Continuous-time Linear Systems Research Articles

On LMI conditions to design robust static output feedback controller for continuous-time linear systems subject to norm-bounded uncertainties

This paper addresses the problem of Static Output Feedback (SOF) stabilisation for continuous-time linear systems subject to norm-bounded parameter uncertainties using the Linear Matrix Inequality (LMI) approach. Usually, this issue leads to the feasibility of a Bilinear Matrix Inequality (BMI), which is difficult to linearise to get non-conservative LMI conditions. We present first, in this paper, some background results on the SOF controller design and that are found to be extended to the case of norm-bounded uncertainties. We show that some restrictions on the feasibility of these results should be guaranteed. Moreover, by means of some technical Lemmas, we transform the BMI into a new LMI with a line search over a scalar variable. An enhanced and less conservative LMI condition with a line search over two scalar variables is also developed. Furthermore, a simplified version of each LMI condition without a priori fixed parameters is also presented. An extensive portfolio of numerical examples is presented in order to evaluate the conservativeness and to show the superiority of the proposed design method to the background results.

Asymptotically optimal consensus-based distributed filtering of continuous-time linear systems

We describe a consensus-based distributed filtering algorithm for linear systems with a parametrized gain and show that when the parameter becomes large the error covariance at each node becomes arbitrarily close to the error covariance of the optimal centralized Kalman filter. The result concerns distributed estimation over a connected un-directed or directed graph and for static configurations it only requires to exchange the estimates among adjacent nodes. A comparison with related approaches confirms the theoretical results and shows that the method can be applied to a wide range of distributed estimation problems.

Consensus of Continuous-Time Linear Multiagent Systems With Discrete Measurements.

This article concerns the robust consensus problem of continuous-time linear multiagent systems (MASs) with uncertainty and discrete-time measurement information, where the output measurement information is in the data-sampled form. Distributed output-feedback protocol with or without controller interaction is proposed for each agent. Specifically, the output-feedback protocol runs in continuous time with an output error correction term mixed with the discrete-time measurement information. The concrete algorithm is given for the construction of the feedback matrices. Then, by using the delay-input approach, sufficient conditions are provided for the robust consensus of this kind of MASs interacting over networks described by the directed graphs. Finally, numerical simulations are given to illustrate the theoretical results.

Robust decentralized switched controller design for linear continuous-time systems

Abstract The paper is devoted to obtain original equivalent subsystem method to design of decentralized controller for linear large scale systems. On the theoretical example a new robust decentralized PID switched controller design procedure is obtained for linear time-varying (gain scheduled plant model) uncertain complex system with decentralized output and input structure. Controller design procedure to decentralized controller design performs on the subsystem level. The designed decentralized switched controller ensures the robust stability of closed-loop complex polytopic system with performance H 2 quadratic cost function (QSR). The proposed practical examples with ideal or non-ideal switch of switching parameters show the effectiveness of equivalent subsystem approach.

Simultaneous fault detection and containment control design for multi-agent systems with multi-leaders

The problem of simultaneous fault detection and containment control of linear continuous-time multi-agent systems with multi-leaders is firstly addressed in this paper. Distributed detection filters are designed using relative output information and local input information among agents to detect the actuator faults and sensor faults in agents, and H∞/H− formulation for this problem is presented in the finite frequency domain. In order to solve the complexity problems caused by multiple leaders in the system, the influence of leaders on the residual signals of followers is taken into account in the design of the proposed detection filter, and through the reasonable design of the controller, the containment control of the followers can also be achieved simultaneously. The controller and filter parameters are obtained by solving the optimization problem with linear matrix inequalities(LMI) constraints. Furthermore, faults are allowed to occur in leaders in this paper, and an effective algorithm is proposed to detect and isolate faults in the system. At last, an adjustable filter structure is proposed to prevent the followers from being influenced by the failure leader, which is of great practical significance. Finally, to illustrate the effectiveness of the proposed methodology, a simulation study is provided.

Robust finite‐frequency filter design for linear uncertain systems using polynomially parameter‐dependent approach

This study investigates the problem of robust filter design for continuous-time linear uncertain systems affected by external noises in finite-frequency (FF) ranges. The objective is to design an admissible filter guaranteeing asymptotic stability of the filtering error system with a prescribed FF noise attenuation level. To this end, homogenous polynomially parameter-dependent Lyapunov functions are used to assess the stability of the filtering error system, and robust FF bounded realness lemma is used to describe the noise attenuation specifications. Polynomially parameter-dependent matrices and some scalars are introduced to facilitate filter design and reduce conservatism. Moreover, a sequential convex approximation method is used to replace the non-convex constrained set with a convex feasible one. On this basis, an iterative algorithm is developed to calculate the filter parameters. It is demonstrated that the proposed robust FF filter can achieve a better noise attenuation performance in the FF range than the existing ones in the literature. An example about F-18 aircraft model is given to illustrate the effectiveness of the proposed method.

On construction of sampling patterns for preserving observability/controllability of linear sampled-data systems

ABSTRACT In this paper, we design a new nonequidistant sampling pattern such that the property of controllability and observability of the linear continuous time systems is preserved during the procedure of the sample and zero-order hold operation. The idea behind the strategy comes from Kronecker's theorem and it is proven that new construction is more effective by requiring less number of sampling times which guarantees controllability and observability of sampled-data system in two dimensional space. Moreover, we provide one application in the concept of controllability of the pendulum in two dimensional space and obtain corresponding digital controls which make the corresponding sampled-data system controllable. Finally, the research is extended to different type of sampling patterns which are compared with the constructed sampling pattern separately.

Parallel control for continuous-time linear systems: A case study

In this paper, a new parallel controller is developed for continuous-time linear systems. The main contribution of the method is to establish a new parallel control law, where both state and control are considered as the input. The structure of the parallel control is provided, and the relationship between the parallel control and traditional feedback controls is presented. Considering the situations that the systems are controllable and incompletely controllable, the properties of the parallel control law are analyzed. The parallel controller design algorithms are given under the conditions that the systems are controllable and incompletely controllable. Finally, numerical simulations are carried out to demonstrate the effectiveness and applicability of the present method.

An exploration of the Razumikhin stability theorem with applications in stabilization of delay systems

In this paper, we establish a new evolution property of Lyapunov functions that satisfy the Razumikhin Stability Theorem conditions. The derived property is shown to be a valid alternative criterion to prove the asymptotic stability of a delay system without calculating the derivative of the Lyapunov function, and it is especially useful in solving stabilization problems for time delay systems when the delay is unknown. As an example application of this derived property, we investigate the stabilization of continuous-time linear systems with an unknown time-varying delay, by proposing a switched low gain feedback control technique that utilizes the derived property. Existing literature on stabilization of continuous time delay systems all require some knowledge, such as an upper bound, of the delay. In this work, by proposing a switched low gain feedback control law utilizing the derived property, we are able to achieve asymptotic stabilization of the system without any knowledge of the delay. Furthermore, the proposed switched low gain feedback law also significantly improves the closed-loop system performance in terms of overshoot and convergence speed, in comparison with the traditional fixed gain low gain feedback design. Simulation study verifies the theoretical results.

On Assignment of the Upper Bohl Exponent for Linear Time-Invariant Control Systems in a Hilbert Space by State Feedback

We consider a linear continuous-time control system with time-invariant linear bounded operator coefficients in a Hilbert space. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary assignment for the upper Bohl exponent by state feedback control. We prove that if the open-loop system is exactly controllable then one can shift the upper Bohl exponent of the closed-loop system by any pregiven number with respect to the upper Bohl exponent of the free system. This implies arbitrary assignability of the upper Bohl exponent by linear state feedback. Finally, an illustrative example is presented.

Superstabilization of Descriptor Continuous-Time Linear Systems via State-Feedback Using Drazin Inverse Matrix Method

In this paper the descriptor continuous-time linear systems with the regular matrix pencil ( E , A ) are investigated using Drazin inverse matrix method. Necessary and sufficient conditions for the stability and superstability of this class of dynamical systems are established. The procedure for computation of the state-feedback gain matrix such that the closed-loop system is superstable is given. The effectiveness of the presented approach is demonstrated on numerical examples.

Supervisory Nonlinear State Observers for Adversarial Sparse Attacks.

This article investigates the problem of secure state estimation for continuous-time linear systems in the presence of sparse sensor attacks. Compared with the existing results, the attacked sensor set can be changed by adversaries against secure estimation. To address the more erratic attacks, a novel supervisory state observer is proposed, which employs a bank of candidate nonlinear subobservers and a switching logic administrated by a monitoring function to select the active subobserver at every instant of time. Based on the stability analysis of switched systems, it is proven that the supervisory observer asymptotically converges to a neighborhood of the true system state in the presence of the sensor attacks and bounded disturbances. A simulation example is given to substantiate the theoretical results.

Adaptive Dynamic Programming for Model-Free Global Stabilization of Control Constrained Continuous-Time Systems.

This article addresses the problem of global stabilization of continuous-time linear systems subject to control constraints using a model-free approach. We propose a gain-scheduled low-gain feedback scheme that prevents saturation from occurring and achieves global stabilization. The framework of parameterized algebraic Riccati equations (AREs) is employed to design the low-gain feedback control laws. An adaptive dynamic programming (ADP) method is presented to find the solution of the parameterized ARE without requiring the knowledge of the system dynamics. In particular, we present an iterative ADP algorithm that searches for an appropriate value of the low-gain parameter and iteratively solves the parameterized ADP Bellman equation. We present both state feedback and output feedback algorithms. The closed-loop stability and the convergence of the algorithm to the nominal solution of the parameterized ARE are shown. The simulation results validate the effectiveness of the proposed scheme.

Event-triggered observer design for continuous-time linear networked systems

In this paper, a novel event-triggered observer is proposed to decrease the information exchange in continuous-time linear networked systems. The structure of the proposed observer consists of two main parts. The first one, called the event-triggered mechanism (ETM), is placed by the system and is used to determine when the measured output of the system must be sent through the network. The ETM consists of a traditional Luenberger observer, an event-triggered local observer, and an event-detector. The second part is a copy of the event-triggered local observer which is used to estimate the system state using the transmitted data from the sensors. Under the observability assumption, it is proved that the error between the system state and its estimation is asymptotically limited to a predetermined bound. Moreover, it is guaranteed that the Zeno behaviour is avoided. Numerical simulations are provided to demonstrate the design procedure of the proposed event-triggered observer.

Adaptive Rational Orthogonal Basis Functions for Identification of Continuous-Time Systems

In this article, we present two algorithms for identification of continuous-time linear time-invariant systems in frequency domain. These algorithms are constructed in terms of continuous rational orthogonal basis functions. First, a two-stage algorithm is developed through one-by-one selection of poles for the basis functions, and such consecutive selection is easy to realize. Next, a direct algorithm is proposed to select poles for the rational orthogonal basis functions. Different poles result in different bases and these selections guarantee that better approximations can be reached. For different systems, there are different sequences of poles selected for basis functions, which shows the adaptivity of the proposed algorithms. A numerical example is given to show that the proposed algorithms are useful. Also, in this example, comparison is made with the method that uses rational orthogonal basis functions in which all poles of the basis functions are true poles of the original system.

Reinforcement learning for adaptive optimal control of continuous-time linear periodic systems

This paper studies the infinite-horizon adaptive optimal control of continuous-time linear periodic (CTLP) systems, using reinforcement learning techniques. By means of policy iteration (PI) for CTLP systems, both on-policy and off-policy adaptive dynamic programming (ADP) algorithms are derived, such that the solution of the optimal control problem can be found without the exact knowledge of the system dynamics. Starting with initial stabilizing controllers, the proposed PI-based ADP algorithms converge to the optimal solutions under mild conditions. Application to the adaptive optimal control of the lossy Mathieu equation demonstrates the efficacy of the proposed learning-based adaptive optimal control algorithm.

Improved delay-dependent stability criteria for linear systems with multiple time-varying delays

This paper concentrates on the stability analysis of continuous-time linear systems with multiple time-varying delays. The time-delays are assumed to be bounded and have or not lower and upper bounds on their time-derivative. Based on the Lyapunov-Krasovskii theory the proposed results formulated as linear matrix inequality conditions can verify the delay-intervals for which the system stability is preserved. The main results follow from developing new augmented Lyapunov-Krasovskii functionals that exploit the benefits of the Bessel-Legendre-based integral inequality and the reciprocally convex approach. Also, the proposed functionals allow us to take advantage of the relationships between the current and delayed states. As a result, the proposed stability conditions outperformed similar existing ones in all numerical tests accomplished in this paper.

Event-triggered output consensus for linear multi-agent systems via adaptive distributed observer

This paper investigates the distributed event-triggered cooperative output regulation problem for heterogeneous linear continuous-time multi-agent systems (MASs). To eliminate the requirement of continuous communication among interacting following agents, an event-triggered adaptive distributed observer is skillfully devised. Furthermore, a class of closed-loop estimators is constructed and implemented on each agent such that the triggering times on each agent can be significantly reduced while at the same time the desired control performance can be preserved. Compared with the existing open-loop estimators, the proposed estimators can provide more accurate state estimates during each triggering period. It is further shown that the concerned cooperative output regulation problem can be effectively resolved under the proposed control scheme and the undesirable Zeno behavior can be excluded. Finally, the effectiveness of the proposed results is verified by numerical simulations.

Optimal preview repetitive control with application to permanent magnet synchronous motor drive system

In this paper, the problem of optimal preview repetitive control (OPRC) for continuous-time linear systems is proposed. First, an augmented dynamic system including the derivative of the state vector and the output of the basic repetitive controller is constructed by using lift technique. Then, a new performance index is denoted, which transforms the tracking problem into a linear quadratic regulation problem. Based on the optimal control theory, the explicit optimal preview repetitive controller is obtained, which combination of the state feedback, the integral action related to the tracking error, the basic repetitive controller and the preview compensation. Finally, the developed design techniques are applied to permanent magnet synchronous motor (PMSM) speed control problem. The simulation results demonstrate the validity of the proposed method.

Observer-based robust exponential stabilization for linear systems with parameter uncertainties

This paper is concerned with the robust observer-based control problem for a class of continuous-time linear systems with parameter uncertainties. A less conservative approach is provided to deal with the quadratic terms involving in the Lyapunov stability analysis for the estimation error dynamic system. Sufficient conditions that ensure the exponential stability of the closed-loop system are established and formulated in terms of linear matrix inequalities. The obtained results are less conservative than some existing ones in recent literature. Finally, two numerical examples are given to show the validity and superiority of the proposed observer-based robust controller.