Section: New Results

Set-Theoretic Methods of Control And Estimation

Interval and ellipsoidal estimations can be regarded as particular finite-time algorithms, since they provide guaranteed estimates of the values from the intial time. We develop these tools for some years now.

• An approach to interval observer design for Linear Parameter-Varying (LPV) systems is proposed in [20] . Stability conditions are expressed in terms of matrix inequalities. Applying L1/L2 framework the robustness and estimation accuracy with respect to model uncertainty are analyzed.

• New delay-dependent conditions of positivity for linear systems with time-varying delays are introduced in [56] . These conditions are applied to interval observer design for systems with time-varying delays in the state equations and in the measurements. In [28] the problem of interval observer design is addressed for a class of descriptor linear systems with time delays. An interval observation for any input in the system is provided. The control input is designed together with the observer gains in order to guarantee interval estimation and stabilization simultaneously. Efficiency of the proposed approach is illustrated by numerical experiments with Leontief delayed model.

• The work [29] is devoted to interval observers design for discrete-time Linear Parameter-Varying (LPV) systems under the assumption that the vector of scheduling parameters is not available for measurements. Two problems are considered: a pure estimation problem and an output stabilizing feedback design problem where the stability conditions are expressed in terms of Linear Matrix Inequalities (LMIs).

• The paper [48] investigates the interval observer design for a class of nonlinear continuous systems, which can be represented as a superposition of a uniformly observable nominal subsystem with a Lipschitz nonlinear perturbation. It is shown in this case there exists an interval observer for the system that estimates the set of admissible values for the state consistent with the output measurements. In [77] similar methodology is extended to singular systems.

• A finite-time version, based on Implicit Lyapunov Functions, for the Attractive Ellipsoid Method is developed in [65] . Based on this, a robust control scheme [36] is presented to ensure finite-time convergence of the solutions of a chain of integrators with bounded output perturbations to a minimal ellipsoidal set. The control parameters are obtained by solving a minimization problem of the " size " of the ellipsoid subject to a set of Linear Matrix Inequalities, and by applying the implicit function theorem.

• In [78] we consider a problem of sliding mode control design for LTI systems with multiplicative disturbances of the input and noisy measurements of the output. We apply the minimax observer to provide the best possible estimate of the system's state. Then we solve a problem of optimal reaching for the observer: we design sub-optimal control algorithms generating continuous and discontinuous feedback controls that steer the observer as close as possible to a given sliding hyperplane in a finite time.