On the $L^p$-stabilization of the double integrator subject to input saturation

Chitour, Yacine

On the

L^{p}

-stabilization of the double integrator subject to input saturation

Chitour, Yacine

ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 291-331

Résumé

We consider a finite-dimensional control system $(Σ) \dot{x} (t) = f (x (t), u (t))$ , such that there exists a feedback stabilizer $k$ that renders $\dot{x} = f (x, k (x))$ globally asymptotically stable. Moreover, for $(H, p, q)$ with $H$ an output map and $1 \leq p \leq q \leq \infty$ , we assume that there exists a $𝒦_{\infty}$ -function $α$ such that $∥ H (x_{u}) ∥_{q} \leq {α (∥ u ∥}_{p})$ , where $x_{u}$ is the maximal solution of ${(Σ)}_{k} \dot{x} (t) = f (x (t), k (x (t)) + u (t))$ , corresponding to $u$ and to the initial condition $x (0) = 0$ . Then, the gain function $G_{(H, p, q)}$ of $(H, p, q)$ given by

G_{(H, p, q)} (X) \overset{def}{=} sup_{{∥ u ∥}_{p} = X} {∥ H (x_{u}) ∥}_{q},

is well-defined. We call profile of

k

for

(H, p, q)

any

𝒦_{\infty}

-function which is of the same order of magnitude as

G_{(H, p, q)}

. For the double integrator subject to input saturation and stabilized by

k_{L} (x) = - {(1 1)}^{T} x

, we determine the profiles corresponding to the main output maps. In particular, if

σ_{0}

is used to denote the standard saturation function, we show that the

L_{2}

-gain from the output of the saturation nonlinearity to

u

of the system

\ddot{x} = σ_{0} (- x - \dot{x} + u)

with

x (0) = \dot{x} (0) = 0

, is finite. We also provide a class of feedback stabilizers

k_{F}

that have a linear profile for

(x, p, p)

,

1 \leq p \leq \infty

. For instance, we show that the

L_{2}

-gains from

x

and

\dot{x}

to

u

of the system

\ddot{x} = σ_{0} (- x - \dot{x} - {(\dot{x})}^{3} + u)

with

x (0) = \dot{x} (0) = 0

, are finite.

MR Zbl

Classification : 93D15, 93D21, 93D30
Keywords: nonlinear control systems, $L^p$-stabilization, input-to-state stability, finite-gain stability, input saturation, Lyapunov function

@article{COCV_2001__6__291_0,
     author = {Chitour, Yacine},
     title = {On the $L^p$-stabilization of the double integrator subject to input saturation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {291--331},
     year = {2001},
     publisher = {EDP Sciences},
     volume = {6},
     mrnumber = {1824105},
     zbl = {0996.93082},
     language = {en},
     url = {https://www.numdam.org/item/COCV_2001__6__291_0/}
}

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Chitour, Yacine. On the $L^p$-stabilization of the double integrator subject to input saturation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 291-331. https://www.numdam.org/item/COCV_2001__6__291_0/

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