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This article focuses on the design of proportional‐derivative (PD) controllers for positive linear systems in the continuous‐time domain, which is a well‐known open problem in positive systems theory. The main objective is to design a PD controller subject to interval gain variations, which simultaneously preserves the closed‐loop system stability and positivity. The provided results fill the literature gap by considering both the PD controller structure and positivity. Although various feedback control strategies of positive systems have been developed recently, the work provides, for the first time, a systematic and tractable framework for finding nonfragile PD controller gains for positive stabilization. Finally, the theory and algorithm performance are evaluated and illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

This article considers the problem of designing lead, lag, and lead-lag compensators for multi-input–multi-output (MIMO) linear systems under the $H_{\infty }$ performance measure. This is the first time that the lead, lag and lead-lag compensators are generalized to the MIMO cases by preserving their classical compensator structures. Theoretical results on the stability analysis and synthesis of MIMO systems under the $H_{\infty }$ control performance with the proposed lead, lag, and lead-lag compensators are obtained. Then relevant algorithms for designing lead, lag, and lead-lag compensators are provided to determine the compensator parameters. Differing from traditional design methods, which mostly rely on some trial-and-error procedures, the proposed methods are algorithmic and the compensators can be synthesized systematically. Illustrative examples are used to demonstrate the effectiveness and advantages of the proposed methods. [ABSTRACT FROM AUTHOR]

This article addresses the real‐time reachable set problem on nonlinear time‐delay switched singular systems with state jump, which is subject to initial state and peak‐bound disturbance. Initially, based on the system state decomposition method, a state jump lemma is proposed, which combines the nonlinear function and the system state at the switching instant. Then, a real‐time bounded lemma is developed, premised on using the idea of average dwell‐time and analyzing subinterval function changes. Afterwards, the sufficient conditions for estimating the real‐time and nonreal‐time reachable set are established, which ensures that the reachable state of system can be restricted within a real‐time or nonreal‐time closed set. Finally, simulation results of the typical Direct Current boost chopper are performed to exhibit the effectiveness of the devised estimation scheme for the real‐time reachable set of system. [ABSTRACT FROM AUTHOR]

In this brief, the problem of the $L_{1}/L_{-}$ fault detection for positive systems is revisited. In the existing literature, the $L_{1}$ -gain and $L_{-}$ index for positive systems are often characterized separately, and thus their linear programming descriptions involve different Lyapunov vectors. This casts the fault detection filter design as a bilinear optimization problem. To circumvent this obstacle, we first show that, for an externally positive system, the $L_{1}$ -gain and $L_{-}$ index are determined, respectively, by the largest and smallest column sums of the static gain matrices. Based on this fact, an exact characterization is given for the $L_{1}/L_{-}$ index for positive systems in terms of a linear program with equality constraints. The new characterization only involves one single Lyapunov vector, and thus renders the fault detection filter design problem convex. In addition, we find that the maximum fault sensitivity (characterized by the $L_{-}$ index from the fault to the residual) that can be achieved by the filter design approach is proportional to the required upper bound on the $L_{1}$ -gain from the disturbance to the residual. Finally, an illustrative example of a positive electric circuit is presented to show the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this brief, we study the decentralized H 2 state-feedback control problem for networked discrete-time systems with positivity constraint. This problem (for a single positive system), raised recently in the area of positive systems theory, is known to be challenging due to its inherent nonconvexity. In contrast to most works, which only provide sufficient synthesis conditions for a single positive system, we study this problem within a primal-dual scheme, in which necessary and sufficient synthesis conditions are proposed for networked positive systems. Based on the equivalent conditions, we develop a primal-dual iterative algorithm for solution, which helps prevent from converging to a local minimum. In the simulation, two illustrative examples are employed for verification of our proposed results.