Theory and Background

This section provides the mathematical and theoretical background for the Standard Process Control Library.

Control Theory Fundamentals

Transfer Functions

A transfer function represents the relationship between the input and output of a linear time-invariant system in the Laplace domain:

G(s) = Y(s)/U(s) = (b_m*s^m + ... + b_1*s + b_0)/(a_n*s^n + ... + a_1*s + a_0)

where s is the Laplace variable, Y(s) is the output, and U(s) is the input.

Common Process Models:

  1. First-Order Plus Dead Time (FOPDT):

    G(s) = K * exp(-θ*s) / (τ*s + 1)

    Where: - K = Process gain - τ = Time constant - θ = Dead time

  2. Second-Order System:

    G(s) = K * ωn² / (s² + 2*ζ*ωn*s + ωn²)

    Where: - ζ = Damping ratio - ωn = Natural frequency

PID Control

The PID controller implements proportional, integral, and derivative control actions:

Time Domain:

u(t) = Kp*e(t) + Ki*∫[0 to t]e(τ)dτ + Kd*de(t)/dt

Frequency Domain:

C(s) = Kp + Ki/s + Kd*s

Practical PID (with derivative filter):

C(s) = Kp + Ki/s + (Kd*s)/(τd*s + 1)

where τd is the derivative time constant (typically τd = Kd/(8*Kp)).

Process Modeling

Chemical Reactor Models

Continuous Stirred Tank Reactor (CSTR):

The CSTR model combines material and energy balances:

Material Balance:

V * dCA/dt = q*(CA,in - CA) - V*rA

Energy Balance:

V*ρ*Cp * dT/dt = q*ρ*Cp*(Tin - T) + (-ΔHr)*V*rA + Q

where: - rA = Reaction rate (typically Arrhenius: k0*exp(-E/RT)*CA^n) - Q = Heat transfer rate: UA*(Tcool - T)

Tank Level Control:

For a gravity-drained tank:

A * dh/dt = qin - Cv*√h

where: - A = Tank cross-sectional area - h = Liquid height - Cv = Valve coefficient

Linearization

For nonlinear systems ẋ = f(x,u), linearization around operating point (x0, u0) gives:

Δẋ = A*Δx + B*Δu

where:

A = ∂f/∂x|_(x0,u0),    B = ∂f/∂u|_(x0,u0)

Example - Tank Linearization:

For the tank equation around h0, qin,0:

A = -Cv/(2*√h0),    B = 1/Atank

Frequency Domain Analysis

Bode Plots

Bode plots show the frequency response of a system:

Magnitude Plot:

|G(jω)| in dB = 20*log10(|G(jω)|)

Phase Plot:

∠G(jω) in degrees

Key Features:

  • Gain crossover frequency (ωgc): Where |G(jω)| = 1 (0 dB)

  • Phase crossover frequency (ωpc): Where ∠G(jω) = -180°

Stability Analysis

Gain Margin (GM):

GM_dB = -20*log10(|G(jωpc)|)

Phase Margin (PM):

PM = 180° + ∠G(jωgc)

Stability Criteria:

  • System is stable if GM > 0 dB AND PM > 0°

  • Good stability: GM > 6 dB, PM > 30°

Controller Tuning Methods

Ziegler-Nichols Tuning

Based on process reaction curve (step response):

  1. Identify FOPDT parameters from step response

  2. Apply tuning rules:

    Controller Type

    Kp

    Ti

    Td

    P

    τ/(K*θ)

    —

    —

    PI

    0.9*τ/(K*θ)

    3.3*θ

    —

    PID

    1.2*τ/(K*θ)

    2*θ

    0.5*θ

AMIGO Tuning

Advanced Method for Integrating and General Oscillatory processes:

For FOPDT processes:

Kp = (1/K) * (0.15 + 0.35*τ/(τ + θ))

Ti = 0.35*τ + (13*τ*θ)/(τ + 12*θ)

Td = (0.5*τ*θ)/(τ + 0.5*θ)

Optimization Theory

Linear Programming

Standard form:

min  c^T * x
 x

subject to:  A*x ≤ b,  x ≥ 0

Solved using: Simplex method, Interior point methods

Nonlinear Programming

General form:

min  f(x)
 x

subject to:  gi(x) ≤ 0,  hj(x) = 0

Solution methods: - Sequential Quadratic Programming (SQP) - Interior Point Methods - Gradient-based methods

Model Predictive Control

MPC Formulation

At each time step, solve:

min   Σ[i=1 to Np] ||y(k+i|k) - r(k+i)||²Q + Σ[i=0 to Nc-1] ||Δu(k+i)||²R
Δu

Subject to:

x(k+i+1|k) = A*x(k+i|k) + B*u(k+i)
y(k+i|k)   = C*x(k+i|k)
umin ≤ u(k+i) ≤ umax
ymin ≤ y(k+i|k) ≤ ymax
|Δu(k+i)| ≤ Δumax

where: - Np = Prediction horizon - Nc = Control horizon - Q, R = Weighting matrices

Batch Process Scheduling

State-Task Networks

Mathematical Model:

Binary variables: - W(i,t) = 1 if task i starts at time t

Continuous variables: - B(i,t) = Batch size of task i starting at time t - S(s,t) = Amount of state s at time t

Objective:

max  Σs price(s)*S(s,T) - Σi,t cost(i)*B(i,t)

Constraints:

Material balances:

S(s,t) = S(s,t-1) + Σi ρ(s,i)*B(i,t-τi) - Σi ρ(i,s)*B(i,t)

Resource constraints:

Σi Σt'=max(1,t-τi+1)^t W(i,t') ≤ 1   ∀ equipment unit, t

Advanced Topics

Robust Control

Uncertainty Models: - Parametric uncertainty: G(s,θ) where θ ∈ Θ - Multiplicative uncertainty: G(s) = G0(s)*(1 + W(s)*Δ(s))

H∞ Control: Minimize worst-case performance over all uncertainties.

Adaptive Control

Model Reference Adaptive Control (MRAC): Adjust controller parameters to make the closed-loop system behave like a reference model.

Self-Tuning Regulators: Online parameter estimation combined with controller design.

Implementation Considerations

Discretization

For digital implementation, continuous controllers must be discretized:

Tustin’s method (bilinear transform):

s = (2/Ts) * (z-1)/(z+1)

where Ts is the sampling period.

Practical Guidelines: - Sampling period: Ts ≤ τ/10 (where τ is dominant time constant) - Anti-aliasing filters for noisy measurements - Integral windup protection

Real-Time Implementation

Key considerations: - Computational delay - Measurement noise filtering - Actuator saturation - Communication delays in distributed systems

References

  1. Seborg, D.E., Edgar, T.F., Mellichamp, D.A., Doyle III, F.J. (2016). Process Dynamics and Control, 4th Edition.

  2. Stephanopoulos, G. (1984). Chemical Process Control: An Introduction to Theory and Practice.

  3. Bequette, B.W. (2003). Process Control: Modeling, Design, and Simulation.

  4. Marlin, T.E. (2000). Process Control: Designing Processes and Control Systems for Dynamic Performance.

  5. Kantor, J.C. Chemical Process Control. https://jckantor.github.io/CBE30338/

Mathematical Notation

Symbols: - s - Laplace variable - t - Time - ω - Frequency (rad/time) - K - Process gain - τ - Time constant - θ - Dead time - ζ - Damping ratio - ωn - Natural frequency

Subscripts: - in - Inlet/input - out - Outlet/output - ss - Steady state - gc - Gain crossover - pc - Phase crossover