st
1 HYCON PhD School on Hybrid Systems
www.unisi.it
www.ist-hycon.org
Model Predictive Control
of Hybrid Systems
Alberto Bemporad
University of Siena, Italy
bemporad@dii.unisi.it
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HYSCOM
IEEE CSS Technical Committee on Hybrid Systems
Siena, July 19-22, 2005 - Rectorate of the University of Siena 9
Outline
Model Predictive Control
of Hybrid Systems
• Model Predictive Control (MPC) concepts
Alberto Bemporad
• Hybrid models for MPC
Dept. Information Engineering - University of Siena, Italy
• MPC of hybrid systems
http://www.dii.unisi.it/~bemporad
http://www.dii.unisi.it/~bemporad
• Explicit MPC (multiparametric programming)
• Optimization-based reachability analysis
• Examples
University
University of Siena
(founded in 1240)
EIHS – European Institute
of Hybrid Systems
1st HYCON PhD School on Hybrid Systems (Siena, 20/7/05)
2
Model Predictive Control
Optimizer
Model Predictive Control
Plant
Reference
Input
r(t)
u(t)
Output
y(t)
Measurements
• MPC concepts
• MODEL: a model of the plant is needed to predict
the future behavior of the plant
• PREDICTIVE: optimization is based on the
predicted future evolution of the plant
• CONTROL: control complex constrained
multivariable plants
• Linear MPC
• Matlab tools for linear MPC
3
Receding Horizon Philosophy
• At time t :
– minimize
Receding Horizon - Example
y(t+k)
r(t)
Solve an optimal
control problem over a
finite future horizon N :
4
Predicted outputs
Manipulated
Inputs
t t+1
• MPC is like playing chess !
u(t+k)
t+N
– subject to constraints
t+1 t+2
t+N+1
• Only apply the first optimal move
• Get new measurements, and repeat the optimization
at time t+1
Advantage of on-line optimization: FEEDBACK!
5
6
Constrained Optimal Control
Constrained Optimal Control
• Optimization problem:
• Linear Model:
By substituting
, we get
(quadratic)
• Constraints:
(linear)
• Constrained optimal control problem
(quadratic performance index):
Convex QUADRATIC
PROGRAM (QP)
7
8
Model Predictive Control Toolbox
MPC of Linear Systems
past
future
• MPC Toolbox 2.0 (Bemporad, Ricker, Morari, 1998-2004):
Predicted
outputs y(t+k|t)
–
–
–
–
Manipulated
Inputs u(t+k)
t t+1
At time t:
t+N
Object-oriented implementation (MPC object)
MPC Simulink Library
MPC Graphical User Interface
RTW extension (code generation)
• Get/estimate the current state x(t)
• Solve the QP problem
and let U={u*(0),...,u*(N-1)} be the solution
(=finite-horizon constrained open-loop optimal control)
• Apply only
optimal inputs
and discard the remaining
Only linear models are handled
• Go to time t+1
9
Example: AFTI-16
http://www.mathworks.com
10
Example: AFTI-16
• Linearized model:
see demo afti16.m
(MPC-Tbx)
11
12
Convergence
Convergence Proof
Lyapunov function
(Keerthi and Gilbert, 1988)(Bemporad et al., 1994)
Proof: Use value function as Lyapunov function
13
Global optimum is not needed to prove convergence !
Convergence Proof
14
Convergence Proof
15
16
Outline
MPC and LQR
• Consider the MPC control law:
9 Model Predictive Control (MPC) concepts
Jacopo Francesco
Riccati (1676 - 1754)
• Hybrid models for MPC
• MPC of hybrid systems
• Explicit MPC (multiparametric programming)
• Optimization-based reachability analysis
• In a polyhedral region around the origin the MPC control
law is equivalent to the constrained LQR controller with
(Chmielewski, Manousiouthakis, 1996)
weights Q,R.
• Examples
(Scokaert and Rawlings, 1998)
MPC ≡ constrained LQR
• The larger the horizon, the larger the region where MPC=LQR
17
18
Hybrid Systems
Hybrid Models for MPC
B
• Discrete Hybrid Automata (DHA)
Control
Theory
Finite
state
machines
Continuous
dynamical
systems
Hybrid systems
A
• Mixed Logical Dynamical (MLD) Systems
Computer
Science
B
C
• Piecewise Affine (PWA) Systems
u(t)
B
system
y(t)
C
C
A
19
“Intrinsically Hybrid” Systems
20
Cruise Control Problem
continuous inputs
GOAL:
command gear ratio, gas
pedal, and brakes to track
a desired speed and minimize
consumptions
discrete input
Discrete input
(1,N,2,3,4)
+
Continuous inputs
(brakes, gas, clutch)
Continuous
+
dynamical states
CHALLENGES:
(velocities, torques,
• continuous and discrete inputs
• dynamics depends on gear
• nonlinear torque/speed maps
air-flows, fuel level)
21
Key Requirements for Hybrid Models
• Descriptive enough to capture the behavior of the system
22
Piecewise Affine Systems
state+input space
– continuous dynamics (physical laws)
– logic components (switches, automata, software code)
– interconnection between logic and dynamics
• Simple enough for solving analysis and synthesis problems
(Sontag 1981)
?
linear systems
nonlinear systems
linear hybrid systems
“Make everything as simple as possible, but not simpler.”
— Albert Einstein
• Can approximate nonlinear/discontinuous dynamics
arbitrarily well
24
Switched Affine System
Discrete Hybrid Automaton
(Torrisi, Bemporad, 2004)
Event
Generator
Switched
Affine
System
1
Finite State
Machine
2
time or
event
counter
Mode Selector
s
The affine dynamics depend on the current mode i(k):
mode
25
26
Mode Selector
Event Generator
Event variables are generated by linear threshold conditions over
continuous states, continuous inputs, and time:
The active mode i(k) is selected by a Boolean function of the
current binary states, binary inputs, and event variables:
Example:
0
0
Example: [δ=1] ↔ [xc(k)≥0]
1
27
Finite State Machine
1
the system has 3 modes
28
Logic and Inequalities
Glover 1975, Williams 1977
The binary state of the finite state machine evolves
according to a Boolean state update function:
Switched
Affine System
Finite State
Machine
Mode Selector
1
Event
Generator
2
Example:
s
29
30
Logic → Inequalities: Symbolic Approach
Switched Affine Systems
Switched
Affine System
1
The state-update equation can be rewritten as a
combination of affine terms and if-then-else
conditions:
0. Given a Boolean statement
2
1. Convert to Conjunctive Normal Form (CNF):
s
2. Transform into inequalities:
polyhedron
where
.
If-then-else conditions are converted to inequalities.
Output equations yc(t)=Cixc(t)+Diuc(t)+gi admit a similar
transformation.
Any logic proposition can be translated into linear integer ineq.
31
Logic → Inequalities: Symbolic Approach
32
Logic → Inequalities: Geometric Approach
Example: logic “AND”
F (X1, X2, X3) = [X3 ↔ X1 ∧ X2]
Example: F (X1, X2, X3) = [X3 ↔ X1 ∧ X2]
1. Convert to Conjunctive Normal Form (CNF):
(see e.g.: http://www.oursland.net/aima/propositionApplet.html
T:
or just search [CNF + applet] on Google ...)
(X3 ∨ ¬X1 ∨ ¬X2) ∧ (X1 ∨ ¬X3) ∧ (X2 ∨ ¬X3)
X1
0
0
1
1
X2
0
1
0
1
X3
0
0
0
1
Key idea:
White points cannot be
in the hull of black points
δ3
(1,1,1)
δ2
(0,1,0)
δ1
(0,0,0)
(1,0,0)
2. Transform into inequalities:
⎧
⎪
⎨ δ3 + (1 − δ 1) + (1 − δ 2) ≥ 1
δ1 + (1 − δ3) ≥ 1
⎪
⎩
δ2 + (1 − δ3) ≥ 1
Convex hull algorithms: cdd, lrs, qhull, chD, Hull, Porto
33
34
Mixed Logical Dynamical Systems
A Simple Example
Discrete Hybrid Automaton
• System:
• Associate
and transform
HYSDEL
(Torrisi, Bemporad, 2004)
Mixed Logical Dynamical (MLD) Systems
(Bemporad, Morari 1999)
• Then
Continuous and
binary variables
• Rewrite as a linear equation
• Computationally oriented (mixed-integer programming)
• Suitable for controller synthesis, verification, ...
35
36
Hybrid Toolbox for Matlab
HYSDEL
(Bemporad, 2003-2004)
Features:
• Hybrid model (MLD and PWA) design and simulation
(HYbrid Systems DEscription Language)
• Describe hybrid systems:
• Control design for linear systems w/ constraints
and hybrid systems (on-line optimization via QP/MILP/MIQP)
–
–
–
–
–
• Explicit control (via multiparametric programming)
• C-code generation
• Simulink
Automata
Logic
Lin. Dynamics
Interfaces
Constraints
(Torrisi, Bemporad, 2004)
• Automatically generate MLD models in Matlab
Download:
http://www.dii.unisi.it/hybrid/toolbox
http://www.dii.unisi.it/hybrid/toolbox
Reference: http://control.ethz.ch/~hybrid/hysdel
37
Example: Bouncing Ball
38
Bouncing Ball – Time Discretization
x
x
-mg
N
F= m g
x
How to model this system in MLD form?
39
Systems Theory for
DiscreteDiscrete-Time Linear Hybrid Systems
HYSDEL - Bouncing Ball
SYSTEM bouncing_ball {
INTERFACE {
/* Description of variables and constants */
STATE { REAL height [-10,10];
REAL velocity [-100,100];
PARAMETER {
REAL g;
REAL dissipation;
REAL Ts; }
}
IMPLEMENTATION {
AUX {
REAL z1;
REAL z2;
BOOL negative;
–
–
–
–
–
–
}
negative = height <= 0; }
DA {
z1 = {
z2 = {
• Analysis
}
/* 0=elastic, 1=completely anelastic */
AD {
40
IF negative THEN height-Ts*velocity
ELSE height+Ts*velocity-Ts*Ts*g};
IF negative THEN -(1-dissipation)*velocity
ELSE velocity-Ts*g}; }
Well-posedness
Realization & Transformation
Reachability (=Verification)
Observability
Stability
Identification
• Synthesis
– Control
– State estimation
CONTINUOUS {
height = z1;
velocity=z2;}
}}
go to demo /demos/hybrid/bball.m
41
42
Well-posedness
Equivalences of Hybrid Models
Are state and output trajectories defined ?
Uniquely defined ? Persistently defined ?
• MLD well-posedness :
î(t) = F(x(t), u(t))
x1(k), y1(k)
Σ1
{x(t), u(t)} → {x(t + 1)}
{x(t), u(t)} → {y(t)}
z(t) = G(x(t), u(t))
are single valued
u(k)
x1(0)=x2(0)
x2(k), y2(k)
Σ2
Numerical test based on mixed-integer programming available
(Bemporad, Morari, Automatica, 1999)
43
44
MLD and PWA Systems
Example: Room Temperature
Theorem MLD systems and PWA systems are equivalent
AC system
(Bemporad, Ferrari-Trecate, Morari, IEEE TAC,2000)
•
Proof is constructive: given an MLD system it returns its
equivalent PWA form
•
Drawback: it needs the enumeration of all possible
combinations of binary states, binary inputs, and δ variables
ucold
uhot
Heater
•
Most of such combinations lead to empty regions
•
Efficient algorithms are available for converting MLD models
into PWA models avoiding such an enumeration:
Hybrid Dynamics
• #1 turns the heater (air conditioning) on whenever he is cold (hot)
• If #2 is cold he turns the heater on,
unless #1 is hot
Tamb
T2
T1
• If #2 is hot he turns the air conditioning on, unless #2 is cold
T2
T1
• Otherwise, heater and air conditioning are off
• A. Bemporad, “Efficient Algorithms for Converting Mixed Logical
Dynamical Systems into an Equivalent Piecewise Affine Form”, IEEE
Trans. Autom. Contr., 2004.
• Ṫ 1 = −α1 (T1 − T amb) + k 1 (uhot − ucold)
(body temperature dynamics of #1)
• T. Geyer, F.D. Torrisi and M. Morari, “Efficient Mode Enume-
• Ṫ 2 = −α2 (T2 − T amb) + k 2 (uhot − ucold)
(body temperature dynamics of #2)
ration of Compositional Hybrid Models”, HSCC’03
45
HYSDEL Model
go to demo /demos/hybrid/heatcool.m
46
Hybrid MLD Model
• MLD model
Hybrid Toolbox
for Matlab
(Bemporad, 2003-2005)
(temperatures T1,T2)
• 2 continuous states:
http://www.dii.unisi.it/hybrid/toolbox
• 1 continuous input:
(room temperature Tamb)
• 2 auxiliary continuous vars:
• 6 auxiliary binary vars:
(power flows uhot, ucold)
(4 thresholds + 2 for OR condition)
• 20 mixed-integer inequalities
>>S=mld('heatcoolmodel',Ts)
get the MLD model in Matlab
>>[XX,TT]=sim(S,x0,U);
simulate the MLD model
Possible combination of integer variables: 26 = 64
47
48
Hybrid PWA Model
Simulation in Simulink
• PWA model
50
• 2 continuous states:
(temperatures T1,T2)
Temperature T
2 (deg C)
40
• 1 continuous input:
(room temperature Tamb)
• 5 polyhedral regions
(partition does not depend on input)
30
20
10
0
-10
-10
>>P=pwa(S);
0
10
20
30
Temperature1T(deg C)
uhot = 0
ucold = 0
40
50
uhot = 0
uhot = Ū H
ucold = ŪC ucold = 0
49
50
Equivalence Results
Other Existing Hybrid Models
• Linear complementarity (LC) systems
x(t + 1) = Ax(t) + B1 u(t) + B2 w(t)
y(t) = Cx(t) + D1 u(t) + D2 w(t)
v(t) = E1 x(t) + E2 u(t) + E3 w(t) + e4
0 ô v(t)⊥w(t) õ 0
?
(Heemels, 1999)
?
Ex: mechanical systems
circuits with diodes etc.
MMPS
PWA
?
LC
?
?
?
DHA
MLD
• Extended linear complementarity (ELC) systems
?
ELC
?
(De Schutter, De Moor, 2000)
Generalization of LC systems
• Min-max-plus-scaling (MMPS) systems
(De Schutter, Van den Boom, 2000)
x(t + 1) = Mx(x(t), u(t), d(t))
y(t) = My (x(t), u(t), d(t))
0 õ Mc(x(t), u(t), d(t))
?
Theorem All the above six classes of discrete-time hybrid
models are equivalent (possibly under some additional
assumptions, such as boundedness of input and state variables)
(Heemels, De Schutter, Bemporad, Automatica, 2001)
MMPS function: defined by the grammar
(Torrisi, Bemporad, IEEE CST, 2003)
(Bemporad and Morari, Automatica, 1999)
(Bemporad, Ferrari-T.,Morari, IEEETAC, 2000)
M := x i|ë| max(M1 , M2 )| min(M1 , M2 )|M1 + M2 |ìM1
Example: x(t + 1) = 2 max(x(t), 0) + min(à 12u(t), 1)
Theoretical properties and analysis/synthesis tools
can be transferred from one class to another
Used for modeling discrete-event systems (t=event counter)
51
MLD and LC Systems
52
Hybrid System Identification
(Heemels, De Schutter, Bemporad, Automatica, 2001)
•
Sometimes a hybrid model of the process (or of a part of it)
cannot be derived manually from available knowledge.
•
Therefore, a model must be either
– Estimated from data (model unknown)
– or hybridized before it can be used for control/analysis (model
known but nonlinear)
Proof:
x(t + 1) = Ax(t) + B1 u(t) + B2 î(t) + B3 z(t)
y(t) = Cx(t) + D1 u(t) + D2 î(t) + D3 z(t)
E 2 î(t) + E 3 z(t) ô E 4 x(t) + E 1 u(t) + E 5
x(t + 1) = Ax(t) + B1 u(t) + B2 w(t)
y(t) = Cx(t) + D1 u(t) + D2 w(t)
v(t) = E1 x(t) + E2 u(t) + E3 w(t) + e4
0 ô v(t)⊥w(t) õ 0
•
If a linear model is enough, no problem: several algorithms are
available (e.g.: use Ljung’s ID TBX)
•
If switching modes are known and data can be generated for
each mode, no problem: we identify one linear model per
mode (e.g.: use Ljung’s ID TBX)
•
If modes & dynamics must be identified together, we need
For each complementarity pair vi(t), w i(t) introduce a binary variable îi(t) ∈ {0, 1}
[îi(t) = 1] → [vi(t) = 0, w i(t) õ 0]
[îi(t) = 0] → [vi(t) õ 0, w i(t) = 0]
w i(t)
vi(t)
w i(t)
vi(t)
ô
ô
õ
õ
Mîi(t)
M(1 à îi(t))
0
0
hybrid system identification
Set zi(t) = w i(t) and substitute v(t) = E 1 x(t) + E 2 u(t) + E 3 w(t) + e4
53
54
Outline
PWA Identification Problem
Estimate from data both the parameters of the
affine submodels and the partition of the PWA map
9 Model Predictive Control (MPC) concepts
9 Hybrid models for MPC
Example Let the data be generated by the PWARX system
• MPC of hybrid systems
• Explicit MPC (multiparametric programming)
• Optimization-based reachability analysis
• Examples
55
56
Hybrid Control Problem
hybrid process
continuous
states
continuous
inputs
MPC of Hybrid Systems
binary
states
binary
inputs
• Problem setup
on-line decision maker
• Convergence properties
• Computational aspects
desired behavior
• Matlab tools for hybrid MPC
constraints
57
Model Predictive Control
of Hybrid Systems
58
MPC for Hybrid Systems
past
future
Predicted
outputs y(t+k|t)
MLD model
Manipulated
Inputs u(t+k)
t t+1
Controller
Reference
r(t)
t+T
• At time t solve with respect to
Hybrid System
Input
u(t)
Model
Predictive (MPC)
Control
the finite-horizon open-loop, optimal control problem:
Output
y(t)
Measurements
•
•
•
MODEL: use an MLD or PWA model of the plant to predict
the future behavior of the hybrid system
PREDICTIVE: optimization is still based on the predicted
future evolution of the hybrid system
CONTROL: the goal is to control the hybrid system
• Apply only u(t) =uã(t) (discard the remaining optimal inputs)
• Repeat the whole optimization at time t+1
59
60
Closed-Loop Convergence
Convergence Proof
Lyapunov function
(Bemporad, Morari 1999)
Proof: Easily follows from standard Lyapunov arguments
More stability results: see (Lazar, Heemels, Weiland, Bemporad, 2005)
61
Hybrid MPC - Example
62
Hybrid MPC - Example
PWA system:
Constraint:
Note: Global optimum not needed for convergence !
Closed loop:
Open loop behavior
go to demo /demos/hybrid/bm99sim.m
63
Hybrid MPC – Temperature Control
64
Hybrid MPC – Temperature Control
>>refs.x=2;
% just weight state #2
>>Q.x=1;
>>Q.rho=Inf; % hard constraints
>>Q.norm=2;
% quadratic costs
>>N=2;
% optimization horizon
>>limits.xmin=[25;-Inf];
>>C=hybcon(S,Q,N,limits,refs);
>> C
Hybrid controller based on MLD model S <heatcoolmodel.hys>
2
0
0
1
0
state measurement(s)
output reference(s)
input reference(s)
state reference(s)
reference(s) on auxiliary continuous z-variables
20 optimization variable(s) (8 continuous, 12 binary)
46 mixed-integer linear inequalities
sampling time = 0.5, MILP solver = 'glpk'
Type "struct(C)" for more details.
>>
>>[XX,UU,DD,ZZ,TT]=sim(C,S,r,x0,Tstop);
65
66
MIQP Formulation of MPC
(Bemporad, Morari, 1999)
Optimal Control of Hybrid Systems:
Computational Aspects
Mixed Integer Quadratic Program (MIQP)
67
68
(Bemporad, Borrelli, Morari, 2000)
• Mixed-Integer Programming is NP-hard
Phase transitions have been found in
computationally hard problems.
• Introduce slack variables:
ïxk
ïuk
• Set
õ kQy(t + k|t)k∞
õ kRu(t + k)k∞
ø,[ïx1 ,
. . .,
ïxN , ïu1 ,
y
min |x|
ïxk
ïxk
ïuk
ïuk
BUT
min ï
s.t. ï õ x
ïõ àx
[Qy(t + k|t)]i
õ
õ à [Qy(t + k|t)]i
õ [Ru(t + k))]i
õ à [Ru(t + k))]i
i = 1, . . ., p,
i = 1, . . ., p,
i = 1, . . ., m,
i = 1, . . ., m,
Cost of Computation
Mixed-Integer Program Solvers
MILP Formulation of MPC
4000
50 var
40 var
20 var
3000
2000
1000
0
2
3
4
5
6
7
8
Ratio of Constraints to Variables
(Monasson et al., Nature, 1999)
k = 1, . . ., T à 1
k = 1, . . ., T à 1
k = 0, . . ., T à 1
k = 0, . . ., T à 1
• General purpose Branch & Bound/Branch & Cut solvers available
for MILP and MIQP (CPLEX, Xpress-MP, BARON, GLPK, ...)
More solvers and benchmarks: http://plato.la.asu.edu/bench.html
. . ., ïTà1 , U, î, z]
min J(ø, x (t)) =
Mixed Integer Linear Program
(MILP)
ø
Tà1
X
ïxi + ïui
• No need to reach global optimum (see proof of the theorem),
although performance deteriorates
k=0
s.t.Gø ô W + Sx (t)
69
70
Mixed-Integer Programming Solvers
Mixed-Integer Programming Solvers
Main drawbacks:
Main drawbacks:
• Loss of the original discrete structure (Boolean formulas)
• Loss of the original discrete structure (Boolean formulas)
• On-line combinatorial optimization
• On-line combinatorial optimization
Good for large sampling times (e.g., 1 h) / expensive hardware …
Good for large sampling times (e.g., 1 h) / expensive hardware …
… but not for fast sampling (e.g. 10 ms) / cheap hardware !
… but not for fast sampling (e.g. 10 ms) / cheap hardware !
71
72
Exploiting Logic Structures
“Hybrid” Solvers
Main drawbacks when using Mixed-Integer Programming for
Combine MIP and Constraint Satisfaction (CSP) techniques to
exploit the discrete structure of the problem
implementing hybrid MPC control laws:
1.
Why CSP ?
Loss of the original Boolean structure
y More flexible modeling than MIP
(e.g.: constraint logic programming (CLP) and Satisfiability of
Boolean formulas (SAT))
y Structure is kept and exploited to direct the search.
FALSE
TRUE
1
0
x`(k + 1) = x` (k) ∧ u` (k)
Why MIP ?
y Specialized techniques for highly structured problems
(e.g. LP problems); Better for handling continuous vars
y A wide range of tight relaxations are available
Efficiency of MIP solver usually not good when continuous
LP/QP relaxations are not tight
Why a combined approach ?
Performance increase already shown in other application domains
(Harjunkoski, Jain, Grossmann, 2000)74
73
SAT-Based Branch&Bound
Computations: MILP vs. LBLB-B&B
The basic modeling framework has the following form:
(Bockmayr, Kasper,1998)
Convex cost function
Continuous constraints
Mixed constraints
Purely logic constraints
Computation time and # LP solved
for finding an optimal control sequence
optimization variables
SAT-based B&B
Pure B&B
SAT-based B&B “ingredients”
(Bemporad, Giorgetti, 2003)
A relaxed convex problem
A SAT feasibility problem
(pure logic constraints dropped)
(for logic constraints only)
Convex solver (e.g.: LP/QP)
#nodes=11
SAT solver
75
#nodes=6227
Pentium IV 1.8GHz SAT solver: zCHAFF 2003.07.22
76
Mixed-Integer Programming Solvers
SAT vs. MILP
Uniform Random 3CNF benchmarks (from http://www.satlib.org)
All 3SAT instances are in the phase transition region.
Main drawbacks:
• Loss of the original discrete structure (Boolean formulas)
• On-line combinatorial optimization
Good for large sampling times (e.g., 1 h) / expensive hardware …
All clauses are passed to CPLEX new MILP solver as logic
constraints (we exploit the new feature of release 9.0).
PC: P4 2.8GHz + 1GB RAM
Solvers:
CPLEX 9.0
zCHAFF 2003.12.04
77
… but not for fast sampling (e.g. 10 ms) / cheap hardware !
78
Outline
9 Model Predictive Control (MPC) concepts
9 Hybrid models for MPC
Explicit Model Predictive Control
9 MPC of hybrid systems
• Explicit MPC (multiparametric programming)
• Multiparametric quadratic programming
• Optimization-based reachability analysis
• Explicit linear MPC
• Examples
• Explicit hybrid MPC
• Matlab tools for explicit MPC
79
80
OnOn-Line vs. OffOff-Line Optimization
MPC: Pros and Cons
• PRO: systematic design approach
–
–
–
–
Multivariable systems
State and Input Constraints
Stability guarantees
Reference preview
• On-line optimization: given x(t) solve the problem
at each time step t (the control law u=u(x) is implicitly
defined by the QP solver)
Quadratic Program (QP)
• CON: computation complexity !
– Large sampling time/fast hardware
– Software reliability
• Off-line optimization: solve the QP for all x(t) to find
the control law u=u(x) explicitly
multi-parametric Quadratic Program (mp-QP)
81
82
Linearity of the Solution
Multiparametric Quadratic Programming
(Bemporad et al., 2002)
x0∈ X
solve QP to find
identify active constraints at
form matrices
by collecting
active constraints:
KKT
optimality
conditions:
From (1) :
• Objective: solve the QP for all
From (2) :
• Assumptions:
(always satisfied if QP problem
originates from optimal control problem)
(can be easily satisfied, e.g. by
choosing positive input weights)
83
In some neighborhood of x0, λ and U are explicit
affine functions of x !
84
Neighboring Regions
Determining a Critical Region
(Tøndel, Johansen, Bemporad, 2003)
• Impose primal and dual feasibility:
x2
x2
x2
linear inequalities in x !
CR
CR
CR
• Remove redundant constraints:
(this requires solving LP’s)
x1
critical region CR0
• x0
CR0
x-space
x1
x1
The active set of a neighboring region is found by using
the active set of the current region + knowledge of the
type of hyperplane we are crossing:
X
⇒ The corresponding constraint is added to the
active set
• CR0 is the set of all and only parameters x for which
is the optimal combination of active constraints at the optimizer
⇒ The corresponding constraint is withdrawn from
the active set
85
86
Mp-QP Properties
Complexity Reduction
2.5
2.5
CR1
CR2
CR3
CR4
CR5
CR6
CR7
CR8
CR9
2
1.5
1
1
0.5
0.5
0
0
-0.5
continuous,
piecewise affine
-0.5
-1
-1
-1.5
-1.5
-2
-2.5
-2.5
CR1
CR2
CR3
CR4
CR5
CR6
CR7
2
1.5
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
convex, continuous,
piecewise quadratic,
C1 (if no degeneracy)
Corollary: The linear MPC controller is a continuous
piecewise affine function of the state
Regions where the first component of the solution is the same
can be joined (when their union is convex).
(Bemporad, Fukuda, Torrisi, Computational Geometry, 2001)
87
Double Integrator Example
88
mp-QP solution
• System:
sampling + ZOH
Ts=1 s
Nu=2
• Constraints:
• Control objective: minimize
• Optimization problem: for Nu=2
(cost function is
normalized by
max svd(H))
go to demo /demos/linear/doubleintexp.m
89
(Hyb-Tbx)
90
Complexity
Example: AFTI-16
• Linearized model:
Nu=5
Nu=4
Nu=3
Section with x=[0 0 0 0]’, u=[0 0]’
10
8
CPU time
40
Nu=6
4
2
0
# Regions
6
20
0
5
10
15
r2
20
0
300
-2
200
-4
100
0
-6
0
5
10
# free moves
15
20
Explicit controller: 8 parameters, 51 regions
-8
-10
-10
-8
-6
-4
-2
0
2
4
6
8
10
r1
go to demo linear/afti16.m (Hyb-Tbx)
(is the number of regions finite for Nu→∞ ?)
91
92
Explicit Hybrid MPC (MLD)
Multiparametric MILP
min
ø={ø c ,ø d}
øc ∈ Rn
f 0øc + d0ød
ø d ∈ {0, 1} m
s.t. Gøc + Eød ô W + Fx
• mp-MILP can be solved (by alternating MILPs and mp-LPs)
(Dua, Pistikopoulos, 1999)
• On-line optimization: solve the problem for each given x(t)
• Theorem: The multiparametric solution
ã
ø (x) is piecewise affine
Mixed-Integer Linear Program (MILP)
• The MPC controller is piecewise affine in x,r
• Off-line optimization: solve the MILP for all x(t) in advance
(x,r)-space
4
2
M
1
5
63
multi-parametric Mixed Integer Linear Program (mp-MILP)
93
Explicit Hybrid MPC (PWA)
94
Explicit Hybrid MPC (PWA)
Method A:
(Borrelli, Baotic, Bemporad, Morari, Automatica, 2005)
Use a combination of DP (dynamic programming) and mpLP
(1-norm, ∞-norm), or mpQP (quadratic forms)
Method B:
(Bemporad, Hybrid Toolbox, 2003) (Alessio, Bemporad, 2005) (Mayne, ECC 2001)
1 - Use backwards (=DP) reachability analysis for enumerating all
feasible mode sequences I={i(0),i(1),…,i(T-1)};
• Solution u(x,r) found via a combination of
- Dynamic programming or enumeration of feasible mode
sequences, multiparametric linear or quadratic programming,
(Borrelli, Baotic, Bemporad, Morari, 2003)
and polyhedral computation.
(Mayne, ECC 2001)
• The MPC controller is piecewise affine in x,r
3 - Case 1/∞-norm: Compare value functions and split regions.
Quadratic case: keep overlapping regions (possibly eliminate
overlaps that are never optimal) and compare on-line (if needed).
(x,r)-space
4
2
2 - For each fixed sequence I, solve the explicit finite-time
optimal control problem for the corresponding linear
time-varying system (mpQP or mpLP);
M
1
5
Note: in the 2-norm case the partition may not be fully polyhedral
63
Note: in the 2-norm case, the fully
explicit partition may not be polyhedral
95
96
Hybrid Control - Example
PWA system:
Hybrid Control Example
(Revisited)
Closed loop:
Constraints:
Objective:
Open loop
behavior:
HybTbx: /demos/hybrid/bm99sim.m
97
98
Explicit PWA Controller
Hybrid MPC - Example
Closed loop:
Section with r=0
10
1
2
3
4
5
8
6
4
x2
2
0
-2
-4
-6
-8
-10-10 -8
-6
-4
-2
0
x12
4
6
8
10
ñ MPC law
PWA law
HybTbx: /demos/hybrid/bm99sim.m
(CPU time: 1.51 s, Pentium M 1.4GHz)
99
100
Explicit MPC – Temperature Control
Explicit PWA Regulator
>>E=expcon(C,range,options);
Objective:
>> E
Prediction Horizon N=1
Prediction Horizon N=2
10
CR1
CR2
CR3
CR4
CR5
CR6
CR7
CR8
8
6
4
x2
2
10
10
8
8
6
6
4
4
2
0
-2
-4
-4
-4
-6
-6
-8
-8
-8
-10
-10
-10
-10
-6
-4
-2
0
2
x1
4
6
8
10
Type "struct(E)" for more details.
>>
x2
0
-2
-8
The controller is for hybrid systems (tracking)
This is a state-feedback controller.
2
x2
0
-2
-10
-10
Explicit controller (based on hybrid controller C)
3 parameter(s)
1 input(s)
12 partition(s)
sampling time = 0.5
Prediction Horizon N=3
-6
-8
-6
-4
-2
0
x1
2
4
6
8
10
-8
-6
-4
-2
0
x1
2
4
6
8
10
Section in the (T1,T2)-space for Tref = 30
HybTbx: /demos/hybrid/bm99benchmark.m
101
102
Explicit MPC – Temperature Control
Comments on Explicit Solutions
• Alternatives: either (1) solve an MIP on-line
or (2) evaluate a PWA function
• For problems with many variables and/or long
horizons: MIP may be preferable
• For simple problems (short horizon/few constraints):
- time to evaluate the control law is shorter than MIP
Generated
C-code
- control code is simpler (no complex solver must be
included in the control software !)
utils/expcon.h
- more insight in controller’s behavior
103
104
Hybrid Toolbox for Matlab
Hybrid Control Design Flow
(Bemporad, 2003-2004)
Features:
physical
system
hybrid
model
control objectives
& constraints
simulation
and validation
optimal control
setup
(on-line MIP)
• Hybrid model (MLD and PWA) design and simulation
• Control design for linear systems w/ constraints
and hybrid systems (on-line optimization via QP/MILP/MIQP)
• Explicit control (via multiparametric programming)
simulation
and validation
• C-code generation
multiparametric
solver
• Simulink
PWA control law
too
complex
?
yes
implementation
(C code) and
experiments
no
105
http://www.dii.unisi.it/hybrid/toolbox
106
State Estimation / Fault Detection
• Problem: given past output measurements and inputs, estimate the current
state/faults
• Solution: Use Moving Horizon Estimation for MLD systems (dual of MPC)
Augment the MLD model with:
n
• Input disturbances
ø ∈ R
• Output disturbances ð ∈ R p
State Estimation / Fault Detection
At each time t solve
min
the optimization:
T
P
k yê(t à k|t) à y(t à k)k 2 + . . .
MHE optimization = MIQP
Convergence can be guaranteed
Fault detection:
107
and get estimates x
ê (t)
k=1
(Bemporad, Mignone, Morari, ACC 1999)
(Ferrari-T., Mignone, Morari, 2002)
augment MLD with unknown binary distubances þ ∈ {0, 1}p
108
Example: Three Tank System
Example: Three Tank System
COSY Benchmark problem, ESF
COSY Benchmark problem, ESF
• þ1
: leak in tank 1
for 20s ô t ô 60s
• þ1
: leak in tank 1
for 20s ô t ô 60s
• þ2
: valve V1 blocked
for t õ 40s
• þ2
: valve V1 blocked
for t õ 40s
• Add logic constraint
[h 1 ô h v ] ⇒ þ2 = 0
109
110
Outline
9 Model Predictive Control (MPC) concepts
9 Hybrid models for MPC
Optimization-based
Reachability Analysis
9 MPC of hybrid systems
9 Explicit MPC (multiparametric programming)
(Verification of Safety Properties
via Mixed-integer Programming)
• Optimization-based reachability analysis
• Examples
111
Hybrid Verification Problem
Verification Algorithm
hybrid process
continuous
disturbances
binary
disturbances
112
• QUERY: Is the target set Xf reachable after N steps from some
initial state x0 ∈ X0 for some input profile u ∈ U ?
continuous
states
• Computation: Solve the mixed-integer linear program (MILP)
binary
states
verification algorithm
counterexample
answer
always safe !
safety
query
with respect to u(0),δ(0),z(0),…, u(N-1),δ(N-1),z(N-1),x(0)
set of possible
initial states
• Alternative solutions:
• Exploit the special structure of the problem and use polyhedral
set of possible
disturbances
computation.
113
• Use abstractions (LPs) + SAT solvers
(Torrisi, 2003)
(Giorgetti, Pappas, 2005)
114
Verification Example
Verification Example
• MLD model: room temperature system
•
(set of unsafe states)
•
(set of initial states)
•
(set of possible inputs)
•
N=10 (time horizon)
>>[flag,x0,U]=reach(S,N,Xf,X0,umin,umax);
115
116
Verification Example
Verification Example
• MLD model: semiactive suspension system
(set of unsafe states)
•
•
(set of initial states)
(set of possible inputs)
•
•
N=10
(time horizon)
>>[flag,x0,U]=reach(S,N,Xf,X0,umin,umax);
117
118
Integral Switched Affine System
(Bemporad, Di Cairano, Julvez, 2005)
Event
Generator
Integral dynamics:
Integral
Switched
Affine System
Finite State
Machine
1
time
2
s
Mode Selector
Event-Driven / Continuous-Time
Hybrid Automata
mode
k = event index
tk = event instant
Change of variables:
119
Same as discrete-time dynamics !
120
Event Generator
Event
Generator
Asynchronous Finite State Machine
Event
Generator
Events occur at time t if:
Integral
Switched
Affine System
(1)
or
(2)
or
1
Finite State
Machine
time
2
Integral
Switched
Affine System
Finite State
Machine
1
time
2
s
s
(3)
mode
Mode Selector
mode
Mode Selector
Asynchronous FSM dynamics:
x c (t)
hj1 x c1 + hj2 x c2 > w j
hj
hj1 x c1 + hj2 x c2 < w j
x c (tk )
1x
c1 +
hj
2x
c2 =
Example:
Discrete state transitions, events, and mode switches might
occur at any (continuous) time t !
w
j
for some threshold index j
121
122
Hybrid Automaton with Integral
Continuous()
Continuous(-time) Dynamics (icHA
(icHA)
Outline
9 Model Predictive Control (MPC) concepts
Event
Generator
Wrapping up:
Integral
Switched
Affine System
event-driven (asynchronous) FSM
Finite State
Machine
+
9 Hybrid models for MPC
1
tim
e
2
s
integral dynamics
=
Mode Selector
9 MPC of hybrid systems
mod
e
9 Explicit MPC (multiparametric programming)
same as discrete hybrid automaton
9 Optimization-based reachability analysis
• Examples
states
inputs
All numerical control/verification techniques developed for DHA
can be immediately applied to icHA !
123
124
Hybrid Control Problem
Hybrid Control Example:
Cruise Control System
GOAL:
command gear ratio, gas pedal,
and brakes to track a desired
speed and minimize consumption
125
126
Hybrid Model
Hybrid Model
+
à Cà
e (ω) ô M ô Ce (ω)
• Engine torque
• Vehicle dynamics
mẍ = F e à F b à ìxç
C+
e (ω)
• Max engine torque
xç =
vehicle speed
Fe =
traction force
Fb =
brake force
180
160
discretized with sampling time T s = 0.5 s
140
• Transmission kinematics
ω=
Rg(i)
xç
ks
Fe =
Rg(i)
M
ks
ω =
engine speed
M=
engine torque
i =
Piecewise-linearization:
120
(PWL Toolbox, Julián, 1999)
100
requires: 4 binary aux variables
4 continuous aux variables
60
1000
2000
3000
4000
Cà
e (ω) = ë1 + ì 1 ω
• Min engine torque
gear
80
127
128
Hybrid Model
• Gear selection:
Hysdel Model
for each gear #i,
gi ∈ {0, 1}
define a binary input
• Gear selection (traction force):
Fe =
Rg(i)
M
ks
depends on gear #i
define auxiliary continuous variables:
IF gi = 1 THEN F ei =
Rg(i)
M
ks
ELSE 0
F e = F eR + F e1 + F e2 + F e3 + F e4 + F e5
• Gear selection (engine/vehicle speed):
ω=
Rg(i)
xç
ks
similarly, also requires 6 auxiliary continuous variables
129
Hybrid Model
• Max-speed controller
x(t + 1) = Ax(t) + B1 u(t) + B2 î(t) + B3 z(t)
y(t) = Cx(t) + D1 u(t) + D2 î(t) + D3 z(t)
E 2 î(t) + E 3 z(t) ô E 4 x(t) + E 1 u(t) + E 5
x, v
• 2 continuous inputs: M, F b
• 6 binary inputs: gR, g1 , g2 , g3 , g4 , g5
• 1 continuous output: v
• 16 auxiliary continuous vars:
• 4 auxiliary binary vars:
130
Hybrid Controller
• MLD model
• 2 continuous states:
go to demo /demos/cruise/init.m
max J(u t, x(t)),v(t + 1|t)
ut
ú
MLD model
subj. to
x(t|t) = x(t)
Objective: maximize speed
(to reproduce max acceleration plots)
250
200
(vehicle position and speed)
(engine torque, brake force)
MILP optimization problem
(gears)
Linear constraints
Continuous variables
Binary variables
Parameters
Time to solve mpMILP (Sun Ultra 10)
Number of regions
(vehicle speed)
(6 traction force, 6 engine speed,
4 PWL max engine torque)
(PWL max engine torque breakpoints)
v(t)
96
18
10
1
45 s
11
150
100
50
0
x(t)
(x(t)
is irrelevant)
• 96 mixed-integer inequalities
131
(Parameters: Renault Clio 1.9 DTI RXE)
132
Hybrid Controller
Hybrid Controller
• Max-speed controller
Velocity (km/h)
• Tracking controller
Gear
200
Fraction of Max Torque (Nm)
Brakes (Nm)
5
min J(u t, x(t)),|v(t + 1|t) à vd (t)| + ú|ω|
ut
ú
MLD model
subj. to
x(t|t) = x(t)
1
1
150
4
100
3
50
2
0
1
0.5
0.5
0
50
100
0
0
-0.5
-0.5
-1
0
Road Slope (deg)
50
100
Engine speed (rpm)
1
200
-1
0
50
100
0
Engine Torque (Nm)
6000
50
MILP optimization problem
100
Linear constraints
Continuous variables
Binary variables
Parameters
Time to solve mp-MILP
(PC 850Mhz)
Number of regions
Power (kW)
200
60
5000
50
0.5
150
4000
0
40
3000
100
30
2000
20
-0.5
250
98
19
10
2
vd(t)
150
100
50
00
43 s
49
40
80
120
160
200
v(t)
50
1000
-1
10
0
0
50
Time (s)
100
0
0
50
Time (s)
100
0
0
50
Time (s)
100
0
50
Time (s)
100
go to demo /demos/cruise/init_exp.m
133
Hybrid Controller
• Tracking controller
Velocity (km/h), Desired velocity (km/h)
120
Hybrid Controller
min J(u t, x(t)),|v(t + 1|t) à vd(t)| + ú|ω|
• Smoother tracking controller
ut
Gear
Brakes (Nm)
Fraction of Max Torque (Nm)
5
10000
min J(u t, x(t)),|v(t + 1|t) à vd(t)| + ú|ω|
ut
⎧
⎨ |v(t + 1|t) à v(t)| < Tsamax
subj. to
MLD model
⎩
x(t|t) = x(t)
1
100
8000
4
0.5
80
6000
60
3
134
0
4000
40
2
250
-0.5
2000
20
0
1
0
100
200
-1
0
100
200
0
200
0
100
200
0
100
MILP optimization problem
200
ú = 0.001
Engine Torque (Nm)
Engine speed (rpm)
Road Slope (deg)
6
6000
4
5000
2
4000
0
3000
-2
2000
-4
1000
100
100
50
0
0
-100
-6
0
200
-50
-200
0
100
19
10
2
47 s
54
150
vd(t)
100
50
00
40
80
120
160
200
v(t)
0
100
100
-100
0
200
Time (s)
Time (s)
Linear constraints
Continuous variables
Binary variables
Parameters
Time to solve mp-MILP
(PC 850Mhz)
Number of regions
Power (kW)
200
100
200
0
100
200
Time (s)
Time (s)
135
136
Hybrid Controller
• Smoother tracking controller
Velocity (km/h), Desired velocity (km/h)
120
Gear
Brakes (Nm)
Fraction of Max Torque (Nm)
5
10000
1
100
8000
4
0.5
80
6000
60
3
0
4000
40
2
-0.5
0
1
0
100
200
-1
0
100
0
200
0
6000
4
5000
2
4000
0
3000
-2
2000
100
200
-6
0
100
200
Power (kW)
200
100
100
50
0
(joint work with F. Borrelli,
Borrelli, M. Fodor, D. Hrovat)
Hrovat)
0
-100
-4
0
Engine Torque (Nm)
Engine speed (rpm)
Road Slope (deg)
6
Traction Control System
2000
20
-50
1000
-200
0
100
Time (s)
200
0
100
Time (s)
200
-100
0
100
Time (s)
200
0
100
200
Time (s)
137
138
Tire Force Characteristics
Vehicle Traction Control
Improve driver’s ability to control a vehicle
under adverse external conditions (wet or icy roads)
Longitudinal Force
al
ce
Tire Forces
t er
La
r
Fo
Slip Target
Zone
Maximum
Braking
Model
nonlinear, uncertain,
constraints
Maximum
Cornering
Controller
suitable for real-time
implementation
Maximum
Acceleration
Steer Angle
Tire Slip
Longitudinal
Force
Lateral
Force
MLD hybrid framework + optimization-based control strategy
139
140
Hybrid Model
Simple Traction Model
• Mechanical system
Nonlinear tire torque τt =f(Δω , μ)
Torque
ò
ó
üt
1
üc à beωe à
gr
Je
üt
vç v =
mvrt
ωç e =
μ
Slip
PWA Approximation
• Manifold/fueling dynamics
(PWL Toolbox, Julian, 1999)
Torque
üc = biüd(t à üf)
• Tire torque τt is a function of
slip Δω and road surface
μ
adhesion coefficient μ
Mixed-Logical
HYSDEL
Slip
(Hybrid Systems
Description Language)
ωe vv
Δω = à
gr rt
Dynamical (MLD)
Wheel slip
Hybrid Model
(discrete time)
Δω
141
142
Performance and Constraints
MLD Model
• Control objective:
min
PN
k=0
subj. to.
State x(t)
4 variables
Input u(t)
1 variable
Aux. Binary vars δ(t)
1 variable
Aux. Continuous vars z(t)
3 variables
Mixed-integer inequalities
14
|Δω(k|t) à Δωdes|
MLD Dynamics
• Constraints:
• Limits on the engine torque: à 20Nm ô ü d ô 176Nm
• Logic Constraint:
• Hysteresis
The MLD matrices are automatically
generated in Matlab format by HYSDEL
go to demo /demos/traction/init.m
143
144
Experimental Results
Experiment
Wheel Speeds, [rad/s]
20
15
10
5
0
Engine Torque Command, [Nm]
200
150
100
50
0
0
3
6
9
12
30
Controller Region
25
• ≈500 regions
• 20ms sampling time
• Pentium 266Mhz +
Labview
20
15
10
5
0
0
3
6
9
12
Time, [s]
145
146
DISC Engine Control Problem
Goal: Develop a controller for a Direct-Injection
Stratified Charge (DISC) engine.
Main features of DISC engines:
•
•
•
•
Hybrid Control of a DISC Engine
(joint work with N. Giorgetti, I. Kolmanovsky and D. Hrovat)
Direct injection of fuel (as Diesel engines);
Two operating modes (homogeneous/stratified);
Reduction of consumptions up to 15%;
Highly complex post-treatment system of exhaust
gas;
•
Different control objectives and constraints that
depend on the operating mode
(Photo: Courtesy Mitsubishi)
Strategy:
Develop a hybrid model that captures the two operating modes;
Design a constrained MPC controller based on the hybrid model
147
148
DISC Engine
DISC Engine
Two distinct regimes:
Regime
fuel
injection
Homogeneous intake
combustion
stroke
Stratified
combustion
pm
States/Controlled outputs:
Intake manifold pressure (pm);
Air-to-fuel ratio (λ);
Engine brake torque (τ);
air-tofuel ratio
λ=14.64
compression λ
stroke
λ
Wth
Inputs (continuous):
14.64
τ
ρ
Wf
Air mass flow rate through throttle (Wth);
Mass flow rate of fuel (Wf);
Spark timing (δ);
• Mode is switched by changing fuel injection timing (late / early)
• Better fuel economy during stratified mode
Inputs (binary):
Periodical cleaning of the aftertreatment system needed
(λ=14.00, homogeneous regime)
Constraints on:
δ
ρ = regime of combustion (homogeneous/stratified);
Air-to-fuel ratio (due to engine roughness, misfiring, smoke emiss.)
normal
purge
the stratified operation
can only be sustained in a
restricted part of the engine
operating range
Spark timing (to avoid excessive engine roughness)
Mass flow rate on intake manifold (constraints on throttle)
149
•
•
Dynamic equations are nonlinear
Dynamics and constraints depend on regime ρ
150
DISC Dynamics
Hybridization of DISC Model
Nonlinear model of the engine developed
and validated at Ford Research (Kolmanovsky, Sun, …)
DYNAMICS (intake pressure, air-to-fuel ratio, torque):
Assumptions:
- no EGR (exhaust gas recirculation) rate,
- engine speed=2000 rpm.
• Definition of two operating points;
• Linearization of nonlinear dynamics;
• Time discretization of the linear models.
ρ-dependent dynamic
equations
• Intake manifold pressure:
CONSTRAINTS on:
• Air-to-Fuel Ratio: λmin(ρ)·λ(t)·λmax(ρ);
• Mass of air through the throttle: 0 · Wth · K;
• Spark timing: 0·δ(t)· δmbt(λ, ρ)
• In-cylinder Air-to-Fuel ratio:
• Engine torque:
with
ρ−dependent
constraints
functions of pm
where θa, θb, δmbt are functions of
λ, δ and ρ.
9 Good for simulation
Hybrid system with 2 modes (switching affine system)
8 Not suitable for controller synthesis
151
MPC for Hybrid Systems
Integral Action
past future
Predicted
outputs y(t+k|t)
Manipulated
Inputs u(t+k)
t t+1
152
Model
Predictive (MPC)
Control
Integrators on torque error and air-to-fuel ratio error
are added to obtain zero offsets in steady-state:
t+T
= sampling time
• At time t solve with respect to
the finite-horizon open-loop, optimal control problem:
brake torque and air-to-fuel references
Simulation based on nonlinear model
confirms zero offsets in steady-state
• Apply only
(discard the remaining optimal inputs);
(despite the model mismatch)
• Repeat the whole optimization at time t+1
153
MPC of DISC Engine
154
DISC Engine - HYSDEL List
N = control horizon
x(t) = current state
where:
and:
155
SYSTEM hysdisc{
INTERFACE{
STATE{
REAL pm
[1, 101.325];
REAL xtau
[-1e3, 1e3];
REAL xlam
[-1e3, 1e3];
REAL taud
[0,
100];
REAL lamd
[10,
60];
}
OUTPUT {
REAL lambda; /* [10,
50]; */
REAL tau;
/* [0,
100]; */
}
INPUT{
REAL Wth
[0,38.5218];
REAL Wf
[0,
2];
REAL delta
[0,
40];
BOOL rho;
}
PARAMETER{
REAL Ts, pm1, pm2;
REAL l01, l02, l0c;
REAL l11,l12,l1c;
REAL t01,t02,t03,t04,t05;
REAL t11,t12,t13,t14,t15;
}
}
IMPLEMENTATION{
AUX{
REAL lam, taul;
REAL lmin,lmax;
REAL dmbt; }
DA{
lam={ IF rho THEN l11*pm+l12*Wf+l1c
ELSE
l01*pm+l02*Wf+l0c};
taul={IF rho THEN
t11*pm+t12*Wf+t13*delta+t14*lam+t15
ELSE
t01*pm+t02*Wf+t03*delta+t04*lam+t05 };
lmin={IF rho THEN 13 ELSE 19};
lmax ={IF rho THEN 21 ELSE 38};
dmbt ={IF rho THEN -28.74+3.1845*lam
ELSE 14.0877+0.2810*lam};
}
CONTINUOUS{
pm=pm1*pm+pm2*Wth;
xtau=xtau+Ts*(taud-taul);
xlam=xlam+Ts*(lamd-lam);
taud=taud;
lamd=lamd;
}
OUTPUT {
lambda=lam;
tau=taul;
}
MUST{
lmin-lam
<=0;
lam-lmax
<=0;
delta-dmbt
<=0;
}
}}
156
Simulation Results
MPC – Torque Control Mode
Solve
MIQP problem
(mixed-integer
quadratic
program)
to compute u(t)
Weights:
pm
λ
(prevents
unneeded
chattering)
qλ
sε
τ
sε
(Purge
Lean
NOx Trap)
Time (s)
Time (s)
Simulation in torque control mode:
• Control horizon N=1;
homogeneous
• Simulation time T=2.5 s;
τ
ρ
qτ
14
Combustion mode
Wth
rρ
Air-to-Fuel Ratio
[Nm]
Engine Brake Torque
Wf
• Sampling time Ts=10 ms;
• CPU time for simulation ' 8 s;
(on an Intel Centrino 1.2 GHz, 640 Mb
RAM with Cplex 9.01)
δ
λ
main emphasis
on torque
' 32 ms per time step
stratified
MPC
Time (s)
157
Multiparametric (Explicit) Solution
Not directly implementable !
158
Explicit MPC Controller
Mixed
Integer
Quadratic
Program
Explicit control law:
N=1 (control horizon)
75 partitions
where:
• On-line optimization: given x(t), solve an MIQP at each t.
• Off-line optimization: solve the MIQP for all x(t) off line
Cross-section by the τref-λref plane
- find feasible switching sequences using reachability analysis
- solve sequence of multi-parametric quadratic programs
(using the Hybrid Toolbox)
4
2
• Sampling time Ts=10 ms;
• CPU time for simulation:
0.52 s
M
1
5
Time to compute the
explicit MPC: 3.4750 s;
• Simulation time T=2.5 s;
ρ=0
(x,r)-space
The MPC controller is piecewise affine in x,r
•
ρ=1
' 10 μs per time step
63
159
160
Microcontroller Implementation
Explicit MPC Controller (N=2)
• C-code automatically generated
by the Hybrid Toolbox
Explicit control law:
5637 partitions
where:
• Microcontroller Motorola MPC 555
(custom made for Ford)
Engine Brake Torque
• 43 Kb memory available
N=2 (control horizon)
Air-to-Fuel Ratio
• Floating point arithmetic
≈ 3ms execution time
sampling period = 10ms
Implementable !
• Further reduction of number of partitions possible
(Alessio, Bemporad, 2005)
• C-code can be further optimized
Closed-loop N=2
161
Closed-loop N=1
Adequate!
162
Conclusions
Key References
• Hybrid systems are a modeling framework for automotive
control problems where continuous switched dynamics
and logic are relevant (and linear models are not enough!)
[1] A. Bemporad, “Modeling, control, and reachability analysis of discrete-time
hybrid systems,” Lecture Notes - DISC School on Hybrid Systems, Mar.
2003.
• MPC control design handle all performance specs and constraints
in a natural and direct way. Quite complex systems can be
controlled using on-line optimization
• Piecewise affine MPC controllers can be synthesized, off-line,
and implemented as look-up tables of linear gains
[5] W.P.H.M Heemels, B. de Schutter, and A. Bemporad, “Equivalence of
hybrid dynamical models,” Automatica, vol. 37, no. 7, pp. 1085-1091, July
2001
M
1
5
[3] A. Bemporad and M. Morari, “Control of systems integrating logic,
dynamics, and constraints,” Automatica, vol. 35, no. 3, pp. 407-427, Mar.
1999.
[4] A. Bemporad, “Efficient conversion of mixed logical dynamical systems into
an equivalent piecewise affine form,” IEEE Trans. Automatic Control, vol.
49, no. 5, pp. 832-838, 2004.
(x,r)-space
4 2
[2] F.D. Torrisi and A. Bemporad, “HYSDEL - A tool for generating
computational hybrid models,” IEEE Trans. on Control Systems
Technology, vol. 12, no. 2, pp. 235-249, Mar. 2004
63
• Matlab tools available to assist the whole design process (models,
simulation, MPC design, code generation):
MPC Toolbox (linear), Hybrid Toolbox (hybrid, explicit),
Multi-Parametric Toolbox (PWA, explicit)
163
Key References
[6] A. Bemporad, M. Morari, V. Dua, and E.N. Pistikopoulos, “The explicit
linear quadratic regulator for constrained systems,” Automatica, vol. 38, no.
1, pp. 3-20, 2002.
more on http://www.dii.unisi.it/~bemporad/publications
The End
[7] A. Bemporad, F. Borrelli, and M. Morari, “Piecewise linear optimal
controllers for hybrid systems,” In Proc. American Control Conference,
2000, pp. 1190-1194.
[8] F. Borrelli, M. Baotic, A. Bemporad, and M. Morari, “Dynamic
programming for constrained optimal control of discrete-time linear hybrid
systems,” Automatica, vol. 41, no. 10, Oct. 2005.
[9] A. Bemporad and N. Giorgetti, “Logic-based methods for optimal control of
hybrid systems,”IEEE Trans. Autom. Control, 2005, to appear
[10] A. Bemporad, D. Mignone, and M. Morari, “Moving horizon estimation for
hybrid systems and fault detection,” in Proc. American Control Conf., 1999,
Chicago, IL, pp. 2471-2475.
[11] A. Bemporad, A. Garulli, S. Paoletti, and A. Vicino, “A bounded-error
approach to piecewise affine system identification,” IEEE Trans. Automatic
Control, 2005, to appear.
[12] A. Bemporad, “Hybrid Toolbox – User’s Guide,’’ Dec. 2003,
http://www.dii.unisi.it/hybrid/toolbox.
[13] M. Lazar, M. Heemels, S. Weiland, A. Bemporad, “Stability of Hybrid
Model Predictive Control,” IEEE Trans. Automatic Control, 2005,
submitted.
more on http://www.dii.unisi.it/~bemporad/publications
MPC controller - SIMO
DC-Servomotor
Hybrid Toolbox
165
http://www.dii.unisi.it/hybrid/toolbox
164