Signals, Systems and Control

Continuous/discrete, periodic/aperiodic. Overlay candidate labels on a signal and fix its kind with two independent questions.

Impulse, step, exponential. The step is the running integral of the impulse and the impulse is the step’s derivative. Sweep the upper limit and watch the step build.

Time shift, scaling, reflection. Slide two parameters to lay a signal onto a target and learn to read x(at − b) from the inside out.

Every signal splits uniquely into an even part and an odd part. Fold the signal against its reflection to extract the two pieces, and tell energy signals from power signals.

Linear, time-invariant, causal. Lower the nonlinearity to find when two paths overlap, see that linearity is superposition, and arrive at LTI.

An LTI system is fully fixed by its response h(t) to a single impulse. Move the impulse’s amplitude and position to see the output is always a copy of h.

Convolution flips the response, slides it, and measures the overlap area with the input. Move the slide position and watch the overlap trace out the output curve.

Split time into intervals at the breakpoints and integrate the product on each. See two rectangular pulses convolve into a triangle, piece by piece.

Stability and causality are written in the shape of the impulse response h. Move the decay rate and see that stability is when the running |h| area hits a ceiling.

The step response is the running integral of the impulse response, and differentiating returns it. Increase time and watch s settle to the final value (DC gain).

A periodic signal is a sum of harmonics at integer multiples of the fundamental. Add harmonics and watch the partial sum rebuild a square wave, with the Gibbs phenomenon.

The spectrum views a signal on the frequency axis. Slide a sinusoid’s frequency to move its spectral line and see that time and frequency are one pair.

Stretch the period to infinity and discrete harmonic lines melt into a continuous spectrum. Move the pulse width to see the rect ↔ sinc reciprocity between time and frequency.

Convolution in time becomes multiplication in frequency. Move h’s width to see the convolution theorem: y=x∗h and Y=X·H change together.

The frequency response H(ω) has magnitude as gain and phase as delay. Raise the input frequency to see the output sinusoid shrink and lag — passband and stopband.

A Bode plot draws magnitude in dB and frequency on a log axis, straightening the curve into asymptotes. Move the corner frequency to see the −20 dB/decade slope.

Filtering multiplies the spectrum by a filter. Use type buttons and a cutoff slider to match the passband to a target band, seeing bandwidth and the four filters.

The Laplace transform adds a convergence factor e^(−σt) to Fourier, widening jω into s=σ+jω. Move σ to see the ROC where a growing signal converges.

A pole’s position on the s-plane is the system’s behavior. Move it with σ and ω sliders to see left half stable, right half diverging.

The transfer function H(s)=output/input is the system itself. Move a coefficient to see the ODE, H(s), the pole, and the impulse response move as one.

Series multiplies, parallel adds, feedback gives H/(1+HG). Raise the feedback gain to move an unstable plant’s pole into the left half and stabilize it.

Pole position fixes the response shape. First order is an exponential with time constant τ; for second order, ζ and ωn split it into over-, critically, and underdamped. Tune ζ to settle fastest with no overshoot.

To rebuild a continuous signal from samples, the sampling rate must exceed twice the highest frequency (Nyquist). Move the sampling rate to see aliasing and the condition for recovery.

The folding frequency fs/2 is a mirror. A true frequency above it folds to the lower phantom |f − fs·round(f/fs)|. Raise f to see the folding triangle and the wheel, moiré, and pitch effects.

The Laplace transform of a discrete signal. A one-step delay = multiply by z⁻¹, so a pole z = r·e^(jθ) sets the mode rⁿ cos(θn). Stability boundary is the unit circle (|z|<1). Move the z-plane pole.

A discrete LTI system is fixed by one impulse response h[n], and the output is the discrete convolution Σ h[k]x[n−k]. Stretch h[n] with the smoothing strength to shave noise. FIR vs IIR and unit-circle stability.

The DTFT is a discrete signal’s spectrum (the z-transform on the unit circle); the DFT samples it at N points for computation (FFT). On a bin you get a clean spike; off a bin, spectral leakage.

Decide stability from the characteristic polynomial’s coefficients alone, without solving for poles. The first-column sign changes of the Routh table equal the number of right-half-plane poles. Move the gain K to find where poles cross the imaginary axis.

The path the closed-loop poles (roots of 1+KG=0) trace on the s-plane as the gain K varies, from open-loop poles toward zeros, crossing the imaginary axis. Land the poles on a target ζ=0.5 line to read the design gain.

On the open-loop Bode plot, read the distance to instability (the −1 condition) as the phase margin and gain margin. Move the gain K and watch both reach zero at K=6 (the same boundary as F1 and F2).

Plot the open loop L(jω) in the complex plane and judge stability by encirclements of −1 (Z=N+P). Move the gain K to see the −K/6 crossing pass −1 and the curve encircle it (the K=6 boundary).

Wrap a PID controller u=Kp·e+Ki·∫e+Kd·de/dt around the plant G=1/((s+1)(s+2)) and shape the step response. Proportional alone leaves a steady-state error, the integral drives it to zero, and the derivative tames the ringing. Move the three gains to tune a fast, accurate, smooth response (closed-loop characteristic s³+(3+Kd)s²+(2+Kp)s+Ki).

Insert a compensator C(s)=(1+αTs)/(1+Ts) into the same plant as F1 to F4 and shape the phase and gain of the Bode plot. Lead (α>1) lifts the phase by φ_max=asin((α−1)/(α+1)) at ω_m=1/(T√α) to raise the phase margin (like the derivative term); lag (α<1) raises the low-frequency gain to cut steady-state error (like the integral term). Move α and ω_m to seat the bump on the crossover and pull PM from about 11° up to a healthy value.

Read the error a closed loop is left with from the final value theorem e_ss=lim_{s→0} s·R/(1+L). With input (step, ramp, parabola) and system-type (type 0, 1, 2 = poles at the origin) buttons and a gain K, shape the tracking error e(t): when type beats input order the error is zero, equal gives finite (1/(1+Kp), 1/Kv, 1/Ka), lower diverges. The type-1 K/(s(s+1)(s+2)) is the F1 to F6 plant. The control chapter finale.