Section: New Results

Interval control and estimation

In many cases due to parametric and/or signal uncertainties presented in a plant model it is not possible to design a conventional observer, which provides a point-wise estimate of state in a finite time or asymptotically. In this case it is still frequently possible to design interval observers, which generate an estimate on the interval of the admissible values of the state at the current instant of time. The recent new results in this field are listed below:

• The work [49] is devoted to interval observer design for Linear Parameter-Varying (LPV) systems under assumption that the vector of scheduling parameters is not available for measurements. Stability conditions are expressed in terms of matrix inequalities, which can be solved using standard numerical solvers. Robustness and estimation accuracy with respect to model uncertainty is analyzed. Two solutions are proposed for nonnegative systems and for a generic case. The efficiency of the proposed approach is demonstrated through computer simulations.

• Development of interval observers for time invariant [55] and time-varying [21] discrete-time systems has been presented by the members of the team.

• Interval estimation for uncertain systems with time-varying delays has been considered in [22] , [56] . A reduced-order interval observer has been designed, stability and robustness conditions have been obtained.

• The paper [24] is devoted to design of interval observers for Linear Time Varying (LTV) systems and a class of nonlinear time-varying systems in the output canonical form. An interval observer design is feasible if it is possible to calculate the observer gains making the estimation error dynamics cooperative and stable. It has been shown in [24] that under some mild conditions the cooperativity of an LTV system can be ensured by a static linear transformation of coordinates.

• The problem of output stabilization of a class of nonlinear systems subject to parametric and signal uncertainties has been studied in [25] . First, an interval observer has been designed estimating the set of admissible values for the state. Next, it has been proposed to design a control algorithm for the interval observer providing convergence of interval variables to zero, that implies a similar convergence of the state for the original nonlinear system. An application of the proposed technique shows that a robust stabilization can be performed for linear time-varying and LPV systems without assumption that the vector of scheduling parameters is available for measurements.

• The paper [26] deals with the problem of joint state and parameter estimation based on a set adaptive observer design. The problem is formulated and solved for an LPV system. The resolution methodology avoids the exponential complexity obstruction usually encountered in the set-membership parameter estimation.

• The output stabilization problem for a linear system with an unknown bounded time-varying input delay has been considered in [34] , [76] . The interval observation technique has been applied in order to obtain guaranteed interval estimate of the system state. The procedure of the interval observer synthesis uses lower and upper estimates of the unknown delay and requires to solve a special Silvester's equation. The interval predictor has been introduced in order to design a linear stabilizing feedback. The control design procedure is based on LMIs.

• The paper [37] describes a robust set-membership-based Fault Detection and Isolation (FDI) technique for a particular class of nonlinear systems, the so-called flat systems. The proposed strategy consists in checking if the expected input value belongs to an estimated feasible set computed using the system model and the derivatives of the measured output vector. The output derivatives are computed using a numerical differentiator. The set-membership estimator design for the input vector takes into account the measurement noise thereby making the consistency test robust.

• The objective of the work [82] is to develop some design methods of interval observers for a class of nonlinear continuous-time systems. It has been assumed that the estimated system can be represented as a superposition of the nominal subsystem (belonged to the class of uniformly observable systems) and a Lipschitz nonlinear perturbation vanishing at the origin. Then it has been shown that there exists an interval observer for the system that estimates the set of admissible values for the state consistent with the output measurements.