Discrete-time Linear Systems Research Articles

Learning Minimax-Optimal Terminal State Estimators and Smoothers

We develop the first model-free policy gradient (PG) algorithm for the minimax state estimation of discrete-time linear dynamical systems, where adversarial disturbances could corrupt both dynamics and measurements. Specifically, the proposed algorithm learns a minimax-optimal solution for three fundamental tasks in robust (minimax) estimation, namely terminal state filtering, terminal state prediction, and smoothing, in a unified fashion. We further establish convergence and finite sample complexity guarantees for the proposed PG algorithm. Additionally, we propose a model-free algorithm to evaluate the attenuation (robustness) level of any estimator or smoother, which serves as a model-free solution to identify the maximum size of the disturbance under which the estimator will still be robust. We demonstrate the effectiveness of the proposed algorithms through extensive numerical experiments.

Learning Safety Filters for Unknown Discrete-Time Linear Systems

A learning-based safety filter is developed for discrete-time linear time-invariant systems with unknown models subject to Gaussian noises with unknown covariance. Safety is characterized using polytopic constraints on the states and control inputs. The empirically learned model and process noise covariance with their confidence bounds are used to construct a robust optimization problem for minimally modifying nominal control actions to ensure safety with high probability. The optimization problem relies on tightening the original safety constraints. The magnitude of the tightening is larger at the beginning since there is little information to construct reliable models, but shrinks with time as more data becomes available.

Data-Driven Control Design With LMIs and Dynamic Programming

The goal of this paper is to study model-free data-driven control evaluation and design strategies for discrete-time linear time-invariant systems, where the system model is unknown. In particular, our main contribution is twofold: 1) new state-input exploration and data collection schemes from experiences; 2) new data-driven linear matrix inequalities and dynamic programming methods for stabilization and optimal control problems. The proposed exploration and data collection schemes theoretically guarantee to acquire sufficient information from the system’s state-input trajectories that can solve the underlying control design problems. We prove that under mild assumptions, as more and more data is accumulated, the collected data can solve the problems with higher probability along with the proposed algorithms.

Innerization and Singular LQ Control Problem for Linear Discrete-Time Systems

In this paper, we discuss the LQ problem for discrete-time linear systems with singular weight. The interactor with all-pass property is used in order to make a given strictly proper system be proper. In our approach, the type of Riccati equation to be solved is different from the result for a square transfer function matrix case, where the input weight equals to zero. The relation between our approach and the singular weight case is discussed. Unlike the square case, the explicit solution to the problem cannot be obtained and a recursive method to solve the Riccati equation is shown. An application to the finite settling-time control is also discussed.

The $ l_\infty $-induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysis

<abstract><p>This paper develops a method for computing the $ l_{\infty} $-induced norm of a multivariable discrete-time linear system, for which an infinite-dimensional matrix should be intrinsically concerned with. To make such a computation feasible, we treat the infinite-dimensional matrix in a truncated fashion, and an upper bound and a lower bound on the $ l_\infty $-induced norm of the original multivariable discrete-time linear system are derived. More precisely, the matrix $ \infty $-norm of the (infinite-dimensional) tail part can be approximately computed by deriving its upper and lower bounds, while that of the (finite-dimensional) truncated part can be exactly obtained. With these values, an upper bound and a lower bound on the original $ l_\infty $-induced norm can be computed. Furthermore, these bounds are shown to converge to each other within an exponential order of $ N $, where $ N $ is the corresponding truncation parameter. Finally, some numerical examples are provided to demonstrate the theoretical validity and practical effectiveness of the developed computation method.</p></abstract>

Interval Observer for Switching Discrete-Time Linear Systems with Unknown Switching Sequences

The present paper deals with interval observers in the discrete-time case for switching linear systems. When the discrete state is known, several interval observers have been proposed. In the present paper, such an observer is proposed in the case there is no assumption nor knowledge on the switching parameter, and when the switching system considered is affected by bounded perturbations.

Learning-Based Adaptive Optimal Output Regulation of Discrete-Time Linear Systems

In this paper, we address the problem of model-free optimal output regulation of discrete-time systems that aims at achieving asymptotic tracking and disturbance rejection without the knowledge of the system parameters. Insights from reinforcement learning and adaptive dynamic programming are used to solve this problem. An interesting discovery is that the model-free discrete-time output regulation differs from the continuous-time counterpart in terms of the persistent excitation condition required to ensure the uniqueness and convergence of the policy iteration. In this work, we carefully establish the persistent excitation condition to ensure the uniqueness and convergence properties of the policy iteration.

Stabilization using shifted equilibria for saturated discrete-time linear systems⋆

We discuss controller designs for asymmetric saturated discrete-time linear systems. Under the assumption that a locally stabilizing controller of the origin is known, we augment the original controller with an additional term that vanishes in a neighborhood of the origin. The augmented controller outperforms the original controller in terms of the estimate of the region of attraction. The paper translates the results discussed in Braun et al. (2022a,b), from the continuous-time setting to the discrete-time setting, and numerically verifies that the results derived for continuous-time systems are recovered if the discrete-time system is obtained through an Euler discretization of a continuous-time system with a sufficiently small sampling rate.

Tighter Interval Estimation for Discrete-Time Linear Systems With a New Dynamic Triggering Approach

Tighter Interval Estimation for Discrete-Time Linear Systems With a New Dynamic Triggering Approach

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.

Long Duration Stochastic MPC With Mission-Wide Probabilistic Constraints Using Waysets

A stochastic Model Predictive Control (MPC) formulation is presented for systems operating for a finite time subject to constraints on the Mission-Wide Probability of Safety (MWPS). For linear discrete-time systems subject to unknown disturbances, the goal is to formulate an MPC controller to achieve a desired probability of mission success, where the entire closed-loop state and input trajectories stay within the state and input constraint sets and the final state reaches the desired terminal set. To enable longer missions under greater uncertainty, a wayset-based approach is proposed that allows for the prediction horizon of the MPC to be significantly shorter than the length of the mission. Using a scenario-based approach to stochastic MPC, the use of constrained zonotopes makes the computation of these waysets efficient and practical. Numerical results demonstrate the utility of the waysets for increasing the feasibility of MWPS constraints to longer missions and that the percentage of successful missions asymptotically converges above the desired probability.

A Short Note on Some Observable Outputs of Some Linear Discrete Time Systems

In this paper, we study the problem of controllable and observable output of the type: On the other hand, we present some new result about the formula of this linear discrete-time system and about controllable out put&

On LQ Control Problem for Linear Discrete-Time Systems with Singular Weightings

On LQ Control Problem for Linear Discrete-Time Systems with Singular Weightings

Multi-Symmetric Lyapunov Equations

This letter studies multi-symmetric Lyapunov equations and their application to stochastic control. It is shown how Lyapunov recursions can be used to efficiently compute the tensor-valued cumulants of the transient- and limit distributions of stochastic linear systems. This analysis is based on the assumption that the process noise distribution admits a moment expansion but, apart from this, all derivations and numerical algorithms are kept entirely general—without introducing any further assumptions on the distribution. Moreover, it is shown both theoretically and numerically how to exploit these recursions to construct accurate approximations of a rather general class of stochastic optimal control problems for linear discrete-time systems with conditional-value-at-risk constraints.

Delay-Dependent Invariance of Polyhedral Sets for Discrete-Time Linear Systems

In this paper we propose a delay-dependent analysis of the positive invariance property with respect to linear discrete-time systems with delayed states. An appropriate model transformation is employed, together with a matrix parametrization, which allow the derivation of delay-dependent invariance conditions of polyhedral sets with respect to the transformed model. We then show that such conditions imply the confinement of the state trajectories of the original system in the set, as long as the initial states satisfy additional constraints related to the system dynamics. The characterization of this set of admissible initial conditions gives rise to the proposition of a less conservative definition of set-invariance. We illustrate through numerical examples the fact that, under the proposed definition, confinement of state trajectories in the set can be achieved even though it is not invariant according to the classical definition.

Data-Based Control Design for Output-Error Linear Discrete-Time Systems With Probabilistic Stability Guarantees

Data-Based Control Design for Output-Error Linear Discrete-Time Systems With Probabilistic Stability Guarantees

On the Design of Robust Four Degree of Freedom Controllers for Linear Discrete-Time Systems

On the Design of Robust Four Degree of Freedom Controllers for Linear Discrete-Time Systems

Design of Finite Time Reduced Order H∞ Controller for Linear Discrete Time Systems

The current article gives a new approach that is efficient for the design of a low-order H∞ controller over a finite time interval. The system under consideration is a linear discrete time system affected by norm bounded disturbances. The proposed method has the advantage that takes into account both robustness aspects and desired closed-loop characteristics, reducing the number of variables in Linear Matrix Inequalities (LMIs). Thus, reduced order H∞ controller parameters are given to guarantee a finite time H∞ bound (FTB-H∞) for a closed-loop system. The method of the finite time stability, that is proven in this paper by the Lyapunov theory, can be applied to a wide range of process models. Numerical examples demonstrating the effectiveness of the results developed are presented at the end of this paper.

Markov parameters identification and adaptive iterative learning control for linear discrete-time MIMO systems with higher-order relative degree

For a kind of linear discrete-time-invariant multi-input-multi-output systems with a higher-order relative degree that repetitively operates within a finite time length, the paper exploits a Markov parameters identification method by making use of the multi-operation inputs and outputs obeying a criterion. Simultaneously, an adaptive iterative learning control scheme is architected by formulating the compensator with the sequentially identified Markov parameters and the tracking error in minimizing a performance index consisting of the quadratic tracking error of the next iteration and the compensation cost. Algebraic manipulations including the singular value decomposition of a matrix and the eigenvalues estimation conduct that the identification error of the Markov parameters is monotonically declining as the iteration goes on and a smaller identification ratio in the criterion delivers a faster decline rate. Meanwhile, a rigorous derivation achieves that under the assumption that the initial identification error is within an appropriate range the tracking error is monotonously convergent for the case when the relative degree is unit whilst the tracking error is asymptotically bounded for a positive level for the case where the relative degree is higher. Numerical simulations illustrate the validity and efficiency.

Stabilization of discrete-time linear systems with infinite distributed input delays

This paper investigates the stabilization problem of discrete-time linear systems with infinite distributed input delays. A novel framework is adopted to analyze the stability of the concerned systems. Under this framework, two truncated predictor feedback controllers are developed for two classes of discrete-time linear systems with infinite distributed input delays via the low gain method respectively. It is shown that under the designed controllers, those two classes systems are globally exponentially stable. To the best of our knowledge, this is the first time that the stabilization problem of discrete-time linear systems with infinite distributed input delays is considered. Two simulation examples are provided to illustrate the effectiveness of the proposed controllers.