update

Author: wanxinjin

_build/.doctrees/environment.pickle

@@ -169,12 +169,21 @@ $$\frac{\dot{\Theta}(s)}{V_a(s)}=\frac{k_m}{1+T_ms}$$
 
 
 
+```{admonition} Minimal material on control basics 
 
 
-```{admonition} Minimal Control Background
+Please read my hand-writing note on control basics, in order to help you understand the following content of this chapter.
 
-For ease of the reader, I willl provide the minimal background of
-control system. A typical closed-loop control diagram is shown below.
+[Minimal notes on control basics](./control_basics.pdf)
+
+
+```
+
+
+
+```{admonition}  Minimal Introduction to Closed-Loop (Feedback) Control Systems
+
+A typical closed-loop control diagram is shown below.
 Here, $\theta_r$ is the reference input; $\theta_m$ is the output; $D$
 is the disturbance; $G(s)$ is the transfer function of the plant to be
 controlled; $C(s)$ is the controller; $H(s)$ is the backward pass
@@ -227,19 +236,18 @@ $$1+C(s)G(s)H(s)=0$$
 ```
 
 
-```{admonition} Final value theorem
 
 
-If a continuous signal $f(t)$ has its Laplace transformation $F(s)$,
-then the final value theorem states
 
-$$\lim_{t\rightarrow\infty}f(t)=\lim_{s\rightarrow 0}sF(s)$$
-```
+
+
+
+
 
 </br></br>
-# Single-Joint Position Feedback Control
+# Position Feedback Control Design
 
-The single joint motor system in {numref}`motor_model3` is the plant $G(s)$ we want to design the controller $C(s)$ for.
+The single joint system in {numref}`motor_model3` is the plant $G(s)$ we want to design the controller $C(s)$ for.
 
 The plant
 transfer function $G(s)$ is {eq}`equ.motortf`, rewritten below
@@ -286,20 +294,25 @@ $$
 
 
 
+
 The closed-loop input/output transfer function is
 
 $$\frac{\Theta_{m}(s)}{\Theta_{r}(s)}=\frac{C_P(s)G(s)}{1+C_P(s)G(s)H(s)}=\frac{{k_{m}K_P(1+T_Ps)}}{k_{TP}k_{m}K_P(1+T_Ps)+s^2(1+sT_m)},$$(equ.closed_loop)
 
 
-To analyze the control stability of the closed-loop control system {eq}`equ.closed_loop`, we look at the roots (also called poles) of its characteristic equation, which can be factorized into the following
+</br>
+
+### Input-to-output stability analysis
+
+
+To analyze the input-to-output stability of the closed-loop control system {eq}`equ.closed_loop`, we look at the roots (also called poles) of its characteristic equation, which can be factorized into the following
 form
 
 $${k_{TP}k_{m}K_P(1+T_Ps)+s^2(1+sT_m)}
 =
 {\left({\omega_{n}^{2}}+{2 \zeta }{\omega_{n}}s+{s^{2}}\right)(1+s \tau)}=0$$(equ.cha_equ)
 
-where $\omega_{n}$ and $\zeta$ are the natural frequency and damping
-ratio of the complex poles. The roots (also called poles) to the characteristics equation {eq}`equ.cha_equ` is 
+The roots (also called poles) to the characteristics equation {eq}`equ.cha_equ` is 
 
 
 $$
@@ -310,18 +323,21 @@ s_3&=-1 / \tau
 \end{aligned}
 $$
 
+Here,  $\omega_{n}$ and $\zeta$ are the natural frequency and damping
+ratio for the complex roots $s_1$ and $s_2$, and $s_3$ is a real pole.
+Please see my [Minimal notes on control basics](./control_basics.pdf) for explaination of how the values of $\omega_{n}$, $\zeta$ and $s_3$ effect the time-domain performance of the closed-loop control system.
+
+
 
-The location of the above poles $(s_1, s_2, s_3)$ on $s-$plane depends on the of the open-loop gain 
+The location of the above poles $(s_1, s_2, s_3)$ on s-plane depends on the of the open-loop gain 
 
 $$\frac{k_{m} K_{P} k_{T P} T_{P}}{T_{m}}$$(equ.ol_gain)
 
 
 
-The root locus (i.e., the trajectory of the above poles on $s-$plane)  can be drawn by taking different open loop gain value {eq}`equ.ol_gain`. Here, we derive into two cases
-to plot the root locus.
+The root locus (i.e., the trajectory of the above poles on $s-$plane)  can be drawn by taking different open loop gain value {eq}`equ.ol_gain`. Here, we derive into two cases:
 
--   If $T_{P}<T_{m}$, the root locus is shown. Because there is always a pole (root) living on the right-half "s-plane" regardless of the choice of $\frac{k_{m} K_{P} k_{T P} T_{P}}{T_{m}}$, the closed-loop control system {eq}`equ.closed_loop` thus
-    is inherently unstable.
+-   If $T_{P}<T_{m}$, the root locus is shown. Because there is always a pole (root) living on the right-half s-plane regardless of the choice of $\frac{k_{m} K_{P} k_{T P} T_{P}}{T_{m}}$, the closed-loop control system {eq}`equ.closed_loop` thus is inherently unstable.
 
     :::{figure} ../lec19/control/p_control_rl_1.jpg
     ---
@@ -334,9 +350,8 @@ to plot the root locus.
 
 
 
--   If $T_{P}>T_{m}$, the root locus is shown as below. First of all, all poles can be located on the left half of the $s$-plane, closed-loop control system {eq}`equ.closed_loop` is stable. Also,  as you can see As $T_{P}$
-    increases, the absolute value of the real part of the two complex poles, ($s_1$ and $s_2$) tending towards the asymptotes increases too, and the
-    system has faster time response.
+-   If $T_{P}>T_{m}$, the root locus is shown as below. First, all poles can be located on the left half of s-plane, thus closed-loop control system {eq}`equ.closed_loop` is stable. Also,  as $T_{P}$
+    increases, the absolute value of the real part of the two complex poles, ($s_1$ and $s_2$) tending towards the asymptotes increases too, and the     system has faster time response (please see my [Minimal notes on control basics](./control_basics.pdf) for explaination)
 
 
     :::{figure} ../lec19/control/p_control_rl_2.jpg
@@ -350,12 +365,27 @@ to plot the root locus.
 
 
 
+</br>
+
+### Disturbance-to-output performance
+
+
+
+```{admonition} Final value theorem
+
+
+If a continuous signal $f(t)$ has its Laplace transformation $F(s)$,
+then the final value theorem states
+
+$$\lim_{t\rightarrow\infty}f(t)=\lim_{s\rightarrow 0}sF(s)$$
+```
+
 
 The closed-loop disturbance-to-output transfer function is
 
 $$\frac{\Theta_{m}(s)}{D(s)}=-\frac{\frac{R_a}{K_t}G(s)}{1+C(s)G(s)H(s)}=-\frac{\frac{R_a}{K_t}k_ms}{k_{TP}k_{m}K_P(1+T_Ps)+s^2(1+sT_m)}$$
 
-If $\theta_r(t)=0$ (i.e., no input signal), based on the final value thoerem, we have
+If $\theta_r(t)=0$ (i.e., no input signal), based on the final value thoerem (see the above), we have
 
 
 $$\lim_{t\rightarrow \infty} {\theta_{m}(t)}= \lim_{s\rightarrow 0} s{\Theta_{m}(s)}=\lim_{s\rightarrow 0} s\frac{\Theta_{m}(s)}{D(s)}{D(s)}$$
@@ -377,6 +407,9 @@ higher-order) disturbance. Increasing $K_{P}$ can help with reducing the
 effect of $D$, but too high $K_P$ can lead to unacceptable oscillations
 of the output, as implied from the root locus.
 
+
+
+
 <!-- 
 </br></br>
 # Single-Joint Position and Velocity Feedback Control

_build/.doctrees/lec20/djc.doctree

@@ -385,7 +385,11 @@ <h2> Contents </h2>
 </ul>
 </li>
 <li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#single-joint-control-diagram">Single-Joint Control Diagram</a></li>
-<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#single-joint-position-feedback-control">Single-Joint Position Feedback Control</a></li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#position-feedback-control-design">Position Feedback Control Design</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#input-to-output-stability-analysis">Input-to-output stability analysis</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#disturbance-to-output-performance">Disturbance-to-output performance</a></li>
+</ul>
+</li>
 </ul>
 
             </nav>
@@ -517,10 +521,14 @@ <h1>Single-Joint Control Diagram<a class="headerlink" href="#single-joint-contro
 <p>The  input (<span class="math notranslate nohighlight">\(v_a\)</span>) to output velocity (<span class="math notranslate nohighlight">\(\dot{\theta}_m\)</span>) transfer function is</p>
 <div class="math notranslate nohighlight">
 \[\frac{\dot{\Theta}(s)}{V_a(s)}=\frac{k_m}{1+T_ms}\]</div>
-<div class="admonition-minimal-control-background admonition">
-<p class="admonition-title">Minimal Control Background</p>
-<p>For ease of the reader, I willl provide the minimal background of
-control system. A typical closed-loop control diagram is shown below.
+<div class="admonition-minimal-material-on-control-basics admonition">
+<p class="admonition-title">Minimal material on control basics </p>
+<p>Please read my hand-writing note on control basics, in order to help you understand the following content of this chapter.</p>
+<p><a class="reference download internal" download="" href="../_downloads/af66a2a39b5480f30ad4e93bc7990bf7/control_basics.pdf"><span class="xref download myst">Minimal notes on control basics</span></a></p>
+</div>
+<div class="admonition-minimal-introduction-to-closed-loop-feedback-control-systems admonition">
+<p class="admonition-title">Minimal Introduction to Closed-Loop (Feedback) Control Systems</p>
+<p>A typical closed-loop control diagram is shown below.
 Here, <span class="math notranslate nohighlight">\(\theta_r\)</span> is the reference input; <span class="math notranslate nohighlight">\(\theta_m\)</span> is the output; <span class="math notranslate nohighlight">\(D\)</span>
 is the disturbance; <span class="math notranslate nohighlight">\(G(s)\)</span> is the transfer function of the plant to be
 controlled; <span class="math notranslate nohighlight">\(C(s)\)</span> is the controller; <span class="math notranslate nohighlight">\(H(s)\)</span> is the backward pass
@@ -561,18 +569,11 @@ <h1>Single-Joint Control Diagram<a class="headerlink" href="#single-joint-contro
 <div class="math notranslate nohighlight">
 \[1+C(s)G(s)H(s)=0\]</div>
 </div>
-<div class="admonition-final-value-theorem admonition">
-<p class="admonition-title">Final value theorem</p>
-<p>If a continuous signal <span class="math notranslate nohighlight">\(f(t)\)</span> has its Laplace transformation <span class="math notranslate nohighlight">\(F(s)\)</span>,
-then the final value theorem states</p>
-<div class="math notranslate nohighlight">
-\[\lim_{t\rightarrow\infty}f(t)=\lim_{s\rightarrow 0}sF(s)\]</div>
-</div>
 <p></br></br></p>
 </section>
-<section class="tex2jax_ignore mathjax_ignore" id="single-joint-position-feedback-control">
-<h1>Single-Joint Position Feedback Control<a class="headerlink" href="#single-joint-position-feedback-control" title="Permalink to this heading">#</a></h1>
-<p>The single joint motor system in <a class="reference internal" href="#motor-model3"><span class="std std-numref">Fig. 75</span></a> is the plant <span class="math notranslate nohighlight">\(G(s)\)</span> we want to design the controller <span class="math notranslate nohighlight">\(C(s)\)</span> for.</p>
+<section class="tex2jax_ignore mathjax_ignore" id="position-feedback-control-design">
+<h1>Position Feedback Control Design<a class="headerlink" href="#position-feedback-control-design" title="Permalink to this heading">#</a></h1>
+<p>The single joint system in <a class="reference internal" href="#motor-model3"><span class="std std-numref">Fig. 75</span></a> is the plant <span class="math notranslate nohighlight">\(G(s)\)</span> we want to design the controller <span class="math notranslate nohighlight">\(C(s)\)</span> for.</p>
 <p>The plant
 transfer function <span class="math notranslate nohighlight">\(G(s)\)</span> is <a class="reference internal" href="#equation-equ-motortf">(68)</a>, rewritten below</p>
 <div class="math notranslate nohighlight">
@@ -606,14 +607,16 @@ <h1>Single-Joint Position Feedback Control<a class="headerlink" href="#single-jo
 <p>The closed-loop input/output transfer function is</p>
 <div class="math notranslate nohighlight" id="equation-equ-closed-loop">
 <span class="eqno">(69)<a class="headerlink" href="#equation-equ-closed-loop" title="Permalink to this equation">#</a></span>\[\frac{\Theta_{m}(s)}{\Theta_{r}(s)}=\frac{C_P(s)G(s)}{1+C_P(s)G(s)H(s)}=\frac{{k_{m}K_P(1+T_Ps)}}{k_{TP}k_{m}K_P(1+T_Ps)+s^2(1+sT_m)},\]</div>
-<p>To analyze the control stability of the closed-loop control system <a class="reference internal" href="#equation-equ-closed-loop">(69)</a>, we look at the roots (also called poles) of its characteristic equation, which can be factorized into the following
+</br>
+<section id="input-to-output-stability-analysis">
+<h2>Input-to-output stability analysis<a class="headerlink" href="#input-to-output-stability-analysis" title="Permalink to this heading">#</a></h2>
+<p>To analyze the input-to-output stability of the closed-loop control system <a class="reference internal" href="#equation-equ-closed-loop">(69)</a>, we look at the roots (also called poles) of its characteristic equation, which can be factorized into the following
 form</p>
 <div class="math notranslate nohighlight" id="equation-equ-cha-equ">
 <span class="eqno">(70)<a class="headerlink" href="#equation-equ-cha-equ" title="Permalink to this equation">#</a></span>\[{k_{TP}k_{m}K_P(1+T_Ps)+s^2(1+sT_m)}
 =
 {\left({\omega_{n}^{2}}+{2 \zeta }{\omega_{n}}s+{s^{2}}\right)(1+s \tau)}=0\]</div>
-<p>where <span class="math notranslate nohighlight">\(\omega_{n}\)</span> and <span class="math notranslate nohighlight">\(\zeta\)</span> are the natural frequency and damping
-ratio of the complex poles. The roots (also called poles) to the characteristics equation <a class="reference internal" href="#equation-equ-cha-equ">(70)</a> is</p>
+<p>The roots (also called poles) to the characteristics equation <a class="reference internal" href="#equation-equ-cha-equ">(70)</a> is</p>
 <div class="math notranslate nohighlight">
 \[\begin{split}
 \begin{aligned}
@@ -622,14 +625,15 @@ <h1>Single-Joint Position Feedback Control<a class="headerlink" href="#single-jo
 s_3&amp;=-1 / \tau
 \end{aligned}
 \end{split}\]</div>
-<p>The location of the above poles <span class="math notranslate nohighlight">\((s_1, s_2, s_3)\)</span> on <span class="math notranslate nohighlight">\(s-\)</span>plane depends on the of the open-loop gain</p>
+<p>Here,  <span class="math notranslate nohighlight">\(\omega_{n}\)</span> and <span class="math notranslate nohighlight">\(\zeta\)</span> are the natural frequency and damping
+ratio for the complex roots <span class="math notranslate nohighlight">\(s_1\)</span> and <span class="math notranslate nohighlight">\(s_2\)</span>, and <span class="math notranslate nohighlight">\(s_3\)</span> is a real pole.
+Please see my <a class="reference download internal" download="" href="../_downloads/af66a2a39b5480f30ad4e93bc7990bf7/control_basics.pdf"><span class="xref download myst">Minimal notes on control basics</span></a> for explaination of how the values of <span class="math notranslate nohighlight">\(\omega_{n}\)</span>, <span class="math notranslate nohighlight">\(\zeta\)</span> and <span class="math notranslate nohighlight">\(s_3\)</span> effect the time-domain performance of the closed-loop control system.</p>
+<p>The location of the above poles <span class="math notranslate nohighlight">\((s_1, s_2, s_3)\)</span> on s-plane depends on the of the open-loop gain</p>
 <div class="math notranslate nohighlight" id="equation-equ-ol-gain">
 <span class="eqno">(71)<a class="headerlink" href="#equation-equ-ol-gain" title="Permalink to this equation">#</a></span>\[\frac{k_{m} K_{P} k_{T P} T_{P}}{T_{m}}\]</div>
-<p>The root locus (i.e., the trajectory of the above poles on <span class="math notranslate nohighlight">\(s-\)</span>plane)  can be drawn by taking different open loop gain value <a class="reference internal" href="#equation-equ-ol-gain">(71)</a>. Here, we derive into two cases
-to plot the root locus.</p>
+<p>The root locus (i.e., the trajectory of the above poles on <span class="math notranslate nohighlight">\(s-\)</span>plane)  can be drawn by taking different open loop gain value <a class="reference internal" href="#equation-equ-ol-gain">(71)</a>. Here, we derive into two cases:</p>
 <ul>
-<li><p>If <span class="math notranslate nohighlight">\(T_{P}&lt;T_{m}\)</span>, the root locus is shown. Because there is always a pole (root) living on the right-half “s-plane” regardless of the choice of <span class="math notranslate nohighlight">\(\frac{k_{m} K_{P} k_{T P} T_{P}}{T_{m}}\)</span>, the closed-loop control system <a class="reference internal" href="#equation-equ-closed-loop">(69)</a> thus
-is inherently unstable.</p>
+<li><p>If <span class="math notranslate nohighlight">\(T_{P}&lt;T_{m}\)</span>, the root locus is shown. Because there is always a pole (root) living on the right-half s-plane regardless of the choice of <span class="math notranslate nohighlight">\(\frac{k_{m} K_{P} k_{T P} T_{P}}{T_{m}}\)</span>, the closed-loop control system <a class="reference internal" href="#equation-equ-closed-loop">(69)</a> thus is inherently unstable.</p>
 <figure class="align-default" id="p-control-rl-1">
 <a class="reference internal image-reference" href="../_images/p_control_rl_1.jpg"><img alt="../_images/p_control_rl_1.jpg" src="../_images/p_control_rl_1.jpg" style="width: 90%;" /></a>
 <figcaption>
@@ -638,9 +642,8 @@ <h1>Single-Joint Position Feedback Control<a class="headerlink" href="#single-jo
 </figcaption>
 </figure>
 </li>
-<li><p>If <span class="math notranslate nohighlight">\(T_{P}&gt;T_{m}\)</span>, the root locus is shown as below. First of all, all poles can be located on the left half of the <span class="math notranslate nohighlight">\(s\)</span>-plane, closed-loop control system <a class="reference internal" href="#equation-equ-closed-loop">(69)</a> is stable. Also,  as you can see As <span class="math notranslate nohighlight">\(T_{P}\)</span>
-increases, the absolute value of the real part of the two complex poles, (<span class="math notranslate nohighlight">\(s_1\)</span> and <span class="math notranslate nohighlight">\(s_2\)</span>) tending towards the asymptotes increases too, and the
-system has faster time response.</p>
+<li><p>If <span class="math notranslate nohighlight">\(T_{P}&gt;T_{m}\)</span>, the root locus is shown as below. First, all poles can be located on the left half of s-plane, thus closed-loop control system <a class="reference internal" href="#equation-equ-closed-loop">(69)</a> is stable. Also,  as <span class="math notranslate nohighlight">\(T_{P}\)</span>
+increases, the absolute value of the real part of the two complex poles, (<span class="math notranslate nohighlight">\(s_1\)</span> and <span class="math notranslate nohighlight">\(s_2\)</span>) tending towards the asymptotes increases too, and the     system has faster time response (please see my <a class="reference download internal" download="" href="../_downloads/af66a2a39b5480f30ad4e93bc7990bf7/control_basics.pdf"><span class="xref download myst">Minimal notes on control basics</span></a> for explaination)</p>
 <figure class="align-default" id="p-control-rl-2">
 <a class="reference internal image-reference" href="../_images/p_control_rl_2.jpg"><img alt="../_images/p_control_rl_2.jpg" src="../_images/p_control_rl_2.jpg" style="width: 90%;" /></a>
 <figcaption>
@@ -650,10 +653,21 @@ <h1>Single-Joint Position Feedback Control<a class="headerlink" href="#single-jo
 </figure>
 </li>
 </ul>
+</br>
+</section>
+<section id="disturbance-to-output-performance">
+<h2>Disturbance-to-output performance<a class="headerlink" href="#disturbance-to-output-performance" title="Permalink to this heading">#</a></h2>
+<div class="admonition-final-value-theorem admonition">
+<p class="admonition-title">Final value theorem</p>
+<p>If a continuous signal <span class="math notranslate nohighlight">\(f(t)\)</span> has its Laplace transformation <span class="math notranslate nohighlight">\(F(s)\)</span>,
+then the final value theorem states</p>
+<div class="math notranslate nohighlight">
+\[\lim_{t\rightarrow\infty}f(t)=\lim_{s\rightarrow 0}sF(s)\]</div>
+</div>
 <p>The closed-loop disturbance-to-output transfer function is</p>
 <div class="math notranslate nohighlight">
 \[\frac{\Theta_{m}(s)}{D(s)}=-\frac{\frac{R_a}{K_t}G(s)}{1+C(s)G(s)H(s)}=-\frac{\frac{R_a}{K_t}k_ms}{k_{TP}k_{m}K_P(1+T_Ps)+s^2(1+sT_m)}\]</div>
-<p>If <span class="math notranslate nohighlight">\(\theta_r(t)=0\)</span> (i.e., no input signal), based on the final value thoerem, we have</p>
+<p>If <span class="math notranslate nohighlight">\(\theta_r(t)=0\)</span> (i.e., no input signal), based on the final value thoerem (see the above), we have</p>
 <div class="math notranslate nohighlight">
 \[\lim_{t\rightarrow \infty} {\theta_{m}(t)}= \lim_{s\rightarrow 0} s{\Theta_{m}(s)}=\lim_{s\rightarrow 0} s\frac{\Theta_{m}(s)}{D(s)}{D(s)}\]</div>
 <p>thus,</p>
@@ -781,6 +795,7 @@ <h1>Single-Joint Position Feedback Control<a class="headerlink" href="#single-jo
 can be interpreted as the disturbance rejection factor for velocity (or
 higher-order) disturbance. Increasing $K_{P}$ and $K_V$ can help with
 reducing the effect of $D$. -->
+</section>
 </section>
 
     <script type="text/x-thebe-config">
@@ -855,7 +870,11 @@ <h1>Single-Joint Position Feedback Control<a class="headerlink" href="#single-jo
 </ul>
 </li>
 <li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#single-joint-control-diagram">Single-Joint Control Diagram</a></li>
-<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#single-joint-position-feedback-control">Single-Joint Position Feedback Control</a></li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#position-feedback-control-design">Position Feedback Control Design</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#input-to-output-stability-analysis">Input-to-output stability analysis</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#disturbance-to-output-performance">Disturbance-to-output performance</a></li>
+</ul>
+</li>
 </ul>
 
   </nav></div>

_build/html/_downloads/af66a2a39b5480f30ad4e93bc7990bf7/control_basics.pdf

@@ -169,12 +169,21 @@ $$\frac{\dot{\Theta}(s)}{V_a(s)}=\frac{k_m}{1+T_ms}$$
 
 
 
+```{admonition} Minimal material on control basics 
 
 
-```{admonition} Minimal Control Background
+Please read my hand-writing note on control basics, in order to help you understand the following content of this chapter.
 
-For ease of the reader, I willl provide the minimal background of
-control system. A typical closed-loop control diagram is shown below.
+[Minimal notes on control basics](./control_basics.pdf)
+
+
+```
+
+
+
+```{admonition}  Minimal Introduction to Closed-Loop (Feedback) Control Systems
+
+A typical closed-loop control diagram is shown below.
 Here, $\theta_r$ is the reference input; $\theta_m$ is the output; $D$
 is the disturbance; $G(s)$ is the transfer function of the plant to be
 controlled; $C(s)$ is the controller; $H(s)$ is the backward pass
@@ -227,19 +236,18 @@ $$1+C(s)G(s)H(s)=0$$
 ```
 
 
-```{admonition} Final value theorem
 
 
-If a continuous signal $f(t)$ has its Laplace transformation $F(s)$,
-then the final value theorem states
 
-$$\lim_{t\rightarrow\infty}f(t)=\lim_{s\rightarrow 0}sF(s)$$
-```
+
+
+
+
 
 </br></br>
-# Single-Joint Position Feedback Control
+# Position Feedback Control Design
 
-The single joint motor system in {numref}`motor_model3` is the plant $G(s)$ we want to design the controller $C(s)$ for.
+The single joint system in {numref}`motor_model3` is the plant $G(s)$ we want to design the controller $C(s)$ for.
 
 The plant
 transfer function $G(s)$ is {eq}`equ.motortf`, rewritten below
@@ -286,20 +294,25 @@ $$
 
 
 
+
 The closed-loop input/output transfer function is
 
 $$\frac{\Theta_{m}(s)}{\Theta_{r}(s)}=\frac{C_P(s)G(s)}{1+C_P(s)G(s)H(s)}=\frac{{k_{m}K_P(1+T_Ps)}}{k_{TP}k_{m}K_P(1+T_Ps)+s^2(1+sT_m)},$$(equ.closed_loop)
 
 
-To analyze the control stability of the closed-loop control system {eq}`equ.closed_loop`, we look at the roots (also called poles) of its characteristic equation, which can be factorized into the following
+</br>
+
+### Input-to-output stability analysis
+
+
+To analyze the input-to-output stability of the closed-loop control system {eq}`equ.closed_loop`, we look at the roots (also called poles) of its characteristic equation, which can be factorized into the following
 form
 
 $${k_{TP}k_{m}K_P(1+T_Ps)+s^2(1+sT_m)}
 =
 {\left({\omega_{n}^{2}}+{2 \zeta }{\omega_{n}}s+{s^{2}}\right)(1+s \tau)}=0$$(equ.cha_equ)
 
-where $\omega_{n}$ and $\zeta$ are the natural frequency and damping
-ratio of the complex poles. The roots (also called poles) to the characteristics equation {eq}`equ.cha_equ` is 
+The roots (also called poles) to the characteristics equation {eq}`equ.cha_equ` is 
 
 
 $$
@@ -310,18 +323,21 @@ s_3&=-1 / \tau
 \end{aligned}
 $$
 
+Here,  $\omega_{n}$ and $\zeta$ are the natural frequency and damping
+ratio for the complex roots $s_1$ and $s_2$, and $s_3$ is a real pole.
+Please see my [Minimal notes on control basics](./control_basics.pdf) for explaination of how the values of $\omega_{n}$, $\zeta$ and $s_3$ effect the time-domain performance of the closed-loop control system.
+
+
 
-The location of the above poles $(s_1, s_2, s_3)$ on $s-$plane depends on the of the open-loop gain 
+The location of the above poles $(s_1, s_2, s_3)$ on s-plane depends on the of the open-loop gain 
 
 $$\frac{k_{m} K_{P} k_{T P} T_{P}}{T_{m}}$$(equ.ol_gain)
 
 
 
-The root locus (i.e., the trajectory of the above poles on $s-$plane)  can be drawn by taking different open loop gain value {eq}`equ.ol_gain`. Here, we derive into two cases
-to plot the root locus.
+The root locus (i.e., the trajectory of the above poles on $s-$plane)  can be drawn by taking different open loop gain value {eq}`equ.ol_gain`. Here, we derive into two cases:
 
--   If $T_{P}<T_{m}$, the root locus is shown. Because there is always a pole (root) living on the right-half "s-plane" regardless of the choice of $\frac{k_{m} K_{P} k_{T P} T_{P}}{T_{m}}$, the closed-loop control system {eq}`equ.closed_loop` thus
-    is inherently unstable.
+-   If $T_{P}<T_{m}$, the root locus is shown. Because there is always a pole (root) living on the right-half s-plane regardless of the choice of $\frac{k_{m} K_{P} k_{T P} T_{P}}{T_{m}}$, the closed-loop control system {eq}`equ.closed_loop` thus is inherently unstable.
 
     :::{figure} ../lec19/control/p_control_rl_1.jpg
     ---
@@ -334,9 +350,8 @@ to plot the root locus.
 
 
 
--   If $T_{P}>T_{m}$, the root locus is shown as below. First of all, all poles can be located on the left half of the $s$-plane, closed-loop control system {eq}`equ.closed_loop` is stable. Also,  as you can see As $T_{P}$
-    increases, the absolute value of the real part of the two complex poles, ($s_1$ and $s_2$) tending towards the asymptotes increases too, and the
-    system has faster time response.
+-   If $T_{P}>T_{m}$, the root locus is shown as below. First, all poles can be located on the left half of s-plane, thus closed-loop control system {eq}`equ.closed_loop` is stable. Also,  as $T_{P}$
+    increases, the absolute value of the real part of the two complex poles, ($s_1$ and $s_2$) tending towards the asymptotes increases too, and the     system has faster time response (please see my [Minimal notes on control basics](./control_basics.pdf) for explaination)
 
 
     :::{figure} ../lec19/control/p_control_rl_2.jpg
@@ -350,12 +365,27 @@ to plot the root locus.
 
 
 
+</br>
+
+### Disturbance-to-output performance
+
+
+
+```{admonition} Final value theorem
+
+
+If a continuous signal $f(t)$ has its Laplace transformation $F(s)$,
+then the final value theorem states
+
+$$\lim_{t\rightarrow\infty}f(t)=\lim_{s\rightarrow 0}sF(s)$$
+```
+
 
 The closed-loop disturbance-to-output transfer function is
 
 $$\frac{\Theta_{m}(s)}{D(s)}=-\frac{\frac{R_a}{K_t}G(s)}{1+C(s)G(s)H(s)}=-\frac{\frac{R_a}{K_t}k_ms}{k_{TP}k_{m}K_P(1+T_Ps)+s^2(1+sT_m)}$$
 
-If $\theta_r(t)=0$ (i.e., no input signal), based on the final value thoerem, we have
+If $\theta_r(t)=0$ (i.e., no input signal), based on the final value thoerem (see the above), we have
 
 
 $$\lim_{t\rightarrow \infty} {\theta_{m}(t)}= \lim_{s\rightarrow 0} s{\Theta_{m}(s)}=\lim_{s\rightarrow 0} s\frac{\Theta_{m}(s)}{D(s)}{D(s)}$$
@@ -377,6 +407,9 @@ higher-order) disturbance. Increasing $K_{P}$ can help with reducing the
 effect of $D$, but too high $K_P$ can lead to unacceptable oscillations
 of the output, as implied from the root locus.
 
+
+
+
 <!-- 
 </br></br>
 # Single-Joint Position and Velocity Feedback Control