Wide Class Of Nonlinear Systems Research Articles

New Control Design via Switching Approach with Higher Order Derivatives of Lyapunov Function for Takagi-Sugeno Fuzzy Systems

This paper proposes a new control design method via two techniques to decrease the conservativeness in control design conditions for Takagi-Sugeno fuzzy systems that can represent a wide class of nonlinear systems. The first technique we employ is to use higher order derivatives of Lyapunov function that is shown to reduce the conservativeness in the literature. The second one is a switching approach that takes advantages of a property of the membership function. In this paper, we provide control design methods and their less conservative conditions based on the above-mentioned techniques. The proposed control design conditions are independent of the membership function and are written in a linear matrix inequality form. Finally, Numerical examples illustrate the effectiveness of the proposed control methods by comparing with the existing methods.

Design of Robust Model Reference Adaptive Controller for a Wider Class of Nonlinear Systems

Design of Robust Model Reference Adaptive Controller for a Wider Class of Nonlinear Systems

Functional observer design for nonlinear systems with incremental quadratic constraints

We present a functional observer design method for a class of nonlinear systems with output nonlinearities and incremental quadratic constraints. The system under consideration includes a wider class of nonlinear systems, such as, Lipschitz, one-sided Lipschitz, nondecreasing nonlinearity system as special cases. Based on Moore–Penrose pseudo-inverse and linear matrix inequality theory, the existence conditions of gain matrices of a functional observer are obtained. Moreover, exponential convergence of the estimate errors is achieved. Through the Lorenz system, it is shown that the proposed approach provides a smaller order of observer than that of the conventional approaches, while maintaining satisfactory performances.

Disturbance Rejection of Nonlinear Boiler–Turbine Unit Using High-Order Sliding Mode Observer

It is interesting and challenging to develop advanced controller for industrial boiler–turbine units due to their genuine nonlinearities, serious couplings among state variables, physical constraints imposed on control inputs, and in particular, various types of unknown uncertainties/disturbances. In this paper, a nonlinear disturbance rejection control method is investigated in a composite design manner for an oil-fired drum-type boiler–turbine unit. A baseline exponentially stable feedback controller is first designed, and then a high-order sliding mode observer is utilized to estimate and thus to compensate unknown lumped disturbances. It is shown that the obtained results for the boiler–turbine unit can be extended to a wide class of nonlinear systems. More interestingly, the finite-time stability of the closed-loop systems can be rendered in several cases. Finally, some numerical simulation scenarios are conducted on the boiler–turbine unit to demonstrate the claimed control performance.

Attractive Ellipsoid Sliding Mode Observer Design for State of Charge Estimation of Lithium-Ion Cells

This work investigates the real-time estimation of the state-of-charge (SoC) of Lithium-ion (Li-ion) cells for reliable, safe and efficient utilization. A novel attractive ellipsoid based sliding-mode observer (AESMO) algorithm is designed to estimate the SoC in real-time. The algorithm utilizes standard equivalent circuit model (ECM) of a Li-ion cell and provides reliable and efficient SoC estimate in the presence of bounded uncertainties in the battery parameters as well as exogenous disturbances. The theoretical framework of the observer design is not limited to the SoC estimation problem of Li-ion cell but applicable to a wider class of nonlinear systems with both matched and mismatched uncertainties. The main advantage of the proposed observer is to provide a fast and optimal SoC estimate based on minimization over the uncertainty bound. The proposed method is experimentally tested and evaluated for a range of temperatures using the hybrid pulse power characterization test (HPPC), EPA's Federal Test Procedure (FTP75) and Supplemental Test Procedure (US06) data, which demonstrate its effectiveness and feasibility.

A general model system related to affine stochastic differential equations

We link a general method for modeling random phenomena using systems of stochastic differential equations (SDEs) to the class of affine SDEs. This general construction emphasizes the central role of the Duffie–Kan system [Duffie and Kan, A yield-factor model of interest rates, Math. Finance 6 (1996) 379–406] as a model for first-order approximations of a wide class of nonlinear systems perturbed by noise. We also specialize to a two-dimensional framework and propose a direct proof of the Duffie–Kan theorem which does not passes through the comparison with an auxiliary process. Our proof produces a scheme to obtain an explicit representation of the solution once the one-dimensional square root process is assigned.

An analytical approach to optimal control of nonlinear systems with input constraints

In this paper, a novel optimal control method with the analytical solution strategy is developed for a wide class of nonlinear systems with input constraints. First, an equivalent constrained optimisation problem is formulated by performing a quadratic performance index including the control inputs and the predicted tracking errors. Then, the problem is analytically solved by using the Kerush-Kuhn-Tucker (KKT) theorem to find the optimal control law. For comparison, a computational technique based on calling the genetic algorithm (GA) is also provided to online solving the developed optimisation problem. The proposed control method with two solutions is applied on a mathematical example and a chemical reactor which is a multi-input multi-output (MIMO) system. The results show that the proposed KKT-based predictive controller is effective from different aspects. Importantly, it is very fast, easy to solve and suitable for online implementation compared with the conventional nonlinear model predictive control (NMPC) method.

An analytical approach to optimal control of nonlinear systems with input constraints

In this paper, a novel optimal control method with the analytical solution strategy is developed for a wide class of nonlinear systems with input constraints. First, an equivalent constrained optimisation problem is formulated by performing a quadratic performance index including the control inputs and the predicted tracking errors. Then, the problem is analytically solved by using the Kerush-Kuhn-Tucker (KKT) theorem to find the optimal control law. For comparison, a computational technique based on calling the genetic algorithm (GA) is also provided to online solving the developed optimisation problem. The proposed control method with two solutions is applied on a mathematical example and a chemical reactor which is a multi-input multi-output (MIMO) system. The results show that the proposed KKT-based predictive controller is effective from different aspects. Importantly, it is very fast, easy to solve and suitable for online implementation compared with the conventional nonlinear model predictive control (NMPC) method.

State‐limiting PID controller for a class of nonlinear systems with constant uncertainties

SummaryProportional Integral Derivative (PID) controllers still represent the core control method for achieving output regulation of either linear or nonlinear systems in the majority of industrial applications. However, conventional PID control cannot guarantee specific state constraint requirements for the plant, when the system introduces uncertainties. In this paper, a novel nonlinear PID control that achieves output regulation and guarantees a desired state limitation below a given value for a wide class of nonlinear systems with constant uncertainties is proposed. Using nonlinear ultimate boundedness theory, it is shown that the proposed state‐limiting PID (sl‐PID) control maintains a given bound for the desired system states at all times, ie, even during transients, whereas an analytic method for selecting the controller gains is also presented to ensure closed‐loop system stability and convergence at the desired equilibrium. Two nonlinear engineering examples that include an electric motor and a dc/dc converter are investigated using the conventional PID and the proposed sl‐PID to validate the superiority of the proposed controller in achieving the desired output regulation with a given bounded state requirement.

General framework for the analysis of imperfections in nonlinear systems.

In this Letter, we derive a framework to understand the effect of imperfections on the phase-matching spectrum of a wide class of nonlinear systems. We show that this framework is applicable to many physical systems, such as waveguides or fibers. Furthermore, this treatment reveals that the product of the system length and magnitude of the imperfections completely determines the phase-matching properties of these systems, thus offering a general rule for system design. Additionally, our framework provides a simple method to compare the performance of a wide range of nonlinear systems.

Iterative Learning of Optimal Control for Nonlinear Processes With Applications to Laser Additive Manufacturing

Our target application is a laser additive manufacturing (LAM) process that suffers from too wide end points. To overcome this drawback, we propose to extend an algorithm of iterative learning of optimal control (ILOC) to nonlinear systems. We demonstrate its application for a LAM process, not only by simulations but also by comparing the results of 3-D metallic printing with and without learning. Extending previously published results, we provide proof of convergence of an ILOC algorithm for a wide class of nonlinear systems.

An extension of a lemma of Dayawansa and its application in the design of extended observers for nonlinear systems

In this paper we show how, by extending a classical Lemma due to Dayawansa, it is possible to enhance the theory of robust extended observers so as to cover a wide class of nonlinear systems, namely those systems whose normal form is characterized by “time-dependent gains” in the “string of integrators between input and output”. Such class includes helicopters, quadrotors, robot arms and electric drives.

Neural network model for path-finding problems with the self-recovery property.

The large-scale synchronization of neural oscillations is crucial in the functional integration of brain modules, but the combination of modules changes depending on the task. A mathematical description of this flexibility is a key to elucidating the mechanism of such spontaneous neural activity. We present a model that finds the loop structure of a network whose nodes are connected by unidirectional links. Using this model, we propose a path-finding system that spontaneously finds a path connecting two specified nodes. The solution path is represented by phase-synchronized oscillatory solutions. The model has the self-recovery property: that is, it is a system with the ability to find a new path when one of the connections in the existing path is suddenly removed. We show that the model construction procedure is applicable to a wide class of nonlinear systems arising in chemical reactions and neural networks.

Learning an Approximate Model Predictive Controller With Guarantees

A supervised learning framework is proposed to approximate a model predictive controller (MPC) with reduced computational complexity and guarantees on stability and constraint satisfaction. The framework can be used for a wide class of nonlinear systems. Any standard supervised learning technique (e.g. neural networks) can be employed to approximate the MPC from samples. In order to obtain closed-loop guarantees for the learned MPC, a robust MPC design is combined with statistical learning bounds. The MPC design ensures robustness to inaccurate inputs within given bounds, and Hoeffding's Inequality is used to validate that the learned MPC satisfies these bounds with high confidence. The result is a closed-loop statistical guarantee on stability and constraint satisfaction for the learned MPC. The proposed learning-based MPC framework is illustrated on a nonlinear benchmark problem, for which we learn a neural network controller with guarantees.

Robust predictive synchronization of uncertain fractional-order time-delayed chaotic systems

In this paper, a novel robust predictive control strategy is proposed for the synchronization of fractional-order time-delay chaotic systems. A recurrent non-singleton type-2 fuzzy neural network (RNT2FNN) is used for the estimation of the unknown functions. Additionally, another RNT2FNN is used for the modeling of the tracking error. A nonlinear model-based predictive controller is then designed based on the proposed fuzzy model. The asymptotic stability of the approach is derived based on the Lyapunov stability theorem. A number of simulation examples are presented to verify the effectiveness of the proposed control method for the synchronization of two uncertain fractional-order time-delay identical and nonidentical chaotic systems. The proposed control strategy is also employed for high-performance position control of a hydraulic actuator. In this example, the nonlinear mechanical model of the hydraulic actuator, instead of a mathematical model, is simulated. The example demonstrates that the proposed control strategy can be applied to a wide class of nonlinear systems.

Polynomial fuzzy model‐based approach for underactuated surface vessels

The main goal of this study is to introduce a new polynomial fuzzy model-based structure for a class of marine systems with non-linear and polynomial dynamics. The suggested technique relies on a polynomial Takagi–Sugeno (T–S) fuzzy modelling, a polynomial dynamic parallel distributed compensation and a sum-of-squares (SOS) decomposition. The new proposed approach is a generalisation of the standard T–S fuzzy models and linear matrix inequality which indicated its effectiveness in decreasing the tracking time and increasing the efficiency of the robust tracking control problem for an underactuated surface vessel (USV). Additionally, in order to overcome the USV control challenges, including the USV un-modelled dynamics, complex nonlinear dynamics, external disturbances and parameter uncertainties, the polynomial fuzzy model representation is adopted. Moreover, the USV-based control structure is formulated in terms of SOS conditions. Unlike the conventional approaches, the proposed approach can be easily implemented in the real-time and applied to a reasonably wide class of non-linear systems. Finally, the extensive studies and hardware-in-the-loop simulations are presented, which prove the effectiveness of the novel proposed method.

The effects of linear and nonlinear characteristic parameters on the output frequency responses of nonlinear systems: The associated output frequency response function

In the present study, a new concept known as the Associated Output Frequency Response Function (AOFRF) is introduced to facilitate the analysis of the effects of both linear and nonlinear characteristic parameters on the output frequency responses of nonlinear systems. Based on the AOFRF concept, the study has shown, for the first time, that the output frequency responses of a wide class of nonlinear systems that are described by the NARX (Nonlinear Auto Regressive with eXogenous input) model can be represented by a polynomial function of both the system linear and nonlinear characteristic parameters of interests to the system analysis. Moreover, an efficient algorithm is derived to determine the structure and coefficients of the AOFRF based representation for system output frequency responses. Finally, a case study is provided to demonstrate the effectiveness and advantages of the new AOFRF based representation and the implication of the result to the analysis and design of nonlinear systems in the frequency domain.

Strong convergence of the partially truncated Euler–Maruyama method for a class of stochastic differential delay equations

This paper establishes the convergence of a class of highly nonlinear stochastic differential delay equations without the linear growth condition replacing by Khasminskii-type condition, so the convergence criteria here may cover a wider class of nonlinear systems. Our aim is to propose the partially truncated Euler–Maruyama method for stochastic differential delay equations d y ( t ) = f ( y ( t ) , y ( t − τ ) ) d t + g ( y ( t ) , y ( t − τ ) ) d w ( t ) and consider the strong- L q convergence for 2 ≤ q < p under the local Lipschitz condition plus Khasminskii-type condition, and p is a parameter in Khasminskii-type condition.

Application of adaptive sliding mode control for nonlinear systems with unknown polynomial bounded uncertainties

In this paper, we discuss the application of a novel switching integral-exponential-adaptation-law-based adaptive sliding mode control design for a wide class of nonlinear systems with unknown polynomial bounds on the uncertainty norm. A robust finite time convergence, i.e. finite stability, is obtained with low chatter on control actions and a fast-transient performance for adaptive sliding mode control handling the multi-input multi-output nonlinear systems with uncertainties of amplitudes bounded within unknown polynomials in the state vector norm. The exponential term of the proposed adaptation law targets the reduction of the chatter levels of the sliding mode by significantly reducing the gain overestimation while simultaneously suppressing the overshoot by speeding up the system response to the uncertainties. It also prevents the instability issues which encounters the classic integral-gain-law-based adaptive sliding mode control when underestimating its initial gain or gain rate parameter. A simple example illustrates the motivation and feasibility of the proposed adaptive sliding mode control. The applications on a nonlinear mass–spring system and on a two degree of freedom electromechanical rotative plant demonstrate the effectiveness of the proposed design.

Relaxing the conditions for parameter estimation-based observers of nonlinear systems via signal injection

Parameter estimation-based observers are a new kind of state reconstruction methods where the state observation task is translated into an on-line parameter estimation problem. A key step for its application is the transformation of the system dynamics into a particular cascade form, which involves the solution of a partial differential equation that, moreover, should satisfy some injective requirement. In this note we use a recently proposed technique of signal injection to generate new outputs and simplify these tasks. In this way, we make this observer applicable to a wider class of nonlinear systems—even with indistinguishable states. The application of the proposed approach is illustrated with the design of a novel sensorless controller for magnetic levitation systems.