Numerical Results

In this section we apply the theoretical results presented before to a simple exam-ple. This section is divided into two parts: In the first one we follow our road map and compute concrete values of the minimal stabilizing horizon for our test example.

Afterwards, we compare these quantitative results with the minimal stabilizing hori-zon we observed in a numerical simulation of the closed loop system. In the second part we use our theoretical results for a qualitative analysis in order to describe the dependence of the stabilizing horizon on different parameters and stage costs.

Our test example in this section is given by the linear one dimensional heat equation yt(x, t) =yxx(x, t) +µy(x, t) +u(x, t) in (0, L)×(0,∞) (3.12a)

y(0, t) =y(L, t) = 0 in (0,∞) (3.12b)

y(x,0) =y₀(x) in (0, L) (3.12c)

with the corresponding stage cost l(y(n), u(n)) = 1

2ky(·, nT)k²L²(0,L)+ λ

2ku(·, nT)k²L²(0,L) (3.13) and

l(y(n), u(n)) = 1

2ky_x(·, nT)k²L²(0,L)+λ

2ku(·, nT)k²L²(0,L) (3.14) respectively. The parameters in this example are given by the reaction valueµ, the interval length L, the regularization parameter λ and the sampling time T.

Quantitative analysis

In Section 3.1.1 we derived the exponential constants C and σ for a prescribed control u. Thus, two steps of our road map are already done. In the last step we use formula (1.35) in order to determine the smallest horizon that guarantees stability. The procedure is as follows: We insert the K-dependent constants σ(K) andC(K) in (1.35) and obtainαN(K) =αN(C(K), σ(K), N). If there existsK ∈R such thatαN(K)>0 Theorem 1.20 guarantees stability with horizonN. Therefore, the minimal stabilizing horizon is the smallest horizon ¯N with maxKαN¯(K) > 0.

Note that the optimization over K is an easy one dimensional problem. In the implementation for the example we use Maple to solve this problem.

Furthermore, we determine the smallest horizon where we observe stability in the numerical simulation of the MPC closed loop system. The arising control problems are solved with the optimization algorithms presented in Chapter 4.

First, we look at the 1D heat equation (3.12) with stage cost (3.13), domain Ω = (0,1) and sampling time T = 0.01. In Table 3.1 we display the minimal stabilizing horizonNT computed by the results of Section 3.1.1 and the horizon observed in the numerical simulation NP. Obviously, the values of NT are always equal or greater than those ofNP. Thus, the theoretical results provide indeed upper bounds for the

3.1 Distributed Control λ= 0.001 λ = 0.005 λ= 0.01

µ N_T N_P N_T N_P N_T N_P

10 2 2 2 2 2 2

11 2 2 4 2 9 3

12 2 2 8 3 18 4

13 2 2 12 3 27 4

14 3 2 16 4 34 5

15 3 2 20 4 42 6

Table 3.1: Minimal stabilizing horizon computed from theory NT and observed in the numerical simulation N_P

stabilizing horizons. However, it can be seen that the computed horizons are quite conservative for this example. This observation is true regardless of the parameter setting and the type of stage cost (but much less for small values of λ andµ). This behaviour is caused by different reasons: First, we have to note that the condition αN > 0 is only sufficient to guarantee stability. For αN ≤ 0 no statement about stability of a concrete example is possible (see also Remark 1.21). Furthermore, Theorem 1.20 only requires the values of the exponential constants C and σ but no information about the underlying dynamical system. It seems to be reasonable that tighter estimates can be obtained by taking the special structure of the PDE and the stage cost into account (see Section 3.6). Finally, it is important to note that the chosen control is not the optimal one and this fact can also produce conservatism.

It should be mentioned that the described problems arise from the power of the used method: It can be applied to very general systems and the knowledge of an optimal control is not required. Since the conservatism of the results is also present for boundary control and different stage cost, in the remaining chapter we will focus on a qualitative analysis.

Remark 3.7

It should be mentioned that the derived exponential constants C and σ can be used to gain improved stability results. In [9] we reduce the conservatism of the estimates by replacing the controllability condition (1.35) by a boundedness condition on the finite horizon optimal value functions. For details we refer to [99].

Qualitative analysis

In the quantitative analysis we have seen that the theoretically computed horizons are in general quite conservative for a concrete example. Nevertheless, in the follow-ing we will show that the presented methods are very well suited for a qualitative analysis concerning the stability behaviour depending on parameters and different stage costs.

In the first step we want to analyse the impact of the different stage costs (3.2)

3 Minimal Stabilizing Horizons

and (3.9) on the minimal stabilizing horizon. Obviously, we obtain the same de-cay rate σ = e^{−2T(λ1−M+K)} for both costs (see Theorem 3.4 and 3.6). Thus, we can only explain the different behaviour through the overshoot constant C. The eigenvalue λ1 influences not only the decay rate but also the overshoot bound in (3.9). It is important to note that λ1 solely depends on the domain Ω. Thus, in order to investigate the difference between the stage costs in detail it is reasonable to look at a problem with different domains. In view of our test example this means that we consider (3.12) with varying interval length L. For the numerical simula-tion we choose the destabilizing reacsimula-tion valueµ≡15, the regularization parameter λ= 0.01 and the sampling timeT = 0.01. The smallest eigenvalue for this problem is given by λ1 = (π/L)², cf. [88]. Thus, the origin is an unstable equilibrium for L≥1 (λ1 = (π/L)² <15 =µ).

The overshoot constants are given by C = (1 +λK²) for stage cost (3.2) and by C = (1 + ^λK_λ1²) for (3.9). Since a smaller constant C leads to a shorter horizon we expect to observe shorter horizons for (3.9) than for (3.2) if λ1(L) < 1 and vice versa for λ1(L) > 1. In order to demonstrate that this behaviour in fact occurs, we consider the minimal stabilizing horizon of (3.12) with varying interval length L. The results of the numerical simulation are presented in Table 3.2. The first two

L λ1(L) N_kyk N_k∇yk

1 π² 6 2

2 π²/4 8 5

3 π²/9 8 7

π 1 8 8

4 π²/16 8 10

5 π²/25 8 12

Table 3.2: Minimal stabilizing optimization horizons for the reaction-diffusion equa-tion (3.12) with T = 0.01 and λ = 0.01 determined by numerical MPC closed loop simulations

rows show the varying interval length L with the corresponding eigenvalue λ1. The minimal stabilizing horizon observed in the numerical example is denoted by N_kyk

for stage cost (3.2) and by N_k∇yk for stage cost (3.9). Obviously, the values N_k∇yk

are smaller than N_kyk up to L = π . This behaviour changes to the opposite for L > π. The observation corresponds exactly to the theoretical results presented in Theorem 3.4 and Theorem 3.6: For L < π there is λ1 >1 and the value C for the stage cost (3.9) is smaller than the corresponding C for (3.2). The smaller value of C leads to a shorter horizon (see Figure 1.3). Note that the chosen initial func-tion y0(x) = sin(^π_Lx) is the corresponding eigenfunction to the least eigenvalue and therefore the Friedrichs inequality in the theorems is tight. This explains why the turning point λ1 = 1 is tight. For different initial functions it is possible to obtain shorter horizons for (3.9) even if L > π.

3.2 Introduction to Backstepping