15.1 Process Dynamics and Transfer Functions

Process Control Fundamentals and Laplace Transforms

Chemical manufacturing facilities are highly dynamic systems subjected to continuous external disturbances, such as fluctuations in feed rate, composition, ambient temperature, and utility pressures. Process control is the engineering discipline that monitors and manipulates process variables to ensure safe, stable, and economically optimal plant operation.

To analyze how a chemical process responds to changes in inputs, dynamic models are developed using fundamental conservation laws (mass, energy, and momentum balances). For linear systems, or non-linear systems linearized around a steady-state operating point using Taylor series expansion, the Laplace transform converts time-domain ordinary differential equations (ODEs) into algebraic equations in the s-domain. The transfer function, $G(s)$, is defined as the ratio of the Laplace transform of the output variable $Y(s)$ to the Laplace transform of the input variable $U(s)$ under the assumption of zero initial conditions: $G(s) = \frac{Y(s)}{U(s)}$ In process control, variables are typically expressed as deviation variables, which represent the difference between the absolute value of the variable and its initial steady-state value (e.g., $y(t) = Y_{absolute}(t) - Y_{ss}$). This ensures that all initial conditions are zero, simplifying the Laplace transformation.

---

First-Order Process Dynamics

A first-order system represents a process with a single capacity for storing mass, energy, or momentum and a single resistance to flow. Physical examples include a liquid level surge tank with a discharge valve, a thermometer bulb sensing temperature, or a well-mixed heating tank. The general dynamic behavior is governed by the following first-order ODE: $\tau_p \frac{dy(t)}{dt} + y(t) = K_p u(t)$ where $y(t)$ is the output deviation variable, $u(t)$ is the input deviation variable, $K_p$ is the process gain, and $\tau_p$ is the process time constant. Applying the Laplace transform to both sides under zero initial conditions ($y(0) = 0$) yields the standard first-order transfer function: $G(s) = \frac{Y(s)}{U(s)} = \frac{K_p}{\tau_p s + 1}$

Parameters and Response Characteristics

• Process Gain ($K_p$): A measure of the steady-state sensitivity of the system. It represents the ratio of the ultimate change in the output to the step change in the input: $K_p = \lim_{s \to 0} G(s) = \frac{\Delta y_{ss}}{\Delta u_{ss}}$
• Process Time Constant ($\tau_p$): A measure of the speed of the system response. It has units of time. If a step change of magnitude $M$ is applied to the input ($u(t) = M$, which transforms to $U(s) = M/s$), the output response is: $Y(s) = \frac{K_p M}{s(\tau_p s + 1)}$ Taking the inverse Laplace transform yields: $y(t) = K_p M \left(1 - e^{-t/\tau_p}\right)$
  • When $t = \tau_p$, $y(\tau_p) = K_p M (1 - e^{-1}) \approx 0.632 K_p M$. Thus, the time constant is the time required for the process variable to reach $63.2%$ of its ultimate steady-state change.
  • When $t = 3\tau_p$, the response is $95.0%$ complete.
  • When $t = 5\tau_p$, the response is $99.3%$ complete.

Worked Example: Temperature Sensor Response

A thermometer is modeled as a first-order system with a time constant of $10\text{ seconds}$ and a gain of $1.0$. The thermometer is initially at a steady-state temperature of $25^\circ\text{C}$. At $t=0$, it is suddenly placed in a water bath maintained at $75^\circ\text{C}$. Determine the thermometer reading at $t=15\text{ seconds}$. Solution: The step change magnitude in the input is $M = 75 - 25 = 50^\circ\text{C}$. Using the first-order response equation: $T'(t) = K_p M \left(1 - e^{-t/\tau_p}\right) = (1.0)(50)\left(1 - e^{-15/10}\right) = 50\left(1 - e^{-1.5}\right) \approx 50(1 - 0.2231) = 38.8^\circ\text{C}$ The actual temperature reading $T(t)$ is the sum of the initial steady state and the deviation variable: $T(t) = T_{initial} + T'(t) = 25 + 38.8 = 63.8^\circ\text{C}$

---

Second-Order Process Dynamics

Second-order systems are common in chemical processes. They arise from two first-order systems in series (e.g., two stirred tanks where the output of the first tank feeds the second) or from systems with inherent physical inertia (e.g., pneumatic control valves, liquid flow in a U-tube manometer). The standard transfer function is: $G(s) = \frac{Y(s)}{U(s)} = \frac{K_p}{\tau^2 s^2 + 2\zeta\tau s + 1}$ where $\tau$ is the characteristic time parameter (natural period of oscillation $\tau = 1/\omega_n$), $\omega_n$ is the natural frequency, and $\zeta$ is the dimensionless damping factor. The roots of the characteristic denominator polynomial represent the poles of the system. The value of $\zeta$ determines the dynamic behavior:

• Overdamped ($\zeta > 1$): Real and distinct poles. The response is sluggish and returns to steady state without oscillation.
• Critically Damped ($\zeta = 1$): Real and equal poles. This represents the fastest possible response without overshoot or oscillation.
• Underdamped ($0 < \zeta < 1$): Complex conjugate poles. The response oscillates with decaying amplitude.
• Undamped ($\zeta = 0$): Purely imaginary poles. The system oscillates indefinitely with constant amplitude.

Transient Response of Underdamped Systems

For an underdamped system subjected to a step change in input of magnitude $M$, the transient response has several key metrics:

• Overshoot ($OS$): The ratio of the maximum peak value above steady state to the steady-state change: $OS = \exp\left(-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}\right)$
• Decay Ratio ($DR$): The ratio of the heights of two successive peaks. The decay ratio is the square of the overshoot: $DR = (OS)^2 = \exp\left(-\frac{2\pi\zeta}{\sqrt{1-\zeta^2}}\right)$ Industrial controllers are often tuned to achieve a decay ratio of $0.25$ (quarter-amplitude decay).
• Peak Time ($t_p$): The time required to reach the first peak: $t_p = \frac{\pi\tau}{\sqrt{1-\zeta^2}}$
• Period of Oscillation ($P$): The time between two successive peaks: $P = \frac{2\pi\tau}{\sqrt{1-\zeta^2}}$
• Settling Time ($t_s$): The time required for the response to decay and remain within a specified band around the final value (typically $\pm 2%$). For a $2%$ criterion, it is estimated as: $t_s \approx \frac{4\tau}{\zeta}$

---

Block Diagram Algebra

Process control loops are represented visually via block diagrams. The diagram shows the path of signals through individual components, summing junctions (which add or subtract signals), and branch points. To find the overall relationship between any input (e.g., setpoint, disturbance) and output, block diagram reduction rules are applied:

• Series Reduction: If blocks $G_1(s)$ and $G_2(s)$ are in series, the equivalent transfer function is their product: $G_{series}(s) = G_1(s) G_2(s)$
• Parallel Reduction: If blocks are in parallel and their outputs are summed, the equivalent transfer function is their sum: $G_{parallel}(s) = G_1(s) \pm G_2(s)$
• Closed-Loop Feedback: For a standard loop with forward path $G(s) = G_c G_v G_p$ and feedback path $H(s)$, the closed-loop transfer function under negative feedback is: $\frac{Y(s)}{R(s)} = \frac{G(s)}{1 + G(s)H(s)} = \frac{G_c(s)G_v(s)G_p(s)}{1 + G_c(s)G_v(s)G_p(s)H(s)}$ The denominator expression $1 + G_{OL}(s) = 1 + G_c(s)G_v(s)G_p(s)H(s)$ is fundamental to stability analysis.

---

Closed-Loop Stability Analysis

A control system must be stable to operate safely. Under the Bounded-Input Bounded-Output (BIBO) definition, a system is stable if its output remains bounded for any bounded input.

Closed-loop stability is determined by the poles of the closed-loop transfer function, which are the roots of the characteristic equation: $1 + G_c(s) G_v(s) G_p(s) H(s) = 0$

• Stability Rule: If any root of the characteristic equation has a positive real part, it lies in the Right-Half s-Plane (RHP). In the time domain, this corresponds to an exponentially growing term ($e^{at}$ where $a > 0$), resulting in an unstable system. For a system to be stable, all roots must lie strictly in the Left-Half s-Plane (LHP) (negative real parts).
• Routh-Hurwitz Stability Criterion: An algebraic method to determine stability without solving for roots. For a polynomial $a_n s^n + a_{n-1} s^{n-1} + \dots + a_0 = 0$, a Routh array is constructed. The number of sign changes in the first column of the Routh array represents the number of unstable RHP roots. For stability, all elements in the first column must be positive (assuming $a_n > 0$).