Discrete-time Linear Dynamical Systems Research Articles

Modeling, Analysis, and Control of Autonomous Mobility-on-Demand Systems: A Discrete-Time Linear Dynamical System and a Model Predictive Control Approach

Modeling, Analysis, and Control of Autonomous Mobility-on-Demand Systems: A Discrete-Time Linear Dynamical System and a Model Predictive Control Approach

Distributed set-based observers using diffusion strategies

We propose two distributed set-based observers using strip-based and set-propagation approaches for linear discrete-time dynamical systems with bounded modeling and measurement uncertainties. Both algorithms utilize a set-based diffusion step, which decreases the estimation errors and the size of estimated sets, and can be seen as a lightweight approach to achieve partial consensus between the distributed estimated sets. Every node shares its measurement with its neighbor in the measurement update step. In the diffusion step, the neighbors intersect their estimated sets using our novel lightweight zonotope intersection technique. A localization example demonstrates the applicability of our algorithms.

Privacy-preserving set-based estimation using partially homomorphic encryption

The set-based estimation has gained a lot of attention due to its ability to guarantee state enclosures for safety-critical systems. However, collecting measurements from distributed sensors often requires outsourcing the set-based operations to an aggregator node, raising many privacy concerns. To address this problem, we present set-based estimation protocols using partially homomorphic encryption that preserve the privacy of the measurements and sets bounding the estimates. We consider a linear discrete-time dynamical system with bounded modeling and measurement uncertainties. Sets are represented by zonotopes and constrained zonotopes as they can compactly represent high-dimensional sets and are closed under linear maps and Minkowski addition. By selectively encrypting parameters of the set representations, we establish the notion of encrypted sets and intersect sets in the encrypted domain, which enables guaranteed state estimation while ensuring privacy. In particular, we show that our protocols achieve computational privacy using the cryptographic notion of computational indistinguishability. We demonstrate the efficiency of our approach by localizing a real mobile quadcopter using ultra-wideband wireless devices.

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.

Tight computationally efficient approximation of matrix norms with applications

We address the problems of computing operator norms of matrices induced by given norms on the argument and the image space. It is known that aside of a fistful of ``solvable cases,'' most notably, the case when both given norms are Euclidean, computing operator norm of a matrix is NP-hard. We specify rather general families of norms on the argument and the images space (``ellitopic'' and ``co-ellitopic,'' respectively) allowing for reasonably tight computationally efficient upper-bounding of the associated operator norms. We extend these results to bounding ``robust operator norm of uncertain matrix with box uncertainty,'' that is, the maximum of operator norms of matrices representable as a linear combination, with coefficients of magnitude $\leq1$, of a collection of given matrices. Finally, we consider some applications of norm bounding, in particular, (1) computationally efficient synthesis of affine non-anticipative finite-horizon control of discrete time linear dynamical systems under bounds on the peak-to-peak gains, (2) signal recovery with uncertainties in sensing matrix, and (3) identification of parameters of time invariant discrete time linear dynamical systems via noisy observations of states and inputs on a given time horizon, in the case of ``uncertain-but-bounded'' noise varying in a box.

Bayesian computational methods for state-space models with application to SIR model

The state-space model is a powerful statistical tool to estimate linear or non-linear discrete-time dynamic systems. This model naturally leads to the estimation problem of the time-varying parameters of the discovery-time demographic version of the susceptible-infected-recovered (SIR) model that we consider. In this paper, we consider computational methods to perform Bayesian inference on state-space models for analysing time-series data. We compare the three popular Bayesian computational methods for state-space models: the adaptive Metropolis-within-Gibbs algorithm, Liu and West's algorithm and variational approximation method based on Gaussian distributions. The performances of the three methods are compared based on synthetic datasets. Furthermore, we analyse the trend of the spread of COVID-19 in South Korea to point out the limitations of existing methods and derive meaningful results.

Passivity-based Analysis of the ADMM Algorithm for Constraint-Coupled Optimization

We propose a novel, system theoretic analysis of the Alternating Direction Method of Multipliers (ADMM) applied to a convex constraint-coupled optimization problem. The resulting algorithm can be interpreted as a linear, discrete-time dynamical system (modeling the multiplier ascent update) in closed loop with a static nonlinearity (representing the minimization of the augmented Lagrangian). When expressed in suitable coordinates, we prove that the discrete-time linear dynamical system has a discrete positive-real transfer function and is interconnected in closed loop with a static, passive nonlinearity. This readily shows that the origin is a stable equilibrium for the feedback interconnection. Finally, we also show global asymptotic stability of the origin for the closed-loop system and, thus, global asymptotic convergence of ADMM to the optimal solution of the optimization problem.

Measurement-outlier robust Kalman filter for discrete-time dynamic systems

This paper proposes a recursive filter for discrete-time linear dynamic systems subject to output outliers or heavy-tailed noises. First, we introduce a weight matrix in the conventional MAP estimation. It is shown that this matrix plays an influential role in the innovation whitening and asymptotic variance of the modified MAP estimation and, consequently, can be used in outlier detection. Then, we propose two different constrained optimization problems to obtain this weight. These constraints, stemming from environmental noise characteristics, help to obtain the weight matrix more precisely, which increases the filtering performance significantly. In the first approach, we introduce a convex optimization problem to minimize the estimation upper bound of the error covariance matrix. The second approach converts the modified MAP estimation to a min–min optimization problem with a concave cost function. Consequently, to reduce the effect of outliers in estimation, a semidefinite program (SDP) is proposed for outlier detection. At last, simulation results show the effectiveness and verify the performance of the proposed filter for dynamic systems in the presence of measurement outliers.

Metric entropy limits on recurrent neural network learning of linear dynamical systems

One of the most influential results in neural network theory is the universal approximation theorem [1], [2], [3] which states that continuous functions can be approximated to within arbitrary accuracy by single-hidden-layer feedforward neural networks. The purpose of this paper is to establish a result in this spirit for the approximation of general discrete-time linear dynamical systems—including time-varying systems—by recurrent neural networks (RNNs). For the subclass of linear time-invariant (LTI) systems, we devise a quantitative version of this statement. Specifically, measuring the complexity of the considered class of LTI systems through metric entropy according to [4], we show that RNNs can optimally learn—or identify in system-theory parlance—stable LTI systems. For LTI systems whose input-output relation is characterized through a difference equation, this means that RNNs can learn the difference equation from input-output traces in a metric-entropy optimal manner.

Distributed secure state estimation with a priori sparsity information

This paper addresses the problem of distributed secure state estimation of a discrete-time linear dynamical system in an adversarial environment. Much of the recent work on this problem has imposed redundancy requirements on measurement and communication networks for state estimation in the presence of attacks. In this paper, the authors aim to relax this redundancy requirement by using prior information about the state to be estimated. A scenario where sparsity information of the state is given is especially considered, and it is shown that the a priori sparsity information alleviates the redundancy requirement. An algorithm for distributed secure state estimation is then derived and a sufficient condition on the convergence of the algorithm is given. Numerical simulations finally illustrate the effectiveness of the algorithm.

O-Minimal Invariants for Discrete-Time Dynamical Systems

Termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Already for the simplest variants of linear loops the question of termination relates to deep open problems in number theory, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this article, we introduce the class of o-minimal invariants , which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel’s conjecture is transcendental number theory.

What’s decidable about linear loops?

We consider the MSO model-checking problem for simple linear loops, or equivalently discrete-time linear dynamical systems, with semialgebraic predicates (i.e., Boolean combinations of polynomial inequalities on the variables). We place no restrictions on the number of program variables, or equivalently the ambient dimension. We establish decidability of the model-checking problem provided that each semialgebraic predicate either has intrinsic dimension at most 1, or is contained within some three-dimensional subspace. We also note that lifting either of these restrictions and retaining decidability would necessarily require major breakthroughs in number theory.

Computing Different Realizations of Linear Dynamical Systems with Embedding Eigenvalue Assignment

In this paper we investigate realizability of discrete time linear dynamical systems (LDSs) in fixed state space dimension. We examine whether there exist different Θ = (A,B,C,D) state space realizations of a given Markov parameter sequence Y with fixed B, C and D state space realization matrices. Full observation is assumed in terms of the invertibility of output mapping matrix C. We prove that the set of feasible state transition matrices associated to a Markov parameter sequence Y is convex, provided that the state space realization matrices B, C and D are known and fixed. Under the same conditions we also show that the set of feasible Metzler-type state transition matrices forms a convex subset. Regarding the set of Metzler-type state transition matrices we prove the existence of a structurally unique realization having maximal number of non-zero off-diagonal entries. Using an eigenvalue assignment procedure we propose linear programming based algorithms capable of computing different state space realizations. By using the convexity of the feasible set of Metzler-type state transition matrices and results from the theory of non-negative polynomial systems, we provide algorithms to determine structurally different realization. Computational examples are provided to illustrate structural non-uniqueness of network-based LDSs.

Online Convex Optimization for Data-Driven Control of Dynamical Systems

We propose an algorithm based on online convex optimization for controlling discrete-time linear dynamical systems. The algorithm is data-driven, i.e., does not require a model of the system, and is able to handle a priori unknown and time-varying cost functions. To this end, we make use of a single persistently exciting input-output sequence of the system and results from behavioral systems theory which enable it to handle unknown linear time-invariant systems. Moreover, we consider noisy output feedback instead of full state measurements and allow general economic cost functions. Our analysis of the closed loop reveals that the algorithm is able to achieve sublinear regret, where the measurement noise only adds an additional constant term to the regret upper bound. In order to do so, we derive a data-driven characterization of the steady-state manifold of an unknown system. Moreover, our algorithm is able to asymptotically exactly estimate the measurement noise. The effectiveness and applicational aspects of the proposed method are illustrated by means of a detailed simulation example in thermal control.

Optimal measurement budget allocation for Kalman prediction over a finite time horizon by genetic algorithms

In this paper, we address the problem of optimal measurement budget allocation to estimate the state of a linear discrete-time dynamical system over a finite horizon. More precisely, our aim is to select the measurement times in order to minimize the variance of the estimation error over a finite horizon. In addition, we investigate the closely related problem of finding a trade-off between number of measurements and signal to noise ratio.First, the optimal measurement budget allocation problem is reduced to a deterministic combinatorial program. Then, we propose a genetic algorithm implementing a count preserving crossover to solve it. On the theoretical side, we provide a one-dimensional analysis that indicates that the benefit of using irregular measurements grows when the system is unstable or when the process noise becomes important. Then, using the duality between estimation and control, we show that the problem of selecting optimal control times for a linear quadratic regulator can be reduced to our initial problem.Finally, numerical implementations demonstrate that using measurement times optimized by our genetic algorithm gives better estimate than regularly spaced measurements. Our method is applied to a discrete version of a continuous-time system and the impact of the discretization time step is studied. It reveals good convergence properties, showing that our method is well suited to both continuous-time and discrete-time setups.

Controllability of a Linear System With Nonnegative Sparse Controls

This paper studies controllability of a discrete-time linear dynamical system using nonnegative and sparse inputs. These constraints on the control input arise naturally in many real-life systems where the external influence on the system is unidirectional, and activating each input node adds to the cost of control. We derive the necessary and sufficient conditions for controllability of the system, without imposing any constraints on the system matrices. Unlike the well-known Kalman rank based controllability criteria, the conditions presented in this paper can be verified in polynomial time, and the verification complexity is independent of the sparsity level. The proof of the result is based on the analytical tools concerning the properties of a convex cone. Our results also provide a closed-form expression for the minimum number of control nodes to be activated at every time instant to ensure controllability of the system using positive controls.

Statistical and Nonstatistical Methods for Estimating the Impulse Characteristic of a Dynamical System

We consider the problem of estimating the impulse characteristic of a linear time-invariant discrete-time dynamical system, using data on its input and output signals. We propose versions of estimation methods for asymptotically stable dynamical systems in the context of various approaches to the presentation of parametric uncertainty, including approaches based on the interval analysis and the fuzzy set theory. A comparative analysis and numerical illustration of the methods are presented.

Discrete-Time Networked Dynamic Systems

In this paper, we provides a general framework for the analysis of a class of linear discrete-time networked dynamic systems (DNDS). We focus our attention on DNDS where the underlying connection topology couples the agents at their outputs. A distinction is made between DNDS with homogeneous agent dynamics and DNDS with heterogeneous agent dynamics. It is emphasized that developing efficient solution methods for the design of such systems involve connecting and interpreting results from graph theory and convex optimization in a systems-theoretic context

First-Order Orbit Queries

Orbit Problems are a class of fundamental reachability questions that arise in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. Instances of the problem comprise a dimension $d\in \mathbb {N}$ , a square matrix $A\in \mathbb {Q}^{d\times d}$ , and a query regarding the behaviour of some sets under repeated applications of A. For instance, in the Semialgebraic Orbit Problem, we are given semialgebraic source and target sets $S,T\subseteq \mathbb {R}^{d}$ , and the query is whether there exists $n\in {\mathbb {N}}$ and x ∈ S such that Anx ∈ T. The main contribution of this paper is to introduce a unifying formalism for a vast class of orbit problems, and show that this formalism is decidable for dimension d ≤ 3. Intuitively, our formalism allows one to reason about any first-order query whose atomic propositions are a membership queries of orbit elements in semialgebraic sets. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory—Baker’s theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of $\mathbb {R}^{d}$ for which membership is decidable. On the other hand, previous work has shown that in dimension d = 4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

Controllability of Linear Dynamical Systems Under Input Sparsity Constraints

In this article, we consider the controllability of a discrete-time linear dynamical system with sparse control inputs. Sparsity constraints on the input arises naturally in networked systems, where activating each input variable adds to the cost of control. We derive algebraic necessary and sufficient conditions for ensuring controllability of a system with an arbitrary transfer matrix. The derived conditions can be verified in polynomial time complexity, unlike the more traditional Kalman-type rank tests. Further, we characterize the minimum number of input vectors required to satisfy the derived conditions for controllability. Finally, we present a generalized Kalman decomposition-like procedure that separates the state-space into subspaces corresponding to sparse-controllable and sparse-uncontrollable parts. These results form a theoretical basis for designing networked linear control systems with sparse inputs.