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Provision of Global Stability
Published in Boris J. Lurie, Paul J. Enright, Classical Feedback Control with Nonlinear Multi-Loop Systems, 2020
Boris J. Lurie, Paul J. Enright
Nonlinearities in the actuator, feedback path, and plant are reviewed. The following concepts are developed: limit cycles, stability of linearized systems, conditional stability, global stability, and absolute stability. The Popov criterion is discussed and applied to control system analysis and design. Nonlinear dynamic compensators (NDCs), which ensure absolute stability without penalizing the available feedback, are introduced.
Stability of Non-Linear Systems
Published in T. Thyagarajan, D. Kalpana, Linear and Non-Linear System Theory, 2020
The Popov criterion is a stability criterion by Vasile M. Popov. It illustrates the asymptotic stability of non-linear systems whose nonlinearity is strictly under sector condition. Popov criterion is applicable to sector-type nonlinearity and linear time-invariant system only. It can also be applied for systems with delay and higher-order systems.
Frequency conditions for stable networked controllers with time-delay
Published in International Journal of Control, 2019
Johannes Nygren, Torbjörn Wigren, Kristiaan Pelckmans
NCSs are as of now fairly well investigated, at least for the case of linear plants controlled over bandlimited channels. The data-rate theorem of NCSs, for example, states the minimum data rate that is needed to stabilise a plant, a result which supports data channel dimensioning, see Baillieul (2002), Nair and Evans (2000). Specific schemes to encode control signals have been developed in Goodwin, Quevedo, and Silva (2008). Many applications have been reported over the years. The references Lee, Chen, and Chen (2004) and Quevedo and Wigren (2012) describe power control schemes for wireless systems, while Liu, Xia, Rees, and Hu (2007) and Wigren (2016) focus on the effect of delay on data flow control between network nodes. In Wigren (2015), a stability analysis inspired by the wireless flow control problem of Wigren (2016) was performed. A system with general dynamics, subject to long delays and a saturation in the feedback loop was studied. It was proven that unless the static loop gain is bounded and constrained by a relation between the open-loop zeros, the open-loop poles and the slope of nonlinearity, then stability does not follow from the classical input–output Popov criterion, see, for example, Vidyasagar (1978). This asymptotic result holds when the loop delay tends to infinity. The very important implication is that a stable integrating control may not be feasible for loops dominated by delay in NCS. Rather leaky integration or lead–lag type controllers need to be used, as illustrated, for example, in Wigren (2016).
ILC-adapted parameter optimization of cross-coupled single-input fuzzy tracking controllers for an X-Y positioning table
Published in Journal of the Chinese Institute of Engineers, 2020
The intelligent control schemes also offer flexible tracking behavior. The Conventional-Fuzzy-Logic-Controller (CFLC) is a model-free and linguistic-based scheme that reaches robust control decisions by using multiple qualitative logical rules (Abid and Toumi 2016). It mimics the process of human decision-making to control the systems with parametric uncertainties. Generally, the CFLCs aggregate the real-time variations in system’s error and error-derivative on the basis of an empirically defined two-dimensional rule-base (Jie et al. 2014). The rule-base is defined such that it effectively compensates for the disturbances caused by environmental indeterminacies. Despite their resilience, the CFLCs also put excessive computational burden on the embedded processor, owing to their dependence on a large two-dimensional rule-base for making accurate inferences in real-time. Due to this deficiency, the utilization of two-input CFLC is avoided in real-time control applications with small sampling intervals. The Single-Input FLC (or SIFLC) offers a tenable solution to the aforementioned problem (Lee and Shih 2012a, 2012b). As the name implies, it depends on a single input-variable that is derived by unifying the error and error-derivative signals. It simplifies the inference mechanism by transforming the two-dimensional rule-base into a one-dimensional rule-vector (Saleem, Shami, and Mahmood‐ul‐Hasan 2019). Consequently, fewer rules are required to make the desired inference, which reduces the execution-time while yielding an equally robust control effort. The Popov criterion has been used to prove the absolute stability of SIFLC (Choi, Kwak, and Kim 2000). Choosing a unique set of center locations for fuzzy Membership-Functions (MFs) is an ill-posed problem. Well-postulated MFs enhance the controller’s degrees-of-freedom to compensate the system’s un-modeled intrinsic nonlinearities (Bhatti et al. 2018).