Page 1
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 1
Robust MPC design,
Future and Practical
Applications
Eduardo F. Camacho
Universidad de Sevilla
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 2
Outline
1. Model Predictive Control
2. Stability and robustness for MPC
3. Min max MPC
4. Fault tolerant MPC
5. Conclusions
Page 2
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 3
Woody Allen:
“I took a speed reading course and read
'War and Peace' in twenty minutes. …
….. It involves Russia.”
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 4
MPC successful in industry.
Many and very diverse and successful
applications:
Petrochemical, polymers,
Semiconductor production,
Air traffic control
Clinical anesthesia,
….
Life Extending of Boiler-Turbine Systems via
Model Predictive Methods, Li et al (2004)
Many MPC vendors
Page 3
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 5
MPC successful in Academia
Many MPC sessions in control conferences
(2/12 at this symposium) and control journals,
MPC workshops.
MPC in other research areas: industrial
electronics, chemical engineering, energy,
transport …
4/8 finalist papers for the IFAC journal CEP
best paper award were MPC papers (2/3
finally awarded were MPC papers)
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
IFAC Pilot Industry Committee
Chaired by Tariq Samad (Honeywell), 28 total: 15 industry, 12
academia, 1 gov’t;
Members asked to assess impact of several advanced control
technologies:
Q1 Responses [23 responses]
• PID control: 23 High-impact
• Model-predictive control: 18 High-impact; 2 No/Lo impact
• System identification: 14 High-impact; 2 No/Lo impact
• Process data analytics: 14 High-impact; 4 No/Lo impact
• Soft sensing: 12 High-impact; 5 No/Lo impact
• Fault detection and identification [22]: 11 High-impact; 4 No/Lo impact
• Decentralized and/or coordinated control: 11 High-impact; 7 No/Lo impact
• Intelligent control: 8 High-impact; 7 No/Lo impact
• Discrete-event systems [22]: 5 High-impact; 7 No/Lo impact
• Nonlinear control: 5 High-impact; 8 No/Lo impact
• Adaptive control: 4 High-impact; 10 No/Lo impact
• Hybrid dynamical systems: 3 High-impact; 10 No/Lo impact
• Robust control: 3 High-impact; 10 No/Lo impact
6
Page 4
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 7
Why is MPC so successful ?
MPC is Most general way of posing the control problem in the time domain: Optimal control
Stochastic control
Known references
Measurable disturbances
Multivariable
Dead time
Constraints
Uncertainties
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 8
Real reason of success: Economics
MPC can be used to optimize operating points (economic objectives). Optimum usually at the intersection of a set of constraints.
Obtaining smaller variance and taking constraints into account allow to operate closer to constraints (and optimum).
Repsol reported 2-6 months payback periods for new MPC applications.
P1 P2
Tmax
Page 5
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 9
Flash
Línea 2
Línea 1Lavado
Contacto 1
Contacto 3
Contacto 2
EXHAUST GAS PIPING
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 10
Page 6
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 11
-5
-4
-3
-2
-1
0
1
2
1 100 199 298 397 496 595 694 793 892
Serie1
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 12
Page 7
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 13
Benefits
Yearly saving of more that 1900 MWh
Standard deviation of the mixing chamber
pressure reduced from 0.94 to 0.66
Operator’s supervisory effort: percentage
of time operating in auto mode raised
from 27% to 84%.
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 14
A little bit of history: the
beginning
Kalman, LQG (1960)
Propoi, “Use of LP methods ...” (1963)
Richalet et al, Model Predictive Heuristic Control (MPHC)
IDCOM (1976, 1978) (150.000 $/year benefits because of
increased flowrate in the fractionator application)
Cutler & Ramaker, DMC (1979,1980)
Cutler et al QDMC (QP+DMC) (1983)
Clarke et al GPC (1987)
First book: Bitmead et al, (1990)
Page 8
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 15
The impulse of the 90s. A renewed
interest from Academia (stability)
Stability was difficult to prove because of the finite horizon and the
presence of constraints (non linear controller, no explicit solution, …)
A breakthrough produced in the field. As pointed out by Morari: ”the
recent work has removed this technical and to some extent
psychological barrier (people did not even try) and started wide
spread efforts to tackle extensions of this basic problem with the
new tools”. (Rawlings & Muske, 1993)
Many contributions to stability and robustness of MPC: Allgower,
Campo, Chen, Jaddbabaie, Kothare, Limon, Magni, Mayne, Michalska,
Morari, Mosca, de Nicolao, de Olivera, Scattolini, Scokaert…
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 16
MPC now
Linear MPC is a mature discipline. More than 30.000 industrial applications.
The number of applications seems to duplicate every 4 years.
Some vendors have NMPC products: Adersa (PFC), Aspen Tech (Aspen Target), Continental Control (MVC), DOT Products (NOVA-NLC), Pavilon Tech. (Process Perfecter)
Efforts to develope MPC for more difficult situations: Multiple and logical objectives (Morari, Floudas)
Hybrid processes (Morari, Bemporad, Borrelli, De Schutter, van den Boom …)
Nonlinear (Alamir, Alamo, Allgower, Biegler, Bock, Bravo, Chen, De Nicolao, Findeisen, Jadbadbadie, Limon, Magni, …)
Fast MPC (Bemporad, Löfberg, Fikar, ...)
Challenge: Incorporate stability and robustness issues in industrial MPC design.
Page 9
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 17
MPC strategy
At sampling time t the future control sequence is compute so that the future sequence of predicted output y(t+k/t) along a horizon N follows the future references as best as possible.
The first control signal is used and the rest disregarded.
The process is repeated at the next sampling instant t+1
t t+1 t+2 t+N
Control actions
Setpoint
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 18
tt+1t+2
t+N t t+1 t+2 …….. t+N
u(t)
Only the first
control move is
applied
Errors minimized over
a finite horizon
Constraints
taken into
account
Model of
process used
for predicting
Page 10
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 19
t+2 t+1
t+N
t+N+1
t t+1 t+2 …….. t+N t+N+1
u(t)
Only the first
control move is
applied again
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 20
PID: u(t)=u(t-1)+g0 e(t) + g1 e(t-1) + g2 e(t-2)
MPC vs. PID
Page 11
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 21
MPC strategy
Consider a nonlinear invariant discrete time system:
x+=f(x,u), x Rn, u Rm
The system is subject to hard constraints
x X, u U
Let u={u(0),...,u(N-1) } be a sequence of N control inputsapplied at x(0)=x,
the predicted state at i is
x(i)=(i;x, u)=f(x(i-1), u(i-1) )
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 22
MPC strategy
1. Optimization problem PN(x,):
u*= arg minu (i=0,...,N-1) l(x(i),u(i)) + F(x(N))
Operating constaints .
x(i) X, u(i) U, i=0,...,N-1
Terminal constraint (stability): x(N)
2. Apply the receding horizon control law: KN(x)=u*(0).
Page 12
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 23
Linear MPC
f(x,u) is an affine function (model)
X,U, are polyhedra (constraints)
l and F are quadratic functions (or 1-norm
or -norm functions)
QP or LP
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 24
Otherwise
If f(x,u) is not an affine function
Or any of X,U, are not polyhedra
Or any of l and F are not quadratic functions
(or 1-norm or -norm functions)
Non linear MPC (NMPC)
Non linear (non necessarily convex) optimization problem much more difficult to solve.
Page 13
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 25
MPC stability and constraints
Stability was difficult to prove because of the finite horizon
and the presence of constraints (non linear controller, no
explicit solution, …)
Manipulated variables can always be kept in bound by the
controller by clipping the control action or by the actuator.
Output constraints are mainly due to safety reasons, and
must be controlled in advance because output variables
are affected by process dynamics.
Not considering contraints properly may lead to unstability
Gunter Stein: “Respect the unstable”
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 26
Stability and constraints
y(t+1)=1.2 y(t)+0.2 u(t-2) with -4 < u(t) < 4, N=5
Page 14
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 27
MPC stability
Infinite horizon. Keerthi and Gilbert (J. Optim.Theory Appl., 1988) the objective function can be considered a Lyapunov function, providing nominal stability. Cannot be implemented: an infinite set of decision variables.
Terminal state equality constraint. Clarke and Scattolini (IEE, 1991)
x(k+N)= xS
difficult to implement in practice.
xS
x(t)
x(t+1)x(t+2)
x(t+N)
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 28
MPC stability (2)
Dual control. Michalska and Mayne
(1993) x(N)
Once the state enters the
controller switches to a previously
computed stable linear strategy.
Quasi-infinite horizon. Chen and Allgower (1998). Terminal region and stabilizing control, but only for the computation of the terminal cost. The control action is determined by solving a finite horizon problem without switching to the linear controller even inside the terminal region. The term (|| x(t+N)||P)2 added to the cost function and approximates the infinite- horizon one.
x(t)
x(t+1)x(t+2)
x(t+N)
Page 15
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 29
MPC stability (3)
Asymptotic stability theorem (Mayne 2001)
The terminal set is a control invariant set.
The terminal cost F(x) is an associated Control
Lyapunov function such that
min{u U} {F(f(x,u))-F(x) + l(x,u) | f(x,u)} ≤0 x
Then the closed loop system is asymptotically
stable in XN( )
How robust is the stable MPC ?
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 30
stable MPC is Input to State Stable
D. Limón, T. Alamo and E.F. Camacho, Input to State Stable MPC for Constrained Discrete-time Nonlinear
Systems with Bounded Additive Uncertainties, 2012, Las Vegas.
D. Limon. T. Alamo, D.M. Raimondo, D. Muñoz de la. Peña. J.M. Bravo and E.F. Camacho, Input-to-state
stability: a unifying framework for robust model predictive control, Nonlinear Model Predictive Control Lecture
Notes in Control and Information Sciences Volume 384, 2009, pp 1-26
MPC is inherently robust
under mild conditions:
continuity of f(x,u,d,w)
Page 16
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
Uncertainties in MPC
Past and present:
Model
State
Future
Model
Process load
References and Control objectives
31
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
Robustness in MPC
Robust stability.
Robust constraint satisfaction.
Robust performance.
Robustness to failures.
32
0 1 2 3 4 5 6-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Page 17
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 33
Robustness
Uncertain system: x+=f(x,u, ), x Rn, u Rm Rp
With bounded uncertainties and subject to hard constraints x X, u U
The uncertain evolution sets or reachable sets (tube):
X(i)=(i;x, u)= {z Rn | , y X(i-1), z=f(y, u(i-1), )}
and X(0)=x
t t+1 t+2 … t+N
y(t)
u(t)
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 34
Robust stability
The stability conditions has to be satisfied for all possible values of the uncertainties.
The terminal set is a robustcontrol invariant set. (i.e. x,
u U | f(x,u, ))
The terminal cost F(x) is an associated Control Lyapunov function such that
min{u U} {F(f(x,u,))-F(x) + l(x,u) |
f(x,u,)} ≤0 x,
Page 18
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 35
Computation of reachable sets and invariant sets for robust constraint satisfaction or robust stability
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 36
Computation of reachable sets and invariant sets for robust constraint satisfaction or robust stability
• Reachable sets are difficult to compute.
• Approximations and bounding based on:
• Ellipsoids
• Linealization
• Lipschitz continuity
• Interval Arithmetic
• Zonotopes
• DC Programming
Page 19
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 37
Illustrative example
-1.2 -1.1 -1 -0.9 -0.80.8
0.9
1
1.1
1.2
-0.8 -0.75 -0.7 -0.65 -0.6 -0.55
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
Interval arithmetics
Zonotope inclusion
DC-programmingOne step set
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 38
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Illustrative example Sequence of reachable sets
Zonotope inclusion
DC-programming
Page 20
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
MPC for tracking (motivation)
• Most stability – robustness results for the origin.
• What if your setpoints change ?
Moving the invariant
set to the new
setpoint may not
work in the presence
of constraints
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
Robust MPC for Tracking
Consider the following discrete time LTI system with additive bounded
uncertainties:
Objective: Given any admissible setpoint s, design a control law such that:
y(k) tends to the neighbourhood of yt when k→
x(k) and u(k) are admissible for all k ≥ 0 and all possible realizations of
Problem description
The system is constrained to:
Linear
Processu xRobust MPC
for tracking
target
yt
Page 21
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
Robust MPC for Tracking
Lemma (Langson 2004)
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
• Considering the tighter set of constraints for the nominal system
The tube: (Langson 2004 ; Bertsekas 1972)
Robust MPC for Tracking
(Mayne et al., 2005)
Page 22
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
MPC vs Robust MPC for tracking
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
Robust MPC for Tracking
Then, for any feasible initial state i.e., x N and any reachable target, the
uncertain system is steered asymptotically to the set
for all possible realization of the disturbances, satisfying the constraints
Theorem: Consider that
is such that is stable
Q>0, R>0, and P such that:
at is an admissible invariant set for tracking for the nominal system
subject to the following constraints
K is such that (A+BK) is stable and are not empty sets
Let be the feasibility region of the optimization problem
(Alvarado, Limon, Camacho, 2010)
Page 23
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
PSA solar plant
Located in Taberna desert (Almeria, Spain).
Hot oil that can be used to produce steam to produce electricity or for
a desalination plant
The control goal is to keep the oil’s temperature close to the reference.
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
Metal: ρm CmAm∂Tm/ ∂t= ηo I G – G Hl(Tm-Ta) – LHt(Tm-Tf)
Fluid: ρf CfAf∂Tf/ ∂t + ρf Cf q ∂Tm/ ∂x = LHt(Tm-Tf)
Process model
Simulink model can be downloaded from:
E.F. Camacho, et al. Control of Solar Energy Systems, Springer, 2014
http://www.esi2.us.es/~eduardo/libro-s/libro.html
46
Page 24
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
Solar field
Parabolic trough
Outlet temp.Oil flow
Solar
radiation
Air
Temp.
Inlet oil
Temp.
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
PSA trough solar plant
12.8 13 13.2 13.4 13.6 13.8 14 14.2 14.4 14.6 14.8150
160
170
180
190
200
210
local time (h)
tem
pera
ture
s (
oC
)
FFref
tsal
tmodel
12.8 13 13.2 13.4 13.6 13.8 14 14.2 14.4 14.6 14.820
40
60
80
100
120
A2=8.685678e-001, B
2=1.280501e-001, C
2=1, D
2=0
local time (h)
Envirom
ent
Irr/10
Tamb
Tin
The first order model
Identification:
Page 25
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
PSA trough solar plant
Identification:To determine the set W the output of the model is compared with the real output
for a big set of data.
14 14.2 14.4 14.6 14.8 15 15.2 15.4 15.6 15.8 16150
160
170
180
190
200
local time (h)
tem
pera
ture
s (
oC
)
FF
ref
tsal
tmodel
14 14.2 14.4 14.6 14.8 15 15.2 15.4 15.6 15.8 16
-2
0
2
Maximal error = 2.681207e+000
local time (h)
Err
or
Model
Error
14 14.2 14.4 14.6 14.8 15 15.2 15.4 15.6 15.8 160
50
100
150
Environment variables
local time (h)
Envirom
ent
Irr/10
Tamb
Tin
The constraints sets are:
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
Plant controlled by robust MPC for tracking
Controller parameters:
ACUREXFFRMPCT FFref flow
Tin Tamb Irr
Tout
First order model
s
11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13220
230
240
250
260
Local time
MPC#1
Tout
Tref
Trefart
11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13220
240
260
280
Local time
Control Action
Tff
11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13
-50
0
50
100
Local time
Disturbances
Rad/10
Radcor
/10
west 10
Tin
/10
Pyrometer sensor error
Page 26
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
UKF MPC
Control Predictivo de Sistemas de Energ´ıa Solar Distribuidos 80 / 81
Conclusiones y trabajos futuros
Lista de Publicaciones
1 A.J.Gallego, E.F. Camacho (2012,2013)
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
UKF NMPC
Page 27
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
PSA Acurex
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
Spain (650 MWe)• Solucar (3x50MWe)
• Helioenergy (2x50MWe)
• Solacor (2x50MWe)
• Helios (2x50MWe)
• Solaben (4x50MWe)
USA (560 MWe)• Solana (280 MWe)
• Mojave (2x140 MWe)
South Africa (100 MWe)• Kaxu (100 MWe)
Arabs Emirates (100
MWe)• Sham1 (100 MWe)
Argelia (20 MW)• Hassi R'Mel (co-generation)
Abengoa trough plants
Page 28
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
Solucar (3x50MWe)
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015
Solucar (3x50MWe)
Page 29
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 57
Outline
1. Model Predictive Control
2. Stability and robustness for MPC
3. Min max MPC
4. Fault tolerant MPC
5. Conclusions
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 58
Why Min-Max Model Predictive Control ?
0 2 4 6 8 10-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1MPC
segundos
y
0 2 4 6 8 10-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1MMMPC
segundos
y
Better performance against uncertainties More Robustness
Valladolid'2012
Eduardo F.
Camacho
MPC:
Stability and
Robustness
Issues
Page 30
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 59
Why Min-Max Model Predictive Control ?
4 5 6 710
-1
100
101
102
103
horizonte de predicción
Tie
mpo (
seg.)
In spite of advantages, the numberof reported applications is very low
¿?
Computational
burden
Berenguel et al. 1997Kim et al. 1998Álvarez et al. 2003
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 60
Open loop vs close loop
prediction
MPC with open-loop prediction: The sequence of control actions is computed with the information available at time t.• 1987 (Campo and Morari). • Min-max over real numbers• Conservatism.• Techniques available.
MPC with close-loop prediction: The controller considersthat the value of the disturbances will be known in the future.• 1997 (Lee and Yu) and 1998 (Scokaert et al)• Min-max over control laws.• Less conservative• Greater computational burden (not any single reported application to
a real process).
Page 31
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 61
Robust Model Predictive Control
System model:
Bounded additive uncertainties
Two strategies to consider u(t):
Open-loop predictions:
Semi-feedback predictions:Computed bythe controller
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 62
Robust MPC
Outputpredictions
VariablesManipulated
t t+1 t+N
futurepast
u(t+k)
y(t+k)
min-max
MPCPlant C
K
w v ux
y
-+
The inner loop pre-stabilizesthe nominal system
Page 32
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 63
Min-max MPC open loop (1-norm)
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 64
Min-max MPC open loop (1-norm)
LP (with many constraints: the vertices of the uncertainty polytope)
Page 33
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 65
Multiparametric min-max MPC
N=3
N=5
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 66
Reduction of computational burden
Set of active verticesis very small
Page 34
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 67
Feedback PT-326•Scaled laboratory process•2nd order system•Fast dynamics•Ts = 0.4 s.
(Explicit solution)
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 68
Comparisons
MPC
MMMPC
MMMPCwith linear feedback
Different positions of the inlet throttle from 20o to 100o
Page 35
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 69
Outline
1. Model Predictive Control
2. Stability and robustness for MPC
3. Min max MPC
4. Fault tolerant MPC
5. Conclusions
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 70
Modelling bounded uncertainties:
Difference inclussion
•Model to confine the sucessor state into a set
•The function f(.,.,.) and the set W provide a
difference inclussion for the system if for any pair
(x,u), there is a w in W such that
Page 36
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 71
Difference Inclussions
Example:
Suppose that we obtain a nominal linear model around an
operation point :
Suppose that we are able to bound the discrepancy between the
nominal model and the actual behaviour of the system:
Thus we obtain the following difference inclussion:
This can be rewritten using the Minkowski sum notation:
ρ
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 72
Consistent state set
Page 37
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 73
Determination of the compatible output set
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 74
A simplified case
Page 38
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 75
Non Detectability
Consistent state set for model 1
Consistent state set for model 2
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 76
Non detectable faults:
Multimodel MPC
Suppose a series of M model compatible with the last set
of measurements.
xj(t+1)=Aj (t) xj (t) + Bj u(t) + ej(t)
y(t) = Cj xj(t) + vj(t)
When a control sequence U is applied, the prediction
equation for each active model (i.e. j=1,2, …M)
Yj= Fj xj(t) + Guj U + Gwj Wj
Each model is constrained (incuding stability and/or
robustness constraints) by
Rj U ≤ bj + dj xj(t) + fj Wj
Page 39
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 77
Multimodel MPC
y
y1
u
y2
y3
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 78
Multimodel MPC
min J*(U, x1 (t), x2 (t), … xM (t))
s.t.
Rj U ≤ bj + dj xj(t) + fj Wj j = 1, … , M
J*= max J(U, x1 (t), x2 (t), … xM (t), W1 , W2 , … WM )
J*= E[J(U, x1 (t), x2 (t), … xM (t), W1 , W2 , … WM )]
QP problem !!!
U
W1 , W2 , … WM
Page 40
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 79
Hypothesis on future faults
Example: actuator jamming
Define: Uk=[u(t),u(t+1),…u(t+k-1),0, …0]
min J*(U, x1 (t), x2 (t), … xM (t))U
s.t. Rj Uk ≤ bj + dj xj(t) + fj Wj j = 1, … , M, k= 1,…,N
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 80
Hypothesis on future faults
u(k)
u(k+3)u(k+2)
u(k+1)
u(k)u(k)
u(k)
u(k+1)
u(k+1)
u(k+2)
Terminal set
Page 41
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 81
Hypothesis on future faults
Robust MPC scenario
u(k)
u(k+3)u(k+2)
u(k+1)
u(k)u(k)
u(k)
u(k+1)
u(k+1)
u(k+2)
Robust terminal set
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 82
Conclusions
1. Nominal stable MPC shown to be input to state stable
2. Although there are robust (or stable) MPC design
techniques developed in the academia, these are not
used in industry.
3. Number of difficulties: modelling uncertainties,
determining invariant regions, computing reach sets,
solving optimization problem…
4. Efforts needed to simplify robust design techniques
Simpler models ? >> bigger uncertainties bound.
Heuristics.
Can stability be guarantied 100% ?
Probabilistic approaches ?
Page 42
E.F. Camacho Robust MPC design, Future and Practical Applications Rocond'2015 83
[email protected]