Disturbance cost

Be able to generate the C R measures relative-gain array (RGA), and disturbance cost (DC), given the matrices P 5 and P s), describing the effects of the manipulated variables and disturbances on the process outputs, using MATLAB. 

All of the linear C R measures use the approximations, F s and which describe the effects of the control variables and disturbances, respectively, on the process outputs. A commonly used controllability measure is the relative-gain array (RGA Bristol, 1966), which relies only on F s. The disturbance condition number (DCN Skogestad and Morari, 1987) and the disturbance cost (DC Lewin, 1996) are resiliency measures that require a disturbance model, in addition to P s. These C R measures are especially useful in... 

Using the Disturbance Cost to Assess Resiliency to Disturbances... 

The next example shows the utility of the disturbance cost for predicting the ease of rejecting disturbances, as applied to the operation of a distillation tower. 

The time units in this model are minutes, and both manipulated variables and disturbances are in the range 0.5. After scaling, the inputs (both distiurbances and manipulated variables) are in the range 1. First, the disturbance cost at steady state is computed for various disturbance vectors. For d = [1, -1] ... 

Table 21.4 Input Changes and Disturbance Cost for the HEN without Bypasses...

The impact of the bypass fraction on the resiliency of the HEN is examined next. The manipulated variable values and the disturbance cost are computed for dismrbances of 5% in F, and 5 F in Tq. Table 21.5 shows the changes in the control variables, AFj, AFj, and A<(> (assuming perfect control), and the disturbance cost, for four disturbance vectors, = [F -b AF, Tq + ATg]. Note that for the worst-case disturbance (AF, = —5% and ATg = +5°F), the scaled change in the bypass fraction is A<1) = 12.3, which far exceeds unity. To avoid this, the nominal bypass fraction is increased further to account for the expected disturbance levels, noting that heat exchanger E-102 must be resized. 

In Eq. (8), the constraints are the state and output equations. This NLP permits the simultaneous design of the process and its control system, subject to worst-case disturbance scenarios (e.g. see [7]). Note that this formulation supports the definition of a partial control strategy [8, 9], where the output variables are required to lie inside a hypercube, Y, rather than meet specific setpoints. Eq. (8) can also be formulated in the steady-state mode, by expressing the first constraint in its stationary form. Of particular note is the use of the nonlinear disturbance cost (DC, [9]) to guide the designer to a process that can maintain output targets inside the partial control hypercube. 

Table 1 shows the changes in the control variables, AF2 and AF3 (assuming perfect control), the disturbance cost, and the resulting change in Ts, computed using Eq. (22) for four disturbance vectors. The results indicate that perfect disturbance rejection is achieved for 02 and 04 with negligible control effort. However, the uncontrolled temperature, T3, is significantly perturbed, with the worst-case disturbance where AF and ATq are in opposite directions. Variations of 5% in Fi and + 5 F in To lead to variations of approximately 4 F in T3. 

Types of disturbances for each equipments are categorized in Table 6.7a. It is clear from the table that halts and locks (freeze) of equipments are about half of the total disturbances, and those including malfunctions (not operated too) reach 70% of the total disturbances. The aforementioned disturbances of control equipments in Table 6.1a result in troubles of power supply and system operation as in Figure 6.5, which shows that more than 10% of the control circuit disturbances cause power system disturbances. As far as a control equipment disturbance is closed within a control circuit, the disturbance costs not... 

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