Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 878146, 9 pages

http://dx.doi.org/10.1155/2013/878146

Research Article

A Constructive Positive Realization with Sparse Matrices for a Continuous-Time Positive Linear System

Kyungsup Kim

Department of Computer Engineering, Chungnam National University, 99 Daehak-ro, Yuseong-gu, Daejeon 305-764, Republic of Korea

Received 9 September 2013; Revised 22 November 2013; Accepted 10 December 2013

Academic Editor: Yong-Kui Chang

Copyright © 2013 Kyungsup Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper discusses a computational method to construct positive realizations with sparse matrices for continuous-time positive linear systems with multiple complex poles. To construct a positive realization of a continuous-time system, we use a Markov sequence similar to the impulse response sequence that is used in the discrete-time case. The existence of the proposed positive realization can be analyzed with the concept of a polyhedral convex cone. We provide a constructive algorithm to compute positive realizations with sparse matrices of some positive systems under certain conditions. A sufficient condition for the existence of a positive realization, under which the proposed constructive algorithm works well, is analyzed.

1. Introduction

We address the positive realization problem of a positive linear time-invariant system with a single input and a single output using the polyhedral convex cone introduced in [1]. Nonnegative constraints can be encountered in some applications, such as in probability theory [2], fiber-optic filters [3] and compartmental models in pharmacokinetics [4]. Several types of implementation of constructive methods of positive realizations for general transfer functions in discrete-time have been proposed [5–8]. Most of the constructive methods that have been proposed to solve the positive realization problem in minimal dimensions have focused on discrete systems [6, 9, 10]. The positive realization of a continuous-time linear system can be obtained by finding a possibly minimal generator of polyhedral cone intervening reachability and observability. Finding a general minimal positive realization in a positive linear system is an open and difficult problem. Therefore, we instead try to find a positive realization of a positive system with a special structure and lower order by applying some constraints.

We note that constructive methods for positive realization in the continuous-time case have been studied less frequently than those in the discrete-time case. We will consider an efficient constructive method to compute positive realizations with sparse matrices for some given continuous-time positive systems with (possibly multiple) complex poles. We consider the relationship between positive realizations of the discrete-time systems introduced by [10, 11] and those of the continuous-time systems. A positive realization with sparse matrices is of practical interest, because it allows a simpler and cheaper realization in real system implementations. For example, we can construct the oriented weighted graphs that correspond to the computed positive realizations in applications such as circuits and fiber optic filters [3, 10, 12]. By reducing the number of incoming and outgoing branches of the weighted graph in a sparse positive realization, we can reduce the cost of the system implementation. In addition, because positive realizations are not unique, it is better to choose a sparse positive realization with fewer free parameters and a lower dimension. We also investigate the positive realization of a transfer function with multiple complex poles and present the sufficient conditions for the positive realization of this case.

In Section 2, we introduce the preliminary concepts for the analysis of a continuous-time positive linear system. In Section 3, we discuss the positive realization of a transfer function with simple poles. In Section 4, we consider a positive realization of a transfer function with multiple complex or real poles. We present the numerical results in Section 5.

2. Preliminary Concepts and Problem Formulation

The notations used throughout the paper are introduced here. is the set of real numbers , and is the set of positive real numbers. A convex cone is denoted by the smallest convex cone of a set , which consists of all finite nonnegative linear combinations of elements of the set . The dual cone, , of a cone is defined by . A convex cone is said to be a polyhedral cone if is spanned by a finite number of vector sets such that ; that is, and is a polyhedral generator of [13]. Hereafter, is also denoted by the matrix with columns ; that is, , as long as it is not ambiguous. Given a general matrix , we write if at least some entry .

Consider a singleinput, single output linear time-invariant system of the form:

where is an matrix, is an matrix, and is an matrix. Define the transfer function as the Laplace transform of the impulse response function .

The linear system in (1) is a positive linear system if for any non-negative input and non-negative initial state, the state trajectory , and the output are always non-negative. We provide a formal statement of the positive realization problem for continuous-time positive systems. A matrix is defined by a Metzler matrix if all of its off-diagonal elements are positive. A triple is said to be a positive realization of the transfer function if is a Metzler matrix, , and .

The positive realization problem consists of both the existence and the minimality problems. The existence problem is to find conditions and provide an algorithm for the existence of the positive realization. The minimality problem is to determine the minimal order allowed by the positive realization. It is known that in the absence of any nonnegative restriction, every proper rational function has a realization whose minimal dimension coincides with the McMillan degree [14]. There is no guarantee that properties such as existence and minimality hold for the positive realization problem [1, 9, 15]. The minimal order of the positive realization is generally larger than or equal to the McMillan degree. Our objective is to find a positive realization with the lower order from the given positive system in the absence of non-negative constraints.

Theorem 1 (see [1]). Let be a polyhedral cone in and . Thus, for any if and only if there is a such that for all .

To prove the sufficiency of this problem, if for some , for all . Let . For some and , . Hence . Therefore, . To prove the necessity of this theorem, set . Let and be polyhedral generators of and , respectively. Let for and otherwise. Let . Thus, . The interested reader can refer to [1] for more details. We observe that if for a sufficiently large , then there exists a positive matrix such that . Therefore, we obtain a Metzler matrix .

A reachable set is the set of all points to which the states approach from the origin by non-negative inputs within finite time. It is wellknown that the reachable set consists of a closure of the non-negative linear combination set of the impulse response functions [1]. In Lemma 1 in [1], it was showed that , where is a closure set of . An observable set is the set of initial states in which the output is non-negative for all . is defined by . Assume that for some . Because , . Thus, we can verify that . It is well-known that a positive realization problem of a given transfer function is reduced to the problem of finding an appropriate polyhedral cone in a room sandwiched by the reachability and observability cones [1, 9]. The following theorem is presented as in [1] without proof.

Theorem 2 (see [1]). Let a transfer function be a strictly proper rational function with degree n, whose (minimal) realization is given by . Then, has a positive realization if and only if there exists a generator matrix and such that a polyhedral cone satisfies

(1), (2),

where , , is a reachable set and is an observable set.

From this theorem, and constitute necessary and sufficient conditions for the existence of a positive realization with a Metzler matrix in a continuous-time positive system. We note that the equation is related to the conditions of the existence of a positive realization in discrete time [9, 10]. For the discrete-time case, several constructive methods to compute the polyhedral cones have been presented [6–8, 10, 11]. We propose a constructive method to compute a positive realization with sparse matrices for a continuous-time positive system by modifying and extending the discrete version presented in [11].

Consider the relationship between the eigenvalues of in the state space realization and the poles of the transfer function . Let be an eigenvalue set of . A dominant eigenvalue of is defined by a maximal real eigenvalue in and denoted by . The spectral radius of is denoted by ; that is, . If is a Metzler matrix, then the dominant eigenvalue of is a unique real eigenvalue and the real parts of the other eigenvalues of are less than [16, 17].

A positive system is defined as positive realizable if we can find an equivalent positive realization of . For the continuous-time positive system, if with realization is positive realizable, then we can find a positive realization of the positive system . The transfer function of an impulse response function for all is denoted by where and . We note that has similar properties to the transfer function in the discrete-time domain. From Theorem 1, if is positive realizable in the continuous-time domain, then is positive realizable in the discrete-time domain. Let be defined as a disk with a center 0 and a radius . The function is analytical outside for properly large . Therefore, is interpreted as a transfer function of a discrete-time domain on the disk .

Set and let for each be an eigenvalue of . The sequence of with for all is denoted by the Markov sequence introduced in [18, 19]. We can observe that for all . is described by . We note that the Markov sequence is similar to an impulse response sequence in the discrete-time domain. We define a transfer function that corresponds to the shifted Markov sequence (i.e., ) and for a given .

Lemma 3. Assume that is decomposed into

and the -shifted transfer function has a positive realization with dimension . Then, has a positive realization with dimension such that

where the realization of is given by

Proof. We observe that the transfer function with a non-negative finite Markov sequence has an -dimensional positive realization . The positive realization of the transfer matrix with dimension is given by . The following combination of the two realizations yields a non-negative realization of :

Therefore, has a positive realization of dimension .

A rational transfer function is primitive if the maximum modulus pole of is the only maximum real eigenvalue of , which is similar to concepts presented in discrete-time versions [5, 20]. Thus, a transfer function is defined by a primitive transfer function, if is a primitive transfer function for some . Using partial fractional expansion of the rational transfer function , we can define the primitive positive transfer function as

where is the maximum number of the multiplicity of , which we call the Jordan-form realization. An Jordan-form matrix , an matrix , and an matrix are given, respectively, as

for [14]. A direct sum is defined as the block matrix with proper zero matrices . The vector represents a column vector of dimension with the th entry equal to 1 and all other entries equal to 0. has a canonical Jordan-form realization [14] such that

where is the transpose matrix of .

Theorem 4. A primitive transfer function is defined by (6). is decomposed into . Assume that is positive realizable and such that and are defined by

Then, a canonical Jordan-form realization of is a positive realization, a canonical Jordan-form realization of is a positive realization and satisfies

for all with and some for .

Proof. Because is positive realizable, we have the Markov parameter sequence . However, is necessary to obtain a positive Markov parameter sequence (refer to [5]). The Jordan-form realization of given by (4) is a positive realization and is represented by . We have a realization of . Then is computed by

where is the number of -combinations from a given set with elements. For , there is a sufficiently large such that for all with . We can find a sufficiently large such that . Therefore, the canonical Jordan-form realization of is a positive realization. Because increases asymptotically as , we can choose a proper such that for for all and for .

As increases, increases asymptotically. For sufficiently large and , we obtain the transfer functions in (10) with non-negative Markov sequences. Using Theorem 4, and in (2) and (9), respectively, also have non-negative Markov sequences. Therefore, the positive realizations of and are obtained. A primitive function with a simple dominant pole can be divided into

Therefore, we need to find a positive realization for for all . In the continuous-time domain, we obtain by substituting . In the next section, we will discuss how to find a positive realization of for each .

3. Positive Realizations for Systems with Simple Complex Poles

Definition 5. Let for denote the set of points in the complex plane that lie in the interior of a regular polygon with edges with one vertex at point and a zero point at 0. For , the polygon is defined by a subset in through the following inequalities:

for all with , where and .

The complex number corresponds to the point in the Cartesian plane . The polygon set on the complex plane can be mapped onto . A matrix that consists of the vertices of a polygon in the complex plane is defined as

for a given . A cone generator matrix is defined as

where represents an row vector with all entries equal to 1; the vectors are extreme points in a polyhedral cone, . The real part of is denoted by and the imaginary part of is denoted by . The angle of is denoted by . We consider a simple third-order positive proper transfer function with a pair of strictly conjugate complex poles and a dominant real pole in partial fractional form such that

where , is a complex pole with , is a complex conjugate of , and is complex number. A minimal real canonical realization of the transfer function is given by

where is a direct sum and is a real pole with . A complex pole is defined as with and another complex pole is a conjugate of (i.e., ). Then, , and .

Theorem 6. Assume that a transfer function is defined by (17) where , and an extreme generator is constructed as described by (16). If

then there is a positive realization of with order .

Proof. According to the assumption made in (19), the angle is less than . As shown in Figure 1, there is a sufficiently large positive real number such that the eigenvalues of are located in . is projected onto by dividing by . This means that the set is an eigenvalue set of . As reported for the discrete-time cases in [6, 10], we can construct the polyhedral cone generator as described in (16) such that . By Theorem 1, implies that . We can demonstrate the invariance of for . Therefore, we obtain a Metzler matrix .

We note that an invariant polytope argument is equal to an -invariant polyhedral cone. We will show that and . Because , there exists such that . It follows that . We also know that . The inequality leads to the following sufficient condition: . To satisfy the above inequality, we impose a sufficient condition such that for sufficiently large .

Figure 1: The geometric interpretation of using .

This result is similar to the original results of Dmitriev and Dynkin with respect to the Kolmogorov equation [21], which represents the earliest example of an invariant polytope argument. The invariance polytope approach, which is similar to the polyhedral convex cone, is adapted for a phase-type distribution [2]. If the conditions of Theorem 6 are satisfied, then we obtain a proper matrix such that . However, the solutions for are not unique. Furthermore, we wish to avoid a dense matrix for for real system implementation. Therefore, we consider how to obtain a proper sparse matrix with elements. The form of this realization matrix is related to circular Toeplitz matrices. Circular Toeplitz matrices have several useful properties (i.e., refer to [22]). For a given vector , a circular Toeplitz matrix is defined as

where each element is with ; here, is a residue of the integer modulo .

Theorem 7. Assume that the conditions of Theorem 6 are satisfied. Then there exists a sparse positive circular Toeplitz matrix with elements such that for a proper , and is a Metzler matrix.

Proof. For , the eigenvalue set of is . Thus, . We choose column vectors from as in Figure 1 such that . We also obtain , where , with . Let be defined as . For ,

We can obtain a sparse circular Toeplitz matrix such that a vector is defined by , , and and the other values of are zero. Thus, the matrix is a sparse Toeplitz positive matrix.

4. Positive Realizations for Systems with Multiple Poles

Let us consider the properties of primitive transfer functions with a simple dominant pole and multiple complex poles (i.e., set for some in (13)). Let us consider a transfer function with a single pair of multiple complex conjugate poles:

where the pole and the coefficients are complex, and is defined as the conjugate of . The transfer function with the partial fractional expansion in (23) has a canonical Jordan-form realization such that

There exists a similarity transformation matrix such that a block Jordan-form realization is given by

where . Let . Using a block diagonal matrix as a similarity transformation, we obtain a new real block Jordan-form realization such that

where the entries of are defined by , for each and .

We consider a positive realization of transfer functions with a multiple complex conjugate pole pair. Let us define for a given and . We propose a constructive method to generate a positive realization for a positive system with multiple poles, by modifying the discrete time version of [11], and provide the sufficient condition necessary for the proposed method in the next theorem.

Theorem 8. Assume that a positive system has a partial fractional expansion of the form

with one dominant real pole with a simple order and two complex poles and with multiple orders , where , and is defined as in (23). Set and . One assumes that the following conditions are satisfied:

for a given sufficiently large and all . Then, there exists a positive realization of the transfer function where is a Metzler matrix with order .

Proof. We obtain a real block Jordan-form realization of the transfer function in (27) as follows:

where , , and are defined as in (26). We use the fact that there exists an such that there exists a -invariant cone with edges (i.e., ) to generalize the concept of the cone generator introduced in [10] for the case of multiple complex poles. To formulate a polyhedral cone generator with the -invariant property, a block shift matrix and a matrix are defined as

where is an identity matrix, is a zero matrix with the proper dimension, and the matrix is defined as in (15). A cone generator matrix is defined as , where is defined as for , such that represents an row vector in which all entries equal 1 and . The columns of matrix represent the extreme vertices of a finite generated cone in (i.e., ) and are positive independent. The polyhedral cone is generated by , that is; for each . We will prove that is -invariant (i.e., ). Let be the cone generated from the column vectors of (i.e., ). For , we observe that . Note that for has nonzero values for the 1st, th, and + 1th block components

The invariance property requires that . Set . The matrix has eigenvalues with . Thus, the positive realization problem with respect to is closely related to that of the discrete-time cases reported in [6, 10]. Choose such that

for each , where the values satisfy , and all of the entries in the first row of are equal to 1. A sufficient condition for a feasible solution should satisfy two inequalities, and , and an equality for a given . By rearranging the above conditions, we obtain the inequality . From this result, we observe that and are parameters that can be tuned to obtain a positive matrix. The polyhedral cone is -invariant under the condition described by (28). We observe that , which is -invariant, is equal to by Theorem 1. We obtain a sufficient condition as described by (28) to guarantee that is -invariant for all . When the sufficient conditions are satisfied, there exists a positive such that . Therefore, is a Metzler matrix.

Next, let us consider and . Because , it follows trivially that . Finally, we will prove that . The entries and of are given by , where and . An inequality for each leads to the sufficient condition in (29) as follows:

To satisfy the above inequality, we impose a sufficient condition such that for sufficiently large and each . We show that . Therefore, there exists a positive realization with order such that , and satisfy , and . Therefore, we obtain a Metzler matrix .

From Theorem 8, we obtain a proper matrix such that in the case of a transfer function with multiple poles and provide a sufficient condition for the existence of a positive realization. By the Previous theorem, we extended the positive realization problem to a continuous-time version based on the discrete-time version that was presented in [11]. As an additional contribution, we will show that a sparse positive realization can be constructed under the given conditions of Theorem 8. In the next theorem, we show how to obtain a proper sparse matrix with elements.

Theorem 9. Assume that the conditions of Theorem 8 are satisfied. Then, there exists a sparse circular Toeplitz-type matrix with at most non-zero elements such that for a proper . A sparse Toeplitz-type matrix is defined by

where , , and . Vector has at most three non-zero elements and has at most two non-zero elements. Finally, one obtains a sparse Metzler matrix .

Proof. Consider , where is the th-column of . For sufficiently large , we observe that a polyhedral convex cone that includes can be obtained using a similar procedure to that outlined in Theorem 7. From (33), we note that, for , is generated by the first column vector of and is generated by the convex combination of two column vectors of . Then, is positive linearly combined by the three vectors chosen from and in Theorem 8. Therefore, we obtain a sparse Toeplitz-type matrix as described by (35) and a Metzler matrix .

Let denote the sum of the orders of the (possibly multiple) non-negative poles in , except for a dominant pole. Then the order of the positive realization is . Let be the number of (possibly multiple) negative real poles in , except for a dominant pole. Let () denote the total number of complex conjugate pairs of (possibly multiple) complex poles of the transfer functions in the form that belong to the region , except for a dominant pole. The order of the upper bound of the positive realization is bounded by for each by Theorem 8. We obtain a positive realization of the primitive transfer function with upper bound

If a transfer function is positive realizable in a continuous-time linear system, then is a primitive transfer function because the dominant eigenvalue of is real. Then, is primitive on for a sufficiently large , according to Theorem 1.

We now provide a summary of the process to construct the positive realization of a given transfer function in a continuous-time domain.

(1)Reasonably, we assume that a primitive transfer function has a partial fractional expansion form as described by (6).

(i)Initially, a canonical Jordan-form realization of is precomputed in the form presented in (8).(ii)Then, a minimum that maximizes and minimizes for all , is computed to satisfy the condition in (28).

(2)Using Theorem 4, we choose a proper such that the realizations of and are positive realizations, and for each satisfies the condition in (11), where and .

(i)A minimum is chosen such that for all and satisfies condition (11).(ii)A positive realization of , which is denoted by , is obtained by (4).(iii)A positive realization of with order , which is denoted by , is a canonical Jordan-form realization of from Theorem 4.

(3)We compute a positive realization of the primitive transfer function . For each , we compute of the positive realization of as follows.

(i) is properly divided into to satisfy the conditions in (11) and  (29).(ii)For each , a sparse positive realization of each primitive transfer function , which is denoted by , can be computed by Theorems 8 and 9.

(4)Then a positive realization of the combination is integrated as

(5)A positive realization of the transfer function is integrated using (3) of Lemma 3. The Metzler matrix is set to obtain the positive realization of .

5. Numerical Results

In this section, we present two typical examples to verify that the proposed method is able to compute the positive realizations of given transfer functions. In these examples, we used the MATLAB toolbox for numerical computations to generate the results. First, we consider the positive realization of a transfer function with simple complex poles.

Example 1. Consider the transfer function of the form

with poles equal to , . In this example, does not have a three-dimensional positive realization [1]. We observe that the poles of are , , , and . Define a polyhedral generator as

From Theorem 7, and are sufficiently large to generate a positive realization. We choose and , which satisfies the condition of . Then, we can compute a positive realization such that , , and .

Example 2. Consider a primitive transfer function with multiple poles. The eigenvalue set of is such that , , and . The multiple orders for the poles are , and . Using a canonical statespace realization of the transfer function, we can precompute a canonical Jordan-form realization , where , and and are defined in (8), where , , and . Using Theorem 4, we choose a proper number that satisfies the conditions to compute a realization of such that , , and . We observe that a canonical Jordan-form realization of in the form

and are positive-valued. We compute a positive realization of . We choose a proper and to satisfy conditions in (29) and (28), respectively. Therefore, we set a sufficiently large and choose for and for to obtain . Using the proposed method, we obtain the positive matrix with non-zero elements . First, a matrix is computed as described by (35) with and , where , , and . Then, and . Another positive realization can be obtained using similar methods. can be obtained using (37) and (3). Therefore, we obtain a Metzler matrix to obtain the positive realization .

6. Conclusion

In this paper, we have presented a method to construct a positive realization with sparse matrices for a continuous-time positive system with complex multiple poles. In this approach, we use a constructive computational algorithm to obtain a positive realization with sparse matrices with the lowest order possible. We provided the conditions to construct a positive realization of sparse matrices with a bounded order for a positive continuous-time system. However, the positive realization is not guaranteed to be optimally minimal or sparse for a general positive system without any constraints. It remains an open problem to find sparse positive realizations with an optimal minimal dimension for general positive systems.

Acknowledgments

The author is indebted to the referee and Yong-Kui Chang for their careful work and useful suggestions, which improved the presentation of the results. This work was supported by research fund of Chungnam National University.

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