Quadratic Difference Forms Research Articles

A Behavioral Approach to Data-Driven Control With Noisy Input–Output Data

This paper deals with data-driven stability analysis and feedback stabilization of linear input-output systems in auto-regressive (AR) form. We assume that noisy input-output data on a finite time-interval have been obtained from some unknown AR system. Data-based tests are then developed to analyse whether the unknown system is stable, or to verify whether a stabilizing dynamic feedback controller exists. If so, stabilizing controllers are computed using the data. In order to do this, we employ the behavioral approach to systems and control, meaning a departure from existing methods in data driven control. Our results heavily rely on a characterization of asymptotic stability of systems in AR form using the notion of quadratic difference form (QDF) as a natural framework for Lyapunov functions of autonomous AR systems. We introduce the concepts of informative data for quadratic stability and quadratic stabilization in the context of input-output AR systems and establish necessary and sufficient conditions for these properties to hold. In addition, this paper will build on results on quadratic matrix inequalities (QMIs) and a matrix version of Yakubovich's S-lemma.

A Data-Driven Predictive Control Structure in the Behavioral Framework

This paper presents a data-driven predictive control (DPC) algorithm for linear time-invariant (LTI) systems in the behavioral framework. The system is described by the parametrization of the Hankel matrix constructed from its measured trajectories. The proposed structure follows a two-step procedure. The existence of a controlled behavior is firstly verified from the perspective of dissipativity with the aid of quadratic difference forms (QdFs), then the controlled trajectory is selected from the original uncontrolled behavior through optimization. An illustrative example is presented to demonstrate the effectiveness of the proposed approach.

Data-driven stabilizing control of DC–DC converters with unknown active loads

We address the issue of stabilizing control design of DC–DC power converters with active loads, e.g. distributed generation systems, cascade converters, converters feeding DC buses, etc. We test the results in the presence of potential destabilizers, such as constant power loads (CPLs). The proposed approach is particularly devised for those situations where a full network dynamic model is either not available or it has an unsuitable level of complexity—in terms of number of state–space variables and equations, e.g. multiple loads, DC distribution networks, micro grids, etc. This is due to its model-free approach based on a data-driven strategy, which is able to guarantee stability using a LMI-Lyapunov approach. Elements of behavioral system theory such as linear difference systems, quadratic difference forms and Lyapunov theory, are crucial for the development of the proposed stabilizing control schemes. We also provide simulation and experimental results to corroborate the practical advantages of the proposed framework.

Fault diagnosis based on dissipativity property

In this paper, a novel fault diagnosis scheme for linear process systems using dissipativity theory is developed. Dissipativity (supply rate) of a process is an input/output property, which may not be valid when a fault occurs. For a given process, dissipativity is not a unique property, with different dissipative supply rates reflecting different aspects of its dynamics. In this approach, the dissipativity of a process is “shaped” such that it is fault-sensitive (i.e., no longer valid when faults occur) and fault-selective (i.e., no longer valid when one particular fault occurs). By adopting the storage functions and supply rates in the quadratic difference form (QdF), the dissipativity conditions are represented as quadratic functions of the input/output trajectories of the process, which captures much more detailed dynamical features compared to conventional dissipativity (e.g., QSR-type supply rates). These dissipativity properties are determined offline by solving an optimization problem with linear matrix inequality constraints. The online diagnosis algorithm involves checking of inequalities on input/output trajectories, which is much simpler compared to the diagnosis approaches based on observers or parameter estimation. The proposed approach is illustrated using a case study of fault diagnosis of a heat exchanger.

On Lyapunov functions and data-driven dissipativity

Our contribution in this paper is twofold. In the first part, we study Lyapunov functions when a plant is interconnected with a dissipative stabilizing controller. In the second, we present results on data-driven approach to dissipative systems. In particular, we provide conditions under which an observed trajectory can be used to determine whether a system is dissipative with respect to a given supply rate. Our results are based on linear difference systems for which the use of quadratic difference forms play a central role for dissipativity and Lyapunov theory.

Reconfigurable distributed model predictive control

An approach to reconfigurable distributed model predictive control based on reconfigurable controller dissipativity properties is developed. The dissipativity properties of the controllers are updated online to reconfigure themselves for changes in the process network topology, which may be due to changing product specifications, feedstock type or scheduled or unscheduled maintenance; allowing for more flexible and agile manufacturing processes. The use of dissipative systems theory allows for the interaction effects between individual processes to be taken into account in control design to achieve high levels of plant-wide performance. Plant-wide performance and stability bounds are developed based on dissipative systems theory, which in turn are translated into the dissipative trajectory conditions on each local controller. This approach is enabled by the use of dynamic supply rates in quadratic difference form parameterised as linear functions of the process network structure.

Multi-rate dissipativity based control of process networks

An approach to multi-rate distributed control design for process networks is presented, where the local measurements, local control and controller communication are allowed to operate at different sampling rates. Dissipative systems theory is used to facilitate stability and performance analysis of the process network, based upon dynamic supply rates which have been lifted into a global sampling rate. Quadratic difference forms are used as supply rates and storage functions, leading to less conservative stability and performance conditions as compared to classical types of supply rates. These theoretical results are illustrated by a case study of a heat exchanger network.

Multi-Rate Dissipative Control of Large-Scale Systems

An approach to distributed multi-rate control for large-scale systems, and in particular process networks, is presented. Where the local measurements, local control and controller communication are allowed to operate at different sampling rates. Dissipative systems theory is used to facilitate stability and performance analysis of the process network, based upon dynamic supply rates which have been lifted into a global sampling rate. Quadratic difference forms are used as supply rates and storage functions, which facilitate less conservative stability and performance conditions as compared to classical types of supply rates.

Integrated Process Identification and Control Design based on Dissipativity

An approach to the identification of the dissipativity properties of process systems from input-output data is presented, this approach lends itself to analysis and control design for multivariable process systems. In particular, the dissipativity based approach facilitates the design of decentralized or distributed control systems. The proposed approach has the intuitive advantage that only information that is used in the proceeding analysis and design steps is gathered in the system identification step, and not any extraneous information. Quadratic difference forms are used as supply rates and storage functions for the processes. Extensions, including applications to data driven control and thermodynamic properties of nonlinear systems are discussed.

Distributed model predictive control based on dissipativity

A noncooperative approach to plant‐wide distributed model predictive control based on dissipativity conditions is developed. The plant‐wide process and distributed control system are represented as two interacting process and controller networks, with interaction effects captured by the dissipativity properties of subsystems and network topologies. The plant‐wide stability and performance conditions are developed based on global dissipativity conditions, which in turn are translated into the dissipative trajectory conditions that each local model predictive control MPC must satisfy. This approach is enabled by the use of dynamic supply rates in quadratic difference forms, which capture detailed dynamic system information. A case study is presented to illustrate the results. © 2012 American Institute of Chemical Engineers AIChE J, 59: 787–804, 2013

A Unified Approach to Plant-wide Dissipative Model Predictive Control

An approach to plant-wide model predictive control based on dissipativity theory is presented. Dissipativity theory facilitates the decomposition of the plant-wide system into interacting process and controller networks. The controller may take a decentralized, distributed or fully centralized structure without significantly changing the complexity of analysis. This approach to dissipativity based control is enabled by the use of dynamic supply rates as quadratic difference forms which capture more dynamic system information than constant supply rates and decrease the conservativeness of the approach. Plant-wide dissipativity properties ensure stability of the plant-wide system and minimum performance.

Lyapunov stability analysis of higher-order 2-D systems

In this paper we prove a necessary and sufficient condition for the asymptotic stability of a 2-D system described by a system of higher-order linear partial difference equations. We show that asymptotic stability is equivalent to the existence of a vector Lyapunov functional satisfying certain positivity conditions together with its divergence along the system trajectories. We use the behavioral framework and the calculus of quadratic difference forms based on four-variable polynomial algebra.

ビヘイビアアプローチに基づく２‐Ｄ離散時間システムのＬｙａｐｕｎｏｖ安定解析

This paper is concerned with a Lyapunov stability analysis of a two-dimensional (2-D) discrete-time system described by a high-order partial difference equation in the behavioral setting. We first formulate the autonomy and asymptotic stability of 2-D discrete-time behavior in terms of the characteristic set over Z2. A sufficient condition for asymptotic stability is derived in terms of quadratic difference forms. This condition can be numerically checked by solving a certain four-variable polynomial matrix equation. It also turns out that the present result generalizes some existing stability conditions for the Fornasini-Marchesini state-space model to the behavioral setting.

A characterization of solutions of the discrete-time algebraic Riccati equation based on quadratic difference forms

This paper is concerned with a characterization of all symmetric solutions to the discrete-time algebraic Riccati equation (DARE). Dissipation theory and quadratic difference forms from the behavioral approach play a central role in this paper. Along the line of the continuous-time results due to Trentelman and Rapisarda [H.L. Trentelman, P. Rapisarda, Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation, SIAM J. Contr. Optim. 40 (3) (2001) 969–991], we show that the solvability of the DARE is equivalent to a certain dissipativity of the associated discrete-time state space system. As a main result, we characterize all unmixed solutions of the DARE using the Pick matrix obtained from the quadratic difference forms. This characterization leads to a necessary and sufficient condition for the existence of a non-negative definite solution. It should be noted that, when we study the DARE and the dissipativity of the discrete-time system, there exist two difficulties which are not seen in the continuous-time case. One is the existence of a storage function which is not a quadratic function of state. Another is the cancellation between the zero and infinite singularities of the dipolynomial spectral matrix associated with the DARE, due to the infinite generalized eigenvalues of the associated Hamiltonian pencil. One of the main contributions of this paper is to demonstrate how to resolve these difficulties.

二次差分形式に基づく離散時間システムに対する一般化Lyapunov安定定理

In this paper, we consider the generalized Lyapunov stability analysis for a discrete-time system described by a high order difference-algebraic equation. In the behavioral approach, a Lyapunov function is characterized in terms of a quadratic difference form. As a main result, we derive a generalized Lyapunov stability theorem that the asymptotic stability of a behavior is equivalent to the solvability of the two-variable polynomial Lyapunov equation (TVPLE) whose solution induces the Lyapunov function. Moreover, we derive another asymptotic stability condition by using a polynomial matrix solution of the one-variable dipolynomial Lyapunov equation which is reduced from the TVPLE.

DISCRETE TIME DISSIPATIVENESS ON THE DIPOLYNOMIAL RING

In this paper, we consider discrete time dissipativeness on quadratic difference forms and two-variable dipolynomial matrices. We show that if a system is dissipativeness for a supply rate on the dipolynomial ring, then there also exist storage functions and dissipation rates on the dipolynomial ring. Moreover, we clarify whether extremal storage functions for a given supply rates on the dipolynomial ring. This generalization is effective to extend dissipation theory in discrete time from the theoretical points of view.

Discrete-time average positivity and spectral factorization in a behavioral framework

Quite recently, Willems and Trentelman (SIAM J. Control Opt. 36 (1998) 1703–1749) have proposed quadratic differential forms for system analysis and synthesis from the behavioral point of view. In this paper, we develop their discrete-time version, called quadratic difference forms, and some related fundamental concepts with respect to dissipativeness for a dynamical system which interacts with its environment along square summable trajectories. Moreover, we provide an algorithm for spectral factorization of polynomial matrices as one example of quadratic difference forms. A numerical example will be given to show the validity of our result.