Discrete transfer function

In equation (13) b and d represent the time delays in the system, (p, q) is the order of the noise model, (r, s) is the order of the deterministic model, and z and b are shift operators defined as z iyt = Biyt = Yt-i The component et or at represents uncorrelated white noise which is passed through a transfer function to describe the noise or disturbance model. ut is the input and yt is the output. Gp and Gn are discrete transfer functions. 

This is a first order discrete transfer function of the sampled process. In general, the dynamic behaviour between a single input variable X and an output variable y in most polymer reactor systems can be adequately modelled at the sampling instants by a difference equation model of the form y = < iyt l + 2yt-2 + ... 

Cameron, R.G., Marcos, R.L. and de Prada, C. (1998). Model validation of discrete transfer functions using the distortion method. Mathematical tmd Computer Modelling of Dytm-mica Systems, 4. 58-72. 

Equation (29.5) yields the discrete transfer function of a first-order digital filter. The noise-free signal (the output of the filter) is given by... 

It can be easily shown that for N discrete systems in series (Figure 29.3a) the overall discrete transfer function is the product of the discrete transfer functions of the individual systems ... 

For a discrete system with multiple inputs and outputs, we define the discrete transfer function matrix D(z) as follows ... 

Also, for the proportional digital controller the discrete transfer function is... 

Consider a discrete system with a discrete transfer function given by... 

From eqs. (30.2) and (30.3) we can easily derive the discrete transfer functions for the velocity form of PID and PI control algorithms. Thus we take ... 

Consider a controller with the following discrete transfer function ... 

Any negative pole of the controller s discrete transfer function [eq. (30.7)] will cause ringing. 

Bode diagram, 330-31, 334-37 frequency response, 323-24 interacting capacities, 197-200 noninteracting capacities, 194-96 pulse transfer function, 619 Multiple-input multiple-output system, 20 discrete-time model, 586 discrete transfer function, 612 input-output model, 83-85, 163-68 linearization, 121-26 transfer-function matrix, 164, 166 Multiple loop control systems, 394-409 Multiplexer, 560, 564 Multivariable control systems, 461-62 alternative configurations, 467-84 decoupling of loops, 503-8 design questions, 461-62 interaction of loops, 487-94 selection of loops, 494-503 Multivariable process (see Multiple-input multiple-output system)... 

Equation (29.3) yields the discrete transfer function of the velocity PID control algorithm. For given changes in the sampled values of the error signal, the resulting discrete-time control action can be found from the inverse z-transform ... 

In the preceding section the analysis was centered around the response of the discrete components in a direct digital control (DDC) loop with characteristic representative the control algorithm. The use of z-transforms allowed easy and straightforward development of simple input-output models through the discrete transfer functions. 

Consider the block diagram of a direct digital feedback control loop shown in Figure 29.9. Such loops contain both continuous- and discrete-time signals and dynamic elements. Three samplers are present to indicate the discrete-time nature of the set point j/Sp( ), control command c(z), and sampled process output y(z). The continuous signals are denoted by their Laplace transforms [i.e., y(s), Jn(s), and d(s)]. Furthermore, the continuous dynamic elements (e.g., hold, process, disturbance element) are denoted by their continuous transfer functions, H(s), Gp(s), and GAs), respectively. For the control algorithm, which is the only discrete element, we have used its discrete transfer function, D(z). 

The controller is a digital PI algorithm in the velocity form. Its discrete transfer function will be developed in Section 30.1, but for the time being consider it is known and given by... 

What is a discrete transfer function, and what is it needed for Develop the discrete transfer function for (a) a proportional control algorithm, (b) the velocity form of a PI control algorithm, and (c) a second-order digital filter. 

VII.20 Find the discrete transfer functions of the following discrete dynamic systems. 

Calculate unity-feedback closedioop discrete transfer function [ncl,dclj-clo() Xnurud kc,dend. - i) printsys(ncl,dci z ... 

The discrete transfer function has three parameters that need to be identified nk, b, and a I. Of course, if we are identifying an unknown plant, we do not know what the real order of the systems is. A crucial part of the identification problem is the determination of what model structure yields the best fit to the real plant. In addition to finding the deadtime (nk), we must find the number of terms (na) and the number of bk terms (nb) to use in the model. 

The transfer function between the actual feed and the peak resultant cutting force is usually modeled as a first-order lag in a stable cutting process. The peak force is detected usually at one spindle period in order to avoid run-out effects in multipoint cutting operations such as drilling and milling. The overall discrete transfer function between the feed command (fc) and peak force (Fp) is considered to represent the combined dynamics of CNC, feed drives, and cutting process, and may be represented as follows ... 

In this chapter, two transformation techniques will be introduced that are useful for model analysis. Continuous as well as discrete transfer functions, which are derived by using these transformations, will be discussed and conversion between these domains will be illustrated. 

Equations, such as Eqn. (5.45), are not very convenient for representation in a control block diagram. The z-transform is an elegant way of representing discrete transfer functions. The backward shift operator z is defined as ... 

Equation 17-28 defines the discrete transfer function G(z) of the first-order difference equation, which is analogous to the transfer function obtained by applying Laplace transforms to a first-order linear differential equation. If the input t/(z) is known, then an expression for the output Y(z) can be found by multiplying G(z) times U(z). 

To calculate the response of a discrete transfer function, which corresponds to the response of the equivalent difference equation, we can use direct simulation of the difference equation based on the specified input. Alternatively, the output z-transform can be calculated using long division, which is a power series expansion in terms of z. We will illustrate this calculation in Examples 17.2 and 17.3. 

The ratio of polynomials in the discrete transfer function, G(z), can be derived by algebraic manipulations... 

Figure 17.8 Time-domain responses for different locations of the pole a, indicated by an x, of a first-order discrete transfer function and a pulse input at /r = 0.

Chapter 4 addressed the notion of physical realizability for continuous-time transfer functions. An analogous condition can be stated for a difference equation or its transfer function—namely, that a discrete-time model cannot have an output signal that depends on future inputs. Otherwise, the model is not physically realizable. Consider the ratio of polynomials given in Eq. 17-39. The discrete transfer function will be physically realizable as long as aq assuming that G(z) has been reduced so that common factors in the numerator and denominator have been canceled. To show this property, examine Eq. 17-36. If uq = 0, the difference equation is... 

Derive the discrete transfer function for the parallel form of a PID controller. 

Table 17.1 Discrete Transfer Functions Obtained Using a Zero-Order Hold...

In (17-59), Gc(z) is the discrete transfer function for the digital controller. A digital controller is inherently a discrete-time device, but with the zero-order hold, the discrete-time controller output is converted to a continuous signal that is sent to the final control element. So the individual elements of G are inherently continuous, but by conversion to discrete-time we compute their values at each sampling instant. The discrete closed-loop transfer function in (17-59) provides a framework to perform closed-loop analysis and controller design, as discussed in the next section. Additional material on closed-loop analysis for discrete-time systems is available elsewhere (Ogata, 1994 Seborg et al., 1989). 

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