Input-output Linearizable Systems 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.

Active disturbance rejection control for discrete systems with zero dynamics

AbstractIn this paper, a discrete active disturbance rejection control (DADRC) scheme is proposed to manipulate linear input–output systems with zero dynamics. The DADRC is formulated according to the relative degree of the system output and is designed to track a reference and cancel both internal uncertainties and external disturbances. A parameter tuning method is also presented based on a part of the model information. The robustness and effectiveness of the designed scheme are demonstrated via numerical simulations.

Model Order Reduction Based on Approximate Cross Gramian and Laguerre Series for Linear Input-Output Systems

In this paper, we first focus on the topic of calculation of cross gramians for linear square systems, which are constructed approximately from the Laguerre series expansion coefficient vectors that are obtained by solving a recurrence formula instead of solving Sylvester equations directly. Then based on such approximate cross gramian, the reduced-order models (ROMs) are produced by truncating the states that are associated with the smaller approximate Hankel singular values (HSVs). In addition, combining with the idea of dominant subspace projection method, we modify our proposed algorithm to obtain a ROM that preserves the stability. What's more, our algorithms are extended to non-square case successfully. The main properties of ROMs are discussed as well. Finally, some numerical simulations are provided to illustrate the effectiveness of our proposed algorithms in the views of accuracy and computational cost.

People and Nature in an Urban World

Humanity is facing an existential contradiction: we are building an urban future for ourselves, yet urbanization in its current form is threatening the very future of humanity and the natural world. Urban growth is a seemingly unstoppable worldwide process. Much has been written about this historic trend, but the environmental impacts of an urbanizing world are barely discussed.

Cross-Gramian-based dominant subspaces

A standard approach for model reduction of linear input-output systems is balanced truncation, which is based on the controllability and observability properties of the underlying system. The related dominant subspaces projection model reduction method similarly utilizes these system properties, yet instead of balancing, the associated subspaces are directly conjoined. In this work, we extend the dominant subspace approach by computation via the cross Gramian for linear systems, and describe an a-priori error indicator for this method. Furthermore, efficient computation is discussed alongside numerical examples illustrating these findings.

Data-Driven Coordination of Distributed Energy Resources for Active Power Provision

In this paper, we propose a framework for coordinating distributed energy resources (DERs) connected to a power distribution system, the model of which is not completely known, so that they collectively provide a specified amount of active power to the bulk power system as quantified by the power exchange between both systems at the bus interconnecting them, while respecting distribution line capacity limits. The proposed framework consists of (i) a linear time-varying input-output (IO) system model that represents the relation between the DER active power injections (inputs), and the total active power exchanged between the distribution and bulk power systems (output); (ii) an estimator that aims to estimate the IO model parameters, and (iii) a controller that determines the optimal DER active power injections so the power exchanged between both systems equals to the specified amount at a minimum generating cost. We formulate the estimation problem as a quadratic program with box constraints and solve it using the projected gradient descent algorithm. To resolve the potential issue of collinearity in the measurements used by the estimator, we introduce random perturbations in the DER active power injections during the estimation process. Using the estimated IO model, the optimal DER coordination problem to be solved by the controller can be formulated as a convex optimization problem, which can be solved easily. The effectiveness of the framework is validated via numerical simulations using the IEEE 123-bus distribution test feeder.

Properties of eventually positive linear input–output systems

In this study, the author considers systems with trajectories originating in the non-negative orthant and becoming non-negative after some finite time transient. These systems are called eventually positive and the results are based on recent theoretical developments in linear algebra. The author considers dynamical systems (i.e. fully observable systems with no inputs), for which they compute forward-invariant cones and Lyapunov functions. They then extend the notion of eventually positive systems to the input–output system case. The extension is performed in such a manner, that some valuable properties of classical internally positive input–output systems are preserved. For example, their induced norms can be computed using linear programming and the energy functions have non-negative derivatives. The author illustrates the theoretical results on numerical examples.

Step-by-step implicit discrete super-twisting differentiator for input-output linearizable nonlinear systems

The aim of this study was to design a step-by-step numerical differentiator for a class of discretized version of a n-th chain-of-integrator system based on the application of the implicit discretization of the super-twisting algorithm (STA). The high-order differentiator reconstructed the non-measurable states of an input-output linearizable systems. The gains of the observer were calculated based on the application of a class of generalized functions. The convergence analysis showed that all the states in the linearized system achieved an invariant set with a predefined size. The existence of this invariant set depends on the uniqueness of the solution for the generalized function. A simple interpretation of a possible exact reconstruction of the non-measurable states of the implicit discretized form. The application of the implicit discretized STA served to design a class of observer for the microalgae culture system.

Constrained control of input–output linearizable systems using control sharing barrier functions

Control barrier functions (CBFs) have been used as an effective tool for designing a family of controls that ensures the forward invariance of a set. When multiple CBFs are present, it is important that the set of controls satisfying all the barrier conditions is non-empty. In this paper, we investigate such a control-sharing property for multiple CBFs and provide sufficient and necessary conditions for the property to hold. Based on that, we study the tracking control design problem of an input–output linearizable system with multiple time-varying output constraints, where the output constraints are encoded as CBFs and the barrier conditions are expressed as hard constraints in a quadratic program (QP) whose feasibility is guaranteed by the control-sharing property of the CBFs. With the controller generated from the QP, the output constraints are always satisfied and the tracking objective is achieved when it is not conflicting with the constraints. The effectiveness of our control design method is illustrated by two examples.

Model order reduction based dynamic equivalence of a wind farm

This paper presents dynamic equivalence of a wind farm based on the model order reduction (MOR) methods. A doubly-fed induction generator (DFIG) with complete mechanical, electrical and control components is the basis for model development. For the purpose of wind fluctuation simulation, the dynamic model of DFIG can be seen as a linear input–output system with the incoming wind speed as the input and the generated active power as the output. Linear model reduction techniques, dominant pole based modal analysis (DPMA) and balanced truncation (BT), combined with classical aggregation skills are applied to obtain a new low-order system representing a wind farm. Simulations and comparisons are carried out in the test system to validate the ability of the reduced order model in matching the active power generated by the detailed wind farm model.

Cross‐gramian‐based dynamic equivalence of wind farms

This study presents dynamic equivalence of wind farms using the cross-gramian (CG)-based model order reduction method. For the purpose of wind fluctuation simulation, the dynamic model of wind farms can be described as a high-order linear input–output system, whose input is the wind speed and output is the active power injected from point of common coupling (PCC). The main input–output behaviours of the system can be measured by its CG. The right and left eigenspaces associated with large eigenvalues of CG are used to obtain a low-order model. Frequency-domain analysis of wind farms consisted of doubly-fed induction generators shows that the proposed method is rapid and numerically stable. The non-linear dynamic simulation also confirms that the reduced model is capable of retaining the dynamic relationship between the input and output of original wind farms. The proposed method offers a potential solution to dynamic simulation of the power system with large-scale wind farms.

System Identification for Passive Linear Quantum Systems

System identification is a key enabling component for the implementation of quantum technologies, including quantum control. In this paper, we consider the class of passive linear input-output systems, and investigate several basic questions: (1) which parameters can be identified? (2) Given sufficient input-output data, how do we reconstruct the system parameters? (3) How can we optimize the estimation precision by preparing appropriate input states and performing measurements on the output? We show that minimal systems can be identified up to a unitary transformation on the modes, and systems satisfying a Hamiltonian connectivity condition called "infecting" are completely identifiable. We propose a frequency domain design based on a Fisher information criterion, for optimizing the estimation precision for coherent input state. As a consequence of the unitarity of the transfer function, we show that the Heisenberg limit with respect to the input energy can be achieved using non-classical input states.

Time Domain Model Order Reduction of General Orthogonal Polynomials for Linear Input-Output Systems

For a class of large linear input-output systems, we present a new model order reduction algorithm based on general orthogonal polynomials in the time domain. The main idea of the algorithm is first to expand the unknown state variables in the space spanned by orthogonal polynomials, then the coefficient terms of polynomial expansion are calculated by a recurrence formula. The basic procedure is to use the coefficient terms to generate a projection matrix. Many classic methods with orthogonal polynomials are special cases of the general approach. The proposed approach has a good computational efficiency and preserves the stability and passivity under certain condition. Numerical experiments are reported to verify the theoretical analysis.

Event-based control of input-output linearizable systems

AbstractThis paper proposes a new event-based control method for nonlinear systems that are input-output linearizable. The control input generator uses a copy of a continuous reference system to generate an exponential control input that keeps the state of the disturbed plant in a bounded surrounding of the setpoint. An upper bound for the deviation of the event-based control loop to the reference system is derived, which depends on the event threshold of the event generator. Hence, by appropriately choosing the event threshold, the event-based control can be made to mimic the continuous control with arbitrary accuracy. The proposed event-based control method is applied to a cooling process.

Krylov type subspace methods for matrix polynomials

We consider solving eigenvalue problems or model reduction problems for a quadratic matrix polynomial Iλ 2 − Aλ − B with large and sparse A and B. We propose new Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear input–output system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree.

Diagnosability Study of Multistage Manufacturing Processes Based on Linear Mixed-Effects Models

Automatic in-process data collection techniques have been widely used in complicated manufacturing processes in recent years. The huge amounts of product measurement data have created great opportunities for process monitoring and diagnosis. Given such product quality measurements, this article examines the diagnosability of the process faults in a multistage manufacturing process using a linear mixed-effects model. Fault diagnosability is defined in a general way that does not depend on specific diagnosis algorithms. The concept of a minimal diagnosable class is proposed to expose the “aliasing” structure among process faults in a partially diagnosable system. The algorithms and procedures needed to obtain the minimal diagnosable class and to evaluate the system-level diagnosability are presented. The methodology, which can be used for any general linear input–output system, is illustrated using a panel assembly process and an engine head machining process.

A Singular Perturbation Approach to a Recursive Deconvolution Problem

Following the approach described by A. V. Kryazhimskii and Yu. S. Osipov, we present a recursive algorithm which can be applied to solve the deconvolution problem of linear finite dimensional input output systems. The method gives an on line approximation of the unknown input, based on approximate samples of the output. Key features of this approach are the introduction of an associated singularly perturbed system and the use of a quasi canonical form due to Morse.

PARAMETRIC IDENTIFICATION OF LINEAR NOISY INPUT-OUTPUT SYSTEMS

An efficient method is proposed for identification of linear noisy input-output systems. Central to this method is that the variances of the input and output noises, which determine the bias in the ordinary least-squares (LS) estimator, are estimated in the way of increasing the degrees of both the denominator and the numerator of the system transfer function by one, but with no need to evaluate the average LS errors. While achieving estimation unbiasedness, the proposed method exhibits algorithmic advantages over the LS-based algorithms recently developed. Performance comparisons with other existing estimation algorithms based upon computer simulations are given.

Control of a class of nonlinear systems subject to periodic exogenous signals

This paper examines the control of single-input feedback linearizable nonlinear systems that are either subject to periodic disturbances or tracking periodic reference trajectories with known periods. The key concept is the combination of differential geometric techniques with the internal model principle resulting in a nonlinear repetitive control strategy. A formulation is presented for the case of input-state linearizable and input-output linearizable systems in continuous time. The potential benefits of the nonlinear repetitive controller are given. It is shown that while the standard nonlinear control techniques can be made robust to disturbances, the nonlinear repetitive technique is more robust to specific types of disturbances in that it does not require knowledge of the disturbance magnitude and does not need high gains to accomplish robustness. An experiment on the active vibration isolation of a single degree of freedom magnetic levitation testbed shows the effectiveness of the approach.

Exponential possibility regression analysis by identification method of possibilistic coefficients

This paper introduces exponential possibility regression analysis by the proposed identification method. To identify the exponential possibility distribution, first we solve all possible different sets of the input-output linear systems from the given data and obtain the corresponding coefficient vectors. Then, we select minimum number of coefficient vectors which represent all the given data. Using the selected coefficient vectors, we apply the identification method to determine a possibility distribution of coefficients in the fuzzy linear system. As a criterion to select minimum coefficient vectors, we use the inner product. By the proposed approach, the smaller area of possibility distribution can be obtained. This is the main characteristic of our proposed regression, comparing with the other regression analyses.