effinet - initial presentation
DESCRIPTION
Presentation of initial ideas for the EU project EFFINET.TRANSCRIPT
March 2013
EFFINET A fusion of the spectrum of control technologies
Pantelis Sopasakis, Post-Doctoral Fellow
About EFFINET | MARCH 18-‐21, 2013
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
The End!
Thank you for your attention.
EFFINET | MARCH 18-‐21, 2013