Control Of Nonlinear Dynamical Systems Research Articles

CONSTRAINED CONTROLLABILITY OF SEMILINEAR SYSTEMS WITH MULTIPLE DELAYS IN CONTROL

In the present paper finite-dimensional dynamical control systems described by semilinear ordinary differential state equations with multiple point delays in control are considered. It is generally assumed, that the values of admissible controls are in a convex and closed cone with vertex at zero. Using generalized open mapping theorem, sufficient conditions for constrained local relative controllability near the origin are formulated and proved. Roughly speaking, it will be proved that under suitable assumptions constrained global relative controllability of a linear associated approximated dynamical system implies constrained local relative controllability near the origin of the original semilinear dynamical system. This is a generalization to constrained controllability case some previous results concerning controllability of linear systems with multiple point delays in the control and with unconstrained controls. Moreover, necessary and sufficient conditions for constrained global relative controllability of an associated linear dynamical system with multiple point delays in control are discussed. Simple numerical example which illustrates theoretical considerations is also given. Finally, some remarks and comments on the existing results for controllability of nonlinear dynamical systems are also presented.

Intellectual Method for Computer Construction of Border of Envelope of Inclusion Cone for Nonlinear Dynamic Control System

Intellectual Method for Computer Construction of Border of Envelope of Inclusion Cone for Nonlinear Dynamic Control System

Neural input selection—A fast model-based approach

Neural input selection is an important stage in neural network configuration. For neural modeling and control of nonlinear dynamic systems, the inputs to the neural networks may include any system variable of interest with various time lags. To choose a set of significant inputs is a combinational problem, and the selection procedure can be very time consuming. In this paper, a model-based neural input selection method is proposed. Essentially, the neural input selection is transformed into the problem of identifying the significant terms for a linear-in-the-parameters model. A fast method is then proposed to identify significant nonlinear terms or functions, from which the neural inputs are grouped and selected. Both theoretic analysis and simulation examples demonstrate the effectiveness and efficiency of the proposed model-based approach.

Geometric Identification and Control of Nonlinear Dynamic Systems Based on Floating Basis Vector Representation

In this paper a simple adaptive controller is outlined that creates only temporal and situation-dependent system model. It may be a plausible alternative of the more sophisticated soft computing approaches that aim the identification of permanent and complete models. The temporal model can be built up and maintained step-by-step on the basis of slow elimination of fading information by the use of simple updating rules consisting of finite algebraic steps of lucid geometric interpretation. It may be used for filling in the “lookup tables” or rule bases of the more sophisticated representations experimentally. The method applies simple elimination of the the casual algebraic singularities the occurrence of which cannot be evaded in the practice. The operation of the method is illustrated by the control of a 2 Degrees Of Freedom dynamic system as a typical paradigm via simulation.

T–S fuzzy model with nonlinear consequence and PDC controller for a class of nonlinear control systems

In this paper a new Takagi–Sugeno (T–S) fuzzy model with nonlinear consequence (TSFMNC) is presented which can approximate a class of smooth nonlinear systems , nonlinear dynamical systems and nonlinear control systems . It is also proved that Takagi–Sugeno fuzzy controller with nonlinear consequence (TSFCNC) can be used to approximate a class of nonlinear state-feedback controllers using the so-called parallel distributed compensation (PDC) method. The inverted pendulum problem has been simulated with TSFCNC and compared with Takagi–Sugeno fuzzy controller with linear consequence (TSFCLC) and the results show that TSFCNC performs better than TSFCLC. A real-life example of dynamic positioning of ship is simulated and the results also show that TSFCNC performs better than TSFCLC.

An LMI and fuzzy model approach to H ∞ PI controller design

A Linear Matrix Inequality (LMI) approach for designing the H ∞ Proportional‐Integral (PI) controller for nonlinear dynamic systems is studied. The whole operating range of a nonlinear system is partitioned into several regimes. A local linear model containing time‐varying norm‐bounded uncertain parameters is identified with parameter uncertainties for each region. These local linear models are then integrated as a norm‐bounded Tagaki‐Sugeno (TS) nonlinear fuzzy model. The robust PI control design problem based on these norm‐bounded uncertain linear models is then transformed into a series of standard H ∞ control problems, where the latter is further formulated as LMIs. By adopting the LMI expressions, a symmetric positive definite matrix with guaranteed overall system stability can be easily determined and then be further used to infer the robust multiple PI controller parameters. One chemical process, a double‐effect evaporator, is illustrated to demonstrate the effectiveness of the proposed LMI‐based H ∞ PI controller design method for nonlinear dynamic processes.

Modified descend curvature based fixed form fuzzy optimal control of nonlinear dynamical systems

In this study, a fuzzy rule-based optimal controller is designed for nonlinear dynamical systems. The direct second order method (or direct-descend-curvature algorithm) with a modification called “modified descend controller (MDC)” is used for calculating the parameters of the fuzzy feedback controller. The optimal control problem defined here has dynamic constraints of nonlinear system states and static constraint of a known form of fuzzy controller. The form used here is a standard fuzzy system that uses singleton fuzzifier, product inference engine, center average defuzzifier, and with the Gaussian membership functions of the system states to be controlled. The design is developed by minimizing a quadratic performance index selected for the desired operating conditions. Successful simulation results of controlling the temperature of a continuous stirred tank reactor (CSTR) and a bioreactor are given.

Supervisory grey-based tracking control for non-linear dynamical systems

Because the dynamic characteristics of many physical systems are non-linear and time varying, it is difficult to evaluate appropriate control effort to track the desired trajectory. To deal with this problem, a novel supervisory state feedback controller for unknown non-linear dynamical systems is proposed in this paper to achieve high-precision tracking performance. The proposed scheme comprises a grey uncertainty predictor and a state feedback linearization controller, which evaluates the control effort according to the forecasted data on line. Moreover, to stabilize the system states around a predefined bound region, a supervisory controller is added to adjust the control effort. The design process, theoretical and stability analysis are discussed in detail. Finally, three applications are provided to demonstrate the effectiveness of the proposed strategy. Compared with other methods, the proposed scheme possesses the salient advantages of a simple framework, free from chattering, stable tracking control performance, and robust to uncertainty.

On the Stability and Control of Nonlinear Dynamical Systems via Vector Lyapunov Functions

Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In this paper, we extend the theory of vector Lyapunov functions by constructing a generalized comparison system whose vector field can be a function of the comparison system states as well as the nonlinear dynamical system states. Furthermore, we present a generalized convergence result which, in the case of a scalar comparison system, specializes to the classical Krasovskii-LaSalle invariant set theorem. In addition, we introduce the notion of a control vector Lyapunov function as a generalization of control Lyapunov functions, and show that asymptotic stabilizability of a nonlinear dynamical system is equivalent to the existence of a control vector Lyapunov function. Moreover, using control vector Lyapunov functions, we construct a universal decentralized feedback control law for a decentralized nonlinear dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. Furthermore, we establish connections between the recently developed notion of vector dissipativity and optimality of the proposed decentralized feedback control law. Finally, the proposed control framework is used to construct decentralized controllers for large-scale nonlinear systems with robustness guarantees against full modeling uncertainty.

An approximate stability analysis of nonlinear systems described by Universal Learning Networks

Stability is one of the most important subjects in control systems. As for the stability of nonlinear dynamical systems, Lyapunov's direct method and linearized stability analysis method have been widely used. But, it is generally recognized that finding an appropriate Lyapunov function is fairly difficult especially for the nonlinear dynamical systems, and also it is not so easy for the linearized stability analysis to find the locally asymptotically stable region. Therefore, it is crucial and highly motivated to develop a new stability analysis method, which is easy to use and can easily study the locally asymptotically stable region at least approximately, if not exactly. On the other hand, as for the calculation of the higher order derivative, Universal Learning Networks (ULNs) are equipped with a systematic mechanism that calculates their first and second order derivatives exactly. So, in this paper, an approximate stability analysis method based on @h approximation is proposed in order to overcome the above problems and its application to a nonlinear dynamical control system is discussed. The proposed method studies the stability of the original trajectory by investigating whether the perturbed trajectory can approach the original trajectory or not. The above investigation is carried out approximately by using the higher order derivatives of ULNs. In summarizing the proposed method, firstly, the absolute values of the first order derivatives of any nodes of the trajectory with respect to any initial disturbances are calculated by using ULNs. If they approach zero at time infinity, then the trajectory is locally asymptotically stable. This is an alternative linearized stability analysis method for nonlinear trajectories without calculating Jacobians directly. In the method, the stability analysis of time-varying systems with multi-branches having any sample delays is possible, because the systems are modeled by ULNs. Secondly, the locally asymptotically stable region, where asymptotical stability is secured approximately, is obtained by finding the area where the first order terms of Taylor expansion are dominant compared to the second order terms with @h approximation assuming that the higher order terms more than the third order are negligibly small in the area. Simulations of an inverted pendulum balancing system are carried out. From the results of the simulations, it is clarified that the stability of the inverted pendulum control system is easily analyzed by the proposed method in terms of studying the locally asymptotically stable region.

ADAPTIVE BACKSTEPPING SLIDING-MODE CONTROL WITH APPLICATION TO A FLEXIBLE-JOINT MANIPULATOR

This paper is concerned with the adaptive backstepping sliding-mode control of nonlinear dynamic systems with model uncertainties. The proposed control method suitable for mechatronic systems combines the advantages of sliding-mode control and backstepping methodology, such that the requirement of the matching condition is removed, which seriously clams the application of sliding-mode control. In the control scheme, networks of Gaussian radial basis functions with variable weights are used to compensate the model uncertainties. The adaptive law developed using the Lyapunov synthesis approach guarantees the stability of the control scheme. The performance is illustrated by experimental studies with a flexible-joint manipulator.

A nonlinear model reference adaptive inverse control algorithm with pre-compensator

In this paper, the reduced-order modeling (ROM) technology and its corresponding linear theory are expanded from the linear dynamic system to the nonlinear one, and H ∞ control theory is employed in the frequency domain to design some nonlinear system s pre-compensator in some special way. The adaptive model inverse control (AMIC) theory coping with nonlinear system is improved as well. Such is the model reference adaptive inverse control with pre-compensator (PCMRAIC). The aim of that algorithm is to construct a strategy of control as a whole. As a practical example of the application, the numerical simulation has been given on matlab software packages. The numerical result is given. The proposed strategy realizes the linearization control of nonlinear dynamic system. And it carries out a good performance to deal with the nonlinear system.

Multi-Input Fuzzy control of an inverted pendulum using an armature controlled DC motor

This paper presents a design methodology to stabilize a class of multi-variant nonlinear system after a high disturbance occurs. It investigates application of Takagi-Sugeno type fuzzy controller (T-S-FC) to an inverted pendulum mechanism, actuated by an armature-controlled DC electrical motor.Fuzzy controllers use heuristic information in developing design methodologies for control of non-linear dynamic systems. This approach eliminates the need for comprehensive knowledge and mathematical modeling of the system, and in cases of more complex systems, approximation and simplifications in order to achieve feasible mathematical model is not required.The paper presents the stages of development of the Fuzzy Controller for an inverted pendulum by developing a two-input, Mamdani type system. It evaluates the performance of the system. Then a four-input T-S-FC type is developed. The research compares performances of each controller and presents the result of tests. A model for a DC motor is developed in this study, in order to measure the effect of time delays and response time caused by inherent properties of the physical system. The final part will demonstrate the complete operational system with the DC electrical motor included in the test system.

On Categories and Foundations of the Theory of Nonlinear Control Dynamical Systems: IV

On Categories and Foundations of the Theory of Nonlinear Control Dynamical Systems: IV

Three Mathematica supported proposals for the discretization of nonlinear dynamical control systems

In this paper, we present a library of functions in Mathematica allowing to perform some experiments on sampling and interpolation in dynamical control systems. Some of these functions implement fundamental techniques in dynamical system theory: the Euler polygon, and the Picard iteration method, while others allow to calculate approximate solutions using nonstandard algorithms exploiting the advantages of both Euler's and Picard's method. Using Taylor polynomial approximations of vector-fields may speed up the calculation of Picard iterated solutions, even though this procedure involved double approximations.

CMAC Neural Network and Its Use in Problems of Identification and Control of Nonlinear Dynamic Objects

Efficient use of CMAC (Cerebellar Model Articulation Controller) for identification and real-time control of nonlinear dynamical systems is demonstrated. An on-line weight training algorithm is proposed. The results of modelling and controlling nonlinear objects with unknown dynamics testify to the efficiency of this network.

Linear stabilization of programmed motions of non-linear controlled dynamical systems

A non-linear controlled dynamical system (NCDS), describing the dynamics of a wide class of non-linear mechanical and electromechanical systems, is considered. A technique is proposed for synthesizing control laws with linear feedback according to state, which describe the stabilization of programmed motions of such systems. A non-singular linear transformation of the state space is constructed, bringing the initial NCDS in deviations (from its programmed motion and programmed control) to a certain NCDS of a special form, which is convenient for analysing and synthesizing control laws governing the motion of the system. A NCDS of canonical form is separated out from the initial NCDS in deviations. The aforementioned non-singular linear transformation of coordinates of the state space and the method of Lyapunov functions are used to synthesize control laws with linear state feedback, which guarantee global symptotic stability of an equilibrium position of a NCDS of canonical form and asymptotic stability in the large of a NCDS of special form and of the initial NCDS in deviations. Estimates are given for the domain of asymptotic stability in the large of the equilibrium positions of a NCDS of special form, of the initial NCDS in deviations, and of programmed motions of the initial NCDS, closed by the synthesized stabilizing controls.

Design and Analysis of Optimal Controller for Fuzzy Systems With Input Constraint

In this paper, we present a design method of the optimal and robust controller subject to the constraint on control inputs for continuous-time Takagi-Sugeno (TS) fuzzy systems. In order to establish this design method, we consider an optimal and robust control problem for nonlinear dynamic systems. For this problem, we present an analytic way which can provide the optimal controller for nonlinear dynamic systems by the dynamic programming approach and the inverse optimal approach. Moreover, we analyze the robustness property of the proposed optimal controller with respect to a class of input uncertainties by the passivity approach. Then, based on the theoretical results presented in this paper, we formulate the design problem of the optimal and robust controller with input constraint for continuous-time TS fuzzy systems as the semidefinite programming problem, and find the controller by solving it. The usefulness of the proposed approach is illustrated by considering the three-axis attitude stabilization problem of rigid spacecraft.

On categories and foundations of the theory of nonlinear control dynamical systems: III

On categories and foundations of the theory of nonlinear control dynamical systems: III

Performance of a Generalized Neuron-Based PSS in a Multimachine Power System

An artificial neural network can work as an intelligent controller for nonlinear dynamic systems through learning, as it can easily accommodate the nonlinearities and time dependencies. In dealing with complex problems, most common neural networks have some drawbacks of large training time, large number of neurons and hidden layers. These drawbacks can be overcome by a nonlinear controller based on a generalized neuron (GN) which retains the quick response of neural net. Results of studies with a GN-based power system stabilizer on a five-machine power system show that it can provide good damping over a wide operating range and significantly improve the dynamic performance of the system.