Class Of Discrete-time Linear Systems Research Articles

A Performance Evaluation of Repetitive and Iterative Learning Algorithms for Periodic Tracking Control of Functional Electrical Stimulation System

Functional electrical stimulation (FES) is a medical device that delivers electrical pulses to the muscle, allowing patients with spinal cord injuries to perform activities such as walking, cycling, and grasping. It is critical for the FES to generate stimuli with the appropriate controller so that the desired movements can be precisely tracked. By considering the repetitive nature of the movements, the learning-based control algorithms are utilized for regulating the FES. Iterative learning control (ILC) and repetitive control (RC) are two learning algorithms that can be used to accomplish accurate repetitive motions. This study investigates a variety of ILC and RC designs with distinct learning functions; this constitutes our contribution to the field. The FES model of ankle angle, which is in a class of discrete-time linear systems is considered in this study. Two learning functions, i.e., proportional, and zero-phase learning functions, are simulated for the second-order FES model running at a sampling time of 0.1 s. The results indicate the superior performance of the ILC and RC in terms of convergence rate using the zero-phase learning function. ILC and RC with a zero-phase learning function can reach a zero root-mean-square error in two iterations if the model of the plant is correct. This is faster than proportional-based ILC and RC, which takes about 40 iterations. This indicates that the zero-phase learning function requires two iterations to ensure that the patient's ankle angle precisely tracks the intended trajectory. However, the tracking performance is degraded for both control methods, especially when the model is subject to uncertainties. This specific problem can lead to future research directions.

Parameter-Optimal-Gain-Arguable Iterative Learning Control for Linear Time-Invariant Systems with Quantized Error

In this paper, a parameter optimal gain-arguable iterative learning control algorithm is proposed for a class of linear discrete-time systems with quantized error. Based on the lifting model description for ILC systems, the iteration time-variable derivative learning gain in the algorithm is optimized by resolving a minimization problem regarding the tracking error energy and the learning effort amplified by a weighting factor. Further, the tracking error can be monotonically convergent to zero when the condition is guaranteed and the rate of convergence can be adjusted by scaling the weighting factor of an optimization problem. This algorithm is more innovative when compared with the existing iterative learning control algorithm for quantization systems. The innovations of this algorithm are as follows: (i) this optimization-based strategy for selecting learning gains can improve the active learning ability of the control mechanism and avoid the passivity of existing selective learning gains; (ii) the algorithm of POGAILC with data quantization can improve the convergence performance of tracking errors and reduce the negative effects of data quantization on the control performance of the logarithmic quantizer; and (iii) we provide a rigorous algorithm convergence analysis by deriving the existence of the unique solution for the optimal learning-gain vector under a singular and nonsingular tracking-error diagonalized matrix. Finally, numerical simulations are used to demonstrate the effectiveness of the algorithm.

Stabilization of linear discrete-time systems over resource-constrained networks under dynamical multiple description coding scheme

In this paper, the stabilization problem is investigated for a class of linear discrete-time systems in a resource-constrained network environment where only the data with finite bits is permitted to transmit. The data transmissions from the sensors to the controller are subject to random packet dropouts. To improve the reliability of the data transmission, a novel dynamical multiple description coding scheme is developed with guaranteed convergence of the decoding error. By recurring to the matrix inequality theory and the stochastic analysis technique, sufficient conditions are established to ensure the stochastic stability of the closed-loop system. Furthermore, a parametric relationship is discovered that characterizes the influences from the packet-dropout probabilities and the encoder parameters on the stability of the closed-loop system. Finally, a numerical example is provided to illustrate the validity of the proposed control scheme.

Dynamic Event-Based Distributed Piecewise Filtering Over Sensor Networks Under Periodic DoS Attacks

In this article, the problem of distributed filtering for a class of linear discrete-time systems under periodic denial-of-service (DoS) attack is investigated. First, a cyclical and repeatable variable is introduced to characterize the behavior of a periodic DoS attack. Based on the introduced variable, a novel dynamic event-triggered scheme is designed to reduce the network transmission burden. Second, a cyclic switching estimation error model, which includes stable subsystems in the absence of a DoS attack and unstable subsystems in the presence of a DoS attack, is constructed. The global exponential stability with <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 is synthesized by utilizing multiple piecewise Lyapunov functions, and the gain matrix of the distributed filtering is derived in piecewise form. Third, the codesign algorithm of the dynamic event-triggered scheme and distributed piecewise filtering is proposed to make a tradeoff between network transmission burden and the performance of distributed filtering. Finally, an example is applied to verify the effectiveness of the proposed distributed piecewise filtering.

A New Adaptive Suboptimal Controller for a Class of Linear Discrete-Time Systems with Unknown Bounded Disturbances

This paper deals with adaptive suboptimal control of linear discrete-time, time-invariant, minimum phase, scalar systems in the presence of arbitrary bounded unmeasurable disturbances whose upper and lower bounds are a priori unknown. The distinguishing feature of the systems to be controlled is that there is no information concerning a priori knowledge of a bounded set to which the unknown parameters of these systems belong. Novelty of this paper is that, instead of unknown a priori membership set of parameters, the peculiar a posteriori membership sets are designed via the use of the measured signals. The adaptive suboptimal control algorithm utilizing this approach is proposed. The properties of the adaptive controller including the convergence of the adaptation procedure and also the ultimate boundedness of system's signals are established. To demonstrate an efficiency of this controller and support the theoretical study, simulation results are presented.

Energy-to-Peak State Estimation With Intermittent Measurement Outliers: The Single-Output Case.

This article is concerned with the energy-to-peak state estimation problem for a class of linear discrete-time systems with energy-bounded noises and intermittent measurement outliers (IMOs). In order to capture the intermittent nature, two sequences of step functions are introduced to model the occurrence of the IMOs. Furthermore, two special indices (i.e., minimum and maximum interval lengths) are adopted to describe the "occurrence frequency" of IMOs. Different from the considered energy-bounded noises, the outliers are assumed to have their magnitudes larger than certain thresholds. In order to achieve a satisfactory performance constraint on the energy-to-peak state estimation under the addressed kind of measurement outliers, a novel parameter-dependent (PD) state estimation strategy is developed to guarantee that the measurements contaminated by outliers would be removed in the estimation process. The proposed PD state estimation method is essentially a two-step process, where the first step is to examine the appearing and disappearing moments for each IMO by using a dedicatedly constructed outlier detection scheme, and the second step is to implement the state estimation task according to the outlier detection results. Sufficient conditions are obtained to ensure the existence of the desired estimator, and the gain matrix of the desired estimator is then derived by solving a constrained optimization problem. Finally, a simulation example is presented to illustrate the effectiveness of our developed PD state estimation strategy.

Optimal output feedback control of a class of linear systems with quasi-colored control-dependent multiplicative noise

This paper addresses the mean-square optimal control problem for a class of discrete-time linear systems with a quasi-colored control-dependent multiplicative noise via output feedback. The noise under study is novel and shown to have advantage on modeling a class of network phenomena such as random transmission delays. The optimal output feedback controller is designed using an optimal mean-square state feedback gain and two observer gains, which are determined by the mean-square stabilizing solution to a modified algebraic Riccati equation (MARE), provided that the plant is minimum-phase and left-invertible. A necessary and sufficient condition for the existence of the stabilizing solution to the MARE is explicitly presented. It shows that the separation principle holds in a certain sense for the optimal control design of the work. The result is also applied to the optimal control problems in networked systems with random transmission delays and analog erasure channels, respectively.

A reference governor for linear systems with polynomial constraints

The paper considers reference governor design for a class of linear discrete-time systems with constraints given by polynomial inequalities and constant reference commands. For such systems, we propose a novel algorithm to compute the maximal output admissible set. The reference governor solves a constrained nonlinear minimization problem at the initialization and then uses a bisection algorithm at the subsequent time steps. The effectiveness of the method is demonstrated by two numerical examples.

Fault Detection and Isolation for Linear discrete-time Delayed Systems based on L∞ Observer and Zonotopic Analysis

This paper considers actuator fault detection and isolation (FDI) problem for a class of linear discrete-time systems with constant time-delay. Based on delay-free representation, a novel fault detection and isolation scheme based on a bank of L∞ fault detection observers (FDOs) is proposed. Under the assumption that the disturbances and measurement noise are unknown but bounded, L∞ performance is introduced to reduce the effects of disturbances. Meanwhile, the FDOs are designed to be sensitive to the fault by means of fault detectability indices and matrix. The design conditions of the proposed FDOs are derived in terms of Linear Matrix Inequalities (LMIs). Furthermore, a zonotopic analysis method is presented to achieve the residual evaluation. Simulation results are provided to demonstrate the performance of the proposed approach.

Adaptive optimal output regulation of linear discrete-time systems based on event-triggered output-feedback

This paper presents novel event-triggered control approaches to solve the adaptive optimal output regulation problem for a class of linear discrete-time systems. Different from most existing research on output regulation problems, the developed adaptive optimal control approaches are based on (1) output-feedback instead of full-state or partial-state feedback, (2) adaptive dynamic programming (ADP) which provides approximate solutions of the optimal control problem without requiring the precise knowledge of the plant dynamics, and (3) an event-triggering mechanism that reduces the communication between the controller and the plant. It is shown that the system in closed-loop with the developed controllers is asymptotically stable at an equilibrium of interest, and the tracking errors asymptotically converge to zero. Moreover, the suboptimality of the closed-loop system is directly determined by the relative threshold, which is a ratio between the triggering threshold and the actual state. A numerical simulation example is employed to verify the effectiveness of the proposed methodologies.

Discrete networked dynamic systems with mode separation: state and output synchronization

This paper examines the impact of coupling in a class of linear discrete-time systems with mode separation over networks. We provide a complete analysis and design treatment allowing at extending the stabilization control design to the corresponding synchronization problem. We deploy a time-scale separation of the whole network dynamics to synthesize the controls that synchronize both the slow and fast states. We show that the design implementation is cast as feedback simultaneous stabilization framework and cast as feasibility problem over finite number of linear matrix inequalities (LMIs). As demonstrated by simulation studies, the ensuing results provide designers with a network-centric technique guarantee stability and enhance the performance of considered coupled systems.

A Memoryless Delay-Adaptive Feedback Law for the Regulation of Discrete-Time Linear Systems

It is challenging to regulate a linear system with input delay when the delay information is absent. A solution is proposed in this paper to regulate the class of discrete-time linear systems whose open loop poles are located inside or on the unit circle. With the delay below an established upper bound, a memoryless delay-adaptive feedback design achieves the convergence of the system state and input to zero as time approaches infinity. In the event that the system has all its open loop poles strictly inside the unit circle or at $z=1$, regulation is achieved for any, arbitrarily large, bounded delay without resorting to any knowledge of the delay. The proposed delay-adaptive control design utilizes only the state at the current time instant for feedback, resulting in a memoryless feedback law convenient for implementation.

$\mathcal{H}_{\infty}$ PID Control With Fading Measurements: The Output-Feedback Case

This paper is concerned with the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> proportional-integral-derivative (PID) control problem for a class of linear discrete-time systems with fading measurements. The fading measurements are governed by the Rice fading model whose coefficients are hypothesized to be a series of independent and identically distributed Gaussian variables. By utilizing the received measurements subject to fading phenomena, a novel output-feedback PID controller is proposed where the integral-loop (accumulation sum-loop for the discrete-time case) is equipped with the limited time-window in order to reduce the computational burden. The main objective of the addressed problem is to design a desired PID controller such that both the exponentially mean-square stability and the prescribed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance are guaranteed for the closed-loop system in the presence of fading measurements. With the help of Lyapunov stability theory, a sufficient condition is obtained to guarantee the desired performance and, on the basis of such a condition, the synthesis issue of the PID controller is subsequently discussed, where the orthogonal decomposition combined with a free matrix is introduced to facilitate the controller design. Finally, a numerical example is exploited to demonstrate the usefulness and effectiveness of the presented control scheme.

Low-complexity stabilizing PWA controllers for linear systems with parametric uncertainties

Explicit MPC often results in a large number of irregular partitions in the feasible region as the dimension of the system increases and the storage requirement for the control and region parameters often limit its applications. In this paper, we consider a class of discrete-time linear systems with polytopic parametric uncertainties and provide a robust control Lyapunov based synthesis method to obtain robust low-complexity PWA controllers on regular partitionings. By implementing a refinement procedure, we can fit the PWA feedback control law in each regular partitioning based on feasibility of linear programming problems, which preserves stability, constraint satisfaction, and certain performance requirement. Numerical examples will demonstrate the effectiveness of the approach.

Recursive state estimation for linear systems with lossy measurements under time-correlated multiplicative noises

This paper is concerned with the recursive state estimation problem for a class of linear discrete-time systems with lossy measurements and time-correlated multiplicative noises (TCMNs). The lossy measurements result from one-step transmission delays and packet dropouts. Different from the traditional white multiplicative noises, TCMNs are included in the measurement model in order to reflect engineering practice. Utilizing the state augmentation approach, the system under investigation is first converted into a stochastic parameter system, and some new recursive terms (including the estimation for the product of state and multiplicative noises) are introduced to handle the difficulties caused by the TCMNs. Then, by the well-known projection theorem, recursive state estimation algorithms are developed in the sense of minimum mean-square error, which facilitate the design of the filter, the multi-step predictor and the smoothers. The proposed algorithms are explicitly dependant on the key system parameters including the covariances of the TCMNs, the occurrence probabilities of the transmission delays and the packet losses. Finally, simulation results illustrate the effectiveness of the presented estimation algorithms.

Stability and passivity analysis of discrete-time linear systems with time-varying delay

This paper focuses on the stability and passivity analysis for a class of discrete-time linear systems. A novel general free-matrix-based summation inequality is proposed, and two improved stability criteria in terms of linear matrix inequalities (LMIs) are derived by employing the new summation inequality. Furthermore, when an external disturbance appears, a sufficient condition is established to ensure the considered linear system to be passive. Three numerical examples are provided to demonstrate the improvement of the proposed method.

Constrained Adaptive Model-Predictive Control for a Class of Discrete-Time Linear Systems With Parametric Uncertainties

In this technical note, an adaptive model-predictive control (MPC) is proposed for a class of discrete-time linear systems with constant parametric uncertainties and control constraint. The proposed adaptive MPC originates from the principle of min–max optimization, which cannot be solved in a direct numerical way. An adaptive strategy is proposed to estimate the uncertain parameters, such that the estimated error converges, and the optimization in the MPC can be transferred into a solvable simple structure. Feasibility of the optimization and stability of the closed-loop system are proved theoretically, and a simulation example is presented to illustrate the theoretical result.

Robust model predictive control with randomly occurred networked packet loss in industrial cyber physical systems

for a class of linear discrete-time systems that is subject to randomly occurred networked packet loss in industrial cyber physical systems, a novel robust model predictive control method with active compensation mechanism was proposed. The probability distribution of packet loss is described as the Bernoulli distributed white sequences. By using the Lyapunov stability theory, the existing sufficient conditions of the controller are derived from solving a group of linear matrix inequalities. Moreover, dropout-rate with uncertainty and unknown dropout-rate are also considered, which can greatly reduce the conservativeness of the controller. The designed robust model predictive control method not only efficiently eliminates the negative effects of the networked data loss in industrial cyber physical systems but also ensures the stability of closed-loop system. Two examples were provided to illustrate the superiority and effectiveness of the proposed method.

On the Optimal Speed Problem for the Class of Linear Autonomous Infinite-Dimensional Discrete-Time Systems with Bounded Control and Degenerate Operator

The optimal speed problem for the class of linear discrete-time systems with the infinite-dimensional state vector and degenerate operator is solved. Some properties of convex sets are formulated and proved. Necessary and sufficient conditions under which this problem has a solution in the case of the zero point located on the boundary of the reachability set are established. The optimality conditions are written as a discrete-time maximum principle. For the inner points, the degenerate character of the maximum principle is demonstrated. For an inner point, the optimal speed problem is solved by developing an algorithm with reduction to the admissible case of the boundary point. Some examples are given.

Distributed MPC-based adaptive control for linear systems with unknown parameters

This paper is concerned with the adaptive control problem for a class of linear discrete-time systems with unknown parameters based on the distributed model predictive control (MPC) method. Instead of using the system state, the state estimate is employed to model the distributed state estimation system. In this way, the system state does not have to be measurable. Furthermore, in order to improve the system performance, both the output error and its estimation are considered. Moreover, a novel Lyapunov functional, comprised of a series of distributed traces of estimation errors and their transposes, has been presented. Then, sufficient conditions are obtained to guarantee the exponential ultimate boundedness of the system as well as the asymptotic stability of the error system by solving a nonlinear programming (NP) problem subject to input constraints. Finally, the simulation examples is given to illustrate the effectiveness and the validity of the proposed technique.