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260109_html/SysReg_crashcourse_TimeDomain_BlockA_01_Basics.html renamed to .html/BlockA_01_Basics.html

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<h3 id="Block-diagrams">Block diagrams<a class="anchor-link" href="#Block-diagrams"></a></h3><p>        <em>Pixel perfect ways to visualise systems</em></p>
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<p><img alt="Slide about block diagram elements" src="../figures/BlockDiagEls.png"/></p>
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<p>It's a nice sanity check that any fundamental block scheme of an $n$-th order system has $n$ integrators.</p>
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<h2 id="Equilibrium-points">Equilibrium points<a class="anchor-link" href="#Equilibrium-points"></a></h2><p>        <em>Forever and unchanging</em></p>
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<p>Often, systems will have equilibria, meaning there are states where system will remain over time. These state-input point pairs are denoted as $(x_e, u_e)$ or $(\bar x, \bar u)$. I'll use the latter since this is more traditional in control and we have to appease our control elders. So equilibria remain unchanging, so the derivative of the state is zero. In mathematicians' language</p>

260109_html/SysReg_crashcourse_TimeDomain_BlockA_02_StateSpace.html renamed to .html/BlockA_02_StateSpace.html

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260109_html/SysReg_crashcourse_TimeDomain_BlockA_03_ControlSynthesis.html renamed to .html/BlockA_03_ControlSynthesis.html

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<h2 id="Feedback-control">Feedback control<a class="anchor-link" href="#Feedback-control">¶</a></h2><p>WOW ARE WE GOING TO DO CONTROL NOW??? <em>FINALLY</em>???? Yes! No... There's one more difference we need to explain. There are two types of feedback control, output and state feedback. The difference is in the name and here you also see the difference in a block diagram for a plant/process/system, $P$, and a controller, $C$.</p>
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<p>Output feedback is stuff like PID, or in other terms: disgusting. We'll look at state feedback now! <em>The real stuff.</em></p>
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<p>However, regulating to zero is boring, I want to be able to tell my system to do a backflip! We need to add a reference, and I'll spoil that we also need a reference gain $k_f$:</p>
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<p>So how does this work then... Same substitution as before!</p>
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<p>$$ \dot x = Ax + Bu = Ax + B(k_f r - Kx) = (A-BK) x + Bk_f r.$$</p>
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<p>Now to determine $k_f$, we want to prevent steady state errors, so $\bar y = r$. Steady state also means</p>
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<h2 id="Integral-action">Integral action<a class="anchor-link" href="#Integral-action">¶</a></h2><p>        <em>The theory of everything is not found yet, so our model is <strong>always</strong> terrible</em></p>
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<p>Most likely the model we have is not a perfect representation of the real world. "Neglect air friction, neglect static/dynamic friction and hysteresis, assume rigid bodies, assume perfect efficiency, ..." All lies we tell ourselves to make our lives easier and to blind ourselves to the harsh reality.</p>
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<p>So if out model is incorrect, what does that mean for our controller? Simply said, it's not exactly what we want, but <em>it's close</em>. To get it perfect however, we need to add an output feedback part to our state feedback controller. An integrator of the reference-output-error to be precise. This integrator wil drive the integral of the error, $z$, to zero, meaning that the error will be zero. I'll stop talking now and show the block diagram so you actually understand.</p>
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<p>Sooooo maths time. Substitute everything into everything, yada yada, this is largely what control engineers do. Also assume D=0. Lets start with plant equations and work our way backwards through the block diagram.</p>
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<p>$$\left\{\begin{aligned}\dot x &amp;= Ax + Bu = Ax + B(k_fr-Kx-k_iz) = (A-BK)x + Bk_fr-Bk_iz \\ y &amp;= Cx\end{aligned}\right.$$</p>
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<p>and</p>
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