model predictive control based on reduced-order models

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

−5

0

5

10

−10

−5

0

5

10

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

0.4

0.6

0.8

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

0.8

1

k

u

0 10 20−10

−8

−6

−4

−2

0

2

4

6

8

10

k

x

0 10 20

−4

−3

−2

−1

0

1

2

3

k

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

−4

−2

0

2

4

6

8

10

k

x

0 10 20

−4

−3

−2

−1

0

1

2

3

k

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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