# Spatio-temporal Transfer Function Conditions of Positive Realness for Translation Invariant Lattice Networks of Interacting Linear Systems

###### Abstract

This paper is concerned with networks of interacting linear systems at sites of a multidimensional lattice. The systems are governed by linear ODEs with constant coefficients driven by external inputs, and their internal dynamics and coupling with the other component systems are translation invariant. Such systems occur, for example, in finite-difference models of large-scale flexible structures manufactured from homogeneous materials. Using the spatio-temporal transfer function of this translation invariant network, we establish conditions for its positive realness in the sense of energy dissipation. The latter is formulated in terms of block Toeplitz bilinear forms of the input and output variables of the composite system. We also discuss quadratic stability of the network in isolation from the environment and phonon theoretic dispersion relations.

Translation invariant networks, spatio-temporal frequency domain, energy balance relations.

MSC Codes — 93C05, 90B10, 37L60, 37K05, 93C80.

## I Introduction

Complex physical systems can be viewed as a large number of relatively simple subsystems whose collective behaviour is a cumulative effect of their interaction rather than a particular individual structure. Spatially homogeneous states of matter are modelled as identical building blocks which interact with each other in a translationally invariant fashion. A natural example of large-scale composite systems with translational symmetry is provided by crystalline solids, where spatially periodic arrangements of constituent particles result from their interaction and play an important role in their thermodynamic and mechanical properties (including the heat transfer and wave propagation) studied in the phonon theory [10].

Modern engineering exploits translation invariant interconnections in drone swarming, vehicle platooning and artificially fabricated metamaterials [12], such as split ring resonator arrays with unusual electrodynamic characteristics (a negative refraction index). Nontrivial input-output properties of such networks of systems (natural or artificial) are not merely a “sum” of individual internal dynamics of their constituent blocks and come from a specific structure of energy flows through the translation invariant interaction.

The energy balance relations, which reflect the energy conservation and dissipation in isolated and open systems (for example, due to electrical resistance and mechanical friction), significantly affect the behaviour of physical systems and play an increasingly important role in control design [4, 5, 11]. These equations involve the internal energy and the work done on the system (which are represented in the dissipativity theory [14] in terms of storage and supply rate functions). Work is modelled by using a bilinear form of the input and output variables, which are interpreted as the generalised force and velocity respectively. For linear time-invariant systems with a finite-dimensional internal state, the properties of being passive, positive real or negative imaginary (in the case of position variables instead of the velocity as the output) admit criteria in the form of linear matrix inequalities for the transfer functions in the frequency domain or the state-space matrices themselves [7, 15].

The present paper is concerned with similar conditions for networks of interacting linear systems at sites of a multidimensional lattice. The composite system is governed by an infinite set of linear ODEs with constant coefficients driven by external inputs, and their internal dynamics and coupling with the other component systems are translation invariant. These ODEs have block Toeplitz state-space matrices and can be represented in the spatio-temporal frequency domain by using appropriately modified transfer functions of several variables. Such systems arise, for example, as finite-difference approximations of PDEs for large-scale flexible structures made of spatially homogeneous materials. Using the spatio-temporal transfer function of this translation invariant network, we establish conditions for its positive realness in the sense of energy dissipation. The latter is formulated in terms of block Toeplitz bilinear forms of the input and output variables of the composite system. The multivariate Laplace and Fourier transform techniques, which are used for this purpose, are similar to those for distributed control systems in the classical and quantum settings [9, 13].

The paper is organised as follows. Section II describes the class of translation invariant networks under consideration. Section III represents the network dynamics in terms of the spatial Fourier transforms of its signals. Section IV discusses energy balance relations in the case of bilinear supply rate and quadratic storage functions. Section V establishes conditions for passivity of the network in terms of its spatio-temporal transfer function, and also discusses quadratic stability bounds for a dissipative network in isolation from the environment. Section VI considers phonon theoretic dispersion relations for the isolated network. Section VII provides concluding remarks.

## Ii Translation Invariant Networks

We consider a translation invariant network of coupled linear systems at sites of the -dimensional integer lattice . For any spatial index , the th system is endowed with an -valued vector of time-varying state variables (for example, the generalised positions and velocities). Associated with the th lattice site are vectors and of external input and output variables which take values in and , respectively, and also vary in time (unless indicated otherwise, vectors are organised as columns). The states and outputs of these systems are governed by an infinite set of coupled ODEs

(1) | ||||

(2) |

for all , where is the time derivative (the time arguments are often omitted for brevity). Their right-hand sides are organised as convolutions, with the matrices , , , depending on the relative location of the lattice sites in accordance with the translation invariance of the individual dynamics of the systems and their coupling. By assembling the inputs, states and outputs into the infinite-dimensional vectors , , , the set of ODEs (1) and (2) can be written as

(3) | ||||

(4) |

where the matrices , , , are block Toeplitz in the sense of the additive group structure of the lattice . The ODE (3) is understood as a Volterra integral equation (of the second kind)

(5) |

whose solution is given by

(6) |

For completeness, we note that (3)–(6) can be endowed with a rigorous meaning as follows. To this end, the state-space matrices in (1) and (2) are assumed to be square summable:

(7) |

where is an arbitrary matrix norm (whose particular choice is irrelevant in this case). The fulfillment of (7) allows the Fourier series

(8) |

to be defined (in a blockwise fashion) for almost all as the -limit of appropriate partial sums over finite subsets of the lattice forming an exhausting sequence. Since the matrix-valued functions , , , (with values in , , , , respectively) are -periodic with respect to each of their arguments, then, without loss of generality, they can be considered on a -dimensional torus (where can be identified with the interval ). These functions are Hermitian in the sense that (and similarly for the other functions , , ) for all , where is the complex conjugate. Furthermore, let , , denote the norms of the input, state and output of the network at a fixed but otherwise arbitrary moment of time in the corresponding Hilbert spaces

(9) |

of appropriately dimensioned square summable real vector-valued functions and on the lattice with the inner product . Now, suppose the state-space matrices in (3) and (4) specify bounded linear operators , , , . This is equivalent to their -induced operator norms being finite:

(10) | ||||

(11) | ||||

(12) | ||||

(13) |

where the essential supremum is applied to the operator norms of the matrices in (8). In particular, if the matrices , , , vanish for all with large enough (so that each of the component systems in (1) and (2) is coupled with a finite number of the other systems in the network), then the functions , , , in (8) are multivariate trigonometric polynomials, and the conditions (10)–(13) are satisfied in this case. In general, the fulfillment of (10) and (11) guarantees that the operator exponential and the product in (6) are bounded block Toeplitz operators acting on the Hilbert space and from to , respectively, with

(14) |

for any , in view of the submultiplicativity of the operator norm and the fact that block Toeplitz matrices form an algebra. Therefore, if the initial network state is square summable, that is,

(15) |

and the network input is locally absolutely integrable with respect to time in the sense that

(16) |

then these properties are inherited by the subsequent states of the network. Indeed, a combination of (6) with (14)–(16) leads to

(17) |

for all , and hence,

(18) |

for any . Together with (4), (12), (13), the property (17) implies that the output norm is finite for almost all since so is in view of (16). Furthermore, due to (18), the network output is also locally absolutely integrable with respect to time:

(19) |

for any . In Section IV, we will replace (16) with a stronger condition on the network inputs in order to guarantee time-local square integrability for the outputs instead of (19).

## Iii Network Dynamics in the Spatial Frequency Domain

The preservation of the spatial square summability (17) (provided the input satisfies (16)) allows the network dynamics (3) and (4) to be represented in the spatial frequency domain as

(20) | ||||

(21) |

Here, similarly to (8), the functions , , on with values in , , are the Fourier transforms of the input, state and output of the network:

(22) | ||||

(23) | ||||

(24) |

for almost all at time . For any given , the equations (20) and (21) describe an autonomous system (which is independent of the other systems in this parametric family with different values of ) with a finite-dimensional internal state . Their solution can be represented in terms of the Laplace transform over the time variable as

(25) | ||||

(26) |

where

(27) |

Here,

(28) |

is the spatio-temporal transfer function of the network with values in . In view of (10), the relations (25)–(28) are valid for all satisfying

(29) |

where denotes the spectral radius of a square matrix (so that is the largest real part of the eigenvalues of , with denoting its spectrum). The presence of the upper limit in (29) makes the integral in (27) convergent in the Hilbert space due to the Parseval identity for any . In the case when the input vanishes, so that the network is effectively isolated from the environment, (20) reduces to a homogeneous linear ODE . Its solution , considered for different values of , admits a direct analogy (formulated in system theoretic terms) with the phonon theory [10] of collective oscillations in spatially periodic arrangements of atoms in crystalline solids. More precisely, for a given , let be an eigenvector of the matrix associated with its eigenvalue . Then the functions

(30) |

satisfy the ODEs (1) with . Indeed, substitution of the latter equality and (30) into the right-hand side of (1) yields . For purely imaginary eigenvalues of the matrix (with ), the solutions (30) describe persistent oscillations of frequency in the network, which are organised as (hyper) plane waves of length in . Their wavefronts are orthogonal to the vector and move (along ) at constant phase velocity , with the direction depending on the sign of . Similarly to the dispersion relations of the phonon theory, the spectral structure of such oscillations in the network is represented by the multi-valued map (of the wave vector to the frequency), which will be discussed in Section VI.

## Iv Supply Rate and Energy Balance

A generalised model for the work done by the input on the network is provided by a supply rate [14] at time in the form

(31) |

where the Parseval identity is applied to the Fourier transforms and from (22) and (24). The quantity is a bilinear function of the current input and output of the network which is specified by a block Toeplitz matrix , where are given matrices satisfying

(32) |

with the Fourier transform

(33) |

It is assumed that the matrix describes a bounded linear operator from the output space in (9) to the input space in (9), so that

(34) |

similarly to the operator norms in (10)–(13). For example, the condition (34) holds in the case of a standard supply rate, when and consist of the corresponding force and velocity variables (with ) and is the identity operator. Returning to the general case, substitution of (21) into (31) leads to

(35) |

where

(36) |

is the Fourier transform of the augmented state-input pair of the network at time . Now, let be a symmetric block Toeplitz matrix, which is specified by a square summable matrix-valued function on the lattice satisfying . Then the corresponding Fourier transform

(37) |

takes values in the subspace of complex Hermitian matrices of order and satisfies

(38) |

Under the additional condition

(39) |

the matrix describes a self-adjoint operator on the Hilbert space in (9). This gives rise to a quadratic form of the network state at time :

(40) |

where

(41) |

Here, the Parseval identity is applied to the Fourier transform from (23), whose contribution to at a given is quantified by the real-valued quantity in (41). The spatial frequency domain representation (20) of the network state dynamics allows the time derivative of in (40) to be computed as

(42) |

with

(43) |

where use is made of (36). If in (40) describes the internal energy (usually referred to as the Hamiltonian) of the network at the current moment of time , then the difference

(44) |

can be interpreted as the energy dissipation rate. This quantity is part of the supply rate in (31) which is not accounted for by the rate of change of the internal energy in (42). The complex Hermitian matrix

(45) |

of order in (44) is obtained by using (35), (42) and (43), and satisfies

(46) |

similarly to (38). In order for the network model to correspond to a real physical system, the dissipation rate in (44) has to be nonnegative in view of the total energy conservation, and hence,

(47) |

A sufficient condition for this inequality to hold for arbitrary admissible inputs and initial network states is positive semi-definiteness of the matrix (45):

(48) |

This condition is also necessary under an additional controllability assumption.

###### Theorem 1

In view of (44), the fulfillment of (48) implies (47) regardless of the controllability of , thus proving the sufficiency of (48), mentioned above. In order to show the necessity, let be a measurable function which is Hermitian (that is, for any ) and such that is a unit eigenvector associated with the smallest eigenvalue

(49) |

of the Hermitian matrix in (45), so that, in accordance with (46),

(50) |

Note that, due to (46), the function in (49) is symmetric. Therefore, if the above controllability condition is satisfied, then for any time horizon , there exists an admissible network input on the time interval such that the vector (36) satisfies

(51) |

for any . For such an input, in view of (49)–(51), the energy dissipation rate (44) takes the form

(52) |

If the property (48) does not hold, then the set is of positive Lebesgue measure, and the right-hand side of (52) is negative, thus contradicting (47). This proves that, under the controllability assumption on the pair , the condition (48) is necessary for the network dissipativity.

If the self-adjoint operator is positive semi-definite (which is equivalent to that the corresponding Fourier transform in (37) satisfies for almost all ), and the network is initialized at zero state , then the integration of both parts of (47) over a time interval leads to

(53) |

for any . These relations hold for any network input such that is a locally square integrable function of time in the sense that

(54) |

The condition (54) is stronger than (16) and guarantees finiteness of the work by such an input on the network over any bounded time interval. More precisely, (31) implies that

(55) |

in view of the Cauchy-Bunyakovsky-Schwarz inequality and the boundedness (34) of the operator . Here, the time-local square integrability of the input allows (19) to be enhanced as

(56) |

which is obtained by applying the inequality in an arbitrary Hilbert space to the right-hand side of (4) and using the boundedness (12) and (13) of the operators and . Also, (17) implies that

(57) |

where the Cauchy-Bunyakovsky-Schwarz inequality is used again. The local square integrability of as a function of time follows from (57) in view of (15) and (54). In combination with (56), this ensures the time-local square integrability of the network output: for any . From the last property, (54) and (55), it follows that the supply rate is indeed locally integrable with respect to time.

## V Network Passivity Conditions in the Spatial Frequency Domain

The relations (53) imply that the network is passive in the sense that any time-locally square integrable input in (54) performs a nonnegative work:

(58) |

The passivity can also be considered irrespective of a specific internal energy (storage) function (provided the initial network state is zero). For what follows, the network input is assumed to be time-space square summable in the sense that

(59) |

where the norm is associated with the inner product for real or complex vector-valued functions and of the time and space variables. By the Parseval identity, (59) yields , where the function is given by (27). Although (59) is a stronger condition than (54), the input produces the same work in (58) over a given time interval as its “truncated” version

(60) |

(with being an orthogonal projection operator for any ). This property follows from causality of the network as an input-output operator and from the fact that the supply rate in (31) is a bilinear function of and .

###### Theorem 2

Suppose the network, governed by (1) and (2), satisfies (7) and (10)–(13), and is endowed with the supply rate (31) subject to (32) and (34). Also, suppose the matrix in (8) is Hurwitz for any . Then the network is passive in the sense of (58) for zero initial states and arbitrary time-locally square integrable inputs in (54) if and only if its spatio-temporal transfer function (28) satisfies

(61) |

for almost all . Here, the function is given by (33).

Assuming that the network is initialized at zero, consider the work up until a given time horizon . Since an arbitrary admissible input (in the sense of (54)) can be replaced with in (60) without affecting the work, we will assume, without loss of generality, that vanishes beyond the time interval , and hence, so also does the supply rate in (31). For any such input, substitution of (31) into (58) leads to

(62) |

where the function in (28) is well-defined on the set (with the imaginary axis) due to the matrix being Hurwitz for all . Here, in view of (29), the Parseval identity is used together with the Fourier transforms (26), (27) and the Hermitian matrix in (61). In view of (62), the positive semi-definiteness of almost everywhere in implies that for all admissible inputs vanishing outside the time interval . The necessity of the matrix inequality in (61) for the network passivity can be obtained by letting and considering all possible time-space square summable inputs satisfying (59).

Note that in the case of and the standard supply rate mentioned above, in (33) is the identity matrix, and (61) reduces to

(63) |

which is a network counterpart of the positive real property. On the other hand, in an extended setting, the block Toeplitz matrix can be formed from differential operators with respect to time (whose entries are, for example, polynomials of ). In this case, Theorem 2 is modified by replacing the function in (61), given by (33), with the Fourier-Laplace transform

(64) |

In particular, if and the extended operator acts on the network output as , then (64) yields . In accordance with the structure of the supply rate (31), this describes the setting when and consist of the corresponding force and position (rather than velocity) variables. In this case, (61) takes the form

(65) |

Since the spatio-temporal transfer function in (28) satisfies for all , , and the complex conjugation of a Hermitian matrix preserves positive semi-definiteness, then (65) is equivalent to

(66) |

Similarly to (63), the condition (66) is a network version of the negative-imaginary property [7, 15]. In the case when the matrix in (8) is Hurwitz for all (as assumed in Theorem 2), there exists a positive definite Hermitian -valued function in (37) on the torus satisfying (38), (39) and such that

(67) |

In isolation from the environment (when ), the quadratic stability of the network can be formulated by enhancing the positive definiteness as

(68) |

Then the corresponding block Toeplitz matrix describes a positive definite self-adjoint operator on whose inverse is also such an operator and its norm is related to the quantity in (68) by

(69) |

Furthermore, the Hamiltonian , associated with by (40), admits the following bounds in terms of the standard -norm of the network state:

(70) |

Similarly to finite-dimensional settings, the dissipativity (47) (or a nonstrict version of the inequality (67)) implies that the Hamiltonian of the isolated network does not increase with time (that is, ), which, in combination with (70) leads to

(71) |

(similar bounds, arising from quadratic Hamiltonians, are also used for quantum harmonic oscillators [6, Eq. (22)]). In view of (69), the factor on the right-hand side of (71) is the condition number of the operator (which quantifies how far is from scalar operators).

## Vi Phonon Theoretic Dispersion Relations

We will now consider a class of translation invariant networks, which, in isolation from the environment, manifest phonon-like dynamics mentioned in Section III. Suppose the state vectors of the component systems consist of the conjugate position and momentum variables which are assembled into -dimensional vectors and , respectively (with being even):

(72) |

Accordingly, the network is assumed to have the following Hamiltonian:

(73) |

Here, is a real positive definite symmetric mass matrix of order , and use is made of a weighted Euclidean norm of a vector specified by such a matrix . Also, is a symmetric block Toeplitz stiffness operator whose blocks specify the potential energy part of the Hamiltonian. In view of (72) and (73), the blocks of the corresponding block Toeplitz matrix in (40) are given by

(74) |

where is the Kronecker delta. In application to the isolated network, the Hamiltonian equations of motion take the form

(75) |

for all , where