Laguerre series approximation of infinite dimensional systems

Laguerre-Fourier approximations of stable systems are shown to exhibit many desirable properties for various classes of infinite dimensional systems. Specifically, time domain supremum and L¹ norm convergence results, and frequency domain H^∞ norm convergence results, are given for Laguerre-Fourier approximations. It is also shown that the theory of Laguerre polynomials solves explicitly the problem of determining Laguerre-Fourier approximations for a large class of delay systems. Furthermore, it is believed that these results are important for the study of orthonormal series identification as a general technique for identification of infinite dimensional systems.     

Robust H ∞ Control of Linear Uncertain Systems: Its Limiting Performance

The concept of a H norm is introduced for linear control systems with uncertain bounded parameters. A procedure for determining the lower estimate of the minimal robust H norm in the linear state feedback class is designed. An example of a system in which the estimate coincides with the minimal robust norm is given.     

A least distance algorithm for a smooth strictly convex norm

An algorithm for approximating a non negative solution of inconsistent systems of linear equations is presented. We define a best approximate solution of a system Ax = b x≥0 to be the vector x≥0 which minimizes the norm of the residual r(x) = b ? Ax, for a smooth and strictly convex norm. The algorithm is shown to be feasible and globally convergent. The special case of the ? ^p norm is included. In particular, the method converges for 1 < p < 2. A generalization of this algorithm is also given. Numerical results are included.     

Complexity of identification of linear systems with rational transfer functions

We study the complexity of worst-case time-domain identification of linear time-invariant systems using model sets consisting of degree-n rational models with poles in a fixed region of the complex plane. For specific noise level δ and tolerance levels τ, the number of required output samples and the total sampling time should be as small as possible. In discrete time, using known fractional covers for certain polynomial spaces (with the same norm), we show that the complexity isO(n ²) for theH ^∞ norm,O(n) for the ℓ² norm, and exponential inn for the ℓ¹ norm, for each δ and τ. We also show that these bounds are tight. For the continuous-time case we prove analogous results, and show that the input signals may be compactly supported step functions with equally spaced nodes. We show, however, that the internodal spacing must approach 0 asn increases.     

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.     

Residual mode filters and H ∞ control for a class of infinite-dimensional discrete-time systems

An H _∞, control problem with measurement feedback.for infinite-dimensional discrete-time (IDDT) systems whose homogeneous parts are described by Riesz-spectra operators is considered. The aim is to construct a finite-dimensional stabilizing controller for the IDDT system that makes the H _∞ norm of the closed-loop transfer function less than a given positive number δ. For that purpose, we first formulate the IDDT system as an IDDT system in l² and derive a finite-dimensional reduced-order system for the IDDT system in l². A stabilizing controller that makes the H _∞ norm of the closed-loop transfer function less than another positive number is then constructed for the reduced-order model. The finite-dimensional controller together with a residual mode Jilter plays a role of a finite-dimensional stabilizing controller that makes the H _∞ norm of the closed-loop transfer function less than δ for the original IDDT system, if the order of the residual mode filter is chosen suficiently large.     

Robust H∞ Control Problem For General Nonlinear Systems With Uncertainty

In this paper, both state and output feedback robust H^∞ control problems for general nonlinear systems with norm‐bound uncertainty are considered. Sufficient conditions for the existence of robust output feedback H^∞ controller are provided. State space formulas for robust H^∞ output controller are provided.     

On robust stability with structured time-invariant perturbations

We consider the problem of robust stability for uncertain systems described in terms of multiple scalar structured time-invariant perturbations, which are norm-bounded. The issue we address is the role that the type of perturbation norm plays in the robust stability conditions. To this end, the L^p-induced norm (for 1 ≤ p ≤ ∞) on the perturbations is considered, and it is shown that the structured singular value is the necessary and sufficient condition for robust stability for all these norms. We conclude that for time-invariant perturbations, in contrast to the time-varying case, robust stability conditions are independent of the type of perturbation norms.     

Composite State Feedback of the Finite Frequency H∞ Control for Discrete‐Time Singularly Perturbed Systems

This paper mainly is concerned with the finite frequency H_∞ control for the discrete‐time singularly perturbed systems. A state feedback controller is designed to stabilize the whole system and to satisfy the desired design specifications. The generalized Kalman–Yakubovich–Popov (GKYP) lemma is used to convert the related frequency domain inequalities in finite frequency ranges to feasible linear matrix inequalities. Based on the Lyapunov stability method, stable conditions are obtained for discrete‐time singularly perturbed systems. A bounded real lemma then is derived, which characterizes the H_∞ norm performance in specific frequency ranges. Furthermore, the approach for the design of a composite state feedback controller is put forward combined with the unique frequency characteristics of singularly perturbed systems. Detailed analysis of the performance achieved by the piecewise composite controller is provided when it is applied to the original system, and the effectiveness and merits of the proposed controller are illustrated with a numerical result.     

Practical multi-constraint H∞ controller synthesis from time-domain data

This paper presents a practical algorithm for MIMO controller design with multiple H^∞ norm constraints. The plant is described by its sampled-data impulse response matrix, which could be determined directly from measurements; the controller design parameters are the tap weights of an FIR discretization of the Q-(or Youla-) parameter. We approximate the H^∞ norm by the maximum transfer matrix 2-norm on an even frequency sampling over θ = [0, π]. The control design algorithm uses the Newton method to minimize two cost functions sequentially: the first determines multiple H^∞ feasibility, and the second minimizes a generalized entropy. As a design example, we control the acoustic radiation from a (mathematically modelled) submerged spherical shell. The plant-model impulse response matrix has McMillan degree 8 800. We specify and synthesize a controller that simultaneously achieves ten H^∞ constraints: one on the radiation reduction, one for stability robustness, and eight on actuator authority.     

Robust performance with fixed and worst-case signals for uncertain time-varying systems

In this paper we focus our attention on the determination of upper bounds of the l^∞ norm of the output of a linear discrete-time dynamic system driven by a step input, in the presence of both persistent unknown, but, l^∞ bounded disturbances and memoryless time-varying model uncertainty. For the same type of systems we also analyze the transient behavior of the step response in terms of its overshoot. The problem is solved in a constructive way by determining appropriate invariant sets contained in a given convex region. Finally, we show how to extend these results to continuous-time systems.     

H∞-type control for discrete-time stochastic systems

In this paper we consider discrete-time, linear stochastic systems with random state and input matrices which are subjected to stochastic disturbances and controlled by dynamic output feedback. The aim is to develop an H^∞-type theory for such systems. For this class of systems a stochastic bounded real lemma is derived which provides the basis for a linear matrix inequality (LMI) approach similar to, but more general than the one presented in Reference 1 for stochastic differential systems. Necessary and sufficient conditions are derived for the existence of a stabilizing controller which reduces the norm of the closed-loop perturbation operator to a level below a given threshold γ. These conditions take the form of coupled nonlinear matrix inequalities. In the absence of the stochastic terms they get reduced to the linear matrix inequalities of deterministic H^∞-theory for discrete time systems. Copyright © 1999 John Wiley & Sons, Ltd.     

Approximate solution of a class of optimal control problems in domains of arbitrary form

A combination of both fictitious domain and net methods is used for solution of optimal-control problems for elliptic systems. The proposed difference scheme has an order of accuracy ofO(h ^1/2) in the net norm L₂. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 138–146, January–February, 2000.     

Robust absolute stability of continuous systems

The problem of stability of continuous-time nonlinear systems with nonparametric uncertainty of a linear part and sector uncertainty of its nonlinear part is considered. The proposed robust stability criterion is a natural extension of the circle criterion for absolute stability. The situation with the Popov criterion is also discussed; it is demonstrated to be nonrobust for perturbations of a linear part, bounded in H^∞ norm.     

Computing the frequency response of linear systems with parametric perturbation

The problem of computing the frequency response of linear systems with parametric perturbation is addressed. Under the assumption that the coefficients of the system transfer function depend on the perturbation parameters in a linear manner, we provide results for plotting the frequency response of the perturbed transfer function. These results are useful in determining the H∞ norm, gain margin and phase margin and in improving the diagonal dominance of multi-input multi-out-put uncertain systems. An illustrative example is provided to demonstrate the results.     

General optimal attenuation of harmonic disturbance with unknown frequencies

We present algorithms for optimal harmonic disturbance attenuation in standard discrete-time control structure, based on a parametrisation of (marginally) stabilising controllers. The Frobenius norm and the spectral norm of the closed-loop transfer matrix at the disturbance frequencies are minimised. If there is only one frequency of the disturbance, the controller has an observer–based form, which we obtain by solving a static output feedback (SOF) stabilisation control problem. Although the SOF stabilisation problem is hard, the generical case of nonsquare matrix G ₂₂ is solved by linear algebra methods. Numerical simulation results are presented. As a corollary, we transform the control problem with unit circle invariant zeros into a ?_∞ control problem without such zeros. The elimination of the unit circle invariant zeros is based on the fact that matrix Y(zI???A?+?BF)^?1 is stable, where (Y,?F) with Y?≥?0 is a solution of a discrete-time algebraic Riccati system.     

Computing the L2-induced norm of a compression operator

A new computationally viable approach is derived for computing the induced norm of a state space compression operator; that is an integral operator defined on the finite length space L₂[0,h]. Determining this norm is a crucial component in the control analysis of both delay systems and sampled-data systems. The technique developed may have application to a wider variety of integral operators.     

Internal positivity preserved model reduction

This article studies model reduction of continuous-time stable positive linear systems under the Hankel norm, H _∞ norm and H ₂ norm performance. The reduced-order systems preserve the stability as well as the positivity of the original systems. This is achieved by developing new necessary and sufficient conditions of the model reduction performances in which the Lyapunov matrices are decoupled with the system matrices. In this way, the positivity constraints in the reduced-order model can be imposed in a natural way. As the model reduction performances are expressed in linear matrix inequalities with equality constraints, the desired reduced-order positive models can be obtained by using the cone complementarity linearisation iterative algorithm. A numerical example is presented to illustrate the effectiveness of the given methods.     

High‐order internal model‐based iterative learning control design for nonlinear distributed parameter systems

This article deals with the problem of iterative learning control algorithm for a class of nonlinear parabolic distributed parameter systems (DPSs) with iteration‐varying desired trajectories. Here, the variation of the desired trajectories in the iteration domain is described by a high‐order internal model. According to the characteristics of the systems, the high‐order internal model‐based P‐type learning algorithm is constructed for such nonlinear DPSs, and furthermore, the corresponding convergence theorem of the presented algorithm is established. It is shown that the output trajectory can converge to the desired trajectory in the sense of (L²,λ) ‐norm along the iteration axis within arbitrarily small error. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.