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

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. ℝ is the set of real numbers (−∞,∞), and ℝ₊ is the set of positive real numbers. A convex cone 𝒳=cone(𝑋) is denoted by the smallest convex cone of a set 𝑋, which consists of all finite nonnegative linear combinations of elements of the set 𝑋. The dual cone, 𝒳^∗, of a cone 𝒳 is defined by 𝒳^∗={𝑦∣𝑥^𝑇𝑦≥0,∀𝑥∈𝒳}. A convex cone 𝒳 is said to be a polyhedral cone if 𝒳 is spanned by a finite number of vector sets 𝑋={𝑥₁,…,𝑥_𝑚} such that 𝑥_𝑖∈ℝ^𝑛; that is, 𝒳=cone({𝑥₁,…,𝑥_𝑛}) and 𝑋 is a polyhedral generator of 𝒳 [13]. Hereafter, 𝑋 is also denoted by the matrix with columns 𝑥_𝑖∈𝑋; that is, 𝑋=[𝑥₁𝑥₂…𝑥_𝑚], as long as it is not ambiguous. Given a general matrix 𝐴=[𝑎_𝑖𝑗], we write 𝐴≥0 if at least some entry 𝑎_𝑖𝑗≥0.

Consider a singleinput, single output linear time-invariant system of the form: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),𝑦(𝑡)=𝐶𝑥(𝑡), (1) where 𝐴 is an 𝑛×𝑛 matrix, 𝐵 is an 𝑛×1 matrix, and 𝐶 is an 1×𝑛 matrix. Define the transfer function 𝐻(𝑠)=𝐶(𝑠𝐼−𝐴)⁻¹𝐵 as the Laplace transform of the impulse response function ℎ(𝑡)=𝐶exp(𝐴𝑡)𝐵.

The linear system in (1) is a positive linear system if for any non-negative input 𝑢(𝑡) and non-negative initial state, the state trajectory 𝑥(𝑡), and the output 𝑦(𝑡) are always non-negative. We provide a formal statement of the positive realization problem for continuous-time positive systems. A matrix 𝐴_∗∈ℝ^𝑚×𝑚 is defined by a Metzler matrix if all of its off-diagonal elements are positive. A triple (𝐴_∗,𝐵₊,𝐶₊) is said to be a positive realization of the transfer function 𝐻(𝑠) if 𝐴_∗ is a Metzler matrix, 𝐵₊≥0, and 𝐶₊≥0.

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 (𝐴_∗,𝐵₊,𝐶₊) with the lower order from the given positive system (𝐴,𝐵,𝐶) in the absence of non-negative constraints.

Theorem 1 (see [1]). Let 𝒫 be a polyhedral cone in ℝ^𝑛 and 𝐴∈ℝ^𝑛×𝑛. Thus, exp(𝐴𝑡)𝒫⊂𝒫 for any 𝑡≥0 if and only if there is a 𝜆>0 such that (𝐴+𝜂𝐼)𝒫⊂𝒫 for all 𝜂≥𝜆.

To prove the sufficiency of this problem, if (𝐴+𝜆𝐼)𝒫⊂𝒫 for some 𝜆, (𝐴+𝜂𝐼)𝒫=(𝐴+𝜆𝐼)𝒫+(𝜂−𝜆)𝒫⊂𝒫 for all 𝜂≥𝜆. Let 𝑥_𝑘=(𝑡/𝑘)(𝐼+𝐴𝑘/𝑡)^𝑘𝑥. For some 𝑡 and 𝑘/𝑡≥𝜆, 𝑥_𝑘∈𝒫. Hence lim_𝑘→∞𝑥_𝑘=exp(𝐴𝑡)𝑥∈𝒫. Therefore, exp(𝐴𝑡)𝒫⊂𝒫. To prove the necessity of this theorem, set 𝒬=𝒫^∗. Let 𝑃 and 𝑄 be polyhedral generators of 𝒫 and 𝒬, respectively. Let 𝜆_𝑖𝑗=−𝑞^𝑇_𝑖𝐴𝑝_𝑗/𝑞^𝑇_𝑖𝑝_𝑗 for 𝑞^𝑇_𝑖𝐴𝑝_𝑗<0 and 𝜆_𝑖𝑗=0 otherwise. Let 𝜆=max(𝜆_𝑖𝑗). Thus, (𝐴+𝜆𝐼)𝒫⊂𝒫. The interested reader can refer to [1] for more details. We observe that if (𝐴+𝜂𝐼)𝒫⊂𝒫 for a sufficiently large 𝜂, then there exists a positive matrix 𝐴₊≥0 such that (𝐴+𝜂𝐼)𝑃=𝑃𝐴₊. Therefore, we obtain a Metzler matrix 𝐴_∗=𝐴₊−𝜆𝐼.

A reachable set ℛ 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 ℛ=cl[cone{𝑥∣𝑥=exp(𝐴𝑡)𝐵,𝑡≥0}], where cl(𝑆) is a closure set of 𝑆. An observable set 𝒮 is the set of initial states in which the output is non-negative for all 𝑡≥0. 𝒮 is defined by 𝒮={𝑥∣𝐶exp(𝐴𝑡)𝑥≥0,∀𝑡≥0}. Assume that 𝑥=exp(𝐴𝑡₀)𝐵∈ℛ for some 𝑡₀≥0. Because 𝐶exp(𝐴𝑡)𝑥=𝐶exp(𝐴(𝑡+𝑡₀))𝐵≥0, 𝑥∈𝒮. Thus, we can verify that ℛ⊂𝒮. 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 𝐻(𝑠) be a strictly proper rational function with degree n, whose (minimal) realization is given by (𝐴,𝐵,𝐶). Then, 𝐻(𝑠) has a positive realization (𝐴_∗,𝐵₊,𝐶₊) if and only if there exists a generator matrix 𝑃 and 𝜆≥0 such that a polyhedral cone 𝒫=cone(𝑃) satisfies (1)(𝐴+𝜆𝐼)𝒫⊂𝒫, (2)ℛ⊂𝒫⊂𝒮, where 𝑃∈ℝ^𝑛×𝑚, 𝑛≤𝑚, ℛ is a reachable set and 𝒮 is an observable set.

From this theorem, (𝐴+𝜂𝐼)𝑃=𝑃𝐴₊ and ℛ⊂𝒫⊂𝒮 constitute necessary and sufficient conditions for the existence of a positive realization with a Metzler matrix 𝐴_∗=𝐴₊−𝜂𝐼 in a continuous-time positive system. We note that the equation (𝐴+𝜂𝐼)𝑃=𝑃𝐴₊ 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 𝐴 in the state space realization and the poles of the transfer function 𝐻(𝑠). Let 𝜆(𝐴) be an eigenvalue set of 𝐴. A dominant eigenvalue of 𝐴 is defined by a maximal real eigenvalue in 𝜆(𝐴) and denoted by 𝜆₀. The spectral radius of 𝐴 is denoted by 𝜌(𝐴); that is, 𝜌(𝐴)=max|𝜆(𝐴)|. If 𝐴 is a Metzler matrix, then the dominant eigenvalue 𝜆₀ of 𝐴 is a unique real eigenvalue and the real parts of the other eigenvalues of 𝐴 are less than 𝜆₀ [16, 17].

A positive system 𝐻(𝑠) is defined as positive realizable if we can find an equivalent positive realization of 𝐻(𝑠). For the continuous-time positive system, if 𝐻(𝑠) with realization (𝐴,𝐵,𝐶) is positive realizable, then we can find a positive realization (𝐴_∗,𝐵₊,𝐶₊) of the positive system 𝐻(𝑠). The transfer function of an impulse response function ℎ₁(𝑡)=ℎ(𝑡)exp(𝜂𝑡) for all 𝑡≥0 is denoted by 𝐻(𝑧)=𝐶(𝑧𝐼−𝐴)⁻¹𝐵 where 𝐴=𝐴+𝜂𝐼 and 𝑧=𝑠−𝜂. We note that 𝐻(𝑧) has similar properties to the transfer function in the discrete-time domain. From Theorem 1, if (𝐴,𝐵,𝐶) is positive realizable in the continuous-time domain, then (𝐴+𝜂,𝐵,𝐶) is positive realizable in the discrete-time domain. Let 𝔻(𝑟) be defined as a disk 𝔻(𝑟) with a center 0 and a radius 𝑟>0. The function 𝐻(𝑧) is analytical outside 𝔻(𝑟) for properly large 𝑟. Therefore, 𝐻(𝑧) is interpreted as a transfer function of a discrete-time domain on the disk 𝔻(𝑟).

Set 𝐴₊=𝜂𝐼+𝐴_∗ and let 𝜆_𝑘≜𝜆_𝑘+𝜂 for each 𝑘 be an eigenvalue of 𝐴=𝜂𝐼+𝐴. The sequence {𝑚_𝑖} of 𝐻(𝑧) with 𝑚_𝑖=𝐶𝐴^𝑖−1𝐵 for all 𝑖≥1 is denoted by the Markov sequence introduced in [18, 19]. We can observe that 𝐶𝐴^𝑖−1𝐵=𝐶₊𝐴₊^𝑖−1𝐵₊ for all 𝑖. 𝐻(𝑧) is described by 𝐻(𝑧)=∑^∞_𝑘=0𝑚_𝑘𝑧^−𝑘. We note that the Markov sequence {𝑚_𝑘} is similar to an impulse response sequence in the discrete-time domain. We define a transfer function 𝐻_[𝐿](𝑧) that corresponds to the shifted Markov sequence {𝑚_𝑘}_{𝑘≥𝐿+1} (i.e., 𝐻_[𝐿](𝑧)=∑^∞_𝑗=1𝑚_𝑗+𝐿𝑧^−𝑗) and 𝐻^𝑐(𝑧)=∑^𝐿_𝑗=1𝑚_𝑗𝑧^−𝑗 for a given 𝐿.

Lemma 3. Assume that 𝐻(𝑧) is decomposed into 𝐻(𝑧)=𝐻^𝑐(𝑧)+𝑧^−𝐿𝐻_[𝐿](𝑧), (2) and the 𝐿-shifted transfer function 𝐻_[𝐿](𝑧) has a positive realization (𝐴₂,𝐵₂,𝐶₂) with dimension 𝐾. Then, 𝐻(𝑧) has a positive realization (𝐴₊,𝐵₊,𝐶₊) with dimension 𝐾+𝐿 such that 𝐴₊=𝐴₁0𝐵₂0𝐴₂,𝐵₊=𝐵₁0,𝐶₊=𝐶₁𝐶₂, (3) where the realization (𝐴₁,𝐵₁,𝐶₁) of 𝐻^𝑐(𝑧) is given by 𝐴₁=010⋯0001⋯0⋮⋱⋮000⋯1000⋯0,𝐵₁=00⋮01,𝐶₁=𝑚_𝐿𝑚_𝐿−1⋯𝑚₂𝑚₁. (4)

Proof. We observe that the transfer function 𝐻^𝑐(𝑧) with a non-negative finite Markov sequence has an 𝐿-dimensional positive realization (𝐴₁,𝐵₁,𝐶₁). The positive realization of the transfer matrix 𝐻_[𝐿](𝑧) with dimension 𝐾 is given by (𝐴₂,𝐵₂,𝐶₂). The following combination of the two realizations yields a non-negative realization (𝐴,𝐵,𝐶) of 𝐻(𝑧): 𝐴𝐴=¹0𝐵₂0𝐴₂𝐵,𝐵=¹0,𝐶=𝐶₁𝐶₂. (5) Therefore, 𝐻(𝑧) has a positive realization of dimension 𝐾+𝐿.

A rational transfer function 𝐻(𝑧) is primitive if the maximum modulus pole of 𝐻(𝑧) is the only maximum real eigenvalue of (𝐴+𝜂𝐼), which is similar to concepts presented in discrete-time versions [5, 20]. Thus, a transfer function 𝐻(𝑠) is defined by a primitive transfer function, if 𝐻(𝑧) is a primitive transfer function for some 𝜂. Using partial fractional expansion of the rational transfer function 𝐻(𝑧), we can define the primitive positive transfer function 𝐻(𝑧) as 𝐻(𝑧)=^𝑛0_𝑘=1𝛽_𝑘⁽⁰⁾𝑧−𝜆₀^𝑘+^𝜏𝑛_𝑗=1^𝑗_𝑘=1𝛽_𝑘^(𝑗)𝑧−𝜆_𝑗^𝑘, (6) where 𝑛_𝑘 is the maximum number of the multiplicity of 𝜆_𝑘=𝜆_𝑘+𝜂, which we call the Jordan-form realization. An 𝑛_𝑗×𝑛_𝑗 Jordan-form matrix 𝐴_𝑗=𝐽(𝜆_𝑗), an 𝑛_𝑗×1 matrix 𝐵_𝑖, and an 1×𝑛_𝑗 matrix 𝑐_𝑖 are given, respectively, as 𝐽𝜆_𝑗≜𝜆_𝑗0𝜆1⋯00_𝑗𝜆⋯00⋮⋮⋱⋮00⋯_𝑗100⋯0𝜆_𝑗,𝐵_𝑗=𝐞_1,𝑛𝑗,𝐶_𝑗=𝛽_𝑛^(𝑗)_𝑗𝛽_𝑛^(𝑗)_𝑗−1⋯𝛽₁^(𝑗) (7)

for 1≤𝑗≤𝑟 [14]. A direct sum 𝐴⊕𝐵 is defined as the block matrix 𝐴=^𝐴𝑂_𝑂𝐵 with proper zero matrices 𝑂. The vector 𝐞_𝑖,𝑛 represents a column vector of dimension 𝑛 with the 𝑖th entry equal to 1 and all other entries equal to 0. 𝐻(𝑧) has a canonical Jordan-form realization (𝐴,𝐵,𝐶) [14] such that 𝐴=^𝜏_𝑗=0𝐴_𝑖,𝐵=𝐵^𝑇₀𝐵^𝑇₁𝐵^𝑇₂⋯𝐵^𝑇_𝜏^𝑇,𝐶=𝐶₀𝐶₁𝐶₂⋯𝐶_𝜏, (8) where 𝐴^𝑇 is the transpose matrix of 𝐴.

Theorem 4. A primitive transfer function 𝐻(𝑧) is defined by (6). 𝐻(𝑧) is decomposed into 𝐻(𝑧)=𝐻^𝑐(𝑧)+𝑧^−𝐿𝐻_[𝐿](𝑧). Assume that 𝐻(𝑧) is positive realizable and 𝐻_[𝐿](𝑧)=𝐹_[𝐿](𝑧)+𝐺_[𝐿](𝑧) such that 𝐹_[𝐿](𝑧) and 𝐺_[𝐿](𝑧) are defined by 𝐹_[𝐿](𝑧)=∑^𝑛0_𝑘=2𝛽_𝑘⁽⁰⁾𝑧−𝜆₀^𝑘𝐺_[𝐿],(𝑧)=^𝜏_𝑗=1𝛽_0,𝑗𝑧−𝜆₀+^𝜏𝑛_𝑗=1^𝑗_𝑖=1𝛽_𝑖^(𝑗)𝑧−𝜆_𝑗^𝑖. (9)(10) Then, a canonical Jordan-form realization of 𝐻^𝑐(𝑧) is a positive realization, a canonical Jordan-form realization of 𝐹_[𝐿](𝑧) is a positive realization and 𝛽_0,𝑗>0 satisfies 𝛽₁⁽⁰⁾=^𝜏_𝑗=1𝛽_0,𝑗,𝛽_0,𝑗≥𝛿𝛽_𝑖^(𝑗) (11) for all 𝑖 with 1≤𝑖≤𝑛_𝑗 and some 𝛿>0 for 𝐺_[𝐿](𝑧).

Proof. Because 𝐻(𝑧) is positive realizable, we have the Markov parameter sequence 𝑚_𝑘≥0. However, 𝛽_𝑛⁽⁰⁾₀>0 is necessary to obtain a positive Markov parameter sequence 𝑚_𝑘≥0 (refer to [5]). The Jordan-form realization of 𝐻^𝑐(𝑧) given by (4) is a positive realization and 𝐻_[𝐿](𝑧) is represented by (𝐴,𝐵,𝐶𝐴^𝐿). We have a realization (𝐴₀,𝐵₀,𝐶₀𝐴^𝐿₀) of 𝐹_[𝐿](𝑧). Then 𝐶₀𝐴^𝐿₀=[𝛽_𝑛⁽⁰⁾₀𝛽_𝑛⁽⁰⁾_𝑗−1⋯𝛽₁⁽⁰⁾] is computed by𝛽_𝑘⁽⁰⁾=𝐿𝑛₀𝛽−1^𝑛(0)0𝜆^𝐿−𝑛0₀⁺¹+∑^𝑛0−2_𝑖=0𝐿𝑖𝛽_𝑖⁽⁰⁾𝜆₀^𝐿−𝑖, (12) where (^𝑛_𝑘) is the number of 𝑘-combinations from a given set with 𝑛 elements. For 𝜖>0, there is a sufficiently large 𝐿 such that ^𝐿_𝑛0−1/^𝐿_𝑛0−𝑘≥𝜖(𝐿−𝑛₀+1) for all 𝑘 with 𝑛₀>𝑘≥0. We can find a sufficiently large 𝐿 such that 𝛽_𝑘⁽⁰⁾>0. Therefore, the canonical Jordan-form realization of 𝐹_[𝐿](𝑧) is a positive realization. Because 𝛽₁⁽⁰⁾ increases asymptotically as 𝐿→∞, we can choose a proper 𝐿 such that 𝛽₁⁽⁰⁾=∑^𝜏_𝑗=1𝛽_0,𝑗 for 𝛽_0,𝑗>𝛿|𝛽_𝑖^(𝑗)| for all 𝑖 and 𝛽_0,𝑗>0 for 𝐺_[𝐿](𝑧).

As 𝐿 increases, 𝛽₁⁽⁰⁾ increases asymptotically. For sufficiently large 𝐿 and 𝛽₁⁽⁰⁾, we obtain the transfer functions 𝐺_[𝐿](𝑧) in (10) with non-negative Markov sequences. Using Theorem 4, 𝐻^𝑐(𝑧) and 𝐹_[𝐿](𝑧) in (2) and (9), respectively, also have non-negative Markov sequences. Therefore, the positive realizations of 𝐻^𝑐(𝑧) and 𝐹_[𝐿](𝑧) are obtained. A primitive function 𝐺_[𝐿](𝑧) with a simple dominant pole can be divided into 𝐺_𝑗(𝑧)=𝛽_0,𝑗𝑧−𝜆₀+^𝑛𝑗_𝑖=1𝛽_𝑗^(𝑖)𝑧−𝜆_𝑗^𝑖. (13) Therefore, we need to find a positive realization for 𝐺_𝑗(𝑧) for all 𝑗. In the continuous-time domain, we obtain 𝐻(𝑠)=𝐻(𝑠−𝜂)=𝐻^𝑐(𝑠−𝜂)+(𝑠−𝜂)^−𝐿𝐻_[𝐿](𝑠−𝜂) by substituting 𝑧=𝑠−𝜂. In the next section, we will discuss how to find a positive realization of 𝐺_𝑗(𝑠−𝜂) for each 𝑗.

3. Positive Realizations for Systems with Simple Complex Poles

Definition 5. Let 𝔓_𝑚(𝜌) for 𝑚≥1 denote the set of points in the complex plane that lie in the interior of a regular polygon with 𝑚 edges with one vertex at point 𝜌 and a zero point at 0. For 𝑚≥2, the polygon 𝔓_𝑚(𝜌) is defined by a subset in ℂ through the following inequalities:𝔓_𝑚𝜌=()𝜋𝑥+𝑖𝑦∈ℂ∣𝑟cos2𝑘+1𝑚𝜋−𝜑≤𝜌cos𝑚, (14) for all 𝑘 with 0≤𝑘≤𝑚, where 𝑥=𝑟cos𝜑 and 𝑦=𝑟sin𝜑.

The complex number 𝑥+𝑖𝑦 corresponds to the point (𝑥,𝑦) in the Cartesian plane ℝ². The polygon set 𝔓_𝑚(𝜌) on the complex plane can be mapped onto ℝ². A 2×𝑚 matrix Ψ that consists of the vertices of a polygon in the complex plane is defined as Ψ=1cos2𝜋𝑚cos4𝜋𝑚()⋯cos2𝜋𝑚−1𝑚0sin2𝜋𝑚sin4𝜋𝑚()⋯sin2𝜋𝑚−1𝑚 (15) for a given 𝑚. A cone generator matrix 𝑃∈ℝ^3×𝑚 is defined as 𝐄Ψ𝑃=≜𝑝₁𝑝₂⋯𝑝_𝑚, (16) where 𝐄 represents an 1×𝑚 row vector with all entries equal to 1; the vectors 𝑝_𝑖 are extreme points in a polyhedral cone, cone(𝑃). The real part of 𝛼 is denoted by Re(𝛼) and the imaginary part of 𝛼 is denoted by Im(𝛼). The angle of 𝛼 is denoted by arg(𝛼). We consider a simple third-order positive proper transfer function 𝐻(𝑠) with a pair of strictly conjugate complex poles and a dominant real pole in partial fractional form such that 𝐻(𝑠)=𝛽⁽⁰⁾𝑠−𝜆₀+𝛽₁𝑠−𝜆₁+𝛽₁𝑠−𝜆₁, (17) where 𝜆₀<0, 𝜆₁ is a complex pole with Re(𝜆₁)<𝜆₀, 𝜆₁ is a complex conjugate of 𝜆₁, and 𝛽₁ is complex number. A minimal real canonical realization of the transfer function is given by 𝜆𝐴=⁰000𝑥−𝑦0𝑦𝑥≜𝜆₀110,𝑐⊕Γ,𝐵=𝐶=₁𝑐₂𝑐₃, (18) where ⊕ is a direct sum and 𝜆₀ is a real pole with 𝜆₀<0. A complex pole is defined as 𝜆₁=𝑥+𝑖𝑦 with 𝑥<𝜆₀ and another complex pole 𝜆₂ is a conjugate of 𝜆₁ (i.e., 𝜆₂=𝜆₁). Then, 𝛽⁽⁰⁾>0, 𝑐₂=2Re(𝛽₁) and 𝑐₃=2Im(𝛽₁).

Theorem 6. Assume that a transfer function 𝐻(𝑠) is defined by (17) where 𝜆₀>Re(𝜆₁), and an extreme generator 𝑃 is constructed as described by (16). If −𝜋2+𝜋𝑚≤arg𝜆₁−𝜆₀≤𝜋2−𝜋𝑚,𝛽₁≤𝛽⁽⁰⁾2, (19)(20) then there is a positive realization (𝐴_∗,𝐵₊,𝐶₊) of 𝐻(𝑠) with order 𝑚.

Proof. According to the assumption made in (19), the angle arg(𝜆₁−𝜆₀) is less than (𝜋/2)−(𝜋/𝑚). As shown in Figure 1, there is a sufficiently large positive real number 𝜂>0 such that the eigenvalues of 𝜂𝐼+𝐴 are located in 𝔓_𝑚(𝜆₀+𝜂). 𝔓_𝑚(𝜆₀+𝜂) is projected onto 𝔓_𝑚 by dividing by 𝜆₀+𝜂. This means that the set {1,(𝜆₁+𝜂)/(𝜆₀+𝜂),(𝜆₁+𝜂)/(𝜆₀+𝜂)}⊂𝔓_𝑚(1) is an eigenvalue set of (𝜂𝐼+𝐴)/(𝜆₀+𝜂). As reported for the discrete-time cases in [6, 10], we can construct the polyhedral cone generator 𝑃 as described in (16) such that (𝜂𝐼+𝐴)/(𝜂+𝜆₀)𝑃⊂cone(𝑃). By Theorem 1, (𝜂𝐼+𝐴)𝑃=𝑃𝐴₊ implies that exp(𝐴𝑡)𝑃⊂cone(𝑃). We can demonstrate the invariance of exp(𝐴𝑡) for 𝑃. Therefore, we obtain a Metzler matrix 𝐴_∗=𝐴₊−𝜂𝐼.
We note that an invariant polytope argument is equal to an exp(𝐴𝑡)-invariant polyhedral cone. We will show that ℛ⊂𝒫 and 𝒫⊂𝒮. Because 𝐵∈𝑃, there exists 𝐵₊ such that 𝑃𝐵₊=𝐵. It follows that ℛ⊂𝒫. We also know that [𝑐₂𝑐₃]=[2Re(𝛽₁)2Im(𝛽₁)]. The inequality 𝐶𝑃≥0 leads to the following sufficient condition: 𝛽⁽⁰⁾𝐄+[2Re(𝛽₁)2Im(𝛽₁)]Ψ≥0. To satisfy the above inequality, we impose a sufficient condition such that |𝛽₁|≤𝛽⁽⁰⁾/2 for sufficiently large 𝛽⁽⁰⁾.

Figure 1 

The geometric interpretation of (𝜂𝐼+𝐴)𝑃⊂𝒫 using 𝔓𝑚(𝜌).

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 𝐴₊ such that (𝜂𝐼+𝐴)𝑃=𝑃𝐴₊. However, the solutions for 𝐴₊ are not unique. Furthermore, we wish to avoid a dense matrix for 𝐴₊ for real system implementation. Therefore, we consider how to obtain a proper sparse matrix 𝐴₊ with 2𝑛 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 𝐭⃗=(𝑡₀,𝑡₁,…,𝑡_𝑚−1), a circular Toeplitz matrix 𝑇(𝐭)⃗ is defined as 𝑇𝐭⃗=𝑡₀𝑡₁𝑡₂⋯𝑡_𝑚−1𝑡_𝑚−1𝑡₀𝑡₁⋯𝑡_𝑚−2𝑡_𝑚−2𝑡_𝑚−1𝑡₀⋯𝑡_𝑚−3𝑡⋮⋮⋮⋱⋮₁𝑡₂𝑡₃⋯𝑡₀, (21) where each element 𝑡_𝑖,𝑗 is 𝑡_𝑖,𝑗=𝑡_𝑘 with 𝑘=mod(𝑗−𝑖,𝑚); here, mod(𝑛,𝑚) is a residue of the integer 𝑛 modulo 𝑚.

Theorem 7. Assume that the conditions of Theorem 6 are satisfied. Then there exists a sparse positive circular Toeplitz matrix 𝐴₊≥0 with 3𝑚 elements such that (𝜂𝐼+𝐴)𝑃=𝑃𝐴₊ for a proper 𝜂>0, and 𝐴_∗=𝐴₊−𝜂𝐼 is a Metzler matrix.

Proof. For 𝜖=𝜆₀+𝜂>0, the eigenvalue set of (𝐴+𝜂𝐼)/𝜖 is {1,𝑧₁,𝑧₁}. Thus, 𝑧₁,𝑧₁∈𝔓_𝑚. We choose column vectors {𝜓_𝑖1,𝜓_𝑖2,𝜓_𝑖3} from Ψ as in Figure 1 such that 𝑧₁∈cone{𝜓_𝑖1,𝜓_𝑖2,𝜓_𝑖3}. We also obtain (𝐴+𝜂𝐼)𝑝₁=𝑎₁𝑝_𝑖1+𝑎₂𝑝_𝑖2+𝑎₃𝑝_𝑖3, where ∑_𝑘𝑎_𝑘=1, with 𝑎_𝑘≥0. Let [𝑛] be defined as [𝑛]=1+mod(𝑛−1,𝑚). For 𝑘≥1, 𝐴+𝜂𝐼𝑝_1+𝑘=𝑎₁𝑝_[𝑖1+𝑘]+𝑎₂𝑝_[𝑖2+𝑘]+𝑎₃𝑝_[𝑖3+𝑘]. (22) We can obtain a sparse circular Toeplitz matrix 𝐴₊=𝑇(𝐭)⃗ such that a vector 𝐭⃗ is defined by 𝑡_[𝑖1+𝑘]=𝑎₁, 𝑡_[𝑖2+𝑘]=𝑎₂, and 𝑡_[𝑖3+𝑘]=𝑎₃ and the other values of 𝑡_𝑘 are zero. Thus, the matrix 𝐴₊ 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 𝐻(𝑠)=𝐺_𝑗(𝑠) for some 𝑖 in (13)). Let us consider a transfer function 𝐻₁(𝑠) with a single pair of multiple complex conjugate poles: 𝐻₁(𝑠)=^𝑛_𝑘=1𝛽_𝑘𝑠−𝜆₁^𝑘+𝛽_𝑘𝑠−𝜆₁^𝑘, (23) where the pole 𝜆₁ and the coefficients 𝛽_𝑖 are complex, and 𝛽_𝑖 is defined as the conjugate of 𝛽_𝑖. The transfer function 𝐻₁(𝑠) with the partial fractional expansion in (23) has a canonical Jordan-form realization (𝐴,𝐵,𝐶) such that 𝐴=𝐽𝜆_𝑗⊕𝐽𝜆_𝑗𝐞,𝐵=^𝑛,𝑛𝐞_𝑛,𝑛,𝐶=𝛽_𝑛𝛽_𝑛−1⋯𝛽₁𝛽_𝑛⋯𝛽₁. (24) There exists a similarity transformation matrix 𝑄 such that a block Jordan-form realization is given by 𝐴₁=𝑄⁻¹Γ𝐴𝑄=⁰𝐼𝑂⋯𝑂𝑂Γ₀𝐼⋯𝑂𝑂𝑂Γ₀⋯𝑂⋮⋱⋮𝑂𝑂𝑂⋯Γ₀,𝐵₁=0⋮011,𝐶₁=𝛽_𝑛𝛽_𝑛⋯𝛽₁𝛽₁, (25) where Γ₀=𝜆₁⊕𝜆₁. Let Δ=^1−𝑖_1𝑖. Using a block diagonal matrix 𝑄₁=𝑤^−𝑛+1(⨁^𝑛_𝑘=1𝑤^𝑘−1Δ) as a similarity transformation, we obtain a new real block Jordan-form realization (𝐽(𝜆₁,𝑤),𝐵₂,𝐶₃) such that 𝐽𝜆₁,𝑤=𝑄₁⁻¹𝐴₁𝑄₁=,𝐵Γ𝑤𝐼𝑂⋯𝑂𝑂Γ𝑤𝐼⋯𝑂𝑂𝑂Γ⋯𝑂⋮⋱⋮𝑂𝑂𝑂⋯Γ₂=0⋮010,𝐶₂=𝛽_2𝑛𝛽_2𝑛−1⋯𝛽₁, (26) where the entries of 𝐶₂ are defined by 𝛽_2𝑘=2𝑤^𝑘−𝑛Re(𝛽_𝑘), 𝛽_2𝑘−1=2𝑤^𝑘−𝑛Im(𝛽_𝑘) for each 𝑘 and Γ=^𝑥−𝑦_𝑦𝑥.

We consider a positive realization of transfer functions with a multiple complex conjugate pole pair. Let us define 𝛾_𝑚(𝑧)=max{𝑟̂∣(𝑟̂cos𝜃,𝑟̂sin𝜃)∈𝔓_𝑚,𝑟̂>0} for a given 𝑧=𝑟cos𝜃+𝑖𝑟sin𝜃 and 𝑚. 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 𝐻(𝑠)=𝛽⁽⁰⁾𝑠−𝜆₀+𝐻₁(𝑠) (27) with one dominant real pole 𝜆₀ with a simple order and two complex poles 𝜆₁ and 𝜆₁ with multiple orders 𝑛≥1, where 𝛽⁽⁰⁾>0, Re(𝜆₁)<𝜆₀ and 𝐻₁(𝑠) is defined as in (23). Set 𝑧₁=(𝜂+𝜆₁)/(𝜂+𝜆₀) and 𝑧₂=1/(𝜂+𝜆₀). One assumes that the following conditions are satisfied: 𝑧0<𝑤≤1−¹𝛾_𝑚𝑧₁𝛾_𝑚𝑧₂𝑧₂𝛽_𝑘,≤𝛽⁽⁰⁾(2𝑤)^𝑘−𝑛 (28)(29) for a given sufficiently large 𝜂>0 and all 1≤𝑘≤𝑛. Then, there exists a positive realization (𝐴_∗,𝐵₊,𝐶₊) of the transfer function 𝐻(𝑠) where 𝐴_∗ is a Metzler matrix with order 𝑚𝑛.

Proof. We obtain a real block Jordan-form realization (𝐴,𝐵,𝐶) of the transfer function 𝐻(𝑠) in (27) as follows: 𝐴=𝜆₀⊕𝐽𝜆₁1𝐵,𝑤,𝐵=₂,𝐶=1𝐶₂, (30) where 𝐽(𝜆₁,𝑤), 𝐵₂, and 𝐶₂ are defined as in (26). We use the fact that there exists an 𝜂>0 such that there exists a (𝜂𝐼+𝐴)-invariant cone 𝒫 with 𝑚𝑛 edges (i.e., (𝜂𝐼+𝐴)𝒫⊂𝒫) 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 (𝜂𝐼+𝐴)-invariant property, a block shift matrix 𝑍∈ℝ^2𝑛×2𝑛 and a matrix 𝐾∈ℝ^2𝑛×𝑚 are defined as Ψ𝑂𝑂⋮𝑂𝑍=,𝑂𝑂⋯𝑂𝑂𝐼𝑂𝑂𝑂𝑂𝐼⋯𝑂𝑂⋮⋮⋱⋮⋮𝑂𝑂⋯𝐼𝑂,𝐾= (31) where 𝐼 is an identity matrix, 𝑂 is a zero matrix with the proper dimension, and the 2×𝑚 matrix Ψ is defined as in (15). A cone generator matrix 𝑃∈ℝ^{(2𝑛+1)×𝑚𝑛} is defined as 𝑃=[𝑃₁𝑃₂⋯𝑃_𝑛], where 𝑃_𝑗 is defined as 𝑃_𝑗=^𝐄_𝜗𝑗 for 1≤𝑗≤𝑛−1, such that 𝐄 represents an 1×𝑚 row vector in which all entries equal 1 and 𝜗_𝑗=𝑍^𝑗−1𝐾. The columns of matrix 𝑃 represent the extreme vertices of a finite generated cone 𝒫 in ℝ^2𝑛+1 (i.e., cone(𝑃)=𝒫) and are positive independent. The polyhedral cone 𝒫_𝑗 is generated by 𝑃_𝑗, that is; 𝒫_𝑗=cone(𝑃_𝑗) for each 𝑗. We will prove that 𝒫 is 𝐴-invariant (i.e., 𝐴𝒫⊂𝒫). Let 𝒫_𝑗 be the cone generated from the column vectors of 𝑃_𝑗 (i.e., 𝒫_𝑗=cone(𝑃_𝑗)). For 𝜆₁∈𝔓_𝑚(𝜆₀+𝜂), we observe that 𝐴𝑃₁∈𝒫₁. Note that (𝐴+𝜂𝐼)𝑃_𝑗 for 𝑗>1 has nonzero values for the 1st, 𝑗th, and 𝑗 + 1th block components 𝐴+𝜂𝐼𝑃_𝑗=1𝑗𝐄⋮𝑂Ψ𝑂⋮𝑗+1𝑤Ψ𝜂𝐼+Γ. (32) The invariance property requires that ((𝐴+𝜂𝐼)/(𝜆₀+𝜂))𝑃_𝑗∈𝒫. Set 𝐴=(𝐴+𝜂𝐼)/(𝜆₀+𝜂). The matrix 𝐴 has eigenvalues {1,𝑧₁,𝑧₁} with |𝑧₁|<1. Thus, the positive realization problem with respect to 𝐴 is closely related to that of the discrete-time cases reported in [6, 10]. Choose {𝑊_𝑗−1,𝑊_𝑗} such that 𝐴+𝜂𝐼𝜆₀𝑃+𝜂^𝑗=𝛼₁𝑊_𝑗−1+𝛼₂𝑊_𝑗 (33) for each 𝑗, where the values 𝛼_𝑗 satisfy 𝛼_𝑗≥0, 𝛼₁+𝛼₂=1 and all of the entries in the first row of 𝑊_𝑗∈𝒫_𝑗 are equal to 1. A sufficient condition for a feasible solution (𝛼₁,𝛼₂) should satisfy two inequalities, 𝑤|𝑧₂|/𝛾_𝑚(𝑧₂)≤𝛼₁ and |𝑧₁|/𝛾_𝑚(𝑧₁)≤𝛼₂, and an equality 𝛼₁+𝛼₂=1 for a given 𝑤. By rearranging the above conditions, we obtain the inequality 0<𝑤≤(1−|𝑧₁|/𝛾_𝑚(𝑧₁))(𝛾_𝑚(𝑧₂)/|𝑧₂|). From this result, we observe that 𝑤 and 𝜂 are parameters that can be tuned to obtain a positive matrix. The polyhedral cone 𝒫 is 𝜂𝐼+𝐴-invariant under the condition described by (28). We observe that 𝒫, which is 𝜂𝐼+𝐴-invariant, is equal to exp(𝐴𝑡)𝒫⊂𝒫 by Theorem 1. We obtain a sufficient condition as described by (28) to guarantee that 𝒫 is exp(𝐴𝑡)-invariant for all 𝑡>0. When the sufficient conditions are satisfied, there exists a positive 𝐴₊ such that (𝐴+𝜂𝐼)𝑃=𝑃𝐴₊. Therefore, 𝐴_∗=𝐴₊−𝜂𝐼 is a Metzler matrix.
Next, let us consider ℛ⊂𝒫 and 𝒫⊂𝒮. Because 𝐵∈𝑃, it follows trivially that ℛ⊂𝒫. Finally, we will prove that 𝐶=[𝛽⁽⁰⁾𝐶₂]∈𝒫^∗. The entries 𝑗₁=𝑛−2𝑗+1 and 𝑗₁+1 of 𝐶₂ are given by [𝑐̂_𝑗1𝑐̂_𝑗1+1]=[𝛽_2𝑗𝛽_2𝑗−1](𝑤)^𝑗−𝑛, where 𝛽_2𝑗−1=2Im(𝛽_𝑖) and 𝛽_2𝑗=2Re(𝛽_𝑖). An inequality 𝐶𝑃_𝑘≥0 for each 𝑗≥1 leads to the sufficient condition in (29) as follows: 𝛽⁽⁰⁾𝐄+2Re𝛽_𝑗2Im𝛽_𝑗(𝑤)^𝑗−𝑛Ψ≥0. (34) To satisfy the above inequality, we impose a sufficient condition such that |𝛽_𝑗|≤𝛽⁽⁰⁾/(2𝑤)^𝑗−𝑛 for sufficiently large 𝛽⁽⁰⁾ and each 1≤𝑗≤𝑛. We show that 𝐶𝑃_𝑗≥0. Therefore, there exists a positive realization (𝐴₊,𝐵₊,𝐶₊) with order 𝑛𝑚 such that 𝐴₊=𝐴+𝜂𝐼∈ℝ₊^{𝑛𝑚×𝑛𝑚}, 𝐵₊∈ℝ₊^𝑛𝑚 and 𝐶^𝑇₊∈ℝ₊^𝑛𝑚 satisfy (𝐴+𝜂𝐼)𝑃=𝑃𝐴₊, 𝐵=𝑃𝐵₊ and 𝐶₊=𝐶𝑃. Therefore, we obtain a Metzler matrix 𝐴_∗=𝐴₊−𝜂𝐼.

From Theorem 8, we obtain a proper matrix 𝐴₊ such that (𝜂𝐼+𝐴)𝑃=𝑃𝐴₊ 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 𝐴₊ with 3𝑚𝑛 elements.

Theorem 9. Assume that the conditions of Theorem 8 are satisfied. Then, there exists a sparse circular Toeplitz-type matrix 𝐴₊ with at most 3𝑛𝑚 non-zero elements such that (𝜂𝐼+𝐴)𝑃=𝑃𝐴₊ for a proper 𝜂>0. A sparse Toeplitz-type matrix 𝐴₊ is defined by 𝐴₊=𝑇₁𝛼₁𝐼0⋯00𝑇₂𝛼₁𝐼⋯000𝑇₂⋯0⋮⋮⋮⋱⋮000⋯𝑇₂, (35) where 𝑇₁=𝑇(𝐭₁), 𝑇₂=𝑇(𝐭₂), and 𝛼₁>0. Vector 𝐭₁ has at most three non-zero elements and 𝐭₂ has at most two non-zero elements. Finally, one obtains a sparse Metzler matrix 𝐴_∗=𝐴₊−𝜂𝐼.

Proof. Consider (𝐴+𝜂𝐼)𝑝_1,1=𝑎₁𝑝_1,𝑖1+𝑎₂𝑝_1,𝑖2+𝑎₂𝑝_1,𝑖3, where 𝑝_1,𝑘 is the 𝑘th-column of 𝑃₁. For sufficiently large 𝜂, we observe that a polyhedral convex cone 𝒫₁ that includes (𝐴+𝜂𝐼)𝑃₁ can be obtained using a similar procedure to that outlined in Theorem 7. From (33), we note that, for 𝑘>1, 𝑊_𝑘−1∈𝒫_𝑘−1 is generated by the first column vector of 𝑃_𝑘−1 and 𝑊_𝑘∈𝒫_𝑘 is generated by the convex combination of two column vectors of 𝑃_𝑘. Then, 𝐴𝑃_𝑘 is positive linearly combined by the three vectors chosen from 𝑃_𝑘 and 𝑃_𝑘−1 in Theorem 8. Therefore, we obtain a sparse Toeplitz-type matrix 𝐴₊ as described by (35) and a Metzler matrix 𝐴_∗=𝐴₊−𝜂𝐼.

Let 𝑁₁ denote the sum of the orders of the (possibly multiple) non-negative poles in 𝔓₁, except for a dominant pole. Then the order of the positive realization is 𝑁₁+1. Let 𝑁₂ be the number of (possibly multiple) negative real poles in 𝔓₂, except for a dominant pole. Let 𝑁_𝑚 (𝑚≥3) denote the total number of complex conjugate pairs of (possibly multiple) complex poles of the transfer functions in the form 𝐺_𝑘(𝑧) that belong to the region 𝔓_𝑚−⋃^𝑚−1_𝑗=1𝔓_𝑗, except for a dominant pole. The order of the upper bound of the positive realization is bounded by 𝑚𝑁_𝑚 for each 𝑚≥2 by Theorem 8. We obtain a positive realization of the primitive transfer function 𝐻(𝑧) with upper bound𝑁=𝐿+𝑁₀+1+𝑁₁+_𝑗≥2𝑗𝑁_𝑗. (36) If a transfer function 𝐻(𝑠) is positive realizable in a continuous-time linear system, then 𝐻(𝑠) is a primitive transfer function because the dominant eigenvalue of 𝐴 is real. Then, 𝐻(𝑧) is primitive on 𝔻(𝜂+𝜆₀) for a sufficiently large 𝜂, according to Theorem 1.

We now provide a summary of the process to construct the positive realization of a given transfer function 𝐻(𝑠) in a continuous-time domain.(1)Reasonably, we assume that a primitive transfer function 𝐻(𝑧) has a partial fractional expansion form as described by (6).(i)Initially, a canonical Jordan-form realization (𝐴,𝐵,𝐶) of 𝐻(𝑠) is precomputed in the form presented in (8).(ii)Then, a minimum 𝜂 that maximizes 𝑤 and minimizes 𝑚 for all 𝜆_𝑖, 𝑖≥1 is computed to satisfy the condition in (28).(2)Using Theorem 4, we choose a proper 𝐿 such that the realizations of 𝐻^𝑐(𝑧) and 𝐹_[𝐿](𝑧) are positive realizations, and 𝐺_𝑗(𝑧) for each 𝑗 satisfies the condition in (11), where 𝐻(𝑧)=𝐻^𝑐(𝑧)+𝑠^−𝐿𝐻_[𝐿](𝑧) and 𝐻_[𝐿](𝑧)=𝐹_[𝐿](𝑧)+∑^𝜏_𝑗=1𝐺_𝑗(𝑧).(i)A minimum 𝐿 is chosen such that 𝛽_𝑘⁽⁰⁾>0 for all 𝑘 and 𝛽₁⁽⁰⁾ satisfies condition (11).(ii)A positive realization of 𝐻^𝑐(𝑧), which is denoted by (𝐴₊,𝐵₊,𝐶₊), is obtained by (4).(iii)A positive realization of 𝐹_[𝐿](𝑧) with order 𝑁₀, which is denoted by (𝐴₀₊,𝐵₀₊,𝐶₀₊), is a canonical Jordan-form realization of 𝐹_[𝐿](𝑧) from Theorem 4.(3)We compute a positive realization of the primitive transfer function 𝐺_[𝐿](𝑠−𝜂). For each 𝑘, we compute of the positive realization of 𝐺_𝑘(𝑠−𝜂) as follows.(i)𝛽₁⁽⁰⁾ is properly divided into {𝛽_0,𝑘} to satisfy the conditions in (11) and  (29).(ii)For each 𝜆_𝑘, a sparse positive realization of each primitive transfer function 𝐺_𝑘(𝑠−𝜂), which is denoted by (𝐴_𝑘+,𝐵_+𝑘,𝐶_+𝑘), can be computed by Theorems 8 and 9.(4)Then a positive realization (𝐴₊,𝐵₊,𝐶₊) of the combination 𝐹_[𝐿](𝑧)+𝐺_[𝐿](𝑧) is integrated as 𝐴̆₊=^𝜏_𝑘=0𝐴_𝑘+,𝐵̆₊=𝐵^𝑇₀₊⋯𝐵^𝑇_𝜏+^𝑇,𝐶̆₊=𝐶₀₊⋯𝐶_𝜏+. (37)(5)A positive realization (𝐴₊,𝐵₊,𝐶₊) of the transfer function 𝐻(𝑧) is integrated using (3) of Lemma 3. The Metzler matrix 𝐴_∗=𝐴₊−𝜂𝐼 is set to obtain the positive realization (𝐴_∗,𝐵₊,𝐶₊) of 𝐻(𝑠).

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 𝐻(𝑠) of the form 𝐻(𝑠)=1.5+𝑠+10.25+0.25𝑗+𝑠+2−𝑗0.25−0.25𝑗𝑠+2+𝑗 (38) with poles equal to −1, −2±𝑗. In this example, 𝐻(𝑠) does not have a three-dimensional positive realization [1]. We observe that the poles of 𝐻(𝑠) are 𝜆₀=−1, 𝜆₁=−2+𝑗, 𝛽⁽⁰⁾=1.5, and 𝛽⁽¹⁾=0.25+0.25𝑗. Define a polyhedral generator 𝑃 as 𝑃=111110−10010−1. (39) From Theorem 7, 𝜂 and 𝑚 are sufficiently large to generate a positive realization. We choose 𝜂=2 and 𝑚=4, which satisfies the condition of (𝐴+𝜂𝐼)𝑃⊂𝒫. Then, we can compute a positive realization (𝐴_∗,𝐵₊,𝐶₊) such that 𝐴_∗=𝑇(−2,0,0,1), 𝐵₊=[1000]^𝑡, and 𝐶₊=[2112].

Example 2. Consider a primitive transfer function 𝐻(𝑠) with multiple poles. The eigenvalue set of 𝐴 is {𝜆₀,𝜆₁,𝜆₂} such that 𝜆₀=−0.2, 𝜆₁=−1.8+0.5𝑖, and 𝜆₂=−2+0.3𝑖. The multiple orders for the poles are 𝑛₀=3, 𝑛₁=3 and 𝑛₂=3. Using a canonical statespace realization of the transfer function, we can precompute a canonical Jordan-form realization (𝐴,𝐵,𝐶), where 𝐴=𝐽(𝜆₀)⊕𝐽(𝜆₁)⊕𝐽(𝜆₁)⊕𝐽(𝜆₂)⊕𝐽(𝜆₁), and 𝐵 and 𝐶 are defined in (8), where 𝐶₀=[3−11], 𝐶₁=𝐶₂=[1−10], and 𝐶₃=𝐶₄=[−11−1]. Using Theorem 4, we choose a proper number 𝐿=2 that satisfies the conditions to compute a realization (𝐴₊,𝐵₊,𝐶₊) of 𝐻^𝑐(𝑧) such that 𝐴₊=⁰¹₀₀, 𝐵₊=[01]^𝑇, and 𝐶₊=[3.00004.8000]. We observe that a canonical Jordan-form realization (𝐴₀₊,𝐵₀₊,𝐶₀₊) of 𝐹_[𝐿](𝑧) in the form 𝐴₀₊=,𝐶1.8001.0000001.8001.0000001.800₀₊=3.24000.36000 (40) and 𝐵₀₊=𝐞_3,3 are positive-valued. We compute a positive realization (𝐴₁₊⊕𝐴₂₊,^𝐵1+_𝐵2+,𝐶₁₊𝐶₂₊) of 𝐺_[𝐿](𝑧). We choose a proper 𝜂 and 𝑤 to satisfy conditions in (29) and (28), respectively. Therefore, we set a sufficiently large 𝜂=2 and choose 𝑤₁=0.7340 for 𝜆₁ and 𝑤₂=1.2804 for 𝜆₂ to obtain (𝑚₁,𝑚₂)=(3,3). Using the proposed method, we obtain the positive matrix 𝐴₁₊⊕𝐴₂₊ with non-zero elements 3𝑚₁𝑛₁+3𝑚₂𝑛₂=54. First, a 9×9 matrix 𝐴₁₊ is computed as described by (35) with 𝑇₁=𝑇(𝐭₁) and 𝑇₂=𝑇(𝐭₂), where 𝐭₁=[0.73330.24470.8220], 𝐭₂=[0.488700.5774], and 𝛼₁=0.7340. Then, 𝐵₁₊=𝐞_9,9 and 𝐶₁₊=[3.33103.3310⋯2.5554]. Another positive realization (𝐴₂₊,𝐵₂₊,𝐶₂₊) can be obtained using similar methods. (𝐴₊,𝐵₊,𝐶₊) can be obtained using (37) and (3). Therefore, we obtain a Metzler matrix 𝐴_∗=𝐴₊−𝜂𝐼 to obtain the positive realization (𝐴_∗,𝐵₊,𝐶₊).

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.