Continuous-time Linear Systems Research Articles

Robust implementable regulator design of linear systems with non-vanishing measurements

Robust implementable output regulator design approaches are studied for linear continuous-time systems with periodically sampled measurements, consisting of both the regulation errors and extra measurements that are generally non-vanishing in steady state. A digital regulator is first developed via the conventional emulation-based approach, rendering the regulation errors asymptotically bounded with a small sampling period. We then develop a hybrid design framework by incorporating a generalized hold device, which transforms the original problem into the problem of designing an output feedback controller fulfilling two conditions for a discrete-time system, given any (large) non-pathological sampling period. In our hybrid design framework, we show that such a controller can always be obtained by designing a discrete-time internal model, a discrete-time washout filter, and a discrete-time output feedback stabilizer. As a result, the regulation errors are shown to be globally exponentially convergent to zero. This design framework is further developed for a multi-rate digital regulator with a large sampling period of the measurements and a small control execution period.

Event-based state estimation under constrained bit rate: An encoding–decoding approach

This paper is concerned with the state estimation problem for a class of linear continuous-time systems under bit rate constraints. The data transmission between the sensors and the estimator is implemented through a digital communication network with limited bandwidth. By resorting to the singular value decomposition technique, an event-based encoding strategy is developed to encode the measurement signals into 1-bit codewords so as to reduce network resource consumption. The Zeno phenomenon is first proven to be excluded, and then a sufficient condition is established under which the decoding error is ultimately bounded. Subsequently, a necessary condition is proposed to derive a lower bound of the bit rate below which the decoding error diverges. Moreover, the explicit expression of the desired state estimator is parameterized and the boundedness of the estimation errors is also analyzed. Finally, two simulation examples are presented to validate the effectiveness of the proposed theoretical results.

Practical Prescribed-Time Sampled-Data Control of Linear Systems With Applications to the Air-Bearing Testbed

This article studies the practical prescribed time sampled-data control problem for input-constrained linear time invariant continuous time systems. By using the solution to a parametric Lyapunov equation, a bounded linear time-varying (LTV) sampled-data controller is designed to solve such a problem. In addition, stability analysis and performance analysis of the closed-loop sampled-data control system are presented by using the methodology of scalarization. Finally, the developed LTV sampled-data state controller is utilized to the whole physical simulation system of air-bearing testbed and experimental results verify the effectiveness of the proposed approach.

Global Cooperative Output Regulation of Linear Multiagent Systems With Limited Bandwidth

This article investigates the global cooperative output regulation problem of continuous-time linear multiagent systems (MASs), under the communication constraint of limited bandwidth. With a novel scaling function strategy, a distributed protocol based on the sampled and quantized data is designed to solve this problem. Compared with the existing works on quantized control of MASs, not only the boundedness assumption on initial conditions is removed, but also the communication graph is not restricted to be balanced or strongly connected. Moreover, the proposed quantized cooperative output regulation protocol does not rely on any global information of the communication graph. A simulation example is finally presented to illustrate the effectiveness of the proposed quantized cooperative output regulation protocol.

Robust boundary of stability and design of robust controller in uncertain polytopic linear systems

Abstract This paper is devoted to design of a robust controller for an uncertain continuous-time linear polytopic system. The design procedure will take place in two steps. In the first step, we calculate the robust stability boundary of a closed loop system. Then on the basis of the result obtained in this way, we select a robust controller design method that accepts the calculated stability boundary. This procedure allows better implementation of the second step of the designed robust controller. Two examples of application of the proposed method follow finally, which illustrate its effectiveness.

Distributed Sampled-Data Nonfragile Consensus Filtering Over Sensor Networks With Topology Switching and Transmission Delay

This article concerns the distributed sampled-data nonfragile consensus filter design for a continuous-time linear system with respect to exogenous disturbance. A network of sensor nodes is employed to monitor the plant. The information from each sensor node and its neighbors is aperiodically sampled while collaboratively transmitted in the network, where the phenomena of switching directed topologies and network-induced transmission delay are unavoidably existent. The estimated state of the plant at each node is updated by a consensus of state estimates from its neighbors. By acquiring the sensor output measurement, a Luenberger-type filter is constructed not only to provide robustness against some level of filter gain perturbations, but also to guarantee the asymptotic stability of the resultant filtering error system with an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> performance requirement. Based on algebraic graph theory, nonfragile synthesis technique and Lyapunov–Krasovskii functional method, the filter parameters are characterized in terms of some feasible solutions to certain linear matrix inequalities. The theoretical analysis is validated by numerical simulations.

Consensus error analysis for linear continuous-time multi-agent systems with additive noises

This paper investigates the distributed consensus problem for linear continuous-time multi-agent systems subject to additive process noises. Due to the existence of noises, the asymptotic consensus cannot be achieved. Hence, a distributed consensus protocol is proposed such that all the agents can achieve consensus to some degree. Furthermore, based on Lyapunov theory, we characterize the consensus error under undirected and directed topologies. Numerical examples are presented to verify the effectiveness of two results.

Sampled-data distributed state estimation with multiple transmission channels under denial-of-service attacks

This paper is concerned with the problem of sampled-data distributed state estimation for a linear continuous-time system with multiple transmission channels under denial-of-service attacks. To solve this problem, sampled-data distributed observers, which only have access to their respective measurable output variables and the information obtained from their neighbors, are designed, where a decomposition form of the coefficient matrices of the system and a time interval division-based method are proposed to deal with the complex interaction of the sampled-data schemes and the attack schemes among different transmission channels. Different from the previous results, the transmission channels from the sensors to the remote observers do not share the same sampled-data scheme and are allowed to be compromised by different adversaries. An example is provided to validate the effectiveness of the proposed method.

Robust [formula omitted] and [formula omitted] control for positive continuous-time uncertain linear systems

This paper addresses the problems of H2 and H∞ control design for positive continuous-time linear systems with uncertain parameters belonging to a polytope. Although it is well known that the optimal H∞ state-feedback gain can be obtained from the bounded real lemma condition with a diagonal Lyapunov matrix without any conservatism for precisely known positive linear systems, the same is not true for the H2 norm, neither for output-feedback control. When uncertainties affect the plant, a fixed diagonal structure for the Lyapunov or any other matrix variable involved in the conditions can produce poor results in terms of H2 and H∞ guaranteed costs, specially if the matrix constrained to be diagonal is involved in the computation of the control gain. In this sense, sufficient linear matrix inequalities (LMIs) for the existence of a stabilizing gain and an H2 (or H∞) guaranteed cost are proposed. As main characteristic, the gain and the Lyapunov matrix can be treated as independent optimization variables in the conditions. Using relaxations in the stability of the system, an algorithm is constructed to iteratively solve the LMIs. Therefore, no change of variable is necessary and, differently from other LMI-based methods for positive systems from the literature, there is no need to impose a diagonal structure on the matrix used to recover the gain to enforce closed-loop positivity. Moreover, static output- or state-feedback controllers, including bounded entries or decentralized structures, can be computed in a straightforward way. With an appropriate initialization, the iterative procedure, implemented in two phases, is assured to end up, at the first step, with a finite bound for the real part of the eigenvalues of the closed-loop positive system. Whenever a stabilizing gain is obtained, the second step provides non-increasing H2 (or H∞) guaranteed costs along the iterations. Numerical examples show that the proposed technique provides good estimates for the H2 and H∞ norms when compared with other methods available in the literature.

Theoretical and practical aspects of the convergence of the SRIVC estimator for over-parameterized models

The Simplified Refined Instrumental Variable method for Continuous-time systems (SRIVC) is one of the most popular direct methods for linear continuous-time system identification. Only recently, some optimal asymptotic properties of this method have been proven. Here we provide a comprehensive analysis of the convergence of the SRIVC method for over-parameterized models. The results we derive are used to discuss practical aspects of the method and to analyze the behavior of the normalized error variance norm, which is a central part in Young’s Information Criterion for model order selection. The theoretical findings are validated through simulation examples.

High-gain interval observer for continuous–discrete-time systems using an LMI design approach

In this paper, a high-gain interval observer (HGIO) for a class of partially linear continuous-time systems with sampled measured outputs in the presence of bounded noise and additive disturbances is proposed. The design of the HGIO is formulated in the Linear Matrix Inequality (LMI) framework. The gain of the HGIO is designed to satisfy the cooperative property using a time-varying change of coordinates based on pole placement in separate LMI regions. Moreover, a procedure for designing the HGIO gain to minimise the effect of the noise and disturbance in the estimation is provided. The stability of the proposed HGIO is also guaranteed. The proposed approach is assessed in simulation using a numerical example.

Zero-order hold discretization of general state space systems with input delay

Abstract Plethora of applications in physics and engineering are dealing with systems that are subject to input delays. Despite the scientific focus, previous research investigated only state space systems with input delays. Here, we investigate the discretization of generalized state space system with input delay by using the zero-order hold method. Firstly, the solution of a continuous time, generalized state space system with input delay is addressed. By applying the appropriate zero-order hold sampling method, we transform a continuous time linear system into the equivalent discrete time system, while the discrete time solution of the equivalent system is analytically presented. Finally, we use local error metric to estimate the difference between the continuous time and discrete time solutions.

Synchronization analysis of linear time-varying matrix-weighted coupled systems and its applications

In this paper, we study the synchronization problem of continuous-time linear systems under time-varying matrix-weighted coupling. With the help of the notion of persistent excitation and an adapted stability result in adaptive control, we establish two sufficient exponential synchronization conditions using the connectivity of the persistent graph and the joint connectivity of the instantaneous time-varying graphs. The conditions recover those for the single-integrator consensus model with scalar-valued coupling. The results are further applied to solve the synchronization problem of continuous-time linear systems with time-varying and nonidentical coupling matrices, extending the existing conclusions in the literature. Finally, numerical examples are given to verify the derived results.

Inverse variational principles for control systems

Inspired by Katok’s idea in defining measure-theoretic entropy, we introduce several measure-theoretic invariance entropies in control systems via spanning sets. Properties of them are investigated. Especially, we show that the measure-theoretic invariance entropies of a given set can be characterized by its relatively invariant subsets. More importantly, we obtain two inverse variational principles connecting measure-theoretic invariance entropy and topological invariance entropy, which help us to establish two formulae for the measure-theoretic invariance entropies of control sets for linear control systems, including linear continuous-time and discrete-time systems.

Bidirectional fragmentation approach on the stability analysis of sampled-data linear systems

In this paper, a novel stability analysis method for continuous-time linear systems with aperiodic sampling is presented by using a partitioned looped-functional approach. Based on the Lyapunov function with a new looped-functional, the stability condition is derived by using a bidirectional fragmentation of a sampling interval from to , which is divided into two sides forward and backward directions based on time t. For the bidirectional regions, the new looped functional is constructed by taking into account relations between states defined at the fragmented time and the sampling time. The improvement using the proposed scheme is shown through the comparisons with existing methods.

Model predictive control of switching continuous‐time systems with stochastic jumps: Application to an electric current source

This paper proposes an extension of the model predictive control framework for switching continuous-time linear systems. The switching times follow a stochastic process with limited statistical information. At each switching time, the controller knows the system state, but it is blind with respect to the switching continuous-time subsystems. In this setting, the paper's main contribution is to show how to compute the model predictive control gain. The paper also illustrates the implications of our approach for applications. The approach was used in practice to control an electric current source that supplied a switching load. The experimental data support the usefulness of our approach.

A Probability Theory Approach to Stability Analysis of Networked Sampled-Data Systems With Consecutive Packet Dropouts

In this brief note, a probability theory approach is proposed to investigate the stability analysis issue for a class of networked sampled-data systems with packet dropouts. The continuous-time linear system under consideration is first transformed into a discrete-time system, where the system matrix of the discrete-time system is characterized by stochastic characteristic. Then, by the matrix exponential computation, an equivalent discrete-time stochastic system is established. Based on this, a controller is designed by the law of total expectation, which guarantees the stochastic stability of the closed-loop stochastic system. Finally, a numerical example and an example using power grid with IEEE 4-bus line are used to verify the effectiveness and applicability of the proposed design algorithm.

Homotopic policy iteration-based learning design for unknown linear continuous-time systems

Recent results have emerged that policy iteration is a powerful reinforcement learning tool in designing a stabilizing control policy for continuous-time systems with unknown system dynamics. Policy iteration involves a model-based initialization stage, i.e., seeking an initial stabilizing control policy, which is, however, dependent on the full system dynamics including the drift dynamics and system input matrix. To remove such model requirements, this paper utilizes a homotopy-based initialization strategy for policy iteration, wherein a stabilizing control policy for continuous-time systems is obtained by gradually moving a stable system to the original system. We propose two homotopic policy iteration-based stabilizing control schemes, namely, a model-based design and a model-free design using system data, which are proved to place unstable poles into a stable region. The effectiveness of the proposed designs is validated through an illustrative example.

Value iteration and adaptive optimal output regulation with assured convergence rate

In this paper, we investigate the learning-based adaptive optimal output regulation problem with convergence rate requirement for disturbed linear continuous-time systems. An adaptive optimal control approach is proposed based on reinforcement learning and adaptive dynamic programming to learn the optimal regulator with assured convergence rate. The above-mentioned problem is successfully solved by tackling a static optimization problem to find the optimal solution to the regulator equations, and a dynamic and constrained optimization problem to obtain the optimal feedback control gain. Without requiring on the accurate system dynamics or a stabilizing feedback control gain, a novel online value iteration algorithm is proposed, which can learn both the optimal feedback control gain and the corresponding feedforward control gain using measurable data. Moreover, the output of the closed-loop system is guaranteed to converge faster or equal to a predefined convergence rate set by user. Finally, the numerical analysis on a LCL coupled inverter-based distributed generation system shows that the proposed approach can achieve desired disturbance rejection and tracking performance.

Dynamic logarithmic state and control quantization for continuous-time linear systems

AbstractThis paper addresses logarithmic quantizers with dynamic sensitivity design for continuous-time linear systems with a quantized feedback control law. The dynamics of state quantization and control quantization sensitivities during “zoom-in”/“zoom-out” stages are proposed. Dwell times of the dynamic sensitivities are co-designed. It is shown that with the proposed algorithm, a single-input continuous-time linear system can be stabilized by quantized feedback control via adopting sensitivity varying algorithm under certain assumptions. Also, the advantage of logarithmic quantization is sustained while achieving stability. Simulation results are provided to verify the theoretical analysis.