Linear Copositive Lyapunov Function Research Articles

Robust exponential stabilization of positive uncertain switched neural networks with actuator saturation and sensor faults

This article focuses on the robust exponential stabilization of positive uncertain switched neural networks subject to actuator saturation and sensor faults. Given the existence of interval uncertainty and the constraint concerning positivity of the original system, a new positive state-bounding observer is constructed to guarantee the coinstantaneous estimation of system state and sensor faults. To deal with actuator saturation, the convex hull scheme is employed. By designing the state-feedback controller and utilizing the multiple time-varying linear co-positive Lyapunov function, sufficient conditions for the robust exponential stability on the studied system are established under dwell-time switching. Furthermore, for optimizing the observer matrix, an iterative algorithm is developed. Eventually, a numerical example is exploited to illuminate the feasibility and effectiveness of both the deduced results and the proposed approaches.

Controller synthesis for discrete‐time positive switched T–S fuzzy systems with partially controllable subsystems

AbstractThe controller synthesis problem for discrete‐time positive switched T–S fuzzy systems (PSTSFSs) is investigated in this paper. Unlike previous literature, subsystems are assumed to be partially controllable in this paper. A novel slow–fast combined mode‐dependent average dwell time (SFCMDADT) approach is proposed to obtain sufficient conditions for the stability of general positive switched nonlinear systems with partially stable subsystems. Then based on the obtained results, a parallel distributed compensation (PDC)‐based fuzzy controller is designed, and sufficient conditions for the stability of PSTSFSs with partially controllable subsystems are developed with the aid of a time‐scheduled multiple linear copositive Lyapunov function. Furthermore, sufficient conditions for the stability of the positive switched T–S fuzzy system are given respectively for the two cases that the subsystems are all controllable and all uncontrollable. Finally, two simulation examples are provided to illustrate the effectiveness of the proposed method.

Stability and [formula omitted]-gain characterization of positive linear systems on time scales

The stability and Lp-gain (p∈{1,∞}) analysis problems are investigated for positive linear systems on arbitrary time scales in this paper. Firstly, by virtue of the linear copositive Lyapunov function, the uniform stability and uniformly exponential stability conditions are presented for positive linear time-varying systems on time scales. Then, we conclude that the uniformly exponential stability condition of positive linear time-invariant systems is independent of time scales. Meanwhile, it is further proved that the Lp-gain (p∈{1,∞}) conditions of these systems are also independent of time scales and can be fully characterized by system matrices. Finally, a numerical example is given to demonstrate the validity of our conclusions.

Guaranteed Cost and Finite-Time Non-fragile Control of Fractional-Order Positive Switched Systems with Asynchronous Switching and Impulsive Moments

This paper considers the problem of guaranteed cost and finite-time non-fragile control for a class of fractional-order positive switched systems with asynchronous switching and impulsive moments. Firstly, a novel cost function is presented. The sufficient conditions for the guaranteed cost and finite-time stability of the considered systems are derived via linear programming, using linear co-positive Lyapunov functions, the average dwell time method, and the average impulsive interval approach. Secondly, the finite-time and finite-time non-fragile controllers are designed to ensure that the corresponding closed-loop system is finite-time stable with a certain cost upper bound. Finally, an example of a fractional-order electrical circuit is provided, proving the proposed method’s feasibility and effectiveness.

Stochastic stability of positive Markov jump linear systems with fixed dwell time

This paper studies the stochastic stability of positive Markov jump linear systems with a fixed dwell time. By constructing an auxiliary system that originated from the initial system with state jumps, sufficient and necessary conditions of stochastic stability for positive Markov jump linear systems are obtained with both exactly known and partially known transition rates. The main idea in the latter case is applying a convex combination to convert bilinear programming into linear programming problems. On this basis, multiple piecewise linear co-positive Lyapunov functions are provided to achieve less conservative results. Then state feedback controller is designed to stabilize the positive Markov jump linear systems by solving linear programming problems. Numerical examples are presented to illustrate the viability of our conclusions.

Filter for Positive Stochastic Nonlinear Switching Systems With Phase-Type Semi-Markov Parameters and Application

In this article, the issue of positive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {L}_{1}$ </tex-math></inline-formula> filter design is investigated for positive nonlinear stochastic switching systems subject to the phase-type semi-Markov jump process. Many complicated factors, such as semi-Markov jump parameters, positivity, T–S fuzzy strategy, and external disturbance, are taken into consideration. Practical systems under positivity constraint conditions and unpredictable structural changes are characterized by positive semi-Markov jump systems (S-MJSs). First, by the key properties of the supplementary variable and the plant transformation technique, phase-type S-MJSs are transformed into Markov jump systems (MJSs), which means that, to an extent, these two kinds of stochastic switching systems are mutually represented. Second, with the help of the normalized membership function, the associated nonlinear MJSs are transformed into the local linear MJSs with specific T–S fuzzy rules. Third, by choosing the linear copositive Lyapunov function (LCLF), stochastic stability (SSY) criteria are given for the corresponding system with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {L}_{1}$ </tex-math></inline-formula> performance. Some solvability conditions for positive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {L}_{1}$ </tex-math></inline-formula> filter are constructed under a linear programming framework. Finally, an epidemiological model illustrates the effectiveness of the theoretical findings.

Improved reachable set estimation for positive systems: A polyhedral approach

This paper revisits the problem of estimating the set of states of a positive system, that are reachable from the origin, by constrained exogenous inputs. More specifically, we seek the smallest polyhedron that bounds the reachable set of a positive system, subject to inputs whose column sums have a bounded peak value. By resorting to a particular auxiliary function, the maximum of linear functions of state variables, a sufficient condition is given such that the reachable set is bounded by a prescribed polyhedron. This auxiliary function is not necessarily positive definite even for nonnegative state variables and, therefore, the face normals of the resulting bounding polyhedron may not be positive vectors. This greatly improves the existing results using linear copositive Lyapunov functions, and also paves the way for the reachable set estimation of unstable positive systems. Moreover, we provide an approach to compute the bounding hyper-rectangle, which turns out to be nonconservative for the single-input case. The obtained bounding polyhedron depends on the time delays, while the bounding hyper-rectangle does not. On the other hand, for a positive system subject to inputs with the integral of their column sums bounded by a prescribed value, we give an estimate for the reachable set using the convex hull of hyper-rectangles, as well as a bounding polyhedron whose outward face normals are not necessarily positive vectors. These two categories of bounding sets are both delay independent. Finally, we show via numerical examples the superiority of the developed results.

Estimation of Domain of Attraction for Discrete-Time Positive Interval Type-2 Polynomial Fuzzy Systems With Input Saturation

This article focuses on expanding the estimation of the domain of attraction (DOA) for discrete-time positive nonlinear systems subject to input saturation and parameter uncertainties. To facilitate analysis and design, the interval type-2 (IT2) polynomial fuzzy model is used to represent the nonlinear plant and capture uncertainties. Combining with the IT2 polynomial fuzzy controller, the discrete-time positive IT2 polynomial fuzzy-model-based (PIT2PFMB) control system is formed to facilitate analysis. To enlarge the estimation of DOA of the discrete-time PIT2PFMB system, polyhedron is used to characterize the DOA with the help of linear copositive Lyapunov function (LCLF). Referring to the nonconvex conditions derived by LCLF, an effective convexification method is proposed in this article. For comparison purposes, the saturation-dependent-Lyapunov-function-based method is extended to the PIT2PFMB control system by adding the corresponding positivity conditions. In addition, this article attempts to enlarge the estimation of the DOA by improving the IT2 membership-function-dependent (IT2MFD) method and extending it to all conditions, including the stability conditions and the DOA estimation conditions. Finally, an example with simulation results is given to verify the effectiveness of all the methods proposed in this article for expanding the estimation of the DOA.

Hybrid [formula omitted]-performance analysis and control of linear time-varying impulsive and switched positive systems

Recent works have shown that the L1 and L∞-gains are natural performance criteria for linear positive systems as they can be exactly characterized by linear programs. Those performance measures have also been extended to linear positive impulsive and switched systems through the concept of hybrid L1×ℓ1-gain. For LTI positive systems, the L∞-gain is known to coincide with the L1-gain of the transposed system and, as a consequence, one can use linear copositive Lyapunov functions for characterizing the L∞-gain of LTI positive systems. Unfortunately, this does not hold in the time-varying setting and one cannot characterize the hybrid L∞×ℓ∞-gain of a linear positive impulsive system in terms of the hybrid L1×ℓ1-gain of the transposed system. To solve this, an approach based on the use of linear copositive max-separable Lyapunov functions is proposed. We first prove very general necessary and sufficient conditions characterizing the exponential stability and the L∞×ℓ∞- and L1×ℓ1-gains using linear max-separable copositive and linear sum-separable copositive Lyapunov functions. These two results are then connected together using operator theoretic results and the notion of adjoint system. Results characterizing the stability and the hybrid L∞×ℓ∞-gain of linear positive impulsive systems under arbitrary, constant, minimum, and range dwell-time constraints are then derived from the previously obtained general results. These conditions are then exploited to yield constructive convex stabilization conditions via state-feedback. By reformulating linear positive switched systems as impulsive systems with multiple jump maps, stability and stabilization conditions are also obtained for linear positive switched systems. It is notably proven that the obtained conditions generalize existing ones of the literature. As all the results are stated as infinite-dimensional linear programs, sum of squares programming is used to turn those optimization problems into sufficient tractable finite-dimensional semidefinite programs. Interestingly, the relaxation becomes necessary if we allow the degrees of the polynomials to be arbitrarily large. Several particular cases of the approach such as LTV positive systems and periodic positive systems are also discussed for completeness. Examples are given for illustration.

Exponential stability analysis for singular switched positive systems under dwell-time constraints

The article addresses the exponential stability analysis for singular switched positive systems(SSPSs) under dwell-time constraints which include mode-dependent minimum dwell time(MDMDT), mode-dependent constant dwell time(MDCDT) and mode-dependent ranged dwell time(MDRDT) constraints. For SSPSs in both delay-free case and time-varying delay case, a sufficient exponential stability condition is proposed with MDMDT, MDCDT and MDRDT constraints, respectively, and the exponential decay rate can be set as a free parameter based on diverse circumstances. To analyze the dwell-time stability, a novel discretized linear copositive Lyapunov function(DLCLF) approach is introduced in the article, and compared with the general copositive and homogeneous Lyapunov function approach, the main advantage of the DLCLF technique is itself affine dependence on conditions of system matrix, which will be extended and utilized to systems with uncertainties or/and time-varying parameters quite easily. Meanwhile, the proposed condition of exponential stability under MDMDT(or MDCDT or MDRDT) constraint will be degenerated into the one under minimum dwell time(or constant dwell time or ranged dwell time) constraint for some specific situations, which implies that the considered MDMDT(or MDCDT or MDRDT) case is more general and practical. Finally, the validity and significance of the results are illustrated by seven examples.

Mode-dependent impulsive control of positive switched systems: Stability and L1-gain analysis

Abstract This paper studies the stability and L1-gain problems for impulsive positive switched systems (IPSS) consisting of both unstable and stable subsystems, and proposes a mode-dependent impulsive control (MDIC) method for the first time. By means of a piecewise linear co-positive Lyapunov function (PLCLF) and using average dwell time (ADT) approach, some sufficient criteria for global exponential stability (GES) and standard L1-gain are established in terms of linear programming (LP), which can be verified conveniently by Matlab. A special impulsive controller is designed to guarantee the positivity and stability of the addressed system. Several numerical examples with simulations are provided to demonstrate the effectiveness of the obtained results.

Event-Triggered Filter Design of Positive Systems With State Saturation

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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -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.

Reliable control for positive switched systems with random nonlinearities

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.

Robust ℓ₁-Controller Design for Discrete-Time Positive T–S Fuzzy Systems Using Dual Approach

In this article, a new approach is proposed for stability analysis and controller design of nonlinear discrete-time positive systems by means of the Takagi&#x2013;Sugeno fuzzy model. The closed-loop stability and the positivity constraint are guaranteed by synthesizing a linear co-positive Lyapunov function and by applying the parallel distributed compensation controller. In contrast to the state-of-the-art approaches for ensuring the <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-stability of the positive system which are based on bilinear matrix inequalities, the proposed optimal robust control design under <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-induced performance is derived based on linear programming framework. It has been shown that the computational complexity of the proposed optimization problem can be effectively reduced. Finally, a numerical example and the Leslie population model are adopted to show the capabilities of the proposed method.

Event-triggered Control of Positive Systems With State Saturation Using Linear Programming

This paper proposes the event-triggered control of positive systems with state saturation both in discrete-time and continuous-time cases. A 1-norm based event-triggered mechanism is established for positive systems. Under the event-triggered mechanism, the error term between actual and sample states is transformed into interval uncertain form. Together with the properties of saturation, the systems with state saturation are transformed into interval uncertain systems and the corresponding lower and upper bounds of the system matrices are obtained, respectively. Using a linear co-positive Lyapunov function, the event-triggered controller and the auxiliary gain matrix of the domain of attraction are designed in terms of linear programming, respectively. Then, the systems with state and input saturation are also described via interval uncertain systems. An event-triggered controller is designed and thus the closed-loop systems are positive and stable under the designed controller. Furthermore, the presented event-triggered control approach is extended to the continuous-time case. Compared to existing control approaches, the event-triggered control can reduce energy consumption and increase the practicability. Finally, several numerical examples are given to illustrate the effectiveness of the proposed design.

Distributed model predictive control of positive Markov jump systems

This paper proposes a new distributed model predictive control (DMPC) for positive Markov jump systems subject to uncertainties and constraints. The uncertainties refer to interval and polytopic types, and the constraints are described in the form of 1-norm inequalities. A linear DMPC framework containing a linear performance index, linear robust stability conditions, a stochastic linear co-positive Lyapunov function, a cone invariant set, and a linear programming based DMPC algorithm is introduced. A global positive Markov jump system is decomposed into several subsystems. These subsystems can exchange information with each other and each subsystem has its own controller. Using a matrix decomposition technique, the DMPC controller gain matrix is divided into nonnegative and non-positive components and thus the corresponding stochastic stability conditions are transformed into linear programming. By virtue of a stochastic linear co-positive Lyapunov function, the positivity and stochastic stability of the systems are achieved under the DMPC controller. A lower computation burden DMPC algorithm is presented for solving the min-max optimization problem of performance index. The proposed DMPC design approach is extended for general systems. Finally, an example is given to verify the effectiveness of the DMPC design.

Stability Analysis of Switched Positive Systems with an Impulse Interval

The stability analysis issue of switched positive systems (SPSs) with an impulse interval is discussed in this paper for the first time. All subsystems are allowed to be unstable. Unlike previous studies, the impulse is restricted to an interval, that is, the impulse interval. By dividing the state space on the nonnegative orthant and establishing multiple linear copositive Lyapunov functions, sufficient linear programming conditions are obtained to ensure that SPSs with an impulse interval are asymptotically stable. A hysteresis state-dependent switching law is designed to prevent the chattering behavior caused by frequent switching. Finally, two numerical examples are given to validate the theoretical findings.

New Results on Stability Analysis and Estimator Design for Switched Positive Linear Systems: A Reverse-Timer-Dependent Linear Co-Positive Lyapunov Function Approach

In this brief, the stability and estimator design issues are addressed for switched positive linear systems (SPLSs) with average dwell time (ADT). Based on a reverse timer which starts timing at the end of the activated time of a subsystem, a reverse-timer-dependent linear co-positive Lyapunov function (RLCLF) is constructed. With the help of the RLCLF approach, a new stability criteria is derived. Then, based on the stability criteria, a sufficient condition checking the existence of state estimator is derived for SPLSs with ADT. All the results can be relaxed into sum of square (SOS) program by using a SOS approximation approach. It's shown that the results obtained by RLCLF approach are less conservative compared with those of the literature. Finally, the advantages of the results are illustrated within two examples.

Finite-time interval observer design for discrete-time switched systems: A linear programming approach

This paper deals with the finite-time interval observer design method for discrete-time switched systems subjected to disturbances. The disturbances of the system are unknown but bounded. The framework of the finite-time interval observer is established and the sufficient conditions are derived by the multiple linear copositive Lyapunov function. Furthermore, the conditions which are expressed by the forms of linear programming are numerically tractable by standard computing software. One example is simulated to illustrate the validity of the designed observer.

Stabilization of switched positive system with impulse and marginally stable subsystems: A mode-dependent dwell time method

Abstract In this article, the problem of stabilization for the switched positive linear system (SPLS) with impulse and marginally stable subsystems is studied, in which both the continuous time case and the discrete time case are taken into account. By the usage of a piecewise weak linear copositive Lyapunov function (LCLF) and the mode-dependent dwell time (MDT), time-dependent switching control laws are designed to guarantee the asymptotic stability of the system. Furthermore, based on the comparison principal for positive systems, the main results are generalized to the time-variant SPLS. Two examples are worked out to indicate that our results have advantages over some known results in the literature.