Discrete-time Linear Dynamical Systems Research Articles

Data-Driven Robust Control of Discrete-Time Uncertain Linear Systems via Off-Policy Reinforcement Learning.

This paper presents a model-free solution to the robust stabilization problem of discrete-time linear dynamical systems with bounded and mismatched uncertainty. An optimal controller design method is derived to solve the robust control problem, which results in solving an algebraic Riccati equation (ARE). It is shown that the optimal controller obtained by solving the ARE can robustly stabilize the uncertain system. To develop a model-free solution to the translated ARE, off-policy reinforcement learning (RL) is employed to solve the problem in hand without the requirement of system dynamics. In addition, the comparisons between on- and off-policy RL methods are presented regarding the robustness to probing noise and the dependence on system dynamics. Finally, a simulation example is carried out to validate the efficacy of the presented off-policy RL approach.

Feedback control using a strategic sensor

ABSTRACT A dynamic estimation and control problem with a strategic sensor is considered. The strategic sensor may provide corrupted messages about the state measurements of a discrete-time linear time-invariant dynamical system to the system operator (or the controller). The system operator then uses this information to construct an estimate of the state of the system (and perhaps private variables of the sensor). The estimate is used to control the system to achieve the operator's desired objective. The problem is formulated as a game, which might be conflicting to that of the strategic sensor. An equilibrium of the game is computed and its properties are investigated.

Linear system security—Detection and correction of adversarial sensor attacks in the noise-free case

We address the problem of attack detection and attack correction for multi-output discrete-time linear time-invariant dynamical systems under sensor attack. More specifically, we focus on the situation where adversarial attack signals are added to some of the system’s output signals. Algorithms for sensor attack detection and correction are presented using the notion of ‘security index’ which characterizes the system’s vulnerability against such attacks. The results are illustrated by examples involving multiple sensors.

A trust-region method for optimal H2 model reduction of discrete-time dynamical systems

ABSTRACTIn this paper, we present an optimal model reduction method on the basis of the trust-region technique for linear time-invariant discrete-time dynamical systems. First, based on the poles and residues, the error norm for single-input and single-output discrete-time systems is investigated, which leads to the error gradient and Hessian. Next, for multiple-input and multiple-out discrete-time systems, the gradient and Hessian of the error norm are accordingly derived. Then, the error gradient and Hessian are employed to establish the trust-region method for optimal model reduction. Moreover, it is shown that the proposed method can produce a decreasing sequence. The construction of the state space realization of the reduced order system is studied concerning the divisions of the resulting residues. Model reduction is investigated for nonlinear discrete-time system. Finally, an illustrative example is used to demonstrate the performance.

Distributed Quickest Detection of Cyber-Attacks in Smart Grid

In this paper, online detection of false data injection attacks and denial of service attacks in the smart grid is studied. The system is modeled as a discrete-time linear dynamic system and state estimation is performed using the Kalman filter. The generalized cumulative sum algorithm is employed for quickest detection of the cyber-attacks. Detectors are proposed in both centralized and distributed settings. The proposed detectors are robust to time-varying states, attacks, and set of attacked meters. Online estimates of the unknown attack variables are provided, that can be crucial for a quick system recovery. In the distributed setting, due to bandwidth constraints, local centers can only transmit quantized messages to the global center, and a novel event-based sampling scheme called level-crossing sampling with hysteresis is proposed that is shown to exhibit significant advantages compared with the conventional uniform-in-time sampling scheme. Moreover, a distributed dynamic state estimator is proposed based on information filters. Numerical examples illustrate the fast and accurate response of the proposed detectors in detecting both structured and random attacks and their advantages over existing methods.

A distributed Kalman filtering algorithm with fast finite-time convergence for sensor networks

This paper proposes a new distributed algorithm for Kalman filtering. It is assumed that a linear discrete-time dynamic system is monitored by a network of sensors with some being active and some idle. The goal of distributed state estimation is to devise a distributed algorithm such that each node can independently compute the optimal state estimate by using its local measurements and information exchange with its neighbours. The proposed algorithm applies to acyclic network graphs (i.e., tree graphs) with fast finite-time convergence, but is also applicable to cyclic graphs by combining it with a distributed loop removal algorithm. The proposed algorithm enjoys low complexities, robustness against transmission adversaries and asynchronous implementability. The proposed distributed algorithm also applies to maximum likelihood estimation and weighted least-squares estimation, as special cases. With simple modifications, the proposed algorithm also applies to an important problem in signal processing called distributed field estimation.

Algorithm for Solving of Two-Level Hierarchical Minimax Adaptive Control Problem in a Linear Discrete-Time Dynamical System

Abstract In this paper we consider a discrete-time dynamical system consisting from two controllable objects. The dynamics of each object is described by the corresponding vector linear discrete-time recurrent relation. In this dynamical system there are two levels of control. The quality of process implementation at each level of the control system is estimated by the corresponding terminal linear functional. For the dynamical system under consideration, a mathematical formalization of a two-level hierarchical minimax adaptive control problem in the presence of perturbations, and an algorithm for its solving are proposed. The construction of this algorithm can be implemented as a finite sequence of solutions of a linear mathematical programming problems, and a finite discrete optimization problems.

Change detection via affine and quadratic detectors

The goal of the paper is to develop a specific application of the convex optimization based hypothesis testing techniques developed in A. Juditsky, A. Nemirovski, Hypothesis testing via affine detectors, Electronic Journal of Statistics 10:2204--2242, 2016. Namely, we consider the Change Detection problem as follows: given an evolving in time noisy observations of outputs of a discrete-time linear dynamical system, we intend to decide, in a sequential fashion, on the null hypothesis stating that the input to the system is a nuisance, vs. the alternative stating that the input is a signal, with both the nuisances and the nontrivial signals modeled as inputs belonging to finite unions of some given convex sets. Assuming the observation noises zero mean sub-Gaussian, we develop computation-friendly sequential decision rules and demonstrate that in our context these rules are provably near-optimal.

The Effect of Time-Varying Jamming Interference on Networked Stabilization

The effect of jamming attacks on the networked control of a linear discrete-time dynamical system is investigated. In this networked control system, a wireless communication channel is used for the transmission of the control input packets from the controller to the plant. The probability of transmission failures on this channel depends on the power of the interference signals emitted by the jamming attacker. We explore the case where the attacker's interference power is time-varying but bounded in average. We obtain almost sure asymptotic stability conditions for the networked control system. In our stability analysis, we utilize nondecreasing and concave functions of the attacker's interference power that upper-bound the transmission failure probability. We also obtain upper and lower bounds on the probability of the state being outside a given level set at a given time. Finally, we present numerical examples to demonstrate the efficacy of our results under various scenarios.

The Polynomial Approach to the LQ non-Gaussian Regulator Problem through Output Injection

In this paper, an improved approach for the solution of the regulator problem for linear discrete-time dynamical systems with non-Gaussian disturbances and quadratic cost functional is proposed. It is known that a suboptimal recursive control can be derived from the classical linear quadratic Gaussian (LQG) solution by substituting the linear filtering part with a quadratic, or in general polynomial, filter. However, we show that when the system is not asymptotically stable the polynomial control does not improve over the classical LQG solution, due to the lack of the internal stability of the polynomial filter. In order to enlarge the class of systems that can be controlled, we propose a new method based on a suitable rewriting of the system by means of an output injection term. We show that this allows us to overcome the problem and to design a polynomial optimal controller also for non asymptotically stable systems. Numerical results show the effectiveness of the method.

A Probabilistic Characterization of Random and Malicious Communication Failures in Multi-Hop Networked Control

The control problem of a linear discrete-time dynamical system over a multi-hop network is explored. The network is assumed to be subject to packet drops by malicious and nonmalicious nodes as well as random and malicious data corruption issues. We utilize asymptotic tail-probability bounds of transmission failure ratios to characterize the links and paths of a network as well as the network itself. This probabilistic characterization allows us to take into account multiple failures that depend on each other, and coordinated malicious attacks on the network. We obtain a sufficient condition for the stability of the networked control system by utilizing our probabilistic approach. We then demonstrate the efficacy of our results in different scenarios concerning transmission failures on a multi-hop network.

Модифицированный алгоритм построения областей достижимости линейных дискретных динамических систем

The paper considered the modified general recursion algebraic method for the linear discrete-time dynamic systems reachable sets computation. The algorithms of application programs for reachable sets constructing is described. The comparative analysis of original (Shorikov A.F. [Minimax Estimation and Control in Discrete-Time Dynamic Systems]. Yekaterinburg. Ural State University Publ., 1997. 242 p. (in Russ.)) and suggest algorithms is provided by giving of numerical example.

Shannon Entropy And Tracking Dynamic Systems Over Noisy Channels

This paper is concerned with the estimation of state trajectory of linear discrete timedynamic systems subject to parametric uncertainty over the compound erasure channelthat uses feedback channel intermittently. For this combined system and channel,using the data processing inequality and a robust version of the Shannon lower bound,a necessary condition on channel capacity for estimation of state trajectory at thereceiver giving almost sure asymptotically zero estimation error is presented. Then,an estimation technique over the compound erasure channel that includes an encoder,decoder and a sucient condition under which the estimation error at the receiveris asymptotically zero almost surely is presented. This leads to the conclusion thatover the compound erasure channel, a condition on Shannon capacity in terms ofthe rate of expansion of the Shannon entropy is a necessary and sucient conditionfor estimation with uniform almost sure asymptotically zero estimation error. Thesatisfactory performance of the proposed technique is illustrated using simulation.

An Improved Approach to the LQ non-Gaussian Regulator Problem

In this paper, an improved approach for the solution of the regulator problem for linear discrete-time dynamical systems with non-Gaussian disturbances and quadratic cost function is proposed. It is known that a sub-optimal control can be derived from the classical LQG solution by substituting the linear filtering part with a quadratic optimal filter. However, classical quadratic filters have some critical drawbacks when the system is not asymptotically stable and, as a consequence in that case, there is no guarantee on the stochastic stability of the controlled system. In order to enlarge the class of systems that can be controlled we will make use of the Feedback Quadratic Filter and a quadratically optimal controller is designed also for non asymptotically stable systems. Numerical results show the performance of these methods.

Leading impulse response identification via the Elastic Net criterion

This paper deals with the problem of finding a low-complexity estimate of the impulse response of a linear time-invariant discrete-time dynamic system from noise-corrupted input–output data. To this purpose, we introduce an identification criterion formed by the average (over the input perturbations) of a standard prediction error cost, plus an ℓ1 regularization term which promotes sparse solutions. While it is well known that such criteria do provide solutions with many zeros, a critical issue in our identification context is where these zeros are located, since sensible low-order models should be zero in the tail of the impulse response. The flavor of the key results in this paper is that, under quite standard assumptions (such as i.i.d. input and noise sequences and system stability), the estimate of the impulse response resulting from the proposed criterion is indeed identically zero from a certain time index nl (named the leading order) onwards, with arbitrarily high probability, for a sufficiently large data cardinality N. Numerical experiments are reported that support the theoretical results, and comparisons are made with some other state-of-the-art methodologies.

Chaotic behavior of discrete-time linear inclusion dynamical systems

Given any finite family of real d-by-d nonsingular matrices {S1,…,Sl}, by extending the well-known Li–Yorke chaos of a deterministic nonlinear dynamical system to a discrete-time linear inclusion or hybrid or switched system:xn∈{Skxn−1;1≤k≤l},x0∈Rdandn≥1,we study the chaotic dynamics of the state trajectory (xn(x0, σ))n ≥ 1 with initial state x0∈Rd, governed by a switching law σ:N→{1,…,l}. Two sufficient conditions are given so that for a “large” set of switching laws σ, there exhibits the scrambled dynamics as follows: for all x0,y0∈Rd,x0≠y0,lim infn→+∞∥xn(x0,σ)−xn(y0,σ)∥=0andlim supn→+∞∥xn(x0,σ)−xn(y0,σ)∥=∞.This implies that there coexist positive, zero and negative Lyapunov exponents and that the trajectories (xn(x0, σ))n ≥ 1 are extremely sensitive to the initial states x0∈Rd. We also show that a periodically stable linear inclusion system, which may be product unbounded, does not exhibit any such chaotic behavior. An explicit simple example shows the discontinuity of Lyapunov exponents with respect to the switching laws.

Multisensory Prediction Fusion of Nonlinear Functions of the State Vector in Discrete-Time Systems

We propose two new multisensory fusion predictors for an arbitrary nonlinear function of the state vector in a discrete-time linear dynamic system. Nonlinear function of the state (NFS) represents a nonlinear multivariate functional of state variables, which can indicate useful information of the target system for automatic control. To estimate the NFS using multisensory information, we propose centralized and decentralized predictors. For multivariate polynomial NFS, we propose an effective closed-form computation procedure for the predictor design. For general NFS, the most popular procedure for the predictor design is based on the unscented transformation. We demonstrate the effectiveness and estimation accuracy of the fusion predictors on theoretical and numerical examples in multisensory environment.

System Identification of Heat-Transfer Process of Frequency Induction Furnace for Melting Copper Based on Particle Swarm Algorithm

An adaptive evolutionary strategy in standard particle swarm optimization is introduced. Adaptive evolution particle swarm optimization is constructed to improve the capacity of global search. A method based on adaptive evolution particle swarm optimization for identification of continuous system with time delay is proposed. The basic idea is that the identification of continuous system with time delay is converted to an optimization of continuous nonlinear function. The adaptive evolution particle swarm optimization is utilized to find an optimal solution of continuous nonlinear function. Convergence conditions are given by the convergence analysis based on discrete time linear dynamic system theory. Numerical simulation results show that the proposed method is effective for a general continuous system with time delay and the system of heat-transfer process of frequency induction furnace for melting copper.

$\mathcal{H}_\infty$ Control Problem for Discrete-Time Algebraic Dynamical Systems

This paper considers the suboptimal $\mathcal{H}_\infty$ control problem for linear discrete-time algebraic dynamical systems. Such systems can be formally described as linear and time-invariant discrete-time systems, whose transfer function matrix is allowed to be improper or polynomial. The parametrization of output feedback controllers is given in a realization-based setting involving two generalized algebraic Riccati equations and features the same elegant simplicity of the standard (proper) case. Two relevant numerical examples prove the effectiveness of our approach.

Event-Triggered Output Feedback Control Resilient Against Jamming Attacks and Random Packet Losses

A networked output feedback control problem for linear discrete-time dynamical systems is explored. The communication channel used for transmission of output information packets from the plant to the controller is assumed to be subject to malicious jamming attacks and random packet losses due to unreliability of the network. We propose an event-triggered observer-based control strategy that guarantees almost sure asymptotic stabilization. We provide a numerical method for finding observer gains as well as parameters for our proposed event-triggering mechanism and illustrate the efficacy of our approach through a numerical example.