反推控制 （backstepping）也称为反演控制 或反步法 ，是一种控制理论 的技术，在约1990年时由Petar V. Kokotovic [ 1] [ 2] 非线性 动力系统 设计可以稳定系统 的控制器。此一系统是由许多子系统一层一层组成，最内层的子系统不可再简化，可以由其他方式稳定最内层的系统。由于此系统的递归 结构，设计者可以以最内层可稳定的系统为启始点，反推新的控制器来稳定较外层的子系统，此程序会一直进行到处理到最外层的外部控制命令为止。因此此方式称为是“反推控制”[ 3] 。
反推控制的设计方式可以针对严格回授型式 递归 方式使其在原点 可以稳定 。考虑以下型式的动力系统 [ 3]
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{\displaystyle {\begin{aligned}{\begin{cases}{\dot {\mathbf {x} }}&=f_{x}(\mathbf {x} )+g_{x}(\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}}\end{aligned}}}
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纯量 。u
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{\displaystyle f_{x},f_{1},f_{2},\ldots ,f_{i},\ldots ,f_{k-1},f_{k}}
原点 处数值为零 （也就是说
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{\displaystyle g_{1},g_{2},\ldots ,g_{i},\ldots ,g_{k-1},g_{k}}
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另外假设系统
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{\displaystyle {\dot {\mathbf {x} }}=f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )u_{x}(\mathbf {x} )}
在原点处有李雅普诺夫稳定性 ，可以用某种已知的控制方式
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李亚普诺夫函数
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{\displaystyle V_{x}}
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在x
反推控制的控制输入u
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{\displaystyle z_{n}}
状态
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{\displaystyle z_{n}}
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{\displaystyle z_{n-1}}
此程序会继续，使每一个状态
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反推控制会确认用
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x x u
递归控制的作法如下
假设较小（低阶）的子系统
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{\displaystyle {\dot {\mathbf {x} }}=f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )u_{x}(\mathbf {x} )}
已经可以用一些控制方式
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{\displaystyle u_{x}(\mathbf {x} )}
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{\displaystyle u_{x}}
李亚普诺夫函数
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{\displaystyle V_{x}}
会设计控制信号
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{\displaystyle u_{1}(\mathbf {x} ,z_{1})}
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{\displaystyle {\dot {z}}_{1}=f_{1}(\mathbf {x} ,z_{1})+g_{1}(\mathbf {x} ,z_{1})u_{1}(\mathbf {x} ,z_{1})}
稳定，让
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{\displaystyle V_{1}(\mathbf {x} ,z_{1})=V_{x}(\mathbf {x} )+{\frac {1}{2}}(z_{1}-u_{x}(\mathbf {x} ))^{2}}
此控制信号
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会设计控制信号
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{\displaystyle u_{2}(\mathbf {x} ,z_{1},z_{2})}
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{\displaystyle {\dot {z}}_{2}=f_{2}(\mathbf {x} ,z_{1},z_{2})+g_{2}(\mathbf {x} ,z_{1},z_{2})u_{2}(\mathbf {x} ,z_{1},z_{2})}
稳定，让
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{\displaystyle V_{2}(\mathbf {x} ,z_{1},z_{2})=V_{1}(\mathbf {x} ,z_{1})+{\frac {1}{2}}(z_{2}-u_{1}(\mathbf {x} ,z_{1}))^{2}}
此控制信号
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会反复此一程序，一直到其实际u 真实控制信号u
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{\displaystyle z_{k}}
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{\displaystyle u_{k-1}}
虚拟控制信号
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{\displaystyle u_{k-1}}
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{\displaystyle z_{k-1}}
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{\displaystyle u_{k-2}}
虚拟控制信号
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{\displaystyle u_{k-2}}
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{\displaystyle z_{k-2}}
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{\displaystyle u_{k-3}}
...
虚拟控制信号
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虚拟控制信号
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{\displaystyle u_{1}}
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{\displaystyle z_{1}}
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{\displaystyle u_{x}}
虚拟控制信号
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{\displaystyle u_{x}}
x 此程序称为反推（backstepping），因为是从内部稳定的子系统开始，渐渐反推到较外围的系统，同时维持每一步的稳定性。因
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给定控制信号
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{\displaystyle u_{x}(\mathbf {0} )=0}
因此所得的系统在原点（
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{\displaystyle z_{k-1}=0}
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{\displaystyle z_{k}=0}
全域渐进稳定 。
