Improved Transient Performance by Lyapunov-based Reset of Dynamic Controllers

Abstract

Many control applications implemented in the industry are optimized for use in normal conditions. However, special or extreme conditions may occur, i.e. sudden changes in the environmental parameters or the controller set-points. These incidents may lead to unsatisfactory performance of the closed-loop system. A dynamic controller may, in the first place, interpret this as measurement noise to be ignored, as a result of being optimized for steady-state noise performance. Adaptive controllers are often designed in order to achieve asymptotic stability (AS) requirements, hence the transient performance may only be satisfactory for slow varying environmental parameters and controller set-points.

In this thesis, a framework for improving controller performance in such situations is presented, followed by three application examples.

The framework is presenting a controller strategy using a Lyapunov-based resetting mechanism, changing one or more of the controller states to a different value when an extreme situation is detected. The controller states are reset to a different value if this leads to a drop in the Lyapunov function value. An appropriately selected Lyapunov function is expected to lead to a controller with increased transient performance. In this way, the Lyapunov function is used both as a part in the controller algorithm and as a tool for proving stability for the whole controller system.

The system performance may then be divided into two separate regimes, having two separate sets of specifications and tuning variables being decoupled from each other. Performance close to steady-state, being one of the regimes, is typically tuned in order to handle normal conditions. In the other regime, however, a separate part of the controller is having a fast reacting performance, triggered by sudden changes in the environmental parameters or the controller set-points. This regime is referred to as the systems transient regime. The fast reacting performance in this regime typically leads to reduced measurement noise suppressions, but if appropriately tuned, this will only be triggered when severe changes in the environmental parameters or the controller set-points is measured. In this way, the steady-state noise performance of the overall controller system may be improved, without reducing the transient performance.

The main challenge encountered developing a controller using this strategy, is the search for a suitable Lyapunov function needed in the resetting strategy. One problem is to find a Lyapunov function, another is to make sure this is a suitable measure for the system’s transient energy. The first problem may be solved in some situations where development of the dynamic controller also yields a Lyapunov function for the closed-loop system. The other problem is only briefly discussed. Optimization theory is, however, being motivated as a promising solution, tried out on a simple example system.

A fast observer is needed in order to estimate the Lyapunov function value, since this is not available for measurement in practice. Hence, additional precautions need to be undertaken when implementing the strategy. In particular, filtering effects are studied in detail for a Lyapunovbased resetting mechanism applied on an adaptive backstepping controller for parametric-strictfeedback systems. Tuning guidelines are also provided in order to prevent erroneous resets due to the filtering effects. Simulations of a car braking on a road, partly covered with ice or water, is given as the first example application presented in this thesis. Improved transient performance is shown in these cases of sudden changes in the road/tyre friction parameter.

Another challenge in using this resetting strategy is how to ensure parameter convergence of the fast observer described above. In addition to assume that the exciting signal is sufficiently rich during the transient period where typically a reset takes place, some monitoring of the richness of this signal is proposed followed by a suitable action to be taken.

This first example system is tested using simulations, hence being able to compare estimated and real values of the plant parameters. The other two examples presented in this thesis are not having this quality, due to plant parameters being unknown, but they illustrate other aspects being important when performing experimental testings.

Transient conditions, may also happen for ships operating in rough seas. There is a considerable loss in propeller thrust in case of ventilation and in-and-out-of-water effects. The today’s industrial standard for electrically driven thrusters are shaft speed proportional-integral (PI) controllers, optimized for dynamic positioning (DP) and low speed manoeuvering in normal sea conditions. In extreme sea conditions, however, the propeller may start to spin when severe thrust losses occur. Hence, a Lyapunov-based controller state resetting strategy for the integrator state in the PI-controller is proposed. Increased transient performance in propeller speed is shown by experimental tests, carried out in a test basin, being the second example application presented in this thesis.

The reset strategy is also augmented to a thruster controller for transit operations. Additional losses due to operation in transit need to be handled. This is managed by modifying the plant model and upgrading the PI-controller used in the DP case. Experimental tests in a towing tank are carried out in order to show the increased transient performance in the propeller speed, constituting this thesis’ third example application. In this case, also the power fluctuations are reduced, leading to reduced risk of blackout in the vessel power distribution system, in extreme sea conditions.