# A Constructive Positive Realization with Sparse Matrices for

Authors: Kyungsup Kim
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 Section2, 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∣xTy≥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,…,xm} such that xi∈ℝn; that is, 𝒳=cone({x1,…,xn}) and X is a polyhedral generator of 𝒳 [13]. Hereafter, X is also denoted by the matrix with columns xi∈X; that is, X=[x1x2…xm], as long as it is not ambiguous. Given a general matrix A=[aij], we write A≥0 if at least some entry aij≥0.Consider a singleinput, single output linear time-invariant system of the form:(1)x˙(t)=Ax(t)+Bu(t),y(t)=Cx(t),
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*∈ℝm×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]).
Let𝒫 be a polyhedral cone in ℝn and A∈ℝn×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 xk=(t/k)(I+Ak/t)kx. For some t and k/t≥λ, xk∈𝒫. Hence limk→∞xk=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 λij=-qiTApj/qiTpj for qiTApj<0 and λij=0 otherwise. Let λ=max(λij). 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 setℛ 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 functionH(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∈ℝn×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 ofA 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 systemH(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).SetA+=ηI+A* and let λ~k≜λk+η for each k be an eigenvalue of A~=ηI+A. The sequence {mi} of H~(z) with mi=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=0∞‍mkz-k. We note that the Markov sequence {mk} 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 {mk}k≥L+1 (i.e., H[L](z)=∑j=1∞‍mj+Lz-j) and Hc(z)=∑j=1L‍mjz-j for a given L.Lemma 3.
Assume thatH~(z) is decomposed into
(2)H~(z)=Hc(z)+z-LH[L](z),
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
(3)A+=[A10[B20]A2],B+=[B10],C+=[C1C2],
where the realization (A1,B1,C1) of Hc(z) is given by
(4)A1=[010⋯0001⋯0⋮⋱⋮000⋯1000⋯0],B1=[00⋮01],C1=[mLmL-1⋯m2m1].Proof.
We observe that the transfer functionHc(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):
(5)A=[A10[B20]A2],B=[B10],C=[C1C2].
Therefore, H~(z) has a positive realization of dimension K+L.A rational transfer functionH~(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
(6)H~(z)=∑k=1n0‍βk(0)(z-λ~0)k+{∑j=1τ‍∑k=1nj‍βk(j)(z-λ~j)k},
where nk is the maximum number of the multiplicity of λ~k=λk+η, which we call the Jordan-form realization. An nj×nj Jordan-form matrix A~j=J(λ~j), an nj×1 matrix Bi, and an 1×nj matrix ci are given, respectively, as
(7)J(λ~j)≜[λ~j1⋯000λ~j⋯00⋮⋮⋱⋮00⋯λ~j100⋯0λ~j],Bj=e1,nj,Cj=[βnj(j)βnj-1(j)⋯β1(j)]for1≤j≤r [14]. A direct sum A⊕B is defined as the block matrix A=[AOOB] with proper zero matrices O. The vector ei,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
(8)A~=⨁j=0τA~i,B=[B0TB1TB2T⋯BτT]T,C=[C0C1C2⋯Cτ],
where AT is the transpose matrix of A.Theorem 4.
A primitive transfer functionH~(z) is defined by (6). H~(z) is decomposed into H~(z)=Hc(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
(9)F[L](z)=∑k=2n0‍β^k(0)(z-λ~0)k,(10)G[L](z)=∑j=1τ‍β^0,jz-λ~0+∑j=1τ‍∑i=1nj‍β^i(j)(z-λ~j)i.
Then, a canonical Jordan-form realization of Hc(z) is a positive realization, a canonical Jordan-form realization of F[L](z) is a positive realization and β^0,j>0 satisfies
(11)β^1(0)=∑j=1τ‍β^0,j,β^0,j≥δ|β^i(j)|
for all i with 1≤i≤nj and some δ>0 for G[L](z).Proof.
BecauseH~(z) is positive realizable, we have the Markov parameter sequence mk≥0. However, βn0(0)>0 is necessary to obtain a positive Markov parameter sequence mk≥0 (refer to [5]). The Jordan-form realization of Hc(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~0L) of F[L](z). Then C0A~0L=[β^n0(0)β^nj-1(0)⋯β^1(0)] is computed by(12)β^k(0)=(Ln0-1)βn0(0)λ~0L-n0+1+∑i=0n0-2‍(Li)βi(0)λ~0L-i,
where (nk) is the number of k-combinations from a given set with n elements. For ϵ>0, there is a sufficiently large L such that (Ln0-1)/(Ln0-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 β^1(0) increases asymptotically as L→∞, we can choose a proper L such that β^1(0)=∑j=1τ‍β^0,j for β^0,j>δ|β^i(j)| for all i and β^0,j>0 for G[L](z).AsL increases, β^1(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, Hc(z) and F[L](z) in (2) and (9), respectively, also have non-negative Markov sequences. Therefore, the positive realizations of Hc(z) and F[L](z) are obtained. A primitive function G[L](z) with a simple dominant pole can be divided into
(13)Gj(z)=β^0,jz-λ~0+∑i=1nj‍β^j(i)(z-λ~j)i.
Therefore, we need to find a positive realization for Gj(z) for all j. In the continuous-time domain, we obtain H(s)=H~(s-η)=Hc(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 Gj(s-η) for each j.


## 3. Positive Realizations for Systems with Simple Complex Poles

Definition 5.
Let𝔓m(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 𝔓m(ρ) is defined by a subset in ℂ through the following inequalities:(14)𝔓m(ρ)={x+iy∈ℂ∣rcos[(2k+1)πm-φ]≤ρcosπm},
for all k with 0≤k≤m, where x=rcosφ and y=rsinφ.The complex numberx+iy corresponds to the point (x,y) in the Cartesian plane ℝ2. The polygon set 𝔓m(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
(15)Ψ=[1cos2πmcos4πm⋯cos2π(m-1)m0sin2πmsin4πm⋯sin2π(m-1)m]
for a given m. A cone generator matrix P∈ℝ3×m is defined as
(16)P=[EΨ]≜[p1p2⋯pm],
where E represents an 1×m row vector with all entries equal to 1; the vectors pi 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
(17)H(s)=β(0)s-λ0+β1s-λ1+β¯1s-λ¯1,
where λ0<0, λ1 is a complex pole with Re(λ1)<λ0, λ¯1 is a complex conjugate of λ1, and β1 is complex number. A minimal real canonical realization of the transfer function is given by
(18)A=[λ0000x-y0yx]≜λ0⊕Γ,B=[110],C=[c1c2c3],
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=λ¯1). Then, β(0)>0, c2=2Re(β1) and c3=2Im(β1).Theorem 6.
Assume that a transfer functionH(s) is defined by (17) where λ0>Re(λ1), and an extreme generator P is constructed as described by (16). If
(19)-π2+πm≤arg(λ1-λ0)≤π2-πm,(20)|β1|≤β(0)2,
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 𝔓m(λ0+η). 𝔓m(λ0+η) is projected onto 𝔓m by dividing by λ0+η. This means that the set {1,(λ1+η)/(λ0+η),(λ¯1+η)/(λ0+η)}⊂𝔓m(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 anexp(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 t→=(t0,t1,…,tm-1), a circular Toeplitz matrix T(t→) is defined as
(21)T(t→)=[t0t1t2⋯tm-1tm-1t0t1⋯tm-2tm-2tm-1t0⋯tm-3⋮⋮⋮⋱⋮t1t2t3⋯t0],
where each element ti,j is ti,j=tk 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 Theorem6 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.
Forϵ=λ0+η>0, the eigenvalue set of (A+ηI)/ϵ is {1,z1,z¯1}. Thus, z1,z¯1∈𝔓m. We choose column vectors {ψi1,ψi2,ψi3} from Ψ as in Figure 1 such that z1∈cone{ψi1,ψi2,ψi3}. We also obtain (A+ηI)p1=a1pi1+a2pi2+a3pi3, where ∑k‍ak=1, with ak≥0. Let [n] be defined as [n]=1+mod(n-1,m). For k≥1,
(22)(A+ηI)p1+k=a1p[i1+k]+a2p[i2+k]+a3p[i3+k].
We can obtain a sparse circular Toeplitz matrix A+=T(t→) such that a vector t→ is defined by t[i1+k]=a1, t[i2+k]=a2, and t[i3+k]=a3 and the other values of tk 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., setH(s)=Gj(s) for some i in (13)). Let us consider a transfer function H1(s) with a single pair of multiple complex conjugate poles:
(23)H1(s)=∑k=1n‍{βk(s-λ1)k+β¯k(s-λ¯1)k},
where the pole λ1 and the coefficients βi are complex, and β¯i is defined as the conjugate of βi. The transfer function H1(s) with the partial fractional expansion in (23) has a canonical Jordan-form realization (A,B,C) such that
(24)A=J(λj)⊕J(λ¯j),B=[en,nen,n],C=[βnβn-1⋯β1β¯n⋯β¯1].
There exists a similarity transformation matrix Q such that a block Jordan-form realization is given by
(25)A1=Q-1AQ=[Γ0IO⋯OOΓ0I⋯OOOΓ0⋯O⋮⋱⋮OOO⋯Γ0],B1=[0⋮011],C1=[βnβ¯n⋯β1β¯1],
where Γ0=λ1⊕λ¯1. Let Δ=[1-i1i]. Using a block diagonal matrix Q1=w-n+1(⨁k=1nwk-1Δ) as a similarity transformation, we obtain a new real block Jordan-form realization (J(λ1,w),B2,C3) such that
(26)J(λ1,w)=Q1-1A1Q1=[ΓwIO⋯OOΓwI⋯OOOΓ⋯O⋮⋱⋮OOO⋯Γ],B2=[0⋮010],C2=[β~2nβ~2n-1⋯β~1],
where the entries of C2 are defined by β~2k=2wk-nRe(βk), β~2k-1=2wk-nIm(βk) for each k and Γ=[x-yyx].We consider a positive realization of transfer functions with a multiple complex conjugate pole pair. Let us defineγm(z)=max{r^∣(r^cosθ,r^sinθ)∈𝔓m,r^>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(27)H(s)=β(0)s-λ0+H1(s)
with one dominant real pole λ0 with a simple order and two complex poles λ1 and λ¯1 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:
(28)0<w≤(1-|z1|γm(z1))γm(z2)|z2|,(29)|βk|≤β(0)(2w)k-n
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:
(30)A=λ0⊕J(λ1,w),B=[1B2],C=[1C2],
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∈ℝ2n×2n and a matrix K∈ℝ2n×m are defined as
(31)Z=[OO⋯OOIOOOOI⋯OO⋮⋮⋱⋮⋮OO⋯IO],K=[ΨOO⋮O],
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⋯Pn], where Pj is defined as Pj=[Eϑj] for 1≤j≤n-1, such that E represents an 1×m row vector in which all entries equal 1 and ϑj=Zj-1K. The columns of matrix P represent the extreme vertices of a finite generated cone 𝒫 in ℝ2n+1 (i.e., cone(P)=𝒫) and are positive independent. The polyhedral cone 𝒫j is generated by Pj, that is; 𝒫j=cone(Pj) for each j. We will prove that 𝒫 is A-invariant (i.e., A𝒫⊂𝒫). Let 𝒫j be the cone generated from the column vectors of Pj (i.e., 𝒫j=cone(Pj)). For λ1∈𝔓m(λ0+η), we observe that AP1∈𝒫1. Note that (A+ηI)Pj for j>1 has nonzero values for the 1st, jth, and j + 1th block components
(32)(A+ηI)Pj=1jj+1(E⋮OwΨ(ηI+Γ)ΨO⋮).
The invariance property requires that ((A+ηI)/(λ0+η))Pj∈𝒫. Set A~=(A+ηI)/(λ0+η). The matrix A~ has eigenvalues {1,z1,z¯1} 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 {Wj-1,Wj} such that
(33)(A+ηI)λ0+ηPj=α1Wj-1+α2Wj
for each j, where the values αj satisfy αj≥0, α1+α2=1 and all of the entries in the first row of Wj∈𝒫j are equal to 1. A sufficient condition for a feasible solution (α1,α2) should satisfy two inequalities, w|z2|/γm(z2)≤α1 and |z1|/γm(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|/γm(z1))(γm(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 considerℛ⊂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 [c^j1c^j1+1]=[β~2jβ~2j-1](w)j-n, where β~2j-1=2Im(βi) and β~2j=2Re(βi). An inequality CPk≥0 for each j≥1 leads to the sufficient condition in (29) as follows:
(34)β(0)E+[2Re(βj)2Im(βj)](w)j-nΨ≥0.
To satisfy the above inequality, we impose a sufficient condition such that |βj|≤β(0)/(2w)j-n for sufficiently large β(0) and each 1≤j≤n. We show that CPj≥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 Theorem8, 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 Theorem8 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
(35)A+=[T1α^1I0⋯00T2α^1I⋯000T2⋯0⋮⋮⋮⋱⋮000⋯T2],
where T1=T(t→1), T2=T(t→2), and α^1>0. Vector t→1 has at most three non-zero elements and t→2 has at most two non-zero elements. Finally, one obtains a sparse Metzler matrix A*=A+-ηI.Proof.
Consider(A+ηI)p1,1=a1p1,i1+a2p1,i2+a2p1,i3, where p1,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, Wk-1∈𝒫k-1 is generated by the first column vector of Pk-1 and Wk∈𝒫k is generated by the convex combination of two column vectors of Pk. Then, APk is positive linearly combined by the three vectors chosen from Pk and Pk-1 in Theorem 8. Therefore, we obtain a sparse Toeplitz-type matrix A+ as described by (35) and a Metzler matrix A*=A+-ηI.LetN1 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 Nm (m≥3) denote the total number of complex conjugate pairs of (possibly multiple) complex poles of the transfer functions in the form Gk(z) that belong to the region 𝔓m-⋃j=1m-1‍𝔓j, except for a dominant pole. The order of the upper bound of the positive realization is bounded by mNm for each m≥2 by Theorem 8. We obtain a positive realization of the primitive transfer function H(z) with upper bound(36)N=L+N0+(1+N1+∑j≥2‍jNj).
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 functionH(s) in a continuous-time domain.(1)
Reasonably, we assume that a primitive transfer functionH~(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 minimumη that maximizes w and minimizes m for all λi, i≥1 is computed to satisfy the condition in (28).(2)
Using Theorem4, we choose a proper L such that the realizations of Hc(z) and F[L](z) are positive realizations, and Gj(z) for each j satisfies the condition in (11), where H~(z)=Hc(z)+s-LH[L](z) and H[L](z)=F[L](z)+∑j=1τ‍Gj(z).
(i)
A minimumL is chosen such that β^k(0)>0 for all k and β^1(0) satisfies condition (11).(ii)
A positive realization ofHc(z), which is denoted by (A^+,B^+,C^+), is obtained by (4).(iii)
A positive realization ofF[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 functionG[L](s-η). For each k, we compute of the positive realization of Gk(s-η) as follows.
(i)
β^1(0) is properly divided into {β^0,k} to satisfy the conditions in (11) and  (29).(ii)
For eachλk, a sparse positive realization of each primitive transfer function Gk(s-η), which is denoted by (Ak+,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
(37)A˘+=⨁k=0τAk+,B˘+=[B0+T⋯Bτ+T]T,C˘+=[C0+⋯Cτ+].(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 functionH(s) of the form
(38)H(s)=1.5s+1+0.25+0.25js+2-j+0.25-0.25js+2+j
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
(39)P=[111110-10010-1].
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 functionH(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(λ¯1)⊕J(λ2)⊕J(λ¯1), 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 Hc(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
(40)A0+=[1.8001.0000001.8001.0000001.800],C0+=[3.24000.36000]
and B0+=e3,3 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(t→1) and T2=T(t→2), where t→1=[0.73330.24470.8220], t→2=[0.488700.5774], and α^1=0.7340. Then, B1+=e9,9 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.



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*Source: 878146-2013-12-24.xml*