Frequency Response & State Variable Analysis

1. 5 Key Ideas: Frequency Dynamics & State-Space Formulations

Idea 1: The Sinusoidal Transfer Function & Frequency-Domain Specifications

When a stable linear time-invariant (LTI) system with transfer function $G(s)$ is excited by a sinusoidal input $r(t) = A \sin(\omega t)$, the steady-state output response $c_{ss}(t)$ is an identical sinusoid of the exact same frequency $\omega$, but with modified amplitude and shifted phase angle: $c_{ss}(t) = A |G(j\omega)| \sin(\omega t + \angle G(j\omega))$

Where the Sinusoidal Transfer Function is obtained by substituting $s = j\omega$: $G(j\omega) = |G(j\omega)| e^{j \phi(\omega)} = \text{Re}\{G(j\omega)\} + j \text{Im}\{G(j\omega)\}$

For the canonical second-order closed-loop prototype: $T(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} \implies T(j\omega) = \frac{1}{\left(1 - \frac{\omega^2}{\omega_n^2}\right) + j 2\zeta\left(\frac{\omega}{\omega_n}\right)}$

Let normalized frequency $u = \frac{\omega}{\omega_n}$. The magnitude ratio $M(u)$ is: $M(u) = |T(j\omega)| = \frac{1}{\sqrt{\left(1 - u^2\right)^2 + 4\zeta^2 u^2}}$

This defines three critical frequency-domain performance specifications:

1. Resonant Peak ($M_r$): The maximum value of the closed-loop magnitude ratio $|T(j\omega)|$. For $\zeta < \frac{1}{\sqrt{2}} \approx 0.707$: $M_r = \frac{1}{2\zeta\sqrt{1 - \zeta^2}}$ Notice: $M_r$ depends strictly on damping ratio $\zeta$. A higher resonant peak directly correlates with larger transient overshoot $\%M_p$ in the time domain.
2. Resonant Frequency ($\omega_r$): The frequency at which the maximum magnitude $M_r$ occurs: $\omega_r = \omega_n \sqrt{1 - 2\zeta^2}$ (For $\zeta > 0.707$, no resonant peak exists; $|T(j\omega)|$ monotonically decreases from $1$ as $\omega$ increases).
3. Closed-Loop Bandwidth ($\omega_{BW}$): The frequency at which the magnitude $|T(j\omega)|$ drops by $3\text{ dB}$ ($1/\sqrt{2} \approx 0.707$ of its DC value): $\omega_{BW} = \omega_n \sqrt{\left(1 - 2\zeta^2\right) + \sqrt{4\zeta^4 - 4\zeta^2 + 2}}$ Bandwidth measures the speed of response (shorter rise time $t_r$) and frequency-tracking capability.

Idea 2: The Logarithmic Bode Plot & Relative Stability Margins

The Bode Plot represents the frequency response using two separate graphs with a common logarithmic frequency horizontal axis ($\omega$ in rad/s across decades):

1. Bode Magnitude Plot: Logarithmic gain in decibels: $|G(j\omega)|_{\text{dB}} = 20 \log_{10} |G(j\omega)|$
2. Bode Phase Plot: Phase angle $\angle G(j\omega)$ linearly in degrees.

Asymptotic Construction Factors

Any rational transfer function factors into 4 elementary building blocks:

1. Constant Gain $K$: Horizontal line at $20\log_{10} K\text{ dB}$, Phase $= 0^\circ$ (or $-180^\circ$ if $K < 0$).
2. Integrator/Differentiator $(j\omega)^{\pm 1}$: Straight line with constant slope $\pm 20\text{ dB/decade}$ passing through $0\text{ dB}$ at $\omega = 1\text{ rad/s}$. Phase is constant $\pm 90^\circ$.
3. Simple First-Order Factor $(1 + j\omega T)^{\pm 1}$:
   • Corner (break) frequency: $\omega_c = \frac{1}{T}\text{ rad/s}$.
   • Below $\omega_c$: Asymptote is $0\text{ dB}$.
   • Above $\omega_c$: Asymptote rolls off at $\pm 20\text{ dB/decade}$.
   • Exact magnitude at $\omega = \omega_c$ is $\pm 3\text{ dB}$ ($20\log_{10}\sqrt{2}$).
   • Phase shifts from $0^\circ$ to $\pm 90^\circ$, passing through $\pm 45^\circ$ at $\omega = \omega_c$.
4. Quadratic Complex Factor $\left[1 + j 2\zeta\left(\frac{\omega}{\omega_n}\right) - \left(\frac{\omega}{\omega_n}\right)^2\right]^{\pm 1}$:
   • Corner frequency: $\omega = \omega_n$.
   • High-frequency asymptote rolls off at $\pm 40\text{ dB/decade}$.
   • Phase shifts through $180^\circ$, passing through $\pm 90^\circ$ at $\omega = \omega_n$.

Robustness Metrics: Gain Margin & Phase Margin

To prevent closed-loop instability, the open-loop transfer function $G(j\omega)H(j\omega)$ must not pass through the critical point $-1 + j0$ (which corresponds to $|GH| = 1 = 0\text{ dB}$ and $\angle GH = -180^\circ$):

• Gain Crossover Frequency ($\omega_{gc}$): The frequency where the magnitude is unity ($0\text{ dB}$): $|G(j\omega_{gc})H(j\omega_{gc})| = 1 \iff |G(j\omega_{gc})H(j\omega_{gc})|_{\text{dB}} = 0\text{ dB}$
• Phase Margin (PM): The additional phase lag required at $\omega_{gc}$ to bring the system to the verge of instability: $\text{PM} = 180^\circ + \angle G(j\omega_{gc})H(j\omega_{gc})$
• Phase Crossover Frequency ($\omega_{pc}$): The frequency where the phase angle is $-180^\circ$: $\angle G(j\omega_{pc})H(j\omega_{pc}) = -180^\circ$
• Gain Margin (GM): The reciprocal of the open-loop magnitude at $\omega_{pc}$, expressed in dB: $\text{GM} = -20 \log_{10} |G(j\omega_{pc})H(j\omega_{pc})| = 20 \log_{10}\left(\frac{1}{|G(j\omega_{pc})H(j\omega_{pc})|}\right)\text{ dB}$

Stability Criteria via Bode Margins:

• Stable: $\omega_{gc} < \omega_{pc} \iff \text{GM} > 0\text{ dB}$ and $\text{PM} > 0^\circ$.
• Marginally Stable: $\omega_{gc} = \omega_{pc} \iff \text{GM} = 0\text{ dB}$ and $\text{PM} = 0^\circ$.
• Unstable: $\omega_{gc} > \omega_{pc} \iff \text{GM} < 0\text{ dB}$ and $\text{PM} < 0^\circ$.

Idea 3: Polar Plots & The Nyquist Stability Encirclement Criterion

The Polar Plot (or Nyquist Plot) graphs the trajectory of the complex phasor $G(j\omega)H(j\omega)$ directly in the complex plane as frequency $\omega$ sweeps from $-\infty$ to $+\infty$.

Cauchy's Principle of the Argument

Let a closed contour $\Gamma_s$ in the $s$-plane enclose $Z$ zeros and $P$ poles of a rational function $F(s) = 1 + G(s)H(s)$. If the contour does not pass through any poles or zeros, then as $s$ traverses $\Gamma_s$ clockwise, the mapped contour $\Gamma_F$ in the $F(s)$-plane encircles the origin $N$ times clockwise, where: $N = Z - P$

Because $F(s) = 1 + G(s)H(s)$, encirclements of the origin by $F(s)$ correspond identically to encirclements of the critical point $-1 + j0$ by the open-loop contour $G(s)H(s)$.

The Nyquist Stability Theorem

Let:

• $P$: Number of open-loop poles of $G(s)H(s)$ in the Right-Half $s$-Plane (RHP).
• $N_{\text{cw}}$: Number of clockwise encirclements of the critical point $-1 + j0$ by the complete Nyquist plot of $G(s)H(s)$ ($\omega \in (-\infty, +\infty)$).
• $Z$: Number of closed-loop poles in the RHP (zeros of $1 + G(s)H(s)$).

The fundamental stability formula states: $Z = P + N_{\text{cw}} \quad \text{or equivalently} \quad N = P - Z$ (where $N = -N_{\text{cw}}$ is counterclockwise encirclements).

Condition for Closed-Loop Stability: For the closed-loop system to be strictly stable, there must be zero closed-loop RHP poles ($Z = 0$): $N_{\text{cw}} = -P \iff N_{\text{ccw}} = P$

• If the open-loop plant is already stable ($P = 0$), then the closed-loop system is stable if and only if the Nyquist plot does NOT encircle $-1 + j0$ ($N = 0$).
• If the open-loop plant has $P$ unstable poles ($P > 0$), the Nyquist plot MUST encircle $-1 + j0$ exactly $P$ times in the COUNTERCLOCKWISE direction for the closed-loop feedback system to achieve stability.

Idea 4: State-Space Representation & Transfer Function Inversion

Classical control relies on input-output transfer functions $T(s) = Y(s)/U(s)$ under zero initial conditions. Modern state-space control represents an $n$-th order dynamic system as a set of $n$ coupled first-order differential equations: $\mathbf{\dot{x}}(t) = \mathbf{A}\mathbf{x}(t) + \mathbf{B}\mathbf{u}(t) \quad \text{(State Equation)}$ $\mathbf{y}(t) = \mathbf{C}\mathbf{x}(t) + \mathbf{D}\mathbf{u}(t) \quad \text{(Output Equation)}$

Where:

• $\mathbf{x}(t) \in \mathbb{R}^n$: State Vector (minimal set of internal dynamic variables, e.g., position, velocity, inductor current, capacitor voltage).
• $\mathbf{u}(t) \in \mathbb{R}^m$: Input Vector (control inputs).
• $\mathbf{y}(t) \in \mathbb{R}^p$: Output Vector (measured sensor outputs).
• $\mathbf{A} \in \mathbb{R}^{n \times n}$: System Matrix (governs internal unforced dynamics).
• $\mathbf{B} \in \mathbb{R}^{n \times m}$: Input Matrix (couples control inputs to state derivatives).
• $\mathbf{C} \in \mathbb{R}^{p \times n}$: Output Matrix (couples states to observed sensor measurements).
• $\mathbf{D} \in \mathbb{R}^{p \times m}$: Direct Transmission (Feedthrough) Matrix (instantaneous throughput).

Phase-Variable (Controllable Canonical) Form

Given an $n$-th order differential equation $\frac{d^n y}{dt^n} + a_{n-1}\frac{d^{n-1}y}{dt^{n-1}} + \dots + a_1\frac{dy}{dt} + a_0 y = u(t)$, define state variables as successive derivatives: $x_1 = y$, $x_2 = \dot{y}$, $x_3 = \ddot{y}$, $\dots$, $x_n = y^{(n-1)}$. The companion matrix takes the canonical form: