Balanced model order reduction method for systems ...

IntroductionBalanced realization procedure

Power series expansion for the balanced realizationApplication: a system of masses and springs

Conclusions

Balanced model order reduction method forsystems depending on a parameter

Carles Batlle1

Nestor Roqueiro2

1 Departament de Matematiques and Institute of Control and IndustrialEngineering, Universitat Politecnica de Catalunya — BarcelonaTech, Vilanova i la

Geltru2 Departamento de Automacao e Sistemas, Universidade Federal de Santa

Catarina, Florianopolis

[email protected] [email protected]

Florianopolis, Brazil, April 23-26, 2019 DYCOPS 2019



Conclusions

Introduction

Order reduced models are useful to simulate very large modelsusing less computational resources, allowing, for instance, theexploration of parameter regions.

They should be easily computable, preserving some of thestructural properties of the full model and yielding an errorwith respect to the original model that can be bounded interms of the complexity of the approximating model.

For linear time-invariant MIMO systems, model orderreduction (MOR) based on the truncation of balancedrealizations preserves the stability, controllability andobservability of the full model, and provides bounds for thenorm of the error system.




Conclusions

Computation of a balanced realization for a linear systemrelies on numerical linear algebra algorithms, and does notallow for the presence of symbolic parameters in the model.

Our goal is to present a balanced realization MOR for systemsdepending on a parameter, with a final reduced order modelwhich depends also on the parameter, up to a certain power.

We deduce expressions to arbitrary order in the parameter,except for the last step of the procedure, where we only haveexplicit results up to second order.

As a by-product, we obtain results for the second ordercorrection to the singular subspaces of the Singular ValueDecomposition (SVD).




Conclusions

The rest of the presentation is organized as follows

Review of the balanced realization procedure.

Power series expansion for the balanced realization.

Application: a system of masses and springs.

Conclusions.




Conclusions

Detailed computations can be found in the proceedings paperand in Batlle, C. and Roqueiro, N., Balanced model orderreduction for systems depending on a parameter,http://arxiv.org/abs/1604.08086.

Extensive presentations of model order reduction techniquesand its applications can be found in

Antoulas, A.C. (2005), Approximation of large-scale dynamicalsystems, Advances in Design and Control. SIAM,Schilders, W.H.A., van der Vorst, H.A., and Rommes, J. (eds.)(2008), Model order reduction: theory, research aspects andapplications, volume 13 of Mathematics in industry; TheEuropean Consortium for Mathematics in Industry, Elsevier,Benner, P., Gugercin, S., and Willcox, K. (2015), A Survey ofProjection-Based Model Reduction Methods for ParametricDynamical Systems, SIAM Review, 57(4), 483–531,

which contain references to the original literature.


http://arxiv.org/abs/1604.08086



Conclusions

Balanced realization procedure

Consider the nonlinear control system

x = f(x) + g(x)u,

y = h(x),

with x ∈ RN , u ∈ RM , y ∈ RP and f(0) = 0.

The controllability function Lc(x) is the solution of theoptimal control problem

Lc(x) = infu∈L2((−∞,0),RM )

1

2

∫ 0

−∞||u(t)||2dt

subject to x(−∞) = 0, x(0) = x and x = f(x) + g(x)u.

Roughly speaking, Lc(x) measures the minimum 2-norm ofthe input signal necessary to bring the system to the state xfrom the origin.




Conclusions

The observability function Lo(x) is proportional to the 2-normof the output signal obtained when the system is relaxed fromthe state x

Lo(x) =1

2

∫ ∞0||y(t)||2dt =

1

2

∫ ∞0||h(x(t))||2dt,

with x(0) = x and subjected to x = f(x), that is, with u = 0.

For linear control systems,

x = Ax+Bu, y = Cx,

assumed to be observable, controllable and Hurwitz, bothLc(x) and Lo(x) are quadratic functions

Lc(x) =1

2xTW−1c x,

L0(x) =1

2xTWox.




Conclusions

The matrices Wc > 0 and Wo > 0, the controllability andobservability Gramians, are the solutions to the matrixLyapunov equations

AWc +WcAT +BBT = 0, ATWo +WoA+ CTC = 0.

They are explicitly given by

Wc =

∫ ∞0

eAtBBT eAT t dt, Wo =

∫ ∞0

eAT tCTCeAt dt,

but the numerical solution of the Lyapunov equations ifcomputationally preferred.

Notice that Wc > 0 and Wo > 0 means that theuncontrollable or unobservable subspaces have been alreadydropped from the system (Kalman decomposition).




Conclusions

The matrix Wc provides information about the states that areeasy to control, in the sense that signals u of small norm canbe used to reach them.

Wo allows to find the states that are easily observable, i.e.they produce outputs of large norm.

From the point of view of the input-output map associated tothe linear control system, one would like to select the statesthat score well on both counts, and this leads to the conceptof balanced realization, for which Wc = Wo.

The balanced realization is obtained by means of a lineartransformation x = Tz, with T computed in a series of steps.

This is a well known algorithm, implemented for instance withthe Matlab command balred or, with greater flexibility andwith some state-of-the-art improvements, in the MORLABtoolbox.




Conclusions

1 Solve the Lyapunov equations for Wc > 0, Wo > 0.

2 Perform Cholesky factorizations of the Gramians:

Wc = XXT , Wo = Y Y T .

Notice that X > 0 and Y > 0.

3 Compute the SVD of Y TX:

Y TX = UΣV T ,

with U and V orthogonal and

Σ = diag(σ1, . . . , σN ), σ1 ≥ · · · ≥ σN > 0.

4 The balancing transformation is given then by

T = XV Σ−1/2, with T−1 = Σ−1/2UTY T .




Conclusions

5 The balanced realization is given by the linear system

A = T−1AT, B = T−1B, C = CT.

6 In the new coordinates,

Wc = T−1WcT−T = Σ, Wo = T TWoT = Σ,

and the controllability and observability functions are

Lc(z) =1

2

N∑i=1

z2iσi

=1

2zTΣ−1z,

Lo(z) =1

2

N∑i=1

σiz2i =

1

2zTΣz,

so that the state with only nonzero coordinate zi is botheasier to control and easier to observe than the statecorresponding to zi+1, for i = 1, 2, . . . , N − 1.




Conclusions

If, for a given r, 1 ≤ r < N , one has σr � σr+1, it may besensible to keep just the states corresponding to thecoordinates z1, z2, . . . , zr.

The σi are the Hankel singular values of the control system.

H∞-norm lower and upper error bounds of the balancedtruncation method are given by

σr+1 ≤ ‖G(s)−Gr(s)‖H∞≤ 2

N∑i=r+1

σi.

From these inequalities it follows that, in order to get thesmallest error for the truncated system, one should disregardthe states associated with the smallest Hankel singular values.




Conclusions

If we denote by Ar the upper-left square block of A formed bythe first r rows and columns, and by Br and Cr the matricesobtained from the first r rows or columns of B or C,respectively, the reduced system of order r obtained bybalanced truncation is given by

Zr = ArZr + Bru, y = CrZr,

with Zr = (z1, . . . , zr).




Conclusions

Power series expansion for the balanced realization

The balanced realization algorithm relies on numerical algebraand hence cannot be carried out as described if the systemcontains a symbolic parameter.

In this paper we address this issue, assuming that the linearsystem is given by matrices A(m), B(m) and C(m) whichdepend analytically on the parameter m.

The parameter m may represent an uncertain physicalcoefficient, or it may appear by considering an unspecifiedworking point in the linearisation of a non-linear system.

Goal: to develop a power series expansion in m of thebalanced model order reduction algorithm for the linearinput/output system given by A(m), B(m), C(m).




Conclusions

Assume that A(m), B(m) and C(m) are analytic in m,

A(m) =

∞∑k=0

Akmk,

B(m) =

∞∑k=0

Bkmk,

C(m) =

∞∑k=0

Ckmk,

Our goal is to find power expansions for the matrices Wc(m),Wo(m), X(m), Y (m), U(m), V (m), Σ(m) and T (m) whichappear in the balanced realization MOR procedure.




Conclusions

For instance, for Wc(m) one has Wc(m) =∑∞

k=0Wckm

k, andsubstituting into the Lyapunov equation for the controllabilityGramian one gets the set of Lyapunov equations

A0Wc0 +W c

0AT0 +B0B

T0 = 0,

A0Wcr +W c

rAT0 + Pr = 0, r = 1, 2, . . . ,

with Pr = B0BTr +

∑r−1s=0

(Ar−sW

cs +W c

sATr−s +Br−sB

Ts

).

These equations can be solved recursively to the desired order,starting with the zeroth order Lyapunov equationA0W

c0 +W c

0AT0 +B0B

T0 = 0.

Notice that the internal dynamics is always given by A0, andthat it is only the effective control term Pr the one thatchanges with the order.




Conclusions

Similarly, one can obtain recursive relations for Wo(m),X(m), Y (m) and R(m) = Y T (m)X(m).

The final step in the algorithm is the power expansion of thematrices of the SVD of R(m).

This is a more involved step, and in the paper we only providethe results up to second order in m.

Let

R(m) = R0 +mR1 +m2R2,

U(m) = U0 +mU1 +m2U2,

V (m) = V0 +mV1 +m2V2,

Σ(m) = Σ0 +mΣ1 +m2Σ2,

and let u(k)j denote the jth column vector of Uk, and v

(k)j the

one of Vk, and σ(k)j the jth element of the diagonal matrix Σk.




Conclusions

The first order corrections to the singular values are given by

σ(1)i =

⟨u(0)i , R1v

(0)i

⟩=⟨v(0)i , RT

1 u(0)i

⟩, i = 1, . . . , N,

and they can be computed from the zeroth order SVD and the firstorder data R1. Here 〈 , 〉 denotes the Euclidean inner product.

The first order correction to the singular vectors uj and vj are the

solutions u(1)i and v

(1)i to the systems(

R0RT0 − (σ

(0)i )2I

(u(0)i )T

)u(1)i =

(Q

(1)i

0

), i = 1, . . . , N,(

RT0 R0 − (σ

(0)i )2I

(v(0)i )T

)v(1)i =

(P

(1)i

0

), i = 1, . . . , N,

with Q(1)i and P

(1)i column vectors that can again be obtained

from the zeroth order solution and the first order data R1.




Conclusions

The second order correction to the singular value σi, i = 1, . . . , N ,is given by

σ(2)i =

1

2σ(0)i

(||u(1)i ||

2−||v(1)i ||2)

+⟨u(0)i , R1v

(1)i +R2v

(0)i

⟩,

with the right-hand side depending only on data from the zeroth andfirst order approximations, plus the second order perturbation R2.

Finally, the second order correction to the singular vectors are thesolutions of the systems(

R0RT0 − (σ

(0)i )2I

(u(0)i )T

)u(2)i =

(Q

(2)i

− 12 ||u

(1)i ||2

),(

RT0 R0 − (σ

(0)i )2I

(v(0)i )T

)v(2)i =

(P

(2)i

− 12 ||v

(1)i ||2

).




Conclusions

The matrices appearing on the left hand-sides are the same asthose of the first order correction, and the data on theright-hand side can be computed from the first order solutionand the second order perturbation data.To our knowledge, this result about the second orderperturbation of the singular vectors has not been reported inthe litrature.Using the above results,one can compute

T (m) = X(m)V (m)Σ−1/2(m),

T−1(m) = Σ−1/2(m)U(m)TY T (m)

up to second order in m, and then the matrices A2(m),B2(m), C2(m) which yield the balanced realization of thesystem again up to powers m2.From the balanced realization, a suitable order reduced systemcan then be obtained.




Conclusions

Application: a system of masses and springs

We consider a system of N masses mi and (linear)springswith constants ki and natural lengths di, so that the ithspring lies between masses mi and mi+1, i = 1, . . . , N − 1,and the last spring connects mass mN to a fixed wall. We alsoadd a linear damper to each mass, with coefficients γi and,furthermore, act with an external force F on the first mass.

After redefining the coordinates to absorb the lengths di andintroducing the canonical momenta pi = xi/mi, the systemcan be put in the first order form

X =

0N×N diag(1/m1, . . . , 1/mN )

KN×N −diag(γ1/m1, . . . , γN/mN )

X +BF,




Conclusions

The state is X = (x1, . . . , xN , p1, . . . , pN )T . and the inputvector is

B = (0, . . . , 0︸︷︷︸N

, 1, 0, . . . , 0)T .

K is the stiffness matrix

K =

−k1 k1 · · · 0 0k1 −(k1 + k2) · · · 0 00 k2 · · · 0 0...

......

...0 0 · · · −(kN−2 + kN−1) kN−1

0 0 · · · kN−1 −(kN−1 + kN )




Conclusions

If we measure the velocity of the first mass, we have theoutput y = CZ with C

C =(

1 0 · · · 0 0 0 · · · 0).

In order to obtain a test of our whole algorithm, we considerthe set of physical constants given by

ki = 100(i+ 1), i = 1, . . . , N,

mi = i(1 +m), i = 1, . . . , N,

γi = 1, i = 1, . . . , N,

with m the parameter of the Taylor expansion. We setN = 10, which yields a system with 20 states, and considerreduced systems with four states.




Conclusions

Our procedure, which we have implemented entirely in Matlab,yields the reduced system, parametrized by m, given by

A4 =

−0.28m2 + 0.255m− 0.218 0.504m2 − 0.84m+ 2.06−0.504m2 + 0.84m− 2.06 −0.0393m2 + 0.0548m− 0.07990.198m2 − 0.193m+ 0.181 1.01m2 − 1.05m+ 1.070.648m2 − 0.745m+ 0.862 0.0653m2 − 0.0808m+ 0.103

0.198m2 − 0.193m+ 0.181 −0.648m2 + 0.745m− 0.862−1.01m2 + 1.05m− 1.07 0.0653m2 − 0.0808m+ 0.103−0.143m2 + 0.149m− 0.155 1.39m2 − 2.14m+ 4.91−1.39m2 + 2.14m− 4.91 −0.106m2 + 0.119m− 0.134

,

B4 =

−0.0362m2 + 0.0505m− 0.143

3.95 · 10−4m2 + 0.00639m− 0.08130.0135m2 − 0.0239m+ 0.102

0.00731m2 − 0.0167m+ 0.0922

,




Conclusions

and

C4 =

−0.0362m2 + 0.0505m− 0.143

−3.95 · 10−4m2 − 0.00639m+ 0.08130.0135m2 − 0.0239m+ 0.102−0.00731m2 + 0.0167m− 0.0922

T

.

The next figure shows a detail of the Bode diagrams form = 0.5 computed using the polynomial approximations ofdegree zero (black), one (blue) and two (red), together withthe exact reduced system (green).

It is clearly seen that the results improve as the order of thepolynomial approximation is increased.

Notice that the zeroth order polynomial approximation isequivalent to considering m = 0.




Conclusions

−80

−60

−40

−20

Ma

gn

itu

de

(d

B)

100

101

−180

−90

0

90

Ph

ase

(d

eg

)Bode Diagram

Frequency (rad/s)




Conclusions

Conclusions

We have developed a parameter dependent model order reductionalgorithm based on the balanced realization approximation. Thealgorithm yields a reduced order model which can be used to designa controller valid for a range of values of the parameter. As aby-product, we have obtained an expression for the second orderperturbation of the singular subspaces.

Some extensions of our work include considering several parametersinstead of just one, or computing some further higher ordercorrections of the parametrized SVD.

We have not addressed the issue of the estimation of the error ofthe reduced model. This error involves both the truncation errors ofthe different steps of the algorithm and the error which comes fromthe truncation of the balanced realization.

In its present form, our algorithm can only be applied to stable

systems.


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