Discrete-time LTI Systems Research Articles

Some Results on the Properties of Discrete-Time LTI State-Space Systems

Abstract This work brings forward several interesting facts about system properties of linear time-invariant discrete-time systems like output controllability, state controllability and state observability, trackability, and input and state observability. Particularly, the work brings about the inter-relationships between these properties while taking a state-space formulation-based approach to come up with simple facts that are backed up by proofs using preliminary linear algebra.

Undiscounted reinforcement learning for infinite-time optimal output tracking and disturbance rejection of discrete-time LTI systems with unknown dynamics

This paper proposes a novel control structure to solve the infinite-time linear quadratic tracking (LQT) problem. The major challenge in the LQT problem is the boundedness issue of the cost function in an infinite time framework. In many studies, a discount factor is utilised to overcome the challenge. However, it can affect the stability of the closed-loop system and the steady-state error. This paper proposes an optimal control structure that guarantees zero steady-state error with bounded cost function without utilising the discount factor. The optimal gains of the proposed control structure are computed via model-based and model-free reinforcement learning (RL) algorithms. As a novelty in model-based RL algorithms, a model predictive RL algorithm is proposed to reduce the number of iterations in the learning phase. A model-free reinforcement learning algorithm is utilised to obtain optimal control for tracking the reference online and without any knowledge of system dynamics. Finally, the simulation results verify the advantages of the proposed optimal control structure.

A data based system representation

The paper proposes a system representation formed by a minimal collection of sufficiently long restricted trajectories generated by an observable discrete time LTI system. Conditions are given under which such a collection is a system representation and also an exhaustive parametrization of these representations is provided. These can be also interpreted as a generalized persistency condition which complements the results encountered for the controllable case. In terms of the proposed representation some system properties are investigated and a controllable-autonomous decomposition is given. Finally it is shown how the representation associated to the inverse system, to the parallel and cascade connection, respectively, can be derived.

Deep Learning of a Communication Policy for an Event-Triggered Observer for Linear Systems

The problem of learning a communication policy is investigated in this paper for the design of an event-triggered observer for discrete-time LTI systems. Firstly, the event-triggered observer problem is formulated as an optimisation problem. The existence of a solution to this problem (communication policy) is investigated and it is verified if this solution still preserves the stability of the estimation error dynamics. Secondly, an algorithm is provided to approximate this optimal solution using neural networks and deep learning. Simulation examples are provided to illustrate the effectiveness of the learned communication policies.

Distributed State Estimation Over Time-Varying Graphs: Exploiting the Age-of-Information

We study the problem of designing a distributed observer for an LTI system over a time-varying communication graph. The limited existing work on this topic imposes various restrictions either on the observation model or on the sequence of communication graphs. In contrast, we propose a single-time-scale distributed observer that works under mild assumptions. Specifically, our communication model only requires strong-connectivity to be preserved over nonoverlapping, contiguous intervals that are even allowed to grow unbounded over time. We show that under suitable conditions that bound the growth of such intervals, joint observability is sufficient to track the state of any discrete-time LTI system exponentially fast, at any desired rate. We also develop a variant of our algorithm that is provably robust to worst-case adversarial attacks, provided the sequence of graphs is sufficiently connected over time. The key to our approach is the notion of a “freshness-index” that keeps track of the age-of-information being diffused across the network. Such indices enable nodes to reject stale estimates of the state, and, in turn, contribute to stability of the error dynamics.

LMI Framework for Set Reachability Inclusion in Discrete-Time LTI Systems

In this article, we present a convex optimization framework to verify the reachability of a desired set for discrete-time linear time-invariant systems. Given elliptically bounded inputs, the set of reachable states in <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> time steps is the Minkowski sum of a finite number of ellipsoids. We formulate the inclusion verification problem as a chain of constraints in the form of linear matrix inequalities. As the time horizon grows, the number of constraints becomes unwieldy, and we present a technique to achieve a similar level of accuracy with far fewer terms, significantly reducing the computational cost of the method. Numerical examples presented in this article show that the method is highly adaptable.

A graph-oriented approach to address generically flat outputs in structured LTI discrete-time systems

This paper addresses difference flatness for structured LTI discrete-time systems. Two forms of necessary and sufficient conditions for an output to be a structural flat output are given. First, a preliminary result algebraically defines a flat output in terms of invariant zeros regardless whether an LTI system is structured or not. Next, the conditions are expressed in terms of graphical conditions to define a structural flat output. Checking for the graphical conditions calls for algorithms that have polynomial-time complexity and that are commonly used for digraphs. The tractability of the conditions is illustrated on several examples.

On Discrete-Time Output Negative Imaginary Systems

This letter introduces the notion of linear Discrete-time Output Negative Imaginary (D-ONI) systems. The D-ONI class is defined in the z-domain and it includes the systems having poles on the unit circle. The proposed definition involves a real parameter δ ≥ 0, which indicates the strictness properties. δ > 0 specifies the strict subset, Discrete-time Output Strictly Negative Imaginary (D-OSNI), within the stable D-ONI class. Interestingly, the new D-ONI class captures the existing D-NI systems while restricted to discrete-time LTI systems having a real, rational and proper transfer function. However, the D-OSNI systems are not identical to the existing strictly D-NI (D-SNI) subset. Instead, these two subsets intersect each other. An LMI-based state-space characterisation is derived to check the strict/non-strict D-ONI properties of a given system relying on the value of δ. The paper also establishes the connections between the discrete-time Passive and discrete-time NI systems. Finally, a closed-loop stability result is proposed for a positive feedback interconnection of two D-ONI systems without poles at z = -1 and z = +1.

Identifying Resonant Poles by Visual Inspection of Pole-Zero Plots

We derive the s- and z-plane pole regions for continuous-time and discrete-time LTI systems to yield resonance. In this way, resonance can be identified by visual inspection of the pole-zero plot, without the need for calculations.

Formula omitted] induced norm analysis of discrete-time LTI systems for nonnegative input signals and its application to stability analysis of recurrent neural networks

In this paper, we focus on the “positive” l2 induced norm of discrete-time linear time-invariant systems where the input signals are restricted to be nonnegative. To cope with the nonnegativity of the input signals, we employ copositive programming as the mathematical tool for the analysis. Then, by applying an inner approximation to the copositive cone, we derive numerically tractable semidefinite programming problems for the upper and lower bound computation of the “positive” l2 induced norm. This norm is typically useful for the stability analysis of feedback systems constructed from an LTI system and nonlinearities where the nonlinear elements provide only nonnegative signals. As a concrete example, we illustrate the usefulness of the “positive” l2 induced norm for the stability analysis of recurrent neural networks with activation functions being rectified linear units.

Tracking control with aperiodic sampling over networks with delay and dropout

This paper introduces a discrete-time technique for output feedback tracking control with aperiodic sampling over networks suffering delay and dropout. Introducing the concept of pseudo sampling, NCS is modelled as a linear parameter varying (LPV) system. Two methods are explained to obtain a convex approximation of the uncertainty region. It is clarified how to perform the approximation to have the uncertainty space as small as possible. Using this uncertainty space, a polytopic approximation is obtained for the LPV model. The approximated model is used to synthesise a controller for set-point tracking and disturbance rejection. An observer is also synthesised based on this model. The simplicity of the model together with a small uncertainty space, results in less conservative LMI conditions. Compared to the existing results, an improved tracking performance is obtained. The final value theorem for discrete-time LTI systems with constant sampling period is extended to time varying sampling periods. This theorem is used to design a feedforward gain for zero steady-state tracking of staircase reference signals.

Discrete-time SISO LTI Systems with Monotonic Closed-loop Step Responses: Analysis and Control Based on Impulse Response Models

An explicit relation between the open- and closed-loop system responses is established directly in the time domain by means of Bell polynomials. This relation is used to derive necessary and sufficient conditions on the open-loop impulse response that guarantees that its closed-loop impulse response under unity feedback is non-negative in a given time range. Based on these conditions, a new design technique for PID controllers is presented that guarantees a monotonic closed-loop step response over a desired time interval. The results hold for any (possibly infinite-dimensional) discrete-time LTI system with a known impulse response. Numerical examples are provided to assess the contribution.

Observability of Discrete-Time LTI Systems Under Unknown Piece-Wise Constant Inputs

This letter is on observability of discrete-time LTI systems under unknown piece-wise constant inputs with sufficiently slow, but arbitrary update times. Assuming knowledge of the update times, we characterize the unobservable subspace and show that with sufficiently many measurements in each inter-update interval of the input, the unobservable subspace remains fixed. We explore the implications of the result for privacy in event-triggered control through an illustrative example.

On the resilience of a class of Correntropy-based state estimators

This paper deals with the analysis of a class of offline state estimators for LTI discrete-time systems in the presence of an arbitrary measurement noise which can potentially take any value. The considered class of estimators is defined as the solution of an optimization problem involving a performance function which can be interpreted as a generalization of cost functions used in the Maximum Correntropy Criterion. The conclusion of the analysis is that if the system is observable enough, then the considered class of estimators is resilient, which means that the obtained estimation error is independent from the highest values of the measurement noise. In the case of systems with a bounded process noise, the considered class of estimators provides a bounded estimation error under the appropriate conditions despite not being designed for this scenario.

Subspace identification of individual systems in a large-scale heterogeneous network

This paper considers the identification of a network consisting of discrete-time LTI systems that are interconnected by their unmeasurable states. For a large-scale network, the computational burden prevents a centralized solution. To cope with this problem, a subspace-based local identification approach using local observations is presented, which consists of subspace intersection operations in both the temporal and spatial domains. Sufficient conditions are provided for the consistent identification of the presented identification approach. Finally, the implementation of this approach on 1D network is specially investigated and numerical simulations are provided to show its effectiveness.

Construction of Externally Positive Systems and Order Reduction for Discrete-Time LTI System Analysis

This paper is concerned with the analysis of discrete-time LTI systems via construction of associated externally positive systems. Recently, the authors established a construction method of an externally positive system whose impulse response is given by the square of the original discrete-time LTI SISO system to be analyzed. This externally positive system allows us to characterize the H2 norm of the original system by means of the closed-form l∞-induced norm characterization of externally positive systems. It is nonetheless true that, for the original system of order n, the order of the resulting externally positive system is n2, incurring a drastic increase in computational burden of computer-aided analysis and synthesis. With this important issue in mind, in this paper, we show that the order can be reduced down to n(n+1)/2 by using the elimination and duplication matrices that are intensively studied by J. R. Magnus in the 80’s. In addition to the computational complexity reduction for the aforementioned H2 analysis, we show that such construction of externally positive systems with reduced order is quite effective in semidefinite-programming-based peak value analysis of impulse responses of general LTI systems.

Functional observer-based sliding mode controller design for discrete-time stochastic LTI systems with unmatched uncertainty

ABSTRACTDiscrete-time sliding mode control is one of the most prominent robust control technique to stabilise systems. However, the assumption of full state feedback being available is seldom satisfied in practice. This paper presents the functional observer-based sliding mode controller (SMC) design for discrete-time stochastic systems with unmatched uncertainty. A disturbance-dependent sliding function method is proposed such that the effect of unmatched uncertainty of the system is minimised. Furthermore, SMC is calculated using the functional observer method. Finally, a simulation example is given to show the effectiveness of the proposed method.

Data-driven robust fault tolerant linear quadratic preview control of discrete-time linear systems with completely unknown dynamics

In this paper, we propose a data-driven robust fault tolerant preview control scheme for discrete-time LTI systems with completely unknown dynamics. In general, this work consists of three key design parts. Firstly, the Markov parameters sequence is identified to extract system feature from data. Then, robust linear quadratic preview control policy composed of finite data feedback, integral operation and preview action is designed through establishing data-space characterisation and solving the related convex optimisation problem. Thirdly, an active fault tolerant method composed of fault detection, isolation, estimation and corresponding compensation is further constructed to accommodate fault influence. Note that all these involved designs are performed at Markov parameters sequence level. Thus above integrated preview control scheme can avoid the priori knowledge requirements including system matrices, order, and state variables. The effectiveness of the obtained results is finally verified via a case study of the injection moulding machine.

State space realizations robust to overloading for discrete-time LTI systems

Abstract In digital signal processing, overloading of a value occurs if the amplitude of a signal exceeds a predetermined limit. The overloading should be avoided, since it may cause a catastrophic effect on the system. To mitigate the overloading, the l2 norm scaling has been utilized for many years. However, the l2 norm scaling does not always work well, since it is not directly related to the amplitude of the signal. This paper presents a design method of state space realizations for digital discrete-time LTI systems that are robust to overloading. Upper bounds of the amplitudes of state variables are evaluated by using the invariant set theory. When different numbers of bits are allocated to represent the values of state variables, our realization is given by minimizing the volume of an ellipsoidal invariant set. On the other hand, when an equal number of bits is assigned to each state variable, an upper bound on the magnitude of the state variables is minimized and the optimal realization is obtained by the spherical invariant set. Numerical examples are provided to show that our realizations are robust to overloading.

Construction of Externally Positive Systems for Discrete-Time LTI System Analysis

Abstract This paper is concerned with analysis techniques of discrete-time LTI systems via construction of relevant externally positive systems. Recently, the author established a construction method of an externally positive system whose impulse response is given by the square of the original discrete-time LTI SISO system to be analyzed. This externally positive system allows us to characterize the H2 norm of the original system by means of the l∞-induced norm characterization of externally positive systems. It is nonetheless true that, for the original system of order n, the order of the resulting externally positive system is n2 and hence this leads to the increase of associated computational burden. With this important issue in mind, in this paper, we show that the order can be reduced down to n(n + 1)/2 by using the elimination and duplication matrices that are intensively studied by Jan R. Magnus in the 80’s. In addition to the computational complexity reduction for the aforementioned H2 analysis, we show that such construction of externally positive systems with reduced order is quite effective in semidefinite-programming-based peak value analysis of impulse responses of general LTI systems.