Discrete-time Sliding Mode Control S. Janardhanan

Rethink: Quasi –Sliding mode Advantage : Is in discrete-time, more practical Disadvantage : Is Quasi. Not Exactly Sliding Mode There is always chattering. Even in theory.

Why the chattering ? QSMC is derived from ‘discretizing’ continuous SMC logic. The signum term is required in CSMC

Chattering .. Why ? In discrete-time system Result : Always chattering The sign(s) changes abruptly near s=0 Control cannot be changed at any time Result : Always chattering

Fresh Approach Let us again see the aim of sliding mode control. But, now in discrete-time systems Aim : To get the system to the sliding surface and maintain the state on the surface To get to the surface : s(k+1)=0 (with s(k)  0) To maintain : Again s(k+1)=0

Discrete-time sliding mode So, apparently the reaching law is simply s(k+1)=0 For this the control would be

Problem The control action Problem Solution Brings the system to sliding mode in one step Keeps RP on sliding surface thereafter. Problem Probability of too much control in the ‘one step’ Solution GO SLOW But, be sure you are going forward

Mathematically speaking … Let the maximum applicable control be u0 Use the control as

To go in right direction The control u1 tries to make s(k+1)=0 But, it may require too much ‘effort’. So, with limited resources we can apply a maximum of only u0. Now, to be sure this is ‘enough’. Ie., we are moving forward, we get the condition

The condition Thus, the condition for this logic to work is See : G. Bartolini, A. Ferrara and V. Utkin, “Adaptive sliding mode control in discrete-time systems”, Automatica, Vol. 31, No. 5, pp. 769-773, May 1995.

Extension to Uncertain Systems The same algorithm can be extended to systems with matched uncertainty. In this case, the condition would become

Another Algorithm Another approach to the DSMC problem is to design a control algorithm that would inherently ‘go slow’. Instead of trying to reach the surface in one step. Try to reach it in say k* steps.

The Problem Statement Consider the design of DSMC for system Bartoszewicz proposed the reaching law cTx=s(k) is the desired sliding surface See : A. Bartoszewicz, “Discrete-time quasi-sliding mode control strategies”, IEEE Trans. Ind. Electron., Vo. 45, No. 1, pp. 633-637, 1998

Bartoszewicz QSMC … sd is an a priori known function such that  sd(k) is always of the same sign

Feasible Sd(k) A feasible function for sd(k) that satisfies the above said conditions can be given as

Bartoszewicz Control Law The control input is of the form

Proof of Convergence The Bartoszewicz control law satisfies the reaching condition For hence and to s(k)=0 in deterministic systems atleast in theory.

Proposed Algorithm Combines Bartoszewicz’s approach and FOS to get multirate output feedback based QSMC control strategy that does not have switching function and hence no chatter. Reaching Law Modified

Proposed Control Law The Control is of the form

Proof of Convergence The proposed control law satisfies the reaching condition For Hence and to s(k)=0 in deterministic systems atleast in theory.