Approximating Arbitrary Impulse Response Functionswith Prony Basis Functions

Technical Report UIUC-ESDL-2019-01

Daniel R. Herber∗Colorado State University

James T. Allison†

University of Illinois at Urbana-Champaign

October 21, 2019

Abstract

In this report, we are concerned with approximating the input-to-output behavior of a typeof scalar convolution integral given its so-called impulse response function by constructing anappropriate linear time-invariant state-space model. Such integrals frequently appear in themodeling of hydrodynamic forces, viscoelastic materials, among other applications. First, lin-ear systems theory is reviewed. Next, Prony basis functions, which are exponentially decayingcosine waves with phase delay and variable amplitude, are described as potential objects to beused to approximate a given impulse response function. Then it is shown how a superposi-tion of Prony basis functions can be directly mapped back to an equivalent linear state-spacemodel. Also, it is directly shown that both the Golla-Hughes-McTavish model and Prony series(generalized Maxwell model) are special cases of the considered Prony basis function. Severalnonlinear optimization (�tting) problems are then described to determine the value of the modelparameters that result in the desired approximation. Finally, a few numerical examples are pre-sented to demonstrate that Prony basis functions can approximation a diverse set of impulseresponse behaviors.

∗Assistant Professor, Department of Systems Engineering, daniel.herber@colostate.edu†Associate Professor, Department of Industrial and Enterprise Systems Engineering, jtalliso@illinois.edu

©2019 Daniel R. Herber

ii

Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiList of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

1 Introduction 1

2 Impulse Response Function as a LTI State-Space System 22.1 Time-Domain Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Frequency-Domain Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 State-Space Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 Some Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Prony Basis Functions 63.1 Prony Basis Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Reduction to the Golla-Hughes-McTavish Model . . . . . . . . . . . . . . . . . . . . 103.3 Reduction to Prony Series (Generalized Maxwell Model) . . . . . . . . . . . . . . . 123.4 Comparison between the Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Fitting 154.1 Optimization Problem with Companion Form . . . . . . . . . . . . . . . . . . . . . 154.2 Optimization Problem with Prony Basis Functions . . . . . . . . . . . . . . . . . . . 164.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Numerical Examples 225.1 Models for Some Impulse Response Functions . . . . . . . . . . . . . . . . . . . . . 225.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Supporting Code 28

References 29

List of Figures

3.1 Comparison between the transfer function coverage of the di�erent representations. 145.1 Fitted impulse response functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Extended response of the �tted impulse response functions. . . . . . . . . . . . . . 245.3 Basis function contributions to the �tted impulse response functions. . . . . . . . . 255.4 Transfer functions of the �tted impulse response functions. . . . . . . . . . . . . . . 265.5 Simulation results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

iii

List of Tables

3.1 Summary of the considered representations. . . . . . . . . . . . . . . . . . . . . . . 144.1 Summary of the considered �tting problem formulations. . . . . . . . . . . . . . . . 215.1 Relative computational expense for the simulations. . . . . . . . . . . . . . . . . . . 27

List of Symbols and Abbreviations

Symbol Description PageBIBO bounded-input, bounded-output . . . . . . . . . . . . . . . . . . . . . . . . 8GHM Golla-Hughes-McTavish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

IRF impulse response function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1LTISS linear time-invariant state-space . . . . . . . . . . . . . . . . . . . . . . . . 2

PBF Prony basis function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6PS Prony Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

SISO single-input, single-output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2TF transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3A state matrix with size = × = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2a polynomial coe�cients for* with size (= − 1) × 1 . . . . . . . 3B input matrix with size = × 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2b polynomial coe�cients for . with size (= − 1) × 1 . . . . . . . . 3C output matrix with size 1 × = . . . . . . . . . . . . . . . . . . . . . . . . . . . 2� error function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15n small number parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

GSS general state space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3I identity matrix of size = × = . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 indexing variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

=(·) complex part of complex number . . . . . . . . . . . . . . . . . . . . . . . 209 imaginary unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 impulse response function in frequency domain . . . . . . . . . . 3: impulse response function in time domain . . . . . . . . . . . . . . . 1

L{·} Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Λ multiplicity of a complex-valued pole . . . . . . . . . . . . . . . . . . . 4_ multiplicity of a real-valued pole . . . . . . . . . . . . . . . . . . . . . . . . 4< number of discrete time points . . . . . . . . . . . . . . . . . . . . . . . . . 15µ model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15# number of basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7N number of basis functions in Prony’s method . . . . . . . . . . . . 19= number of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2l frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9p poles of (B), roots of* (B) of size (= − 1) × 1 . . . . . . . . . . . . 3

iv

Symbol Description PageΦ basis function in the frequency domain . . . . . . . . . . . . . . . . . . 7q basis function in the time domain . . . . . . . . . . . . . . . . . . . . . . . 6Ψ distinct complex-valued pole . . . . . . . . . . . . . . . . . . . . . . . . . . . 4k distinct real-valued pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

<(·) real part of complex number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4{d1, d2, d3} scaling parameters relating : and : . . . . . . . . . . . . . . . . . . . . . 5

B frequency-domain independent variable . . . . . . . . . . . . . . . . . 3C time-domain independent variable . . . . . . . . . . . . . . . . . . . . . . 1t time grid with< points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15g variable of integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1\1 amplitude parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\2 decay rate parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\3 frequency parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\4 phase shift parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6� collection of all basis function parameters . . . . . . . . . . . . . . . 7

{o1, o2, o3} alternative model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 10* denominator of (B), polynomial of degree = . . . . . . . . . . . . 3D single input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1w weights (�tting or quadrature) . . . . . . . . . . . . . . . . . . . . . . . . . . 15ξ states with size = × 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. numerator of (B), polynomial of degree = − 1 . . . . . . . . . . . 3~ single output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1z zeros of (B), roots of . (B) of size (= − 2) × 1 . . . . . . . . . . . . 3◦ Hadamard or entrywise product . . . . . . . . . . . . . . . . . . . . . . . . 15� complex conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4|�| magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4� scaled quantity (e.g., : is related to : through scaling laws) 5� approximated function (e.g., : approximates :) . . . . . . . . . . . 1�2 related to complex-valued part . . . . . . . . . . . . . . . . . . . . . . . . . 4�0 initial value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2‖�‖? ?-norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15�A related to real-valued part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4[�]t evaluated at the points in t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15�† inner-loop optimal value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16�+ Moore-Penrose pseudoinverse . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Acknowledgments

We would like to thank Dr. Randy Ewoldt and the students of Dr. Ewoldt’s and Dr. Allison’sresearch groups for their feedback on this report.

1

Section 1

Introduction

Consider the following equation:

~ (C) =∫ C

−∞: (C − g)D (g)3g (1.1)

where ~ (C) is termed the output, D (C) is the input, and : (C) is the impulse response function (IRF)1

and each is a scalar-valued function. The output then is the convolution between : (C) and D (C). Inthis report, we will present methods that approximate the behavior of Eq. (1.1) with the following:

~ (C) =∫ C

−∞: (C − g)D (g)3g (1.2)

where : (C) is the approximated IRF, ~ (C) is the approximated output, and the input remains thesame. The e�cacy of these methods relies on the assumption that if : (C) ≈ : (C), then ~ (C) ≈ ~ (C).The usefulness of the approximation is due to a number of factors including e�cient simulation [1–6], control-system analysis [2, 4–7], and design studies [8], among other reasons.

There has been signi�cant work performed in creating suitable approximations. A basic ap-proach is to obtain the direct solution of the convolution integral whenever the value of the outputis needed (see Refs. [1–3] for a more thorough discussion). Typically, this method is described ascomputationally expensive because the the integral must be calculated for each instance of di�er-ent input data, but there are some techniques for reducing computational expense.

The primary alternative to the direct solution of the convolution integral is approximation by a(linear) state space. State-space models are potentially advantageous because they have the Markovproperty, i.e., at any time instant, the value of the states contains all the past information of thesystem and the entire sequence of previous events is not needed for accurate computation [4].There are two main categories for determining the state-space model (sometimes called realizationmethods). First are methods that �t the response in the frequency domain (see Refs. [2, 4, 7] for anoverview). Second are the methods that �t the response in the time domain, i.e., methods that tryto approximate : (C) (see Refs. [1, 2, 4, 7] for an overview). This report focuses on the latter group:approximating in the time domain.

Many time-domain methods seek to minimize some error metric between the original IRF andthe approximate one. Prony’s estimation [1, 7] is one such approach, and other methods use al-ternative model representations [8, 9]. The �nal type of time-domain method reviewed here isrealization theory based on the minimal realization from its Markov parameters generated fromthe impulse response data, typically using the Hankel singular value decomposition of a sampled: (C) [2, 4, 5, 7].

1In some communities, : (C ) is called the relaxation kernel or scalar memory function.

2

Section 2

Impulse Response Function as a LTIState-Space System

In this section we present the connections between linear time-invariant state-space (LTISS) mod-els and their corresponding IRFs. This theory is presented in the time domain, frequency domain,and canonical forms along with additional topics on model reduction and scaling.

2.1 Time-Domain Representation

Here we seek a LTISS model [10, p. 64] for the following single-input, single-output (SISO) equa-tion:

~ (C) ≈ ~ (C) =∫ C

−∞: (C − g)D (g)3g =

{¤ξ(C) = Aξ(C) +BD (C)~ (C) = Cξ(C)

(2.1)

where ξ are the= additional states associated with the LTISS system and the matrices {A=×= ,B=×1,C1×=} comprise the LTISS system that is exact with : (C). To ensure output parity between the twosystems, we enforce an equivalence condition at some initial time point C0:

limC→C0

~ (C) = limC→C0

~ (C) = limC→C0

Cξ(C0) (2.2)

The solution for the output represented as a di�erential-algebraic equation in Eq. (2.1) at somestarting point C0 is:

~ (C) = C4A[C−C0 ]ξ(C0) +∫ C

C0

C4A[C−g ]BD (g)3g (2.3)

where 4AC is the matrix exponential 4AC =∑∞8=0 [A8C8/8!] [10, p. 66]. Therefore, the approximate

IRF and the LTISS representation are related by the so-called impulse response:

: (C) = C4ACB (2.4)

It can be shown that ~ (C) = : (C) when the impulse conditions are applied1.1The impulse conditions are ξ (0) = 0 and D follows the unit impulse function with n → 0:

D (C ) ={

1/n, 0 < C < n

0, otherwise(2.5)

3

2.2 Frequency-Domain Representation

The Laplace transform (denoted L{·}) of : provides another form in the frequency or B-domain:

(B) = L{: (C)} = C [BI −A]−1B =.=−1 (B)*= (B)

(2.6a)

= (B, b,a) = 1=−1B=−1 + · · · + 11B + 10

B= + 0=−1B=−1 + · · · + 01B + 00(2.6b)

= (B, z,p, 0) = 0

∏=−18=1 [B − I8 ]∏=8=1 [B − ?8 ]

(2.6c)

where (B) is termed the transfer function (TF) [10, p. 72] and is a rational function, . (B) is an(= − 1)-degree polynomial with roots z termed zeros, * (B) is an =-degree polynomial with rootsp termed poles, and I is the identity matrix of size = × =.

2.3 State-Space Canonical Forms

For a speci�ed SISO system, there are in�nite possible state-space models that will give identi-cal input/output dynamics [10, p. 92]. However, there are certain useful standardized state-spacemodel structures, and these are referred to as canonical forms [10, p. 159]. These canonical formsrepresent all real-valued LTISS systems and are termed GSS for general state space. Two suchcanonical forms are now discussed.

2.3.1 Companion FormFirst we will consider the companion [10, p. 25] or observer canonical form:

A =

0 0 0 · · · 0 −001 0 0 · · · 0 −010 1 0 · · · 0 −020 0 1 · · · 0 −03...

......

. . ....

...

0 0 0 · · · 1 −0=−1

=×=(2.7a)

B =

10111213...

1=−1

=×1

(2.7b)

C =[0 0 0 · · · 0 1

]1×= (2.7c)

where the parameters are the coe�cients of . and * in Eq. (2.6b). One advantage of this formis the reduction of the number of parameters required. If each entry in the LTISS matrices areparameters, then there would be =2 + 2= parameters in the generic form. However, the companionform only requires 2= parameters without loss of generality.

4

2.3.2 Modal FormModal or Jordan form is an another canonical form characterized by diagonal submatrices de�nedby the poles of the TF [10]. Modal form is represented by the complex partial fraction decomposi-tion of (B) [11]:

(B) ==A∑8=1

_8∑ℓ=1

U8,ℓ

[B +k8 ]ℓ+

=2∑8=1

Λ8∑ℓ=1

V8,ℓB + W8,ℓ[B2 + 2<(Ψ8 )B + |Ψ8 |2]ℓ

(2.8)

wherek 8 is one of the=A distinct real-valued poles, Ψ8 is one of the=2 distinct complex-valued poles,_8 is the multiplicity of the pole k8 , Λ8 is the multiplicity of the pole Ψ8 , and the other parametersare real valued. Note that = = =A + 2=2 and:

p = [k1,k2, · · · ,k=A ,Ψ1,Ψ2, · · · ,Ψ=2 , Ψ1, Ψ2, · · · , Ψ=2 ]

where � represents the complex conjugate.There are two types of diagonal submatrices, one for k8 and another for Ψ8 . For the distinct

real-valued poles, the submatrices are:

AA,8 =

k8 1 0 0 . . . 00 k8 1 0 . . . 0...

. . .. . .

. . .. . .

...

0 · · · 0 k8 1 0

_8×_8(2.9a)

BA,8 =

0...

01

_8×1

(2.9b)

CA,8 =[U8,1 U8,2 · · · U8,_8

]1×_8 (2.9c)

The submatrices for repeated complex-valued poles, denoted {A2,8 ,B2,8 ,C2,8 }, are relatively in-volved. However, the submatrices for distinct complex-valued poles are simpler and shown inSec. 3.1.

Each of set of submatrices can be combined to form the complete LTISS system:

A = diag(AA,1, · · · ,AA,=A ,A2,1, · · · ,A2,=2

)=×= (2.10a)

B =

BA,1...

BA,=A

B2,1...

B2,=2

=×1

(2.10b)

C =[CA,1 · · · CA,=A C2,1 · · · C2,=2

]=×1 (2.10c)

From a numerical analysis perspective, there are a few general properties of the LTISS systemthat can both reduce errors and computation time. These properties include states having similarmagnitudes (ideally near unity), a smaller condition number for matrix A, and sparser matrices(i.e., more zero entries). Some of these properties are especially important for the success of somesolution methods for dynamic optimization such as direct transcription [8]. Modal form can beuseful in this regard but its e�ectiveness will be problem speci�c.

5

2.4 Model Reduction

The goal of model reduction is to reduce the number of states in a state-space realization whilekeeping the system input-output properties approximately the same [12, p. 153]. Fewer statescan prevent over�tting and improve simulation times. A common approach is to remove statesbased on their Hankel singular values in a balanced realization. Relatively small Hankel singularvalues have a relatively minimal contribution to the value of the output; thus, are removed. Modelreduction in the context of IRF �tting is discussed in Refs. [5–7]. Model reduction can be useful forstate-space models created with any of the methods that will be discussed.

2.5 Scaling

Consider two IRFs : and : related by:

: (C) = d1: (d2C + d3) (2.11)

where {d1, d2, d3} are the scaling parameters; d1 is termed the amplitude scale, d2 > 0 is termedthe time scale, and d3 is termed the time shift. We can apply this transformation to the outputequation in Eq. (2.3):∫ C

C0

: (C − g)D (g)3g =∫ C

C0

d1: (d2 [C − g] + d3)D (g)3g (2.12a)

= d1C4Ad34Ad2 [C−C0 ]ξ(C0) +

∫ C5

C0

d1C4Ad34Ad2 [C−g ]BD (g)3g (2.12b)

= C4A[C−C0 ]ξ(C0) +∫ C5

C0

C4A[C−g ]BD (g)3g (2.12c)

where A = d2A, B = B, and C = d1C4Ad3 . We note that the equivalence condition in Eq. (2.2)

is still satis�ed. Therefore, we can obtain an equivalent scaled LTISS model from a related IRF.

2.6 Some Advantages

The key advantage of the LTISS system approximation is the Markovian characteristic of the model,i.e., at any time instant, the current value of the states is all that is needed to predict future behaviorand all other previous time history is unnecessary [3, 4, 7]. This approximation has the result thata system of integro-di�erential equations is be transformed into a system of ordinary di�erentialequations. Therefore, standard simulation and dynamic optimization techniques can be readilyapplied. Furthermore, this approximation adds additional linear dynamics. If the original dynamicmodel, excluding the convolution integral, was linear, then the dynamic model will remain linearwith this approximation. Therefore, linear systems theory, as well as other techniques suitable forlinear systems, can be readily applied (such as control analysis [2, 4–7]).

6

Section 3

Prony Basis Functions

In this section we describe a particular basis function q (C) used to construct both : (C) and theLTISS system. The basis functions serve as basic building blocks for approximating the originalIRF through a superposition of their responses:

: (C) ≈ : (C) =#∑8=1

q8 (C) (3.1)

The selected basis function is the same one used in Prony’s method [13], and has a number ofdesirable properties for �tting IFRs. It is motivated by the partial fraction decomposition [10] of theTF in Eq. (2.8), the linearity property of the unilateral Laplace transform [14, p. 138], and simpleLaplace transforms. In Secs. 3.2 and 3.3, we show that the chosen representation encompassesthe existing Prony series method (generalized Maxwell model) [15] and Golla-Hughes-McTavishmodel [9,16]. In other words, the Prony basis functions shown here represent a more general classof IRFs than the Prony series or Golla-Hughes-McTavish models (but this �exibility comes withsome caveats). However, it does not capture all real-valued TFs as is discussed at the end of thesection.

3.1 Prony Basis Function

The method described in this section will be referred to as the Prony basis function (PBF) method.

3.1.1 Single Basis Function

Time-Domain Representation

Here we will consider the following basis function q in a few di�erent forms:

q (C, \ ) = \14−\2C cos(\3C + \4) (3.2a)

= \14−\2C [cos(\4) cos(\3C) − sin(\4) sin(\3C)] (3.2b)

=12\1

[4 9\44 [−\2+9\3 ]C + 4−9\44 [−\2−9\3 ]C

](3.2c)

= \14−\2C

[4 9 [\3C+\4 ] + 4−9 [\3C+\4 ]

2

](3.2d)

which is an exponentially-decaying cosine wave with phase delay and variable amplitude, where9 is the imaginary unit. The parameter \1 determines the amplitude, \2 determines the decay rate,\3 determines the frequency, and \4 determines the phase shift. Without loss of generality, we willassume \1 > 0, \3 > 0, and 0 ≤ \4 ≤ 2c . We will limit the representation with \2 > 0 to ensuresome properties that will be discussed shortly.

7

Frequency-Domain Representation

Now consider the Laplace transform [14, p. 134] of q (C), denoted Φ(B):

Φ(B) = L{q (C)} = \1

[cos(\4)

B + \2

[B + \2]2 + \ 23− sin(\4)

\3

[B + \2]2 + \ 23

](3.3a)

= \1cos(\4)B + \2 cos(\4) − \3 sin(\4)

B2 + 2\2B + \ 22 + \ 2

3(3.3b)

= \1 cos(\4)B + I1 (\2, \3, \4)

[B + ?1 (\2, \3)] [B + ?2 (\2, \3)](3.3c)

The rational function in Eq. (3.3c) has the following zero and poles (when \4 ≠ c/2):

I1 (\2, \3, \4) = \2 − \3 tan(\4) (3.4a)?1,2 (\2, \3) = \2 ± 9\3 (3.4b)

State-Space Representation

The LTISS representation for q (C) will require exactly two states as a consequence of the degreeof the denominator of Φ(B). Now it can be shown that the following matrices produce the desiredimpulse response, i.e., q (C) = Cq4AqCBq :

Aq =

[−\2 \3−\3 −\2

](3.5a)

Bq = \1

[sin(\4)cos(\4)

](3.5b)

Cq =[0 1

](3.5c)

Noting that:

4Aq =

[4−\2C cos(\3C) 4−\2C sin(\3C)−4−\2C sin(\3C) 4−\2C cos(\3C)

](3.6)

3.1.2 Combining Basis Functions

Time-Domain Representation

Here we will consider approximated IRFs that are de�ned by the sum of # basis functions:

: (C) =#∑8=1

q8 =

#∑8=1

q (C,�8 ) (3.7)

where � is the collection of all basis function parameters de�ned as:

� =

�1�2...

�#

=

[θ1 θ2 θ3 θ4

]=

\1,1 \2,1 \3,1 \4,1\1,2 \2,2 \3,2 \4,2...

......

...

\1,# \2,# \3,# \4,#

(3.8)

8

Frequency-Domain Representation

Using the linearity property of the unilateral Laplace transform [14, p. 138], the approximation canbe written as follows in the B-domain:

(B) =#∑8=1

Φ8 =#∑8=1

Φ(B,�8 ) =.2#−1 (B,�)*2# (B,�)

(3.9)

which is a rational function that can be mapped to a LTISS model as discussed in Sec. 2.

State-Space Representation

IfA = diag (A1,A2, . . . ,A# ) and all {A1,A2, . . . ,A# } are square, then [17, p. 819]:

4AC = diag(4A1C , 4A2C , . . . , 4A# C

)(3.10)

This property of the matrix exponential allows us to combine the individual state-space systemsin Eq. (3.5) to create a complete realization of : :

AΦ = diag(Aq1 ,Aq2 , . . . ,Aq#

)=

Aq1 0 0 0

0 Aq2 0 0

0 0. . . 0

0 0 0 Aq#

2#×2#

(3.11a)

BΦ =

Bq1

Bq2...

Bq#

2#×1

(3.11b)

CΦ =[Cq1 Cq2 · · · Cq#

]1×2# (3.11c)

Note that this realization is essentially modal form with distinct complex-valued poles as discussedin Sec. 2.3.2.

3.1.3 Properties of this Representation

There are some properties of : (C) and (B) that are frequently desired to ensure the approximatedsystem is useful. Please refer to Refs. [2,4,7] for more details on these properties and why they arediscussed.

BIBO Stability

(B) will be bounded-input, bounded-output (BIBO) stable if all \2 > 0 because of exponentialdecay. This property is equivalent to:

limC→∞

: (C) = 0 (3.12)

Passivity

A SISO system is said to be passive if there exists an n ≥ 0 such that [18]:∫ C

0~ (g)D (g) ≥ −n for all C ≥ 0 (3.13)

9

i.e., if the absorbed energy by the system is greater than the stored energy in the system over thesame time horizon. Naturally-occurring physical systems are passive because they do not generateenergy on their own [19, p. 234]. For a LTISS system, the equivalent condition is [20]:

<( ( 9l)

)≥ 0, ∀l (3.14)

Because of the linearity of the chosen representation, the condition is equivalent to:

#∑8=1< (Φ8 ( 9l)) ≥ 0, ∀l (3.15)

Therefore, this condition will not generally be satis�ed unless some additional constraints areplaced on the parameters. One option is ensuring that each basis function represents a passivesystem. The condition for a single basis function Φ can be shown1 to simplify to I1 > 0:

\2 ≥ \3 tan(\4) (3.16)

Then this condition could be included as a constraint to ensure passivity. Alternatively, since\2 > 0 and \3 > 0, a simple condition would be c/2 ≤ \4 ≤ c or 3c/2 ≤ \4 ≤ 2c becausetan(\4) ≤ 0. This would limit the representation power of the basis functions but would guaranteepassivity of the realization. Another simple condition is \4 = − tan−1 (\2/\3) which is required inSec. 3.2.

Initial-Time Value

The initial-time value of : (C) is:

limC→0+

: (C) = :0 =

#∑8=1

\1,8 cos(\4,8 ) (3.17)

which generally implies that a speci�c value of :0 cannot be set.

Low-Frequency Asymptotic Value

The low-frequency asymptotic value of (B) is:

liml→0

( 9l) = 0 =

#∑8=1

\1,8\2,8 cos(\4,8 ) − \3,8 sin(\4,8 )

\ 22,8 + \ 2

3,8(3.18)

which generally implies that a speci�c value of 0 cannot be set.

High-Frequency Asymptotic Value

(B) is strictly proper since . has degree 2# − 1 and* has degree 2# . This property is equivalentto:

liml→∞

( 9l) = 0 (3.19)1To obtain this result multiply the complex fraction by the complex conjugate of the denominator, then simplify the

numerator only because the denominator is positive real. Now the real part is quadratic in l so the discriminant must benegative to ensure no real roots. The discriminant condition simpli�es to I1 > 0, which is the positivity condition. Notethat the more general condition is \2 cos(\4) ≥ \3 sin(\4) .

10

3.2 Reduction to the Golla-Hughes-McTavish Model

3.2.1 Basis Function

Frequency-Domain Representation

The frequency-domain basis function used in the Golla-Hughes-McTavish (GHM) model [9,16] is:

��" (B, o1, o2, o3) = o1B + 2o2o3

B2 + 2o2o3B + o23

(3.20)

where the coe�cients {o1, o2, o3} are all positive. To show the agreement between the Φ��" (B)and Φ(B), we need to assume the following condition:

\4 = − tan−1(\2

\3

)(3.21)

Now the relationships between the two sets of model parameters are:

o1 =\1\3√\ 2

2 + \ 23

, o1 ≥ 0 if \1 ≥ 0 and \3 ≥ 0 (3.22a)

o2 =\2√\ 2

2 + \ 23

, o2 ≥ 0 if \2 ≥ 0 (3.22b)

o3 =

√\ 2

2 + \ 23 (3.22c)

where we note that the poles and zeros are the same between the Φ��" and Φ so two states areneeded per basis function.

This alternative parameterization is frequently utilized because of its connection to the poles. o2is known as the natural damping ratio and o3 is known as the natural frequency. If o2 > 1 then wehave two distinct real-valued poles, and equality implies one distinct real-valued pole. If o2 < 1,then we have a single complex-valued pole pair. Therefore, so-called overdamped systems (o2 > 1)simply have two distinct real-valued poles and can be represented by the Prony Series in Sec. 3.3.This parameterization could be used with the PBFs, but its usefulness has not been explored in thisreport.

Time-Domain Representation

In the time domain, the GHM model is:

q��" (C, \1, \2, \3) = \14−\2C cos

(\3C − tan−1

(\2

\3

))(3.23)

or using the alternative parameterization:

q��" (C, o1, o2, o3) = o14−o2o3C

©­­«o2√o2

2 − 1sinh

(o3

√o2

2 − 1C)+ cosh

(o3

√o2

2 − 1C)ª®®¬ (3.24)

11

State-Space Representation

Applying the conditions of the GHM model to Eq. (3.5), the LTISS matrices for an individual basisfunction are:

Aq =

[−\2 \3−\3 −\2

](3.25a)

Bq =\1√\ 2

2 + \ 23

[−\2\3

](3.25b)

Cq =[0 1

](3.25c)

The matrices representing the individual basis functions can be combined in the same mannerpresented in Sec. 3.1.2.

3.2.2 Properties of this Representation

BIBO Stability

Similar to (B), ��" (B) will be BIBO stable if all \2 > 0.

Passivity

Since the GHM model has a direct mechanical analogy, it is expected that the system is passive.Observing the passivity condition in Eq. (3.15), we see that this condition is always satis�ed foreach individual q��" using the assumption in Eq. (3.21). Therefore, the GHM model is alwayspassive since each individual basis function is passive.

Initial-Time Value

Similar to : (C), a speci�c value of :0 cannot be set as the initial-time value is:

limC→0+

:��" (C) = :0 =

#∑8=1

\1,8\3,8√\ 2

2,8 + \ 23,8

(3.26)

Low-Frequency Asymptotic Value

Similar to (B), a speci�c value of 0 cannot be set as the low-frequency asymptotic value is:

liml→0

��" ( 9l) = 0 =

#∑8=1

\1,82\2,8\3,8(

\ 22,8 + \ 2

3,8

)3/2 (3.27)

High-Frequency Asymptotic Value

Similar to (B), ��" (B) is strictly proper.

12

3.3 Reduction to Prony Series (Generalized Maxwell Model)

3.3.1 Basis Function

Time-Domain Representation

The time-domain basis function used in the Prony Series (PS) model [15] is:

q%( (C, o1, o2) = o14−o2C (3.28)

To show the agreement between the PS model and q , we need to assume the following conditions:

\3 = 0 (3.29a)\4 = 0 (3.29b)

Now the relationships between PS and q model parameters are:

o1 = \1 (3.30a)o2 = \2 (3.30b)

Frequency-Domain Representation

In the frequency domain, there is a pole/zero cancellation so TF for the PS model is:

Φ%( (B) = \11

B + \2(3.31)

where there is only one real-valued pole per basis function (and therefore only one state is neededper basis function).

State-Space Representation

A state-space realization for the PS model can be readily obtained from Eqs. (3.5) and (3.11):

A%( = diag(−\2,1,−\2,2, · · · ,−\2,#

)=

−\2,1 0 · · · 0

0 −\2,2 · · · 0...

.... . .

...

0 0 · · · −\2,#

#×#(3.32a)

B%( =

\1,1...

\1,#

#×1

(3.32b)

C%( =[1 · · · 1

]1×# (3.32c)

3.3.2 Properties of this Representation

BIBO Stability

Similar to (B), %( (B) will be BIBO stable if all \2 > 0.

13

Passivity

Since the PS model has a direct mechanical analogy, it is expected that the system is passive. Ob-serving the passivity condition in Eq. (3.15), we see that this condition is always satis�ed for eachindividual q%( . Therefore, the PS model is always passive since each individual basis function ispassive.

Initial-Time Value

Similar to : (C), a speci�c value of :0 cannot be set as the initial-time value is:

limC→0+

:%( (C) = :0 =

#∑8=1

\1,8 (3.33)

However, a linear constraint could be included to ensure a speci�c value of :0 and reduce thenumber of parameters.

Low-Frequency Asymptotic Value

Similar to (B), a speci�c value of 0 cannot be set as the low-frequency asymptotic value is:

liml→0

%( ( 9l) = 0 =

#∑8=1

\1,8

\2,8(3.34)

High-Frequency Asymptotic Value

Similar to (B), %( (B) is strictly proper.

3.4 Comparison between the Models

Four di�erent representations for : have been presented: 1) the general LTISS system (e.g., com-panion or Modal form) in Sec. 2.3, 2) the Prony basis function system in Sec. 3.1, 3) the Golla-Hughes-McTavish model in Sec. 3.2, and 4) the Prony series (or generalized Maxwell model) inSec. 3.3. Each was abbreviated as GSS, PBF, GHM, and PS. As illustrated in Fig. 3.1, the methodshave the following hierarchy GSS⊃PBF⊃GHM⊃PS, indicating GSS is the least restrictive whilePS is the most. A summary of the considered representations is shown in Table. 3.1. Finally, allbasis functions are shown below and can provide insights into what IRFs can be accurately ande�ciently modeled by the chosen representation:

q�((,8 (C) =[Λ8−1∑ℓ=0

\1,ℓCℓ

] [4−\2C cos(\3C + \4)

](3.35a)

q8 (C) = \14−\2C cos(\3C + \4) (3.35b)

q��",8 (C) = \14−\2C cos

(\3C − tan−1

(\2

\3

))(3.35c)

q%(,8 (C) = \14−\2C (3.35d)

where basis function for GSS for repeated complex-valued roots is found in Ref. [21, p. 197].

14

TFs with repeated poles

TFs with distinct real poles

GSS

PBFGHM

PS

TFs with distinct complex poles

Figure 3.1: Comparison between the transfer function coverage of the di�erent representations.

Table 3.1: Summary of the considered representations.

Method : (C) |µ| Poles Repeated Poles BIBO Stability PassivityGSS C4ACB 4# C Yes No NoPBF

∑#8=1 q8 (C) 4# C No Yes No

GHM∑#8=1 q��",8 (C) 3# C No Yes Yes

PS∑#8=1 q%(,8 (C) 2# R No Yes Yes

15

Section 4

Fitting

The di�erent LTISS parameterizations are only useful if we can �nd values of the model parameterssuitably approximate the provided IRFs. Fitting here refers to �nding the values of the approximatemodel parameters µ that minimize some !? norm (? ≥ 1) of an error function �. The natural errorfunction is the di�erence between the original impulse function and the approximate one:

minµ

(∫ ∞

0|� (C,µ) |1/? 3C

)?= min

µ

(∫ ∞

0

���: (C) − : (C,µ)���? 3C )1/?(4.1)

Here we make the familiar choice of ? = 2 and cancel the outer 1/? exponent. Only for thesimplest analytical forms of : and : can the integral be evaluated analytically. An alternative is(weighted) numerical integration. Here we consider some �nite maximum value Cmax and a timegrid t consisting of < points between 0 and Cmax. With a proper selection of the weights w, thesolution using numerical integration will be similar to the original �tting problem:

minµ

∫ ∞

0

���: (C) − : (C,µ)���2 3C ≈ minµ

w ◦ [: (C) − : (C,µ)

]t

2

2(4.2)

While the appropriate selections of Cmax, t, and w = [F1,F2, · · · ,F<]T are critical to ensuring a

suitable �nal �t, they will not be discussed in this report.Most of the �tting problems that will be presented are in the class of nonlinear least-squares

problems. Due to the general nonconvex nature of these �tting problems, it will be importantto perform a global search procedure to improve the quality of the �nal solution, such as withmultistart methods [22, p. 364].

First, some optimization problem variations for companion form are shown in Sec. 4.1. Next,similar optimization problems are discussed with the Prony basis functions and related methodsin Sec. 4.2. This includes direct comparisons to the original Prony’s method [13, 23].

4.1 Optimization Problem with Companion Form

The model parameters when using companion form are the coe�cients of the TF in Eq. (2.6a), aand b. Then the optimization problem is:

mina,b

w ◦ [: (C) −C4A(a)CB(b)

]t

2

2(4.3)

In this form, there are 2= optimization variables and no useful bounds for their values withoutcompromising the allowable TFs. Also, the matrix exponential has to be computed for every pointin t. This method is denoted canon.direct.

4.1.1 Parameter Reduction using Linear Least SquaresWe note that the coe�cients b have a linear dependence in the impulse response. Least-squaresmodel �tting with linear models is relatively easy, motivating an alternative optimization strategywhere we solve for the optimal values of b for a given a. This is a nested form of the original

16

problem in Eq. (4.3) and is only attractive if there are e�ective optimization methods that couldnot be used in the simultaneous form. Fortunately, the solution of the inner-loop optimizationproblems can be found readily using linear least-squares estimation methods [22, p. 42]. Thisequivalent formulation is:

mina,b

w ◦ [: (C) −C4A(a)CB(b)

]t

2

2= min

a

w ◦ [:]t −X (a)B† (a) 22 (4.4)

where:

X (a)<×= = w ◦[C4A(a)C

]t

(4.5a)

B† (a)=×1 = w ◦X+ [:]t (4.5b)

The number of parameters reduces from 2= to =. This method is denoted canon.direct.ls.

4.1.2 Directly Fitting the Poles and ZerosIn Eq. (2.6a), the equivalence between the companion form and the poles and zeros of the TF wasshown. We will generally assume complex conjugate pairs for both the poles and zeros, althoughthis need not be the case (i.e., a mixed distribution of the real and complex roots). Therefore, thepoles are de�ned by p =

[pA + p2 9 pA − p2 9

]and zeros similarly by z =

[zA + z2 9 zA − z2 9

].

Then an alternative strategy is to �t the poles and zeros directly:

minzA ,z2 ,pA ,p2

w ◦ [: (C) −C4A(pA ,p2 )CB(zA , z2 )

]t

2

2(4.6)

where there are generally 2= parameters. The primary advantage of this representation is theability to place limits on the magnitude of poles, such as all ?A < 0 ensuring BIBO stability. Thismethod is denoted canon.roots.

In the same manner as canon.direct.ls in Sec. 4.1.1, parameter reduction using linear leastsquares is possible for canon.roots. The number of parameters reduces from 2= to = and theconstraints on the poles can still be readily included. This method is denoted canon.roots.ls.

4.2 Optimization Problem with Prony Basis Functions

The approximate model parameters when using the PBF basis functions are � and the optimizationproblem is:

min�

w ◦[: (C) −

#∑8=1

q (C,�8 )] 2

2

(4.7)

In this form, we do not need to compute the matrix exponential at any point. There are 4# = 2=optimization variables. This method is denoted basis.pbf.

Similar optimization problems can be formed using the GHM and PS basis functions and theirappropriate model parameters. Using the GHM basis functions there are 3# = 3=/2 optimizationvariables, while for PS, there are 2# = 2=. These methods are denoted basis.ghm and basis.ps,respectively.

17

4.2.1 ConstraintsWe also can impose some constraints on the model parameters. Some of the constraints are basedon the desired properties discussed above, some are natural constraints, and some are to bias thesearch process to a general region of interest:

(amplitude) \1,min ≤\1 ≤ \1,max (4.8a)(decay rate) 0 < \2,min ≤\2 ≤ \2,max (4.8b)(frequency) 0 ≤\3 ≤ \3,max (4.8c)

(phase shift) 0 ≤\4 ≤ c (4.8d)

The speci�c choice of the bounds is problem speci�c. For example, the value of \2,min can beselected such that the : decays to (near) zero in a certain amount of time, which frequently is adesirable property. Additional nonlinear constraints could also be included such as the passivityconstraint in Eq. (3.15).

For basis.ps, the frequency and phase shift constraints are not necessary. For basis.ghm,constraints on the amplitude, decay rate, and frequency can be readily included as the relationshipbetween the two sets of model parameters was shown in Eq. (3.22). Alternatively, simple boundconstraints on {o1, o2, o3} is possible and can ensure many of the same properties as the onesshown in Eq. (4.8). Additionally, a nonlinear constraint on the phase shift could be included usingEq. (3.21).

4.2.2 GradientIn this section, we analyze the gradient of the objective in Eq. (4.7). Providing this information tothe optimization routine reduces the number of function calls of : (C) and improves the accuracy ofthe derivatives (as opposed to a �nite di�erencing scheme) [22]. Since the error terms are combinedthrough linear summation, we will consider the gradient for any C :

m

m�[� (C,�)]2 = 2� (C,�)

[m

m�� (C,�)

](4.9a)

= −2

[#∑8=1

q (C,�8 )] [

m

m�

#∑8=1

q (C,�8 )]

(4.9b)

We note that : (C,�) is needed in every derivative but is already computed when evaluating �.Focusing on the other term, we note the linear summation of the basis functions. Since the param-eters �8 only appear in the 8-th basis function, the derivatives for the ℓ-th basis function can besimpli�ed as:

m

m�ℓ

#∑8=1

q (C,�8 ) =m

m�ℓq (C,�ℓ ) (4.10a)

=

[mq (C,�ℓ )m\1,ℓ

mq (C,�ℓ )m\2,ℓ

mq (C,�ℓ )m\3,ℓ

mq (C,�ℓ )m\4,ℓ

](4.10b)

18

The requested derivatives for each of the four parameters are then:

mq (C,�ℓ )m\1,ℓ

= 4−\2,ℓ C cos(\3,ℓC + \4,ℓ ) (4.11a)

mq (C,�ℓ )m\2,ℓ

= −\1,ℓC4−\2,ℓ C cos(\3,ℓC + \4,ℓ ) (4.11b)

mq (C,�ℓ )m\3,ℓ

= \1,ℓC4−\2,ℓ C sin(\3,ℓC + \4,ℓ ) (4.11c)

mq (C,�ℓ )m\4,ℓ

= \1,ℓ4−\2,ℓ C sin(\3,ℓC + \4,ℓ ) (4.11d)

For basis.ghm, the derivatives using the {\1, \2, \3} parameterization are (dropping the subscriptℓ for conciseness):

mq��" (C)m\1

= 4−\2C cos(\3C − tan−1

(\2

\3

))(4.12a)

mq��" (C)m\2

= \14−\2C

[ [\3

\ 22 + \ 2

3

]sin

(\3C − tan−1

(\2

\3

))− C cos

(\3C − tan−1

(\2

\3

))](4.12b)

mq��" (C)m\3

= −\14−\2C

[\2

\ 22 + \ 2

3+ C

]sin

(\3C − tan−1

(\2

\3

))(4.12c)

Similar derivatives can be found for the alternative representation in Eq. (4.13), but are quite in-volved.

For basis.ps, the derivatives are:

mq%( (C,�ℓ )m\1,ℓ

= 4−\2,ℓ C (4.13a)

mq%( (C,�ℓ )m\2,ℓ

= −\1,ℓC4−\2,ℓ C (4.13b)

4.2.3 Parameter Reduction using Linear Least SquaresConsider Eq. (3.2b) when \2 and \3 are �xed. Then we have:

q (C, \ ) =[4−\2C cos(\3C) 4−\2C sin(\3C)

] [\1 cos(\4)−\1 sin(\4)

](4.14a)

=[4−\2C cos(\3C) 4−\2C sin(\3C)

] [\5\6

](4.14b)

= i (C, \2, \3)[\5\6

](4.14c)

where we now note that the additional parameters \5 and \6 have linear dependence in i . We canreadily obtain the original model parameters with:

\1 = sign(\5)√\ 2

5 + \ 26 (4.15a)

\4 = 2 tan−1 ©­­«\5 − sign(\5)

√\ 2

5 + \ 26

\6

ª®®¬ (4.15b)

19

Least-squares model �tting with linear models is relatively easy, motivating an alternative opti-mization strategy where we solve for the optimal values of all \5 and \6 for given θ2 and θ3. Thisis a nested form of the original problem in Eq. (4.7), and is only attractive if there are e�ectiveoptimization methods that could not be used in the simultaneous form. Fortunately, the solutionof the inner-loop optimization problems can be found readily using least-squares estimation meth-ods [22, p. 42].

The optimization formulation is now:

min�

‖w ◦ [:]t −X (θ2, θ3) β (θ1, θ4)‖22 = minθ2,θ3

w ◦ [:]t −X (θ2, θ3) β† (θ2, θ3) 2

2 (4.16)

where:

X (θ2, θ3)<×2# = w ◦[i (C, \2,1, \3,1) i (C, \2,2, \3,2) · · · i (C, \2,# , \3,# )

]t

(4.17a)

β† (θ2, θ3)2#×1 = w ◦X+ [:]t (4.17b)

The number of parameters reduces from 4# to 2# . This method is denoted basis.pbf.ls. Notethat this method is only directly optimizing the roots of the system ({θ2, θ3}).

For basis.ghm, recall that \4 is a function of \2 and \3. Therefore, we can utilize a similarleast-squares procedure for \1. Now the number of parameters is 2N, and this method is denotedbasis.ghm.ls. Note that this method is only directly optimizing the roots of the system, similarto basis.pbf.ls, but with the additional assumption in Eq. (3.21). For basis.ps, we can utilizea similar least-squares procedure for \1. Now the number of parameters is # and this method isdenoted basis.ps.ls.

4.2.4 Prony’s MethodProny’s method (or Prony analysis) uses the PBF and seeks to approximate : (C) as the sum of Nbasis functions [13, 23]:

: (C) =N∑8=1

q8 (C), where: q (C, \ ) = \14−\2C cos(\3C + \4) (4.18)

There are some similarities and di�erences between Prony’s method and the di�erent �tting pro-cedures described in this section. Prony’s method is denoted prony and is now described.

1. Select a number of basis functions N , time grid points<, and a �nite maximum time valueCmax such that the condition< ≥ 2N is satis�ed (so that we do not have an underdeterminedsystem)1. The time grid tmust be constructed using equidistant points with sampling period)B = Cmax/(< − 1).

2. The summation of complex exponentials is the homogeneous solution to a linear di�erenceequation. To determine the homogeneous solution, we can �nd the coe�cients of the asso-ciated characteristic equation [24, p. 75]. First construct a system of linear equations using[:]t. Then �nd the least-squares solution to this system of equations, denoted c, which arethe required coe�cients.

: (CN) : (CN−1) · · · : (C1): (CN+1) : (CN) · · · : (C2)

......

. . ....

: (C<−1) : (C<−2) · · · : (C<−N)

(<−N)×N2122...

2N

N×1

= −

: (CN+1): (CN+2)

...

: (C<)

(<−N)×1

(4.19)

1If we select< = 2N, then Prony’s method interpolates the data points (i.e., the values exactly match).

20

3. Now, �nd the roots of the N -degree polynomial formed using c as the coe�cients whichdetermines the homogeneous solution:

roots(/N + 21/

N−1 + · · · + 2N)=

A1...

AN

= 4)Bp =

4)B [−\2,1+9\3,1 ]

...

4)B [−\2,N+9\3,N ]

(4.20)

where p are the (distinct) poles discussed in Sec. 2.3.2 potentially containing both real-valuedand complex-valued poles. Complex-valued poles will be present in complex conjugate pairs.These roots are then related to the model parameters by:

\2,8 =<(log(A8 ))

)B(4.21a)

\3,8 ==(log(A8 ))

)B(4.21b)

4. Finally, construct a system of linear equations using [:]t, θ2, and θ3. Find the least-squaressolution to this system of equations2 denoted d:

4 [−\2,1+9\3,1 ]C1 · · · 4 [−\2,# +9\3,# ]C1

.... . .

...

4 [−\2,1+9\3,1 ]C< · · · 4 [−\2,# +9\3,# ]C<

<×#31...

3#

#×1

=

: (C1)...

: (C<)

<×1

(4.22)

The solution to this system of equations is then related to the model parameters by:

\1,8 = |38 | (4.23a)

\4,8 = tan−1(=(38 )<(38 )

)(4.23b)

because 38 = \1,849\4,8 .

There will be between N/2 and N distinct basis functions depending on the number of dis-tinct real-valued poles. If all the poles are complex-valued, then we have # = N/2 distinct basisfunctions because the parameters from these pole pairs can be combined into a single equivalentbasis function. In any case, there will be 2# states (unless there are any sets of degenerate modelparameters such as \1 = 0). Since the same basis functions are used, the techniques in Sec. 3.1.2can be utilized. Therefore, prony can be used to construct real-valued LTISS approximations.

The prony method requires no nonlinear optimization problems, but only the solution of twolinear least-squares problems and an eigenvalue problem. It also theoretically covers the same TFsas the methods that use the PBF. However, prony requires a �xed time grid, cannot include theweightsw, cannot easily include any constraints on the parameters, and cannot easily specify anyproperties of the realization.

4.3 Summary

Here we summarize the problem formulations presented in this section for �tting a LTISS modelto speci�ed IRFs in Table 4.1.

2Note that this step is functionally the same as the process in Sec. 4.2.3, i.e., construct a linear least-squares problemwhen θ2 and θ3 are �xed. The only di�erence is the procedure used in Prony’s method operates on complex-valued linearsystems, while the method described in Sec. 4.2.3 uses real-valued linear systems. The results should be the same.

21

The method full directly uses the entries of the state-space matrices as optimization variablesand has no real advantages over the other methods. The method canon.direct was described inSec. 4.1 using the companion form and greatly reduces the number of parameters without loss ofgenerality. The method canon.direct.ls further reduces the number of parameters through anested linear least-squares procedure and was described in Sec. 4.1.1. The methods canon.rootsand canon.roots.ls were described in Sec. 4.1.2, and use the poles parameterization of the de-nominator of the TF, allowing for some useful constraints to be placed on the values of the modelparameters. All of these methods would require computing the matrix exponential during theoptimization procedure.

The method prony was described in Sec. 4.2.4 using the PBF, but a di�erent �tting procedureconsisting of two systems of linear equations and an eigenvalue problem. Because of this approach,no useful constraints can be placed on the model parameters among other potential issues. It doeshave the advantage that there is no longer a need to compute the matrix exponential. Next arethe basis.X methods which were described in Sec. 4.2, and each use a speci�c basis function,respectively. Each can include the useful constraints and have analytic gradients. The main trade-o�s are related to what TFs can be approximated and various properties of the LTISS model. Finallyare the basis.X.ls methods which were described in Sec. 4.2.3, and use the linear-least squaresproblem to reduce the number of free model parameters. Thus, the methods have a reduction inthe number of parameters, but no longer have a straightforward analytic gradient.

Table 4.1: Summary of the considered �tting problem formulations.

Method : µ |µ|/= Gradient Constraintsfull C4ACB Matrix entries = + 2 No Nocanon.direct C4ACB a, b 2 No Nocanon.direct.ls C4ACB a 1 No Nocanon.roots C4ACB pA ,p2 , zA , z2 2 No Yescanon.roots.ls C4ACB pA ,p2 1 No Yesprony

∑N8=1 q8 (C) θ1, θ2, θ3, θ4 2 − No

basis.pbf∑#8=1 q8 (C) θ1, θ2, θ3, θ4 2 Yes Yes

basis.pbf.ls∑#8=1 q8 (C) θ2, θ3 1 No Yes

basis.ghm∑#8=1 q��",8 (C) θ1, θ2, θ3 3/2 Yes Yes

basis.ghm.ls∑#8=1 q��",8 (C) θ2, θ3 1 No Yes

basis.ps∑#8=1 q%(,8 (C) θ1, θ2 2 Yes Yes

basis.ps.ls∑#8=1 q%(,8 (C) θ2 1 No Yes

Recall the pA = −θ2 and p2 = θ3 if the distribution of poles is assumed to be only consisting of complex conjugate pairs.

22

Section 5

Numerical Examples

In this section, a few numerical examples are provided to illustrate the potential use of PBFs to ap-proximate a variety of di�erent IRFs. This includes Maxwell materials, wave forces, and arbitrarily-decreasing functions. Note that the found realization may not be optimal with respect to the chosennumber of states. Additional studies and tool development is needed to be able to make such claimswith con�dence.

5.1 Models for Some Impulse Response Functions

Below are six IRFs that are approximated using PBFs:

(Maxwell material) :1 (C) = 0.14−C/2 + 0.24−2C + 0.54−3C (5.1a)

(Power law) :2 (C) = min([C + 0.99]−1/4, 1

)(5.1b)

(Wave forces) :3 (C) = #3 in MarinSemi097.mat from Ref. [20] (5.1c)(Step-like) :4 (C) = −0.5 tanh(14[C − 0.5]) + 0.5 (5.1d)

(Arbitrarily decreasing) :5 (C) = −0.5 tanh(50[C − 0.25]) − 0.5 tanh(5[C − 0.75]) + · · · (5.1e)− tanh(20[C − 2]) + 2;

(Inverse Gamma) :6 (C) =1

Γ(C) (5.1f)

Some of the selected : (C) are based on IRFs found in the literature, while others are created arbi-trarily to illustrate interesting IRFs approximations.

The model parameters were found using the code described in App. 6. Both : (C) and [:]t areshown in Figs. 5.1 and 5.2. The individual basis function contributions are shown in Fig. 5.3. Finally,all (B) are shown in Fig. 5.4.

23

(a) :1: Maxwell material with = = 3. (b) :2: Power law with = = 13.

(c) :3: Wave forces with = = 6. (d) :3: Wave forces with = = 10.

(e) :4: Step-like with = = 6. (f) :4: Step-like with = = 10.

(g) :5: Arbitrarily decreasing with = = 12. (h) :6: Inverse Gamma with = = 4.

Figure 5.1: Fitted impulse response functions.

24

(a) :1: Maxwell material with = = 3. (b) :2: Power law with = = 13.

(c) :3: Wave forces with = = 6. (d) :3: Wave forces with = = 10.

(e) :4: Step-like with = = 6. (f) :4: Step-like with = = 10.

(g) :5: Arbitrarily decreasing with = = 12. (h) :6: Inverse Gamma with = = 4.

Figure 5.2: Extended response of the �tted impulse response functions.

25

(a) :1: Maxwell material with = = 3. (b) :2: Power law with = = 13.

(c) :3: Wave forces with = = 6. (d) :3: Wave forces with = = 10.

(e) :4: Step-like with = = 6. (f) :4: Step-like with = = 10.

(g) :5: Arbitrarily decreasing with = = 12. (h) :6: Inverse Gamma with = = 4.

Figure 5.3: Basis function contributions to the �tted impulse response functions.

26

(a) :1: Maxwell material with = = 3. (b) :2: Power law with = = 13.

(c) :3: Wave forces with = = 6. (d) :3: Wave forces with = = 10.

(e) :4: Step-like with = = 6. (f) :4: Step-like with = = 10.

(g) :5: Arbitrarily decreasing with = = 12. (h) :6: Inverse Gamma with = = 4.

Figure 5.4: Transfer functions of the �tted impulse response functions.

27

5.2 Simulation Results

One of the primary motivators for constructing the LTISS approximations is to improve simulationspeed. In this section, a few di�erent IRFs from Sec. 5.1 are used with the following di�erentialequation:

¥G (C) = −10G (C) −∫ C

−∞: (C − g) ¤G (g)3g + sin(C) (5.2)

Four simulation variations are tested: 1) no convolution integral (CI) as baseline, 2) using theLTISS model from Sec. 5.1, 3) using the trapezoidal rule to compute the CI, and 4) using an accurateadaptive quadrature method to compute the CI. The simulation computational expenses relative tothe using no CI are shown in Table 5.1. The simulation results for ¤G (C) are shown in Fig. 5.5. Notethe di�erent responses when : changes. We see a substantial reduction in computational expenseusing the LTISS approximations and from Fig. 5.5, the dynamic responses are nearly the same.

Table 5.1: Relative computational expense for the simulations.

: = no CI LTISS CI trapz CI direct CI:1 3 1.0 2.2 68.8 789.0:4 6 1.0 1.9 46.1 476.2:4 10 1.0 2.1 56.4 517.1:6 4 1.0 1.1 50.9 456.9

(a) :1: Maxwell material with = = 3. (b) :4: Step-like with = = 6.

(c) :4: Step-like with = = 10. (d) :6: Inverse Gamma with = = 4.

Figure 5.5: Simulation results.

28

Section 6

Supporting Code

The results shown in Sec. 5 were found using impulse2LTI, a MATLAB-based LTISS �tting pro-cedure using PBFs. It is available at:

https://github.com/danielrherber/impulse-2-lti

Alternative software for creating approximations for IRFs include SS_Fitting [7,20] and MSS FDItoolbox [25].

29

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