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Page 1: EFFINET - Initial Presentation

March  2013  

EFFINET  A  fusion  of  the  spectrum  of  control  technologies  

Pantelis Sopasakis, Post-Doctoral Fellow

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About  EFFINET  |  MARCH  18-­‐21,  2013  

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The  Closed-­‐Loop  

Energy Price Water Demand

Potable Water Network Model Predictive

Controller (running on GPUs+CPUs)

Online Measurements

Flow Pressure

Quality

Precipitation

Price of water

EFFINET  |  MARCH  18-­‐21,  2013  

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Today’s  PresentaFon  

Outline of the presentation: o  Summary of WP2 requirements o  Formulation of the MPC problem o  Solution approaches

²  Hierarchical MPC ² Model Reduction ²  Newton methods ²  Dual Projection Algorithms ²  Decomposition methods

o  Implementation o  Open Problems and Directions

EFFINET  |  MARCH  18-­‐21,  2013  

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WP2  Requirements  

Requirements of WP2: Involved Partners: IMTL, IRI, AASI, SGAB, WBL •  Construct models for MPC based on mass-balance

equations accompanied by constraints, •  Define risk-sensitive cost functions to be optimised, •  Devise stochastic models for the water demand, •  Develop stochastic models for the energy prices in

the day-ahead market.

Implementation: •  Prototype application in MATLAB/Simulink, •  Control-Oriented models available in MATLAB.

EFFINET  |  MARCH  18-­‐21,  2013  

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Mass balance equations:

⇢Adh

dt= F

i

� Fo

(h)Fo

(h) =h

R

Simple linear correlation:

Bernoulli and Haagen-Poisseuille:

Fo

(h) = �phInflux

Level

Fo

(h) ' �(h� h0) +O((h� h0)2)

* Modelling error

Control-­‐Oriented  Modelling  EFFINET  |  MARCH  18-­‐21,  2013  

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Control-­‐Oriented  Modelling  The mass-balance equations of the water network yield an LTI dynamical model in the following form:

xk+1 = Axk +Buk +Dwk

yk = Cxk

wk|k = wk

wk+j|k = wk+j|k + ek+j|k

ek+j|k ⇠ D

Disturbance Model (Stochastic):

Note: The uncertainty is considered to be bounded and possibly discrete.

The demand requirements can be cast either as (hard) equality constraints:

Muk +Nwk = 0

Or can be introduced in the cost function (soft constraints). The state and input variables are bounded in convex sets:

xk 2 X, 8k 2 Nuk 2 U, 8k 2 N

Alternatively, we may impose bounds on the probability of cosntraints’ violation, e.g.,

Prob(x

k

/2 X) ↵

x

, 8k 2 NProb(u

k

/2 U) ↵

u

, 8k 2 N

EFFINET  |  MARCH  18-­‐21,  2013  

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Control-­‐Oriented  Modelling  The mass-balance equations of the water network yield an LTI dynamical model with parametric uncertainty:

xk+1 =Axk +Buk +Dwk

yk = Cxk

Parametric Uncertainty arises from modelling errors:

(A,B) ⇠ D supp(D)where is compact, or

(A,B) 2 co {�i}i2N[1,K]

EFFINET  |  MARCH  18-­‐21,  2013  

Note: We can treat the quantisation of input as uncertainty:

xk+1 = Axk +Bq(uk) q(uk) = uk + �kwith

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Risk-­‐SensiFve  Cost  FuncFons  

Goal: Introduce Cost Functions so as to: o  Minimise the total energy consuption o  Minimise variations of the control signal

(A motor consumes 6~8 times its nominal operating currect on startup)

o  Optimise the performance of the water network

o  Penalise violation of (soft) constraints.

`

e(xk, pk) , kpkukk1

`�(�uk) , �u0kS�uk

Energy cost:

Startup/(Shutdown) cost:

Performance index:

V

N

(xk

, w

k

, p

k

, x

sp

k

,⇡

k

) = V

f

(xk

, w

k

, p

k

, x

sp

k

)+X

k2N[0,N�1]

`

e(xk

, p

k

) + `

�(�u

k

) + `

x(xk

, x

sp

k

)

MPC Optimisation problem:

* We may also use a quadratic form

`(xk, xspk ) , ⇠

0kQ⇠k

⇠k , xk � x

spk

Reference signal

Terminal Cost

EFFINET  |  MARCH  18-­‐21,  2013  

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FormulaFon  of  the  MPC  Problem  

Our MPC problem amounts to solving the following optimisation problem:

⇡ = {uk}k2N[0,N�1]

Subj. to:

x0 = x

w0 = w

p0 = p

V

?N (x,w, p, xsp) = min

⇡2RmNEV (x,w, p, xsp

,⇡)

And the initial conditions:

xk 2 X, 8k 2 N[1,N�1]

uk 2 U, 8k 2 N[0,N�1]

xk+1 = Axk +Buk +Dwk, 8k 2 N[0,N�1]

wk+1 ⇠ ⌦(wk, uk), 8k 2 N[1,N�1]

pk+1 ⇠ ⇥(pk), 8k 2 N[1,N�1]

xN 2 Xf

* There exist various other ways in which the problem can be formulated

These probability distributions may well be dicrete.

EFFINET  |  MARCH  18-­‐21,  2013  

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 The  MPC  OpFmisaFon  Problem  

Remarks: i.  Proper conditions on the terminal cost and the terminal

set should be imposed for the mean-square stability of the closed loop,

ii.  Recursive feasibility should be enforced and iii.  Constraints that involve probabilities may be imposed. iv.  Discrete distributions call for scenario reduction

methods.

Take away: i.  Large-scale optimisation problem! ii.  We need distributed computational methods to solve it

efficiently.

k k +NE

k k +NE

D. Bernardini and A. Bempoad, “Scenario-based Model Predictive Control of Stochastic Constrained Linear Systems,” proc. Joint 48th IEEE Conf. Decision & Control, 28th Chinese Control Conf., Shangai, China, 2013, pp. 6333-8.

EFFINET  |  MARCH  18-­‐21,  2013  

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Hierarchical  MPC  

Remarks: •  Upper & Lower Layers run at

different sampling rates •  The LCL steers the plant’s state

towards the prescribed set-point •  The UCL sets the references and

takes care about the satisfaction of constraints.

EFFINET  |  MARCH  18-­‐21,  2013  

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 Reduced-­‐Order  MPC  

Large-Scale Systems

xk+1 = A11xk +A12wk +B1uk,

wk+1 = A21xk +A22wk +B2uk

Dominant Dynamics

Neglected Dynamics

Constraints:

xk 2 X, 8k 2 N,uk 2 U, 8k 2 N.

Nominal system: zk+1 = A11zk +B1vk

where uk = vk +K · (xk � zk)| {z }ek

And we know that: w0 2 W

P. Sopasakis, D. Bernardini, A. Bemporad, “Constrained Model Predictive Control Based on Reduced-Order Models,” in proc. 51st CDC conf., 2013, submitted.

Assumption 1. A22 is Hurwicz and there is an ε such that:

kA22k "

Notice that wk 2 Wk , where:

Wk = Ak22W�

k�1X

j=0

Aj22(A21X�B2U),

and notice that: Wk ✓ W, 8k 2 N

where:

W=W�(I�A22)�1(A21X�B2U)

(ellipsoid)

EFFINET  |  MARCH  18-­‐21,  2013  

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 Reduced-­‐Order  MPC  

Idea: Exploit online information to estimate the whereabouts of the neglected variables. Define:

Hk|k , A12Wk|k

Resides in a low-dimensional space…

Result: If and Hk|k ! H?

S? , (I �AK)�1H?

then the set is exponen-tially stable for the system:

S? ⇥ {0}

zk+1 = A11zk +B1vk

xk+1 = A11xk +B1uk +A12wk

EFFINET  |  MARCH  18-­‐21,  2013  

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 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

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

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

Full Order Model/Full state feedback. Solution time: 14.3 ± 1.8(95%)s

Reduced-Order MPC. Only the dominant variables are measured Solution time: 8.4 ± 2.6(95%)ms

“Speedup” 1700 (!)

EFFINET  |  MARCH  18-­‐21,  2013  

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 Newton-­‐Based  MPC  

P. Patrinos, P. Sopasakis, H. Sarimveis, “A global piecewise smooth Newton method for fast large-scale model predictive control,” Automatica 47 (2011), pp. 2016-2022.

Primal Space: •  Constraints are complicated •  Smooth optimisation

Dual Space: •  Constraints are simple and manageable, thus •  Most algorithms are based on the dual problem which is •  unconstrained and involves a PW-smooth function, •  The Hessian is positive semi-definite.

Interior-Point Active Set

Large number of cheap computations

Few expensive iterations

Newton-Based

min

⇢1

2u0Mu+ c0u | b

min

Gu bmax

mid(l, u; y) = max{min{y, u}, l}

�⌧,mid(y) , ⌧Gu�mid(⌧b

min

, ⌧bmax

; ⌧Gu+ y) = 0

* No duality gap…

•  Global Q-Quadratic convergence •  Excellent scale-up •  Exact Line Search

EFFINET  |  MARCH  18-­‐21,  2013  

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 Newton-­‐Based  MPC  

Algorithm: 1.  Let

2.  If stop

3.  Pick a

4.  Solve the system

5.  Update

y0 2 Rm

k�⌧,mid(yk)k ✏

Hk 2 @�⌧,mid(yk)

Hkrk = ��⌧,mid(yk)

yk+1 = yk + rk, k k + 1

Notes: i.  The Hessian is positive semi-definite ii.  Regularised Cholesky Factorisation iii.  Cholesky Updates at every iteration

EFFINET  |  MARCH  18-­‐21,  2013  

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 Newton-­‐Based  MPC  

Characteristics: i.  Outperforms all existing fast MPC

approaches (especially for high horizons) ii.  Scales-up well with the dimensions of the

problem iii.  In practise converges after just a few

iterations iv.  No easy way to calculate error bounds for

large problems.

EFFINET  |  MARCH  18-­‐21,  2013  

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 Accelerated  Dual-­‐Gradient  ProjecFon  

P(x) : V ?(x) = minz2Z(x)

{V (z) | g(z) 0}An MPC problem can be written as (primal form):

where

Z(x) =

⇢z 2 Rn

����x0 = x, 8k 2 N[0,N�1] :xk+1 = Axk +Buk + f

The dual problem is:

D(x) : ?(x) = max

y�0 (x, y)

, where (x, y) = minz2Z(x)

L(z, y)

and L(z, y) = V (z) + y0g(z)

P. Patrinos and A. Bemporad, “An Accelerated Dual-Gradient Projection Algorithm for Embedded Linear Model Predictive Control,” 2013, Submitted for publication.

Equality Constraints

Danskin’s Theorem: r (y) = g(zy), zy , argminz2Z L(z, y)

The Dual QP has much simpler constraint set (orthant)!

EFFINET  |  MARCH  18-­‐21,  2013  

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 Accelerated  Dual-­‐Gradient  ProjecFon  

P. Patrinos and A. Bemporad, “An Accelerated Dual-Gradient Projection Algorithm for Embedded Linear Model Predictive Control,” 2013, Submitted for publication.

Primal suboptimality & Dual Infeasibility:

V (z)� V ? "V��[g(z)]+��1 "g

Let Ψ be LΨ-smooth. The following algorithm converges to an suboptimal solution:

("V , "g)

Idea: Apply a standard fast gradient projection algorithm to solve the dual problem.

Strong Duality

Solution of the primal problem!

Additionally

Primal convergence, infeasibili-ty, suboptimality, propagation of error.

Only simple algebraic operations!

EFFINET  |  MARCH  18-­‐21,  2013  

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 Accelerated  Dual-­‐Gradient  ProjecFon  

P. Patrinos and A. Bemporad, “An Accelerated Dual-Gradient Projection Algorithm for Embedded Linear Model Predictive Control,” 2013, Submitted for publication.

Primal suboptimality & Dual Infeasibility of a solution:

V (z)� V ? "V��[g(z)]+��1 "g

Let Ψ be LΨ-smooth. The following algorithm converges to an suboptimal solution:

("V , "g)

Dual Infeasibility Bound:

Let z(⌫) , #�1⌫

⌫X

i=0

✓�1i z(i)

Then:

* Averaged Sequence

���⇥g(z(⌫))

⇤+

���1

8L (⌫ + 2)2

ky0 � y?k

EFFINET  |  MARCH  18-­‐21,  2013  

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 Accelerated  Dual-­‐Gradient  ProjecFon  

P. Patrinos and A. Bemporad, “An Accelerated Dual-Gradient Projection Algorithm for Embedded Linear Model Predictive Control,” 2013, Submitted for publication.

Primal Suboptimality Bound:

Let z(⌫) , #�1⌫

⌫X

i=0

✓�1i z(i)

Then the following bound holds:

* Averaged Sequence

� 8L (⌫ + 2)2

ky(0) � y?k · ky?k V (z(⌫))� V ? 2L (⌫ + 2)2

(ky(0)k2 + ky?k2)

Hence: We can compute complexity certificates = number of iterations/operations needed to reach an - neighbourhood of the solution. ("V , "g)

EFFINET  |  MARCH  18-­‐21,  2013  

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 Accelerated  Dual-­‐Gradient  ProjecFon  

Characteristics: i.  GPAD does not propagate round-off

errors (works even on an Arduino Uno, 8bit PLC)

ii.  It is very fast – it requires few cheap iterations

iii.  Converges quadratically (with respect to the primal problem)

iv.  Complexity Certification (Necessary for embedded applications),

v.  Primal suboptimality bounds are known. Directions: i.  A C/MATLAB toolbox is under preparation. ii.  On-chip implementation of the algorithm

and demo applications.

EFFINET  |  MARCH  18-­‐21,  2013  

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DecomposiFon  Methods  

Decomposition: Large-scale optimisation problems need to be decomposed so as to be solved in a distributed fashion. Examples: •  Direct Methods

•  Cutting Plane •  Regularised (Smoothened)

Cutting Plane methods •  Nested Decomposition

•  Dual Methods •  Augmented Lagrangian

Decomposition •  Splitting methods

•  Stochastic Methods

Andrzej Ruszuński, “Decomposition methods in stochastic programming,” Mathematical Programming, 79 (1997), pp. 333-353.

Research  Direc:on:  Fast  MPC  methods  coupled  with  decomposiFon  methods…  

EFFINET  |  MARCH  18-­‐21,  2013  

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ImplementaFon  

GPU programming because: •  A CPU core can execute 4 to 8 32-

bit instructions per clock (IPC32)

•  A GPU can execute >3200 IPC32.

•  GPUs are good at doing the same thing, but they’re not good at switching from one job to the other.

1100  paint-­‐guns  

A Success Story:

EFFINET  |  MARCH  18-­‐21,  2013  

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The  End!  

Thank you for your attention.

EFFINET  |  MARCH  18-­‐21,  2013