Sliding Mode Observer Design for a Parabolic PDE in the Presence of Unknown Inputs

Abstract

This paper considers observer design for systems modeled by linear partial differential equations (PDEs) of parabolic type, which may be subject to unknown inputs. The system is assumed to have only one spatial dimension, over which it is discretised to obtain what is referred to as the lattice system, which is a set of linear time invariant (LTI) ordinary differential equations (ODEs) having a canonical Toeplitz-like structure with a specific sparsity pattern. This lattice structure is shown to be particularly appropriate for step-by-step sliding mode observer design that can reconstruct the state estimates at the points of discretisation and estimate the unknown input. Simulation results for both stable and unstable PDEs show that accurate state estimates can be provided at the points of discretisation. An approach to reconstruct the unknown input is demonstrated.

A Concise Review of State Estimation Techniques for Partial Differential Equation Systems

“…In the design process of the controller, a sliding mode-based state derivative observer is constructed, which estimates the derivative of the spatial variable. More recently, several works [80][81][82] have expanded these results and suggested sliding mode observers for a specific class of DPSs considering only a limited number of online available measurements. In this connection, the assessment of the asymptotic convergence of the observation error using Lyapunov arguments plays a fundamental role.…”

Output feedback stabilization of cascaded ODE‐Wave equations with time delay in observation

“…In control system, cascaded PDE-ODE and PDE-PDE systems describe fundamental laws of physics and mechanic, such as ODE-Schrödinger equation [1][2][3], ODE-Heat equation [4][5][6][7], ODE-Wave equation [6,8,9], coupled strings [10], coupled Timoshenko beam [11] etc. The stabilization of systems utilizing PDEs subject to time delay is drawn more attention [12][13][14][15][16][17].…”

“…, there are two positive constants M 1 and M 2 > 0 such thatM 1 ||(X(t), w(•, t), w t (•, t))|| ≤ ||(X(t), u(•, t), u t (•, t))|| ≤ M 2 ||(X(t), w(•, t), w t (•, t))||. (69)This together with (68) yields||(X(t), u(•, t), u t (•, t))|| ≤ M 2 ||(X(t), w(•, t), w t (•, t))|| ≤ M 2 C e − t [||(X( ), w(•, ), w t (•, ))|| +C 0 ||B(Z(0), (•, 0), s (•, 0))|| 2 ] , ≤ M 2 C e − t [ M −1 1 ||(X 0 , u 0 , u 1 )|| +C 0 ||B(Z(0), (•, 0), s (•, 0))|| 2 ], which gets(5). The proof is complete.Remark When we take initial condition of observer(17):(X 0 ,û 0 ,û 1 ) = (X 0 , u 0 , u 1 ), then (Z(0), (•, 0), s (•, 0)) = (0, 0, 0), and (51) reduces to the usual exponential stability:||(X(t), u(•, t), u t (•, t))|| ≤ M 2 M −1 1 C e − t ||(X 0 , u 0 , u 1 )||.…”

Backstepping design for a class of coupled parabolic PDEs with spatially varying coefficient

“…Next, an adaptive boundary control is developed for vibration suppression of an anaxially moving accelerated or decelerated belt system in [21]. Also, sliding mode observer design for systems modelled by parabolic PDEs equation with unknown input is proposed in [22].…”