AMC | Advanced Modeling and Control

Cascade control

The feedback control configuration involves one measurement (output) and one manipulated variable in a single loop.

A disadvantage of conventional feedback control is that corrective action for disturbances does not begin until after the controlled variable deviates from the set point.

Feedforward control offers large improvements over feedback control for processes that have large time constants or time delays.

However, feedforward control requires that the disturbances be measured explicitly, and that a steady-state or dynamic model be available to calculate the controller output.

An alternative approach that can significantly improve the dynamic response to disturbances employs a secondary measured variable and a secondary feedback controller.

In cascade control, we have one manipulated variable and more than one measurement.

How Cascade Control Works?

Cascade control system consists of at least 2 controllers with 1 primary loop and 1 secondary loop.

Requires 2 measurements – 1 primary measurement and 1 secondary measurement

Cascade control is primarily aimed to improve disturbance rejection or regulatory control performance.

Provides early compensation of input disturbance via the secondary controller.

Key features:
1. The disturbance must has an effect on the secondary measurement
2. Causal (cause-and-effect) relationship between the secondary measurement and manipulated variable
3. Causal relationship between the manipulated variable, and between secondary and primary measurements.
4. Secondary loop must be faster than the primary loop

Series cascade control

Parallel cascade control

Advantages and Disadvanteges

Advantages

Removes effects of disturbances and improves disturbance rejection performance
Reduces the negative effect of process nonlinearity
Improves control performance and stability of a process with long time-delay
Uses traditional PID-type controllers

Disadvantages

Requires more than 1 measurements and sensors – increased cost
More tuning parameters to handle – increased tuning task
Potentially more wear and tear as the the inner loop is tuned aggressively

Analysis of Cascade Control System

Process models

Open loop stable $G_p = \frac{K_p e^{-\theta s}}{\tau s + 1}$
Integrating $G_p = \frac{K_p e^{-\theta s}}{s}$
Open loop unstable $G_p = \frac{K_p e^{-\theta s}}{\tau s - 1}$

PID controllers

PI controller $G_c = K_c \left( 1 + \frac{1}{\tau_I s} \right)$
PID controller (Ideal) $G_c = K_c \left( 1 + \frac{1}{\tau_I s} + \tau_D s\right)$
PID controller (Parallel) $G_c = K_c + I \frac{1}{s} + D \frac{N}{1 + N \frac{1}{s}}$

Stability analysis

Primary process $G_{p1} = \frac{K_{p1} e^{-\theta_1 s}}{\tau_1 s + 1}$
Secondary process $G_{p2} = \frac{K_{p2} e^{-\theta_2 s}}{\tau_2 s + 1}$

Primary controller $G_c = K_c \left( 1 + \frac{1}{\tau_I s} \right)$
Secondary controller $G_c = K_c$

Inner loop analysis

Setpoint transfer function $H_{r2} = \frac{G_{C2} G_{P2}}{1 + G_{C2} G_{P2}}$
Characteristic equation $1 + G_{C2} G_{P2} = 0$

$1 + \frac{K_{C2} K_{P2} e^{-\theta_2 s}}{\tau_2 s + 1} = 0$

Let Loop gain $K_{L2} = K_{C2} K_{P2}$
Delay: $e^{-\theta_2 s} \approxeq 1 - \theta_2 s$
Characteristic Polynomial $\tau_2 s + 1 + K_{L2}(1 - \theta_2 s) = 0$

$\underbrace{\left(\tau_2-K_{L 2} \theta_2\right)}_{a_1} s+\underbrace{\left(1+K_{L 2}\right)}_{a_0}=0$

Inner loop analysis

Necessary stability criterion: $a_1 > 0, a_0 > 0$
Upper limit on the loop gain

$a_1 = \tau_2 - K_{L2}\theta_2 > 0 ; \therefore K_{L2} = \frac{\tau_2}{\theta_2}$
lower limit on the loop gain

$a_0 = 1 + K_{L2} > 0 ; \therefore K_{L2} > -1$
Since the lower limit is negative, due to practical reason the minimum value of loop gain should be above 0 but lower than its upper limit. Thus, for stability the loop gain is given as

$K_{L2} = R_{p2} \left( \frac{\tau_2}{\theta_2} \right); 0 < R_{p2} < 1$
The parameter $R_{P2}$ can be used to tune the controller gain as: $K_{C2} = \frac{R_{p2}}{K_{p2}} \frac{\tau_2}{\theta_2}$

Inner loop analysis

Simplify the setpoint transfer function $H_{r2}$ as

$H_{r 2}=\frac{K_O \exp \left(-\theta_2 s\right)}{\tau_{c 2} s+1}; \text{where, } K_o=\frac{K_{L 2}}{1+K_{L 2}}, \quad \tau_{c 2}=\frac{\tau_2}{1+K_{L 2}}$

Notice that $K_{L2} = R_{p2}\frac{\tau_2}{\theta_2}$
Therefore, overall gain and closed-loop time constant can be written as $K_o=\frac{R_{p2} \tau_2}{\theta_2 + R_{p2} \tau_2}; \tau_{c2} = \frac{\theta_2 \tau_2}{\theta_2 + R_{p2} \tau_2}$

To increase the speed of response of secondary controller, increase the value of $R_{p2}$ but keep the value below 1 to ensure stability.

Primary loop analysis

Augmented primary process

$G_{p a}=H_{r 2} G_{p 1} \cong \frac{K_o K_{p 1} e^{-\left(\theta_1+\theta_2+\tau_{c 2}\right) s}}{\tau_1 s+1}$

$G_{pa}$ is used to design or tune the primary controller. This means that the secondary controller should be designed first, as the primary design depends on the $H_{r2}$ .

Primary loop analysis

The effect of input disturbance is given by $H_{d2}$

$H_{d2} = \frac{K_{D0}}{\tau_{c2} s + 1}; K_{D0} = \frac{1}{1 + K_{L2}} = \frac{\theta_2}{\theta_2 + R_{p2} \tau_2}$
Primary setpoint transfer function $H_{r1}$ :

$H_{r1} = \frac{ G_{c1} H_{r2} G_{p1}}{1 + G_{c1} H_{r2} G_{p1}}$
Characteristic equation (CE):

$1 + G_{c1} H_{r2} G_{p1} = 0; 1 + \frac{K_{c1} K_0 K_{p1} (\tau_I s + 1) e^{-\theta_t s}} { \tau_I s (\tau_1 s + 1)} = 0$

where, $\theta_t = \theta_1 + \theta_2 + \tau_{c2}$

Primary loop analysis

Let loop gain $K_{L1} = K_{C1} K_0 K_{p1}$ and $e^{-\theta_t s} \approxeq 1 - \theta_t s$
Simplifying CE to polynomial

$\tau_I s (\tau_1 s + 1) + K_{L1} (\tau_I s + 1)(1 - \theta_t s) = 0$

$\underbrace{\tau_I \left(\tau_1-K_{L 1} \theta_t\right)}_{a_2} s^2 +\underbrace{\tau_I + K_{L 1}\left(\tau_I - \theta_t\right)}_{a_1} s +\underbrace{K_{L 1}}_{a_0}=0$
Necessary stability criterion requires $a_2 > 0, a_1 > 0$ , and $a_0 > 0$
These provide the limits for loop gain $K_{L1}$
To ensure stability, the loop gain must be bounded between its minimum upper limit and maximum lower limit
Provides tuning parameters for the controller

Choice of Secondary Measured Variables

There should be a well-defined relation between the primary and secondary measured variables.
Essential disturbances should act in the inner loop.
The inner loop should be faster than the outer loop. The typical rule of thumb is that the average residence times should have a ratio of at least five.
It should be possible to have a high gain in the inner loop.

Summary

Cascade control can be used when there are several measurement signals and one control variable.
It is particularly useful when there are significant dynamics, e.g., long dead times or long time constants, between the control variable and the process variable.
Tighter control can then be achieved by using an intermediate measured signal that responds faster to the control signal.