Uncertain Nonlinear Dynamical Systems Research Articles

Observer-based controller for constrained uncertain stochastic nonlinear discrete-time systems

Summary In this paper, an observer-based control approach is proposed for uncertain stochastic nonlinear discrete-time systems with input constraints. The widely used extended Kalman filter (EKF) is well known to be inadequate for estimating the states of uncertain nonlinear dynamical systems with strong nonlinearities especially if the time horizon of the estimation process is relatively long. Instead, a modified version of the EKF with improved stability and robustness is proposed for estimating the states of such systems. A constrained observer-based controller is then developed using the state-dependent Riccati equation approach. Rigorous analysis of the stability of the developed stochastically controlled system is presented. The developed approach is applied to control the performance of a synchronous generator connected to an infinite bus and chaos in permanent magnet synchronous motor. Simulation results of the synchronous generator show that the estimated states resulting from the proposed estimator are stable, whereas those resulting from the EKF diverge. Moreover, satisfactory performance is achieved by applying the developed observer-based control strategy on the two practical problems. Copyright © 2015 John Wiley & Sons, Ltd.

Design Second Order Vibration Reduction

Design of a robust controller for multi input-multi output (MIMO) nonlinear uncertain dynamical system can be a challenging work. This paper focuses on the design and analysis of a high performance adaptive baseline sliding mode control for second order nonlinear uncertain system, in presence of uncertainties to reduce the vibration. In this research, sliding mode controller is a robust and stable nonlinear controller which selected to control of robot manipulator. The proposed approach effectively combines of design methods from switching sliding mode controller, adaptive model-free baseline controller and linear Proportional-Derivative (PD) control to improve the performance, stability and robustness of the sliding mode controller. Sliding mode controller has two important subparts, switching and equivalent. Switching part (discontinuous part) is very important in uncertain condition but it causes chattering phenomenon. To solve the chattering, the most common method used is linear boundary layer saturation method, but this method lost the stability. To reduce the chattering with respect to stability and robustness; linear controller is added to the switching part of the sliding mode controller. The linear controller is to reduce the role of sliding surface slope and switching (sign) function. The nonlinearity term of the sliding mode controller is used to eliminate the decoupling and nonlinear term of link’s dynamic parameters. However nonlinearity term of sliding mode controller is very essential to reliability but in uncertain condition or highly nonlinear dynamic systems it can cause some problems. To solve this challenge the baseline controller is used as online tune or adaptive controller. This controller improves the stability and robustness, reduces the chattering as well and reduces the level of energy due to the torque performance as well.

A Computational Method for Robust Bifurcation Analysis and Its Application to Biomolecular Systems

We consider a general uncertain nonlinear dynamical system defined in a certain model set, and reformulate a problem of robustness bifurcation analysis (RBA), which has been originally formulated in our previous work. As such, we develop an efficient computational method for the RBA, which can be used for quantitative evaluation of bifurcation robustness in uncertain dynamical systems. Specifically, we first linearize the uncertain system properly and then apply a feedback transformation technique to reduce the RBA problem to a linear robustness analysis one, which can be solved using μ-analysis, a common analysis technique in robust control theory. Finally, we provide robustness analysis of a gene regulatory network model where oscillatory behavior appears according to Hopf bifurcation. We give quantitative evaluation of the bifurcation robustness using the RBA method proposed here.

Intelligent switching adaptive control for uncertain nonlinear dynamical systems

In this paper, we aim at proposing a switching adaptive control scheme using a Hopfield-based dynamic neural network (SACHNN) for nonlinear systems with external disturbances. In our proposed scheme, an auxiliary direct adaptive controller (DAC) ensures the system stability when the indirect adaptive controller (IAC) is failed; that is, gˆ(x) approaches to zero, where gˆ(x) is the denominator of an indirect adaptive control law. The IAC's limitation of gˆ(x)>ε then can be solved by simply switching the IAC to the DAC, where ɛ is a positive desired value. The Hopfield dynamic neural network (HDNN) is used to not only design DAC but also approximate the unknown plant nonlinearities in IAC design. The designed simple structure of HDNN keeps the tracking performance well and also makes the practical implementation much easier because of the use of less and fixed number of neurons.

Robust adaptive control scheme for uncertain non‐linear model reference adaptive control systems with time‐varying delays

The problem of model reference adaptive control is considered for a class of uncertain dynamical systems with non-linear delayed state perturbations. It is assumed that the time-varying delays are any non-negative continuous and bounded functions. In particular, it is only assumed that the non-linear delayed state perturbations are bounded in any non-negative non-linear functions which are not required to be known for the system designer. A new method is presented whereby a class of continuous memoryless robust adaptive controllers with a rather simple structure can be constructed. It is also shown that the proposed robust adaptive control schemes can guarantee that the current state of uncertain non-linear time-delay dynamical systems can track the state of a given delay-free reference model as far as possible. Finally, the simulations of a numerical example are provided to demonstrate the validity of the theoretical results.

Response and reliability analysis of nonlinear uncertain dynamical structures by the probability density evolution method

The paper deals with the response and reliability analysis of hysteretic or geometric nonlinear uncertain dynamical systems of arbitrary dimensionality driven by stochastic processes. The approach is based on the probability density evolution method proposed by Li and Chen (Stochastic dynamics of structures, 1st edn. Wiley, London, 2009; Probab Eng Mech 20(1):33–44, 2005), which circumvents the dimensional curse of traditional methods for the determination of non-stationary probability densities based on Markov process assumptions and the numerical solution of the related Fokker–Planck and Kolmogorov–Feller equations. The main obstacle of the method is that a multi-dimensional convolution integral needs to be carried out over the sample space of a set of basic random variables, for which reason the number of these need to be relatively low. In order to handle this problem an approach is suggested, which reduces the number of basic random variables to merely a single one. Correspondingly, the response and reliability problems reduce to the solution of one-dimensional quadratures.

Design Robust Artificial Intelligence Model-base Variable Structure Controller with Application to Dynamic Uncertainties OCTAM VI Continuum Robot

Design a robust artificial intelligent nonlinear controller for second order nonlinear uncertain dynamical systems is one of the most important challenging works. This paper focuses on the design of a robust chattering free mathematical model-base artificial intelligence (fuzzy inference system) variable structure controller (MFVSC) for highly nonlinear dynamic continuum robot manipulator, in presence of uncertainties. In order to provide high performance nonlinear methodology, variable structure controller is selected. Pure variable structure controller can be used to control of partly known nonlinear dynamic parameters of continuum robot manipulator. In order to reduce/eliminate the chattering, this research is used the artificial intelligence (fuzzy logic) theory. The results demonstrate that the model base fuzzy variable structure controller with switching function is a model-based controllers which works well in certain and partly uncertain system. Lyapunov stability is proved in mathematical modelbased fuzzy variable structure controller with switching (sign) function. This controller has acceptable performance in presence of uncertainty (e.g., overshoot=1%, rise time=0.9 second, steady state error = 1.6e-8 and RMS error=4.8e-8).

A constrained extremum‐seeking control approach

SummaryThis paper proposes an extremum‐seeking control approach for a class of uncertain nonlinear dynamical systems subject to constraints. It is assumed that the objective function along with the constraints are available for measurement. The contribution of the paper is to formulate the extremum‐seeking problem as a time‐varying estimation problem. The constraints are enforced by using an interior point strategy that preserves feasibility of the systems' trajectories. Simulation examples are used to illustrate the effectiveness of the proposed technique. Copyright © 2014 John Wiley & Sons, Ltd.

A CSP Versus a Zonotope-Based Method for Solving Guard Set Intersection in Nonlinear Hybrid Reachability

Computing the reachable set of hybrid dynamical systems in a reliable and verified way is an important step when addressing verification or synthesis tasks. This issue is still challenging for uncertain nonlinear hybrid dynamical systems. We show in this paper how to combine a method for computing continuous transitions via interval Taylor methods and a method for computing the geometrical intersection of a flowpipe with guard sets, to build an interval method for reachability computation that can be used with truly nonlinear hybrid systems. Our method for flowpipe guard set intersection has two variants. The first one relies on interval constraint propagation for solving a constraint satisfaction problem and applies in the general case. The second one computes the intersection of a zonotope and a hyperplane and applies only when the guard sets are linear. The performance of our method is illustrated on examples involving typical hybrid systems.

Adaptive Fuzzy Hierarchical Sliding-Mode Control for the Trajectory Tracking of Uncertain Underactuated Nonlinear Dynamic Systems

The trajectory tracking of uncertain underactuated nonlinear dynamic systems is tackled by an adaptive fuzzy hierarchical sliding-mode control (AFHSMC). First, one of the subsystems is assigned as the first layer sliding surface. Next, a second layer sliding surface from the first layer sliding surface and the sliding surface of another subsystem is constructed. In this paper, the nth layer is supposed to be the top layer (or hierarchical layer) for including the sliding surfaces of all subsystems. Because two nonlinear system functions and the time-varying external disturbance of each subsystem are supposed to be unknown, different online fuzzy models are employed to approximate these nonlinear system functions and the upper bounded functions of external disturbances. Moreover, the upper bound of uncertainties caused by these fuzzy modeling errors is estimated online. Based on these learning fuzzy models and the estimated upper bound of these modeling errors, an AFHSMC is developed. The stability analysis and tracking performance of the closed-loop system are verified by Lyapunov stability theory. Finally, two simulation examples including different amplitudes of external disturbance and comparison with hierarchical sliding-mode control confirm the effectiveness and robustness of the proposed control.

Adaptive Control for Modified Projective Synchronization-Based Approach for Estimating All Parameters of a Class of Uncertain Systems: Case of Modified Colpitts Oscillators

A method of estimation of all parameters of a class of nonlinear uncertain dynamical systems is considered, based on the modified projective synchronization (MPS). The case of modified Colpitts oscillators is investigated. Through a suitable transformation of the dynamical system, sufficient conditions for achieving synchronization are derived based on Lyapunov stability theory. Global stability and asymptotic robust synchronization of the considered system are investigated. The proposed approach offers a systematic design procedure for robust adaptive synchronization of a large class of chaotic systems. The combined effect of both an additive white Gaussian noise (AWGN) and an artificial perturbation is numerically investigated. Results of numerical simulations confirm the effectiveness of the proposed control strategy.

Design of an adaptive fuzzy sliding mode control for uncertain discrete-time nonlinear systems based on noisy measurements

This paper presents the design of an adaptive fuzzy sliding mode control (AFSMC) for uncertain discrete-time nonlinear dynamic systems. The dynamic systems are described by a discrete-time state equation with nonlinear uncertainties, and the uncertainties include the modelling errors and the external disturbances to be unknown but nonlinear with the bounded properties. The states are measured by the restriction of measurement sensors and the contamination with independent measurement noises. The nonlinear uncertainties are approximated by using the fuzzy IF-THEN rules based on the universal approximation theorem, and the approximation error is compensated by adding an adaptive complementary term to the proposed AFSMC. The fuzzy inference approach based on the extended single input rule modules is proposed to reduce the number of the fuzzy IF-THEN rules. The estimates for the un-measurable states and the adjustable parameters are obtained by using the weighted least squares estimator and its simplified one. It is proved that under some conditions the estimation errors will remain in the vicinity of zero as time increases, and the states are ultimately bounded subject to the proposed AFSMC. The effectiveness of the proposed method is indicated through the simulation experiment of a simple numerical system.

Robust PID TS fuzzy control methodology based on gain and phase margins specifications

This paper proposes the analysis and design of TS fuzzy robust PID control based on gain and phase margins specifications for uncertain nonlinear dynamic systems with time delay. The input-output d...

Robust PID TS fuzzy control methodology based on gain and phase margins specifications

This paper proposes the analysis and design of TS fuzzy robust PID control based on gain and phase margins specifications for uncertain nonlinear dynamic systems with time delay. The input-output data set of the uncertain nonlinear time delay dynamic system is decomposed into several input-output spaces by Gustafson-Kessel GK clustering algorithm, which are used to compute several linear submodels by least squares algorithm, and grouped in a Takagi-Sugeno TS fuzzy inference system. From the gain and phase margins specifications, in the frequency domain, analytical formulas are derived for fuzzy model based robust PID control design via Paralel and Distributed Compensation PDC strategy, considering the influence of the time delay. The main contribution of the paper is the proposal of two theorems related to necessary and sufficient conditions for fuzzy robust control design. The first theorem shows that the robust PID controller in the i-th rule of the fuzzy robust PID controller guarantees the gain and phase margins specifications for the corresponding linear model. The second theorem shows that the robust PID controller in the i-th rule of the fuzzy robust PID controller guarantees the stability for all linear models. A simulation example illustrates the efficiency of the fuzzy controller for control of a single link robotic manipulator when compared to others control methods. The experimental results for real-time fuzzy robust PID control of a termic process are obtained to demonstrate the effectiveness and practical viability of the proposed strategy.

Model-free controller with an observer applied in real-time to a 3-DOF helicopter

In this paper, a model-free controller with an observer is presented for a class of uncertain continuous-time multiinput-multioutput nonlinear dynamic systems. The proposed model-free control law consists of 2 parts: the first part is a linear control term used to specify the dynamics of the closed loop system, and the second part is a compensator of the effects of uncertainties and external disturbances. The compensator is synthesized from an estimator of the effects of uncertainties and disturbances based on the Lyapunov approach. In order to estimate the unavailable states of the controlled system, a linear state observer is designed. All of the signals in the closed-loop system are proved to be uniformly ultimately bounded using the Lyapunov stability theory. The effectiveness and feasibility of the proposed control strategy are examined in a real-time application for a helicopter with 3 degrees of freedom.

Variable Torque Control of Offshore Wind Turbine on Spar Floating Platform Using Advanced RBF Neural Network

Offshore floating wind turbine (OFWT) has been a challenging research spot because of the high-quality wind power and complex load environment. This paper focuses on the research of variable torque control of offshore wind turbine on Spar floating platform. The control objective in below-rated wind speed region is to optimize the output power by tracking the optimal tip-speed ratio and ideal power curve. Aiming at the external disturbances and nonlinear uncertain dynamic systems of OFWT because of the proximity to load centers and strong wave coupling, this paper proposes an advanced radial basis function (RBF) neural network approach for torque control of OFWT system at speeds lower than rated wind speed. The robust RBF neural network weight adaptive rules are acquired based on the Lyapunov stability analysis. The proposed control approach is tested and compared with the NREL baseline controller using the “NREL offshore 5 MW wind turbine” model mounted on a Spar floating platform run on FAST and Matlab/Simulink, operating in the below-rated wind speed condition. The simulation results show a better performance in tracking the optimal output power curve, therefore, completing the maximum wind energy utilization.

Design and experimentation of a self-tuning PID control applied to the 3DOF helicopter

Abstract The paper presents design and experimental validation of a stable self-tuning PID controller for three degrees of freedom (3-DOF) helicopter. At first, it is proposed a self-tuned proportional-integral-derivative (PID) controller for a class of uncertain second order multiinput multi-output nonlinear dynamic systems to which the 3-DOF helicopter dynamic model belongs. Within this scheme, the PID controller is employed to approximate unknown ideal controller that can achieve control objectives. PID controller gains are the adjustable parameters and they are updated online with a stable adaptation mechanism designed to minimize the error between the unknown ideal controller and the used by PID controller. The stability analysis of the closed-loop system is performed using Lyapunov approach. It is proven that all signals in the closed-loop system are uniformly ultimately bounded. The proposed approach can be regarded as a simple and effective model-free control since the mathematical model of the system is assumed unknown. Experimental results are presented to verify the effectiveness of the proposed controller.

Design Artificial Intelligence-Based Switching PD plus Gravity for Highly Nonlinear Second Order System

Refer to this research, an intelligent fuzzy parallel switching Proportional-Derivative (PD) p lus gravity controller is proposed for highly nonlinear continuum robot manipulator. Design a nonlinear controller for second order nonlinear uncertain dynamical systems is one of the most impo rtant challenging works. In order to provide high performance in nonlinear systems, switching partly slid ing mode plus gravity controller is selected. Pure switching partly slid ing mode p lus gravity controller can be used to control of partly known nonlinear dynamic parameters of continuum robot manipulator. Conversely, this method is used in many applications; it must to solve chattering phenomenon which it can cause some problems such as saturation and heat the mechanical parts of continuum robot man ipulators or drivers. In order to solve the chattering phenomenon, implement easily and avoid mathemat ical model base controller, Mamdani’s performance/erro rbased fuzzy logic methodology with two inputs and one output and 49 rules is parallel applied to pure switching partly sliding mode plus gravity controller. The results demonstrate that this method is a model-free controllers which works well in certain and partly uncertain system.

Evaluation Performance of IC Engine: Linear Tunable Gain Computed Torque Controller vs. Sliding Mode Controller

Design a nonlinear controller for second order nonlinear uncertain dynamical systems (e.g., internal combustion engine) is one of the most important challenging works. This paper focuses on the comparative study between two important nonlinear controllers namely; computed torque controller (CTC) and sliding mode controller (SMC) and applied to internal combustion (IC) engine in presence of uncertainties. In order to provide high performance nonlinear methodology, sliding mode controller and computed torque controller are selected. Pure SMC and CTC can be used to control of partly known nonlinear dynamic parameters of IC engine. Pure sliding mode controller and computed torque controller have difficulty in handling unstructured model uncertainties. To solve this problem applied linear error-based tuning method to sliding mode controller and computed torque controller for adjusting the sliding surface gain ( ) and linear inner loop gain ( ). Since the sliding surface gain ( ) and linear inner loop gain ( ) are adjusted by linear error-based tuning method. In this research new and new are obtained by the previous and multiple gains updating factor . The results demonstrate that the error-based linear SMC and CTC are model-based controllers which works well in certain and uncertain system. These controllers have acceptable performance in presence of uncertainty.

Distributed fault diagnosis for continuous-time nonlinear systems: The input–output case

In this paper, some new results on distributed fault diagnosis of continuous-time nonlinear systems with partial state measurements are proposed. By exploiting an overlapping decomposition framework, the dynamics of a nonlinear uncertain large-scale dynamical system is described as the interconnections of several subsystems. Each subsystem is monitored by a Local Fault Diagnoser: a set of local estimators, based on the nominal local dynamic model and on an adaptive approximation of the interconnection and of the fault function, allows to derive a local fault decision. A consensus-based protocol is used in order to improve the detectability and the isolability of faults affecting variables shared among different subsystems because of the overlapping decomposition. A sufficient condition ensuring the convergence of the estimation errors is derived. Finally, possibly non-conservative time-varying threshold functions guaranteeing no false-positive alarms and theoretical results dealing with detectability and isolability sufficient conditions are presented.