Control Of Nonlinear Dynamical Systems Research Articles

Robustifying nonlinear systems using high-order neural network controllers

A robust control methodology for affine control of nonlinear dynamical systems is developed in this paper. A correction control signal is added to a nominal controller (designed to guarantee a desired performance for the corresponding nominal system), to render the actual system uniformly and ultimately bounded. The control signal is smooth and does not require a priori knowledge of an upper bound on the modeling error and/or optimal weight values. Simulations performed on a simple nonlinear system illustrate the approach.

Application of adaptive matching pursuit to adaptive control of nonlinear dynamic systems

A common problem in the adaptive control of nonlinear systems is the selection of an appropriate trade-off between performance and computational complexity. A reduced-complexity recursive algorithm is proposed for the identification of nonlinear auto-regressive with exogenous inputs (NARX) models of a nonlinear system. This algorithm, which follows the matching pursuit algorithm, is used for the adaptive control of nonlinear dynamic systems, where quadratic spline functions are used as the basic building blocks. The adaptive controller is applied to several nonlinear systems, previously solved by other nonlinear control methods.

Local Learning for Nonlinear Control

This paper presents a local method for modeling and control of nonlinear dynamical systems, when only a limited amount of input-output data is available. The proposed methodology couples a local model identification inspired by the lazy learning technique, with a control strategy based on linear optimal control theory. The local modeling procedure uses a query-based approach to select the best model configuration by assessing and comparing different alternatives. The control method combines the linearization provided by the local learning techniques with optimal linear control theory, to control nonlinear systems in far from equilibrium configurations. Simulation results of the control of a complex nonlinear system (the bioreactor) are presented

Decomposition of Control for Nonlinear Lagrangian Systems

A feedback control for nonlinear dynamical systems described by the Lagrangian equations of motion is proposed. The system is subjected to bounded control forces and uncertain but bounded disturbances. The design of feedback control is based on the decomposition of the original system with many degrees of freedom into subsystems with one degree of freedom each. Under certain assumptions, the nonlinearities are estimated, and the feedback control is designed which counteracts the influence of nonlinearities and disturbances for each subsystem. Thus, the decomposition of the original system is accomplished, and explicit formulas for the feedback control in the original system are obtained. This control brings the nonlinear system from a given initial state to the prescribed terminal state in finite time. The time of control is estimated from above. Other versions of the decomposition approach are discussed. Applications of the described approach to control of robots are presented.

Stable Neural Controllers for Nonlinear Dynamic Systems

In this paper, a stability based approach is introduced to design neural controllers for nonlinear systems. The requisite control input is generated as the output of a neural network, which is trained off-line such that the time derivative of a positive definite function of the state variables becomes negative at all points. By using the successfully trained networks as controllers, the closed-loop system can be made stable. The stability framework introduced permits the generation of more efficient algorithms that lead to a larger region of stability for a wide class of nonlinear systems.

On-line tuning of fuzzy-neural network for adaptive control of nonlinear dynamical systems

The adaptive fuzzy-neural controllers tuned online for a class of unknown nonlinear dynamical systems are proposed. To approximate the unknown nonlinear dynamical systems, the fuzzy-neural approximator is established. Furthermore, the control law and update law to tune on-line both the B-spline membership functions and the weighting factors of the adaptive fuzzy-neural controller are derived. Therefore, the control performance of the controller is improved. Several examples are simulated in order to confirm the effectiveness and applicability of the proposed methods in this paper.

Adaptive control of nonlinear dynamic systems using θ-adaptive neural networks

The adaptive control of dynamic systems with nonlinear parametrization is considered. An algorithm based on a neural network, similar to the TANN algorithm proposed in Annaswamy and Yu (1996), is suggested for adjusting the control parameters. The resulting adaptive controller is shown to lead to stability of the closed-loop system. How the neural network is trained off-line in order to lead to closed-loop stability is described in detail. The resulting improvement in performance using this neural controller over other methods proposed in the literature including extended Kalman filter, linear adaptive control, and other neural strategies is demonstrated through simulation studies.

Natural tracking with a prediction-error based parameter estimator

Current trends in control applications are oriented toward control algorithms which, to the least extent, depend on the system's mathematical model and/or its parameters. The recently disclosed concept of high gain natural tracking is one such approach to control of nonlinear dynamic systems. This law is independent of the plant's internal dynamics, but dependent on the system parameters. The work presented in this article demonstrates how the natural tracker can be applied to a nonlinear system, such as a robot, with unknown parameters. The parameters can be identified online using the prediction-error based estimator and the linearly parametrized mathematical model. The persistent excitation condition is satisfied with external high-frequency and low-amplitude dither signals which are superimposed on the control inputs. The identification is based on the knowledge of the system's total energy and under the assumption that the acceleration measurements are not available. This methodology, however, requires the use of additional filtering of the plant's input signals.

Identification and Control for Nonlinear Dynamic Systems by RBFNN

Identification and control for nonlinear dynamic systems still remains a challenging issue. Neural networks based models and controllers have often been effective in representing fields. In this paper, identification capability of radial basis function neural networks (RBFNN) is integrated into nonlinear internal model control to obtain a new control strategy. The model of inverse of the process is identified by training RBFNN. The NIMC controller consists of a model inverse controller and a robustness filter. At last, the simulation results for a continuous stirred tank reactor process demonstrate the advantage of the performance of the new strategy over that of a conventional PID controller.

Identification and Adaptive Control of Dynamic Systems Using Self-Organized Distributed Networks

Abstract An adaptive control technique, using system identification based on Self-Organized Distributed Networks (SODNs), is presented for a class of discrete nonlinear dynamic systems having unknown dynamics. The SODN belongs to the category of distributed local learning networks and is composed of two main networks called the learning network and the distribution network. The learning network consists of subnets each responsible for a subproblem. The distribution network is responsible for input space decomposition. The learning of the SODN is fast and precise because of the local learning mechanism. In this paper, methods for identification and indirect adaptive control of nonlinear systems using the SODN are presented. Through extensive simulation, the SODN is shown to be effective both for identification and adaptive control of nonlinear dynamic systems.

CMAC neural networks for control of nonlinear dynamical systems: Structure, stability and passivity

The cerebellar model articulation computer (CMAC) neural network (NN) has advantages over fully connected NNs due to its increased structure. While its advantages over conventional control techniques have been recognized in the literature, it has primarily been used in system identification and pattern recognition, but not in control applications. This paper attempts to provide a comprehensive treatment of CMAC NNs in closed-loop control applications. Novel weight-update laws are derived that guarantee the stability of the closed-loop system. The passivity properties of the CMAC under the specified tuning laws are examined and the relationship between passivity and closed-loop stability is derived.

The construction of bounded game-theoretic controls for non-linear dynamical systems

The problem of transferring a non-linear dynamical system, subject to perturbations, to the null equilibrium position in a finite time by means of a bounded control is considered. Only the levels of uncontrollable perturbations are known, and are not assumed to be small. Sufficient conditions are obtained which ensure that the problem has a guaranteed solution for the given domain of initial conditions. An estimate of the guaranteed control time is obtained. The construction of the control can be reduced to the construction of game strategies for auxiliary linear game-theoretic problems. To estimate the “auxiliary noise” in the resulting linear system, the principle of “prescribing and subsequent confirmation” of noise levels is put forward. On the basis of this principle, these estimates are checked on the set of states of the auxiliary linear systems, where the control is also subsequently estimated. As a result, an iterative algorithm for solving the original non-linear problem is obtained. Within the framework of the method proposed a new solution of the game-theoretic problem of the reorientation of an asymmetric rigid body in the presence of noise is given.

Constrained Controllability of Nonlinear Systems

In this paper infinite-dimensional dynamical systems described by nonlinear abstract differential equations are considered. Using the generalized open mapping theorem sufficient conditions for constrained exact local controllability are formulated and proved. It is generally assumed that the values of controls are in a convex and closed cone with vertex at zero. As an illustrative example a constrained exact local controllability problem for a nonlinear delayed dynamical system is solved in detail. Some remarks and comments on the existing results for controllability of nonlinear dynamical systems are also presented.

Static-state feedback laws for output regulation of non-linear systems

In this paper static-state feedback laws are considered which address the problem of output stabilization of non-linear dynamic control systems without drift. It is shown that as long as a dynamic-state control law exists which input-output decouples a given system, it is possible to construct a static-state control which stabilizes the system's output. The design procedure sheds light on the inherent difficulties in designing internally stable control laws for non-linear systems. To demonstrate the theory, it is applied to the task of designing an output stabilizing feedback for a simple model of a mobile robot.

A new method for identification and control of nonlinear dynamic systems

This paper presents a new model for nonlinear systems which consists of two parts: a linear part and a static nonlinear output part. The linear part is a linear combination of the model's outputs, where the static nonlinear function maps the output of the linear part to the model's output. This model can be applied to represent a relatively large class of nonlinear dynamic systems with fading memory. Based on this model, one can directly introduce the corresponding output feedback control law. The identification and control method is applied to two simulation examples of a discrete-time system and the dynamics of a complicated missile to demonstrate the performance and efficiency of the proposed method.

Discrete-time model reference adaptive control of nonlinear dynamical systems using neural networks

A multilayer discrete-time neural net (NN) controller is presented for the direct model reference adaptive control of a class of multi-input multi-output (MIM0) nonlinear dynamical systems. The nonlinear dynamical system is assumed to be controllable and its state vector is available for measurement. The NN controller exhibits learning-while-functioning features instead of learning-then-control so that control is immediate with no explicit learning phase needed. The tracking error between the output of a nonlinear plant and an ideal linear model converges within a very short time. Persistence of excitation (PE) is not needed, linearity in the parameters is not required, and certainty equivalence is not used. This overcomes several limitations of standard adaptive control. The novel weight tuning paradigm for the NN controller is based on the well-known delta rule but includes a modification to the learning rate parameter plus a correction term. It guarantees tracking as well as bounded NN weights in non-ideal situations so that a persistency of excitation conditions on the internal signals is not needed. It is found that the constant learning rate parameter at a given layer should decrease with the number of hidden-layer neurons present in that particular layer; this is a major drawback often documented in the NN control literature. This drawback is easily overcome by employing a projection algorithm at each layer. Theoretical analysis is justified through extensive simulation studies.

Designing nonlinear controllers using connectionist networks

In this paper we focus on the use of adaptive connectionist networks as a tool for designing controllers for nonlinear dynamic systems. A general interrelation between representation, control and learning is set up. We have developed controllers for nonlinear systems using three control techniques. This provides a step towards the goal of generating systematic control design techniques for nonlinear systems.

Control of nonlinear dynamical systems using neural networks. II. Observability, identification, and control

For pt. I see ibid., vol. 4 (1993). This paper considers the problems of regulation and tracking of a dynamical system when the state variables of the dynamical system are not accessible. The existence of the nonlinear maps describing the identifier and controller are first established and the implications for neural network realizations are described. Simulation results are included to complement the theoretical discussions.

Dynamic neural network methods applied to on-vehicle idle speed control

The application of neural network techniques to the control of nonlinear dynamical systems has been the subject of substantial interest and research in recent years. In our own work, we have concentrated on extending the dynamic gradient formalism as established by Narendra and Parthasarathy (1990, 1991), and on employing it for applications in the control of nonlinear systems, with specific emphasis on automotive subsystems. The results we have reported to date, however, have been based exclusively upon simulation studies. In this paper, we establish that dynamic gradient training methods can be successfully used for synthesizing neural network controllers directly on instances of real systems. In particular we describe the application of dynamic gradient methods for training a time-lagged recurrent neural network feedback controller for the problem of engine idle speed control on an actual vehicle, discuss hardware and software issues, and provide representative experimental results.

Nonlinear system identification and adaptive control using polynomial networks

This work introduces a new nonlinear computational model called polynomial network for identification and adaptive control of nonlinear dynamical systems. The network approximates the Volterra systems or recursive polynomial systems. The approximation properties of this network are compared with the sigmoid networks. The results show that the polynomial network constructs a simpler and smaller model and requires less training data. Also, the model realized by the polynomial network is mathematically tractable. The feasibility of using this model for direct model reference adaptive control of a class of nonlinear systems is demonstrated.