1. Multirate Output Feedback
S. Janardhanan

2. The Problem Given a system
Give the system required degree of stability
Implementation of control is generally by computer (now a days)

3. Discrete-time Control
IF Computer being used in control implemetation
Closed loop is discrete-time
Better use discrete-time control rather than continuous time control

4. Controller - Options
For given (state space representation) system , State feedback based control would give best performance .
Problem : Availability of state information
Static Output Feedback
Good : Output is available
Problem : No guarantee of success for a controllable and observable system.
V. L. Syrmos, C. T. Abdallah, P. Dorato and K. Grigoriadis, “Static output feedback-A survey”, Automatica, Vol 33, No. 2, pp , Feb. 1997

5. Options … Dynamic Output Feedback
Good : Output + Stability Guaranteed
Problem : More dynamics in closed loop
My Option - Multirate Output Feedback  Controller input (or system output) and controller output (control input) are sampled at different rates.

6. Multirate Output Feedback
Control input sampled faster than System output.
Periodic Output Feedback
System output sampled faster than control input.
Fast Output Sampling

7. On Periodic Output Feedback
A. B. Chammas and C. T. Leondes, “Pole placement by Piecewise constant output feedback”, International Journal of Control, pp

8. Periodic Output Feedback Control Law
Plant described by equations
(A,B) controllable and (A,C) observable
Output sampled at a rate of  sec
Input applied at a rate = /N sec
Control Input is derived as

9. Periodic Output Feedback – Control Law Deduction
Let the system sampled at rate 1/ be represented {,,C}, then the closed loop system is
N>controllability index of system is sufficient condition for existence of a K
Solution for K is found by solving K=G
G is the Output injection gain such that the dual system
is stable

10. Periodic Output Feedback – Control Law (Contd.)
Linear equation
has solution if

11. Periodic Output Feedback – LMI Formulation
Solving may give a gain that is high in magnitude, amplifying noise in practical system.
Hence impose condition for noise reduction and for stability during controller design
Restrictions posed as LMI problem

12. On Fast Output Sampling

13. Some Literature
The concept of multirate output feedback (output faster than input) is quite old.
G. M. Kranc, “Input-output analysis of multirate feedback systems”, IRE Trans. Auto. Contr., Vol. 3, No. 1, pp , Nov
E. Jury, “A note on multirate sampled data systems”, IRE Trans. Auto. Contr., Vol. 12, No. 3, pp , Jun
P. T. Kabamba, “Control of linear systems using generalized sampled data hold functions”, IEEE Trans. Auto. Contr. , Vol. 32, No. 9, pp , Sept
T. Hagiwara and M. Araki, “Design of a stable state feedback controller based on multirate sampling of plant output”, IEEE Trans. Auto. Contr., Vol. 33, No. 9, pp , Sept
H. Werner and K. Furuta, “Simultaneous stabilization based on output measurement”, Kybernetika, Vol. 31, pp , 1995.

14. Visualization

15. A Multirate Output Sampled System
Let (,,C) denote the system sampled at interval  and (,,C) for sampling interval . System is then represented by
Assume the existence of a SFB gain F such that (+F) is stable with no eigenvalues at origin

16. Fast Output Sampling – Controller Deduction Contd.
Define fictitious matrix C(F,n)=(C0+D0F)(+F)-1 which satisfies the measurement equation yk=CXk
Feedback law can be interpreted as uk=Lyk
where L = [L0 L1 .. LN-1] is the FOS controller gain and
yk = [y(k- ) y(k- +) .. y(k- )]T the last N output samples.
This requires L to satisfy F = LC
For 0<t  , uk =FX0
For t >  , uk =Lyk
Define fictitious measurement matrix --THIS--
satisfying Eqn
Feedback law u_{k} = L y_{k}
L=[..] is FOS controller Y_{k}=[..]
for this F=LC
Application--
u=FX t<Tau
u=Ly t>Tau

17. Initial State Estimation Consideration
Initial Control Signal u=FX0
X0 is assumed and control input calculated
Error due to assumption causes error in input u(k)
Input error Governed by the dynamics
(L)=LD0-F should have eigenvalues within unit disc for stability
For faster decay eigenvalues should be close to zero.
Constraint incorporated in design of L as

18. Noise Considerations L derived from LC=F
May give control signal of small magnitude even if L’s elements are large in magnitude Theoretically.
Does not imply good control as Noise Amplification is imminent in practical case
Therefore, Constraint to be considered while designing L

19. LMI Formulation of the Design Problem
Approximate Solution LCF
helps satisfying above constraints
Effect on stability is minimal
LC F can be put as
Constraints posed as LMI problem as

20. i.e.,
Choose a sampling time  sec, at which Output is sampled and =N  sec during which the Input is held constant
Control signal u(t) for k  < t (k+1)  is calculated as linear combination of past N Outputs
Controller Gains are adjusted so as to take care of
Noise amplification
Initial Control Error
Closeness to desired State Feedback

21. Modified Multirate Output Feedback Technique
What if control is not a linear state feedback ?
New algorithm

22. Improved Algorithm : Multirate Output Feedback
Consider the output equation
From this the state can be computed as
This computed state can be used for implementing any state feedback based control

23. Visualization

24. Advantage Output feedback based
Inherent control computation time. Useful for fast systems
As compared to FOS
No restriction on F. In fact, no restriction on u=Fx !
Initial control error does not propagate.

25. NEXT
Multirate Output Feedback based Sliding Mode Control