Control of Hybrid Systemsmaler/TCC/Morari-tcc.pdfConstrained Finite Time Optimal Control of PWA...
Transcript of Control of Hybrid Systemsmaler/TCC/Morari-tcc.pdfConstrained Finite Time Optimal Control of PWA...
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Automatic Control Laboratory, ETH Zürichwww.control.ethz.ch
Control of Hybrid SystemsTheory, Computation & Applications
Manfred Morari
M. Baotic, F. Christophersen,T. Geyer, P. Grieder, C. Jones. M. Kvasnica,
U. Mäder, D. Niederberger, G. Papafotiou
Outline
1. Basics• Hybrid Modeling• Controller Computation
2. Low Complexity Controllers• The Three Levers of Complexity• Minimum-Time Controller• N-Step Controller
3. Industrial Applications• Control of a DC-DC Converter• Vibration Damping
4. Conclusions
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Outline
1. Basics• Hybrid Modeling• Controller Computation
2. Low Complexity Controllers• The Three Levers of Complexity• Minimum-Time Controller• N-Step Controller
3. Industrial Applications• Control of a DC-DC Converter• Vibration Damping
4. Conclusions
•Piecewise affine (PWA) systems.•Polyhedral partition of the state space.•Affine dynamics in each region.
PWA Hybrid Models
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Equivalence of Hybrid Models Equivalent classes of discrete-time hybrid models:
• Mixed Logical Dynamical (MLD)• Piecewise affine (PWA)• Linear Complementarity (LC)• Extended Linear Complementarity (ELC)• Max-Min-Plus-Scaling (MMPS)
The above discrete-time hybrid models can be proven to beequivalent (under some mild assumptions). Equivalenceimplies that for all feasible initial states and for all feasibleinput trajectories, the models yield the same state and outputtrajectories.
[Heemels, De Schutter, Bemporad, 2001]
Outline
1. Basics• Hybrid Modeling• Controller Computation
2. Low Complexity Controllers• The Three Levers of Complexity• Minimum-Time Controller• N-Step Controller
3. Industrial Applications• Control of a DC-DC Converter• Vibration Damping
4. Conclusions
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System• Discrete PWA Dynamics• Constraints on the state x(k) 2 X• Constraints on the input u(k) 2 U
Objectives• Stability (feedback is stabilizing)• Feasibility (feedback exists for all time)• Optimal Performance
Optimal Control forConstrained PWA Systems
Constrained Finite Time OptimalControl of PWA Systems
Algebraicmanipulation
Mixed Integer Linear Program (MILP)
Linear Performance Index (p=1,1)
Constraints
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Constrained Finite Time OptimalControl of PWA Systems
Algebraicmanipulation
Linear Performance Index (p=1,1)
Constraints
Receding Horizon Control
Mixed Integer Linear Program (MILP)
PLANT output y
plant state x
apply u0*
OptimizationProblem (MILP)obtain U*(x)
Receding Horizon ControlOn-Line Optimization
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OptimizationProblem (mpMILP)
Explicit Solution
u* = f(x)
PLANT output y
plant state xcontrol u*
(=Look-Up Table)
Receding Horizon Policy: Off-Line Opt.Multi-Parametric Programming
off-line
Characteristics of mp-MILP Solution
k Qxk{1, 1 } CR is polyhedral
• The optimizer z*(x) is piecewise affine• The feasible set X* is partitioned into critical regions
(CR).• The value function J*(x) is piecewise affine.
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Receding Horizon ControlStability Guarantee
Linear Performance Index (p=1,1)
Stability Guarantee
• Choose horizon T= ∞=> Constrained Infinite Time Optimal Control
or
• Choose ||PxT||p to be a (Control) Lyapunov Function
Receding Horizon ControlStability Guarantee
Linear Performance Index (p=1,1)
Stability GuaranteeChoose ||PxT||p to be a (Control) Lyapunov Function => Simple LP or algebraic test Christophersen, 2006
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Multi-parametric controllersAlgorithms have been developed for over 5 years:
…Minimization of linear and quadratic objectives (Baotic, Baric, Bemporad, Borrelli, De Dona, Dua, Goodwin, Grieder,
Johansen, Mayne, Morari, Pistikopoulos, Rakovic, Seron, Toendel)
…Minimum-Time controller computation(Baotic, Grieder, Kvasnica, Mayne, Morari, Schroeder)
…Infinite horizon controller computation(Baotic, Borrelli, Christophersen, Grieder, Morari, Torrisi)
…Computation of robust controllers(Borrelli, Bemporad, Kerrigan, Grieder, Maciejowski, Mayne, Morari, Parrilo, Sakizlis)
) Advanced computation schemes are available !
PROs:– Easy to implement– Fast on-line evaluation– Analysis of closed-loop system possible
CONs:– Number of controller regions can be large– Off-line computation time may be prohibitive– Computation scales badly.
) controller complexity is the crucial issue
Multi-parametric controllers
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Outline
1. Basics• Hybrid Modeling• Controller Computation
2. Low Complexity Controllers• The Three Levers of Complexity• Minimum-Time Controller• N-Step Controller
3. Industrial Applications• Control of a DC-DC Converter• Vibration Damping
4. Conclusions
ControllerConstruction
Partition Complexity
PLANT Output y
State xControl u*
3 Complexity Levers ofReceding Horizon Control
off-line
on-line
1
3
2
Region Identification
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ObjectiveCompute controller partition as quickly as possible.
Why is computation time an issue?– For LTI systems, controller computation time correlates to
controller complexity– For PWA systems, controller computation time may not
correlate to controller complexity
However…– Controller complexity and runtime grows exponentially with
problem size
) Controller computation is not a bottleneck
1st Lever forComplexity Reduction
Optimal merging:hyperplane arrangmt: 34 secmerging (boolean min.): 5 sec
2nd Lever forComplexity Reduction - Region Merging
189 polyhedra
252 polyhedra
39 polyhedra
Greedy merging:merging based on LPs: 17 sec
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ObjectiveFor a given partition, identify controller region as quickly aspossible.
AlgorithmsIdentification of region in time logarithmic in the number ofregions is possible (Bemporad, Grieder, Johansen, Jones, Rakovic,Toendel)
However…– Pre-processing for most schemes is very expensive– Usually not applicable to general partitions (e.g. overlapping
regions, holes, non-convex partitions, etc.)
3rd Lever forComplexity Reduction
IdeaConstruct an interval tree based onbounding boxes. The search thenreduces to answering stabbingqueries in O(log(N)) time.
Advantage– Very cheap pre-processing.– Applicable to any type of partitions (not nec. polyhedral).
On-line strategy– The tree selects possible candidate regions.– Locally search the list of candidates.
Region Identification usingBounding Boxes
(Christophersen et. al, submitted to CDC 2006)
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Control Objective vs.Partition Complexity
Objective– Compute controller partition with as few regions as possible– Guarantee stability and constraint satisfaction
Observation Complex objectives yield complex controllers
Approach Use simpler objectives to obtain simpler controllers
) Controller complexity is a bottleneck
Outline
1. Basics• Hybrid Modeling• Controller Computation
2. Low Complexity Controllers• The Three Levers of Complexity• Minimum-Time Controller• N-Step Controller
3. Industrial Applications• Control of a DC-DC Converter• Vibration Damping
4. Conclusions
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Numerical ExampleN-Step ControlOptimal Control
• All results and plots were obtained with the MPTtoolbox
http://control.ethz.ch/~mpt
• MPT is a MATLAB toolbox that provides efficientcode for– (Non)-Convex Polytope Manipulation– Multi-Parametric Programming– Control of PWA and LTI systems
Conclusions
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Outline
1. Basics• Hybrid Modeling• Controller Computation
2. Low Complexity Controllers• The Three Levers of Complexity• Minimum-Time Controller• N-Step Controller
3. Industrial Applications• Control of a DC-DC Converter• Vibration Damping
4. Conclusions
Furuta Pendulum• Rotating inverted
pendulum• Explicit controller
with 5 regions• Results equivalent to [Åkesson and Åström, 2001]
[Akesson and Aström, 2001]
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Switch-mode DC-DC Converter
d
unregulated DC voltage low-pass filter load
dually operated switches
Switched circuit: supplies power to load with constant DC voltage
Illustrating example: synchronous step-down DC-DC converter
regulated DC voltage
S1
S2
Operation Principle
k-th switching period
duty cycle
• Length of switching period Ts constant (fixed switching frequency)• Switch-on transition at kTs, k2N • Switch-off transition at (k+d(k))Ts (variable pulse width)• Duty cycle d(k) is real variable bounded by 0 and 1
S1 = 1S2 = 0
S1 = 0S2 = 1
kTs (k+1)Ts(k+d(k))Ts
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S1 = 1S2 = 0
S1 = 0S2 = 1
kTs (k+1)Ts(k+d(k))Ts
duty cycle d(k)
Mode 1 and 2
mode 1: mode 2:
d
S1
S2
Control Objective
unregulated DC input voltagedisturbance
duty cyclemanipulated variable
regulated DC output voltagecontrolled variable
inductor currentstate
capacitor voltagestate
Regulate DC output voltage by appropriate choice of duty cycle
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State-feedback Controller:Polyhedral Partition
Colors correspond to the 121 polyhedra
PWA state-feedbackcontrol law:
computed in 100s using theMPT toolbox
121 polyhedra after simplificationwith optimal merging
Outline
1. Basics• Hybrid Modeling• Controller Computation
2. Low Complexity Controllers• The Three Levers of Complexity• Minimum-Time Controller• N-Step Controller
3. Industrial Applications• Control of a DC-DC Converter• Vibration Damping
4. Conclusions
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Smart Damping Materials
– Device suppresses vibration– External power source for operation is not required– Weight and size of the device have to be kept to a
minimum
• Demands
• Idea– Switched Piezoelectric (PZT)
Patches• Problem– What is the optimal switching law for optimal vibration
suppression?
PZTPZT
Niederberger, Moheimani
Application:Brake Squeal Reduction
• Friction induced vibration in brakes– Strong vibration radiates unwanted noise– One frequency, small bandwidth– Frequency can vary
Neubauer, Popp
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Brake Squeal Reductionusing Shunt Control
• Vibration reduction– Piezoelectric actuator between
brake pad and calliper– Switching shunt control
• Advantages– Tracks resonance frequency– Cheap solution– No electrical power required
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• Control / Scheduling of Cogeneration Power Plants
• Control / Scheduling of Cement Kilns and Mills
• Electronic Throttle Control / Traction Control
• Control of Thermal Print Heads
• Control of Voltage Stability for Electric PowerNetworks
• Direct Torque Control of 3 phase Induction Motors
• Control of Anaesthesia (Inselspital Bern)
• Adaptive Cruise Control
Other Applications ofHybrid Systems Control Tools
• All results and plots were obtained with the MPTtoolbox
http://control.ethz.ch/~mpt
• MPT is a MATLAB toolbox that provides efficientcode for– (Non)-Convex Polytope Manipulation– Multi-Parametric Programming– Control of PWA and LTI systems
Conclusions
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Facts about MPT
4000+ downloads
Rated 4.5 / 5 on mathworks.com
Conclusions
• Foundations of a theoretical framework forfor practical controller design for PWAsystem have been established
• Complexity reduction and robustness aremain research issues
• Applications in industry are beginning