Comprehensive Study Notes on Transfer Functions, Stability, and System Dynamics

Fundamentals of Transfer Functions

• Definition: A transfer function is the ratio of the Laplace transform of the output to the Laplace transform of the input for a given system.

• Initial Conditions: When calculating a transfer function, all initial conditions are assumed to be zero.

• Mathematical Expression: If $x(t)$ is the input and $y(t)$ is the output, let $L$ be the Laplace operator. The transforms are represented as follows:

  • Input: $L[x(t)] = X(s)$

  • Output: $L[y(t)] = Y(s)$

  • Transfer Function: $G(s) = \frac{Y(s)}{X(s)}$

• Process Representation: In a control block diagram, the input $X(s)$ enters the plant represented by $G(s)$, resulting in the output $Y(s)$.

Poles and Zeros of a Transfer Function

• Zeros:

  • Zeros are the roots of the numerator polynomial of a transfer function.

  • A zero is defined as the frequency at which the system output becomes zero ($0$).

  • Mathematically, a zero is a value of $s$ that makes the numerator of the transfer function equal to zero ($N(s) = 0$).

• Poles:

  • Poles are the roots of the denominator polynomial of a transfer function.

  • A pole is the frequency at which the system response becomes infinite. This is the origin of the term "pole," signifying a point where the field is infinite.

  • Mathematically, a pole is a value of $s$ that makes the denominator of the transfer function equal to zero ($D(s) = 0$).

Calculation Example: Poles and Zeros

• Given Transfer Function:     $G(s) = \frac{2s + 5}{s^2 + 4s + 3}$

• Calculating Zeros:

  • Set the numerator to zero: $2s + 5 = 0$

  • Solve for $s$: $s = -2.5$

  • The system has one zero at $s = -2.5$.

• Calculating Poles:

  • Set the denominator to zero: $s^2 + 4s + 3 = 0$

  • Factorize the quadratic equation: $(s + 1)(s + 3) = 0$

  • Solve for $s$: $s = -1$ and $s = -3$

  • The system has two poles: $s_1 = -1$ and $s_2 = -3$.

Mathematical Modeling of a Stirred Tank Heater

• Deviation Variables: The model uses deviation variables $T'$, $T'_i$, and $T'_{st}$.

• Differential Equation: The physical model is defined by the following equation:     $\frac{dT'}{dt} + aT' = \frac{1}{\tau}T'_i + KT'_{st}$

• Parameters:

  • $a = \frac{1}{\tau} + K$

  • $K = \frac{UA_s}{V\rho C_p}$

  • $a = \frac{1}{\tau} + \frac{UA_s}{V\rho C_p}$

• Laplace Transformation: Taking the Laplace transform of both sides of the differential equation yields:     $(s + a)T'(s) = \frac{1}{\tau}T'_i(s) + KT'_{st}(s)$

• Decoupled Transfer Functions:

  • The output response is separated into two specific transfer functions:         $T'(s) = \frac{1/\tau}{s + a}T'_i(s) + \frac{K}{s + a}T'_{st}(s)$

  • $G_1(s) = \frac{T'(s)}{T'_i(s)} = \frac{1/\tau}{s + a}$

  • $G_2(s) = \frac{T'(s)}{T'_{st}(s)} = \frac{K}{s + a}$

• Total Response: The final expression for the deviation variable in the Laplace domain is:     $T'(s) = G_1(s)T'_i(s) + G_2(s)T'_{st}(s)$

Dynamic Response and Pole Locations

• Pole Indications: The location of the poles of a transfer function dictates the qualitative dynamic behavior of the system (where $a, b, c, d$ are positive constants).

• Stability and the Complex Plane:

  • The Complex Plane: Behavior is analyzed based on the Real ($Re$) axis and Imaginary ($Im$) axis.

  • Exponential Decay: Occurs when poles are located on the negative real axis (Left Half Plane). This reflects a stable system where the output returns to equilibrium over time.

  • Damped Sinusoidal: Occurs when complex conjugate poles are located in the Left Half Plane ($LHP$). The response oscillates but eventually decays.

  • Exponentially Growing Response: Occurs when poles are located in the Right Half Plane ($RHP$). This indicates an unstable system where the output increases without limit.

  • Sinusoidal Behavior (Unstable): Occurs when poles are located on the imaginary axis or in the right half plane, leading to sustained or growing oscillations.

Stability Analysis and Case Studies

• Stable System Example:

  • Transfer Function: $G_1(s) = \frac{1}{s + 3}$

  • Pole: $s = -3$ (located in the Left Half Plane).

  • Time Domain Response: Applying the inverse Laplace transform $L^{-1}$ results in $y(t) = e^{-3t}u(t)$.

  • Outcome: The exponential term $e^{-3t}$ decays toward zero as time increases.

• Unstable System Example:

  • Transfer Function: $G_2(s) = \frac{1}{s - 3}$

  • Pole: $s = 3$ (located in the Right Half Plane).

  • Time Domain Response: Applying the inverse Laplace transform $L^{-1}$ results in $y(t) = e^{3t}u(t)$.

  • Outcome: The exponential term $e^{3t}$ grows without bound as time increases, indicating instability.

• General Rules for Stability:

  • A process is referred to as unstable if the output grows without bound for a bounded input.

  • If the real portion of any pole of a transfer function is positive, the process is unstable.

  • If any pole is located in the Right Half Plane ($RHP$), the process is unstable because the solution to the dynamic equations will contain increasing exponential terms.

Characteristics of Zeros in Transfer Functions

• Filtering Effects: Zeros have the ability to block specific frequency components of the input, preventing them from reaching the output.

• Non-minimum Phase Behavior: Zeros can cause initial inversions where the output initially moves in the opposite direction of the final steady state.

• Inverse Response:

  • The zeros of a transfer function are the values of $s$ where $N(s) = 0$.

  • If any zeros are positive (located in the $RHP$), it indicates the system will exhibit an inverse response.

• Overshoot: If all zeros are negative, the system may still experience overshoot in specific configurations.