Course on Model Predictive Control Part I ... - Centro E Piaggio

<strong>Course</strong> on Model Predictive ControlPart I – IntroductionGabriele PannocchiaDepartment of Chemical Engineering, University of Pisa, ItalyEmail: g.pannocchia@diccism.unipi.itFacoltà di Ingegneria, Pisa.July 9th, 2012, Room A12G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 1 / 33

Outline1 Introduction to MPC: motivations and history2 Comparison with conventional feedback control3 Simple example and typical industrial architecture4 Some reminders of linear systems theory and optimal control/estimationG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 2 / 33

Brief history of Model Predictive ControlOrigins and motivationsModel Predictive Control (MPC) algorithms were born inindustrial environments (mostly refining companies) during the70’s:▶ DMC (Shell, USA) [Cutler and Ramaker, 1979]▶ IDCOM (Adersa-Gerbios, France) [Richalet et al., 1978]Necessity to satisfy the more stringent production requests, e.g.:▶▶▶economic optimizationmaximum exploitation of production capacitiesminimum variability in product qualitiesG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 3 / 33

Brief history of Model Predictive Control (cont’d)Industry and academiaNowadays, most complex plants especially in refining and(petro)chemical industries use MPC systemsAfter an initial reluctance, the academia “embraced” MPCcontributing to:▶▶establish theoretical foundationsdevelop new algorithmsG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 4 / 33

Commercial product evolutionCommercial Products (partial list updated to 1996)[DMC] Dynamic Matrix Control ⇒ DMC Corporation (USA)[SMCA] (ex IDCOM) Multivariable Control Architecture ⇒ Set-Point,Inc.(USA)[PCT] (RMPCT) Predictive Control Technology ⇒ Honeywell - Profimatics(USA)[OPC] Optimum Predictive Control ⇒ Treiber Controls, Inc. (Canada)[MVPC] Multivariable Predictive Control ⇒ ABB Ind. System Corp. (USA)[IDCOM-Y] ⇒ Johnson Yokogawa Corp. (USA)[MVC] Multivariable Control ⇒ Continental Control, Inc. (USA)[C-MCC] Contas-Multivariable Constrained Control ⇒ CONTAS s.r.l. (Italy)G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 5 / 33

. Control algorithms are emphasized here becausetively little information is available on the developtof Commercial industrial identification product technology. The evolutionfollow-desired steady state. It was found that the solutionthis problem, known as the linear quadratic Gaussub-sections describe key algorithms on the MPC(LQG) controller, involves two separate steps. At tutionary tree.interval k; the output measurement y k is first usedobtain an optimal state estimate #x kjk :MergesLQG#x kjk 1 ¼ A#x k 1jk 1 þ Bu k 1 ; ðIn middle of the 90’s, many acquisitions and merges occurredhe development of modern control concepts can be #x kjk ¼ #x kjk 1 þ K f ðy k C#x kjk 1 Þ: ðed to theThe worksituation of Kalmanbecame et al. in quite the early steady 1960s with two main competitors (DMC+ andThen the optimal input u k is computed using an optilman, 1960a, RMPCT) b). A greatly and other simplified less description diffused technologies ofproportional (Connoisseur, state controller: SMOC, PFC, etc.)r results will be presented here as a reference pointthe discussion to come. In the discrete-time context, u k ¼ K c #x kjk :From [Qin and Badgwell, 2003]2000DMC+RMPCT4th generationMPC1990SMOCSMCAIDCOM-MHIECONPFCPCTRMPC3rd generationMPC1980ConnoisseurQDMCDMCIDCOM2nd generationMPC1st generationMPC1970LQG1960Fig. 1. Approximate genealogy of linear MPC algorithms.G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 6 / 33

KeywordsMPC keywordsIn most commercial product acronyms we find several important keywords thatdefine the MPC technologiesControlModelPredictiveMultivariableRobustnessConstraintsOptimizationIdentificationAnalysis of such characteristic features in comparison with conventional controlschemesG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 7 / 33

Conventional feedback control (PID)Essential featuresControl action based on the tracking error, e(t) = y sp (t) − y(t) (no prediction)Fixed structure regulator (e.g., PID)u(t) = K c e(t) + K ∫ tcτ I0e(τ)dτ + K c τ DdedtConstraints: only on the manipulated variable (absolute or incremental)u min ≤ u(t) ≤ u max ,du∣dt∣ ≤ ∆u maxProcess model: “sometimes” used to define the tuning parameters K c , τ I , τ DOptimization: no direct optimization is achieved (only by tuning)G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 8 / 33

Shortcomings of conventional feedback control (PID)IssuesConventional feedback controllers are not able to face:Interactions from each manipulated variable to allcontrolled variablesDirectionalityCertain combinations of control actions have amuch larger (20-200 times) effect on the controlledvariables than other combinations of the samecontrol actions. Thus:Perturbations in the former directions are rejected muchmore easily than perturbations in the latter directions.Constraints on the controlled variables (e.g., productqualities)Optimization of the overall plant (nonsquare systems)G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 9 / 33

Conventional multivariable controlTypical structureFeatures and limitationsThe decoupler structure (model based) determines the achievableperformances (interactions and directionality)Decoupler robustness is an issueWhen # CV ≠ # MV (nonsquare systems) different alternatives are necessary(split-range, selective control, etc.)G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 10 / 33

Main features of MPCMPC became a successful technology due to the following features:ease of handling multivariable systemsease of handling complicated dynamics (e.g., delays, inverseresponse, ramps, etc.)ease of handling constraints on controlled and manipulatedvariables (pushing the plant towards its limits)straightforward applicability to feedforward information(measurable disturbances)G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 11 / 33

Industrial applications [Qin and Badgwell, 2003]Area Aspen Honeywell Adersa PCL MDC TotalTechnology Hi-SpecRefining 1200 480 280 25 1985Petrochemicals 450 80 – 20 550Chemicals 100 20 3 21 144Pulp and Paper 18 50 – – 68Air & Gas – 10 – – 10Utility – 10 – 4 14Mining/Metallurgy 8 6 7 6 37Food Processing – – 41 10 51Polymer 17 – – – 17Furnaces – – 42 3 45Aerospace/Defense – – 13 – 13Automotive – – 7 – 7Unclassified 40 40 1045 26 450 1601Total 1833 696 1438 125 450 4542First App. DMC:1985 PCT:1984 IDCOM:1973 PCL: SMOC:IDCOM-M:1987 RMPCT:1991 HIECON:1986 1984 1988OPC:1987Largest App 603×283 225×85 – 31×12 –G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 12 / 33

Hierarchy of an optimization and control systemG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 13 / 33

Typical example: a distillation unit−Q00000001111111000000111111 00000001111111000000111111000000111111000000111111000000111111LDF000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111V+QBG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 14 / 33

Conventional decentralized control of a distillation unitPCFCLCLACDFCFFCLCBVACG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 15 / 33

MPC of a distillation unitPCFCLCLAIDFCFMPCFCLCBVAIG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 16 / 33

MPC: basic ideaManual control of a furnace temperatureT% valve1234Use the process model (DC gain)Feedback information is thedifference between actual andpredicted process outputActions are iterated based onfeedback informationTimeG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 17 / 33

MPC frameworkPast reconciliationFuture predictiondataestimateSensorsypredictionInput seq.ActuatorsutTimeG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 18 / 33

General structure of an MPC algorithmTuning parametersDynamicOptimizationu(k)ˆx − (k + 1)ˆ d − (k + 1)Processy(k)ˆx(k), ˆ d(k)z −1x s (k), u s (k)Steady-stateOptimizationˆ d(k)EstimatorTuning parametersMPCG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 19 / 33

Linear dynamic models: continuous-timeState-space formulation (LTV)dx= A(t)x(t) + B(t)u(t)dt xy(t) = C (t)x(t) + D(t)u(t)x(0) = x 0∈ Rnu ∈ R my ∈ R pIn most applications D(t) = 0State-space formulation (LTI)dx= Ax(t) + Bu(t)dty(t) = C x(t) + Du(t)Solution:x(t) = e At x 0 +∫ t0e A(t−τ) Bu(τ)dτG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 20 / 33

Linear dynamic models: discrete-timeState-space formulation (LTV)x(k + 1) = A(k)x(k) + B(k)u(k)y(k) = C (k)x(k) + D(k)u(k)x(0) = x 0x ∈ R nu ∈ R my ∈ R pState-space formulation (LTI)or simply:x(k + 1) = Ax(k) + Bu(k)y(k) = C x(k) + Du(k)x + = Ax + Buy = C x + DuSolution: x(k) = A k k−1 ∑x 0 + A k−j −1 Bu(j )j =0G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 21 / 33

Linear Quadratic Regulation problemProblem setupDiscrete-time LTI system x + = Ax + BuConsider N time steps into the future, collect input sequenceDefine the cost function:V N (x(0), u) = 1 2u = {u(0), u(1), ,...,u(N − 1)}N−1 ∑k=0[x(k) ′ Qx(k) + u(k) ′ Ru(k) ] + 1 2 x(N )′ P f x(N )subject to: x + = Ax + BuOptimal LQ control problemmin uV N (x(0), u)G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 22 / 33

Optimizing multi-stage functionsBasic ideaSolve the following problem of three variables (x, y, z):min f (w, x) + g (x, y) + h(y, z), w x,y,zfixedRewrite as three single-variable problems:[[] ]min f (w, x) + min g (x, y) + minh(y, z)x y zIterative strategySolve the most inner problem first: h 0 (y) = min z h(y, z)Proceed to the intermediate problem: g 0 (x) = min y g (x, y) + h 0 (y)Solve the most outer problem: f 0 (w) = min x f (x, y) + g 0 (x)G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 23 / 33

Dynamic programming solution of the LQR problemPrinciple of dynamic programming applied to LQR problemLet l(x,u) = 1/2(x ′ Qx + u ′ Ru) and l N (x) = 1/2x ′ P f xOptimize over u(N − 1) and x(N )minN−2 ∑u(0),x(1),...,u(N−2),x(N−1)k=0l(x(k),u(k))+min l(x(N − 1),u(N − 1)) + l N (x(N ))u(N−1),x(N )} {{ }Solve this first s.t. x(N ) = Ax(N − 1) + Bu(N − 1)Obtain:u 0 (N−1) = K N (N−1)x(N−1), with K N (N−1) = −(B ′ P f B+R) −1 B ′ P f ARepeat to obtain the (backward) Riccati recursions:u 0 (k) = K N (k)x(k), with K N (k) = −(B ′ Π(k + 1)B + R) −1 B ′ Π(k + 1)AΠ(k − 1) = Q + A ′ Π(k)A − A ′ Π(k)B(B ′ Π(k)B + R) −1 B ′ Π(k)A,Π(N ) = P fG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 24 / 33

Infinite horizon LQR problemA quote from [Kalman, 1960]In the engineering literature it is often assumed (tacitly and incorrectly)that a system with optimal control law is necessarily stable.Closed-loop with finite-horizon LQRConsider the optimal finite-horizon (N ) control law: u = K N (0)xClosed-loop system: x + = Ax + Bu = (A + BK N )xExamples for which (see e.g. [Rawlings and Mayne, 2009]):max |eig(A + BK N )| ≥ 1and hence the origin is not asymptotically stableInfinite horizon LQR: let N → ∞ and solve the Riccati equationΠ = Q + A ′ ΠA − A ′ ΠB(B ′ ΠB + R) −1 B ′ ΠA ⇒ K = −(B ′ ΠB + R) −1 B ′ ΠAG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 25 / 33

ControllabilityDefinition [Sontag, 1998]A system x + = Ax +Bu is controllable if for any pair of state y, z in R n , there exists afinite input sequence {u(0), u(1),...,u(N − 1)} such that x(0) = y implies x(N ) = zTests for controllabilityVia controllability matrix: C = [ B AB ··· A n−1 B ](A,B) is controllable iff rank(C ) = nVia Hautus Lemma – conceptual:rank [ λI − A B ] = n for all λ ∈ CVia Hautus Lemma – practical:rank [ λI − A B ] = n for all λ ∈ eig(A)Infinite-horizon LQR and controllabilityFor (A,B) controllable and Q,R positive definite, there exists a positive definitesolution of the Riccati equation, and the matrix (A + BK ) is strictly HurwitzG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 26 / 33

Stochastic linear systemsDiscrete-time LTI systemsx + = Ax +Gwy = C x + vx(0) = x 0x(0), w and v are random variablesGaussian assumptionWe often make the following assumption:x(0) ∼ N ( ¯x(0),P(0)), w ∼ N (0,Q), v ∼ N (0,R)Notation: x ∼ N ( ¯x,P) means that the random variable x is normally distributedwith mean ¯x and covariance PG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 27 / 33

Linear optimal state estimationPreliminary results on normally distributed random variablesIf x and y are n.d. and (statistically) independent, i.e.x ∼ N (m x ,P x ) and y ∼ N (m y ,P y ), then the joint density is[ xy] ( [ mx] [ ])Px 0∼ N m y , 0 P yIf x and y are jointly n.d., i.e. [ x] ( [ mx] [ Pxy ∼ N m y ,the conditional density of x given y, (x|y), is:((x|y) ∼ N m x +P x y Py−1If x ∼ N (m x ,P) and y = C x, then:y ∼ N (Cm x ,CPC ′ )P x yP x ′ y P y(y − m y ),P x −P x y Py−1 P x ′ yIf x ∼ N (m x ,P), v ∼ N (0,R) and y = C x + v, then:y ∼ N (Cm x ,CPC ′ + R)]), then)G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 28 / 33

Linear optimal state estimation (cont.’d)Deriving the Kalman filter...Assume prior knowledge: x(k) ∼ N ( ˆx − (k),P − (k))]Obtain measurement y(k) that satisfies:[x(k)y(k)[x(k)y(k)= [ I 0C ISince x(k) and v(k) are independent, there holds:]] [ ])∼ N , P − (k) P − (k)C ′([ˆx − (k)C ˆx − (k)CP − (k) CP − (k)C ′ +RConditional density (x(k)|y(k)) ∼ N ( ˆx(k),P(k)) with:ˆx(k) = ˆx − (k) + L(k) ( y(k) −C ˆx − (k) )L(k) = P − (k)C ′ (CP − (k)C ′ + R) −1P(k) = P − (k) − P − (k)C ′ (CP − (k)C ′ + R) −1 P − (k)C ′Forecast using x(k + 1) = Ax(k) +Gw(k)x(k + 1) ∼ N ( A ˆx(k) , AP(k)A ′ +GQG ′ )} {{ } } {{ }ˆx − (k+1) P − (k+1)] [ ]x(k)v(k)G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 29 / 33

Linear optimal state estimation (cont.’d)Convergence of the state estimatorConsider the noise-free system:x(k + 1) = Ax(k) + Bu(k),y = C x(k)Given an incorrect initial estimate ˆx − (0), we use a time-varyingKalman filter L(k). Is ˆx − (k) → x(k) as k → ∞?Estimation error and steady-state Kalman filterDefine the state estimation error: e(k) = x(k) − ˆx − (k)We obtain:e(k + 1) = (A − AL(k)C )e(k)Thus, e(k) → 0 as k → ∞ if (A − ALC ) is strictly Hurwitz, where:L = ΠC ′ (CΠC ′ + R) −1Π = AΠA ′ − AΠC ′ (CΠC ′ + R) −1 ΠC ′ A ′ +GQG ′G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 30 / 33

ObservabilityDefinition [Sontag, 1998]A system x + = Ax +Bu with measured output y = C x, is observable if there exists afinite N such that for any (unknown) initial state x(0) and N measurements{y(0), y(1),..., y(N − 1)}, the initial state x(0) can be determined uniquelyTests for observabilityVia observability matrix: O = ⎣(A,C ) is observable iff rank(O ) = nVia Hautus[Lemma]– conceptual:λI − Arank = n for all λ ∈ CC⎡C⎤C A. ⎦.C A n−1Via Hautus[Lemma]– practical:λI − Arank = n for all λ ∈ eig(A)CG. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 31 / 33

Regulator vs estimatorRegulator:x + = Ax + Bu, y = C x, V ∞ (x(0), u) = 1 ∑ ∞[2 k=0 x(k) ′ C ′ QC x(k) + u(k) ′ Ru(k) ]Estimator:x + = Ax +Gw, y = C x + vDualityRegulatorEstimatorR > 0, Q > 0 R > 0, Q > 0(A,B) controllable (A,C ) observable(A,C ) observable (A,G) controllableA A ′B C ′C G ′ΠΠK −(AL) ′A + BK (A − ALC ) ′x e ′G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 32 / 33

ReferencesC. R. Cutler and B. L. Ramaker. Dynamic matrix control – a computer algorithm. In AIChE 86th NationalMeeting, Houston, TX, 1979.R. E. Kalman. Contributions to the theory of optimal control. Bull. Soc. Math. Mex., 5:102–119, 1960.S. J. Qin and T. A. Badgwell. A survey of industrial model predictive control technology. Cont. Eng.Practice, 11:733–764, 2003.J. B. Rawlings and D. Q. Mayne. Model Predictive Control: Theory and Design. Nob Hill Publishing,Madison, WI, 2009.J. Richalet, J. Rault, J. L. Testud, and J. Papon. Model predictive heuristic control: applications toindustrial processes. Automatica, 14:413–1554, 1978.E. D. Sontag. Mathematical Control Theory. Springer, 1998.G. Pannocchia <strong>Course</strong> on Model Predictive Control. Part I – Introduction 33 / 33