Constrained Model Predictive Control Based onReduced-Order Models
Pantelis Sopasakis, Daniele Bernardini, andAlberto Bemporad
IMT Institute for Advanced Studies Lucca
December 13, 2013
52nd IEEE Conf. Decision & Control,Florence, Italy, 2013.
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Model Reduction: Motivation
Challenge:
A lot of systems are modelledwith (∞-dimensional) PDEsand whose approximations com-prise tens of thousands of states.MPC faces its limitations as thestate dimension goes into suchorders of magnitude.
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Model Reduction: Motivation
Examples:
1. Sloshing of liquids (Ardakani & Bridges, 2011)
2. Distribution of anti-tumour drugs (Jackson & Byrne, 2000)
3. HVAC systems (Moukalled et al., 2011)
4. Seismic excitation of buildings (Banerji & Samanta, 2011)
5. Control of Flexible Structures (Rao et al., 1990)
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Model Reduction
Consider a linear time-invariant control system in the form:
xk+1 = A11xk +A12wk +B1uk
wk+1 = A21xk +A22wk +B2uk,
where:
1. xk ∈ Rnx is the measured (dominant) state,
2. wk ∈ Rnw is the unmeasured (neglected) state
Assumption 1. The pair (A11, B1) is stabilizable and K is stabi-lizing gain for it.
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Model Reduction
We impose the following state and input constraints:
xk ∈ X , uk ∈ U , ∀k ∈ N,
and we assume that we have some information about the positionof the initial value of the neglected variables:
w0 ∈ W , {w ∈ Rnw |w′W−1w ≤ 1}.
Assumption 2 (Reduced Model). There exists an ε ∈ (0, 1) sothat A22W ⊆ εW (can be written as ε2W −A22WA′22 � 0).
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The Nominal System
We consider the following nominal system:
zk+1 = A11zk +B1vk,
with zk ∈ Rnx , v ∈ Rnu . Define AK , A11 +B1K and e , x− zand apply the feedback
uk = vk︸︷︷︸MPC control action
+ Kek︸︷︷︸Error feedback
.
The error dynamics is given by:
ek+1 = AKek︸ ︷︷ ︸Stable
+ A12wk︸ ︷︷ ︸Disturbance
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An Invariance Result
Reduced-Order MPC architecture.
Define the set
S(∞)K , T1W ⊕ T2X ⊕ T3U ,
where
T1 , (I −AK)−1A12
T2 , T1(I −A22)−1A21
T3 , T1(I −A22)−1B2.
If xk ∈ X , uk ∈ U for all k ∈ Nand e0 ∈ S(∞)
K , then ek ∈ S(∞)K
for all k ∈ N.
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Ellipsoid+Polytope=?
Reduced-Order MPC architecture.
Notice that:
S(∞)K , T1W︸ ︷︷ ︸
Ellipsoid
⊕T2X ⊕ T3U︸ ︷︷ ︸Polytope
,
Solution: The polytope ΓB∞,with Γ = (T1WT ′1)1/2 is aminimum-volume outboundingparallelotope for T1W.
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The MPC formulation
Let us introduce the sets
Z , X S(∞)K
V , U KS(∞)K ,
Along the prediction horizon N we impose the constraints:
zk ∈ Z, ∀k ∈ N[1,N−1]
vk ∈ V, ∀k ∈ N[0,N−1]
and the terminal constraint
zN ∈ Zf .
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The MPC optimization problem is:
PN (z) : V ?N (z) = min
v∈V(z)VN (z,v),
where the cost function is given by:
VN (z,v) , z′NPzN +
N−1∑k=0
z′kQzK + v′kRvk,
and V(z) is the following multi-valued mapping:
V(z),
v
∣∣∣∣∣∣∣∣z0=z,zk+1=A11zk+B1vk, ∀k∈N[1,N−1],
zk ∈ Z, ∀k∈N[1,N−1],
vk ∈ V, ∀k∈N[0,N−1], zN ∈ Zf
.
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Exponential Robust Stability
Let κN be the MPC control action and
κN (z, x) = K(x− z) + κN (z).
The set S(∞)K × {0} is exponentially stable for the system
xk+1 = A11xk +B1(κN (zk, xk)) +A12wk
zk+1 = A11zk +B1κN (zk),
(with state variable [ xz ]) over the domain of attraction
(ZN ⊕ S(∞)K )×ZN .
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Can we do better?
1. The estimation wk ∈ W for all k ∈ N can be veryconservative for k 6= 0,
2. Online measurements of xk and uk can be used to estimatethe whereabouts of wk+i|k by sets Wk+i|k – then:
ek+j|k ∈ Sk+j|k =
j⊕i=0
AiKA12Wk+i|k.
3. All online operations must be carried out in lowdimensional spaces.
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Set Membership Estimator
A set membership estimator for wk consists of a correction anda prediction step concisely written as:
Wk−1|k ={w ∈ Wk−1|k−1|A12w=xk−A11xk−1−B1uk−1
}Wk|k = A21xk−1 +B2uk−1 ⊕A22Wk−1|k,
while along the prediction horizon we have:
Wk+j|k = A21Xk+j−1|k ⊕B2U ⊕A22Wk+j−1|k,
where
Xk+j|k = X ∩ (A11Xk+j−1|k ⊕B1U ⊕A12Wk+j−1|k).
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Lightweight Set Membership Estimator
We compute sets Hk+j|k so that
hk+j|k , A12wk ∈ Hk+j|k,
with
H0|0 = A12W0|0
W0|0 = W ← Polytopic
Overapprox. of W.
The set membership estimator is given by:
Hk|k = A12Ak22W ⊕A12
k−1∑j=0
A222(A21xj +B2uj)
⊆ εkA12W ⊕A12
k−1∑j=0
T (xj , uj , εj)B∞
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Lightweight Set Membership Estimator
Along the prediction horizon we have:
Hk+j|k = A12A21Xk+j−1|k⊕A12B2U⊕εHk+j−1|k,
Xk+j|k = X ∩ (A11Xk+j−1|k ⊕Hk+j−1|k ⊕B1U).
The error is then bound in
Sk+j|k =
j⊕i=0
AiKHk+i|k
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Model Predictive Control
Reduced-Order MPC
architecture in presence of a
set-membership estimator.
The MPC problem becomes...
V ?N (zk,Hk|k) = min
v∈V(zk,Hk|k)VN (zk,v),
where the set of constraints encom-passes:
zk+j|k ∈ Zk+j|k,XSk+j|k
Vk+j|k,U KSk+j|k
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Exponential Robust Stability
Reduced-Order MPC
architecture in presence of a
set-membership estimator.
Stability Result:
Assume that Hk|k → H? and let S? ,(I −AK)−1H?. The set
S? × {0}
is exponentially stable for the dynam-ics of [ xz ].
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Simulation Example
Our case study: 2 Inputs, 3 Measured States, 500 NeglectedVariables and ε = 0.012.
−10
−5
0
5
10
−10
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xy
z
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
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0.6
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1
xy
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Simulation Example
Comparison with Full-Order MPC
0 10 20−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
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1
k
u
0 10 20−10
−8
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k
x
0 10 20
−4
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w
Reduced-Order MPC
0 10 20−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
ku
0 10 20−10
−8
−6
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10
k
x
0 10 20
−4
−3
−2
−1
0
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w
Full-Order MPC
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Simulation Example
Reduced-Order MPC is of course way faster...
Table : Computational times
Reduced-Order Full-OrderMPC MPC
Computation of P , K, Z, V, Zf 1.3s 14.4sSolution of the MPC problem (avg.) 8.4ms 14297msSolution of the MPC problem (st. dev) ±0.42ms ±859ms
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Thank you for your attention!
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References
1. H. A. Ardakani and T. J. Bridges, “Shallow-water sloshing in vesselsundergoing prescribed rigid-body motion in three dimensions,” J.Fluid Mechanics, vol. 667, pp. 474519, 2011.
2. T. L. Jackson and H. M. Byrne, “A mathematical model to study theeffects of drug resistance and vasculature on the response of solidtumors to chemotherapy,” Math. Biosc., vol. 164, no. 1, pp. 17 38,2000.
3. F. Moukalled, S. Verma, and M. Darwish, “The use of CFD forpredicting and optimizing the performance of air conditioningequipment,” Int. J. Heat and Mass Transfer, vol. 54, no. 13, pp. 549563, 2011.
4. P. Banerji and A. Samanta, “Earthquake vibration control ofstructures using hybrid mass liquid damper,” Engineering Structures,vol. 33, no. 4, pp. 1291 1301, 2011.
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References
5. S. Rao, T. Pan, and V. Venkayya, “Modeling, control, and design offlexible structures: A survey,” Appl. Mech. Rev., vol. 43, no. 5, 1990.
6. T.Bui-Thanh,K.Willcox,O.Ghattas,andB.vanBloemenWaanders,Goal-oriented, model-constrained optimization for reduction of large-scale systems, Journal of Computational Physics, vol. 224, no. 2, pp.880 896, 2007.
7. T. L. Jackson and H. M. Byrne, “A mathematical model to study theeffects of drug resistance and vasculature on the response of solidtumors to chemotherapy,” Math. Biosc., vol. 164, no. 1, pp. 17 38,2000.
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