Elena N. Gryazina

Stability regions in the parameter space: D-decomposition revisited

The challenging problem in linear control theory is to describe the total set of parameters (controller coefficients or plant characteristics) which provide stability of a system. For the case of one complex or two real parameters and SISO system (with a characteristic polynomial depending linearly on these parameters) the problem can be solved graphically by use of the so-called D-decomposition. Our goal is to extend the technique and to link it with general M-@D framework. In this way we investigate the geometry of D-decomposition for polynomials and estimate the number of root invariant regions. Several examples verify that these estimates are tight. We also extend D-decomposition for the matrix case, i.e. for MIMO systems. For instance, we partition real axis or complex plane of the parameter k into regions with invariant number of stable eigenvalues of the matrix A+kB. Similar technique can be applied to double-input double-output systems with two parameters.

D-decomposition technique state-of-the-art

It is a survey of recent extensions and new applications for the classical D-decomposition technique. We investigate the structure of the parameter space decomposition into root invariant regions for single-input single-output systems linear depending on the parameters. The D-decomposition for uncertain polynomials is considered as well as the problem of describing all stabilizing controllers of the certain structure (for instance, PID-controllers) that satisfy given H∞-criterion. It is shown that the D-decomposition technique can be naturally linked with M-Δ framework (a general scheme for analysis of uncertain systems) and it is applicable for describing feasible sets for linear matrix inequalities. The problem of robust synthesis for linear systems can be also treated via D-decomposition technique.

Mixed LMI/randomized methods for static output feedback control design

This paper addresses the problem of stabilization of LTI systems via static output feedback (sof). The objective is not only to compute a stabilizing sof but rather to compute a discrete set of stabilizing sof. Two complementary mixed LMI/randomized algorithms are defined for this purpose. The main idea is to combine a particular relaxed LMI parametrization of stabilizing sof with high efficiency of Hit-and-Run method for generating random points in a given domain. Their respective relevance is analysed on several examples of the COMPleib library which is intended to be the reference library for evaluating performance of reduced-order controller synthesis algorithms. Finally, the paper additionally provides an extensive evaluation of the different relevant instances of the COMPleib library.

RACT: Randomized Algorithms Control Toolbox for MATLAB

Abstract This paper introduces a new M atlab package, R act , aimed at solving a class of probabilistic analysis and synthesis problems arising in control. The package offers a convenient way for defining various types of structured uncertainties as well as formulating and analyzing the ensuing robustness analysis tasks from a probabilistic point of view. It also provides a full-featured framework for LMI-formulated probabilistic synthesis problems, which includes sequential probabilistic methods as well as scenario methods for robust design. The R act package is freely available at http://ract.sourceforge.net , and only requires the Y almip toolbox to be installed in the M atlab environment.

ON THE ROOT INVARIANT REGIONS STRUCTURE FOR LINEAR SYSTEMS

Abstract D -decomposition technique is targeted to describe the stability domain in parameter space for linear systems, depending on parameters. The technique is very simple and effective for the case of one or two parameters. However the geometry of the arising parameter space decomposition into root invariant regions has not been studied in detail; it is the purpose of the present paper. We prove that the number of stability intervals for one real parameter is no more than n /2 ( n being the degree of the characteristic polynomial) and provide an example, where this number is achieved. For one complex or two real parameters we estimate the number of root invariant regions (equal n 2 – 2 n + 3 for complex and 2 n 2 – 2 n + 3 for real case) and demonstrate that this upper bound is tight. The example with n – 1 simply connected stability regions in 2D parameter plane is analyzed.

Hit-and-Run: new design technique for stabilization, robustness and optimization of linear systems

Abstract New randomized algorithms for stabilization and optimal control for linear systems are proposed. They are based on Hit-and-Run method, which allows generating random points in convex or nonconvex domains. These domains are either stability domain in the space of feedback controllers, or quadratic stability domain, or robust stability domain, or level set for a performance specification. By generating random points in the prescribed domain one can optimize some additional performance index. The approach demonstrated its high efficiency for numerous classical examples of design problems.

Randomized methods based on new Monte Carlo schemes for control and optimization

We address randomized methods for control and optimization based on generating points uniformly distributed in a set. For control systems this sets are either stability domain in the space of feedback controllers, or quadratic stability domain, or robust stability domain, or level set for a performance specification. By generating random points in the prescribed set one can optimize some additional performance index. To implement such approach we exploit two modern Monte Carlo schemes for generating points which are approximately uniformly distributed in a given convex set. Both methods use boundary oracle to find an intersection of a ray and the set. The first method is Hit-and-Run, the second is sometimes called Shake-and-Bake. We estimate the rate of convergence for such methods and demonstrate the link with the center of gravity method. Numerical simulation results look very promising.

Markov Chain Monte Carlo method exploiting barrier functions with applications to control and optimization

In previous works the authors proposed to use Hit-and-Run (H&R) versions of Markov Chain Monte Carlo (MCMC) algorithms for various problems of control and optimization. However the results are unsatisfactory for “bad“ sets, such as level sets of ill-posed functions. The idea of the present paper is to exploit the technique developed for interior-point methods of convex optimization, and to combine it with MCMC algorithms. We present a new modification of H&R method exploiting barrier functions and its validation. Such approach is well tailored for sets defined by linear matrix inequalities (LMI), which are widely used in control and optimization. The results of numerical simulation are promising.

Multidimensional stability domain of special polynomial families

Consideration was given to the characteristic polynomials with special affine uncertainty. For this family, the stability domain in the parameter space was shown to be a union of polyhedra. For continuous-time and discrete-time systems, a simple method was proposed to single out the stability domain and determine the stability radius for different norms of uncertainty. Efficiency of this method was corroborated by examples.

Design of the low-order controllers by the H∞ criterion: A parametric approach

Consideration was given to the problem of describing all stabilizing controllers of a given structure (for example, the the PID-controllers) satisfying the H∞ criterion. Controllers of a certain family were defined by the parameters k, and in the parameter space a domain corresponding to the desired criteria was specified. Two approaches were proposed where (i) the desired domain is represented as an intersection of the admissible sets or (ii) its boundary is determined analytically. The two-parameter case is of special importance because it allows one to make use of the graphical mathematics.