Model Predictive Control: On-line optimization versus explicit precomputed controller

1
Model Predictive Control On-line optimization
versus explicitprecomputed controller

• Espen Storkaas
• Trondheim, 7.6. 2005

2
Outline

• Introduction
• Brief history
• Linear MPC
• Theory, feasibility, stability, performance
• Derivation of explicit MPC
• Nonlinear and hybrid MPC
• Applications
• Future directions
• Conclusions

3
Introduction

• Control problem
• Find stabilizing control strategy that
• Minimize objective functional
• Satisfies constraints
• is robust towards uncertainty

4
Solution strategies

• Closed loop optimal control
• Feedback uk(x)
• s.t. closed loop trajectories satisfying
  optimality
• Advantages
• Feedback
• Uncertainty
• Disturbances
• Unstable systems
• Drawbacks
• Find k(x)?

• Open loop optimal control
• Input trajectory uu(t,x0)
• solving optimization problem
• Advantages
• Computationally feasible
• Drawbacks
• No feedback
• Disturbances?
• Unstable systems
• Uncertainty

5
Possible solution 1 MPC with online optimization

• Solve optimization problem over finite horizon
• Implement optimal input for ?2t,td
• Re-optimize at next sample (feedback)
• Optimal control inputs implicitly via
  optimalization

6
MPC with online optimization
(Allgöwer, 2004)
7
Solution strategies

• Close loop optimal control
• Feedback uk(x)
• s.t. closed loop trajectories satisfying
  optimality
• Advantages
• Feedback
• Uncertainty
• Disturbances
• Unstable systems
• Drawbacks
• Find k(x)?

• Open loop optimal control
• Input trajectory uu(t,x0)
• solving optimization problem
• Advantages
• Computationally feasible
• Drawbacks
• No feedback
• Disturbances?
• Unstable systems
• Uncertainty

8
Possible solution 2 Explicit MPC(Bemporad et
al., 2002, Tøndel et al., 2003)

• Solve optimization problem offline for all x2X
• For linear systems multiparametric QP (mp-QP)
  with solution
• Piecewise affine controller
• Exactly identical to implicit solution (via
  online optimization)

9
Model Predictive Control (MPC)Brief history(Qin
Badgwell, 2003)

• LQR (Kalman, 1964)
• Unconstrained infinite horizon
• Constrained finite horizon MPC (Richalet et
  al., 1978, Cutler Ramaker,1979)
• Driven by demands in industry
• Defined MPC paradigm
• Posed as quadration program (QP) (Cutler et al.
  1983)
• Constraints appear explicitly
• Academic research (919 papers in 2002! (Allgöwer,
  2004))
• Stability
• Performance
• Explicit MPC (Bemporad et al. 2002, Tøndel et al.
  2003)

10
Linear MPC Problem formulation(Scokaert
Rawlings, 1998, Bemporad et al, 2002)

• Linear time-invariant discrete model
• Objective
• Constraints

11
Linear MPC Unconstrained case

• Problem

• Classical LQR solution (Kalman 1960)
• K calculated from algebraic Ricatti equation
• Assymptotically stabilizing

12
Linear MPC Infinite horizon (Constrained
LQR)

• Problem

• Infinite number of decision variables
• Stability proved by Rawlings Muske (1993)
• Computationally feasible (Scokeart Rawlings,
  1998)
• Computationally expensive

13
Linear MPC Finite input horizon

• Problem

• Achieved solution

• Stabilizing for K0 and KKLQ provided N large
  enough

14
Important aspects
x(t?)
x(t)

• Feasibility
• Slack on output constraints
• Feasible region for unstable systems under input
  constraints
• Closed loop stability
• Contraction constraint
• Terminal constraint (x(kN)0)
• Stable for control horizon N large enough
• Performance
• Implemented control trajectory may differ
  significantly from computed open-loop optimal
• May lead to infeasibility
• Solution Long enough control horizon
• On-line computational requirements

15
Derivation of explicit MPC (Bemporad et al.,
2002)

• Rewrite constrained LQR problem

• QP parameterized in initial state x(t)
• Solution for all x(t) by multi-parametric
  quadratic program (mp-QP)
• Solve mp-QP offline to find optimal solution
  UtU(x(t))
• Optimal input given by

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Derivation of explicit MPC (2)

• With
• From Karush-Kuhn-Tucker optimality conditions and
  assuming linearly independent active constraints
• KKT conditions gives partitioning of feasible
  regions into polyhedra
• Inherits properties of optimization problem

17
Partitioning of state spaceOffline computations

• Typical Algorithm
• Choose initial active set
• Find control law for active set
• Find critical region correspond to active set
• Systematic exploration of remaining parameter
  space
• (Build search tree/reduce complexity)

Bemproad et al. 2002
Tøndel et al. 2003
18
Explicit MPCOnline computations

• Determine critical region
• Sequential search
• Binary search tree
• Implement optimal control
• Complexity of partition increses with
  states/parameters

Binary search tree
Sequential search
19
Properties of explicit MPC

• Dimensional explosion
• max 5-7 states/parameter with current formulation
  
• Disturbance rejection, reference tracking and
  soft/variable constraints can be included, but
  increases complexity
• Greatly simplified code vs. online optimization
• Safety-critical systems

20
Nonlinear MPC

• Based on nonlinear process model and/or
  constraints to improve forcasting
• Requires solution of NLP, generally non-convex
• Stability and performance issues more important
• There are no analysis methods available that
  permit to analyze close loop stability based on
  knowledge of plant model, objective functional
  and horizon lengths (Allgöwer et al.,1999)
• Approaches
• Infinite horizon NMPC
• Zero state terminal equality constraint
• Dual mode NMPC
• Contractive NMPC
• Quasi-infinite horizon NMPC

21
Nonlinear explicit MPC

• Exact solution cannot be represented as PWA
  control law
• Approximative PWA solutions with user-specified
  tolerance can be found (Johansen, 2004)
• Solution of NLPs offline
• k-d tree partitioning of state space
• Joint convexity of obejctive functional and
  constraints assumed
• Complexity similar to linear explicit MCP
• Guaranteed stability under assumptions on
  tolerance
• Larger potential than linear EMPC?

22
Hybrid MPC

• Applications to broad class of systems including
• Linear hybrid dynamical systems
• Piecewise linear systems (including
  approximations of nonlinear systems
• Linear systems with constraints
• Modeled as mixed logical dynamical systems
  (Bemporad Morari, 1999)
• MPC problem is MILP/MIQP
• Difficult to solve online in available time
• Explicit Hybrid MPC is PWA (Bemporad et al. 2002,
  Dua et al. 2002)
• Identical to implicit solution found by online
  optimization

23
Application areas
Linear Nonlinear/Hybrid
Online optimization Reconfigurable Proven technology Slow processes Not safety critical Refinery Important nonlinearities/discret events Reconfigureable Slow processes Not safety critical Polymer reactor
Explicit precomputed Safety critical Low-cost hardware High sampling rate Low order Fixed configuration ESP for cars Safety critical Low-cost hardware High sampling rate Low order Fixed configuration Compressor Anti-surge
24
Future directions

• Linear MPC
• Improved models / adaptive formulations
• Multi-objective, prioritized constraints etc.
• Nonlinear/Hybrid MPC
• Computational efficiency
• Guaranteed stability/performance
• Explicit MPC
• Reduction of complexity vs degree of
  suboptimality
• Reconfigurability
• Exploit structure of problem

25
Concluding remarks

• Online optimization MPC for
• Slow systems
• Large systems
• Explicit precomputed MPC for
• Small systems with high sampling rate
• Safety critical
• Dedicated hardware (controller on a chip)

Acknowledgements
Thanks to Tor Arne Johansen, Petter Tøndel and
Olav Slupphaug for invaluable help with preparing
this presentation
26
Selected References

• Allgöwer, F. (2004), Model Predictive Control A
  Success Story Continues, APACT04, Bath,April
  26-28, 2004
• Allgöwer, F., Badgwell, T.A., Qin, S.J.,
  Rawlings, J.B. and Wright, S.J., (1999).
  Nonlinear predictive control and moving horizon
  estimationan introductory overview. In Frank,
  P.M., Editor, , 1999. Advances in control
  highlights of ECC 99, Springer,
• Berlin. Bemporad, A., Morari, M., Dua, V. and
  Pistikopoulos, E.N. (2002), The explicit linear
  quadratic regulator for constrained systems.
  Automatica 38 1, pp. 320, 2002.
• Bemporad A, Borrelli F, Morari M, (2002). On the
  optimal control law for linear discrete time
  hybrid systems, Lecture notes in computer science
  2289 105-119 2002
• Bemporad A, Morari M, (1999), Control of systems
  integrating logic, dynamics and constraints,
  Automatica 35 (3) 407-427 MAR 1999
• Cutler, C. R., Ramaker, B. L. (1979). Dynamic
  matrix controla computer control algorithm.
  AICHE national meeting, Houston, TX, April 1979.
• Cutler, C., Morshedi, A., Haydel, J. (1983). An
  industrial perspective on advanced control. In
  AICHE annual meeting, Washington, DC, October
  1983
• Dua V, Bozinis NA, Pistikopoulos EN. (2002), A
  multiparametric approach for mixed-integer
  quadratic engineering problems, Computers
  Chemical Engineering 26 (4-5) 715-733 MAY 15
  2002

27
Selected References

• Kalman, R. (1964), When is a linear control
  system optimal?, Journal of Basic Engineering
  Transactions on ASME Series D, 51-60,
• Johansen, T.A., Approximate Explicit Receding
  Horizon Control of Constrained Nonlinear Systems,
  Automatica, Vol. 40, pp. 293-300, 2004
• Qin, SJ., Badgwell, TA., A survey of industrial
  model predictive control technology, Control
  Engineering practice 11 (7) 733-764, 2003
• Rawlings, J.B. and Muske, K.R., 1993. Stability
  of constrained receding horizon control. IEEE
  Transactions on Automatic Control 38 10, pp.
  15121516
• Richalet, J., Rault, A., Testud, J.L. and Papon,
  J., Model predictive heuristic control
  Applications to industrial processes. Automatica
  14, pp. 413428, 1978
• Scokaert, P.O.M. and Rawlings, J.B., Constrained
  linear quadratic regulation. IEEE Transactions on
  Automatic Control 43 8, pp. 11631169, 1998
• Tøndel, P., Johansen, T.A. and Bemporad,
  A.(2003), An algorithm for multi-parametric
  quadratic programming and explicit MPC solutions.
  Automatica 39, 2003
• Tøndel, P., Johansen, T.A. and Bemporad, A
  (2003). Evalution of piecewise affine control via
  binary search tree. Automatica 39, 2003

28
Ting som ikke er nevnt

• Robusthet
• Practical implementations

29
Thank you for your attention!
30
Functional spec. in modern MPC

• Prevent violation of input and output constraints
• Drive CVs to steady state optimal values (or
  within bounds)
• Drive MVs to steady state optimal values (or
  within bounds)
• Prevent excessive use of MVs
• In case of signal or actuator failure, control as
  much of the plant as possible as possible

31
Modern industrial MPC algorithmOverview

• Read MV, CV, DV
• Output feedback
• Determination of controlled sub-process
• Removal of ill-condisioned plant
• Local steady state optimization
• Dynamical optimization
• MVs to process

32
Modern industrial MPC algorithmOutput feedback

• Process states and kalman filter seldom used
• Ad-hoc biasing scheemes with challenges regarding
• Extra measurements ?
• Linear combinations of states?
• Unmeasured disturbances models?
• Measurements noise?
• Implications
• Sluggish input disturbance rejection
• Poor control of integrating and unstable systems

33
Modern industrial MPC algorithmDynamic
optimization
Deviations from output trajectory
Output slack variables
Input deviations
Input moves
Process model
Output constraints
Input constraints
34
Modern industrial MPC algorithmDynamic
optimization (2)

• Solved as a sequence according to prioritized
  constraints and targets
• Hard constraint on MV rate of change (always)
• Hard constraint on MV magnitude
• Sequential high priority soft constraints on CVs
• Set point control
• Sequencial low priority soft constraints on CVs
  and MVs

35
Limitations with modern MPC algorithms
36
Pros/Cons
37
Road Ahead
38
Plan

• Introduction
• General control problem formulation
• Goal
• Constraints-ARW or MPC
• Uncertainty
• Etc.
• control hierachy
• MPC
• History
• Drivers (industry, academia)
• Development
• State of the art
• Theorethical status
• Fuctionality
• Industrial Practice
• Limitations
• Theory
• Explicit MPC
• History

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Optimal operation of constrained processes

• Control of exothermal reaction
• Maximize throughput
• Quality requirements
• Limited cooling capacity
• Variable feed composition and temperature