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The problem of constructing reachable and null-controllable sets for stationary linear discrete-time systems with a summary constraint on the scalar control is considered. For the case of quadratic constraints and a diagonalizable matrix of the system, these sets are built explicitly in the form of ellipsoids. In the general case, the limit reachable and null-controllable sets are represented as fixed points of a contraction mapping in the metric space of compact sets. On the basis of the method of simple iteration, a convergent procedure for constructing their external estimates with an indication of the a priori approximation error is proposed. Examples are given. [ABSTRACT FROM AUTHOR]

This paper proposes a novel approach to suppressing bounded exogenous disturbances in a linear discrete-time control system by a static state- or output-feedback control law. The approach is based on reducing the original problem to a nonconvex matrix optimization problem with the gain matrix as one variable. The latter problem is solved by the gradient method; its convergence is theoretically justified for several important special cases. An example is provided to demonstrate the effectiveness of the iterative procedure proposed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Stability estimates are obtained for delayed linear periodic discrete-time systems. Bounds on the decay of the solution are derived via a suitable Lyapunov–Krasovskii-type functional and the solvability of some periodic discrete-time Lyapunov equations. [ABSTRACT FROM AUTHOR]

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.

We consider the problem of optimal control of a linear time-invariant discrete-time system by inaccurate measurements of its output signals subject to guaranteed satisfaction of geometric constraints on the output signals. We study the case in which a minimal realization of the system in the state space is known and the case where the parametric model of the system is not known. A novel method is proposed for solving the problem in the case of an unknown model. The method is based on a single observed trajectory of the input and output signals of the system and allows omitting the stage of parametric identification of the system. [ABSTRACT FROM AUTHOR]

Robust stability conditions are established for a family of linear discrete-time systems subjected to uncertainties. The traditional approach, which involves the construction of a common quadratic Lyapunov function for the entire family of systems with uncertainty, often leads to the problem of conservatism. In this connection, constructing parametric quadratic Lyapunov functions seems promising. The main tools of the proposed approach are the apparatus of linear matrix inequalities and a modification, presented here, of the well-known Petersen's lemma. A simple approach to finding the robust quadratic stability radius of the family in question is proposed in the paper as well. The corresponding optimization problems have the form of semidefinite programming and one-dimensional minimization, which can be easily solved numerically. The efficiency of our approach is demonstrated via a numerical example. The results obtained can be generalized to design problems for linear discrete-time systems subjected to uncertainties, to other robust statements, and to the case of exogenous disturbances. [ABSTRACT FROM AUTHOR]

In this paper, a novel robust fault detection (FD) approach called parity space vector machine (PSVM) is proposed for linear discrete-time systems. Aiming to achieve a tradeoff between false alarm rate (FAR) and FD rate (FDR) simultaneously, we focus our study on an integrated design of parity space-based FD in the context of residual generation and residual evaluation. Without a prior knowledge of the distribution of the unknown inputs, we propose to construct a PSVM model and formulate the underlying FD problem as a distribution-free Bayes optimal classifier, where the FAR and FDR indicate the worst-case classification accuracies of future residuals for the fault free case and faulty case. Then a bank of parity space vectors and corresponding thresholds can be designed integratedly by applying the techniques of the minimum error minimax probability machine and, at the same time, an optimal tradeoff between FAR and FDR is achieved. Finally, the effectiveness of the proposed approach is demonstrated on a longitudinal control system of unmanned aerial vehicle and further comparison with a traditional parity space-based FD is also addressed. [ABSTRACT FROM AUTHOR]

In this paper, the augment Kalman filter (AKF) approach is discussed under the infinity condition. It is proved that the AKF in infinity condition is equivalent to the minimum-variance unbiased (MVU) estimation proposed in Gillijns & De Moor [(2007). Unbiased minimum-variance input and state estimation for linear discrete-time systems. Automatica, 43(1), 111–116]. This result can be regarded as a new contribution in the field of joint input and state estimation. Meanwhile, this result gives another way to estimate the state and unknown input simultaneously. Finally, the theoretical result is validated by simulations. [ABSTRACT FROM AUTHOR]

The aim of this work is to show that interval positive fractional discrete-time linear systems are asymptotically stable if and only if the respective lower and upper bound systems are asymptotically stable. The classical Kharitonov theorem is extended to interval positive fractional linear systems.

This study proposes an interval estimation method of actuator fault for linear discrete‐time system. An augmented descriptor system, which is equivalent to the original system, is formulated by letting the actuator fault term be an auxiliary state. Then, an H∞ fault estimation observer is constructed to attenuate the effect of unknown but bounded disturbances and measurement noise. By using interval analysis technique, the interval estimation of actuator fault can be obtained. Finally, a DC servo‐motor example is given to show the effectiveness and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates the robust model predictive control (RMPC) problem for a class of linear discrete‐time systems subject to saturated inputs and randomly occurring uncertainties (ROUs). Due to limited bandwidth of the network channels, the networked transmission would inevitably lead to incomplete measurements and subsequently unavoidable network‐induced phenomenon that include saturated inputs as a special case. The saturated inputs are assumed to be sector‐bounded in the underlying system. In addition, the ROUs are taken into account to reflect the difficulties in precise system modelling, where the norm‐bounded uncertainties are governed by certain uncorrelated Bernoulli‐distributed white noise sequences with known conditional probabilities. Based on the invariant set theory, a sufficient condition is derived to guarantee the robust stability in the mean‐square sense of the closed‐loop system. By employing the convex optimization technique, the controller gain is obtained by solving an optimization problem with some inequality constraints. Finally, a simulation example is employed to demonstrate the effectiveness of the proposed RMPC scheme. [ABSTRACT FROM AUTHOR]

Abstract: This work focuses on the iterative learning control (ILC) for linear discrete‐time systems with unknown initial state and disturbances. First, multiple high‐order internal models (HOIMs) are introduced for the reference, initial state, and disturbances. Both the initial state and disturbance consist of two components, one strictly satisfies HOIM and the other is random bounded. Then, an ILC scheme is constructed according to an augmented HOIM that is the aggregation of all HOIMs. For all known HOIMs, an ILC design criterion is introduced to achieve satisfactory tracking performance based on the 2‐D H ∞ theory. Next, the case with unknown HOIMs is discussed, where a time‐frequency‐analysis (TFA)‐based ILC algorithm is proposed. In this situation, it is shown that the tracking error inherits the unknown augmented HOIM that is an aggregation of all unknown HOIMs. Then, a TFA‐based method, e.g., the short‐time Fourier transformation (STFT), is employed to identify the unknown augmented HOIM, where the STFT could ignore the effect of the random bounded initial state and disturbances. A new ILC law is designed for the identified unknown augmented HOIM, which has the ability to reject the unknown the initial state and disturbances that strictly satisfy HOIMs. Finally, a gantry robot system with iteration‐invariant or slowly‐varying frequencies is given to illustrate the efficiency of the proposed TFA‐based ILC algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we aim to propose a data-based method to testify whether the system is a normal linear system or a generalised linear system and then analyse its controllability via the measured state data and the control input satisfying certain condition. For this purpose, we first describe a linear discrete-time system in a general form and derive a necessary and sufficient condition for the equivalent condition of complete controllability with a normal linear discrete time system. Second, we check whether the system is normal or singular, and then construct the controllability matrices only based on the input and measured state data. Third, the controllability of the corresponding system is investigated thoroughly based on available data without identifying system parameters. Finally, a numerical example and a stock price example are used to show the effectiveness and feasibility of the proposed data-based method. [ABSTRACT FROM AUTHOR]

Two description forms of a linear fractional-order discrete system are considered. The first one is by a fractional-order difference equation, whereas the second by a fractional-order state-space equation. In relation to the two above-mentioned description forms, stability domains are evaluated. Several simulations of stable, marginally stable and unstable unit step responses of fractional-order systems due to different values of system parameters are presented.

In this paper, a data-based method is used to analyse the controllability of discrete-time linear time-delay systems. By this method, one can directly construct a controllability matrix using the measured state data without identifying system parameters. Hence, one can save time in practice and avoid corresponding identification errors. Moreover, its calculation precision is higher than some other traditional approaches, which need to identify unknown parameters. Our methods are feasible to the study of characteristics of deterministic systems. A numerical example is given to show the advantage of our results. [ABSTRACT FROM AUTHOR]

In this paper, stochastic optimal strategy for unknown linear discrete-time system quadratic zero-sum games in input-output form with communication imperfections such as network-induced delays and packet losses, otherwise referred to as networked control system (NCS) zero-sum games, relating to the H8 optimal control problem is solved in a forward-in-time manner. First, the linear discrete-time zero sum state space representation is transformed into a linear NCS in the state space form after incorporating random delays and packet losses and then into the input-output form. Subsequently, the stochastic optimal approach, referred to as adaptive dynamic programming (ADP), is introduced which estimates the cost or value function to solve the infinite horizon optimal regulation of unknown linear NCS quadratic zero-sum games in the presence of communication imperfections. The optimal control and worst case disturbance inputs are derived based on the estimated value function in the absence of state measurements. An update law for tuning the unknown parameters of the value function estimator is derived and Lyapunov theory is used to show that all signals are asymptotically stable (AS) and that the estimated control and disturbance signals converge to optimal control and worst case disturbances, respectively. Simulation results are included to verify the theoretical claims. [ABSTRACT FROM AUTHOR]

Abstract: This work proposes a robust fault detection and isolation (FDI) scheme for linear discrete-time systems subject to faults, bounded additive disturbances and norm-bounded structured uncertainties. FDI is achieved by computing, on-line, upper and lower bounds on the fault signal such that a fault is regarded as having occurred when its upper bound is smaller than zero or lower bound is larger than zero. Linear Matrix Inequality (LMI) optimization techniques are used to obtain the bounds. Furthermore, a subsequent-state-estimation technique, together with an estimation horizon update procedure, is proposed, which allows the on-line FDI process to be repeated in a moving horizon procedure. The approach is also extended to solve the fault detection (FD) problem of obtaining lower bounds on the total fault signal energy within the estimation horizon. The scheme gives the best estimates of the fault signals given the information available and is sufficiently flexible to incorporate other information that may be available, such as bounds on the disturbance energy. Thus our scheme is immune to false alarms if the system and disturbance are within the uncertainty description. Moreover, we propose a new robustness result to obtain the bounds, which is an extension of current techniques for handling model uncertainties. Finally, the approach is verified using two numerical examples. [Copyright &y& Elsevier]

This study extends the problem of fault detection (FD) for linear discrete‐time systems with unknown input to non‐linear systems. Moreover, based on physical consideration, the constraints of state are considered. A non‐linear recursive filter is developed where the constrained state and the input are interconnected. Constraints which can improve the quality of estimation are imposed on individual updated sigma points as well as the updated state. The advantage of algorithm is that it is able to incorporate arbitrary constraints on the states during the estimation procedure. Unknown input which can be any signal is obtained by least‐squares unbiased estimation and the state estimation problem is transformed into a standard unscented Kalman filter problem. By testing the mean of the innovation process, a real‐time FD approach is proposed. Simulations are provided to demonstrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

Recursive formula for the terminal cost distribution in a scalar linear discrete-time system with disturbance and noise corrupted measurements is obtained. The system is subject to a linear saturated control strategy. The distributions of the initial state and the estimator error are assumed to be known. The disturbance is independent of the state/control and its distribution is known. The general result is applied to an interception problem with different types of disturbance. An illustrative numerical example confirms that the analytical method can replace extensive Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

Abstract: This paper develops a model-free control design algorithm for unknown linear discrete-time systems by using Q-learning, which is a reinforcement learning method based on an actor-critic structure. In model-free design, there is no known dynamical model of the system. Thus, one has no information on the system matrices, but can access the state variables and input variables. The paper derives an iterative solution algorithm for control design that is based on policy iteration. The algorithm is expressed in the form of linear matrix inequalities (LMI) that do not involve the system matrices, but only require data measured from the system state and input. It is shown that, for sufficiently rich enough disturbance, this algorithm converges to the standard control solution obtained using the exact system model. Two numerical examples are given to show the effectiveness in obtaining the control without any using knowledge of the system dynamics matrices, and the examples show that the results converge to the ones obtained with the exact system dynamics matrices. [ABSTRACT FROM AUTHOR]

We study robustness of 𝒟-stability of linear difference equations under multiperturbation and affine perturbation of coefficient matrices via the concept of 𝒟-stability radius. Some explicit formulae are derived for these 𝒟-stability radii. The obtained results include the corresponding ones established earlier in Hinrichsen and Son and Ngoc and Son as particular cases. [ABSTRACT FROM AUTHOR]