Grüne, Lars

Schaller, Manuel

Schiela, Anton

EPub Bayreuth

Sensitivity Analysis of Optimal Control Motivated by Model PredictiveControl∗Lars Gru¨ne1, Manuel Schaller1 and Anton Schiela1Abstract— We analyze the sensitivity of the extremal equa-tions that arise when concluding first order optimality condi-tions for time dependent optimization problems. More specif-ically, we consider parabolic PDEs with a linear quadraticperformance criterion. We prove the solutions boundednesswith respect to the right-hand side of the first order optimalitycondition which includes initial data. As a consequence, it canbe shown that the influence of a perturbation at at certaintime decays exponentially in the temporal distance to the timeof perturbation. Moreover, a quantitative turnpike theorem canbe derived.Index Terms— Sensitivity Analysis, Turnpike Property,Model Predictive ControlAMS MSC 2000: 49K20, 49K40, 93D20I. INTRODUCTIONModel Predictive Control (MPC) is a control method inwhich the solution of optimal control problems on infiniteor indefinitely long horizons is split up into the successivesolution of optimal control problems (OCPs) on finite timehorizons. Only a first part with given length of each finitehorizon solution is implemented as a control for the longer,possibly infinite horizon. This means that in a numericalsolution of the optimal control problem only the first partof the optimal control must be computed accurately.Motivated by this observation, we will study the effect ofperturbations and discretization errors near the end of theoptimization horizon on the initial part of the control. It willturn out that, considering linear quadratic optimal controlproblems, their influence decays exponentially in time. Thus,they are indeed negligible if the horizon is long enough.An in depth introduction to Model Predictive Control canbe found in [1]. A central assumption for the approximationof the infinite horizon solution by a solution controlled withan MPC-feedback is that the dynamics of the optimal controlproblem exhibits turnpike behavior, i.e. the time dependentsolution of the optimal control problem remains close to aconstant value for the majority of the time [2]. This valueis called the turnpike of the OCP. A quantitative turnpikeresult was proven in [3] for nonlinear finite dimensionalproblems. This was extended to a general Hilbert spacesetting in [4], where the assumptions made to show turnpikebehavior are stabilizability and detectability of the dynamics.The interested reader is also referred to [5], [6].*This work was supported by the DFG Grants GR 1569/17-1 and SCHI1379/5-11 Chair of Applied Mathematics, Department of Mathematics, Universityof Bayreuth, 95447 Bayreuth, Germany.lars.gruene, manuel.schaller andanton.schiela@uni-bayreuth.deAs depicted in Fig. 1, one might conjecture that forsystems showing turnpike behaviour, perturbations at the endof the trajectory do not affect the initial part if T is large.t = Tt = 0εexact solutionperturbed solutionturnpikeFig. 1. Turnpike behavior of the OCP’s solution and possible perturbationε at the end of the optimization horizon.In this paper, we will show that this property can be provenrigorously for linear-quadratic optimal control problems withstabilizable and detectable dynamics.II. PRELIMINARIES AND SETTINGLet (V, ‖ · ‖V ) be a separable and reflexive Banachspace and [0, T ] a bounded proper interval on R. ByL2(0, T ;V ) we denote the space of Bochner integrablefunctions f :]0, T [→ V , see [7] and [8, Section 23.2ff] forbasic results on these spaces. It holds that L2(0, T ;V )? ∼=L2(0, T ;V?).Let H be a Hilbert space with scalar product 〈·, ·〉H andV ↪→ H continuously and densely. The spaces form a socalled Gelfand triple (or evolution triple) V ↪→ H ↪→ V ?.Moreover, by W [0, T ] we denote the space of L2(0, T ;V )functions with weak derivative in L2(0, T ;V ?).We will consider the parabolic PDE in weak form. Forthis, we define the operatorD[0,T ] : W ([0, T ])→ L2(0, T ;V )? ×H?,where(D[0,T ]w)(v, v0) :=T∫0w′(t)(v(t)) dt+ 〈w(0), v0〉H ,which constitutes a time derivative operator with initialvalue testing. Additionally, we introduce operatorsΛ[0,T ] : L2(0, T ;V ) → L2(0, T ;V ?) and Λ : V → V ?linear, bounded with(Λ[0,T ]v)w =T∫0(Λv)w dt,where Λ could be e.g. the weak form of the Laplaceoperator. For the influence of the control, we define B[0,T ] :L2([0, T ] × Ω) → L2(0, T ;H?), B : L2(Ω) → H? linear,bounded with(B[0,T ]u)v =T∫0(Bu)v dt.For details on the solution theory of parabolic PDEs formu-lated as operator equations, see [9]. In this paper, we willstudy the sensitivity of the first order optimality condition ofthe optimal control problemminy,u12‖C(y − yd)‖2L2(0,T ;H) +α2‖R(u− ud)‖2L2([0,T ]×Ω)(1)subject to the parabolic PDED[0,T ]y − Λ[0,T ]y −B[0,T ]u = 〈y0, ·〉for linear bounded operators C : L2(0, T ;H)→ L2(0, T ;H)and R : L2([0, T ]×Ω)→ L2([0, T ]×Ω) additionally elliptic.If we denote the Riesz isomorphisms of the state space byMy : L2(0, T ;H)→ L2(0, T ;H)?, the first order optimalitycondition is given by the following equation in dual spacesM(y(λ, λ0))=(0〈y0, ·〉)+(C?MyCydB[0,T ]ud), (2)whereM :=(C?MyC (D[0,T ] − Λ[0,T ])?(D[0,T ] − Λ[0,T ]) −B[0,T ]Q−1B?[0,T ]). (3)Here we eliminated the control via u = Q−1B?[0,T ]λ + ud,where Q := αR?MuR and Mu : L2([0, T ] × Ω) →L2([0, T ] × Ω)? is the Riesz isomorphism of the controlspace.III. ANALYSIS OF THE EXTREMAL EQUATIONS’SOLUTION OPERATOROur main analysis consists of proving the bounded invert-ibility of the linear operator M defined in (3) with constantsindependent of T . The approach taken is inspired by thestability estimate performed in [4, Lemma 2] and [5, Lemma3.5]. The main result in this section reads as follows.Theorem 1: Let (Λ, C) satisfy an exponential detectabilityand (Λ, B) satisfy an exponential stabilizability assumption.Then there exists a constant c ≥ 0 independent of T , suchthat‖M−1‖(L2(0,T ;H?)×H?)2→L2(0,T ;H)2 ≤ c. (4)The constant c is indirectly proportional to the grade ofexponential stabilizability and exponential detectability. Theassumptions of this theorem are similar to standard assump-tions under which turnpike behavior can be shown in infinitedimension, cf. [4].One will observe that the range of the adjoint equation wasrestricted. Originally, the equation for the adjoint variableis posed in W [0, T ]?. However, this results in an adjointstate tuple (λ, λ0) ∈ L2(0, T ;V ) × H . By restricting therange of the adjoint operator D?[0,T ] to L2(0, T ;H?) ×H?,one can show that λ ∈ W ([0, T ]) and λ0 = λ(0) (see [9,Proposition 3.8]). As a result, we can treat the variables y andλ in the same way. In the remainder of this paper, the twovariables (λ, λ0) will therefore be denoted by one variableλ, as λ0 = λ(0).IV. SENSITIVITY WITH RESPECT TO L2 - PERTURBATIONSA. Theoretical considerationsAn application of the above analysis is the property thatthe effect of perturbations decays exponentially in time.In the following we consider sensitivity of solutions of(2) with respect to perturbations of the right hand side. Inaddition to the exact solution (y, λ) of (2) we consider aperturbed solution (y˜, λ˜) that solvesM(y˜λ˜)=(0〈y0, ·〉)+(C?MyCydB[0,T ]ud)+(ε1ε2).The perturbation variables ε1 and ε2 model the error per-formed by e.g. a discretization in time or space. Hence,defining the difference of exact and perturbed solution(δy, δλ) := (y˜ − y, λ˜− λ) it follows using linearity thatM(δyδλ)=(ε1ε2). (5)Theorem 1 directly yields an estimate of (δy, δλ) in terms of‖(ε1, ε2)‖L2(0,T ;H?)2 . In the following result, we will refinethis estimate by introducing an exponential scaling in time.Theorem 2: Assume (δy, δλ) ∈W ([0, T ])2 solves (5) andlet the assumptions of Theorem 1 hold.Furthermore assume that for ρ ≥ 0 and 0 ≤ µ < 1‖M−1‖‖e−µtε1(t)‖L2(0,T ;H?) ≤ ρ‖e−µtε2(t)‖L2(0,T ;H?) ≤ ρ.(6)Then there exists a constant c1 ≥ 0 independent of T , suchthat‖e−µtδy‖Y + ‖e−µtδu‖U + ‖e−µtδλ‖P ≤ c1ρ,where Y = P = L2(0, T ;H) and U = L2([0, T ]× Ω).For µ > 0, all inequalities involve weighted norms, wherethe functions are scaled by e−µt. We interpret this result forthe error in the state δy = y˜ − y: Assume we solve for thestate and adjoint with perturbations that are bounded by (6).Then, ‖e−µtδy‖ is bounded by a constant independent of T .This implies that at the beginning of the time horizon, i.e.for small t, ‖δy(t)‖ ' ρ, where ρ is the scaling factor ofthe perturbations bound in (6). Therefore, only local (in atemporal sense) perturbations play a role and perturbationse.g. at the end of the horizon only slightly affect the initialpart, even if they grow exponentially in time.The proof of Theorem 2 utilizes scaling of the variablesby e−µt and the identity(D[0,T ]δy)e−µtv = ((µI +D[0,T ])e−µtδy)v.This implies that the scaled variables satisfy a modifiedequation with M replaced by M + µP , whereP =(0 −II 0).Since M + µP = (I + µPM−1)M , a Neumann-seriesargument implies invertibility of this operator for sufficientlysmall µ, depending only on ‖M−1‖. The smaller ‖M−1‖,the larger µ can be chosen, the faster the exponential decay.B. Numerical ResultsIn this subsection, the statement of Theorem 2 will beillustrated by a numerical example.We take a look at solutions of the optimization problemmin(y,u)12‖(y − yd)‖2L2([0,T ]×[0,1]2) +14‖u‖2L2([0,T ]×[0,1]2)subject to the constraintD[0,T ]y + 0.1T∫0∫[0,1]2∇y∇ · dωdt−T∫0∫[0,1]2u · dωdt = 0for different temporal grids. We choose V = H10 (Ω) forthe adjoint and state space and H = L2(Ω) for the initialvalue. The constraint constitutes a linear heat equation withdistributed control and zero initial condition and it obviouslysatisfies the stabilizability and detectability assumption ofTheorem 1, as ∇ ·∇· is elliptic in H10 (Ω) and hence stable.yd ∈ H10 (Ω) is a reference temperature which is depicted inFig. 2.Fig. 2. Plot of the constant (w.r.t. time) reference temperature yd on [0, 1]2.We compute two perturbed solutions, the first triple with auniform grid Guni consisting of 11 grid points and the secondwith a grid Gexp with meshsize increasing exponentially intime as depicted in Fig. 3.To compute the absolute error ‖δy(t)‖L2(Ω), ‖δu(t)‖L2(Ω)and ‖δλ(t)‖L2(Ω) as functions of the time t, we will considera solution triple (y, u, λ) on a fine grid with 101 grid points in[0, 30] as a reference. Fig. 4 shows the norm of the perturbedand exact solution triple’s state, control and adjoint. Notethat, as stated in the introduction, Model Predictive Controlonly implements an initial part of the trajectory and the0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30time tGuniGexpFig. 3. Grids used for the computations.00.050.10.15exact solutionexponential griduniform grid00.10.20 5 10 15 20 25 300.512time tFig. 4. From top to bottom: Norm of the state ‖y(t)‖L2(Ω), norm of thecontrol ‖u(t)‖L2(Ω) and norm of the adjoint ‖λ(t)‖L2(Ω) for every gridpoint.remaining part is discarded. Therefore, we are especiallyinterested in this initial part. Additionally, we observe, thatthe exact solution stays close to a certain quantity. Thisfeature is called turnpike property and will be discussed inSec. V. Moreover, we note the accuracy of the solution onGexp in the first part of the horizon, even though it does notfollow the so called leaving arc close to the end time.The absolute error is illustrated in Fig 5. We observe thatdue to the fine meshsize of the exponential grid Gexp at thebeginning of the time interval (9 of 11 grid points are usedin [0, 5]), the error is lower by several orders of magnitudein comparison to the uniform grid Guni. For both solutions,we observe an increase of the error when approaching timezero and time T = 30. This stems from the faster dynamics10−610−3100exponential griduniform grid10−710−410−10 5 10 15 20 25 3010−810−410−1time tFig. 5. Absolute error of the perturbed solutions. From top to bottom:‖δy(t)‖L2(Ω), ‖δu(t)‖L2(Ω) and ‖δλ(t)‖L2(Ω).of the system in this part.These results motivate the construction of algorithms foradaptive grid generation which is subject to future research.In this example, only the time grid was altered. The spatialgrid was kept constant over all timesteps. However, to fullyexploit the results of Theorem 2, the spatial grids can followthe same refinement pattern, leading to fine spatial grids onthe initial part with meshsize increasing in time. This willlead to a fast solution of the problems arising in applicationsof Model Predictive Control for PDEs.V. A QUANTITATIVE TURNPIKE THEOREMA second application of the results presented in the firstpart of this paper is a turnpike result for infinite dimensionallinear quadratic optimization problems. The proof in the ref-erences put forward in the Introduction includes the solutionof the Algebraic Riccati Equation. We present a differentapproach, using only the assumptions of Theorem 1.We introduce the solution of a steady state optimizationproblem, namely (y¯, u¯, λ¯) that solves the operator equation(C?C −Λ?−Λ −BQ−1B?)(y¯λ¯)=(C?CydBud),where Q = αR?R and u¯ = Q−1B?λ¯ + ud. This resem-bles the first order optimality condition of the optimizationproblemminy¯,u¯12‖C(y¯ − yd)‖2H +α2‖R(u¯− ud)‖2L2(Ω)s.t. Λy¯ −Bu¯ = 0.Similar to the perturbation result presented in the previoussection, we consider the distance of the solution of (2) tothis variable: (δy, δλ) := (y − y¯, λ − λ¯). A straightforwardcomputation shows that this variable solvesM(δyδλ)=( −〈λ¯, ·〉〈y0 − y¯, ·〉). (7)We conclude a result similar to Theorem 2.Theorem 3: Assume (δy, δλ) solves (7) and let the as-sumptions of Theorem 1 hold. Then, there exists µ > 0 andc2 ≥ 0 independent of T , such that‖ 1e−µt + e−µ(T−t)δy‖Y + ‖ 1e−µt + e−µ(T−t)δu‖U+‖ 1e−µt + e−µ(T−t)δλ‖P ≤ c2(‖y0 − y¯‖H + ‖λ¯‖H),(8)where Y = P = L2(0, T ;H) and U = L2([0, T ]× Ω).The scaling factor’s denominator is near zero for times t inthe middle of the optimization horizon. Therefore, to fulfillinequality (8), the solution triple (y, u, λ) has to be closeto the solution (y¯, u¯, λ¯) of the static problem in the middlepart of the optimization horizon. As the constants involvedare independent of T , this resembles a turnpike property.Fig. 4 shows this effect for the three variables. Althoughthe variables are scaled by a factor different from that inTheorem 2, the proof is very similar.REFERENCES[1] L. Gru¨ne and J. Pannek, Nonlinear Model Predictive Control: Theoryand Algorithms. Springer, 2016.[2] L. Gru¨ne, “Approximation properties of receding horizon optimalcontrol,” Jahresbericht Deutsche Mathematiker-Vereinigung, vol. 118,no. 1, pp. 3–37, 2016.[3] E. Tre´lat and E. Zuazua, “The turnpike property in finite-dimensionalnonlinear optimal control,” Journal of Differential Equations, vol. 258,no. 1, pp. 81–114, 2015.[4] E. Tre´lat, C. Zhang, and E. Zuazua, “Steady-state and periodic exponen-tial turnpike property for optimal control problems in hilbert spaces,”2016, 0.[5] A. Porretta and E. Zuazua, “Long time versus steady stateoptimal control,” SIAM Journal on Control and Optimization,vol. 51, no. 6, pp. 4242–4273, 2013. [Online]. Available:”http://dx.doi.org/10.1137/130907239”[6] ——, “Remarks on long time versus steady state optimal control,”Springer INdAM Series, p. 6789, 2016. [Online]. Available:”http://dx.doi.org/10.1007/978-3-319-39092-5 5”[7] H. Gajewski, K. Gro¨ger, and K. Zacharias, NichtlineareOperatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag Berlin, 1974, vol. 67, no. 22. [Online]. Available:http://dx.doi.org/10.1002/mana.19750672207[8] E. Zeidler, Nonlinear Functional Analysis and its Applications- II/A:Linear Monotone Operators. Springer-Verlag New York, 1990.[9] A. Schiela, “A concise proof for existence and uniqueness of solutionsof linear parabolic PDEs in the context of optimal control,” System &Control Letters, vol. 62, no. 10, pp. 895–901, 2013.

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Sensitivity analysis of optimal control motivated by model predictive control

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