Linear Matrix Inequality-Based Design of Structured Sparse Feedback Controllers for Sensor and Actuator Networks

Abstract

A sensor and actuator network (SAN) is a control system where many sensors and actuators are connected through a communication network. In a SAN with redundant sensors and actuators, it is important to consider choosing sensors and actuators used in control design. Depending on applications, it is also important to consider not only the choice of sensors/actuators but also that of communication channels in which some sensors/actuators are connected. In this paper, based on a linear matrix inequality (LMI) technique, we propose a design method for structured sparse feedback controllers. An LMI technique is one of the fundamental tools in systems and control theory. First, the sparse reconstruction problems for vectors and matrices are summarized. Next, two design problems are formulated, and an LMI-based solution method is proposed. Finally, two numerical examples are presented to show the effectiveness of the proposed method.

1. Introduction

A cyber-physical system (CPS) is a system where physical and software components are deeply intertwined through communication networks [1,2,3]. A CPS has several applications such as smart grid [4,5,6], medical monitoring [7,8,9], and automobile systems [10,11,12]. In order to realize these applications, it is important to develop a control method of CPSs. A CPS is constructed by many sensors and actuators. A sensor and actuator network (SAN) is one of the typical CPSs [13,14,15,16,17]. In SANs, a plant, sensors, and actuators are connected through a communication network. In the case where many sensors and actuators are used, it is important to choose sensors and actuators used in control. Moreover, it is also important to reduce the number of used communication channels in which some sensors/actuators are connected.

On the other hand, sparse modeling methods (sparsity methods, sparse reconstruction methods) have been widely studied. Sparse modeling is a method where the original information is reconstructed from a small data set, and has several applications such as image processing, signal processing, and machine learning (see, e.g., [18,19,20,21,22,23]). Theoretical methods in sparse modeling is based on sparse solutions of simultaneous linear equations. Sparse modeling is also applied to systems and control theory [24]. In [25,26,27,28], sparse modeling has been applied to maximum hands-off control and self-triggered control. In [29], sparse modeling has been applied to event-triggered control. In [30], structured sparse feedback design has been studied. By structured sparse feedback design, redundant actuators/sensors can be eliminated. Therefore, this design method is important in design of SANs. However, in [30], only simple controllers such as state-feedback controllers and static output-feedback controllers have been considered. Although redundant actuators/sensors have been focused on, redundant communication channels have not been focused in [30].

In this paper, based on sparsity methods, we propose a design method of structured sparse output-feedback controllers for SANs. A class of controllers studied in this paper is a dynamic output-feedback controller, which is more generalized than a state-feedback controller and a static output-feedback controller. Since only simple controllers were considered in [30], a dynamic output-feedback controller cannot be handled by the method in [30]. In this paper, we consider two design problems. In the first problem, we use the sparse reconstruction problem, and we focus on reducing the number of actuators and sensors. In the second problem, we use the block-sparse reconstruction problem [31,32], and we focus on reducing the number of communication channels. Both problems are reduced to an LMI optimization problem. An LMI technique is one of the fundamental tools in systems and control theory, and is widely used for stabilization, $H_{\infty }$ control, and so on [33,34]. In networked control systems, it is used for event-triggered control [35,36,37], self-triggered control [38,39], control of time delay systems [40,41], and so on. Since an LMI optimization problem is reduced to a convex optimization problem, which can be solved in polynomial time, we can easily solve it using a suitable solver.

This paper is organized as follows. In Section 2, the sparse reconstruction problems for vectors and matrices are summarized. The block-sparse reconstruction problems are also summarized. In Section 3, two design problems are formulated. In Section 4, solution methods for two design problems are derived. In Section 5, two examples are presented to show the effectiveness of the proposed method. One is a simple example. The other is the benchmark model in [42]. In Section 6, we conclude this paper.

Finally, the main contributions of this paper are highlighted as follows:

(i)

For SANs, the design problems of structured sparse output-feedback controllers are formulated using sparse reconstruction and block-sparse reconstruction.

(ii)

Two design problems are reduced to an LMI optimization problem.

(iii)

The effectiveness of the proposed method is clarified through numerical examples.

Notation: Let $\mathcal{R}$ denote the set of real numbers. Let $I_n$ and $0_{m\times n}$ denote the $n\times n$ identity matrix and the $m\times n$ zero matrix, respectively. For simplicity, we sometimes use the symbol 0 instead of $0_{m\times n}$, and the symbol I instead of $I_n$. Let $M\succ 0$ denote that the matrix M is positive-definite. For the vector x, let ${\parallel x\parallel }_i$ denote the $l_i$ norm of x. The symmetric matrix $\left[\begin{array}{cc}A& B^{\top }\\ B& C\end{array}\right]$ is denoted by $\left[\begin{array}{cc}A& *\\ B& C\end{array}\right]$. For the matrix $M\in {\mathcal{R}}^{m\times n}$ and the index set $\mathcal{I}=\{i_1,i_2,\dots \}\subseteq \{1,2,\dots ,n\}$, the matrix ${\mathrm{Col}}_{\mathcal{I}}\left(X\right)$ is defined by

$${\mathrm{Col}}_{\mathcal{I}}\left(X\right)=[{\mathrm{Col}}_{i_1}\left(X\right){\mathrm{Col}}_{i_2}\left(X\right)\cdots ],$$

where ${\mathrm{Col}}_i\left(X\right)$ is the i-th column of X. In a similar way, for the index set $\mathcal{I}=\{i_1,i_2,\dots \}\subseteq \{1,2,\dots ,m\}$, the matrix ${\mathrm{Row}}_{\mathcal{I}}\left(X\right)$ is defined by

$${\mathrm{Row}}_{\mathcal{I}}\left(X\right)=\left[\begin{array}{c}{\mathrm{Row}}_{i_1}\left(X\right)\\ {\mathrm{Row}}_{i_2}\left(X\right)\\ ⋮\end{array}\right],$$

where ${\mathrm{Row}}_i\left(X\right)$ is the i-th row of X.

2. Sparse Reconstruction

In this section, the sparse reconstruction problem and its solution method are summarized. See, e.g., [30,43] for further details.

2.1. Vector Case

Consider the problem of finding a vector $x\in {\mathcal{R}}^n$ that satisfies

$$Ax=b,$$

(1)

where $A\in {\mathcal{R}}^{m\times n}$ and $b\in {\mathcal{R}}^m$ are given. If $m\ge n$, there exist a unique solution except for the rare case. We can derive a reasonable solution using, e.g., the least-squares method. If $m<n$, no unique solution exists. If we have prior knowledge that x is sparse (i.e., almost elements of x are zero), then x may be uniquely determined. The problem of finding a spare vector x from (1) is called the sparse reconstruction problem. The sparse reconstruction problem is equivalently rewritten as the following problem:

$$\underset{x\in {\mathcal{R}}^n}{min}{\parallel x\parallel }_{0}s.t.Ax=b,$$

(2)

where ${\parallel x\parallel }_0$ is the number of nonzero elements of x. A sparse solution can be obtained by solving (2). Since Problem (2) is NP-hard, we consider the following relaxed problem:

$$\underset{x\in {\mathcal{R}}^n}{min}{\parallel x\parallel }_{1}s.t.Ax=b,$$

(3)

which can be equivalently rewritten as a linear programming problem. A solution to (3) is equal to that to (2), except for in rare cases [43].

Next, we consider the block-sparse reconstruction problem. Suppose that the vector x is decomposed to p subvectors, i.e., $x={[x_{\left(1\right)}^{\top }{x}_{\left(2\right)}^{\top }\cdots x_{\left(p\right)}^{\top }]}^{\top }$ ($x_{\left(i\right)}\in {\mathcal{R}}^{n_i}$, ${\sum }_i{n}_i=n$). We may permutate x in advance. The vector x may be uniquely determined, based on prior knowledge that x is block-sparse (i.e., almost subvectors of x are zero, and all elements of a few subvectors may be non-zero). Then, a convex relaxation of the block-sparse reconstruction problem is given by the following problem:

$$\underset{x\in {\mathcal{R}}^n}{min}\sum _{i=1}^p{\parallel x_{\left(i\right)}\parallel }_{2}s.t.Ax=b.$$

(4)

In the case where there are priorities for subvectors, we may add weights as follows:

$$\underset{x\in {\mathcal{R}}^n}{min}\sum _{i=1}^p{w}_{\left(i\right)}{\parallel x_{\left(i\right)}\parallel }_{2}s.t.Ax=b,$$

(5)

where $w_{\left(i\right)}\ge 0$ is a given weight. See, e.g., [31,32] for further details.

2.2. Matrix Case

The sparse reconstruction problem for vectors is extend to that for matrices [30]. For the matrix $X\in {\mathcal{R}}^{n\times m}$, row and column norms are defined by

$${\parallel X\parallel }_{r_1}=\sum _{i=1}^n\underset{1\le j\le m}{max}|x_{ij}{|,\parallel X\parallel }_{c_1}=\sum _{j=1}^m\underset{1\le i\le n}{max}\left|x_{ij}\right|,$$

where $x_{ij}$ is the $(i,j)$-th element. Consider the following problem:

$$\underset{X\in {\mathcal{R}}^{n\times m}}{min}{\parallel X\parallel }_{c_1}s.t.AX=B,$$

(6)

where $A\in {\mathcal{R}}^{q\times n}$ and $B\in {\mathcal{R}}^{q\times m}$. If this problem is feasible, then there exists a solution such that a few columns are nonzero [30]. In a similar way, we can consider the problem of minimizing ${\parallel X\parallel }_{r_1}$ subject to $AX=B$. Hereafter, it is stated that the matrix X is row (column) sparse if only a few rows (columns) of X are nonzero.

Also, for row and column norms, we can consider the block-sparse reconstruction problem. In the case of the row norm, suppose that the matrix X is decomposed to p submatrices, i.e.,

$$X=\left[\begin{array}{c}{X}_{\left(1\right)}^r\\ X_{\left(2\right)}^r\\ ⋮\\ X_{\left(p\right)}^r\end{array}\right],X_{\left(i\right)}^r\in {\mathcal{R}}^{n_i\times m},\sum _i{n}_i=n.$$

Then, a convex relaxation of the block-sparse reconstruction problem for matrices is given by the following problem:

$$\underset{X\in {\mathcal{R}}^{n\times m}}{min}\sum _{i=1}^p{w}_{\left(i\right)}{\parallel X_{\left(i\right)}^r\parallel }_{r_1}s.t.AX=B,$$

(7)

where $w_{\left(i\right)}\ge 0$ is a given weight. In the case of the column norm, suppose that the matrix X is decomposed to p submatrices, i.e.,

$$X=\left[\begin{array}{cccc}{X}_{\left(1\right)}^c& X_{\left(2\right)}^c& \cdots & X_{\left(p\right)}^c\end{array}\right],X_{\left(i\right)}^c\in {\mathcal{R}}^{n\times m_i},\sum _i{m}_i=m.$$

In this case, a convex relaxation of the block-sparse reconstruction problem is given by the following problem:

$$\underset{X\in {\mathcal{R}}^{n\times m}}{min}\sum _{i=1}^p{w}_{\left(i\right)}{\parallel X_{\left(i\right)}^c\parallel }_{c_1}s.t.AX=B.$$

(8)

3. Problem Formulation

In this section, we formulate two design problems of sparse feedback controllers for discrete-time linear systems.

3.1. Mathematical Model of Plants and Feedback Controller

First, we explain the outline of SANs. A SAN is generally a system that through a communication network integrates sensors and actuators to enable autonomous data sensing, processing, and actuation. Figure 1 shows an example of SANs. Sensors and actuators are located in a distributed way. Routers are frequently used. There are several technical issues depending on applications. In this paper, we focus on the reduction of redundant sensors/actuators as one of the technical issues. In other words, we do not focus on the technical issues related to communications such as delays.

As a mathematical model of the plant in a SAN, consider the following discrete-time linear system:

$$\left\{\begin{array}{c}x(k+1)=Ax\left(k\right)+Bu\left(k\right),\hfill \\ y\left(k\right)=Cx\left(k\right),\hfill \end{array}\right.$$

(9)

where $x\left(k\right)\in {\mathcal{R}}^n$ is the state of the system, $u\left(k\right)\in {\mathcal{R}}^m$ is the control input which represents m actuators, $y\left(k\right)\in {\mathcal{R}}^p$ is the measured output which represents p sensors, $A\in {\mathcal{R}}^{n\times n}$, $B\in {\mathcal{R}}^{n\times m}$, and $C\in {\mathcal{R}}^{p\times n}$ are given matrices, and $k\in \{0,1,2,\dots \}$ is the discrete time. We suppose that the control input and the measured output include redundant actuators and sensors, respectively.

As a controller, we consider the following dynamic output-feedback controller:

$$\left\{\begin{array}{c}\widehat{x}(k+1)=A_c\widehat{x}\left(k\right)+B_{c}y\left(k\right),\hfill \\ u\left(k\right)=C_c\widehat{x}\left(k\right),\hfill \end{array}\right.$$

(10)

where $\widehat{x}\left(k\right)\in {\mathcal{R}}^n$ is the state of the controller, and $A_c\in {\mathcal{R}}^{n\times n}$, $B_c\in {\mathcal{R}}^{n\times m}$, and $C_c\in {\mathcal{R}}^{p\times n}$ are design parameters.

3.2. Design Problem of Structured Sparse Feedback Controllers

First, we formulate the design problem of structured sparse feedback controllers.

We suppose that there exist redundant actuators (elements of the control input) and sensors (elements of the measured output). Then, it is desirable that a given plant is controlled by as few actuators and sensors as possible. Based on this idea, we formulate the following problem.

Problem 1.

For System (9), find a feedback controller such that the closed-loop system is asymptotically stable and the number of actuators and sensors is as few as possible.

3.3. Design Problem of Structured Block-Sparse Feedback Controllers

First, we formulate the design problem of structured block-sparse feedback controllers.

In SANs, actuators and sensors are located in a distributed way. Some actuators and sensors may be shared a communication channel. In reduction of the number of times of communications, we focus on communication channels. For example, in Figure 1, communication channels are expressed as routers, and the number of communication channels is three. It is desirable that the number of communication channels used in control is smaller.

We consider using the block-sparse reconstruction problem for matrices.

We suppose that the numbers of actuators and sensors are m (the number of elements in the control input) and p (the number of elements in the measured output), respectively. Let ${\mathcal{I}}^a=\{1,2,\dots ,m\}$ and ${\mathcal{I}}^s=\{1,2,\dots ,p\}$ denote the sets of actuators and sensors, respectively. The sets of actuators and sensors assigned to the communication channel $i\in \{1,2,\dots ,q\}$ are denoted by ${\mathcal{I}}_i^a$ and ${\mathcal{I}}_i^s$, respectively, where ${\mathcal{I}}_i^a\subseteq {\mathcal{I}}^a$, ${\mathcal{I}}_i^a\cap {\mathcal{I}}_j^a=\varnothing$, ${\mathcal{I}}_i^s\subseteq {\mathcal{I}}^s$, and ${\mathcal{I}}_i^s\cap {\mathcal{I}}_j^s=\varnothing$.

In the case of Figure 1, q is given by $q=3$. For the communication channel 1 (router 1), we set ${\mathcal{I}}_1^a=\{1,2\}$ and ${\mathcal{I}}_1^s=\{1,2\}$. For the communication channel 2 (router 2), we set ${\mathcal{I}}_2^a=\left\{3\right\}$ and ${\mathcal{I}}_2^s=\{3,4\}$. For the communication channel 3 (router 3), we set ${\mathcal{I}}_3^a=\left\{4\right\}$ and ${\mathcal{I}}_3^s=\{5,6\}$.

Under the above preparations, consider the following problem.

Problem 2.

For System (9), find a feedback controller (10) such that the closed-loop system is asymptotically stable, and the number of communication channels is as few as possible.

4. Solution Method

4.1. Solution Method for Problem 1

First, we derive a solution method for Problem 1.

From the system (9) and the controller (10), the closed-loop system is given by

$$\overline{x}(k+1)=\overline{A}\overline{x}\left(k\right),$$

(11)

where

$$\overline{x}=\left[\begin{array}{c}x\\ \widehat{x}\end{array}\right],\overline{A}=\left[\begin{array}{cc}A& B{C}_c\\ B_{c}C& A_c\end{array}\right].$$

To guarantee the stability of the closed-loop system, we introduce the following quadratic Lyapunov function:

$$V\left(k\right)={\overline{x}}^{\top }\left(k\right)P\overline{x}\left(k\right),$$

(12)

where $P=P^{\top }\in {\mathcal{R}}^{2n\times 2n}$ is a positive-definite matrix. To guarantee the stability of the closed-loop system, consider the problem of finding a controller such that

$$V(k+1)-V\left(k\right)<-\beta V\left(k\right),$$

(13)

where $\beta \in [0,1)$ is a given parameter.

We have the following theorem.

Theorem 1.

Problem 1 is reduced to the following LMI optimization problem.

Problem 3.

$$\begin{array}{cc}\mathrm{find}\hfill & X\succ 0,Y\succ 0,W_1,W_2,W_3\hfill \\ \min \hfill & {∥\left[\begin{array}{cc}{W}_1& W_2^{\top }\end{array}\right]∥}_{c_1}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill subject to& \left[\begin{array}{cccc}\overline{\beta }X& *& *& *\\ \overline{\beta }I& \overline{\beta }Y& *& *\\ XA+W_{1}C& W_3& X& *\\ A& AY-B{W}_2& I& Y\end{array}\right]\succ 0,\hfill \end{array}$$

(15)

where $\overline{\beta }=1-\beta$.

Using the solution to Problem 3, the coefficient matrices $A_c$, $B_c$, and $C_c$ in the controller (10) are derived as

$$\begin{array}{cc}\hfill A_c& =Z^{-1}(XAY+Z{B}_{c}CY-XB{C}_{c}Y-W_3)Y^{-1},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill B_c& =Z^{-1}{W}_1,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill C_c& =W_2{Y}^{-1},\hfill \end{array}$$

(18)

where $Z=X-Y^{-1}$.

Proof. 

First, without of loss of generality, the positive definite matrix P can be replaced with

$$P=\left[\begin{array}{cc}X& Z\\ Z& Z\end{array}\right]$$

(see [44]). By applying the Schur complement [33] to $P\succ 0$, we can obtain $X-Z\succ 0$. Then, the matrix $Y:={(X-Z)}^{-1}\succ 0$ is defined.

Next, from (13), we can obtain

$$\overline{\beta }P-{\overline{A}}^{\top }P\overline{A}\succ 0.$$

(19)

By applying the Schur complement to (19), we can obtain

$$\left[\begin{array}{cc}\overline{\beta }P& *\\ P\overline{A}& P\end{array}\right]\succ 0.$$

(20)

We define the matrix T as follows:

$$T:=\left[\begin{array}{cc}{I}_n& 0\\ Y& -Y\end{array}\right].$$

Pre-multiplying by $\mathrm{block-diag}(T,T)$, and post-multiplying by $\mathrm{block-diag}(T^{\top },T^{\top })$, we can obtain

$$\left[\begin{array}{cc}\overline{\beta }TP{T}^{\top }& *\\ TP\overline{A}{T}^{\top }& TP{T}^{\top }\end{array}\right]\succ 0,$$

(21)

where

$$\begin{array}{cc}\hfill TP{T}^{\top }& =\left[\begin{array}{cc}X& I\\ I& Y\end{array}\right],\hfill \\ \hfill TP\overline{A}{T}^{\top }& =\left[\begin{array}{cc}XA+W_{1}C& W_3\\ A& AY-B{W}_2\end{array}\right]\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill W_1& =Z{B}_c,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill W_2& =C_{c}Y,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill W_3& =XAY+Z{B}_{c}CY-XB{C}_{c}Y-Z{A}_{c}Y.\hfill \end{array}$$

(25)

From (21), we can obtain (15). From (15), $P\succ 0$ is guaranteed. In addition, from (22)–(24), we can obtain (16)–(18).

Finally, from (17), if the matrix $W_1$ is column-sparse, then the matrix $B_c$ is also column-sparse. In a similar way, from (18), if the matrix $W_2$ is row-sparse, then the matrix $C_c$ is also row-sparse. Hence, we introduce the objective function (14), which includes decision variables.

This completes the proof. □

4.2. Solution Method for Problem 2

Next, we derive a solution method for Problem 2.

To reduce the number of communication channels, we use the block-sparse reconstruction problem for matrices. To realize that the number of communication channels is as few as possible, we consider finding a controller (10), minimizing

$$\sum _{i=1}^q{w}_{\left(i\right)}{∥\left[{\mathrm{Col}}_{{\mathcal{I}}_i^s}\left(B_c\right){\left({\mathrm{Row}}_{{\mathcal{I}}_i^a}\left(C_c\right)\right)}^{\top }\right]∥}_{c_1}$$

(26)

under the condition that the closed-loop system is asymptotically stable, where $w_{\left(i\right)}\ge 0$ is a given weight that represents priorities of communication channels.

Then, we can obtain the following theorem.

Theorem 2.

Problem 2 is reduced to the LMI optimization problem in which the objective function in Problem 3 is replaced with

$$\sum _{i=1}^q{w}_{\left(i\right)}{∥\left[{\mathrm{Col}}_{{\mathcal{I}}_i^s}\left(W_1\right){\left({\mathrm{Row}}_{{\mathcal{I}}_i^a}\left(W_2\right)\right)}^{\top }\right]∥}_{c_1}$$

(27)

Proof. 

From (17), if the matrix $W_1$ is column-sparse, then the matrix $B_c$ is also column-sparse. In a similar way, from (18), if the matrix $W_2$ is row-sparse, then the matrix $C_c$ is also row-sparse. Hence, the objective function (25) can be rewritten as (26), which includes decision variables. □

5. Numerical Example

We present two numerical examples to demonstrate the proposed method.

5.1. Example 1

Suppose that the coefficient matrices A, B, and C in (9) are given by

$$\begin{array}{cc}\hfill A& =\left[\begin{array}{cccc}1.03& 0.6& 0.2& 0.2\\ 0& 0.7& 0& 0.8\\ 0.5& 0.4& 1.01& 0.2\\ 0.1& 0.2& 0.3& 0.8\end{array}\right],B=\left[\begin{array}{cc}1.1& 0.9\\ 0.6& -0.6\\ -0.5& 1.2\\ 0.3& 0.6\end{array}\right],\hfill \\ \hfill C& =\left[\begin{array}{cc}{I}_2& 0_{2,2}\end{array}\right].\hfill \end{array}$$

The parameter $\beta$ in (13) is given by $\beta =0.1$. Suppose also that q (the number of communication channels) is given by $q=2$, and ${\mathcal{I}}_1^a=\left\{1\right\}$, ${\mathcal{I}}_1^s=\left\{1\right\}$, ${\mathcal{I}}_2^a=\left\{2\right\}$, and ${\mathcal{I}}_2^s=\left\{2\right\}$ are given. In other words, the communication channel 1 is used in transmission of the first input and output. The communication channel 2 is used in transmission of the second input and output.

First, consider solving Problem 1. Then, we can obtain

$$\begin{array}{cc}\hfill X& =1.0\times {10}^3\left[\begin{array}{cccc}8.48& -1.13& -3.72& -5.48\\ -1.13& 4.79& -1.06& -0.00\\ -3.72& -1.06& 3.41& 0.26\\ -5.48& -0.00& 0.26& 8.27\end{array}\right],\hfill \\ \hfill Y& =1.0\times {10}^3\left[\begin{array}{cccc}6.54& -3.81& -0.91& -0.33\\ -3.81& 9.39& -3.06& -3.44\\ -0.91& -3.06& 7.91& -1.81\\ -0.33& -3.44& -1.81& 4.12\end{array}\right],\hfill \\ \hfill W_1& =\left[\begin{array}{cc}-1.44& -0.00\\ -1.44& -0.00\\ -1.44& -0.00\\ -1.44& -0.00\end{array}\right],\hfill \\ \hfill W_2& =\left[\begin{array}{cccc}0.00& 0.00& 0.00& 0.00\\ 1.44& 1.44& 1.44& 1.44\end{array}\right],\hfill \\ \hfill W_3& =\left[\begin{array}{cccc}490.06& -980.10& 40.52& 473.63\\ -477.73& 930.80& -9.54& -460.47\\ -341.44& 700.31& -7.04& -357.93\\ 255.72& -526.64& -45.31& 301.49\end{array}\right]\hfill \end{array}$$

From these matrices, we can obtain coefficient matrices of the controller (10):

$$\begin{array}{cc}\hfill A_c& =\left[\begin{array}{cccc}-0.50& 0.23& -0.08& -0.26\\ -0.34& 1.08& 0.29& 1.27\\ -1.39& -0.11& 0.62& -0.43\\ -0.87& -0.05& 0.11& 0.49\end{array}\right],\hfill \\ \hfill B_c& =\left[\begin{array}{cc}-1.25& -0.00\\ -0.63& -0.00\\ -1.51& -0.00\\ -0.78& -0.00\end{array}\right],\hfill \\ \hfill C_c& =\left[\begin{array}{cccc}0.00& 0.00& 0.00& 0.00\\ 0.40& 0.54& 0.41& 0.66\end{array}\right].\hfill \end{array}$$

From this result, we see that $B_c$ is column-sparse and $C_c$ is row-sparse. It is theoretically guaranteed that the obtained controller can stabilize the closed-loop system. However, in implementing the controller, the second column of $B_c$ and the first row of $C_c$ must be replaced with a zero vector and a zero row vector, respectively. Then, by a numerical simulation from time 0 to 80 (the state almost converges to the origin by time 80), we can confirm that owing to the truncation error, (13) with $\beta =0.1$ does not hold, and (13) with $\beta =0.99$ holds. Although a sparse controller can be obtained, two communication channels must be used.

Next, consider solving Problem 2. We set $w_{\left(1\right)}=1$ and $w_{\left(2\right)}=10$ (i.e., the channel 1 is used preferentially). Then, we can obtain

$$\begin{array}{cc}\hfill X& =1.0\times {10}^4\left[\begin{array}{cccc}1.11& -0.24& -0.60& -0.43\\ -0.24& 0.73& -0.02& -0.17\\ -0.60& -0.02& 0.68& -0.34\\ -0.43& -0.17& -0.34& 1.48\end{array}\right],\hfill \\ \hfill Y& =1.0\times {10}^4\left[\begin{array}{cccc}1.06& -0.16& -0.55& -0.17\\ -0.16& 0.72& -0.54& -0.15\\ -0.55& -0.54& 1.49& -0.14\\ -0.17& -0.15& -0.14& 0.31\end{array}\right],\hfill \\ \hfill W_1& =\left[\begin{array}{cc}-1.26& -0.00\\ -1.26& -0.00\\ -1.26& -0.00\\ -1.26& -0.00\end{array}\right],\hfill \\ \hfill W_2& =\left[\begin{array}{cccc}1.26& 1.26& 1.26& 1.26\\ 0.00& 0.00& 0.00& 0.00\end{array}\right],\hfill \\ \hfill W_3& =1.0\times {10}^3\left[\begin{array}{cccc}-0.97& 1.65& 0.46& -1.03\\ 1.04& -1.32& -0.87& 0.98\\ 1.30& -1.91& -0.98& 1.36\\ -1.78& 2.09& 1.85& -1.75\end{array}\right]\hfill \end{array}$$

From these matrices, we can obtain coefficient matrices of the controller (10),

$$\begin{array}{cc}\hfill A_c& =\left[\begin{array}{cccc}-0.68& -0.01& -0.26& -0.54\\ -0.88& 0.36& -0.25& 0.39\\ -0.66& 0.86& 1.36& 0.77\\ -0.78& 0.07& 0.21& 0.65\end{array}\right],\hfill \\ \hfill B_c& =\left[\begin{array}{cc}-1.25& -0.00\\ -0.63& -0.00\\ -1.51& -0.00\\ -0.78& -0.00\end{array}\right],\hfill \\ \hfill C_c& =\left[\begin{array}{cccc}0.49& 0.65& 0.49& 0.80\\ 0.00& 0.00& 0.00& 0.00\end{array}\right].\hfill \end{array}$$

From this result, we see that $B_c$ is column-sparse and $C_c$ is row-sparse. In implementing the controller, the second column of $B_c$ and the second row of $C_c$ are replaced with a zero vector and a zero row vector, respectively. Then, by a numerical simulation from time 0 to 80, we can confirm that (13) with $\beta =0.1$ holds. In this case, the effect of the truncation error is sufficiently small. In addition, only one communication channel is required. Thus, communication channels used in control can be reduced by using block-sparse reconstruction.

5.2. Example 2

Consider the benchmark model HE3 in [42] as a more large-scale example. Discretizing this model, we can obtain A, B, and C in (9) as follows:

$$\begin{array}{cc}\hfill A& =\left[\begin{array}{cccccccc}0.99& 0.00& -0.01& -0.00& -0.03& -0.00& -0.98& -0.00\\ -0.01& 0.94& 0.03& -0.00& -0.02& 0.05& 0.00& -0.00\\ 0.00& -0.00& 0.97& 0.00& 0.02& 0.00& -0.00& 0.00\\ 0.00& -0.00& -0.03& 0.99& 0.00& 0.02& -0.00& 0.97\\ 0.00& -0.00& -0.07& -0.00& 0.92& 0.01& -0.00& -0.00\\ -0.00& -0.00& 0.01& 0.00& -0.02& 0.92& 0.00& 0.00\\ 0.00& -0.00& 0.09& 0.00& 0.00& 0.00& 1.00& 0.00\\ 0.00& -0.00& -0.00& -0.00& 0.09& 0.00& -0.00& 0.99\end{array}\right],\hfill \\ \hfill B& =\left[\begin{array}{cccc}0.00& 0.01& -0.00& -0.00\\ -0.10& 0.01& 0.00& -0.00\\ -0.00& -0.00& 0.00& -0.00\\ -0.00& 0.00& 0.01& 0.01\\ -0.00& 0.00& 0.02& 0.01\\ 0.01& 0.00& 0.00& -0.03\\ -0.00& -0.00& 0.00& -0.00\\ -0.00& 0.00& 0.00& 0.00\end{array}\right],\hfill \\ \hfill C& =\left[\begin{array}{cc}{I}_6& 0_{6,2}\end{array}\right].\hfill \end{array}$$

The parameter $\beta$ in (13) is given by $\beta =0.15$. We suppose that the structure of a SAN is given by Figure 1. Then, as explained in Section 3.3, q is given by $q=3$. We also set the following: ${\mathcal{I}}_1^a=\{1,2\}$, ${\mathcal{I}}_1^s=\{1,2\}$, ${\mathcal{I}}_2^a=\left\{3\right\}$, ${\mathcal{I}}_2^s=\{3,4\}$, ${\mathcal{I}}_3^a=\left\{4\right\}$, and ${\mathcal{I}}_3^s=\{5,6\}$.

First, consider solving Problem 1. Then, we can obtain

$$\begin{array}{cc}\hfill A_c& =\left[\begin{array}{cccccccc}0.56& -0.07& -0.05& -0.03& -0.16& -0.15& -0.87& -0.20\\ 0.05& 0.88& 0.18& 0.00& -0.35& 0.13& -1.03& -0.88\\ 0.04& 0.01& 0.81& 0.01& 0.05& 0.02& -0.31& 0.08\\ -0.05& 0.14& -0.29& 0.62& 0.27& 0.13& -0.25& 1.44\\ -0.00& 0.00& -0.09& -0.03& 0.73& -0.00& -0.21& -0.49\\ 0.05& -0.02& 0.13& 0.01& 0.04& 0.88& 0.06& 0.06\\ 0.07& 0.01& 0.16& 0.01& 0.02& 0.02& 1.10& 0.03\\ -0.02& 0.02& -0.06& -0.05& 0.15& 0.01& -0.06& 1.11\end{array}\right],\hfill \\ \hfill B_c& =\left[\begin{array}{cccccc}-0.49& -0.03& 0.00& -0.00& -0.00& 0.00\\ 0.00& -0.07& -0.00& 0.04& 0.00& -0.00\\ 0.03& 0.00& -0.00& -0.00& 0.00& -0.00\\ -0.03& 0.12& 0.00& -0.40& -0.00& 0.00\\ -0.02& 0.00& 0.00& -0.00& -0.00& 0.00\\ 0.03& -0.00& -0.00& 0.01& 0.00& -0.00\\ 0.09& 0.01& -0.00& 0.00& 0.00& -0.00\\ -0.01& 0.02& 0.00& -0.06& -0.00& 0.00\end{array}\right],\hfill \\ \hfill C_c& =\left[\begin{array}{cccccccc}0.87& 0.06& -2.42& -0.11& -1.60& 0.73& -15.10& -4.52\\ 1.83& 0.27& -28.12& 0.86& 2.89& 1.15& -50.71& 11.44\\ -1.38& 0.05& 0.60& 1.33& 7.75& 1.05& 13.55& 20.94\\ 0.90& -0.31& 0.82& -0.05& 1.82& -0.96& -4.71& 2.28\end{array}\right].\hfill \end{array}$$

From this result, we see that $B_c$ is column-sparse but $C_c$ is not row-sparse. Also in this example, it is theoretically guaranteed that the obtained controller can stabilize the closed-loop system. In implementing the controller, the third, fifth, and sixth columns of $B_c$ are replaced with a zero vector. Then, by a numerical simulation from time 0 to 80, we can confirm that (13) with $\beta =0.15$ holds. In this case, the effect of the truncation error is sufficiently small. However, since $C_c$ is not row-sparse, all communication channels must be used.

Next, consider solving Problem 2. We suppose that using the channel 1 is avoided as much as possible. Based on this purpose, we set $w_{\left(1\right)}=100,000$ and $w_{\left(2\right)}=w_{\left(3\right)}=1$. Then, we can obtain

$$\begin{array}{cc}\hfill A_c& =\left[\begin{array}{cccccccc}1.05& 0.01& -22.22& -0.24& -37.37& 12.90& -2.78& 0.34\\ 0.00& 0.94& 1.43& 0.19& 0.16& -0.72& -0.59& 0.08\\ 0.01& -0.00& 0.62& -0.02& -0.07& 0.09& -0.17& 0.00\\ -0.08& -0.03& 5.69& 0.69& 0.46& -0.26& 4.85& -0.40\\ 0.15& -0.05& 3.26& -0.07& 0.56& 0.05& 0.90& -1.49\\ 0.67& 0.00& -12.57& 0.18& 0.25& 0.80& -18.62& 2.53\\ -0.00& -0.00& 0.96& 0.01& 2.10& -0.62& 1.13& -0.02\\ -0.01& -0.00& 0.38& -0.03& 0.19& -0.06& 0.04& 0.93\end{array}\right],\hfill \\ \hfill B_c& =\left[\begin{array}{cccccc}-0.00& -0.00& -20.96& -0.27& -37.32& 12.91\\ -0.00& -0.00& 1.78& 0.18& 0.19& -0.77\\ -0.00& 0.00& -0.28& -0.02& -0.08& 0.09\\ 0.00& 0.00& 1.03& -0.23& 0.51& -0.30\\ -0.00& -0.00& 0.02& -0.00& -0.15& 0.04\\ 0.00& 0.00& -0.04& -0.04& 0.49& -0.08\\ 0.00& 0.00& 0.77& 0.01& 2.10& -0.62\\ 0.00& 0.00& 0.22& -0.02& 0.11& -0.06\end{array}\right],\hfill \\ \hfill C_c& =\left[\begin{array}{cccccccc}0.00& -0.00& -0.00& 0.00& -0.00& -0.00& -0.00& 0.00\\ 0.00& 0.00& -0.00& 0.00& -0.00& -0.00& -0.00& 0.00\\ -16.93& 2.17& 42.85& -0.28& 13.03& 0.78& 243.70& 27.99\\ 15.84& 0.30& -316.42& 5.19& -4.65& -0.99& -456.53& 66.43\end{array}\right].\hfill \end{array}$$

From this result, we see that $B_c$ is column-sparse and $C_c$ is row-sparse. In implementing the controller, the first and second columns of $B_c$ and the first and second rows of $C_c$ are replaced with a zero vector and a zero row vector, respectively. Then, by a numerical simulation from time 0 to 80, we can confirm that (13) with $\beta =0.15$ holds. In the obtained controller, it is not necessary to use the channel 1. Thus, we can obtain a controller satisfying the purpose.

6. Conclusions

In this paper, we proposed an LMI-based design method for structured sparse output feedback controllers. We considered two types of controllers: structured sparse controllers and structured block-sparse controllers. Through two numerical examples, we presented the effectiveness of the proposed method. The proposed method is useful for control of SANs with many actuators and sensors.

In this paper, based on sparse modeling, we focused on reducing the number of sensors and actuators and the number of communication channels. The proposed method enables us to design appropriate structured controllers. On the other hand, in SANs, there are several technical issues. From the viewpoint of control design, it is important to reduce communications from/to the controller to/from actuators/sensors. In future works, it is important to combine the proposed method with event-triggered and self-triggered control methods [45], which are typical control method for reducing communications. It is also important to consider switching control such as PWM controllers. Moreover, applying the proposed method to real large-scale systems such as smart grids and autonomous driving systems is an important future effort.