Unit 3: Time Domain analysis and Stability

1. Standard Input Signals (Test Signals)

In control systems, we use standard test signals to analyze the performance of a system. The response of the system to these inputs provides a clear picture of its characteristics.

2. Time Response of a First-Order System

A first-order system is one whose dynamics are described by a first-order differential equation.

Standard Form of Transfer Function:
The transfer function of a generic first-order system is:
C(s) / R(s) = G(s) = K / (τs + 1)
where:

• K is the DC Gain (final value of the output for a unit step input).
• τ (tau) is the Time Constant of the system.

Unit Step Response Analysis:
Let's analyze the response for a unit step input, R(s) = 1/s.
The output in the Laplace domain is C(s):
C(s) = G(s) * R(s) = (K / (τs + 1)) * (1/s)

Using partial fraction expansion:
C(s) = A/s + B/(τs + 1)
Solving for A and B gives A=K and B=-Kτ.
C(s) = K/s - Kτ/(τs + 1) = K/s - K/(s + 1/τ)

Taking the Inverse Laplace Transform to get the time-domain response c(t):
c(t) = K(1 - e^(-t/τ)) for t ≥ 0

Key Characteristics of the First-Order Step Response:

• Time Constant (τ): This is the most important characteristic. It represents the time it takes for the system's response to reach 63.2% of its final (steady-state) value.
  • At t = τ, c(τ) = K(1 - e^(-1)) = K(1 - 0.368) = 0.632K.
• Response Curve: The response is an exponential rise from 0 towards the final value K. It has no oscillations and no overshoot.
• Settling Time (Ts): The time required for the response to reach and stay within a certain percentage (usually 2% or 5%) of its final value.
  • Ts ≈ 4τ (for 2% tolerance, since e⁻⁴ ≈ 0.0183)
  • Ts ≈ 3τ (for 5% tolerance, since e⁻³ ≈ 0.0498)
• Rise Time (Tr): Time taken to rise from 10% to 90% of the final value. For a first-order system, Tr ≈ 2.2τ.
• Steady-State Error (ess): For a unit step input, the final value is c(∞) = K. The error is e(∞) = r(∞) - c(∞) = 1 - K. If the DC gain K=1, the steady-state error is zero.

(A sketch of c(t) would show an exponential curve starting at the origin, rising quickly at first, then more slowly, and asymptotically approaching the line y=K. The point (τ, 0.632K) would be marked on the curve.)

3. Time Response of a Second-Order System

A second-order system's dynamics are described by a second-order differential equation. These systems are common in practice (e.g., RLC circuits, mass-spring-damper systems).

Standard Form of Transfer Function:
Consider a unity feedback system. The standard form of the closed-loop transfer function is:
C(s) / R(s) = ωn² / (s² + 2ζωn s + ωn²)
where:

• ωn (omega-n) is the Undamped Natural Frequency. It is the frequency at which the system would oscillate if there were no damping.
• ζ (zeta) is the Damping Ratio. It is a dimensionless quantity describing how oscillations in a system decay after a disturbance.

The behavior of the system is entirely determined by the values of ζ and ωn. The roots of the characteristic equation (s² + 2ζωn s + ωn² = 0) are the closed-loop poles:
s₁,₂ = -ζωn ± ωn√(ζ² - 1)

Unit Step Response Analysis (based on Damping Ratio ζ):

a) Case 1: Undamped System (ζ = 0)

• Poles: s₁,₂ = ±jωn. The poles are purely imaginary and lie on the jω-axis.
• Response c(t): c(t) = 1 - cos(ωn t)
• Characteristics: The response is a sustained oscillation around the final value of 1, with a constant amplitude and a frequency of ωn. The system is marginally stable.

b) Case 2: Underdamped System (0 < ζ < 1)

• Poles: s₁,₂ = -ζωn ± jωn√(1 - ζ²). The poles are a complex conjugate pair in the left-half of the s-plane.
• Let ωd = ωn√(1 - ζ²), which is the Damped Frequency of Oscillation.
• Response c(t): c(t) = 1 - [e^(-ζωn t) / √(1 - ζ²)] * sin(ωd t + φ) where φ = cos⁻¹(ζ).
• Characteristics: This is the most common practical case. The response oscillates, but the amplitude of oscillations decays exponentially. The response overshoots the final value and then settles.

c) Case 3: Critically Damped System (ζ = 1)

• Poles: s₁,₂ = -ωn. The poles are real, repeated, and negative.
• Response c(t): c(t) = 1 - e^(-ωn t) * (1 + ωn t)
• Characteristics: The response approaches the final value as quickly as possible without any overshoot. This is often considered an "optimal" response.

d) Case 4: Overdamped System (ζ > 1)

• Poles: s₁,₂ = -ζωn ± ωn√(ζ² - 1). The poles are real, distinct, and negative.
• Response c(t): The response is the sum of two decaying exponential terms. It is sluggish and slow.
  c(t) = 1 + [ωn / (2√(ζ²-1))] * [(e^(-s₁t)/s₁) - (e^(-s₂t)/s₂)]
• Characteristics: The response does not oscillate and does not overshoot. It takes longer to reach the final value than a critically damped system.

(A sketch of these four responses on the same plot would show: (a) Undamped as a continuous sine wave, (b) Underdamped as a wave that overshoots 1 and then settles, (c) Critically damped as a fast curve that smoothly reaches 1, and (d) Overdamped as a slower curve that smoothly reaches 1.)

4. Time-Domain Specifications

For the underdamped (0 < ζ < 1) second-order system, we define several performance metrics from its unit step response.

(These are typically visualized on the underdamped response curve.)

(Description of an image illustrating the following terms on a typical underdamped step response curve)

1. Delay Time (td): The time required for the response to reach 50% of its final value for the first time.
   td ≈ (1 + 0.7ζ) / ωn

2. Rise Time (tr): The time required for the response to rise from 0% to 100% of its final value for the first time.
   tr = (π - φ) / ωd = (π - cos⁻¹(ζ)) / (ωn√(1 - ζ²))

3. Peak Time (tp): The time required for the response to reach its first peak (the peak of the overshoot).
   tp = π / ωd = π / (ωn√(1 - ζ²))

4. Peak Overshoot (Mp): The maximum amount by which the response overshoots its final value, expressed as a percentage.
   %Mp = (c(tp) - c(∞)) / c(∞) * 100
%Mp = e^(-ζπ / √(1 - ζ²)) * 100
   Note: This value depends only on the damping ratio ζ.

5. Settling Time (ts): The time required for the response to reach and stay within a specified tolerance band (commonly 2% or 5%) of the final value.

   • For 2% tolerance: ts ≈ 4 / (ζωn) = 4τ
   • For 5% tolerance: ts ≈ 3 / (ζωn) = 3τ
     (Here, the "time constant" of the second-order system's envelope is τ = 1 / (ζωn))

5. Steady-State Error (ess)

The steady-state error is the difference between the input (command) and the output of a system as time approaches infinity (t → ∞). It is a measure of the system's accuracy.

Calculation using Final Value Theorem:
ess = lim(t→∞) e(t) = lim(s→0) sE(s)

For a unity feedback system, the error signal E(s) is given by:
E(s) = R(s) - C(s) = R(s) - E(s)G(s)
E(s) [1 + G(s)] = R(s)
E(s) = R(s) / (1 + G(s))

Therefore, the steady-state error is:
*`ess = lim(s→0) [s R(s) / (1 + G(s))]** *(For non-unity feedback with transfer function H(s), the denominator becomes1 + G(s)H(s)`.)*

The value of ess depends on two factors:

1. The type of the input signal (step, ramp, parabolic).
2. The type of the system.

System Type: The type of a system is defined as the number of pure integrators (poles at s=0) in the open-loop transfer function G(s)H(s).
G(s)H(s) = K(s+z₁)(s+z₂)... / sⁿ(s+p₁)(s+p₂)...
Here, the system type is n.

6. Static Error Coefficients

These coefficients provide a convenient way to determine the steady-state error for standard inputs without computing the full limit expression each time.

a) Position Error Constant (Kp)

• Associated Input: Unit Step (R(s) = 1/s)
• Formula: Kp = lim(s→0) G(s)H(s)
• Steady-State Error: ess = 1 / (1 + Kp)

b) Velocity Error Constant (Kv)

• Associated Input: Unit Ramp (R(s) = 1/s²)
• Formula: Kv = lim(s→0) sG(s)H(s)
• Steady-State Error: ess = 1 / Kv

c) Acceleration Error Constant (Ka)

• Associated Input: Unit Parabolic (R(s) = 1/s³)
• Formula: Ka = lim(s→0) s²G(s)H(s)
• Steady-State Error: ess = 1 / Ka

Summary Table of Steady-State Errors:

Key Takeaway: To track an input with zero steady-state error, the system type must be at least one higher than the input type (e.g., a Type 1 system can track a ramp input with a finite error, while a Type 2 system can track it with zero error).

7. Concept of Stability

Definition (BIBO Stability): A system is Bounded-Input, Bounded-Output (BIBO) stable if every bounded input results in a bounded output.

Stability in the s-Plane:
The stability of a Linear Time-Invariant (LTI) system is determined by the location of the poles of its closed-loop transfer function.

• Stable System: All closed-loop poles lie in the Left-Half Plane (LHP) of the s-plane.
  • The real parts of all poles are negative.
  • The time response will decay to zero (or to the steady-state value) as t → ∞.
• Unstable System: At least one closed-loop pole lies in the Right-Half Plane (RHP) of the s-plane.
  • The real part of at least one pole is positive.
  • The time response will grow without bound as t → ∞.
• Marginally Stable System:
  1. One or more non-repeated (single) poles lie on the imaginary axis (jω-axis).
  2. No poles lie in the RHP.
     • The time response will have sustained oscillations that neither grow nor decay.
     • Note: If there are repeated poles on the jω-axis, the system is unstable.

8. Absolute and Relative Stability

• Absolute Stability: This is a binary concept. It answers the question: "Is the system stable or not?" A system is either stable or unstable. The Routh-Hurwitz criterion is a tool to determine absolute stability.

• Relative Stability: This is a measure of how stable a system is. It indicates how close the system is to becoming unstable. It is determined by the proximity of the dominant closed-loop poles to the imaginary axis.

  • Poles far from the jω-axis in the LHP correspond to a highly stable system with a fast-decaying transient response.
  • Poles close to the jω-axis correspond to a less stable system with a slowly decaying, oscillatory response.
  • In the context of a second-order system, a higher damping ratio ζ implies greater relative stability.
  • Frequency domain concepts like Gain Margin and Phase Margin are quantitative measures of relative stability.

9. Routh-Hurwitz Criterion

This is an analytical method to determine the absolute stability of a system without having to calculate the exact locations of the closed-loop poles. It only requires the characteristic equation polynomial.

Characteristic Equation: Q(s) = a_n s^n + a_{n-1} s^{n-1} + ... + a_1 s + a_0 = 0

Procedure:

1. Necessary Condition: Check the polynomial Q(s). For a stable system, it is necessary (but not sufficient) that:

   • All coefficients (a_n, a_{n-1}, ..., a_0) are present.
   • All coefficients have the same sign (e.g., all positive).
   • If either of these conditions is not met, the system is unstable or at best marginally stable, and no further analysis is needed.

2. Sufficient Condition (Routh Array): If the necessary condition is met, construct the Routh Array.

   sⁿ	a_n	a_{n-2}	a_{n-4}	...
   sⁿ⁻¹	a_{n-1}	a_{n-3}	a_{n-5}	...
   sⁿ⁻²	b₁	b₂	b₃	...
   sⁿ⁻³	c₁	c₂	c₃	...
   ...	...	...	...
   s⁰	...

   Where the coefficients b, c, etc., are calculated as follows:

   • b₁ = (a_{n-1} * a_{n-2} - a_n * a_{n-3}) / a_{n-1}
   • b₂ = (a_{n-1} * a_{n-4} - a_n * a_{n-5}) / a_{n-1}
   • c₁ = (b₁ * a_{n-3} - a_{n-1} * b₂) / b₁
   • ...and so on, until the array is complete down to the s⁰ row.

3. Stability Criterion: The number of closed-loop poles in the RHP is equal to the number of sign changes in the first column of the Routh array. For a system to be stable, there must be no sign changes in the first column.

Special Cases:

• Case 1: A Zero in the First Column

  • If a zero appears in the first column, but the rest of the row is not all zeros.
  • Method: Replace the zero with a small positive number, ε (epsilon), and continue calculating the array. Then, analyze the signs in the first column by taking the limit as ε → 0⁺.
  • The number of sign changes as ε → 0⁺ still indicates the number of RHP poles.

• Case 2: An Entire Row of Zeros

  • This indicates that there are poles that are symmetrically located about the origin of the s-plane (e.g., poles on the jω-axis, or real poles like s = ±σ). This often happens in marginally stable systems.
  • Method:
    1. Form an Auxiliary Polynomial A(s) from the coefficients of the row just above the row of zeros. The powers of s in this polynomial will be even.
    2. Differentiate the auxiliary polynomial with respect to s: dA(s)/ds.
    3. Replace the row of zeros with the coefficients of dA(s)/ds.
    4. Continue constructing the rest of the array as usual.
  • The number of sign changes in the modified first column still indicates the number of RHP poles.
  • Crucially, the roots of the auxiliary equation A(s) = 0 are the actual symmetric poles of the system. This allows us to find the frequency of oscillation in marginally stable systems.