Error Dynamic System Research Articles

Dissipativity-Based Filtering for Fuzzy Switched Systems With Stochastic Perturbation

In this technical note, the problem of the dissipativity-based filtering problem is considered for a class of T-S fuzzy switched systems with stochastic perturbation. Firstly, a sufficient condition of strict dissipativity performance is given to guarantee the mean-square exponential stability for the concerned T-S fuzzy switched system. Then, our attention is focused on the design of a filter to the T-S fuzzy switched system with Brownian motion. By combining the average dwell time technique with the piecewise Lyapunov function technique, the desired fuzzy filters are designed that guarantee the filter error dynamic system to be mean-square exponential stable with a strictly dissipative performance, and the corresponding solvability condition for the fuzzy filter is also presented based on the linearization procedure approach. Finally, an example is provided to illustrate the effectiveness of the proposed dissipativity-based filter technique.

Data‐driven optimal tracking control for a class of affine non‐linear continuous‐time systems with completely unknown dynamics

In this study, the optimal tracking control problem (OTCP) for affine non-linear continuous-time systems with completely unknown dynamics is addressed based on data by introducing the reinforcement learning (RL) technique. Unlike existing methods to the OTCP, the proposed data-driven policy iteration (PI) method does not need to have or identify any knowledge of the system dynamics, including both drift dynamics and input dynamics. To carry out the proposed method, the original OTCP is pre-processed to construct an augmented system composed of the error system dynamics and the desired trajectory dynamics. Then, based on the augmented system, a data-driven PI, which introduces discount factor to solve the OTCP, is implemented on an actor–critic neural network (NN) structure by only using system data rather than the exact knowledge of system dynamics. Two NNs are used in the structure to generate the optimal cost and optimal control policy, respectively, and the weights are updated by a least-square approach which minimises the residual errors. The proposed method is an off-policy RL method, where the data can be arbitrarily sampled on the state and input domain. Finally, simulation results are provided to show the effectiveness of the proposed method.

State estimation based on fractional order sliding mode observer method for a class of uncertain fractional-order nonlinear systems

State estimation of dynamic systems is quite significant in many research areas, such as state-based control and stabilization, state-based monitoring and fault detection. This paper is concerned with the problem of observer-based state estimation for a class of fractional-order (FO) nonlinear dynamic systems. The objective is to consider the state estimation of FO nonlinear system with the sliding mode control (SMC) technique, and conduct the asymptotic stability analysis for the estimate error dynamic system. Firstly, the considered FO nonlinear system models with constant and uncertain parameters are both presented. Fractional order sliding mode observer (FOSMO) structures are established for the FO nonlinear system models. Then, the asymptotic stability of the estimate error dynamic systems are analyzed via employing the Lyapunov stability analysis method for FO systems, and the sufficient conditions of asymptotic stability are derived. FOSMOs design for FO nonlinear systems of Caputo׳s and Riemann–Liouville׳s differential operators are both investigated, and the corresponding asymptotic stability sufficient conditions of the error dynamic systems are presented to insure the estimation accuracy of FOSMOs. Finally, multiple simulation examples are provided to demonstrate the effectiveness of the presented FOSMOs.

Incipient Fault Detection Using an Associated Adaptive and Sliding-Mode Observer for Quadrotor Helicopter Attitude Control Systems

An associated adaptive and sliding-mode observer (AASMO) design is proposed to detect and estimate the incipient actuator faults of a quadrotor. The incipient faults considered are physical structure aging and quadrotor leakage. First, disturbances and nonlinear parameters are considered in system formulation for a realistic mathematical model of the quadrotor. Its fault model is also introduced. Second, the decomposed subsystems are obtained through coordinate transformations to separate the incipient faults from the disturbances. For the subsystem with no disturbance, the adaptive observer can estimate the incipient faults. For the subsystem with disturbances, the sliding-mode observer has strong robustness against the disturbances. Dynamic error convergence and system stability can also be guaranteed by Lyapunov stability theory. Finally, the simulation results of quadrotor helicopter attitude systems validate the efficiency of the proposed AASMO-based incipient fault detection algorithm.

Fault estimation and adaptive fault tolerant control for dynamic systems based on the second-order sliding mode observer

This article is concerned with observer-based fault estimation and adaptive fault tolerant control for a class of continuous-time dynamical systems. First, a second-order sliding mode observer is proposed, which is capable to eliminate chattering effects for fault estimation caused by the conventional sliding mode observer. Second, in order to avoid the cumbersome stability proving process by the geometric method, a Lyapunov function is adopted to prove the observer error dynamical system. Third, based on the estimated fault information by the proposed observer, an adaptive fault tolerant control law is developed to guarantee the stability of the faulty system. Finally, the proposed method is applied to the Boeing 747-100/200 flight control system, and the simulation results verify the effectiveness of the proposed scheme.

Full‐Order and Reduced‐order Observer Design for a Class of Fractional‐order Nonlinear Systems

AbstractThe paper is concerned with problem of the full‐order and reduced‐order observer design for a class of fractional‐order one‐sided Lipschitz nonlinear systems. By introducing a continuous frequency distributed equivalent model and using indirect Lyapunov approach, the sufficient condition for asymptotic stability of the full‐order observer error dynamic system is presented. Furthermore, the proposed design method was extended to reduced‐order observer design for fractional‐order nonlinear systems. All the stability conditions are obtained in terms of LMI, which are less conservative than some existing ones. Finally, a numerical example demonstrates the validity of this approach.

Robust Fault Estimation for a Class of T-S Fuzzy Singular Systems with Time-Varying Delay via Improved Delay Partitioning Approach

The problem of delay-dependent robust fault estimation for a class of Takagi-Sugeno (T-S) fuzzy singular systems is investigated. By decomposing the delay interval into two unequal subintervals and with a new and tighter integral inequality transformation, an improved delay-dependent stability criterion is given in terms of linear matrix inequalities (LMIs) to guarantee that the fuzzy singular system with time-varying delay is regular, impulse-free, and stable firstly. Then, based on this criterion, by considering the system fault as an auxiliary disturbance vector and constructing an appropriate fuzzy augmented system, a fault estimation observer is designed to ensure that the error dynamic system is regular, impulse-free, and robustly stable with a prescribedH∞performance satisfied for all actuator and sensor faults simultaneously, and the obtained fault estimates can practically better depict the size and shape of the faults. Finally, numerical examples are given to show the effectiveness of the proposed approach.

Development of a Fractional Order Chaos Synchronization Dynamic Error Detector for Maximum Power Point Tracking of Photovoltaic Power Systems

In recent years, the photovoltaic (PV) power generation system has been widely discussed and researched. Research on electric energy focuses on the development of Maximum Power Point Tracking (MPPT) technology, and many methods have been proposed. However, these studies have a common defect: the tracking continues near the maximum power point (MPP), so that the waveform of output power jitters, thus causing power loss and rapid wearing of electronic modules. In order to remedy this defect, this paper proposes a new type of fractional order chaos synchronization dynamic error detector for the MPPT design of a PV power system. In this study, the Sprott chaos synchronization dynamic error system was used to control the pulse width duty cycle of PWM and optimize the power oscillation of a PV power system during steady-state response. The simulation and experimental results showed that the voltage detector proposed in this paper can reduce the power oscillation of a PV power system during steady-state response, and increase the overall system efficiency. From the steady-state responses of MPPT, it can be seen that about 0.2 vibration amplitude can be suppressed with control action. Therefore, about 4% of steady-state vibration energy can be saved.

Fault diagnosis and fault tolerant tracking control for the non-Gaussian singular time-delayed stochastic distribution system with PDF approximation error

In this paper, fault diagnosis and fault tolerant tracking control algorithms are proposed for the non-Gaussian singular time-delayed stochastic distribution control system. The square root B-spline model is used to approximate the output probability density function (PDF), and the PDF approximation error is taken into consideration. A fault diagnosis approach based on the adaptive observer is constructed to diagnose the size of fault in the singular stochastic distribution control (SDC) system. When fault occurs, in order to track the expected PDF, a distribution tracking error dynamic system is established. The purpose of fault tolerant tracking control is transformed into making the distribution tracking error at each time instant satisfy a certain upper bound beyond a limited time. Then the fault tolerant controller is designed using the fault diagnosis information and other measurable information. A simulation example is included to illustrate the effectiveness of the proposed algorithms and encouraging results have been obtained.

Adaptive estimator design for unstable output error systems: A test problem and traditional system identification based analysis

A key open question in adaptive estimator design is how to assure that the parameters of the proposed algorithms are converging to their almost correct solutions; hence, the learning algorithm is unbiased. Moreover, determining the speed of parameter convergence is important as it provides insight about the performance of the learning algorithms. The main contributions of the article are fourfold: the first one is that the article, initially, introduces an adaptive estimator to learn the discounted Q-function and approximate optimal control policy without requiring linear, discrete time, unstable output error system dynamics, but using only the noisy system measurements. The simulation results show that the adaptive estimator minimizes the stochastic cost function and temporal difference error and also learns the approximate Q-function together with the control policy. The second one is consideration of a different approach by taking a simple test problem to investigate issues associated with the Q-function’s representation and parametric convergence. In particular, the terminal convergence problem is analyzed with a known optimal control policy where the aim is to accurately learn only the Q-function. It is parameterized by terms which are functions of the unknown plant’s parameters and the Q-function’s discount factor, and their convergence properties are analyzed and compared with the adaptive estimator. The third one is to show that even though the adaptive estimator with a large Q-function discount factor yields larger control feedback gains, so that faster state converges upright, the learning problem is badly conditioned; hence, the parameter convergence is sluggish, as the Q-function discount factor approaches the inverse of the dominant pole of the unstable system. Finally, the fourth one is comparison of the state output learned by the adaptive estimator with the ones obtained from traditional system identification algorithms. Simulation result for a higher order unstable output error system shows that the adaptive estimator closely follows the real system output whereas the system identification algorithms do not.

An Enhanced Error Model for EKF-Based Tightly-Coupled Integration of GPS and Land Vehicle's Motion Sensors.

Reduced inertial sensor systems (RISS) have been introduced by many researchers as a low-cost, low-complexity sensor assembly that can be integrated with GPS to provide a robust integrated navigation system for land vehicles. In earlier works, the developed error models were simplified based on the assumption that the vehicle is mostly moving on a flat horizontal plane. Another limitation is the simplified estimation of the horizontal tilt angles, which is based on simple averaging of the accelerometers’ measurements without modelling their errors or tilt angle errors. In this paper, a new error model is developed for RISS that accounts for the effect of tilt angle errors and the accelerometer’s errors. Additionally, it also includes important terms in the system dynamic error model, which were ignored during the linearization process in earlier works. An augmented extended Kalman filter (EKF) is designed to incorporate tilt angle errors and transversal accelerometer errors. The new error model and the augmented EKF design are developed in a tightly-coupled RISS/GPS integrated navigation system. The proposed system was tested on real trajectories’ data under degraded GPS environments, and the results were compared to earlier works on RISS/GPS systems. The findings demonstrated that the proposed enhanced system introduced significant improvements in navigational performance.

Hybrid synchronization behavior in an array of coupled chaotic systems with ring connection

In this paper, we investigate the hybrid synchronization behavior in an array of coupled chaotic systems with ring connection, of which means complete synchronization (CS) and anti-synchronization (AS) could coexist. First, the anti-synchronization controllers are designed, which can transform the synchronization error dynamic system into a nonlinear system with an antisymmetric structure. Second, we investigate the complete synchronization behavior in such a chaotic system under the anti-synchronization control. After that, the stability conditions are given for reaching hybrid synchronization. Finally, numerical examples and simulation results are presented to verify and validate the hybrid synchronization behavior in coupled chaotic system.

Synchronization of fuzzy bidirectional associative memory neural networks with various time delays

The problem of synchronization for fuzzy bidirectional associative memory (BAM) neural networks (NNs) with various time delays is formulated and investigated. The various delays consist of discrete delays, unbounded distributed delays and constant delay in the leakage term (i.e. ‘leakage delay’) are considered. Then, some sufficient conditions are presented to guarantee the global asymptotic stability of the error dynamical system by using Lyapunov–Krasovskii functional and linear matrix inequality (LMI) method. As a result, we achieved synchronization of master-slave (drive-response) fuzzy BAM NNs systems. Even, if there is no leakage delay, the obtained results are less restrictive to some known sufficient conditions. Moreover, the proposed results do not require the boundedness, differentiability and monotonicity of the activation functions, which can also be easily checked via the LMI solver in robust control toolbox in Matlab. Furthermore, using a parameter-dependent Lyapunov function approach the synchronization problem for polytopic uncertain fuzzy BAM NNs with various time delays is considered. The parameter uncertainties under consideration are assumed to belong to a fixed convex polytope. Finally, two numerical examples and simulations are given to illustrate the effectiveness of the derived theoretical results.

Pre-specified performance based model reduction for time-varying delay systems in fuzzy framework

This paper attempts to provide a new solution to the model approximation problem for dynamic systems with time-varying delays under the fuzzy framework. For a given high-order system, our focus is on the construction of a reduced-order model, which approximates the original one in a prescribed error performance level and guarantees the asymptotic stability of the corresponding error system. Based on the reciprocally convex technique, a less conservative stability condition is established for the dynamic error system with a given error performance index. Furthermore, the reduced-order model is eventually obtained by applying the projection approach, which converts the model approximation into a sequential minimization problem subject to linear matrix inequality constraints by employing the cone complementary linearization algorithm. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.

Global stable tracking control of underactuated ships with input saturation

In this paper, tracking control of underactuated ship in the presence of input saturation is addressed. By dividing the tracking error dynamic system into a cascade of two subsystems, the torques in surge and yaw axes are designed separately using the backstepping technique. More specifically, we design the yaw axis torque in such a way that its corresponding subsystem is finite time stable, which makes it to be de-coupled from the second subsystem after a finite time. This enables us to design the torque in the surge axis independently. It is shown that the closed-loop system is stable and the mean-square tracking errors can be made arbitrarily small by choosing design parameters. Simulation results also verify the effectiveness of the proposed scheme.

Adaptive synchronization of Cohen–Grossberg neural network with mixed time-varying delays and stochastic perturbation

In this paper, based on the LaSalle invariant principle of stochastic differential delay equations and the stochastic analysis theory as well as the adaptive feedback technique, several sufficient conditions ensuring the adaptive synchronization of Cohen–Grossberg neural network with mixed time-varying delays and stochastic perturbation are derived. In particular, the synchronization criterion considered globally is the almost surely asymptotic stability of the error dynamical system. Our synchronization criterion is easily verified and does not solve any linear matrix inequality. These results generalized a few previous known results. At last, a numerical example and its simulations are provided to demonstrate the effectiveness and advantage of the theoretical results.

A novel method for fault detection in singularly perturbed systems via the finite frequency strategy

This paper addresses the fault detection synthesis for a class of linear singularly perturbed systems with unknown disturbances. The observer is designed via a novel method, in which singularly perturbed systems could be processed within a uniform framework instead of using the classical slow–fast decomposition. The problem of robust fault detection can be converted into a standard H∞ model-matching. Based on the generalized Kalman–Yakubovich–Popov lemma and parameter-dependent Lyapunov functions, a full-order observer is designed such that the corresponding error dynamic system is asymptotically stable and satisfies a prescribed finite-frequency H_/H∞ performance index. A novel three-step design procedure is then proposed to extract the fault feature from strong background disturbances. An illustrative example is given to demonstrate the validity and applicability of the proposed approaches in the simulation part.

Robustly adaptive synchronization for stochastic Markovian neural networks of neutral type with mixed mode-dependent delays

A robust adaptive feedback controller is designed to achieve complete synchronization of a class of neutral-type stochastic Markovian jump neural networks with mode-dependent and mixed delays. The mixed delays consist of time-varying discrete delays and continuously distributed delays. By employing Wirtinger-based integral inequality, a reciprocally convex combination technique and a generalized LaSalle-type invariance principle for stochastic Markovian differential delay equations, this paper establishes two globally almost surely asymptotical stability conditions of the error dynamical system in the mean-square sense for all admissible uncertainties. An example and its numerical simulation are given to demonstrate the effectiveness of the theoretical results.

An approach to achieve modified projective synchronization between different types of fractional-order chaotic systems with time-varying delays

In this paper, modified projective synchronization (MPS) for different fractional-order chaotic systems with variable time delays is investigated. By using active control method, and analysis of the error dynamical systems, a suitable controller for achieving the modified projective synchronization is given. Finally the proposed method illustrated by two examples in two conditions and performance of proposed method is shown.

Output Feedback Adaptive Control with Low-Frequency Learning and Fast Adaptation

Although adaptive control has been used in numerous applications to achieve system performance without excessive reliance on system models, the necessity of high-gain learning rates for achieving fast adaptation can be a serious limitation of adaptive controllers. Specifically, in safety-critical systems involving large system uncertainties and abrupt changes in system dynamics, fast adaptation is required to achieve stringent tracking performance specifications. However, fast adaptation using high-gain learning rates can cause high-frequency oscillations in the control response, resulting in system instability. This paper develops an output feedback adaptive control framework for continuous-time minimum-phase multivariable dynamical systems for output stabilization and command following to address the problem of achieving fast adaptation using high-gain learning rates for systems with partial state information. The proposed framework uses a controller architecture involving a modification term in the update law that filters out the high-frequency content in the control response while preserving uniform boundedness of the system error dynamics. The approach is based on a nonminimal state-space realization that generates an expanded set of states using the filtered inputs and filtered outputs, as well as their derivatives, of the original system, and requires knowledge of only the open-loop system’s relative degree and a bound on the system’s order.