optimization-based control of hybrid dynamical...
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Optimization-based Control of Hybrid Dynamical Systems
Lisbon, September 11, 2006
Alberto Bemporad
COHES GroupControl and Optimization of Hybrid and Embedded Systems
http://www.dii.unisi.it/~bemporad
University of Siena(founded in 1240)
Department of
Information Engineering
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Contents
• Models of hybrid systems
• Model predictive control of hybrid systems
• Explicit reformulation
• Automotive applications
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Hybrid systems
Hybrid Systems
Computer
Science
ControlTheory
Finite
state
machines
Continuous dynamical systems
systemu(t) y(t)
AB
C
C
A
B
B
C
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Hybrid Systems
Computer
Science
ControlTheory
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continuous dynamical system
discrete inputs
Embedded Systems
symbolssymbols
continuous states
continuous inputs
automaton / logic
interface
• Consumer electronics
• Home appliances
• Oce automation
• Automobiles
• Industrial plants
• ...
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Motivation: “Intrinsically Hybrid”
• Transmission
Discrete command
(R,N,1,2,3,4,5)
• Four-stroke engines
Automaton,
dependent on
power train motion
Continuous
dynamical variables
(velocities, torques)+
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Key Requirements for Hybrid Models
• Descriptive enough to capture the behavior of the system
– continuous dynamics (physical laws)
– logic components (switches, automata, software code)
– interconnection between logic and dynamics
• Simple enough for solving analysis and synthesis problems
“Make everything as simple as possible, but not simpler.”! — Albert Einstein
linear hybrid systems
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Piecewise Ane Systems
Can approximate nonlinear and/or discontinuous dynamics arbitrarily well
(Sontag 1981)
state+input space
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Discrete Hybrid Automaton(Torrisi, Bemporad, 2004)
Event Generator
Finite State Machine
Mode Selector
Switched Ane System
1
2
s
mode
time orevent counter
continuous
discrete
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Switched Ane System
The ane dynamics depend on the current mode i(k):
continuous
discrete
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Event Generator
Event variables are generated by linear threshold conditions over continuous states, continuous inputs, and time:
Example: ["(k)=1] ! [xc(k)"0]
continuous
discrete
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Finite State Machine
The binary state of the nite state machine evolves according to a Boolean state update function:
Example:
continuous
discrete
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The active mode i(k) is selected by a Boolean function of the current binary states, binary inputs, and event variables:
Mode Selector
Example: 0
1
0 1
the system has 3 modes
continuous
discrete
The mode selector can be seen as the output function of the discrete dynamics
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(Glover 1975, Williams 1977,Hooker 2000)
Logic and Inequalities
Finite State Machine
Mode Selector
Switched Ane System
1
2
s
Event Generator
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Mixed Logical Dynamical Systems
(Bemporad, Morari 1999)Mixed Logical Dynamical (MLD) Systems
HYSDEL(Torrisi, Bemporad, 2004)
Discrete Hybrid Automaton
• Computationally oriented (mixed-integer programming)
• Suitable for controller synthesis, verication, ...
Continuous and binary variables
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Hybrid Toolbox for MatlabFeatures:
• Hybrid model (MLD and PWA) design, simulation, verication
• 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
(Bemporad, 2003-2006)
Support:
http://www.dii.unisi.it/hybrid/toolbox
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Mixed-Integer Models in OR
Translation of logical relations into linear inequalities is heavilyused in operations research (OR) for solving complex decision problems by using mixed-integer (linear) programming (MIP)
Example: Timetable generation (for demanding professors …)
CPU time: 0.2 s
Example: Optimal multi-period investments formaintenance and upgrade of electrical energy distribution networks
(Bemporad, Muñoz, Piazzesi, 2006)
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Major Advantages of Linear Hybrid Models
Many problems of analysis:
– Stability
– Safety / Reachability
– Observability
– Passivity
Many problems of synthesis:
– Controller design
– Robust control design
– Filter design (state estimation/fault detection)
(However, all these problems are NP-hard !)
(Johansson, Rantzer, 1998)
(Torrisi, Bemporad, 2001)
(Bemporad, Ferrari-Trecate, Morari, 2000)
(Bemporad, Bianchini, Brogi, 2006)
(Pina, Botto, 2006)
(Bemporad, Mignone, Morari, 1999)
(Bemporad, Morari, 1999)
(Ferrari-Trecate, Mignone, Morari, 2002)
can be solved through mathematical programming
(Silva, Bemporad, Botto, Sá da Costa, 2003)
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Contents
! Models of hybrid systems
• Model predictive control of hybrid systems
• Explicit reformulation
• Automotive applications
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Hybrid Control Problem
binary inputs
continuous inputs
binary states
continuous states
on-line decision maker
desired behavior
constraints
hybrid process
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• Apply only u(t)=u*(t) (discard the remaining optimal inputs)
• At time t solve with respect to the nite-horizon open-loop, optimal control problem:
MPC for Hybrid Systems
Predictedoutputs
Manipulated
y(t+k|t)
Inputs
t t+1 t+T
futurepast
u(t+k)
• Repeat the whole optimization at time t+1
ModelPredictive (MPC)Control
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Closed-Loop Convergence
Proof: Easily follows from standard Lyapunov arguments
(Bemporad, Morari 1999)
More stability results: see (Lazar, Heemels, Weiland, Bemporad, 2006)
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Open loop behavior
PWA system:
Hybrid MPC - Example
Constraint:
/demos/hybrid/bm99sim.m
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Hybrid MPC - Example
Closed loop:
Performance index:
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Hybrid MPC - Extensions
1
2
3
e, p12e
e, p13e¬e(Bemporad, Di Cairano, HSCC’05)
Stochastic Finite State Machine (sFSM)
Discrete-time Hybrid Stochastic Automaton (DHSA)
Event-based Continuous-time Hybrid Automaton (icHA)
Switched integral dynamics
(Bemporad, Di Cairano, Julvez, CDC05 &HSCC-06)
k = event counterAsynchronous FSM
k = discrete-time counter
All MPC techniques described earlier can be applied !
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Optimal Control of Hybrid Systems: Computational Aspects
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MPC for Linear Systems
This is a (convex) Quadratic Program (QP)
Quadratic performance index
Constraints
Linear model
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Mixed Integer Quadratic Program (MIQP)
MIQP Formulation of MPC(Bemporad, Morari, 1999)
• Optimization vector:
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MILP Formulation of MPC(Bemporad, Borrelli, Morari, 2000)
Mixed Integer Linear Program (MILP)
• Optimization vector:
• Basic trick: introduce slack variables:
• Generalization:
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BUT
• Mixed-Integer Programming is NP-hard
Mixed-Integer Program Solvers
• Extremely rich literature in Operations Research (still very active)
• No need to reach the global optimum for stability of MPC
(see proof of the theorem), although performance deteriorates
MILP/MIQP is nowadays a technology (CPLEX, Xpress-MP, BARON,
GLPK, see e.g. http://plato.la.asu.edu/bench.html for a comparison)
• Possibility of combining symbolic + numerical solvers Example: SAT + linear programming
(Bemporad, Giorgetti, IEEE TAC 2006)
#nodes=11 #nodes=6227
SAT-based B&B Pure B&B
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Contents
! Models of hybrid systems
! Model predictive control of hybrid systems
• Explicit reformulation
• Automotive applications
• Hybrid MPC over wireless sensor networks
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On-Line vs O-Line Optimization
• On-line optimization: given x(t) solve the problem at each time step t.
multi-parametric programming
Mixed-Integer Linear/Quadratic Program (MILP/MIQP)
• O-line optimization: solve the MILP/MIQP for all x(t)
" Good for large sampling times (e.g., 1 h) / expensive hardware …
… but not for fast sampling (e.g. 10 ms) / cheap hardware !
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-60 -40 -20 0 20 40 60-60
-40
-20
0
20
40
60
CRf2;3g
CRf1;4g
CRf1;2;3g
CRf1;3g
x1
x2
Example of Multiparametric SolutionMultiparametric LP ( )
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MPC for Linear Systems
Quadratic performance index
Constraints
Linear model
Objective: solve the QP for all (o-line)
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;;
Properties of multiparametric-QP
continuous,piecewise ane
convexcontinuous, piecewise quadratic, C1 (if no degeneracy)
Corollary: The linear MPC controller is a continuous piecewise ane function of the state
Optimizer
Valuefunction
Feasiblestate set convex polyhedral
(Bemporad et al., 2002)
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Polyhedral Invariant Sets for Closed-loop Linear MPC Systems
• Convexity of value function implies convexity of the piecewise
ellipsoidal sets and
• Any polyhedron P contained in between is a positively
invariant set for the closed-loop MPC system
(Note: explicit form of MPC not required)
• By changing # invariant polyhedra of arbitrary size can be
constructed for the closed-loop MPC system
(Alessio, Bemporad, Lazar, Heemels, CDC’06)
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Explicit Hybrid MPC (PWA)
• The MPC controller is piecewise ane in x,r
Note: in the 2-norm case the partition may not be fully polyhedral
(x,r)-space
1
2
3
M4
5 6
(Borrelli, Baotic, Bemporad, Morari, Automatica, 2005)
(Mayne, ECC 2001) (Alessio, Bemporad, ADHS 2006)
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Open loopbehavior:
PWA system:
Hybrid Control - Example
Constraints:
Closed loop:
HybTbx: /demos/hybrid/bm99sim.m
Objective:
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Explicit PWA Controller
PWA law MPC law
(CPU time: 1.51 s, Pentium M 1.4GHz)
HybTbx: /demos/hybrid/bm99sim.m
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10Section with r=0
12345
x1
x2
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Hybrid MPC - ExampleClosed loop:
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Explicit PWA Regulator
Prediction Horizon N=1
HybTbx: /demos/hybrid/bm99benchmark.m
Objective:
Prediction Horizon N=2 Prediction Horizon N=3
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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
- control code is simpler (no complex solver must be included in the control software !)
- more insight in controller’s behavior
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Contents
! Models of hybrid systems
! Model predictive control of hybrid systems
! Explicit reformulation
• Automotive applications
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44
Hybrid Control of a DISC Engine
(joint work with N. Giorgetti, G. Ripaccioli, I. Kolmanovsky, and D. Hrovat)
(Photo: Courtesy Mitsubishi)
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45
DISC EngineTwo distinct regimes:
Regime fuel injection
air-to-fuel ratio
Homogeneouscombustion
intake stroke
$=14.64
Stratiedcombustion
compression stroke
$ >
14.64
Objective: Design a controller for the engine that
• Automatically choose operating mode (homogeneous/stratied)
• Can cope with nonlinear dynamics
• Handles constraints (on A/F ratio, air-ow, spark)
• Achieves optimal performance
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46
Hybridization of DISC Dynamics
DYNAMICS (intake pressure, air-to-fuel ratio, torque):
CONSTRAINTS on:
• Denition of two operating points;
• Numerical linearization of nonlinear dynamics;
• Time discretization of the linear models.
• Air-to-Fuel Ratio: $min(%)#$(t)#$max(%);
• Mass of air through the throttle: 0 # Wth # K;
• Spark timing: 0#"(t)# "mbt($, %)
%-dependent dynamic
equations
%&dependent constraints
Hybrid system with 2 modes (switched ane system)
• Proprietary nonlinear model of the DISC engine developed and validated at Ford Research Labs (Dearborn)(Kolmanovsky, Sun, …)
• Model good for simulation, not good for control design!
MODEL HYBRIDIZATION
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47
Integral Action
Integrators on torque error and air-to-fuel ratio error added to obtain zero osets in steady-state:
Simulation based on nonlinear model conrms zero osets in steady-state
(despite the model mismatch)
brake torque and air-to-fuel references
= sampling time
Reference values are automatically generated from 'ref and $ref
by numerical computation based on the nonlinear model
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48
DISC Engine - HYSDEL List
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, tau, ddelta;
}
INPUT{
REAL Wth [0,38.5218];
REAL Wf [0, 2];
REAL delta [0, 40];
BOOL rho;
}
PARAMETER{
REAL Ts, pm1, pm2;
…
}
}
IMPLEMENTATION{
AUX{
REAL lam,taul,dmbtl,lmin,lmax;
}
DA{
lam={IF rho THEN l11*pm+l12*Wth...
+l13*Wf+l14*delta+l1c
ELSE l01*pm+l02*Wth+l03*Wf...
+l04*delta+l0c };
taul={IF rho THEN tau11*pm+...
tau12*Wth+tau13*Wf+tau14*delta+tau1c
ELSE tau01*pm+tau02*Wth...
+tau03*Wf+tau04*delta+tau0c };
dmbtl ={IF rho THEN dmbt11*pm+dmbt12*Wth...
+dmbt13*Wf+dmbt14*delta+dmbt1c+7
ELSE dmbt01*pm+dmbt02*Wth...
+dmbt03*Wf+dmbt04*delta+dmbt0c-1};
lmin ={IF rho THEN 13 ELSE 19};
lmax ={IF rho THEN 21 ELSE 38};
}
CONTINUOUS{
pm=pm1*pm+pm2*Wth;
xtau=xtau+Ts*(taud-taul);
xlam=xlam+Ts*(lamd-lam);
taud=taud; lamd=lamd;
}
OUTPUT{
lambda=lam-lamd;
tau=taul-taud;
ddelta=dmbtl-delta;
}
MUST{
lmin-lam <=0;
lam-lmax <=0;
delta-dmbtl <=0;
}
}
}
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49
MPC of DISC Engine
pm
'
$
Wth
Wf "
%
MPC
Weights:
Solve MIQP problem (mixed-integerquadratic program)to compute u(t)
q' q$
r%
s(' s($
(prevents unneeded chattering)
main emphasis on torque
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50
Simulation Results (nominal engine speed)[N
m]
Time (s)
Air-to-Fuel Ratio
Time (s)
Engine Brake Torque
Time (s)
Combustion mode
• Control horizon N=1;
• Sampling time Ts=10 ms;
• PC Xeon 2.8 GHz + Cplex 9.1
$ 3 ms per time step
14
(PurgeLean NOx Trap)
homogeneous
stratied
) = 2000 rpm
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51
Simulation Results (varying engine speed)[N
m]
Time (s)
Air-to-Fuel Ratio
Time (s)
Engine Brake Torque
Time (s)
Engine speed
Hybrid MPC design is quite robust with respect to engine speed variations
20 s segment of the Europeandrive cycle (NEDC)
Control code too complex (MILP) % not implementable !
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52
Explicit MPC Controller
N=1 (control horizon)
75 partitions
Cross-section by the 'ref-$ref plane
• Time to compute the explicit MPC: 3.4750 s;
• Sampling time Ts=10 ms;
• PC Xeon 2.8 GHz + Cplex 9.1
$ 8 *s per time step
Explicit control law:
where:
%=0
%=1
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53
Microcontroller Implementation
Implementable !
• C-code automatically generated by the Hybrid Toolbox
• Microcontroller Motorola MPC 555 (custom made for Ford)
• 43 Kb memory available
• Floating point arithmetic
• Further reduction of number of partitions possible
• C-code can be further optimized
& 3ms execution time
sampling period = 10ms
(Alessio, Bemporad, 2005)
(Tøndel, Johansen, Bemporad, 2003)
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54
Conclusions• Hybrid systems as a framework for new applications, where both logic and continuous dynamics are relevant
y(t)u(t)
Plant
OutputInput
Measurements
• Piecewise Linear MPC Controllers can be synthesized o-line via multiparametric programming for fast-sampling applications
• Supervisory MPC controllers schemes can be synthesized via on-line mixed-integer programming (MILP/MIQP)
Hybrid modelingand MPC design
Multiparametricprogramming
C-code download& testing
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55
Hybrid MPC & Wireless Sensor Networks
• Measurements acquired and sent to base station (MPC) by wireless sensors
• MPC computes the optimal plan when new measurements arrive
• Optimal plan implemented by local controller if received in time, otherwise previous plan still kept
Packet loss possible along both network links, delayed packets must be discarded (out-of-date data)
network links
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Challenges in Wireless MPC
• Synchronization schemes must ensure correct prediction in spite of packet loss
• MPC algorithm must be robust w.r.t. packet loss % stochastic hybrid MPC, robust hybrid MPC
• Wireless sensors must be interfaced to optimization tools
Network
measurements
optimal plan
network link
Stabilized plant
plant
local control
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Demo Application in Wireless Automation
• Telos motes provide wireless temperature feedback in Matlab
• Hybrid MPC algorithm adjust belt speed and coordinate linear motors (via Simulink/xPC-Target link)
(Automatic Control Lab, Univ. Siena)
Telos motes
(Bemporad, Di Cairano, Henriksson)
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Acknowledgements
A. Alessio, M. Baotic, F. Borrelli, B. De Schutter, S. Di Cairano, G. Ferrari-Trecate, K. Fukuda, N. Giorgetti, M. Heemels, D. Hrovat, J. Julvez, I. Kolmanovski, M. Lazar, L. Ljung, D. Mignone, M. Morari, D. Munoz de la Pena, S. Paoletti, G. Ripaccioli, J. Roll, F.D. Torrisi
• 10th “Hybrid Systems: Computation and Control” Conference (Pisa, Italy, April 3-5, 2006). Submission: Oct. 9, 2006
• 2nd PhD School on Hybrid Systems, Siena, 2007
Announcements
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The End
MPC controller - SIMODC-ServomotorHybrid Toolbox
http://www.dii.unisi.it/hybrid/toolbox