Trajectory design for a nonlinear system to insure observability

In this work, the idea of coupling between actuation and observation is used to design a control policy which guarantees the observability of a nonlinear system. The Lie algebra observability matrix has been used as a tool for evaluating the observability. Different notions of observability are introduced base on Lie algebra observability matrix. In the case of some simple forms of the nonlinear systems, the control policy required for observability is obtained by looking at the geometry of the vector fields. Then, for the case of single input control-affine nonlinear systems, the Lie algebra observability condition is used to extract the control to construct an observability matrix with nearly orthogonal rows. In case of given control, an optimization problem is presented to tune the given trajectory to insure the observability. Results are demonstrated in simulation.