Control Of Nonlinear Dynamical Systems Research Articles

A Pseudo-rigid model for the inverse dynamics of an Euler beam

Inversion technique has been very successful in the tracking control of nonlinear dynamical systems. However, when applied to manipulators constructed with elastic links, inverse dynamics through direct integration in temporal space causes unbounded controller command. It has been suggested that seeking an inverse dynamics solution for a given tip trajectory with given initial conditions is an ill-posed problem. It has also been suggested that increasing model accuracy by including more terms in a truncated beam model worsens the controller’s ability to stabilize the system dynamics. In this paper, we seek to understand the nature of the inverse dynamics instability and to find an alternative solution. We appeal to the notion of a pseudo-rigid model which describes the beam deflection by a homogeneous displacement field. Particularly, we derive the mode shape in order to yield a bounded inverse dynamics solution. Different from most of the existing solutions where solution stability was achieved through modifying the output function, we modified the inverse dynamics model. A bounded inverse solution and model simplicity provide much needed ease in the design and implementation of an inversion controller. Numerical simulations and experiments have both been conducted to prove the validity of the proposed method.

A multi-lobe stereo phase portrait of a three-dimensional dynamic system and visualization of the control for objects with a multiplicity of stationary states

The problem of predictor-corrector control of nonlinear dynamic systems based on a preliminarily constructed set of their three-dimensional phase portraits corresponding to the set of control vectors is formulated. The definition of a stereo portrait of a three-dimensional dynamic system is introduced, and a mechanism for its creation is presented; this mechanism consists in achieving some animation effect, which helps the spatial perception of images on the visual control panel. The definition of the domain of holding states is introduced for a particular class of three-dimensional models with multiple stationary states, and algorithms for the approximation of the boundaries of these domains in the extended space of phase variables and controls are developed. A description of the operation of a man-machine visual monitoring system and a step-wise control designated for a wide range of state variables are given.

Identification and Adaptive Control of Dynamic Nonlinear Systems Using Sigmoid Diagonal Recurrent Neural Network

The goal of this paper is to introduce a new neural network architecture called Sigmoid Diagonal Recurrent Neural Network (SDRNN) to be used in the adaptive control of nonlinear dynamical systems. This is done by adding a sigmoid weight victor in the hidden layer neurons to adapt of the shape of the sigmoid function making their outputs not restricted to the sigmoid function output. Also, we introduce a dynamic back propagation learning algorithm to train the new proposed network parameters. The simulation results showed that the (SDRNN) is more efficient and accurate than the DRNN in both the identification and adaptive control of nonlinear dynamical systems.

Fault Tolerant Model Predictive Control of Quad-Rotor Helicopters with Actuator Fault Estimation

AbstractModel predictive control (MPC) at each time step minimizes a cost function subject to dynamical constraints to obtain a stabilizing control signal. Further, MPC is one of the few methodologies that can be used to design feedback control for nonlinear dynamical systems taking into consideration of actuator saturations. It can thus serve as a suitable fault tolerant control approach for quad-rotor helicopter governed by nonlinear dynamics. However, MPC needs a relatively accurate model of the post-failure system to calculate a stabilizing control signal. The problem becomes more critical where the system dynamics is described by a nonlinear model, because there exist few effective nonlinear parameter estimators with reasonable online computation time. To address this issue, for online actuator fault estimation, this paper investigates Moving Horizon Estimation (MHE) and Unscented Kalman Filter (UKF) as two methods for nonlinear parameter estimation. A framework is then formulated for integrating MHE/UKF based fault estimator with MPC to form an active fault tolerant control system for systems with nonlinear constrained dynamics. Performance and computation requirement of both algorithms are also investigated.

Strong and weak invariance for nonlinear impulsive control systems

AbstractThis paper is devoted to the study of invariance for nonlinear dynamical control systems with trajectories of bounded variation and impulsive controls. The definitions and proximal characterization of strong and weak invariance adapted to the nonlinear impulsive control systems are presented.

Interval arithmetic techniques for the design of controllers for nonlinear dynamical systems with applications in mechatronics. II

In this paper, we give an overview of interval arithmetic techniques for both the offline and online verification of robust control and observer strategies. In Part 1 of this paper, we have introduced basic interval arithmetic techniques focusing on offline applications while we deal with their online application in this Part 2. In contrast to the offline tasks considered in Part 1, in which guaranteed enclosures of all admissible solutions to a specific control problem are computed, real-time requirements have to be considered in online tasks. This means that we have to ensure to find a guaranteed admissible solution within a limited computing time. Therefore, the adapted goal in the online design of controllers and observers must be to determine at least one solution, and not necessarily all, such that feasibility under the influence of uncertainties is proven. After a summary of the basic computational procedures, first experimental results for the real-time application of interval tools for feedforward control and disturbance estimation are presented for the temperature control of a distributed heating system.

Nonsingular fast terminal sliding‐mode control for nonlinear dynamical systems

AbstractThis paper investigates fast finite‐time control of nonlinear dynamics using terminal sliding‐mode (TSM) scheme. Some new norms of fast TSM strategies are proposed, and a faster convergence rate is established in comparison with the conventional fast TSM. A novel concept of nonsingular fast TSM, which is able to avoid the possible singularity during the control phase, is adopted in the robust high‐precision control of uncertain nonlinear systems. Numerical simulation on a two‐link rigid robot manipulator demonstrates the effectiveness of the proposed algorithm. Copyright © 2010 John Wiley & Sons, Ltd.

Interval arithmetic techniques for the design of controllers for nonlinear dynamical systems with applications in mechatronics

Interval arithmetic techniques for the design of controllers for nonlinear dynamical systems with applications in mechatronics

Differential equations for ellipsoidal estimates for reachable sets of a nonlinear dynamical control system

The problem of estimating trajectory tubes of a nonlinear control dynamical system with uncertainty in initial data is considered. It is assumed that the dynamical system has a special structure, in which nonlinear terms are quadratic in phase coordinates and the values of the uncertain initial states and admissible controls are subject to ellipsoidal constraints. Differential equations are found that describe the dynamics of the ellipsoidal estimates of reachable sets of the nonlinear dynamical system under consideration. To estimate reachable sets of the nonlinear differential inclusion corresponding to the control system, we use results from the theory of ellipsoidal estimation and the theory of evolution equations for set-valued states of dynamical systems under uncertainty.

Linear Methods in Observability and Controllability of Nonlinear Systems

The paper is devoted to the development of approach to analyze observability and controllability of nonlinear dynamic systems by linear methods. The main idea of this approach is as follows: observability (controllability) of the initial nonlinear system linear part is checked and if it is unobservable (uncontrollable), one examines whether or not the nonlinear term violates this conclusion. Sufficient conditions for unobservability and uncontrollability are obtained.

Constructing subsystems for nonlinear controlled systems

We consider the problem of constructing subsystems representing simplified models of nonlinear controlled dynamical systems. Subsystems define a part of phase trajectories of the original controlled system.

Polynomial chaos expansions for optimal control of nonlinear random oscillators

The polynomial chaos decomposition of stochastic variables and processes is implemented in conjunction with optimal polynomial control of nonlinear dynamical systems. The procedure is demonstrated on a base-excited system whereby ground motion is modeled as a stochastic process with a specified correlation function and is approximated by its Karhunen–Loeve expansion. An adaptive scheme for stochastic approximation with polynomial chaos bases is proposed which is based on a displacement–velocity norm and is applied to the identification of phase orbits of nonlinear oscillators. This approximation is then integrated in the design of an optimal polynomial controller, allowing for the efficient estimation of statistics and probability density functions of quantities of interest. Numerical investigations are carried out employing the polynomial chaos expansions and the Lyapunov asymptotic stability condition based control policy. The results reveal that the performance, as gaged by probabilistic quantities of interest, of the controlled oscillators is greatly improved. A comparative study is also presented against the classical stochastic optimal control, whereby statistical linearization based LQG is employed to design the optimal controller. It is remarked that the proposed polynomial chaos expansion is a preferred approach to the optimal control of nonlinear random oscillators.

Adaptive Control of Non-linear System Using Neural Network

The paper deals with one of possible methods for control of non-linear dynamical system, namely with the adaptive neural PS controller (ANPSC). Its principle is in on-line refinement of the model system and then in the adaptive adjustment of the controller parameters. The model of the plant is realized with a two-layer feedforward neural network. The control part of the ANPSC has the structure of the classical PS controller, and it is also created in the form of the neural network (simple perceptron). The simulation calculations were performed in MATLAB.

PID control model and method based on process neural network identification

Concerning PID process control of nonlinear dynamic system,a control model and its method of PID parameter adaptive tuning based on Process Neural Network(PNN) identification were proposed in the paper.Using the learning mechanism of PNN for time-varying input-output signals of dynamic system,the PID control with parameter adaptive matching was implemented by tracing change sensitivity information of the output with the controlling input of the controlled object by identifying the controlled object under certain optimal control rule.The PID control system structure and corresponding realization mechanism based on PNN identification were presented in the paper and the experimental results verify the effectiveness of the model and algorithm.

Analysis of observability and controllability of nonlinear dynamic systems by linear methods

The problem of finding sufficient unobservability and uncontrollability criteria for nonlinear dynamic systems (including systems with nondifferentiable nonlinearities) is considered on the basis of known unobservability and uncontrollability criteria for linear systems and additional linear conditions. Duality relations supplementing the well-known duality principle for linear systems are established.

Verification Techniques for Sensitivity Analysis and Design of Controllers for Nonlinear Dynamic Systems with Uncertainties

Verification Techniques for Sensitivity Analysis and Design of Controllers for Nonlinear Dynamic Systems with UncertaintiesControl strategies for nonlinear dynamical systems often make use of special system properties, which are, for example, differential flatness or exact input-output as well as input-to-state linearizability. However, approaches using these properties are unavoidably limited to specific classes of mathematical models. To generalize design procedures and to account for parameter uncertainties as well as modeling errors, an interval arithmetic approach for verified simulation of continuoustime dynamical system models is extended. These extensions are the synthesis, sensitivity analysis, and optimization of open-loop and closed-loop controllers. In addition to the calculation of guaranteed enclosures of the sets of all reachable states, interval arithmetic routines have been developed which verify the controllability and observability of the states of uncertain dynamic systems. Furthermore, they assure asymptotic stability of controlled systems for all possible operating conditions. Based on these results, techniques for trajectory planning can be developed which determine reference signals for linear and nonlinear controllers. For that purpose, limitations of the control variables are taken into account as further constraints. Due to the use of interval techniques, issues of the functionality, robustness, and safety of dynamic systems can be treated in a unified design approach. The presented algorithms are demonstrated for a nonlinear uncertain model of biological wastewater treatment plants.

A Sum-of-Squares Approach to Modeling and Control of Nonlinear Dynamical Systems With Polynomial Fuzzy Systems

This paper presents a sum of squares (SOS) approach for modeling and control of nonlinear dynamical systems using polynomial fuzzy systems. The proposed SOS-based framework provides a number of innovations and improvements over the existing linear matrix inequality (LMI)-based approaches to Takagi-Sugeno (T-S) fuzzy modeling and control. First, we propose a polynomial fuzzy modeling and control framework that is more general and effective than the well-known T--S fuzzy modeling and control. Secondly, we obtain stability and stabilizability conditions of the polynomial fuzzy systems based on polynomial Lyapunov functions that contain quadratic Lyapunov functions as a special case. Hence, the stability and stabilizability conditions presented in this paper are more general and relaxed than those of the existing LMI-based approaches to T-S fuzzy modeling and control. Moreover, the derived stability and stabilizability conditions are represented in terms of SOS and can be numerically (partially symbolically) solved via the recently developed SOSTOOLS. To illustrate the validity and applicability of the proposed approach, a number of analysis and design examples are provided. The first example shows that the SOS approach renders more relaxed stability results than those of both the LMI-based approaches and a polynomial system approach. The second example presents an extensive application of the SOS approach in comparison with a piecewise Lyapunov function approach. The last example is a design exercise that demonstrates the viability of the SOS-based approach to synthesizing a stabilizing controller.

H∞ control of non-linear dynamic systems: a new fuzzy delay partitioning approach

The problem of robust H infin control for Takagi-Sugeno (T-S) fuzzy systems with norm-bounded parameter uncertainties and a time delay in the state is investigated. Attention is focused on the design of robust H infin controllers via the parallel distributed compensation scheme such that the closed-loop fuzzy time-delay system is asymptotically stable and the H infin disturbance attenuation is below a prescribed level. By utilising the instrumental idea of delay partitioning, a new Lyapunov-Krasovskii functional is introduced, and some novel ideas for achieving delay dependence and basis dependence have been employed, which guarantee the obtained conditions to be much less conservative than most existing results in the literature. These conditions are formulated in the form of linear matrix inequalities (LMIs), based on which, the controller design is cast into a convex optimisation problem subject to LMI constraints. Finally, two examples are illustrated to show the feasibility and effectiveness of the obtained results.

Non-linear control by input–output state variable feedback pole assignment

This article considers pole assignment control of non-linear dynamic systems described by state-dependent parameter (SDP) models. The approach follows from earlier research into linear proportional-integral-plus methods but, in SDP system control, the control coefficients are updated at each sampling instant on the basis of the latest SDP relationships. Alternatively, algebraic solutions can be derived off-line to yield a practically useful control algorithm that is relatively straightforward to implement on a digital computer, requiring only the storage of delayed system variables, coupled with straightforward arithmetic expressions in the control software. Although the analysis is limited to the case when the open-loop system has no zeros, time delays are handled automatically. This article shows that the closed-loop system reduces to a linear transfer function with the specified (design) poles. Hence, assuming pole assignability at each sample, global stability of the non-linear system is guaranteed at the design stage.

A new approach to fuzzy modeling and control for nonlinear dynamic systems: Neuro-fuzzy dynamic characteristic modeling and adaptive control mechanism

The study on nonlinear control system has received great interest from the international research field of automatic engineering. There are currently some alternative and complementary methods used to predict the behavior of nonlinear systems and design nonlinear control systems. Among them, characteristic modeling (CM) and fuzzy dynamic modeling are two effective methods. However, there are also some deficiencies in dealing with complex nonlinear system. In order to overcome the deficiencies, a novel intelligent modeling method is proposed by combining fuzzy dynamic modeling and characteristic modeling methods. Meanwhile, the proposed method also introduces the low-level learning power of neural network into the fuzzy logic system to implement parameters identification. This novel method is called neuro-fuzzy dynamic characteristic modeling (NFDCM). The neuro-fuzzy dynamic characteristic model based overall fuzzy control law is also discussed. Meanwhile the local adaptive controller is designed through the golden section adaptive control law and feedforward control law. In addition, the stability condition for the proposed closed-loop control system is briefly analyzed. The proposed approach has been shown to be effective via an example.