ONE WAY TO DESIGN THE CONTROL LAW OF A MINI-UAV Projet c o l e - PowerPoint PPT Presentation

ONE WAY TO DESIGN THE CONTROL LAW OF A MINI-UAV Projet É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Plan  Introduction  Model of the μ Drone  Control design of the MAV  Choice of the adjustment parameters  Conclusion É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Introduction  The ENSMM μ Drone This MAV uses six propellers to fly : - Two are counter-rotating and provide the lift - Two provide the trim stabilization - Two are used for the propulsion  Approximation of the MAV behaviour The model is a MIMO linear time-invariant system  Control design of the MAV The LQ state feedback regulator design is applied to a « standard model » To compute the weighting matrices of quadratic criterions we use a « partial observability gramian » The great advantage of this method is due to the use of only three scalars to synthesize a robust control law. É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Model of the μ Drone The μ Drone is a « gyrodyne » with six propellers We worked on the assumption that it can be described as seven solids in interaction  Model of the actuators Let denote the thrust produced by the ith actuator and the associated F M i i moment. If denotes the control input of the actuator, we assume that u i  Global model With this assumption and after linearization of the mecanical model in the vicinity of a horizontal trim, the following MIMO linear time-invariant system is an approximation of the MAV behaviour : É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Control design of the MAV  Standard model We introduce a constant perturbation vector : Initial model is then augmented :  Extraction of an observable subsystem É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 State estimator • Gain matrix is chosen as the solution of the LQ problem : A positive scalar is to be designed so that k o k o and the weighting matrices where is a partial observability gramian É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

Partial observability gramian : where State estimator is then : É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Uncontrollable modes influence rejection • Controllability staircase form of x uncontrollable modes perturb controllable ones nc • Rejection with a linear transformation leading to : É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

It is often possible to find such as : with these conditions : A state feedback is then possible… É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Control feedback • New LQ problem choice of the weighting matrices define another positive scalar , a positive control horizon k T c c such as : and the partial observability gramian : É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

• Feedback control elaboration with this feedback control the system is : and we want at permanent rate : If that matrix is invertible, the solution is : É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

Then, the control vector is : and the use of transformation matrix leads to T oc • Regulator equations with É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Choice of the adjustment parameters  First step , Assign a fixed value to Adjust filtering horizon so that is a stability matrix. T A o reg  Second step (quantify the stability robustness) The regulator, without reference, is converted to a transfer matrix with É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

• Similarly, initial model of the MAV is converted to • Now define - Loop transfers - Static margin - Dynamic margin • When is a stability matrix, feedback is robust in relation to : A reg and É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Last step Now it is not difficult to compute static and dynamic margins for different values of the three parameters : - Filtering horizon T o - Shape factor k o - Recovery factor k c The reasons why this strategy is often efficient is due to : - The natural robustness of a LQ problem solution - The notion of loop transfer recovery (LTR) É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Implementation on the μ Drone T 1.2 s o Static margin : M 42.2% st k 1.2 o Dynamic margin : M 32.7 ms dyn k 3.2 c É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Conclusion  The proposed method is quite easy to use Only three positive scalars have to be adjusted  Tricky problems Rejection of uncontrollable modes is not always possible Internal stability of regulator is sometimes difficult to reach  Philippe de Larminat’s recent book in which this method is explained in detail is of great interest. É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s