Feedback Linearization of Continuous and Discrete Multidimensional Systems

One of the most common methods of synthesis of the nonlinear control systems is the method of a feedback linearization (FL). The idea of this method consists in conversion of the original nonlinear system into a linear one by means of a state feedback and coordinate transformation. Then, the methods of control theory for the linear systems are used for the system design. If the original nonlinear system cannot be linearized exactly by the state feedback, the method of the approximate feedback linearization (AFL) is used. The essence of AFL method lies in the feedback linearization only of a certain part of the original nonlinear system (not of the entire system). In this paper, the author proposes a method of an approximate feedback linearization control of the nonlinear singularly perturbed (SP) systems. The proposed method is based on a decomposition of the original SP system and construction of AFL control in the form of composite FL controls for the slow and fast subsystems. In general, a nonlinear SP system cannot be easily separated into slow and fast subsystems, because the conditions of Tikhonov theorem are not complied. In order to overcome this, the author proposes to perform the feedback linearization method at first for the system's part, which describes the fast state variables. Thus, a fast control is chosen, so that the conditions of Tikhonov theorem would be met. Then, using a standard singular perturbation technique, we obtain a slow subsystem. Further the problem of FL control for a slow subsystem is solved. The resulting AFL control is obtained in the form of a composite control. Application of the proposed approach is illustrated with two examples.

In this work we study the design and analysis for nonlinear discrete time active control system with boring charter. It is shown that most analysis result for continuous time nonlinear system can be extended to the discrete time case. In previous studies, a method of nonlinear Model Following Control System (MFCS) was proposed by Okubo (1985). In this study, the method of nonlinear MFCS will be extended to nonlinear discrete time system with boring charter. Nonlinear systems which are dealt in this study have the property of norm constraints ║ƒ (v (k))║≤α + β║v (k)║^γ , where α≥0, β≥0, 0≤γ≤1. When 0≤γ<1. It is easy to extend the method to discrete time systems. But in the case γ = 1 discrete time systems, the proof becomes difficult. In this case, a new criterion is proposed to ensure that internal states are stable. We expect that this method will provide a useful tool in areas related to stability analysis and design for nonlinear discrete time systems as well.

This paper studies complete stable and unstable periodic solutions for n-dimensional nonlinear discrete dynamical systems. The positive and negative iterative mappings of discrete systems are used to develop mapping structures of the stable and unstable periodic solutions. The complete bifurcation and stability analysis are presented for the stable and unstable periodic solutions which are based on the positive and negative mapping structures. A comprehensive investigation on the Henon map is carried out for a better understanding of complexity in nonlinear discrete systems. Given is the bifurcation scenario based on positive and negative mappings of the Henon map, and the analytical predictions of the corresponding periodic solutions are achieved. The corresponding eigenvalue analysis of the periodic solutions is presented. The Poincare mapping sections relative to the Neimark bifurcations of periodic solutions are presented. A parameter map for periodic and chaotic solutions is developed. The complete unstable and stable periodic solutions in nonlinear discrete systems are presented for the first time. The results presented in this paper provide a new idea for one to rethink the current existing theories.

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

This article is devoted to the problem of composite control design for continuous nonlinear singularly perturbed (SP) system using approximate feedback linearization (AFL) method. The essence of AFL method lies in the feedback linearization only of a certain part of the original nonlinear system. According to AFL approach, we suggest to solve feedback linearization problems for continuous nonlinear SP system by reducing it to two feedback linearization problems for slow and fast subsystems separately. The resulting AFL control is constructed in the form of asymptotic composition (composite control). Standard procedure for the composite control design consists of the following steps: 1) system decomposition, 2) solution of control problem for fast subsystem, 3) solution of control problem for slow subsystem, 4) construction of the resulting control in the form of the composition of slow and fast controls. The main difficulty during system decomposition is associated with dynamics separation condition for nonlinear SP system. To overcome this, we propose to change the sequence of the design procedure: 1) solving the control problem for fast state variables part, 2) system decomposition, 3) solving the control problem for slow state variables part, 4) construction of the resulting composite control. By this way, fast feedback linearizing control is chosen so that the dynamics separation condition would be met and the fast subsystem would be stabilizable. The application of the proposed approach is illustrated through several examples.

In this chapter, as in Luo (2019, 2020), discrete systems with infinite-fixed-points are discussed through the local analysis. A method for fixed-points in nonlinear discrete systems is developed. The generalized normal forms of nonlinear discrete systems at fixed-points are presented for a better understanding of singularity in nonlinear discrete systems. The dynamics of infinite-fixed-point discrete systems is discussed for the complexity and singularity of nonlinear discrete systems. A few examples are presented for complexity and singularity in infinite-fixed-point discrete systems.

Reduced order modelling of two-time-scale discrete non-linear systems

The paper considers the issues of control and stabilization of nonlinear dynamic systems described by a set of differential and difference equations, the latter of which contain a control vector. The states of these systems have both continuous and discrete components, so such systems are called continuous-discrete or hybrid. Necessary and sufficient features of controllability of nonlinear hybrid systems with a constant discretization step are established, which imply a transition from these systems to equivalent, in the natural sense, nonlinear discrete dynamic systems. A transformation is presented that allows reducing a linear discrete system to the canonical Brunovsky form and constructing a stabilizing control on its basis for the corresponding continuous-discrete system with scalar control. An algorithm for reducing a first approximation system of a nonlinear discrete system with scalar control to the canonical Brunovsky form and an algorithm for constructing a stabilizing control for nonlinear hybrid systems with scalar control are developed and illustrated with examples. Sufficient signs of stabilization of nonlinear hybrid systems are presented both without and with the feedback controller.

In this paper, we discuss the quantitative characterization problems of the exponential stability and trajectory convergence property of nonlinear discrete dynamical systems on a bounded set of the state space. Through introducing two new concepts-the Lip constant of a nonlinear operator and strongly equivalent metrics of a norm-we show that the nonlinear discrete dynamical systems are exponentially stable on the bounded set if and only if the corresponding nonlinear operator is contractive under some strongly equivalent metrics, or if and only if the Lip constant of the nonlinear operator is less than one. In the latter case, we further show that the infimum of the exponential bounds of trajectories of the system equals exactly to the Lip constant. Based on the obtained results, we clearly explain how trajectory convergence properties of the systems are determined quantitatively by the Lip constant and strongly equivalent metrics. The obtained results not only are of importance in understanding the essence of exponential stability and trajectory convergence properties of nonlinear discrete dynamical systems on bounded sets of the state space but also provide some new, useful criteria of testing exponential stability and estimating convergence speed of trajectories of the systems.

The feedback linearization method can accurately linearize the nonlinear system. However, feedback linearization needs the exact dynamic of nonlinear systems, it is difficult to apply to unknown nonlinear systems. To be capable to perform the feedback linearization of the continuous-time nonlinear system containing uncertain parameters, a model reference adaptive control (MRAC) scheme is introduced in this paper. First, we construct a state feedback controller by the knowledge of the system model structure and form it into an adjustable system together with the nonlinear object. The reference model is decided upon to be a standard linear system. Second, based on the output errors of reference model and adjustable system, we adaptively modify the state feedback controller’s unknown parameters using the gradient descent approach. Finally, the simulations and real-world experiments on a first-order inverted pendulum system are carried out to assess the effectiveness of given methods.

This paper proposes a novel stabilizing decentralized model predictive control for nonlinear continuous systems.We extend the method proposed by [1] from nonlinear discrete time systems to nonlinear continuous systems. The key idea of this method is adding a contractive constraint to the optimal control problem of each subsystems.By adding the contractive constraint the whole system can be stabilizable within certain degree of interaction. The degree of the interaction is also represented by inequalities. Finally a numerical simulation is used to prove the effectiveness of the method and the method is also applied to a four-tank benchmark problem.

On linear models for nonlinear systems

MTN optimal control of MIMO non-affine nonlinear time-varying discrete systems for tracking only by output feedback

Robust model predictive control of discrete nonlinear systems with time delays and disturbances via T–S fuzzy approach

This paper presents a method of designing optimal dynamic compensator for a class of typical chemical reaction process-CSTR systems. Firstly, the model of nonlinear CSTR system is provided, Secondly, by using feedback linearization method, the nonlinear system is converted into a linear system, then according to the LQR theory, the methods of designing dynamic compensator is designed, and we analyze the asymptotic stability of the optimal regulator which has been compensated. At last, the simulation results show the effectiveness of the method.