Formula Sheet: Feedback Systems

Table of Contents
1. Basic Feedback System Concepts
2. Stability Analysis
3. Time-Domain Analysis
4. Steady-State Error Analysis
5. Frequency-Domain Analysis
6. Compensation and Controller Design
7. State-Space Analysis
8. Discrete-Time Control Systems
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Basic Feedback System Concepts

Block Diagram Elements

• Open-Loop Transfer Function: \(G(s)\) - Forward path transfer function
• Feedback Transfer Function: \(H(s)\) - Feedback path transfer function
• Loop Transfer Function: \[L(s) = G(s)H(s)\]
• Closed-Loop Transfer Function (Negative Feedback): \[T(s) = \frac{C(s)}{R(s)} = \frac{G(s)}{1 + G(s)H(s)}\]
  • Where \(C(s)\) = output (controlled variable)
  • \(R(s)\) = input (reference)
• Closed-Loop Transfer Function (Positive Feedback): \[T(s) = \frac{G(s)}{1 - G(s)H(s)}\]
• Unity Feedback System: \(H(s) = 1\), therefore \[T(s) = \frac{G(s)}{1 + G(s)}\]

Error Transfer Function

• Error Signal: \(E(s) = R(s) - H(s)C(s)\) for negative feedback
• Error Transfer Function: \[\frac{E(s)}{R(s)} = \frac{1}{1 + G(s)H(s)}\]
• Sensitivity to Parameter Variations: \[S_G^T = \frac{\partial T/T}{\partial G/G} = \frac{1}{1 + G(s)H(s)}\]
  • Lower sensitivity indicates better performance

Stability Analysis

Characteristic Equation

• Characteristic Equation: \[1 + G(s)H(s) = 0\]
  • Roots of this equation are the closed-loop poles
  • System is stable if all roots have negative real parts
• General Form: \[a_n s^n + a_{n-1} s^{n-1} + \cdots + a_1 s + a_0 = 0\]

Routh-Hurwitz Stability Criterion

• Routh Array Construction: For characteristic equation \(a_n s^n + a_{n-1} s^{n-1} + \cdots + a_1 s + a_0 = 0\)
  • Row \(s^n\): \(a_n, a_{n-2}, a_{n-4}, \ldots\)
  • Row \(s^{n-1}\): \(a_{n-1}, a_{n-3}, a_{n-5}, \ldots\)
  • Subsequent rows computed using: \[b_1 = \frac{a_{n-1}a_{n-2} - a_n a_{n-3}}{a_{n-1}}\]
• Stability Condition: Number of sign changes in first column = number of right-half-plane poles
• Necessary Condition: All coefficients \(a_i > 0\) (for stable system)
• Special Case - Zero in First Column: Replace with small positive \(\varepsilon\) and take limit as \(\varepsilon \to 0\)
• Special Case - Entire Row of Zeros: Use auxiliary equation from previous row

Root Locus Method

• Root Locus Equation: \[1 + K\frac{G(s)}{1} = 0\]
  • \(K\) = variable gain parameter (0 ≤ \(K\) <>
• Magnitude Condition: \[|KG(s)H(s)| = 1\]
• Angle Condition: \[\angle G(s)H(s) = \pm 180°(2k + 1), \quad k = 0, 1, 2, \ldots\]
• Number of Branches: Equal to number of poles of \(G(s)H(s)\)
• Asymptotes:
  • Number of asymptotes = \(n - m\), where \(n\) = number of poles, \(m\) = number of zeros
  • Asymptote angles: \[\theta_a = \frac{180°(2k + 1)}{n - m}, \quad k = 0, 1, 2, \ldots, (n-m-1)\]
  • Centroid (asymptote intersection): \[\sigma_a = \frac{\sum \text{poles} - \sum \text{zeros}}{n - m}\]
• Breakaway/Break-in Points: Solutions to \[\frac{dK}{ds} = 0\]
• Departure Angle (from complex pole): \[\theta_d = 180° - \sum \angle \text{(zeros)} + \sum \angle \text{(other poles)}\]
• Arrival Angle (at complex zero): \[\theta_a = 180° + \sum \angle \text{(poles)} - \sum \angle \text{(other zeros)}\]

Nyquist Stability Criterion

• Nyquist Criterion: \[Z = N + P\]
  • \(Z\) = number of closed-loop poles in right-half plane
  • \(N\) = number of clockwise encirclements of -1 + j0 point
  • \(P\) = number of open-loop poles in right-half plane
  • For stability: \(Z = 0\), therefore \(N = -P\)
• Gain Margin (GM): \[GM = \frac{1}{|G(j\omega_{pc})H(j\omega_{pc})|}\]
  • \(\omega_{pc}\) = phase crossover frequency where \(\angle G(j\omega)H(j\omega) = -180°\)
  • In dB: \(GM_{dB} = -20\log_{10}|G(j\omega_{pc})H(j\omega_{pc})|\)
  • GM > 1 (or > 0 dB) for stable system
• Phase Margin (PM): \[PM = 180° + \angle G(j\omega_{gc})H(j\omega_{gc})\]
  • \(\omega_{gc}\) = gain crossover frequency where \(|G(j\omega)H(j\omega)| = 1\)
  • PM > 0° for stable system
  • Typical design specification: PM ≥ 30° to 60°

Time-Domain Analysis

Standard Test Inputs

• Unit Step Input: \[r(t) = u(t), \quad R(s) = \frac{1}{s}\]
• Unit Ramp Input: \[r(t) = t \cdot u(t), \quad R(s) = \frac{1}{s^2}\]
• Unit Parabolic Input: \[r(t) = \frac{t^2}{2} u(t), \quad R(s) = \frac{1}{s^3}\]
• Unit Impulse Input: \[r(t) = \delta(t), \quad R(s) = 1\]

First-Order System Response

• Standard Form: \[T(s) = \frac{K}{\tau s + 1}\]
  • \(K\) = DC gain (steady-state gain)
  • \(\tau\) = time constant (seconds)
• Pole Location: \(s = -\frac{1}{\tau}\)
• Step Response: \[c(t) = K\left(1 - e^{-t/\tau}\right)u(t)\]
• Time Constant: Time to reach 63.2% of final value
• Settling Time (2% criterion): \[t_s = 4\tau\]
• Settling Time (5% criterion): \[t_s = 3\tau\]
• Rise Time: \[t_r = 2.2\tau\]

Second-Order System Response

• Standard Form: \[T(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\]
  • \(\omega_n\) = natural frequency (rad/s)
  • \(\zeta\) = damping ratio (dimensionless)
• Pole Locations: \[s_{1,2} = -\zeta\omega_n \pm \omega_n\sqrt{\zeta^2 - 1}\]
• For Underdamped Case (0 < \(\zeta\)=""><> \[s_{1,2} = -\zeta\omega_n \pm j\omega_n\sqrt{1 - \zeta^2} = -\sigma \pm j\omega_d\]
  • \(\sigma = \zeta\omega_n\) = damping coefficient
  • \(\omega_d = \omega_n\sqrt{1 - \zeta^2}\) = damped natural frequency

Underdamped Step Response Characteristics

• Step Response (0 < \(\zeta\)=""><> \[c(t) = 1 - \frac{e^{-\zeta\omega_n t}}{\sqrt{1-\zeta^2}}\sin(\omega_d t + \phi)\]
  • where \(\phi = \cos^{-1}(\zeta)\)
• Peak Time: \[t_p = \frac{\pi}{\omega_d} = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}}\]
• Percent Overshoot: \[PO = e^{-\pi\zeta/\sqrt{1-\zeta^2}} \times 100\%\]
  • Also written as: \(M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}}\) (overshoot ratio)
• Settling Time (2% criterion): \[t_s = \frac{4}{\zeta\omega_n} = \frac{4}{\sigma}\]
• Settling Time (5% criterion): \[t_s = \frac{3}{\zeta\omega_n} = \frac{3}{\sigma}\]
• Rise Time (0% to 100%): \[t_r \approx \frac{1.8}{\omega_n}\] (approximation for \(0.3 < \zeta=""><>
• Delay Time (0% to 50%): \[t_d \approx \frac{1 + 0.7\zeta}{\omega_n}\]

Damping Classification

• Overdamped: \(\zeta > 1\) - Two real, distinct poles; no overshoot
• Critically Damped: \(\zeta = 1\) - Two real, repeated poles; fastest response without overshoot
• Underdamped: \(0 < \zeta="">< 1\)="" -="" complex="" conjugate="" poles;="" overshoot="">
• Undamped: \(\zeta = 0\) - Pure imaginary poles; continuous oscillation

Relationship Between Pole Location and Transient Response

• Damping Ratio from Pole Angle: \[\zeta = \cos(\theta)\]
  • where \(\theta\) = angle from negative real axis to pole location
• Natural Frequency from Pole Magnitude: \[\omega_n = |s|\]
• Overshoot Constraint Line Angle: \[\theta = \cos^{-1}(\zeta)\]

Steady-State Error Analysis

Error Constants

• Position Error Constant: \[K_p = \lim_{s \to 0} G(s)H(s)\]
• Velocity Error Constant: \[K_v = \lim_{s \to 0} s \cdot G(s)H(s)\]
• Acceleration Error Constant: \[K_a = \lim_{s \to 0} s^2 \cdot G(s)H(s)\]

System Type and Open-Loop Transfer Function

• System Type N: Number of integrators (poles at origin) in open-loop transfer function \[G(s)H(s) = \frac{K(s + z_1)(s + z_2)\cdots}{s^N(s + p_1)(s + p_2)\cdots}\]
  • \(N\) = system type

Steady-State Error for Unity Feedback (H(s) = 1)

• General Formula (Final Value Theorem): \[e_{ss} = \lim_{t \to \infty} e(t) = \lim_{s \to 0} s \cdot E(s) = \lim_{s \to 0} \frac{s \cdot R(s)}{1 + G(s)}\]

Step Input (\(R(s) = 1/s\))

• Type 0: \[e_{ss} = \frac{1}{1 + K_p}\]
• Type 1 or higher: \[e_{ss} = 0\]

Ramp Input (\(R(s) = 1/s^2\))

• Type 0: \(e_{ss} = \infty\)
• Type 1: \[e_{ss} = \frac{1}{K_v}\]
• Type 2 or higher: \[e_{ss} = 0\]

Parabolic Input (\(R(s) = 1/s^3\))

• Type 0 or 1: \(e_{ss} = \infty\)
• Type 2: \[e_{ss} = \frac{1}{K_a}\]
• Type 3 or higher: \[e_{ss} = 0\]

Steady-State Error for Non-Unity Feedback

• General Error: \[e_{ss} = \lim_{s \to 0} \frac{s \cdot R(s)}{1 + G(s)H(s)}\]
• For Step Input: \[e_{ss} = \frac{1}{1 + G(0)H(0)}\]

Disturbance Rejection

• Error Due to Disturbance: For disturbance \(D(s)\) entering after \(G_1(s)\) and before \(G_2(s)\): \[E(s) = -\frac{G_2(s)}{1 + G_1(s)G_2(s)H(s)} D(s)\]

Frequency-Domain Analysis

Frequency Response Basics

• Transfer Function Evaluation: Replace \(s\) with \(j\omega\): \[G(j\omega) = |G(j\omega)|e^{j\angle G(j\omega)}\]
• Magnitude: \[|G(j\omega)| = \sqrt{\text{Re}^2[G(j\omega)] + \text{Im}^2[G(j\omega)]}\]
• Phase Angle: \[\angle G(j\omega) = \tan^{-1}\left(\frac{\text{Im}[G(j\omega)]}{\text{Re}[G(j\omega)]}\right)\]
• Magnitude in Decibels: \[|G(j\omega)|_{dB} = 20\log_{10}|G(j\omega)|\]

Bode Plot Asymptotic Approximations

Basic Elements

• Constant Gain K:
  • Magnitude: \(20\log_{10}K\) dB (horizontal line)
  • Phase: 0° if \(K > 0\), -180° if \(K <>
• Pole at Origin \(1/s\):
  • Magnitude: -20 dB/decade slope
  • Phase: -90°
• Zero at Origin \(s\):
  • Magnitude: +20 dB/decade slope
  • Phase: +90°
• \(N\) Poles/Zeros at Origin \(1/s^N\) or \(s^N\):
  • Magnitude slope: \(\mp 20N\) dB/decade
  • Phase: \(\mp 90N\)°

Simple Real Pole

• Form: \[\frac{1}{1 + j\omega\tau} = \frac{1}{1 + j\omega/\omega_c}\]
  • Corner frequency: \(\omega_c = 1/\tau\)
• Magnitude Asymptotes:
  • For \(\omega \ll \omega_c\): 0 dB
  • For \(\omega \gg \omega_c\): -20 dB/decade
  • At \(\omega = \omega_c\): -3 dB (exact)
• Phase:
  • At \(\omega = 0.1\omega_c\): -6°
  • At \(\omega = \omega_c\): -45°
  • At \(\omega = 10\omega_c\): -84°

Simple Real Zero

• Form: \(1 + j\omega\tau\)
• Magnitude Asymptotes:
  • For \(\omega \ll \omega_c\): 0 dB
  • For \(\omega \gg \omega_c\): +20 dB/decade
  • At \(\omega = \omega_c\): +3 dB (exact)
• Phase:
  • At \(\omega = 0.1\omega_c\): +6°
  • At \(\omega = \omega_c\): +45°
  • At \(\omega = 10\omega_c\): +84°

Complex Conjugate Poles

• Form: \[\frac{\omega_n^2}{(j\omega)^2 + 2\zeta\omega_n(j\omega) + \omega_n^2} = \frac{1}{1 - (\omega/\omega_n)^2 + j(2\zeta\omega/\omega_n)}\]
• Magnitude Asymptotes:
  • For \(\omega \ll \omega_n\): 0 dB
  • For \(\omega \gg \omega_n\): -40 dB/decade
• Resonant Peak (for \(\zeta <> \[M_r = \frac{1}{2\zeta\sqrt{1-\zeta^2}}\]
  • Occurs at: \(\omega_r = \omega_n\sqrt{1 - 2\zeta^2}\)
• At \(\omega = \omega_n\): Magnitude = \(-20\log_{10}(2\zeta)\) dB, Phase = -90°
• Phase: Ranges from 0° to -180° through resonance

Complex Conjugate Zeros

• Magnitude Asymptotes:
  • For \(\omega \ll \omega_n\): 0 dB
  • For \(\omega \gg \omega_n\): +40 dB/decade
• Phase: Ranges from 0° to +180°

Bandwidth and Frequency Response Specifications

• Bandwidth (\(\omega_{BW}\)): Frequency at which closed-loop magnitude is -3 dB from DC value \[|T(j\omega_{BW})| = \frac{1}{\sqrt{2}}|T(0)| = 0.707|T(0)|\]
• Resonant Frequency (\(\omega_r\)): Frequency at which peak magnitude occurs (for underdamped systems)
• Resonant Peak (\(M_r\)): Maximum value of closed-loop frequency response magnitude
• Cutoff Rate: Slope of magnitude plot near cutoff frequency (dB/decade or dB/octave)

Relationship Between Time and Frequency Domain

• Bandwidth and Rise Time: \[\omega_{BW} \cdot t_r \approx 1.8 \text{ to } 2.0\]
• Bandwidth and Settling Time: \[\omega_{BW} \approx \frac{3 \text{ to } 5}{t_s}\]
• Resonant Peak and Damping Ratio: \[M_r \approx \frac{1}{2\zeta} \quad \text{(for small } \zeta\text{)}\]
• Phase Margin and Damping Ratio (approximation): \[\zeta \approx \frac{PM(\text{degrees})}{100}\]
  • More accurate: \(\zeta \approx 0.01 \cdot PM\) for PM in degrees
• Phase Margin and Percent Overshoot: \[PO \approx e^{-\pi\zeta/\sqrt{1-\zeta^2}} \times 100\%\]
  • where \(\zeta \approx PM/100\)

Compensation and Controller Design

PID Controller

• Ideal PID Transfer Function: \[G_c(s) = K_p + \frac{K_i}{s} + K_d s\]
  • \(K_p\) = proportional gain
  • \(K_i\) = integral gain
  • \(K_d\) = derivative gain
• Alternative Form: \[G_c(s) = K_p\left(1 + \frac{1}{T_i s} + T_d s\right)\]
  • \(T_i = K_p/K_i\) = integral time constant
  • \(T_d = K_d/K_p\) = derivative time constant
• Parallel Form: \[G_c(s) = K_p + \frac{K_i}{s} + K_d s = \frac{K_d s^2 + K_p s + K_i}{s}\]
• Series Form: \[G_c(s) = K_c\left(1 + \frac{1}{T_i s}\right)(1 + T_d s)\]

Individual Controller Actions

• Proportional (P) Controller: \(G_c(s) = K_p\)
  • Increases system bandwidth
  • Reduces but does not eliminate steady-state error
  • May reduce stability margins
• Integral (I) Controller: \(G_c(s) = K_i/s\)
  • Eliminates steady-state error for step inputs
  • Increases system type by 1
  • Adds -90° phase lag, reducing phase margin
  • Slows response
• Derivative (D) Controller: \(G_c(s) = K_d s\)
  • Improves transient response
  • Increases damping
  • Adds +90° phase lead, improving phase margin
  • Amplifies high-frequency noise
• Proportional-Integral (PI) Controller: \[G_c(s) = K_p + \frac{K_i}{s} = K_p\frac{s + K_i/K_p}{s}\]
  • Zero at \(s = -K_i/K_p\)
  • Pole at origin
  • Eliminates steady-state error
  • May reduce stability
• Proportional-Derivative (PD) Controller: \[G_c(s) = K_p + K_d s = K_p(1 + T_d s)\]
  • Zero at \(s = -K_p/K_d = -1/T_d\)
  • Improves damping and stability
  • Does not affect steady-state error

Lead Compensation

• Lead Compensator Transfer Function: \[G_c(s) = K_c \frac{s + z}{s + p} = K_c \frac{1 + \tau s}{1 + \alpha\tau s}, \quad 0 < \alpha="">< 1\]="">
  • \(z = 1/\tau\) = zero location
  • \(p = 1/(\alpha\tau)\) = pole location
  • \(p > z\) (pole is further left than zero)
• Maximum Phase Lead: \[\phi_m = \sin^{-1}\left(\frac{1-\alpha}{1+\alpha}\right)\]
  • Occurs at: \(\omega_m = \frac{1}{\tau\sqrt{\alpha}}\)
• Magnitude at Maximum Phase: \[|G_c(j\omega_m)| = \frac{1}{\sqrt{\alpha}}\]
  • In dB: \(20\log_{10}(1/\sqrt{\alpha})\)
• Design Steps:
  • Place \(\omega_m\) at desired gain crossover frequency
  • Select \(\alpha\) to provide required phase lead (plus 5° to 12° margin)
  • Adjust \(K_c\) for desired gain crossover

Lag Compensation

• Lag Compensator Transfer Function: \[G_c(s) = K_c \frac{s + z}{s + p} = K_c \frac{1 + \tau s}{1 + \beta\tau s}, \quad \beta > 1\]
  • \(z = 1/\tau\) = zero location
  • \(p = 1/(\beta\tau)\) = pole location
  • \(z > p\) (zero is further left than pole)
• Maximum Phase Lag: \[\phi_m = -\sin^{-1}\left(\frac{\beta-1}{\beta+1}\right)\]
  • Occurs at: \(\omega_m = \frac{1}{\tau\sqrt{\beta}}\)
• DC Gain Increase: Factor of \(\beta\) (or \(20\log_{10}\beta\) dB)
• Design Steps:
  • Place pole and zero at least one decade below gain crossover frequency
  • Select \(\beta\) to achieve required steady-state error reduction
  • Verify minimal phase reduction at crossover frequency

Lead-Lag Compensation

• Lead-Lag Compensator: \[G_c(s) = K_c \frac{(s+z_1)(s+z_2)}{(s+p_1)(s+p_2)}\]
  • Combines lead (for transient improvement) and lag (for steady-state improvement)
  • Lead section: \(p_1 > z_1\)
  • Lag section: \(z_2 > p_2\)

State-Space Analysis

State-Space Representation

• State Equation: \[\dot{\mathbf{x}}(t) = \mathbf{A}\mathbf{x}(t) + \mathbf{B}\mathbf{u}(t)\]
• Output Equation: \[\mathbf{y}(t) = \mathbf{C}\mathbf{x}(t) + \mathbf{D}\mathbf{u}(t)\]
  • \(\mathbf{x}(t)\) = \(n \times 1\) state vector
  • \(\mathbf{u}(t)\) = \(m \times 1\) input vector
  • \(\mathbf{y}(t)\) = \(p \times 1\) output vector
  • \(\mathbf{A}\) = \(n \times n\) state matrix
  • \(\mathbf{B}\) = \(n \times m\) input matrix
  • \(\mathbf{C}\) = \(p \times n\) output matrix
  • \(\mathbf{D}\) = \(p \times m\) feedforward matrix

Transfer Function from State-Space

• Transfer Function Matrix: \[\mathbf{G}(s) = \mathbf{C}(s\mathbf{I} - \mathbf{A})^{-1}\mathbf{B} + \mathbf{D}\]
  • \(\mathbf{I}\) = identity matrix
• For SISO Systems: \[G(s) = \frac{Y(s)}{U(s)} = C(sI - A)^{-1}B + D\]

Characteristic Equation and Eigenvalues

• Characteristic Equation: \[\det(sI - A) = 0\]
• Eigenvalues: Roots of characteristic equation = system poles
• Stability Condition: All eigenvalues must have negative real parts

Controllability and Observability

• Controllability Matrix: \[\mathcal{C} = \begin{bmatrix} B & AB & A^2B & \cdots & A^{n-1}B \end{bmatrix}\]
  • System is controllable if \(\text{rank}(\mathcal{C}) = n\)
• Observability Matrix: \[\mathcal{O} = \begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^{n-1} \end{bmatrix}\]
  • System is observable if \(\text{rank}(\mathcal{O}) = n\)

State Feedback Control

• State Feedback Law: \[\mathbf{u}(t) = -\mathbf{K}\mathbf{x}(t) + \mathbf{r}(t)\]
  • \(\mathbf{K}\) = \(m \times n\) feedback gain matrix
  • \(\mathbf{r}(t)\) = reference input
• Closed-Loop System: \[\dot{\mathbf{x}}(t) = (\mathbf{A} - \mathbf{BK})\mathbf{x}(t) + \mathbf{B}\mathbf{r}(t)\]
• Closed-Loop Characteristic Equation: \[\det(sI - A + BK) = 0\]
• Pole Placement: Select \(\mathbf{K}\) to place closed-loop poles at desired locations (system must be controllable)

Observer Design

• Full-Order Observer: \[\dot{\hat{\mathbf{x}}}(t) = \mathbf{A}\hat{\mathbf{x}}(t) + \mathbf{B}\mathbf{u}(t) + \mathbf{L}[\mathbf{y}(t) - \mathbf{C}\hat{\mathbf{x}}(t)]\]
  • \(\hat{\mathbf{x}}(t)\) = estimated state vector
  • \(\mathbf{L}\) = observer gain matrix
• Observer Error Dynamics: \[\dot{\mathbf{e}}(t) = (\mathbf{A} - \mathbf{LC})\mathbf{e}(t)\]
  • \(\mathbf{e}(t) = \mathbf{x}(t) - \hat{\mathbf{x}}(t)\)
• Observer Characteristic Equation: \[\det(sI - A + LC) = 0\]
• Observer Pole Placement: Select \(\mathbf{L}\) to place observer poles at desired locations (system must be observable)

Discrete-Time Control Systems

Z-Transform

• Z-Transform Definition: \[X(z) = \mathcal{Z}\{x[k]\} = \sum_{k=0}^{\infty} x[k]z^{-k}\]
• Sampling Relationship: \[z = e^{sT}\]
  • \(T\) = sampling period (seconds)
  • \(f_s = 1/T\) = sampling frequency (Hz)
• Unit Delay: \[\mathcal{Z}\{x[k-1]\} = z^{-1}X(z)\]
• Unit Advance: \[\mathcal{Z}\{x[k+1]\} = zX(z) - zx[0]\]

Discrete Transfer Function

• Pulse Transfer Function: \[G(z) = \frac{Y(z)}{U(z)}\]
• General Form: \[G(z) = \frac{b_0 z^m + b_1 z^{m-1} + \cdots + b_m}{z^n + a_1 z^{n-1} + \cdots + a_n}\]
  • Typically \(n \geq m\) (proper or strictly proper)

Stability in Z-Domain

• Stability Condition: All poles must lie inside unit circle \(|z| <>
• Characteristic Equation: \[1 + G(z)H(z) = 0\]
• Marginal Stability: Poles on unit circle \(|z| = 1\)
• Instability: Any pole outside unit circle \(|z| > 1\)

Mapping Between S-Plane and Z-Plane

• Left-Half S-Plane: Maps to interior of unit circle in Z-plane
• Right-Half S-Plane: Maps to exterior of unit circle in Z-plane
• Imaginary Axis: Maps to unit circle \(|z| = 1\)
• Constant Damping Lines: Map to logarithmic spirals in Z-plane
• Constant Frequency Lines: Map to radial lines from origin

Sampling Theorem

• Nyquist Sampling Criterion: \[f_s \geq 2f_{max}\]
  • \(f_s\) = sampling frequency
  • \(f_{max}\) = highest frequency component in signal
• Practical Sampling Rate: \[f_s \geq (6 \text{ to } 10) \times \omega_n/(2\pi)\]
  • where \(\omega_n\) is the natural frequency of the system

Discrete-Time PID Controller

• Position Form: \[u[k] = K_p e[k] + K_i T \sum_{j=0}^{k} e[j] + K_d \frac{e[k] - e[k-1]}{T}\]
• Velocity Form: \[\Delta u[k] = u[k] - u[k-1]\]
• Z-Domain Transfer Function: \[G_c(z) = K_p + K_i T \frac{z}{z-1} + K_d \frac{z-1}{Tz}\]

Bilinear Transformation (Tustin's Method)

• S to Z Mapping: \[s = \frac{2}{T}\frac{z-1}{z+1}\]
• Z to S Mapping: \[z = \frac{1 + sT/2}{1 - sT/2}\]
• Discrete Equivalent: Replace \(s\) with \(\frac{2}{T}\frac{z-1}{z+1}\) in continuous transfer function

Zero-Order Hold (ZOH)

• ZOH Transfer Function: \[G_{h}(s) = \frac{1 - e^{-sT}}{s}\]
• Discretization with ZOH: \[G(z) = (1 - z^{-1})\mathcal{Z}\left\{\frac{G(s)}{s}\right\}\]

Steady-State Error in Discrete Systems

• Final Value Theorem (Z-domain): \[e_{ss} = \lim_{k \to \infty} e[k] = \lim_{z \to 1} (z-1)E(z)\]
• Position Error Constant: \[K_p = \lim_{z \to 1} G(z)H(z)\]
• Velocity Error Constant: \[K_v = \lim_{z \to 1} \frac{z-1}{T} G(z)H(z)\]
• Acceleration Error Constant: \[K_a = \lim_{z \to 1} \frac{(z-1)^2}{T^2} G(z)H(z)\]