Model Predictive Control (MPC) - Control Automatic Control

Model Predictive Control (MPC)Bo Bernhardsson and Karl Johan ÅströmDepartment of Automatic Control LTH,Lund UniversityBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

Model Predictive Control (MPC)MPC Problem Setup and ParametersOptimization ProblemMPC ToolsHow to get Integral ActionExample - Quad TankExplicit MPC and CVXGENMaterial:Rawlings (2000), Tutorial overview of model predictive controlÅkesson (2006), Manual to MPC tools 1.0Bo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

Basic Idea of MPC: Receding Horizon Control12At time k solve an open loop optimal control problem over apredefined horizon and apply the first inputAt time k 1 repeat the same procedure (the previous optimalsolution can be used as initial guess)Bellman’s principle of optimalityBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

Model Predictive ControlModel Predictive Control (MPC)Uses models explicitly to predict future plant behaviourConstraints on inputs, outputs, and states are respectedControl sequence is determined by solving an (often convex)optimization problem each sampleCombined with state estimationBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

MPCRefineries (Shell in 1970s, "Dynamic Matrix Control")(Bellman), Propoi (1963), Richalet, Prett, Morari, . . .Main reasons for successSuitable for multivariable systemsRespects actuator limitations: no integrator wind-up, saturationproblems handled explicitlyProcess can be run close to constraints, which is oftencost-effectiveBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

MPC Optimization CriterionControl changes u(k i) u(k i) u(k i 1) areoptimization variablesHp Hw 1J(k) X ẑ(k i k) r(k i) 2Q HXu 1 u(k i) 2Ri 0i HwEvaluated signals z Cz x Dz uHw : system time-delayHu : cover typical closed loop time constantHp : somewhat larger than HuBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

MPC Prediction Horizons (with Hw 0)r(k)ŷ(k)y(k)u(k)u(k)kk Huk HptLarge Hu and Hp give better performance but more computations andnumerical problemsBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

u(k) vs u(k)Why penalize u(k) instead of u(k)?Reference r 6 0 requires u 6 0 to avoid static errorNo need to guess and specify urStill possible to penalize u(k), just include u in z -vectorTAT: Do we get integral action by penalizing u instead of u?Consider y u d, do we get y r in stationarity if d 6 0?Bo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

MPC Block Diagram (typical)evryyuMPCPLANTdv, d, e disturbances, v available for feedforwardr reference valueBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

Assumed Dynamicsxk 1 Axk Bu uk Bv vk Bw wkyk Cxk Dv vk Dw wk"#dkwhere vk is a measured disturbance and wk .ekBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

MPC Prediction AlgorithmsFuture trajectories starting at time k over the prediction horizon HpAssuming v is not known in advance yk Hp k C AHp xk Hp 1XAHp 1 B(uk 1 j 0Bo Bernhardsson and Karl Johan ÅströmjXi 0 ui ) Dv vHpModel Predictive Control (MPC)

MPC Prediction AlgorithmsAll Hp predicted time steps can be summarized as yk 1 yk 2 . . yk Hp uk uk 1 Sx xk Su 1 uk 1 Su . . Bo Bernhardsson and Karl Johan Åström uk Hp 1 vk vk 1 Hv . . Model Predictive Control (MPC)vk Hp

MPC Prediction Algorithms.where CA 2 CA Sx RHp ny nx. . . Hp CASu 1 CBu RHp nx nuCAB CB uu P Hp 1 jj 0CA BuBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

MPC Prediction Algorithms. and CBu0.0 CBu CABuCBu.0 RHp ny Hp nu.Su . . . . PHp 1 PHp 1jj. . . CBuj 0 CA Buj 0 CA Bu CBvD0. 0 CABvCBvD0 RHp ny (Hp 1)nvHv . . . CAHp 1 Bv CAHp 2 Bv CAHp 3 Bv . . . DBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

MPC Blocking FactorsAt each step the following cost function is minimizedHp 1J(k) X ŷ(k i k) r(k i k) 2Q i kHXu 1 u(k i k) 2Ri kBlocking factors can be used to ease the computationalrequirements. This means that constraints and cost is onlyevaluated at certain time steps, contained in sets Ip and Iu .J(k) X ŷ(k i k) r(k i k) 2Q i IpX u(k i k) 2Ri IuMove blocking restrictions can also be imposed on the controlsignal, so that u 0 at certain time stepsBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

MPC Optimization VariablesIntroduce the optimization variables, if we have reference controlsignals urkeu uk uk 1 . . uk Hp 1ande u uk uk 1 . . uk Hu 1 , urk urk 1 . . ruk Hp 1 yk 1 y k 2 ey . . yk Hpfor control moves and reference trackingBo Bernhardsson and Karl Johan Åström , rk 1 rk 2 . . Model Predictive Control (MPC)rk Hp

MPC Optimization VariablesFor unconstrained optimization, the cost function is2J(z) eTu Wu2 eu eT u W ue u eTy Wy2 eywhere the first term penalizes control signals, the second penalizescontrol moves and the last term penalizes tracking error.Matrices Wu , W u , and Wy are design parameters.Alternative notation (here without eu )Hp 1J(k) X ŷ(k i k) r(k i ki k) 2Q HXu 1 u(k i k) 2Ri kMatrices Q and R are (possibly time-varying) design parameters.Bo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

ConstraintsThe MPC controller should respect the constraints umin u(k) umaxumin u(k) umaxzmin zc (k) zmaxSome variables might be constrained, but have no reference valuesIf a constrained variable is not measured, the constraints will be put onan estimate insteadBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

Soft or Hard Constraints?Always need to find a u. Problem if constraints unfeasible.Replace hard constraints with soft constraints.Minimize slack variable ǫ 0 so that ymin max y yk 1k 1 k 1 . . . . . ymin max yk H k Hpyk Hpp umin umax uk k 1k 1 . . ǫV min . . uk H. 1 umin max upk Hp k Hp max u umin k uk k . . . . max . min uk Hu 1 u uk H 1uk Hu 1 max ǫV Relaxation vectors V min 0, V max 0 are user definedparameters. Larger V , more relaxed constraintBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

MPC Optimization ConstraintsFor the constrained case, a suitable MPC cost criterion to beminimized is2J(z, ǫ) eTu Wu2 eu eT u W ue u eTy Wy2 ey ρǫ ǫ2subject to dynamics and constraintsA large value of ρǫ penalizes constraint violation harder.In practice, softening of output constraints is often a good idea.Bo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

MPC Design ParametersInternal Dynamic Model(State Estimation Parameters)Prediction- and Control HorizonsBlocking factorsWeighting MatricesConstraint ParametersSample rateIntegral action? alt. disturbance modelBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

Quadratic OptimizationThe optimization problem is often posed as quadraticprogrammingminx1 T2 x Hx fTxsubject to Ax bIn M ATLAB: ’quadprog’Advantages with QP: Fast, reliable softwareBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

Disturbance ModelsWhen solving the optimization problem, predicted future values ofthe disturbances are neededWith some knowledge about the nature of the disturbances theprediction can be improvedOften Gaussian noise and Kalman filter is usedCan also formulate the prediction problem as a QP problem, e.g.maximizing log-likelihood of measurementsOptimization approach to estimation can e.g. give robustness tooutliersBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

State Estimation - Error Update 0 0 x̂k k x̂k k 1 x̂d d x̂ K (ykm ŷkm ) k kk k 1 m m x̂k kx̂k k 1d : known disturbancem : reference modelBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

MPC Internal Dynamic Model 0 x k 1 d x k 1 m xk 1B0 Cd 0 x0k Bu d Ad0 x0 uk k m xk0Am0 A 0 0 0 Bv B0 Dd 0 v k Bd 0 ykm C0 Cd0 0 Bu00Bm 0 0 xk D d xCm mk mBo Bernhardsson and Karl Johan Åström wd k Bv wkm 0 u w k 0 v wkD 0xkModel Predictive Control (MPC) d wk m wk 0 u w k wkv

State Estimation - Temporal Update 0 x̂k 1 k d x̂ k 1 k m x̂k 1 kx̂0k k 1 B0 Cd 0 Bu d x̂Ad0 0uk k k 1 v0Am0x̂k k 1 A0 0 0 Bv B0 Dd 0 Bd vk 0ŷkm C00Cd wkd 0 Bu Bv wkm 000 u w k Bm 00 v wk 0 x̂k k 1 x̂d Cm k k 1 m Bo Bernhardsson and Karl Johan Åström x̂k k 1Model Predictive Control (MPC)

SoftwareMPCtools - Developed by Johan Åkesson, Automatic Control,LundSeparates measured/constrained/controlled outputsIntegral action by means of disturbance estimationDifferent QP-solvers for the optimization problemUsed in Lab 3 i Predictive controlMathworks - Model Predictive Control ToolboxGUI or text-based control designIntegrated observer design and basic disturbance modelingMPT - Multi-Parametric Toolbox for Matlab from ETHSupports linear, nonlinear, and hybrid system descriptionsExplicit MPCCVX CVXGen: MPC code generator, runs typically on micro ormillisec for small size problems. Nice project!Bo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

MPC ToolboxModel assumptions in MPC toolboxx(k 1) Ax(k) Bu(k)y(k) Cy x(k)z(k) Cz x(k) Dz u(k)zc (k) Cc x(k) Dc u(k)Measured outputs yControlled outputs zConstrained outputs zc(no known disturbances)Bo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

MPCTool - Control ProblemCost functionHp Hw 1J(k) Xkẑ(k i k) r(k i k)k2Q i HwHXu 1k u(k i k)k2Ri 0Prediction Horizon, HpControl Horizon, HuFirst sample to be included, HwBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)(1)

Solving the Quadratic Program (QP)One can rewrite the optimization criterion on the formmin J(k) U T H U U T G E T QEsubject to Ω U ωwhere U, H, G, E, Q, Ω, ω are large vectors/matrices, used to stackup the equations above for all time indices.Details are in the MPC toolbox manual, but will not be neededConvex problem - efficient algorithmsBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

State EstimationUse a traditional Kalman filter, having the formx̂(k 1) Ax̂(k) Bu(k) K(y(k) Cy x̂(k)).Gain matrix K obtained by solving a Riccati equation(information about state constraints are not utilized in MPCTool)Bo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

Error-free tracking - Integral Action 1One option is to use a disturbance observerA step disturbance is assumed to act on the input, the followingextended model is then used (when r 0): xk 1 A 0 B xk B 0 ukv0I0 v k 1 k dk 1z yzya0 Cz Ca0I0dk 0 0 xTk I 0 xTkvkTvkT TdTk TdTk Using an observer with this model structure will introduce integralaction giving z 0 in stationarity.If more outputs than inputs, one must introduce constant outputdisturbances vk on the outputs ya that shouldn’t get integral actionIf nr inputs nr outputs one doesn’t need ya and vkBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

Error-free tracking - Integral Action 2Another option is to add integrator states in the model"# "#"#x(k 1)xi (k 1)A0 Cz I y(k) z(k) hhCy 0Cz 0x(k) B0u(k) "0I#r(k)iiA stabilizing feedback is calculated using the extended state. Note thatthe state xi need not be estimated, since it is known perfectly by thecontrollerPolynomial design interpretation to the two methodsA(q 1)R′ BS Am Ao B where either Ao (previous slide) or Am (this slide) has an increasedorder compared to the minimal orderBo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)

Linear Properties of the MPC ControllerThe MPC controller is nonlinear, because of constraints on state andcontrolHowever, if the constraints are not active, the controller is linearThe minimizing solutions of the unconstrained QP is then U(k) (ΘT QΘ R) 1 ΘT QE(k) . for some vector K̄s r(k) K̄s u(k 1) x̂(k)Bo Bernhardsson and Karl Johan ÅströmModel Predictive Control (MPC)