Lecture 15 - Model Predictive Control Part 2: Industrial MPC

EE392m - Spring 2005Gorinevsky

Control Engineering 15-1

Lecture 15 - Model Predictive ControlPart 2: Industrial MPC

• Prediction model• Control optimization• Receding horizon update• Disturbance estimator - feedback • IMC representation of MPC



Control HierarchyCascade loops

Path optimiz

PID

Loop shaping

Across optimiz

Local optimiz

V&V sim



MPC Setup

Plant structure:• CV - controlled variables – y

– plant outputs, output errors

• MV - manipulated variables – u– control inputs

• DV - disturbance variables - v– disturbances and setpoints

Plant

MV: u

DV: v

CV: y



Models for MPC

• FIR (Finite Impulse Response) model– is used in some formulation

• FSR (Finite Step Response) model– Broadly used in process control

dtvstusty DU +∆+∆= ))(*())(*()(

dktvkSktukStyn

k

Dn

k

U +−∆+−∆= ∑∑==

)()()()()(11

11 −−=∆ z

;UU sh ∆=uhusus UUU *** =∆=∆

• Compact notation



Finite Step Response Model

FSR model

dktvkStyn

k

+−∆=∑=

)()()(1

• Ignores anything that happened more than nsteps in the past

• This is attributed to a constant disturbance d



MPC Model Identification

• Identification is a part of most industrial MPC packages• Step (bump test) or PRBS

• Step responses are directly used as models

FIR model ID



MPC Process Model Example

MV, DV

CV



Prediction Model

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+∆

+∆=

)(

)1()(

Ntu

tutU M

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

+=

)(

)1()(

Nty

tytY M

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−∆

∆=

)1(

)()(

ntu

tutU M

Discuss later

FSR model

n - memory depth

N – prediction horizon



System State for FSR Model• MPC Concept

• FSR model state

• FSR model in state-space form

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−∆

∆=

)1(

)()(

ntu

tutU M

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−∆

∆=

)1(

)()(

ntv

tvtV M

Past control moves

Past disturbances

[ ] )(Solver ProblemMPC)( tutx →→

⎥⎦

⎤⎢⎣

⎡=)()(

)(tVtU

tx

)()()()1( tvBtuBtAxtx DU ∆+∆+=+dtCxty += )()( Exercise: what are the

matrices ?CBBA DU ,,,



Prediction Modeldtvstusty DU +∆+∆= ))(*())(*()(

DdtVtUUY D +Φ+⋅Φ+Ψ= )()(

Hankel matrix

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

1

1MD

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

=Ψ

0)1()(

0)1()2(00)1(

L

MOMM

L

L

NSNS

SSS

UU

UU

U

Past DV

Toeplitz matrix

CV Prediction

Future MV Hankel matrixPast MV

Toeplitz matrix Hankel matrix

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

+

=Φ

00)1(

0)2()3()1()1()2(

L

MOMM

L

L

nS

SSnSSS

D

DD

DDD

D

Unmeasured disturbance

Future impact of the disturbance



Optimization of Future Inputs

• MPC optimization problem

4444 34444 21)(*

)()()()(tY

D DdtVtUtUtY +Φ+⋅Φ+Ψ=

min)()()()( →+= tRUtUtYQtYJ TYT

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=u

u

y

y

Y

R

RR

Q

QQ

L

MOM

L

L

MOM

L

0

0,

0

0

)(* tY - free response of the system

• Caveat: the problem might be ill-conditioned, then there is a need for regularization (an additional quadratic term in J)



Optimization Constraints

• MV constraints

• CV constraints

• Terminal constraint:

maxmax )( utuu ∆≤∆≤∆−

)()()( maxmin tytyty ≤≤

maxmin )( utuu ≤≤

∑=

∆+=+k

j

tutuktu2

)()()(

pkktukty ≥=+∆=+ for 0)(;0)(

maxmin

maxmax

uCUuuUu

≤−Σ≤∆≤≤∆−

maxmin YYY ≤≤



QP Solution

• QP Problem:

min21 →+=

=≤

qfQqqJ

bqAbAq

TT

eqeq

• Standard QP codes can be used

⎥⎦

⎤⎢⎣

⎡=YU

q Decision vector:predicted MVs, CVs



Control Update• System dynamics as an equality constraint in optimization

• Update of the system state

)(* tYUY +Ψ=

[ ] )()(* tdtxY D +⋅ΦΦ=

• Optimization problem solution at step t :

{ } ( )[ ]YUJYU OPTOPT ,minarg, =

)()()()1( tvBtuBtAxtx DU ∆+∆+=+

Forced response Free response



Receding Horizon Control

• Use the first computed control value only• Repeat at each t

)(]001[)1( tUtu OPT⋅=+∆ K



State Update and Estimation• FSR model state update – delay line

• Disturbance estimator

• Disturbance estimator = Integrator feedback

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−∆

∆=

)1(

)()(

ntu

tutU M

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−∆

∆=

)1(

)()(

ntv

tvtV M⎥

⎦

⎤⎢⎣

⎡=)()(

)(tVtU

tx

)1( +∆ tu )1( +∆ tv

( ))()()()1( tytyktdtd mI −+=+

Actually observed CV

Unmeasured disturbance

CV Prediction based on the current state x(t).

Computed at the previous step



MPC as IMC• MPC with disturbance estimator is a special case of IMC

Plant

Prediction model

Optimizerreference output

disturbance

∆y-

zeros poles

Integrator)()()1( tyktdtd I ∆+=+

IMC

• Closed-loop dynamics (filter dynamics) – integrator in the disturbance estimator– 2n poles z = 0 in the FSR model update



Optimization detail• Setpoint• Zone• Trajectory• Funnel

• Soft constraints (quadratic penalties) and hard constraints for MV, CV

• Regularization– penalties – singular value thresholding



Soft Constraints• Consider a QP Problem

min21 →+= xfQxxJ TT

bAx ≤

subject to:soft constraint, might be infeasible

• Slack variable formulation: w = 0 yields the original problem

wbxA ss ≤−w≤0

min21

21 →++= xfSwwQxxJ TTT

⎥⎦

⎤⎢⎣

⎡wx new augmented

decision vector

Large penalty for constraint violation

subject to:

ss bxAbAx

≤≤

slack



Industrial MPC Features

• Industrial strength products that can be used for a broad range of applications

• Flexibility to plant size, automated setup• Based on step response/impulse response model • On the fly reconfiguration if plant is changing

– MV, CV, DV channels taken off control or returned into MPC – measurement problems (sensor loss), actuator failures

• Systematic handling of multi-rate measurements and missed measurement points – do not update the disturbance estimate d if the data is missing



Technical detail

• Tuning of MPC feedback control performance is an issue. – Works in practice, without formal analysis– Theory requires

• Large (infinite) prediction horizon or• Terminal constraint

• Additional tricks for– a separate static optimization step– integrating and unstable dynamics – active constraints– regularization– shape functions for control– different control horizon and prediction horizon– ...

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Lecture 15 - Model Predictive Control Part 2: Industrial MPC