Model Predictive Control: Basic Concepts - SYSMA@IMT Lucca

Model Predictive Control: Basic Concepts

© 2009 by A. Bemporad

Controllo di Processo e dei Sistemi di Produzione ‐ A.a. 2008/09

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Model Predictive Control (MPC) model‐based  optimizer reference

r(t)

process input

output

u(t)

y(t)

measurements

A model of the process is used to predict the future evolution  of the process to optimize the control signal 



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Receding horizon philosophy • At time t: solve an optimal control      problem over a finite future horizon      of N steps:

yt+k

r(t)

Predicted outputs Manipulated Inputs

t+N

t t+1

t+1

ut+k

t+2

t+N+1

• Only apply the first optimal move • At time t+1: Get new measurements, repeat the optimization. And so on …        Advantage of repeated on‐line optimization: FEEDBACK! © 2009 by A. Bemporad


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Receding Horizon ‐ Examples • MPC is like playing chess !

• “Rolling horizon” policies are    also used frequently in finance 



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Receding Horizon ‐ Examples ‐ prediction model

how vehicle moves on the map

‐ constraints

drive on roads, respect one‐way roads, etc.

‐ disturbances

mainly driver’s inattention !

‐ set point

desired location

‐ cost function

minimum time,  minimum distance, etc.

‐ receding horizon mechanism event‐based  (optimal route re‐planned when path is lost) © 2009 by A. Bemporad

x = GPS position u = navigation commands

Fastest route Shortest route Avoid motorways Walking route Bicycle route Limited speed


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Good Models for (MPC) Control Note: computational complexity and theoretical properties (e.g.  stability) depend on chosen model/objective/constraints Good models for MPC: • Descriptive enough to capture the most significant 

   dynamics of the system

F F O   E D A TR • Simple enough for solving the optimization problem “Make everything as simple as possible, but not simpler.”   — Albert Einstein © 2009 by A. Bemporad


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MPC in Industry • History: Computer control (“Manual” MPC)

Fluid catalytic cracking (courtesy of Shell / M. Morari)

Fluid catalytic cracking (FCC) is  the most important conversion  process used in petroleum  refineries. It is widely used to  convert the high‐boiling  hydrocarbon fractions of  petroleum crude oils to more  valuable gasoline, olefinic gases  and other products (http://en.wikipedia.org/wiki/Catalytic_cracking)



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MPC in Industry • History: 1979 Dynamic Matrix Control (DMC) by Shell     (Motivation: multivariable, constrained) • Present Industrial Practice • linear impulse/step response models • sum of squared errors      objective function • executed in supervisory mode

• Particularly suited for problems with • many inputs and outputs • constraints on inputs, outputs, states • varying objectives and limits on variables     (e.g. because of faults) © 2009 by A. Bemporad


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MPC in Industry Hierarchy of control system functions:  Conventional

MPC

(Qin, Badgewell, 1997) © 2009 by A. Bemporad


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MPC in Industry

(snapshot survey conducted in mid‐1999)

(Qin, Badgewell, 2003)

“For us multivariable control is predictive control ” Tariq Samad, Honeywell (IEEE Control System Society, President) (1997) © 2009 by A. Bemporad


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MPC in Industry Results from a recent survey (November 7, 2005) about the use of MPC techniques /  real‐time optimization in a set of US industries: Do you see your use of MPC accelerating,  staying constant, or declining?

Industrial area of respondents to the survey: Pharmaceuticals Machinery

0.7 % 0.7 %

Food & Beverage Aerospace Automotive

0.7 % 0.7 % 1.5 %

Decreasing

Constant

1.5 % 2.2 % 2.2 %

Mining Cement & Glass Metals

Increasing

2.9 %

Plastics & Rubber Electronics Power

0

5.1 % 5.1 %

69.8 %

20

40

60

80

100

Does your company use MPC ?

5.1 %

Pulp & Paper

28.8 %

11.0 %

Other Oil & Gas

15.4 %

19.0 %

No

20.6 %

Chemicals Refining

24.3 % 0

5

10

15

20

25

30

35

35.2 %

Routinely

45.8 %

Sometimes courtesy:


0

10

20

30


40

50

60

70

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Model Predictive Control Toolbox Model Predictive Control Toolbox • MPC Toolbox 3.0 (Bemporad, Ricker, Morari, 1998‐today): – Object‐oriented implementation (MPC object) – MPC Simulink Library – MPC Graphical User Interface – RTW extension (code generation)    [xPC Target, dSpace, etc.] – Linked to OPC Toolbox v2.0.1

Only linear models are handled http://www.mathworks.com/products/mpc/ © 2009 by A. Bemporad


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MPC Simulink Library

• Single MPC and multiple swicthed MPC blocks supported • Reference/disturbance preview and time‐varying limits supported © 2009 by A. Bemporad


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MPC Graphical User Interface



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MPC Tuning Advisor



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Hybrid Toolbox for MATLAB Features:

(Bemporad, 2003‐2009)

• Hybrid models: design, simulation, verification • Control design for linear systems w/ constraints      and hybrid systems (on‐line optimization via QP/MILP/MIQP) • Explicit MPC control (via multi‐parametric programming) • C‐code generation • Simulink library

2450+ download requests since October 2004

http://www.dii.unisi.it/hybrid/toolbox © 2009 by A. Bemporad


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Basics of Constrained Optimization



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Mathematical Programming

In general, problem is difficult to solve use software tools © 2009 by A. Bemporad


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Optimization Software • Taxonomy of most known solvers, for different classes of  optimization problems: http://www-fp.mcs.anl.gov/otc/Guide/SoftwareGuide/

• Network Enabled Optimization Server (NEOS) for remotely  solving optimization problems: http://neos.mcs.anl.gov/neos/solvers/

• Comparison on benchmark problems: http://plato.la.asu.edu/bench.html

• Good open‐source optimization software http://www.coin-or.org/ © 2009 by A. Bemporad


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Convex sets

Convex set


Nonconvex set


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Convex function



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Convex Optimization Problem f and C convex

• Very efficient numerical algorithms exist • Global solution attained • Extensive useful theory • Often occurring in engineering problems • Tractable in theory and practice Excellent reference textbook: “Convex Optimization” by S. Boyd  http://www.stanford.edu/~boyd/cvxbook/ and L. Vandenberghe © 2009 by A. Bemporad


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Polyhedra • A convex polyhedron is the intersection of a finite     set of halfspaces of Rd • A convex polytope is a bounded convex polyhedron

A

A2

A

  2 b = x 2

1

A A3x=b3 

1 x=

A3

b1  

• Hyperplane representation:



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Linear Program George Dantzig  (1914 ‐ 2005)

Standard form: ‐f

f’x Slack variables



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Linear Program trasformation from max to min:

Change inequality direction:

It is always possible to formulate LP problems using “min” and “·” inequalities © 2009 by A. Bemporad


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Quadratic Program

• Convex optimization if P º 0 • Hard problem if P ² 0 © 2009 by A. Bemporad


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Mixed‐Integer Linear Program

• Some variables are continuous, some are discrete (0/1) • In general, it is a NP‐hard problem • Rich variety of algorithms/solvers available



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Modeling languages • MOSEL, associated with commercial package Xpress‐MP • OPL (Optimization Programming Language), associated with     commercial package Ilog‐CPLEX  • AMPL (A Modeling Language for Mathematical Programming)     most used modeling language, supports several solvers • GAMS (General Algebraic Modeling System) is one of the first     modeling languages • LINGO, modeling language of Lindo Systems Inc. • GNU MathProg, a subset of AMPL associated with the     free package GLPK (GNU Linear Programming Kit) • FLOPC++ open source modeling language (C++ class library) • CVX matlab‐based modeling language (from Stanford) • YALMIP another matlab‐based modeling language © 2009 by A. Bemporad


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Linear MPC



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Unconstrained Optimal Control • Linear model:

• Goal: find                                         that minimize

                                                                           is the input sequence that steers the  state to the origin “optimally”



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[computation of cost function]



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Unconstrained Optimal Control

The optimum is obtained by zeroing the gradient

batch least  squares

and hence

Alternative approach: use dynamic programming to find U* (Riccati iterations) © 2009 by A. Bemporad


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Constrained Optimal Control • Linear model:

• Constraints:

• Constrained optimal control problem (quadratic performance index):



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Constrained Optimal Control • Optimization problem:

(quadratic) (linear) Convex QUADRATIC PROGRAM (QP)



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Linear MPC Algorithm past

future Predicted outputs y(t+k|t) Manipulated Inputs u(t+k) t t+1

 At time t:

t+N

• 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                             and discard the remaining optimal inputs • Repeat optimization at time t+1. And so on ... © 2009 by A. Bemporad


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Unconstrained Linear MPC • Assume no constraints

r(t)

yt+k

Predicted outputs

• Problem to solve on‐line:

Manipulated ut+k Inputs t t+1

t+N

• Solution:

Unconstrained linear MPC is nothing else than a standard  linear state‐feedback law ! © 2009 by A. Bemporad


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Double Integrator Example • System: sampling +  ZOH  Ts=1 s

• Constraints:

• Control objective: min

• Optimization problem matrices:

cost: constraints:



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Double Integrator Example

go to demo  /demos/linear/doubleint.m see also mpcdoubleint.m © 2009 by A. Bemporad


(Hyb‐Tbx) (MPC‐Tbx) 38 /94

Double Integrator Example • Add a state constraint:

• Optimization problem matrices:

cost: constraints: © 2009 by A. Bemporad


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Double Integrator Example



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Linear MPC ‐ Tracking • Objective: make the output y(t) track a reference signal r(t) • Idea: parameterize the problem using input increments

• Extended system: let xu(t)=u(t‐1)

Again a linear system with states x(t), xu(t) and input Δu(t) © 2009 by A. Bemporad


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Linear MPC ‐ Tracking • Optimal control problem (quadratic performance index): optimization  vector:

• Note: same formulation as before (W=Cholesky factor of weight matrix Q) • Optimization problem: Convex  Quadratic  Program (QP) © 2009 by A. Bemporad


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Linear MPC ‐ Example • Plant:

• Sampling time:

• Model:

go to demo linear/example3.m © 2009 by A. Bemporad

(Hyb‐Tbx)


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Linear MPC ‐ Example • Performance index:

• Closed‐loop MPC:

go to demo linear/example3.m © 2009 by A. Bemporad

(Hyb‐Tbx)


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Linear MPC ‐ Example • Constraint 0.8 · u(t) · 1.2 (amplitude)

• Constraint ‐0.2 · Δu(t) · 0.2 (slew‐rate)



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Anticipative Action

• Future reference samples (partially)    known in advance (anticipating action):

go to demo  mpcpreview.m © 2009 by A. Bemporad

• Reference not known in advance   (causal):

(MPC‐Tbx)


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Measured and Unmeasured Disturbances Manipulated Variables

u(k)

Measured Disturbances

v(k)

nd(k) Disturbance

d(k)

Model

x (k) d

Plant Model

y(k) Outputs

Unmeasured Disturbances x(k)

Linear model for MPC optimization

(nd(k) = white Gaussian noise, nd(t+k|t)=0 over the prediction horizon)

more details about disturbance models later on … © 2009 by A. Bemporad


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Soft Constraints • To prevent QP infeasibility, relax output constraints: 

 ² = “panic” variable  Vmin, Vmax = vectors with entries ¸0 (the larger the entry, the  relatively softer the corresponding constraint)

• Infeasibility can be due to: – modeling errors  – disturbances – wrong MPC setup (e.g., prediction horizon is too short) © 2009 by A. Bemporad


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Delays – Method 1 • Linear model w/ delays:

• Map delays to poles in z=0:

• Apply MPC to the extended system © 2009 by A. Bemporad


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Delays – Method 2 • Linear model w/ delays:

• Delay‐free model:

• Design MPC for delay‐free model: • Compute the predicted state

• Compute MPC action accordingly:

u(t) = fMPC(¯ x(t + τ )) © 2009 by A. Bemporad

For better closed‐loop performance  one can predict x(t+¿) with a much  more complex model than (A,B,C) !


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MPC vs. Conventional Control  Single input/single output control loop w/ constraints:          equivalent performance can be obtained with other     simpler control techniques (e.g.: PID + anti‐windup) HOWEVER  MPC allows (in principle)   UNIFORMITY  (i.e. same technique for wide range of problems) – reduce training – reduce cost – easier design maintenance  Satisfying control specs and walking on water is similar …  both are not difficult if frozen ! © 2009 by A. Bemporad


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MPC Features • Multivariable constrained “non‐square” systems     (i.e. #inputs and #outputs are different) • Delay compensation • Anticipative action for future reference changes • “Integral action”, i.e. no offset for step‐like inputs

Price to pay: • Substantial on‐line computation • For simple small/fast systems other techniques dominate      (e.g. PID + anti‐windup) • New possibilities for MPC: explicit piecewise linear forms © 2009 by A. Bemporad


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MPC Theory • Historical Goal: Explain the success of DMC • Present Goal: Improve, simplify, and extend industrial algorithms • Areas: • Linear MPC:     linear model • Nonlinear MPC: nonlinear model • Robust MPC:    uncertain (linear) model • Hybrid MPC:     model integrating logic, dynamics,                         and constraints • Issues: – Feasibility – Stability (Convergence) – Computations (Mayne, Rawlings, Rao, Scokaert, 2000) © 2009 by A. Bemporad


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Feasibility QUADRATIC  PROGRAM (QP)

• Feasibility: Guarantee that the QP problem is feasible at all sampling times t • Input constraints only: no feasibility issues ! • Hard output constraints: • When N

Model Predictive Control: Basic Concepts - SYSMA@IMT Lucca

Model Predictive Control: Basic Concepts - SYSMA@IMT Lucca

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