Numerical results

wheree_k denotes thek-th unit vector. The matrix B₀ ∈R^N×N is the so-called background covariance matrix. The construction of this matrix is described in [22].

The minimization problem (5.4.1) is equivalent to the following linear least squares problem

A_lsqδx₀−y_lsq²₂ = min! (5.4.3) with

A_lsq =

2 66 66 66 64

B₀⁻¹² C CA

... CA^r−1

3 77 77 77 75

, y_lsq =

2 66 66 66 64

0_N y₀ y₁ ... y_r−1

3 77 77 77 75

,

where 0_N denotes a zero vector of sizeN. To be able to apply model reduction within this data assimilation problem, we follow [22] and consider the under-lying dynamical system

δx_i+1 = Aδx_i+B₀¹²u_i,

y_i = Cδx_i. (5.4.4)

The order of this system can be reduced by projection Π =V_nW_n^∗: δˆx_i+1 = W_n^∗AV_nδxˆ_i+W_n^∗B₀¹²u_i,

ˆ

y_i = CV_nδˆx_i. (5.4.5)

Then instead of (5.4.3), the smaller least squares problem

Aˆ_lsqδˆx₀−yˆ_lsq²₂ = min!, (5.4.6) with

Aˆ_lsq =

2 66 66 66 64

(W_n^∗B₀W_n)⁻¹² CV_n (CV_n)(W_n^∗AV_n)

...

(CV_n)(W_n^∗AV_n)^r−1

3 77 77 77 75

,yˆ_lsq =

2 66 66 66 64

0_n y₀ y₁ ... y_r−1

3 77 77 77 75

has to be solved. In order to compare the solutionδxˆ₀ ∈Rⁿ with original one, δx₀ ∈R^N, we have to liftδxˆ₀ ∈Rⁿ back to the full space:

liftedδxˆ₀ :=V_nδxˆ₀.

Several different model reduction techniques could be used to construct a pro-jection Π =V_nW_n^∗,such that the reduced-order model (5.4.5) is a good overall approximant of the original system (5.4.4). As we already mentioned, in both

5.4. NUMERICAL RESULTS 105 examples considered in this section the resulting models (5.4.4) are unstable.

Therefore, one can use the concept developed in this thesis and apply one of the existing model reduction techniques to the asymptotically stable system obtained byα-shift of unstable butα-bounded original system. Another possi-bility is to divide the given system into stable and unstable subsystems and then apply reduction only to the stable part. In [22], it has been shown that approximate incremental 4D-Var by means of balanced truncation applied to α-shifted system gives very good results. In what follows, we show that in

0 500 1000 1500

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4

Vector components

Solution

SW1500: Solution Plot δ x₀ liftedδ x

0 BT liftedδ x

0 1.1−BT liftedδ x₀^1.1−MIRIAm

(a) Reduced dimesion=750

0 500 1000 1500

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

Vector components

Solution

SW1500: Solution Plot δ x₀ liftedδ x

0 BT liftedδ x

0 1.1−BT liftedδ x₀^1.1−MIRIAm

(b) Reduced dimension=160

Figure 5.2: The comparison of reduced-order approaches by means of balanced truncation (lifted δˆx^BT₀ ), 1.1-shifted balanced truncation (lifted δˆx^1.1−BT_o ) and MIRIAm method for α = 1.1 (lifted δˆx^1.1−MIRIAm₀ ) with original full-order solutionδx₀ for the SW1500 test model: solution plot for different dimensions of the reduced models.

some cases the quality of approximation is significantly worse when the second approach to model reduction of unstable systems is used. These results shall also be compared with approximate incremental 4D-Var method by means of MIRIAm algorithm.

Figures 5.2-5.4 illustrate the results of the experiments with shallow water model SW1500. The original system is only slightly unstable. The absolute values of all poles lying outside the unit circle are very close to one, specifically they all lie inside the disk D⁻_1.01 := {s ∈ C : |s| < 1.01}. Hence, the system is 1.01-bounded and we can use α ≥ 1.01 to shift the system to asymptoti-cally stable one. We reduced the dimension of the given system (5.4.4) by means of balanced truncation as well as by MIRIAm method. We applied ba-lanced truncation to the original, unstable system as well as to asymptotically stable 1.1-shifted system. We used standard MATLAB commands: balreal and modred in order to perform the reduction. We note that in the case of

200 400 600 800 1000 1200 1400 10⁻¹⁰

10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰

Vector components

Error

SW1500: Error Plot

e^BT=δ x₀−liftedδ x₀^BT e^1.1−BT=δ x₀−liftedδ x₀^1.1−BT e^1.1−MIRIAm=δ x

0−liftedδ x 0 1.1−MIRIAm

Figure 5.3: The comparison of reduced-order approaches by means of balanced truncation (liftedδxˆ^BT₀ ), 1.1-shifted balanced truncation (liftedδxˆ^1.1−BT_o ) and MIRIAm method for α = 1.1 (lifted δxˆ^1.1−MIRIAm₀ ) with original full-order solutionδx₀ for the SW1500 test model: error plot for reduced dimension=750.

unstable systems, the MATLAB procedure first divides the system into stable and unstable part, and performs balancing only on the stable one. The problem is that the number of unstable poles is quite large, it reaches 400, and therefore reduction to the dimension smaller than 400 means that we simply truncate the whole stable subsystem and a part of unstable one. We will show that applying balanced truncation to 1.1-shifted system gives better results. In case of MIRIAm algorithm alsoα = 1.1 was chosen.

In Figure 5.2 the solutionsδx₀ of the large problem (5.4.3) are plotted against the lifted solutions δˆx^BT₀ , δˆx^1.1−BT₀ and δxˆ^1.1−MIRIAm₀ of the simplified min-imization problem (5.4.6). The results are presented for reduced dimension n = 750 in Figure 5.2(a) and n = 160 in Figure 5.2(b). In the latter case, one can easily observe that the liftedδˆx^BT₀ is not a good approximation of the true solution δx₀. In order to have a better visualization of these results, the corresponding errors,

e^BT := δx₀−liftedδˆx^BT₀ , e^1.1−BT := δx₀−liftedδˆx^1.1−BT₀ , e^1.1−MIRIAm := δx₀−liftedδˆx^1.1−MIRIAm₀ , are plotted in Figures 5.3 and 5.4.

5.4. NUMERICAL RESULTS 107

0 200 400 600 800 1000 1200 1400

10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰

Vector components

Error

SW1500: Error Plot

e^BT=δ x₀−liftedδ x₀^BT e^1.1−BT=δ x₀−liftedδ x₀^1.1−BT e^1.1−MIRIAm=δ x

0−liftedδ x 0 1.1−MIRIAm

Figure 5.4: The comparison of reduced-order approaches by means of balanced truncation (lifted δˆx^BT₀ ), 1.1-shifted balanced truncation (lifted δˆx^1.1−BT_o ) and MIRIAm method for α = 1.1 (lifted δˆx^1.1−MIRIAm₀ ) with original full-order solutionδx₀for the SW1500 test model: error plot for reduced dimension=160.

On the one hand, Figure 5.3 illustrates that the approximate incremental 4D-Var by means of the reduced-order systems of dimension n = 750 is, on average, of the same order of magnitude for all three considered reduction techniques. On the other hand, see Figure 5.4, when we decreased the size of reduced model to 160, the difference between the approximation errors e^BT, e^1.1−BT and e^1.1−MIRIAm is remarkable. We observe that the appro- xi-mation error e^BT of balanced truncation applied to unstable system without shifting it to asymptotically stable one is significantly larger than the appro-ximation errors for the MIRIAm method and α-shifted balanced truncation.

These numerical results support the theory that, in case of unstable but α-bounded systems, α-dependent methods such as h_2,α-optimal model reduc-tion or α-shifted balanced truncation or other model reduction techniques applied toα-shifted systems, should be considered. We made also more experi-ments, reducing the size of the system ton ∈ {160,200,300,400,500,600,750} for α∈1.0,1.01,1.05,1.1 and for each of them we computed the relative error norms

nrm^α−BT(n) := ^{δx0−lifted}_δx^{δˆxα−BT0 2}

02 ,

nrm^α−MIRIAm(n) := ^{δx0−lifted}_δx^{δˆxα−MIRIAm0 2}

02 .

In Table 5.1 we present a comparison of these errors between the solutionsδx₀ of the large least squares problem (5.4.3) and its approximation obtained by

n α= 1.0 α = 1.01 α = 1.05 α = 1.1 160 0.1994 0.0582 0.0457 0.0311 200 0.1144 0.0376 0.0190 0.0147 300 0.0157 0.0132 0.0103 0.0088 400 0.0114 0.0107 0.0083 0.0068 500 0.0086 0.0082 0.0068 0.0056 600 0.0072 0.0067 0.0057 0.0049 750 0.0057 0.0051 0.0044 0.0038

Table 5.1: The comparison of error norms for the reduced-order method by means of α-shifted balanced truncation for α ∈ {1.0,1.01,1.05,1.1} for the SW1500 example.

n α= 1.01 α = 1.05 α= 1.1

160 0.0471−0.1189 0.0367−0.0666 0.0284−0.0725 200 0.0209−0.0524 0.0181−0.0354 0.0137−0.0206 300 0.0113−0.0116 0.0099−0.0102 0.0085−0.0088 400 0.0086−0.0116 0.0078−0.0122 0.0068−0.0090 500 0.0081−0.0089 0.0068−0.0079 0.0056−0.0069 600 0.0063−0.0094 0.0053−0.0092 0.0045−0.0069 750 0.0051−0.0060 0.0043−0.0052 0.0038−0.0045

Table 5.2: The comparison of error norms for the reduced-order method by means of MIRIAm algorithm with different starting data and α∈ {1.01,1.05,1.1} for the SW1500 example.

applying model reduction via balanced truncation for different dimensions of reduced model and α ∈ {1.0,1.01,1.05,1.1}. The result of analogous experi-ments, where MIRIAm method was chosen as reduction technique are summa-rized in Table 5.2. We made experiments with different starting points and tangential directions and consequently we obtained different values for error norms. Therefore the table contains the lowest and the highest values of errors, when 5−20 iterations of MIRIAm algorithm were performed. One can easily observe that the quality of approximation via α-shifted balanced truncation and MIRIAm algorithm is, on average, equally good. From Table 5.1 and 5.2 as well as from experiments withα varying from 1.0 to 25.0, we conclude that the results depend onα. Therefore further research is necessary to investigate this dependence and develop a criterion for a good selection of parameterα.

We turn our attention to the second model considered in this chapter, referred to as SW400, describing the flow of fluid over an obstacle without rotation.

The system matrices of the dynamical system (5.4.4) arising from the model SW400 are of dimension 400 with 400 inputs and 200 outputs. The model

5.4. NUMERICAL RESULTS 109 n α = 1.0 α= 1.01 α= 1.05 α= 1.1

70 0.2327 0.2302 0.2223 0.2159 80 0.1726 0.1724 0.1707 0.1708 90 0.1015 0.1007 0.0985 0.0972 100 0.0623 0.0617 0.0599 0.0567 150 0.0134 0.0135 0.0142 0.0144 200 0.0027 0.0027 0.0025 0.0024

Table 5.3: The comparison of error norms for the reduced-order method by means α-shifted balanced truncation for α ∈ {1.0,1.01,1.05,1.1} for the SW400 example.

n α = 1.01 α= 1.05 α= 1.1

70 0.2310−0.2382 0.2244−0.2327 0.2164−0.2273 80 0.1604−0.1881 0.1594−0.1706 0.1572−0.1772 90 0.1007−0.1142 0.0991−0.1074 0.0984−0.1156 100 0.0627−0.0739 0.0610−0.0724 0.0578−0.0704 150 0.0144−0.0157 0.0142−0.0156 0.0148−0.0172 200 0.0026−0.0027 0.0024−0.0025 0.0024

Table 5.4: The comparison of error norms for the reduced-order method by means of MIRIAm algorithm with different starting data and α∈ {1.01,1.05,1.1} for the SW400 example.

is also unstable, but only very few eigenvalues do not lie inside unit circle.

Therefore, standard approach to unstable systems will give analogous results to those which we obtain via α-related techniques. Tables 5.3 and 5.4 sum-marize these error norms for α ∈ {1.0,1.01,1.05,1.1} and different sizes of the reduced-order models constructed via balanced truncation and MIRIAm method, respectively. We see that the errors obtained using the different re-duction methods, i.e. MIRIAm and balanced truncation, give approximately the same results. For balanced truncation, the quality of approximation de-pends only on the size of reduced-order model and on α. In case of MIRIAm, not only the parameter α but also the starting data influence the convergence of method and the accuracy of solutions. The need for further research is evident.

In Figure 5.5, we present the comparison between solution δx₀ of the original least squares problem (5.4.3) and lifted solutions of the simplified minimiza-tion problem (5.4.6) obtained by means of balanced truncaminimiza-tion, 1.1-shifted balanced truncation and MIRIAm method for α = 1.1 for the SW400 test model. These experiments have been performed for the reduced-order models of dimensions n = 200 (see Figure 5.5(a)) and n = 80 (see Figure 5.5(b)).

Since the error between different low order methods is hard to observe, we

0 50 100 150 200 250 300 350

−6

−4

−2 0 2 4 6

Vector components

Solution

SW400: Solution Plot

δ x0 liftedδ x₀^BT liftedδ x0 1.1−BT liftedδ x0

1.1−MIRIAm

(a) Reduced dimesion=200

0 50 100 150 200 250 300 350 400

−6

−4

−2 0 2 4 6 8

Vector components

Solution

SW400: Solution Plot

δ x0 liftedδ x₀^BT liftedδ x0 1.1−BT liftedδ x0

1.1−MIRIAm

(b) Reduced dimension=80

Figure 5.5: The comparison of reduced-order approaches by means of balanced truncation (liftedδxˆ^BT₀ ), 1.1-shifted balanced truncation (liftedδxˆ^1.1−BT_o ) and MIRIAm method for α = 1.1 (lifted δxˆ^1.1−MIRIAm₀ ) with original full-order solutionδx₀ for the SW400 test model: solution plot for different dimensions of the reduced models.

50 100 150 200 250 300 350 400

10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

Vector components

Error

SW400: Error Plot

e^BT=δ x₀−liftedδ x₀^BT e^1.1−BT=δ x₀−liftedδ x₀^1.1−BT e^1.1−MIRIAm=δ x

0−liftedδ x 0 1.1−MIRIAm

Figure 5.6: The comparison of reduced-order approaches by means of balanced truncation (liftedδxˆ^BT₀ ), 1.1-shifted balanced truncation (liftedδxˆ^1.1−BT_o ) and MIRIAm method for α = 1.1 (lifted δxˆ^1.1−MIRIAm₀ ) with original full-order solutionδx₀ for the SW400 test model: error plot for reduced dimension=200.

5.4. NUMERICAL RESULTS 111

50 100 150 200 250 300 350 400

10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

Vector components

Error

SW400: Error Plot e^BT=δ x

0−liftedδ x 0 BT

e^1.1−BT=δ x₀−liftedδ x₀^1.1−BT e^1.1−MIRIAm=δ x

0−liftedδ x 0 1.1−MIRIAm

Figure 5.7: The comparison of reduced-order approaches by means of balanced truncation (lifted δˆx^BT₀ ), 1.1-shifted balanced truncation (lifted δˆx^1.1−BT_o ) and MIRIAm method for α = 1.1 (lifted δˆx^1.1−MIRIAm₀ ) with original full-order solutionδx₀ for the SW400 test model: error plot for reduced dimension=80.

plot it explicitly in Figures 5.6 and 5.7. It is easy to observe that the errors e^BT, e^α−BT ande^α−MIRIAm are approximately of the same order of magnitude.

For this example, the standard balanced truncation, for which α = 1.0 gives equivalently good approximation results as the other methods. As it was al-ready explained, it is due to the fact that system SW400 is 1.01-bounded and has only very few unstable poles.

In various numerical experiments with shallow water models SW1500 and SW400, we have demonstrated the potential of the new approach to model reduction of α-bounded systems. In case of systems with large number of small unstable poles, the method has been shown to be better than the other currently used approach involving the application of model reduction only to the stable part. The comparison of h_2,α-optimal model reduction method and balanced truncation for α-shifted systems has also been given, and both methods demonstrated almost the same level of accuracy. These and many other experiments with aforementioned model reduction techniques lead us to the conclusion that MIRIAm method could also be considered as very accurate approximation to balanced truncation. Nevertheless, further research needs to be done in order to improve the efficiency and convergence of MIRIAm method.

Chapter 6 Summary

In this thesis, model reduction techniques for continuous-time and discrete-time LTI dynamical systems have been studied. Most of the current methods are designed for approximating asymptotically stable systems. However, some processes such as weather development are unstable. A standard approach is to divide these systems into stable and unstable subsystems and apply model reduction only to the stable part. In contrast, in this thesis new interpolation-based methods have been proposed that aim to compute an optimal reduced-order model for stable as well as for unstable systems. The key observation here is that for all LTI dynamical models considered in this thesis, there exist an α such that the system is α-bounded. Hence, we have concluded that the H2,α-norm for continuous-time and h_2,α-norm for discrete-time LTI systems exist. By means of basic results from the theory of complex functions, we have derived new explicit formulas for these norms. Based on the preceding observation, the problem of minimizing the H2,α-norm of the error between transfer functions of the original large-scale continuous-time LTI system and its reduced-order approximant has been formulated. Analogous optimization task, in the sense of h_2,α-norm, has been also considered for discrete-time systems. For these optimization problems we have derived the tangential interpolation-based first order necessary optimality conditions. On the ba-sis of the established theory, we have proposed a MIMO Iterative Rational Interpolation Algorithm (MIRIAm) which, if it converges, provides a reduced-order system that satisfies the aforementioned first reduced-order necessary conditions.

By an example of the CD player, the accuracy of the new method has been illustrated and compared with other existing methods. The benefit of the new approach to model order reduction of unstable systems have also been de-monstrated in several numerical experiments with shallow water test models.

In case of systems with large number of small unstable poles, the method has been shown to be better than the other currently used approach involving the application of model order reduction only to the stable part.

113

A special case of the problem discussed in this thesis, applicable only to the asymptotically stable systems, has been extensively investigated in the last forty years. Several different sets of the first order necessary H2- and h₂ -norm optimality conditions have been developed. Among them, the gramian-based conditions given by Wilson and Hyland-Bernstein have been of special interest. We have introduced a generalization of these conditions toα-bounded continuous-time and discrete-time LTI dynamical systems and proved that they are equivalent to the tangential interpolation-based first order necessary conditions.

In this thesis, we have given theoretical results, which form a basis for H2,α -and h_2,α-optimal model reduction. However, for numerical treatment of the considered problems, there are many aspects left to be desired. The first approach to numerical methods, MIRIAm, has been introduced but more so-phisticated numerical methods for constructing the projection matrices as well as for computing the eigenvalues and eigenvectors have to be implemented.

Moreover, further research has to be done in order to ensure the preservation of α-boundedness of the original model in its reduced-order approximant. It has been observed that the convergence of MIRIAm algorithm as well as the quality of approximation strongly depend on the selection of starting data and on the choice of parameterα. These dependences have to be investigated in more details. The physical interpretation of the approach presented here could be very helpful in further research. Nevertheless, the results presented here indicate that the H2,α- and h_2,α-optimal model reduction methods have the potential to give an improvement over other existing approaches to model reduction of dynamical systems, particularly the unstable ones.

115 Errata

Page Line Reviewed version Published version

4 12 are lost. can be lost. To maintain the bound for this case an adaptation would be needed taking into account the error in the approximation.

10 13 x_k=x(k)∈K^N,u_k=u(k)∈K^m x_k=x(t_k)∈K^N,u_k=u(t_k)∈K^m andy_k=y(t_k)∈K^p andy_k=y(k)∈K^p

12 13 Conditions for BIBO-stability in Removed as not relevant for the thesis.

18 Theorem 2.1.1 and Theorem 2.1.2

13 15 is called the reachability matrix is called the (infinite) reachability matrix 13 16 rank of the reachability matrix rank of the (infinite) reachability matrix

14 16 satisfies the Lyapunov equation is the unique symmetric positive definite solution of the Lyapunov equation

14 20 satisfies the Stein equation is the unique symmetric positive definite solution of the Stein equation

16 3 Proposition 2.1.2 Proposition 2.1.2 [4]

19 14 Proposition 2.1.5 Proposition 2.1.5 [4]

19 15 initial conditionx(0) = 0 initial conditionx(0) = ¯x

21 18 Throughout the thesis, we only consider systems

with strictly proper transfer functions.

22 6 We note that throughout the thesis we only consider

systems with simple, i.e. pairwise disctinct, poles.

We make this assumption for original as well as for reduced-order models.

22 21 IfH(s) has a removable singularity Removed as not consistent with Definition 2.2.4 atλ, then Res (H(s), s=λ) =γ₋₁= 0.

25 16 Ais nonsingular and making use of Ais nonsingular and, for suitableη,making use of 36 17 by state-space projection of (3.1.1) by state-space projection Π =V_nW_n^∗of (3.1.1) 37 6 subsystem ˜Σ = ( ˜A,B,˜ C).˜ subsystem ˜Σ = ( ˜A,B,˜ C˜). It is easy to see that this

reduced-order model is constructed by projection Π =V_nW_n^∗,whereV_n=V(:,1 :n) andW_n^∗=V⁻¹(1 :n,:).

39 25 strong subsystem with easily reachable strong subsystem, i.e. to a subsystem with easily and easily observable states. reachable and easily observable states.

39 27 to a weak subsystem. to a weak subsystem, i.e. to a subsystem with states which are difficult to reach and difficult to observe.

40 10 ˆΣ is balanced and asymptotically stable. ˆΣ is asymptotically stable.

46 10 continuous–time systems: continuous-time systems. One possible [(3.3.36)] (3.3.24)⇐⇒V_n:=P12P₂₂⁻¹, choice of projection matricesV_nandW_nis [(3.3.35)](3.3.23)⇐⇒W_n:=−Q12Q⁻¹₂₂ V_n:=P12P₂₂⁻¹,

and [(3.3.34)](3.3.22)⇐⇒W_n^∗V_n=I_n. W_n:=−Q12Q⁻¹₂₂.

55 10 directions, respectively. Let directions, respectively. Assume thatσ_k, k= 1,. . ., N are not eigenvalues ofAand let

62 5 ^H

γ^(R)

F(s)ds= ^PN

k=1

Res (F(s), s=λ_k). ^H

γ^(R)

F(s)ds= 2πi ^PN

k=1

Res (F(s), s=λ_k).

65 14 Assume, without loss of generality, that Assume that the state matricesAand ˆA the state matricesAand ˆA

67 9 are the poles of Σ and ˆΣ, respectively, are the poles of ˆΣ,

72 24 Without loss of generality, In this thesis we only consider original and reduced-order systems with simple poles. Therefore, without loss of generality,

76 5 λ_k< αand consequentlyλ^α_k<0 for all (λ_k)< αand consequently(λ^α_k)<0 for all 83 12 can be, in general, not very beneficial. can be not very beneficial.

88 13 Assume, without loss of genarality, that Assume that the state matricesAand ˆAare diago-the state matricesAand ˆAare diagonal nal

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