Discrete-time Positive Systems Research Articles

Novel consensus framework on discrete-time positive multi-agent systems

This paper investigates the consensus of discrete-time positive multi-agent systems. A framework on the consensus of positive multi-agent systems is constructed. By introducing a finite point and a self-feedback term, a novel consensus protocol is proposed to drive all states to common nonnegative finite values. An improved error model is established for positive multi-agent systems. Based on the consensus error model, the consensus of the agents is transformed into the stability of the error systems. Under the proposed consensus protocol and the new consensus error model, the consensus of agents is realized. A copositive Lyapunov function is chosen to derive the consensus of the multi-agent systems. Such a consensus approach is extended for time-delay multi-agent systems. All positivity and consensus conditions can be solved by virtue of linear programming. The main novelties of this paper lie in that: (i) A novel consensus error model is established, (ii) A finite value consensus is achieved, and (iii) Linear programming-based conditions are presented for the consensus design. Finally, two illustrated examples are given to verify the validity of the theoretical findings.

Stability Analysis and Event-Triggered Control for IT2 Discrete-Time Positive Polynomial Fuzzy Networked Control Systems With Time Delay

Stability Analysis and Event-Triggered Control for IT2 Discrete-Time Positive Polynomial Fuzzy Networked Control Systems With Time Delay

Optimization of exponential state estimates for positive discrete-time systems with delays and disturbances

Optimization of exponential state estimates for positive discrete-time systems with delays and disturbances

Exponential stability for discrete-time impulsive positive singular system with time delays

This paper investigates the impulsive stability analysis issues of discrete-time positive singular systems with time delay. First, the paper addresses the positivity problem of the system by providing sufficient conditions. Next, a new method based on state transformations is presented to derive a new delay-dependent criterion for the exponential stability of impulsive positive singular systems. Finally, the effectiveness of the proposed conditions is validated through three numerical examples.

Proportional-Derivative Control of Discrete-Time Positive Systems: A State-Space Approach

The proportional-derivative (PD) control is investigated for discrete-time positive linear systems in this paper, which is a fundamental research problem in positive systems theory. Based on positive systems theory and Lyapunov theory, this paper proposes a systematic approach on the PD controller design for positive stabilization. The analyses and methods presented in this paper preserve both the necessity and sufficiency of the control problem, and the convergence of the algorithm is also analyzed. Finally, a numerical example on pest population control is employed in the simulation to verify the results.

Exponential decay of transient values in discrete-time positive nonlinear systems

Exponential decay of transient values in discrete-time positive nonlinear systems

Reachable set estimation for positive linear discrete-time systems under the constrained disturbance

This paper studies the problem of estimating the reachable set boundary of positive discrete-time systems with specific inputs in hyperspace. When the L1-norm or L∞-norm of the particular disturbance input to a positive linear discrete-time system is bounded by a constant, by using linear copositive Lyapunov functional method based on S-lemma, the reachable set could be constrained in a hyper-pyramid under some matrix inequality conditions. In an innovative way, by the method of directly calculating the dynamic trajectories of the system state, the reachable set can also be constrained in a hyper-rectangle. The approach of shrinking the size of the reachable set region is also developed. Finally, the validity of the results obtained is verified by examples of passenger flow in and out of customs.

Observer-based leader-following consensus for positive multi-agent systems over time-varying graphs

This paper addresses the leader-following consensus problem for discrete-time positive multi-agent systems over time-varying graphs. We assume that the followers may have mutually different positive dynamics that can also be different from the leader. Compared with most existing positive consensus works for homogeneous multi-agent systems, the formulated problem is more general and challenging due to the interplay between the positivity requirement and high-order heterogeneous dynamics. To solve the problem, we present an extended version of existing observer-based design for positive multi-agent systems. By virtue of the common quadratic Lyapunov function technique, we show the followers will maintain their state variables in the positive orthant and finally achieve an output consensus specified by the leader. A numerical example is used to verify the efficacy of our algorithms.

On positively invariant polyhedrons for discrete-time positive linear systems

In this paper, necessary and sufficient conditions for the polyhedron set to be a positively invariant polyhedron of a discrete-time positive linear system subject to external disturbances are established. By solving a set of inequalities, which is also a linear programming, necessary and sufficient conditions for the existence of positive invariant polyhedra for discrete-time positive linear systems are proposed, and the relationship between Lyapunov stability and positively invariant polyhedron is also investigated, numerical examples illustrate our results.

Non-fragile PD control of linear time-delay positive discrete-time systems

This paper addresses the issue of proportional-derivative (PD) controllers design for positive linear systems in the discrete-time domain, which still remains a challenging problem in positive systems theory. The specific aim is to design a PD controller for a system with constant time delay, which simultaneously ensures closed-loop system stability and preserves positivity. Moreover, additive gain variation of the controller is considered in the synthesis process. Systematic formulation and tractable algorithms are developed to find the PD controller gains for positive stabilization. The performance of such methods is validated by numerical examples.

Parameterized Synthesis of Discrete-Time Positive Linear Systems: A Geometric Programming Perspective

This letter focuses on optimization problems of discrete-time positive linear systems. To this end, the synthesis problem is presented by introducing parameterized system coefficient matrices and optimizing system parameters directly. Based on results concerning positive linear systems and nonnegative matrix theory, we demonstrate that optimization problems of minimizing the parameter tuning cost while satisfying the H2 norm, H∞ norm, and ℓ1/ℓ∞ Hankel norm constraints can be reduced to corresponding geometric programming problems. In turn, by imposing reasonable assumptions on system matrices, these geometric programming problems can be further transformed into convex optimization problems owing to the convexity of the logarithm transformation on posynomials. Finally, simulation experiments on a numerical example and epidemic spreading process example are used to show the validity of the main results.

Asymptotic Tracking of Discrete-Time Systems Subject to Uniform Output Quantization

This letter explores the internal model principle in order to achieve asymptotic tracking of discrete-time systems subject to output quantization. It is shown that, by artificially quantizing the reference signal, asymptotic tracking can be achieved for a class of systems and references: discrete-time positive real systems, and persistent references in the form of sums of sinusoids. Given the importance of discrete time positive real functions, a systematic way of constructing them is also provided as a contribution. The resulting controller relies on frequency domain tools and may be designed by traditional linear techniques.

Finite-Time Control for Discrete-Time Positive Systems Subject to Event-Triggered Scheme and Markov Jump Parameters

This brief investigates the finite-time control problem for discrete-time positive systems subject to event-triggered scheme and Markov jump parameters. Switching transition probability is used to extend the fixed transition probability, which is more suitable to describe the performance of Markov jump systems (MJSs). On the basis of the 1-norm, a developed event-triggered scheme is proposed to be associated with the sample signal and the actual signal. Furthermore, a linear Lyapunov function and an average dwell time approach are adopted to obtain sufficient conditions for guaranteeing stochastic finite-time stability of the underlying systems. Additionally, a matrix decomposition strategy is constructed to design a finite-time event-triggered control law such that the corresponding MJSs are positive and stochastically finite-time bounded. All conditions are proposed on the basis of linear programming. Finally, a local railway transportation model is provided to verify the practicability of the strategy.

Positivity-Preserving H∞ Model Reduction for Discrete-Time Positive Systems via a Successive Convex Optimization Algorithm

This paper considers the positivity-preserving model reduction for discrete-time positive systems. Given a stable high-order positive system, we aim to find a reduced-order model such that the approximation error is minimized within a prescribed H∞ performance and positivity is preserved. Regarding the bounded real lemma, the sufficient and necessary condition for the existence of a reduced-order model is established in terms of bilinear matrix inequality and convex semi-definite constraint, which ensures that the reduced-order system is positive and the resulted error system is stable and has an H∞ performance level. Based on the inner-approximation strategy, we approximate the bilinear constraints with convex ones, under which an iterative procedure is provided to calculate the desired reduced-order model. Finally, an example is provided to demonstrate the effectiveness and potential benefits of the presented results.

Asynchronous Control of Switched Discrete-Time Positive Systems With Delay

This article considers the stabilization issue of switched discrete-time positive systems (SDPSs) with delay by using asynchronous control. Combining transition probability (TP) with mode-dependent average dwell time (MDADT), called TP-based MDADT switching, the SDPSs are more practical than classical models with average dwell-time (ADT) switching. With the aid of a co-positive Lyapunov–Krasovskii functional (CLKF), sufficient conditions ensuring the exponential stability almost surely (ES a.s.) of the SDPSs without control is studied, where the mode is not necessary to be stable. After that, a mode-dependent controller with switching delay is designed to stabilize the SDPSs in the case that there are unstable subsystems. An algorithm is provided to design the control gains. It is discovered that the mode in the closed-loop SDPSs is not required to be stable on synchronous and asynchronous switching intervals. Numerical simulations verify the merits of the new results.

Static output-feedback peak-to-peak gain control of discrete-time positive linear systems with diverse interval delays

In this article, the problem peak-to-peak gain control (also known as [Formula: see text]-gain control) via static output-feedback scheme is studied for discrete-time positive linear systems with diverse interval delays. By novel comparison techniques involving steady states of upper- and lower-scaled systems with peak values of exogenous disturbances, a characterization for [Formula: see text]-induced norm of the input–output operator is formulated. The obtained characterization is then utilized to derive necessary and sufficient conditions subject to [Formula: see text]-induced performance with prescribed attenuation level. On the basis of the analysis results, and based on a vertex optimization technique, a complete solution to the synthesis problem of a static output-feedback controller that minimizes the worst case amplification from disturbances to regulated outputs subject to peak-to-peak gain is addressed. The derived stabilization conditions are formulated in terms of tractable linear programming conditions, which can be effectively solved by various convex algorithms. Numerical examples and simulations are given to illustrate the effectiveness of the proposed method.

Global Stability of Positive Discrete-time Time-varying Nonlinear Feedback Systems

The global stability of positive discrete-time time-varying nonlinear systems with time-varying scalar feedbacks is investigated. Sufficient conditions for the asymptotic stability of discrete-time positive time-varying linear systems are given. The new conditions are applied to discrete-time positive time-varying nonlinear systems with time-varying feedbacks. Sufficient conditions are established for the global stability of the discrete-time positive time-varying nonlinear systems with feedbacks.

Quadratic Stabilization of Linear Uncertain Positive Discrete-Time Systems

The paper provides extended methods for control linear positive discrete-time systems that are subject to parameter uncertainties, reflecting structural system parameter constraints and positive system properties when solving the problem of system quadratic stability. By using an extension of the Lyapunov approach, system quadratic stability is presented to become apparent in pre-existing positivity constraints in the design of feedback control. The approach prefers constraints representation in the form of linear matrix inequalities, reflects the diagonal stabilization principle in order to apply to positive systems the idea of matrix parameter positivity, applies observer-based linear state control to assert closed-loop system quadratic stability and projects design conditions, allowing minimization of an undesirable impact on matching parameter uncertainties. The method is utilised in numerical examples to illustrate the technique when applying the above strategy.

A robust distributed secure interval observation approach for uncertain discrete-time positive systems under deception attacks

This paper deals with the network-based robust distributed state estimation problem of a class of discrete-time positive systems with bounded parameter perturbation and bounded disturbance. The system outputs are measured by multiple groups of sensors and transmitted to observers through communication networks suffering from deception attacks. To attain real-time monitoring and interval estimation of the system state and achieve a satisfactory estimation accuracy, a distributed secure interval observation approach is presented, where the interval observer is composed of an upper-bounding observer and a lower-bounding observer. The construction of each interval observer is based on the system positivity and the bounds of parameter perturbation, disturbance input and attack signals. Using the stochastic finite-time boundedness and l1-gain analysis method together with the linear programming technique, some sufficient conditions on analysis, design and optimization of the distributed interval observers for the uncertain positive system against deception attacks are established. The practicability and advantage of the proposed approach is checked by a numerical example, which comes from the remote monitoring of power distribution in smart grid.

On the similarity to nonnegative and Metzler Hessenberg forms

Abstract We address the issue of establishing standard forms for nonnegative and Metzler matrices by considering their similarity to nonnegative and Metzler Hessenberg matrices. It is shown that for dimensions n 3, there always exists a subset of nonnegative matrices that are not similar to a nonnegative Hessenberg form, which in case of n = 3 also provides a complete characterization of all such matrices. For Metzler matrices, we further establish that they are similar to Metzler Hessenberg matrices if n 4. In particular, this provides the first standard form for controllable third order continuous-time positive systems via a positive controller-Hessenberg form. Finally, we present an example which illustrates why this result is not easily transferred to discrete-time positive systems. While many of our supplementary results are proven in general, it remains an open question if Metzler matrices of dimensions n 5 remain similar to Metzler Hessenberg matrices.