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Q. 15.9

Design a feedforward-feedback control system for the blending system in Example 15.5, for a situation in which an improved sensor is available that has a smaller time delay of $0.1 min$. Repeat parts (b), (c), and (d) of Example 15.5. For part (c), approximate $G_{v} G_{p} G_{m}$ with a first-order plus time-delay transfer function, and then use a PI controller with ITAE controller tuning for disturbances (see Table 12.4). For the feedforward controller in Eq. 15-34, use $\alpha=0.1$.

$G_{f}(s)=-4.17 \frac{1.0833 s+1}{\alpha(1.0833) s+1}$             (15-34)

Develop a Simulink diagram for feedforward-feedback control and generate two graphs similar to those in Fig. 15.13.

 Table 12.4 Controller Design Relations Based on the ITAE Performance Index and a First-Order-plus-Time-Delay Model (Lipták, 2006)*,† Type of Input Type of Controller Mode A B Disturbance PI P 0.859 −0.977 I 0.674 −0.680 Disturbance PID P 1.357 −0.947 I 0.842 −0.738 D 0.381 0.995 Set point PI P 0.586 −0.916 I 1.03† −0.165† Set point PID P 0.965 −0.85 I 0.796† −0.1465† D 0.308 0.929

${ }^{*}$ Design relation: $Y=A(\theta / \tau)^{B}$ where $Y=K K_{c}$ for the proportional mode, $\tau / \tau_{I}$ for the integral mode, and $\tau_{D} / \tau$ for the derivative mode.

${ }^{†}$For set-point changes, the design relation for the integral mode is $\tau / \tau_{I}=A+B(\theta / \tau)$.

Verified Solution

The block diagram for the feedforward-feedback control system is shown in Fig. $15.12 .$

(a) Not required

(b) Feedforward controllers

From Example 15.5,

\begin{aligned}&G_{I P}=K_{I P}=0.75 psi / mA \\&G_{v}(s)=\frac{K_{v}}{\tau_{v} s+1}=\frac{250}{0.0833 s+1}\end{aligned}

Since the measurement time delay is now $0.1 min$, it follows that:

$G_{t}(s)=G_{m}(s)=K_{t} e^{-\theta s}=32 e^{-0.1 s}$

The process and disturbance transfer functions are:

$\frac{X^{\prime}(s)}{W_{2}^{\prime}(s)}=\frac{2.6 x 10^{-4}}{4.71 s+1}, \quad \frac{X^{\prime}(s)}{X_{1}^{\prime}(s)}=\frac{0.65}{4.71 s+1}$

The ideal dynamic feedforward controller is given by Eq. 15-21:

$G_{f}=-\frac{G_{d}}{K_{I P} G_{t} G_{v} G_{p}}$       (15-21)

Substituting the individual transfer functions into Eq. 15-21 gives,

$G_{f}(s)=-0.417(0.0833 s+1) e^{+0.1 s}$           (1)

The static (or steady-state) version of the controller is simply a gain, $K_{f}$ :

$K_{f}=-0.417$          (2)

Note that $G _{f(s)}$ in (1) is physically unrealizable. In order to derive a physically realizable dynamic controller, the unrealizable controller in (1) is approximated by a lead-lag unit, in analogy with Example 15.5 :

$G_{f}(s)=-0.417 \frac{0.1833 s+1}{0.01833 s+1}$              (3)

Equation 3 was derived from (1) by: (i) omitting the time delay term, (ii) adding the time delay of $0.1 min$ to the lead time constant, and (iii) introducing a small time constant of $\alpha x 0.1833=0.01833$ for $\alpha=0.1$.

(c) Feedback controller

Define $G$ as,

$G=G_{I P} G_{v} G_{p} G m=(0.75)\left(\frac{25}{0.0833 s+1}\right)\left(\frac{2.6 \times 10^{-4}}{4.71 s+1}\right)\left(32 e^{-0.1 s}\right)$

First, approximate $G$ as a FOPTD model, $\tilde{G}$ using Skogestad’s half-rule method in Section $6.3$ :

\begin{aligned}&\tau=4.71+0.5(0.0833)=4.75 min \\&\theta=0.1+0.5(0.0833)=0.14 min\end{aligned}

Thus,

$\tilde{G}=\frac{0.208 e^{-0.14 s}}{4.75 s+1}$

The ITAE controller settings are calculated as:

\begin{aligned}&K K_{c}=0.859\left(\frac{\theta}{\tau}\right)^{-0.977}=0.859\left(\frac{0.14}{4.752}\right)^{-0.977} \quad \Rightarrow K_{c}=134 \\&\frac{\tau}{\tau_{I}}=0.874\left(\frac{\theta}{\tau}\right)^{-0.680}=0.674\left(\frac{0.14}{4.752}\right)^{-0.680} \quad \Rightarrow \tau_{I}=0.642 min\end{aligned}

(d) Combined feedforward-feedback control

This control system consists of the dynamic feedforward controller of part (b) and the PI controller of part (c).

The closed-loop responses to $a +0.2$ step change in $x_{1}$ for the two feedforward controllers are shown in Fig. S15.9a. The dynamic feedforward controller is superior to the static feedforward controller because both the maximum deviation from the set point and the settling time are smaller. Figure S15.9b shows that the combined feedforward-feedback control system provides the best control and is superior to the PI controller. A comparison of Figs. S15.9a and S15.9b shows that the addition of feedback control significantly reduces the settling time due to the very large value of $K_{c}$ that can be employed because the time delay is very small. (Note that $\theta / \tau=0.14 / 4.75=0.0029 .$ )