General Nonlinear Systems Research Articles

Constraint-adaptive MPC for large-scale systems: Satisfying state constraints without imposing them

Abstract Model Predictive Control (MPC) is a successful control methodology, which is applied to increasingly complex systems. However, real-time feasibility of MPC can be challenging for complex systems, certainly when an (extremely) large number of constraints have to be adhered to. For such scenarios with a large number of state constraints, this paper proposes two novel MPC schemes for general nonlinear systems, which we call constraint-adaptive MPC. These novel schemes dynamically select at each time step a (varying) set of constraints that are included in the on-line optimization problem. Carefully selecting the included constraints can significantly reduce, as we will demonstrate, the computational complexity with often only a slight impact on the closed-loop performance. Although not all (state) constraints are imposed in the on-line optimization, the schemes still guarantee recursive feasibility and constraint satisfaction. A numerical case study illustrates the proposed MPC schemes and demonstrates the achieved computation time improvements exceeding two orders of magnitude without loss of performance.

Probability-Guaranteed Distributed Secure Estimation for Nonlinear Systems Over Sensor Networks Under Deception Attacks on Innovations

This paper studies the distributed secure state estimation problem for a class of general nonlinear systems over sensor networks under unknown deception attacks on innovations. At each sensing node, an estimator is designed to generate the state estimate by making use of the local measurements in combination with the neighbours’ information shared via the communication network. During the transmission of innovations among nodes, the data are maliciously falsified by adversaries in a random way. A neural-network-based mechanism is put forward to approximate the unknown falsified innovations with the aim to mitigate the effects on the estimation performance. The objective of the addressed problem is to develop a distributed estimation approach to jointly estimate the system states as well as the unknow deception attacks, ensuring that the state estimation errors at each sensing node reside within required ellipsoidal regions with a pre-specified probability. With the help of certain convex optimization methods, we obtain sufficient conditions for the solvability of the addressed problem and the desired estimator gains can be iteratively computed by solving a series of matrix inequalities. On basis of the proposed framework, some optimization problems are presented to determine sub-optimal estimator parameters from different perspectives. Finally, the applicability of the developed algorithms is validated via a numerical simulation example.

First-passage reliability of high-dimensional nonlinear systems under additive excitation by the ensemble-evolving-based generalized density evolution equation

The reliability analysis of high-dimensional stochastic dynamical systems subjected to random excitations has long been one of the major challenges in civil and various engineering fields. Despite great efforts, no satisfactory method with high efficiency and accuracy has been available as yet for high-dimensional systems even when they are linear systems, not to mention generic nonlinear systems. In the present paper, a novel method by imposing appropriate absorbing boundary condition on the newly developed ensemble-evolving-based generalized density evolution equation (EV-GDEE) combined with a feasible numerical method is proposed to capture the time-variant first-passage reliability of high-dimensional systems enforced by additive white noise excitation. In the proposed method, the equivalent drift coefficients in EV-GDEE can be estimated by analytical expression or captured by some representative deterministic dynamic analyses. Further, imposing the absorbing boundary condition and then solving the EV-GDEE, a one-or two-dimensional partial differential equation (PDE), yield the remaining probability density of the response of interest. Consequently, by integrating the remaining probability density, the numerical solution of time-variant first-passage reliability can be obtained. Several numerical examples are illustrated to verify the efficiency and accuracy of the proposed method. Compared to the Monte Carlo simulation, the proposed method is of much higher efficiency. Problems to be further studied are finally discussed.

Stability analysis for time-varying nonlinear systems

This paper is concerned with practical asymptotic and exponential stability analysis of time-varying nonlinear systems. With the help of the new notion of practical stable functions, some differential Lyapunov inequalities based necessary and sufficient conditions are derived for testing global uniform practical asymptotic stability and practical exponential stability of general time-varying nonlinear systems. Therefore, we generalised some works which are already made in the literature. Furthermore, some illustrative examples are presented.

Adaptive control of a two-link robot using batch least-square identifier

We design a regulation-triggered adaptive controller for robot manipulators to efficiently estimate unknown parameters and to achieve asymptotic stability in the presence of coupled uncertainties. Robot manipulators are widely used in telemanipulation systems where they are subject to model and environmental uncertainties. Using conventional control algorithms on such systems can cause not only poor control performance, but also expensive computational costs and catastrophic instabilities. Therefore, system uncertainties need to be estimated through designing a computationally efficient adaptive control law. We focus on robot manipulators as an example of a highly nonlinear system. As a case study, a 2-DOF manipulator subject to four parametric uncertainties is investigated. First, the dynamic equations of the manipulator are derived, and the corresponding regressor matrix is constructed for the unknown parameters. For a general nonlinear system, a theorem is presented to guarantee the asymptotic stability of the system and the convergence of parameters' estimations. Finally, simulation results are discussed for a two-link manipulator, and the performance of the proposed scheme is thoroughly evaluated.

On the Semi-Global Stability of an EK-Like Filter

This letter proposes to apply the Kalman-like observer paradigm to general nonlinear systems by linearization along the estimated trajectory, similarly to an Extended Kalman Filter. The main difference is that the quadratic Riccati equation is replaced by a linear Lyapunov equation which can be solved and explicitly related to a determinability Gramian. This allows to show by Lyapunov analysis and without any ad-hoc assumption on the Riccati solution, that the resulting observer, called Extended Kalman-like Filter, can be made semi-globally convergent if the input is actively used to a) stabilize the (unknown) true trajectory, b) sufficiently excite the determinability of the linearized systems along the (known) estimated trajectory. A class of systems where this compromise can be reached is provided.

Feedforward boundary control of 2[formula omitted]2 nonlinear hyperbolic systems with application to Saint-Venant equations

Because they represent physical systems with propagation delays, hyperbolic systems are well suited for feedforward control. This is especially true when the delay between a disturbance and the output is larger than the control delay. In this paper, we address the design of feedforward controllers for a general class of 2×2 hyperbolic systems with a single disturbance input located at one boundary and a single control actuation at the other boundary. The goal is to design a feedforward control that makes the system output insensitive to the measured disturbance input. We show that, for this class of systems, there exists an efficient ideal feedforward controller which is causal and stable. The problem is first stated and studied in the frequency domain for a simple linear system. Then, our main contribution is to show how the theory can be extended, in the time domain, to general nonlinear hyperbolic systems. The method is illustrated with an application to the control of an open channel represented by Saint-Venant equations where the objective is to make the output water level insensitive to the variations of the input flow rate. Finally, we address a more complex application to a cascade of pools where a blind application of perfect feedforward control can lead to detrimental oscillations. A pragmatic way of modifying the control law to solve this problem is proposed and validated with a simulation experiment.

Mean-Nonovershooting Control of Stochastic Nonlinear Systems

We present mean-nonovershooting tracking control designs for stochastic strict-feedback nonlinear systems. Compared with the existing stochastic nonlinear tracking designs, the advantage of our design is that arbitrary reference trajectories can be tracked “from below.” We first consider a special class of stochastic nonlinear systems where the diffusion terms satisfy the linear growth condition and the drift terms are in special triangular forms. A new controller is designed to guarantee that the mean of the system output can asymptotically track a given trajectory without overshooting while keeping all of the states mean-square bounded. In this case, an appealing feature in our design is that strict mean-nonovershooting tracking is achieved while the existing results on nonlinear systems with deterministic disturbance can only achieve practical nonovershooting tracking. Then, we design a new controller for general stochastic strict-feedback nonlinear systems to ensure that the mean of overshoot can be tuned to be arbitrarily small while all the states of the closed-loop system remain bounded in probability. In this case, the new design ingredient is that our control gains are independent of the initial conditions of system states, and the initial conditions of derivatives of the reference trajectory. This is in contrast with the existing results on corresponding deterministic systems whose control gains heavily rely on these initial conditions. Finally, two simulation examples are given to illustrate the stochastic nonlinear nonovershooting tracking designs.

Two-stage physics-based Wiener process models for online RUL prediction in field vibration data

Due to most failure mechanisms, such as fatigue crack growth and fatigue spall, the degradation process of rotating machinery commonly exhibits two-stage features in engineering practice. Other minor factors are also the key issues affecting the health evolution process, including the component structure, assembly accuracy, and working environment. Ignoring such a mechanism may lead to imprecise in degradation modeling, life prognostic, and ultimately lead to safety risk. Besides, achieving high accuracy of prognostic emphasizes the influence of random effect in the degradation process. The contribution of this study lies in addressing this issue by proposing two-stage physics-based Wiener process models integrating: (a) fatigue crack mechanism and crack growth law, and (b) other minor factors. A general prognostic framework is formulated by jointly employing the online change point detection, parameter estimation, and remaining useful life (RUL) prediction, which has good statistic inference and applicability in two general nonlinear systems, i.e., power-law and exponential-law. A joint implement of offline two-step parameter estimation method and the online Bayesian update method is executed, making full advantage of historical and in-service data, based on which the RUL prediction transcends into an imperative PHM module. A practical case study on the vibration dataset of wheel treads demonstrates the practically implement ability of the proposed method in achieving high accuracy of RUL prediction.

Oscillation behavior driven by processing delay in diffusively coupled inactive systems: Cluster synchronization and multistability.

Couplings involving time delay play a relevant role in the dynamical behavior of complex systems. In this work, we address the effect of processing delay, which is a specific kind of coupling delay, on the steady state of general nonlinear systems and prove that it may drive the system to Hopf bifurcation and, in turn, to a rich oscillatory behavior. Additionally, one may observe multistable states and size-dependent cluster synchronization. We derive the analytic conditions to obtain an oscillatory regime and confirm the result by numerically simulated experiments on different oscillator networks. Our results demonstrate the importance of processing delay for complex systems and pave the way for a better understanding of dynamical control and synchronization in oscillatory networks.

Robust [formula omitted] filtering for nonlinear discrete-time stochastic systems

This paper mainly discusses the H∞ filtering of general nonlinear discrete time-varying stochastic systems. A nonlinear discrete-time stochastic bounded real lemma is firstly obtained by means of the smoothness of the conditional mathematical expectation, and then, based on the given stochastic bounded real lemma and a stochastic LaSalle-type theorem, a sufficient condition for the existence of the H∞ filtering of general nonlinear discrete time-varying stochastic systems is presented via a newly introduced Hamilton–Jacobi inequality (HJI), which is easily verified. When the worst-case disturbance {vk∗}k∈N is considered, the suboptimal H2∕H∞ filtering is studied. Three examples including a practical engineering example show the effectiveness of our main results.

Approximate iterative solutions of Hamilton–Jacobi equations for nonlinear systems

In this paper, we present computational methods for approximate solution to the time-dependent first-order Hamilton–Jacobi equations (HJEs) arising in deterministic optimal control of nonlinear systems in state-space, as well as other areas of science and engineering. We establish convergence of the methods for optimal control problems and for the case of general nonlinear systems under fairly mild assumptions. Examples are then solved to demonstrate the effectiveness of the methods and the special case of affine nonlinear systems is also considered. Results are then extended to more general initial value problems of the HJE arising in other areas of science and engineering.

On the identifiability of interaction functions in systems of interacting particles

We address a fundamental issue in the nonparametric inference for systems of interacting particles: the identifiability of the interaction functions. We prove that the interaction functions are identifiable for a class of first-order stochastic systems, including linear systems with general initial laws and nonlinear systems with stationary distributions. We show that a coercivity condition is sufficient for identifiability and becomes necessary when the number of particles approaches infinity. The coercivity is equivalent to the strict positivity of related integral operators, which we prove by showing that their integral kernels are strictly positive definite by using Müntz type theorems.

Parallel control for optimal tracking via adaptive dynamic programming

This paper studies the problem of optimal parallel tracking control for continuous-time general nonlinear systems. Unlike existing optimal state feedback control, the control input of the optimal parallel control is introduced into the feedback system. However, due to the introduction of control input into the feedback system, the optimal state feedback control methods can not be applied directly. To address this problem, an augmented system and an augmented performance index function are proposed firstly. Thus, the general nonlinear system is transformed into an affine nonlinear system. The difference between the optimal parallel control and the optimal state feedback control is analyzed theoretically. It is proven that the optimal parallel control with the augmented performance index function can be seen as the suboptimal state feedback control with the traditional performance index function. Moreover, an adaptive dynamic programming ( ADP ) technique is utilized to implement the optimal parallel tracking control using a critic neural network ( NN ) to approximate the value function online. The stability analysis of the closed-loop system is performed using the Lyapunov theory, and the tracking error and NN weights errors are uniformly ultimately bounded ( UUB ) . Also, the optimal parallel controller guarantees the continuity of the control input under the circumstance that there are finite jump discontinuities in the reference signals. Finally, the effectiveness of the developed optimal parallel control method is verified in two cases.

Event‐triggered H∞ filtering for nonlinear discrete‐time stochastic systems with application to vehicle roll stability control

SummaryIn this article, the problem of event‐triggered H∞ filtering for general discrete‐time nonlinear stochastic systems is investigated. An event‐triggered mechanism is introduced to reduce the communication burden by judging whether the measured data should be transmitted. First, two results about the event‐triggered H∞ filtering for general discrete nonlinear stochastic systems are presented. Second, when the linear discrete‐time stochastic system is considered, the corresponding event‐triggered H∞ filtering can be designed via linear matrix equality approach. Finally, a numerical example and a practical example about the vehicle roll angle estimation are presented to verify the effectiveness of the proposed new design method.

Fuzzy Modeling Control and Simulation of a Class of Complex Dynamical Systems

In this paper, an equivalent fuzzy financial dynamic system model is established by using T-S fuzzy modeling method. According to Lyapunov stability theory, a fuzzy feedback controller is designed to realize chaos control. The results of theoretical proof and numerical simulation show that the fuzzy control scheme is effective. The fuzzy controller can simulate human’s experience knowledge and is simple and easy to implement. It can also be used to control general nonlinear chaotic systems.

Adaptive Actuator Failure Compensation Control Schemes for Uncertain Noncanonical Neural-Network Systems.

In this article, direct adaptive actuator failure compensation control is investigated for a class of noncanonical neural-network nonlinear systems whose relative degrees are implicit and parameters are unknown. Both the state tracking and output tracking control problems are considered, and their adaptive solutions are developed which have specific mechanisms to accommodate both actuator failures and parameter uncertainties to ensure the closed-loop system stability and asymptotic state or output tracking. The adaptive actuator failure compensation control schemes are derived for noncanonical nonlinear systems with neural-network approximation, and are also applicable to general parametrizable noncanonical nonlinear systems with both unknown actuator failures and unknown parameters, solving some key technical issues, in particular, dealing with the system zero dynamics under uncertain actuator failures. The effectiveness of the developed adaptive control schemes is confirmed by simulation results from an application example of speed control of dc motors.

The general magneto-static model and Maxwell-Stokes system with topological parameters

In this paper we study a nonlinear magneto-static model on a general domain in R3 which is multiply-connected and has holes, and under a nonlinear relation between the magnetic induction B and the magnetic field H. The equation of the model contains a Neumann field h1∈H1(Ω) and a Dirichlet field h2∈H2(Ω), which represent the effects of domain topology. In the case of a general electric current, the equation contains also an unknown gradient ∇p(x), which represents the electric field. Existence results of solutions of the boundary value problems of this model and of a more general nonlinear Maxwell-Stokes system with topological parameters h1 and h2 are proved, which exhibit the effects of domain topology on the electromagnetic fields and on the nonlinear systems involving curl.

Distributed Maximum Correntropy Filtering for Stochastic Nonlinear Systems Under Deception Attacks.

This article focuses on the distributed maximum correntropy filtering issue for general stochastic nonlinear systems subject to deception attacks. The considered nonlinear functions consist of a determined one and a stochastic one, and the stochastic signals sent by deception attacks with identified statistic characteristics could be non-Gaussian. The corresponding calculation formulas of both the filter gains and the upper bound of the filter error covariance are proposed by means of the Taylor series expansion and the fixed-point iterative update rule, where the weighted maximum correntropy criterion is utilized to take the place of traditional minimum covariance indexes. Such an upper bound is only dependent on the local information, neighbor information, and the identified statistics of deception attacks and, therefore, the developed filtering scheme realizes the requirement of distributed calculation. Furthermore, a simplified version is obtained by removing weights in the correntropy criterion. Finally, an illustrative example is given to verify the effectiveness of developed distributed maximum correntropy filtering subject to deception attacks.

A General Framework for Optimal Control of Fractional Nonlinear Delay Systems by Wavelets

An iterative procedure to find the optimal solutions of general fractional nonlinear delay systems with quadraticperformance indices is introduced. The derivatives of state equations are understood in the Caputo sense. By presenting and applying a general framework, we use the Chebyshev wavelet method developed for fractional linear optimal control to convert fractional nonlinear optimal control problems as a sequence of quadratic programming ones. The concepts and computational procedure that are used for fractional linear optimal control are applied on fractional nonlinear optimal control. Different types of nonlinear optimal control problems with fractional or integer order can be solved. To see this, some numerical examples are solved. Another operational property of Chebyshev wavelets is presented.