Continuous-time Positive Systems Research Articles

Novel Proportional–Integral–Derivative Control Framework on Continuous-Time Positive Systems Using Linear Programming

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.

Positive Consensus of Fractional-Order Multiagent Systems Over Directed Graphs.

This article investigates the positive consensus problem of a special kind of interconnected positive systems over directed graphs. They are composed of multiple fractional-order continuous-time positive linear systems. Unlike most existing works in the literature, we study this problem for the first time, in which the communication topology of agents is described by a directed graph containing a spanning tree. This is a more general and new scenario due to the interplay between the eigenvalues of the Laplacian matrix and the controller gains, which renders the positivity analysis fairly challenging. Based on the existing results in spectral graph theory, fractional-order systems (FOSs) theory, and positive systems theory, we derive several necessary and/or sufficient conditions on the positive consensus of fractional-order multiagent systems (PCFMAS). It is shown that the protocol, which is designed for a specific graph, can solve the positive consensus problem of agents over an additional set of directed graphs. Finally, a comprehensive comparison study of different approaches is carried out, which shows that the proposed approaches have advantages over the existing ones.

L 1/L – Observer-Based Fault Detection for 2-D Delayed Positive Continuous-Time Systems

This study focuses on the design of fault detection observer for continuous 2D delayed positive systems. Firstly, the L1-gain and L–index of the 2D positive delayed systems are calculated by 2D Laplace transform; secondly, necessary and sufficient conditions for the system to satisfy the L1/L-performance are derived; finally, sufficient conditions for the existence of a mixed L1/L-observer are derived. The solution to the observer gain is given by an iterative linear programming algorithm. An example is given to demonstrate the validity of the proposed theoretical results.

Linear Quadratic Control of Positive Systems: A Projection-Based Approach

This paper investigates the nonfragile design of linear quadratic regulator (LQR) for continuous-time positive linear systems subject to gain variations. Based on positive systems theory and Lyapunov theory, the solvability and optimality of the positivity-preserving LQR problem are analyzed through the lens of optimization, and two projection theorems are derived for single-input and multi-input positive systems, respectively, which paves the way for developing the projected gradient descent (PGD) algorithm for computation. The proposed results fill the literature gap by considering both the optimality and positivity in LQR design, and finally numerical simulations are provided to verify the effectiveness of the results.

Consensus of Positive Networked Systems on Directed Graphs.

This article addresses the distributed consensus problem for identical continuous-time positive linear systems with state-feedback control. Existing works of such a problem mainly focus on the case where the networked communication topologies are of either undirected and incomplete graphs or strongly connected directed graphs. On the other hand, in this work, the communication topologies of the networked system are described by directed graphs each containing a spanning tree, which is a more general and new scenario due to the interplay between the eigenvalues of the Laplacian matrix and the controller gains. Specifically, the problem involves complex eigenvalues, the Hurwitzness of complex matrices, and positivity constraints, which make analysis difficult in the Laplacian matrix. First, a necessary and sufficient condition for the consensus analysis of directed networked systems with positivity constraints is given, by using positive systems theory and graph theory. Unlike the general Riccati design methods that involve solving an algebraic Riccati equation (ARE), a condition represented by an algebraic Riccati inequality (ARI) is obtained for the existence of a solution. Subsequently, an equivalent condition, which corresponds to the consensus design condition, is derived, and a semidefinite programming algorithm is developed. It is shown that, when a protocol is solved by the algorithm for the networked system on a specific communication graph, there exists a set of graphs such that the positive consensus problem can be solved as well.

Interval state estimation for positive linear systems under DoS attacks

This paper is concerned with the interval state estimation problem for continuous-time positive linear systems under intermittent denial-of-service (DoS) attacks. To solve the problem, two types of estimate strategies are proposed. One is using the interval observer at all times, the other is using the interval observer in the absence of attacks but using, instead, the interval predictor otherwise. To facilitate the analysis, the interval state estimation problem is reformulated into the positivity and stability analysis of the associated error system. Then, stability conditions and disturbance attenuation characterization of the error systems for the two strategies are established via a mode-dependent Lyapunov approach. Roughly speaking, it is shown that the interval estimation accuracy of the former strategy is higher than the latter when the open loop system is stable. Finally, several numerical examples are provided to illustrate the ascendancy of the proposed estimation strategies.

Sign Stability of Dual Switching Linear Continuous-Time Positive Systems

This paper investigates 1-moment exponential stability and exponential mean-square stability (EMS stability) under average dwell time (ADT) and the preset deterministic switching mechanism of dual switching linear continuous-time positive systems when a numerical realization does not exist. The signs of subsystem matrices, but not their structures of magnitude, are key information that causes a qualitative concept of stability called sign stability. Both 1-moment exponential stability and EMS stability, which are the traditional stability concepts, are generalized intrinsically. Hence, both 1-moment exponential sign stability and EMS sign stability are introduced and are proven based on sign equivalency. It is shown that they are symmetrically and qualitatively stable. Notably, the notion of stability can be checked quantitatively using some examples.

Global Consensus of Positive Edge System With Sector Input Nonlinearities

This paper investigates the positive edge consensus problems for undirected networks, where each edge is modeled by a continuous-time positive system with sector input nonlinearities. Based on a given nodal network, the corresponding edge network is presented, and some relationships between the given nodal network and its edge network are given. The observer-based protocol for non-negative consensus is proposed. Rigorous theoretical analysis which considers both the input nonlinearities and the positivity constraint is introduced to achieve a global consensus result. Simultaneous stabilization problems are constructed for edge-based non-negative consensus using the maximum and the minimum nonzero eigenvalues of the edge Laplacian matrix. Furthermore, the robustness of the positive edge consensus to parameter perturbation is illustrated. Sufficient non-negative consensus conditions related to the bounds of system matrices are given, and the non-negative edge consensus conditions for star nodal networks are also given only using the vertex number. Iterative linear matrix inequalities (LMIs)/LMI algorithm is presented to obtain the feedback matrix and the observer matrix. The given theoretical results are finally illustrated by some simulations.

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.

Homogenized first-moment analysis of two-time-scale positive Markov jump linear systems

We address in this paper the mean stability and L∞ performance of continuous-time Positive Markov Jump Linear Systems (PMJLS). The distinguishing aspect of our approach vis-à-vis the existing literature is that the underlying Markov jump process is a two-time-scale Markov chain, and we consider the singular perturbation setup which arises when a small parameter (which determines the time-scale separation) goes to zero. The interest in this limiting scenario stems from large-scale situations, where complexity reduction is a central issue. To achieve this, we carry out a convergence analysis involving the semigroup that describes the first moment dynamics of the system state. This analysis allows us to subsequently characterize homogenized notions of stability and L∞ performance, and we show how these can be connected with linear programming methods. A numerical example, regarding a version of the Foschini-Miljanic algorithm for power allocation in a mobile communication system, illustrates the proposed results.

Finite-Time L2-Gain Asynchronous Control for Continuous-Time Positive Hidden Markov Jump Systems via T-S Fuzzy Model Approach.

This article investigates the finite-time asynchronous control problem for continuous-time positive hidden Markov jump systems (HMJSs) by using the Takagi-Sugeno fuzzy model method. Different from the existing methods, the Markov jump systems under consideration are considered with the hidden Markov model in the continuous-time case, that is, the Markov model consists of the hidden state and the observed state. We aim to derive a suitable controller that depends on the observation mode which makes the closed-loop fuzzy HMJSs be stochastically finite-time bounded and positive, and fulfill the given L2 performance index. Applying the stochastic Lyapunov-Krasovskii functional (SLKF) methods, we establish sufficient conditions to obtain the finite-time state-feedback controller. Finally, a Lotka-Volterra population model is used to show the feasibility and validity of the main results.

Scalable Control of Positive Systems

In this article, we first present some foundational results about the stability and positive stabilization of continuous-time positive systems. Necessary and sufficient conditions for achieving stability are provided, together with some desired performance in terms of disturbance attenuation. These conditions are expressed in terms of linear programming and scale well with the system size. We then discuss the interconnection of positive subsystems by means of a static output feedback that preserves positivity, and propose conditions to achieve both stability and the asymptotic alignment of the closed-loop output to a desired vector. Finally, we describe some results for a class of parameterized positive systems. The second part of the article presents some interesting applications of the results presented in the first part. Specifically, control problems for heating networks, formation control, power control in wireless communication, and the evolutionary dynamics of cancer and HIV are formalized and solved as optimal control problems for positive systems.

On positively invariant polyhedrons for continuous-time positive linear systems

This paper is concerned with the determination of positively invariant polyhedron for positive linear systems subject to external disturbances whose (∞, 1)-norm or (∞, ∞)-norm are bounded by a prescribed constant. Necessary and sufficient conditions for the existence of a positively invariant polyhedron are derived in terms of a set of inequalities which can be solved by linear programming, and the link between Lyapunov stability and positively invariant polyhedron is also revealed.

Impulsive finite-time observer design for uncertain positive linear systems with L2-gain analysis

This paper is concerned with the design of a finite-time positive observer (FTPO) for continuous-time positive linear systems, which is robust regarding the L2-gain performance. In positive observers, the estimation of the state variables is always nonnegative. In contrast to previous positive observers with asymptotic convergence, an FTPO estimates positive state variables in a finite time. The proposed FTPO observer, using two Identity Luenberger observers and based on the impulsive framework, estimates exactly the state variables of positive systems in a predetermined time interval. Furthermore, sufficient conditions are given in terms of linear matrix inequalities (LMIs) to guarantee the L2-gain performance of the estimation error. Finally, the performance and robustness of the proposed FTPO are validated using numerical simulations.

An LMI-based algorithm for static output-feedback stabilization of continuous-time positive polytopic linear systems

This paper presents a new technique to design static output-feedback controllers for continuous-time positive uncertain linear systems. The design is performed through an iterative algorithm based on parameter-dependent linear matrix inequality conditions, solved by means of relaxations, with local convergence guaranteed. A qualified feasible solution provides a stabilizing output-feedback controller that also assures the positivity of the closed-loop system. The main advantage of the proposed methodology is that the control gain is handled directly as an optimization variable, that is, no change of variables is needed to recover the gain and no particular structure (e.g., diagonal) is imposed on the Lyapunov or slack variable matrix to guarantee closed-loop positivity. This particular feature also facilitates the design of decentralized or element-wise bounded gains, as illustrated by numerical experiments.

Constrained model predictive control for positive systems

This article is devoted to the problem of model predictive control (MPC) design for discrete-time and continuous-time positive systems with state and input constraints. The proposed controllers are so designed that the closed-loop constrained systems are positive and stable, meanwhile, linear infinite horizon cost functions through their upper bounds are minimised. In the discrete-time case, the performance of the control system compared to existing studies is remarkably improved. Moreover, in the continuous-time case, the proposed MPC is such that can be directly applied to the continuous-time positive system without discretisation. The merit of this method is that the sampling interval has nothing to do with the stability of the system, just a shorter sampling period results in better optimality and performance. In addition, by defining a slack variable and accounting it in the minimisation problems, a fast rate of convergence will be obtained. In order to solve the optimisation problem of MPC, linear programming (LP) is used which needs to be solved at each iteration. All conditions are derived in the form of LP. Finally, to demonstrate the effectiveness of the proposed method, comparisons with the existing studies are presented through practical and numerical examples.

Exponential stability of homogeneous impulsive positive delay systems of degree one

This paper investigates the global exponential stability of homogeneous impulsive positive delay systems of degree one. By using the max-separable Lyapunov functions, a sufficient criterion is obtained for exponential stability of continuous-time homogeneous impulsive positive delay systems of degree one. We also provide the corresponding counterpart for discrete-time homogeneous impulsive positive delay systems of degree one. Our results show that a stable impulse-free system can keep its original stability property under certain destabilising impulsive perturbations. It should be noted that it's the first time that the exponential stability results for homogeneous impulsive positive delay systems of degree one are given. Numerical examples are provided to demonstrate the effectiveness of the derived results.

Stability analysis for positive singular systems with distributed delays

This paper is concerned with the stability analysis for continuous-time positive singular systems with distributed delays. By introducing an auxiliary system, a necessary and sufficient positivity condition is proposed for singular systems with distributed lags. Based on this condition, we present a sufficient stability condition for positive singular systems with distributed delays and extend the result to systems with distributed delays over a bounded time-varying interval. In addition, this stability condition is also necessary when strictly positive initial conditions are available. Numerical simulations are utilized to illustrate the effectiveness of our results.

Stability Analysis for Positive Singular Systems With Time-Varying Delays

This paper is concerned with the stability analysis for continuous-time positive singular systems with time-varying delays. First, an auxiliary system is introduced to establish the positivity condition for singular time-delay systems. Based on the positivity condition, the stability criterion is obtained for the positive singular systems with constant delays. By analyzing the monotonic property of the system trajectory, we extend the stability condition to the cases with time-varying delays. A numerical simulation is employed to illustrate the effectiveness of our results.

${\mathcal{H}}_2$ State Feedback Control Design of Continuous-Time Positive Linear Systems

This technical note focuses on state feedback control design of positive linear systems in continuous-time domain. The main goal is to analyze stability and performance preserving the closed-loop system positivity. In some sense, the provided results fill the literature gap by considering the ${\mathcal{H}}_2$ performance index. Indeed, although ${\mathcal{H}}_\infty$ state feedback control design of positive systems has received remarkable attention in the recent years, we believe that, for the first time, a numerical method able to provide a local optimal solution to this class of nonconvex control problems is addressed by means of a simple (since no line search is required) convergent sequence of convex programming problems. The theory and the algorithm behavior are illustrated and evaluated by numerical examples borrowed from the recent literature.