Cheatsheet Control Systems - Control Systems - Electrical Engineering (EE)

1. Basic Concepts

1.1 System Definitions

1.2 Block Diagram Algebra

1.3 Signal Flow Graphs

• Mason's Gain Formula: T = (Σ Pₖ Δₖ)/Δ
• Pₖ = gain of kth forward path
• Δ = 1 - (sum of individual loop gains) + (sum of gain products of two non-touching loops) - ...
• Δₖ = value of Δ for part of graph not touching kth forward path
• Path: continuous succession of branches in direction of arrows
• Forward Path: path from input to output without passing any node twice
• Loop: closed path that starts and ends at same node

2. Time Domain Analysis

2.1 Standard Test Signals

2.2 First Order Systems

• Transfer Function: G(s) = K/(τs + 1)
• Time Constant: τ seconds
• DC Gain: K
• Step Response: c(t) = K[1 - e^(-t/τ)]
• Settling Time (2%): ts = 4τ
• Rise Time (10–90%): tr ≈ 2.2τ
• No overshoot in first order systems

2.3 Second Order Systems

2.3.1 Standard Form

• G(s) = ωₙ²/(s² + 2ζωₙs + ωₙ²)
• ωₙ = natural frequency (rad/s)
• ζ = damping ratio (dimensionless)
• ωd = ωₙ√(1 - ζ²) = damped natural frequency

2.3.2 Response Types

2.3.3 Performance Specifications

2.4 Steady State Errors

2.4.1 Error Constants

2.4.2 Steady State Error by System Type

• Open-loop transfer function for steady-state error analysis: followed by same expression.
  G(s)H(s) = K(1 + Tas)(1 + Tbs)... / [s^N(1 + T₁s)(1 + T₂s)...]
• N = system type (number of poles at origin)

3. Stability Analysis

3.1 Routh-Hurwitz Criterion

• Characteristic Equation: aₙsⁿ + aₙ₋₁sⁿ⁻¹ + ... + a₁s + a₀ = 0
• Necessary Condition: All coefficients present and same sign
• Sufficient Condition: All elements in first column of Routh array positive
• Number of sign changes in first column = number of RHP poles

3.1.1 Routh Array Construction

• b₁ = (aₙ₋₁·aₙ₋₂ - aₙ·aₙ₋₃)/aₙ₋₁
• b₂ = (aₙ₋₁·aₙ₋₄ - aₙ·aₙ₋₅)/aₙ₋₁

3.1.2 Special Cases

• Zero in first column (others non-zero): Replace with small ε, continue
• Entire row zero: Form auxiliary equation from previous row, differentiate, use coefficients
• Auxiliary equation roots occur as symmetric pairs about the imaginary axis.

3.2 Root Locus

3.2.1 Construction Rules

• Number of branches = n (number of open-loop poles)
• Locus starts at open-loop poles (K = 0), ends at open-loop zeros or ∞ (K = ∞)
• Locus is symmetric about real axis
• Real axis locus: point on real axis is on locus if total number of poles and zeros to its right is odd
• Asymptotes: (n - m) branches approach ∞ along asymptotes
• Asymptote angles: φ = (2q + 1)180°/(n - m), q = 0, 1, 2, ..., (n - m - 1)
• Centroid: σ = (Σ poles - Σ zeros)/(n - m)
• Breakaway/Break-in points: dK/ds = 0
• Angle of departure from complex pole: 180° + φ =180° +Σ(angles from zeros) - Σ(angles from other poles)
• Angle of arrival at complex zero: 180° + φ =180° +Σ(angles from poles) - Σ(angles from other zeros)
• Intersection with imaginary axis: Use Routh criterion

3.2.2 Angle and Magnitude Conditions

• Angle Condition: Σ∠(s - zi) - Σ∠(s - pi) = (2q + 1)180°, q = 0, ±1, ±2, ...
• Magnitude Condition: K = ∏|s - pi| / ∏|s - zi|

4. Frequency Domain Analysis

4.1 Bode Plots

4.1.1 Standard Transfer Function Form

• G(s) = K(1 + sT₁)(1 + sT₂)... / [s^N(1 + sT₃)(1 + sT₄)...(1 + 2ζs/ωₙ + s²/ωₙ²)...]
• Magnitude in dB: 20log₁₀|G(jω)|
• Phase in degrees: ∠G(jω)

4.1.2 Basic Factor Contributions

4.1.3 Stability Criteria

4.2 Polar Plots

• G(jω) plotted in complex plane as ω varies from 0 to ∞
• Starting point (ω = 0): magnitude and phase from low frequency behavior
• Ending point (ω = ∞): magnitude and phase from high frequency behavior
• Type 0: starts at K∠0°, ends at origin
• Type 1: starts at ∞∠-90°, ends at origin
• Type 2: starts at ∞∠-180°, ends at origin

4.3 Nyquist Criterion

• Z = N + P, where Z = closed-loop RHP poles, P = open-loop RHP poles, N = encirclements of -1
• For stability: Z = 0, so N = -P
• If P = 0 (open-loop stable), no encirclement of -1 for closed-loop stability
• Counter-clockwise encirclements are positive; clockwise encirclements are negative.
• Nyquist path: contour enclosing entire RHP in s-plane
• GM and PM can be read from Nyquist plot distance from -1 point

4.4 Nichols Chart

• Log magnitude vs phase plot
• Superimposed with M and N circles (constant magnitude and phase contours)
• Closed-loop frequency response obtained using Nichols chart contours from open-loop G(jω)H(jω).
• Resonant peak Mr and resonant frequency ωr can be read

5. Controllers and Compensation

5.1 Controller Types

5.2 Compensation Techniques

5.2.1 Lead Compensator

• Gc(s) = K(s + z)/(s + p) where p > z (pole farther from origin)
• Gc(s) = K·α(1 + sT)/(1 + sαT) where α > 1
• Maximum phase lead: φm = sin⁻¹[(1 - α)/(1 + α)]
• Frequency of φm: ωm = 1/(T√α)
• Increases bandwidth, improves transient response, increases PM
• Used for phase margin improvement

5.2.2 Lag Compensator

• Gc(s) = K(s + z)/(s + p) where z > p (zero farther from origin)
• Gc(s) = K·β(1 + sT)/(1 + sβT) where β > 1
• Maximum phase lag magnitude : φm = sin⁻¹[(β - 1)/(β + 1)]
• Provides attenuation at high frequencies
• Improves steady-state accuracy with minimal effect on transient
• Used for error constant improvement

5.2.3 Lead-Lag Compensator

• Gc(s) = K(s + z₁)(s + z₂)/[(s + p₁)(s + p₂)]
• Combines lead and lag: improves both transient and steady-state
• Two separate corner frequencies for lead and lag sections

6. State Space Analysis

6.1 State Space Representation

• State Equation: ẋ = Ax + Bu
• Output Equation: y = Cx + Du
• x = state vector (n × 1), u = input vector (r × 1), y = output vector (m × 1)
• A = system matrix (n × n), B = input matrix (n × r), C = output matrix (m × n), D = feedforward matrix (m × r)

6.2 Transfer Function from State Space

• G(s) = C(sI - A)⁻¹B + D
• Characteristic Equation: det(sI - A) = 0
• Eigenvalues of A = poles of system

6.3 Canonical Forms

6.3.1 Controllable Canonical Form

• Transfer Function: G(s) = (bₙ₋₁sⁿ⁻¹ + ... + b₁s + b₀)/(sⁿ + aₙ₋₁sⁿ⁻¹ + ... + a₁s + a₀)
• A matrix: companion form with last row [-a₀, -a₁, ..., -aₙ₋₁]
• B = [0, 0, ..., 0, 1]ᵀ
• C = [b₀, b₁, ..., bₙ₋₁]

6.3.2 Observable Canonical Form

• A matrix: transpose of controllable canonical form
• B = [b₀, b₁, ..., bₙ₋₁]ᵀ
• C = [0, 0, ..., 0, 1]

6.3.3 Diagonal/Jordan Canonical Form

• A = diag[λ₁, λ₂, ..., λₙ] for distinct eigenvalues
• Jordan blocks for repeated eigenvalues

6.4 Controllability and Observability

6.5 State Transition Matrix

• φ(t) = e^(At) = L⁻¹[(sI - A)⁻¹]
• Properties: φ(0) = I, φ⁻¹(t) = φ(-t), φ(t₁ + t₂) = φ(t₁)φ(t₂)
• Solution: x(t) = φ(t)x(0) + ∫₀ᵗ φ(t - τ)Bu(τ)dτ

6.6 Pole Placement

• Control law: u = -Kx + r
• Closed-loop: ẋ = (A - BK)x + Br
• K chosen such that eigenvalues of (A - BK) are at desired locations
• Ackermann's Formula: K = [0, 0, ..., 0, 1]Qc⁻¹φ(A) where φ(s) = desired characteristic polynomial and Qc is controllability matrix

6.7 State Observer

• Full-order observer: x̂̇ = Ax̂ + Bu + L(y - ŷ) where ŷ = Cx̂
• Observer error: e = x - x̂
• Error dynamics: ė = (A - LC)e
• L chosen such that eigenvalues of (A - LC) are at desired locations
• Separation principle: controller and observer can be designed independently

7. Nonlinear Systems

7.1 Common Nonlinearities

7.2 Describing Function

• N = fundamental component of output / sinusoidal input
• N(A, ω) for amplitude and frequency dependent nonlinearities
• Predicts limit cycles when -1/N intersects G(jω) locus
• Stability: if Nyquist plot encircles -1/N point, limit cycle unstable

7.3 Phase Plane Analysis

• Plot of dx/dt vs x (state velocity vs state)
• Trajectory: path in phase plane
• Singular point: equilibrium where dx/dt = 0.
• Isoclines: curves where slope dx/dt is constant

7.3.1 Singular Point Types

7.4 Lyapunov Stability

• Consider ẋ = f(x) with equilibrium at x = 0
• Lyapunov Function: V(x) scalar function, V(0) = 0, V(x) > 0 for x ≠ 0
• Stable: if V̇(x) ≤ 0
• Asymptotically Stable: V̇(x) < 0 for x ≠ 0.
• Unstable: if V̇(x) > 0

7.4.1 Linear System Lyapunov Equation

• For ẋ = Ax, choose V(x) = xᵀPx
• V̇(x) = xᵀ(AᵀP + PA)x
• Lyapunov Equation: AᵀP + PA = -Q
• If Q > 0 and unique P > 0 exists, system is asymptotically stable

8. Digital Control Systems

8.1 Sampling and Reconstruction

• Sampling Period: T seconds
• Sampling Frequency: fs = 1/T Hz
• Nyquist Theorem: fs ≥ 2fmax for signal reconstruction
• Aliasing: frequency components above fs/2 fold back
• Zero-Order Hold (ZOH): holds sampled value until next sample

8.2 z-Transform

8.2.1 Definition and Properties

• Z{f(kT)} = F(z) = Σ f(kT)z⁻ᵏ, k = 0 to ∞
• z = e^(sT) relates z-plane to s-plane
• Left half s-plane maps to inside unit circle in z-plane
• jω axis maps to unit circle |z| = 1
• Right half s-plane maps to outside unit circle

8.2.2 Important z-Transform Pairs

8.2.3 z-Transform Theorems

8.3 Pulse Transfer Function

• G(z) = Z{G(s)} for system with ZOH and sampling
• With ZOH: G(z) = (1 - z⁻¹)Z{G(s)/s}
• Closed-loop: C(z)/R(z) = G(z)/[1 + G(z)H(z)]

8.4 Stability in z-Domain

8.4.1 Jury Stability Test

• Characteristic Equation: a₀zⁿ + a₁zⁿ⁻¹ + ... + aₙ₋₁z + aₙ = 0
• Necessary conditions: F(1) > 0, (-1)ⁿF(-1) > 0, |aₙ| <a₀
• Jury array similar to Routh array for discrete systems

8.4.2 Bilinear Transformation

• w = (z - 1)/(z + 1) maps unit circle to jω axis
• Allows use of Routh-Hurwitz in w-plane
• Inside unit circle in z-plane maps to LHP in w-plane

8.5 Digital Controller Design

8.5.1 Discretization Methods

8.5.2 Deadbeat Controller

• Achieves zero error in minimum finite settling time
• All closed-loop poles at origin
• D(z) = G⁻¹(z)·F(z)/[1 - F(z)] where F(z) = desired closed-loop transfer function

9. Additional Topics

9.1 Sensitivity

• Sensitivity of T to parameter P: S = (∂T/T)/(∂P/P)
• Closed-loop sensitivity reduced by factor [1 + G(s)H(s)]
• Lower sensitivity at low frequencies where loop gain is high

9.2 Disturbance Rejection

• Effect of disturbance D at output: Y/D = Gd/(1 + GH)
• High loop gain reduces disturbance effect
• Integral control eliminates constant disturbances

9.3 Cascade and Feedback Compensation

9.4 Performance Indices

9.5 Relationship Between Time and Frequency Domain

• Bandwidth (ωBW) ≈ natural frequency (ωₙ) for second-order systems (for ζ ≈ 0.7)
• Higher bandwidth → faster response
• Resonant peak Mr relates to damping: Mr ≈ 1/(2ζ√(1 - ζ²)) for small ζ
• Phase margin PM ≈ 100ζ degrees (approximation for 0.1<ζ <0.6
• Gain margin ≥ 6 dB, Phase margin ≥ 30° for good stability

9.6 Standard Second Order Approximation

• Higher-order systems can be approximated by dominant second-order poles
• Dominant poles: closest to jω axis (slowest)
• Other poles should be 5-10 times farther from jω axis