Linear Sampled-data Systems Research Articles

Standardizability of linear periodic sampled-data systems

Due to periodic sampling, linear sampled-data system are a subclass of linear continuous periodic systems, even in the case, when all other elements are time invariant. Owing to the great practical importance of sampled-data control systems, various approaches for the rigorous description of those systems are known. Moreover, a lot of methods have been developed, that are able to yield rigorous solutions for numerous control and optimization problems. The application of those methods need representations in certain standard forms. Often, it is not clear, whether a given system can be transformed into a standard form or not. The paper considers this question for the standard sampled-data system, for which numerous methods and tools are available, if the system belongs to the subclass with model structure. The paper provides necessary and sufficient conditions for a SD system with standard structure to belong to this important subclass. At hand of a simple SD system, it is shown that the standard structure does not imply model standardizability.

Standardizability of multivariable linear sampled-data systems with delay

Due to periodic sampling, linear sampled-data systems are a subclass of linear continuous periodic systems, even in the case, when all other elements are time invariant. Therefore, the well-established methods for LTI systems cannot be applied. Owing to the great practical importance of sampled-data control systems, various approaches for the rigorous description of those systems are known. Moreover, a lot of methods have been developed, that are able to yield rigorous solutions for numerous control and optimization problems. The application of those methods need representations in certain standard forms. Often, it is not clear, whether a given system can be transformed into a standard form or not. The paper considers this question for the standard sampled-data system with delay, for which numerous methods and tools are available, if the system belongs to the subclass with model structure. The paper provides necessary and sufficient conditions for a SD system with delay in standard structure to belong to this important subclass. The proof is given on basis of the parametric transfer matrix concept, which is at present the only one that can handle problems, where both difficulties -sampling and arbitrary time-delay - occur at the same time.

On an LMI approach to optimal sampled-data state feedback control design

The literature presents a large number of interesting and useful results on sampled-data systems. In this paper, we deal with control design problems for this class of systems following a different route. The novelty is that the optimal and state feedback control design problems for sampled-data linear systems are solved in a unified and direct manner through linear matrix inequalities. To this purpose, the solution to a special two-point boundary value problem is used to verify the stability and to evaluate the performance of the closed-loop system. These results are then generalised to cope with non-uniform data rates. The theory is illustrated by means of simple examples borrowed from the literature.

Sampling-interval-dependent stability for linear sampled-data systems with non-uniform sampling

This paper is concerned with the sampling-interval-dependent stability of linear sampled-data systems with non-uniform sampling. A new Lyapunov-like functional is constructed to derive sampling-interval-dependent stability results. The Lyapunov-like functional has three features. First, it depends on time explicitly. Second, it may be discontinuous at the sampling instants. Third, it is not required to be positive definite between sampling instants. Moreover, the new Lyapunov-like functional can make use of the information fully of the sampled-data system, including that of both ends of the sampling interval. By making a new proposition for the Lyapunov-like functional, a sampling-interval-dependent stability criterion with reduced conservatism is derived. The new sampling-interval-dependent stability criterion is further extended to linear sampled-data systems with polytopic uncertainties. Finally, examples are given to illustrate the reduced conservatism of the stability criteria.

From Fault-Diagnosis and Performance Recovery of a Controlled System to Chaotic Secure Communication

Chaotic systems are often applied to encryption on secure communication, but they may not provide high-degree security. In order to improve the security of communication, chaotic systems may need to add other secure signals, but this may cause the system to diverge. In this paper, we redesign a communication scheme that could create secure communication with additional secure signals, and the proposed scheme could keep system convergence. First, we introduce the universal state-space adaptive observer-based fault diagnosis/estimator and the high-performance tracker for the sampled-data linear time-varying system with unanticipated decay factors in actuators/system states. Besides, robustness, convergence in the mean, and tracking ability are given in this paper. A residual generation scheme and a mechanism for auto-tuning switched gain is also presented, so that the introduced methodology is applicable for the fault detection and diagnosis (FDD) for actuator and state faults to yield a high tracking performance recovery. The evolutionary programming-based adaptive observer is then applied to the problem of secure communication. Whenever the tracker induces a large control input which might not conform to the input constraint of some physical systems, the proposed modified linear quadratic optimal tracker (LQT) can effectively restrict the control input within the specified constraint interval, under the acceptable tracking performance. The effectiveness of the proposed design methodology is illustrated through tracking control simulation examples.

Formula omitted]-induced norm analysis of sampled-data systems via piecewise constant and linear approximations

This paper deals with the L1 analysis of linear sampled-data systems, by which we mean the computation of the L∞-induced norm of linear sampled-data systems. Two computation methods based on piecewise constant and piecewise linear approximations are provided through fast-lifting, by which the sampling interval [0,h) is divided into M subintervals with an equal width. Even though the central part of the method with the former approximation essentially coincides with a conventional method via fast-sample/fast-hold (FSFH) approximation after all, we show that both methods successfully lead to the upper and lower bounds of the L∞-induced norm, whose gap converges to 0 at the rate of 1/M in the former approximation and 1/M2 in the latter extended approximation. Such achievements are in sharp contrast with an existing result on the former (i.e., FSFH) approximation, which only shows the convergence rate of the error in the resulting estimate of the L∞-induced norm, without providing any readily computable upper and lower bounds. A numerical example is given to illustrate the effectiveness of these methods.

Stability and stabilization of linear sampled-data systems with multi-rate samplers and time driven zero order holds

In multi-rate sampled-data systems, a continuous-time plant is controlled by a discrete-time controller which is located in the feedback loop between sensors with different sampling rates and actuators with different refresh rates. The main contribution of this paper is to propose sufficient Krasovskii-based stability and stabilization criteria for linear sampled-data systems, with multi-rate samplers and time driven zero order holds. For stability analysis, it is assumed that an exponentially stabilizing controller is already designed in continuous-time and is implemented as a discrete-time controller. For each sensor (or actuator), the problem of finding an upper bound on the lowest sampling frequency (or refresh rate) that guarantees exponential stability is cast as an optimization problem in terms of linear matrix inequalities (LMIs). Furthermore, sufficient conditions for controller synthesis are formulated as LMIs. It is shown through examples that choosing the right sensors (or actuators) with adequate sampling frequencies (or refresh rates) has a considerable impact on stability of the closed-loop system.

Realization of causal current output-based optimal full/reducedorder observer and tracker for the linear sampled-data system with a direct transmission term

SUMMARY Realization of causal current output-based optimal full/reduced-order observer and tracker for the linear sampled-data system with a direct transmission term from input to output is proposed in this paper. First, a high-gain optimal linear quadratic analog tracker based on the reduced-order observer is proposed for the system model with a direct transmission term from input to output, so that it can effectively induce a high quality performance on state estimation and trajectory tracking design theoretically. Then, the prediction-based digital redesign method enables it to make the digitally controlled system to track the desired trajectory closely. However, there induces a causal problem on the realization of current output-based optimal full/reduced-order observer and tracker for the sampled-data system with a direct transmission term form input to measured output. To overcome this problem, a realization of the causal observer-based sampled-data tracker is newly proposed in this paper. An illustrative example is presented to demonstrate the effectiveness of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.

MPC for Sampled-Data Linear Systems: Guaranteeing Constraint Satisfaction in Continuous-Time

Model Predictive Controllers (MPC) designed for sampled-data systems can be shown to violate the constraints in continuous time. A reformulation of the initial problem will guarantee constraint satisfaction throughout the intersample period. Polytopic inclusions of the continuous trajectory are used in this technical note to establish additional constraints leading to a linearly constrained quadratic optimization problem. Continuous time asymptotic stability and continuous-time positive invariance are proven for the reformulated problem.

Control synthesis problem for networked linear sampled-data control systems with band-limited channels

This paper addresses the control synthesis problem of networked linear sampled-data control systems (NLSCSs) with band-limited channels. First, since there exist packet dropouts when the information is transmitted through the band-limited channels, a novel compression compensation (C–C) method is proposed to ensure the integrity of transmission. Based on this method, a compound control strategy is established to deal with the data transmission synchronization problem under the band-limited channels. Then, a nonlinear switched system model with uncertainty is constructed, in which both the inter-sampler behavior and the packet dropouts behavior are considered. Furthermore, some more general stability results of the NLSCSs with band-limited channels are obtained by using a packet dropouts dependent Lyapunov functional method, and then the compound state feedback controller can be designed. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed results.

H∞Controller Design for Asynchronous Multirate Sampled-Data Systems

This paper considers the analysis and synthesis of a linear discrete asynchronous multirate sampled-data system. AnH∞controller based on an observer is proposed, which guarantees the stability of the closed system and makes theH∞norm of the closed system less than a given attenuation level. To improve the performance further, a tradeoff strategy is applied. That is, the exogenous signals sampled at different rates are lifted to an appropriate signal rate, while the endogenous signals are not lifted for avoiding the causal constraint and the dimension multiplied again. The controller is obtained by solving the corresponding matrix inequality, which can be calculated by Matlab. Finally, an example is presented to demonstrate the validity of these methods.

On exponential stability of linear networked control systems

This paper addresses exponential stability of linear networked control systems. More specifically, the paper considers a continuous-time linear plant in feedback with a linear sampled-data controller with an unknown time varying sampling rate, the possibility of data packet dropout, and an uncertain time varying delay. The main contribution of this paper is the derivation of new sufficient stability conditions for linear networked control systems taking into account all of these factors. The stability conditions are based on a modified Lyapunov–Krasovskii functional. The stability results are also applied to the case where limited information on the delay bounds is available. The case of linear sampled-data systems is studied as a corollary of the networked control case. Furthermore, the paper also formulates the problem of finding a lower bound on the maximum network-induced delay that preserves exponential stability as a convex optimization program in terms of linear matrix inequalities. This problem can be solved efficiently from both practical and theoretical points of view. Finally, as a comparison, we show that the stability conditions proposed in this paper compare favorably with the ones available in the open literature for different benchmark problems.

Hierarchical Least Squares Identification for Linear SISO Systems With Dual-Rate Sampled-Data

This technical note studies identification problems for dual-rate sampled-data linear systems with noises. A hierarchical least squares (HLS) identification algorithm is presented to estimate the parameters of the dual-rate ARMAX models. The basic idea is to decompose the identification model of a dual-rate system into several sub-identification models with smaller dimensions and fewer parameters. The proposed algorithm is more computationally efficient than the recursive least squares (RLS) algorithm since the RLS algorithm requires computing the covariance matrix of large sizes, while the HLS algorithm deals with the covariance matrix of small sizes. Compared with our previous work, a detailed study of the HLS algorithm is conducted in this technical note. The performance analysis and simulation results confirm that the estimation accuracy of the proposed algorithm are close to that of the RLS algorithm, but the proposed algorithm retains much less computational burden.

Output Feedback Sliding Mode Control for Sampled-Data Systems

We address the output feedback sliding mode control problem for a sampled data linear system with external disturbances of the matching type. By taking into account the disturbance compensation, a deadbeat high gain output feedback control strategy with additional dynamics is able to attenuate the disturbances. In the framework of singular perturbation analysis, it is shown that the closed loop system exhibits good robustness against exogenous disturbances.

Stability analysis and stabilization of networked linear systems with random packet losses

This paper is concerned with the stability analysis and stabilization of networked discrete-time and sampled-data linear systems with random packet losses. Asymptotic stability, mean-square stability, and stochastic stability are considered. For networked discrete-time linear systems, the packet loss period is assumed to be a finite-state Markov chain. We establish that the mean-square stability of a related discrete-time system which evolves in random time implies the mean-square stability of the system in deterministic time by using the equivalence of stability properties of Markovian jump linear systems in random time. We also establish the equivalence of asymptotic stability for the systems in deterministic discrete time and in random time. For networked sampled-data systems, a binary Markov chain is used to characterize the packet loss phenomenon of the network. In this case, the packet loss period between two transmission instants is driven by an identically independently distributed sequence assuming any positive values. Two approaches, namely the Markov jump linear system approach and randomly sampled system approach, are introduced. Based on the stability results derived, we present methods for stabilization of networked sampled-data systems in terms of matrix inequalities. Numerical examples are given to illustrate the design methods of stabilizing controllers.

Stability Analysis of Networked Sampled-Data Linear Systems With Markovian Packet Losses

We consider the stability properties of sampled-data networked linear systems with Markovian packet losses. A binary Markov chain is used to characterize the packet loss phenomenon of the network. We show that the sampled-data system under consideration can be considered as a randomly sampled system with an i.i.d. random sampling period. Necessary and sufficient conditions for the stochastic stability properties are established. Those conditions are based on the relationships of stability properties between the systems evolving in deterministic continuous time, deterministic discrete time, and random discrete time. In addition, the asymptotic stability of the system is also studied by using Lyapunov exponent method.

Control Design and Robustness Analysis of Linear Time-Periodic Systems

Abstract This paper proposes a new design approach for control of periodically time-varying systems. The approach is based on the point-mapping technique to obtain an equivalent linear time-invariant sampled-data system for the linear periodically time-varying system with a piecewise parametrization of the control vector. This allows the known control design techniques for sampled-data systems to be applied. The proposed approach is then extended for analysis of robustness of the control design with respect to plant parametric uncertainties. This is achieved by computation of approximate discrete-time dynamics of the perturbed system by truncated point-mappings. By computing an upper norm bound on the error due to the truncated approximations, the robustness analysis of the system with respect to the parametric uncertainties is then formulated as a discrete-time structured singular value problem. Two numerical examples are considered to illustrate the approach and the extension of the approach for robust stability analysis.

Stabilizing Sampled-data Linear Systems with Markovian Packet Losses and Random Sampling

We consider the stability properties and stabilizing problem of sampled-data networked controlled linear systems with Markovian packet losses. A binary Markov chain is used to characterize the packet loss phenomenon of the network. Then with the Markovian packet loss assumption, we obtain a discrete-time augmented Markov jump linear system which describes the continuous-time linear system evolving in deterministic discrete time. Furthermore, we show that the sampled-data system under consideration can also be considered as a randomly sampled system with an i.i.d. random sampling period. A number of necessary and sufficient conditions for the stochastic stability properties are established by using the known results of Markov jump linear systems and randomly sampled systems. Those conditions are based on the relationships of stability properties between the systems evolving in deterministic continuous time, deterministic discrete time, and random discrete time. In addition, the asymptotic stability of the system is also studied by using Lyapunov exponent method. Numerical examples are used to illustrate the main results of the paper.

H∞ control for sampled-data linear systems with two Markov processes

This paper is concerned with the design problem of H∞ digital switching control for linear continuous systems with Markovian jumping parameters. The controller is digital and monitored by the jumping parameters of the plant. The closed-loop system is a hybrid one defined on a hybrid time space (composed of a continuous-time and a discrete-time) and a sample space. The sample space is specified by two separable continuous-time discrete-state Markov processes, one appearing in the open-loop system, and the other appearing in control action, which is different with the traditional Markovian jumping process. Our attention is focused on designing digital output feedback controllers for the system with two Markovian jumping processes such that both stochastic stability and a prescribed H∞ performance are achieved. The problem of robust H∞ control for systems with parameters uncertainties is also studied. It is shown that the sampled-data control problems for linear Markovian jumping systems with and without parameter uncertainties can be solved in terms of the solutions to a set of intercoupled matrix inequalities. Two numerical examples are given to show the design procedures. Copyright © 2005 John Wiley & Sons, Ltd.

Optimal design of 2-DOF digital controller for ship course control system

A two-degrees-of-freedom (2-DOF) controller is designed for a ship course control systems using direct design methods based on the frequency-domain theory of linear sampled-data systems. The feedback controller was optimized to ensure minimal course fluctuation and rudder activity under stochastic wave disturbances, while the reference controller provides a quick transient response when the desired course is changed. The obtained 2-DOF controller is compared with an existing PID-controller, and its advantages over the latter are demonstrated by simulation