2021-03-30 严反馈系统
In control theory, dynamical systems are in strict-feedback form when they can be expressed as
{x˙=f0(x)+g0(x)z1z˙1=f1(x,z1)+g1(x,z1)z2z˙2=f2(x,z1,z2)+g2(x,z1,z2)z3?z˙i=fi(x,z1,z2,…,zi?1,zi)+gi(x,z1,z2,…,zi?1,zi)zi+1for?1≤i<k?1?z˙k?1=fk?1(x,z1,z2,…,zk?1)+gk?1(x,z1,z2,…,zk?1)zkz˙k=fk(x,z1,z2,…,zk?1,zk)+gk(x,z1,z2,…,zk?1,zk)u\left\{\begin{array}{l} \dot{\mathbf{x}}=f_{0}(\mathbf{x})+g_{0}(\mathbf{x}) z_{1} \\ \dot{z}_{1}=f_{1}\left(\mathbf{x}, z_{1}\right)+g_{1}\left(\mathbf{x}, z_{1}\right) z_{2} \\ \dot{z}_{2}=f_{2}\left(\mathbf{x}, z_{1}, z_{2}\right)+g_{2}\left(\mathbf{x}, z_{1}, z_{2}\right) z_{3} \\ \vdots \\ \dot{z}_{i}=f_{i}\left(\mathbf{x}, z_{1}, z_{2}, \ldots, z_{i-1}, z_{i}\right)+g_{i}\left(\mathbf{x}, z_{1}, z_{2}, \ldots, z_{i-1}, z_{i}\right) z_{i+1} \quad \text { for } 1 \leq i<k-1 \\ \vdots \\ \dot{z}_{k-1}=f_{k-1}\left(\mathbf{x}, z_{1}, z_{2}, \ldots, z_{k-1}\right)+g_{k-1}\left(\mathbf{x}, z_{1}, z_{2}, \ldots, z_{k-1}\right) z_{k} \\ \dot{z}_{k}=f_{k}\left(\mathbf{x}, z_{1}, z_{2}, \ldots, z_{k-1}, z_{k}\right)+g_{k}\left(\mathbf{x}, z_{1}, z_{2}, \ldots, z_{k-1}, z_{k}\right) u \end{array}\right. ????????????????????????????x˙=f0?(x)+g0?(x)z1?z˙1?=f1?(x,z1?)+g1?(x,z1?)z2?z˙2?=f2?(x,z1?,z2?)+g2?(x,z1?,z2?)z3??z˙i?=fi?(x,z1?,z2?,…,zi?1?,zi?)+gi?(x,z1?,z2?,…,zi?1?,zi?)zi+1??for?1≤i<k?1?z˙k?1?=fk?1?(x,z1?,z2?,…,zk?1?)+gk?1?(x,z1?,z2?,…,zk?1?)zk?z˙k?=fk?(x,z1?,z2?,…,zk?1?,zk?)+gk?(x,z1?,z2?,…,zk?1?,zk?)u?
where
- x∈Rn\mathbf{x} \in \mathbb{R}^{n}x∈Rn with n≥1n \geq 1n≥1
- z1,z2,…,zi,…,zk?1,zkz_{1}, z_{2}, \ldots, z_{i}, \ldots, z_{k-1}, z_{k}z1?,z2?,…,zi?,…,zk?1?,zk? are scalars,
- uuu is a scalar input to the system, - f0,f1,f2,…,fi,…,fk?1,fkf_{0}, f_{1}, f_{2}, \ldots, f_{i}, \ldots, f_{k-1}, f_{k}f0?,f1?,f2?,…,fi?,…,fk?1?,fk? vanish at the origin (i.e., fi(0,0,…,0)=0)\left.f_{i}(0,0, \ldots, 0)=0\right)fi?(0,0,…,0)=0)
- g1,g2,…,gi,…,gk?1,gkg_{1}, g_{2}, \ldots, g_{i}, \ldots, g_{k-1}, g_{k}g1?,g2?,…,gi?,…,gk?1?,gk? are nonzero over the domain of interest (i.e., gi(x,z1,…,zk)≠0g_{i}\left(\mathbf{x}, z_{1}, \ldots, z_{k}\right) \neq 0gi?(x,z1?,…,zk?)?=0 for 1≤i≤k)\left.1 \leq i \leq k\right)1≤i≤k)
Here, strict feedback refers to the fact that the nonlinear functions fif_{i}fi? and gig_{i}gi? in the z˙i\dot{z}_{i}z˙i? equation only depend on states x,z1,…,zix, z_{1}, \ldots, z_{i}x,z1?,…,zi? that are fed back to that subsystem. [1]^{[1]}[1] That is, the system has a kind of lower triangular form.
Reference: https://en.wikipedia.org/wiki/Strict-feedback_form
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