SWITCHING CONTROL OF UNCERTAIN SYSTEMS

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SWITCHING CONTROL OF UNCERTAIN SYSTEMS
Daniel Liberzon
Coordinated Science Laboratory and Dept. of
Electrical Computer Eng., Univ. of Illinois at
Urbana-Champaign U.S.A.
DISC HS, June 2003
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HYBRID CONTROL
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REASONS for SWITCHING

• Nature of the control problem

• Sensor or actuator limitations

• Large modeling uncertainty

• Combinations of the above

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REASONS for SWITCHING

• Nature of the control problem

• Sensor or actuator limitations

• Large modeling uncertainty

• Combinations of the above

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MODELING UNCERTAINTY
Adaptive control (continuous tuning) vs.
supervisory control (switching)
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EXAMPLE
Scalar system
, otherwise unknown
(purely parametric uncertainty)
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SUPERVISORY CONTROL ARCHITECTURE
Supervisor
candidate controllers
u1
Controller
y
Plant
u
u2
Controller
. . .
um
. . .
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TYPES of SUPERVISION
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TYPES of SUPERVISION
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OUTLINE

• Basic components of supervisor

• Design objectives and general analysis

• Achieving the design objectives

• Examples

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OUTLINE

• Basic components of supervisor

• Design objectives and general analysis

• Achieving the design objectives

• Examples

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SUPERVISOR
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EXAMPLE
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EXAMPLE
disturbance
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STATE SHARING
produces the same signals
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SUPERVISOR
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EXAMPLE
can use state sharing
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SUPERVISOR
Justification?
(certainty equivalence)
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SUPERVISOR
Basic idea
assume
, controllers
Justification?
Plant
gives stable closed-loop system
gt
gt
gt
plant likely in
small
small
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DETECTABILITY
Want this system to be detectable
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SUPERVISOR
This is switched detectability
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DETECTABILITY under SWITCHING
plant in closed loop with
Switched system
view as output

• slow switching (on the average)

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SUMMARY of BASIC PROPERTIES
Multi-estimator
Candidate controllers
Switching logic
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SUMMARY of BASIC PROPERTIES
Multi-estimator
Candidate controllers
Switching logic
4. Switched closed-loop system is detectable
w.r.t. provided this is true for every frozen
value of
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SUMMARY of BASIC PROPERTIES
Multi-estimator
Candidate controllers
Switching logic
4. Switched closed-loop system is detectable
w.r.t. provided this is true for every frozen
value of
Analysis
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OUTLINE
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SUMMARY of BASIC PROPERTIES
Candidate controllers
2. For each , closed-loop system is
detectable w.r.t.
Switching logic
3. is bounded in terms of the smallest
4. Switched closed-loop system is detectable
w.r.t. provided this is true for every frozen
value of
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MULTI-ESTIMATOR
Linear systems general results available
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MULTI-ESTIMATOR
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SUMMARY of BASIC PROPERTIES
Multi-estimator
1. At least one estimation error ( ) is small

• when
• is bounded for bounded

Candidate controllers
2. For each , closed-loop system is
detectable w.r.t.
Switching logic
3. is bounded in terms of the smallest
4. Switched closed-loop system is detectable
w.r.t. provided this is true for every frozen
value of
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CANDIDATE CONTROLLERS
fixed
Plant
Controller
Multi- estimator
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CANDIDATE CONTROLLERS
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CANDIDATE CONTROLLERS
Linear overall system is detectable w.r.t.
if

• system inside the box is stable
• plant is detectable

Nonlinear same result holds if stability and
detectability are interpreted in the ISS sense
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CANDIDATE CONTROLLERS
Linear overall system is detectable w.r.t.
if

• system inside the box is stable
• plant is detectable

Nonlinear same result holds if stability and
detectability are interpreted in the
integral-ISS sense
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CANDIDATE CONTROLLERS
Linear overall system is detectable w.r.t.
if

• system inside the box is output-stable
• plant is minimum-phase

Nonlinear version also possible
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SUMMARY of BASIC PROPERTIES
Multi-estimator
1. At least one estimation error ( ) is small

• when
• is bounded for bounded

Candidate controllers
2. For each , closed-loop system is
detectable w.r.t.
Switching logic
3. is bounded in terms of the smallest
4. Switched closed-loop system is detectable
w.r.t. provided this is true for every frozen
value of
for 3, want to switch to for 4, want to
switch slowly or stop
conflicting
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DWELL-TIME SWITCHING LOGIC
Not suitable for nonlinear systems (finite escape)
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HYSTERESIS SWITCHING LOGIC
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HYSTERESIS SWITCHING LOGIC
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HYSTERESIS SWITCHING LOGIC
Similar results hold
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OUTLINE
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LINEAR OBSERVER-BASED DESIGN
Switching logic hysteresis
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REASONS for SWITCHING

• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above

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PARKING PROBLEM
p
1
p
1
p
Unknown parameters correspond to
the radius of rear wheels and distance between
them
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NONHOLONOMY OBSTRUCTION to STABILIZATION
Solution move away first
Hybrid candidate controllers!
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SIMULATION
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REFERENCES
Morse, Hespanha