A New Approach to the Internally Positive ... - Semantic Scholar

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A New Approach to the Internally Positive Representation of Linear MIMO Systems Filippo Cacace, Alfredo Germani, Costanzo Manes, and Roberto Setola, Senior Member, IEEE

Abstract—The problem of representing linear systems through combination of positive systems is relevant when signal processing schemes, such as filters, state observers, or control laws, are to be implemented using “positive” technologies, such as Charge Routing Networks and fiber optic filters. This problem, well investigated in the SISO case, can be recasted into the more general problem of Internally Positive Representation (IPR) of systems. This paper presents a methodology for the construction of such IPRs for MIMO systems, based on a suitable convex positive representation of complex vectors and matrices. The stability properties of the IPRs are investigated in depth, achieving the important result that any stable system admits a stable IPR of finite dimension. A main algorithm and three variants, all based on the proposed methodology, are presented for the construction of stable IPRs. All of them are straightforward and are characterized by a very low computational cost. The first and second may require a large state-space dimension to provide a stable IPR, while the third and the fourth are aimed to provide stable IPRs of reduced order. Index Terms—Internally positive representation, linear systems, positive systems, state-space realization.

I. INTRODUCTION

A

POSITIVE system is a system whose state and output evolutions are always nonnegative, provided that both the initial state and the input sequence are nonnegative [18], [22]. Positive systems are common in the mathematical modeling of biological processes (e.g., cell growth [3]), economical (e.g., market behaviour [16]), social (e.g., population dynamics [17], [23], epidemic spread [2]), technological (e.g., critical infrastructures [15]), and other kinds of systems. The problem of positive state-space realization of positive filters (i.e., with nonnegative inpulse response) has been widely investigated in the literature [1], [8], with the aim to allow their implementation using technological frameworks that efficiently handle nonnegative quantities, such as Charge Routing Networks (CRN) [13], and fiber optic filters [7]. However, it has been shown that the positivity constraint in the design of a filter may significantly limit the performance, as discussed in [4], [5], and [24]. One way to overcome such a drawback is to design a signal processing

Manuscript received August 27, 2009; revised July 16, 2010; accepted April 01, 2011. Date of publication May 31, 2011; date of current version December 29, 2011. This work supported in part by the Italian Ministry for Education, University and Research (MIUR). Recommended by Associate Editor J. J. Winkin. F. Cacace and R. Setola are with Università Campus Bio-Medico di Roma, 00128 Roma, Italy (e-mail: [email protected]; [email protected]). A. Germani and C. Manes are with Dipartimento di Ingegneria Elettrica e dell’Informazione, Università degli Studi dell’Aquila, 67040 L’Aquila, Italy (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2011.2158170

scheme without any positivity constraint, and to realize it by means of combination of positive systems. The case of Single Input/Single Output (SISO) transfer functions with simple poles has been accurately investigated in [6] and [7]. The extension to SISO transfer functions with multiple poles is studied in [10], [19], and [20]. All the proposed realization schemes are well suited for SISO systems, because they are based on the difference of the outputs of two scalar positive systems, and require the input signal to be nonnegative to ensure that the entries of the state of each positive subsystem remain nonnegative. In this paper the concept of Internally Positive Representation (IPR) of systems is introduced as an extension of the idea of combination of positive systems proposed in [6], and a methodology for the construction of IPRs is developed. The main features of the proposed approach are as follows. • It is naturally suited for MIMO systems. • It is a direct method, and has a low computational cost. • It provides IPRs made of sparse matrices (this feature is of practical interest, because it allows simpler and cheaper circuits layouts, with few interconnections and operational components, such as the costly optical amplifiers in fiber optic filters). • It provides stable IPRs of any stable system. • The input sequences are not restricted to be nonnegative (the approach can be applied even in the case of complex input/output sequences). The work here presented extends the ideas in [11] and [12], where a simple representation scheme was proposed, exploiting the positive and negative parts of the system matrices and of the input sequences. The main drawback of the approach in [11] and [12] is that the stability of the IPR can be guaranteed only under some restrictive conditions. The new approach presented in this paper allows to overcome such a limitation by exploiting a suitable convex positive representation of complex matrices and vectors. The key idea is to represent a complex number by means of positive combinations of the roots of the unity. The paper is organized as follows. • In Section II a preliminar example is reported, aimed to give the reader a rough idea about IPRs and on the proposed IPR construction method. • In Section III a new method of positive representation of complex vectors and matrices, based on the roots of unity, is introduced, and the basic algebraic tools are provided. • Section IV formally introduces the concept of Internally Positive Representation (IPR) of a system, and presents a methodology of construction of IPRs, which is based on the theory developed in Section III. As a first step, a general method of IPR construction of real and complex systems

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is presented (Theorem 5). Then, refinements of the proposed method are developed for cases of particular interest, like systems with block-diagonal structure (Theorem 6), real systems (Theorem 7), and systems in Jordan canonical form (Theorem 9). • In Section V the stability properties of the IPRs provided by the proposed methods are investigated, and two preliminary algorithms are described for the construction of stable IPRs: Algorithm 0 is a naive method, and may not succeed in finding a stable IPR, while Algorithm 1 always provides a stable IPR, although the dimension of the state space of the resulting IPR can be large. • In Section VI, two further algorithms are presented for the construction of low-order stable IPRs: Algorithm 2 deals with general systems (real and complex), while Algorithm 3 is specialized for real systems, and is the most efficient among the presented algorithms. • Numerical examples are provided in Section VII, while Section VIII reports some concluding remarks. For the reader’s ease, the proofs of some of the Theorems stated in the paper are reported in the Appendix. A. Notations In this paper , and denote the sets of integer, real and is the set of nonnegcomplex numbers, respectively, while , and denote the sets of real, ative real numbers. nonnegative and complex -dimensional vectors, respectively, while , and denote the set of real, nonis the identity negative and complex matrices, respectively. , is the zero matrix in , and matrix in is a column vector with all entries 1. For a given , the symbol denotes its componentwise absolute value. The and in denotes the positive and negative parts vectors of , respectively, defined as (1) From these and . Similarly, and denote the positive and negative parts of a matrix . The symbol denote the 1-norm of a , i.e., . vector denotes the integer Given a real number , the symbol denotes part of . Given two integers and , the symbol the remainder of the integer division of by ( modulo ). If , the symbol denotes the set . , the symbol denotes For a given complex number its phase, and and its its magnitude, real and imaginary parts, respectively. The symbol indicates componentwise matrix conjugation (without transposition). The symbol denotes the Kronecker matrix product. The symbol denotes the spectrum of , i.e., the set of all the eigendenotes the spectral radius of , i.e., values of , while the largest among the magnitudes of its eigenvalues. An eigen. A square value of is said to be dominant if matrix is said to be stable if .

II. MOTIVATING EXAMPLE This section is aimed to give the reader a rough idea about IPRs (Internally Positive Representations) of discrete-time linear systems, and about the IPR construction method presented in this paper. The simple case of a single-input-single-output system described by a transfer funcis tion is worked out here. An IPR of a transfer function a positive state-space representation, made up of four positive , plus input and output transformations matrices and , such that, for any input sequence , with -transform , the output can be equivalently computed as

where

(2)

The method in [11] and [12] provides a stable IPR of a system, provided that all the poles of the transfer matrix lie inside , the open square inscribed in the unit circle of the complex plane , (see Fig. 2), whose vertices, denoted (4th roots are the four solutions of the equation ). Note that if and only if of unity: . The new method of IPR construction presented in this paper allows to overcome such a limitation, as it is illustrated in the following example. Example 1: Consider the following transfer function:

(3) Since the two poles

and

of are outside , because , the procedure described in [11], [12] does not provide a stable IPR. The new method takes advantage of the fact that the two poles are in , the open equilateral triangle inscribed in the unit circle of the complex plane, are the solutions of the equation whose vertices (3rd roots of unity: ). The poles and can be written as convex combinations of the two verof tices

(4) with . Since any complex number can be obtained as a positive combination of two suitably chosen , also the residuals and can be written as vertices of positive, although nonconvex, combinations

(5) (note that and are outside , because ). The new methodolgy presented in this paper suitably exploits the diagonal pole-residual complex state-space realization of in (3), and the representations (4) and (5) of the the poles and residuals as positive combinations of the vertices , for the IPR costruction. Following the Stable

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IPR-Algorithm 3 of Section VI-B, an IPR of of the following positive matrices:

is made up

-representation of

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if it is such that (9)

(6) plus the following input and output transformations: (7) Following the nomenclature and notations of Section III, the matrix in (6) is the min-positive circulant 3-representation of the pole , i.e., (see Definition 3), and its comis ponents are the positive coefficients in (4). The matrix [ is defined in (99)] and its componens are the coefficients of the residual in (5) and their sum. The matrix is [ is defined in (35)] and its components are the magnitudes of the real parts of , , 1, 2. coincide with the poles Two of the three eigenvalues of , and . The third eigenvalue is the sum of the of coefficients of and in (4), i.e., . Being and in , then , and therefore the IPR (6), (7) is stable. This simple example shows how a transfer function can be realized by means of an IPR, i.e., by means of a positive system plus input and output transformations (for systems given in state-space form, also state transformations are needed), and gives a rough idea of the role of the roots of the unity in the construction of a stable IPR. III. POSITIVE -REPRESENTATION OF COMPLEX VECTORS AND MATRICES In this section, a new formalism is introduced for the representation of real and complex vectors and matrices by means of larger vectors and matrices of nonnegative reals. Such a formalism extends the one proposed in [11] and [12], restricted to real vectors and matrices. A. Real and Positive Using a Generator

-Representation of Complex Numbers

For a given positive integer , let , with , denote the th roots of unity in the complex plane, i.e., , which are also the vertices of , the solutions of sides inscribed in the unit circle the regular polygon with centered at the origin of the complex plane (see Fig. 2). When the integer is clear from the context, the simpler notation will be used instead of . Let be the row vector (8) For , is said to be a positive generator of the can be represented as complex plane , because any a linear combination with nonnegative coefficients of the ele. ments of and an integer (or, given Definition 1: Given and ) a vector is said to be a positive

Remark 1: Neither nor are positive generators of . For this reason the Definition 1 has been given when is complex, and for when is real. for Remark 2: (Linearity and Convexity of the Representation): Given any pair of complex numbers and , with positive -representations and , respectively, for any pair of nonnegative numbers and , the linear combination is a positive -representation of the sum Moreover, if and are two positive -representations of , then the same complex number , i.e., the vector is a positive for any -representation of . Obviously, a complex number admits infinite positive -rep, while any real number admits infinite resentations for . All positive 2-represenpositive -representations for tations of a given real number are of the type (10) , trivially, , ). The simplest positive 2-representation of the real is , and will be named the min-positive 2-repthe one with , i.e., resentation of and denoted with the symbol (being

(11) is nonzero. Note that at most one component of The concept of min-positive representation can be extended and to complex numbers, by choosing posito the case tive -representations characterized by a minimal number of nonzero components (at most two). In this work we are interested in minimal positive -representations where the two and are considnonzero components are consecutive ( ered consecutive). , the min-positive Definition 2: For a given integer -representation of a real or complex number is the unique such that at most two compositive -representation ponents are nonzero and consecutive. The notation for the min. positive -representation of is is such that for some By definition, it is , , and , : (12) It is possible to give a geometrical interpretation of the minroots of positive -representation (see Fig. 1). Indeed, the , allow to partition the complex plane unity , into disjoint sectors, denoted , numbered from 0 to , defined as for some (13) Note that

, while

, and (14)

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is a sufficient It is worth noting that . A necessary condition is given by Lemma condition for 2. then necessarily Lemma 2: If (16) In Appendix A an algorithm is given for the computation of min-positive representations of complex numbers. B. Positive -Representations of the Product of Complex Numbers , let For matrices, i.e., such that

Fig. 1. Min-positive 5-representations of complex numbers z and v .

denote circular shift and (17)

, and Definition 3: Let ) and let be a positive ). The following associated to

.. . Fig. 2. Sequence P

asymptotically covers the open unit disk.

In Fig. 1, and . Thus, , . and is nonlinear. Indeed, given Remark 3: The mapping is a positive complex numbers and , the sum -representation of the sum , but in general it is not its minpositive -representation, i.e., (nonlinearity), unless and belong to the same sector. Even is linear (the function so, the function is in fact the identity operator in , because by (12) it is , ). , the -sided open regular polygon For a given integer with vertices , ( th roots of unity), of its frontier points can be defined as follows: and the set

, (or let and -representation of (i.e., Toeplitz circulant matrix

.. .

..

.

.. .

(18)

is denoted positive circulant -representation of . If is a , then min-positive -representation of , i.e., is the min-positive circulant -representation of , and will be denoted as . Proposition 3: Given two complex numbers and , let be a positive -representation of , and let be a positive circulant -representation of . Then, the product is a positive -representation of the product . The proof of this Proposition is reported in the Appendix B, and is achieved by proving the identity (19) Remark 4: is a sparse matrix, in that at components are nonzero. most Remark 5: If and are real numbers, the min-positive circulant 2-representation of the multiplier and the min-positive 2-representations of the multiplicand and of the product are

(15) , and therefore , while , so that . The following theorem will be useful in the stability analysis of IPRs in Section V. be such that . Then, Theorem 1: Let for all . is that belongs Proof: A sufficient condition for , whose radius is to the interior of the circle inscribed in . Since the condition is equivalent , the theorem is proved. to The main consequence of Theorem 1 is that if , then for large enough . This property is visually demonstrated in Fig. 2. In Fig. 1,

(20) (recall From (20) it is readily checked that ). This is the kind of representation used in that [11] and [12]. is nonlinear (see Remark 3), for Remark 6: Although and positive-circulant -representation of a given , the function is linear. This fact can be proved exploiting the identity (19), where is replaced by , to represent both sides of the identity , for any pair of real or complex and .

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C. Positive

-Representations of Vectors

and an integer (or , Consider a vector ). Let , , denote some positive -representations of the components of the vector . are such By definition, the vectors , and can be written as a linear combination that as of the roots of unity if

(21)

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circulant -representation of and a positive -representais a positive -representation tion of . Then, the product . of the product The proof of Proposition 4, reported in Appendix C, is achieved by showing that (28) In particular, An interesting property of paper, is the following:

. , that will be used later in the

Definition 4: The vector

(29) (22)

where the subvectors , , are defined in , (21), is called a positive -representation of . If , the vector is said to be the min-positive representation of . will be applied indifferently From now on, the symbol , then . to scalars and vectors, so that if The sum (21) can be written in the compact form (23) , i.e.,

similar to (9), where

(24)

Identity (29) can be proved by comparing (28) with the identity , obtaining (30) , then the vectors (26) are Remark 7: When , such that the min-positive -representations of the terms at most two components are nonzero. It follows that the minpositive circulant -representations are sparse matrices: given , then has at most nonzero componets out of . Remark 8: The min-positive 2-representations of a real vector , and the min-positive circulant 2-representation of a real matrix have the following simple expressions (see also [11] and [12]): (31)

D. Positive -Representation of Products of Matrices With Vectors As it has been done for vectors in Section III-C, the positive -representations of the components of a matrix can be organized into matrices , such that (25) where the matrices

are such that, for each pair of , the vector (26)

. is a positive -representation of and Definition 5: Given a complex matrix (or and ), and given positive matrices such that (25) holds, the following Toeplitz block-circulant matrix:

.. .

.. .

...... ...

.. .

is a nonlinear map, as discussed Remark 9: Although in Remark 3, following the same reasoning of Remark 6, it can is be readily proved that the function linear. E. Positive -Representation of Products Between Complex Matrices and Real Vectors , a real vector , Consider a complex matrix , let be a positive circuand, for a given is a positive lant -representation of . Then, -representation of the product . It is interesting . to find a method for the computation of as a function of This can be done by exploiting the following simple formula that as a linear transformation of : provides (32) where this and from

is defined in (99), Appendix D. From it follows where

(27)

is called a positive circulant -representation of . If the elements of the matrices are the min-positive -representations of the components of , then is the min-positive circu. lant -representation of , and is indicated as , and an integer Proposition 4: Consider (or , and ). Let be a positive

Note that because

(33)

has less columns than

,

.

F. Computation of the Real Part of a Complex Product Using Positive -Representations and . For a given integer Let a positive circulant -representation positive -representation of , so that

of

consider and a .

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Let denote the real part of the product . A positive 2-representation of can be readily computed as (34) where the matrix

is defined as Fig. 3. Block diagram of an IPR (all the quantities inside the dashed block are nonnegative).

(35) state, and output sequences, respectively. An IPR of is a together with Positive System four transformations

The proof of (34) is given in the Appendix E. Note that . IV. INTERNALLY POSITIVE REPRESENTATIONS (IPRS) OF LINEAR SYSTEMS In this paper the symbol a discrete-time state-space system of the form

(40) denotes

(36) where

are the input, state, and output spaces. is a Complex System if , , , ; , , . is a Real System if , , , ; , , . is a Positive System if , , , ; , , . and a nonGiven a real system , with inverse , let denote singular the system , denoted Complex Representation of a Real System. Note that, given a real system , such that the matrix has real eigenvalues and pairs of conjugate eigenvalues, counted with their algebraic multiplicity, so that , there always exists an invertible transformation with the following block structure: (37) where system is

and

, such that the transformed with

the positive input, state, such that, denoting with , and output sequences of , for any pair chosen as initial state and time for the system , and for any starting at , by setting input sequence and in , then (41) and The superscripts and in the state transformations stand for forward and backward, respectively. For consistency, they must be such that (42) Fig. 3 depicts an IPR. Note that Definition 6 ensures that the IPR and of the original system have the same input-output behavior. A.

-IPRs of Real and Complex Systems

In this section it is shown that a straightforward application of the theory of positive -representations presented in Section III can easily provide IPRs for real and complex systems. Theorem 5 ( -IPR): Consider a discrete time real system and an integer (or a and an integer complex system ). Let be positive circulant -represen. Then, the set tations of the system matrices , together with the four transformations

(38) , , and , , . The state of has the , , and . block structure The output of the system is real, and can be written as follows: with

,

(39) When considering systems in Jordan canonical the submatrices and are block-diagonal. be a system Definition 6: Let denote the input, (real or complex), and let

(43) defines an IPR of . and denote Proof: Let the input, state, and output sequences of systems and , re, initial state spectively. For any choice of initial time , and input sequence , , the sequences and , for , obey (36). The proof consists in showing that the sequences and iteratively provided by for

(44)

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where and , are posand , respectively, so that itive -representations of and . For the state sequence the proof is obtained by induction. Since, by assumption, is a positive -representations of , it is sufficient to is a positive -representations of then show that if provided by is a positive -rep. This is true because resentation of and are positive -representations of and , respectively, and by linearity (see Remark 2), the is a positive -representation of the sum . Similarly, is a positive sum -representation of , and the theorem is proved. For a given choice of the integer , the positive representation presented in Theorem 5 is denoted an -IPR of system . Note that a positive system is an IPR of itself. However, a and real system with nonnegative system matrices and , is not an IPR of itself. Remark 10: The approach described in [11] and [12] for the construction of IPRs coincides with the . In this case method presented in Theorem 5 for , i.e.,

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A system with the block-digonal structure given in (47) can be independent subsystems regarded as the parallel of , driven by the same input

(48) In the case of block-diagonal systems, for each subsystem an -IPR can be computed according to Theorem 5. The of resulting IPR is characterized by a set integers, as stated in the following theorem. Theorem 6: ( -IPR) Consider a discrete time real (or complex system system ), with block-diagonal structure ( (47). For a given set of integers for real systems, for complex systems), let the matrices be positive circulant -representations of the , for . Then, the system matrices , where

(49)

(45) and the four transformations (43) are , and

where , together with the four transformations

,

,

, i.e., (50) (46)

Remark 11: For a given system, an infinity of IPRs can be constructed using the method of Theorem 5. First of all, there is the freedom in the choice of the integer . Moreover, for a given , complex vectors and matrices admit infinite positive -representations, as discussed in Section III. A further degree of freedom is the possibility to arbitrarily choose a nonsingualr to change the system coordinates. The following issues arise: for a given stable system , there exists at least one stable IPR among the infinite IPRs provided by Theorem 5? If the answer is yes, what is the right choice of and that provide a stable IPR? These points will be investigated in Section V. In order to increase the degrees of freedom for the construction of IPRs of systems, a variation of the method of Theorem 5 is described below, that can be applied when the matrix is , where are block-diagonal, i.e., matrices, real or complex. In this case the system matrices and the state admit a block partition with compatible sizes

(47)

define an IPR of the system . Proof: The state-space equations of the IPR defined by matrices (49) can be interpreted as the parallel of the following subsystems, where :

(51) of the Each one of these subsystems is an IPR of dimension corresponding subsystem (48) of . Thus, the dimension of the . Thanks to Theorem 5, state space of the IPR is it follows that each output is such that , and from the summation in (48) it follows

(52) This completes the proof. Note that the dimensions and of the input and output spaces of an -IPR can be large. For Real Systems and for Complex Representations of Real Systems, where the input and output sequences are real, smaller

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dimensions can be obtained for the input and output spaces, as described in the following theorem: Theorem 7: Let , with , be a Complex Representation of a Real System. For a given , let the matrices , and be positive circulant -representations of , and , respectively. Then, the positive system where

matrices

, for where

. Then, the system and

(56) together with the four transformations

(53) [the matrix is defined in (99), while (35)] together with the four transformations

is defined in (57)

(54) define an IPR of the system . , Proof: It will be shown that for any , the recursive equations input sequence

define an IPR of the system . IPRs of reduced size can be constructed for complex representations of real systems with block structure of the type (38). The more general case of system matrices with structure

, and (58)

(55) initialized with , provide nonnegative sequences and such that and , for all , where and are the state and output sequences provided by the (36) of the system where . As shown in Section III-E, is a positive -representation of the product , and this justifies the choice in (53). As a consequence, if is a positive -representation of , then given . Being by (55) is a positive -representation of a positive -representation of , then, for all is a positive -representation of . Consider now the output equation in (55). As it has been shown in Section III-C, if and are positive -representations of and , respectively, then is a 2-representation of the real part of the . On the other hand, is real because, by product is a complex representation of a assumption, the system real system, and therefore is a 2-representation of . Moreover, being the positive circulant 2-represenis a 2-representation of tation of , then the product . It follows that in (55) is a positive the real vector , that is the output of 2-representation of the original system , and this implies that . in (54). This justifies the choice of the transformation The results of Theorems 6 and 7 can be easily combined in the following theorem. , with Theorem 8: Let , be a Complex Representation of a Real System, with block-diagonal structure (47). For a given , with set of integers ( if the triple is real), let matrices be positive circulant -representations of the

has only real eigenvalues, is considered, where has only complex eigenvalues. while each The following compatible state partition is also considered

where

(59)

Theorem 9: (2- -IPR) Consider a Complex Representa, with tion of a Real System , with block-diagonal structure (58), (59). Let (as in Remark , 10), while, for a given set of integers , let be positive circuwith -representations of the matrices . lant Then, the system where and

(60) together with the four transformations defined as follows

(61) define an IPR of the system . Proof: Note that the knowledge of the subset of state comis necessary and sufficient ponents

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for the knowledge of the whole state (59) of the original system. For this reason, the IPR (60) is made of a -IPR of the real subsystem and of the IPR of only one half of the complex subsystem, so that the state of the IPR is made of the 2-repreof the real state and of vectors sentation , which are the positive -representations . This explains also the choice of the complex substates and of the forward and backward state transformations in (61). Note that

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Stable IPR-Algorithm 0 (or Given a stable real system ) complex system (or set , if is a complex system); 1. set 2. compute the -IPR using to the method of Theorem 5, ; with , then exit the algorithm (stable -IPR), else 3. if increment and go to step 2.

(62) The choice of and in (60) is readily explained by considering (39), that, when extended to systems of the type (58), takes the form

(63) Remark 12: The Example 1 of Section II presents a 3-IPR constructed according to Theorem 9, exploiting the diagonal pole-residual state-space representation of

This algorithm does not guarantee to terminate or to produce a minimal . This section provides some results that allow to improve the naïve solution of the Algorithm 0. The main questions addressed are the following. • Does any stable system admit a stable -IPR for some integer ? • May similarity transformations (i.e., linear change of coordinates) help in achieving a stable -IPR? Our aim is to provide constructive procedures to give a conclusive answer to both questions. The first step is to establish a relationship between the spectral radius of a (real or complex) matrix and the spectral radius of a positive circulant -representation . To this aim, the following matrix associated to plays an important role: (64)

so that

and

(one pair of complex poles).

V. STABILITY ANALYSIS OF AN

-IPR

Theorem 5 presents a basic method for the construction of IPRs of real and complex systems. Theorems 6, 7, 8, and 9 present variations of the basic method, best suited for systems with particular structures. Whatever the method, the stability of an IPR depends on the eigenvalues of the transition matrix . Note that is always larger than , and therefeore has more eigenvalues. Thus, the IPR always has more natural modes than the original system, although, by Definition 6, they must in have the same input-output behavior (like the eigenvalue the example of Section II). It follows that necessarily all the modes of the original system are also modes of the IPR, i.e., , and that an IPR always has modes that were not present in the original system. This fact arises the issue of the stability of such modes. For instance, the IPR construction method presented in [11] and [12] for real systems, provides . This property ima 2-IPR such that poses that the method can be applied only to those cases where . This section investigates the problem to find a stable circulant -representation of a stable matrix , or of a similar matrix , for some, preferably small, integer ( , if is complex), or set of integers . A naive method, that employs the result of Theorem 5 to solve the problem, is to try increasing values of , starting with ( if the system is complex) until a stable positive circulant -representation of is found. Recall that, for a given integer , there are infinite positive circulant -representations of a matrix. A possible choice is to try only min-positive circulant -representations, as in the following basic algorithm:

where the matrices are the blocks that make up , as defined in (27). The main result, useful for the stability analysis of IPRs, is the following: and ( Theorem 10: Consider and ), and let be a positive circulant -representation and . of . Then The proof of this Theorem, although interesting, requires a number of preliminary definitions and results, and therefore is reported in the Appendix. Let and be two positive circulant -representations of and (i.e., such that two similar matrices for some nonsingular ). Although , and, from and , in general Theorem 10, and . It follows that the choice of similarity transformations plays a main role in the construction of a stable -IPR of a given system. Among similar matrices, the Jordan canonical form (65) are the Jordan blocks associated to the distinct where , is of particular eigenvalues of , denoted , interest for the stability analysis of -IPRs. (or ) be a stable matrix Lemma 11: Let . in the Jordan canonical form (65), with (or ) Then, for any given integer (66) (67) The proof of Lemma 11 is reported in Appendix G.

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Theorem 12: Let be a stable matrix in the Jordan such that . canonical form (65), and let is stable. Then, Proof: Note first that, by the stability assumption, , for . Thus, thanks to Theorem 1, there (large enough) such that . always exists implies that Thus, it is sufficent to recall that [see (15)]. Equation (67) of Lemma 11 , i.e., the stability of . implies that Thanks to Theorem 12, the following algorithm of stable IPR construction can be given. Stable IPR-Algorithm 1 (or Given a stable real system ) complex system such that ; • find the smallest such that is in the Jordan • find canonical form; • using to the method of Theorem 5, compute the -IPR (or of ) (If is real, the method of Theorem 7 can be used, to reduce the dimensions of the input and output matrices.) Remark 13: The result of Theorem 12 can be considered a generalization of the result of [11] and [12] where a method has been presented for the computation of stable IPRs for systems with eigenvalues in . VI. STABLE IPR OF LOW DIMENSION The dimension of the state-space of the IPR provided by the Stable IPR-Algorithm 1 is times larger than the dimension of , where denotes the dimension the original system, i.e., it is of the original system. The problem is that, to ensure stability, must be large enough so that all the eigenvalues of the tran. In this section, two sition matrix are inside the polygon algorithms, that produce stable IPRs with lower dimension, are presented. A. Reducing the Size of IPRs Through Block-Diagonalization The size of an IPR of a system can be reduced by suitably exploiting block-diagonal system representations of the type (47) and a multi-index -IPR of the type presented in Theorems 6 (or Theorem 8 in the case of real systems). To this aim, the following definitions are useful. , : Polygon unions (68) Polygon innovations

,

belong to and are outside i.e., the eigenvalues of . Note that the block only exall polygons , for ists if some eigenvalues are real. A way to construct the above described block-diagonal form of a matrix is to compute the Jordan canonical form first, and then reordering and grouping the Jordan blocks into blocks, so that the eigenvalues of each block satisfy condition (70). Note that, in the Jordan block-diare complex, except for , when agonal form, all matrices present. The IPR construction method presented in the Theorem 6 can be used to provide an efficient -IPR of a system in block, the integer is diagonal form: for each used in the Theorem chosen equal to , so that the set . In this case, the dimension of the state 6 is space of the -IPR is (71) -IPR obthat is smaller than the dimension of the , tained using the Stable IPR-Algorithm 1, which is , and , for all where (this is true because is the smallest integer , for all ). such that Stable IPR-Algorithm 2 Given a stable real system (or ) complex system • find the change of coordinates that put the transition matrix of in the Jordan canonical form; that satisfy • reorder and group Jordan blocks in blocks conditions (70); • compute the IPR of the block-diagonal system using to . (If the method of Theorem 6 with is real, the method of Theorem 8 can be used, to reduce the dimensions of the input and output matrices.) Thanks to Theorem 12, if this method is applied to a stable system in Jordan canonical form, it provides a stable -IPR. Example 2: Consider a system with the following transition matrix: (72)

, . The eigenvalues are is the minimal value for which For this system, , and therefore the dimension of the stable IPR computed . according to the Stable IPR-Algorithm 1 is and : However,

: (69)

Any stable square matrix (real or complex), after a suitable similarity transformation, can be put in a block-diagonal form , where of the type , and the are matrices such that (70)

(73) hence,

and , with and . Thus , and the dimension of the -IPR is , which is an improvement over the dimension of

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the -IPR obtained using Algorithm 1 (it was min-positive representations of the eigenvalues are

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). The

be the pairs of eigenvalues in the sets values of , and let , for with (all eigenvalues are counted with their algebraic multiplicity, so that ). Then there exists an IPR of the system of dimension

(74)

(77)

The state of the IPR obtained using the Stable IPR-Algorithm , where and 2 has the block structure The transition matrix of the {2,5}-IPR of the system , with in Jordan form is

(75)

B. Stable Reduced IPR of Real Systems The construction of stable IPRs of real systems takes advantage of the presence of conjugate eigenvalues, by exploiting the IPR construction method of Theorem 9, by considering a complex representation of a stable real system in the Jordan canonical form. The Jordan blocks can be organized and grouped as in the previous section, so that the transition matrix takes the form

Remark 14: It is interesting to note that the IPRs provided by the Stable IPR-Algorithms 1, 2, and 3, are made of sparse matrices. This is true because they exploit the Jordan canonical form (which is a sparse matrix) of the system transition matrix, and employ the min-positive circulant -representations of the system matrices, which is sparse too (see Remarks 7 and 4). In Jordan matrix, which has at most particular, a nonzero components (the diagonal and the superdiagonal elemin-positive circulant -represenments), admits a nonzero components ( for tation, with at most for the superdiagonal) out of . the diagonal, Example 3: Considering again the matrix of the Example and , then the matrix can be put 2. Since (one real eigenvalue) in the Jordan canonical form with (one pair of complex eigenvalues in ), so that and . In this case the dimension of the 2- -IPR provided , much by the Stable IPR-Algorithm 3 is smaller than the dimensions provided by the Stable IPR-Algoand Stable IPR-Algorithm 2 . The ritm 1 , transition matrix of the stable 2-{5}-IPR is is the same of (75), and where

C. Comments on the Minimality of the IPR (76) Thanks to Theorem 12, the construction method of Theorem 9 applied to a system with transition matrix in the Jordan canon, provides a ical form (76), with the choice stable 2- -IPR of dimension Thus, the following algorithm can be given. Stable IPR-Algorithm 3 Given a stable real system : • find the change of coordinates that put the transition matrix of in the Jordan canonical form (76); • compute the IPR of the block-diagonal system according to the method of Theorem 9, by choosing (2- -IPR). Note the Stable IPR-Algorithm 3 has a very low computational cost, limited to the block diagonalization of the original state transition matrix. The construction of the Stable IPR-Algorithm 3 can be taken as a proof of the following theorem. Theorem 13: Given a stable real system , with transition , let denote the number of real eigenmatrix

The Stable-IPR Algorithm 3 provides a stable IPR of the smallest size among the IPRs provided by the other algorithms. Although the size reported in Theorem 13 is not proved to be the minimal size, among all stable IPRs of a given stable system, it can be considered as an upper bound of such a minimal size. Two issues may be further investigated: the minimality of IPRs of systems by exploiting representations different from the Jordan canonical form; and the relationship between the minimal size of an IPR and the minimal size provided by means of other methods. Due to space reasons we do not elaborate further on this issue, that can be investigated in the light of the Karpelevich Theorem [8], [14], that completely characof points in the complex plane terizes the regions matrices with specthat are eigenvalues of nonnegative and the regions tral radius . The marked similarity between suggests that there is a connection between them that deserves further investigation. VII. EXAMPLES In this section we give two examples of IPR construction using the proposed procedure, although, due to space constraints, the first example can not be fully developed. In these examples, the algorithm described in Appendix A is used for

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the construction of min-positive representations of complex numbers. Example 4: The following example has been considered in [10], and consists in the positive realization of a low-pass Chebyshev filter of order 3:

(78) and The poles of the transfer functions are , so that (one real (one pair of conjugate poles in ). Thus, pole) and in the Algorithm 3, that means . The resulting . This size matches 2- -IPR has size the value obtained in [10] with a different approach. Notice that the original method in [6] generates a size of 48. Hence, in this example the performance of our approach matches the most efficient algorithms for SISO systems. Example 5: Consider the following real MIMO system, with , ,

The IPR is stable because (note that

, thanks to Theorem 12).

VIII. CONCLUDING REMARKS The matrix (

has a pair complex conjugated stable eigenvalues , and ), with algebraic multiplicity 2, and geometric mul(no real eigenvalues) and (two tiplicity 1, thus pairs of conjugate eigenvalues, counted with algebraic multiplicity). For this system the IPR-Algorithm 0 has been applied , and in all cases the resulting IPR was for unstable. Using the algorithm in Appendix A it can be seen , and the min-positive 3-reprethat and sentations are , with (recall that implies ) so that the Stable IPR-Al. The blocks gorithm 3 can be applied, with and of the matrix that transforms in the Jordan canonical form, and of its inverse are (79) so that

, with (80)

The Stable IPR-Algorithm 3 provides the following IPR of di, where : mension

The approach for the construction of IPRs of linear systems presented in this paper is based on a new methodology of nonnegative representation of complex matrices and vectors. The proposed technique is well suited for real and complex systems, and is capable of providing a stable IPR of any stable system, independently of the position of its poles within the open unit disk, and of their multiplicity. The algorithms presented, which indifferently apply to SISO and MIMO systems, are straightforward methods characterized by a very low computational cost, unlike other methods available in the literature, where the matrices of the IPR are computed by numerically solving suitable optimization problems. In particular, among the presented algorithms, the Stable IPR-Algorithm 3, suited for real systems, provides the IPR with the smallest state-space dimension (the size depends on the geometrical position of the eigenvalues in the complex plane). Although the issue of the minimality of the IPR deserves further theoretical investigation, it has been shown on some examples that the proposed approach provides IPRs whose size matches the most efficient existing methods. Another interesting feature of the presented algorithms, is that the positive matrices that make up the IPR of a system are sparse, as it has been pointed out in Remark 14. This property is of practical interest in the physical implementation, in that simple and cheap circuits layouts can be obtained, with few interconnections and operational components (such as the costly optical amplifiers in fiber optic filters). Although all the IPR theory has been presented for state-space representations of systems, the methodology can be equally applied to transfer-function representations, as illustrated in Examples 1 and 4. The presented approach for IPR construction can also be extended to classes of nonlinear

CACACE et al.: NEW APPROACH TO THE INTERNALLY POSITIVE REPRESENTATION OF LINEAR MIMO SYSTEMS

systems, like polynomial systems [9], for which, at present, it is the only viable approach.

Thanks to identity (85) and to Definition (18) of

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:

APPENDIX A. Computation of Min-Positive

-Representations

The case of and is considered here [the case and , is trivially solved in (11)]. The sectors defined in (13) are disjoint, and such that , so that any nonzero complex number belongs to one and . Equivalent conditions for to only one of the sectors be in are the following: or

(87)

C. Proof of Proposition 4 of Section III-D Proof: The first step is to prove the following identity, anal: ogous to (85), for

(81) (88)

By Definition 2, if such that , and

then

is . Thus, considering that , it follows

where is the circular shift matrix defined in (17). By direct computations

(82) Being

and (89)

Equation (83) can be solved for lowing algorithm.

and

(83)

where identity (85) has been used. The product can be rewritten as , so that the Kronecker product property

, as in the fol-

(90) can be applied, giving

Computation of • If then set • If let for set

,

(91)

; , so that and

, and that is the identity (88). Consider a vector and its -representation and the representation (25) of matrix . The product written as

, can be

(84)

(92)

B. Proof of Proposition 3 of Section III-B Proof: The thesis is achieved by proving the identity (19), i.e., . Consider first the following property of the circular shift matrices:

Since is scalar, and therefore , the terms in the summation, thanks (90), can be rewritten as

(85) This property is easily verified by direct computation of , using the definition (8) and the identities . Now, using the representations and , compute the product as (86)

(93)

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Substituting (93) in the summation in (92) gives

and are positive When 2-representation of is

-representations, a positive

(94) and this proves identity (28).

that is (34). Note that the coefficients , (35) are, for

D. Positive -Representation of Products Between Complex Matrices and Real Vectors of (32), that allows the computation of The matrix as a linear transformation of , takes different forms, depending on whether is even or odd. Recall that the are defined in such a way that and sectors . For this reason, for any the positive half line belongs to , while the negative half line belongs to when is even, and to if is odd. When is even, it is , and therefore

and

(101) in

(102)

F. Proof of Theorem 10 of Section V Some preliminary definitions and results are needed before to : give the proof. Consider the following vectors in (103)

(95) where , is odd, note that

, is the canonical basis of

where, as usual, transposition does not include conjugation. It is known that these vectors form an orthogonal basis of . Note that and . The following identities are easy to verify, for all and :

. When

(104) (96) Defining so that (note that, being

, it is

which can be also written as . Note that (104) generalizes (85). Now, consider the following matrices associated to defined in (27):

(97) (105)

) we have

(98)

Lemma 14: For all hold:

the following identities (106)

Defining the matrices

as Proof: By direct computation, using definition (27) and the properties (90) of the Kronecker product, we have

(99) the identity (32), i.e.,

, is readily verified.

E. Computation of the Real Part of a Complex Product Using Positive -Representations Let be a positive circulant -representation of , so that and a positive -representation of . The real part of the product , i.e., be computed as

, can

(100)

(107)

CACACE et al.: NEW APPROACH TO THE INTERNALLY POSITIVE REPRESENTATION OF LINEAR MIMO SYSTEMS

From this, by using the defintion (105), the identity (106) follows. Remark 15: Note that from definition (105) it follows that (108) and that, by definition (104), it is , so that the identities (106) are a generalization of the identity in (29). Moreover, if is a positive block-circulant -representation of , then all are nonnegative, and nonnegative is also the matrix

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) and therefore necessarily . This concludes the proof. and , with min-positive Remark 16: For and . 2-representation , it is , as already proved in [11] and [12]. Thus, Now the Proof of Theorem 10 of Section V can be given. (see Remark Proof of Theorem 10: Recalling that is a trivial consequence of The15), the property , note first that orem 15. In order to prove that , as it follows from (105) and (64). By definitions (105)

(109) with the The following theorem relates the spectrum of spectrum of matrices . Theorem 15: Consider and (or and ), and let be a positive circulant -representation of , as defined in (27). The matrices in (105) are such that

(114) so that p. 619]) that

(110) Proof: First, we show that . This is equivalent to show that any eigenvalue of is also an eigen, and let value of . For a given , let be an eigenvalue of the associated left-eigenvalue, so that . Thanks to the identity (106) it is easy to shown that is also an is the associated eigenvalue for , and that left-eigenvalue. This is obtained by direct computation

(componentwise inequality) for all . It is known (see, for example, [21, ch. 7, implies , so that . Thanks to Theorem 15, .

G. Proof of Lemma 11 of Section V Proof: Let , matrices that make up (64), so that

, denote the positive , as in (27), and let as in

(115) The matrices

are such that, for each

(111) , the pair is an eigenvalue-eigenvector pair Being for the matrix . Now we prove that . This is equivalent , for to show that any eigenvalue of is also an eigenvalue of . Let be an eigenvalue of , and let some be the associated right-eigenvector. By direct computation, again thanks to the identity (106), it is easy to show that is also an eigenvalue of one of the matrices and that is the associated right-eigenvector

(112) , then the pair is an Thus, if for some it is eigenvalue-eigenvector pair for the matrix . It is sufficient to note that, being , there must exist at least one such that . This is proved by showing that , where the square matrix is .. .

.. .

(113)

Since the row vectors are independent (and orthogonal), then the matrix is nonsingular (remember that

(116) From this,

, and by (115) (117)

By noting that and , the has the identity (117) proves that the matrix same Jordan structure of . Moreover, being the eigenvalues on the diagonal of , the eigenvalues of are on its diagonal, and by (117) it is (118) , and Thanks to Theorem 10, it is this proves the inclusion (66). Moreover, by the identity (118), it , and, by Theorem is . This proves (67). 10, it is ACKNOWLEDGMENT The authors thank the anonymous reviewers for their careful reading of this paper and for their valuable comments that concretely contributed to improve the quality of this work.

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REFERENCES [1] B. D. O. Anderson, M. Deistler, L. Farina, and L. Benvenuti, “Nonnegative realization of a linear system with nonnegative impulse response,” IEEE Trans. Circuits Syst. I, vol. 43, no. 2, pp. 134–142, 1996. [2] R. M. Anderson and R. M. May, “Population biology of infectious diseases: Part I,” Nature, vol. 280, pp. 361–367, 1979. [3] A. Asachenkov, G. Marchuk, R. Mohler, and S. Zuev, Disease Dynamics. Boston, MA: Birkhauser, 1994. [4] L. Benvenuti and L. Farina, “On the class of linear filters attainable with charge routing networks,” IEEE Trans. Circ. Syst.-II, vol. 43, pp. 618–622, 1996. [5] L. Benvenuti and L. Farina, “Discrete-time filtering via charge routing networks,” Signal Process., vol. 49, pp. 207–215, 1996. [6] L. Benvenuti, L. Farina, and B. D. O. Anderson, “Filtering through combination of positive filters,” IEEE Trans. Circ. Syst.-I: Fundamental Theory and Applications, vol. 46, no. 12, pp. 1431–1440, 1999. [7] L. Benvenuti and L. Farina, “The design of fiber-optic filters,” J. Lightw. Technol., vol. 19, no. 9, pp. 1366–1375, 2001. [8] L. Benvenuti and L. Farina, “A tutorial on the positive realization problem,” IEEE Trans. Autom. Control, vol. 49, no. 5, pp. 651–664, 2004. [9] F. Cacace, A. Germani, and C. Manes, “Representation of a class of polynomial MIMO systems via positive realizations,” in Proc. Eur. Control Conf. (ECC09), Budapest, Hungary, Aug. 2009. [10] W. Czaja, P. Jaming, and M. Matolsci, “An efficient algorithm for positive realizations,” Syst. Control Lett., vol. 57, no. 5, pp. 436–441, 2008. [11] A. Germani, C. Manes, and P. Palumbo, “State representation of a class of MIMO systems via positive systems,” in Proc. IEEE Conf. Decision and Control, New Orleans, LA, Dec. 2007. [12] A. Germani, C. Manes, and P. Palumbo, “Representation of a class of MIMO systems via internally positive realization,” Eur. J. Control, vol. 16, no. 3, pp. 291–304, 2010. [13] A. Gersho and B. Gopinath, “Charge-routing networks,” IEEE Trans. Circ. and Syst, vol. 26, no. 2, pp. 81–92, 1979. [14] F. I. Karpelevich, “On the characteristic roots of matrices with nonnegative elements,” in Eleven Papers Translated From Russian. Providence, RI: American Math. Soc., 1988, vol. 140. [15] Y. Y. Haimes, B. M. Horowitz, J. H. Lambert, J. R. Satos, L. Chenyang, and K. G. Crowther, “Inoperability input-output model for interdependent infrastructure sectors. I: Theory and methodology,” J. Infrastruct. Syst., vol. 11, no. 2, pp. 67–79, 2005. [16] W. W. Leontieff, The Structure of the American Economy 1919-1939. New York: Oxford Univ. Press, 1951. [17] P. H. Leslie, “On the use of matrices in certain population mathematics,” Biometrika, vol. 33, pp. 183–212, 1945. [18] D. G. Luenberger, Introduction to Dynamic Systems. Hoboken, NJ: Wiley, 1979. [19] B. Nagy and M. Matolcsi, “Minimal positive realizations of transfer functions with nonnegative multiple poles,” IEEE Trans. Autom. Contr., vol. 50, pp. 1447–1450, 2005. [20] B. Nagy, M. Matolcsi, and M. Szilvási, “Order bound for realization of a combination of positive filters,” IEEE Trans. Autom. Contr., vol. 52, pp. 724–729, 2007. [21] C. D. Mayer, Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: SIAM, 2000. [22] S. Rinaldi and L. Farina, Positive Linear Systems: Theory and Applications. Hoboken, NJ : Wiley, 2000. [23] V. Volterra, “Variations and fluctuations of the number of individuals in animal species living together,” in Animal Ecology. New York: McGraw-Hill, 1931. [24] L. Yuzhe and P. H. Bauer, “Frequency domain limitations in the design of nonnegative impulse response filters,” IEEE Trans. Signal Process., vol. 58, no. 9, pp. 4535–4546, Sep. 2010. Filippo Cacace graduated in electronic engineering at Politecnico di Milano, Milano, Italy, in 1988, where he received the Ph.D. degree in computer science in 1992. Since 2003 he has been working with Centro Integrato di Ricerca at University Campus Biomedico of Rome, Italy, where he is currently an Assistant Professor. His current research interests include wireless and heterogeneous computer networks and network applications, nonlinear systems, and system identification in connection with applications in the field of systems biology.

Alfredo Germani received the Laurea degree in physics and the Postdoctoral degree in computer and system engineering from the University of Rome “La Sapienza,” Rome, Italy, in 1972 and 1974, respectively. From 1975 to 1986, he was a Researcher at the Istituto di Analisi dei Sistemi e Informatica “A. Ruberti” of the Italian National Research Council (IASICNR), Rome, where from 1986 to 2007 he has been the Coordinator of the System and Control Theory Research Group. In 1978 and 1979, he was a Visiting Scholar with the Department of System Science, University of California at Los Angeles, (UCLA). From 1986 to 1987, he was a Professor of Automatic Control at the University of Calabria, Italy. Since 1987, he has been Full Professor of System Theory at the University of L’Aquila, Italy, where from 1989 to 1992 he has been the Chairperson of the School of Electronic Engineering. At the University of L’Aquila, from 1995 to 2000 he led the Department of Electrical Engineering, and from 1996 to 2009 he has been the Coordinator of the Ph.D. School in Electronics. In 2001, he was President of the Committee for the Institution of the new Laurea Degree in “Ingegneria Informatica e Automatica.” Since 2001 he has been a member of the Executive Committe of the Research Center of Exellence (DEWS), and since 2007 he is a member of the Advisory Board of the Engineering School. In 2007, he was the President of the Organizing Committe of the second-level Master on Optimization Methods and Data Mining. His areas of interest include systems theory, systems identification and data analysis, nonlinear, stochastic, and optimal control theory, modeling and control of environmental systems, distributed and delay systems, finitely additive white noise theory, approximation theory, optimal polynomial filtering for non-Gaussian systems, image processing and restoring, and mathematical modeling for biological processes. Costanzo Manes received the Laurea degree in electronic engineering and the Research Doctorate degree in systems engineering from University of Rome “La Sapienza,” Rome, Italy, in 1987 and 1992, respectively. Since 1992, he has been working with the Department of Electrical Engineering, University of L’Aquila, Italy, where he is currently an Associate Professor. Since 1998 he has also been a Research Associate at the Istituto di Analisi dei Sistemi e Informatica “A. Ruberti,” of the Italian National Research Council (CNR), Rome. His current research interests are in the field of system estimation and identification, in both deterministic and stochastic settings, with application to nonlinear systems, delay systems, singular systems, and switching systems.

Roberto Setola ( SM’07) received the Laurea degree in electronic engineering in 1992 and Ph.D. degree in electronic engineering and computer science in 1996 from the University of Naples “Federico II,” Naples, Italy. Since 2004, he has been with University Campus Biomedico of Rome where currently he is an Associate Professor of automatic control, Director of the Complex System and Security Laboratory, and the Director of the second-level Master in Homeland Security. From 1999 to 2004 he worked at the Italian Prime Minister’s Office, and from 2006 he was the Secretary of the Italian Association of Critical Infrastructures’ Experts. He has written three books about simulation of dynamic systems, was the editor of three books on Homeland Security and Critical Infrastructures, and a guest editor of three special issue in international journals. He is author of more than 100 scientific papers related with modeling and control of complex systems (electro-mechanical, biological and social) and critical infrastructure protection.