Transcript Part 1
Part 1
Stability Analysis of Linear
Switched Systems:
An Optimal Control Approach
Michael Margaliot
School of Elec. Eng.
Tel Aviv University, Israel
Joint work with: Gideon Langholz (TAU),
Daniel Liberzon (UIUC), Michael S. Branicky
(CWRU), Joao Hespanha (UCSB).
1
Overview
Switched systems
Global asymptotic stability
The edge of stability
Stability analysis:
An optimal control approach
A geometric approach
An integrated approach
Conclusions
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Switched Systems
Systems that can switch between
several possible modes of operation.
Mode 1
Mode 2
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Example 1
x1 a1 (t ) C
x2 a2 (t )
a1 (t )
a2 ( t )
x1
x2
C
server
x1 a1 (t )
x2 a2 (t ) C
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Example 2
Switched power converter
100v
linear filter
50v
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Example 3
A multi-controller scheme
+
plant
controller1
switching logic
controller2
Switched controllers are stronger than
“regular” controllers.
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More Examples
Air traffic control
Biological switches
Turbo-decoding
……
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Synthesis of Switched Systems
Driving: use mode 1 (wheels)
Braking: use mode 2 (legs)
The advantage: no compromise
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Linear Systems
x Ax.
Solution:
x(t ) exp( At ) x(0).
Definition: The system is globally asymptotically
x(0).
stable if lim x(t ) 0,
t
Theorem:
stability Re( λ) 0, λ eig( A).
A is called a Hurwitz matrix.
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Linear Switched Systems
Two (or more) linear systems:
x(t ) A1 x(t ),
x(t ) A2 x(t ).
A system that can switch between them:
x(t ) Aσ (t ) x(t ),
σ (t )
2
1
σ : R {1,2}.
...
t1
t1 t2
t
x(T ) ...exp(t4 A1 )exp(t3 A2 )exp(t2 A1 )exp(t1 A2 ) x0 .
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Stability
Linear switched system: x(t ) Aσ (t ) x(t ),
σ : R {1,2}.
Definition: Globally uniformly asymptotically
x(t ) 0,
x(0), σ .
stable (GUAS):
In other words,
lim x(t ) lim(...exp(t3 A2 ) exp(t2 A1 ) exp(t1 A2 ) x0 ) 0
t
t
for any t1 , t2 ,.... 0
AKA, “stability under arbitrary switching”.
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11
A Necessary Condition for GUAS
The switching law σ (t ) 1 yields x(t ) A1 x(t ).
Thus, a necessary condition for GUAS
is that both A1 , A2 are Hurwitz.
Then instability can only arise due to
repeated switching.
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12
Why is the GUAS problem
difficult?
Answer 1:
The number of possible switching laws
is huge.
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13
Why is the GUAS problem difficult?
Answer 2: Even if each linear subsystem is
stable, the switched system may not be GUAS.
0 1
x
x
2 1
14
0 1
x
x
12 1
14
Why is the GUAS problem difficult?
Answer 2: Even if each linear subsystem is
stable, the switched system may not be GUAS.
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15
Stability of Each Subsystem is
Not Enough
A multi-controller scheme
+
plant
controller1
switching logic
controller2
Even when each closed-loop is stable,
the switched system may not be GUAS.
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Easy Case #1
A trajectory of the switched system:
x(T ) ...exp(t4 A1 )exp(t3 A2 )exp(t2 A1 )exp(t1 A2 ) x0 .
Suppose that the matrices commute:
A1 A2 A2 A1.
Then x(T ) exp((T s) A1 )exp(sA2 ) x0 ,
and since both matrices are Hurwitz, the
switched system is GUAS.
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17
Easy Case #2
Suppose that both matrices are upper
triangular:
1 1
x
x,
0 2
3 7
x
x.
0 2.5
Then x 2 x , x 2.5x , so
2
Now
2 2
2
| x (t ) | exp(2t ) | x (0) |.
2
2
x x x , x 3x 7 x so x (t ) 0.
1
1
1 2 1
1 2
This proves GUAS.
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18
Optimal Control Approach
Pioneered by E. S. Pyatnitsky (1970s).
Basic idea:
(1) A relaxation: linear switched
system → bilinear control system
(2) characterize the “most
destabilizing” control u *
(3) the switched system is GUAS iff
x * (t ) 0
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19
Optimal Control Approach
Relaxation: the switched system:
x Aσ x,
σ : R {1,2},
→ a bilinear control system:
x ( A1 ( A2 A1 )u ) x,
u U ,
where U is the set of measurable functions
taking values in [0,1].
u0
x A1 x
u 1/ 2
u 1
x (1/ 2)( A1 A2 ) x x A2 x
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Optimal Control Approach
The bilinear control system (BCS)
x ( A1 ( A2 A1 )u ) x, u U ,
is globally asymptotically stable (GAS) if:
x(t ) 0,
x(0), u U .
Theorem The BCS is GAS if and only if the
linear switched system is GUAS.
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Optimal Control Approach
The most destabilizing control:
x ( A1 ( A2 A1 )u ) x, u U ,
x(0) x0 .
Fix a final time t f 0.. Let J (u) | x(t f ; u) |.
Optimal control problem: find a control
that maximizes J (u).
u*U
Intuition: maximize the distance to the origin.
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Optimal Control Approach
and Stability
Theorem The BCS is GAS iff
x *(t f ) 0.
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Edge of Stability
The BCS: x ( A1 Bu) x,
B A2 A1.
Consider x ( A1 Bk u ) x,
Bk ( A2 A1 )k .
k 0
x ( A1 0u ) x
GAS
k 1
k ε0
x ( A1 Bεu ) x,
GAS
x ( A1 B1u ) x,
original BCS
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Edge of Stability
The BCS: x ( A1 Bu) x,
B A2 A1.
Consider x ( A1 Bk u ) x,
Bk ( A2 A1 )k .
k 0
x ( A1 0u ) x
GAS
k 1
k ε0
x ( A1 Bεu ) x,
GAS
x ( A1 B1u ) x,
original BCS
Definition: k* is the minimal value of k>0
such that GAS is lost.
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Edge of Stability
The BCS: x ( A1 Bu) x,
B A2 A1.
Consider x ( A1 Bk u ) x,
Bk ( A2 A1 )k .
Definition: k* is the minimal value of k>0
such that GAS is lost.
The system x ( A1 Bk*u) x is said to be on
the edge of stability.
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Edge of Stability
The BCS: x ( A1 Bu) x,
B A2 A1.
Consider x ( A1 Bk u ) x,
Bk ( A2 A1 )k .
Definition: k* is the minimal value of k>0
such that GAS is lost.
0
k* 1
k
0
1 k*
k
Proposition: our original BCS is GAS iff
k*>1.
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Edge of Stability
The BCS: x ( A1 Bu) x,
B A2 A1.
Consider x ( A1 Bk u ) x,
Bk ( A2 A1 )k .
Proposition: our original BCS is GAS iff
k*>1.
→ we can always reduce the problem of
analyzing GUAS to the problem of
determining the edge of stability.
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Edge of Stability When n=2
Consider
x ( A1 Bk u ) x.
The trajectory x* corresponding to u*:
k k*
k k*
k k*
x0
x0
A closed
periodic trajectory
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Solving Optimal Control Problems
2
| x(t f ; u ) | is a functional:
x(t f ; u) F (u(t), t [0, t f ])
Two approaches:
1. The Hamilton-Jacobi-Bellman (HJB)
equation.
2. The Maximum Principle.
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Solving Optimal Control Problems
1. The HJB equation.
Intuition: there exists a function V (, ) : R R
n
R
V (t , x *(t )) const,
and V can only decrease on any other
trajectory of the system.
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The HJB Equation
Find V (, ) : R R
n
R such that V (t f , y) || y ||2 / 2,
d
MAX V (t , x(t )) 0.
u [0,1] dt
(HJB)
Integrating: V (t f , x(t f )) V (0, x(0)) 0
2
| x(t f ) | / 2 V (0, x(0)).
or
2
An upper bound for | x(t f ) | / 2,
obtained for the u * maximizing (HJB).
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The HJB for a BCS:
0 max{Vt Vx x}
u
max{Vt Vx (uAx (1 u ) Bx )}
u
max{Vt Vx Bx uVx ( A B ) x}
u
Hence,
1, Vx ( A B) x 0,
u* 0, Vx ( A B) x 0,
?, V ( A B) x 0.
x
In general, finding V is difficult.
Note: u* depends on Vx only.
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The Maximum Principle
2
Let (t ) : Vx (t , x *(t )). Then, (t f ) x (t f ) / 2 x(t f ).
x
Differentiating 0 Vt Vx x, we get
0 Vtx Vxx x Vx (uA (1- u ) B)
d
dt
Vx
Vx (uA (1- u ) B)
(uA (1- u ) B)
A differential equation for (t ), with a
boundary condition at t f .
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Summarizing,
T
(uA (1- u) B) , (t f ) x(t f )
x (uA (1- u) B) x,
x(0) x0
The WCSL is the u * maximizing
Vt Vx x Vt T (uA (1 u) B) x
that is,
1, T (t )( A B) x(t ) 0,
u *(t )
T
0,
(t )( A B) x(t ) 0.
We can simulate the optimal solution
backwards in time.
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Result #1 (Margaliot & Langholz, 2003)
An explicit solution for the HJB equation,
when n=2, and {A,B} is on the “edge of
stability”.
This yields an easily verifiable necessary
and sufficient condition for stability of
second-order switched linear systems.
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Basic Idea
The HJB eq. is:
Thus,
0= max{Vx Bx uVx ( A B ) x}.
u
u 0 0=Vx Bx
u 1 0=Vx Ax
Let
H A : R2 R
x(t ) Ax(t ),
be a first integral of
that is,
d A
0 H ( x(t )) H xA Ax.
dt
Then V is a concatenation of two
first integrals H A ( x) and H B ( x).
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1
1
0
0
Example: A 2 1 B 2 k 1
x Ax
1
7 x1
2
A
T
H ( x) x P0 x exp(
arctan(
))
x1 2 x2
7
1
x Bx
H B ( x) xT Pk x exp(
2 k 1/ 2
where Pk
1
1/ 2
7 4k x1
2
arctan(
))
x1 2 x2
7 4k
and k * 6.985...
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Nonlinear Switched Systems
x { f ( x), f ( x)}
1
with
2
(NLDI)
x f ( x) GAS.
i
Problem: Find a sufficient condition
guaranteeing GAS of (NLDI).
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Lie-Algebraic Approach
For the sake of simplicity, we present
the approach for LDIs, that is,
x { Ax , Bx}
and
x(t ) ...exp( Bt2 ) exp( At1 ) x(0).
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Commutation and GAS
Suppose that A and B commute,
AB=BA, then
x(t ) ...exp( At3 )exp( Bt2 )exp( At1 ) x(0)
exp( A(... t3 t1 ))exp( B(... t4 t2 )) x(0)
Definition: The Lie bracket of Ax and
Bx is [Ax,Bx]:=ABx-BAx.
Hence, [Ax,Bx]=0 implies GAS.
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Lie Brackets and Geometry
x { Ax, Ax, Bx, Bx}.
Consider
x Ax
x ( 0)
x Bx
x ( 4 )
x Bx
x Ax
A calculation yields:
x(4ε ) x(0) ε 2[ A, B]( x(0)).
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Geometry of Car Parking
This is why we can park our car.
The 2 term is the reason it takes
so long.
Bx
Ax
Ax
Bx
[ A, B]( x)
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Nilpotency
We saw that [A,B]=0 implies GAS.
What if [A,[A,B]]=[B,[A,B]]=0?
Definition: k’th order nilpotency all Lie brackets involving k terms
vanish.
[A,B]=0
→ 1st order nil.
[A,[A,B]]=[B,[A,B]]=0 → 2nd order nil.
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Nilpotency and Stability
We saw that 1st order nilpotency
Implies GAS.
A natural question:
Does k’th order nilpotency
imply GAS?
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Some Known Results
Switched linear systems:
k=2 implies GAS (Gurvits,1995).
k order nilpotency implies GAS
(Liberzon, Hespanha, and Morse, 1999).
(The proof is based on Lie’s Theorem)
Switched nonlinear systems:
k=1 implies GAS.
An open problem: higher orders of k?
(Liberzon, 2003)
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A Partial Answer
Result #2 (Margaliot & Liberzon, 2004)
3rd order nilpotency implies GAS.
Proof: Consider the WCSL
1, T (t )( A B) x(t ) 0
u *(t )
T
0, (t )( A B) x(t ) 0
Define the switching function
m(t ) : (t )Cx(t ), C A B
T
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Differentiating m(t) yields
m(t ) (t )Cx(t ) (t )Cx(t )
T
T
(t )[C, A]x(t ).
T
2nd order nilpotency m 0 m(t ) const
no switching in the WCSL!
Differentiating again, we get
T
T
m
[
C
,
A
]
x
[C , A] x
T [[C , A], A] x uT [[C , A], B ] x
3rd order nilpotency m
0 m(t ) at b
up to a single switching in the WCSL. 48
Singular Arcs
If m(t)0, then the Maximum Principle
provides no direct information.
Singularity can be ruled out using
the auxiliary system.
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Summary
Parking cars is an underpaid job.
Switched systems and differential
inclusions are important in various
scientific fields, and pose
interesting theoretical questions.
Stability analysis is difficult.
A natural and useful idea is to
consider the worst-case trajectory.
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Summary: Optimal Control Approach
Advantages:
reduction to a single control u *
leads to necessary and sufficient conditions
for GUAS
allows the application of powerful tools
(high-order MPs, HJB equation, Liealgebraic ideas,….)
applicable to nonlinear switched systems
Disadvantages:
requires characterizing u *
explicit results for particular cases only
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More Information
1. Margaliot. “Stability analysis of switched systems using
variational principles: an introduction”, Automatica,
42: 2059-2077, 2006.
2. Sharon & Margaliot. “Third-order nilpotency, nice
reachability and asymptotic stability”, J. Diff. Eqns.,
233: 136-150, 2007.
3. Margaliot & Branicky. “Nice reachability for planar
bilinear control systems with applications to planar
linear switched systems”, IEEE Trans. Automatic
Control, 54: 1430-1435, 2009.
Available online:
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www.eng.tau.ac.il/~michaelm
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