Tracking of LTI Systems with Unstable Zero Dynamics using Sliding Mode Control S. Janardhanan

Problem Statement Consider the nth order system with relative degree of r, What is relative degree?

Two-Part Control – Nominal Control Assuming the system output is following the reference trajectory Thus the nominal control is

Two-Part Control – Incremental Control The incremental input Du(k) is designed in such a way as to stabilize the error dynamics, and bring the states to the reference. A error based incremental DSMC control Du(k) = FDx(k) + gsign(s(k)) s(k)=cT Dx(k) is used here

Two Part Control Thus, the total control input (state feedback based) would be

Determination of Nominal Zero Dynamics Trajectory (x2,0) When the reference is being tracked perfectly, the system dynamics is of the form without knowledge of initial state of x2,0, the future value cannot be calculated

Diagonal Transformation Define a transformation T2 such that the following are satisfied.

Reference Computation Thus we calculate the zero dynamics reference as

Assumptions, Restrictions … It is assumed that z(k)=0, k<0 It is assumed that z(k) is such that it keeps the marginally stable dynamics bounded It is assumed that the ‘flow’ of z(k) is pre-determined and known. The zero dynamics reference are null in both infinite past and infinite future.

Bounded Preview The calculation of the xu dynamics requires z(k) up to infinity : impractical Algorithm is modified for a ‘bounded preview’. This keeps error in zero dynamics reference.

Output Feedback Version The multirate output feedback version of the aforementioned control algorithm is of the form

Numerical Example to track

Simulation Results : Control Input

Simulation Results : Reference and Response

Simulation Results : Tracking Error

Simulation Results : Bounded Zero Dynamics