Identifying limits of linear control design validity in nonlinear systems: a continuation-based approach

Abstract

It is well known that a linear-based controller is only valid near the point from which the linearised system is obtained. The question remains as to how far one can move away from that point before the linear and nonlinear responses differ significantly, resulting in the controller failing to achieve the desired performance. In this paper, we propose a method to quantify these differences. By appending a harmonic oscillator to the equations of motion, the frequency responses at different operating points of a nonlinear system can be generated using numerical continuation. In the presence of strong nonlinearities, subtle differences exist between the linear and nonlinear frequency responses, and these variations are also reflected in the step responses. A systematic way of comparing the discrepancies between the linear and the nonlinear frequency responses is presented, which can determine whether the controller performs as predicted by linear-based design. We demonstrate the method on a simple fixed-gain Duffing system and a gain-scheduled reduced-order aircraft model with a manoeuvre-demand controller; the latter presents a case where strong nonlinearities exist in the form of multiple attractors. The analysis is then expanded to include actuator rate saturation, which creates a limit-cycle isola, coexisting multiple solutions (corresponding to the so-called flying qualities cliff), and chaotic motions. The proposed method can infer the influence of these additional attractors even when there is no systematic way to detect them. Finally, when severe rate saturation is present, reducing the controller gains can mitigate—but not eliminate—the risk of limit-cycle oscillation.

Isola in a linear one‐degree‐of‐freedom feedback system with actuator rate saturation

“…Closed-loop systems with rate limiting can better reflect real-world setups, although this presents a new set of challenges to researchers. As far as the authors of the current manuscript are concerned, feedback systems with rate limiting can show complex dynamics with bifurcations 9,10 and isolas, 11,12 although this has only been observed in unforced systems. This short communication aims to facilitate further discussion of isolas in a more realistic setting.…”

Frequency, Phase, and Magnitude Difference Locked-Loop Based Linear Synchronization Scheme for Islanded Inverters and Microgrids

“…As seen from ( 3), the equation is highly nonlinear due to both (ω 1 − ω s ) t and the sine function. So, linear controllers such as PI may not stabilize nonlinear PSL [9], [27], [57], [91], [93], [162]. In addition, a limited range in between ±1 rad is the case for the conventional synchronization method, accordingly, error signal, e qc , oscillates between the limits of ±1 rad under large-signal transients [104], [105].…”

“…Furthermore, all those studies use linear Proportional-Integral (PI) controllers in PSL and VSL, ignoring this nonlinearity [10], [74], [86], [91], [92]. It is a well-known fact in literature; it is not possible for linear controllers used in nonlinear systems to be analytically designed, and the resulting control system is always subject to stability and performance issues [93], [94], [95]. In studies published in the literature [10], [19], [20], [21], [22], [23], [57], [74], [86], [91], [92] no paper presents a solution coping with the nonlinearity mentioned above.…”

Synergetic control of micro positioning stage piezoelectric actuator

“…Which characterize their functioning and the specific requirements of their control circuits make the control of these systems difficult [16]. Indeed, modelling and control are essential to achieve the objectives of high precision movement [17].…”