Brockett's criterion on control theory gives a necessary condition to stabilize a system by continuous time-invariant state feedback. It requires that the mapping (x,u) -> f(x,u) be open at zero...
(see https://books.google.com.br/books?id=-j5rg_0HPksC&lpg=PA165 and https://arxiv.org/pdf/1810.01368.pdf)
Brockett's criterion for continuously differentiable systems
editConsider the nonlinear control system described the differential equation
where f : X x U -> R^n is C1 class and f(0,0) = 0.
If the system is locally asymptotically stabilizable by a stationary C1 feedback law, then it is necessary that f is open at (0,0).
Brockett's criterion for continuous systems
editConsider the nonlinear control system described the differential equation
where f : X x U -> R^n is continuous and f(0,0) = 0.
If the system is locally asymptotically stabilizable by a stationary continuous feedback law, then it is necessary that f is open at (0,0).
Coron's criterion
editSee [1].
TODO
edit- Add shopping cart figure (see https://ieeexplore.ieee.org/document/4939307)
- Add region of attraction of Brockett non-holonomic integrator (see 9.4 of https://arxiv.org/pdf/1208.1751.pdf)
Brockett non-holonomic integrator
editThe Brockett integrator or nonholonomic integrator models a three-wheeled shopping cart
u_1 corresponds to the 'drive' command and u_2 is a steering command.
See also
editReferences
edit- ^ Coron, Jean-Michel (1990). "A necessary condition for feedback stabilization". Systems & Control Letters. 14 (3): 227–232. doi:10.1016/0167-6911(90)90017-O.
- ^ Christopherson, Bryce A.; Jafari, Farhad; Mordukhovich, Boris S. (2020-01-23). "Stabilization of Nonlinear Control Systems via Composition Operators". arXiv:2001.08671 [math].