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Automatic Control Laboratory, ETH Zürich
www.control.ethz.ch
Application of Set Membership
Identification to Controller Design
Marko Tanaskovic
Doctoral Exam 24.11.2015
www.control.ee.ethz.ch

Controller design
• Create mathematical control law for physical system mathematical model required
u y
ydes
f(y; ydes)

Controller design
• Create mathematical control law for physical system mathematical model required
• Derive mathematical model from measurements (system identification)
?
u y
u y
ydes
f(y; ydes)

System identification
• Measurements corrupted by noise and disturbance identified model uncertain
Probabilistic identification Set membership identification
Assumption on Statistical properties Bound
? +
u
e
y
e

Parameter estimate
Probability density function
Optimal estimate
maximizes likelihood
Set of possible parameters
Optimal estimate minimizes
worst case estimation error
? +
u y
e
e

Parameter estimate
Optimal estimate
? +
u y
Application to
controller design
Probabilistic objectives Non-probabilistic objectives
e
e

Parameter estimate
Optimal estimate
? +
u y
e
Application to
controller design
SM identification topic of the thesise
Probabilistic objectives Non-probabilistic objectives

Controller design classification
Identification Design
Indirect Direct
Off-lineOn-line
ModelData Controller
Controller identification
Data Controller
Closed loop operation
Data Controller

Indirect Direct
Off-lineOn-line
• SM useful for robust design (mature theory)
• Only few results on experiment design
• Experiment design for constrained MISO
systems
Data Controller
Data Controller

Indirect Direct
Off-lineOn-line
Data Controller
• SM useful for handling constraints
• No computationally tractable algorithms
• Adaptive model predictive control (MPC)
for constrained MIMO systems
Data Controller
systems

Indirect Direct
Off-lineOn-line
• SM leads to formal stability and
performance guarantees
• Efficient algorithms based on SM
exist
Data Controller
Data Controller
systems

Indirect Direct
Off-lineOn-line
Data Controller
Data Controller
• SM could lead to formal stability and
• No algorithm based on SM exists
• On-line direct controller design for
nonlinear systems
• SM leads to formal stability and
• Efficient algorithms based on SM
exist
systems

Thesis contributions
• Worst-case experiment design for constrained linear systems
• Adaptive model predictive control based on set membership identification
• On-line direct data driven control design based on set membership identification

Set membership identification
je(t)j · ²; 8t
• System model linear in parameters, disturbance magnitude bounded
• Collection of input-output data
y(t) = '(t)Tµ + e(t)
yN = fy(t)gN
t=1 ; 'N = f'(t)gN
t=1

je(t)j · ²; 8t
• Feasible parameter set: all parameters consistent
with assumptions and collected data
y(t) = '(t)Tµ + e(t)
yN = fy(t)gN
t=1 ; 'N = f'(t)gN
t=1
FPS =
n
µ : jy(t) ¡ '(t)T µj · ²; t = 1; : : : ; N
o

je(t)j · ²; 8t
• Feasible parameter set: all parameters consistent
with assumptions and collected data
• Optimal estimate
• Worst case estimation error (radius of information) measure of uncertainty
y(t) = '(t)Tµ + e(t)
yN = fy(t)gN
t=1 ; 'N = f'(t)gN
t=1
FPS =
n
µ : jy(t) ¡ '(t)T µj · ²; t = 1; : : : ; N
o
^µ¤
p = arg inf
v
sup
µ2FPS
jjv ¡ µjjp
Rp(FPS) = sup
µ2FPS
jjµ¤
p ¡ µjjp
Rp(FPS)
^µ¤
p

• Worst case radius of information upper bound on
• Experiment design: find input sequence that minimizes
• Existing results for SISO systems, FIR model and input magnitude bound
Worst case experiment design
Depends on the
applied inputs only
Rp(FPS)
Rp('N) = sup
µ;je(t)j·²;t=1;:::;N
Rp(FPS)
Rp('N) R
¤
p

• Worst case radius of information upper bound on
• Experiment design: find input sequence that minimizes
• Existing results for SISO systems, FIR model and input magnitude bound
• We consider MISO systems subject to polytopic constraints
Worst case experiment design
Depends on the
applied inputs only
'(t) = A'(t ¡ 1) + Bu(t ¡ 1)
Constraints form a convex compact set that contains origin
controllable, stableA; B A
Rp(FPS)
Rp('N) = sup
µ;je(t)j·²;t=1;:::;N
Rp(FPS)
y(t) = '(t)Tµ + e(t)
Cu(t) · o
L¢u(t) · q
Rp('N) R
¤
p

• Set of vectors orthogonal to the faces of norm ball
Minimizing worst-case radius of
information
V 1 =
8
>>><
>>>:
2
6
6
6
4
1
0
...
0
3
7
7
7
5
;
2
6
6
6
4
0
1
...
0
3
7
7
7
5
; : : : ;
2
6
6
6
4
0
0
...
1
3
7
7
7
5
9
>>>=
>>>; ®1 = minf®1; ®2; : : : ; ®mg
V 1
®ivi• set of admissible regressors
regressor vectors that can be reached from origin
1¡
polytope
©
©
©
®i = max
kvi2©
k

• Set of vectors orthogonal to the faces of norm ball
Minimizing worst-case radius of
information
V 1 =
8
>>><
>>>:
2
6
6
6
4
1
0
...
0
3
7
7
7
5
;
2
6
6
6
4
0
1
...
0
3
7
7
7
5
; : : : ;
2
6
6
6
4
0
0
...
1
3
7
7
7
5
9
>>>=
>>>; ®1 = minf®1; ®2; : : : ; ®mg
The minimal worst case radius of information is achieved with an input
sequence that generates regressor vectors ®1
vi;8vi 2 V 1
V 1
®ivi
R
¤
1
• set of admissible regressors
regressor vectors that can be reached from origin
1¡
polytope
©
©
©
®i = max
kvi2©
k

Designing the input sequence
• Design input sequences that creates all regressors of interest
• Ideally shortest possible input sequence hard combinatorial problem
• Our solution input sequence that visits all points computationally tractable

Designing the input sequence
• Design input sequences that creates all regressors of interest
• Ideally shortest possible input sequence hard combinatorial problem
• Our solution input sequence that visits all points computationally tractable
• A sequence of inputs that brings to in steps has to satisfy'0 'T k
• Requires solution of the linear program (LP)
• Greedy algorithm always find the closest regressor vector
'(0) = '0
'(k) = 'T
'(t) = A'(t ¡ 1) + Bu(t)
Cu(t) · o
L¢u · q
9
>=
>;
8t = 1; : : : ; k

Numerical example
• Compare with random input sequence
• Large number of experiments with same disturbance for both input types
• In 13.7% of the experiments with random input actual radius > worst case bound
radius of information
Empirical probability density functionsRadius of information values
experiment realization
optimalinputrandominput
optimalinputrandominput

Adaptive control
• Adaptive control automatically improve controller performance over time
• Useful for time varying or uncertain time invariant systems
Controller Plant
System
identification

Standard adaptive control
• Certainty equivalence adaptive control
• Cannot guarantee output constraint satisfaction
Provide a point estimate of the system model
Set the controller as if there is no uncertainty
in the identified model
Plant estimate - point
Controller Plant
System
identification

• MPC is method of choice for constrained systems
• Combine MPC with recursive set membership identification
Adaptive control with constraints
Use MPC to satisfy output constraints
for all the models in the set
Provide a set of models consistent with initial
assumptions and measured data
Plant estimate - set
Plant estimate - point
Controller Plant
System
identification

Problem definition
• Discrete, open-loop stable, LTI,
SISO system, FIR model
Can generalize to MIMO systems, basis
transfer function parameterization
'(t) = [u(t ¡ 1); : : : ; u(t ¡ m)]T
Length of the impulse response
y(t) =
mX
k=1
#(k)u(t ¡ k) + e(t)
= '(t)T µ + e(t)
µ = [#(1); : : : ; #(m)]T
m

Problem definition
'(t) = [u(t ¡ 1); : : : ; u(t ¡ m)]T
y(t) =
mX
k=1
#(k)u(t ¡ k) + e(t)
= '(t)T µ + e(t)
µ = [#(1); : : : ; #(m)]T
• Goal: track a reference and satisfy constraints
Cu(t) · o
L¢u(t) · q
Hy(t) · p
; 8t 2 Z Convex compact set
that contains origin
m

2 4 6 8 10 12 14
-1.5
-1
-0.5
0
0.5
1
1.5
Problem definition
• Known bounds on disturbance and
noise signal magnitude
• True system unknown, but its impulse
response coefficients lie in a known set
time step
'(t) = [u(t ¡ 1); : : : ; u(t ¡ m)]T
polytope
y(t) =
mX
k=1
#(k)u(t ¡ k) + e(t)
= '(t)T µ + e(t)
µ = [#(1); : : : ; #(m)]T
je(t)j · ²; 8t 2 Z
µ 2 FPS(0)
FPS(0)
• Goal: track a reference and satisfy constraints
Cu(t) · o
L¢u(t) · q
Hy(t) · p
; 8t 2 Z Convex compact set
that contains origin
impulse response
coefficient
magnitude
m

• Bounded complexity polytopic update prevents the unlimited growth of face number
Recursive set membership
identification
• Recursive identification algorithm:
At each time step
add 2 new inequalities to FPS(t)
FPS(t) =FPS(t ¡ 1)

n
µ : jy(t) ¡ #(t)T µj · ²
o

• Bounded complexity polytopic update prevents the unlimited growth of face number
• Nominal model of the plant:
Calculate by solving an LP
Recursive set membership
identification
Center of maximum volume
norm ball inscribed inl2
• Recursive identification algorithm:
At each time step
add 2 new inequalities to FPS(t)
FPS(t) =FPS(t ¡ 1)

n
µ : jy(t) ¡ #(t)T µj · ²
o
FPS(t)
µc(t)
µc(t)

MPC algorithm
Estimate of the current
disturbance
• Finite horizon optimal control problem:
cost function based on the nominal model, prediction horizon:
subject to:
N ¸ m
min
U
t+N¡1X
k=t
(^y(kjt) ¡ ydes(k))T
Q (^y(kjt) ¡ ydes(k))T
+ u(kjt)T Su(kjt) + ¢u(kjt)T R¢u(kjt)
^y(k + 1jt) = '(kjt)tµc(t) + ^e(t)
^e(t) = y(t) ¡ '(t)T µc(t)

last m predicted control moves constant
MPC algorithm
Estimate of the current
disturbance
• Finite horizon optimal control problem:
cost function based on the nominal model, prediction horizon:
subject to:
N ¸ m
robust satisfaction of output constraints
• Can be formulated as a Quadratic Program (QP) of moderate size
• Recursive feasibility and integral action guaranteed
u(t + N ¡ mjt) = : : : = u(t + N ¡ 1jt)
min
U
t+N¡1X
k=t
(^y(kjt) ¡ ydes(k))T
Q (^y(kjt) ¡ ydes(k))T
+ u(kjt)T Su(kjt) + ¢u(kjt)T R¢u(kjt)
^y(k + 1jt) = '(kjt)tµc(t) + ^e(t)
^e(t) = y(t) ¡ '(t)T µc(t)
Cu(kjt) · o L¢u(t) · q
H'(kjt)T µ · p; 8µ 2 FPS(t)

Application case studies
• Quad-tank experimental results
• Non-minimum phase MIMO system
• Goal: track reference and enforce
constraints
• Building climate control simulation results
• Typical office room model
• Goal: minimize energy consumption and
enforce occupant comfort constraints

Quad-tank experimental example
Non-adaptive MPC
Adaptive MPC with set membership
Adaptive MPC with least squares
• Adaptive MPC with least squares:
Constraint violation during adaptation
• Adaptive MPC with set membership
No constraint violation

Conclusion and outlook
• Set membership identification can bring a lot of benefits to controller design
Deterministic and accurate uncertainty description
Can deal with hard constraints
Can make derivation of stability and performance guarantees easier
• Has to be used in a smart way prevent excessive conservativeness and
computational complexity
• Further research required to exploit all its benefits in controller design

Thank you!
• Supervisor
Prof. Manfred Morari
• Committee
Prof. Mario Milanese
• Collaborators
Dr. Lorenzo Fagiano
Prof. Sébastien Mariéthoz
Prof. Carlo Novara
Prof. Roy Smith
Dr. David Sturzenegger
Dr. Paul Goulart
Damian Frick
Oliver Schultes
• Students
Lawrence Minnetian
Isik Ilber Sirmatel
(and many others)
• Colleagues
George Xiaojing
Dr. Claudia Fisher
Dr. Manfred Quack
Alexander Liniger
Dr. Robert Nguyen
Tony Wood
(and many others)
• My family and friends

Questions ?
Thank you for your attention

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