On polyhedral control synthesis for dynamical discrete-time systems under uncertainties and state constraints

Abstract

We deal with a problem of target control synthesis for dynamical bilinear discrete-time systems under uncertainties (which describe disturbances, perturbations or unmodelled dynamics) and state constraints. Namely we consider systems with controls that appear not only additively in the right hand sides of the system equations but also in the coefficients of the system. We assume that there are uncertainties of a set-membership kind when we know only the bounding sets of the unknown terms. We presume that we have uncertain terms of two kinds, namely, a parallelotope-bounded additive uncertain term and interval-bounded uncertainties in the coefficients. Moreover the systems are considered under constraints on the state ("under viability constraints"). We continue to develop the method of control synthesis using polyhedral (parallelotope-valued) solvability tubes. The technique for calculation of the mentioned polyhedral tubes by the recurrent relations is presented. Control strategies, which can be constructed on the base of the polyhedral solvability tubes, are proposed. Illustrative examples are considered.2010 Mathematics Subject Classification. Primary: 93C41, 93C55, 93B52, 93C05, 93C10; Secondary: 93B40, 52B12.

On Solving an Enhanced Evasion Problem for Linear Discrete–time Systems

“…We say that a parallelotope P is nondegenerate if m = n and det P = 0. Note that each parallelepiped P(p, P , π) is a parallelotope P[p, P ], where P = P •diag π, and each nondegenerate parallelotope P[p, P ] is a parallelepiped P(p, P , π) with Recall that in [11,12] the solutions to terminal target polyhedral approach problems are given even for more general classes of systems, namely, for systems (2.1) with uncertainties / controls in the matrices A[k] and with state constraints. There the families of the tubes…”

“…via control strategies v; v; u respectively. To construct these controls v; v; u we will use the solutions to Problem 2 ′ ; to Problem 2; to the terminal target approach problem from [11,12] through construction of several tubes P +,α [•]; P β [•]; P −,γ [•] from parametric families of the tubes described in [14, Theorem 1] and also in Corollary 1; in Theorem 2; in [11,12] respectively (see [14, the end of Sec. III] about using the families of the tubes for more details).…”

“…Note that in Test 2 the aim A 1 u is not opposite to A 2 v (constructing u with the aim A 2 u opposite to A 2 v is out of the scope of this paper). It is possible to use several formulas to construct u basing on tubes P −,γ [•] (see, for example, [12]). Here we have applied the formulas which are similar to [13,Formula (13)].…”

“…Of the many works, we only indicate as examples [1-15, 17-22, 24], including those based on estimation of sets by more simple sets such as ellipsoids [1,2,[4][5][6][17][18][19][20] and parallelepipeds/parallelotopes [9-15, 21, 22]. In particular, for discretetime systems, ellipsoidal and polyhedral solution schemes have been developed for the terminal target approach problem and the terminal evasion problem at a given final time (see, for example, [2,11,12,20] and [14] for both problems respectively). But two methods presented in [14] guarantee the evasion from the given set only at the given final time.…”

Control design with guaranteed transient performance: An approach with polyhedral target tubes

“…They show that the set of feedbacks is convex, but the approach requires exact knowledge of all past states and inputs and an accurate model of the system. The work [26] considers the problem for a class of uncertain discrete-time systems under the assumption of parallelepipeds in the problem data. An iterative algorithm to solve the control synthesis problem is then presented.…”