which enable one to fully characterize the H_∞-stability of systems with transfer function , where h>0 and p,q,r are real polynomials in the variable s^μ for 0<μ<1.As an application of this, in the case r(s)=1 and degp=degq=1, families of H_∞-stabilizing controllers are given and a complete parametrization of all H_∞-stabilizing controllers is obtained when .     

Synthesis of stable H controller via the chain scattering framework

A sufficient condition for the existence of suboptimal stable stabilizing H^∞ controllers is given. By exploiting the free parameter in the parameterization of stabilizing controllers and using the chain scattering framework, we reformulate the H^∞ strong stabilization problem as an equivalent H^∞ optimization problem which can be solved via only one algebraic Riccati equation. A parameterization of all suboptimal stable stabilizing H^∞ controllers is also given.     

Dynamic behavior for a system of neutral functional differential equations

Consider the system of neutral functional differential equations

where r>0, F, . It is shown that if F is nondecreasing on , and some additional assumptions hold, then the ω limit set of every bounded solution of such a system with some initial conditions is composed of 2r-periodic solutions. Our results are new and complement some corresponding ones already known.     

Some new paranormed sequence spaces

Maddox defined the sequence spaces ℓ_∞(p), c(p) and c₀(p) in [Proc. Camb. Philos. Soc. 64 (1968) 335, Quart. J. Math. Oxford (2) 18 (1967) 345]. In the present paper, the sequence spaces a₀^r(u,p) and a_c^r(u,p) of non-absolute type have been introduced and proved that the spaces a₀^r(u,p) and a_c^r(u,p) are linearly isomorphic to the spaces c₀(p) and c(p), respectively. Besides this, the α-, β- and γ-duals of the spaces a₀^r(u,p) and a_c^r(u,p) have been computed and their basis have been constructed. Finally, a basic theorem has been given and later some matrix mappings from a₀^r(u,p) to the some sequence spaces of Maddox and to some new sequence spaces have been characterized.     

Global stability for separable nonlinear delay differential equations

Consider the following separable nonlinear delay differential equation

, where we assume that, there is a strictly monotone increasing function f(x) on (−∞, +∞) such that

In this paper, to the above separable nonlinear delay differential equation, we establish conditions of global asymptotic stability for the zero solution. In particular, for a special wide class of f(x) which contains a case of f(x) = e^x−1, we give more explicit conditions. Applying these, we offer conditions of global asymptotic stability for solutions of nonautonomous logistic equations with delays.     

Achievable H performance in sampled-data smoothing: beyond the -barrier

The lifting technique is a powerful tool for handling the periodically time-varying nature of sampled-data systems. Yet all known solutions of sampled-data H^∞ problems are limited to the case when the feedthrough part of the lifted system, , satisfies , where γ is the required H^∞ performance level. While this condition is always necessary in feedback control, it might be restrictive in signal processing applications, where some amount of delay or latency between measurement and estimation can be tolerated. In this paper, the sampled-data H^∞ fixed-lag smoothing problem with a smoothing lag of one sampling period is studied. The problem corresponds to the a-posteriori filtering problem in the lifted domain and is probably the simplest problem for which a smaller than performance level is achievable. The necessary and sufficient solvability conditions derived in the paper are compatible with those for the sampled-data filtering problem. This result extends the scope of applicability of the lifting technique and paves the way to the application of sampled-data methods in digital signal processing.     

On the pole location of a system designed by robust stability-degree assignment

In this paper, we examine the pole location of the feedback system composed of the nominal plant and the H_∞ central controller designed by the robust stability-degree assignment. Namely, the exact pole location at γ=∞ and the behavior near the infimum of γ are clarified where γ is the upper bound of the H_∞ norm constraint. The original design goal is to stabilize the plant against additive perturbations with the regional pole placement condition Re s<−α, and the design problem is reduced to the one-block H_∞ control problem.     

(J, J′)-lossless factorization based on conjugation

This paper is concerned with a derivation of the state-space form of the (J, J′)-lossless factorization which contains both the inner-outer factorization and the spectral factorization of positive matrices as special cases. Also, the (J, J′)-lossless factorization gives a unified framework of H^∞ control theory. We use the method of conjugation which makes the derivation much simpler than the previous literature, most of which used the technique of (J, J′)-spectral factorization. A necessary and sufficient condition is represented in terms of two Riccati equations one of which is degenerated.     

H∞ control design in bilinear systems: a tensor formal series approach

In this paper, the H_∞ disturbance attenuation problem of bilinear system is discussed. Dynamic game theory is used to solve this bilinear minimax problem. The solvability of H_∞ disturbance attenuation in bilinear system is also discussed. The techniques of tensor products and formal power series are employed to solve the nonlinear Bellman-Isaac differential equation. Furthermore, the convergence for the tensor formal series approach of this H_∞ control problem is discussed, and the radius of convergence for this H_∞ control to be well defined is also obtained.     

control for fast sampling discrete-time singularly perturbed systems

This paper is concerned with the H_∞ control problem via state feedback for fast sampling discrete-time singularly perturbed systems. A new H_∞ controller design method is given in terms of solutions to linear matrix inequalities (LMIs), which eliminates the regularity restrictions attached to the Riccati-based solution. A method for evaluating the upper bound of singular perturbation parameter with meeting a prescribed H_∞ performance bound requirement is also given. Furthermore, the results are extended to robust controller design for fast sampling discrete-time singularly perturbed systems with polytopic uncertainties. Numerical examples are given to illustrate the validity of the proposed methods.     

Asynchronous hybrid systems with jumps – analysis and synthesis methods

In this paper we show that the H_∞ synthesis problem for a class of linear systems with asynchronous jumps can be reduced to a purely discrete-time synthesis problem. The system class considered includes continuous-time systems with discrete jumps, or discontinuities, in the state. New techniques are developed for the analysis of asynchronous time-varying hybrid systems which allow a particularly simple treatment, and provide an elementary proof for the sampled-data H_∞ problem.     

Nonlinear robust H∞ control via dynamic output feedback

The problem on robust H_∞ control for a class of nonlinear systems with parameter uncertainty is studied. Sufficient conditions for the existence of the dynamic output feedback controller are obtained. Under these conditions, the closed-loop systems have robust H_∞-performance. A numerical example is given to illustrate the design of a robust controller using the proposed approach.     

A unified presentation of some classes of meromorphically multivalent functions

The authors introduce and investigate various properties of a general class

U_k[p,a,β,A,B]

p, k ε N {1, 2, 3,…,}; 0 α < p; β 0

;

−1 A < B 1; 0 < B 1)

,which unifies and extends several (known or new) subclasses of meromorphically multivalent functions. The properties and characteristics of this general class, which are presented here, include growth and distortion theorems; they also involve Hadamard products (or convolution) of functions belonging to the class U_k[p,α,β,A,B].     

On improving control-loop robustness of model-matching controllers

This note first characterizes the class of all stabilizing controllers for a two-degree-of-Freedom control system which achieve a prescribed achievable transfer function. The characterization is in terms of an arbitrary proper stable transfer function. With this characterization, robust model matching is formulated as a standard H^∞-optimization problem. This means that standard controller designs for a nominal plant, such as linear-quadratic Gaussian ones, can be enhanced to give improved robustness properties using H^∞-design techniques.