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Backstepping design with local optimality matching. (English) Zbl 1007.93025

The authors study the nonlinear ${H}^{\infty }$-optimal control design for strict-feedback nonlinear systems with disturbance inputs:

$\left\{\begin{array}{cc}\hfill {\stackrel{˙}{x}}_{1}& ={x}_{2}+{f}_{1}\left({x}_{1}\right)+{h}_{1}^{\text{'}}\left({x}_{1}\right)w,\hfill \\ \hfill {\stackrel{˙}{x}}_{2}& ={x}_{3}+{f}_{2}\left({x}_{1},{x}_{2}\right)+{h}_{2}^{\text{'}}\left({x}_{1},{x}_{2}\right)w,\hfill \\ & \cdots \hfill \\ \hfill {\stackrel{˙}{x}}_{n}& =u+{f}_{n}\left({x}_{1},\cdots ,{x}_{n}\right)+{h}_{n}^{\text{'}}\left({x}_{1},\cdots ,{x}_{n}\right)u,\hfill \end{array}\right\$

where $x=\left({x}_{1},\cdots ,{x}_{n}\right)$ is the state variable with $x\left(0\right)=0$; $u$ is the scalar control input; $w$ is the $q$-dimensional disturbance input generated by some adversary player according to $w\left(t\right)=\nu \left(t,x\right)$, where $\nu :\left[0,\infty \right)×{ℝ}^{n}\to {ℝ}^{p}$ is piecewise continuous in $t$ and locally Lipschitz in $x$. The authors construct globally stabilizing control laws to match the optimal control law up to any desired order and to be inverse optimal with respect to some computable cost functional. The recursive construction of a cost functional and the corresponding solution to the Hamilton-Jacobi-Isaacs equation employs a new concept of nonlinear Cholesky factorization. When the value function for the system has a nonlinear Cholesky factorization, the backstepping design procedure can be tuned to yield the optimal control law.

##### MSC:
 93B36 ${H}^{\infty }$-control 93D21 Adaptive or robust stabilization 93C73 Perturbations in control systems 93C10 Nonlinear control systems