Interval-based sliding mode observer design for nonlinear systems with bounded measurement and parameter uncertainty

Abstract

The estimation of non-measurable state variables as well as the reliable identification of unknown system parameters are important prerequisites for the design and implementation of controllers for nonlinear dynamic systems. However, these tasks are often impeded by the nonlinearity of dynamic system models as soon as observer techniques are sought for, which can be used for large operating ranges. Moreover, parameters and measured data are typically only known within given tolerance bounds. Such uncertainty makes the proof of the asymptotic stability of the error dynamics of classical state observers quite difficult. Therefore, a novel interval-based sliding mode observer providing point-valued estimates is presented in this paper which is designed in such a way that asymptotic stability can be guaranteed by means of an online evaluation of a suitable Lyapunov function. Furthermore, an efficient strategy for the adaptation of the switching amplitude of the observer's variable structure part is presented to reduce the amplification of measurement noise as far as possible if time-varying state variables and time-invariant system parameters are estimated simultaneously. An illustrative example, describing the longitudinal dynamics of a vehicle, is presented to highlight the practical applicability of the observer.

Optimal input design for online state and parameter estimation using interval sliding mode observers

“…In this section, a summary of the interval-based sliding mode observer is given, which is used for the simultaneous estimation of non-measured state variables as well as for the online identification of uncertain system parameters [6]. For this purpose, the nonlinear dynamic systeṁ…”

“…is taken into account, where x(t) ∈ R n x is the state vector, u(t) ∈ R p is a vector of control inputs, A as well as B are constant system as well as input matrices, and the product S · w (x(t), u(t)) contains all nonlinearities of the system model [6]. These inputs are either determined by a closedloop controller or -as described in Sec.…”

“…IV -by an optimization procedure aiming at an improvement of the identifiability of system parameters by means of exciting the relevant system dynamics. According to [6], the observer parallel model for the system (1) is then defined bŷ…”

“…These three optimization procedures are validated for an interval sliding mode observer providing point-valued estimates. This observer is designed in such a way that asymptotic stability can be guaranteed despite bounded uncertainties in every time step using a suitable Lyapunov function [6]. The efficiency of this approach is shown for a test rig, which consists of an electric drive that is connected to a brake via a toothed belt.…”

Reliable NonLinear Model-Predictive Control via Validated Simulation

“…Interval analysis (IA) has been already used in control field [6]. More precisely, IA can be a powerful way to develop MPC methods such as in [7,8] for discrete-time plant or as in [2,9,10,11,12] for continuous-time model of the plant, see Section 3 for more details.…”