Linear Time-invariant Discrete-time Research Articles

Controller switching with the anti-bump adapter

Switching on a new or retuned controller while keeping the control loop engaged tends to generate transient “bumps” in the commands and plant outputs. For a linear time-invariant (LTI) controller in state–space form, these bumps can be mitigated or even suppressed by resetting the controller state to an appropriate non-zero value adapted to the pre-switching trajectories of the feedback loop signals. This paper proposes a simple way to compute this adapted controller state by minimizing the difference between the commands actually applied immediately before switching and the virtual command trajectory which the new controller would have generated if it had been activated in parallel. This adapted state can be recursively computed as the output of an LTI system in standard state–space form, the anti-bump adapter. Computing the anti-bump adapter involves only standard matrix algebra and control concepts (e.g., observability and controllability matrices/Gramians). Calculations are very simple and can be translated into compact computer code. This bump suppression procedure is applicable to any discrete-time or continuous-time LTI SISO or MIMO controller, without any assumption on or knowledge of either the previously active controller or the plant. Illustrative applications to continuous-time and discrete-time switching problems and a Matlab® code for computing the anti-bump adapter are presented.

Optimization Landscape of Policy Gradient Methods for Discrete-Time Static Output Feedback.

In recent times, significant advancements have been made in delving into the optimization landscape of policy gradient methods for achieving optimal control in linear time-invariant (LTI) systems. Compared with state-feedback control, output-feedback control is more prevalent since the underlying state of the system may not be fully observed in many practical settings. This article analyzes the optimization landscape inherent to policy gradient methods when applied to static output feedback (SOF) control in discrete-time LTI systems subject to quadratic cost. We begin by establishing crucial properties of the SOF cost, encompassing coercivity, L -smoothness, and M -Lipschitz continuous Hessian. Despite the absence of convexity, we leverage these properties to derive novel findings regarding convergence (and nearly dimension-free rate) to stationary points for three policy gradient methods, including the vanilla policy gradient method, the natural policy gradient method, and the Gauss-Newton method. Moreover, we provide proof that the vanilla policy gradient method exhibits linear convergence toward local minima when initialized near such minima. This article concludes by presenting numerical examples that validate our theoretical findings. These results not only characterize the performance of gradient descent for optimizing the SOF problem but also provide insights into the effectiveness of general policy gradient methods within the realm of reinforcement learning.

Distributed State Estimation Under Jointly Connected Switching Networks: Continuous-Time Linear Systems and Discrete-Time Linear Systems

In this paper, we address the distributed state estimation problem for both continuous-time linear time-invariant (LTI) systems and discrete-time LTI systems under switching networks. The observed system is jointly observable, i.e., each agent can only access a part of the measurement output of the observed system and cannot recover the full state by itself. The full state estimation has to be achieved by network communication of neighboring agents. In contrast to existing works, the salient feature of this work is that the developed approach can deal with the jointly connected switching network and thus is more resilient to unreliable communication. First, we propose a new observability decomposition method for linear systems in modal canonical form. Then, we design distributed observers for both the continuous-time and the discrete-time system. Based on the common Lyapunov function approach, we show that the switched estimation error system is asymptotically stable and thus, the full state estimation can be achieved under jointly connected switching networks.

Modeling microgrid power flow dynamics with covariance based realization

Reducing transient effects of power quality in the distribution network of a microgrid requires a better understanding of the dynamics of power flow and distributed energy resources (DERs) present within the microgrid. Instead of labor intensive and detailed physics-based distribution network and DER modeling, this paper shows how power flow dynamics can also be modeled in a dynamic equivalent with a data-driven approach that exploits the time dependent covariance of power flow at different locations within a microgrid. The proposed approach results in multi-input multi-output (MIMO) linear time invariant (LTI) discrete-time filter concatenated by a static nonlinearity to model both dynamics and power losses. Estimation of the MIMO LTI dynamics is done by the Covariance Based Realization Algorithm (CoBRA) that can handle noisy perturbed measurements and provide a low order MIMO LTI filter with a normalized steady-state gain. The additional static nonlinearity is found by the estimation of the admittance matrix of the distribution network within the microgrid. The main contribution of this paper is proposing a physics-informed data-driven approach to modeling microgrid power flow dynamics, where the CoBRA method is implemented to learn parameters from data while observing a priori physical constraints. The approach is illustrated on experimental data from a microgrid with multiple DERs and it is shown that an excellent agreement in power flow dynamics is obtained over a large operating range, which demonstrates its advantage over the N4SID method.

Theoretical and Algorithmic Study of Inverses of Arbitrary High-Dimensional Multi-Input Multi-Output Linear-Time-Invariant Systems

Nowadays, systems need to be built and/or characterized to handle exceptional circumstances or adapt to a world itself more complex. A typical phenomenon can often be found that systems are required to accommodate high-dimensional inputs and outputs. Although the theories and methods for inverting a single-input single-output (SISO) linear-time-invariant (LTI) system have been well established, the generalized framework (consisting of theories and algorithms) for extending to arbitrary high-dimensional multi-input multi-output (MIMO) scenarios is still unsubstantial in the existing literature as this extension is far from trivial. In this work, we would like to develop such a new framework for governing the inversion of arbitrary high-dimensional discrete-time MIMO LTI systems, where any individual transfer function from a certain input to a certain output may have the infinite-impulse-response (IIR) characteristics. We propose two new inversion algorithms to invert the transfer-function tensors (TFTs) of arbitrary MIMO LTI systems. The pertinent computational complexities are also investigated for our proposed two TFT-inversion algorithms. The approximation of the inverse of an arbitrary TFT by a finite-impulse-response (FIR) TFT is studied and the corresponding approximation-error analysis is derived as well. Finally, numerical evaluations are presented to study the computational complexities with respect to different TFT dimensions and ranks.

On the impulse response peak of discrete-time uncertain LTI systems*

This paper addresses the problem of determining the peak of the admissible impulse responses of a discrete-time uncertain linear time-invariant (LTI) system. The uncertainty is represented by a vector of parameters constrained into a semialgebraic set that affect polynomially the matrices of the system. It is shown that an upper bound of the sought peak can be obtained by solving a semidefinite program (SDP) constructed by introducing a novel approach, which consists of splitting the responses into two temporal parts: head and tail. The head of the responses is analyzed via inspection of the response formula, while the tail of the responses is analyzed via quadratic Lyapunov functions polynomially parameterized by the uncertainty. As shown by some numerical examples, the conservatism of the upper bound can be reduced by increasing the length of the head or the degree of the Lyapunov functions in the uncertainty.

State Observer Under Multi-Rate Sensing Environment and Its Design Using l2-Induced Norm

Since the sensing period of a signal generally depends on the used sensor, it may be different from another, even in a single control system if it involves multiple sensors. This paper investigates a design problem of state observers for linear time-invariant discrete-time plants under a multi-rate sensing environment. The sensing periods of the sensors in the plant are assumed as mutually rational ratios. First, we characterize a state observer for a plant with multi-rate sensing as a periodically time-varying state observer. Then, we discuss the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l_{2}$ </tex-math></inline-formula> performance analysis of a state estimation error with the given periodically time-varying state observer. A linear matrix inequality (LMI) condition is provided for the analysis. By extending the LMI condition for analysis, we also provide that for multi-rate observer synthesis. Finally, we numerically illustrate the effectiveness of the proposed multi-rate state observer. Even if all the sensors have the same period, the sensing timing is not unique. Therefore, we numerically analyze whether the performance changes when the observation timing between multiple sensors is different.

Generalized framework for computing the [formula omitted]-induced norm of sampled-data systems

This paper develops a generalized framework for computing the L∞-induced norm of sampled-data systems, by which we mean those consisting of a continuous-time linear time-invariant (LTI) plant and a discrete-time LTI controller, and such systems have a linear periodically time-varying (LPTV) nature. In the authors’ preceding studies, the input/output relation of sampled-data systems was described in an operator-theoretic fashion, and the kernel and input functions of an input operator and the hold function of an output operator were approximated by piecewise lth order polynomials with l=0,1. This leads to an approximate computation of this norm, where the computation errors were shown to converge to 0 in the order of 1/Ml+1 as the approximation parameter M becomes larger. Along this line, we aim at improving the computation performance compared to the preceding studies by considering the freedom in the point around which the corresponding functions are expanded to a Taylor series. This leads to a generalization of the piecewise lth order approximations, and we show that taking the central point for the Taylor expansion leads to quantitatively improved accuracy than that of our preceding results that take an edge point.Finally, the effectiveness of the developed method is verified through a numerical example.

Phases of discrete-time LTI multivariable systems

In contrast to the well-developed gain analysis for multi-input-multi-output (MIMO) linear time-invariant (LTI) systems, the research on the phase analysis does not share the same status. In this paper, we introduce the phase response of a class of discrete-time (DT) LTI multivariable systems by exploiting a definition of matrix phases based on the numerical range. Positive frequency joint sectorial systems are also defined, which can generalize the positive real and negative imaginary systems. The interpretation of system phases is given in terms of the input and output signals of the system. A sectored real lemma is obtained to characterize the phase information from a state–space realization. Motivated by finding a phasic counterpart to the small gain theorem, we develop a small phase theorem for the internal stability of a closed-loop system. The phase properties are also investigated for parallel and feedback interconnected systems.

An Event-Triggered Watermarking Strategy for Detection of Replay Attacks

The problem of detecting replay attacks in linear time-invariant discrete-time systems is considered in this paper. In the same spirit of watermarking techniques that apply a distinctive signature to the plant\’s signals, we propose an event-triggered control scheme, that is purposely designed to generate a unique sequence of switching intervals, by computing an appropriate input value to be held constant while the communication is not triggered. We provide a detailed undetectability characterization in the time domain and design a controller that achieves the desired behavior. Our proposed method results in a control scheme that makes it hard for an attacker to satisfy undetectability conditions, and, as a result, a standard observer-based residual generator can be employed to reveal replay attacks. We finally validate the method using a numerical example.

The Entropy Gain of Linear Systems and Some of Its Implications.

We study the increase in per-sample differential entropy rate of random sequences and processes after being passed through a non minimum-phase (NMP) discrete-time, linear time-invariant (LTI) filter G. For LTI discrete-time filters and random processes, it has long been established by Theorem 14 in Shannon’s seminal paper that this entropy gain, , equals the integral of . In this note, we first show that Shannon’s Theorem 14 does not hold in general. Then, we prove that, when comparing the input differential entropy to that of the entire (longer) output of G, the entropy gain equals . We show that the entropy gain between equal-length input and output sequences is upper bounded by and arises if and only if there exists an output additive disturbance with finite differential entropy (no matter how small) or a random initial state. Unlike what happens with linear maps, the entropy gain in this case depends on the distribution of all the signals involved. We illustrate some of the consequences of these results by presenting their implications in three different problems. Specifically: conditions for equality in an information inequality of importance in networked control problems; extending to a much broader class of sources the existing results on the rate-distortion function for non-stationary Gaussian sources, and an observation on the capacity of auto-regressive Gaussian channels with feedback.

Complexity of constrained sensor placement problems for optimal observability

This article studies two problems related to observability and efficient constrained sensor placement in linear time-invariant discrete-time systems with partial state observations: (i) We impose the condition that both the set of outputs and the state that each output can measure are pre-specified. We establish that for any fixed k>2, the problem of placing the minimum number of sensors/outputs required to ensure that the structural observability index is at most k, is NP-complete. Conversely, we identify a subclass of systems whose structures are directed trees with self-loops at every state vertex, for which the problem can be solved in linear time. (ii) Assuming that the set of states that each given output can measure is given, we prove that the problem of selecting a pre-assigned number of outputs in order to maximize the number of states of the system that are structurally observable (i.e., to maximize the size of the observable subgraph) is also NP-hard. As an application, we identify suitable conditions on the system structure under which there exists an efficient greedy strategy, which we provide, to obtain a (1−1e)-approximate solution. An illustration of the techniques developed for this problem is given on the benchmark IEEE 118-bus power network containing roughly 400 states in its linearized model.

Set-Valued Feedback Control and Its Application to Event-Triggered Sampled-Data Systems

Motivated by challenges in feedback control with event-triggered control input updates, this article studies an ellipsoidal set-valued feedback control problem for linear time-invariant (LTI) discrete-time (DT) systems, and its application to event-triggered control systems. First, a state feedback controller based on the set-valued information is designed for LTI DT systems with disturbances. Then, the stability of the closed-loop system is analyzed, and the state of the closed-loop system is shown to be uniformly ultimately bounded in an ellipsoidal set. Based on the proposed set-valued feedback controller, we consider the case when the event-triggering condition (ETC) is described by an ellipsoidal set for an event-triggered control system. To ensure the requirements on control performance and communication rates simultaneously, an optimal ETC design problem is formulated and its optimal solution is provided. Finally, simulation examples are given to show the effectiveness of the proposed method.

Static output feedback stabilization of discrete time linear time invariant systems based on approximate dynamic programming

This study proposes a method for the static output feedback (SOF) stabilization of discrete time linear time invariant (LTI) systems by using a low number of sensors. The problem is investigated in two parts. First, the optimal sensor placement is formulated as a quadratic mixed integer problem that minimizes the required input energy to steer the output to a desired value. Then, the SOF stabilization, which is one of the most fundamental problems in the control research, is investigated. The SOF gain is calculated as a projected solution of the Hamilton-Jacobi-Bellman (HJB) equation for discrete time LTI system. The proposed method is compared with several examples from the literature.

Optimal Finite-Horizon Control for Networked Control Systems in the Presence of Random Delays and Packet Losses

An optimal control law for networked control systems with a discrete-time linear time-invariant (LTI) system as plant and networks between sensor and controller as well as between controller and actuator is proposed. This controller is designed by solving an optimization problem that is a generalization of the optimization problem used to obtain the Linear Quadratic Regulator (LQR) for deterministic discrete-time LTI systems. The networks are represented by random delays and drop outs of transmitted data packets.

Computation of Least-Conservative State-Constraint Sets for Decentralized MPC With Dynamic and Constraint Coupling

We address the problem of synthesizing state-constraint sets for a fully decentralized Model Predictive Control (MPC) scheme. We consider linear time-invariant discrete time systems, with subsystems possibly coupled in both dynamics and state constraints. For each individual subsystem we employ a set-based framework to compute the state-constraint sets, that are used to synthesize local tube-based MPC controllers. The offline problem that computes the constraint sets explicitly ensures that the feasible regions of the MPC controllers are non-empty, and whenever the controllers are feasible, the overall system constraints are satisfied with the least conservativeness possible. We demonstrate the closed-loop performance of the decentralized scheme, assessed with respect to centralized MPC, using a numerical example.

Data-based Analysis Methods for the State Controllability and State Observability of Discrete-time LTI Systems with Time-delays

We present a couple of data-based methods to analyze the state controllability and state observability of discrete-time Linear time-Invariant (LTI) systems with time-delays, which have unknown parameter matrices. They first augment the system into a high dimensional LTI model, then apply the measured state and output data to directly build the controllability and observability matrices respectively of this high dimensional model, whose ranks are used as the criteria of the corresponding properties of the system before augmentation. These data-based methods have low computational load and calculation complexity.

Computation of Invariant Tubes for Robust Output Feedback Model Predictive Control

This paper presents an algorithm to calculate tightened invariant tubes for output feedback model predictive controllers (MPC). We consider discrete-time linear time-invariant (DLTI) systems with bounded state and input constraints and subject to bounded disturbances. In contrast to existing approaches which either use pre-defined control and observer gains or compute the control and observer gains that optimize the volume of the invariant sets for the estimation and control errors separately, we consider the problem of optimizing the volume of these sets simultaneously. The nonlinearities associated with computing the control and observer gains are circumvented by the application of Farkas’ Theorem and an extended Elimination Lemma, to convert the nonconvex optimization problem into a convex semidefinite program. An update algorithm is then used to reduce the volume of the invariant tube through a finite number of iterations. Numerical examples are provided to illustrate the effectiveness of the proposed algorithm.

Phase Analysis for Discrete-time LTI Multivariable Systems

In contrast to the well-developed gain analysis for multi-input-multi-output (MIMO) linear time-invariant (LTI) systems, the research on the phase analysis does not share the same status. In this paper, we introduce the phase response of a class of discrete-time (DT) LTI multivariable system by exploiting a definition of matrix phases based on the numerical range. Half-sectorial transfer matrices are defined, which can generalize the positive real and negative imaginary systems. A sectored real lemma is obtained to characterize the phase information of a half-sectorial system from a state-space realization. Motivated by finding a phasic counterpart to the small gain theorem, we develop a small phase theorem for the internal stability of closed-loop systems.

Decoupling of Discrete-time Dynamical Systems Through Input-Output Blending

This paper presents a subsystem decoupling method for Linear Time Invariant Discrete-time systems. The aim is to control a selected subsystem, while not affecting the remaining dynamics. The paper extends earlier continuous time results to discrete time systems over a finite frequency interval. Decoupling is achieved by suitable input and output blend vectors, such that they maximize the sensitivity of the selected subsystem, while at the same time they minimize the transfer through the undesired dynamics. The proposed algorithm is based on an optimization problem involving Linear Matrix Inequalities, where the H- index of the controlled subsystem is maximized, while the transfer through the dynamics to be decoupled is minimized by a sparsity like criteria. The present approach has the advantage that it is directly applicable to stable and unstable subsystems also. Numerical examples demonstrate the effectiveness of the method.