Review of " An optimal sliding mode-like regulator for nonlinear polynomial systems"

COSTANZA, V

Mathematical Reviews

American Mathematical Society

Lugar: Providence, RI; Año: 2012 p. 1 - 1

0025-5629

    In this paper a sliding-mode control strategy is proposed to treat an optimal control problem involving a special class of nonlinear, nonautonomous systems, namely those of the form  x=f(x,t)+B(t)u, where f is a polynomial in the state variable x with time-dependent coefficients, and the control variable u is unbounded.  The finite-horizon cost objective to be minimized includes a Lagrangian and a final penalization which are both linear in the state norm, and a quadratic term for the control energy.  This makes the problem different from the well-known linear-quadratic (LQR) case, but falls into the class problems that can be treated with the help of Lyapunov theory (see for instance Sontag, E. D.:  Mathematical Control Theory, 2nd. edition (1998), Springer, page 392).        It is not totally clear the importance of studying this particular problem (other than the fact that its solution apparently leads to sliding-mode type of strategies).  The authors claim that the problems at hand appear in Classical Mechanics, in situations where the state velocity needs not be minimized.  However, in the worked-out example the state variable x©ü is basically a forced version of the velocity of x©û, and not only x©û but also x©ü is penalized in the Lagrangian, making unconvincing their statement.  Also, the fact that the Hamiltonian may become nonsmooth by the appearing of the sign function in the costate equation is not adequately taken into account during the proof sketched in the Appendix.  Last but not least, the final-condition ODE derived for the Riccati-like matrix Q is an inconvenience in trying to present the solution as a strict feedback law, since the state is needed in calculating Q.  Resorting to shooting strategies to determine Q offline does not seem too attractive in practical terms.     In the reviewer´s opinion, the paper lacks some clearness in the motivating exposition, as well as deepness in the discussion of the technical details involved in the proof of the main result.