Singular Value Analysis Of Nonlinear Symmetric Systems

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 9, SEPTEMBER 2011 2073

Singular Value Analysis Of NonlinearSymmetric Systems

Tudor C. Ionescu, Kenji Fujimoto, Member, IEEE, and Jacquelien M. A. Scherpen, Senior Member, IEEE

Abstract—In this paper, we introduce the notions of state-spacesymmetry for nonlinear systems and of the cross operator as a non-trivial natural extension of the linear symmetric case, in terms ofthe controllability and observability operators associated to it. Wegive a characterization of a symmetric nonlinear system in terms ofthe cross operator and a coordinate transformation. Then we an-alyze the use of the cross operator for solving the Hankel singularvalue problem of the system. The result is a new and simpler char-acterization of the solutions of this problem in terms of the crossoperator and a metric.

Index Terms—Balancing, cross operator, Hankel, nonlinear sys-tems, singular value, symmetry.

I. INTRODUCTION

M ODEL order reduction is the tool to simplify complexmathematical models of systems. The reduced model

is easier to use for analysis and simulation (as in the electricalnetwork theory) and/or for control design. There are several ap-proaches such as: singular perturbation, center manifold theory,Krylov subspace methods, proper orthogonal decompositionand balanced truncation, see, e.g., [1] and the references thereinfor an overview. For control systems, one of the most popular,attractive and used techniques is the balanced approximationintroduced by Moore [2] for the linear time-invariant systemscase and naturally extended in [3], [4] for the case of non-linear systems. In a nutshell, balancing a stable, reachable andzero-state observable system means finding a coordinate trans-formation, such that in the new coordinates, one can measurethe gain between the input effort to reach a state and the outputenergy of that state. These gain measures are called (coordinatefree) axis singular value functions, see, e.g., [4], [5].

Truncating, i.e., getting rid of the states corresponding to thesmaller singular value functions, yields a lower order model thatis stable again. From a computational point of view the balancingprocedure involves solving two Hamilton-Jacobi equations in

Manuscript received April 13, 2009; revised January 19, 2010 and September03, 2010; accepted December 30, 2010. Date of publication March 14, 2011;date of current version September 08, 2011. Recommended by Associate EditorH. Ito.

T. C. Ionescu is with the Department of Electrical and Electronical En-gineering, Imperial College London, SW7 2AZ, London, U.K. (e-mail:[email protected]).

K. Fujimoto is with the Department of Mechanical Science and Engineering,Nagoya University Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan (e-mail:[email protected]).

J. M. A. Scherpen is with the Faculty of Mathematics and Natural SciencesITM, University of Groningen, Nijenborgh 4, 9747 AG, Groningen, The Nether-lands (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAC.2011.2126630

order to obtain the controllability and observability functions ofthe to-be-balanced system. One way to overcome this hindranceis to exploit and/or preserve the structure of the system and toadapt the balancing procedure to it, e.g., [6]–[11]. For instance,in [9]–[11] the symmetry of systems is taken into account. Thisproperty is encountered especially in (reciprocal) electrical net-works, as well as in electromechanical systems. Symmetry hasseveral definitions which in the linear case are proven to be equiv-alent.Geometricdefinitions fornonlinearsystemscanbefoundin[12]–[14] where a symmetric system is a system that possesses astructure with respect to a not necessarily positive definite metric.In the frameworkof (port-)Hamiltoniansystemssomedefinitionssuitable for model reduction and control are found in [15]–[17].A definition in the behavioral framework is given in [18], i.e., adynamical system is symmetric if the system remains unchangedwhen subjected to a group of transformations. From a model re-duction, state-space point of view a notion of symmetric systemswas introduced by Fernando and Nicholson in [19]–[21], i.e., alinear system is symmetric if its transfer matrix is symmetric. Fora minimal LTI system this definition is equivalent to the ones pre-viously mentioned, e.g., there exists a linear metric such that thesystemand itsdual are equivalent (seealso [22]). In this case thereis no need to compute both the controllability and observabilityGramians, but only a so called cross Gramian that contains infor-mation about both the controllability and the observability of thesystem. In general, for any linear system, the cross Gramian is de-fined as the solution of a Sylvester equation. Furthermore there isadirect relationbetweentheeigenvalues of thecrossGramianandthe eigenvalues of the Hankel operator of the system. Moreover,the eigenvalues of the cross Gramian are similarity invariants, yetthere is no connection to the Hankel singular values in the generalcase. For the symmetric systems case, the remarkable propertyis that the absolute value of the eigenvalues of the cross Gramianare the Hankel singular values of the system. Projecting onto theeigenspace corresponding to the larger eigenvalues of the crossGramian results in an almost balanced reduced order model, asin [23]. The term “almost” means that the transfer function ob-tained based on the cross Gramian method is equal to the transferfunction obtained by balanced truncation, yet the state space re-alization in the former case need not be balanced. Further recentdevelopments show that this reduction method based on the crossGramian eigenvalue analysis, can be used for large scale systemswhich are not necessarily symmetric, just as in, e.g., [24] wherethe case of large data-sparse systems is considered.

In the nonlinear case, the first steps towards a definition of thecross Gramian were made in [25], [26], for the class of gradientsystems. The starting point of this work was the characterization

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2074 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 9, SEPTEMBER 2011

of a gradient system as a system whose prolongation and gra-dient extension exhibit the same behavior as described in [14].Hence, the notion of cross Gramian was defined for the prolon-gation as the product between the inverse of the gradient metricand the observability function of the prolongation. This crossGramian is the solution of a nonlinear Sylvester-like equation.However, a connection between the eigenvalue functions of thecross Gramian and the singular value functions of the gradientsystem was not found, due to the missing link between the crossGramian and the controllability of the system.

The first goal of this paper is to find an extension of the crossGramian to a cross operator for nonlinear asymptotically stablesystems in general, as the solution of a nonlinear Sylvester-likeequation. Secondly, we want to find a natural extension of thestate space symmetry property as described in [19]. The sym-metry stems from the controllability and observability prop-erties of the system. A motivating example for this approachis provided by the study of sensitivity analysis in systems bi-ology, see [27]. The sensitivity analysis technique is used inthe study of biochemical reaction networks which contain manyuncertain parameters. The dynamics of these systems are non-linear; large and quite sensitive to changes in the parameters.The sensitivity analysis is related to the controllability and ob-servability analysis of single input single output systems. Hence,the cross Gramian is employed, and computed based on an em-pirical Gramian algorithm, where parameter choices are randomand restricted. However, a more suitable analysis could be ap-proached through the proposed notion of nonlinear symmetricsystems. For the symmetric systems case we find a similar prop-erty as in the linear case, e.g., the (axis) singular value functionsto be obtained using the cross operator and simplifying the bal-ancing procedure. In order to achieve the aforementioned goals,we notice first from the linear case, that the symmetry property,the cross Gramian and the controllability and observability op-erators, associated to the system, are closely related. Using thedefinitions of the nonlinear controllability and observability op-erators, as well as their derivatives, and based on the extensionof the linear connection with the cross Gramian to the nonlinearcase, we come up with a new notion of nonlinear cross operator,as the solution of a nonlinear Sylvester-like equation. Equiva-lently, we write the cross operator in terms of the controllabilityand observability operators, just like in the linear case. Further-more, extending the linear relation between the controllabilityand observability operators and symmetry we come up with anotion of nonlinear state-space symmetry, which is closely con-nected to the cross operator. Using this newly defined symmetryproperty and cross operator, we give and solve the symmetricsystems version of the nonlinear problem of finding the sin-gular value functions of the Hankel operator. This result couldbe used to replace the empirical cross Gramian analysis pre-sented in [27], since no choice of inputs is required.

The paper is organized as follows: in Section II, we give anoverview of the linear systems case, the definition of the crossGramian and its properties and present the relation of the crossGramian with the controllability and observability operators. InSection III, we treat the nonlinear case, where we first give anoverview of the nonlinear balancing technique related notionsand a property of the controllability operator and its derivative.

We briefly mention the gradient systems and the notion of non-linear cross Gramian for this class of symmetric systems. Thenwe give the definition of the cross operator as the solution of anonlinear Sylvester-like equation, as well as the relation of thiscross operator with the controllability and observability opera-tors. In Section IV, we present the definition of symmetry fornonlinear systems and connect it to the notion of cross operatorresulting in necessary and sufficient conditions for symmetry.In Section V, we use the cross operator to solve the nonlinearHankel differential singular value problem for a nonlinear sym-metric system that yields the singular value functions. We con-clude the paper with an academic example in Section VI.

Notation: Let andbe two signals. Their inner

product is written as . The normof is given by .

is the set of signals with positive support and fi-

nite -norm, defined as . Theinner product of and on is described by

. Let andbe a differentiable function, then we write

and, .

II. LINEAR SYSTEMS AND THE CROSS GRAMIAN

In this section we give an overview of the notion of crossGramian for a linear system and of the notion of symmetric sys-tems. First, we define the cross Gramian for the general caseof an asymptotically stable system and describe the propertiesof this matrix in relation with the Hankel operator. Namely, thenonzero eigenvalues of the Hankel operator are the eigenvaluesof the cross Gramian. Further, the notion of linear symmetricsystems is presented. For this class of systems the cross Gramianshows remarkable properties, the most important being the di-rect relationship between the Hankel singular values and theeigenvalues of the cross Gramian.

A. Preliminaries

Let , be thetransfer function of a linear system, with a minimal, asymptoti-cally stable, associated realization

(1)

For system (1) we define the controllability Gramian asand the observability Gramian as

. Since (1) is assumed minimal and asymp-totically stable, and are positive definite and furthermore,they are the unique solutions of the following Lyapunov equa-tions, respectively

(2a)

(2b)

According to, e.g., [1], [2], the square root of the eigenvalues ofmatrix are similarity invariants, called the Hankel singularvalues of system (1), denoted by .

IONESCU et al.: SINGULAR VALUE ANALYSIS OF NONLINEAR SYMMETRIC SYSTEMS 2075

The Hankel operator of (1) is defined as

(3)

where is the impulse response

of (1). The Hankel singular values, ’s, are also singularvalues of the Hankel operator, i.e., they are the solutions ofthe following problem: , called the Hankelsingular value problem associated to system (1). is calleda singular vector of the Hankel operator corresponding tothe singular value . The squared largest singular value,represents the Hankel norm of (1), i.e., , where

.

B. Cross Gramian

We turn our attention to the definition of the cross Gramianof a linear square system, i.e., assume .

Definition 1: Let (1) be such that . Then the crossGramian of system (1) is defined as the solution of theSylvester equation

(4)

If the system is asymptotically stable, then the cross Gramiancan be equivalently written as

For the nonsymmetric case, the cross Gramian possesses in-teresting properties being related to the Hankel operator and theHankel singular values.

Lemma 1: For square linear systems the nonzero eigenvaluesof the cross Gramian are the nonzero eigenvalues of theHankel operator associated to the system.

Remark 1: Let be a system (1) with the asso-ciated cross Gramian . If is an invertiblematrix, then the cross Gramian of the transformed system

is . Furthermore, the eigen-values of the cross Gramian are similarity invariants for (1).

In general, the relation between the eigenvalues of the crossGramian and the Hankel singular values is not straightforward,see [28] for further details.

C. Linear Symmetric Systems

In this subsection, we give the definition of linear symmetricsystems, since for this class of systems, there exists a direct re-lation between eigenvalues of the cross Gramian and the Hankelsingular values.

Definition 2: [19], [28], [29] Let be a square ,bounded-input-bounded-output stable transfer matrix of a linearsystem. The system is called symmetric if .

Let (1) be a minimal, asymptotically stable realization of. Consider its dual

(5)

Then we give the following characterization.

Proposition 1: [19], [28] Assume (1) is minimal. System (1)is symmetric if and only if there exists an invertible symmetricmatrix (i.e., ), such that

(6)

i.e., the linear system and its dual counterpart are state equiva-lent via the coordinate transformation .

These definitions for a symmetric linear system can be con-sidered particular cases of the more general geometric and be-havioral notions, as in, e.g., [12], [18]. From a geometric pointof view, system (1) is symmetric if there exists a pseudo-Rie-mannian metric, i.e., not necessarily positive definite, such thatthe vectorfields of the system are written as the gradient with re-spect to this metric of a potential function. In the linear case, themetric is represented by the matrix and the potential functionis . In the behavioral framework a system is symmetric ifthere exists a group of transformations mapping the system toa similar behavior, i.e., this transformation does not change thesystem. In the LTI case, the transformation is defined by .

D. Properties of the Cross Gramian of Symmetric Systems

In the symmetric case, see for instance [19], [21], [28], thecross Gramian exhibits some remarkable properties in rela-tion with the controllability Gramian and the observabilityGramian , which are summarized in the following theorem.

Theorem 1: [19], [28] Let (1) be a square symmetric systemand let be the coordinate transformation from (6).Denote by a Hankel singular value of (1) and byan eigenvalue of the cross Gramian , . Then thefollowing relations are equivalent:

1) ;2) ;3) , for all , i.e., the Hankel singular

values of (1) are the eigenvalues of the cross Gramian, inabsolute value.

E. Symmetry in Terms of Controllability, Observability andthe Hankel Operators

The properties described in Theorem 1 can be rewritten interms of the controllability, observability and Hankel operatorsassociated to (1). We focus our attention on this approach sinceit paves the way for the definitions for nonlinear systems to bepresented in Section III. For the asymptotically stable, minimalsystem (1) we define the controllability operator as

and the observability operator as

The Hankel operator (3) is equivalently written


If is the cross Gramian and and are the controllabilityand observability operators, then

(7)

and furthermore, , for all .Remark 2: For a linear, asymptotically stable, minimal

system, the nonzero eigenvalues of the operator are equalto the eigenvalues of . The proof is based on thedefinitions of the controllability and observability operators, aswell as on the linearity of these operators.

We recall the well known definition of a linear adjoint oper-ator. If is a linear operator such that ,then its Hilbert adjoint operator is satisfying

(8)

The controllability and observability operators admit adjointcounterparts, which can be defined in terms of the dual realiza-tion (5)

and

Hence, symmetry as described in Proposition 1 is equiva-lently characterized as a property of the controllability, observ-ability and their adjoint operators.

Proposition 2: A linear minimal and stable system (1) is sym-metric if and only if there exists an invertible and symmetricmatrix such that

(9)

Proof: If (1) is symmetric then, there exists such thatrelation (6) is satisfied. Then:

. Thesame holds for

. Conversely, if there exists aand (9) is satisfied then we can write

, for all . Since the system is assumedminimal and asymptotically stable is uniquely deter-mined for all . For , we get whichimplies that for all . Thus, we have

, and by Proposition 1, thesystem is symmetric.

We now rewrite the properties given in Theorem 1 in terms of, and their adjoints. Although it might seem trivial, this line

of thinking is important for the nonlinear extension of the crossGramian and its properties in Section IV-B.

Corollary 1: Assume (1) is asymptotically stable, minimaland symmetric. Then, the eigenvalues of the operatorare the squared Hankel singular values.

Remark 3: The result of Corollary 1 can be rewritten in manyforms, due to the symmetry property, e.g., the squared Hankelsingular values of the system are the eigenvalues of the operator

.In the sequel, we will extend the results of Proposition 2 to the

nonlinear case in order to define the state-space symmetry prop-erty and the cross operator for a nonlinear dynamical system.The extension of Proposition 2 will enable us to write the Hankelsingular value problem of nonlinear symmetric systems in a sim-pler form, in terms of the cross operator.

III. NONLINEAR SYSTEMS AND THE CROSS OPERATOR

In this section, we extend the results of Section II to thenonlinear case. First, we briefly overview the definitions andproperties of the controllability and observability functions andthe corresponding operators. Then, based on these definitionswe give the first definition of a cross operator for the case ofnonlinear asymptotically stable systems, as well as the corre-sponding nonlinear Sylvester-like equation. Further, we comeup with a new definition of state-space symmetry for a nonlinearsystem, based on the extension of Proposition 2. We give neces-sary and sufficient condition in terms of the cross operator, fora system to be symmetric.

A. Preliminaries

Consider the following nonlinear square system:

(10)

with and smooth vectorfields anda smooth output mapping. We assume 0 is an equilibrium

of (10), with and, furthermore the system is asymp-totically stable in a neighborhood of 0. We define energy func-tions, operators and their state-space representations accordingto [3]–[5], [30]. The following controllability and observabilityoperators are defined for (10), respectively

(11a)

(11b)

The Hankel operator associated to (10) is,

(12)

The controllability operator defines how the state isreached from 0 by means of inputs from the past and theobservability operator defines the influence of the state onthe output after the input is turned off. Hence, by (12), theHankel operator describes how inputs from the past influencethe outputs in the future. Assume that ,


, is the pseudo-inverse of . Thecontrollability and observability energy functions are

(13a)

(13b)

If exists and is smooth, then it uniquely satisfies theHamilton-Jacobi equation [3]

(14)such that the dynamics

are asymptoticallystable. If the system is asymptotically reachable from 0,1

then . If exists and is smooth, then ituniquely satisfies the nonlinear Lyapunov equation [3]

(15)

If the system is zero-state observable,2 then .Assuming that and are Fréchet differentiable, we write

the state space realizations of the Fréchet derivatives3 and, as well as the state space realizations of their linear adjoints,

i.e., and . We have (see [4, Lemma 1, Lemma 2],[31])

(16a)

(16b)

(16c)

(16d)

B. Relation Between and

We give a relation between from (16c) to.

1System (10) is called asymptotically reachable from 0 if for all � there existsan input � � � �� and � � � such that � � �� .

2System (10) is zero-state observable if �� implies �� .

3The Fréchet derivative is denoted by ��. The Fréchet derivative �� of agiven function � � � , where and � are two Banach spaces, satisfies�� . �� is linear in �.

Proposition 3: Assume that the operators , and existand are continuously differentiable. Moreover, assume that thecontrollability function exists and is smooth. Then

(17)

Proof: First, according to [4], if exists and is con-tinuously differentiable, then has the following state-spacerepresentation

Denote by . The dynamics ofwith respect to is

.For , the optimal input that defines

we get

(18)

Since is assumed to exist and is smooth, it satisfies (14).Taking the derivative with respect to in (14), we get

(19)

Rearranging the terms, we write

(20)

which, by using (18), is equivalent to

Then by the definition of in (16c),, with

, which proves the statement of the proposition.Remark 4: In the linear case, if the controllability Gramianis positive definite, then it satisfies (2a) which is equivalent

to . Then by the definition ofthe pseudo-inverse of the controllability operator, we write

By the definitions of the adjoint and pseudo-inverse of the con-trollability operator, we have .


C. Cross Operator for Nonlinear Systems

In this section, we give the definition of the cross operatorfor a stable nonlinear system. A motivating example for this ap-proach is provided by the study of sensitivity analysis in systemsbiology, see [27]. The sensitivity analysis technique is used inthe study of biochemical reaction networks which contain manyuncertain parameters. The dynamics of these systems are non-linear, large, and quite sensitive to changes in the parameters.The sensitivity analysis is related to the controllability and ob-servability analysis of single input single output systems, andan empirical cross Gramian is studied. Here, we propose an an-alytic cross Gramian definition for nonlinear systems. We startfrom the definitions of the controllability and observability op-erators.

Assumption 1: The operators , exist and are continuous.From the definition of the controllability operator in (11a)

we write , with described by, . Further, from the definition

of the observability operator (11b), we have the signal, where , starting from the ini-

tial condition . If we plug in , thenwith from the equation

. Let , . Then,for all

Assuming is differentiable with respect to , thenand we get the following equation:

(21)

Now we are ready to present the definition of a cross operatorfor nonlinear (asymptotically stable) systems.

Definition 3: Assume a nonlinear square system (10) satisfiesAssumption 1. We call a differentiable function

, , satisfying (21), the cross operator associatedto (10). Furthermore, we call (21) the Sylvester-like nonlinearequation.

Remark 5: If the system is assumed linear, then solving (21)boils down to solving the Sylvester equation (4).

Remark 6: Regarding the observability operator as a signalgenerator (as the input is turned off) in Fig. 1 we can regard

as the moment of the controllability operatorat , according to [32], [33]. Furthermore, under

some assumptions, the moment for all is the steadystate response of the interconnection depicted in Fig. 1.

Using the definitions of the controllability and observ-ability operators, respectively, we have

. Then, we immediately obtain the followingproposition.

Proposition 4: Let (10) be such that Assumption 1 is satisfied.Then the cross operator from Definition 3 satisfies

(22)

Fig. 1. Box representation of the cross operator �.

Remark 7: The result in Proposition 4 motivates the nameof cross operator given to , since it is described by an opera-tion performed with the controllability and observability opera-tors.

Unfortunately, due to the nonlinearity of the operators, wecannot obtain a (direct) relation between the eigenvalues of theHankel operator and the eigenvalues of the cross operator, as anextension of Lemma 1. Still, if one of the operators exhibits alinearity property, then the eigenvalues of the cross operatorare related to the eigenvalues of the Hankel operator .

Lemma 2: Let the controllability, observability and cross op-erators, , and , respectively, exist such that Assumption 1is satisfied. Then the following statements hold:

1) if is linear, then the eigenvalues of are among theeigenvalues of , i.e., . The equality holds if

is linear too.2) if is linear, then the eigenvalues of are among the

eigenvalues of , i.e., . The equality holdsif is linear too.

Proof: We prove statement 1. Assume is an eigen-value of . Then , . Denoting by

, we have . Premultiplying with andexploiting the assumed linearity of the controllability operator,we get , which means that .The converse is true (i.e., the equality takes place) if is lineartoo. The proof of the second statement is similar.

Remark 8: The particular cases presented in Lemma 2 areencountered in the literature in fields like system identificationor signal processing. The case described in statement 1, fits thesituation of Wiener systems, which consist of linear dynamicswith a nonlinear output, see, e.g., [34], [35]. The second state-ment fits the case of Hammerstein systems, where the dynamicsof the system is nonlinear (by introducing a nonlinearity at theinput), but the output is linear, see, e.g., [36].

IV. SYMMETRIC NONLINEAR SYSTEMS

In this section, we describe the notion of nonlinear symmetricsystems as extension of the linear notion. In the case of the mo-tivating example from systems biology in [27] as mentioned inthe previous section, no choice of inputs is given. Hence, pickingan input such that the system is symmetric could be useful toreplace the empirical cross Gramian analysis of [27]. We ap-proach the notion of a nonlinear symmetric system using twodifferent lines of thinking. The first approach is based on thenotion of gradient systems, which are an important class of non-linear systems, endowed with a pseudo-Riemannian metric onthe state-space manifold, such that the nonlinear dynamics are


expressed as the gradient of some potential functions with re-spect to the same pseudo-metric. For this class of systems, thecross Gramian is defined as an extension of the linear notion,but with no relation to controllability or observability, yet. Thesecond approach derives from the results of Section II-E. Wedefine a nonlinear symmetric system as a system for which theset of inputs that steer the states from 0 to is equal to the setof outputs obtained from the system, after turning off the inputs.For these systems, a necessary and sufficient condition for sym-metry is provided in terms of the cross operator.

A. Gradient Approach

A natural nonlinear extension of the notion of a symmetricsystem is the notion of a gradient system. Examples of gra-dient systems include, for instance, nonlinear models of elec-trical circuits, such as the Brayton-Moser model for electrical(nonlinear) RLC networks. In this subsection, we overview thecharacterization of the gradient property as provided in [14] anddefine the notion of cross Gramian as an extension of the secondstatement of Theorem 1. A wider and detailed discussion and re-sults of this approach is found in [26].

Definition 4: [13], [14] Let (10) be such that , whereis a manifold. System (10) is a gradient system if

1) there exists a pseudo-Riemannian metric on , givenas , with ,

smooth functions of , and the matrixinvertible, for all and representing the

tensor product and2) there exists a smooth potential function ,

such that system (10) can be written as

(23)

In local coordinates , the systemcan be written as (see, e.g., [13])

(24)

The property of a system being gradient is described in [14]in terms of a necessary and sufficient condition satisfied by thevariational system and its gradient extension associated to (10).

Let be a system with a state-space realization(10). We associate to it the variational (prolonged) system

(25)

where , are small variations of the input and the outputof the system, respectively. are small variations ofthe state, where is the tangent bundle of the manifold

. The gradient extension is tedious to construct, since thedefinition of a metric on the dual space (cotangent bundle)

is nontrivial, not straightforward and falls beyond the

scope of this paper. For further details we refer the reader to[14]. Let all the geometric terms obtained from writing thegradient with respect to the metric be described bythe function , with

. The are defined by relation [14,(2.8)]. The gradient extension system associated to (10) is

(26)

Theorem 2: [14, Theorem 5.4, Corollary 4.4, Lemma 5.5]System (10) is a gradient control system, as in Definition 4, ifand only if the prolonged system and the gradient extension

have the same input-output behavior. Moreover, there existsa diffeomorphism , where and

satisfy (25) and (26), respectively.In the sequel, we briefly present a definition of the notion of

cross Gramian, based on the metric of the gradient systemand the observability function of the variational part, given in[25], [26]. First, we consider only the variational part of ,denoted by , and the gradient extension partdenoted by . According to Theorem 2, since(10) is assumed gradient, the input-output behavior of is thesame with the input-output behavior of and furthermore,

. The observability function of , can be written as

with symmetric, positive definite with smooth elements.Then, the cross Gramian matrix associated to is

(27)

and satisfies the following nonlinear Sylvester equation (see[26] for details on how it is obtained)

(28)

or, equivalently

(29)

where .Remark 9: In the linear case, (28) becomes

, where is satisfied as inTheorem 1. Thanks to the linearity in and one recognizesthe Sylvester (4).

At this moment, the relation between this notion of crossGramian and the observability and controllability functions/op-erators is not yet clear; hence, a relation to the Hankel singularvalue functions as in [4] is not immediately available.


B. State-Space Approach

In this section, we extend the results obtained in Proposition9 giving a new definition of symmetry, its metric and the crossoperator for a nonlinear dynamical system in terms of relationsbetween the operators (the pseudo-inverse of the controlla-bility operator), its derivative and . Utilizing the cross op-erator, we give a necessary and sufficient condition for a non-linear system to be symmetric. Throughout the rest of the paperwe make the following working assumption.

Assumption 2: Operators and as in (11) exist and arecontinuously (Fréchet) differentiable, with their derivativesand and their linear adjoints and defined by (16).Moreover, assume the pseudo-inverse of , denoted by , ex-ists, and is (Fréchet) differentiable.

We are now ready to give a definition of symmetry as an ex-tension of the linear relation between controllability and observ-ability, as in (9).

Definition 5: Let (10) be such that Assumption 2 holds. Wecall the system symmetric if it satisfies

(30)

Proposition 5: Let (10) be such that as in (10) exists.Then, system (10) is symmetric if and only if one of the fol-lowing holds

1) is invertible and

(31)

2) there exists invertible and

(32)

holds, with .Proof: If the system is symmetric in the sense of Definition

5, then there exist in the domain of and in the domain ofsuch that

(33)

Let , then . From (33) we haveand then , or equivalently . Since Assump-tion 2 holds, then is smooth. (31) is proven immediately. Forthe second part, since is assumed invertible (32) follows. Theonly if part of the proof is immediate.

Since by Definition 5 the inputs necessary to reach the statesfrom an initial condition are the output signals obtained byturning off the input, we have that for this nonlinear symmetricsystems the reachability and observability properties are con-nected through the cross operator .

Remark 10: In the linear case, according to Proposition 2, alinear, minimal system is symmetric if and only if re-lations (9) hold, i.e., . According to Remark4, we get , with .

Definition 6: Let (10) be such that the operators andexist and are continuous. The cross energy function of system(10) is

If the system is assumed symmetric with as inProposition 5, using (31) we have that

or equivalently,

and moreover

(34)

Equivalently, using (32) we getor equiva-

lently, andmoreover

(35)

Remark 11: For a linear, minimal, symmetric system thecross energy function satisfies ,with . We have , with

. Since the system is symmetric, (9) is satisfied, i.e.,and then . In

particular, if and , i.e., , thenwhich, according to [29], is the so called

co-energy of the dynamical linear system.We provide relations between the derivatives of the controlla-

bility and observability operators and the corresponding energyfunctions.

Proposition 6: [4, Lemma 3], [5] Let (10) be such that As-sumption 2 holds. Then and

.Lemma 3: For a nonlinear symmetric system, the following

property holds

(36)

or, equivalently

(37)

Proof: Taking the Fréchet derivative of along the di-rection , we have

Using the adjoints of the linear operators and , sat-isfying (8), together with Proposition 6, we can rewrite the pre-vious equation as

Using (31) we obtain the property. Taking into account the def-inition of and as in Proposition 5, one getsequivalence between (36) and (37).


Fig. 2. Nonlinear (state-space) symmetry: relations between � , �, and ��.

Remark 12: In the case of linear symmetric systems we have, and (36) becomes

which gives for all . This meansthat the controllability Gramian of the equivalent system

is the observability Gramian of. Furthermore, (as in

Theorem 1).Using Definition 5 for symmetry of the nonlinear system and

the property in Proposition 3, we have a set of relations de-scribed by the diagram presented in Fig. 2.

Since the system is assumed symmetric, according to Propo-sition 5 we have . Then, defines a co-ordinate transformation between the state-space realizations of

and , respectively. Also, Proposition 3 establishes acoordinate transformation between the realization of andthe one of the operator , , de-fined as . Then we can define

(38)

as a coordinate transformation betweenand , i.e.,

, orequivalently in coordinates

. This further yields

(39)

Remark 13: In the linear case,, where is the cross Gramian described by Definition

1 or equivalently by Theorem 1, or (7).Proposition 7: A nonlinear system (10) is symmetric

if and only if the cross operator is invertible and. Equivalently, (10) is symmetric

if and only if there exists , invertible such that

and , where.

Remark 14: In the linear case, the first statement of Proposi-tion 7 reads that a system is symmetric if and only if the cross

Gramian satisfies , with from The-orem 1. The second statement is a nonlinear extension of Propo-sition 1: there exists , invertible such that ,

.We give other necessary and sufficient conditions, respec-

tively, for a nonlinear system to be symmetric, based on the non-linear Sylvester (21).

Proposition 8: If a nonlinear system is symmetric then thecross operator satisfies

(40)

where is described by (38).Proof: If the system is symmetric, then (36) holds. Using

Proposition 6 we write

and so (36) becomes

(41)

Multiplying with , and using (15) we get

where we also made use of the definition of from (38). FromProposition 3 we have that the state-space realization of isobtained from the state space realization of , via the coordinatetransformation and

, and thus, we obtain (40) (see also Fig. 2).The converse statement requires an extra assumption made

upon the input vectorfield and the output.Proposition 9: Assume that for the nonlinear system (10)

there exists satisfying (40) and describedby (38) such that holds. Then the system issymmetric in the sense of Definition 5.

Proof: Substituting from (38) in (40), we have


Since the system is assumed asymptotically reachable,satisfies the Hamilton-Jacobi (14). Then we obtain that

andaccording to Proposition 7 the system is symmetric.

Remark 15: In the linear case, if the system is symmetric,then the cross Gramian satisfies

, where . Conversely, if the Sylvester equation is sat-isfied and the relation holds, then we obtain

and according to Remark 14, thesystem is symmetric.

V. SINGULAR VALUE ANALYSIS OF THE HANKEL OPERATOR:THE SYMMETRIC SYSTEMS CASE

A. Hankel Singular Value Problem

In this section, we briefly describe the Hankel singular valueproblem, depicted in Section II-A, for a nonlinear system asshown in the results from [4], [5], necessary for obtaining a bal-anced realization. The Hankel operator is defined by (12) and thestarting point is the investigation of the gain structure of this op-erator. The gain structure problem means examining the largestsingular value, where a singular value is defined as

(42)

The maximizing input that gives the largest singular value isand , . Supposing is

Fréchet differentiable, the following problem must be solved,according to [5]

(43)that characterizes all the critical points of magnitude c, as wellas the optimal one that gives the largest eigenvalue. Accordingto [4], [5], this problem has the alternative formulation: thereexists s.t.

(44)

The problem of finding such that (44) is satisfied, iscalled the Hankel singular value problem for nonlinear systems.An alternative and simpler characterization of (44) is describedin the following.

Theorem 3: [4] Assume that the controllability operator, itspseudo-inverse and the observability operator exist and are con-tinuously differentiable. Moreover, assume that there exists

and satisfying

(45)

Then satisfies the Hankel singular value (44) with the eigen-vector .

Using the definitions of and in (13), (45) can berewritten equivalently as

(46)

Remark 16: In the linear case, (45) is equivalent to, which is similar to , i.e., the squared

Hankel singular values are the eigenvalues of the .There exists a solution for this problem, depicted in the fol-

lowing theorem.Theorem 4: [4] Let (10) be asymptotically stable in a neigh-

borhood of 0. Suppose that the linearization of (10) (around 0)has all Hankel singular values nonzero and distinct. Then, thereexists a neighborhood of 0 and ,such thatholds for all , . Moreover, there exist

, satisfying the following:

(47)

Furthermore, if , the Hankel norm of the system is.

B. Symmetric Case

Under the assumption of symmetry upon the system, theHankel singular value (44) or (46), can be written in termsof the cross operator and the derivative of one of the energyfunctions.

Theorem 5: Assume a nonlinear system (10) is symmetric inthe sense of Definition 5. The Hankel singular value problem(44) becomes

(48)

or equivalently

(49)

where is described as . Moreover is iden-tical to the one in (44).

Proof: We start with the definition of the largest Hankelsingular value subject to the constraint follwing thereasoning in [4]. Due to the symmetry property, from (32), wehave

Then (43) becomes

(50)


Applying the quotient and then the product rule of derivation inthe left term of (50), we have

We divide byand get

where .Further, applying the definition of linear adjoint operators, see

(8), we have

which yields

(51)Applying (36), (51) becomes (46) which is, as in Theorem 3, theHankel singular value equation. Using (41), (51) is equivalent to

(52)

or, using (38) we obtain the first relation in the proposition.Equation (52) can be written in terms of . First, (38) is equiv-alent to . The differen-tial singular value problem becomes

(53)

which according to (41) is equivalent to

(54)

and yields (49).Remark 17: In the linear case, we have:

which is thesame as , consistent with the result of Theorem1.

VI. EXAMPLE

Given a double mass double spring and damper system (seeFig. 3), we compute the cross operator from (21) for an arbitraryinput. Then using the symmetric output, we compute the axissingular value functions from (49).

Fig. 3. Double mass double spring and damper system.

The system is described by the following equations:

(55)

where are the displacements (the left wall is the 0position and the positive direction for both positions andvelocities is to the right), are the masses and

are the corresponding elastic forces,with the dissipation (damping) function and

. The total energy of the systemis ,where is the potential energy given in thesprings. We take and . As-suming the springs linearly depend on the position, fromFig. 3 it follows that

and. Relabeling the

velocities as and we can rewrite the systemin first-order form

(56)

with and the total energy function

(57)

Since , we notice thatdescribed by the relation (57) satisfies the Hamilton-Jacobiequation

meaning

This is consistent with the result given by Lemma 1 and the se-quel in [8] with matrix , that is if the dissipation is in theimage of the input vectorfield, then the total energy of the systemis the controllability function of that system. We first choose theoutput . For (56) with the output , we com-pute the cross operator as the solution of (21). We com-pute an approximate solution using the algorithm of Lukes [37](see also Al’brekht [38]) a third-order Taylor approximation.


Using Maple, we obtain the cross operatoras

Using Proposition 7 (or equivalently Proposition 9), wechoose the (symmetric) output

(58)

The system defined by (56) and (58) is a symmetric system. Thecoordinates can be computed from (38).

Using (34), the observability function of the system is easilycomputed as

(59)

In order to compute the axis singular value functions of thissystem, we apply Theorem 4. The conditions stated in the pre-amble of Theorem 4 are fulfilled, i.e., the linearization around0 is a linear single-input single-output (symmetric) system withdistinct Hankel singular values. In the symmetric case (47) canbe rewritten as (52) in terms of the controllability function andthe symmetry cross operator. We solve (52) in the unknown

, which is equivalentto solving a system of four nonlinear equations in this unknown.We find the following approximations of the ’s

Equivalently, one can write (47) as (49). An approximationsolution is found as .

The axis singular value functions are computed from the same(47) as yielding (a fifth-orderapproximation)


Remark 18: If we choose the output, since

, then(see, e.g., [8] for more details). In this (trivial) case, thesystem is symmetric with the cross operator and

with all the singular values equalto 1.

VII. CONCLUSION

In this paper, a cross operator notion for nonlinear systemsand a new symmetry notion have been defined in terms of thecontrollability and observability operators. This is a nonlinearextension of the cross Gramian and the symmetric linear case,respectively. The cross operator is the solution of a nonlinearSylvester-like equation. We gave necessary and sufficient con-ditions for a nonlinear system to be symmetric utilizing the crossoperator. Then, we studied the Hankel singular value problemfor symmetric systems and we gave a simpler form of the stateequation that characterizes this problem, in terms of the crossoperator and the derivative of the controllability/observabilityfunction. For future work we want to provide a link betweenthe gradient systems and the symmetric systems defined in thispaper. The answer would shed some light on the dimension ofthe class of nonlinear (state-space) symmetric systems, which isan open question. Furthermore, an extension to the passive caseis an open problem.

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Tudor C. Ionescu received the M.Sc. degree in sys-tems and control engineering from Politehnica Uni-versity of Bucharest, Romania, in 2004, and the Ph.D.degree in applied mathematics from the University ofGroningen, The Netherlands, in 2009.

During his Ph.D. work, he performed researchvisits at SUPÉLEC, Gif-sur-Yvette, France, in 2005and 2007, and at Nagoya University, Japan, in 2008.Currently, he is a Research Associate at the Controland Power Group, EEE, Imperial College, London,U.K. His research interests include modeling and

control of nonlinear systems with a focus on modeling and model orderreduction.


Kenji Fujimoto (S’97–A’97–M’02) received theB.Sc. and M.Sc. degrees in engineering and thePh.D. degree in informatics from Kyoto University,Japan, in 1994, 1996, and 2001, respectively.

He is currently an Associate Professor with theGraduate School of Engineering, Nagoya University,Japan. From 1997 to 2004, he was a Research Asso-ciate with the Graduate School of Engineering andthe Graduate School of Informatics, Kyoto Univer-sity. From 1999 to 2000, he was a Research Fellowwith the Department of Electrical Engineering, Delft

University of Technology, Delft, The Netherlands. He held visiting researchpositions at the Australian National University, Australia and Delft Universityof Technology, The Netherlands, in 1999 and 2002, respectively. His researchinterests include nonlinear control theory and model order reduction.

Dr. Fujimoto received the SICE best paper awards in 2000 and 2009, the SICETakeda Award in 2000, the best paper award at the SICE Annual Conferenceon Control Systems 2003, the IFAC Congress Young Author Prize at the IFACWorld Congress 2005, and the SICE Pioneer Award in 2007.

Jacquelien M. A. Scherpen (M’95–SM’04) re-ceived the M.Sc. and Ph.D. degrees in appliedmathematics from the University of Twente, TheNetherlands, in 1990 and 1994, respectively, in thefield of systems and control.

From 1994 to 2006, she was with the Circuits andSystems and Control Engineering Groups of Elec-trical Engineering, Delft University of Technology,The Netherlands. From 2003 to 2006, she was withthe Delft Center for Systems and Control, DelftUniversity of Technology, The Netherlands. Since

September 2006, she has held a Professor position in the Industrial Technologyand Management Department of the Faculty of Mathematics and Natural Sci-ences, University of Groningen, The Netherlands. She has held visiting researchpositions at the Université de Compiegne, France; SUPÉLEC, Gif-sur-Yvette,France, University of Tokyo, Japan; and Old Dominion University, VA. Her re-search interests include nonlinear model reduction methods, realization theory,nonlinear control methods, in particular modeling and control of physicalsystems with applications to electrical circuits, electromechanical systems andmechanical systems, distributed control systems with applications to smartgrids, and industrial and space applications. She has been an associate editor ofthe IEEE TRANSACTIONS ON AUTOMATIC CONTROL and of the InternationalJournal of Robust and Nonlinear Control. Currently, she is an associate editorof the IMA Journal of Mathematical Control and Information and she is on theeditorial board of the International Journal of Robust and Nonlinear Control.

Singular Value Analysis Of Nonlinear Symmetric Systems

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