lecture12 mpc
TRANSCRIPT
EE392m - Winter 2003 Control Engineering 12-1
Lecture 12 - Model Predictive Control
• Prediction model• Control optimization• Receding horizon update• Disturbance estimator - feedback• IMC representation of MPC• Resource:
– Joe Qin, survey of industrial MPC algorithms– http://www.che.utexas.edu/~qin/cpcv/cpcv14.html
EE392m - Winter 2003 Control Engineering 12-2
Control Hierarchy
EE392m - Winter 2003 Control Engineering 12-3
Models for MPC
Plant structure:• CV - controlled variables - y• MV - manipulated variables - u• DV - disturbance variables - v
• FSR - Finite Step Response model
– compact notation
Plant
MV: u
DV: v
CV: y
dtvstusty DU +∆+∆= ))(*())(*()(
dktvkSktukStyN
k
DN
k
U +−∆+−∆= ∑∑==
)()()()()(11
11;
−−=∆∆=
zsh
EE392m - Winter 2003 Control Engineering 12-4
FSR Model
FSR model
dktvkStyn
k+−∆=∑
=
)()()(1
• Ignores anything thathappened more than nsteps in the past
• This is attributed to aconstant disturbance d
EE392m - Winter 2003 Control Engineering 12-5
MPC Process Model Example
MV, DV
CV
EE392m - Winter 2003 Control Engineering 12-6
Prediction Model
+∆
∆=
)(
)()(
ntu
tutU M
+=
)(
)()(
nty
tytY M
+−∆
−∆=
)1(
)1()(
Ntu
tutU M
Discuss later
FSR model
EE392m - Winter 2003 Control Engineering 12-7
Prediction Modeldtvstusty DU +∆+∆= ))(*())(*()(
DdtVtUtUtY D +Φ+⋅Φ+Ψ= )()()()(
Hankel matrix
=
1
1MD
−
=Ψ
0)1()(
00)1(000
L
MOMM
L
L
nSnS
S
UU
U
Past DV
Toeplitz matrix
CV Prediction
Future MV Hankel matrixPast MV
Toeplitz matrix Hankel matrix
=Φ
00)(
0)3()2()()2()1(
L
MOMM
L
L
NS
SSNSSS
D
DD
DDD
D
Unmeasured disturbance
Future impact of the disturbance
EE392m - Winter 2003 Control Engineering 12-8
Optimization of future inputs
• Optimization problem
4444 34444 21)(*
)()()()(tY
D DdtVtUtUtY +Φ+⋅Φ+Ψ=
( ) ( ) min)()()()()()( →+−−= tRUtUtYtYQtYtYJ Td
Td
=
=u
u
y
y
R
RR
Q
L
MOM
L
L
MOM
L
0
0,
0
0
+=
)(
)()(
Nty
tytY
d
d
d M
EE392m - Winter 2003 Control Engineering 12-9
Optimization constraints
• MV constraints
• CV constraints
• Terminal constraint:
maxmax )( utuu ∆≤∆≤∆−
)()()( maxmin tytyty ≤≤
maxmin )( utuu ≤≤
∑=
∆+−=+k
jtutuktu
1
)()1()(
pkktuykty d ≥=+∆=+ for 0)(;)(
maxmin
maxmax
)()(
uCtUuutUu
≤−Σ≤∆≤≤∆−
maxmin )( YtYY ≤≤
EE392m - Winter 2003 Control Engineering 12-10
QP solution
• QP Problem:
min21 →+=
=≤
xfQxxJ
bxAbAx
TT
eqeq
• Standard QP codes are available
=)()(
tYtU
x Predicted MVs, CVs
EE392m - Winter 2003 Control Engineering 12-11
Receding horizon control• Optimization problem solution at step t :
• Use the first computed control value only
• Repeat at each t)(]001[)( tUtu OPT⋅= K
)(tUU OPT=⇒→ min)(UJ
EE392m - Winter 2003 Control Engineering 12-12
Control dynamics• System dynamics as an equality constraint in optimization
• Update of the system state
)()()( * tYtUtY +Ψ=
[ ] )()()(* tdtXtY D +⋅ΦΦ=
• Optimization problem solution at step t :
• Use the first of the computed control values)(]001[)( tUtu OPT⋅= K
)(tUU OPT=⇒→ min))(;( UYUJ
)()()()1( tvBtuBtAXtX D∆+∆+=+
EE392m - Winter 2003 Control Engineering 12-13
State update and estimation
• State update - shift register
• Disturbance estimator (feedback)
• Integrator feedback
+−∆
−∆=
)1(
)1()(
Ntu
tutU M
+−∆
−∆=
)1(
)1()(
Ntv
tvtV M
=)()(
)(tVtU
tX
)(tu∆ )(tv∆
( ))()()()1( tytytdtd m −+=+
Actually measured CV
Unmeasured disturbance
CV Prediction based on the current state X(t)
EE392m - Winter 2003 Control Engineering 12-14
Optimization detail• Setpoint• Zone• Trajectory• Funnels
• Soft constraints (quadraticpenalties) and hardconstraints for MV, CV
• Regularization– penalty– singular value thresholding
EE392m - Winter 2003 Control Engineering 12-15
Advantages and Conveniences
• Industrial strength products that can be used for a broadrange 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 / returned into MPC– measurement problems, actuator failures
• Systematic handling of multi-rate measurements andmissed measurement points– do not update d if no data
EE392m - Winter 2003 Control Engineering 12-16
Technical detail
• Tuning of MPC feedback control performance is an issue.– Works in practice, without formal analysis– Theory requires
• Large (infinite) prediction horizon• 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– ...
EE392m - Winter 2003 Control Engineering 12-17
MPC as IMC
• MPC is a special case of IMC• Closed-loop dynamics (filter dynamics)
– integrator - in disturbance estimator– N poles z=0 - in the FSR model update
Plant
Predictionmodel
Optimizerreference output
disturbance
∆d-
EE392m - Winter 2003 Control Engineering 12-18
Emerging MPC applications
• Vehicle path planning and control– nonlinear vehicle models– world models– receding horizon preview
EE392m - Winter 2003 Control Engineering 12-19
Emerging MPC applications
• Spacecraft rendezvous with space station– visibility cone constraint– fuel optimality
From Richards & How, MIT
EE392m - Winter 2003 Control Engineering 12-20
Emerging MPC applications
• Nonlinear plants– just need a computable model (simulation)
• Hybrid plants– combination of dynamics and discrete mode change
• Engine control• Large scale operation control problems
– operations management– campaign control