PARTIAL DYNAMIC DECOUPLING FOR THE HULBERT AND WOODBURN WET-GRINDING CIRCUIT. Hulbert and Woodburn [3] have reported the control of a wet-grinding circuit whose transfer function model, obtained by experimental means, is given by: [y1 y2 y3] = [119/217s + 1 153/337s + 1 -21/10s + 1

Question

PARTIAL DYNAMIC DECOUPLING FOR THE HULBERT AND WOODBURN WET-GRINDING CIRCUIT.

Hulbert and Woodburn [3] have reported the control of a wet-grinding circuit whose transfer function model, obtained by experimental means, is given by:

[latex]\begin{bmatrix} y_{1}\ y_{2}\ y_{3}\end{bmatrix} = \begin{bmatrix} \frac{119}{217s + 1} & \frac{153}{337s + 1} & \frac{-21}{10s + 1} \ \ \frac{0.00037}{500s + 1} & \frac{0.000767}{33s + 1} & \frac{-0.00005}{10s + 1} \ \ \frac{930}{500s + 1} & \frac{-667e^{-320s}}{166s + 1} & \frac{-1033}{47s + 1} \end{bmatrix} \begin{bmatrix} u_{1}\ u_{2}\ u_{3}\end{bmatrix}[/latex] (22.54)

The time delay and time constants are in seconds; and the indicated process variables are given in terms of deviations from their respective steady-state values as:

[latex]y_{1}[/latex] = Torque required tolu turn the mill (Nm)
[latex]y_{2}[/latex] = Flowrate from the mill (m³/s)
[latex]y_{3}[/latex] = Density of the cyclone feed (kg/m³)
[latex]u_{1}[/latex] = Feedrate of solids to the mill (kg/s)
[latex]u_{2}[/latex] = Feedrate of water to the mill (kg/s)
[latex]u_{3}[/latex] = Feedrate of water to the sump (kg/s)

A relative gain analysis of this system (see Problem (22.2)) recommends the 1-1/2-2/3-3 input/output pairing configuration; it also indicates that if three independent single-loop feedback controllers are used, the system will experience significant loop interactions. A schematic diagram for this process is shown in Figure 22.6.
Since the least sensitive of the output variables for this system is [latex]y_{2}[/latex] (the total flow from the mill), and since from experience, it is known that the greatest amount of interaction is between control Loops 1 and 3, it is desired to design decouplers for only these loops, leaving Loop 2 to operate without decoupling. Obtain the transfer functions required for these decouplers.

Accepted Answer

Under the indicated conditions, the transfer function matrix for the subsystem of interest is now given by:

[latex]\begin{bmatrix} y_{1}\ y_{3}\end{bmatrix} = \begin{bmatrix} \frac{119}{217s + 1} & \frac{-21}{10s + 1} \ \ \frac{930}{500s + 1} &\frac{-1033}{47s + 1} \end{bmatrix} \begin{bmatrix} u_{1}\ u_{3}\end{bmatrix}[/latex] (22.55)

and if the decoupler transfer functions are represented as [latex]g_{I_{1}}[/latex] for Loop 1, and [latex]g_{I_{3}}[/latex] for Loop 3, then using the simplified decoupling approach, we have, from Eq. (22.55):

[latex]g_{I_{1}} = \frac{\frac{21}{10s + 1}}{\frac{119}{217s + 1}}[/latex]

and

[latex]g_{I_{3}} = \frac{\frac{930}{500s + 1}}{\frac{1033}{47s + 1}}[/latex]

which simplify to:

[latex]g_{I_{1}} = \frac{0.176(217s + 1)}{(10s + 1)}[/latex] (22.56)

and

[latex]g_{I_{3}} = \frac{0.9(47s + 1)}{(500s + 1)}[/latex] (22.57)

each implementable by lead/lag units.
Note that this control scheme is equivalent to letting [latex]g_{I_{13}}[/latex] = [latex]g_{I_{1}}[/latex] and [latex]g_{I_{31}}[/latex] = [latex]g_{I_{3}}[/latex] in Figure 22.4 with all the other [latex]g_{I_{ij}}[/latex] = 0.

Question 22.4: PARTIAL DYNAMIC DECOUPLING FOR THE HULBERT AND WOODBURN WET-...

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