Performance analysis of reset control systems

Abstract

In this paper we present a general linear matrix inequality-based analysis method to determine the performance of a SISO reset control system in both the L 2 gain and H 2 sense. In particular, we derive convex optimization problems in terms of LMIs to compute an upperbound on the L 2 gain performance and the H 2 norm, using dissipativity theory with piecewise quadratic Lyapunov functions. The results are applicable to for all LTI plants and linear-based reset controllers, thereby generalizing the available results in the literature. Furthermore, we provide simple though convincing examples to illustrate the accuracy of our proposed L 2 gain and H 2 norm calculations and show that, for an input constrained H 2 problem, reset control can outperform a linear controller designed by a common nonlinear optimization method. of the controller states when the switching surface is reached. The rationale behind reset control can best be understood when considering an integrating controller in particular. An integrator 'sums' the error over time in order to achieve a zero steady-state error. However, when the error becomes zero for the first time, the integrator still has the 'summed' error stored in its states, and the subsequent 'emptying' of the integrator causes the error signal to overshoot. To reduce the overshoot, it might be beneficial to reset the state of this integrator to zero as soon as its input (the error) becomes zero. In this way the integrator state, containing the 'summed' error, is emptied instantaneously, and hence the overshoot is avoided. This concept was indeed validated by simulations in recent publications [5,6], showing a significant decrease of overshoot in the step response.The concept of reset control was first introduced in 1958 by means of the resetting integrator of Clegg [7]. The describing function of the Clegg integrator has

Lyapunov-based hybrid loops for stability and performance of continuous-time control systems

“…See also Beker et al (2004); Nesić et al (2008) where reset controllers are used to decrease the L 2 gain between perturbations and the output. Consider also Aangenent et al (2010) where it is shown that reset con-1 Work supported by the ANR project ArHyCo, ARPEGE, contract number ANR-2008 SEGI 004 01-30011459, and by HYCON2 Network of Excellence "Highly-Complex and Networked Control Systems", grant agreement 257462. christophe.prieur@gipsa-lab.fr sophie.tarbouriech@laas.fr zaccarian@laas.fr trollers may be useful to improve the L 2 or H 2 stability of linear systems. Finally, see Lazar and Heemels (2009) for the design of predictive controllers for the input-tostate stability of hybrid systems.…”

Reset passivation of nonlinear controllers via a suitable time-regular reset map

“…Introducing resets has advantages and disadvantages, as pointed out in Aangenent et al (2010). Indeed, the performance improvements of resets controllers with respect to continuous dynamics branch out even to the possibility of using controller dynamics that exponentially destabilize the continuous-time closed loop without resets (see Section 4).…”

“…This fact might be important to keep in mind when embedding reset passivation rules. Of course, as underlined in Aangenent et al (2010), providing fully constructive synthesis tools for reset controllers is an open research problem.…”

“…FORE received much attention in recent years and have been proven to overcome some intrinsic limitations of linear controllers (Beker, Hollot, & Chait, 2001). Moreover, by relying on Lyapunov approaches, suitable analysis and synthesis tools for the stability of a class of reset systems generalizing control systems with FORE have been proposed in Aangenent, Witvoet, Heemels, van de Molengraft, and Steinbuch (2010), Beker, Hollot, Chait, and Han (2004), Nešić, Zaccarian, and Teel (2008) and the references therein. Moreover, in the recent paper (Carrasco, Banos, & Van der Schaft, 2010) the L 2 stability of reset control systems has been addressed in the passivity context, by showing interesting properties of the reset system under the assumption that the continuous-time part of the reset controller is passive before resets and that a suitable non-increase condition is satisfied by the storage function at jumps.…”

Limit cycles analysis of reset control systems with reset band

“…Thus, stability of reset control systems is a main concern from a theoretical and practical point of view, and several recent works have approached the stability problem. See for example [6][7][8], and the works [9][10][11].…”