Computation of Positive Realizations for Given Impulse Response Matrices of Discrete-Time Systems

Abstract

A method is presented for computation of positive realizations for given impulse response matrices of discrete-time systems. Sufficient conditions for the existence of positive realizations are established and a procedure for computation of positive realizations for given impulse response matrices is proposed.

1. Introduction

In positive systems inputs, state variables and outputs take only nonnegative values. Examples of positive systems are industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems, compartmental systems, and water and atmospheric pollution models. A variety of models having positive linear systems behavior can be found in engineering, management science, economics, social sciences, biology, medicine, and so forth.

Positive linear systems are defined on cones and not on linear spaces. Therefore, the theory of positive systems is more complicated and less advanced. An overview of state of the art in positive systems theory is given in the monographs [7, 11]. Recent developments in positive system theory and some new results are given in [12]. Realizations problem of positive linear systems without time delays has been considered in many papers and books [1–4, 6–11].

Recently, the reachability, controllability, and minimum energy control of positive linear discrete-time systems with time-delays have been considered in [5, 18]. The realization problem for positive multivariable discrete-time systems and continuous-time systems with delays was formulated and solved in [13, 14, 16, 17]. The notion of cone realization of discrete-time systems without delays has been introduced in [15].

The main purpose of this paper is to present a method for computation of positive realizations for given impulse response matrices of discrete-time systems. Sufficient conditions for the existence of positive realizations will be established and a procedure for computation of positive realizations for given impulse response matrices will be proposed.

2. Preliminaries and Problem Formulation

Let 𝑅^𝑛×𝑚 be the set of 𝑛×𝑚 real matrices and 𝑅^𝑚=𝑅^𝑚×1.

Consider the discrete-time system 𝑥_𝑖+1=𝐴𝑥_𝑖+𝐵𝑢_𝑖,𝑖∈𝑍₊𝑦∶={0,1,…},_𝑖=𝐶𝑥_𝑖+𝐷𝑢_𝑖, (2.1a)(2.1b) where 𝑥_𝑖∈𝑅^𝑛, 𝑢_𝑖∈𝑅^𝑚, 𝑦_𝑖∈𝑅^𝑝 are the state, input, and output vectors, respectively, and 𝐴∈𝑅^𝑛×𝑛,𝐵∈𝑅^𝑛×𝑚,𝐶∈𝑅^𝑝×𝑛,𝐷∈𝑅^𝑝×𝑚.

Let 𝑅₊^𝑛×𝑚 be the set of 𝑛×𝑚 real matrices with nonnegative entries and 𝑅^𝑚₊=𝑅₊^𝑚×1.

Definition 2.1 (see [7, 11]). The system (2.1a) and ((2.1b) is called (internally) positive if for every 𝑥₀∈𝑅^𝑛₊ and any input sequence 𝑢_𝑖∈𝑅^𝑚₊,𝑥_𝑖∈𝑅^𝑛₊, and 𝑦_𝑖∈𝑅^𝑝₊ for 𝑖∈𝑍₊.

Theorem 2.2 (see [11]). The system (2.1a) and ((2.1b) is positive if and only if 𝐴∈𝑅₊^𝑛×𝑛,𝐵∈𝑅₊^𝑛×𝑚,𝐶∈𝑅₊^𝑝×𝑛,𝐷∈𝑅₊^𝑝×𝑚. (2.2)The transfer matrix of (2.1a) and (2.1b) is given by 𝑇(𝑧)=𝐶[𝐼_𝑛𝑧−𝐴]⁻¹𝐵+𝐷. (2.3) The impulse response matrix of (2.1a), (2.1b) and the transfer matrix (2.3) are related by 𝐺=𝑍⁻¹[𝑇(𝑧)]=𝑔₀+𝑔₁𝑧⁻¹+𝑔₂𝑧⁻²+⋯, (2.4) where 𝑔_𝑖=𝐷,for𝑖=0,𝐶𝐴^𝑖−1𝐵,for𝑖=1,2,…. (2.5) If the system (2.1a) and (2.1b) is positive, then [11] 𝑔_𝑖∈𝑅₊^𝑝×𝑚,for𝑖∈𝑍₊. (2.6) It is well known [7, 11] that for positive asymptotically stable systems 𝑔_𝑖+1<𝑔_𝑖,𝑖=𝑛,𝑛+1,…, (2.7a) and for unstable systems 𝑔_𝑖+1≥𝑔_𝑖,𝑖=𝑛,𝑛+1,…. (2.7b) The positive realization problem for a given impulse matrix can be stated as follows. Given an impulse response matrix (2.4) satisfying the condition (2.7a) and (2.7b), find a positive realization (2.2) of the discrete-time system (2.1a) and ((2.1b).

In this paper, a procedure will be proposed for computation of a positive realization (2.2) for a given impulse response matrix (2.4).

3. Problem Solution

To simplify the notation, we will consider the single-input single-output (SISO) discrete-time system with transfer function of the form 𝑏𝑇(𝑧)=^𝑛−1𝑧^𝑛−1+⋯+𝑏₁𝑧+𝑏₀𝑧^𝑛−𝑎_𝑛−1𝑧^𝑛−1−⋯−𝑎₁𝑧−𝑎₀+𝐷. (3.1) The transfer function (3.1) can be written in the form 𝑇(𝑧)=𝑔₀+𝑔₁𝑧⁻¹+𝑔₂𝑧⁻²+⋯. (3.2) From (3.1) and (3.2), we have 𝐷=𝑔₀ (3.3) and the strictly proper part of 𝑇(𝑧) is given by 𝑇_𝑠𝑝𝑏(𝑧)=𝑇(𝑧)−𝐷=^𝑛−1𝑧^𝑛−1+⋯+𝑏₁𝑧+𝑏₀𝑧^𝑛−𝑎_𝑛−1𝑧^𝑛−1−⋯−𝑎₁𝑧−𝑎₀=𝑔₁𝑧⁻¹+𝑔₂𝑧⁻²+⋯. (3.4) Knowing 𝑔₀ and using (3.3), we can find D. Therefore, the realization problem has been reduced to finding the matrices 𝐴∈𝑅₊^𝑛×𝑛,𝐵∈𝑅₊^𝑛×𝑚,𝐶∈𝑅₊^𝑝×𝑛 (3.5) for given 𝑔_𝑖≥0for𝑖=1,2,….

In further considerations, it is assumed that the given sequence 𝑔_𝑖,𝑖=0,1,…, satisfies the condition (2.7a) and(2.7b). Solution of the problem is based on the following lemmas.

Lemma 3.1. The values 𝑔₁,𝑔₂,… of the impulse response of the discrete-time system with transfer function (3.4) satisfy the equality 𝑎₀𝑔_𝑘+𝑎₁𝑔_𝑘+1+⋯+𝑎_𝑛−1𝑔_{𝑘+𝑛−1}=𝑔_𝑘+𝑛,for𝑘=1,2,…. (3.6)

Proof. From (3.4) we have 𝑏_𝑛−1𝑧^𝑛−1+⋯+𝑏₁𝑧+𝑏₀=(𝑧^𝑛−𝑎_𝑛−1𝑧^𝑛−1−⋯−𝑎₁𝑧−𝑎₀)(𝑔₁𝑧⁻¹+𝑔₂𝑧⁻²+⋯). (3.7) Comparison of the coefficients at the same power 𝑧^−𝑘 of equality (3.7) yields (3.6).

Using (3.6) for 𝑘=1,2,…,𝑛, we obtain the matrix equation 𝐺_𝑛𝑛𝑎=𝐺_𝑛, (3.8) where 𝐺_𝑛𝑛=𝑔₁𝑔₂⋯𝑔_𝑛𝑔₂𝑔₃⋯𝑔_𝑛+1𝑔⋯⋯⋯⋯_𝑛𝑔_𝑛+1⋯𝑔_2𝑛−1𝑎,𝑎=⁰𝑎₁⋮𝑎_𝑛−1,𝐺_𝑛=𝑔_𝑛+1𝑔_𝑛+2⋮𝑔_2𝑛. (3.9) From (3.4) and (3.6), it follows that rank𝐺_𝑛𝑛=𝑛. (3.10) Thus, solving (3.8) for given 𝑔₁,𝑔₂,…,𝑔_2𝑛, we can find the coefficients 𝑎₀,𝑎₁,…,𝑎_𝑛−1 of the denominator of the transfer function (3.4).

Lemma 3.2. Let the given sequence 𝑔₁,𝑔₂,… satisfy the condition (2.7a) and (2.7b) and 𝐺_{𝑛+1,𝑛+1}=𝐺_𝑛𝑛𝐺_𝑛𝐺^𝑇_𝑛𝑔_2𝑛+1∈𝑅^{(𝑛+1)×(𝑛+1)}(𝑇denotesthetranspose),𝐺_𝑛=𝑔_𝑛+1𝑔_𝑛+2…𝑔_2𝑛. (3.11) Then, rank𝐺_{𝑛+1,𝑛+1}=𝑛 (3.12) and the adjoint matrix 𝐺_{𝑛+1,𝑛+1}=𝐴𝑑𝑗𝐺_{𝑛+1,𝑛+1}=𝐺_𝑛𝑛𝐺_𝑛𝐺^𝑇_𝑛𝑔_{𝑛+1,𝑛+1},𝐺_𝑛𝑛∈𝑅^𝑛×𝑛,𝐺_𝑛∈𝑅^𝑛, (3.13) has the following properties: (i)the columns 𝑔₁,𝑔₂,…,𝑔_𝑛+1 of the matrix (3.13) are proportional, that is, 𝐺_{𝑛+1,𝑛+1}=𝑔₁𝑔₂…𝑔_𝑛+1=[𝑔₁,𝑘₁𝑔₁,…,𝑘_𝑛𝑔₁𝑘],_𝑖>0,for𝑖=1,…,𝑛−1,<0,for𝑖=𝑛; (3.14)(3.15)(ii)the square matrix 𝐺_𝑛𝑛 has negative entries, the column matrix 𝐺_𝑛 has positive entries, and 𝑔_{𝑛+1,𝑛+1} is negative.

Proof. The condition (3.12) follows from equality (3.6). The condition (3.12) implies that the dimension of the null subspace of 𝐺_{𝑛+1,𝑛+1} is equal to 1, that is, dimker𝐺_{𝑛+1,𝑛+1}=1. (3.16) Thus the columns 𝑔₁,𝑔₂,…,𝑔_𝑛+1 of the matrix (3.13) are proportional. The adjoint matrix (3.13) is symmetrical since the matrix (3.11) is symmetrical. The symmetry of the matrix (3.13) implies the condition (3.15).
From definition of the adjoint matrix (3.13) and (3.12), we have 𝐺_{𝑛+1,𝑛+1}𝐺_{𝑛+1,𝑛+1}=𝐺_{𝑛+1,𝑛+1}𝐺_{𝑛+1,𝑛+1}=0(zeromatrix). (3.17) From (3.17) and 𝑔_𝑖>0, 𝑖=1,2,…, it follows that every column and every row of the matrix (3.13) has at least one positive entry and at least one negative entry. Taking into account this fact and the symmetry of the matrix (3.13), it is easy to show that the condition (ii) holds.

Theorem 3.3. Let the given sequence 𝑔₁,𝑔₂,… satisfy the condition (2.7a) and (2.7b) and let there exist a natural number 𝑛 such that 𝑟𝑎𝑛𝑘𝐺_{𝑛+1,𝑛+1}=𝑟𝑎𝑛𝑘𝐺_𝑛𝑛=𝑛. (3.18) Then, there exists a positive realization (3.5) of the form 𝐴=00⋯0𝑎₀10⋯0𝑎₁01⋯0𝑎₂⋯⋯⋯⋯⋯00⋯1𝑎_𝑛−110⋮0,𝐵=,𝐶=𝑔₁𝑔₂⋯𝑔_𝑛 (3.19) or 𝑎𝐴=010⋯0001⋯0⋯⋯⋯⋯⋯000⋯1₀𝑎₁𝑎₂⋯𝑎_𝑛−1𝑔,𝐵=¹𝑔₂⋮𝑔_𝑛,𝐶=10⋯0. (3.20)

Proof. Using the well-known Cramer rule for (3.8), we obtain 𝑎_𝑖−1=det𝐺_𝑛𝑖det𝐺_𝑛𝑛,for𝑖=1,…,𝑛, (3.21) where 𝐺_𝑛𝑖 is the 𝑛×𝑛 matrix obtained from the matrix 𝐺_𝑛𝑛 by replacement of its 𝑖th column by the column 𝐺_𝑛.
From the condition (ii) of Lemma 3.2 and (3.21), it follows that 𝑎_𝑖−1>0,for𝑖=1,…,𝑛, since both det𝐺_𝑛𝑖 and det𝐺_𝑛𝑛 are negative. Hence, the matrices 𝐴 of realizations (3.19) and (3.20) have nonnegative entries.
We will show that the matrices (3.19) and (3.20) are positive realizations of the transfer function (3.4). Using (3.19), we obtain 𝑇(𝑧)=𝐶[𝐼_𝑛𝑧−𝐴]⁻¹𝐵=𝑔₁𝑔₂⋯𝑔_𝑛𝑧0⋯0−𝑎₀−1𝑧⋯0−𝑎₁⋯⋯⋯⋯⋯00⋯−1𝑧−𝑎_𝑛−1⁻¹10⋮0=𝑔₁𝑔₂⋯𝑔_𝑛𝑧^𝑛−𝑎_𝑛−1𝑧^𝑛−1−⋯−𝑎₁𝑧−𝑎₀1−𝑎_𝑛−1−𝑎_𝑛−2⋯−𝑎₂−𝑎₁01−𝑎_𝑛−1⋯−𝑎₃−𝑎₂⋯⋯⋯⋯⋯⋯000⋯1−𝑎_𝑛−1×𝑧000⋯01^𝑛−1𝑧^𝑛−2⋮𝑧1. (3.22) Comparing the coefficients at the same power of z of equality (3.7), we obtain 𝑏_𝑛−1=𝑔₁,𝑏_𝑛−2=𝑔₂−𝑎_𝑛−1𝑔₁,…,𝑏₀=𝑔_𝑛−𝑎_𝑛−1𝑔_𝑛−1−⋯−𝑎₁𝑔₁,𝑔₁𝑔₂⋯𝑔_𝑛1−𝑎_𝑛−1−𝑎_𝑛−2⋯−𝑎₂−𝑎₁01−𝑎_𝑛−1⋯−𝑎₃−𝑎₂⋯⋯⋯⋯⋯⋯000⋯1−𝑎_𝑛−1=𝑏000⋯01^𝑛−1𝑏_𝑛−2⋯𝑏₀. (3.23)(3.24) Substitution of (3.24) into (3.22) yields 𝑏𝑇(𝑧)=^𝑛−1𝑏_𝑛−2⋯𝑏₀𝑧^𝑛−𝑎_𝑛−1𝑧^𝑛−1−⋯−𝑎₁𝑧−𝑎₀𝑧^𝑛−1𝑧^𝑛−2⋮𝑧1=𝑏_𝑛−1𝑧^𝑛−1+⋯+𝑏₁𝑧+𝑏₀𝑧^𝑛−𝑎_𝑛−1𝑧^𝑛−1−⋯−𝑎₁𝑧−𝑎₀. (3.25) The proof for the matrices (3.20) follows immediately from the equality 𝑇(𝑧)=[𝑇(𝑧)]^𝑇={𝐶[𝐼_𝑛𝑧−𝐴]⁻¹𝐵}^𝑇=𝐵^𝑇[𝐼_𝑛𝑧−𝐴^𝑇]⁻¹𝐶^𝑇, (3.26) where upper index T denotes transpose.

From the above considerations, we have the following procedure for finding a positive realization for given 𝑔_𝑖>0,𝑖=0,1,….

Procedure 1. Step 1. Knowing 𝑔₀ and using (3.3), find 𝐷.Step 2. Using 𝑔_𝑖, for 𝑖=1,…,2𝑛, find 𝑛 satisfying the condition (3.18).Step 3. Using (3.21), find the entries 𝑎₀,𝑎₁,…,𝑎_𝑛−1 of the matrix 𝐴.Step 4. Find the matrices 𝐵 and 𝐶 of the positive desired realization.

4. Examples

The procedure will be illustrated by two numerical examples. In the first example, the system will be unstable and in the second one the system will be asymptotically stable.

Example 4.1. Find a positive realization (2.2) for the given values of the impulse function 𝑔₀=2,𝑔₁=1,𝑔₂=3,𝑔₃=5,𝑔₄=11,𝑔₅=25,…. (4.1) Using Procedure 1, we obtain the following.

Step 1. Using (3.3), we get 𝐷=𝑔₀=2. (4.2)Step 2. In this case, 𝑛=2 since 𝑔rank¹𝑔₂g₂g₃g=rank1335=rank¹g₂g₃g₂g₃g₄g₃g₄g₅=rank135351151125=𝑛=2. (4.3)Step 3. Taking into account that 𝐺₂₂=𝑔₁𝑔₂𝑔₂𝑔₃=1335,𝐺₂=𝑔₃𝑔₄=511 (4.4) and using (3.8), we obtain 13𝑎35𝑎⁰₁=5𝑎11₀,𝑎₁=1335⁻¹5=21.11 (4.5)(4.6)Step 4. Using (3.19) and (4.6), we obtain 𝐴=0𝑎₀1𝑎₁=10𝑔0211,𝐵=,𝐶=₁𝑔₂=13. (4.7) The desired positive realization is given by (4.2) and (4.7).
The transfer function of the unstable system has the form 𝑇(𝑧)=𝐶[𝐼_𝑛𝑧−𝐴]⁻¹𝐵+𝐷=13𝑧−2−1𝑧−1⁻¹10+2=2𝑧²−𝑧−2𝑧²−𝑧−2=2+𝑧⁻¹+3𝑧⁻²+5𝑧⁻³+11𝑧⁻⁴+25𝑧⁻⁵+⋯. (4.8)Example 4.2. Find a positive realization (2.2) for the values of the impulse function 𝑔₀=0,𝑔₁=1,𝑔₂=1.1,𝑔₃=0.21,𝑔₄=0.131,𝑔₅=0.0341. (4.9) Using Procedure 1, we obtain the following.Step 1. From (3.3), we have 𝐷=𝑔₀=0. (4.10)Step 2. Using (4.9) and (3.18), we obtain rank𝐺₂₂𝑔=rank¹𝑔₂g₂g₃g=rank11.11.10.21=rank¹g₂g₃g₂g₃g₄g₃g₄g₅=rank11.10.211.10.210.1310.210.1310.0341=𝑛=2. (4.11)Step 3. In this case, (3.8) has the form 11.1𝑎1.10.21𝑎⁰₁=0.210.131 (4.12) and its solution is 𝑎₀𝑎₁=11.11.10.21⁻¹=.0.210.1310.10.1 (4.13)Step 4. Using (3.20) and (4.13), we obtain 𝑎𝐴=01₀𝑎₁=01𝑔0.10.1,𝐵=¹𝑔₂=11.1,𝐶=10. (4.14) The desired positive realization is given by (4.10) and (4.14). The transfer function of the stable system has the form 𝑇(𝑧)=𝐶[𝐼_𝑛𝑧−𝐴]⁻¹𝐵+𝐷=10𝑧−1−0.1𝑧−0.1⁻¹1=1.1𝑧+1𝑧²−0.1𝑧−0.1=𝑧⁻¹+1.1𝑧⁻²+0.21𝑧⁻³+0.131𝑧⁻⁴+0.0341𝑧⁻⁵+⋯. (4.15) The above considerations can be easily extended for multi-input multi-output discrete-time systems.

5. Concluding Remarks

A method for computation of positive realizations for given impulse response matrices of discrete-time systems has been proposed. The method is based on Lemma 3.2 and Theorem 3.3 that formulates sufficient conditions for the existence of positive realizations. A procedure for computation of positive realizations for given impulse responses has been proposed. The procedure has been illustrated by two numerical examples. The details of the proposed method have been given for SISO discrete-time systems but it can be easily extended for multi-input multi-output discrete-time and continuous-time linear systems.

An extension of the method is also possible for linear systems with delays and for 2D linear systems [11].

Acknowledgment

The work was supported by the State Committee for Scientific Research of Poland under Grant no. 3 T11A 006 27.