Zero Dynamics Analysis for Inverse Decoupling Control of Asynchronous Traction Motor

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

An identification method of nonlinear dynamic system is studied in this paper. First, suppose that the original nonlinear dynamic system is described by Hammerstein model. Then, an intermediate model is generated. Next, Parameters of the intermediate model are obtained using a Bacterial Chemotaxis Optimization (BCO) approach. Finally, through the relationships of the parameters of intermediate model and those of Hammerstein model, we derive the parameters of the system. Consequently, the original nonlinear dynamic system is identified. The feasibility and efficiency of the presented algorithm are demonstrated using numerical simulations.

This paper deals with output feedback stabilisation of a class of nonlinear systems with nonlinearity of the unmeasured state variables. The nonlinearity of the unmeasured state complicates the zero dynamics of the nonlinear system and makes the control design more difficult. In this paper, the zero dynamics analysed for the class of nonlinear systems with respect to the original output and then with respect to a virtual output. It is demonstrated that nonlinear systems with unstable zero dynamics may have stable zero dynamics with the virtual output. Based on the analysis, conditions are then identified for nonlinear systems which can be stabilised by control design with estimated state, including the estimated virtual output, and this removes a restriction of a recent result that requires the transformed system has no zero dynamics. Examples are included to demonstrate that the proposed control design can be used to stabilise nonlinear systems with unstable nonlinear zero dynamics.

One of the approaches to sampled-data controller design for nonlinear continuous-time systems consists of obtaining an appropriate model and then proceeding to design a controller for the model. Then, it is important to derive a good approximate sampled-data model because the exact sampled- data model for nonlinear systems is often unavailable to the controller designers. Recently, Yuz and Goodwin have proposed an accurate sampled-data model which includes extra zero dynamics, so-called the sampling zero dynamics, corresponding to the relative degree of the continuous-time nonlinear system. This paper shows that a more accurate sampled-data model is required for a controlled Van der Pol system with the relative degree two. The reason is that the closed-loop system becomes unstable when a controller design method based on cancellation of the zero dynamics is applied, and the phenomenon seems related to the instability of the sampling zero dynamics of the more accurate sampled-data model. Further, this paper derives a more accurate model than that of Yuz and Goodwin for continuous-time nonlinear systems with the relative degree two, and presents a condition which assures the stability of the sampling zero dynamics of the obtained model.

Zero dynamics of nonlinear control systems with symmetries are studied in this paper. It is shown that the zero dynamics of a symmetric system is also symmetric and admits a special form of cascade decomposition. A local zero dynamics is always possible to be extended to the semiglobal case. The relation between the zero dynamics and the quotient system is also discussed.

Stability of Zero Dynamics of Sampled-Data Nonlinear Systems

Zero dynamics are essential to the design and analysis of input-output feedback linearization control, particularly in nonlinear systems. This study revisits an input-output feedback linearization problem through an illustrative example where the zero dynamics admits multiple equilibrium points. The effect of such zero dynamics on system performance and control input has also been investigated. Conventional tools such as 1-contraction and Lyapunov theory are not directly applicable to the analysis of zero dynamics with multiple equilibria. Therefore, the theory of 2-contraction is employed to perform the analysis effectively. The impact of such zero dynamics on the external control input, tasked with regulating the system's output is also analyzed. Extensive numerical simulations are conducted to validate and exemplify the theoretical results.

Recently, we developed a structural decomposition for multiple input multiple output nonlinear systems that are affine in control but otherwise general. This structural decomposition simplifies the conventional backstepping design and allows a new backstepping design procedure that is able to stabilize some systems on which the conventional backstepping is not applicable. In this paper we further exploit the properties of such a decomposition for the purpose of solving the semi-global stabilization problem for minimum phase nonlinear systems without vector relative degrees. By taking advantage of special structure of the decomposed system, we first apply the low gain design to the part of system that possesses a linear dynamics. The low gain design results in an augmented zero dynamics that is locally stable at the origin with a domain of attraction that can be made arbitrarily large by lowering the gain. With this augmented zero dynamics, backstepping design is then applied to achieve semi-global stabilization of the overall system.

A nonlinear damping controller using partial feedback linearization (PFL) technique is designed with an aim to mitigate subsynchronous resonance (SSR) in series-compensated doubly-fed induction generator (DFIG)-based wind farms, which is integrated to simplified Nordic power system. The controller using PFL technique is implementable into a partially linearized form of a system with DFIG wind farm that results in reduced system order and small computational burden. The derivation of control law does not also demand thorough knowledge of nonlinear and ambiguous system dynamics. The proposed controller design approach has four distinguished steps: 1) scrutinizing partial linearizability, 2) performing partial linearization, 3) assessing stability of internal system dynamics, and 4) obtaining control laws. Small-signal stability analysis is performed that includes electrical damping analysis and eigenvalue analysis. The controller is found to demonstrate a superior performance in SSR mitigation. Time-series simulation is carried out to confirm the veracity of the outcomes obtained from the small-signal stability analysis.

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

This work proposes a method to construct a state-shared model for multiple-input multiple-output (MIMO) systems. A state-shared model is defined as a linear time invariant state-space structure that is driven by measurement signals-the plant outputs and the manipulated variables, but shared by different multiple input/output models. The genesis of the state-shared model is based on a particular reduced non-minimal realization. Any such realization necessarily fulfills the requirement that the output of the state-shared model is an asymptotically correct estimate of the output of the plant, if the process model is selected appropriately. The approach is demonstrated on a nonlinear MIMO system — a physiological model of calcium fluxes that controls muscle contraction and relaxation in human cardiac myocytes.

This work proposes a discrete-time non-linear neural observer based on a recurrent high order neural network in parallel model trained with an algorithm based on the extended Kalman filter for discrete-time multiple input multiple output non-linear systems with unknown dynamics and unknown time-delay. To prove the semi-globally uniformly ultimately boundedness of the proposed neural observer the stability analysis based on the Lyapunov approach is included. Applicability of the proposed observer is shown via simulation and experimental results.

SUMMARYIn a recent paper, we developed a structural decomposition for multiple input multiple output nonlinear systems that are affine in control but otherwise general. In this paper, we exploit the structural properties of such a decomposition in the application of the backstepping design technique on multiple input nonlinear systems. In particular, this decomposition simplifies the conventional backstepping design and motivates new backstepping design procedures that are able to stabilize systems on which the conventional backstepping procedure is not applicable. Copyright © 2011 John Wiley & Sons, Ltd.

Control of MIMO nonlinear systems: A data-driven model inversion approach

In this paper, we study the best achievable performance of output regulation for an uncertain nonlinear system under disturbances. The nonlinear system is a stabilizable nonminimum phase system and its zero dynamics consists of a robust input-to-state stable component and an unstable but stabilizable component. It is assumed that the uncertain parameter vector and disturbances only enter the robust input-to-state stable component of the zero dynamics and the linearizable part of the system. In the latter, the uncertain functions satisfy lower triangular structure. The performance of this problem is measured by the energy of output signal of above system. It is shown in this paper that the best achievable performance of output regulation under the worst uncertain parameter vector and disturbances for the nonlinear system is only determined by the minimum energy of stabilizing the unstable part of its zero dynamics.