Supervisory control of hybrid systems within a behavioural framework

This contribution addresses the synthesis of supervisory control for hybrid systems Σ with discrete external signals. Such systems are in general neither l-complete nor can they be represented by finite state machines. We find an l-complete approximation (abstraction) Σ_l for Σ, represent it by a finite state machine, and investigate the control problem for the approximation. If a solution exists, we synthesize the maximally permissive supervisor for Σ_l. We show that it also solves the control problem for the hybrid system Σ. If no solution exists, approximation accuracy can be increased by computing a k-complete abstraction Σ_k, k>l. This paper is entirely set within the framework on Willems’ behavioural systems theory.     

Control of singular problem via differentiation of a Min-Max

For Ω a smooth domain in Rⁿ with boundary Λ = Λ₀Λ₁, we are concerned with the wave equation y″ − Δy = S in Q_T =]0, T[ × Ω with = ∂/∂t, at source term satisfying S, S′ ε L¹(0, T L² (Ω)). A Dirichlet condition is imposed on Λ₀ and we consider an absorbing condition ∂y/∂n + uy′ = 0 in [0, T] × gL₁ where u is the control.parameter. We introduce the cost function. and using the Min-Max formulation of J we by-pasas the sensitivity analysis of u → y and obtain the gradient of J with a usual adjoint problem. We first present an abstract frame for this kind of problems. using the differentiability results of a Min-Max [1,2], which we very shortly deduce here, we show that the well posedness of the adjoint equation implies differentiability of the cost function governed by a linear well posed problem.     

∞ guaranteed cost control for uncertain continuous-time linear systems

This paper deals with the _∞ guaranteed cost control problems for continuous-time uncertain systems. It consist of the determination of a stabilizing state feedback gain which imposes on all possible closed-loop models an _∞ -norm upper bound γ > 0. Assuming that the uncertain domain is convex-bounded and the uncertain system is quadratic-stabilizable with γ disturbance attenuation, it is shown how to determine, by means of a convex programming problem, the global minimum of γ. As a particular and important case, for precisely known linear systems, the last problem reduces to the classical _∞ optimal control problem. The results follow from the definition of a special parameter space on which the above-mentioned problems are convex.     

H∞ filtering for linear periodic systems with parameter uncertainty

This paper focuses on H_∞ filtering for a class of linear periodic systems with a certain type of norm-bounded time-varying parameter uncertainty which appears in both the state and output matrices. The problem addressed is the design of a linear periodic estimator that guarantees both the quadratic stability and and prescribed H_∞ performance on infinite horizon for the estimation error for all admissible parameter uncertainties. A solution to this problem is obtained via a Riccati equation approach.     

On asymptotic stability of continuous-time risk-sensitive filters with respect to initial conditions

In this paper, we consider the problem of risk-sensitive filtering for continuous-time stochastic linear Gaussian time-invariant systems. In particular, we address the problem of forgetting of initial conditions. Our results show that suboptimal risk-sensitive filters initialized with arbitrary Gaussian initial conditions asymptotically approach the optimal risk-sensitive filter for a linear Gaussian system with Gaussian but unknown initial conditions in the mean square sense at an exponential rate, provided the arbitrary initial covariance matrix results in a stabilizing solution of the (H_∞-like) Riccati equation associated with the risk-sensitive problem. More importantly, in the case of non-Gaussian initial conditions, a suboptimal risk-sensitive filter asymptotically approaches the optimal risk-sensitive filter in the mean square sense under a boundedness condition satisfied by the fourth order absolute moment of the initial non-Gaussian density and a slow growth condition satisfied by a certain Radon–Nikodym derivative.     

On the joint nonlinear filtering-smoothing of diffusion processes

The smoothing of diffusions dx_t = f(x_t) dt + σ(x_t) dw_t, measured by a noisy sensor dy_t = h(x_t) dt + dv_t, where w_t and v_t are independent Wiener processes, is considered in this paper. By focussing our attention on the joint p.d.f. of (x_τ x_t), 0 ≤ τ < t, conditioned on the observation path {y_s, 0 ≤ s ≤ t}, the smoothing problem is represented as a solution of an appropriate joint filtering problem of the process, together with its random initial conditions. The filtering problem thus obtained possesses a solution represented by a Zakai-type forward equation. This solution of the smoothing problem differs from the common approach where, by concentrating on the conditional p.d.f. of x_τ alone, a set of ‘forward and reverse’ equations needs to be solved.     

Robust stabilization of infinite dimensional systems by finite dimensional controllers

The problem of robustly stabilizing an infinite dimensional system with transfer function G, subject to an additive perturbation Δ is considered. It is assumed that: G ε ₀(σ) of systems introduced by Callier and Desoer [3]; the perturbation satisfies |W₁ΔW₂| < ε, where W₁ and W₂ are stable and minimum phase; and G and G + Δ have the same number of poles in ⁺. Now write W₁GW₂=G₁ + G₁, where G₁ is rational and totally unstable and G₂ is stable. Generalizing the finite dimensional results of Glover [12] this family of perturbed systems is shown to be stabilizable if and only if ε σ_min (G^*₁)( = the smallest Hankel singular value of G^*₁). A finite dimensional stabilizing controller is then given by where 2 is a rational approximation of G₂ such that

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) and K₁ robustly stabilizes G₁ to margin ε. The feedback system (G, K) will then be stable if |W₁ΔW₂| _∞< ε − Δ.     

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.     

A note on a system theoretic approach to a conjecture by Peller-Khrushchev

Based on the construction of infinite dimensional balanced realizations an alternative solution to the following inverse spectral problem is presented: Given a monotonically decreasing sequence of positive numbers (σ_n)_{n 1}, does there exist a Hankel operator whose sequence of singular values is (σ_n)_{n 1}?     

Discrete J-spectral factorization

This paper deals with the J-spectral factorization for general discrete rational matrices. A simple approach based on the Kalman filtering in Krein space is proposed. The main idea is to construct a stochastic state space filtering model in Krein space such that the spectral matrix of the output is equal to the rational matrix to be factorized. The spectral factor is then easily derived by using the generalized Kalman filtering in Krein space, which is similar to the H₂ spectral factorization. Our approach unifies the treatment of the H₂ spectral factorization and the J-spectral factorization. The applications of the derived results in H_∞ and risk-sensitive estimation for both nonsingular and singular systems are demonstrated.     

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.     

A spectral characterization of H-optimal feedback performance and its efficient computation

We study the spectral properties of a ‘Toeplitz⁺ Hankel’ operator which arises in the context of the mixed-sensitivity H^∞-optimization problem and whose largest eigenvalue characterizes the optimal achievable performance ε₀. The existence of such an operator was first shown by Verma and Jonckheere [26], who also'noted the potential numerical advantage of computing eo through its eigenvalue characterization rather than through the ε-iteration. Here, we investigate this operator in detail, with the objective of efficiency computing its spectrum. We define an ‘adjoint’ linear-quadratic problem that involves the same ‘Toeplitz⁺ Hankel’ operator, as shown by Jonckheere and Silverman [13–16]. Consequently, a finite polynomial algorithm allows ε₀ to be characterized as simply as the largest root of a polynomial. Finally, a computationally more attractive state space algorithm emerges from the H^t8/LQ relationship. This algorithm yields a very good accuracy evaluation of the performance ε₀ by solving just one algebraic Riccati equation. Thorough exploitation of this algorithm results in a drastic computation reduction with respect to the standard e-iteration.     

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.     

Bounds on element order in rings Zm with divisors of zero

If p is a prime, integer ring Z_p has exactly ((p)) generating elements ω, each of which has maximal index I_p(ω) = (p) = p − 1. But, if m = Π^R_{J = 1} p^αJ_J is composite, it is possible that Z_m does not possess a generating element, and the maximal index of an element is not easily discernible. Here, it is determined when, in the absence of a generating element, one can still with confidence place bounds on the maximal index. Such a bound is usually less than (m), and in some cases the bound is shown to be strict. Moreover, general information about existence or nonexistence of a generating element often can be predicted from the bound.     

Robust H∞ estimation of stationary discrete-time linear processes with stochastic uncertainties

The problem of H_∞ filtering of stationary discrete-time linear systems with stochastic uncertainties in the state space matrices is addressed, where the uncertainties are modeled as white noise. The relevant cost function is the expected value, with respect to the uncertain parameters, of the standard H_∞ performance. A previously developed stochastic bounded real lemma is applied that results in a modified Riccati inequality. This inequality is expressed in a linear matrix inequality form whose solution provides the filter parameters. The method proposed is applied also to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H₂/H_∞ filtering for the above system is also treated. The theory developed is demonstrated by a simple tracking example.     

Factorization and feedback stabilization for nonlinear systems

We establish the equivalence of internal input-out stability for two feedback configurations of a nonlinear, time-varying plant P for which a related plant G is assumed to have a factorization G = R with both R and R⁻¹ incrementally stable; this extends a factorization principle for stabilizability previously given only for the linear, time-invariant case. As an application of a special case we recover a version of the Youla parametrization of stabilizing compensators for the nonlinear case previously presented in the literature. We use degree theory to parametrize a collection of solutions of the H_∞-control problem for the case of a 1-gain stable or lossless plant. In the case of a plant G having a J-inner-outer factorization, this last result combined with the above-mentioned factorization principle leads to results on the H_∞-control problem for P.     

H∞ filtering for a class of uncertain nonlinear systems

This paper investigates the problem of H_∞ filtering for a class of uncertain continuous-time nonlinear systems with real time-varying parameter uncertainty and unknown initial state. We develop an infinite horizon H_∞ filtering methodology which provides both robust stability and a guaranteed H_∞ performance for the filtering error irrespective of the parameter uncertainty.     

On stability, -gain and control for switched systems

This paper addresses the issues of stability, L₂-gain analysis and H_∞ control for switched systems via multiple Lyapunov function methods. A concept of general Lyapunov-like functions is presented. A necessary and sufficient condition for stability of switched systems is given in terms of multiple generalized Lyapunov-like functions, which enables derivation of improved stability tests, an L₂-gain characterization and a design method for stabilizing switching laws. A solution to the H_∞ control problem for switched systems is also provided.     

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.