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}
- \caption{Impulse response vs pole location}
- \label{fig:impulse_response_poles}
+ \input{figs/continuous-impulse-response-vs-pole-location}
+ \caption{Continuous impulse response vs pole location}
+ \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

@@ -5,41 +5,41 @@
  \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)
- \drawtimeplot{-2.125cm}{2.5cm}{0.125cm}{0.44375cm}{
+ % Stable: exp(-1.75t) cos(1.75wt) (80/3*w for readability)
+ \drawcontinuoustimeplot{-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
- \drawtimeplot{-2.25cm}{0.75cm}{0.125cm}{0.125cm}{exp(-2 * \x)}
+ % Stable: exp(-2.5t)
+ \drawcontinuoustimeplot{-2.25cm}{0.75cm}{0.125cm}{0.125cm}{exp(-2 * \x)}
  \drawpole{-2cm}{0cm}
 
- % Stable: e^-t
- \drawtimeplot{-1.125cm}{-0.75cm}{0.125cm}{0.125cm}{exp(-\x)}
+ % Stable: exp(-t)
+ \drawcontinuoustimeplot{-1.125cm}{-0.75cm}{0.125cm}{0.125cm}{exp(-\x)}
  \drawpole{-1cm}{0cm}
 
  % Marginally stable: cos(wt) (80/3*w for readability)
- \drawtimeplot{-0.75cm}{1.125cm}{0.125cm}{0.44375cm}{cos(80/3 * deg(\x))}
+ \drawcontinuoustimeplot{-0.75cm}{1.125cm}{0.125cm}{0.44375cm}{cos(80/3 * deg(\x))}
  \drawpole{0cm}{1cm}
 
  % Marginally stable cos(2wt) (80/3*w for readability)
- \drawtimeplot{0cm}{2.75cm}{0.125cm}{0.44375cm}{cos(80/3 * 2 * deg(\x))}
+ \drawcontinuoustimeplot{0cm}{2.75cm}{0.125cm}{0.44375cm}{cos(80/3 * 2 * deg(\x))}
  \drawpole{0cm}{2cm}
 
  % Integrator
- \drawtimeplot{0.25cm}{-0.75cm}{0.125cm}{0.125cm}{1}
+ \drawcontinuoustimeplot{0.25cm}{-0.75cm}{0.125cm}{0.125cm}{1}
  \drawpole{0cm}{0cm}
 
- % Unstable: e^t
- \drawtimeplot{1.125cm}{0.75cm}{0.125cm}{0.125cm}{exp(\x)}
+ % Unstable: exp(t)
+ \drawcontinuoustimeplot{1.125cm}{0.75cm}{0.125cm}{0.125cm}{exp(\x)}
  \drawpole{1cm}{0cm}
 
- % Unstable: e^2t
- \drawtimeplot{2.25cm}{-0.75cm}{0.125cm}{0.125cm}{exp(2 * \x)}
+ % Unstable: exp(2t)
+ \drawcontinuoustimeplot{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)
- \drawtimeplot{1.5cm}{2.25cm}{0.125cm}{0.44375cm}{
+ % Unstable: exp(0.75t) cos(1.75wt) (80/3*w for readability)
+ \drawcontinuoustimeplot{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 +1,95 @@
+\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 integrator
+ \drawdiscretecoordplot{-3cm - 0.166cm}{-0.75cm}{0.125cm}{0.44375cm}
+ {(0,1) (0.05,-1) (0.1,1) (0.15,-1) (0.2,1) (0.25,-1) (0.3,1) (0.35,-1)
+ (0.4,1) (0.45,-1) (0.5,1)}
+ \drawpole{-3cm}{0cm}
+
+ % LHP stable: xₖ₊₁ = −2/3xₖ
+ \drawdiscretecoordplot{-2cm}{0.75cm}{0.125cm}{0.44375cm}
+ {(0,1) (0.05,-0.667) (0.1,0.444) (0.15,-0.296) (0.2,0.198) (0.25,-0.132)
+ (0.3,0.088) (0.35,-0.059) (0.4,0.039) (0.45,-0.026) (0.5,0.017)}
+ \drawpole{-2cm}{0cm}
+
+ % LHP stable: xₖ₊₁ = −1/3xₖ
+ \drawdiscretecoordplot{-1cm + 0.166cm}{-0.75cm}{0.125cm}{0.44375cm}
+ {(0,1) (0.05,-0.333) (0.1,0.111) (0.15,-0.037) (0.2,0.012) (0.25,-0.004)
+ (0.3,0.001) (0.35,0) (0.4,0) (0.45,0) (0.5,0)}
+ \drawpole{-1cm}{0cm}
+
+ % LHP stable: xₖ₊₁ = (0.333 + 0.5i)xₖ
+ \drawdiscretecoordplot{-1cm}{2.25cm}{0.125cm}{0.44375cm}
+ {(0,1) (0.05,-0.333) (0.1,-0.139) (0.15,0.213) (0.2,-0.092) (0.25,-0.016)
+ (0.3,0.044) (0.35,-0.023) (0.4,0) (0.45,0.009) (0.5,-0.006)}
+ \drawpole{-1cm}{1.5cm}
+
+ % LHP unstable: xₖ₊₁ = (−1 − 0.75i)xₖ
+ \drawdiscretecoordplot{-2.25cm}{-2.25cm}{0.125cm}{0.44375cm}
+ {(0,1) (0.05,-1) (0.1,0.4375) (0.15,0.6875) (0.2,-2.059) (0.25,3.043)
+ (0.3,-2.869) (0.35,0.984) (0.4,2.515) (0.45,-6.568) (0.5,9.206)}
+ \drawpole{-3cm}{-2.25cm}
+
+ % 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
+ \drawdiscretetimeplot{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
+ \drawdiscretetimeplot{1cm - 0.166cm}{-0.75cm}{0.125cm}{0.125cm}{exp(-21.97 * \x)}
+ \drawpole{1cm}{0cm}
+
+ % RHP stable: exp(-8.109t)
+ %
+ % a = ln(2/3)/0.05 = -8.109
+ \drawdiscretetimeplot{2cm}{0.75cm}{0.125cm}{0.125cm}{exp(-8.109 * \x)}
+ \drawpole{2cm}{0cm}
+
+ % RHP marginally stable: cos(15.707wt)
+ %
+ % a = ln(√2 + √2i)/0.05 = 15.707i
+ \drawdiscretetimeplot{3 * (0.707cm + 0.25cm)}{3 * 0.707cm + 0.375cm}{0.125cm}{0.44375cm}{cos(15.707 * deg(\x))}
+ \drawpole{3 * 0.707cm}{3 * 0.707cm}
+
+ % RHP integrator
+ \drawdiscretetimeplot{3cm + 0.166cm}{-0.75cm}{0.125cm}{0.125cm}{exp(0 * \x)}
+ \drawpole{3cm}{0cm}
+
+ % RHP unstable: exp(4.463t) cos(-12.87wt)
+ %
+ % a = ln(1 - 0.75i)/0.05 = 4.463 - 12.87i
+ \drawdiscretetimeplot{2.25cm}{-2.25cm}{0.125cm}{0.44375cm}{exp(4.463 * \x) * cos(-12.87 * deg(\x))}
+ \drawpole{3cm}{-2.25cm}
+
+ % LHP and RHP labels
+ \draw (-3.5,1.5) node {LHP};
+ \draw (3.5,1.5) node {RHP};
+
+ % Stable and unstable labels
+ \fill[white] (-0.5,-2.25) rectangle (0.5,-1.75);
+ \draw (0,-2) node {\small Stable};
+ \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}

fundamentals-of-control-theory/control-system-basics/what-is-gain.tex

@@ -14,14 +14,14 @@ \section{What is gain?}
  % \draw [help lines] (-4,-2) grid (4,2);
 
  % Input
- \drawtimeplot{-2.5cm}{0cm}{0.125cm}{0.44375cm}{0.6 * cos(40 * deg(\x))}
+ \drawcontinuoustimeplot{-2.5cm}{0cm}{0.125cm}{0.44375cm}{0.6 * cos(40 * deg(\x))}
  \draw (-2.5,1) node {\small input};
 
  \node [block] (sys) {K};
  \draw (0,1) node {\small system};
 
  % Output
- \drawtimeplot{2.5cm}{0cm}{0.125cm}{0.44375cm}{1.2 * cos(40* deg(\x))}
+ \drawcontinuoustimeplot{2.5cm}{0cm}{0.125cm}{0.44375cm}{1.2 * cos(40* deg(\x))}
  \draw (2.5,1) node {\small output};
 
  % Arrows between input/output and system

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}
- \caption{Impulse response vs pole location}
- \label{fig:impulse_response_eig}
+ \input{figs/continuous-impulse-response-vs-pole-location}
+ \caption{Continuous impulse response vs pole location}
+ \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

@@ -29,37 +29,40 @@ \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}
 
+Figure \ref{fig:disc_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/discrete-impulse-response-vs-pole-location}
+ \caption{Discrete 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
 the sampling frequency) at $(-1, 0)$, the pole wraps around. This is due to
 poles faster than the sample frequency folding down to below the sample
 frequency (that is, higher frequency signals \textit{alias} to lower frequency
 ones).
 
-You may notice that poles can be placed at $(0, 0)$ in the discrete plane. This
-is known as a deadbeat controller. An $\rm N^{th}$-order deadbeat controller
-decays to the \gls{reference} in N timesteps. While this sounds great, there are
-other considerations like \gls{control effort}, \gls{robustness}, and
-\gls{noise immunity}.
+Placing the poles at $(0, 0)$ produces a \textit{deadbeat controller}. An
+$\rm N^{th}$-order deadbeat controller decays to the \gls{reference} in N
+timesteps. While this sounds great, there are 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
-\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
-doesn't have enough samples to reconstruct the continuous \gls{system}'s
-dynamics.
+Poles in the left half-plane cause jagged outputs because the frequency of the
+\gls{system} dynamics is above the Nyquist frequency (twice the sample
+frequency). The \glslink{discretization}{discretized} signal doesn't have enough
+samples to reconstruct the continuous \gls{system}'s dynamics. See figures
+\ref{fig:continuous_oscillations_1p} and \ref{fig:discrete_oscillations_2p} for
+examples.
 \begin{bookfigure}
  \begin{minisvg}{2}{build/\chapterpath/z_oscillations_1p}
  \caption{Single poles in various locations in discrete plane}

preamble/macros.tex

@@ -116,7 +116,7 @@
 % #3: xcoordshift
 % #4: ycoordshift
 % #5: func to plot
-\newcommand{\drawtimeplot}[5] {
+\newcommand{\drawcontinuoustimeplot}[5] {
  \begin{scope}[xshift=#1-0.5cm,yshift=#2-0.5cm]
  \def\xcoordshift{#3,0}
  \def\ycoordshift{0,#4}
@@ -130,12 +130,65 @@
  \draw[->,font=\tiny] (-0.075cm,0cm) -- (0.8125cm,0cm)
  node[label={[below]90:$t$}] {};
 
+ \clip (0cm,-0.45cm) rectangle (1cm,0.55cm);
  \draw[xscale=1.2,yscale=0.3,domain=0:0.5,smooth,variable=\x,line width=0.5]
  plot ({\x},{#5});
  \end{scope}
  \end{scope}
  \end{scope}}
 
+% #1: xshift
+% #2: yshift
+% #3: xcoordshift
+% #4: ycoordshift
+% #5: func to plot
+\newcommand{\drawdiscretetimeplot}[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:$t$}] {};
+
+ \clip (0cm,-0.45cm) rectangle (1cm,0.55cm);
+ \draw[xscale=1.2,yscale=0.3,domain=0:0.5,variable=\x,line width=0.5]
+ plot ({\x},{#5});
+ \end{scope}
+ \end{scope}
+ \end{scope}}
+
+% #1: xshift
+% #2: yshift
+% #3: xcoordshift
+% #4: ycoordshift
+% #5: coordinates to plot
+\newcommand{\drawdiscretecoordplot}[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:$t$}] {};
+
+ \clip (0cm,-0.45cm) rectangle (1cm,0.55cm);
+ \draw[xscale=1.2,yscale=0.3,domain=0:0.5,variable=\x,line width=0.5]
+ plot coordinates {#5};
+ \end{scope}
+ \end{scope}
+ \end{scope}}
+
 % #1: xshift
 % #2: yshift
 \newcommand{\drawpole}[2] {