System Theoretic Properties Research Articles

An Overview of Systems-Theoretic Guarantees in Data-Driven Model Predictive Control

The development of control methods based on data has seen a surge of interest in recent years. When applying data-driven controllers in real-world applications, providing theoretical guarantees for the closed-loop system is of crucial importance to ensure reliable operation. In this review, we provide an overview of data-driven model predictive control (MPC) methods for controlling unknown systems with guarantees on systems-theoretic properties such as stability, robustness, and constraint satisfaction. The considered approaches rely on the fundamental lemma from behavioral theory in order to predict input–output trajectories directly from data. We cover various setups, ranging from linear systems and noise-free data to more realistic formulations with noise and nonlinearities, and we provide an overview of different techniques to ensure guarantees for the closed-loop system. Moreover, we discuss avenues for future research that may further improve the theoretical understanding and practical applicability of data-driven MPC.

Learning passive policies with virtual energy tanks in robotics

AbstractWithin a robotic context, the techniques of passivity‐based control and reinforcement learning are merged with the goal of eliminating some of their reciprocal weaknesses, as well as inducing novel promising features in the resulting framework. The contribution is framed in a scenario where passivity‐based control is implemented by means of virtual energy tanks, a control technique developed to achieve closed‐loop passivity for any arbitrary control input. Albeit the latter result is heavily used, it is discussed why its practical application at its current stage remains rather limited, which makes contact with the highly debated claim that passivity‐based techniques are associated with a loss of performance. The use of reinforcement learning allows to learn a control policy that can be passivized using the energy tank architecture, combining the versatility of learning approaches and the system theoretic properties which can be inferred due to the energy tanks. Simulations show the validity of the approach, as well as novel interesting research directions in energy‐aware robotics.

Alternating projection‐based iterative learning control for discrete‐time systems with non‐uniform trial lengths

AbstractThis article develops a novel framework for iterative learning control (ILC) design of discrete‐time systems with non‐uniform trial lengths by using the method of alternating projections. In contrast to existing results for the non‐uniform trial length problem, this article uses the Hilbert space setting and hence the linear discrete‐time system dynamics with non‐uniform trial lengths can be represented by multiple affine subspaces (or linear varieties). Motivated by the successive projection design between two closed convex sets, the considered ILC problem can be transformed into alternating projections between multiple sets, then the Hilbert space setting is used to establish key systems theoretic properties. Moreover, an optimal ILC design is developed for systems with non‐uniform trial lengths, which is also extended to the case of input constraints. A numerical case study is given to illustrate the applicability of the new design.

A Geometric Approach to Convex Processes: From Reachability to Stabilizability

This paper studies system theoretic properties of the class of difference inclusions of convex processes. We will develop a framework considering eigenvalues and eigenvectors, weakly and strongly invariant cones, and a decomposition of convex processes. This will allow us to characterize reachability, stabilizability and (null-)controllability of nonstrict convex processes in terms of spectral properties. These characterizations generalize all previously known results regarding for instance linear processes and specific classes of nonstrict convex processes.

An overview of structural systems theory

This paper provides an overview of the research conducted in the context of structural (or structured) systems. These are parametrized models used to assess and design system theoretical properties without considering a specific realization of the parameters (which could be uncertain or unknown). The research in structural systems led to a principled approach to a variety of problems, into what is known as structural systems theory. Hereafter, we perform a systematic overview of the problems and methodologies used in structural systems theory since the latest survey by Dion et al. in 2003. During this period, most of the focus seems to be on structural system’s properties related to controllability/observability and decentralized control, in the context of linear time-invariant systems, under the classic assumption that the parameters are independent and belonging to infinite fields. Notwithstanding, it is notable an increase in research in topics that go beyond such scope and underlying assumptions, as well as applications in a variety of domains. Lastly, we provide a compilation of open questions on several settings and we discuss future directions in this field.

A Novel Analytical Modeling Approach for Quality Propagation of Transient Analysis of Serial Production Systems.

Production system modeling (PSM) for quality propagation involves mapping the principles between components and systems. While most existing studies focus on the steady-state analysis, the transient quality analysis remains largely unexplored. It is of significance to fully understand quality propagation, especially during transients, to shorten product changeover time, decrease quality loss, and improve quality. In this paper, a novel analytical PSM approach is established based on the Markov model, to explore product quality propagation for transient analysis of serial multi-stage production systems. The cascade property for quality propagation among correlated sequential stages was investigated, taking into account both the status of the current stage and the quality of the outputs from upstream stages. Closed-form formulae to evaluate transient quality performances of multi-stage systems were formulated, including the dynamics of system quality, settling time, and quality loss. An iterative procedure utilizing the aggregation technique is presented to approximate transient quality performance with computational efficiency and high accuracy. Moreover, system theoretic properties of quality measures were analyzed and the quality bottleneck identification method was investigated. In the case study, the modeling error was 0.36% and the calculation could clearly track system dynamics; quality bottleneck was identified to decrease the quality loss and facilitate continuous improvement. The experimental results illustrate the applicability of the proposed PSM approach.

Cepstral identification of autoregressive systems

Recent research has provided a better understanding of the power cepstrum, which has led to several applications in time series clustering, classification, and anomaly detection. It has also provided a deeper understanding of the theoretical framework that relates the power cepstrum with some system theoretic properties of the underlying dynamics. In this paper, we pursue the intricate connections between the power cepstrum of a signal and the pole polynomial of the underlying generative model. In this way, we develop a simple and extremely efficient method to identify an autoregressive (AR) system, starting from the power cepstrum of its output signal. This general framework uses Newton’s identities to set up a system of elementary symmetric polynomials over the cepstral coefficients and results in an identification algorithm that is independent of the length of the power cepstrum, with computational complexity only linearly dependent on the order of the model. We provide several numerical examples, first on synthetic time series, then on the classical Yule sunspot numbers modeling problem, and finally on a contemporary application involving structural health monitoring. Subsequently, the novel system identification algorithm is employed to provide insight in the results of weighted cepstral clustering, showing that the model estimated from the center of a cluster provides a good estimator for the dynamics in that cluster.

Dynamic modeling and model predictive control of a continuous pulp digester

AbstractThis work explores the design of a model predictive controller of the continuous pulp digester process consisting of the co‐current zone and counter‐current zone modeled by a set of nonlinear coupled hyperbolic partial differential equations (PDEs). The distributed parameter system of interest is not spectral, and slow–fast dynamic separation does not hold. To address this challenge, the nonlinear continuous‐time model is linearized and discretized in time utilizing the Cayley–Tustin discretization framework, which ensures system theoretic properties and structure preservation without spatial discretization or model reduction. The discrete model is used in the full state model predictive controller design, which is augmented by the Luenberger observer design to achieve the output constrained regulation. Finally, a numerical example is provided to demonstrate the feasibility and applicability of the proposed controller designs.

State representations of convolutional codes over a finite ring

In this paper we study finite support convolutional codes over Zpr by means of an input-state-output representation. We show that the set of finite weight input-state-output trajectories associated to this type of representations has the structure of a Zpr-submodule of Zprn and therefore is a (finite support) convolutional code. Fundamental system-theoretical properties such as observability, reachability or minimality, are investigated in this context.

Manifold Calculus in System Theory and Control—Fundamentals and First-Order Systems

The aim of the present tutorial paper is to recall notions from manifold calculus and to illustrate how these tools prove useful in describing system-theoretic properties. Special emphasis is put on embedded manifold calculus (which is coordinate-free and relies on the embedding of a manifold into a larger ambient space). In addition, we also consider the control of non-linear systems whose states belong to curved manifolds. As a case study, synchronization of non-linear systems by feedback control on smooth manifolds (including Lie groups) is surveyed. Special emphasis is also put on numerical methods to simulate non-linear control systems on curved manifolds. The present tutorial is meant to cover a portion of the mentioned topics, such as first-order systems, but it does not cover topics such as covariant derivation and second-order dynamical systems, which will be covered in a subsequent tutorial paper.

Quasi feedback forms for differential-algebraic systems

Abstract We investigate feedback forms for linear time-invariant systems described by differential-algebraic equations. Feedback forms are representatives of certain equivalence classes. For example, state space transformations, invertible transformations from the left and proportional state feedback constitute an equivalence relation. The representative of such an equivalence class, which we call proportional feedback form for the above example, allows to read off relevant system theoretic properties. Our main contribution is to derive a quasi proportional feedback form. This form is advantageous since it provides some geometric insight and is simple to compute, but still allows to read off the relevant structural properties of the control system. We also derive a quasi proportional and derivative feedback form. Similar advantages hold.

Convex model predictive control for collision avoidance

This manuscript proposes a model predictive control for collision avoidance for the regulation problem of deterministic linear systems, which provides a priori guarantees of strong system theoretic properties, such as positive invariance and asymptotic stability, and high computational efficiency. Notion of safe distance sets is introduced, and also utilized as a novel approach to ensure collision avoidance via suitably defined convex constraints. The proposed convex model predictive control for collision avoidance is obtained by employing interactive strategic-tactical structure for overall decision-making. The strategic stage of the overall algorithm employs direct algebraic manipulations in order to construct safe distance sets that ensure collision avoidance. The tactical stage of the overall algorithm employs strictly convex quadratic programs for the optimization of local finite horizon predicted control processes. The dynamically compatible interaction of strategic and tactical stages of the overall algorithm is ensured by construction, which guarantees structural and computational benefits. These novel and unique features effectively enable both real time implementation and real life utilization of model predictive control for collision avoidance.

Convex MPC for exclusion constraints

This article develops model predictive control for exclusion constraints with a priori guaranteed strong system theoretic properties, which is implementable via computationally highly efficient, strictly convex quadratic programming. The proposed approach deploys safe tubes in order to ensure intrinsically nonconvex exclusion constraints via closed polyhedral constraints. A safe tube is constructed by utilizing the separation theorem for convex sets, and it is practically obtained from the solution of a strictly convex quadratic programming problem. A safe tube is deployed to efficiently optimize a predicted finite horizon control process via another strictly convex quadratic programming problem.

Constructing the Singular Roesser State-Space Model Description of 3D Spatio-Temporal Dynamics From the Polynomial System Matrix

This paper considers systems theoretic properties of linear systems defined in terms of spatial and temporal indeterminates. These include physical applications where one of the indeterminates is of finite duration. In some cases, a singular Roesser state-space model representation of the dynamics has found use in characterizing systems theoretic properties. The representation of the dynamics of many linear systems is obtained in terms of transform variables and a polynomial system matrix representation. This paper develops a direct method for constructing the singular Roesser state-space realization from the system matrix description for 3D systems such that relevant properties are retained. Since this method developed relies on basic linear algebra operations, it may be highly effective from the computational standpoint. In particular, spatially interconnected systems of the form of the ladder circuits are considered as the example. This application confirms the usefulness and effectiveness of the proposed method independently of the system spatial order.

Systems theoretic properties of linear RLC circuits

We consider the differential-algebraic systems obtained by modified nodal analysis of linear RLC circuits from a systems theoretic viewpoint. We derive expressions for the set of consistent initial values and show that the properties of controllability at infinity and impulse controllability do not depend on parameter values but rather on the interconnection structure of the circuit. We further present circuit topological criteria for behavioral stabilizability.

Data Informativity: A New Perspective on Data-Driven Analysis and Control

The use of persistently exciting data has recently been popularized in the context of data-driven analysis and control. Such data have been used to assess system-theoretic properties and to construct control laws, without using a system model. Persistency of excitation is a strong condition that also allows unique identification of the underlying dynamical system from the data within a given model class. In this article, we develop a new framework in order to work with data that are not necessarily persistently exciting. Within this framework, we investigate necessary and sufficient conditions on the informativity of data for several data-driven analysis and control problems. For certain analysis and design problems, our results reveal that persistency of excitation is not necessary. In fact, in these cases, data-driven analysis/control is possible while the combination of (unique) system identification and model-based control is not. For certain other control problems, our results justify the use of persistently exciting data, as data-driven control is possible only with data that are informative for system identification.

Stability and performance in transient average constrained economic MPC without terminal constraints

In this paper, we investigate system theoretic properties of transient average constrained economic model predictive control (MPC) without terminal constraints. We show that the optimal open-loop solution passes by the optimal steady-state for consecutive time instants. Using this turnpike property and suitable controllability conditions, we provide closed-loop performance bounds. Furthermore, stability is proved by combining the rotated value function with an input-to-state (ISS) Lyapunov function of an extended state related to the transient average constraints. The results are illustrated with a numerical example.

Bernoulli serial lines with batching machines: Performance analysis and system-theoretic properties

Aiming at Bernoulli serial lines with batching machines and finite buffers, this study develops analytical methods for performance analysis and system-theoretic properties. Batching machines can process several jobs simultaneously as a batch, as long as the number of jobs in a batch does not exceed the machine’s batch capacity. The batch capacities of all machines are not necessarily equal. Batching machines bring about a new characteristic of state transitions depending on batch capacities and machine states. For two-machine lines, the joint impacts of batching machines on state transitions are analyzed. Theoretically, the state transition rules are revealed. Based on the state transition rules, this study proposes a hierarchical state transition diagram, proves the ergodicity condition, and derives analytical formulas to evaluate the performance measures. Then, for multi-machine lines, this study develops a computationally efficient aggregation method with high accuracy. Furthermore, the impacts of system parameters, including machine efficiency pattern, batch capacity pattern, batch capacity mismatch, and system size, on the accuracy are qualitatively analyzed. Finally, this study investigates the reversibility and monotonicity properties. These analytical methods and results help production managers to scientifically evaluate, accurately predict, and continuously improve Bernoulli serial lines with batching machines.

Nonlinear Reference Tracking: An Economic Model Predictive Control Perspective

In this paper, we study the system theoretic properties of a reference tracking model predictive control (MPC) scheme for general reference trajectories and nonlinear discrete-time systems subject to input and state constraints. Contrary to other existing theoretical results for reference tracking MPC, we do not modify the optimization problem with artificial references or terminal ingredients. Instead, we consider a simple, intuitive implementation and derive theoretical guarantees under a stabilizability condition on the system and suitable conditions on the reference trajectory. We provide sufficient conditions for exponential reference tracking for reachable reference trajectories. In addition, we study unreachable reference trajectories and derive sufficient conditions to ensure practical stability of the unknown optimal reachable reference trajectory. For both cases, we characterize the region of attraction. The results build on the notion of incremental stabilizability, strict dissipativity, and results in (economic) MPC without terminal constraints.

Proxy models for caprock pressure and temperature dynamics during steam-assisted gravity drainage process

The usage of first principles-based dynamic models for advanced control, monitoring and optimization of petroleum reservoirs is constrained by the large scale nature of the models. The parametric uncertainty makes the process even more challenging; hence, development of proxy models is an attractive proposition. In this work, proxy models are developed from spatio-temporal data of pressure and temperature in the caprock during steam-assisted gravity drainage operation. The first proxy model addresses the issue of reduced-order dynamic modelling of the caprock pressure and temperature fields based on proper orthogonal decomposition and system identification using data from the first principles model. The second proxy model takes the first step towards dynamic analysis of factor of safety in reservoir management by modelling the evolution of clusters of high, medium and low pressure regions using graph theory and subspace modelling. System theoretic properties of these proxy models and their practical relevance is also analysed.