General Class Of Nonlinear Systems Research Articles

A perspective on conditional spectrum-based determination of response statistics of nonlinear systems

This work focuses on determining the stochastic response properties, in the frequency domain, of a general class of nonlinear systems with polynomial nonlinearities. Specifically, the results are presented in terms of the stationary power spectral densities of the system’s displacement and velocity. This is pursued by revisiting the conditional power spectrum concept, with the assumption that the response process is both ergodic and pseudo-harmonic and characterized by an amplitude, and a phase. A theoretical elucidation of an existing formula for the conditional spectrum is attempted. In particular, this concept is interpreted in conjunction with the time averaging approximation made in the definition of the stationary probability density function of a response amplitude quantity, associated with the original nonlinear system. It is shown that a proper definition of the stationary probability density of the response amplitude, along with a reasonable treatment of the distribution over the frequency domain of the amplitude contribution, lead to an improved approximation of the stationary response power spectral density. The treatment involves the averaging of a population of surrogate spectral densities of stationary random responses conforming with the system responses associated with individual values of the amplitudes of the responses. The semi-analytical results have been quite favourably juxtaposed with a large suite of à propos Monte Carlo simulations, both in terms of the shape and of the range of the involved germane frequencies, even for strongly nonlinear systems.

Iterative control framework with application to guidance and attitude control of rockets

Traditional control methods for nonlinear dynamical systems are predicated on verification of complex mathematical conditions related to the existence of a positive-definite Lyapunov function whose value must strictly decrease with time. Rigorous verification of Lyapunov conditions can be extremely difficult in real-world systems with high-dimensional and complex dynamics. In this paper, we present a novel control logic that can be readily applied to a general class of nonlinear systems irrespective of the complexities in their dynamics. The Iterative Control Framework (ICF) is designed to guarantee the convergence of the closed-loop system state to zero without a priori verification of Lyapunov-like conditions. The underlying computational routine runs in the background in real time and reconfigures the control vector at each time step in such a way that when the control input is applied to the system, the system trajectory reaches closer to the desired state. The technique is applicable to a broad class of complex nonlinear systems but is particularly suitable for systems inherently admitting control action of short duration such as missiles, rockets, satellites, and space vehicles. In this work, we focus on the application of ICF to guidance and attitude control of rockets and missiles where actuation is provided via single-use thrusters.

Zonotopic Distributed Fusion Over Binary Sensor Networks With Bit Rate Allocation: A Coding-Decoding Approach.

In this article, the zonotopic distributed fusion estimation problem is investigated for a class of general nonlinear systems over binary sensor networks subject to unknown-but-bounded (UBB) noises. The network communication from nodes to the fusion center is confined to the limited bit rate. To alleviate the impact from less measurement information of the binary sensor, a modified innovation is constructed to improve the estimation accuracy. Then, a novel coding-decoding approach is proposed to ensure that the decoder has the ability to decode information from each node. Based on the matrix weighting fusion method, a distributed fusion algorithm is put forward under the zonotopic set-membership filtering framework, and the F -radius of the local zonotopic sets are derived and minimized by selecting the filtering gain parameters. Moreover, the bit rate allocation scheme and the weighting coefficients are determined by resolving two optimization problems. In addition, a sufficient condition is established to guarantee the uniform boundedness of the F -radius of the fused zonopotic. Finally, the ballistic object tracking systems is utilized to illustrate the availability of the presented algorithm.

A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

In this paper, a new efficient, and at the same time, very simple and general class of thermodynamically compatible finite volume schemes is introduced for the discretization of nonlinear, overdetermined, and thermodynamically compatible first-order hyperbolic systems. By construction, the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm. A very peculiar feature of our approach is that entropy is discretized directly, while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization. The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs, including both, conservative and non-conservative products, as well as potentially stiff algebraic relaxation source terms, provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law, such as the conservation of total energy density. The proposed family of finite volume schemes is based on the seminal work of Abgrall [1], where for the first time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented. We apply our new approach to three particular thermodynamically compatible systems: the equations of ideal magnetohydrodynamics (MHD) with thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning, the unified first-order hyperbolic model of continuum mechanics proposed by Godunov, Peshkov, and Romenski (GPR model) and the first-order hyperbolic model for turbulent shallow water flows of Gavrilyuk et al. In addition to formal mathematical proofs of the properties of our new finite volume schemes, we also present a large set of numerical results in order to show their potential, efficiency, and practical applicability.

Fault-tolerant prescribed performance control of nonlinear systems with process faults and actuator failures

This paper investigates the fault-tolerant prescribed performance control problem for a class of multiple-input single-output unknown nonlinear systems subject to process faults and actuator failures. In contrast to the related works, we consider a general class of nonlinear systems with both multiplicative nonlinearities and additive nonlinearities corrupted by the process faults; only the boundedness of the process faults and the continuity of the nonlinear functions are required, without the explicit or fixed structures of the fault functions. To conquer this problem, a less-demanding and low-complexity fault-tolerant prescribed performance control approach is proposed. The controller is independent of the specific information of faults or the system model and does not invoke fault diagnosis or neural/fuzzy approximation to acquire such knowledge. It achieves the reference tracking with the predefined rate and accuracy. A comparative simulation on a single-link robot is conducted to illustrate the effectiveness and superiority of the proposed approach.

Koopman-based deep iISS bilinear parity approach for data-driven fault diagnosis: Experimental demonstration using three-tank system

Precise and timely fault diagnosis is crucial in many practical systems and control processes. Particularly due to the increasing amount of available data collected by sensors, data-driven fault diagnosis has been a hot research topic in the prognosis and health management of industrial systems. In this paper, a reliable, accurate, and interpretable data-driven fault detection and isolation method is proposed for the general class of nonlinear systems. The proposed approach is based on the integration of the bilinear Koopman model realization, deep learning, and a bilinear parity-space framework. This work leverages the potential of neural networks to investigate lifting functions and bilinear Koopman realization simultaneously. Furthermore, to enhance the stability of the realized model, the input-to-state stability constraint is enforced on the training algorithm, ensuring the realized model is integral input-to-state stable. This method does not require any prior knowledge regarding the system dynamics and only uses the data collected from the normal operation of the system. In addition, it is capable of diagnosing all types of sensor and actuator faults, whether they are additive or multiplicative. The effectiveness of the proposed method is demonstrated experimentally using a laboratory setup of the three-tank system. Lastly, a comparison is conducted to demonstrate the advantages of the proposed method over another recent data-driven fault diagnosis method.

On implementing the implicit discrete‐time super‐twisting observer on mechanical systems

AbstractIn this paper, an extension of an algorithm for the implicit discretization of a super‐twisting sliding mode observer is presented. Implicit and explicit discretization algorithms for homogeneous differentiators, where no physical model information is considered, are investigated in literature. This article studies the behavior when considering models of a rather general class of nonlinear systems. The discrete equations of the super‐twisting observer are reformulated as generalized equation and an algorithm for the step‐by‐step solution is given. The uniqueness of the derived algorithm is investigated with an equivalent variational inequality formulation which is derived for a class of nonlinear systems. Furthermore, a semi‐implicit predictor‐corrector discretization is presented which is an approximation method for the presented algorithms and allows an explicit implementation in practical applications. Accuracy properties under noise and sampling are given. The algorithm is applied on two mechanical example systems taken from practice.

Suboptimal Nonlinear Moving Horizon Estimation

In this paper, we propose a suboptimal moving horizon estimator for a general class of nonlinear systems. For the stability analysis, we transfer the "feasibility-implies-stability/robustness" paradigm from model predictive control to the context of moving horizon estimation in the following sense: Using a suitably defined, feasible candidate solution based on an auxiliary observer, robust stability of the proposed suboptimal estimator is inherited independently of the horizon length and even if no optimization is performed. Moreover, the proposed design allows for the choice between two cost functions different in structure: the former in the manner of a standard least squares approach, which is typically used in practice, and the latter following a time-discounted modification, resulting in better theoretical guarantees. We apply the proposed suboptimal estimator to a nonlinear chemical reactor process, verify the theoretical assumptions, and show that even a few iterations of the optimizer are sufficient to significantly improve the estimation results of the auxiliary observer. Furthermore, we illustrate the flexibility of the proposed design by employing different solvers and compare the performance with two state-of-the-art fast MHE schemes from the literature.

Emerging robust and data‐driven control methods for uncertain learning systems

Emerging robust and data‐driven control methods for uncertain learning systems

Iterative learning identification: Dynamic parametrization modeling and comparison

This article elaborates the iterative learning mechanism for time-varying system identification, and describes the learning algorithms that could achieve the consistent estimation for time-varying parameters under persistent repetitive-excitation conditions. A dynamic parametrization approach, in this article, is presented for modeling and analysis a general class of nonlinear systems. The derivations are conducted to give linear-in-the-parameters models with time-varying coefficients. The resultant models can be in a unified form, with the aid of the variable difference representation, and the iterative learning least squares algorithm and its variant are applicable for the purpose of parameter estimation. Moreover, a learning control scheme is adopted for demonstrating effectiveness of the dynamically-parametrized modes, which are simulated and fully compared with the presented numerical results.

A Simple Suboptimal Moving Horizon Estimation Scheme With Guaranteed Robust Stability

We propose a suboptimal moving horizon estimation (MHE) scheme for a general class of nonlinear systems. To this end, we consider an MHE formulation that optimizes over the trajectory of a robustly stable observer. Assuming that the observer admits a Lyapunov function, we show that this function is an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {M}$ </tex-math></inline-formula> -step Lyapunov function for suboptimal MHE. The presented sufficient conditions can be easily verified in practice. We illustrate the practicability of the proposed suboptimal MHE scheme with a standard nonlinear benchmark example. Here, performing a single iteration is sufficient to significantly improve the observer’s estimation results under valid theoretical guarantees.

Data‐driven sensor fault detection and isolation of nonlinear systems: Deep neural‐network Koopman operator

This paper proposes a data-driven sensor fault detection and isolation approach for the general class of nonlinear systems. The proposed method uses deep neural network architecture to obtain an invariant set of basis functions for the Koopman operator to form a linear predictor for a nonlinear system. Then, the obtained Koopman predictor has been used in a geometric framework for sensor fault detection and isolation purposes without relying on a priori knowledge about the underlying dynamics as well as requiring faulty data, leading to a data-driven sensor fault detection and isolation framework for nonlinear systems. Finally, the approach's efficacy is demonstrated using simulation case study on a two-degree of freedom robot arm.

Data-driven fault detection and isolation of nonlinear systems using deep learning for Koopman operator

This paper proposes a data-driven actuator fault detection and isolation approach for the general class of nonlinear systems. The proposed method uses a deep neural network architecture to obtain an invariant set of basis functions for the Koopman operator to form a linear Koopman predictor for a nonlinear system. Then, the obtained linear model is used for fault detection and isolation purposes without relying on prior knowledge about the underlying dynamics. Moreover, a recursive method is proposed for fault detection and isolation that is entirely data-driven with the key feature of global validity for the system’s whole operating region due to the Koopman operator’s global characteristic. Finally, the approach’s efficacy is demonstrated using two simulations on a coupled nonlinear system and a two-link manipulator benchmark.

Fault Detection Approach for Nonlinear Systems via Nonlinear Factorization and Fuzzy Models

This brief aims to propose an effective fault detection method for general class of nonlinear systems in the context of the closed-loop system stability. To this end, the nonlinear factorization technique is first used to model the faulty nonlinear systems, which can be represented by the so-called stable kernel representation with a stable parameterization of the system changes triggered by faults. Then, the closed-loop system stability is discussed according to the internal stability definition and the small gain theorem, respectively, to present the design framework of the fault detection system. Different from the traditional fault detection schemes, the proposed fault detection approach focuses on detecting whether the system closed-loop stability is damaged by faults utilizing the online measurable system and controller residual signals. Furthermore, for the implementation of the proposed fault detection framework, Takagi-Sugeno fuzzy models are applied to approximate the nonlinear systems and thus the fault detection system design methods can be provided by taking advantage of the linear matrix inequality technique. Finally, a case study is used to verify the achieved results.

Neural-Network-Based Filtering for A General Class of Nonlinear Systems under Dynamically Bounded Innovations Over Sensor Networks

In this paper, the design problem of distributed filters is studied for nonlinear systems over sensor networks. At each sensor node, filters are designed by making use of both local measurements and innovation information sent from the neighboring nodes via interaction network. To mitigate adverse effects from abnormal data during transmissions, in the constructed local filters, a mechanism is proposed which utilizes a saturation function to constrain the propagated innovations within a dynamically changeable bound. A neural-network-based algorithm is used for the approximation of the nonlinear dynamics, where the weight matrix is co-designed with the filter gains. By resorting to certain convex optimization techniques, sufficient conditions are determined with respect to the solvability of our addressed problem, which ensure that 1) the estimation errors at each sensor node are confined within a pre-specified ellipsoidal region; and 2) the finite-horizon <inline-formula><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> performance specification is achieved. Moreover, within the established framework, an optimal problem is established to determine a locally optimal filter gain. Finally, a simulation example is given to demonstrate the correctness of the proposed filtering algorithm.

Solution for optimal control of general nonlinear systems using generating functions

AbstractIn this paper, we present an analytical local solution for optimal control problems applicable to a wide class of general nonlinear systems. We first introduce the optimal control problem for a general nonlinear system and formulate the associated Hamilton–Jacobi–Bellman equation. Starting from the necessary conditions for optimality represented by the Hamiltonian system of the optimal control problem, we solve the Hamilton–Jacobi equation for the associated generating functions. The generating functions are obtained in the form of series expansions by using canonical transformation theory and the semi‐tensor product of matrices. The coefficients of the generating functions satisfy simply the algebraic Riccati equation and linear algebraic equations for the second‐degree homogeneous component and higher degree components, respectively, from which an algorithm and data structure can be derived. This enables us to obtain analytical optimal feedback controllers and the corresponding Lyapunov function. Through theoretical analysis, it is established that the proposed methodology guarantees stability of the closed‐loop system. This approach is not affected by the complexity of the performance index and state equations, and most importantly, it does not require any initial admissible control law. A simulation example is presented to illustrate the effectiveness and applicability of the proposed approach.

Optimal solution of a general class of nonlinear system of fractional partial differential equations using hybrid functions

Optimal solution of a general class of nonlinear system of fractional partial differential equations using hybrid functions

Iterative Control Framework with Application to Guidance Control of Rockets with Impulsive Thrusters

In contrast with traditional control techniques that require a priori verification of complex mathematical conditions, in this paper, we develop a fundamentally new framework of control logic that can be readily applied to a general class of nonlinear systems. Our approach, called the Iterative Control Framework (ICF), uses a novel control logic to guarantee the stability of a closed-loop system without a priori verification of Lyapunov-like conditions. The underlying idea is in repetitive reconfiguration of the control vector provided by a computational routine running in the background in real time during each iteration. The control approach is applicable to a broad class of complex nonlinear systems but is particularly suitable for systems inherently admitting only control action of finite-time duration such as short- and long-range missiles, satellites, and spacecrafts. In this work, we focus on the application of ICF to guidance control of rockets and missiles where actuation is provided via single-use thrusters.

Incremental Adaptive Control of a Class of Nonlinear Nonaffine Systems

As a class of familiar nonlinear systems, nonaffine systems are frequently encountered in practical applications. Currently, in the context of learning control, there is a lack of research results about such general class of nonlinear systems, especially for the case of performing infinite interval tasks. This article focuses on the incremental adaptive control for nonlinear systems in nonaffine form, without requiring periodicity or repeatability. Instead of using the integral adaptation, incremental adaptive mechanisms are developed and the corresponding control schemes are presented, by which the numerical integration for implementation can be avoided. With the proposed incremental adaptation, the implicit function theorem is introduced to solve the intractability problem of the nonaffine structure. The robustness (robust convergence) of the tracking error is characterized, with the aid of a proposed key lemma, while the boundedness of all the variables is examined. Numerical results are presented to verify the effectiveness of the proposed control design.

Nonlinear output frequency response functions: A new evaluation approach and applications to railway and manufacturing systems’ condition monitoring

The Nonlinear Output Frequency Response Functions (NOFRFs) are an extension of the well-known Frequency Response Function (FRF) of linear systems to the nonlinear case and have recently been applied by many researchers to resolve different engineering problems. However, there exist several issues with current methods that can be used for the evaluation of the NOFRFs of practical systems. These include the difficulty and lack of an effective approach of accurately evaluating the NOFRFs for a general class of nonlinear systems. In the present study, a new concept known as the Generalized Associated Linear Equations (GALEs) is introduced to develop a novel approach to systematically address these problems. By using the GALEs, the NOFRFs of the NARX (Nonlinear Auto Regressive with eXegenous input) model or NDE (Nonlinear Differential Equation) model of nonlinear systems can be accurately evaluated by simply dealing with the solutions to a series of linear difference or differential equations. This can significantly facilitate the NOFRFs based nonlinear system analyses and associated practical applications. Two experimental studies including monitoring fatigue damage of train wheels and inspecting wearing conditions of cutting tools, respectively, demonstrate the significance and potential applications of the new GALEs based NOFRFs evaluation in railway and manufacturing systems’ condition monitoring.