Control Of Discrete-time Linear Systems Research Articles

Event-Triggered Control for Discrete-Time Systems Using a Positive Systems Approach

We provide an output feedback event-triggered controller for discrete-time linear systems. We make novel use of positive systems, interval observers, an event-triggered state estimator, and triggering times that are computed from estimator values. This provides a discrete-time analog of our recent positive systems approach for continuous-time systems. A key novel ingredient in our discrete-time event triggers is their use of vectors of absolute values, instead of the usual Euclidean norm. We illustrate the benefits of our method using a model for event-triggered BlueROV2 underwater vehicles.

Stochastic model predictive control for linear systems affected by correlated disturbances

In this paper, the problem of stability, recursive feasibility and convergence conditions of stochastic model predictive control for linear discrete-time systems affected by a large class of correlated disturbances is addressed. A stochastic model predictive control that guarantees convergence, average cost bound and chance constraint satisfaction is developed. The results rely on the computation of probabilistic reachable and invariant sets using the notion of correlation bound. This control algorithm results from a tractable deterministic optimal control problem with a cost function that upper-bounds the expected quadratic cost of the predicted state trajectory and control sequence. The proposed methodology only relies on the assumption of the existence of bounds on the mean and the covariance matrices of the disturbance sequence.

Anti-windup-like Compensator Synthesis for Discrete-Time Quantized Control Systems

This paper addresses the problem of designing an anti-windup like compensator for discrete-time linear control systems with quantized input. The proposed compensator provides a correction signal proportional to the quantization error that is fed to the controller. The compensator is designed to ensure that solutions to the closed-loop systems converge in finite time into a compact set containing the origin that can be tuned by the designer. A numerically tractable algorithm with feasibility guarantees is provided for the design of the compensator. The proposed results are illustrated on an academic example and an open-loop unstable aircraft system.

Quantized-output feedback model reference control of discrete-time linear systems

This paper proposes a model reference control (MRC) scheme for general discrete-time linear time-invariant (LTI) systems subject to output quantization and saturation. In such a scheme, the quantized-output feedback MRC law is analytically specified only by using external reference input and saturated-and-quantized output. It is proven that, by appropriately designing the sensitivity of the quantizer, the MRC law can ensure that all closed-loop signals are bounded and the output tracking error converges to a certain small residual set in a certain finite time only under the minimum-phase condition. Particularly, the proposed MRC scheme does not rely on the coprimeness of zero and pole polynomials or initial conditions that are commonly used in the related literature. A representative example is given to demonstrate the design procedure and verify the validity of the proposed MRC scheme.

Spectrum assignment and stabilization by static output feedback of linear difference equations with state variable delays

We study the problem of arbitrary eigenvalue spectrum assignment for a linear discrete-time control system with delays defined by a linear difference equation of the nth order with multiple inputs and multiple outputs via static output feedback with delays. Necessary and sufficient conditions are obtained for arbitrary eigenvalue spectrum assignment by linear static output feedback with delays. These conditions are expressed in terms of system coefficients. A corollary on stabilization is derived. An illustrative example is presented.

Invariant cover: Existence, cardinality bounds, and computation

An invariant cover quantifies the information needed by a controller to enforce an invariance specification. This paper investigates some fundamental problems concerning existence and computation of an invariant cover for uncertain discrete-time linear control systems subject to state and control constraints. We develop necessary and sufficient conditions on the existence of an invariant cover for a polytopic set of states. The conditions can be checked by solving a set of linear programs, one for each extreme point of the state set. Based on these conditions, we give upper and lower bounds on the minimal cardinality of the invariant cover, and design an iterative algorithm with finite-time convergence to compute an invariant cover. We further show in two examples how to use an invariant cover in the design of a coder–controller pair that ensures invariance of a given set for a networked control system with a finite communication data rate.

Improved robust gain-scheduling static output-feedback control for discrete-time LPV systems

This paper presents new synthesis conditions for gain-scheduling static output-feedback control of discrete-time linear systems with time-varying parameters. A feature of the proposed condition, unlike most approaches in the literature, is that no structural constraints on the output matrix are imposed, that is, the proposed approach is able to handle variation in the dynamics, input, and output matrices without resorting to similarity transformations or iterative procedures. The stabilization conditions are extended to cope with the H∞ control problem. The proposed conditions are formulated in the form of linear matrix inequalities. Numerical experiments illustrate the potential of the proposed techniques.

Outer invariance entropy for discrete-time linear systems on Lie groups

We introduce discrete-time linear control systems on connected Lie groups and present an upper bound for the outer invariance entropy of admissible pairs (K, Q). If the stable subgroup of the uncontrolled system is closed and K has positive measure for a left invariant Haar measure, the upper bound coincides with the outer invariance entropy.

Linearization of expectation-based inequality conditions in control for discrete-time linear systems represented with random polytopes

A technique of linearizing what we call expectation-based inequality conditions is proposed for control of discrete-time linear systems with stochastic dynamics. In particular, the coefficient random matrices of the systems are assumed to be represented with random polytopes, and the linearization technique is discussed so as to appropriately deal with associated uncertainties. Our expectation-based inequality is an inequality that involves decision variables contained in the expectation operation, and has a unique difficulty in direct linearization, in general. Hence, two key lemmas are provided so that the decision variables can be taken out from the expectation operation. The combinational use of such lemmas and the conventional linear matrix inequality (LMI) techniques, which is nothing but our linearization technique, is expected to be useful for transforming various kinds of expectation-based nonlinear inequality into numerically solvable standard LMIs. As a demonstration, new robust stability conditions are derived with the technique, whose effectiveness is confirmed numerically.

A mean-field formulation for the mean-variance control of discrete-time linear systems with multiplicative noises

ABSTRACT This paper considers the stochastic optimal control of a multi-period mean-variance trade-off performance criterion with and without constraints for discrete-time linear systems subject to multiplicative noises. We adopt a mean-field approach to tackle the problem and obtain a solution for the unconstrained case in terms of a Riccati-like difference equation. From this general result, we obtain a sufficient condition for a closed-form solution for one of the constrained problems considered in the paper. When particularised to the portfolio selection problem, we show that our results retrieve some of the results available in the literature. We conclude the paper by illustrating the obtained optimal controls with a multi-period portfolio selection problem where we minimise the sum of the mean-variance trade-off costs of a portfolio against a benchmark along the time.

Stabilization of Nonlinear Discrete-Time Systems to Target Measures Using Stochastic Feedback Laws

In this article, we address the problem of stabilizing a discrete-time deterministic nonlinear control system to a target invariant measure using time-invariant stochastic feedback laws. This problem can be viewed as an extension of the problem of designing the transition probabilities of a Markov chain so that the process is exponentially stabilized to a target stationary distribution. Alternatively, it can be seen as an extension of the classical control problem of asymptotically stabilizing a discrete-time system to a single point, which corresponds to the Dirac measure in the measure stabilization framework. We assume that the target measure is supported on the entire state space of the system and is absolutely continuous with respect to the Lebesgue measure. Under the condition that the system is locally controllable at every point in the state space within one time step, we show that the associated measure stabilization problem is well-posed. Given this well-posedness result, we then frame an infinite-dimensional convex optimization problem to construct feedback control laws that stabilize the system to a target invariant measure at a maximized rate of convergence. We validate our optimization approach with numerical simulations of two-dimensional linear and nonlinear discrete-time control systems.

Minimal Perturbations for Zero Controllability of Discrete-Time Linear Systems: Complexity Analysis

This article deals with computational complexity of various problems related to the zero controllability of a discrete-time linear time-invariant system, assuming that purely structural conditions at the level of the connections between the system states (i.e., state-connections) and the connections from the inputs to the states (i.e., input-connections) are known. Given a generically zero controllable system, we consider the following problems: i) find a minimal set of input-connections whose removal makes the resulting system not generic zero controllability; ii) identify a minimal cost set of input-connections that must be retained from the given set of input-connections while preserving generic zero controllability property; and iii) given a not generically zero controllable system, find a smallest set of state-connections whose removal makes the resulting system generically zero controllable. Problem i) is polynomially solvable. Problems ii) and iii) are NP-hard and approximation results are provided for them. The results of i) and iii) provide clues to analyze the fragility and hardness involved in modifying a system structure. Problem ii) is useful to ensure an accurate discrete-time linear approximation of a large-scale system by maintaining generic zero controllability of the linear system.

Switched Stackelberg game analysis of false data injection attacks on networked control systems

This paper is concerned with a security problem for a discrete-time linear networked control system of switched dynamics. The control sequence generated by a remotely located controller is transmitted over a vulnerable communication network, where the control input may be corrupted by false data injection attacks launched by a malicious adversary. Two partially conflicted cost functions are constructed as the quantitative guidelines for both the controller and the attacker, after which a switched Stackelberg game framework is proposed to analyze the interdependent decision-making processes. A receding-horizon switched Stackelberg strategy for the controller is derived subsequently, which, together with the corresponding best response of the attacker, constitutes the switched Stackelberg equilibrium. Furthermore, the asymptotic stability of the closed-loop system under the switched Stackelberg equilibrium is guaranteed if the switching signal exhibits a certain average dwell time. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method in this paper.

Event-triggered static/dynamic feedback control for discrete-time linear systems

This paper visits the design of event-triggered static and dynamic feedback controllers for discrete-time linear systems. For the efficiency of energy, we make the first attempt to devise a switching event-triggered mechanism (ETM) which is characterized by a switching between the discrete-time sampled-data control and the common continuous ETM. In this sense, the event detector has a waiting time interval after sending a sampling signal. This method facilitates to enlarge the lower bound of inter-execution intervals so that the number of samplings can be reduced essentially. The issue of actuator saturation is also considered in this paper. By generalized sector condition, a switching Lyapunov functional is constructed to derive the sufficient conditions such that the discrete-time linear systems are local stable. Based on the stability conditions, the calculation methods are provided to solve both the desired control gains and the triggering parameters. Finally, numerical simulations are given to validate the effectiveness and superiority of the proposed method.

Asynchronous Switching Control of Discrete-time Linear System Based on Mode-dependent Average Dwell Time

In this paper, we investigate the problem of asynchronous switching control for discrete-time linear systems with mode-dependent average dwell time (MDADT) switching and take the time delay constraints of mode-dependent controllers into consideration. In practical, owing to signal transmission, system detection, and the like, switching of controllers is often lagged behind their corresponding subsystems. Therefore, we divide the time interval of a subsystem operating into two parts: the controller-matched and the controlle.-mismatched. First, the controller of each subsystem is designed according to its mode-dependent parameters. Then, considering the lag time of controllers, the appropriate MDADT is derived based on Lyapunov stability conditions, and the global exponential stability of the asynchronous switched systems is guaranteed. Finally, a numerical example is provided to demonstrate the potential effectiveness of the proposed method.

Finite-horizon optimal control of discrete-time linear systems with completely unknown dynamics using Q-learning

This paper investigates finite-horizon optimal control problem of completely unknown discrete-time linear systems. The completely unknown here refers to that the system dynamics are unknown. Compared with infinite-horizon optimal control, the Riccati equation (RE) of finite-horizon optimal control is time-dependent and must meet certain terminal boundary constraints, which brings the greater challenges. Meanwhile, the completely unknown system dynamics have also caused additional challenges. The main innovation of this paper is the developed cyclic fixed-finite-horizon-based Q-learning algorithm to approximate the optimal control input without requiring the system dynamics. The developed algorithm main consists of two phases: the data collection phase over a fixed-finite-horizon and the parameters update phase. A least-squares method is used to correlate the two phases to obtain the optimal parameters by cyclic. Finally, simulation results are given to verify the effectiveness of the proposed cyclic fixed-finite-horizon-based Q-learning algorithm.

Consistent Event-Triggered Control for Discrete-Time Linear Systems With Partial State Information

An event-triggered control (ETC) strategy is consistent if it achieves a better closed-loop performance than that of traditional periodic control for the same average transmission rate and does not generate transmissions in the absence of disturbances. In this letter, we propose a consistent ETC strategy for discrete-time linear systems with partial state information and Gaussian noise and disturbances when the performance is measured by an average quadratic cost, just as in the linear quadratic Gaussian (LQG) framework. This strategy incorporates a scheduler determining transmissions based on the error between two state estimates, which are provided by a stationary Kalman filter at the sensors/scheduler side and an estimator at the controller/actuators side relying on previously transmitted data. Through a numerical example, we show that the proposed strategy can achieve impressive performance gains with respect to periodic control for the same average transmission rate.

Variable-structure repetitive control for discrete-time linear systems with multiple-period exogenous signals

Variable-structure repetitive control for discrete-time linear systems with multiple-period exogenous signals

Performance Improvement of Discrete-time Linear Control Systems Subject to Varying Sampling Rates Using the Tikhonov Regularization Method

Methods to stabilize discrete-time linear control systems subject to variable sampling rates, i.e., using state feedback controllers, are well known in the literature. Several recent works address the use of the Tikhonov regularization method, originally designed to attenuate the noise effects on ill-posed problems, with the aim of improving performance and stabilizing approximately controllable dynamical systems. Inspired by these works, we propose the use of a feedback controller designed using the Tikhonov method to regularize discrete-time linear systems subject to varying sampling rates. The goal is to minimize an error function, thus improving the performance of the closed loop system and reducing the possibility of instability. Illustrative examples show the effectiveness of the proposed method.

Hybrid Optimal LQG Tracking Control for Linear Discrete-Time Systems by Simultaneous Optimization Approach of System States and Mode Distributions

In this paper, we study optimal LQG tracking control problems for linear discrete-time hybrid systems. We adopt an approach to prepare some candidates of mode distributions and minimize value functions at each time for the systems and performance indices. By the approach, system states and mode distributions are simultaneously optimized so that performance indices are minimized for causal information of reference signals. We consider optimization problems for averaged systems and averaged performance indices throughout mode distributions. Finally, we give numerical examples and verify that we can obtain both optimal tracking performance and optimal candidate of mode distributions by the algorithm presented in this paper.