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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
EE392m - Spring 2005Gorinevsky
Control Engineering 15-2
Control HierarchyCascade loops
Path optimiz
PID
Loop shaping
Across optimiz
Local optimiz
V&V sim
EE392m - Spring 2005Gorinevsky
Control Engineering 15-3
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
EE392m - Spring 2005Gorinevsky
Control Engineering 15-4
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
EE392m - Spring 2005Gorinevsky
Control Engineering 15-5
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
EE392m - Spring 2005Gorinevsky
Control Engineering 15-6
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
EE392m - Spring 2005Gorinevsky
Control Engineering 15-7
MPC Process Model Example
MV, DV
CV
EE392m - Spring 2005Gorinevsky
Control Engineering 15-8
Prediction Model
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+∆
+∆=
)(
)1()(
Ntu
tutU M
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+
+=
)(
)1()(
Nty
tytY M
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−∆
∆=
)1(
)()(
ntu
tutU M
Discuss later
FSR model
n - memory depth
N – prediction horizon
EE392m - Spring 2005Gorinevsky
Control Engineering 15-9
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 ,,,
EE392m - Spring 2005Gorinevsky
Control Engineering 15-10
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
EE392m - Spring 2005Gorinevsky
Control Engineering 15-11
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
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)
EE392m - Spring 2005Gorinevsky
Control Engineering 15-12
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 ≤≤
EE392m - Spring 2005Gorinevsky
Control Engineering 15-13
QP Solution
• QP Problem:
min21 →+=
=≤
qfQqqJ
bqAbAq
TT
eqeq
• Standard QP codes can be used
⎥⎦
⎤⎢⎣
⎡=YU
q Decision vector:predicted MVs, CVs
EE392m - Spring 2005Gorinevsky
Control Engineering 15-14
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
EE392m - Spring 2005Gorinevsky
Control Engineering 15-15
Receding Horizon Control
• Use the first computed control value only• Repeat at each t
)(]001[)1( tUtu OPT⋅=+∆ K
EE392m - Spring 2005Gorinevsky
Control Engineering 15-16
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
EE392m - Spring 2005Gorinevsky
Control Engineering 15-17
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
EE392m - Spring 2005Gorinevsky
Control Engineering 15-18
Optimization detail• Setpoint• Zone• Trajectory• Funnel
• Soft constraints (quadratic penalties) and hard constraints for MV, CV
• Regularization– penalties – singular value thresholding
EE392m - Spring 2005Gorinevsky
Control Engineering 15-19
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
EE392m - Spring 2005Gorinevsky
Control Engineering 15-20
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
EE392m - Spring 2005Gorinevsky
Control Engineering 15-21
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– ...