反步法设计步骤及仿真验证（非线性系统）

电玩女神 2024-03-31 18:05 85阅读 0赞

**考虑以下单输入，单输出非线性系统：**

**![\\begin\{cases\} \\dot\{x\}\_\{1\}=x\_\{2\}+\\sin\{x\_\{1\}\}\\\\ \\dot\{x\}\_\{2\}=u+x^\{2\}\_\{1\}+x\_\{2\}\\\\ \\end\{cases\}][begin_cases_ _dot_x_1_x_2_sin_x_1_ _dot_x_2_u_x_2_1_x_2_ _end_cases]**

**反步法的设计思想是每一个子系统 ![\\dot\{x\}\_\{1\}=x\_\{2\}+\\sin\{x\_\{1\}\}][dot_x_1_x_2_sin_x_1] 中 ![x\_\{2\}][x_2] 为虚拟控制，通过适当的虚拟反馈![x\_\{2\} = \\alpha\_\{1\}][x_2_ _ _alpha_1],使得系统的前面状态达到渐进稳定。但是系统的解一般不满足![x\_\{2\} = \\alpha\_\{1\}][x_2_ _ _alpha_1]，因此我们引入误差变量，期望通过控制作用，使得 ![x\_\{2\}][x_2] 与虚拟反馈 ![\\alpha\_\{1\}][alpha_1] 间具有某种渐进特性，从而实现整个系统的渐进稳定。**

**定义以下误差坐标变换（反步法的坐标变换）**

**![\\begin\{cases\} e\_\{1\}=x\_\{1\}\\\\ e\_\{2\}=x\_\{2\}-\\alpha\_\{1\} \\end\{cases\}][begin_cases_ e_1_x_1_ e_2_x_2_-_alpha_1_ _end_cases]**

**![\\alpha\_\{1\}][alpha_1] 代表虚拟控制器**

**反步法设计步骤如下(不断构造李雅普诺夫函数使系统稳定)：**

**Step1:**

**定义李雅普诺夫函数为：**

**![V\_\{1\} = \\frac\{1\}\{2\}e^\{2\}\_\{1\}][V_1_ _ _frac_1_2_e_2_1]**

**显而易见 ![V\_\{1\}][V_1] >0, 对 ![V\_\{1\}][V_1] 求导得：**

![\\dot\{V\}\_\{1\}=x\_\{1\}\\dot\{x\}\_\{1\}][dot_V_1_x_1_dot_x_1]

**将 ![\\dot\{x\}\_\{1\} = x\_\{2\} + \\sin\{x\}\_\{1\}][dot_x_1_ _ x_2_ _ _sin_x_1] 带入 ![\\dot\{V\}\_\{1\}][dot_V_1]得：**

![\\dot\{V\}\_\{1\} = e\_\{1\}(- e\_\{1\}+x\_\{1\}+x\_\{2\} + \\sin\{x\}\_\{1\} )][dot_V_1_ _ e_1_- e_1_x_1_x_2_ _ _sin_x_1_]

**根据 坐标变换 ![e\_\{2\}=x\_\{2\}-\\alpha\_\{1\}][e_2_x_2_-_alpha_1] 可知，![x\_\{2\}=e\_\{2\}+\\alpha\_\{1\}][x_2_e_2_alpha_1]**

**![\\dot\{V\}\_\{1\} = e\_\{1\}( -e\_\{1\}+x\_\{1\}+e\_\{2\}+\\alpha\_\{1\} + \\sin\{x\}\_\{1\} )][dot_V_1_ _ e_1_ -e_1_x_1_e_2_alpha_1_ _ _sin_x_1_]**

**为了使系统稳定，让 ![\\alpha\_\{1\} = -\\sin\{x\}\_\{1\}][alpha_1_ _ -_sin_x_1]**

**即：![\\dot\{V\}\_\{1\} = -e^\{2\}\_\{1\} + e\_\{1\}e\_\{2\}][dot_V_1_ _ -e_2_1_ _ e_1_e_2]**

**显然，如果 ![z\_\{2\}][z_2] = 0,（即：![\\alpha\_\{1\} = -\\sin\{x\}\_\{1\}][alpha_1_ _ -_sin_x_1]），由![\\dot\{V\}\_\{1\} = -e^\{2\}\_\{1\} + e\_\{1\}e\_\{2\}][dot_V_1_ _ -e_2_1_ _ e_1_e_2]可知，![e\_\{1\}][e_1]渐进稳定。但是一般情况![e\_\{2\}\\neq 0][e_2_neq 0],此时我们再引入虚拟控制 ![\\alpha\_\{2\}][alpha_2] 使得误差 ![e\_\{2\}=x\_\{2\}-\\alpha\_\{1\}][e_2_x_2_-_alpha_1]****具有期望的渐进性态。为此，我们进行下一步设计**

**Step2:**

**定义李雅普诺夫函数为：**

![V\_\{2\} = \\frac\{1\}\{2\}e^\{2\}\_\{2\}+V\_\{1\}][V_2_ _ _frac_1_2_e_2_2_V_1]

**显而易见 ![V\_\{2\}][V_2] >0, 对 ![V\_\{2\}][V_2]求导得：**

**![\\dot\{V\}\_\{2\} = \\dot\{e\}\_\{2\}e\_\{2\}-e^\{2\}\_\{1\} + e\_\{1\}e\_\{2\}][dot_V_2_ _ _dot_e_2_e_2_-e_2_1_ _ e_1_e_2]**

**将 ![\\dot\{e\}\_\{2\}=\\dot\{x\}\_\{2\}-\\dot\{\\alpha\}\_\{1\}][dot_e_2_dot_x_2_-_dot_alpha_1] 带入上式可得：**

![\\dot\{V\}\_\{2\} =e\_\{2\}(\\dot\{x\}\_\{2\}-\\dot\{\\alpha\}\_\{1\})-e^\{2\}\_\{1\} + e\_\{1\}e\_\{2\}][dot_V_2_ _e_2_dot_x_2_-_dot_alpha_1_-e_2_1_ _ e_1_e_2]

**将![\\dot\{x\}\_\{2\}=u+x^\{2\}\_\{1\}+x\_\{2\}][dot_x_2_u_x_2_1_x_2]带入上式可得：**

![\\dot\{V\}\_\{2\} =e\_\{2\}(u+x^\{2\}\_\{1\}+x\_\{2\}-\\dot\{\\alpha\}\_\{1\})-e^\{2\}\_\{1\} + e\_\{1\}e\_\{2\}][dot_V_2_ _e_2_u_x_2_1_x_2_-_dot_alpha_1_-e_2_1_ _ e_1_e_2]

为了使得![\\dot\{V\}\_\{2\}][dot_V_2]<0,设计控制器如下：

![u=-e\_\{1\} -m\*e\_\{2\}-x^\{2\}\_\{1\}-x\_\{2\}+\\dot\{\\alpha\}\_\{1\}][u_-e_1_ -m_e_2_-x_2_1_-x_2_dot_alpha_1]

其中m为正参数 将 ![u][] 带入 ![\\dot\{V\}\_\{2\}][dot_V_2]得:

![\\dot\{V\}\_\{2\}=-m\*e^\{2\}\_\{2\}\\leqslant 0][dot_V_2_-m_e_2_2_leqslant 0]

仿真验证：

设置初始值为：![x\_\{1\} = 0.1; x\_\{2\} = 0.1;][x_1_ _ 0.1_ x_2_ _ 0.1]参数设置为：![m = 25;][m _ 25]

![x\_\{1\}][x_1] 和 ![x\_\{2\}][x_2]趋于稳定如图所示：

![193b5f6e641e47eab030dea8e145749a.png][]

控制器如图所示：

![80aa00e9de6742be9aed01e15b9c3b4f.png][]

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[80aa00e9de6742be9aed01e15b9c3b4f.png]: https://image.dandelioncloud.cn/pgy_files/images/2024/03/31/55225d038b0947b99ebc2696e89e3266.png