# Strict-feedback Form

Strict-feedback Form

In control theory, dynamical systems are in strict-feedback form when they can be expressed as

$\begin{cases} \dot{\mathbf{x}} = f_0(\mathbf{x}) + g_0(\mathbf{x}) z_1\\ \dot{z}_1 = f_1(\mathbf{x},z_1) + g_1(\mathbf{x},z_1) z_2\\ \dot{z}_2 = f_2(\mathbf{x},z_1,z_2) + g_2(\mathbf{x},z_1,z_2) z_3\\ \vdots\\ \dot{z}_i = f_i(\mathbf{x},z_1, z_2, \ldots, z_{i-1}, z_i) + g_i(\mathbf{x},z_1, z_2, \ldots, z_{i-1}, z_i) z_{i+1} \quad \text{ for } 1 \leq i < k-1\\ \vdots\\ \dot{z}_{k-1} = f_{k-1}(\mathbf{x},z_1, z_2, \ldots, z_{k-1}) + g_{k-1}(\mathbf{x},z_1, z_2, \ldots, z_{k-1}) z_k\\ \dot{z}_k = f_k(\mathbf{x},z_1, z_2, \ldots, z_{k-1}, z_k) + g_k(\mathbf{x},z_1, z_2, \dots, z_{k-1}, z_k) u\end{cases}$

where

• with ,
• are scalars,
• is a scalar input to the system,
• vanish at the origin (i.e., ),
• are nonzero over the domain of interest (i.e., for ).

Here, strict feedback refers to the fact that the nonlinear functions and in the equation only depend on states that are fed back to that subsystem. That is, the system has a kind of lower triangular form.