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‎BlockA/SysReg_crashcourse_TimeDomain_BlockA_01_Basics.py‎

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%matplotlib notebook
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import numpy as np
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import matplotlib.pyplot as plt
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plt.rcParams.update({ 'text.usetex': False, 'mathtext.fontset': 'cm',
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'font.size': 12.0, 'axes.labelsize': 'medium',
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'xtick.labelsize': 'x-small', 'ytick.labelsize': 'x-small',
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'axes.grid': True, 'axes.formatter.limits': [-3, 6],
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'grid.alpha': 0.5, 'figure.figsize': [11.0, 4],
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'figure.constrained_layout.use': True, 'scatter.marker': 'x',
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'animation.html': 'jshtml'})
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from IPython.display import display, Markdown
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from IPython.display import display
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import warnings
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warnings.filterwarnings("ignore")
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import scipy.signal as signal
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import scipy.linalg as sclin
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import numpy.random as rng
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import numpy.linalg as lin
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import control as cm
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from helperFunctions import *
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setPlotStyle()
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# %% [markdown]
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# # Time domain
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# %%
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## Eigenvalues
58-
sig1 = -0.8
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sig2 = 3 + 15j
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sig3 = -0.5 + 5j
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SIG = [ -0.8,
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2 + 15j,
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-0.5 + 5j,
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-0.5 - 5j,
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] # Add or remove any you'd like to see
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###### Plotting #########
63-
fig, ax = plt.subplots(1,3)
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for sig, idx in zip([sig1, sig2, sig3], range(3)):
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t = np.linspace(0, 3/abs(np.real(sig)), num=300) # Adapt to convergence speed
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fig, ax = plt.subplots(1,len(SIG))
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for sig, idx in zip(SIG, range(len(SIG))):
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t = np.linspace(0, 3/abs(sig.real) if sig.real != 0 else 1, 300) # Adapt to convergence speed
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if np.iscomplex(sig): # Plot decomposition
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l1, = ax[idx].plot(t, np.exp(sig.real * t), 'r--', label=f"$e^{r"{"}{sig.real}t{r"}"}$") # Upper envelope
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ax[idx].plot(t, -np.exp(sig.real * t), 'r--') # Lower envelope
@@ -97,17 +85,13 @@
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#
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# It's a nice sanity check that any fundamental block scheme of an $n$-th order system has $n$ integrators.
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#
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#
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# ## Equilibrium points
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#         *Forever and unchanging*
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#
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# 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 (JW). So equilibria remain unchanging, so the derivative of the state is zero. In mathematicians' language
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# $$ \dot{\bar x} = f(\bar x, \bar u) = 0. $$
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# Finding the equilibria states as a function of the equilibria inputs is as simple as solving this equation.
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#
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#
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#
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#
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# %% [markdown]
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# <div style="text-align:center;background-color:tomato;">End of lecture 3</div>

‎BlockA/SysReg_crashcourse_TimeDomain_BlockA_02_StateSpace.py‎

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Large diffs are not rendered by default.

‎BlockA/SysReg_crashcourse_TimeDomain_BlockA_03_ControlSynthesis.py‎

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%matplotlib notebook
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import numpy as np
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import matplotlib.pyplot as plt
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plt.rcParams.update({ 'text.usetex': False, 'mathtext.fontset': 'cm',
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'font.size': 12.0, 'axes.labelsize': 'medium',
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'xtick.labelsize': 'x-small', 'ytick.labelsize': 'x-small',
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'axes.grid': True, 'axes.formatter.limits': [-3, 6],
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'grid.alpha': 0.5, 'figure.figsize': [11.0, 4],
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'figure.constrained_layout.use': True, 'scatter.marker': 'x',
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'animation.html': 'jshtml'})
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from IPython.display import display, Markdown
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from IPython.display import display
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import warnings
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warnings.filterwarnings("ignore")
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@@ -21,6 +12,7 @@
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import numpy.linalg as lin
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import control as cm
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from helperFunctions import *
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setPlotStyle()
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# %% [markdown]
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# ## Textbook input signals
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## impulse response
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impIn = np.zeros_like(T_res)
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impIn[int(-T_res[0]/(T_res[1] - T_res[0]))] = 1./0.02
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impIn[int(-T_res[0]/(T_res[1] - T_res[0]))] = 1./(T_res[1] - T_res[0])
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impResponse = cm.forced_response(P_res,
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T = T_res,
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U = impIn)

‎BlockA/SysReg_crashcourse_TimeDomain_BlockA_04_Observers.py‎

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%matplotlib notebook
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import numpy as np
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import matplotlib.pyplot as plt
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plt.rcParams.update({ 'text.usetex': False, 'mathtext.fontset': 'cm',
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'font.size': 12.0, 'axes.labelsize': 'medium',
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'xtick.labelsize': 'x-small', 'ytick.labelsize': 'x-small',
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'axes.grid': True, 'axes.formatter.limits': [-3, 6],
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'grid.alpha': 0.5, 'figure.figsize': [11.0, 4],
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'figure.constrained_layout.use': True, 'scatter.marker': 'x',
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'animation.html': 'jshtml'})
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from IPython.display import display, Markdown
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from IPython.display import display
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import warnings
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warnings.filterwarnings("ignore")
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import scipy.signal as signal
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import scipy.linalg as sclin
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import numpy.random as rng
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import numpy.linalg as lin
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import control as cm
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from helperFunctions import *
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setPlotStyle()
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# %% [markdown]
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# ## Observe this

‎BlockA/SysReg_crashcourse_TimeDomain_BlockA_05_Summary.py‎

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%matplotlib notebook
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import numpy as np
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import matplotlib.pyplot as plt
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plt.rcParams.update({ 'text.usetex': False, 'mathtext.fontset': 'cm',
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'font.size': 12.0, 'axes.labelsize': 'medium',
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'xtick.labelsize': 'x-small', 'ytick.labelsize': 'x-small',
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'axes.grid': True, 'axes.formatter.limits': [-3, 6],
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'grid.alpha': 0.5, 'figure.figsize': [11.0, 4],
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'figure.constrained_layout.use': True, 'scatter.marker': 'x',
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'animation.html': 'jshtml'})
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from IPython.display import display, Markdown
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from IPython.display import display
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import warnings
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warnings.filterwarnings("ignore")
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import scipy.signal as signal
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import scipy.linalg as sclin
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import numpy.random as rng
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import numpy.linalg as lin
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import control as cm
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from helperFunctions import *
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setPlotStyle()
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# %% [markdown]
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# ## Connecting the dots
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#
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# Time to make the full picture: state feedback control with state estimation! Some important things to note:
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# - Since the state feedback is based on the state estimation, we want the error to go to zero faster than the controller works. If the controller is faster than the observer you're just doing random stuff basically until the estimate converges. Rule of thumb: the observer should be 4 to 10 times faster (larger magnitude poles) than the controller.
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# - Since the state feedback is based on the state estimation, we want the error to go to zero faster than the controller works. If the controller is faster than the observer you're just doing
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# random stuff basically until the estimate converges. Rule of thumb: the observer should be 4 to 10 times faster (larger magnitude poles) than the controller.
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# - Making the observer extremely fast works well in theory, but as soon as your output is noisy, the fast observer will amplify this noise enourmously and your estimate will be terrible.
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# - In the example below I'll make an LQR observer, but since the observer is a regulator type controller, no feedforward gain is needed.
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# - Strictly speaking you can classify this as output feedback

‎BlockB/SysReg_crashcourse_FreqDomain_BlockB_02_BodeNyquist.py‎

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%matplotlib notebook
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import numpy as np
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import matplotlib.pyplot as plt
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plt.rcParams.update({ 'mathtext.fontset': 'cm',
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'font.size': 12.0, 'axes.labelsize': 'medium',
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'xtick.labelsize': 'x-small', 'ytick.labelsize': 'x-small',
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'axes.grid': True, 'axes.formatter.limits': [-3, 6],
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'grid.alpha': 0.5, 'figure.figsize': [11.0, 4],
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'figure.constrained_layout.use': True, 'scatter.marker': 'x',
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'animation.html': 'jshtml'
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})
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from matplotlib.ticker import MultipleLocator
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from matplotlib.gridspec import GridSpec
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import warnings
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warnings.filterwarnings("ignore")
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import control as cm
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from helperFunctions import *
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setPlotStyle()
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# %% [markdown]
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# # Visualising complex numbers
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192183
ax[i].annotate("$z_1$", xy=(np.real(z1), np.imag(z1)),
193184
xytext=(.5,.3), textcoords='offset fontsize', color='red')
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## Animation time! Yeah don't focus too much on this it's fine
186+
## Animation time! Don't focus too much on this it's fine
196187
r = cntr - z1
197188
rm = np.max(np.abs(r))
198189
ax[1].set(aspect='equal', xlabel="$\mathfrak{Re}\{G(s)\}$", ylabel="$\mathfrak{Im}\{G(s)\}$", xlim=[-rm+np.real(z1), rm+np.real(z1)], ylim=[-rm+np.imag(z1),rm+np.imag(z1)])
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332323
ax[2].cla()
333324
ax[2].set(xlabel = r"$\omega$", ylabel = r"$\angle G(s)$ / ${}^\circ$")
334-
# ax[2].semilogx(np.abs(OM_full), ang, 'k')
335325
ax[2].semilogx(np.abs(OM_full)[:OM_full.size//2], ang[:OM_full.size//2], 'k')
336326
ax[2].semilogx(np.abs(OM_full)[OM_full.size//2:], ang[OM_full.size//2:], 'k--')
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358348
anim = animation.FuncAnimation(fig, func=animate, frames=50, interval=150, blit=True)
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360350
display(anim)
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# %% [markdown]
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# Note the relation between the two plots!

‎BlockB/SysReg_crashcourse_FreqDomain_BlockB_03_1st2ndOrder.py‎

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%matplotlib notebook
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import numpy as np
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import matplotlib.pyplot as plt
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plt.rcParams.update({ 'mathtext.fontset': 'cm',
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'font.size': 12.0, 'axes.labelsize': 'medium',
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'xtick.labelsize': 'x-small', 'ytick.labelsize': 'x-small',
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'axes.grid': True, 'axes.formatter.limits': [-3, 6],
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'grid.alpha': 0.5, 'figure.figsize': [11.0, 4],
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'figure.constrained_layout.use': True, 'scatter.marker': 'x',
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'animation.html': 'jshtml'
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})
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from matplotlib.ticker import MultipleLocator
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from matplotlib.gridspec import GridSpec
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import matplotlib.animation as animation
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from IPython.display import display, Markdown
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from IPython.display import display
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import warnings
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warnings.filterwarnings("ignore")
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import control as cm
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from helperFunctions import *
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setPlotStyle()
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# %% [markdown]
2617
# ## Your own transfer function toolbox
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# - a time-delay, $e^{-\tau_1 s}$, that delays with $\tau_1$ time
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# %%
36-
k1, p1, z1 = 1e-2, 0.3, 7. #CHANGEME
27+
k1, p1, z1 = 1e-2, 0.3, 7. #CHANGEME
3728
om_p, zeta_p, om_z, zeta_z = .1, .1, 4., 2. #CHANGEME
38-
tau1 = 3e-3 #CHANGEME
29+
tau1 = 3e-3 #CHANGEME
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OM = np.logspace(-2, 3.5, 400)
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S = OM*1j

‎BlockB/SysReg_crashcourse_FreqDomain_BlockB_04_SystemAnalysis.py‎

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%matplotlib notebook
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import numpy as np
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import matplotlib.pyplot as plt
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plt.rcParams.update({ 'mathtext.fontset': 'cm',
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'font.size': 12.0, 'axes.labelsize': 'medium',
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'xtick.labelsize': 'x-small', 'ytick.labelsize': 'x-small',
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'axes.grid': True, 'axes.formatter.limits': [-3, 6],
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'grid.alpha': 0.5, 'figure.figsize': [11.0, 4],
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'figure.constrained_layout.use': True, 'scatter.marker': 'x',
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'animation.html': 'jshtml'
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})
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from matplotlib.ticker import MultipleLocator
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from matplotlib.gridspec import GridSpec
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from IPython.display import display, Markdown
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from IPython.display import display
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import warnings
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warnings.filterwarnings("ignore")
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import control as cm
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from helperFunctions import *
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setPlotStyle()
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# %% [markdown]
2615
# ## Performance Margins
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119108
# ## Minimum Phase Systems
120109
# Minimum phase systems (MPS) are systems where the phase is the absolute minimum possible value for any given magnitude. Defined the other way around: MPS are systems without phase addition from factors not
121110
# contributing to the magnitude. These factors are only time-delays and RHP zeros. Lets look at the difference between a LHP and RHP zero:
111+
#
122112
# %%
123113
OM = np.logspace(-3, 3, 700)
124114
S = OM*1j

‎BlockB/SysReg_crashcourse_FreqDomain_BlockB_05_ControlSynthesis.py‎

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%matplotlib notebook
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import numpy as np
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import matplotlib.pyplot as plt
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plt.rcParams.update({ 'mathtext.fontset': 'cm',
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'font.size': 12.0, 'axes.labelsize': 'medium',
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'xtick.labelsize': 'x-small', 'ytick.labelsize': 'x-small',
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'axes.grid': True, 'axes.formatter.limits': [-3, 6],
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'grid.alpha': 0.5, 'figure.figsize': [11.0, 4],
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'figure.constrained_layout.use': True, 'scatter.marker': 'x',
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'animation.html': 'jshtml'
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})
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from matplotlib.ticker import MultipleLocator
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from matplotlib.gridspec import GridSpec
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import matplotlib.animation as animation
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from IPython.display import display, Markdown
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from IPython.display import display
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import warnings
209
warnings.filterwarnings("ignore")
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import control as cm
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from helperFunctions import *
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setPlotStyle()
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# %% [markdown]

‎__pycache__/helperFunctions.cpython-314.pyc‎

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