On optimal robust disturbance attenuation

In this article, we give an explicit solution to the nonstandard H^∞ problem which includes the optimal robust disturbance attenuation problem and the two-disc problem. The solution depends on an operator which is analogous to the Sarason operator for the standard H^∞ problem, it allows to quantify the optimal performance of feedback systems and provides an explicit formula for the optimal controller. The nonstandard H^∞ problem is perhaps the simplest control problem which deals with significant plant uncertainty. Surprisingly, relatively little work has been done on it despite the large number of papers in the H^∞ control literature.     

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].     

The noncoercive quasi-variational inequalities related to impulse control problems

This paper deals with the numerical analysis of noncoercive quasi-variational inequalities of impulse control problems. Optimally L^{∞-error-estimates} are derived using qualitative properties of both the continuous and finite element approximation solutions and the notion of subsolutions.     

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.     

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.     

A comment on “L∞ optimal control of SISO continuous-time systems”

The paper “L_∞ optimal control of SISO continuous-time systems” by Wang and Sznaier (Wang & Sznaier (1997). Automatica, 33 (1), 85–90) studies the problem of designing a controller that optimally minimizes the peak absolute value of system output, due to a fixed input signal. With a newly defined function space A_∞, it was claimed that the set of all L_∞-bounded outputs could be parameterized and that the problem could be transformed to a minimal distance problem on L_∞ space. We believe, however, their formulation has essential flaws.     

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.     

When is a linear convolution system stabilizable?

Lack of operator closedness and closability arguments are used to discuss the impossibility of finite ℓ_p/L_p gain stabilization of unstable finite-dimensional linear (convolution operator) systems in the doubly infinite time-axis case for any 1p∞. The presented analysis generalizes and refines a result due to Georgiou and Smith (IEEE Trans. Automat. Control 40 (1995) 516) for the L₂(−∞,∞) setting.     

Characterization of some gain conditions via the Hamilton-Jacobi inequality

This paper gives a characterization of three gain conditions, i.e., L_2∞ gain, L_2∞ gain subject to an L₂ gain constraint, and the Hankel gain, of nonlinear systems via the Hamilton-Jacobi inequalities. It is shown that the obtained results for the L_2∞ gain and the Hankel gain are analog to that derived for the L₂ gain so far.     

An L∞-convergent method to reduce frequency-weighted infinite-dimensional discrete-time systems

In this paper, we present a method based on balanced realizations truncations to reduce frequency-weighted infinite-dimensional discrete-time systems.The method is shown to be L_∞-convergent for a subclass of the nuclear systems' class. As its numerical computation is not easy, we propose a practical method to reduce frequency-weighted infinite-dimensional systems and prove that it is L_∞-convergent for the class of systems whose transfer function has its derivative in the Wiener algebra.     

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.     

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.     

A small gain theorem for linear stochastic systems

A well-known result in linear control theory is the so-called “small gain” theorem stating that if given two plants with transfer matrix functions T₁ and T₂ in H^∞ such that T₁ < γ and T₂ < 1/γ, when coupling T₂ to T₁ such that u₂ = y₁ and u₁ = y₂ one obtains an internally stable closed system. The aim of the present paper is to describe a corresponding result for stochastic systems with state-dependent white noise.     

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.     

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.     

Conditioned asymptotics for tail probabilities in large multiplexers

Consider a buffer whose input is a superposition of L independent identical sources, and which is served at rate sL. Recent work has shown that, under very general circumstances, the stationary tail probabilities for the queue of unfinished work Q in the buffer have the asymptotics P[Q > Lb] ≈ e^−LI(b) for large L. Here the shape function, I, is obtained from a variational expression involving the transient log cumulant generating function of the arrival process.

In this paper, we extend this analysis to cover time-dependent asymptotics for Markov arrival processes subject to conditioning at some instant. In applications we envisage that such conditioning would arise due to knowledge of the queue at a coarse-grained level, for example of the number of current active sources. We show how such partial knowledge can be used to predict future tail probabilities by use of a time dependent, conditioned shape function. We develop some heuristics to describe the time-dependent shape function in terms of a reduced set of quantities associated with the underlying arrivals process and show how to calculate them for renewal arrivals and a class of ON-OFF arrivals. This bypasses the full variational calculation of the shape function for such models.     

A description of the effect of short range ordering in the compound energy formalism

The current state of the compound energy formalism (CEF) to model ordering and the technique to handle short range ordering (SRO) were discussed. Although there are some methods which allow better treatment for SRO, such as the cluster variation method (CVM) or the Monte Carlo method (MC), these methods are difficult to handle in multi-component systems because of their complexity. In the compound energy formalism, SRO is not explicitly described. However it is possible to take the contribution of SRO to the Gibbs free energy into account through the reciprocal parameter, which has the same concentration dependency as ΔG_m^SRO. Introducing the reciprocal parameter of the form, L_i,j:i,j for the two sublattice model and L_{i,j:i,j:^*:^*} for the four sublattice model, one can satisfactorily describe the contribution of SRO to the Gibbs free energy in the CEF.     

Discrete-time system modelling in Lp with orthonormal basis functions

In this paper, model sets for linear time-invariant systems spanned by fixed pole orthonormal bases are investigated. The obtained model sets are shown to be complete in L_p(T) (1<p<∞), the Lebesque spaces of functions on the unit circle T, and in C(T), the space of periodic continuous functions on T. The L_p norm error bounds for estimating systems in L_p(T) by the partial sums of the Fourier series formed by the orthonormal functions are computed for the case 1<p<∞. Some inequalities on the mean growth of the Fourier series are also derived. These results have application in estimation and model reduction.