Читать книгу Control Theory Applications for Dynamic Production Systems - Neil A. Duffie - Страница 23

The production work system illustrated in Figure 2.6 can be represented by a discrete-time model that predicts work in progress (WIP) at instants in time separated by fixed period T. This period could, for example, be one week (T = 7 days), one day (T = 1 day), one shift (T = 1/3 day) or one hour (T = 1/24 day). The modeled values of work in progress then are ww(kT) hours. If it is assumed that work input rate ri(t) hours/day, nominal capacity rp(t) hours/day, and capacity disturbances rd(t) hours/day are constant (or nearly constant) over each period kT ≤ t < (k + 1)T days, the work output rate at time kT days then is

Figure 2.6 Discrete variables in a discrete-time model of a production work system.

and the WIP is

This discrete-time model only represents the values of WIP ww(kT) at instants in time separated by period T; values between these instants are not represented. When the inputs are constant during period kT ≤ t < (k + 1)T, it is clear from the model obtained in Example 2.1 that ww(t) increases or decreases at a constant rate over that period; however, this information is not contained in the discrete-time model.

The dynamic behavior represented by this discrete-time model is illustrated in Figure 2.2 for a case where T = 1 day and there is a capacity disturbance rd(kT) of –10 hours/day that starts at time kT = 0 and lasts until kT = 3 days. The initial WIP is ww(kT) = 30 hours for kT ≤ 0 days. The rate of work input is the same as the nominal production capacity, ri(kT) = rp(kT), and there are no WIP disturbances: wd(kT) = 0. In this case, the difference equation for WIP can be written as

The response of WIP to the capacity disturbance, with the other inputs constant, is shown in Figure 2.7 and calculated recursively in Program 2.2 using the this difference equation. In Figure 2.7, the responses are denoted by the discrete values at times kT as well as by a staircase plot. The latter is used by convention to indicate that no information is present in the discrete-time model regarding the WIP between the instants in time separated by period T.⁴

Figure 2.7 Response of WIP to a 3-day capacity disturbance; each discrete value is denoted with an X.