St' Petersberg

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 Theorems 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 systems transmission zeros.
Thus a firm, detectable system whose
transmission zeros are all unstable, must
be observable.
6
SWITCHING THEOREM
Suppose Fp p 2 P is a closed bounded subset
of matrices in Rmn with the property that for
each p 2 P, ABFp, 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 AKs CBFs
7

SWITCHING THEOREM
Suppose Fp p 2 P is a closed bounded subset
of matrices in Rmn with the property that for
each p 2 P, ABFp, 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 AKs CBFs
Why is this theorem useful?
Why is it true?
8
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
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

n1 or n2


u
y

d
10
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
with stability margin l. Here l is a design
parameter.
11
d
E
s
S
y
P
u
Cs
12
d
E
s
S
y
P
u
Cs
13
d
E
s
S
y
P
u
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 Ps transfer function.
14
d
s
E
S
y
P
u
Cs
Piecewise-constant switching signal taking values
in 1, 2.
15
d
E
s
S
y
P
u
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
AC 2n-dim and stable.
(fC, AC) n-dim, stable, observable
16
y
-
y1
e1
m1
d


E
s
S
y
y
P
u
-
y2
e2
m2


Cs
17
y
-
y1
e1
m1
d

E
s
S
y
y
P
u
-
y2
e2
m2

Cs
18
Design parameter Dwell-time tD gt 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 ss value was
changed. Otherwise, S does nothing.
y1
e1
m1
d


E
s
S
y
y
P
u
-
y2
e2
m2


Cs
19
ASSUME n1 is Ps nominal transfer function
y
-
y1
e1
m1
d


E
s
S
y
y
P
u
-
y2
e2
m2


Cs
20
s(t)2, t 2 tj, tj1)
tD
tj1
tj
m2(T) m1(T), Ttj and T 2 tj tD,
tj1)
21
s(t)2, t 2 tj, tj1)
tD
tj1
tj
e2l Te22T e2l Te12T, Ttj and
T 2 tj tD, tj1)
e2T e1T, Ttj and T 2 tj
tD, tj1)
22
s(t)2, t 2 tj, tj1)
tD
tj1
tj
e2T e1T, Ttj and T 2 tj
tD, tj1)
23
ASSUME n1 is Ps nominal transfer function
y
-
y1
e1
m1
d


E
s
S
y
y
P
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.
24
y
-
y1
e1
m1
d


E
s
S
y
y
P
u
-
y2
e2
m2


Cs
25
For each p 2 1, 2, (c, A BFp) is detectable.


y
Want to stabilize ABFs by output injection
using cx e_2 e_1.
-
y1
e1
m1
For each p 2 1,2 there is an hp which
stabilizes ABFphpc.

d


For tD sufficiently large, ABFshs c is
exponentially stable.

E
s
S
y
y
P
u
-
y2
e2
m2


Cs
26
For each p 2 1, 2, (c, A BFp) is detectable.


y
Want to stabilize ABFs by output injection
using cx e_2 e_1.
-
y1
e1
m1
Each ABFp, B,c is firm.


d


Switching Theorem applies.


E
s
S
For any tD gt 0, no matter how small, there is a
ks which stabilizes ABFsks c.
y
y
P
u
The matrix M ABFs ks(1-yT )c is also
stable because yT is L1.

-
y2
e2
m2


Cs
27
y
-
y1
e1
m1
d


E
s
S
y
y
P
u
The matrix M ABFs ks(1-yT )c is stable.

-
y2
e2
m2


Cs
28
y
-
y1
e1
m1
d


E
s
S
y
y
P
u
-
y2
e2
m2


The matrix M ABFs ks(1-yT )c is stable.

Cs
29
y
-
y1
e1
m1
d


E
s
S
y
y
P
u
-
y2
e2
m2


Cs
y
(1-yT)e2
stable
e1
u
30
y
-
y1
e1
m1
d


E
s
S
y
y
P
u
-
y2
e2
m2


Cs
y
(1-yT)e2
stable
e1

d
h1(s)

u
h2(s)
d(s)
31
y
-
y1
e1
m1
d


E
s
S
y
y
P
u
-
y2
e2
m2


Cs
y
(1-yT)e2
stable
1
e1

d
h1(s)

u
h2(s)
d(s)
32
SWITCHING THEOREM
Suppose Fp p 2 P is a closed bounded subset
of matrices in Rmn with the property that for
each p 2 P, ABFp, 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 AKs CBFs
Why is this theorem useful?
Why is it true?
33
SWITCHING THEOREM
Suppose Fp p 2 P is a closed bounded subset
of matrices in Rmn with the property that for
each p 2 P, ABFp, 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 AKs CBFs
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 gt 0,
there exists and output injection Kp which
exponentially stabilizes As KsC for any
piecewise constant switching signal s0,1)! P
with dwell time no smaller than tD.
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(AKC)t e-l (t-T),
t 0
35
(No Transcript)
36
(No Transcript)
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.