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

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

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

4. 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

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

6. 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

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

8. Reference Computation
Thus we calculate the zero dynamics reference as

9. 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.

10. 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.

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

12. Numerical Example
to track

13. Simulation Results : Control Input

14. Simulation Results : Reference and Response

15. Simulation Results : Tracking Error

16. Simulation Results : Bounded Zero Dynamics