Grüne, Lars

Schaller, Manuel

Schiela, Anton

English

EPub Bayreuth

Exponential sensitivity analysis for ModelPredictive Control of PDEs ?Lars Gru¨ne ∗ Manuel Schaller ∗ Anton Schiela ∗∗ Chair of Applied Mathematics, Department of Mathematics,University of Bayreuth, 95447 Bayreuth, Germany(e-mail: author@ uni-bayreuth.de).Abstract: Model Predictive Control (MPC) is a control method in which the solution ofoptimal control problems on infinite or indefinitely long horizons is split up into the successivesolution of optimal control problems on relatively short finite time horizons. Only a first partof this solution with given length is implemented as a control for the longer, possibly infinitehorizon. Motivated by this application, we analyze the propagation of discretization errors in thecontext of optimal control of abstract evolution equations in infinite dimensional spaces. Usinga particular stability property, one can show that indeed the error decays exponentially in time,leading to very efficient time and space discretization schemes for MPC. In particular, one canrigorously explain the behavior of goal oriented error estimation algorithms used in this context.Furthermore, an exponential turnpike theorem will be derived. We give particular applicationsof this abstract theory to admissible control of hyperbolic equations, nonautonomous andsemilinear parabolic equations. Eventually, we present several numerical examples illustratingthe theoretical findings.Keywords: Control of Distributed Parameter Systems; Optimal Control; Stability.AMS MSC 2010 : 65M50, 49K40, 35Q93.1. INTRODUCTIONModel Predictive Control (MPC) is a feedback controltechnique used in many industrial applications such ase.g. chemical process engineering, electrical engineering,aerospace or automotive engineering, cf. Camacho andBordons (2007); Qin and Badgwell (2003). The goal ofMPC is to solve an in general nonlinear optimal controlproblem (OCP) on an indefinite or infinite horizon in afeedback manner. To this end, the task is split up into thesuccessive solution of OCPs on a finite horizon T , whereonly an initial part of the optimal control of length τ isthen implemented in the plant. Then, after some time,the state of the plant is measured (or observed) and theprocess is repeated. Thus, on the one hand, a nonlinearfeedback law is obtained, and on the other hand, everyevaluation of the feedback law involves the solution ofan OCP. While the former leads to robustness, the latterraises hope that the generated trajectory is in some wayoptimal on the indefinite or infinite horizon. This propertycan rigorously be proven, if the dynamics satisfy a so-calledturnpike property, depicted in Figure 1, meaning that thesolution to the dynamic OCP is close to the solution ofthe corresponding steady state property for the majority ofthe time, cf. Gru¨ne (2016). In-depth mathematical analysisof MPC-schemes can be found in e.g. Gru¨ne and Pannek(2016); Rawlings et al. (2017). In view of MPC, we willstudy two important properties of infinite dimensionaloptimal control problems in a very general manner.? This work was supported by the DFG Grants GR 1569/17-1 andSCHI 1379/5-1.t = Tt = 0 exact solutionnumerical solutionτImplemented in MPC-loopFig. 1. Turnpike behavior of the OCP’s solution, MPC-feedback and depiction of a numerical solution being(only) accurate on the initial part [0, τ ].One the one hand, we will give sufficient conditionsfor the turnpike property, being important to establish(quasi)optimality of the generated closed-loop trajectory.On the other hand, we aim to show that the OCP occur-ring in every MPC-loop can be solved efficiently by goaloriented adaptive gridding methods. To this end, we willshow that the initial part of the optimal trajectory, i.e. theMPC-feedback, is only slightly influenced by perturbationshappening in the far future. This property is sketched inFigure 1 and allows to use grids which are fine on a smallinitial part [0, τ ] and coarsen up towards T , leading toa very efficient discretization for MPC, if τ  T . Thisrigorously explains the behavior of goal oriented error es-timation techniques which mostly only refine the time andspace grid on the initial part [0, τ ] when aiming to reducethe MPC-feedback error. We note that this property is notclear a priori, as the optimization involves a backwards-in-time adjoint equation, which could propagate perturba-tions from close to the end time towards the initial part.In this extended abstract we will recall some of the resultsobtained in Gru¨ne et al. (2019a) and Gru¨ne et al. (2019b)considering the linear quadratic setting and accompanythem with novel results concerning nonlinear dynamicsand numerical examples.2. LINEAR ABSTRACT SCALING THEOREMSWe will first consider the case of a linear quadratic optimalcontrol problem:min12∫ T0‖C(x− xd)‖2Y + ‖R(u− ud)‖2Us.t. x′ = Ax+Bu+ f, x(0) = x0.(1)We assume that A generates a C0-semigroup on a Hilbertspace X with norm denoted by ‖ · ‖, x0 ∈ X, f ∈L1(0, T ;X) and C ∈ L(X,Y ), B ∈ L(U,X), R ∈ L(U,U)with ‖Ru‖U ≥ γ‖u‖2 for γ > 0, where U and Y areHilbert spaces. We will discuss possible generalizations ofthis setting at the end of this section in Remark 1.Formally, after elimination of the control via u =(R∗R)−1B∗λ + ud, the first order optimality conditionsread C∗C − ddt−A∗0 ETddt−A −BQ−1B∗E0 0︸ ︷︷ ︸=:M(xλ)= C∗Cxd0Bud + fx0︸ ︷︷ ︸=:l,where Etx = x(t) is a time evaluation operator. We willbase our analysis on the couple (x, λ) solving this equation.Perturbations decay exponentially. Consider a per-turbed solution, solvingM(x˜λ˜)= l + ε,u = (R∗R)−1B∗λ˜+ ud where ε is a perturbation boundedby ‖e−µtε(t)‖L1(0,T ;X) ≤ ρ with ρ > 0. This perturbationresembles the residual in function space of a discretesolution, i.e. it reflects the local discretization error; if ε islarge at a time instance, this means that the grid is coarseand vice versa. The exponential scaling in the L1(0, T ;X)norm allows for exponentially increasing perturbations εwhich model grid coarsening exponentially towards T .Theorem 1. Let (A,B) be exp. stabilizable and (A,C) beexp. detectable. Then there is c, µ > 0 independent of Tsuch that‖e−µtx(t)− x˜(t)u(t)− u˜(t)λ(t)− λ˜(t) ‖ ≤ cρ for all t ∈ [0, T ].The interpretation of this theorem is as follows: Considerperturbations which e.g. increase exponentially in timewith rate less than µ, i.e. the estimate ‖e−µtε(t)‖L2(0,T ;X) ≤ρ is satisfied. Then, the effect of this exponentially increas-ing perturbation on the optimal triple is of local nature,i.e. for t small, x(t)− x˜(t) is small as the scaling term e−µtapproaches one. This means, that perturbations happeningin the far future have a negligible effect on the MPC-feedback if τ  T .An exponential turnpike property. Under the verysame stabilizability assumptions, an exponential turnpikeproperty can be derived. To this end, to ensure theexistence of optimal steady states, we assume xd, ud and fto be independent of time. We denote by (x¯, u¯) the optimalsteady state solving(C∗C −A∗−A −BQ−1B∗)(x¯λ¯)=(C∗CxdBud + f),and u¯ = (R∗R)−1B∗λ¯+ ud.Theorem 2. Let (A,B) be exp. stabilizable and (A,C) beexp. detectable. Then there is c, µ > 0 independently of Tsuch that‖x(t)− x¯u(t)− u¯λ(t)− λ¯ ‖ ≤ c(eµt + e−µ(T−t)) for all t ∈ [0, T ].Remark 1. The results from Theorem 1 and 2 can berefined by assuming more structure, i.e. A generating ananalytic semigroup, cf. Gru¨ne et al. (2019a). In this case,maximal parabolic regularity can be used to reduce thespatial regularity assumptions on the perturbations andto obtain estimates in Sobolev norms. Second, using aslightly stronger notion of stability, Theorem 1 can beextended to non-autonomous equations, where A = A(t).Third, boundedness of control and observation operatorsB and C can be replaced by a weaker property, namelyadmissibility, cf. (Gru¨ne et al., 2019b, Section 5). Finally,the initial condition on x can be accompanied by aterminal condition x(T ) = xT for xT ∈ X, if (A,B) isexactly controllable, cf. (Gru¨ne et al., 2019b, Section 7).3. NONLINEAR DYNAMICSFirst, we give an alternative interpretation of Theorem 1and 2: Under stabilizability and detectability assumptions,the operator M is invertible in scaled spaces, and theconstants involved are independent of the time horizon. Wewill use this property to generalize the theory to nonlinearparabolic equations. In order to keep the presentation sim-ple, we will introduce nonlinearity to the OCP via addinga monotone decreasing nonlinearity f(x), e.g. f(x) = −x3,leading to the following dynamicsx′ = Ax+ f(x) +Bu x(0) = x0.We will study (1) with these nonlinear dynamics. Derivingfirst order optimality conditions leads, again after elimina-tion of the control to the nonlinear system of equations atthe optimal pair z := (x, λ)C∗Cx− ddtλ−A∗λ− f ′(x)λETλddtx−Ax− f(x)−BQ−1B∗λE0x︸ ︷︷ ︸=:M(z)=C∗Cxd0Budx0︸ ︷︷ ︸=:l. (2)We will now apply an implicit function theorem to thenonlinear functionG(x, ε) := M(z)−l−ε. First, we observethat z solving (2) satisfies G(z, 0) = 0. Analogously tothe linear case, we define a perturbed variable z˜ := (x˜, λ˜)solving G(z˜, ε) = 0. The question on how the differencez˜−z depends on ε is now answered by the implicit functiontheorem, cf. (Zeidler, 1986, Theorem 4.B) in exponentiallyscaled spaces with weight e−µt or 1e−µt+e−µ(T−t) assumingstabilizability and detectability of the linearized equations.In the case of a turnpike property, the linearization point isthe optimal steady state. However, to bound the influenceof exponential perturbations, we would like to linearizearound the exact optimal solution. As a consequence,the equation is intrinsically non-autonomous as A′(x(t))implicitly depends on time. To this end, we assume a differ-ent stability property which also covers non-autonomoussystems, introduced in Gru¨ne et al. (2019a) under thename V -exponential stability. In a nutshell, the stabilityassumption for the semigroup generated by (A+BK) withfeedback operator K occurring in the definition of expo-nential stabilizability of (A,B) is replaced by a coercivityassumption on −(A+ BK) in a Sobolev space. All in all,Theorem 1 can be generalized to the nonlinear case asfollows.Theorem 3. Let (A+ f ′(x), B), (A+ f ′(x), C) be stabiliz-able in the sense of (Gru¨ne et al., 2019a, Definition 3.6).Moreover, assume ‖e−µtε‖L2(0,T ;H−1(Ω)) ≤ ρ with ρ > 0sufficiently small, where Ω is the spatial domain. Thenthere is µ, c > 0 independent of T such that‖e−µtx(t)− x˜(t)u(t)− u˜(t)λ(t)− λ˜(t) ‖ ≤ cρ ∀t ∈ [0, T ].The analogous generalization of the turnpike propertystated in Theorem 2 reads:Theorem 4. Let (A′(x) + f ′(x), B), (A′(x) + f ′(x), C) bestabilizable in the sense of (Gru¨ne et al., 2019a, Definition3.6). Moreover, assume ‖ (λ¯, x0 − x¯) ‖ ≤ ρ with ρ > 0sufficiently small. Then there is µ, c > 0 independent of Tsuch that‖x(t)− x¯u(t)− u¯λ(t)− λ¯ ‖ ≤ cρ(e−µt + e−µ(T−t)) ∀t ∈ [0, T ].4. NUMERICAL RESULTSThe theoretical results of Theorem 1 and 3 justify the useof fine grids on the implementation horizon [0, τ ] whichcoarsen up towards T for the solution of the underlyingoptimal control problems for the evaluation of the MPC-feedback law. This could be done a priori, with the down-side that only qualitative results are at hand which yieldno quantitative information needed for an implementation.For this reason we will use goal oriented a posteriori errorestimation, cf. Meidner and Vexler (2007). This not onlyallows to estimate the error quantitatively, but also toestimate the error in an arbitrary measure, the so calledquantity of interest (QOI). The natural quantity of interestfor error estimation in an MPC-context is a truncatedversion of the cost function, i.e. to define the QOI viaI(x, u) :=12∫ τ0‖C(x− xd)‖2Y + ‖R(u− ud)‖2U . (3)The aim of goal oriented error estimation is now toadaptively construct time and space grids, such that thenumerical solution of the OCP on these grids has a smallerror in the QOI, i.e. |I(x, u) − I(x˜, u˜)| being small. Thequestion we are interested in is now the following: Howdo the grids, being specialized for this QOI look like?Theorem 1 and 3 suggest, that in order to have a high-quality MPC-feedback, i.e. to be accurate on the beginningof the horizon, the grids generated by the goal orientederror estimation are fine close to the beginning of thehorizon, but not towards T . We will illustrate this propertyat the example of optimal control with a 2d linear heatequation on the unit square and time interval [0, 9].min12∫ T0‖x(t)− xref‖2L2(Ω) + α‖u(t)‖2L2(Ω) dt,x˙ = d∆x+ sx+ u, x(0) = 0(4)with homogeneous Dirichlet boundary conditions, xref is atime independent reference, d > 0 is the heat conductivityand s ∈ R an instability parameter.Time refinement. We first consider adaptive time re-finement for the optimal control problem (4), where weset α = 10−1, s = 5 and τ = 0.5.01234‖x(t)‖L2(Ω)Norms of state and control over time(x, u)(xI , uI)(xJ , uJ)0 2 4 6 800.511.52time t√ α‖u(t)‖L2(Ω)10−610−410−2Error indicators over time (J above, I below)0 2 4 6 810−410−2time tFig. 2. Above: Norm of optimal state and control over time.The reference trajectory was computed on a very finetime grid. Below: Error indicators (blue) and gridpoints on time grid (vertical black lines) over time.Indicators for the cost function J(x, u) in the upperand indicators for truncated cost function I(x, u) inthe lower plot respectively.In the upper picture of Figure 2, the optimal state and con-trol on a very fine grid denoted by (x, u), and the optimalpair on the two adaptively refined grids is depicted, whereone refinement is w.r.t. J(x, u) with the solution denotedby (xJ , uJ) and the other for I(x, u), with the solutiondenoted by (xI , uI). We refined the grid until the errorindicators on every time grid interval for the respectiveobjective are below 10−2. We observe the turnpike prop-erty stated in Theorem 2 for all depicted solutions. Thepairs (x, u) and (xJ , uJ) coincide on the whole horizon,whereas (xI , uI) does not exhibit the so called leaving arcat the end of the horizon. The reason for this becomesclear in the lower picture of Figure 2: As I(x, u) onlyinvolves the interval [0, τ ], the grid for (xI , uI) is verycoarse towards T , rendering it impossible to reflect theleaving arc. On the other hand, the grid for (xJ , uJ) is fineat the beginning and at the end of the interval, allowing thesolution to be accurate on the whole interval. However, thisshows, that in order to be accurate on the whole horizon,a remarkably higher amount of time grid point is neededas opposed to being accurate on [0, τ ]. Figure 3 depicts theperformance of using a truncated cost function instead ofthe full cost function for adaptivity in an MPC-loop. Tothis end, we fixed the maximal number of time points forthe refinement procedure. Then, in every MPC iteration,the optimal control problem was solved and adaptive gridrefinement was performed until the maximal number ofgrid points was reached. It can be seen in Figure 3 thatthe closed-loop cost is significantly lower when refining thegrid for the objective I(x, u) as opposed to J(x, u). Thetotal number of MPC iterations performed in this examplewas four.5 8 11 21 31 41100101102103104# time grid pointsCostfunctionalvalueMPC-closed loop costrefined for I(x, u)refined for J(x, u)referenceFig. 3. Comparison of MPC-closed loop cost for refinementvia I(x, u) and J(x, u).Space refinement. Secondly, we will consider spatialrefinement. In this case, we set α = 10−4 and s = 0. Thegrids on every time grid point are refined independently ofeach other. We again first compare the number of degreesof freedom needed to render the indicators for the objec-tives I(x, u) or J(x, u) on every cell below 10−2. In theupper plot of Figure 4 we observe, that the required spatialdegrees of freedom in every time step are significantlyhigher when refining for J(x, u) as opposed to refiningfor I(x, u). This again reflects the theoretical findings ofTheorem 1: In order to have an accurate solution on [0, τ ],the grid only needs to be fine close to [0, τ ]. In the lowerpart of Figure 4, we show how again the cost of the MPC-closed loop trajectory is lower when using I(x, u) as a QOIfor error estimation. We again fix the number of maximaltotal spatial degrees of freedom and in every MPC-loopsolve the OCP with adaptive refinement until the maximalnumber of spatial degrees of freedom is reached.0 2 4 6 850100200time tspatialDOFsSpatial dofs over time after refinementrefined for I(x, u)refined for J(x, u)572 962 1443 1924 2405 2886 3367 38480.10.5123# total space DOFsCostfunctionalvalueMPC-closed loop costrefined for I(x, u)refined for J(x, u)referenceFig. 4. Above: Spatial degrees of freedom over time afterrefinement. The vertical line indicates τ = 0.5. Below:Comparison of MPC-closed loop cost for refinementvia I(x, u) and J(x, u).REFERENCESCamacho, E.F. and Bordons, C. (2007). Model PredictiveControl. Springer-Verlag London, 2nd edition.Gru¨ne, L., Schaller, M., and Schiela, A. (2019a). Sensitiv-ity analysis of optimal control for a class of parabolicPDEs motivated by model predictive control. SIAMJournal on Control and Optimization, 57(4), 2753–2774.doi:10.1137/18M1223083.Gru¨ne, L. (2016). Approximation properties of reced-ing horizon optimal control. Jahresbericht DeutscheMathematiker-Vereinigung, 118(1), 3–37.Gru¨ne, L. and Pannek, J. (2016). Nonlinear Model Pre-dictive Control: Theory and Algorithms. Springer.Gru¨ne, L., Schaller, M., and Schiela, A. (2019b). Exponen-tial sensitivity and turnpike analysis for linear quadraticoptimal control of general evolution equations. Journalof Differential Equations. Available online.Meidner, D. and Vexler, B. (2007). Adaptive space-time finite element methods for parabolic optimizationproblems. SIAM Journal on Control and Optimization,46(1), 116–142.Qin, S.J. and Badgwell, T.A. (2003). A survey of industrialmodel predictive control technology. Control engineer-ing practice, 11(7), 733–764.Rawlings, J.B., Mayne, D.Q., and Diehl, M. (2017). Modelpredictive control: theory, computation, and design, vol-ume 2. Nob Hill Publishing Madison, WI.Zeidler, E. (1986). Nonlinear Functional Analysis and itsApplications- I: Fixed-Point Theorems. Springer-VerlagNew York.

Exponential sensitivity analysis for Model Predictive Control of PDEs

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Exponential sensitivity analysis for Model Predictive Control of PDEs

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