Bilinear Systems Research Articles

Stabilization for a Class of Positive Bilinear Systems

Stabilization for a Class of Positive Bilinear Systems

Stabilization for a Class of Bilinear Systems: A Unified Approach

Stabilization for a Class of Bilinear Systems: A Unified Approach

On a two-dimensional nonlinear system of difference equations close to the bilinear system

<abstract><p>We consider the two-dimensional nonlinear system of difference equations</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ x_n = x_{n-k}\frac{ay_{n-l}+by_{n-(k+l)}}{cy_{n-l}+dy_{n-(k+l)}},\quad y_n = y_{n-k}\frac{{\alpha} x_{n-l}+{\beta} x_{n-(k+l)}}{{\gamma} x_{n-l}+{\delta} x_{n-(k+l)}}, $\end{document} </tex-math></disp-formula></p> <p>for $ n\in{\mathbb N}_0, $ where the delays $ k $ and $ l $ are two natural numbers, and the initial values $ x_{-j}, y_{-j} $, $ 1\le j\le k+l $, and the parameters $ a, b, c, d, {\alpha}, {\beta}, {\gamma}, {\delta} $ are real numbers. We show that the system of difference equations is solvable by presenting a method for finding its general solution in detail. Bearing in mind that the system of equations is a natural generalization of the corresponding one-dimensional difference equation, whose special cases appear in the literature from time to time, our main result presented here also generalizes many results therein.</p></abstract>

Measure of Controllability on SO(3)

The classical notion of controllability provides us with a boolean variable — whether a given system is controllable or not. However, often as engineers we seek a notion of how ‘controllable’ a given system is, so as to classify systems that are ‘easier to control’ from those which are ‘difficult to control’. In this article we derive a quantitative measure of controllability for bilinear systems defined on the Lie group of rotations SO(3). Our controllability measure is based on the worst case cost of transferring the system from a given initial condition to a given final condition arbitrarily chosen over a set of interest. Here we consider a few frequently encountered cost functions such as control energy optimization and time optimal control, and obtain a relation between the worst case cost and the system parameters.

Stability of bimodal planar linear switched systems

We study dynamics of a bimodal planar linear switched system with both subsystems Hurwitz stable. We give dwell time bounds so that the switched system is asymptotically stable. These bounds obtained are in terms of certain smooth functions of the eigenvalues and (generalized) eigenvectors of the subsystem matrices. The results are extended to a special class of symmetric bilinear systems. The results are also extended to a multimodal planar linear switched system in which the switching is governed by an undirected star graph. For such systems, dwell time bounds are obtained as solutions of a minimax optimization problem.

Controllability of discrete-time inhomogeneous bilinear systems with real spectrum: Algebraically verifiable criteria

One of the drawbacks of the controllability theory for nonlinear systems is that most existing controllability criteria are not algebraically verifiable, which makes them difficult to apply especially if the system dimension is high. Thus, it is a significant task to seek algebraically verifiable controllability criteria for nonlinear systems. In this paper, we study controllability of discrete-time inhomogeneous bilinear systems. In the classical results on controllability of such systems, a necessary condition is that the linear part has to be controllable. However, we will show that this condition is in fact not necessary for controllability. Specifically, we first define the spectrum for discrete-time inhomogeneous bilinear systems and reveal that the spectrum is a fundamental property which is very useful in investigating the controllability problems. We then present controllability criteria for the systems with real spectrum, which are algebraically verifiable. Furthermore, we also provide algorithms for the controllable systems to compute the exact or approximated control inputs to achieve the transition between any given pair of states. The presented controllability criteria and algorithms work for the systems in any finite dimension and are easy to implement. More importantly, through our controllability criteria, we reveal that controllability of the linear part is not necessary for discrete-time inhomogeneous bilinear systems to be controllable. Examples are given to illustrate the presented algebraic controllability criteria.

On the Minimum Time Tracking Control of the Active Front End Converters

This article introduces the minimum time tracking control of an active front-end (AFE) converter with excellent transient performance. Since the dynamical model of the AFE converter is a bilinear system, switching control is provided for minimum time tracking with final time-varying trajectories in this system. Due to the complexity of implementing this controller, a suboptimal closed-loop controller is proposed whose dynamic performance is very close to the minimum time tracking control. Simulation and experimental results show that the suboptimal proposed controller has excellent dynamic performance against load disturbances compared with the conventional control methods.

Finite-Data Error Bounds for Koopman-Based Prediction and Control

The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points, for both ordinary and stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these bounds by means of an example with the Ornstein–Uhlenbeck process. Moreover, we extend our analysis to (stochastic) nonlinear control-affine systems. We prove error estimates for a previously proposed approach that exploits the linearity of the Koopman generator to obtain a bilinear surrogate control system and, thus, circumvents the curse of dimensionality since the system is not autonomized by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the bilinear approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.

Fast-Dynamic Grey Wolf Optimizer for solving model order reduction of bilinear systems based on multi-moment matching technique

Grey Wolf Optimizer (GWO) has been known as one of the most popular and powerful metaheuristic search algorithms. It has great advantages such as simplicity, accuracy, and good convergence rate, but its performance degrades when the global optimum is not zero. To address these issues, in this paper, a new formulation is introduced to change its searching and hunting mechanisms, while the updating structure of wolves is also modified. With these changes, a proper compromise is achieved between exploitation and exploration. Therefore, the proposed algorithm, called Fast-Dynamic GWO (FDGWO), is more accurate with a faster convergence rate compared to GWO. Thirty-eight typical benchmark functions are optimized with FDGWO and compared with a set of efficient metaheuristic search algorithms. The Wilcoxon rank-sum test is used to statistically assess the results. With an overall efficacy of 84.2%, FDGWO is the most effective among competing algorithms. Then, the performance of FDGWO for ten real-world constrained engineering problems is compared with some highly accurate recent algorithms such as BP-ϵMAg-ES and COLSHADE. The results show that while FDGWO performs almost as accurate as these algorithms, but its computational complexity is quite less. Next, a new hybrid method based on FDGWO is proposed for model order reduction (MOR) of bilinear systems. First, the order of reduced system is determined via an iterative approach based on H2 norm of the error system. The unknown parameters of reduced model are then determined by minimizing a multi-objective fitness function aiming to match the multi-moments of original and reduced bilinear systems. Some stability constraints based on the Volterra series expansion of the states of the bilinear system are added to the optimization problem, which is solved by FDGWO algorithm. Finally, two bilinear test systems are reduced by the proposed hybrid method and compared with the most common bilinear MOR methods.

Sensitivity Analysis of Krasovskii Passivity-Based Controllers

In the domain of passivity theory, there were major contributions in the last decade, the most recent notion of passivity being the so-called Krasovskii passivity. This framework offers the possibility of designing a controller which ensures the passivity of the resulting closed-loop system. The current paper proposes a solution to design the parameters of Krasovskii passivity-based controllers (K-PBCs) in order to ensure small sensitivity of the closed-loop systems. As such, after the initial construction of the passivity output, the controller parameters are designed in order to impose the dominant eigenvalues of the Jacobian of the resulting closed-loop system with the smallest deviation around the given forced equilibrium point which is, additionally, smaller than a prescribed stability margin. The resulting optimization problem is non-convex by nature, and a metaheuristic approach is proposed to design these parameters. Moreover, in order to impose an extra set of performances, the control system contains an outer loop where dynamical path planning is used to impose the additional requirements. All mentioned results are developed for processes modeled as bilinear systems. In order to illustrate the proposed control method, a numerical example consisting of a single-ended primary inductor DC-DC converter (SEPIC) process is presented.

Control Design on a Non-minimum Phase Bilinear System by Backstepping Method

Control Design on a Non-minimum Phase Bilinear System by Backstepping Method

Hierarchical multi-innovation stochastic gradient identification algorithm for estimating a bilinear state-space model with moving average noise

This paper considers the combined parameter and state estimation problem of a bilinear state space system with moving average noise. There are product terms of state variables and control variables in bilinear systems, which brings difficulties to parameter and state estimation. By designing a bilinear state estimator based on Kalman filter and using input–output data to estimate the state, a hierarchical multi-innovation stochastic gradient (i.e., H-MISG) algorithm based on the state estimator is proposed to jointly estimate unknown states and parameters. In addition, compared with the hierarchical stochastic gradient algorithm, H-MISG algorithm introduces the innovation length parameter, makes full use of the system input and output data information, and improves the accuracy of parameter estimation. Numerical simulation examples verify the effectiveness of the proposed algorithm.

An approximate stochastic dynamics approach for design spectrum based response analysis of nonlinear structural systems with fractional derivative elements

A novel approximate approach is developed for determining, in a computationally efficient manner, the peak response of nonlinear structural systems with fractional derivative elements subject to a given seismic design spectrum. Specifically, first, an excitation evolutionary power spectrum is derived that is compatible with the design spectrum in a stochastic sense. Next, relying on a combination of statistical linearization and stochastic averaging yields an equivalent linear system (ELS) with time-variant stiffness and damping elements. Further, the values of the ELS elements at the most critical time instant, i.e., the time instant associated with the highest degree of nonlinear/inelastic response behavior exhibited by the structural system, are used in conjunction with the design spectrum for determining approximately the nonlinear system peak response displacement. The herein developed approach can be construed as an extension of earlier efforts in the literature to account for fractional derivative terms in the governing equations of motion. Furthermore, the approach exhibits the significant novelty of exploiting the localized time-dependent information provided by the derived time-variant ELS elements. Indeed, the values of the ELS stiffness and damping elements at the most critical time instant capture the system dynamics better than an alternative standard time-invariant statistical linearization treatment. This leads to enhanced accuracy when determining nonlinear system peak response estimates. An illustrative numerical example is considered for assessing the performance of the approximate approach. This pertains to a bilinear hysteretic structural system with fractional derivative elements subject to a Eurocode 8 elastic design spectrum. Comparisons with pertinent Monte Carlo simulation data are included as well, demonstrating a high degree of accuracy.

On characterisations of the input to state stability properties for conformable fractional order bilinear systems

This paper proposes for the first time the theoretical requirements that a fractional-order bilinear system with conformable derivative has to fulfil in order to satisfy different input-to-state stability (ISS) properties. Variants of ISS, namely ISS itself, integral ISS, exponential integral ISS, small-gain ISS, and strong integral ISS for the general class of conformable fractional-order bilinear systems are investigated providing a set of necessary and sufficient conditions for their existence and then compared. Finally, the correctness of the obtained theoretical results is verified by numerical example.

Necessary and Sufficient Conditions for Harmonic Control in Continuous Time

In this article, we revisit the concepts and tools of harmonic analysis and control and provide a rigorous mathematical answer to the following question: When does an harmonic control have a representative in the time domain? By representative we mean a control in the time domain that leads by sliding Fourier decomposition to exactly the same harmonic control. Harmonic controls that do not have such representatives lead to erroneous results in practice. The main results of this article are: A one-to-one correspondence between <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ad hoc</i> functional spaces guaranteeing the existence of a representative, a strict equivalence between the Caratheorody solutions of a differential system and the solutions of the associated harmonic differential model, and as a consequence, a general harmonic framework for linear time periodic systems and bilinear affine systems. The proposed framework allows to design globally stabilizing harmonic control laws. We illustrate the proposed approach on a single-phase rectifier bridge. Through this example, we show how one can design stabilizing control laws that guarantee periodic disturbance rejection and low harmonic content.

Positive-Controllability, Positive-Near-Controllability, and Canonical Forms of Driftless Discrete-Time Bilinear Systems

Positive-Controllability, Positive-Near-Controllability, and Canonical Forms of Driftless Discrete-Time Bilinear Systems

Feedback stabilization of non‐homogeneous bilinear systems with a finite time delay

This paper investigates the feedback stabilization of non‐homogeneous delayed bilinear systems in Hilbert state space. More precisely, under an observability‐like assumption, we prove the exponential and strong stability of the solution by using bounded feedback control. Partial stabilization is discussed as well. The proofs of the main results are based on the decomposition method. The decay estimates of the stabilized solutions are obtained. Finally, some examples are presented.

Feedback stabilization and convergence rate of bilinear systems on time scales

Feedback stabilization and convergence rate of bilinear systems on time scales

Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms

The application of transformations of the state equations of continuous linear and bilinear systems to the canonical form of controllability allows one to simplify the computation of Gramians of these systems. In this paper, we develop the method and obtain algorithms for computation of the controllability and observability Gramians of continuous linear and bilinear stationary systems with many inputs and one output, based on the method of spectral expansion of the Gramians and the iterative method for computing the bilinear systems Gramians. An important feature of the concept is the idea of separability of the Gramians expansion: separate computation of the scalar and matrix parts of the spectral Gramian expansion reduces the sub-Gramian matrices computation to calculation of numerical sequences of their elements. For the continuous linear systems with one output the method and the algorithm of the spectral decomposition of the controllability Gramian are developed in the form of Xiao matrices. Analytical expressions for the diagonal elements of the Gramian matrices are obtained, and by making use of which the rest of the elements can be calculated. For continuous linear systems with many outputs the spectral decompositions of the Gramians in the form of generalized Xiao matrices are obtained, which allows us to significantly reduce the number of calculations. The obtained results are generalized for continuous bilinear systems with one output. Iterative spectral algorithms for computation of elements of Gramians of these systems are proposed. Examples are given that illustrate theoretical results.

It is undecidable whether the growth rate of a given bilinear system is 1

We show that there exists no algorithm that decides for any bilinear system (B,v) if the growth rate of (B,v) is 1. This answers a question of Bui who showed that if the coefficients are positive the growth rate is computable (i.e., there is an algorithm that outputs the sequence of digits of the growth rate of (B,v)). Our proof is based on a reduction of the computation of the joint spectral radius of a set of matrices to the computation of the growth rate of a bilinear system. We also use our reduction to deduce that there exists no algorithm that approximates the growth rate of a bilinear system with relative accuracy ε in time polynomial in the size of the system and of ε. Our two results hold even if all the coefficients are nonnegative rationals.