For an LTI system, the frequency response \( G(j\omega) \): encodes how sinusoidal components are amplified and phase-shifted. Since a transient (e.g., a step response) can be decomposed into many frequency components, constraints in one domain necessarily constrain the other domain:

• \( \omega_{bw} \): larger bandwidth typically implies faster rise time and smaller dominant time constants.
• \( M_r \): a larger resonant peak typically implies stronger ringing and higher overshoot for lightly damped dynamics.
• Increasing damping reduces overshoot and resonance but generally reduces bandwidth (slows the response).

flowchart TD
  A["Design knobs: wn (natural freq), \nzeta (damping)"] --> B["Frequency response"]
  A --> C["Time response"]

  B --> B1["Bandwidth wb: \nlarger -> faster"]
  B --> B2["Resonant peak Mr: \nlarger -> more ringing"]
  B --> B3["Phase slope near crossover: \nrelates to damping"]

  C --> C1["Rise time tr: \nsmaller -> faster"]
  C --> C2["Settling time ts: \nsmaller -> faster"]
  C --> C3["Overshoot Mp: \nsmaller -> less ringing"]

  B2 --> C3
  B1 --> C1
  B1 --> C2
  A --> D["Trade-off: increase wb \noften increases Mr and \nMp unless zeta increased"]
        

In what follows, we make these statements precise via (i) general LTI transform relationships, (ii) a universal uncertainty inequality for signals, and (iii) explicit second-order formulas used throughout control engineering as design approximations.

Let an LTI system have impulse response \( h(t) \): (assume causality: \( h(t)=0 \) for \( t < 0 \):). For input \( u(t) \): the output is convolution \( y(t):

\[ y(t) = (h * u)(t) = \int_{0}^{\infty} h(\tau)\,u(t-\tau)\,d\tau. \]

Consider a complex exponential input \( u(t)=e^{j\omega t} \): (used as an algebraic device). Substitute into convolution:

\[ y(t) = \int_{0}^{\infty} h(\tau)\,e^{j\omega(t-\tau)}\,d\tau = e^{j\omega t}\int_{0}^{\infty} h(\tau)\,e^{-j\omega \tau}\,d\tau. \]

This motivates the definition of the frequency response: \( G(j\omega):

\[ G(j\omega) \triangleq \int_{0}^{\infty} h(t)\,e^{-j\omega t}\,dt, \quad \Rightarrow \quad y(t)=G(j\omega)\,e^{j\omega t}. \]

If the Laplace transform exists, the transfer function is \( G(s)=\int_{0}^{\infty} h(t)\,e^{-st}\,dt \): and the frequency response is the restriction \( G(j\omega)=G(s)\vert_{s=j\omega} \) when that substitution is valid (e.g., stable proper LTI).

Key consequence: broad frequency content in \( h(t) \): (fast impulse response) implies large bandwidth, which tends to produce fast step responses; narrow frequency content implies slower dynamics.

Beyond LTI systems, there is a universal statement for energy signals: you cannot simultaneously localize a signal arbitrarily well in time and in frequency. This is the mathematical backbone of “fast transients require wide frequency content.”

Let \( x(t) \in L^2(\mathbb{R}) \): with Fourier transform \( X(\omega) \): defined by

\[ X(\omega) = \int_{-\infty}^{\infty} x(t)\,e^{-j\omega t}\,dt, \qquad x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(\omega)\,e^{j\omega t}\,d\omega. \]

Define signal energy and the time/frequency “centers”: \( E \): , \( t_0 \): , \( \omega_0 \):

\[ E = \int_{-\infty}^{\infty} |x(t)|^2\,dt, \quad t_0 = \frac{1}{E}\int_{-\infty}^{\infty} t\,|x(t)|^2\,dt, \quad \omega_0 = \frac{1}{2\pi E}\int_{-\infty}^{\infty} \omega\,|X(\omega)|^2\,d\omega. \]

Define the spreads (standard deviations) in time and frequency: \( \Delta t \): and \( \Delta \omega \):

\[ (\Delta t)^2 = \frac{1}{E}\int_{-\infty}^{\infty} (t-t_0)^2\,|x(t)|^2\,dt, \qquad (\Delta \omega)^2 = \frac{1}{2\pi E}\int_{-\infty}^{\infty} (\omega-\omega_0)^2\,|X(\omega)|^2\,d\omega. \]

Theorem (time–frequency uncertainty):

\[ \Delta t\,\Delta \omega \ge \frac{1}{2}. \]

Proof (sketch with full mathematical steps):

Without loss of generality, shift the signal so that \( t_0=0 \) and modulate so that \( \omega_0=0 \) (these operations do not change \( \Delta t \) and \( \Delta\omega \)). Consider the functions \( f(t)=t\,x(t) \): and \( g(t)=\frac{d}{dt}x(t) \):. By Cauchy–Schwarz:

\[ \left(\int_{-\infty}^{\infty} |f(t)|^2 dt\right)\left(\int_{-\infty}^{\infty} |g(t)|^2 dt\right) \ge \left|\int_{-\infty}^{\infty} f(t)\,\overline{g(t)}\,dt\right|^2. \]

The left factor is \( \int t^2|x(t)|^2 dt = E(\Delta t)^2 \). For the second factor, Parseval’s identity and the differentiation property give:

\[ \int_{-\infty}^{\infty}\left|\frac{d}{dt}x(t)\right|^2 dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} \omega^2 |X(\omega)|^2 d\omega = E(\Delta \omega)^2. \]

Now evaluate the cross term by integration by parts (assuming sufficient decay so boundary terms vanish):

\[ \int_{-\infty}^{\infty} t\,x(t)\,\overline{x'(t)}\,dt = \left[t\,x(t)\,\overline{x(t)}\right]_{-\infty}^{\infty} - \int_{-\infty}^{\infty} \left(x(t)\,\overline{x(t)} + t\,x'(t)\,\overline{x(t)}\right)\,dt. \]

The boundary term is zero by decay, and the remaining expression implies

\[ \int_{-\infty}^{\infty} t\,x(t)\,\overline{x'(t)}\,dt = -\int_{-\infty}^{\infty} |x(t)|^2 dt - \int_{-\infty}^{\infty} t\,x'(t)\,\overline{x(t)}\,dt. \]

Taking complex conjugates shows the real part satisfies:

\[ 2\,\Re\left\{\int_{-\infty}^{\infty} t\,x(t)\,\overline{x'(t)}\,dt\right\} = -E. \]

Hence \( \left|\int t\,x(t)\,\overline{x'(t)}\,dt\right| \ge \frac{E}{2} \). Substituting into Cauchy–Schwarz yields

\[ \big(E(\Delta t)^2\big)\big(E(\Delta\omega)^2\big) \ge \left(\frac{E}{2}\right)^2 \quad \Rightarrow \quad \Delta t\,\Delta\omega \ge \frac{1}{2}. \]

Interpretation: making \( \Delta t \) small (fast, localized transient) forces \( \Delta\omega \) large (broad spectral content), which in LTI systems manifests as higher bandwidth and/or stronger high-frequency gain.

The canonical closed-loop (or normalized plant) second-order low-pass model is:

\[ G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}, \qquad 0 < \zeta, \quad \omega_n > 0. \]

Define the damped natural frequency \( \omega_d \):

\[ \omega_d = \omega_n\sqrt{1-\zeta^2}, \qquad 0 < \zeta < 1. \]

4.1 Step response and overshoot

For a unit-step input and \( 0 < \zeta < 1 \): the step response can be written as

\[ y(t) = 1 - \frac{1}{\sqrt{1-\zeta^2}}\,e^{-\zeta\omega_n t}\, \sin\!\big(\omega_d t + \phi\big), \qquad \phi = \arctan\!\left(\frac{\sqrt{1-\zeta^2}}{\zeta}\right). \]

The first peak occurs at \( t_p \):

\[ t_p = \frac{\pi}{\omega_d} = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}}. \]

Proposition (percent overshoot):

\[ M_p = 100\,\exp\!\left(\frac{-\zeta\pi}{\sqrt{1-\zeta^2}}\right)\%. \]

Proof: evaluate \( y(t) \) at \( t=t_p \). Using \( \sin(\omega_d t_p + \phi)=\sin(\pi+\phi)=-\sin(\phi) \) and \( \sin(\phi)=\sqrt{1-\zeta^2} \) for the chosen \( \phi \), we get:

\[ y(t_p)=1+\exp\!\left(\frac{-\zeta\pi}{\sqrt{1-\zeta^2}}\right), \quad \Rightarrow \quad M_p = 100\big(y(t_p)-1\big). \]

A common settling-time approximation (2% band) follows from the envelope \( e^{-\zeta\omega_n t} \):

\[ t_s \approx \frac{4}{\zeta\omega_n} \quad \text{(2% criterion)}. \]

4.2 Frequency response, resonance peak, and resonance frequency

Evaluate at \( s=j\omega \):

\[ G(j\omega) = \frac{\omega_n^2}{\omega_n^2-\omega^2 + j(2\zeta\omega_n\omega)}. \]

Hence the magnitude squared is

\[ |G(j\omega)|^2 = \frac{\omega_n^4}{(\omega_n^2-\omega^2)^2 + (2\zeta\omega_n\omega)^2}. \]

Normalize by letting \( x = \omega/\omega_n \):

\[ |G(j\omega)|^2 = \frac{1}{(1-x^2)^2 + (2\zeta x)^2}. \]

The resonant peak \( M_r \): occurs at the \( \omega \) maximizing \( |G(j\omega)| \). This is equivalent to minimizing the denominator \( D(x)=(1-x^2)^2 + (2\zeta x)^2 \):. Differentiate:

\[ \frac{dD}{dx} = 2(1-x^2)(-2x) + 8\zeta^2 x = 4x\big(x^2 - 1 + 2\zeta^2\big). \]

The nonzero critical point satisfies \( x^2 = 1-2\zeta^2 \): which requires \( \zeta < 1/\sqrt{2} \):. Therefore:

\[ \omega_r = \omega_n\sqrt{1-2\zeta^2}, \qquad \text{valid when } \zeta < \frac{1}{\sqrt{2}}. \]

Substitute \( x^2=1-2\zeta^2 \) into the denominator: \( 1-x^2=2\zeta^2 \) and \( x=\sqrt{1-2\zeta^2} \), yielding

\[ D(x)= (2\zeta^2)^2 + (2\zeta)^2(1-2\zeta^2) = 4\zeta^2(1-\zeta^2). \]

Thus the resonant peak is:

\[ M_r = \max_\omega |G(j\omega)| = \frac{1}{2\zeta\sqrt{1-\zeta^2}}, \qquad \zeta < \frac{1}{\sqrt{2}}. \]

Direct linkage: both \( M_p \) and \( M_r \) are monotone decreasing in \( \zeta \) over the underdamped range, so a frequency-domain resonance constraint imposes a time-domain overshoot constraint, and conversely.

Define the (magnitude) bandwidth \( \omega_{bw} \): as the smallest frequency at which the magnitude falls to \( 1/\sqrt{2} \) of the DC gain. Since \( G(0)=1 \) for the canonical model, the -3 dB condition is:

\[ |G(j\omega_{bw})|^2 = \frac{1}{2}. \]

Using the normalized expression with \( x=\omega/\omega_n \):

\[ \frac{1}{(1-x^2)^2 + (2\zeta x)^2} = \frac{1}{2} \quad \Longleftrightarrow \quad (1-x^2)^2 + (2\zeta x)^2 = 2. \]

Expand and write it as a quadratic in \( x^2 \):

\[ (1 - 2x^2 + x^4) + 4\zeta^2 x^2 - 2 = 0 \quad \Longleftrightarrow \quad x^4 + (4\zeta^2 - 2)x^2 - 1 = 0. \]

Let \( u=x^2 \):. Then:

\[ u^2 + (4\zeta^2 - 2)u - 1 = 0 \quad \Rightarrow \quad u = 1 - 2\zeta^2 + \sqrt{4\zeta^4 - 4\zeta^2 + 2}, \]

taking the positive root. Therefore the -3 dB bandwidth is

\[ \omega_{bw} = \omega_n\sqrt{1 - 2\zeta^2 + \sqrt{4\zeta^4 - 4\zeta^2 + 2}}. \]

Speed linkage: for moderate damping, \( \omega_{bw} \) scales with \( \omega_n \), while time metrics scale roughly as \( 1/(\zeta\omega_n) \). This yields an intrinsic trade-off: increasing \( \omega_n \) tends to increase bandwidth (faster), while increasing \( \zeta \) tends to reduce overshoot and resonance (safer) but can reduce bandwidth.

For the second-order prototype, the core metrics are explicit functions of \( \zeta \) and \( \omega_n \):

\[ M_p(\zeta) = 100\exp\!\left(\frac{-\zeta\pi}{\sqrt{1-\zeta^2}}\right), \quad M_r(\zeta) = \frac{1}{2\zeta\sqrt{1-\zeta^2}}, \]

\[ t_s \approx \frac{4}{\zeta\omega_n}, \quad \omega_{bw} = \omega_n\sqrt{1 - 2\zeta^2 + \sqrt{4\zeta^4 - 4\zeta^2 + 2}}. \]

Design implication A (overshoot constraint fixes damping): given a maximum overshoot \( M_p^\star \), solve for \( \zeta \):

\[ \frac{M_p^\star}{100} = \exp\!\left(\frac{-\zeta\pi}{\sqrt{1-\zeta^2}}\right) \quad \Rightarrow \quad \zeta = \frac{-\ln\!\left(M_p^\star/100\right)}{\sqrt{\pi^2 + \ln^2\!\left(M_p^\star/100\right)}}. \]

Design implication B (speed constraint fixes natural frequency): if a settling-time target \( t_s^\star \) is given, choose

\[ \omega_n \approx \frac{4}{\zeta t_s^\star}. \]

Alternatively, if a bandwidth target \( \omega_{bw}^\star \) is given, use the bandwidth formula to solve for \( \omega_n \):

\[ \omega_n = \frac{\omega_{bw}^\star}{\sqrt{1 - 2\zeta^2 + \sqrt{4\zeta^4 - 4\zeta^2 + 2}}}. \]

Qualitative trade-off (one sentence, fully rigorous): for fixed \( \zeta \), increasing \( \omega_n \) increases \( \omega_{bw} \) and decreases \( t_s \); for fixed \( \omega_n \), increasing \( \zeta \) decreases \( M_p \) and \( M_r \) but can decrease \( \omega_{bw} \), limiting achievable speed without overshoot.

The following implementations compute step-response metrics and frequency-response metrics for the second-order prototype and sweep \( \zeta \) to reveal the speed–overshoot–resonance coupling.

7.1 Python (SciPy)

File: Chapter12_Lesson5.py

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
# Chapter 12 - Lesson 5: Time–Frequency Domain Relationships and Trade-offs
# File: Chapter12_Lesson5.py

import numpy as np
from scipy import signal

def second_order_tf(zeta: float, wn: float) -> signal.TransferFunction:
    num = [wn**2]
    den = [1.0, 2.0 * zeta * wn, wn**2]
    return signal.TransferFunction(num, den)

def step_metrics(t: np.ndarray, y: np.ndarray, tol: float = 0.02):
    y_final = float(y[-1])
    y_peak = float(np.max(y))
    Mp = 0.0 if y_final == 0.0 else max(0.0, (y_peak - y_final) / abs(y_final)) * 100.0

    y10, y90 = 0.1 * y_final, 0.9 * y_final

    def first_cross(level: float):
        idx = np.where(y >= level)[0]
        return float(t[idx[0]]) if len(idx) else float("nan")

    tr = first_cross(y90) - first_cross(y10)

    band_low, band_high = (1.0 - tol) * y_final, (1.0 + tol) * y_final
    outside = np.where((y < band_low) | (y > band_high))[0]
    ts = float(t[outside[-1]]) if len(outside) else 0.0

    return {"y_final": y_final, "Mp_percent": float(Mp), "tr": float(tr), "ts": float(ts)}

def frequency_metrics(sys: signal.TransferFunction, w: np.ndarray):
    w, H = signal.freqresp(sys, w=w)
    mag = np.abs(H)

    Mr = float(np.max(mag))
    wr = float(w[int(np.argmax(mag))])

    dc = float(mag[0])
    target = dc / np.sqrt(2.0)
    idx = np.where(mag <= target)[0]
    wb = float(w[idx[0]]) if len(idx) else float("nan")

    return {"Mr": Mr, "wr": wr, "wb": wb}

def demo_single(zeta=0.3, wn=10.0):
    sys = second_order_tf(float(zeta), float(wn))
    t = np.linspace(0, 3.0, 4000)
    t, y = signal.step(sys, T=t)
    tm = step_metrics(t, y)
    w = np.logspace(-2, 3, 2000) * wn
    fm = frequency_metrics(sys, w)
    return tm, fm

if __name__ == "__main__":
    zeta, wn = 0.35, 12.0
    tm, fm = demo_single(zeta=zeta, wn=wn)

    print("=== Single system ===")
    print(f"zeta={zeta:.3f}, wn={wn:.3f} rad/s")
    print(f"Overshoot Mp = {tm['Mp_percent']:.2f}%")
    print(f"Rise time tr = {tm['tr']:.4f} s (10%->90%)")
    print(f"Settling time ts = {tm['ts']:.4f} s (2%)")
    print(f"Resonant peak Mr = {fm['Mr']:.4f}")
    print(f"Bandwidth wb = {fm['wb']:.4f} rad/s")

7.2 C++ (from-scratch frequency sweep)

File: Chapter12_Lesson5.cpp

/*
Chapter 12 - Lesson 5: Time–Frequency Domain Relationships and Trade-offs
File: Chapter12_Lesson5.cpp
*/

#include <iostream>
#include <vector>
#include <complex>
#include <cmath>
#include <iomanip>
#include <limits>

static inline double sqr(double x){ return x*x; }

struct FreqMetrics { double Mr, wr, wb; };

std::complex<double> G_jw(double w, double zeta, double wn){
    std::complex<double> jw(0.0, w);
    std::complex<double> den = jw*jw + 2.0*zeta*wn*jw + sqr(wn);
    return sqr(wn) / den;
}

FreqMetrics frequency_metrics(double zeta, double wn){
    const int N = 6000;
    double wmin = 1e-2 * wn, wmax = 1e+3 * wn;

    double Mr = 0.0, wr = wmin;
    double dc = std::abs(G_jw(wmin, zeta, wn));
    double target = dc / std::sqrt(2.0);

    double wb = std::numeric_limits<double>::quiet_NaN();
    bool found_bw = false;

    for(int i=0;i<N;i++){
        double a = static_cast<double>(i)/(N-1);
        double w = wmin * std::pow(wmax/wmin, a);
        double mag = std::abs(G_jw(w, zeta, wn));

        if(mag > Mr){ Mr = mag; wr = w; }
        if(!found_bw && mag <= target){ wb = w; found_bw = true; }
    }
    return {Mr, wr, wb};
}

int main(){
    double zeta = 0.35, wn = 12.0;
    auto fm = frequency_metrics(zeta, wn);

    std::cout << std::fixed << std::setprecision(6);
    std::cout << "Mr=" << fm.Mr << ", wr=" << fm.wr << ", wb=" << fm.wb << "\\n";
    return 0;
}

7.3 Java (self-contained Complex arithmetic)

File: Chapter12_Lesson5.java

/*
Chapter 12 - Lesson 5: Time–Frequency Domain Relationships and Trade-offs
File: Chapter12_Lesson5.java
*/

import java.util.*;

public class Chapter12_Lesson5 {
    static class Complex {
        final double re, im;
        Complex(double re, double im){ this.re = re; this.im = im; }
        Complex add(Complex b){ return new Complex(this.re + b.re, this.im + b.im); }
        Complex mul(Complex b){ return new Complex(this.re*b.re - this.im*b.im, this.re*b.im + this.im*b.re); }
        Complex div(Complex b){
            double d = b.re*b.re + b.im*b.im;
            return new Complex((this.re*b.re + this.im*b.im)/d, (this.im*b.re - this.re*b.im)/d);
        }
        double abs(){ return Math.hypot(re, im); }
    }

    static Complex G_jw(double w, double zeta, double wn){
        Complex jw = new Complex(0.0, w);
        Complex jw2 = jw.mul(jw);
        Complex term = new Complex(0.0, 2.0*zeta*wn*w);
        Complex den = jw2.add(term).add(new Complex(wn*wn, 0.0));
        return new Complex(wn*wn, 0.0).div(den);
    }

    public static void main(String[] args){
        double zeta = 0.35, wn = 12.0;
        double wmin = 1e-2*wn, wmax = 1e3*wn;
        int N = 6000;

        double Mr = 0.0, wr = wmin;
        double dc = G_jw(wmin, zeta, wn).abs();
        double target = dc / Math.sqrt(2.0);
        double wb = Double.NaN;
        boolean found = false;

        for(int i=0;i<N;i++){
            double a = (double)i/(N-1);
            double w = wmin * Math.pow(wmax/wmin, a);
            double mag = G_jw(w, zeta, wn).abs();
            if(mag > Mr){ Mr = mag; wr = w; }
            if(!found && mag <= target){ wb = w; found = true; }
        }

        System.out.printf(Locale.US, "Mr=%.6f, wr=%.6f, wb=%.6f%n", Mr, wr, wb);
    }
}

7.4 MATLAB / Simulink

File: Chapter12_Lesson5.m

% Chapter 12 - Lesson 5: Time–Frequency Domain Relationships and Trade-offs
% File: Chapter12_Lesson5.m

clear; clc;

zeta = 0.35;
wn   = 12; % rad/s

s = tf('s');
G = wn^2/(s^2 + 2*zeta*wn*s + wn^2);

info = stepinfo(G);
Mp = info.Overshoot;
tr = info.RiseTime;
ts = info.SettlingTime;

[mag,~,w] = bode(G);
mag = squeeze(mag); w = squeeze(w);
Mr = max(mag);
wr = w(find(mag == Mr, 1, 'first'));
wb = bandwidth(G);

fprintf('Mp=%.2f%%, tr=%.6fs, ts=%.6fs\\n', Mp, tr, ts);
fprintf('Mr=%.6f, wr=%.6f rad/s, wb=%.6f rad/s\\n', Mr, wr, wb);

% Optional scripted Simulink build:
try
  mdl = 'Chapter12_Lesson5_Simulink';
  if bdIsLoaded(mdl); close_system(mdl,0); end
  new_system(mdl); open_system(mdl);

  add_block('simulink/Sources/Step', [mdl '/Step'], 'Position',[50 80 80 110]);
  add_block('simulink/Continuous/Transfer Fcn', [mdl '/G(s)'], 'Position',[150 75 260 115]);
  add_block('simulink/Sinks/Scope', [mdl '/Scope'], 'Position',[320 75 350 115]);

  set_param([mdl '/G(s)'], 'Numerator', mat2str([wn^2]), 'Denominator', mat2str([1 2*zeta*wn wn^2]));
  add_line(mdl, 'Step/1', 'G(s)/1');
  add_line(mdl, 'G(s)/1', 'Scope/1');
  set_param(mdl, 'StopTime', '3');
  sim(mdl);
catch ME
  fprintf('Simulink skipped: %s\\n', ME.message);
end

7.5 Wolfram Mathematica

File: Chapter12_Lesson5.nb

(* Chapter 12 - Lesson 5: Time–Frequency Domain Relationships and Trade-offs *)
(* File: Chapter12_Lesson5.nb *)

Needs["ControlSystems`"];
zeta = 0.35; wn = 12;
s = Unique["s"];
G = TransferFunctionModel[{wn^2},{1, 2 zeta wn, wn^2}, s];

resp = StepResponse[G, {0, 3}][[1]];
t = resp[[1]]; y = resp[[2]];
Print["Overshoot Mp (%) = ", 100 Max[0, (Max[y] - y[[-1]])/y[[-1]]]];

wz = LogSpace[-2, 3, 2000]*wn;
mag = Abs[FrequencyResponse[G, I*#]]& /@ wz;
Mr = Max[mag];
wr = wz[[First@Ordering[mag, -1]]];

bw = Module[{dc = mag[[1]], idx},
  idx = FirstPosition[mag, _?(# <= dc/Sqrt[2] &)];
  If[idx === Missing["NotFound"], Indeterminate, wz[[idx[[1]]]]]
];

Print["Mr = ", Mr];
Print["wr (rad/s) = ", wr];
Print["wb (rad/s) = ", bw];

7.6 Design Workflow (specs to parameters)

flowchart TD
  S["Given specs: Mp_max, ts_max OR wb_min"] --> Z["Solve zeta from Mp_max"]
  Z --> WN1["If ts_max: choose wn ~ 4/(zeta*ts_max)"]
  Z --> WN2["If wb_min: choose wn from wb formula"]
  WN1 --> V["Verify: compute Mr, wb, tr, ts via formulas or simulation"]
  WN2 --> V
  V --> OK{"All constraints satisfied?"}
  OK -->|yes| DONE["Finalize second-order design point \n(zeta, wn)"]
  OK -->|no| ADJ["Adjust: increase zeta to reduce \nMp/Mr OR increase wn to speed up"]
  ADJ --> V
        

Problem 1 (Derive resonance frequency): For \( G(s)=\frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2} \), show that if \( \zeta < 1/\sqrt{2} \), the magnitude \( |G(j\omega)| \) is maximized at \( \omega_r=\omega_n\sqrt{1-2\zeta^2} \).

Solution: Using \( x=\omega/\omega_n \), \( |G(j\omega)|^2 = 1/\big((1-x^2)^2+(2\zeta x)^2\big) \). Maximize \( |G|^2 \) by minimizing \( D(x)=(1-x^2)^2+(2\zeta x)^2 \). Differentiate:

\[ \frac{dD}{dx} = 4x\big(x^2 - 1 + 2\zeta^2\big). \]

Nonzero critical point satisfies \( x^2=1-2\zeta^2 \), which requires \( \zeta < 1/\sqrt{2} \). Thus \( \omega_r=\omega_n x \).

Problem 2 (Resonant peak formula): Show that for \( \zeta < 1/\sqrt{2} \), \( M_r = 1/(2\zeta\sqrt{1-\zeta^2}) \).

Solution: At \( x^2=1-2\zeta^2 \), \( 1-x^2=2\zeta^2 \) and \( D(x)= (2\zeta^2)^2 + (2\zeta)^2(1-2\zeta^2)=4\zeta^2(1-\zeta^2) \). Hence

\[ M_r = \frac{1}{\sqrt{D(x)}} = \frac{1}{2\zeta\sqrt{1-\zeta^2}}. \]

Problem 3 (Overshoot–damping inversion): Given an overshoot limit \( M_p^\star \), derive \( \zeta = \frac{-\ln(M_p^\star/100)}{\sqrt{\pi^2 + \ln^2(M_p^\star/100)}} \).

Solution: From \( M_p/100=\exp\!\left(\frac{-\zeta\pi}{\sqrt{1-\zeta^2}}\right) \), take logs and solve:

\[ \ln\!\left(\frac{M_p^\star}{100}\right) = \frac{-\zeta\pi}{\sqrt{1-\zeta^2}} \quad \Rightarrow \quad \ln^2\!\left(\frac{M_p^\star}{100}\right) = \frac{\zeta^2\pi^2}{1-\zeta^2}. \]

Rearranging gives \( \zeta^2 = \frac{\ln^2(M_p^\star/100)}{\pi^2 + \ln^2(M_p^\star/100)} \), and since \( \zeta > 0 \), take the positive root with the minus sign in the numerator to ensure positivity.

Problem 4 (-3 dB bandwidth derivation): Starting from \( |G(j\omega_{bw})|^2 = 1/2 \), derive \( \omega_{bw} = \omega_n\sqrt{1 - 2\zeta^2 + \sqrt{4\zeta^4 - 4\zeta^2 + 2}} \).

Solution: With \( x=\omega/\omega_n \), the condition is \( (1-x^2)^2 + (2\zeta x)^2 = 2 \), which expands to \( x^4 + (4\zeta^2 - 2)x^2 - 1 = 0 \). Let \( u=x^2 \) and solve the quadratic:

\[ u = \frac{-(4\zeta^2-2) + \sqrt{(4\zeta^2-2)^2 + 4}}{2} = 1 - 2\zeta^2 + \sqrt{4\zeta^4 - 4\zeta^2 + 2}, \]

and \( \omega_{bw}=\omega_n\sqrt{u} \).

Problem 5 (Feasibility check): Suppose you require \( M_p \le 10\% \) and \( t_s \le 0.8\,\text{s} \) (2% criterion) for a second-order prototype. Compute a feasible \( (\zeta,\omega_n) \) and report the implied \( \omega_{bw} \).

Solution: First compute \( \zeta \) from \( M_p^\star=10 \):

\[ \zeta = \frac{-\ln(0.10)}{\sqrt{\pi^2 + \ln^2(0.10)}}. \]

Then choose \( \omega_n \approx 4/(\zeta t_s^\star) \) with \( t_s^\star=0.8 \). Finally compute \( \omega_{bw} \) from the bandwidth formula in Section 5. (The provided code files perform this evaluation numerically.)

We established rigorous bridges between time and frequency viewpoints: (i) frequency response is the transform of the impulse response and predicts sinusoidal steady state; (ii) a universal uncertainty inequality proves that fast time localization requires broad frequency spread; (iii) for the canonical second-order system we derived closed-form relations connecting damping and natural frequency to overshoot, settling time, resonance peak, resonance frequency, and -3 dB bandwidth. These results operationalize the central control-engineering trade-off: increasing speed (bandwidth) tends to increase resonance and overshoot unless damping is increased, which in turn can reduce bandwidth.