  This paper studies the extremum seeking control (ESC) problem for a class of
constrained nonlinear systems. Specifically, we focus on a family of
constraints allowing to reformulate the original nonlinear system in the
so-called input-output normal form. To steer the system to optimize a
performance function without knowing its explicit form, we propose a novel
numerical optimization-based extremum seeking control (NOESC) design consisting
of a constrained numerical optimization method and an inversion based
feedforward controller. In particular, a projected gradient descent algorithm
is exploited to produce the state sequence to optimize the performance
function, whereas a suitable boundary value problem accommodates the
finite-time state transition between each two consecutive points of the state
sequence. Compared to available NOESC methods, the proposed approach i) can
explicitly deal with output constraints; ii) the performance function can
consider a direct dependence on the states of the internal dynamics; iii) the
internal dynamics do not have to be necessarily stable. The effectiveness of
the proposed ESC scheme is shown through extensive numerical simulations.
