Make discrete version of impulse response vs pole location figure

appendices/classical-control-theory/laplace-domain-analysis/laplace-transform.tex

@@ -24,7 +24,7 @@ \subsection{Laplace transform}
 \gls{system response} (the frequency domain) and also the x coordinate
 representing the speed at which that oscillation decays and the \gls{system}
 converges to zero (i.e., a decaying exponential). Figure
-\ref{fig:impulse_response_poles} shows this for various points.
+\ref{fig:cont_impulse_response_poles} shows this for various points.
 
 If we move the component frequencies in the Fmajor4 chord example parallel to
 the real axis to $\sigma = -25$, the resulting time domain response attenuates

appendices/classical-control-theory/transfer-functions/parts-of-a-transfer-function.tex

@@ -43,13 +43,13 @@ \subsubsection{Poles and zeroes}
 The locations of the closed-loop poles in the complex plane determine the
 stability of the \gls{system}. Each pole represents a frequency mode of the
 \gls{system}, and their location determines how much of each response is induced
-for a given input frequency. Figure \ref{fig:impulse_response_poles} shows the
-\glspl{impulse response} in the time domain for transfer functions with various
-pole locations. They all have an initial condition of $1$.
+for a given input frequency. Figure \ref{fig:cont_impulse_response_poles} shows
+the \glspl{impulse response} in the time domain for transfer functions with
+various pole locations. They all have an initial condition of $1$.
 \begin{bookfigure}
-  \input{figs/impulse-response-vs-pole-location}
+  \input{figs/continuous-impulse-response-vs-pole-location}
   \caption{Impulse response vs pole location}
-  \label{fig:impulse_response_poles}
+  \label{fig:cont_impulse_response_poles}
 \end{bookfigure}
 
 Poles in the left half-plane (LHP) are stable; the \gls{system}'s output may

figs/impulse-response-vs-pole-location.tex → figs/continuous-impulse-response-vs-pole-location.tex

@@ -5,16 +5,16 @@
   \draw[->] (-4,0) -- (4,0) node[below] {\small Re};
   \draw[->] (0,-2) -- (0,4) node[right] {\small Im};
 
-  % Stable: e^-1.75t * cos(1.75wt) (80/3*w for readability)
+  % Stable: exp(-1.75t) cos(1.75wt) (80/3*w for readability)
   \drawtimeplot{-2.125cm}{2.5cm}{0.125cm}{0.44375cm}{
     exp(-1.75 * \x) * cos(80/3 * 1.75 * deg(\x))}
   \drawpole{-1.75cm}{1.75cm}
 
-  % Stable: e^-2.5t
+  % Stable: exp(-2.5t)
   \drawtimeplot{-2.25cm}{0.75cm}{0.125cm}{0.125cm}{exp(-2 * \x)}
   \drawpole{-2cm}{0cm}
 
-  % Stable: e^-t
+  % Stable: exp(-t)
   \drawtimeplot{-1.125cm}{-0.75cm}{0.125cm}{0.125cm}{exp(-\x)}
   \drawpole{-1cm}{0cm}
 
@@ -30,15 +30,15 @@
   \drawtimeplot{0.25cm}{-0.75cm}{0.125cm}{0.125cm}{1}
   \drawpole{0cm}{0cm}
 
-  % Unstable: e^t
+  % Unstable: exp(t)
   \drawtimeplot{1.125cm}{0.75cm}{0.125cm}{0.125cm}{exp(\x)}
   \drawpole{1cm}{0cm}
 
-  % Unstable: e^2t
+  % Unstable: exp(2t)
   \drawtimeplot{2.25cm}{-0.75cm}{0.125cm}{0.125cm}{exp(2 * \x)}
   \drawpole{2cm}{0cm}
 
-  % Unstable: e^0.75t * cos(1.75wt) (80/3*w for readability)
+  % Unstable: exp(0.75t) cos(1.75wt) (80/3*w for readability)
   \drawtimeplot{1.5cm}{2.25cm}{0.125cm}{0.44375cm}{
     exp(0.75 * \x) * cos(80/3 * 1.75 * deg(\x))}
   \drawpole{0.75cm}{1.75cm}

figs/discrete-impulse-response-vs-pole-location.tex

@@ -0,0 +1,63 @@
+\begin{tikzpicture}[auto, >=latex']
+  % \draw [help lines] (-4,-4) grid (4,4);
+
+  % Draw main axes
+  \draw[->] (-4,0) -- (4,0) node[below] {\small Re};
+  \draw[->] (0,-4) -- (0,4) node[right] {\small Im};
+
+  % Unit circle
+  \draw[black] (0,0) circle (3cm);
+
+  % Exponent for the given x plot coordinate:
+  %
+  %   exp(at) = x
+  %   at = ln(x)
+  %   a = ln(x)/t
+  %
+  % t = 50 ms
+
+  % LHP stable: exp(-21.97t)
+  %
+  % a = ln(1/3)/0.05 = -21.97
+  \drawtimeplot{1cm - 0.166cm}{-0.75cm}{0.125cm}{0.125cm}{exp(-21.97 * \x)}
+  \drawpole{1cm}{0cm}
+
+  % LHP stable: exp(-15t)
+  %
+  % a = ln(2/3)/0.05 = -8.109
+  \drawtimeplot{2cm}{0.75cm}{0.125cm}{0.125cm}{exp(-8.109 * \x)}
+  \drawpole{2cm}{0cm}
+
+  % RHP stable: exp(-10.192t) cos(19.665wt) (40/3*w for readability)
+  %
+  % a = ln(0.333+0.5i)/0.05 = -10.192 + 19.665i
+  \drawtimeplot{1cm}{2.25cm}{0.125cm}{0.44375cm}{
+    exp(-10.192 * \x) * cos(40/3 * 19.665 * deg(\x))}
+  \drawpole{1cm}{1.5cm}
+
+  % RHP stable: exp(-21.97t)
+  %
+  % a = ln(1/3)/0.05 = -21.97
+  \drawtimeplot{1cm - 0.166cm}{-0.75cm}{0.125cm}{0.125cm}{exp(-21.97 * \x)}
+  \drawpole{1cm}{0cm}
+
+  % RHP stable: exp(-15t)
+  %
+  % a = ln(2/3)/0.05 = -8.109
+  \drawtimeplot{2cm}{0.75cm}{0.125cm}{0.125cm}{exp(-8.109 * \x)}
+  \drawpole{2cm}{0cm}
+
+  % Integrator
+  \drawtimeplot{3cm + 0.166cm}{-0.75cm}{0.125cm}{0.125cm}{exp(0 * \x)}
+  \drawpole{3cm}{0cm}
+
+  % LHP and RHP labels
+  \draw (-3.5,1.5) node {LHP};
+  \draw (3.5,1.5) node {RHP};
+
+  % Stable and unstable labels
+  \draw (2.5,3.5) node {\small Unstable};
+  \draw (-2.5,3.5) node {\small Unstable};
+  \draw (-2.5,-3.5) node {\small Unstable};
+  \draw (2.5,-3.5) node {\small Unstable};
+\end{tikzpicture}

modern-control-theory/continuous-state-space-control/eigenvalues-and-stability.tex

@@ -67,14 +67,14 @@ \section{Eigenvalues and stability}
 \index{stability!poles}
 The eigenvalues of $\mat{A}$ are called \textit{poles}.\footnote{This name comes
 from classical control theory. See subsection \ref{subsec:poles_and_zeroes} for
-more.} Figure \ref{fig:impulse_response_eig} shows the \glspl{impulse response}
-in the time domain for \glspl{system} with various pole locations in the complex
-plane (real numbers on the x-axis and imaginary numbers on the y-axis). Each
-response has an initial condition of $1$.
+more.} Figure \ref{fig:cont_impulse_response_eig} shows the \glspl{impulse
+response} in the time domain for \glspl{system} with various pole locations in
+the complex plane (real numbers on the x-axis and imaginary numbers on the
+y-axis). Each response has an initial condition of $1$.
 \begin{bookfigure}
-  \input{figs/impulse-response-vs-pole-location}
+  \input{figs/continuous-impulse-response-vs-pole-location}
   \caption{Impulse response vs pole location}
-  \label{fig:impulse_response_eig}
+  \label{fig:cont_impulse_response_eig}
 \end{bookfigure}
 
 Poles in the left half-plane (LHP) are stable; the \gls{system}'s output may

modern-control-theory/discrete-state-space-control/continuous-to-discrete-pole-mapping.tex

@@ -29,14 +29,16 @@ \section{Continuous to discrete pole mapping}
 
 \subsection{Discrete system stability}
 
-Eigenvalues of a \gls{system} that are within the unit circle are stable, but
-why is that? Let's consider a scalar equation $x_{k + 1} = ax_k$. $a < 1$ makes
-$x_{k + 1}$ converge to zero. The same applies to a complex number like
-$z = x + yi$ for $x_{k + 1} = zx_k$. If the magnitude of the complex number $z$
-is less than one, $x_{k+1}$ will converge to zero. Values with a magnitude of
-$1$ oscillate forever because $x_{k+1}$ never decays.
+Eigenvalues of a \gls{system} that are within the unit circle are stable. To
+demonstrate this, consider the discrete system $x_{k + 1} = ax_k$ where $a$ is a
+complex number. $|a| < 1$ will make $x_{k + 1}$ converge to zero.
 
 \subsection{Discrete system behavior}
+\begin{bookfigure}
+  \input{figs/discrete-impulse-response-vs-pole-location}
+  \caption{Impulse response vs pole location}
+  \label{fig:disc_impulse_response_eig}
+\end{bookfigure}
 
 As $\omega$ increases in $s = j\omega$, a pole in the discrete plane moves
 around the perimeter of the unit circle. Once it hits $\frac{\omega_s}{2}$ (half
@@ -51,10 +53,9 @@ \subsection{Discrete system behavior}
 other considerations like \gls{control effort}, \gls{robustness}, and
 \gls{noise immunity}.
 
-If poles from $(1, 0)$ to $(0, 0)$ on the x-axis approach infinity, then what do
-poles from $(-1, 0)$ to $(0, 0)$ represent? Them being faster than infinity
-doesn't make sense. Poles in this location exhibit oscillatory behavior similar
-to complex conjugate pairs. See figures \ref{fig:continuous_oscillations_1p} and
+Systems whose discrete poles are in the left half-plane have their outputs
+reflect across the steady-state value
+change sign every timestep. See figures \ref{fig:continuous_oscillations_1p} and
 \ref{fig:discrete_oscillations_2p}. The jaggedness of these signals is due to
 the frequency of the \gls{system} dynamics being above the Nyquist frequency
 (twice the sample frequency). The \glslink{discretization}{discretized} signal

preamble/macros.tex

@@ -136,6 +136,31 @@
     \end{scope}
   \end{scope}}
 
+% #1: xshift
+% #2: yshift
+% #3: xcoordshift
+% #4: ycoordshift
+% #5: coordinates to plot
+\newcommand{\drawcoordplot}[5] {
+  \begin{scope}[xshift=#1-0.5cm,yshift=#2-0.5cm]
+    \def\xcoordshift{#3,0}
+    \def\ycoordshift{0,#4}
+
+    \draw[fill=headingbg] (0,0) rectangle (1,1);
+    \begin{scope}[shift=(\xcoordshift)]
+      % Draw y-axis
+      \draw[->] (0cm,0.05cm) -- (0cm,0.9375cm) node {};
+      \begin{scope}[shift=(\ycoordshift)]
+        % Draw x-axis
+        \draw[->,font=\tiny] (-0.075cm,0cm) -- (0.8125cm,0cm)
+          node[label={[below]90:$t$}] {};
+
+        \draw[xscale=1.2,yscale=0.3,domain=0:0.5,smooth,variable=\x,line width=0.5]
+          plot coordinates {#5};
+      \end{scope}
+    \end{scope}
+  \end{scope}}
+
 % #1: xshift
 % #2: yshift
 \newcommand{\drawpole}[2] {