Safe Control Design for Unknown Nonlinear Systems with Koopman-based Fixed-Time Identification
Mitchell Black and Dimitra Panagou
Submitted to 2023 IFAC World Congress (under review)
We consider the problem of safe control design for a class of nonlinear, control-affine systems subject to an unknown, additive, nonlinear disturbance. Leveraging recent advancements in the application of Koopman operator theory to the field of system identification and control, we introduce a novel fixed-time identification scheme for the infinitesimal generator of the infinite-dimensional, but notably linear, Koopman dynamical system analogous to the nonlinear system of interest. That is, we derive a parameter adaptation law that allows us to recover the unknown, residual nonlinear dynamics in the system within a finite-time independent of an initial estimate. We then use properties of fixed-time stability to derive an error bound on the residual vector field estimation error as an explicit function of time, which allows us to synthesize a provably safe controller using control barrier function based methods. We conduct a quadrotor-inspired case study in support of our proposed method, in which we show that safe trajectory tracking is achieved despite unknown, nonlinear dynamics.
XY paths under the various CBF-QP control laws in the double-integrator example. Only the controllers using the proposed Koopman-based fixed-time identification scheme succeed in preserving safety.
The estimates of the unknown wind gusts. In our scheme, the estimates converge to the true values within the fixed-time T without noise, and converge to a close approximation in the presence of measurement noise
Control inputs for the perturbed double-integrator example.
Evolutions of h1 and h2 for the various controllers considered in the double-integrator example.
