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Novel Approaches to Online Process Optimization underUncertainty
Dinesh Krishnamoorthy
Department of Chemical EngineeringNorwegian University of Science and Technology (NTNU)
07 November 2019
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Outline
1 Research question - Why? and How?
2 Part 1 - Optimal steady-state operation using transient measurements
3 Part 2 - Dynamic optimization - addressing computation time
4 Summary and future outlook
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Outline
1 Research question - Why? and How?
2 Part 1 - Optimal steady-state operation using transient measurements
3 Part 2 - Dynamic optimization - addressing computation time
4 Summary and future outlook
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Research Question - Why?
Why is traditional RTO not commonly used in industry?
Challenge 1 - Cost of developing the model (offline).
Challenge 2 - Model uncertainty, including wrong values of disturbances and parameters (online update).
Challenge 3 - Numerical robustness, including computational issues of solving optimization problems.
Challenge 4 - Frequent grade changes, which makes steady-state optimization less relevant.
Challenge 5 - Dynamic limitations, including infeasibility due to (dynamic) constraint violation.
Challenge 6 - Problem formulation - choosing the right formulation for the right problem.
Human factors - Corporate culture and technical competence
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My Main Contributions - How?
Process
Parameterestimator(Dynamic)
Setpointcontrol
Steady-stateoptimization x
d
ysp
u
y Process
Parameterestimator(Dynamic)
Setpointcontrol
xd
Ju
u
y
x = Ax+ Bu
J = Cx+ Du
Jspu = 0
K1
Kn
...
G1
...
Gn
yi
ynyspn
ysp1
...
selector
u
u1
un
−
−
Hybrid RTO Feedback RTO Optimization using simple
controllers
Process
Setpointcontrol
Ju
u
y
Extremum seeking using
transient measurementsGradientestimator(Dynamic)
J(y)Jspu = 0
x0,1
NrN
x1,1
x1,3
x2,1 x3,1 xN,1
x2,2 x3,2 xN,2...
...
...
...x2,3 x3,3 xN,3
x2,4 x3,4 xN,4
x1,2
x1,4
u1,1 u2,1
u1,2 u2,2
u1,3 u2,3
u1,4 u2,4
u3,1
u3,2
u3,3
u3,4
economic MPC Distributed multistage MPC using
primal decomposition
Challenge 2 & 5 Challenge 2,3 & 5 Challenge 1,2,3 & 5 Challenge 1,2,3 & 5
Challenge 3,4& 5 Challenge 3, 4 & 5
Process
Economic-
NMPC ParameterEstiamtor
(dynamic)
d, x
u
yProcess
Setpointcontrol
csp
u
y
Extremumseeking
J(y)
Challenge 1,2,3 & 5
H
c
Extremum seeking +
Self-optimizing control
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Outline
1 Research question - Why? and How?
2 Part 1 - Optimal steady-state operation using transient measurements
3 Part 2 - Dynamic optimization - addressing computation time
4 Summary and future outlook
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Conventional steady-state RTO
Two-step approach
Process
Steady-state
optimization
Parameterestimator
(Static)Setpoint
control
ysp
d
u
Steady state
detectiony
Chen & Joseph, Ind. Eng. & Chem. Res (1987)
Darby et al, J. Proc. Control (2011)
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Steady-state wait time
Challenge 2 - Model uncertainty, including wrong values of disturbances and parameters (online update of themodel).
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How to address steady-state wait time?
Obvious : Dynamic RTO
Process
Dynamicoptimization
Setpointcontrol
Parameterestimator(Dynamic)
d, x
ysp
u
y
Dynamic RTO has problems - especially the optimization part (Challenge 3!)
Findeisen et al. IFAC World Congress (2004)
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How to avoid steady-state wait time? - Hybrid RTO
If the focus is steady-state optimization, dynamic terms are needed only for the model update.
Dynamic models - Online model update
Steady-state models - Optimization
Process
Dynamicoptimization
Setpointcontrol
Parameterestimator(Dynamic)
d, x
ysp
u
yProcess
Steady-stateoptimization
Parameterestimator
(Static)Setpointcontrol
ysp
d
u
Steady statedetectiony Process
Parameterestimator(Dynamic)
Setpointcontrol
Steady-stateoptimization x
d
ysp
u
y
Krishnamoorthy et al. Comput. & Chem. Eng. (2018)
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Example: Hybrid RTO for oil production optimization
Krishnamoorthy et al. (Submitted to ACC 2020) - 6 gas-lifted wells with 20 differential states, 78 algebraic states and 6 inputs
Krishnamoorthy et al. Comput. & Chem. Eng. (2018)
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Feedback RTO
Challenge 3 - Numerical robustness, including computational issues
Convert the Hybrid RTO problem into a feedback control problem
Process
Parameterestimator(Dynamic)
Setpointcontrol
Steady-stateoptimization x
d
ysp
u
yProcess
Parameterestimator(Dynamic)
Setpointcontrol
xd
Ju
u
y
x = Ax+ Bu
J = Cx+ Du
Jspu = 0
Krishnamoorthy et al. Ind. Eng. Chem. Res (2019)
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Novel gradient estimation using transient measurements
Step 1- Linearize the dynamic model around (x , d)
x = f (x , u, d)
J = h(x , u, d)
⇒ x = Ax + Bu
J = Cx + Du
Step 2 - Set x = 0 †
J = (−CA−1B + D︸ ︷︷ ︸Ju
)u
Process
x = f(x,u,d)y = h(x,u)
x, d
ymeasy
ny
Ju = −CA−1B + D
Jusp = 0
u
d
GradientEstimation
Linearize
model from
u to J
state andparameterestimationA B
C D
FeedbackController(e.g. PID)
Krishnamoorthy et al. Ind. Eng. Chem. Res (2019)†Garcia & Morari, AIChE J. (1980), Bamberger & Isermann, Automatica (1978)
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Comparison with other RTO approaches
Krishnamoorthy et al. Ind. Eng. Chem. Res. (2019)
CSTR process from Economou et. al. (1986)
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Other Case Examples
Steam
Evaporator
CondensateF3
FeedProduct
Separator
oil export
wgli
wpoi
riser
FA + FB
CBiCAi
u1 = FA u2 = FB
A+ B → C
2B → DQ
FAFB
xA, xB , xC
xP , xE , xG
TR
A+ B → CB + C → P + EC + P → G
u0,1
u0,2
u0,3
Bed 1
Bed 2
Bed 3
in
3-bed Ammonia reactor Gas-lift optimization
Evaporator process Isothermal reactor William-Otto reactor
Krishnamoorthy et al., IFAC OOGP, 2018
Krishnamoorthy et al., Control Engineering Practice, 2019
Bonnowitz et al.,Computer Aided Chemical Engineering, 2018
Krishnamoorthy & Skogestad, Ind. Eng. Chem. Res, 2019
Krishnamoorthy et al. PSE Asia, 2019
Krishnamoorthy & Skogestad, Computer-Aided Chemical Engineering, submitted
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Do we always need a model to optimize?
Challenge 1 - Cost of developing the model
Example: Drive from A → B in shortest time
CV: speed → 50km/h
MV: gas pedal
Translate economic objectives into control objectives
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What to control?
Control (in this order)
1 Active constraints gA → 0
2 Self-optimizing variable c = NT∇uJ → 0
N is chosen such that NT∇ugA = 0
Case 1: Fully constrained - active constraint control
Case 2: Fully unconstrained - control cost gradient to zero (N = I )
Case 3: Partially constrained - active constraint control + control linear gradientcombination to zero
Case 4: Over constrained - give up less important constraints
“...the ideal global self-optimizing variable is the gradient Ju .”
- Halvorsen & Skogestad (1997)
But gradients are not measured!
Morari et al, AIChE Journal (1980)
Skogestad, J. Proc. Control (2000)
Krishnamoorthy & Skogestad, Ind. Eng. Chem. res, (2019)
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Gradient Estimation
Model-free approaches Model-based approaches
Finite Difference
Sinusoidal perturbation (Draper & Li, 1951)
Least sqaures estimation (Hunnekens et al, 2014)
Linear Identification ARX/ARMAX (Bamberger &Isermann, 1978)
Nonlinear Identification NARX (Golden & Ydstie, 1989)
Fixed dynamics (Krishnamoorthy & Skogestad)†
Kalman filter (Gelbert et al, 2012)
Polynomial fitted surfaces (Gao & Engell, 2005)
Guassian Process regression (Ferriera et al, 2018/ Matias& Jaschke 2019)
Multiple units (Srinivasan, 2007)
Parameter estimation (Chen & Joseph, 1987 / Adetola &Guay 2007)
Neighboring extremals (Gros et al, 2009)
Nullspace method (Alstad & Skogestad, 2007)
Feedback RTO (Krishnamoorthy et al, 2019)
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Estimate cost gradient without model - Extremum seeking control
J
u
Ju
Prohibitively slow convergence!
Dynamic plant assumed to be static map
Dither signal ≈ 10× slower than plant dynamics
Integral gain ≈ 10× slower than perturbations
Draper & Li, ASME (1951)
Krstic & Wang, Automatica (2000)
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ARX-based Extremum seeking control
Identify local linear dynamic model, instead of local linear static model
Step 1 - Identify black-box ARX model
J(t) + a1J(t − 1) + · · ·+ anaJ(t − na)
= + b′u(t − 1) + · · ·+ b′nbu(t − nb) + e(t)
⇒ Apoly (q)J = Bpoly (q)u
Step 2 - The steady-state part of the locally valid linear model is obtained with q = 1
Ju = A−1polyBpoly ⇔ −CA−1B + D
Bamberger & Isermann, Automatica (1979)
Garcia & Morari, AIChE Journal (1981)
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ARX-based extremum seeking control
Robustness Issues with identifying Apoly (q)J = Bpoly (q)u
Too many parameters to estimate : (na + nb)
Neglected dynamics/unmodeled effects
Consider a Hammerstein class of systems
G (s)h(u)Ju x
Fix the nominal plant dynamics, in order to use transient measurements
Estimate only the steady-state gradient
h(u)G0(q) ≈ Ju
(b1q−1 + · · ·+ bnbq
−nb
1 + a1q−1 + · · ·+ anaq−na
)︸ ︷︷ ︸
fixed nominal dynamics
u
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Illustrative Example
h(u) = d +3∑
i=1
p2i−1 sin(iu) + p2i cos(iu)
G0(s) =1
0.25s + 1
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Illustrative Example - Unmodeled dynamics
G(s) = G0(s)−0.05s + 1
0.0001s + 1G(s) = G0(s)e−0.01s
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Change in active constraint regions
J(u, d1)
J(u, d2)
feasible regiong
J
u
Optimal operation: Need to switch between regions using control system!
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Switching between active constraints
Output-to-output (CV-CV) switching - use selectors
Input to output (CV-MV) switching - Pair MV that optimally saturates with the CV
Input-to-input (MV-MV) switching - use split range control / valve position control 1
Selector block
Used when one input is used to switch between controlling several outputs.
Each output has a separate controller and the selectors chooses which output to use.
C1
0
0
g1
g2
min
select
C2
1Reyes-Lua et al, Computer Aided Chemical Engineering, 2018
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CV-CV switching using selectors
For each MV
at most 1 CV y0 with setpoint control that may be given up
n number of CV constraints yi that may be optimally active
u = mini∈[0,n](ui ) or u = maxi∈[0.n](ui )
K1
Kn
.
.
.
G1
.
.
.
Gn
yi
yn
y limitn
y limit1
.
.
.selector
u
u1
un
−
−
When to use min-selector? when to use max-selector? and when is it not feasible?
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CV-CV switching using selectors
Introduce logic variable y limi for CV constraints
y limi =
{1, for max-constraint−1, for min-constraint
Theorem
CV-CV switching is feasible only if
sgn(Gi )sgn(y limi ) = sgn(Gj )sgn(y lim
j ) ∀i , j ∈ {1, . . . , n}
and,
sgn(Gi )sgn(y limi ) = 1⇒ min selector
sgn(Gi )sgn(y limi ) = −1⇒ max selector
This ensures problem feasibility!
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Case examples
PID
sp : wmaxpg
PID
sp:$gl/$o
PID
sp: 0ν1 − ν2
PID
sp: 0νi−1 − νi
∑nw
i=1wpgi
ν1
wmaxgl
−
∑nw
i=2wgli
min wgl1
wgl2
wglnw
CC
CspA = Cmax
A
FC
F sp
TC
T sp = Tmax
GC
0
∇TiJ
T
F
CA
A ⇀↽ B
CA,i CB,i
Ti
T
CB
F
min
selector
min
selector
Gradient
Estimator
CA
u2 = F
u1 = Ti
FA + FB
CBiCAi
u1 = FA u2 = FB
V
A+ B → C
2B → D
QC
GC
Qsp = Qmax
0
Q
c1
min
select
FC
F sp = Fmax
0
c3
min
selectFA + FB
1GC3
FAFB
xA, xB , xC
xP , xE , xG
TR
xGCxmaxG
xAC GC
max
xmaxA 0
Ju
Exothermic ReactorIsothermal CSTR
Gas-lift Optimization William-Otto reactor
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Illustrative example
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Hierarchical combination of ESC and SOC
Optimal operation is when the plant gradient → 0
Extremum
seeking
controller
SOC Setpoint
Controller
Process
J(y)
u = cs
u
yd
J
ym
c
ny
H
dither
(e.g. PID orMPC)
Krishnamoorthy et al, AIChE Annual Meeting (2017)
Straus et al, J. Proc. Control (2019)
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Hierarchical combination
Model-free
optimization
Model-based
RTO
Self-optimizing
control
Process
H
ModelUpdate(Dynamic)
Steady-state
gradient
estimationSlow timescale
Medium timescale
Fast timescale
J
Ju
d
c
y
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Outline
1 Research question - Why? and How?
2 Part 1 - Optimal steady-state operation using transient measurements
3 Part 2 - Dynamic optimization - addressing computation time
4 Summary and future outlook
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Dynamic RTO / economic MPC
Challenge 4 - Frequent grade changes, which makes steady-state optimization less relevant.
Process
Economic-
NMPC ParameterEstiamtor
(dynamic)
d, x
u
y
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Economic NMPC under uncertainty
Where to handle the uncertainty?
Externally
Process
Economic-
NMPC ParameterEstiamtor
(dynamic)
d, x
u
y
Use external parameter estimator(observable)
˙d = Ke(y − h(x, u, d))
Solve certainty-equivalence MPC with d
Internally
0.2
2
0.4
3
0.6
2
0.8
0 10
-1-2 -2-3
-2
0
2
-3-2
-10
12
3
Needs a-priori information (compact set /PDF)
p ∈ P
Optimize under uncertainty
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Economic NMPC under uncertainty
What is a good problem formulation?
Easy to understand → operator confidence
Low complexity → Maintenance and service by process engineers
Not overly conservative → Management approval
Feedback!
“...optimize over a sequence of control laws, as done in Dynamic Programming, rather than overa sequence of control actions.”
-Mayne, Automatica (2014)
Mayne, Annual Reviews in Control (2015)
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Solution strategies
Open loop
optimizationDynamic optimization
Standard MPC / Recedinghorizon controlMultistage MPC
/Feedback min-max MPC−
Open loop optimization
Closed loop optimization
Open loop implementationClosed loop
implementation
Perfect information Open-loop optimization Closed-loop optimization
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Multistage scenario MPC
One possible formulation: Multistage scenario MPC / Feedback min-max MPC
x0
x1
1
x2
1
x1
2
x2
2
x3
2
x4
2
x1
3
x2
3
x3
3
x4
3
x5
3
x6
3
x7
3
x8
3
u1
1
u2
1
u1
2
u2
2
u3
2
u4
2
u1
3
u2
3
u3
3
u4
3
u5
3
u6
3
u7
3
u7
3
Discrete scenario tree
Introduces notion of feedback
Different control trajectories
Subject tonon-anticipativity/causalityconstraints
Optimization over different control trajectories ≈ Optimization over control policies(desirable vs. achievable)
Scokaert & Mayne, IEEE Trans. Autom. Control (1998)
Lucia et al, J. Proc. Control (2013)
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Why Multistage MPC?
Certainty equivalence
minxk ,uk
N−1∑k=0
`(xk , uk )
s.t.
xk+1 = f(xk , uk , pk ) ∀k ∈ Kg(xk , uk , pk ) ≤ 0 ∀k ∈ Kx0 = x t
xk ∈ X , uk ∈ U ∀k ∈ KS∑
j=1
Multistage MPC
minxk,j ,uk,j
S∑j=1
ωj
N−1∑k=0
`(xk,j , uk,j )
s.t
xk+1,j = f(xk,j , uk,j , pk,j ) ∀j ∈ S, ∀k ∈ K
g(xk,j , uk,j , pk,j ) ≤ 0 ∀j ∈ S, ∀k ∈ K
x0,j = x t ∀j ∈ Sxk,j ∈ X , uk,j ∈ U ∀j ∈ S, ∀k ∈ KS∑
j=1
Ejuj = 0
Easier to implement and maintain
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Why Multistage MPC?
Easier to understand the control actions
Krishnamoorthy et al, Processes (2016)
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Why Multistage MPC?
Less conservative than worst-case MPC
Krishnamoorthy et al, Processes (2016)
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Multistage MPC - Major Drawback
Problem size explodes exponentially S = MN
minxk,j ,uk,j
S∑j=1
ωj
N−1∑k=0
`(xk,j , uk,j )
s.t
xk+1,j = f(xk,j , uk,j , pk,j ) ∀j ∈ S, ∀k ∈ K
g(xk,j , uk,j , pk,j ) ≤ 0 ∀j ∈ S, ∀k ∈ K
x0,j = xt ∀j ∈ S
xk,j ∈ X , uk,j ∈ U ∀j ∈ S, ∀k ∈ K
S∑j=1
Ejuj = 0
p2,1 pN−1,1· · ·
p2,2· · ·
p2,3· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
Nr = 2
S1
S2
S3
S4
S5
S6
S7
S8
S9
p0,1
p0,2
p0,3
p1,1
p1,1
p1,1
p1,2
p1,2
p1,2
p1,3
p1,3
p1,3
x0,j
x1,2
x1,1
x1,3
x1,5
x1,4
x1,6
x1,8
x1,7
x1,3
x2,1
x2,2
x2,3
x2,4
x2,5
x2,6
x2,7
x2,8
x2,9
x3,1
x3,2
x3,3
x2,4
x2,5
x2,6
x2,7
x2,8
x2,9
xN,1
xN,2
xN,3
xN,4
xN,5
xN,6
xN,7
xN,8
xN,9
pN−1,2
pN−1,3
∀j
p2,1 pN−1,1
p2,2
p2,3
pN−1,2
pN−1,3
p2,1 pN−1,1
p2,2
p2,3
pN−1,2
pN−1,3
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
· · ·
u0,j ∀j
u1,1
u1,2
u1,3
u1,4
u1,5
u1,6
u1,7
u1,8
u1,9
u2,1
u2,2
u2,3
u2,4
u2,5
u2,6
u2,7
u2,8
u2,9
uN−1,1
uN−1,2
uN−1,3
uN−1,4
uN−1,5
uN−1,6
uN−1,7
uN−1,8
uN−1,9
S1
S2
S3
S4
S5
S6
S7
S8
S9
Non-anticipativity constraints enable closed-loop implementation
Lucia et al, J. Proc. Control (2013)
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Scenario Decomposition using Dual Decomposition
Relax Non-anticipativity constraints
Non-anticipativity constraints are may not be feasible - (Duality gap) !
· · ·
λ+ = λ + α
S∑
j=1
Ej u∗
j
Scenario 1 Scenario 2 Scenario S
u2∗
uS∗
u1∗
λ+
Γ1(λ) Γ2(λ) ΓS (λ)
N
Predicted optimal control trajectories
Closed-loop implementation fails!!
Marti et al, Comput. & Chem. Eng (2015)
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Scenario decomposition using Primal decomposition
u1,1 u2,1
u1,2 u2,2
u1,3 u2,3
u1,4 u2,4
S1
S2
S3
S4
uN,1
uN,2
uN,3
uN,4
· · ·
· · ·
· · ·
· · ·
u3,1
u3,2
u3,3
u3,4
· · ·
· · ·
· · ·
· · ·
u1,1
u1,2
u1,3
u1,4
u2,1
u2,2
u2,3
u2,4
u3,1
u3,2
u3,3
u3,4
uN,1
uN,2
uN,3
uN,4
t1 t2
t3
Krishnamoorthy et al, IFAC ADCHEM (2018)
Krishnamoorthy et al, J. Proc. Control (2019)
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Scenario Decomposition using Primal Decomposition
Non-anticipativity constraints are always feasible !
scenario 1
Φ(tl)scenario 2
Φ(tl)scenario S
Φ(tl)· · ·
t+l = tl + α
∑λj
λ1 λ2 λS
tl
N
Predicted optimal control trajectories
Closed-loop implementation OK!!
“Approximate solution now is better than accurate solution tomorrow!”
Krishnamoorthy et al, J. Proc. Control (2019)
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Scenario Decomposition using Parametric Sensitivities
Reduce the number of NLPs needed to solve the distributed scenario MPC problem
w∗0 = arg minw
Φ(w, p0)
s.t.
g(w, p0) ≤ 0
w∗0 + ∆w∗ = arg minw
Φ(w, p0 + ∆p)
s.t.
g(w, p0 + ∆p) ≤ 0
Compute ∆w∗ using path-following predictor-corrector QP.
scenario 1
Φ(tl ,p1)scenario 2
Φ(tl ,p2)scenario S
Φ(tl ,pS)· · ·
t+
l = tl + α∑
λj
λ1 λ2 λS
tl
Krishnamoorthy et al, IEEE Contr. Syst. Lett. (2018)
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Illustrative example
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Outline
1 Research question - Why? and How?
2 Part 1 - Optimal steady-state operation using transient measurements
3 Part 2 - Dynamic optimization - addressing computation time
4 Summary and future outlook
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Future Outlook
Understand the uncertainty - Handle only those uncertainties that have an impact.
In the context of multistage NMPC,
How to select the scenarios - use of data analytic tools
Shrink the uncertainty set using data - Online scenario tree update
Different methods work in different timescales, can handle different kinds of uncertainty, and havedifferent advantages and disadvantages!
Model-free
optimization
Model-based
RTO
Self-optimizing
control
Process
H
ModelUpdate(Dynamic)
Steady-state
gradient
estimationSlow timescale
Medium timescale
Fast timescale
J
Ju
d
c
y
Krishnamoorthy et al. IFAC NMPC (2018)
Krishnamoorthy et al. ECC (2019)
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Summary - The Big Picture
Process
Parameterestimator(Dynamic)
Setpointcontrol
Steady-stateoptimization x
d
ysp
u
y Process
Parameterestimator(Dynamic)
Setpointcontrol
xd
Ju
u
y
x = Ax+ Bu
J = Cx+ Du
Jspu = 0
K1
Kn
...
G1
...
Gn
yi
ynyspn
ysp1
...
selector
u
u1
un
−
−
Hybrid RTO Feedback RTO Optimization using simple
controllers
Process
Setpointcontrol
Ju
u
y
Extremum seeking using
transient measurementsGradientestimator(Dynamic)
J(y)Jspu = 0
x0,1
NrN
x1,1
x1,3
x2,1 x3,1 xN,1
x2,2 x3,2 xN,2...
...
...
...x2,3 x3,3 xN,3
x2,4 x3,4 xN,4
x1,2
x1,4
u1,1 u2,1
u1,2 u2,2
u1,3 u2,3
u1,4 u2,4
u3,1
u3,2
u3,3
u3,4
economic MPC Distributed multistage MPC using
primal decomposition
Challenge 2 & 5 Challenge 2,3 & 5 Challenge 1,2,3 & 5 Challenge 1,2,3 & 5
Challenge 3,4& 5 Challenge 3, 4 & 5
Process
Economic-
NMPC ParameterEstiamtor
(dynamic)
d, x
u
yProcess
Setpointcontrol
csp
u
y
Extremumseeking
J(y)
Challenge 1,2,3 & 5
H
c
Extremum seeking +
Self-optimizing control
Thank you!
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ARX-based extremum seeking control
G(s) =1
(τs + 1)
ARX model with na = 1 and nb = 1
J(t) = −a1J(t − 1) + b1u(t − 1)
Ju =b1
1 + a1, τ =
−Ts
ln(−a1)
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Dynamic Extremum seeking control
∆
M
y∆u∆
∆
Gp
y∆u∆
K
yu
⇔
P :=
[0 1
wIG0 G0
], M := wIG0K(1 + G0K)−1
wI (jω) =
∣∣∣∣Gp(jω)− G0(jω)
G0(jω)
∣∣∣∣ =
∣∣∣∣1− (−5jω + 1)
(10jω + 1)
∣∣∣∣
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Dynamic Extremum seeking control - SISO case
SISO ARX model:
J(t) + a1J(t − 1) + · · ·+ anaJ(t − na) = Ju [b1u(t − 1) + b2u(t − 2) + · · ·+ bnbu(t − nb)]
θ = arg minθ‖ψ − Φθ‖2
2
θ = Ju
ψ =
J(N) + a1J(N − 1) + · · ·+ anaJ(N − na)
J(N − 1) + a1J(N − 2) + · · ·+ anaJ(N − 1− na)...
J(n + 1) + a1J(n + 2) + · · ·+ anaJ(n + 1− na)
Φ =
b1u(N − 1) + · · ·+ bnbu(N − nb)
b1u(N − 2) + · · ·+ bnbu(N − 1− nb)...
b1u(n) + · · ·+ bnbu(n − 1− nb)
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Dynamic Extremum seeking control - Multivariable case
MISO ARX model with n inputs
J(t) + a1J(t − 1) + · · ·+ anaJ(t − na) = B1u(t − 1) + · · ·+ Bnbu(t − nb)
with Bi ∈ R1×n
J(t)+a1J(t − 1) + · · ·+ anaJ(t − na)
=n∑
i=1
Jui[b1,iui (t − 1) + b2,iui (t − 2) + · · ·+ bnb,iui (t − nb)
]θ = arg min
θ‖ψ − Φθ‖2
2
θ = [Ju1 , Ju2 , . . . Jun ]T , ψ - same as before
Φ = [Φ1,Φ2, . . .Φn]
Zhu, Multivariable System Identification for Process Control.
Lyung, System Identification: Theory for the user.
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Dynamic Extremum seeking control - Multivariable case
Illustrative example - 2D Gaussian function
J(u1, u2) = exp
(−(
(u1 − 3)2
8+
(u2 − 4)2
18
))
G(s) =1
120s + 1
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Linear gradient combination satisfies KKT conditions
For a steady-state optimization problem with na < nu active constraints gA(u, d)
∇uL(u, d) =∇uJ(u, d) + λTA∇ugA(u, d) = 0
⇒∇uJ(u, d) = −λTA∇ugA(u, d)
Pre-multiplying by NT gives
NT∇uJ(u, d) = −NT∇ugA(u, d)Tλ
Since NT∇ugA(u, d)T = 0, ⇒ NT∇uJ(u, d) = 0
Since na < nu , ∇ugA(u, d) has full row rank and N is well defined.
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CV-CV switching using selectors - Illustrative Example
PC FCmin
pmax
qmax
CV1: Pressure CV2: Flow
MV: Valve
pmax
qmax
PC FCmin
pmin qmax
CV1: Pressure CV2: Flow
MV: Valve
pmin
qmax
When a CV constraint is active, all other CVs must be within its limit. To remain feasible,
when the MV changes, all the CVs must change in the same direction relative to theirconstraints
if sgn(Gi )sgn(y limi ) = 1, then increasing u moves output yi closer to its constraint.
u = min(ui )⇒ feasibility
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CV-CV switching: William-Otto reactor
2 CV constraints; xG ≤ 0.08kg/kg and xA ≤ 0.12kg/kg
FA > 1.5 FA < 1.5
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CV-CV switching: William-Otto reactor
Use max-selector to switch between xmaxA and c
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Parametric Optimization
w∗0 = arg minw
Φ(w, p0)
s.t.
g(w, p0) ≤ 0
w∗0 + ∆w∗ = arg minw
Φ(w, p0 + ∆p)
s.t.
g(w, p0 + ∆p) ≤ 0
- Pure Predictor QP
∆w∗ = arg min∆w
1
2∆wT∇2
wwL∆w + ∆wT∇wpL∆p
s.t
∇wgT ∆w +∇pg
T ∆p ≤ 0 p
w∗
w∗(p0)
w∗(p0 + ∆p)
-Predictor Corrector QP
∆w∗ = arg min∆w
1
2∆wT∇2
wwL∆w + ∆wT∇wpL∆p +∇wΦT ∆w
s.t
∇wgT ∆w +∇pg
T ∆p ≤ 0
Suwartadi et al, Processes (2017)
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Path-following predictor corrector QP
feasible region
infeasible region
w∗(p0)
w∗(p1)
w∗(p0) +∆w
∗
Suwartadi et al, Processes (2017)
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Online update of the scenario tree
x0,1
Nr N
State trajectory
xmeas
. . .
. . .
p1
p1
p2
p1 p1
p2
p1
. . . p1
p2
p1
. . .p2
p2p2
x0,1
. . .
. . .
. . .
. . .
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Feedback versus open-loop optimization
feedback optimization ≡ open-loop optimization with closed-loop implementation
xk+1 = xk + uk+wk
Open-loop optimization with wk = 0
u0(x0) = [−(8/13)x0,−(3/13)x0,−(1/13)x0]
x0(x0) = [x0, (5/13)x0, (2/13)x0, (1/13)x0]
Dynamic Programming
µ := (µ0(·), µ1(·), µ2(·))
µ0 = −(8/13)x0 ⇒ x0(0) = x0 u0(0) = −(8/13)x0
µ1 = −(3/5)x1 ⇒ x0(1) = (5/13)x0 u0(1) = −(3/13)x0
µ2 = −(1/2)x2 ⇒ x0(2) = (2/13)x0 u0(2) = −(1/13)x0
⇒ x0(3) = (1/13)x0
Section3.1.2 - Rawlings, Mayne, Diehl, Model Predictive Control, 2nd ed.
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Feedback versus open-loop optimization
feedback optimization ≈ multistage optimization with closed-loop implementation
xk+1 = xk + uk+wk wk ∈ [−1, 1]
Open-loop
0 1 2 3
-2
0
2
4
Feedback - DP
0 1 2 3
-2
0
2
4
Feedback - multistage
0 1 2 3
-2
0
2
4
Section3.1.2 - Rawlings, Mayne, Diehl, Model Predictive Control, 2nd ed.
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Feedback versus open-loop optimization
feedback optimization ≈ multistage optimization with closed-loop implementation
Open-loop
0 1 2 3 4 5 20 22 24
-25
-23
-21
-2
0
2
4
21
23
25
|xw=−1(N)− xw=1(N)| = 2N
N →∞
Feedback - DP
0 1 2 3 4 5 20 22 24
-2
0
2
4
|xw=−1(N)− xw=1(N)| → 3.24
N →∞
Feedback - multistage
0 1 2 3 4 5 20 22 24
-2
0
2
4
|xw=−1(N)− xw=1(N)| → 3.24
N →∞
Section3.1.2 - Rawlings, Mayne, Diehl, Model Predictive Control, 2nd ed.
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Multistage MPC approximation of DP
...
...
...
...
uN−1,1
uN−1,2
uN−1,3
uN−1,4
u0
u2,2
u1,1
u2,4
u1,2
Control trajectory
u2,1
u2,3
Ideal: Optimize over control policies
µ = (µ0, µ1, µ2 . . . , µN−1)
Sampled control policy
µ0 = {u0}µ1 = {u1,1, u1,2}µ2 = {u2,1, u2,2, u2,3, u2,4}
...
µN−1 = {uN−1,1, uN−1,2, uN−1,3, uN−1,4}
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Multistage MPC approximation of DP
0 1 2 3
-2
0
2
0 1 2 3 4 5 20 22 24
-2
0
2
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Price of robust horizon - what am I getting for the additional CPU time?
Simulation results from Chapter 9.
Partially deterministic approximation (Berstekas, 2019)“When the problem is stochastic, one may consider an approximation to the `-step lookahead, which is based on deterministic
computations. This is a hybrid, partially deterministic approach, whereby at state xk we allow for a stochastic disturbance at the
current stage, but fix the future disturbances... up to the end of the lookahead horizon, to some typical values.”
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Stability properties of AHeNMPC
Lyapunov stability framework
`(x0, u0) + ∆xTNP∆xN − `(xN , uN)− f(xN , uN)TPf(xN , uN) ≤ 0
For descent property, ` must be strictly dissipative ⇒ (xf , uf ) unique global minima2
if stage cost and dynamic model form strong duality ⇒ dissipativity 3
Steady-state optimization problem has a strongly convex Lagrange function ⇒ strongduality4
Regularization of the stage cost ⇒ strong convexity
ψreg (x, u) := `(x, u) +1
2‖(x− xf , u− uf )‖2
Q+ λT(x− f(x, u))
With Q : eig(∇2ψ + Q) > 0∀(x , u) ∈ X × U
2Faulwasser et al Foundations and Trends in System and Control (2018)3Angeli et al, IEEE Trans. Autom. Control(2011)4Huang et al, J. Proc. Control (2018)
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Scenario-based MPC
OL Scenario-based MPC / Sampled averageapproximation/ Randomized MPC
minxk,j ,uk
S∑j=1
ωj
N−1∑k=0
`(xk,j , uk )
s.t
xk+1,j = f(xk,j , uk , pj ) ∀j ∈ S, ∀k ∈ K
g(xk,j , uk , pj ) ≤ 0 ∀j ∈ S, ∀k ∈ K
x0,j = x t ∀j ∈ Sxk,j ∈ X , uk ∈ U ∀j ∈ S, ∀k ∈ K
Multistage scenario-based MPC / Feedbackmin-max MPC
minxk,j ,uk,j
S∑j=1
ωj
N−1∑k=0
`(xk,j , uk,j )
s.t
xk+1,j = f(xk,j , uk,j , pk,j ) ∀j ∈ S, ∀k ∈ K
g(xk,j , uk,j , pk,j ) ≤ 0 ∀j ∈ S, ∀k ∈ K
x0,j = x t ∀j ∈ Sxk,j ∈ X , uk,j ∈ U ∀j ∈ S, ∀k ∈ KS∑
j=1
Ejuj = 0
Schildbach et al, Automatica 2014
Fagiano et al, IEEE Trans. Autom. Control 2014
Campi & Garatti, IEEE Trans. Autom. Control (2012)
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Scenario-based MPC
OL Scenario-based MPC / Sampled averageapproximation/ Randomized MPC
Multistage scenario-based MPC / Feedbackmin-max MPC
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Data analytic tools to select the discrete scenarios
How to select the different scenarios?
Aim: Seek robustness to variability
Use data analytic tools to explain the variability
0 50 100 150 200 250
-1
-0.5
0
0.5
1
0 50 100 150 200 250
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Uncover hidden data structures to select scenarios judiciously !
Krishnamoorthy et al, IFAC NMPC (2018)
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Robust-adaptive framework - updating the uncertainty set
For time-invariant parameters, shrink the uncertainty set by updating the scenarios based onrecursive Bayesian probability.
Recursive Bayes Theorem
Conditional probability for the j th scenario being the true realization,
Pk,j =e−0.5εTk,j Kεk,j Pk−1,j∑S
m=1 e−0.5εT
k,mKεk,mPk−1,m
Wk,j =Pk,j∑S
m=1 Pk,m
Krishnamoorthy et al, ECC (2019)
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Real time optimization
Online Process
Optimization
Datareconciliation
Modelling
Numericalsolvers
DecompositionmethodsUncertainty
EconomicNMPC
Feedback-based
optimization
Domain knowledge(ChemE, MechE,
EE...)
Applied math andORMS
Control theory
Estimation,Identification and
statistics
Ch. 3–6
Ch. 2
Ch. 7,8,11
Ch. 8–10,12,13Ch. 9,10, App. J