Last modified November 27, 2007.

Seminars

Seminar schedule (Fall 2007)

The talk is devoted to the new MATLAB Toolbox, which is called RACT (Randomised Algorithms Control Toolbox). This package implements the randomized algorithm procedures for solving the Linear Matrix Inequalities that appeared in control problems. Some questions and perspectives are discussed.

This is the first seminar of the three lecture minicourse devoted to the new approach to the problems of analysis and synthesis of linear control systems. This approach is based on the technique of invariant ellipsoids and linear matrix inequalities routine. Some problems of control that can be treated with this approach are discussed.

Many robust control problems can be formulated in abstract form as convex feasibility programs, where one seeks a solution vector x that satisfies a set of inequalities of the form F = { f(x,δ) ≤ 0,  δ from D }. This set typically contains an infinite and uncountable number of inequalities, and it has been proved that the related robust feasibility problem is numerically hard to solve in general. In this talk, we discuss a family of cutting plane methods that solve efficiently a probabilistically-relaxed version of the problem. Specifically, under suitable hypotheses, we show that an Analytic Center Cutting Plane scheme based on a probabilistic oracle returns in a finite and pre-specified number of iterations a solution x which is feasible for most of the members of F, except possibly for a subset having arbitrarily small probability measure.

In this talk we present the filtering problem under nonrandom disturbances. The method of invariant ellipsoids is proposed for its solution. The problem of robust filtering is considered. The continuous-time and discrete-time cases are studied. As an example of the approach the robust disturbance rejection problem is considered.

This is the final lecture on the method of invariant ellipsoids in control. The problem of persistent disturbance rejection via output feedback is considered. The algorithm of static state-feedback design is constructed, which minimizes the size of invariant ellipsoids of dynamical system for the output control. For these purposes the Lyuenberger's observer-based state estimate is used. The continuous-time and discrete-time cases are studied.

In this talk Akkerman and Bassa-Gourr formulas are extended to linear MIMO systems. These formulas is used modal synthesis of controllers in state-feedback for linear time-invariant systems. Conditions are derived for the parametrization of feedback laws and possible output sets. Illustrative examples are given that show the effectiveness of new extended formulas.

This seminar is devoted to the parameter estimation problem for linear time-invariant systems under model uncertainty. All uncertainties are assumed to be unknown-but-bounded. The presence of model uncertainty (multiplicative uncertainty) leads to the nonconvexity of feasible parameter sets of the system. The ellipsoidal technique is used as the main tool to estimate the vector of system parameters, i.e. to construct outer-bounding approximations of nonconvex feasible parameter sets. We reformulate this problem as Semi-Definite Programming and use the LMI tools for its solution.

The problem of the (robust) controller design for linear dynamic system guaranteeing the given phase constraints is considered. The constraints are transformed into symmetric-to-origin form (to the form of rectangular parallelepiped). In this case it is shown that the satisfaction of given constraints leads to the conic constraints for matrix rows of closed-loop system. The problem is rewritten in the form of linear equations for controller parameters. Some particular simple cases of the problem are considered.

The problem of reduction of Rozenbroek matrix is considered for the analysis of invariant zeros of linear multi-dimensional MIMO system. The proposed method is based on the so-called zeros divisor. This method is compared with another approaches for Rozenbroek matrix reduction; its advantages are emphasized (simple realization, better conditionality of the problem). Some properties of redused Rozembroek matrix is analysed.

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