Approximate Disturbance Decoupling for a Class of Nonlinear Time Delay Systems

A nonlinear state predictor (NSP) for a class of nonlinear systems with input time delay is proposed. Similar to the extended Kalman filter, the idea of the NSP is first to calculate the future state predictors via the system model, then to adjust the predictors based on the predictive errors between current observations and their corresponding predictors. We first present a state predictive algorithm for a class of pseudo linear systems and then extend it to a class of nonlinear time delay systems. After the detailed NSP algorithm is presented, it is proved that the NSP is locally asymptotically convergent for a class of nonlinear deterministic systems if some sufficient conditions are satisfied. In the presence of measurement noise, it is further proved that the proposed NSP is extended exponentially bounded under certain conditions. Finally, computer simulations with two different nonlinear examples illustrate that the proposed NSP can effectively and accurately predict the future states for a class of nonlinear time-delay systems, no matter whether the state variables change quickly or slowly.

An extended nonlinear state predictor (ENSP) for a classof nonlinear systems with input time delay is proposed. Based on the extended Kalman filter (EKF), the ENSP first estimates the current states according to the previous estimations and estimation errors, next calculates the future state values via the system model, and then adjusts the values based on the current errors. After a state predictive algorithm for a class of linear systems is presented, it is extended to a class of nonlinear time delay systems and the detailed ENSP algorithm is further proposed. Finally, computer simulations with the nonlinear example are presented, which demonstrates that the proposed ENSP can effectively and accurately predict the future states for a class of nonlinear time-delay systems no matter whether the state variables change quickly or slowly.

Observer design for a class of nonlinear systems under a persistent excitation

When calculating the sampled-date representation of nonlinear systems second-order hold (SOH) assumption can be applied to improving the precision of the discretization results. This paper proposes a discretization method based on Taylor series and the SOH assumption for the nonlinear systems with the time delayed non-affine input. The mathematical structure of the proposed discretization method is explored. This proposed discretization method can provide a precise and finite dimensional discretization model for the nonlinear time-delayed non-affine system by keeping the truncation order of the Taylor series. The performance of the proposed discretization method is evaluated by doing the simulation using a nonlinear system with the time-delayed non-affine input. Different input signals, time-delay values and sampling periods are considered in the simulation to investigate the proposed method. The simulation results demonstrate that the proposed method is practical and easy for time-delayed nonlinear non-affine systems. The comparison between SOH assumption with first-order hold (FOH) and zero-order hold (ZOH) assumptions is given to show the advantages of the proposed method.

This paper proposes an indirect adaptive fuzzy neural network (FNN) controller with state observer and supervisory controller for a class of uncertain nonlinear dynamic time delay systems. First, the approximate function of unknown time delay system is inferred by the adaptive time delay FNN system. Next, a state observer is designed to estimate the unknown system states and the indirect adaptive fuzzy controller is constructed. Finally, the closed loop controller is obtained by incorporating the supervisory controller with the indirect adaptive fuzzy controller. Therefore, if the system tends to unstable, i.e., error dynamics is larger than a prescribed constraint which is determined by designer, the supervisory controller will activate to force the state to be stable. The free parameters of the indirect adaptive FNN controller can be tuned online by observer-based output feedback control law and adaptive laws by means of Lyapunov stability criterion. The resulting simulation example shows that the performance of nonlinear time delay chaotic system is fully tracking the reference trajectory. Meanwhile, simulation results show that the adaptive control effort of the proposed control scheme is much less due to the assist of the supervisory controller.

In this study, an adaptive fuzzy control scheme is proposed for a class of non-linear systems with unknown discrete and distributed time-varying delays via dynamic output-feedback approach. Unlike the system with only one unknown control coefficient, the so-called high-frequency gain, the system we will consider is more general. During the controller design procedure, novel Lyapunov–Krasovskii functionals are introduced to compensate for the unknown time-varying delay terms and all unknown functions are lumped into a suitable unknown function which can be approximated by only one fuzzy logic system (FLS). The main advantages of this study are that (i) the output-feedback adaptive fuzzy controller can dispose a class of non-linear systems with unknown time-varying delays, in which the virtual control coefficients are all unknown, (ii) it does not need to know the time delays and their upper bounds and (iii) only one parameter needs to be adjusted online in controller design procedure, which reduces the online computation burden greatly. It is proven that all the signals of the closed-loop system are semi-globally uniformly ultimately bounded, whereas the tracking error converges to a small neighbourhood of the origin. Finally, simulation results are provided to show the effectiveness of the proposed approach.

References

In this paper, an optimal control scheme for a class of nonlinear systems with time delays in both state and control variables with respect to a quadratic performance index function is proposed using a new iterative adaptive dynamic programming(ADP) algorithm.By introducing a delay matrix function, the explicit expression of the optimal control is obtained using the dynamic programming theory and the optimal control can iteratively be obtained using the adaptive critic technique.Convergence analysis is presented to prove that the performance index function can reach the optimum by the proposed method.Neural networks are used to approximate the performance index function, compute the optimal control policy, solve delay matrix function, and model the nonlinear system, respectively, for facilitating the implementation of the iterative ADP algorithm.Two examples are given to demonstrate the validity of the proposed optimal control scheme.

With the development of science and technology, practical systems such as the power systems, traffic systems, robot manipulator systems, etc., have become more complex. Therefore, it is difficult to build practical systems by accurate models. Under the lack of accurate process models, using system data to improve system performance and learn optimal decisions becomes very important. Through the recent years, data-based learning control theories and technologies have widely been investigated, including adaptive dynamic programming, reinforcement learning, iterative learning control, and so on. Data-based methods require the system data instead of the accurate knowledge of system dynamics that can be considered as model-free learning control methods. The data-based methods are effective solutions for the optimal control of nonlinear systems, which motivate this special issue. This special issue aims to collect and present original research dealing with data-based learning and their applications for optimization and control problems. The first group of papers1-7 focuses on data-based control theory, approaches, and applications. A fuzzy model predictive control approach is proposed for stick-slip type piezoelectric actuator to realize the precise control of the end effector.1 A systematic online adaptive dynamic programming control framework is proposed for smart buildings control to ensure hard constraints to be satisfied.2 A multi-verse optimizer tuned PI-type active disturbance rejection generalized predictive control method is described for the motion control problems of ships.3 The sufficient optimality conditions for the optimal controls are established under some convexity assumptions.4 A receding-horizon reinforcement learning algorithm is proposed for near-optimal control of continuous-time systems under control constraints.5 In order to solve the interference compensation control problem of a class of nonlinear systems, a method based on memory data is introduced to suppress interference greatly.6 A new controller design method is proposed for the trajectory tracking problem of robots with imprecise dynamic properties and interference.7 The second group of papers8-12 considers iterative learning identification and iterative learning control. An iterative learning control approach is proposed for linear parabolic distributed parameter systems with multiple actuators and multiple sensors.8 The quantized data-based iterative learning tracking control problem is studied for nonlinear networked control systems with signals quantization and denial-of-service attacks.9 The output tracking problem is considered for a class of nonlinear parabolic distributed parameter systems with moving boundaries.10 A just-in-time learning based dual heuristic programming algorithm is proposed to optimize the control performance of autonomous wheeled mobile robots under faults or disturbances.11 A novel optimal constraint-following controller is proposed for uncertain mechanical systems.12 The third group of papers13-19 focuses on robustness on data-based optimal learning control. A novel Nash game-theoretical optimal adaptive robust control design approach is proposed to address the constraint-following control problem for the uncertain underactuated mechanical systems with fuzzy evidence theory.13 A partial model-free sliding mode control strategy is proposed for a class of disturbed systems.14 A new data-based adaptive dynamic programming algorithm is proposed to solve the optimal control policy for discrete-time systems with uncertainties.15 A method that applies event-triggered mechanism H ∞ $$ {\mathrm{H}}_{\infty } $$ control to continuous-time nonlinear systems with asymmetric constraints based on dual heuristic dynamic programming structure is proposed.16 A novel anti-disturbance inverse optimal controller design method is proposed for a class of high-dimensional chain structure systems with any disturbances, matched, or mismatched.17 A data-driven H ∞ $$ {\mathrm{H}}_{\infty } $$ controller design method is studied for continuous-time linear periodic systems.18 The problem of the post-stall pitching maneuver of an aircraft with lower deflection frequency of control actuator is studied by considering the unsteady aerodynamic disturbances.19 The fourth group of papers20-23 focuses on neural networks and deep neural networks learning methods for optimal control. An optimal tracking control problem for the injection flow front position arising in the filling process in the injection molding machine is considered, and an intelligent real-time optimal control method based on deep neural networks is developed for the online tracking of the flow front position to improve the efficient production process of the plastics.20 An efficient and systematic method is proposed for model-based predictive control synthesis.21 The decentralized control issues of nonlinear large-scale systems are investigated via critic-only adaptive dynamic programming learning methods.22 A singularity-free online neural network-based sliding mode control method is proposed to realize the fixed-wing perch maneuver.23 The fifth group of papers24-27 discusses data-based control for distributed control systems. A mission-driven control scheme, including a consensus-based near-optimal formation controller and a finite-time precise formation controller, is proposed aiming at different requirements of unmanned aerial vehicle swarm.24 The neural network adaptive formation control of a class of second-order nonlinear systems with unmodeled dynamics is investigated, where the control law merely depends on the relative bearings between neighboring agents.25 The neighbor Q-learning based consensus control algorithm is developed for discrete-time multiagent systems.26 The fault-tolerate containment control problem is considered for stochastic nonlinear multiagent systems in the presence of input saturation and sensor faults.27 The sixth group of papers28-30 considers applications of data-based learning methods to industrial processes. A stochastic gradient algorithm based on the minimum Shannon entropy is proposed to identify a type of Hammerstein system with random noise.28 A predictive control strategy based on Hammerstein–Wiener inverse model compensation is proposed aiming at the nonlinearity and large lag of the pH change in wet flue gas desulfurization process.29 An algorithm called the kernel entropy regression is proposed to enhance the interpretability between the fault and the key performance indicator.30 The seventh group of papers31-36 focuses on machine learning, data mining, and practical applications in automation. The performance of a Takagi–Sugeno fuzzy-model-based observer is enhanced by proposing a featured multi-instant united switch-type observer.31 The reinforcement learning theory with deep Q-network is applied for the mobile robot to achieve a collision-free path in an unknown dynamic environment.32 An energy-saving velocity planning algorithm is proposed for rail transit train with running and computation delays.33 A novel COVID-19 transmission model is established by introducing traditional susceptible–exposed–infected–removed disease transmission models into complex network.34 A novel collaborative diagnosis method is presented by combining variational modal decomposition and stochastic configuration network for incipient faults of rolling bearing.35 The linear dependence graph associated with a finite-dimensional vector space is studied.36 In summary, this special issue provides an opportunity to review the most recent developments in data-based learning control for optimization of nonlinear systems, by considering theory, algorithms, and applications.

New sufficient conditions for the global stability of different classes of nonlinear fractional feedback systems are presented. The linear parts of the systems are positive systems with interval state matrices. The nonlinear parts are described by static nonlinear characteristics located in the first and third quarter of the plane. The feedbacks are described in general case by matrices with positive entries. The sufficient conditions for the global stability are given for the following classes of the nonlinear systems: Positive interval continuous-time feedback nonlinear systems; Fractional positive interval continuous-time feedback nonlinear systems; Positive interval discrete-time feedback nonlinear systems; Descriptor nonlinear feedback discrete-time systems and Positive nonlinear electrical circuits. Procedures are given for calculations of gain matrices of the characteristics of nonlinear elements of the systems. The effectiveness of the procedures are demonstrated on numerical examples of nonlinear systems.

The nonlinear system describing self-dual Einstein metrics and its generalizations are discussed from the point of view of integrability. It is shown that these nonlinear systems share a variety of remarkable features (such as the existence of a linear scattering problem, a group-theoretical solution technique similar to the Riemann-Hilbert problem, and a geometric interpretation as dynamical motion in an infinite dimensional Grassmann manifold) with nonlinear integrable systems known until now. Differences of the relevant group-theoretical structures between these two classes of nonlinear systems are also pointed out. These results lead to the conclusion that the nonlinear systems in question do form a new class of nonlinear integrable systems.

This paper is concerned with the discretization of nonlinear continuous time delay systems. Our approach is based on Taylor-Lie series. The main idea aims to minimize the effect of the delay and neglects the importance of nonlinear parameter by the linearization of the system study in an attempt to make its handling and easier programming as possible. We investigate a new method based on the development of new theoretical methods for the time discretization of nonlinear systems with time delay .The performance of these proposed discretization methods was validated by doing the numerical simulation using a nonlinear system with state delay. Some illustrative examples are given to show the effectiveness of the obtained results.

The adaptive control law, the noise estimator and the recursive adaptive parameter predicting algorithm are presented for a class of nonlinear discrete systems with multiple time delays based on dynamic approximate second-order increment full parameter recursive predicting model nonlinearization. They realize the adaptive control for the time-delayed nonlinear systems with larger time delays. The simulation results for several typical nonlinear systems show the correctness and effectiveness of the proposed algorithm.

The increment minimized recursive predicting model, the adaptive control law and the recursive adaptive parameter predicting algorithm unified form are presented for a class of nonlinear discrete-time systems with time delays based on dynamic approximate any higher-order Hammerstein model. They realize the adaptive control for the time-delayed nonlinear systems with larger time delays. The simulation results show the correctness and effectiveness of the proposed algorithm for a class of nonlinear systems.

New instability conditions for some classes of nonlinear dynamical systems of an arbitrary order are considered. These conditions reduce the investigation of instability of the initial nonlinear system to the investigation of instability of the linearized system. In this connection, the considered conditions are similar to the hypothesis of the instability theorem of the first Lyapunov method (the stability investigation by the first approximation), but they are applicable to more complicated nonlinear systems, because it takes into account the non-autonomy of the linearized system and the fact that the nonlinear terms in the right-hand sides of the equations of the initial system belong not only to the class of analytical functions. For different classes of the nonlinear systems, sufficient instability conditions are presented.