## Quick Primer On Cascade Control

__Definition__

The definition of cascade closed loop system uses two or more independent process control loops where there is a ‘slower’ outer control loop and a faster inner control loop and where the output from the slower control loop is combined with another input for the inner control loop is where the controlled variable is measured and this measurement is used to manipulate a process variable..

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__Example__

The easiest way to explain closed loop control is to take a blank sheet of paper and put a dot in the middle of the page….now put your finger on the dot…easy right? This is an example of closed loop control where your eyes provide the feedback information you need to move your finger onto the dot. The target is defined and there is a measurement process loop to control the processing action.

In this example a temperature sensor measures the temperature of the liquid flowing out and that temperature reading is compared to the desired temperature (known as the setpoint) and the controller will increase or decrease the steam valve opening accordingly, affecting the flow of steam. Taking this example a little further, with Cascade Control there are two or more controllers where one controller’s output drives the setpoint of another controller.

The controller that drives the setpoint of the system, in this example that is the output temperature of the process fluid, is considered the primary, or outer, control loop. The second control loop, in this example that is the flow of heating steam, is considered the secondary, or inner, control loop and is the “faster” control loop.

There are several advantages of cascade control and most come down to isolating a dynamically slow control loop from nonlinearities and disturbances in the final control element. Cascade control should always be used if you have a process with relatively slow dynamics (like level, temperature, composition, humidity) and a liquid or gas flow, or some other relatively-fast process, has to be manipulated to control the slow process. As in the above example, where modifying the steam flow rate is used to control heat exchanger outlet temperature, the steam flow control loop is used as the inner loop in a cascade arrangement.

It should be noted that cascade control does have some disadvantages. Firstly, it requires an additional measurement and an additional controller to work and that second controller will require tuning. Also, the control strategy is more complex. These disadvantages have to be weighed against the benefits of the expected improvement in control to decide if cascade control should be implemented.

Cascade control is beneficial only if the dynamics of the inner loop are fast compared to those of the outer loop and, as a rule of thumb, should not be used if the inner loop is not at least three times faster than the outer loop, because the improved performance may not justify the added complexity. Additionally, when the inner loop is not significantly faster than the outer loop, there is a risk of interaction between the two loops that could result in instability – especially if the inner loop is tuned very aggressively.

With these concepts and principles in mind, we can determine the basic criteria for the design and implementation of cascade control.

- Cascade control is desirable when single loop control cannot provide sufficient control performance, and
- When a measurable second variable is available.

If these criteria are met then cascade control can be considered and there are three additional criteria that must now be satisfied.

- The secondary variable must indicate the occurrence of an important disturbance
- There must be a casual relationship between the manipulated and secondary variables
- The secondary variable dynamics must be faster than the primary variable dynamics

__Mathematical Model__

As a starting point, let’s put the example scenario into an engineering line diagram and labelled at each test, measurement and control junction. It should look something like this:

You will note that the stirrer / impeller has been removed since it has no bearing on the cascade control. In the above model, CV represents the controlled variable (CV_{1} is the temperature of the process fluid leaving the reactor and CV_{2} is the flow of steam); MV represents the modified variable; SP represents the different set points for the different controllers; F represents the flow of material at the various measurement points; P represents the pressure at various measurement points; and T represents the temperature at various measurement points.

While it may seem odd to use the two closed loop controllers to achieve the same process goal, considering the degrees of freedom of the system indicates that cascade control is legitimate since

The mass and energy balances for heating loop follow the equation:

The heating flow is related to the valve position (v) according to the general equation:

where we have to assume that the pressures and the coefficient Cv are constant.

The final equations are the two cascade controllers:

Degrees Of Freedom = 5 – 5 = 0

Variables: F_{1} , F_{h} , T , (F_{h})_{sp} , v

External Variables: F_{0 }, T_{0} , T_{hin} , T_{sp}

Parameters:

V = valve stem position (equivalent to the percent open)

ρ = density

C_{p} = heat capacity at constant pressure

ρ_{h } = density of the heating medium

C_{ph} = process capability

Cv = heat capacity at constant volume (a valve characteristic that relates pressure, orifice opening and flow through the orifice)

P_{0 }= pressure at measurement point 0

P_{1} = pressure at measurement point 1

K_{cl} = feedback controller gain for first controller

T_{1l} = integral time in first PID controller

K_{c2} = feedback controller gain for second controller

T_{12} = integral time in second PID controller

I_{Fh} = constant to be determined by the initial condition of the problem

I_{v }= constant to be determined by the initial condition of the problem

The number of degrees of freedom is equal to the number of variables minus the number of equations, as such the system is exactly specified when the outer / primary temperature controller set point is defined.

The number of parameters in a cascade system , which include primary control loop dynamics, secondary control loop dynamics and disturbance dynamics, make general performance correlations difficult to work with. The block diagram below shows the structure of a cascade control system, and this summarizes the “flow” of information throughout the system and can be used to determine key properties such as the stability and frequency response of the individual control loops.

__Block diagram of the standard cascade control structure.__

Transfer functions can be derived from this block diagram for the relationships between the controlled variable, *CV _{1}(s)*, of the primary / outer loop and the secondary disturbance,

*D*, the primary loop disturbance

_{2}(s)*D*, and the primary loop set point,

_{1}(s)*SP*, as follows:

_{1}(s)Where *G(s)* = transfer function for continuous systems. * Note* that the

*(s)*denotes that the system is continuous.

Common transfer functions include:

*G _{c}(s)* = feedback controller function

*G*= disturbance transfer function

_{d}(s)*G*= feedback process transfer function

_{p}(s)*G*= sensor transfer function

_{s}(s)*G*= valve (or final element) transfer function

_{v}(s)As stated earlier, the key factor in cascade control is the relative dynamic behaviour of the secondary and primary processes, with emphasis on the disturbances in the secondary process. If we assume the transfer functions for the sensors and valve are taken as 1.0 and the relative dynamics between the secondary and primary processes are defined by a variable *η,* then the feedback process transfer functions boil down to: