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Analysis Methods
Published in William S. Levine, Control System Fundamentals, 2019
The problem is illustrated by the system of Figure 19.1 with a symmetrical odd single-valued nonlinearity confined to a sector between lines of slope k1 and k2, that is, k1\x < n(x) < k2x for x > 0. For absolute stability, the circle criterion requires satisfying the Nyquist criterion for the locus G(jω) for all points within a circle having its diameter on the negative real axis of the Nyquist diagram between the points (–l/k1, 0) and (–l/k2, 0). as shown in Figure 19.9. On the other hand, because the DF for this nonlinearity lies within the diameter of the circle, the DF method requires satisfying the Nyquist criterion for G(jω) for all points on the circle diameter, if the autonomous system is to be stable.
Analysis Methods
Published in William S. Levine, The Control Handbook: Control System Fundamentals, 2017
The problem is illustrated by the system in Figure 18.1 with a symmetrical odd single-valued nonlinearity confined to a sector between lines of slope k1 and k2, that is, k1x < n(x) < k2x for x > 0. For absolute stability, the circle criterion essentially requires satisfying the Nyquist criterion for the locus G(jω) for all points within a circle having its diameter on the negative real axis of the Nyquist diagram between the points (−1/k1, 0) and (−1/k2, 0), as shown in Figure 18.9. On the other hand, because the DF for this nonlinearity lies within the diameter of the circle, the DF method requires satisfying the Nyquist criterion for G(jω) for all points on the circle diameter, if the autonomous system is to be stable.
Stability of Non-Linear Systems
Published in T. Thyagarajan, D. Kalpana, Linear and Non-Linear System Theory, 2020
Circle criterion states that, the closed loop system is asymptotically stable if the Nyquist plot of G(jω) does not intersect the circle given by the intercepts of x and y, except the fact that, the system undergoes pole-zero cancellation.
Trajectory tracking control for a quadrotor UAV via extended state observer
Published in Systems Science & Control Engineering, 2018
Wendong Gai, Jie Liu, Chengzhi Qu, Jing Zhang
The PD-type trajectory tracking controller with the ESO is proposed to improve the trajectory tracking performance of the quadrotor UAV in the presence of the wind disturbance. The six-degree-of-freedom quadrotor UAV model is built, and the accuracy of the model is verified by the actual fight. In addition, the stability of the closed-loop system with the proposed controller is proved by the circle criterion. The simulation results indicate that the proposed method has the better tracking performance for the attitude and position than the PD tracking controller. And the wind disturbance can be estimated by the ESO with the small estimation error. In the future, the parameters uncertainty and faults will be considered for the trajectory tracking of the quadrotor UAV.
Frequency conditions for stable networked controllers with time-delay
Published in International Journal of Control, 2019
Johannes Nygren, Torbjörn Wigren, Kristiaan Pelckmans
The present paper focuses on further results on -stability for NCSs subject to long delays and a static nonlinearity in the loop. The main contribution of the paper proves the equivalence between a number of statements that, for example, imply the necessity of the Popov- and circle-criteria when the loop gain is uniformly less than one over all frequencies. Although the loop gain assumption is very restrictive performance-wise, the result serves to extend the understanding of, for example, Wigren (2015). Here, it needs to be understood that the results of the present paper and Wigren (2015) cover slightly different cases. The present paper considers finite delays, assuming a loop gain that is small enough for all frequencies, while Wigren (2015) considers the case where the loop delay tends to infinity using conditions only on the static loop gain. This, for example, means that methods that exploit the results in Wigren (2015) as a tuning guideline in case of large loop delays, can be used to design stable controllers with loop gains that extend beyond one. This can be achieved by utilising an off-line pre-computation of the -stability region for finite but large loop delays as in Wigren (2016). Similar loop gain criteria are found in Verriest, Fan, and Kullström (1993), guaranteeing delay-independent stability of linear delay systems. Moreover, while the earlier work of Wigren (2015) concerned saturating nonlinearities and Verriest et al. (1993) concerned linear feedback, the present paper generalises this to nonlinearities satisfying a general sector condition. A further contribution proves that similar necessary and sufficient conditions for input–output stability hold for the circle criterion, and in the linear case also for the Nyquist criterion. The tools of analysis are provided by the input–output stability theory as pioneered in Sandberg (1964), Zames (1966). The results are illustrated by a simulation study.