P. M. Mäkilä

On approximation of stable linear dynamical systems using Laguerre and Kautz functions

Approximation of stable linear dynamical systems by means of so-called Laguerre and Kautz functions, which are the Laplace transforms of a class of orthonormal exponentials, is studied. Since the impulse response of a stable finite dimensional linear dynamical system can be represented by a sum of exponentials (times polynomials), it seems reasonable to use basis functions of the same type. Assuming that the transfer function of a system is bounded and analytic outside a given disc, it is shown that Laguerre basis functions are optimal in a mini-max sense. This result is extended to the “two-parameter” Kautz functions which can have complex poles, while the poles of Laguerre functions are restricted to the real axis. By conformai mapping techniques the “two-parameter” Kautz approximation problem is recast as two Laguerre approximation problems. Thus, the well-developed theory of Laguerre functions can be applied to analyze Kautz approximations. Unilateral shifts are used to further develop the connection between Laguerre functions and Kautz functions. Results on þ2 and þ∞ approximation using Kautz models are given. Furthermore, the weighted L2 Kautz approximation problem is shown to be equivalent to solving a block Toeplitz matrix equation.

Worst-case control-relevant identification

Abstract This paper introduces the reader to several recent developments in worst-case identification motivated by various issues of modelling of systems from data for the purpose of robust control design. Many aspects of identification in H ∞ and l 1 are covered including algorithms, convergence and divergence results, worst-case estimation of uncertainty models, model validation and control relevancy issues.

Approximation of stable systems by laguerre filters

Laguerre methods are used to develop concrete techniques and results for approximating stable infinite dimensional systems. Both Hankel norm and L∞ approximation by Laguerre filters are studied. The infimum of the Hankel norm approximation error when the system is approximated by Laguerre filters of given maximum order is given by the norm of a certain shifted Hankel operator, and the best Laguerre filter to approximate the system is obtained essentially from the associated eigenproblem for the shifted Hankel operator. Hankel-optimal Laguerre filters are here called CF-Laguerre filters as there is a certain connection to the Caratheodory-Fejer interpolation problem, and typically provide near best solutions to related L∞ approximation problems. A time domain compressed Laguerre shift method for solving certain Nehari problems for delay systems is used to determine analytically the achievable Hankel error norms and thus also to obtain bounds for the achievable L∞ error norm for delay systems. A result on a Laguerre filter approximation problem in the Hilbert-Schmidt norm is given as well.

Robust identification and Galois sequences

Abstract Worst-case l1 identification is studied for BIBO stable linear shift-invariant systems. It is shown that the Chebyshev identification method when used with Galois input designs satisfies a certain robust convergence property and provides l1 model error bounds in worst-case identification of BIBO stable systems with a uniformly bounded noise set-up. The robust identification methodology developed is compatible with the modelling requirements of modern robust control design.

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 identification of strongly stabilizable systems

For strongly stabilizable systems for which a strongly stabilizing controller is known approximately, the authors consider system identification in the graph, gap, and chordal metrics using robust H/sub infinity / identification of the closed-loop transfer function in the framework proposed by A.J. Helmicki et al. (1990). Error bounds are derived showing that robust convergence is guaranteed and that the identification can be satisfactorily combined with a model reduction step. Two notions of robust identification of stable systems are compared, and an alternative robust identification technique based on smoothing, which can be used to yield polynomial models directly, is developed. >

Paper: Constrained linear quadratic gaussian control with process applications

Constrained linear quadratic Gaussian control is considered. Important practical design constraints including restrictions in control signal variations and in regulator structure are introduced quantitatively into the control problem formulation. Various topics in the resulting extension of the standard LQG design procedure are discussed, for instance optimality conditions, design of optimal low-order controllers and variance-constrained self-tuning control. Numerical algorithms for solving the constrained LQG control problems are given facilitating the application of the design procedure. Three industrial applications of linear quadratic Gaussian design are described. The examples are taken from the cement industry and from a process for the production of plastic film.

Comments on "Robust, fragile, or optimal?" [with reply]

The issue of fragile controllers produced by using popular robust and optimal control synthesis methods was raised by Keel-Bhattacharyya in the above mentioned paper (ibid. vol.42 (1997)). The commenter points out that the controller fragility problem described in the paper is: 1) well known in the literature; 2) due to the poor choice of examples; 3) can easily be fixed by changing the optimization criteria; and 4) can be fixed by changing the implementation. In reply, Keel-Bhattacharyya disagree with him on all counts and explain their argument.

Computer-aided design procedure for multiobjective LQG control problems

Abstract A procedure for computing a satisfactory non-inferior solution to a multiobjective LQG control problem is given. An efficient computational algorithm is developed by exploiting the structure of the LQG problem. The proposed multiobjective optimization procedure is based on a sequence of constrained min-max control problems, which are efficiently solved by applying Newtons method to an associated dual maximization problem.

Approximation of delay systems—a case study

Abstract The approximation of delay systems is studied. A numerically well-behaved method for computing Hankel optima) rational approximations for delay systems is presented based on certain properties of Hankel matrices and the theory of Laguerre polynomials. The CF method of Trefethen and certain Pade approximations of delay systems are also considered. The importance of a certain Wiener algebra property in the analysis of the rational approximation of an important class of delay systems in the L∞ norm and the Hankel norm is shown, completing certain results presented in the literature for systems satisfying certain nuclearity or absolute continuity conditions. A case study and numerical comparison is presented for the approximation of the parametric family of first-order stable delay systems G(s) = exp ( —τs)/(Ts + 1). Numerical experience indicates that L°° optimized CF approximations based on short truncated Maclaurin series give, in general, somewhat smaller L∞ approximation errors than Hankel optim...