Positive Real Condition Research Articles

Improvement of Commercial Vehicle Seat Suspension Employing a Mechatronic Inerter Element

To further improve the ride comfort of commercial vehicles, a seat ISD (Inerter–Spring–Damper) suspension utilizing a mechatronic inerter is proposed in this paper. Firstly, a five-DOF (degree-of-freedom) commercial vehicle seat ISD model was built. Then, the positive real network constraint conditions of a biquadratic impedance transfer function were determined, and the meta-heuristic intelligent optimization algorithm was used to solve the parameters. According to the solution, the impedance transfer function was obtained and the specific network structure was realized by network synthesis. Lastly, this study compares the vibration isolation performance of the mechatronic ISD suspension of the vehicle seat with that of a passive suspension. In comparison to passive seat suspension, the seat mechatronic ISD suspension reduces seat vibration transmissibility by 16.33% and vertical acceleration by 16.78%. Results indicate that the new suspension system can be an effective improvement in ride comfort.

Support Recovery and Parameter Identification of Multivariate ARMA Systems with Exogenous Inputs

Focusing on identification, support recovery, and parameter estimation of multivariate ARMA systems with exogenous inputs (ARMAX) is considered. It is assumed that sparsity appears not only in the unknown parameter matrix of the system inputs and outputs but also in the coefficients of system noises, and with exogenous inputs the system admits feedback control and thus the observation of the ARMAX system may be nonstationary. A two-stage sparse identification algorithm is introduced. First, the extended least squares (ELS) algorithm is applied to obtain an initial estimate for the unknown parameter matrix of the ARMAX system. Second, a convex optimization criterion is introduced based on norm with regularization, where adaptive weights generated from ELS algorithms are adopted in the optimization variables in the term. Third, it is proved that by optimizing the criterion, the sets of the zero and the nonzero entries of the parameter matrix, i.e., the support set of the ARMAX system, can be correctly identified with a finite number of observations, and estimates of the nonzero parameters converge to the true values with probability one. Fourth, the strictly positive realness (SPR) condition required by the ELS algorithm is relaxed by an overparametrization technique. The performance of the algorithm is testified through simulation examples.

Improving performance of adaptive feedforward noise attenuators using a dynamic adaptation gain

The paper explores in detail the use of dynamic adaptation gain/learning rate (DAG) for improving the performance of adaptive feedforward attenuation schemes. The DAG is an ARMA (poles-zeros) filter embedded in gradient type adaptation/learning algorithms and generalizes the various improved gradient algorithms available in the literature. After introducing the DAG algorithm in the context of adaptive feedforward attenuation schemes and providing relationships with other algorithms, its design is developed. Strictly Positive Real (SPR) conditions play an important role in the design of the DAG. Then the stability issues for adaptive/learning systems using a DAG are discussed. The potential of the DAG is then illustrated by experimental results obtained on a relevant adaptive active noise control system.

Fault diagnosis for a class of nonlinear uncertain systems using deterministic learning approach

In this paper, we propose a fault diagnosis (FD) approach for a class of nonlinear uncertain systems based on the deterministic learning approach (DLA). Specifically, an adaptive learning observer is constructed, in which the adaptive neural networks (NNs) are constructed to approximate the unknown system dynamics under normal and fault modes. Based on the strictly positive real (SPR) condition, the convergence of the state estimation can be guaranteed. When the system is undergoing a periodic or periodic-like (recurrent) motion, the states of the observer will also become recurrent. Thus through DLA, the partial persistent excitation (PE) condition of the associated subvectors of NNs is satisfied. By utilizing the partial (PE) condition, the uniformly completely observable (UCO) property of the identification system is analyzed and the exponential convergence condition of the identification system is derived. Under this condition, the unknown dynamics under normal and fault modes can be accurately identified along the system trajectory. And by utilizing the knowledge obtained in the identification phase, the fault can be detected in the diagnosis phase. The main attraction of this paper lies in the analytical result, which shows that the exponential convergence condition of the learning observer not only depends on the observer gain matrix, but also depends on the PE level of the regressor subvector of NN. Simulation results are included to illustrate the effectiveness of the proposed scheme.

Iterative learning control applied to distributed‐order linear time invariant MIMO systems to achieve learnability

AbstractSufficient and necessary conditions for a distributed‐order linear time invariant system to be positive real are derived in terms of linear matrix inequalities. The positive realness condition is derived for three of the most usual cases presented in literature, in the realm of distributed‐order linear time invariant systems. As an additional product of this paper, the strictly positive realness condition can be derived. In addition, the concept of learnability of fractional‐order multi‐input multi‐output systems is extended to the case of distributed‐order systems, which is approached from the concept of output‐dissipativity by using an iterative learning scheme.

Establishment of Strictly Positive Real Condition for Tuning MIMO Feedforward Control

The two-degree-of-freedom structure enables us to achieve stabilization and tracking independently using feedback (FB) and feedforward (FF) controls, respectively. The latter usually undergoes a model uncertainty, which is resolved by online tuning. A tuning law guarantees zero convergence of the output error for any target signals, under a certain strictly positive real (SPR) condition; however, it is noticeably restrictive. This study proposes a method for guaranteeing the SPR condition by designing an FF filter and modifying the error signal for tuning in the multi-input multi-output case. The proposed method is illustrated through a numerical example.

Design and experimental implementation of observer-based adaptive neural network steering control for automated vehicles

In this paper, an observer-based adaptive neural network controller is developed for the Steer-by-Wire (SbW) system of automated vehicles with uncertain nonlinearity and unmeasured state. An observer is introduced to estimate the angular velocity of the front wheels, so the hardware cost and the complexity of mechanical structure and electronic circuits are reduced. Then, an observer-based adaptive neural network controller is proposed for the SbW system to achieve excellent steering precision. A radial basis function (RBF) neural network is used to model the uncertain nonlinearity, which mainly includes self-aligning torque and unknown friction torque with strong nonlinearity. The adaptive law of the RBF neural network designed by fuzzy basis functions rather than by its filtering is derived by Lyapunov stability theory and the Strictly Positive Real (SPR) condition. The tracking error and the observation error can be guaranteed to converge asymptotically to zero. Simulation and experimental results for two paths highlight the effectiveness of the proposed control algorithm.

Constant-gain nonlinear adaptive observers revisited: an application to chemostat systems

This study deals with constant-gain adaptive observers for nonlinear systems, for which relatively few solutions are available for some particular cases. We introduce an asymptotic observer of constant gain for nonlinear systems that have linear input. This allows the observer design to be formulated within the linear matrix inequality paradigm provided that a strictly positive real condition between the input disturbance and the output is fulfilled. The proposed observer is then applied to a large class of nonlinear chemostat dynamical systems that are widely used in the fermentation process, cell cultures, medicine, etc. In fact, under standard practical assumptions, the necessary change of the chemostat state coordinates exists, allowing use of the constant-gain observer. Finally, the developed theory is illustrated by estimating pollutant concentration in a Spirulina maxima wastewater treatment facility.

A rules‐firing strength‐based neuro‐fuzzy observer for information‐poor systems

In this paper, a neuro-fuzzy observer (NFO) is proposed for estimating the unmeasured states of an information-poor system by relaxing the strictly positive real condition (without using filtered fuzzy basis function (FBF) and filtered output estimation error) and without using high-gain terms. To recover the performance of the observer in the absence of high-gain terms, a concept of weighted fuzzy rules (or strengthened FBF) is proposed. The weighted fuzzy rules are then used to propose the concept of weighted function approximation which is then utilized in the design of NFO to estimate the unknown dynamical function. For reducing computational power and avoiding over-tuning of the weights, a concept of relay-switching is also introduced in the design of the NFO. The stability analysis of the proposed NFO is also presented using Lyapunov approach and the effectiveness of the proposed scheme is demonstrated through a simulation example.

An integrated fault estimation and fault tolerant control method using H∞‐based adaptive observers

SummaryAn integrated fault estimation/fault‐tolerant control (FTC) scheme is developed in this article for nonlinear Lipschitz systems in the presence of external disturbances and actuator failures. To address this problem, coupled uncertainties between the observer error dynamics and the control system are considered, which is conveniently ignored in control approaches based on the separation principle. An H∞‐based adaptive observer is proposed to simultaneously estimate the system states and actuator faults without the restrictive strictly positive realness or persistent excitation conditions. The FTC is constructed by sliding mode control using the estimated states generated by the developed observer. A novel sufficient condition is derived in terms of linear matrix inequality (LMI) including both the system control dynamics and the estimation errors; then, the control parameters and observer gains are simultaneously obtained via solving the mentioned LMI based on the H∞ optimization. Finally, a flexible joint robot is considered to illustrate the effectiveness of the developed method.

Synchronization of Heterogeneous Dynamical Networks via Phase Analysis

This paper is concerned with the synchronization of heterogeneous agents interacting over a dynamical network, where the edge dynamics are heterogeneous, modeling the nonuniform communication environment between the agents. Novel synchronization conditions are obtained from a phasic perspective by utilizing a newly formulated small phase theorem. These conditions have lower conservatism compared to gain-based conditions and generalize positive real and negative imaginary type conditions. They scale well with the size of the network and reveal the trade-off between the phases of node dynamics and edge dynamics.

Positive Real Properties and Physical Realizability Conditions For a Class of Linear Quantum Systems

Theoretical developments in the field of quantum optics and quantum superconducting electrical circuits involving continuous measurement based feedback control as well as coherent control are an important prerequisites for advances in the domain of quantum technology. Within these perspectives, this paper considers positive real properties for a class of quantum systems whose quantum stochastic differential equation model involves annihilation operators only and then relates them to corresponding bounded real properties and consequently to physical realizability conditions developed earlier by the authors. Based on the positive real properties of these quantum systems, it is anticipated that it is possible to use the Brune algorithm in order to find an electrical circuit that can physically implement these quantum systems. This theory, in the case of one-port circuits, may be useful for the implementation of microwave circuits related to quantum filters found in the field of quantum computing.

MIMO Generalization of Feedback Error Learning Control based on Strictly Positive Real Condition

MIMO Generalization of Feedback Error Learning Control based on Strictly Positive Real Condition

A modeling error-based adaptive fuzzy observer approach with input saturation analysis for robust control of affine and non-affine systems

In this paper, a robust control approach is applied for both MIMO/SISO affine/non-affine nonlinear systems based on a modeling error-based adaptive fuzzy observer controller, in the presence of input saturation. In the proposed scheme, non-affine nonlinear systems can be transformed to affine systems and unknown higher-order term of expansion (HOTE) that appears due to the use of this method can be estimated by an adaptive fuzzy technique. Using the modeling error between the system states observer and a serial–parallel estimator model, a modeling error-based adaptive fuzzy observer estimator is proposed that uses the modeling error as the input of fuzzy system to approximate and adaptively compensate the unknown HOTE and also the external disturbance. The proposed scheme is able to hold control performance in the presence of input saturation. An analysis of the controlled system is presented to verify the stability of the system under control. The stability of the closed-loop system is provided based on the strictly positive real condition and Lyapunov theory. The proposed approach is effectual and robust. The simulation results demonstrate the usefulness of the proposed method for both MIMO and SISO systems.

Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

Robust H∞ gain neuro-adaptive observer design for nonlinear uncertain systems

To guarantee convergent state estimates and exact approximations, it is highly desirable that observers can independently dominate the effects of unmodelled dynamics. Based on adaptive nonlinear approximation, this paper presents a robust H∞ gain neuro-adaptive observer (R H∞GNAO) design methodology for a large class of uncertain nonlinear systems in the presence of time-varying unknown parameters with bounded external disturbances on the state vector and on the output of the original system. The proposed R H∞GNAO incorporates radial basis function neural networks (RBFNNs) to approximate the unknown nonlinearities in the uncertain system. The weight dynamics of every RBFNN are adjusted online by using an adaptive projection algorithm. The asymptotic convergence of the state and parameter estimation errors is achieved by using Lyapunov cogitation under a well-defined persistent excitation condition, and without recourse to the strictly positive real condition. The repercussions of unknown disturbances are reduced by integrating an H∞ gain performance criterion into the proposed estimation approach. The condition imposed by this proposed observer approach, such that all estimated signals are uniformly ultimately bounded, is expressed in the form of the linear matrix inequality problem and warrants the demanded performances. To evaluate the performance of the proposed observer, various simulations are presented.

Eigenfunctions of Periodic Differential Operators Analytic in a Strip

Ordinary differential operators with periodic coefficients analytic in a strip act on a Hardy–Hilbert space of analytic functions with inner product defined by integration over a period on the boundary of the strip. Simple examples show that eigenfunctions may form a complete set for a narrow strip, but completeness may be lost for a wide strip. Completeness of the eigenfunctions in the Hardy–Hilbert space is established for regular second order operators with matrix-valued coefficients when the leading coefficient satisfies a positive real part condition throughout the strip.

Combined adaptive feedback and feedforward compensation for active vibration control using Youla–Kučera parametrization

The combination of adaptive feedforward with adaptive feedback disturbance compensation has been suggested since a number of years as a potential solution for improving the performance of active vibration control systems. Unfortunately, as shown in the present paper there is a strong coupling between adaptive feedback and adaptive feedforward which may lead to instabilities and new algorithms which take into account this interaction are proposed in the paper. These algorithms take advantage of the Youla-Kučera parametrization of the feedback controller and feedforward compensator. First, the adaptive feedback algorithm is presented alone. Then, the adaptive feedforward algorithm is designed taking into account the feedback controller's presence in the loop. Stability of the complete scheme is obtained by appropriately filtering the regressor vectors in order to take into account a strictly positive real condition. Experimental results obtained on a relevant test-bench illustrate the performance of this approach to active vibration control. Comparison with adaptive feedback alone, with adaptive feedforward alone and adaptive feedforward + fixed feedback are provided.

Low-Complexity Passive Vehicle Suspension Design Based on Element-Number-Restricted Networks and Low-Order Admittance Networks

This paper is concerned with the low-complexity passive suspension design problem, aiming at improving vehicle performance in the meanwhile maintaining simplicity in structure for passive suspensions. Two methods are employed to construct the low-complexity passive suspensions. Using the first method, the number of each element is restricted to one, and the performance for all networks with one inerter, one damper, and one spring is evaluated, where best configurations for different vehicle settings are identified. Using the second method, low-order admittance networks whose orders of admittance functions are no larger than three are utilized. Design methods are proposed by directly using the positive realness conditions imposed on the admittance functions. The effectiveness of the proposed methods is numerically demonstrated, and the comparison between these two constructing methods is conducted.

Global Dynamics of Certain Mix Monotone Difference Equation

We investigate global dynamics of the following second order rational difference equation x n + 1 = x n x n − 1 + α x n + β x n − 1 a x n x n − 1 + b x n − 1 , where the parameters α , β , a , b are positive real numbers and initial conditions x − 1 and x 0 are arbitrary positive real numbers. The map associated to the right-hand side of this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the corresponding parametric space. In most cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability.