11,221 research outputs found
The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation
The paper is to reveal the direct links between the well known Sylvester
equation in matrix theory and some integrable systems. Using the Sylvester
equation we introduce a scalar
function
which is defined as same as in discrete case. satisfy some
recurrence relations which can be viewed as discrete equations and play
indispensable roles in deriving continuous integrable equations. By imposing
dispersion relations on and , we find the
Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian
Korteweg-de Vries equation and sine-Gordon equation can be expressed by some
discrete equations of defined on certain points. Some special
matrices are used to solve the Sylvester equation and prove symmetry property
. The solution provides function
by . We hope our results can not only
unify the Cauchy matrix approach in both continuous and discrete cases, but
also bring more links for integrable systems and variety of areas where the
Sylvester equation appears frequently.Comment: 23 page
Fault tolerant control of a quadrotor using L-1 adaptive control
Purpose β The growing use of small unmanned rotorcraft in civilian applications means that safe operation is increasingly important. The purpose of this paper is to investigate the fault tolerant properties to faults in the actuators of an L1 adaptive controller for a quadrotor vehicle.
Design/methodology/approach β L1 adaptive control provides fast adaptation along with decoupling between adaptation and robustness. This makes the approach a suitable candidate for fault tolerant control of quadrotor and other multirotor vehicles. In the paper, the design of an L1 adaptive controller is presented. The controller is compared to a fixed-gain LQR controller.
Findings β The L1 adaptive controller is shown to have improved performance when subject to actuator faults, and a higher range of actuator fault tolerance.
Research limitations/implications β The control scheme is tested in simulation of a simple model that ignores aerodynamic and gyroscopic effects. Hence for further work, testing with a more complete model is recommended followed by implementation on an actual platform and flight test. The effect of sensor noise should also be considered along with investigation into the influence of wind disturbances and tolerance to sensor failures. Furthermore, quadrotors cannot tolerate total failure of a rotor without loss of control of one of the degrees of freedom, this aspect requires further investigation.
Practical implications β Applying the L1 adaptive controller to a hexrotor or octorotor would increase the reliability of such vehicles without recourse to methods that require fault detection schemes and control reallocation as well as providing tolerance to a total loss of a rotor.
Social implications β In order for quadrotors and other similar unmanned air vehicles to undertake many proposed roles, a high level of safety is required. Hence the controllers should be fault tolerant.
Originality/value β Fault tolerance to partial actuator/effector faults is demonstrated using an L1 adaptive controller
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