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Background on Dynamic Systems
Published in F.L. Lewis, S. Jagannathan, A. Yeşildirek, Neural Network Control of Robot Manipulators and Nonlinear Systems, 2020
F.L. Lewis, S. Jagannathan, A. Yeşildirek
The additional dynamics neglected in the feedback linearization design are made unobservable by this design procedure; that is, with respect to the design output y(t) = x1 (t) they are unobservable. They are known as the internal dynamics. A different choice of y(t) results in different internal dynamics (see Problems section). The zero dynamics is defined as the internal dynamics when the control input is selected to keep the output y(t) equal to zero. If the zero dynamics are unstable, the system is said to be non-minimum phase. The zero dynamics extends to nonlinear systems the concept of systemzeros. In the LTI case, the zero dynamics define the system zeros. For i/o feedback linearization to function correctly, the internal dynamics must be stable. Then, the controller given in Fig. 2.4.1 performs in an adequate manner.
Optimization for Control Design
Published in Derek A. Linkens, CAD for Control Systems, 2020
David Q. Mayne, Hannah Michalska
Design of controllers for nonlinear systems (or nonlinear controllers for linear systems) is at a less advanced state despite recent spectacular progress due to the complexity and variety of the systems encountered in practice. In practice, of course, controllers are nearly always nonlinear, even if designed using linear theory. Practical requirements, such as necessity to restrict control magnitude, control rate, or certain states (temperature, for example), enforce nonlineari-ties in the controller. The controlled system is, therefore, nonlinear, even if the plant being controlled is linear. The lack of techniques for analyzing general nonlinear systems necessitates the use of extensive simulation to ensure that performance and stability objectives are met. Because design by simulation is relatively inefficient (it is difficult for a person to make an optimal choice of more than two or three variables), it appears almost certain that optimization procedures will eventually be extensively employed in nonlinear controller design. Since frequency- domain techniques are irrelevant, except in a restricted class of problems, performance specifications are expressed in the time domain. An optimization- based design environment would, therefore, consist of a nonlinear simulator to which is coupled a suitable optimizer whose purpose is to adjust the parameters of a controller to ensure stability and satisfaction of time domain performance specifications. As in linear systems, it is important to exploit appropriate system-theoretic advances. Among these is the interesting work on input-output linearization [2] and its use in “inverse” control to make the input-output map linear with a specified input-output transfer function. This approach has been used to good effect in control of vertical takeoff and landing (VTOL) aircraft and control of robot arms (the “computed control torque” method). However, a controller of this type cancels the zero dynamics and can, therefore, only be employed when the zero dynamics are stable (the minimum phase case when the system is linear). In Ref. 6, approximate methods to deal with this problem have been developed. More recently, Isidori and Byrnes [3] have developed a nonlinear analogue of linear regulator theory [7], yielding necessary and sufficient conditions for the local solvability of the state and output feedback regulator problems. This work is an important advance, indicating the possibility of regulation even when the zero dynamics are unstable and providing useful guidance on the structure of the controller. It does, nonetheless, rely on the absence of state and control constraints.
On the zero dynamics of linear input–output models
Published in International Journal of Control, 2022
Sneha Sanjeevini, Syed Aseem Ul Islam, Dennis S. Bernstein
A broader framework within which to understand the implications of zeros is the notion of zero dynamics, which is applicable to both linear and nonlinear systems (Berger et al., 2015; Daoutidis & Kravaris, 1991; Isidori, 2013). Zero dynamics are the ‘dynamics’ of the input assuming that the output is identically zero. Of course, for a state-space model, if the initial condition and input are both zero, then the output is zero. However, the zero dynamics have the interesting property that there exist a nonzero input and a nonzero initial condition such that the output is identically zero; this phenomenon is called output zeroing (Callier & Desoer, 2012; Desoer & Schulman, 1974; Karampetakis, 1998; Karcanias & Kouvaritakis, 1979; Tokarzewski, 2006). Along with basic results on output zeroing, Karcanias and Kouvaritakis (1979) explores the zero structure using zero pencils and describes the geometric properties of the zero structure. Zeros of discrete-time systems are discussed in detail in Tokarzewski (2006). The above-mentioned works on output zeroing consider state-space models; input–output models are not considered.
Performance recovery of a class of uncertain non-affine systems with unmodelled dynamics: an indirect dynamic inversion method
Published in International Journal of Control, 2018
Bowen Yi, Shuyi Lin, Bo Yang, Weidong Zhang
It is well known that zero dynamics (or internal dynamics) exist in numerous physical systems, and they play an important role in the areas of modeling, analysis, and control of nonlinear systems. Without loss of generality, if a nonlinear system has zero dynamics with well-defined structure, the system can be still described by the above-motioned model. To illustrate this point, consider a system with internal dynamics z as follows: subject to unmodelled dynamics as , where . Define a state vector as and the corresponding output of the -subsystem as . Hence, system (3) with unmodelled dynamics is transformed into the form of (1–2), where the ‘extended’ unmodelled subsystem is
New results on robust tracking control for a class of high-order nonlinear time-delay systems
Published in International Journal of Systems Science, 2019
Lingrong Xue, Zhenguo Liu, Zongyao Sun, Wei Sun
In control engineering, plenty of practical models such as the mobile robot (Ding, Li, & Li, 2010), the networked system (Jia, Zhang, Hao, & Zheng, 2009) and so on are inherently nonlinear systems. For such kind of systems, the zero dynamics and the external disturbances can bring serious impacts on the system performance, and sometimes make the system unstable. Besides, time-delay phenomenon also exists extensively in nonlinear systems such as chemical systems, biological systems and economical systems (Hua, Liu, & Guan, 2009). It is one of the instability factors and can severely deteriorate the system performance. Therefore, it is meaningful and necessary to study control problems of nonlinear time-delay systems with zero dynamics and the external disturbances.