Continuous-time Linear Systems Research Articles

Analysis of average consensus of multi-agent systems with time delay using packet selection and synchronization protocol

A stabilization approach for the average consensus of homogeneous multi-agent systems (MASs) entailing non-uniform, asymmetric, and time-varying delays in communication networks is proposed. Agents are modeled using a continuous-time linear dynamical system with a local discrete-time controller. To address communication delays, we propose two logic functions, namely, a packet selection algorithm that constantly selects and updates the latest received control signal packets and a synchronization algorithm that asymmetrically synchronizes these packets to preserve the average value of state variables. Based on the Lyapunov theory, we establish a linear matrix inequality (LMI) condition that sufficiently guarantees the stability and consensus of the overall system. To establish this condition, only the time delay upper bound is required, without any special assumptions on the delay pattern. We thoroughly discuss the proposed algorithm and evaluate its effectiveness via numerical simulations.

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.

Preview repetitive control with equivalent input disturbance for continuous‐time linear systems

This paper investigates the problem of preview repetitive control with equivalent-input-disturbance for a class of continuous-time linear systems in the presence of unknown external disturbances. First, an augmented delay system is constructed by using the nominal state equation with error system and the state equation of a modified repetitive controller, which is then transformed into a non-delayed one by state transformation. Next, by using optimal control theory, a preview repetitive control law is obtained. It is composed of state feedback, tracking error compensation, output of modified repetitive controller and preview compensation. Furthermore, in order to achieve a good disturbance estimation and attenuation performance, by applying the results of the standard equivalent-input-disturbance theory, a preview repetitive control law with equivalent-input-disturbance is offered for the uncertain system with unknown external disturbance. Finally, the numerical simulation example is provided to illustrate the effectiveness of the proposed method.

A complementary learning approach for expertise transference of human-optimized controllers

In this paper, a complementary learning scheme for experience transference of unknown continuous-time linear systems is proposed. The algorithm is inspired in the complementary learning properties that exhibit the hippocampus and neocortex learning systems via the striatum. The hippocampus is modelled as pattern-separated data of a human optimized controller. The neocortex is modelled as a Q-reinforcement learning algorithm which improves the hippocampus control policy. The complementary learning (striatum) is designed as an inverse reinforcement learning algorithm which relates the hippocampus and neocortex learning models to seek and transfer the weights of the hidden expert’s utility function. Convergence of the proposed approach is analysed using Lyapunov recursions. Simulations are given to verify the proposed approach.

Bounded real lemma for singular linear continuous-time fractional-order systems

This paper proposes novel necessary and sufficient strict linear matrix inequalities (LMIs), to characterize admissibility of singular fractional-order linear continuous-time systems with the fractional derivative of order α belonging to 1≤α<2. Then, the problem of the bounded real lemma corresponding to the H∞ norm computation is addressed involving additional variables. Necessary and sufficient conditions are established via a set of LMIs that can be effectively used to design H∞ controllers. Based on the corresponding bounded real lemma, a state feedback control with a prescribed H∞ performance index for the underlying systems is proposed. Finally, numerical examples are provided to show effectiveness of the given results.

Optimal control of unknown continuous time linear periodic systems by adaptive dynamic programming with applications to magnetic attitude control

AbstractThis article studies the data‐driven controller design of continuous‐time linear periodic systems. By treating the optimal control problem as a two‐player zero‐sum game problem, the adaptive dynamic programming technique is utilized. Based on the Newton's method, a one‐loop model‐based policy‐iteration (PI) method is introduced. Since the initial matrix for starting the proposed model‐based PI method is not easy to choose, a data‐driven value iteration method without requiring the knowledge of the initial matrix is then proposed. The convergence of the proposed methods is analyzed and the numerical simulation verifies the effectiveness of the proposed scheme.

A quasi‐differential type event‐triggered model predictive control for perturbed continuous linear systems with constraints

A quasi-differential type event-triggered model predictive control (ET-MPC) framework for continuous-time linear systems with additional disturbances is constructed. Different from the existing ET-MPC, the triggering condition of the proposed method is focused on the differences of the errors between the actual states and the best prediction sequence at two consecutive sampling moments. Its advantage is that the dynamic characteristics of state changes can be better considered, which will achieve a more effective balance between communication load and system control performance. Then, to deal with persistent disturbances, a time-varying tightening robustness state constraint is utilized in this scheme, instead of conventional state constraints. Finally, the new ET-MPC algorithm is designed based on the dual-mode control framework, and its feasibility, the system closed-loop stability, as well as avoidance of Zeno behaviour are rigorously verified. Simulations and comparisons are provided to demonstrate the algorithm effectiveness.

Distributed Observer Design for Linear Systems under Time-Varying Communication Delay

The paper investigates the state estimation problem of general continuous-time linear systems with the consideration of time-varying communication delay. A solution is proposed in terms of the networked distributed observer, which consists of multiple local observers. Each local observer relies on only part of the system output and exchanges information with neighbors through undirected links modeled by a prespecified communication graph. A simple approach for computing observer parameters is presented by solving a parametric algebraic Riccati equation. Furthermore, by the Lyapunov–Krasovskii stability theorem, an upper bound of the delay could be calculated explicitly and together with the conditions of joint observability and connectivity of the communication graph; the resulting distributed observers work coordinately to achieve an asymptotic estimate of the full plant state. An illustrative example is provided to confirm the analytical results.

Identification of the Thermoelectric Cooler Using Hybrid Multi-Verse Optimizer and Sine Cosine Algorithm Based Continuous-Time Hammerstein Model

Abstract This paper presents the identification of the ThermoElectric Cooler (TEC) plant using a hybrid method of Multi-Verse Optimizer with Sine Cosine Algorithm (hMVOSCA) based on continuous-time Hammerstein model. These modifications are mainly for escaping from local minima and for making the balance between exploration and exploitation. In the Hammerstein model identification a continuous-time linear system is used and the hMVOSCA based method is used to tune the coefficients of both the Hammerstein model subsystems (linear and nonlinear) such that the error between the estimated output and the actual output is reduced. The efficiency of the proposed method is evaluated based on the convergence curve, parameter estimation error, bode plot, function plot, and Wilcoxon’s rank test. The experimental findings show that the hMVOSCA can produce a Hammerstein system that generates an estimated output like the actual TEC output. Moreover, the identified outputs also show that the hMVOSCA outperforms other popular metaheuristic algorithms.

Padé-Approximation-Based Preview Repetitive Control for Continuous-Time Linear Systems

This paper concerns a Padé-approximation-based preview repetitive control (PRC) scheme for continuous-time linear systems. Firstly, the state space representation of the repetitive controller is transformed into a nondelayed one by Padé approximation. Then, an augmented dynamic system is constructed by using the nominal state equation with the error system and the state equation of a repetitive controller. Next, by using optimal control theory, a Padé-approximation-based PRC law is obtained. It consists of state feedback, error integral compensation, output integral of repetitive controller, and preview compensation. Finally, the effectiveness of the method is verified by a numerical simulation.

Adaptive gain-scheduling control for continuous-time systems with polytopic uncertainties: An LMI-based approach

We develop a new adaptive gain-scheduling control scheme for continuous-time linear systems with polytopic uncertainties. The gain-scheduled control law is proposed as a convex sum of a fixed set of controller gains, exploiting the polytopic representation of the system uncertainty, which is not possible with classical robust control results in the literature. To realize this scheme, an adaptation law is proposed to adaptively provide the tuning parameter for the gain-scheduling implementation. The admissible domain of the stabilizing control feedback gains, defined by the fixed set of controller gains, can be determined offline by solving a set of linear matrix inequality constraints over a scalar line search. Using Lyapunov-based arguments, the proposed design conditions and the adaptation law ensure that all closed-loop signals are bounded. In particular, if the uncertain parameters are not time-varying, then the system states asymptotically converge to the origin. Theoretical arguments and appropriate numerical illustrations are provided to demonstrate the effectiveness of the proposed control scheme.

Sampled-data-based fault-tolerant consensus control for multi-agent systems: A data privacy preserving scheme

This paper is concerned with the fault-tolerant consensus control problem for a general class of linear continuous-time multi-agent systems (MASs) with the data privacy preserving (DPP) constraints. Rather than employing the traditional cryptographic mechanism, we develop a novel dynamic-quantization-based DPP scheme from a control-theoretic perspective. Under our developed DPP scheme, the sampled, quantized, released, received and recovered data is used to design a fault-tolerant controller. With the adopted controller structure, the consensus of the MASs is guaranteed where the size of the data transmitted among agents can be made finite. A series of numerical examples are given and the comparative analysis is carried out to illustrate the effectiveness of the developed consensus control scheme.

Comparison principle for positive time-delay systems: An extension and its application

This paper proposes an extended comparison principle for continuous-time linear positive time-delay systems. Unlike the existing comparison principle, which uses a constant initial function for the comparison system, instead, we propose a time-varying initial function in order to derive a more general solution comparison. Based on this extended comparison principle, we develop a novel computational method, which exploits more effectively the information of the initial value function, to derive tighter exponential estimates for the state vector of positive time-delay systems. The effectiveness of our developed method is illustrated through two numerical examples.

Derivative-Free Observability Analysis of a Stochastic Dynamical System

This paper proposes a novel approach to analyze the observability of a stochastic linear or non-linear dynamical system. Unlike the traditional observability analysis approach, our approach is fully derivative-free, resulting in a low complexity and an easy implementation. Furthermore, it can not only analyze the observability of each system state, but also it can assess the effect of the observation and the system noise on the system observability, which is not considered by the traditional approach. Specifically, using the generalized polynomial chaos expansion, an observability-coefficient matrix is first calculated, which enables us to judge whether the system is observable or not. Then, the puny and brawny observability indexes are assessed to quantify the degree of observability. The equivalence between the conventional method and the proposed method for a continuous-time linear time-invariant system is proved in the appendix. The effectiveness of the proposed approach is mathematically proved and its good performance is demonstrated in several test cases.

Integral reinforcement learning-based optimal output feedback control for linear continuous-time systems with input delay

In this paper, an integral reinforcement learning (IRL)-based model-free optimal output-feedback (OPFB) control scheme is developed for linear continuous-time systems with input delay, where the input and past output data are employed rather than the system dynamic model. First, the equivalence between the delayed optimal control and delay-free case is analyzed. Subsequently, the system state is constructed with output signal and the Bellman equation is written in the form of past outputs. Therefore, the IRL algorithm is developed to learn the OPFB control policy, where the iterative policy is evaluated and improved simultaneously. It is proved that the obtained optimal OPFB controller gives the same solution as the optimal state-feedback. Finally, the presented simulation results illustrate the effectiveness of the developed control method.

Interval estimation based on the reduced-order observer and peak-to-peak analysis

An interval estimation method based on the reduced-order observer and peak-to-peak analysis is proposed for continuous-time linear time-invariant systems with disturbance and measurement noise. The proposed method consists of two steps. First, a reduced-order observer with performance is designed to obtain point estimation. Second, interval estimation is achieved by integrating the obtained point estimation and the error interval estimation by peak-to-peak analysis. Steady-state gain optimisation in terms of linear matrix inequalities is used in both steps to improve the estimation accuracy of the proposed method. The superiority of the method over the reduced-order interval observer is illustrated through numerical simulations.

Continuous-Time Linear Periodic Systems: A Note on the Controllability Interval Length

Continuous-time linear systems with periodic coefficients of period T are considered in this paper. We study the question of the controllability interval length of periodic systems. It is known that a controllable T-periodic system is controllable over intervals of the length kT where k is the degree of the minimal polynomial of the monodromy matrix. It is proved that this boundary kT is exact and cannot be decreased. It is proved that controllability of a T-periodic system on some interval of the length sT depends on the initial moment, in general.

Exact bounded-error continuous-time linear state estimator

This paper proposes an interval-based method for estimating the state of a linear continuous-time dynamical system. In this work, we assume that the measurements are provided at discrete times and that all errors are bounded. Interval analysis is used to propagate the interval uncertainties continuously over time. The resulting method is guaranteed to never lose any feasible solution and provides an optimal polygonal enclosure of the state trajectory. A reproducible example illustrates the principle of the method.

Modified spectral method for optimal estimation in linear continuous-time stochastic systems

The spectral method to solve estimation problems for linear continuous-time stochastic systems with polynomial measurements is presented. It is based on both the spectral form of mathematical description (the representation of deterministic functions and random processes by orthogonal series) and the particle filter. The main goal of this work is to implement the continuous-time particle filter without a time discretization. The proposed spectral method provides the possibility to solve estimation problems such as filtering, smoothing and prediction.

Commutativity of Sixth-Order Time-Varying Linear Systems

This paper studies commutativity of sixth-order differential systems with zero and nonzero initial conditions (ICs). Given a sixth-order differential system $$A$$ , we find its commutative pair, that is a new differential system $$B$$ of order $$m \le 6$$ ; the obtained result will be used to show the input–output equivalency between the cascaded-connected systems $$AB$$ and $$BA$$ . Necessary and sufficient conditions for commutativity of sixth-order continuous-time linear time-varying systems (CTLTVSs) are derived. Based on our findings, we discovered that only a small group of sixth-order CTLTVSs has commutative pairs other than their feedback conjugates obtained by constant feedback and forward path gains. The explicit results obtained in this paper will help to fill the gap on commutativity theory, system behaviors which has significant application to science and play an important role in engineering. This research will also address the problems of sensitivity due to changes in ICs or parameters, and also disturbance as a result of external noise. The results are well verified by examples as well as validated by Matlab Simulink tool and Wolfram Mathematica 11.