# Data Informativity for Analysis and Design of Positive Systems

Takumi Iwata, Shun-Ichi Azuma, Masaaki Nagahara, Dimitri Peaucelle, Yoshio Ebihara

► To cite this version:

Takumi Iwata, Shun-Ichi Azuma, Masaaki Nagahara, Dimitri Peaucelle, Yoshio Ebihara. Data Informativity for Analysis and Design of Positive Systems. IEEE Control Systems Letters, 2025, 1 - 1, pp.2475-1456. 10.1109/LCSYS.2025.3638814. hal-05394706

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Data Informativity for Analysis and Design of
Positive Systems

Takumi Iwata†

Shun-ichi Azuma‡

Masaaki Nagahara§

Dimitri Peaucelle¶

Yoshio Ebihara||

Abstract

This paper studies data informativity of positive systems using linear programming (LP). The concept called data informativity represents the sufficiency of a given dataset to solve analysis/design problems. We provide the necessary and sufficient conditions for the data-driven analysis and design problems of positive systems to be solvable. Moreover, we clarify that these conditions are characterized by LP problems. We provide numerical examples to demonstrate the effectiveness of our approaches.

**Keywords:** positivity, data informativity, linear programming, positive observation, positive stabilization.

# 1 Introduction

Data informativity is a concept that indicates the sufficiency of a dataset to solve analysis/design problems [1]. This concept leads to direct data-driven analysis/design that does not require system identification. Using the informativity framework, stability analysis, state feedback stabilization, stabilizability and detectability analysis, model reduction, etc., have been studied (see [1, 2, 3]). In [4], the controller design method combining prior information and data has also been developed. Most of the studies give analysis/design approaches based on linear matrix inequalities (LMIs).

This work was supported in part by JST FOREST Program under Grant Number JP-MJFR2123, JST ASPIRE Program under Grant Number JPMJAP2402, and JSPS KAKENHI under Grant Numbers JP24K21314, JP23K26130, JP22H00512, and JP22KK0155.

<footnote>†School of Informatics and Data Science, Hiroshima University, Higashihiroshima 739-8521, Japan; tiwata@hiroshima-u.ac.jp</footnote>

<footnote>‡Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan; sazuma@i.kyoto-u.ac.jp</footnote>

<footnote>§Graduate School of Advanced Science and Engineering, Hiroshima University, Higashihiroshima 739-8521, Japan; nagam@hiroshima-u.ac.jp</footnote>

<footnote>¶LAAS-CNRS, Université de Toulouse, CNRS, F-31400, Toulouse, France; peaucelle@laas.fr</footnote>

<footnote>||Graduate School of Information Science and Electrical Engineering, Kyushu University, Fukuoka 819-0395, Japan; ebihara@ees.kyushu-u.ac.jp</footnote>
On the other hand, data informativity for positive stabilization has been investigated [5, 6]. In particular, [5] has proposed a linear programming (LP) approach to the design of positive stabilizing controllers. In [7], [5] has been extended to localized data-driven distributed positive stabilization.

In this paper, data informativity for analysis and design specific to positive systems [8, 9] is considered. In particular, we propose approaches for determining data informativity in situations where data and prior information regarding the positivity of the system are given. In addition, we give data-driven analysis and design methods based on LP problems. To this end, we adopt the extended Farkas lemma [10].

The novelty of this paper is presented as follows. First, we address the informativity for α-stability [11] of the positive system. We illustrate through a numerical example that combining prior information of positivity enables stability analysis using small data relative to the system size. This implies that positivity can reduce data size required for stability analysis. Such a view point has not been presented in existing works including [1, 5]. Second, based on the above result, we study the informativity for positive observation [12] and positive stabilization [13]. Utilizing a linear Lyapunov inequality for α-stability, we extend the result on Schur stability in [5] to α-stability. In practical, for some cases where data are pre-collected offline, slight differences of system matrices appear between data measurement and observer/controller implementation. Since α-stability provides a stability margin, our methods have the potential effectiveness even in such cases. As a byproduct, we also consider the informativity for state-bounded control [13].

The remainder of this paper is organized as follows. In Section 2, we study the informativity for α-stability of the positive system. In Sections 3 and 4, we discuss the informativity for positive observation and positive stabilization, respectively. Section V concludes this paper.

**Notation:** Let $\mathbf{R}$, $\mathbf{R}_{\ge 0}$, and $\mathbf{R}_{> 0}$ be the real number field, the set of nonnegative real numbers, and the set of positive real numbers, respectively. Let $\mathbf{P}\mathbf{D}_n$ be the set of $n \times n$ positive definite and diagonal matrices. We use $\mathbb{I}_n$ ($\mathbb{I}_{m \times n}$) to denote the $n$-dimensional vector (the $m \times n$ matrix) whose components are all one. Let $I_n$ be the $n \times n$ identity matrix. For a vector $x \in \mathbf{R}^n$, $\|x\|_{\infty}$ denotes its max norm. For a matrix $A \in \mathbf{R}^{m \times n}$, $\mathrm{vec}(A)$ is $[a_1^\mathrm{T} \quad a_2^\mathrm{T} \quad \dots \quad a_n^\mathrm{T}]^\mathrm{T} \in \mathbf{R}^{mn}$, where $a_i \in \mathbf{R}^m$ is the $i$th column vector of $A$. The notation $\mathrm{rank}(A)$ expresses the rank of $A$. We denote by $A \ge 0$ that all components of $A$ are nonnegative. The inequality $A \ge B$ represents that $A - B \ge 0$. We use $A \otimes B$ to denote the Kronecker product of $A$ and $B$. The system $x(t+1) = Ax(t)$ with $x(t) \in \mathbf{R}^n$ and $A \in \mathbf{R}^{n \times n}$ is said to be positive if $x(t) \in \mathbf{R}_{\ge 0}^n$ ($t = 1, 2, \dots$) for any $x(0) \in \mathbf{R}_{\ge 0}^n$.

## 2 Informativity for α-Stability

Consider the system

$$x(t+1) = \hat{A}x(t) + w(t), \quad (1)$$
where $x(t) \in \mathbf{R}^n$ is the state, $\hat{A} \in \mathbf{R}^{n \times n}$ is a constant matrix, and $w(t) \in \mathbf{R}^n$ is an unknown noise term. Given $\varepsilon \in \mathbf{R}_{\ge 0}$, we assume that $\|w(t)\|_{\infty} \le \varepsilon$ ($t = 0, 1, \dots$).

In this section, we investigate the informativity for $\alpha$-stability of the positive system. Suppose that $\hat{A}$ is unknown but the following matrix is given:

$$X = [x(0) \ x(1) \ \dots \ x(T)] \in \mathbf{R}^{n \times (T+1)}, \quad (2)$$

where $X$ is the pre-collected time-series data of the state for (1) at the time instant $t = 0, 1, \dots, T$. Moreover, we suppose that (1) is previously known to be positive.

**Remark 1.** Some systems are previously known to be positive even if the detail models are unknown. For example, transportation, social, and biological network systems have the states whose denote physical quantities taking non-negative values [8, 9]. For some such network systems, the positivity can be known a priori from their physical properties. $\triangleleft$

To formulate our problem, we define

$$\Sigma_D^s := \left\{ A \in \mathbf{R}_{\ge 0}^{n \times n} \mid \begin{array}{l} X_+ = AX_- + W_- \text{ holds for} \\ \text{some } W_- \text{ s.t. } -\varepsilon \mathbb{I}_{n \times T} \le W_- \le \varepsilon \mathbb{I}_{n \times T} \end{array} \right\}, \quad (3)$$

where $X_- := [x(0) \ x(1) \ \dots \ x(T-1)]$ and $X_+ := [x(1) \ x(2) \ \dots \ x(T)]$ are submatrices of $X$ in (2) and $W_- \in \mathbf{R}^{n \times T}$ is an unknown noise matrix. Here, $D = X$ is called the dataset. Referring to [8], $\Sigma_D^s$ is the set of all coefficient matrices of positive systems consistent with $D$.

Now, we give the definition for $\alpha$-stability.

**Definition 1.** Given $\alpha \in \mathbf{R}_{>0}$, $A$ is said to be $\alpha$-stable if its spectral radius is less than $\alpha$. $\triangleleft$

If $\alpha \le 1$, the value of $\alpha$ corresponds to the exponential decay rate of the state of the system. In particular, if $\alpha = 1$, $\alpha$-stability is exactly Schur stability.

Next, for $\alpha \in \mathbf{R}_{>0}$, we consider the following set:

$$\Sigma(\alpha) := \left\{ A \in \mathbf{R}_{\ge 0}^{n \times n} \mid A \text{ is } \alpha\text{-stable} \right\}. \quad (4)$$

This is the set of all nonnegative and $\alpha$-stable matrices.

For $\Sigma_D^s$ and $\Sigma(\alpha)$, we give the following definition.

**Definition 2.** Suppose that $D = X$ and $\alpha \in \mathbf{R}_{>0}$ are given. The dataset $D$ is said to be informative for $\alpha$-stability of the positive system if $\Sigma_D^s \subseteq \Sigma(\alpha)$. $\triangleleft$

Informativity can be interpreted as a robustness property; e.g., if $\Sigma_D^s \subseteq \Sigma(\alpha)$, $\alpha$-stability is guaranteed for all system matrices belonging to $\Sigma_D^s$, as for the definition of robust $\alpha$-stability. The difference is that the set of system matrices to be considered is parameterized by data, not by uncertainties.

Here, we consider the following problem.
**Problem 1.** Consider the system (1). Suppose that $\mathcal{D} = X$, $\alpha \in \mathbf{R}_{>0}$, and the information that (1) is positive are given. Then, determine whether $\mathcal{D}$ is informative for $\alpha$-stability of the positive system or not. $\triangleleft$

To derive a solution to Problem 1, we introduce the following results referenced from [10] and [14].

**Lemma 1** (see [10]). Given $G \in \mathbf{R}^{p \times r}$, $\gamma \in \mathbf{R}^p$, $H \in \mathbf{R}^{q \times r}$, and $\eta \in \mathbf{R}^q$, consider the polyhedral sets $\mathbf{G} := \{x \in \mathbf{R}^r | Gx \le \gamma\}$ and $\mathbf{H} := \{x \in \mathbf{R}^r | Hx \le \eta\}$. Then, $\mathbf{G} \subseteq \mathbf{H}$ if and only if there exists a nonnegative matrix $M \in \mathbf{R}_{\ge 0}^{q \times p}$ such that $MG = H$ and $M\gamma \le \eta$. $\triangleleft$

**Lemma 2** (see [14]). Suppose that $\alpha \in \mathbf{R}_{>0}$ and $A \in \mathbf{R}_{\ge 0}^{n \times n}$ are given. Then, $A$ is $\alpha$-stable if and only if there exists $(\xi, \mu) \in \mathbf{R}_{>0}^n \times \mathbf{R}_{>0}$ such that

$$A\xi \le \alpha\xi - \mu\mathbf{1}_n. \quad (5)$$

$\triangleleft$

Lemma 1 is called the extended Farkas lemma. In Lemma 2, (5) is a linear Lyapunov inequality for $\alpha$-stability. In most cases of stability analysis of positive systems, the strict inequality $A\xi < \alpha\xi$ is considered (e.g., see [9]). However, using the non-strict inequality (5) with $\mu \in \mathbf{R}_{>0}$, we can directly apply Lemma 1 and obtain the following theorem.

**Theorem 1.** Consider Problem 1. Then, $\mathcal{D}$ is informative for $\alpha$-stability of the positive system if and only if there exists $(M, \xi, \mu) \in \mathbf{R}_{\ge 0}^{q \times \tilde{p}} \times \mathbf{R}_{>0}^n \times \mathbf{R}_{>0}$ such that $MG_D^s = H_\xi$ and $M\gamma_D^s \le \eta_{\xi, \mu}$, where

$$G_D^s := \begin{bmatrix} X_T^\top \otimes I_n \\ -X_T^\top \otimes I_n \\ -I_{n^2} \end{bmatrix}, \quad \gamma_D^s := \begin{bmatrix} \operatorname{vec}(\varepsilon \mathbf{1}_{n \times T} + X_+) \\ \operatorname{vec}(\varepsilon \mathbf{1}_{n \times T} - X_+) \\ 0 \end{bmatrix},$$

$$H_\xi := \begin{bmatrix} \xi^\mathrm{T} \otimes I_n \\ -I_{n^2} \end{bmatrix}, \quad \eta_{\xi, \mu} := \begin{bmatrix} \alpha\xi - \mu\mathbf{1}_n \\ 0 \end{bmatrix},$$

$$\check{q} := n + n^2, \text{ and } \check{p} := 2nT + n^2.$$

*Proof.* In (3), $W_- = X_+ - AX_-$ holds for some $W_-$ satisfying $-\varepsilon\mathbf{1}_{n \times T} \le W_- \le \varepsilon\mathbf{1}_{n \times T}$. Accordingly, we have

$$\Sigma_D^s = \{ A \in \mathbf{R}_{\ge 0}^{n \times n} \mid -\varepsilon \mathbf{1}_{n \times T} \le X_+ - AX_- \le \varepsilon \mathbf{1}_{n \times T} \},$$

which implies that

$$\Sigma_D^s = \left\{ A \in \mathbf{R}_{\ge 0}^{n \times n} \mid \begin{array}{l} AX_- \le \varepsilon \mathbf{1}_{n \times T} + X_+, \\ -AX_- \le \varepsilon \mathbf{1}_{n \times T} - X_- \end{array} \right\}.$$

Applying [15, Lemma 4.3.1] to this, we have

$$\Sigma_D^s = \left\{ A \in \mathbf{R}_{\ge 0}^{n \times n} \mid \begin{array}{l} (X_-^\mathrm{T} \otimes I_n) \mathrm{vec}(A) \le \mathrm{vec}(\varepsilon \mathbf{1}_{n \times T} + X_+), \\ -(X_-^\mathrm{T} \otimes I_n) \mathrm{vec}(A) \le \mathrm{vec}(\varepsilon \mathbf{1}_{n \times T} - X_+) \end{array} \right\}.$$
From this, it follows that

$$ \Sigma_D^s = \{ A \in \mathbf{R}^{n \times n} \mid (X_-^T \otimes I_n) \mathrm{vec}(A) \le \mathrm{vec}(\varepsilon \mathbf{1}_{n \times T} + X_+), \\ -(X_-^T \otimes I_n) \mathrm{vec}(A) \le \mathrm{vec}(\varepsilon \mathbf{1}_{n \times T} - X_+), -\mathrm{vec}(A) \le 0 \}. $$

Hence, we conclude that

$$ \Sigma_D^s = \{ A \in \mathbf{R}^{n \times n} \mid G_D^s \mathrm{vec}(A) \le \gamma_D^s \}. \quad (6) $$

On the other hand, applying Lemma 2 to (4), we also have

$$ \Sigma(\alpha) = \{ A \in \mathbf{R}_{\ge 0}^{n \times n} \mid \exists(\xi, \mu) \in \mathbf{R}_{>0}^n \times \mathbf{R}_{>0} \text{ s.t. (5) holds} \}, $$

which leads to

$$ \Sigma(\alpha) = \{ A \in \mathbf{R}^{n \times n} \mid \exists(\xi, \mu) \in \mathbf{R}_{>0}^n \times \mathbf{R}_{>0} \text{ s.t. } H_{\xi} \mathrm{vec}(A) \le \eta_{\xi, \mu} \}. \quad (7) $$

The proof follows from (6), (7), and Lemma 1. □

Theorem 1 shows that Problem 1 can be reduced into the following LP problem:

$$ \text{find} \quad M \in \mathbf{R}_{\ge 0}^{q \times p}, \xi \in \mathbf{R}_{>0}^n, \mu \in \mathbf{R}_{>0} \quad (8a) $$

$$ \text{s.t.} \quad MG_D^s = H_\xi, \quad M\gamma_D^s \le \eta_{\xi, \mu}. \quad (8b) $$

Example 1. Consider the network system (1) as described in Fig. 1, where $x_i(t) \in \mathbf{R}$ is the state of the node $i$ and $\hat{A}_{ij}$ is the $(i, j)$ component of $\hat{A}$, which indicates the weight of the edge $(j, i)$ and each value is given as shown in Fig. 1. Assume that $\hat{A} \in \mathbf{R}^{n \times n}$ is unknown. Instead, we can use $\mathcal{D} = X$ as shown in Fig. 2 and the information that (1) is positive, i.e., $\hat{A} \ge 0$. Here, we assume $\varepsilon = 0$ for simplicity. Suppose that $\alpha = 0.6$ is given. Then, consider Problem 1. Solving (8), we see that $\mathcal{D}$ is informative for $\alpha$-stability of the positive system. In this case, we conclude that the positive system is $\alpha$-stable even though $\mathcal{D}$ has only two time steps and not informative for stability of the general linear system [1, Corollary 11]. $\triangleleft$

# 3 Informativity for Positive Observation

Consider the system

$$ x(t + 1) = \hat{A}x(t) + w(t), \quad (9a) $$

$$ y(t) = \hat{C}x(t) + v(t), \quad (9b) $$

where $x(t) \in \mathbf{R}^n$ is the state, $y(t) \in \mathbf{R}^l$ is the output, $\hat{A} \in \mathbf{R}^{n \times n}$ and $\hat{C} \in \mathbf{R}^{l \times n}$ are constant matrices, and $w(t) \in \mathbf{R}^n$ and $v(t) \in \mathbf{R}^l$ are unknown noise terms.
Given $\varepsilon \in \mathbf{R}_{\ge 0}$ and $\delta \in \mathbf{R}_{\ge 0}$, we assume that $\|w(t)\|_{\infty} \le \varepsilon$ and $\|v(t)\|_{\infty} \le \delta$ ($t = 0, 1, \dots$). For (9), the full-order observer can be given as

$$z(t+1) = (\hat{A} - L\hat{C})z(t) + Ly(t), \quad (10)$$

where $z(t) \in \mathbf{R}^n$ is the estimated state and $L \in \mathbf{R}^{n \times l}$ is the observer gain to be designed. Now, our objective is to find a positive observer [12]. In (10), $L$ is said to be a positive observer gain for $(\hat{A}, \hat{C})$ if $z(t) \in \mathbf{R}_{\ge 0}^n$ ($t = 1, 2, \dots$) for any $z(0) \in \mathbf{R}_{\ge 0}^n$ and $z(t) - x(t) \to 0$ as $t \to \infty$. In [12], $L$ is designed such that $\hat{A} - L\hat{C}$ is Schur. We here consider designing a positive observer gain $L$ for $(\hat{A}, \hat{C})$ such that $\hat{A} - L\hat{C}$ is $\alpha$-stable for $\alpha < 1$, ensuring that $z(t) - x(t)$ converges to zero more rapidly.

We study the informativity for positive observation. Assume that $\hat{A}$ and $\hat{C}$ are unknown but $X$ in (2) and

$$Y_{-} := [y(0) \ y(1) \ \dots \ y(T-1)] \in \mathbf{R}^{l \times T}$$

are available, where $Y_{-}$ is the pre-collected time-series data of the output for (9) corresponding to $X$.

Now, suppose that $\mathcal{D} = (X, Y_{-})$ is given. We first define

$$\Sigma_{\mathcal{D}}^{s/o} := \left\{ (A, C) \in \mathcal{M}^{s/o} \;\middle|\; \begin{bmatrix} X_+ \\ Y_- \end{bmatrix} = \begin{bmatrix} A \\ C \end{bmatrix} X_- + \begin{bmatrix} W_- \\ V_- \end{bmatrix} \text{ holds} \\ \text{for some } (W_-, V_-) \text{ s.t. } \begin{bmatrix} -\varepsilon \mathbb{1}_{n \times T} \\ -\delta \mathbb{1}_{l \times T} \end{bmatrix} \le \begin{bmatrix} W_- \\ V_- \end{bmatrix} \le \begin{bmatrix} \varepsilon \mathbb{1}_{n \times T} \\ \delta \mathbb{1}_{l \times T} \end{bmatrix} \right\},$$

where $\mathcal{M}^{s/o} := \mathbf{R}^{n \times n} \times \mathbf{R}^{l \times n}$ and $W_- \in \mathbf{R}^{n \times T}$ and $V_- \in \mathbf{R}^{l \times T}$ are unknown noise matrices. We also define

$$\Sigma^{\text{obs}}(\alpha, L) := \left\{ (A, C) \in \mathcal{M}^{s/o} \;\middle|\; \begin{array}{l} A - LC \text{ is } \alpha\text{-stable} \\ \text{and } L \text{ is a positive observer gain for } (A, C) \end{array} \right\}.$$

For $\Sigma_{\mathcal{D}}^{s/o}$ and $\Sigma^{\text{obs}}(\alpha, L)$, we define the following.

*Definition 3.* Suppose that $\mathcal{D} = (X, Y_{-})$ and $\alpha \in \mathbf{R}_{>0}$ are given. The dataset $\mathcal{D}$ is informative for positive observation with $\alpha$-stability if there exists a matrix $L \in \mathbf{R}^{n \times l}$ such that $\Sigma_{\mathcal{D}}^{s/o} \subseteq \Sigma^{\text{obs}}(\alpha, L)$. $\triangleleft$

We here formulate the following problem.

*Problem 2.* Consider the system (9) and the observer (10). Suppose that $\mathcal{D} = (X, Y_{-})$ and $\alpha \in \mathbf{R}_{>0}$ are given. Then,

1. determine whether $\mathcal{D}$ is informative for positive observation with $\alpha$-stability or not;

2. find a matrix $L \in \mathbf{R}^{n \times l}$ such that $\Sigma_{\mathcal{D}}^{s/o} \subseteq \Sigma^{\text{obs}}(\alpha, L)$. $\triangleleft$
<figure><img src="image_2.png" /><figcaption>Fig. 1: The network consisting of 8 nodes in Example 1.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2: The dataset given in Example 1.</figcaption></figure>

*Remark 2.* The availability of state data is often considered in a remote estimation scenario [2, 16, 17, 18]. In such a scenario, it may be unsafe or costly to transfer state measurements to observers in real time via a communication network. Instead, it is assumed that the offline historical state data are available and transferred to the observer using a distinct physical method, as in [16, Remark 3]. Indeed, under such assumptions, data-driven observer design and state estimation have been considered in [17] and [18], respectively. ◁

To solve Problem 2, we prepare the following lemma.

*Lemma 3.* Consider the system (9) and the observer (10). Then, there exists $L \in \mathbf{R}^{n \times l}$ such that $L$ is a positive observer gain for $(\hat{A}, \hat{C})$ and $\hat{A} - L\hat{C}$ is
α-stable if and only if there exists $(\Xi, \Phi, \mu) \in \mathbf{PD}_n \times \mathbf{R}^{n \times l} \times \mathbf{R}_{>0}$ such that

$$ (\hat{A}^{\mathrm{T}}\Xi - \hat{C}^{\mathrm{T}}\Phi^{\mathrm{T}})\mathbf{1}_n \leq (\alpha\Xi - \mu I_n)\mathbf{1}_n, \qquad (11) $$

$$ \hat{A}^{\mathrm{T}}\Xi - \hat{C}^{\mathrm{T}}\Phi^{\mathrm{T}} \geq 0, \qquad (12) $$

$$ \hat{C}^{\mathrm{T}}\Phi^{\mathrm{T}} \geq 0. \qquad (13) $$

Moreover, $L = \Xi^{-1}\Phi$ is a positive observer gain for $(\hat{A}, \hat{C})$ and satisfies that $\hat{A} - L\hat{C}$ is α-stable.

*Proof.* First, we show the ‘if’ part of the first statement. Suppose that there exists $(\Xi, \Phi, \mu) \in \mathbf{PD}_n \times \mathbf{R}^{n \times l} \times \mathbf{R}_{>0}$ satisfying (11), (12), and (13). From [12, Lemma 3.1], there exists a positive observer gain $L \in \mathbf{R}^{n \times l}$ for $(\hat{A}, \hat{C})$ with α-stability if and only if $\hat{A} - L\hat{C}$ is α-stable, $\hat{A} - L\hat{C} \geq 0$, and $L\hat{C} \geq 0$. Accordingly, we show that there exists $L \in \mathbf{R}^{n \times l}$ satisfying these three conditions. From (11), we have

$$ (\hat{A}^{\mathrm{T}} - \hat{C}^{\mathrm{T}}L^{\mathrm{T}})\xi \leq \alpha\xi - \mu\mathbf{1}_n, \qquad (14) $$

where $L^T := \Phi^T\Xi^{-1}$ and $\xi := \Xi\mathbf{1}_n$. Note that $\Xi$ is a diagonal matrix with the components of $\xi$ on its diagonal. Applying Lemma 2 to (14), $\hat{A}^T - \hat{C}^T L^T$ is α-stable. This shows that $\hat{A} - L\hat{C}$ is α-stable. Moreover, from (12) and (13),

$$ \hat{A}^{\mathrm{T}} - \hat{C}^{\mathrm{T}}L^{\mathrm{T}} \geq 0, \quad \hat{C}^{\mathrm{T}}L^{\mathrm{T}} \geq 0, \qquad (15) $$

which implies that $\hat{A} - L\hat{C} \geq 0$ and $L\hat{C} \geq 0$. This completes the proof of ‘if’ part of the first statement.

Next, we prove the ‘only if’ part of the first statement. Suppose that there exists a matrix $L \in \mathbf{R}^{n \times l}$ such that $L$ is a positive observer gain for $(\hat{A}, \hat{C})$ and $\hat{A} - L\hat{C}$ is α-stable. Recalling [12, Lemma 3.1], (15) holds. From Lemma 2, there exists $(\xi, \mu) \in \mathbf{R}_{>0}^n \times \mathbf{R}_{>0}$ satisfying (14). Take $\Xi \in \mathbf{PD}_n$ such that $\xi = \Xi\mathbf{1}_n$, i.e., $\Xi$ is a diagonal matrix with the components of $\xi$ on its diagonal. Then, we have (11), where $\Phi^T := L^T\Xi$. Because $\Xi \geq 0$ and $\hat{C}^T L^T \geq 0$, (13) holds. From $\hat{A} - L\hat{C} \geq 0$, (12) is satisfied. This completes the proof of the first statement.

The proof of the second statement can be given by the proof of the first one. □

Note here that, from (12) and (13) in Lemma 3, there exists a positive observer gain $L$ for $(\hat{A}, \hat{C})$ only if $\hat{A} \geq 0$.

Using Lemmas 1 and 3, we give a solution to Problem 2.

*Theorem 2.* Consider Problem 2. Then, $\mathcal{D}$ is informative for positive observation with α-stability if and only if there exists $(M, \Xi, \Phi, \mu) \in \mathbf{R}_{\geq 0}^{q \times p} \times \mathbf{PD}_n \times \mathbf{R}^{n \times l} \times$
$\mathbf{R}_{>0}$ such that $MG_D^{s/o} = H_{\Xi, \Phi}^{\text{obs}}$ and $M\gamma_D^{s/o} \le \eta_{\Xi, \mu}^{\text{obs}}$, where

$$G_D^{s/o} := \begin{bmatrix} I_{n+l} \otimes X_{-}^T \\ -I_{n+l} \otimes X_{-}^T \end{bmatrix},$$

$$\gamma_D^{s/o} := \begin{bmatrix} \text{vec}([\varepsilon \mathbf{1}_{T \times n} + X_+^T \quad \delta \mathbf{1}_{T \times l} + Y_-^T]) \\ \text{vec}([\varepsilon \mathbf{1}_{T \times n} - X_+^T \quad \delta \mathbf{1}_{T \times l} - Y_-^T]) \end{bmatrix},$$

$$H_{\Xi, \Phi}^{\text{obs}} := \begin{bmatrix} (\mathbf{1}_n^T [\Xi - \Phi]) \otimes I_n \\ -[\Xi - \Phi] \otimes I_n \\ [0 - \Phi] \otimes I_n \end{bmatrix}, \quad \eta_{\Xi, \mu}^{\text{obs}} := \begin{bmatrix} (\alpha \Xi - \mu I_n) \mathbf{1}_n \\ 0 \\ 0 \end{bmatrix},$$

$\dot{q} := n + 2n^2$, and $\dot{p} := 2(n + l)T$.

Moreover, $L = \Xi^{-1}\Phi$ satisfies $\Sigma_D^{s/o} \subseteq \Sigma_{\ge 0}^{\text{obs}}(L)$.

*Proof.* To prove the first statement, we first introduce

$$\begin{align*} \tilde{\Sigma}^{\text{obs}}(\alpha, \Xi, \Phi, \mu) &:= \{(A, C) \in \mathcal{M}^{s/o} \mid \begin{array}{l} A^T \Xi - C^T \Phi^T \ge 0, \\ C^T \Phi^T \ge 0, \end{array} (A^T \Xi - C^T \Phi^T) \mathbf{1}_n \le (\alpha \Xi - \mu I_n) \mathbf{1}_n \}. \end{align*}$$

From Lemma 3 and the definition of $\tilde{\Sigma}_{\ge 0}^{\text{obs}}(\alpha, \Xi, \Phi, \mu)$, there exists a matrix $L \in \mathbf{R}^{n \times l}$ such that $\Sigma_D^{s/o} \subseteq \Sigma_{\ge 0}^{\text{obs}}(\alpha, L)$ if and only if there exists $(\Xi, \Phi, \mu) \in \mathbf{P}D_n \times \mathbf{R}^{n \times l} \times \mathbf{R}_{>0}$ such that $\Sigma_D^{s/o} \subseteq \tilde{\Sigma}_{\ge 0}^{\text{obs}}(\alpha, \Xi, \Phi, \mu)$. From [15, Lemma 4.3.1], we have

$$\tilde{\Sigma}^{\text{obs}}(\alpha, \Xi, \Phi, \mu) = \{(A, C) \in \mathcal{M}^{s/o} \mid H_{\Xi, \Phi}^{\text{obs}} \text{vec}([A^T \quad C^T]) \le \eta_{\Xi, \mu}^{\text{obs}}\}. \quad (16)$$

On the other hand, $\Sigma_D^{s/o}$ can be expressed as

$$\Sigma_D^{s/o} = \{(A, C) \in \mathcal{M}^{s/o} \mid \begin{bmatrix} -\varepsilon \mathbf{1}_{n \times T} \\ -\delta \mathbf{1}_{l \times T} \end{bmatrix} \le \begin{bmatrix} X_+ \\ Y_- \end{bmatrix} - \begin{bmatrix} A \\ C \end{bmatrix} X_- \le \begin{bmatrix} \varepsilon \mathbf{1}_{n \times T} \\ \delta \mathbf{1}_{l \times T} \end{bmatrix}\},$$

which implies that

$$\Sigma_D^{s/o} = \{(A, C) \in \mathcal{M}^{s/o} \mid X_-^T \begin{bmatrix} A \\ C \end{bmatrix}^T \le \begin{bmatrix} \varepsilon \mathbf{1}_{n \times T} + X_+ \\ \delta \mathbf{1}_{l \times T} + Y_- \end{bmatrix}^T, \\ -X_-^T \begin{bmatrix} A \\ C \end{bmatrix}^T \le \begin{bmatrix} \varepsilon \mathbf{1}_{n \times T} - X_+ \\ \delta \mathbf{1}_{l \times T} - Y_- \end{bmatrix}^T \}.$$

Applying [15, Lemma 4.3.1] to this, we have

$$\Sigma_D^{s/o} = \{(A, C) \in \mathcal{M}^{s/o} \mid G_D^{s/o} \text{vec}([A^T \quad C^T]) \le \gamma_D^{s/o}\}. \quad (17)$$
The proof follows from (16), (17), and Lemma 1.

The proof of the second statement can be given by Lemma 3 and the proof of the first one. □

Similar to Theorem 1, Theorem 2 shows that Problem 2 can be reduced into the LP problem

$$ \text{find} \quad M \in \mathbf{R}_{\ge 0}^{\dot{q} \times \dot{p}}, \Xi \in \mathbf{P}\mathbf{D}_n, \Phi \in \mathbf{R}^{n \times l}, \mu \in \mathbf{R}_{>0} $$

$$ \text{s.t.} \quad MG_{\mathcal{D}}^{s/o} = H_{\Xi, \Phi}^{\text{obs}}, \quad M\gamma_{\mathcal{D}}^{s/o} \le \eta_{\Xi, \mu}^{\text{obs}} $$

# 4 Informativity for Positive Stabilization

Consider the system

$$ x(t+1) = \hat{A}x(t) + \hat{B}u(t) + w(t), \quad (18) $$

where $x(t) \in \mathbf{R}^n$ is the state, $u(t) \in \mathbf{R}^m$ is the input, $\hat{A} \in \mathbf{R}^{n \times n}$ and $\hat{B} \in \mathbf{R}^{n \times m}$ are constant matrices, and $w(t) \in \mathbf{R}^n$ is an unknown noise term and $\|w(t)\|_{\infty} \le \varepsilon$ ($t = 0, 1, \dots$), given $\varepsilon \in \mathbf{R}_{\ge 0}$. The input $u(t)$ is given by $u(t) = Fx(t)$, where $F \in \mathbf{R}^{m \times n}$ is the state feedback gain to be designed such that the following closed-loop system is positive:

$$ x(t+1) = (\hat{A} + \hat{B}F)x(t). \quad (19) $$

We suppose that $\hat{A}$ and $\hat{B}$ are unknown but

$$ U_{-} := [u(0) \ u(1) \ \dots \ u(T-1)] \in \mathbf{R}^{m \times T}, \quad (20) $$

$$ X := [x(0) \ x(1) \ \dots \ x(T)] \in \mathbf{R}^{n \times (T+1)} \quad (21) $$

are given, where $U_{-}$ is the input sequence applied to (18) and $X$ denotes the state trajectory corresponding to $U_{-}$.

## 4.1 Pole Assignability

In this section, the informativity for pole assignability is discussed. Suppose that $\mathcal{D} = (U_{-}, X)$ is given. We define

$$ \Sigma_{\mathcal{D}}^{i/s} := \left\{ (A, B) \in \mathcal{M}^{i/s} \mid \begin{array}{l} X_+ = AX_- + BU_- + W_- \text{ holds} \\ \text{for some } W_- \text{ s.t. } -\varepsilon \mathbb{1}_{n \times T} \le W_- \le \varepsilon \mathbb{1}_{n \times T} \end{array} \right\} $$

where $X_- := [x(0) \ x(1) \ \dots \ x(T-1)]$ and $X_+ := [x(1) \ x(2) \ \dots \ x(T)]$ are submatrices of $X$ in (21), $\mathcal{M}^{i/s} := \mathbf{R}^{n \times n} \times \mathbf{R}^{n \times m}$, and $W_- \in \mathbf{R}^{n \times T}$ is an unknown noise matrix.

Moreover, given $\alpha \in \mathbf{R}_{>0}$ and $F \in \mathbf{R}^{m \times n}$, we define

$$ \Sigma^{\text{pole}}(\alpha, F) := \left\{ (A, B) \in \mathcal{M}^{i/s} \mid A + BF \text{ is nonnegative and } \alpha\text{-stable} \right\}. $$

For $\Sigma_{\mathcal{D}}^{i/s}$ and $\Sigma^{\text{pole}}(\alpha, F)$, we define the following.
**Definition 4.** Suppose that $\mathcal{D} = (U_-, X)$ and $\alpha \in \mathbf{R}_{>0}$ are given. The dataset $\mathcal{D}$ is said to be informative for $\alpha$-assignability with positivity if there exists $F \in \mathbf{R}^{m \times n}$ such that $\Sigma_D^{i/s} \subseteq \Sigma^{\text{pole}}(\alpha, F)$. $\triangleleft$

Now we consider the following problem.

**Problem 3.** Consider the system (18). Suppose that $\mathcal{D} = (U_-, X)$ and $\alpha \in \mathbf{R}_{>0}$ are given. Then,

1. determine whether $\mathcal{D}$ is informative for $\alpha$-assignability with positivity or not;

2. find a matrix $F \in \mathbf{R}^{m \times n}$ such that $\Sigma_D^{i/s} \subseteq \Sigma^{\text{pole}}(\alpha, F)$. $\triangleleft$

To solve Problem 3, we provide the following lemma.

**Lemma 4.** Consider the system (19). Then, there exists a matrix $F \in \mathbf{R}^{m \times n}$ such that $\hat{A} + \hat{B}F$ is nonnegative and $\alpha$-stable if and only if there exists $(\Xi, \Psi, \mu) \in \mathbf{P}\mathbf{D}_n \times \mathbf{R}^{m \times n} \times \mathbf{R}_{>0}$ such that

$$ (\hat{A}\Xi + \hat{B}\Psi)\mathbf{1}_n \leq (\alpha\Xi - \mu I_n)\mathbf{1}_n, \quad (22) $$

$$ \hat{A}\Xi + \hat{B}\Psi \geq 0. \quad (23) $$

Moreover, $F = \Psi\Xi^{-1}$ is a matrix such that $\hat{A} + \hat{B}F$ is nonnegative and $\alpha$-stable.

**Proof.** We first show the ‘if’ part of the first statement. Suppose that there exists $(\Xi, \Psi, \mu) \in \mathbf{P}\mathbf{D}_n \times \mathbf{R}^{m \times n} \times \mathbf{R}_{>0}$ satisfying (22) and (23). From (23), $\hat{A}\Xi + \hat{B}\Psi = (\hat{A} + \hat{B}F)\Xi \geq 0$, where $F := \Psi\Xi^{-1}$. From $\Xi^{-1} \geq 0$, $\hat{A} + \hat{B}F \geq 0$. Let $\xi := \Xi\mathbf{1}_n$. Then, $\Xi$ is a diagonal matrix whose diagonal components are $\xi$. Now, $(\hat{A}\Xi + \hat{B}\Psi)\mathbf{1}_n = (\hat{A} + \hat{B}F)\xi \leq \alpha\xi - \mu\mathbf{1}_n$. This and Lemma 2 imply that $\hat{A} + \hat{B}F$ is $\alpha$-stable. This completes the proof of ‘if’ part of the first statement.

Next we prove the ‘only if’ part of the first statement. Suppose that there exists $F \in \mathbf{R}^{m \times n}$ such that $\hat{A} + \hat{B}F \geq 0$ and $\hat{A} + \hat{B}F$ is $\alpha$-stable. Then, there exists $(\xi, \mu) \in \mathbf{R}_{>0}^n \times \mathbf{R}_{>0}$ such that $(A + BF)\xi \leq \alpha\xi - \mu\mathbf{1}_n$. Take $\Xi \in \mathbf{P}\mathbf{D}_n$ such that $\Xi\mathbf{1}_n = \xi$, i.e., $\Xi$ is a diagonal matrix with the components of $\xi$ on its diagonal. Then, $(\hat{A} + \hat{B}F)\Xi\mathbf{1}_n = (\hat{A}\Xi + \hat{B}\Psi)\mathbf{1}_n \leq (\alpha\Xi - \mu I_n)\mathbf{1}_n$, where $\Psi := F\Xi$. Because $\hat{A} + \hat{B}F \geq 0$ and $\Xi \geq 0$, $(\hat{A} + \hat{B}F)\Xi = \hat{A}\Xi + \hat{B}\Psi \geq 0$. This completes the proof of ‘only if’ part of the first statement.

The proof of the second statement can be given by the proof of the first one. $\square$

Now, we derive a solution to Problem 3.

**Theorem 3.** Consider Problem 3. Then, $\mathcal{D}$ is informative for $\alpha$-assignability with positivity if and only if there exists $(M, \Xi, \Psi, \mu) \in \mathbf{R}^{q \times p} \times \mathbf{P}\mathbf{D}_n \times \mathbf{R}^{m \times n} \times \mathbf{R}_{>0}$
such that $MG_D^{i/s} = H_{\Xi, \Psi}^{\text{stab}}$ and $M\gamma_D^{i/s} \le \eta_{\Xi, \mu}^{\text{stab}}$, where

$$G_D^{i/s} := \left[ \begin{bmatrix} X_T^T & U_T^T \\ X_T^- & U_T^- \end{bmatrix} \otimes I_n \right], \gamma_D^{i/s} := \begin{bmatrix} \text{vec}(\varepsilon \mathbb{1}_{n \times T} + X_+) \\ \text{vec}(\varepsilon \mathbb{1}_{n \times T} - X_+) \end{bmatrix},$$

$$H_{\Xi, \Psi}^{\text{stab}} := \begin{bmatrix} (\mathbb{1}_n^T [\Xi \quad \Psi^T]) \otimes I_n \\ -[\Xi \quad \Psi^T] \otimes I_n \end{bmatrix}, \eta_{\Xi, \mu}^{\text{stab}} := \begin{bmatrix} (\alpha \Xi - \mu I_n) \mathbb{1}_n \\ 0 \end{bmatrix},$$

$\dot{q} := n + n^2$, and $\dot{p} := 2nT$.

Moreover, $F = \Psi\Xi^{-1}$ satisfies $\Sigma_D^{i/s} \subseteq \Sigma^{\text{pole}}(\alpha, F)$.

*Proof.* We prove the first statement. To this end, let

$$\tilde{\Sigma}^{\text{pole}}(\alpha, \Xi, \Psi, \mu) := \left\{ \begin{array}{l} (A, B) \in \mathcal{M}^{i/s} \\ (A\Xi + B\Psi)\mathbb{1}_n \le (\alpha\Xi - \mu I_n)\mathbb{1}_n, \quad A\Xi + B\Psi \ge 0 \end{array} \right\}.$$

From Lemma 4 and the definition of $\tilde{\Sigma}^{\text{pole}}(\alpha, \Xi, \Psi, \mu)$, there exists a matrix $F \in \mathbf{R}^{m \times n}$ such that $\Sigma_D^{i/s} \subseteq \Sigma^{\text{pole}}(\alpha, F)$ if and only if there exists $(\Xi, \Psi, \mu) \in \mathbf{P}\mathbf{D}_n \times \mathbf{R}^{m \times n} \times \mathbf{R}_{>0}$ such that $\Sigma_D^{i/s} \subseteq \tilde{\Sigma}^{\text{pole}}(\alpha, \Xi, \Psi, \mu)$. From [15, Lemma 4.3.1],

$$\tilde{\Sigma}^{\text{pole}}(\alpha, \Xi, \Psi, \mu) = \left\{ (A, B) \in \mathcal{M}^{i/s} \middle| H_{\Xi, \Psi}^{\text{stab}} \text{vec}([A \quad B]) \le \eta_{\Xi, \mu}^{\text{stab}} \right\} \quad (24)$$

On the other hand, $\Sigma_D^{i/s}$ can be represented as

$$\Sigma_D^{i/s} = \left\{ (A, B) \in \mathcal{M}^{i/s} \middle| -\varepsilon \mathbb{1}_{n \times T} \le X_+ - AX_- - BU_- \le \varepsilon \mathbb{1}_{n \times T} \right\},$$

which implies that

$$\Sigma_D^{i/s} = \left\{ (A, B) \in \mathcal{M}^{i/s} \middle| \begin{array}{l} [A \quad B] \begin{bmatrix} X_- \\ U_- \end{bmatrix} \le \varepsilon \mathbb{1}_{n \times T} + X_+, \\ -[A \quad B] \begin{bmatrix} X_- \\ U_- \end{bmatrix} \le \varepsilon \mathbb{1}_{n \times T} - X_+ \end{array} \right\}.$$

Applying [15, Lemma 4.3.1] to this, we have

$$\Sigma_D^{i/s} = \left\{ (A, B) \in \mathcal{M}^{i/s} \middle| G_D^{i/s} \text{vec}([A \quad B]) \le \gamma_D^{i/s} \right\}. \quad (25)$$

The proof of the first statement follows from (24), (25), and Lemma 1.

The proof of the second statement can be given by Lemma 4 and the proof of the first one. $\square$
<figure><img src="image_4.png" /><figcaption>Fig. 3: The dataset given in Example 2.</figcaption></figure>

Theorem 3 shows that verifying the informativity for α-assignability can be reduced into the LP problem

$$ \text{find} \quad M \in \mathbf{R}_{\ge 0}^{q \times p}, \quad \Xi \in \mathbf{P}\mathbf{D}_n, \quad \Phi \in \mathbf{R}^{m \times n}, \quad \mu \in \mathbf{R}_{>0} \quad (26a) $$

$$ \text{s.t.} \quad MG_D^{i/s} = H_{\Xi, \Phi}^{\text{stab}}, \quad M\gamma_D^{i/s} \le \eta_{\Xi, \mu}^{\text{stab}}. \quad (26b) $$

Example 2. Consider the system (18) with

$$ \hat{A} = \begin{bmatrix} 0.1 & 0.0 & 0.1 \\ 0.2 & 0.3 & 0.1 \\ -0.1 & 0.2 & 0.8 \end{bmatrix}, \quad \hat{B} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}. \quad (27) $$

Suppose that $\hat{A}$ and $\hat{B}$ are unknown but $\mathcal{D} = (U_-, X)$ as shown in Fig. 3 is given, where $\varepsilon = 0.005$. Given $\alpha = 0.8$, note that $\hat{A}$ is neither $\alpha$-stable nor nonnegative. Now, we consider Problem 3. Solving (26), we see that $\mathcal{D}$ is informative for $\alpha$-assignability with positivity. Moreover, we can obtain

$$ F = \Psi \Xi^{-1} = \begin{bmatrix} 0.1588 & -0.1773 & -0.7849 \end{bmatrix} \quad (28) $$

satisfying $\Sigma_D^{i/s} \subseteq \Sigma^{\text{pole}}(\alpha, F)$. Indeed, for $\hat{A}$ and $\hat{B}$ in (27) and $F$ in (28), $\hat{A}+\hat{B}F$ is nonnegative and $\alpha$-stable. $\triangleleft$

## 4.2 State-Bounded Control

In this section, the informativity for state-bounded control is presented. Our objective is to find a matrix $F \in \mathbf{R}^{m \times n}$ such that, for $\bar{x} \in \mathbf{R}_{>0}^n$, (19) is stable and satisfies

$$ x(t) \in \mathbf{B}(\bar{x}) \quad (t = 1, 2, \dots) \quad \forall x(0) \in \mathbf{B}(\bar{x}), \quad (29) $$
where $\mathbf{B}(\bar{x}) := \{x \in \mathbf{R}^n | 0 \le x \le \bar{x}\}$. Using such a matrix $F$, (19) is state-bounded with respect to $\mathbf{B}(\bar{x})$ and stabilized. For the objective, we consider the case where $\hat{A}$ and $\hat{B}$ are unknown but $\mathcal{D} = (U_-, X)$ in (20) and (21) is given.
Now, for $\bar{x} \in \mathbf{R}^n$ and $F \in \mathbf{R}^{m \times n}$, we define

$$ \Sigma^{\text{bd}}(\bar{x}, F) := \left\{ (A, B) \in \mathcal{M}^{i/s} \mid x(t+1) = (A + BF)x(t) \text{ is stable and satisfies (29)} \right\}. $$

For $\Sigma_D^{i/s}$ and $\Sigma^{\text{bd}}(\bar{x}, F)$, we introduce the following.

**Definition 5.** Suppose that $\mathcal{D} = (U_-, X)$ and $\bar{x} \in \mathbf{R}_{>0}^n$ are given. The dataset $\mathcal{D}$ is said to be informative for state-bounded control with respect to $\mathbf{B}(\bar{x})$ if there exists a matrix $F \in \mathbf{R}^{m \times n}$ such that $\Sigma_D^{i/s} \subseteq \Sigma^{\text{bd}}(\bar{x}, F)$. △
Now we consider the following problem.

**Problem 4.** Consider the system (18). Suppose that $\mathcal{D} = (U_-, X)$ and $\bar{x} \in \mathbf{R}_{>0}^n$ are given. Then,

1. determine whether $\mathcal{D}$ is informative for state-bounded control with respect to $\mathbf{B}(\bar{x})$ or not;

2. find a matrix $F \in \mathbf{R}^{m \times n}$ such that $\Sigma_D^{i/s} \subseteq \Sigma^{\text{bd}}(\bar{x}, F)$. △

For Problem 4, we prepare the following lemma.

**Lemma 5.** Consider the system (19). If there exists $(\Xi, \Psi, \mu) \in \mathbf{P}\mathbf{D}_n \times \mathbf{R}^{m \times n} \times \mathbf{R}_{>0}$ such that $\Xi \mathbf{1}_n \le \bar{x}$, $(\hat{A}\Xi + \hat{B}\Psi)\mathbf{1}_n \le (\Xi - \mu I_n)\mathbf{1}_n$, and $\hat{A}\Xi + \hat{B}\Psi \ge 0$, then $F = \Psi\Xi^{-1}$ satisfies (29) and (19) is stable.

*Proof.* The proof follows from Lemma 4 with $\alpha = 1$ and [13, Lemma 4.1]. □

Similar to [13, Theorem 5.1], Lemma 5 shows a sufficient condition for model-based state-bounded control with respect to $\mathbf{B}(\bar{x})$. Based on the model-based result, we can provide the following sufficient condition for solving Problem 4.

**Theorem 4.** Consider Problem 4. If there exists a quadruplet $(M, \Xi, \Psi, \mu) \in \mathbf{R}_{>0}^{q \times \hat{p}} \times \mathbf{P}\mathbf{D}_n \times \mathbf{R}^{m \times n} \times \mathbf{R}_{>0}$ such that $\Xi \mathbf{1}_n \le \bar{x}$, $MG_D^{i/s} = H_{\Xi, \Psi}^{\text{stab}}$, and $M\gamma_D^{i/s} \le \eta_{\Xi, \mu}^{\text{stab}}$ with $\alpha = 1$, then $\mathcal{D}$ is informative for state-bounded control with respect to $\mathbf{B}(\bar{x})$, and $F = \Psi\Xi^{-1}$ is a matrix such that $\Sigma_D^{i/s} \subseteq \Sigma^{\text{bd}}(\bar{x}, F)$.

*Proof.* Using Lemma 5, the proof can be given in a similar manner to Theorem 3. □

From Theorem 4, we see that Problem 4 is solvable if the following LP problem is feasible:

find $M \in \mathbf{R}_{\ge 0}^{q \times \hat{p}}$, $\Xi \in \mathbf{P}\mathbf{D}_n$, $\Phi \in \mathbf{R}^{m \times n}$, $\mu \in \mathbf{R}_{>0}$

s.t. $\Xi \mathbf{1}_n \le \bar{x}$, $MG_D^{i/s} = H_{\Xi, \Phi}^{\text{stab}}$, $M\gamma_D^{i/s} \le \eta_{\Xi, \mu}^{\text{stab}}$.
# 5 Conclusion

In this paper, we investigated data informativity for analysis and design of positive systems. We developed data-driven analysis and design of positive systems using LP. Assuming that the state data with time derivative are available, our results can also be discussed for continuous-time systems (e.g., see [5, 3]). Future work includes developing data-driven scalable analysis and design methods [7].

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