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This paper considers the proportional–integral–derivative (PID) control for continuous-time positive systems. A three-stage strategy is introduced to design the PID controller. In the first stage, the proportional and integral components of the PID control are designed. A matrix decomposition approach is used to describe the gain matrices of the proportional and integral components. The positivity and stability of the closed-loop systems without the derivative component of PID control are achieved by the properties of a Metzler and Hurwitz matrix. In the second stage, a non-negative inverse matrix is constructed to maintain the Metzler and Hurwitz properties of the closed-loop system matrix in the first stage. To deal with the inverse of the derivative component of PID control, a matrix decomposition approach is further utilized to design a non-negative inverse matrix. Then, the derivative component is obtained by virtue of the designed inverse matrix. All the presented conditions can be solved by virtue of a linear programming approach. Furthermore, the three-stage PID design is developed for a state observer-based PID controller. Finally, a simulation example is provided to verify the effectiveness and validity of the proposed design. [ABSTRACT FROM AUTHOR]

This paper presents a new hybrid dynamic event‐triggered control framework on positive Markovian jump systems in continuous‐ and discrete‐time contexts subject to actuator fault. First, a new hybrid dynamic event‐triggering mechanism consists of two types of conditions: One is a dynamic event‐triggering condition associated to dynamic variable and the other one is a static event‐triggering condition. Under the new triggering mechanism, an interval approach is introduced to assist the control design, for which the lower and upper bounds of the original system are obtained. A nonfragile controller is proposed for stochastic stability of the underlying continuous‐time system with actuator fault by virtue of stochastic copositive Lyapunov functions. Under the designed controller, the positivity and stability of the system are achieved by guaranteeing the positivity and stability of the lower and upper bound systems, respectively. Moreover, a hybrid dynamic event‐triggered mechanism is established for discrete‐time systems. A nonfragile hybrid dynamic event‐triggered control is explored for discrete‐time positive Markovian jump systems. All the conditions can be solved by linear programming. Finally, two examples are given to verify the validity of the proposed design. [ABSTRACT FROM AUTHOR]

This paper is concerned with the event-triggered control of positive switched systems with randomly occurring state saturation. First, a vector-based event-triggering condition is established. Thus, interval and polytope approaches are used to deal with the positivity and stability of the systems and the saturation, respectively. An event-triggered controller and the corresponding domain of attraction are designed by decomposing gain matrix into the sum of non-negative and non-positive components. The proposed design approach is developed for the systems with input saturation. Finally, two examples are given to illustrate the effectiveness of the proposed design. [ABSTRACT FROM AUTHOR]

This technique note proposes two classes of functional and disturbance observers for positive systems with structural and non-structural disturbances, respectively. A positive functional observer is first proposed for positive systems by introducing the estimation of disturbance to the observer. By developing the disturbance observer technique, a positive disturbance observer is designed to supply the estimation of disturbance in the functional observer. Then, a new unknown input observer is constructed for positive systems. A matrix decomposition method is employed to design the observer gains. All conditions are described in terms of linear programming. The corresponding algorithms are addressed for computing the presented conditions. Finally, two examples are provided to verify the effectiveness of the theoretical findings. [ABSTRACT FROM AUTHOR]

Summary: This article investigates the asynchronous fault detection filter of nonlinear positive semi‐Markov jump systems with disturbances and actuator faults based on adaptive event‐triggering scheme. First, an adaptive event‐triggering strategy in the linear form is presented and a framework on the fault detection filter is established. Then, a hidden semi‐Markov model is introduced to describe the asynchronous phenomenon between the original system and the filter and thus an augmented fault detection system is constructed. Using stochastic Lyapunov functions and linear programming, the filter gains are determined for guaranteeing the positivity and stochastic stability with hybrid L1$$ {L}_1 $$ performance. Compared with existing results, the novelties of the proposed design lie in: (i) a new adaptive event‐triggered strategy is constructed, which improves the existing static one, (ii) an adaptive event‐triggered asynchronous filter is proposed for the underlying systems, (iii) hybrid gain performances of the filter are obtained with respect to the disturbance and fault, and (iv) a linear programming‐based computation method is given to determine the filter gains. Finally, one illustrate example is provided to verify the validity of the results in this article. [ABSTRACT FROM AUTHOR]

This paper investigates the event-triggered asynchronous filter of positive switched systems with state saturation using linear programming and multiple Lyapunov functions. First, a filter is constructed for continuous-time positive switched systems. Under the asynchronous switching law, an error system is proposed with respect to positive switched systems and their filters, where the state saturation term is described in a polytopic form by virtue of the saturation property. A novel event-triggering condition is addressed based on a 1-norm inequality. Under the event-triggering condition, the error system is transformed into an interval system with lower and upper bounds. By using multiple Lyapunov functions and linear programming, the positivity and stability of the error system are achieved by considering the corresponding properties of the lower and upper bounds, respectively. Then, the event-triggered l1-gain filter and nonfragile filter are also proposed for the systems with disturbances. Moreover, the presented filter framework is extended to the discrete-time case. Finally, two examples are given to verify the effectiveness of the proposed filters. [ABSTRACT FROM AUTHOR]

This article proposes a new random event‐triggering scheme to design filter of positive semi‐Markovian jump systems with intermittent sensor faults. To obtain a balance of saving network resources and improving system performance, a random event‐triggering mechanism consisting of time‐ and event‐triggering modes is constructed. A Bernoulli distribution‐based stochastic variable is introduced to orchestrate the switching of two triggering modes. A concept of hybrid gain performance is presented to guarantee that the filter can attenuate the affect from the disturbance and sensor faults. Then, a random event‐triggered filter is designed for positive semi‐Markovian jump systems subject to intermittent sensor faults. Under the designed filter, hybrid gain performance is achieved. Furthermore, the proposed approach is extended to the systems with stochastically incorrect sensor measurement. A linear programming method is employed for describing and computing the presented conditions. Finally, the validity of the results is verified via two examples. [ABSTRACT FROM AUTHOR]

This paper is concerned with the non-fragile reliable control of positive Markovian jump systems with actuator saturation based on event-triggered mechanism. First, an event-triggering condition is given in the form of 1-norm. Using a stochastic co-positive Lyapunov function, a design approach is proposed for the non-fragile controller gain and the corresponding auxiliary feedback gain. Under the designed controller, the closed-loop system is positive and stochastically stable even if actuator faults occur. A cone is constructed as the invariant set of the systems. All conditions are solvable in terms of linear programming. Finally, two examples are provided to verify the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

This article deals with event-triggered filter design of positive systems subject to state saturation. An event-triggering condition in the form of 1-norm is established based on the error and the measurable output. Under the presented event-triggering condition, the filter systems are transformed into interval uncertain systems. The positivity of the filter and error dynamics is achieved by considering the lower bound of the interval systems. Using a linear copositive Lyapunov function, the stability is guaranteed by considering the upper bound of the interval systems. By virtue of linear programming, an $\ell _1$ -gain filter is designed for the systems subject to exogenous disturbance. Then, multiplicative and additive gain fluctuations are introduced. Two classes of nonfragile filters are proposed, and a cone is constructed to guarantee that the states of the systems with state saturation can keep inside it. Finally, two examples are given to illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This article is concerned with the event‐triggered nonfragile reliable control of nonlinear positive semi‐Markovian jump systems with randomly occurring faults. The randomly occurring actuator fault, describing the phenomenon of the actuator fault appearing in a random way, is assumed to obey a semi‐Markovian process different from the jump process of the systems. The nonlinear function is located in a sector. Using a stochastic copositive Lyapunov function integrated with linear programming, an event‐triggered control strategy is employed to deal with the stochastic stabilization of the underlying systems, where the transition rates satisfy a polytopic uncertain condition. Nonfragile reliable controllers with additive and multiplicative perturbations are proposed for the considered systems, respectively. Compared with existing results, the obtained approach establishes a more general fault description and proposes a more practicable control design. Finally, the novelties of the proposed approach are verified via two examples. [ABSTRACT FROM AUTHOR]

This paper presents a distributed fault detection filter (FDF) for positive semi-Markovian jump systems (PSMJSs) based on an adaptive event-triggered mechanism (AETM) scheme. A novel AETM is first proposed for the systems. A weighted fault model is introduced to assist the estimation of the original fault. Utilizing stochastic copositive Lyapunov function (CLF), a distributed FDF whose parameters are designed by virtue of linear programming (LP), is proposed. Under the designed filter, the positivity and stochastic stability with L 1 / L _ -gain performance of a fault detection system is reached. The contribution of this paper contains three aspects: (i) An AETM is constructed for the considered systems, (ii) A distributed FDF is designed for the systems, and (iii) The hybrid L 1 / L _ -gain performance is achieved with respect to the disturbance and the fault. Two illustrative examples are given for verifying the validity of the results. [ABSTRACT FROM AUTHOR]

This paper proposes an event-triggered control for positive Markov jump systems without/with input saturation. First, an event-triggering condition is established in the form of 1-norm for positive Markov jump systems. Under the event-triggering condition, the term related to error signal between the sample state and the actual state is transformed into a form of interval uncertainty. Then, the positivity and stability of the systems are discussed by using the lower and upper bounds of interval uncertain systems, respectively. An event-triggered control law for the underlying systems is proposed by decomposing controller gain matrix into non-positive and non-negative parts. For positive Markov jump systems with input saturation, a design approach to the controller gain matrices and the corresponding controller auxiliary gain is presented. All conditions are described in terms of linear programming. Furthermore, the obtained approaches are developed for the discrete-time case. Finally, the effectiveness of the proposed design is verified via three examples. [ABSTRACT FROM AUTHOR]

This paper proposes a reliable control of positive switched systems with random nonlinearities which may induce the security problem of the systems. The random nonlinearities are governed by stochastic variables obeying the Bernoulli distribution. A switched linear copositive Lyapunov function is employed for the systems. Using a matrix decomposition approach, the gain matrix of controller is formulated by the sum of nonnegative and non-positive components. A reliable controller is designed for positive switched systems with actuator faults by virtue of linear programming. Under the designed reliable controller, the systems can resist some possible security risks triggered by random nonlinearities and actuator faults. The obtained approach is developed for systems subject to exogenous disturbances. Finally, two examples are provided to verify the validity of the obtained results. • A criterion for the positivity of nonlinear systems with random nonlinearities is addressed. • A novel reliable control approach to positive switched systems with actuator faults is presented. • A linear programming is provided for computing the corresponding controller gain matrices. [ABSTRACT FROM AUTHOR]

Summary: This article investigates the stability analysis and control design of a class of nonlinear positive Markovian jump systems with randomly occurring actuator faults and saturation. It is assumed that the actuator faults of each subsystem are varying and governed by a Markovian process. The nonlinear term is located in a sector. First, sufficient conditions for stochastic stability of the underlying systems are established using a stochastic copositive Lyapunov function. Then, a family of reliable L1‐gain controller is proposed for nonlinear positive Markovian jump systems with actuator faults and saturation in terms of a matrix decomposition technique. Under the designed controllers, the closed‐loop systems are positive and stochastically stable with an L1‐gain performance. An optimization method is presented to estimate the maximum domain of attraction. Furthermore, the obtained results are developed for general Markovian jump systems. Finally, numerical examples are given to illustrate the effectiveness of the proposed techniques. [ABSTRACT FROM AUTHOR]

This study focuses on the problem of event‐triggered control for positive switched systems without/with input saturation in both continuous‐time and discrete‐time contexts. First, a 1‐norm based event‐triggering mechanism is established for continuous‐time positive switched systems. By means of the matrix decomposition technique, an event‐triggered controller for the systems is designed by decomposing controller gain matrix into non‐positive and non‐negative components. Under the designed controller, the resulting closed‐loop systems are positive and stable. For the systems with input saturation, the saturation term is transformed into interval form under the event‐triggering condition. Then, an event‐triggered controller gain matrix and a cone domain of attraction gain matrix are designed for the corresponding interval systems in terms of linear programming, respectively. Furthermore, the presented approaches are extend to discrete‐time positive switched systems without/with input saturation. Compared with existing results on positive switched systems, the designed event‐triggered controller can reduce sampling frequency and save resources. Finally, two numerical examples are provided to verify the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

Summary: This paper presents a nonfragile reliable control approach to positive switched systems with actuator faults and saturation. In order to guarantee the reliability of the controller, the controller gain matrix is chosen as the sum of a normal gain matrix and a gain perturbation matrix. A nonfragile reliable control is first proposed for the considered systems using a gain matrix decomposition technique. Then, the presented nonfragile control design approach is developed for the systems with exogenous disturbances. An approach to compute the normal gain matrix and the gain perturbation matrix is also provided. Under the obtained controller, the resulting closed‐loop systems are positive and L1‐gain stable. Meanwhile, all the states will stay inside a cone. All presented conditions are described via linear programming. Finally, two examples are provided to verify the effectiveness of the theory findings. [ABSTRACT FROM AUTHOR]

Abstract This paper investigates the saturation control for a class of switched nonlinear systems, where the nonlinearity function is restricted in a sector. The proposed approach is based on the theory of positive systems and a criterion is addressed to ensure the positivity property of the considered systems. The system states starting from a cone will remain in another cone. Under an average dwell time condition, a control design methodology is established in terms of linear programming. It is shown in this paper that the proposed approach can be applied for positive systems and also for a general class of nonlinear systems. Then, the saturation control synthesis of switched nonlinear systems with exogenous disturbances is addressed. Under the designed saturation controller, the states of each subsystem start from a cone and remain within it. Thus, the states of the switched systems start from a cone containing the origin and remain in another cone. Finally, the proposed methodology is applied to a nonlinear control system. [ABSTRACT FROM AUTHOR]

This paper presents the S-procedure characterization for the stabilization of positive switched linear systems and establishes the relationship between the S-procedure and its equivalent Lyapunov–Metzler inequalities. First, a piecewise linear co-positive Lyapunov function is constructed for positive switched linear systems. Under the Lyapunov function, the S-procedure stabilization for positive switched linear systems in the continuous-time context is explored under a min state switching law. The S-procedure conditions are formulated in the form of linear programming. Finally, an equivalence relationship between S-procedure and Lyapunov–Metzler inequalities is presented. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the controller design of uncertain positive systems. First, we decompose the feedback gain matrix Km×n into m×n non‐negative components and m×n non‐positive components. For the non‐negative components, each component contains only one positive element and the other elements are zero. Similarly, each non‐positive component contains only one negative element and the other ones are zero. Then, a simple and effective controller design approach of uncertain positive systems is proposed by incorporating the decomposed feedback gain matrix into the resulting closed‐loop systems and further applied to uncertain positive switched systems. It is shown that the designed controller is less conservative compared with those in the literature. Finally, a numerical example is provided to verify the validity of the proposed design. [ABSTRACT FROM AUTHOR]

This study addresses the control synthesis of positive switched systems in both continuous‐ and discrete‐time contexts. First, a novel design approach to the state‐feedback controller of continuous‐time positive switched systems without time delay is presented by virtue of multiple linear copositive Lyapunov functions associated with linear programming. Then, the design approach is extended to discrete‐time positive switched systems. It is shown that the presented approach is less conservative than existing ones through comparisons. Furthermore, the approach is developed to the stabilisation of positive switched systems with time delay. Some discussions on the presented approach are provided to show potential applications in corresponding control issues of positive switched systems. Finally, two examples are given to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper addresses two kinds of dual approaches to stability and stabilisation of uncertain switched positive systems under arbitrary switching and average dwell-time switching, respectively. The uncertainties in systems refer to polytopic ones. A new parameter-dependent switched linear copositive Lyapunov function is first proposed for uncertain switched positive systems. By using the new Lyapunov function associated with arbitrary switching and average dwell-time switching, respectively, sufficient conditions for the stability of the systems are established. Two alternative stability criteria based on two kinds of dual approaches are addressed. It is shown that the alternative criteria hold for not only the primal switched positive system but also its dual system. Then, the stabilisation of primal and dual switched positive systems under arbitrary switching and average dwell-time switching is solved, respectively. All present conditions are solvable in terms of linear programming. By some comparisons with existing results, the less conservativeness of the obtained results is verified. Finally, a practical example is provided to illustrate the effectiveness of the theoretical findings. [ABSTRACT FROM PUBLISHER]

This study investigates the problem of robust model predictive control for positive systems under a new model predictive control framework. A robust model predictive control method is presented in this study for uncertain positive systems. A state‐feedback control law that robustly stabilises the underlying system is designed by using linear programming. Different from the traditional model predictive control technique, the authors' proposed model predictive control framework employs a linear infinite horizon objective function and a linear Lyapunov function rather than quadratic performance indices and quadratic Lyapunov functions commonly used in the literature. Compared with existing design techniques for positive systems, the present approach owns the following advantages: (i) it gives a locally optimal control strategy which approaches to actual operation conditions and the control law is designed by solving a locally optimal control problem at each time step, (ii) it can explicitly deal with constraints of the systems, and (iii) the controller can be easily designed via linear programming without any additional constraints. An practical example is provided to verify the validity of the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper investigates absolute exponential stability and stabilization with L 1 -gain performance of switched nonlinear positive systems with time-varying delay. Nonlinear functions considered in the paper are located in a sector field and the time-varying delay is unknown but bounded. The notion of absolute exponential L 1 stability of switched nonlinear positive systems is first introduced. A nonlinear Lyapunov–Krasovskii functional is constructed for the underlying systems. By using the Lyapunov–Krasovskii functional, a sufficient condition for absolute exponential L 1 stability of the systems is established in terms of linear programming. Then, absolute exponential L 1 control synthesis of the systems is addressed. A feedback control law containing nonlinear functions and a state-feedback control law are designed, respectively. By comparing with existing results, it is shown that the present design is less conservative. Finally, an illustrative example is provided to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

This paper is concerned with finite-time stability analysis and control synthesis of fractional order positive switched systems. By using the linear copositive Lyapunov function integrated with average dwell time switching technique, the finite-time stability of fractional order positive switched systems is first addressed. Then, the finite-time boundedness of fractional order positive switched systems with exogenous input is discussed. Finally, the stabilization of the considered systems is proposed, where a control strategy based on linear programming is designed. Several implemental algorithms are provided to reduce the conservativeness of results. Two numerical examples are given to show the effectiveness of the findings of theory. [ABSTRACT FROM AUTHOR]

This paper investigates the problems ofL1-gain analysis and control synthesis of positive switched systems. Linear supply rates andL1-gain notations are introduced to analyse the performance of the underlying systems. Stability with a weightedL1-gain for autonomous systems are solved by using multiple linear copositive Lyapunov functions incorporated with the average dwell time approach. Then, state-feedback and output-feedback controllers are designed to guarantee the stabilisation with a weightedL1-gain for non-autonomous systems. All present conditions are solvable in terms of linear programming. Finally, a practical example is given to illustrate the validity of theoretical findings. [ABSTRACT FROM PUBLISHER]

This paper is concerned with finite-time control and L1-gain analysis of positive switched systems. First, by using the multiple linear copositive Lyapunov functions approach, a sufficient condition for the finite-time stability of autonomous systems under consideration is established. Second, the finite-time boundedness with a weighted L1-gain performance of autonomous systems with uncertain disturbances is addressed. Furthermore, state-feedback controllers guaranteeing the finite-time stability and the finite-time boundedness of non-autonomous systems are constructed, respectively. Finally, an illustrative example is given to show the validity of the present design. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the problems of absolute exponential stability and stabilization for a class of switched nonlinear systems whose system matrices are Metzler. Nonlinearity of the systems is constrained in a sector field, which is bounded by two odd symmetric piecewise linear functions. Multiple Lyapunov functions are introduced to deal with the stability of such nonlinear systems. Compared with some existing results obtained by the common Lyapunov function approach in the literature, the conservatism of our results is reduced. All present conditions can be solved by linear programming. Furthermore, the absolute exponential stabilization for the considered systems is designed by the state-feedback and average dwell time switching strategy. Two examples are also given to illustrate the validity of the theoretical findings. [Copyright &y& Elsevier]