Positive System Theory Research Articles

Decentralized H2 Control for Discrete-Time Networked Systems With Positivity Constraint.

In this brief, we study the decentralized H2 state-feedback control problem for networked discrete-time systems with positivity constraint. This problem (for a single positive system), raised recently in the area of positive systems theory, is known to be challenging due to its inherent nonconvexity. In contrast to most works, which only provide sufficient synthesis conditions for a single positive system, we study this problem within a primal-dual scheme, in which necessary and sufficient synthesis conditions are proposed for networked positive systems. Based on the equivalent conditions, we develop a primal-dual iterative algorithm for solution, which helps prevent from converging to a local minimum. In the simulation, two illustrative examples are employed for verification of our proposed results.

Secure consensus for positive multi‐agent systems with bumpless transfer control under Denial‐of‐Service attacks

AbstractThis article investigates the secure consensus control problem of positive multi‐agent systems under multiple‐mode Denial‐of‐Service (DoS) attacks. A bumpless transfer (BT) control based secure mode strategy is proposed to improve the switching transient performance due to the system being actively switched to secure mode when attack is detected. Considering possible time delays for agents dealing with the issues of DoS attacks, an asynchronous switching model is established between system models and communication topology. With the stability theory of positive systems, sufficient and necessary conditions are established to guarantee the positivity of multi‐agent systems under the secure mode strategy. Because the switching law caused by external attacks is unknown in advance, the BT control is specifically implemented through the controller gain interpolation technique. By multiple Lyapunov functions (MLFs) approach, secure consensus is achieved with a trade‐off among controller performance, attack parameters, and switching delays. Numerical examples are presented to demonstrate the effectiveness of the obtained results.

Distributed interval estimation for sensor networks with energy-limited attacks and stochastic communication protocols

In this paper, the problem of the distributed interval estimation is investigated for wireless sensor networks under the energy-limited attack and the stochastic communication protocol (SCP). A model with randomness and restrictions is constructed to describe the energy-limited attack, which is considered to occur in two positions, one is the transmission process of sensors and local estimators (LEs), and another is the transmission process of LEs and their neighbor nodes. Furthermore, to mitigate the communication burden, the SCP is adopted to distribute network access rights to sensor nodes. All in all, the objective of this paper is to develop a distributed interval estimator (DIE) which can use local and neighbor information to generate an interval that covers all potential estimation states when SCP, bounded noises, and energy-limited attacks are all present at the same time. To achieve this, the positive system theory and the stability analysis theory are used to obtain some sufficient conditions for the error dynamic system to be positive system and stable, and the needed DIE gains are achieved by solving the linear matrix inequality. Finally, an example is presented to demonstrate the effectiveness and superiority of the offered method.

Flexible Performance-Based Control for Nonlinear Systems Under Strong External Disturbances.

Addressing external disturbances has been a critical issue for control design to ensure reliable operation of systems. This article investigates the tracking control problem for the uncertain nonlinear systems with the strong external disturbance and the prescribed performance. The flexible performance-based control scheme is developed by introducing an external disturbance criterion into the prescribed performance. It is capable of guaranteeing the prescribed performance if the external disturbance is less than a specified threshold and degrading that in light of the user-appointed rule otherwise. Particularly, the disturbance interval observer is synthesized to generate the boundaries of the external disturbances and realize the judgment of that criterion. With the generated boundaries, the interval-type auxiliary system is designed to provide the modified performance functions (MPFs) that characterize performance requirement and degradation rule simultaneously. Based on the positive system theory and the Lyapunov method, it is theoretically shown that the system output can always track the reference signal and satisfy the constraints of MPFs. Finally, both the numerical simulation and the application of flight control design verify that the results are effective and valid.

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.

An Interval Observer for a Class of Cyber–Physical Systems with Disturbance

This paper investigates the problem of interval estimation for cyber–physical systems with unknown disturbance. In order to realize the interval estimation of cyber–physical systems, two technical methods are adopted. The first one requires the observer dynamic error system to be non-negative, and the second one relaxes this limitation by coordinate transformation. The sufficient conditions are established using both Lyapunov stability and positive system theory. Furthermore, according to the Schur complement, the linear matrix inequality is solved to determine the observer gains. Finally, the effectiveness and feasibility of the designed interval observer are verified by one numerical simulation.

Fault-Tolerant Consensus Control of Positive Networked Systems

In this paper, we explore the consensus of positive networked systems with actuator faults. Firstly, the undirected and strongly connected topology is established with graph theory. The positive system theory is used to analyze the positive consensus of the closed-loop networked systems. State feedback gains are derived utilizing Algebraic Riccati Inequalities. Bounded multiplicative faults are regarded as uncertainties in the system matrix, while treating additive faults as external disturbances. Further, this transformation refocuses the analysis on the consensus problem with an L2-gain. Subsequently, the Genetic Algorithm is employed to optimize the L2 performance criteria. Finally, the effectiveness of the proposed theory is validated through simulations involving both single-input electric circuit systems and multi-input networked systems.

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.

Flexible performance constraint-based control of a quadrotor UAV-suspended payload system under input saturation

In this article, a flexible performance constraint-based tracking control scheme is proposed for the anti-swing control of a quadrotor UAV-suspended payload system under external disturbance and input saturation. By applying system coupling relations and positive system theory, a self-regulating auxiliary system is constructed to generate correction signals to achieve the ability to regulate a prescribed performance function. The disturbance interval observer is employed to estimate the unknown disturbance. The interval width of the disturbance is used to design a continuous robust signal, enhancing the system's robustness. Additionally, it is with the intention of designing the controller for the under-actuated system to achieve effective tracking that the backstepping control scheme is used. It can always track the reference signal and satisfy flexible performance constraints based on theoretical analysis. Finally, the effectiveness of the proposed scheme is verified by experimental studies.

Delayed and Switched Deployment of Second-Order Agents on a Line Segment: Delays and Switches Do Not Matter

The problem of the equidistant deployment of mobile agents on a line segment is studied. It is assumed that agent dynamics are modeled by linear second-order models. Two types of control protocols are studied: with and without relative velocity measurements. For any positive damping coefficient of agents we propose a control protocol that ensures the achievement of the control goal and possesses robustness for any communication delays and any switching topology. The designed multi-agent system behavior is illustrated by numerical simulation. The proofs of the results are based on the theory of positive systems.

Event-triggered interval observer design for a class of Euler-Lagrange systems with disturbances

This paper investigates the problem of interval estimation for a class of Euler-Lagrange systems with unknown disturbances. An event-triggered mechanism is applied in the design of interval observers, and unnecessary data communication burden is reduced. The sufficient conditions are derived by both the positive system theory and the Lyapunov stability theory. Moreover, the gains of observer are determined by solving a set of inequalities in the sufficient conditions. At last, the validity of the presented event-triggered interval observers is demonstrated by a numerical simulation.

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.

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.

Outlier-resistant interval observer design for multirate time-delayed systems under the adaptive event-triggered protocols

In this paper, an interval observer has been presented for a type of multirate time-delayed systems subject to measurement outliers under the adaptive event-triggered protocol (AETP). First of all, the multirate sampling system is studied to meet the practical requirements. Meanwhile, for relieving the communication burden, an AETP is considered to schedule the measurement information received by the observer. The augmented system is built at a consistent sampling/update rate by using the lifting technique. Moreover, the occurrence of measurement outliers would pose some serious threats to the reliability of the systems. For the addressed issue, an outlier-resistant interval observer is developed by means of saturation function to weaken the impact of the measurement outliers. A sufficient condition is proposed by using the positive system theory and Lyapunov stability theory to ensure that the interval observer is effective and the error system is input-to-state stable. Besides, the gain matrices of the interval observer are obtained by solving a set of linear matrix inequalities (LMIs). Finally, the effectiveness of the proposed interval observer is verified by a numerical simulation.

Event-Triggered Control Under Unknown Input and Unknown Measurement Delays Using Interval Observers

We provide a new input-to-state stabilizing event-triggered feedback design for linear systems with unknown input delays, unknown measurement delays, and unknown additive disturbances. Our trigger times are computed using only the matrices defining the system and time-lagged sampled state values. We use the theory of positive systems, interval observers, and a vector version of Halanay’s inequality. We illustrate our method using a marine robotic model.

Definitions of Internal/External Positivity of Sampled-Data Systems and the Necessary and Sufficient Conditions

This paper extends the theory of positive systems to sampled-data systems. When considering positivity of LTI systems, where to take the initial time is unimportant. However, since sampled-data systems are periodically time-varying systems from the perspective of the continuous-time input and output signals, it is important to consider which time should be viewed as the initial time. Therefore, this paper studies positivity of sampled-data systems first by considering the case where a sampling instant is viewed as the initial time, and secondly by further considering the case where an arbitrary intersample instant is viewed as the initial time. The two different situations lead to quite relevant issues of what to take as the initial conditions and what to assume on the initial conditions in appropriately defining the internal/external positivity of sampled-data systems. These arguments are combined to derive the necessary and sufficient conditions for ultimate notions of (i.e., initial-time-independent) internal/external positivity of sampled-data systems.

External Positivity of Sampled-data Systems and Their <i>L<sub>q</sub></i>/<i>L</i><sub>∞</sub> Hankel Norm Analysis

This paper extends the theory of positive systems to sampled-data systems. When considering positivity of LTI systems, where to take the initial time is unimportant. However, since sampled-data systems are periodically time-varying from the perspective of continuous-time, introducing an adequate definition for their positivity should be relevant to considering which time to take as the initial time. This paper begins by studying positivity of sampled-data systems first by considering the case where a sampling instant is regarded as the initial time, and secondly by further considering the case where an arbitrary intersample instant is regarded as the initial time. These arguments are combined to derive the necessary and sufficient conditions for a unified notion of (i.e., initial-time-independent) positivity of sampled-data systems. This paper then applies this concept to the analysis of the Lq/L∞ Hankel norm, known as a measure for evaluating the effects of disturbances in dynamical systems. It is shown that positive sampled-data systems lead to concise expressions for the (quasi) Lq/L∞ Hankel norms not only for q = ∞ but also for q = 1,2, where the latter are hard to handle for general sampled-data systems. Some relevant properties for these norms are also shown. The theoretical results derived in this paper are confirmed through numerical examples.

Nonnegative Consensus Tracking of Networked Systems With Convergence Rate Optimization.

This article investigates the nonnegative consensus tracking problem for networked systems with a distributed static output-feedback (SOF) control protocol. The distributed SOF controller design for networked systems presents a more challenging issue compared with the distributed state-feedback controller design. The agents are described by multi-input multi-output (MIMO) positive dynamic systems which may contain uncertain parameters, and the interconnection among the followers is modeled using an undirected connected communication graph. By employing positive systems theory, a series of necessary and sufficient conditions governing the consensus of the nominal, as well as uncertain, networked positive systems, is developed. Semidefinite programming consensus design approaches are proposed for the convergence rate optimization of MIMO agents. In addition, by exploiting the positivity characteristic of the systems, a linear-programming-based design approach is also proposed for the convergence rate optimization of single-input multi-output (SIMO) agents. The proposed approaches and the corresponding theoretical results are validated by case studies.

Teaching Is a Story Whose First Page Matters—Teacher Counselling as Part of Teacher Growth

The starting point of the study is the work counselling process taking place in the induction phase of the students’ teaching career. The work counselling process consists of activities that support and strengthen the students’ teacher hood and their primary function as teachers by helping them to analyse their work and to attach to working life as newly graduated teachers. In this study, we use contextualisation and storytelling in the frameworks of positive pedagogy and systems theory to reflect on the students’ growth process as teachers. We also acknowledge the importance of Vygotsky’s zone of proximal development and, related to that, timely learning support (scaffolding) and a solution-focused approach as part of the supervision process utilising the methods of work counselling. In this study, autoethnography was used as a tool to help us reflect on our actions and work. Methods were chosen based on the thought that in the induction period, the students’ experiences function as mirrors for us supervisors, helping us to reflect on our actions and the supervision process and to form ideas and meanings based on the documented discourse data.

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.