L1-gain Control Research Articles

L1-gain control for 2D delayed positive continuous Markov jumping systems

This study is focused on analyzing the L1-stochastic internal stability and the design of an L1-gain controller for two-dimensional (2D) continuous positive Markov jumping systems (PMJSs) with constraints on the inputs and states. An algorithm is developed to explicitly determine the state feedback control law with the optional L1-gain performance. First, by constructing a co-positive stochastic Lyapunov function for the positive system and establishing the equation for the Markov states' mathematical expectation, several sufficient and necessary conditions for L1-stochastic internal stability are derived. This analysis indicates that 2D PMJSs with directional delays are influenced by sizes of the system matrices and magnitude of delays, and transition matrix. Second, a method for calculating the L1-gain is formulated utilizing linear programming (LP), thus, an L1-gain controller is designed to guarantee that the closed-loop system is positive and L1-stochastically internally stable, while achieving optimal L1-gain performance. Two numerical and practical examples are considered, and the influence of delays and transition probability matrices on the L1-gain is specified to validate the preceding theoretical findings.

Event-Triggered L1-Gain Control of Nonlinear Positive Switched Systems

This paper is concerned with the event-triggered L1-gain control of a class of nonlinear positive switched systems. First, an event-triggering condition in the form of 1-norm is presented for the systems. By virtue of the event-triggering strategy, the original system is transformed into an interval uncertain system. An event-triggered L1-gain controller is designed by decomposing the controller gain matrix into the sum of nonnegative and non-positive components. Under the design controller, the resulting closed-loop systems are positive and L1-gain stable. The obtained approach is developed for the systems subject to input saturation. All presented conditions are solvable in terms of linear programming. Finally, two examples are provided to verify the effectiveness of the design.

Robust finite-time control of impulsive positive systems under ℒ1-gain performance

This paper is concerned with robust finite-time ℒ1-gain control problem for impulsive positive systems (IPSs). By adopting the average impulsive interval technique, sufficient conditions ensuring the finite-time boundedness of IPSs under ℒ1-gain characterization are formulated. The design of a feedback controller is also addressed to make the closed-loop system be positive, finite-time bounded (FTB), and have ℒ1-gain characterization. Results are presented in the form of linear programming (LP) inequalities. Finally, a numerical example is given to demonstrate the efficiency of the proposed design.

On [formula omitted]-gain control of 2-D positive Roesser systems with directional delays: Necessary and sufficient conditions

In this paper, the problem of ℓ1-gain control is addressed for two-dimensional (2-D) positive Roesser systems with directional delays. A complete solution to the design problem is presented for the first time. Specifically, by exploiting a new approach based on 2-D Z-transform, an exact value of ℓ1-induced norm of the input–output operator is obtained. This analytical result is utilized to derive necessary and sufficient conditions under which a 2-D positive system with delays is asymptotically stable with a prescribed ℓ1-gain performance. Then, by utilizing a vertex optimization approach, necessary and sufficient conditions are established for the existence of a state-feedback controller (SFC) that makes the closed-loop system positive and stable subject to ℓ1-induced performance. The proposed synthesis conditions are formulated in terms of linear programming (LP), which can be effectively solved by various convex algorithms to minimize the ℓ1-performance index.

Heart Rate Variability as a Prognostic Factor for Cancer Survival - A Systematic Review

This paper addresses the problems of robust stability analysis and L1-gain controller synthesis for uncertain impulsive positive systems. First, by employing the idea of impulse interval partitioning, an impulse-time-dependent discretized copositive Lyapunov function is proposed to analyze the robust stability and L1-gain performance of the considered system without control inputs, and several stability conditions and L1-gain criteria are respectively derived. Subsequently, a sufficient condition is formulated for the existence of state-feedback controllers with which not only the positivity and robust uniform asymptotic stability of the resulting closed-loop system are guaranteed, but also a prescribed L1-gain performance is satisfied simultaneously. Furthermore, to make the controller synthesis problem numerically tractable, we propose an iterative convex optimization algorithm to compute the desired controller parameters. Finally, three numerical examples and two realistic examples are presented to show the effectiveness and applicability of the proposed methodology.

Robust stability analysis and controller synthesis for uncertain impulsive positive systems under [formula omitted]-gain performance

This paper addresses the problems of robust stability analysis and L1-gain controller synthesis for uncertain impulsive positive systems. First, by employing the idea of impulse interval partitioning, an impulse-time-dependent discretized copositive Lyapunov function is proposed to analyze the robust stability and L1-gain performance of the considered system without control inputs, and several stability conditions and L1-gain criteria are respectively derived. Subsequently, a sufficient condition is formulated for the existence of state-feedback controllers with which not only the positivity and robust uniform asymptotic stability of the resulting closed-loop system are guaranteed, but also a prescribed L1-gain performance is satisfied simultaneously. Furthermore, to make the controller synthesis problem numerically tractable, we propose an iterative convex optimization algorithm to compute the desired controller parameters. Finally, three numerical examples and two realistic examples are presented to show the effectiveness and applicability of the proposed methodology.

Stability, L1-gain analysis and asynchronous L1-gain control of uncertain discrete-time switched positive linear systems with dwell time

In this paper, the stability, L1-gain analysis and asynchronous L1-gain control problems of uncertain discrete-time switched positive linear systems (DSPLSs) with dwell time are investigated. First, several convex and non-convex conditions on dwell time stability of DSPLSs with interval and polytopic uncertainties are presented, and the relation between these conditions is revealed. Then, via a switched dwell-time-dependent co-positive Lyapunov functions (SDTLFs) approach, convex sufficient conditions on L1-gain analysis and asynchronous L1-gain control of DSPLSs with interval uncertainties are derived. Meanwhile, via the switched parameter-dwell-time-dependent co-positive Lyapunov functions (SPDTLFs) approach, the L1-gain analysis and asynchronous L1-gain controller design problems of DSPLSs with polytopic uncertainties are also solved. The stability and L1-gain analysis results are given in terms of linear programming (LP). The controller design results are presented in terms of bilinear programming (BP), which can be solved with the help of iterative algorithm. At last, both numerical and practical examples are provided to show the effectiveness of the results.

Asynchronous L1-gain control of uncertain switched positive linear systems with dwell time

In this paper, dwell time (DT) stability, L1-gain performance analysis and asynchronous L1-gain controller design problems of uncertain switched positive linear systems (SPLSs) are investigated. Via a time-scheduled multiple linear co-positive Lyapunov function (TSMLCLF) approach, convex sufficient conditions of DT stability and L1-gain performance of SPLSs with interval and polytopic uncertainties are presented. Furthermore, by utilizing the feature that the TSMLCLF keeps decreasing even if the controller is running asynchronously with the system, the asynchronous L1-gain controller design problem of SPLSs with interval and polytopic uncertainties is investigated. Convex sufficient conditions of the existence of time-varying asynchronous state-feedback controller which can ensure the closed-loop system's positivity, stability and L1-gain performance are established, and the controller gain matrices can be calculated instantaneously online. The obtained L1-gain in the paper is standard. All the results are presented in terms of linear programming. A practical example is provided to show the effectiveness of the results.

Stability analysis and L1-gain controller synthesis of switched positive T–S fuzzy systems with time-varying delays

Abstract This paper is concerned with the stability analysis and L1-gain controller synthesis problem of continuous-time switched positive T–S fuzzy systems with time-varying delays. Some sufficient conditions guaranteeing the stability and a desired L1-gain performance of an autonomous switched positive T–S fuzzy system are first presented. The sufficient conditions are developed by adopting the mode-dependent average dwell time (MDADT) technique. Then, fuzzy controllers of two classes of switched positive T–S fuzzy systems are designed without using iterative algorithm. A simulation example is presented to show the effectiveness of the obtained theoretical results.

Stability and L1-gain control for positive impulsive switched systems with mixed time-varying delays

Stability and L1-gain control for positive impulsive switched systems with mixed time-varying delays

Investigating the effects of time-delays on stochastic stability and designing l1-gain controllers for positive discrete-time Markov jump linear systems with time-delay

This paper is concerned with stochastic stability, l1-gain performance analysis and positivity-preserving l1-gain controller design for a positive discrete-time Markov jump linear system with time-delay. First, necessary and sufficient conditions for stochastic stability of the system are derived by constructing a linear co-positive stochastic Lyapunov functional and establishing a system equation whose state variables consist of the mathematical expectation of the markovianized states and whose coefficient matrices depend on time-delay and the transition probability. It is revealed that stochastic stability of the positive discrete-time Markov jump linear system with time-delay is influenced by the size of time-delay and it is demonstrated by an example that the effect of time-delay on stochastic stability can be either positive or negative. Second, exact computation on an l1-gain index of a stochastically stable positive discrete-time Markov jump linear system with time-delay is presented, and a necessary and sufficient condition for the l1-gain performance is derived in the form of linear programming. Third, an iterative algorithm is proposed to design a positivity-preserving l1-gain controller, and in single-input case, an optimal controller is obtained analytically such that the closed-loop system achieves the minimal l1-gain performance. Then, a modified pest’s structured population dynamic model is developed to illustrate the effectiveness of the designed method.

State Feedback $$L_1$$ L 1 -Gain Control of Positive 2-D Continuous Switched Delayed Systems Via State-Dependent Switching

This paper investigates the stability and $$L_1$$L1-gain control of two-dimensional (2-D) continuous positive switched delayed systems. Firstly, by constructing an appropriate co-positive Lyapunov---Krasovskii functional, a sufficient condition for asymptotical stability of the system under consideration is derived. Secondly, $$L_1$$L1-gain performance analysis of the underlying system is investigated. Thirdly, a design methodology for state feedback controller is proposed to ensure that the closed-loop system is asymptotically stable with $$L_1$$L1-gain performance. Finally, an example is provided to show the effectiveness of the proposed method.

Stability and [formula omitted]-gain controller design for positive switched systems with mixed time-varying delays

This paper investigates the problems of stability and L1-gain controller design for positive switched systems with mixed time-varying delays. The mixed time-varying delays are presented in the forms of discrete delay and distributed delay. The purpose of this paper is to design a class of switching signals and a state feedback controller for the considered system such that the resulting closed-loop system is exponentially stable with L1-gain performance. By constructing an appropriate co-positive type Lyapunov–Krasovskii functional and using the average dwell time approach, we propose a sufficient condition to ensure the exponential stability with weighted L1-gain performance for the system. On the basis of the obtained results, an effective method for designing such controller is established. Finally, a numerical example is presented to demonstrate the feasibility of the results obtained.