###### 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. sparse reconstruction; block-sparse reconstruction; linear matrix inequality (LMI); output feedback controller; sensor and actuator network (SAN) +
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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:  
1. For SANs, the design problems of structured sparse output-feedback controllers are formulated using sparse reconstruction and block-sparse reconstruction. 2. Two design problems are reduced to an LMI optimization problem. 3. 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 \(\|x\|_{i}\) denote the \(l_{i}\) norm of \(x\). The symmetric matrix \( \begin{bmatrix}A&B^{\top}\\ B&C\end{bmatrix} \) is denoted by \( \begin{bmatrix}A&*\\ B&C\end{bmatrix} \).