EFFINET - Initial Presentation

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Presentation of initial ideas for the EU project EFFINET.

Transcript of EFFINET - Initial Presentation

  • 1. March 2013 EFFINET A fusion of the spectrum of control technologies Pantelis Sopasakis, Post-Doctoral Fellow

2. About EFFINET | MARCH 18-21, 2013 3. The Closed-Loop Energy Price Water DemandPotable WaterNetworkModel PredictiveController(running on GPUs+CPUs)Online MeasurementsFlowPressureQualityPrecipitationPrice ofwaterEFFINET | MARCH 18-21, 2013 4. Todays PresentaFon Outline of the presentation:o Summary of WP2 requirementso Formulation of the MPC problemo Solution approaches Hierarchical MPC Model Reduction Newton methods Dual Projection Algorithms Decomposition methodso Implementationo Open Problems and DirectionsEFFINET | MARCH 18-21, 2013 5. WP2 Requirements Requirements of WP2:Involved Partners: IMTL, IRI, AASI, SGAB, WBL Construct models for MPC based on mass-balanceequations 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 inthe day-ahead market.Implementation: Prototype application in MATLAB/Simulink, Control-Oriented models available in MATLAB.EFFINET | MARCH 18-21, 2013 6. Mass balance equations:Adhdt= Fi Fo(h)Fo(h) =hRSimple linear correlation:Bernoulli and Haagen-Poisseuille:Fo(h) =phInfluxLevelFo(h)(h h0) + O((h h0)2)* Modelling errorControl-Oriented Modelling EFFINET | MARCH 18-21, 2013 7. Control-Oriented Modelling The mass-balance equations of the waternetwork yield an LTI dynamical model inthe following form:xk+1 = Axk + Buk + Dwkyk = Cxkwk|k = wkwk+j|k = wk+j|k + ek+j|kek+j|k DDisturbance Model (Stochastic):Note: The uncertainty is considered tobe bounded and possibly discrete.The demand requirements can be casteither as (hard) equality constraints:Muk + Nwk = 0Or can be introduced in the cost function(soft constraints). The state and inputvariables are bounded in convex sets:xk 2 X, 8k 2 Nuk 2 U, 8k 2 NAlternatively, we may imposebounds on the probability ofcosntraints violation, e.g.,Prob(xk /2 X) x, 8k 2 NProb(uk /2 U) u, 8k 2 NEFFINET | MARCH 18-21, 2013 8. Control-Oriented Modelling The mass-balance equations of the waternetwork yield an LTI dynamical model withparametric uncertainty:xk+1 =Axk + Buk + Dwkyk = CxkParametric Uncertainty arisesfrom 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 9. Risk-SensiFve Cost FuncFons Goal: Introduce Cost Functions so as to:o Minimise the total energy consuptiono Minimise variations of the control signal(A motor consumes 6~8 times itsnominal operating currect on startup)o Optimise the performance of the waternetworko Penalise violation of (soft) constraints.`e(xk, pk) , kpkukk1` ( uk) , u0kS ukEnergy cost:Startup/(Shutdown) cost:Performance index:VN (xk, wk, pk, xspk , k) = Vf (xk, wk, pk, xspk )+Xk2N[0,N 1]`e(xk, pk) + ` ( uk) + `x(xk, xspk )MPC Optimisation problem:* We may also use aquadratic form`(xk, xspk ) , 0kQkk , xk xspkReferencesignalTerminalCostEFFINET | MARCH 18-21, 2013 10. FormulaFon of the MPC Problem Our MPC problem amounts to solving thefollowing optimisation problem: = {uk}k2N[0,N 1]Subj. to:x0 = xw0 = wp0 = pV ?N (x, w, p, xsp) = min2RmNEV (x, w, p, xsp, )And the initialconditions: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 waysin which the problem can beformulatedThese probability distributionsmay well be dicrete.EFFINET | MARCH 18-21, 2013 11. The MPC OpFmisaFon Problem Remarks:i. Proper conditions on the terminal cost and the terminalset should be imposed for the mean-square stability ofthe closed loop,ii. Recursive feasibility should be enforced andiii. Constraints that involve probabilities may be imposed.iv. Discrete distributions call for scenario reductionmethods.Take away:i. Large-scale optimisation problem!ii. We need distributed computational methods to solve itefficiently.k k + NEk k + NED. 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 12. Hierarchical MPC Remarks: Upper & Lower Layers run atdifferent sampling rates The LCL steers the plants statetowards the prescribed set-point The UCL sets the references andtakes care about the satisfactionof constraints.EFFINET | MARCH 18-21, 2013 13. Reduced-Order MPC Large-Scale Systemsxk+1 = A11xk + A12wk + B1uk,wk+1 = A21xk + A22wk + B2ukDominant DynamicsNeglected DynamicsConstraints:xk 2 X, 8k 2 N,uk 2 U, 8k 2 N.Nominal system: zk+1 = A11zk + B1vkwhere uk = vk + K (xk zk)| {z }ekAnd we know that: w0 2 WP. 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 Hurwiczand there is an such that:kA22k "Notice that wk 2 Wk , where:Wk = Ak22Wk 1Xj=0Aj22(A21X B2U),and notice that: Wk W, 8k 2 Nwhere:W=W (I A22) 1(A21X B2U)(ellipsoid)EFFINET | MARCH 18-21, 2013 14. Reduced-Order MPC Idea: Exploit online information toestimate the whereabouts of theneglected variables. Define:Hk|k , A12Wk|kResides in a low-dimensional spaceResult: If andHk|k ! H?S?, (I AK) 1H?then the set is exponen-tially stable for the system:S? {0}zk+1 = A11zk + B1vkxk+1 = A11xk + B1uk + A12wkEFFINET | MARCH 18-21, 2013 15. Reduced-Order MPC 0 10 2010.80.60.40.200.20.40.60.81ku0 10 201086420246810kx0 10 2043210123kw0 10 2010.80.60.40.200.20.40.60.81ku0 10 201086420246810kx0 10 2043210123kwFull Order Model/Full statefeedback.Solution time: 14.3 1.8(95%)sReduced-Order MPC. Only thedominant variables are measuredSolution time: 8.4 2.6(95%)msSpeedup 1700 (!)EFFINET | MARCH 18-21, 2013 16. Newton-Based MPC P. Patrinos, P. Sopasakis, H. Sarimveis, A global piecewise smooth Newton method for fast large-scale model predictivecontrol, Automatica 47 (2011), pp. 2016-2022.Primal Space: Constraints are complicated Smooth optimisationDual 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-PointActive SetLarge number ofcheap computationsFew expensiveiterationsNewton-Basedmin12u0Mu + c0u | bmin Gu bmaxmid(l, u; y) = max{min{y, u}, l},mid(y) , Gu mid(bmin, bmax; Gu + y) = 0* No duality gap Global Q-Quadratic convergence Excellent scale-up Exact Line SearchEFFINET | MARCH 18-21, 2013 17. Newton-Based MPC Algorithm:1. Let2. If stop3. Pick a4. Solve the system5. Updatey02 Rmk ,mid(yk)k Hk2 @ ,mid(yk)Hkrk= ,mid(yk)yk+1= yk+ rk, k k + 1Notes:i. The Hessian is positive semi-definiteii. Regularised Cholesky Factorisationiii. Cholesky Updates at every iterationEFFINET | MARCH 18-21, 2013 18. Newton-Based MPC Characteristics:i. Outperforms all existing fast MPCapproaches (especially for high horizons)ii. Scales-up well with the dimensions of theproblemiii. In practise converges after just a fewiterationsiv. No easy way to calculate error bounds forlarge problems.EFFINET | MARCH 18-21, 2013 19. Accelerated Dual-Gradient ProjecFon P(x) : V ?(x) = minz2Z(x){V (z) | g(z) 0}An MPC problem can be written as (primal form):whereZ(x) =z 2 Rn x0 = x, 8k 2 N[0,N 1] :xk+1 = Axk + Buk + fThe dual problem is:D(x) : ?(x) = maxy 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 PredictiveControl, 2013, Submitted for publication.Equality ConstraintsDanskins Theorem: r (y) = g(zy), zy, argminz2Z L(z, y)The Dual QP has muchsimpler constraint set(orthant)!EFFINET | MARCH 18-21, 2013 20. Accelerated Dual-Gradient ProjecFon P. Patrinos and A. Bemporad, An Accelerated Dual-Gradient Projection Algorithm for Embedded Linear Model PredictiveControl, 2013, Submitted for publication.Primal suboptimality & Dual Infeasibility:V (z) V ? "V[g(z)]+ 1 "gLet be L-smooth. The followingalgorithm converges to ansuboptimal solution:("V , "g)Idea: Apply a standard fastgradient projection algorithm tosolve the dual problem.StrongDualitySolution of the primal problem!AdditionallyPrimal convergence, infeasibili-ty, suboptimality, propagationof error.Only simplealgebraic operations!EFFINET | MARCH 18-21, 2013 21. Accelerated Dual-Gradient ProjecFon P. Patrinos and A. Bemporad, An Accelerated Dual-Gradient Projection Algorithm for Embedded Linear Model PredictiveControl, 2013, Submitted for publication.Primal suboptimality & Dual Infeasibility of asolution:V (z) V ? "V[g(z)]+ 1 "gLet be L-smooth. The followingalgorithm converges to ansuboptimal solution:("V , "g)Dual Infeasibility Bound:Let z() , # 1Xi=0 1i z(i)Then:* Averaged Sequenceg(z())+ 18L( + 2)2ky0 y?kEFFINET | MARCH 18-21, 2013 22. Accelerated Dual-Gradient ProjecFon P. Patrinos and A. Bemporad, An Accelerated Dual-Gradient Projection Algorithm for Embedded Linear Model PredictiveControl, 2013, Submitted for publication.Primal Suboptimality Bound:Let z() , # 1Xi=0 1i z(i)Then the following bound holds:* Averaged Sequence8L( + 2)2ky(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 23. Accelerated Dual-Gradient ProjecFon Characteristics:i. GPAD does not propagate round-offerrors (works even on an Arduino Uno,8bit PLC)ii. It is very fast it requires few cheapiterationsiii. Converges quadratically (with respect tothe primal problem)iv. Complexity Certification (Necessary forembedded applications),v. Primal suboptimality bounds are known.Directions:i. A C/MATLAB toolbox is under preparation