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STATE SWITCHING USING PD-LIKE CONTROL IN MULTISTABLE SYSTEMS
Last modified: 2017-12-04
Abstract
Multistability is a ubiquitous phenomenon in real world, such as biological neurons, climate, chemical reactions, drill-strings in oil well drilling, electronic oscillators, vibrational energy harvesters, optical ring cavity and resonators, etc. Control of multistable systems can increase their efficiency and the exact dynamics of the systems is not needed. In order to control multistable systems, a PD-like controller is proposed. This controller can control multistable systems without destroying their original basins of attraction. Moreover, this controller makes systems to stay in their desired state when it is impacted by sensor noise and external disturbance. In this paper, an impact oscillator without and with a drift for percussive drilling (Figure 1) are investigated for the performance of the proposed controller because they are common multistable systems in oil and gas industry and the real world.
The asymptotic stability of the proposed controller on the impact oscillator and the impact oscillator with a drift, when both the current state and desired state exhibit the same behaviours, namely non-contact, contact without progression and contact with progression, are proved. Moreover, a stability bound of the velocity gain of control law is found and this bound can guarantee the asymptotic stability of the control law. Figure 2 shows that an impact oscillator with a drift exhibiting multiple stable states with different levels of efficiency. Switching the system to state with lower energy consumption can increase system efficiency. Simulations are conducted to show that the proposed controller can switch the impact oscillator with a drift (percussive drilling) from arbitrary state to the desired state successfully as shown in Figure 3. Moreover, no drilling progression is observed before applying the proposed controller as the mass moves but the bottom slider does not and the drill bit stays in the same position (Figure 3(a)). On the other hand, the drilling progresses as the mass moves with the bottom slider (Figure 3(b)).
In summary, the proposed PD-like controller is able to switch multistable systems to desired states with higher efficiency.
The asymptotic stability of the proposed controller on the impact oscillator and the impact oscillator with a drift, when both the current state and desired state exhibit the same behaviours, namely non-contact, contact without progression and contact with progression, are proved. Moreover, a stability bound of the velocity gain of control law is found and this bound can guarantee the asymptotic stability of the control law. Figure 2 shows that an impact oscillator with a drift exhibiting multiple stable states with different levels of efficiency. Switching the system to state with lower energy consumption can increase system efficiency. Simulations are conducted to show that the proposed controller can switch the impact oscillator with a drift (percussive drilling) from arbitrary state to the desired state successfully as shown in Figure 3. Moreover, no drilling progression is observed before applying the proposed controller as the mass moves but the bottom slider does not and the drill bit stays in the same position (Figure 3(a)). On the other hand, the drilling progresses as the mass moves with the bottom slider (Figure 3(b)).
In summary, the proposed PD-like controller is able to switch multistable systems to desired states with higher efficiency.