TUTORIAL on LOGIC-BASED CONTROL
Part I: SWITCHED CONTROL SYSTEMS
Daniel Liberzon
Coordinated Science Laboratory and
Dept. of Electrical & Computer Eng.,
Univ. of Illinois at Urbana-Champaign
MED ’02, Lisbon
OUTLINE
Switched Control Systems
Questions, Break
Stability of Switched Systems
OUTLINE
Switched Control Systems
Questions, Break
Stability of Switched Systems
SWITCHED and HYBRID SYSTEMS
Switched
systems:
continuous
systems with
Hybrid
systems:
interaction
of continuous
anddiscrete
discreteswitchings
dynamics
emphasis on properties of continuous state
Switching can be:
• State-dependent
or
Time-dependent
x  f (x )
where x  f p (x), p  P
is a family of systems
and  : [0, )  P is a
switching signal
• Autonomous or Controlled
SWITCHING CONTROL
Plant:
Classical continuous
feedback paradigm:
u
P
u
y
y
P
C
u
But logical decisions
are often necessary:
y
P
C1
C2
The closed-loop
system is hybrid
logic
REASONS for SWITCHING
• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above
REASONS for SWITCHING
• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above
PARKING PROBLEM
x2
x1  w1 cos

x2  w1 sin 
x1
  w2
Nonholonomic constraint: wheels do not slip
OBSTRUCTION to STABILIZATION
?
Solution: move away first
REASONS for SWITCHING
• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above
OUTPUT FEEDBACK
Example: harmonic oscillator
y  y  u
u  y
u  12 y
switched system
QUANTIZED FEEDBACK
q(x)
u
x
PLANT
QUANTIZER
q(x)
CONTROLLER
x

sensitivity
M values
OBSTRUCTION to STABILIZATION
Assume:
M,  fixed
Asymptotic stabilization is impossible
MOTIVATING EXAMPLES
1. Temperature sensor
normal
too low
2. Camera with zoom
Tracking a golf ball
3. Coding and decoding
too high
VARYING the SENSITIVITY
zoom in
zoom out
Why switch ?
• More realistic
• Easier to design and analyze
• Robust to time delays
LINEAR SYSTEMS
x  Ax  Bu
Assume:
 K s.t. x  Ax  BKx is GAS: V  0
Along solutions of
x  Ax  BK ( x  e)
quantization error
we have
|e|   | x|  V  0
Then can achieve GAS
for some
 0
SWITCHING POLICY
|e|   | x |  V  0
We have | e |   n / 2
level sets of V
| x|   n / 2 
.
NONLINEAR SYSTEMS
x  f ( x, u )
Assume:  k s.t. x  f ( x, k ( x )) is GAS: V  0
Need: along solutions of
x  f ( x, k ( x  e))
|e |   (| x |)  V  0
where

quantization
error
is pos. def., increasing, and unbounded
(this is input-to-state stability wrt measurement error)
Then can achieve GAS
EXTENSIONS and APPLICATIONS
• Arbitrary quantization regions
• Active probing for information
• Output and input quantization
• Relaxing the assumptions
• Performance-based design
• Application to visual servoing
REASONS for SWITCHING
• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above
MODEL UNCERTAINTY
unmodeled
dynamics
 0
parametric
uncertainty
p*  P
Also, noise and disturbances
Adaptive control (continuous tuning)
vs. supervisory control (switching)
SUPERVISORY CONTROL
Supervisor
candidate controllers

Controller 1
Controller 2
...
Controller m
u1
u2
u
um
...

switching signal
Plant
y
STABILITY of SWITCHED SYSTEMS
x  f (x)
unstable
Stable if:
• switching stops in finite time
• slow switching (on the average)
• “locally confined” switching
• common Lyapunov function
REASONS for SWITCHING
• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above
PARKING PROBLEM under UNCERTAINTY
x2
x1  p1 w1 cos
x2  p1 w1 sin 

  p2 w2
x1
Unknown parameters p1 , p 2 correspond to the
radius of rear wheels and distance between them
SIMULATION
OUTLINE
Switched Control Systems
Questions, Break
Stability of Switched Systems
OUTLINE
Switched Control Systems
Questions, Break
Stability of Switched Systems
TWO BASIC PROBLEMS
• Stability for arbitrary switching
• Stability for constrained switching
UNIFORM STABILITY
x  f (x )
x  f p (x), p  P
where
is a family of GAS systems
 : [0, )  P is a piecewise constant switching signal
Want GUAS :
| x(t ) |   ( | x(0) | , t ) t ,  (   KL)
w.r.t. 
GUES:
| x(t ) |  c | x(0) | e  t t , 
COMMON LYAPUNOV FUNCTION
x  f (x)
is GUAS if and only if
V
s.t.
V
f p ( x)   W ( x)  0 x  0, p  P
x
V
x
Example: f p ( x)   p x, P  (0,1]
  (0,1)
Corollary:
is GAS
f p ( x) 0 f px(1
 0
, )pf
is not 
enough
q P
 p, q  P
t
 ( s ) ds
1

0
x(t )  e
x(0)  0 if   L

Usually we take P compact and fp continuous
SWITCHED LINEAR SYSTEMS
x  A x
LAS for every

GUES
 common Lyapunov function
but not necessarily quadratic
COMMUTING STABLE MATRICES => GUES
P  {1, 2}, A1 A2  A2 A1
 1
s1
 2
t1
 1
s2
 2 …
t
t2
A2 tk A1s k … A2 t1 A1s1
x(0)
x(t )  e
e
e
e
A2 (tk  ...  t1) A1 ( s k  ...  s1)
e
e
x(0)  0
quadratic common Lyap fcn:
A1T P1  P1 A1  I
A2T P2  P2 A2  P1
LIE ALGEBRAS and STABILITY
g1  g,
Lie algebra:
g  {Ap , p  P} L A
Lie bracket:
[ A1 , A2 ]  A1 A2  A2 A1
g k 1  [ g , g k ]  g k
g is nilpotent if  k s.t. g k  0

(k )
g(1)  g , g(k 1)  [ g (k ,) g(k ) ]  g (k ) g is solvable if  k s.t. g  0
SOLVABLE LIE ALGEBRA => GUES
g is solvable
Lie’s Theorem: triangular form
 1
Ap   

0

* 


 n

quadratic common Lyap fcn:
x TD x , D diagonal
SOLVABLE + COMPACT => GUES
Levi decomposition:
g  rs
radical (max solvable ideal)
s is compact => GUES
quadratic common Lyap fcn
SOLVABLE + NONCOMPACT => GUES
g  r  s , I  g , s is not compact
•  a set of stable generators for
g
that gives GUES
•  a set of stable generators for g
that leads to an unstable system
Lie algebra doesn’t provide enough information
NONLINEAR SYSTEMS
• Commuting systems
[ f p , f q ]  0 => GUAS
• Linearization
• ???
REMARKS on LIE-ALGEBRAIC CRITERIA
• Checkable conditions
• Independent of representation
• In terms of the original data
• Not robust to small perturbations
SYSTEMS with SPECIAL STRUCTURE
• Triangular systems
Linear => GUES
Nonlinear: need ISS conditions
u
• Feedback systems
Passive:
V  uy , K p  0
-
=> GUAS
Small gain: ||  ||  1, || K p || 1 => GUES
• 2D systems
convex combs of A1, A2 , A11 , A21 stable
quadratic common Lyap fcn

K
y
MULTIPLE LYAPUNOV FUNCTIONS
x  f1 ( x) , x  f 2 ( x)  GAS
V1 , V2  respective Lyapunov functions
V (t ) (t )
x  f (x)
is GAS
 1
 2
 1
 2
t
Very useful for analysis of state-dependent switching
MULTIPLE LYAPUNOV FUNCTIONS
decreasing sequence
V (t ) (t )
x  f (x)
is GAS
decreasing sequence
 1
 2
 1
 2
t
DWELL TIME
The switching times t1 , t 2 , ... satisfy ti 1  ti   D
x  f1 ( x) , x  f 2 ( x)  GES
dwell time
V1
Va11,| Vx2| 
functions
f1 ( x)   1V1 ( x)
 Vrespective
,
1 ( x)  b1 | x |Lyapunov
x
V2
2
2
a2 | x |  V2 ( x)  b2 | x | ,
f 2 ( x)   2V2 ( x)
x
2
2
must be
Need: V1 (t 2 )  V1 (t0 )
1
b1
b1 2 D
V1 (t 2 ) 
V2 (t 2 ) 
e
V2 (t1 )
a2
a2
b1 b2 2 D
b1 b2 (1  2 ) D

e
V1 (t1 ) 
e
V1 (t 0 )
a2 a
a
a
2
1
1
 1
t0
 1
 2
t1
t2
t
AVERAGE DWELL TIME
N (T , t )  N 0 
T t
 AD
# of switches on (t , T )
average dwell time
N 0  0  no switching: cannot switch if T  t   AD
N 0  1  dwell time: cannot switch twice if T  t   AD
x  f (x)
1 (| x | )  V p ( x)   2 (| x | )
Vp
f ( x )   V p ( x )
x p
x  f (x)
V p ( x)   Vq ( x), p, q  P
=>
is GAS
if  AD 
log 

SWITCHED LINEAR SYSTEMS
x  A x
• GUES over all  with large enough
 AD
• Finite induced norms for
x  A x  B u
y  C x
• The case when some subsystems are unstable
STABILIZATION by SWITCHING
x  A1x , x  A2 x  both unstable
Assume: A  A1  (1   ) A2 stable for some   (0,1)
T (1   )( A T P  PA )  0
 ( A1T P  PA1A
)
P  PA  20
2
So for each
x  0:
either xT ( A1T P  PA1 ) x  0 or xT ( A2T P  PA2 ) x  0
UNSTABLE CONVEX COMBINATIONS
Can also use multiple Lyapunov functions
LMIs