1. St. Petersberg
July 5, 2001

2. Thanks to Alexander Kurzhanski and Alexander Fradkov
for inviting me to NOLCOS
and
RUSSIA ….. the home of my ancestors

3. CONTROL OF HIGHLY UNCERTAIN
SYSTEMS USING
FAST SWITCHING
Yale University
A. S. Morse

4. OUTLINE
Definition of a firm linear system
Switching theorem
Application
Outline of Switching Theorem’s proof

5. FIRMNESS
For a linear system S with coefficient matrix triple {An£ n, Bn£ m, Cq£ n}, let R be the
largest (A, B) – controllability subspace in kernel C .
Call S firm if the zero subspace is the only A – invariant subspace contained in R.
Thus S = {A, B, C} is firm if (Q, A) is an observable pair, Q being any matrix
with kernel Q = R.
Every linear system with left-invertible transfer matrix is firm.
Any unobservable eigenvalue of a firm system must be one of the system’s
transmission zeros.
Thus a firm, detectable system whose transmission zeros are all unstable, must be
observable.

6. Suppose {Fp: p 2 P } is a closed bounded subset of matrices in Rm£n
SWITCHING THEOREM
Suppose {Fp: p 2 P } is a closed bounded subset of matrices in Rm£n
with the property that for each p 2 P, {A+BFp, B, C} is the coefficient
triple of a firm, detectable system. Then for each positive number tD,
there is a bounded output-injection function p a Kp which, for any
piecewise constant switching signal s : [0, 1 ) ! P whose
discontinuities are separated by at least tD time units, exponentially
stabilizes the matrix A+Ks C+BFs tD t1 t2 t3
s = p1
s = p2
s = p3

7. Suppose {Fp: p 2 P } is a closed bounded subset of matrices in Rm£n
SWITCHING THEOREM
Suppose {Fp: p 2 P } is a closed bounded subset of matrices in Rm£n
with the property that for each p 2 P, {A+BFp, B, C} is the coefficient
triple of a firm, detectable system. Then for each positive number tD,
there is a bounded output-injection function p a Kp which, for any
piecewise constant switching signal s : [0, 1 ) ! P whose
discontinuities are separated by at least tD time units, exponentially
stabilizes the matrix A+Ks C+BFs
Why is this theorem useful?
Why is it true?

8. P The Underlying Problem
Given a SISO process P with open-loop control input u, disturbance
input d, and sensed output y. Devise a controller, which achieves
“input-to-state” stability with respect to d. d P u y

9. d The Underlying Problem
Given a SISO process P with open-loop control input u, disturbance
input d, and sensed output y. Devise a controller, which achieves
“input-to-state” stability with respect to d. d
nominal transfer functions +
n1 or n2 + + u y + d
norm bounded
unmodelled dynamics

10. np kp CANDIDATE CONTROLLER TRANSFER FUNCTIONS
Take as given candidate controller transfer functions, k1 and k2 ,
designed so that for each p 2 {1, 2}, kp at least stabilizes the loop
nominal transfer function p np kp
with stability margin l. Here l is a design parameter.

11. || || + - y y1 e1 m1 d E S s y
|| || + - y y2 e2 m2 u P Cs

12. || || + - y y1 e1 m1 d E S s y
|| || + - y y2 e2 m2 u P Cs

13. || || + - y y1 e1 m1 d E S s y u P
|| || + - y y2 e2 m2 Cs
Multi-estimator E is a two-input stable linear system with stability margin
l, designed so that for each p 2 {1, 2}, yp would be an asymptotically correct
estimate of y, if d were zero and candidate nominal process transfer function np were P’s transfer function.

14. E S P Cs - s - Piecewise-constant switching
|| || + - y y1 e1 m1 d E S s y u P
|| || + - y y2 e2 m2 Cs
Piecewise-constant switching
signal taking values in {1, 2}.

15. || || + - y y1 e1 m1 d E S s y
|| || + - y y2 e2 m2 u P
Multi-controller Cs designed in such a way so that for each
fixed s = p 2 {1, 2}, Cp
realizes candidate controller transfer function kp and is detectable with stability margin l Cs
type 1
AC 2n-dim and stable.
(fC, AC) n-dim, stable, observable
type 2

16. E S P Cs - s - type 1 type 2 y y1 e1 m1 d y y u y2 e2 m2 || || + || ||
|| || + E S s y P y u y2 - e2 m2
|| || + Cs
type 1
type 2

17. y y1 - e1 m1 d
|| || + E S s y P y u y2 - e2 m2 + Cs
type 1
type 2

18. E S P Cs Design parameter: Dwell-time tD > 0.y
Switching logic S sets s (t) to the index of the smallest mi(t), provided tD time units have elapsed since the last time s’s value was changed. Otherwise,
S does nothing. y1 - e1 m1 d
|| || + E S s y y u P y2 - e2 m2
|| || + Cs
type 1
type 2

19. E S P Cs ASSUME n1 is P’s nominal transfer function - s - type 1d
|| || + E S s y P y u y2 - e2 m2
|| || + Cs
type 1
type 2

20. . s(t)=2, t 2 [tj, tj+1) tD tj+1 tj
m2(T) · m1(T), T=tj and T 2 [tj +tD, tj+1)

21. . s(t)=2, t 2 [tj, tj+1) tD tj+1 tj
m2(T) · m1(T), T=tj and T 2 [tj +tD, tj+1) . tD
tj+1 tj
e2l T||e2||2T · e2l T||e1||2T, T=tj and T 2 [tj +tD, tj+1)
||e2||T · ||e1||T, T=tj and T 2 [tj +tD, tj+1)

22. . s(t)=2, t 2 [tj, tj+1) tD tj+1 tj
For T 2 [tj, tj+1) define signal
||e2||T · ||e1||T, T=tj and T 2 [tj +tD, tj+1)

23. E S P Cs ASSUME n1 is P’s nominal transfer function - s -y y1 - e1 m1 d
|| || + E S s y P y u y2 - e2 m2
|| || + Cs
Multi-estimator E is a two-input stable linear system with the property that
y1 would be an asymptotically correct
estimate of y, if d and d were both zero.
type 1 d e1 +
h1(s) u
h2(s)
d(s)
type 2

24. E S P Cs - s - type 1 d e1 type 2 u y y1 e1 m1 d y y u y2 e2 m2 d(s)
|| || + E S s y u P y y2 - e2 m2
|| || + Cs
type 1 d e1 +
h1(s) u
h2(s)
d(s)
type 2

25. E S P Cs For each p 2 {1, 2}, (c, A +BFp) is detectable.
Want to stabilize A+BFs by output injection using cx = e_2 –e_1. y y1 - e1
For each p 2 {1,2} there is an hp which stabilizes A+BFp+hpc. m1 d
|| || +
For tD sufficiently large, A+BFs+hs c is exponentially stable. E S s y y u P y2 - e2 m2
|| || + Cs d e1 +
h1(s) u
h2(s)
d(s)

26. E S P Cs For each p 2 {1, 2}, (c, A +BFp) is detectable.
Want to stabilize A+BFs by output injection using cx = e_2 –e_1. y y1 -
Each {A+BFp, B,c} is firm. e1 m1 d
|| || +
Switching Theorem applies. E S s
For any tD > 0, no matter how small, there is a ks which stabilizes A+BFs+ks c. y u P y
The matrix M + A+BFs + ks(1-yT )c is also stable because yT is L1. y2 - e2 m2
|| || + Cs d e1 +
h1(s) u
h2(s)
d(s)

27. E S P Cs - s The matrix M + A+BFs + ks(1-yT )c is stable. - d e1 u y
|| || + E S s y P y u
The matrix M + A+BFs + ks(1-yT )c is stable. y2 - e2 m2
|| || + Cs d e1 +
h1(s) u
h2(s)
d(s)

28. E S P Cs - s - The matrix M + A+BFs + ks(1-yT )c is stable. d e1 u y
|| || + E S s y P y u y2 - e2 m2
|| || +
The matrix M + A+BFs + ks(1-yT )c is stable. Cs d e1 +
h1(s) u
h2(s)
d(s)

29. E S P Cs - s - (1-yT)e2 y stable e1 u d e1 u y y1 e1 m1 d y y u y2 e2
|| || + E S s y y u P y2 - e2 m2
|| || + Cs
(1-yT)e2 y
stable e1 u d e1 +
h1(s) u
h2(s)
d(s)

30. E S P Cs - s - (1-yT)e2 y stable e1 d u y y1 e1 m1 d y y u y2 e2 m2
|| || + E S s y y u P y2 - e2 m2
|| || + Cs
(1-yT)e2 y
stable e1 + d
h1(s) + u
h2(s)
d(s)

31. E S P Cs - s - (1-yT)e2 y stable e1 d u y y1 e1 m1 d y y u y2 e2 m2 1
|| || + E S s y y u P y2 - e2 m2
|| || + Cs
(1-yT)e2 y e1 1
stable + d
h1(s) + u
h2(s)
d(s)

32. Suppose {Fp: p 2 P } is a closed bounded subset of matrices in Rm£n
SWITCHING THEOREM
Suppose {Fp: p 2 P } is a closed bounded subset of matrices in Rm£n
with the property that for each p 2 P, {A+BFp, B, C} is the coefficient
triple of a firm, detectable system. Then for each positive number tD,
there is a bounded output-injection function p a Kp which, for any
piecewise constant switching signal s : [0, 1 ) ! P whose
discontinuities are separated by at least tD time units, exponentially
stabilizes the matrix A+Ks C+BFs
Why is this theorem useful?
Why is it true?

33. Suppose {Fp: p 2 P } is a closed bounded subset of matrices in Rm£n
SWITCHING THEOREM
Suppose {Fp: p 2 P } is a closed bounded subset of matrices in Rm£n
with the property that for each p 2 P, {A+BFp, B, C} is the coefficient
triple of a firm, detectable system. Then for each positive number tD,
there is a bounded output-injection function p a Kp which, for any
piecewise constant switching signal s : [0, 1 ) ! P whose
discontinuities are separated by at least tD time units, exponentially
stabilizes the matrix A+Ks C+BFs
Why is this theorem useful?
Why is it true?

34. Let {(C, Ap), p 2 P} be a closed, bounded set of observable matrix pairs. Then for
any tD > 0, there exists and output injection Kp which exponentially stabilizes
As +KsC for any piecewise constant switching signal s:[0,1)! P with dwell time
no smaller than tD.
NOT TRUE with only detectability – consider the case C = 0.
Let {Ap, p 2 P} be a closed, bounded set of constant n £ n matrices. Suppose
each Ap is stable and let Tp and lp be positive numbers for which
Let tD be any number satisfying tD > Tp, p 2 P. Then for any piecewise constant
switching signal s:[0,1)! P with dwell time no smaller than tD, As is exponentially
stable.
Let (C, A) be a fixed, constant, observable matrix pair. For each positive number T there exists a positive number l and a constant output-injection matrix K for which
|e(A+KC)t| · e-l (t-T), t ¸ 0

37. CONCLUDING REMARKS
Stated switching theorem.
Outlined its proof.
Sketched how to use it in analysis of a switched adaptive control system.
More or less clear that theorem can be restated in terms of LMIs.
Extension to interesting class of nonlinear systems likely.