最优控制公式推导（代数里卡提方程，李雅普诺夫方程，HJB方程）
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本文探讨了线性时不变系统（LTI系统）的最优控制问题，特别是线性二次调节器（LQR）问题。通过Hamilton-Jacobi-Bellman (HJB) 方程的推导，求得了系统的最优控制律，并进一步推导了代数里卡提方程（ARE）和李雅普诺夫方程，从而实现系统的最优控制。关键词：线性二次调节器，Hamilton-Jacobi-Bellman方程，代数里卡提方程，李雅普诺夫方程，最优控制律https://blog.csdn.net/weixin_44346182/article/details/140479211 <link href="/image.php?url=https://csdnimg.cn/release/download_crawler_static/css/base.min.css" rel="stylesheet"/><link href="/image.php?url=https://csdnimg.cn/release/download_crawler_static/css/fancy.min.css" rel="stylesheet"/><link href="/image.php?url=https://csdnimg.cn/release/download_crawler_static/89547821/raw.css" rel="stylesheet"/><div id="sidebar" style="display: none"><div id="outline"></div></div><div class="pf w0 h0" data-page-no="1" id="pf1"><div class="pc pc1 w0 h0"><img alt="" class="bi x0 y0 w1 h1" src="/image.php?url=https://csdnimg.cn/release/download_crawler_static/89547821/bg1.jpg"/><div class="c x1 y1 w2 h2"><div class="t m0 x2 h3 y2 ff1 fs0 fc0 sc0 ls0 ws0">1. <span class="ff2">线性时不变系统模型<span class="_ _0"> </span></span><span class="fc1"> </span></div><div class="t m0 x2 h4 y3 ff3 fs1 fc0 sc0 ls0 ws0">考虑一个线性时不变系统，其状态方程和控制输入可以表示为：</div><div class="t m0 x2 h4 y4 ff3 fs1 fc0 sc0 ls0 ws0">其中，<span class="_ _1"> </span><span class="ff4"> </span>是状态向量，<span class="_ _1"> </span><span class="ff4"> </span>是控制向量，<span class="_ _2"> </span><span class="ff4"> </span>和<span class="ff4"> <span class="_ _2"> </span> </span>分别是状态矩阵和输入矩阵。</div><div class="t m0 x2 h3 y5 ff1 fs0 fc0 sc0 ls0 ws0">2. <span class="ff2">线性二次调节器（</span>LQR<span class="ff2">）目标</span> <span class="_ _3"> </span><span class="fc1"> </span></div><div class="t m0 x2 h4 y6 ff4 fs1 fc0 sc0 ls0 ws0">LQR <span class="ff3">问题的目标是找到一个控制输入</span> <span class="_ _1"> </span> <span class="ff3">来最小化以下性能指标（成本函数）：</span></div><div class="t m0 x2 h4 y7 ff3 fs1 fc0 sc0 ls0 ws0">其中，<span class="_ _4"> </span><span class="ff4"> </span>是对称正定的状态权重矩阵，<span class="_ _2"> </span><span class="ff4"> </span>是对称正定的控制权重矩阵。</div><div class="t m1 x2 h5 y8 ff3 fs2 fc0 sc0 ls0 ws0">注意：此时的积分是从</div><div class="t m0 x3 h6 y8 ff5 fs1 fc0 sc0 ls0 ws0"> <span class="_ _5"> </span> </div><div class="t m1 x4 h5 y8 ff3 fs2 fc0 sc0 ls0 ws0">的积分，上下界并不含有</div><div class="t m0 x5 h6 y8 ff5 fs1 fc0 sc0 ls0 ws0">t</div><div class="t m0 x2 h3 y9 ff1 fs0 fc0 sc0 ls0 ws0">3. <span class="ff2">哈密顿</span>-<span class="ff2">雅可比</span>-<span class="ff2">贝尔曼方程</span> <span class="_ _6"> </span><span class="fc1"> </span></div><div class="t m0 x2 h4 ya ff3 fs1 fc0 sc0 ls0 ws0">定义值函数<span class="ff4"> <span class="_ _7"> </span> </span>表示从时刻<span class="ff4"> <span class="_ _8"> </span> </span>开始，到无穷远时刻的最小成本。即：</div><div class="t m0 x2 h4 yb ff3 fs1 fc0 sc0 ls0 ws0">对于线性系统，<span class="ff2">上面的代价函数<span class="ff1"> <span class="_ _9"> </span> </span>可以改写为如下形式</span>：</div><div class="t m0 x2 h4 yc ff3 fs1 fc0 sc0 ls0 ws0">其中，<span class="_ _2"> </span><span class="ff4"> </span>是一个对称正定矩阵<span class="ff4"> <span class="ff1">(<span class="ff2">这是线性系统的非常重要的性质</span>)<span class="ff2">。</span></span></span></div><div class="t m0 x6 h4 yd ff2 fs1 fc2 sc0 ls0 ws0">说明<span class="ff3">：在本质上<span class="_ _a"> </span>与<span class="_ _b"> </span>并不相等，无穷时间的积分</span></div><div class="t m0 x6 h4 ye ff3 fs1 fc2 sc0 ls0 ws0">是不可求导的，但是任意无限大的上界都是可以求导的，即<span class="_ _c"> </span>且<span class="_ _d"> </span>为无限大的时候都可以，所以书本</div><div class="t m0 x6 h4 yf ff3 fs1 fc2 sc0 ls0 ws0">上为了更加严谨，一般写成价值函数为终端时刻的价值函数，即：</div><div class="t m0 x6 h4 y10 ff3 fs1 fc2 sc0 ls0 ws0">其中，<span class="_ _e"> </span>为终端时刻。</div><div class="t m0 x2 h7 y11 ff1 fs1 fc0 sc0 ls0 ws0">1.<span class="ff2">通过动态规划求解</span>HJB<span class="ff2">方程</span></div><div class="t m0 x2 h4 y12 ff3 fs1 fc0 sc0 ls0 ws0">根据动态规划原理，值函数满足以下条件：</div><div class="t m0 x2 h4 y13 ff3 fs1 fc0 sc0 ls0 ws0">考虑<span class="ff4"> <span class="_ _f"> </span> </span>很小的情况下，近似可以写成：</div><div class="t m0 x2 h4 y14 ff3 fs1 fc0 sc0 ls0 ws0">移项并除以<span class="ff4"> <span class="_ _f"> </span></span>，当<span class="ff4"> <span class="_ _7"> </span> </span>时，得到：</div></div><a class="l"><div class="d m2"></div></a><a class="l"><div class="d m2"></div></a><a class="l"><div class="d m2"></div></a></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>