Discrete time control systems

Ogata, K. (1995) Discrete-Time Control Systems, 2nd ed., Prentice-Hall, Inc., Upper Saddle River, NJ. 

Ogata, 0. Discrete-time control systems Prentice-Hall Englewood Cliffs, New Jersey, I987. 

Fig. 7.91. Closed-loop control system with discrete time controller...

Many industrial applications employ discrete time controllers. These operate on the basis of discrete signals rather than on a continuous signal as with analog controllers. They are ideally suited to the digital environment produced by computer control and/or sampled data systems. 

There is a variety of specifications that can be imposed on the system closed-loop response for a given change in set point. These lead to a number of alternative discrete-time control algorithms—the best known of which are the Deadbeat and Dahlin s algorithms. 

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). 

In earlier chapters, Simulink was used to simulate linear continuous-time control systems described by transfer function models. For digital control systems, Simulink can also be used to simulate open- and closed-loop responses of discrete-time systems. As shown in Fig. 17.3, a computer control system includes both continuous and discrete components. In order to carry out detailed analysis of such a hybrid system, it is necessary to convert all transfer functions to discrete time and then carry out analysis using z-transforms (Astrom and Wittenmark, 1997 Franklin et al., 1997). On the other hand, simulation can be carried out with Simulink using the control system components in their native forms, either discrete or continuous. This approach is beneficial for tuning digital controllers. 

This tutorial looks at how MATLAB commands are used to convert transfer functions into state-space vector matrix representation, and back again. The discrete-time response of a multivariable system is undertaken. Also the controllability and observability of multivariable systems is considered, together with pole placement design techniques for both controllers and observers. The problems in Chapter 8 are used as design examples. 

Cadzow, J.A. and Martens, H.R. (1970) Discrete-Time and Computer Control Systems, Prentice-Hall, Inc., Englewood Cliffs, N.J. 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

O.M. Grasselli and S. Longhi. Robust tracking and regulation of bnear periodic discrete-time systems. Int. J. Control, 54 613-633, 1991. 

S. Monaco and D. Normand-Cyrot. Minimum phase nonlinear discrete-time systems and feedback stabilization. In IEEE Conf. Decision and Control (CDC), pages 979-986, Los Angeles, USA, 2002. 

S. Tarbouriech and G. Garcia. Stabilization of Linear Discrete-Time Systems with Saturating Controls and Norm-Bounded Time-Varying Uncertainty. Control of uncertain systems with bounded inputs Lecture Notes in Control and Information Science 221. Springer-Verlag Berlin, 1997. 

We now introduce the concept of the control parameter X (see Section III. A). In the present scheme the discrete time sequence Xk Q transition probability Wt(C C) now depends explicitly on time through the value of an external time-dependent parameter X. The parameter Xk may indicate any sort of externally controlled variable that determines the state of the system, for instance, the value of the external magnetic field applied on a magnetic system, the value of the mechanical force applied to the ends of a molecule, the position of a piston containing a gas, or the concentrations of ATP and ADP in a molecular reaction coupled to hydrolysis (see Fig. 3). The time variation of the control parameter, X = - Xk)/At, is... 

Thus the discrete time form of the control system is less stable than the equivalent continuous case. 

F. Caccavale, F. Pieni, and L. Villani. An adaptive observer for fault diagnosis in nonlinear discrete-time systems. ASME Journal of Dynamic Systems, Measurement and Control, 130 1-9, 2008. 

Pattern recognition self-adaptive controllers exist that do not explicitly require the modeling or estimation of discrete time models. These controllers adjust their tuning based on the evaluation of the system s closed-loop response characteristics (i.e., rise time, overshoot, settling time, loop damp-... 

As shown in the above works, an optimal feedback/feedforward controller can be derived as an analytical function of the numerator and denominator polynomials of Gp(B) and Gn(B). No iteration or integration is required to generate the feedback law, as a consequence of the one step ahead criterion. Shinnar and Palmor (52) have also clearly demonstrated how dead time compensation (discrete time Smith predictor) arises naturally out of the minimum variance controller. These minimum variance techniques can also be extended to multi-variable systems, as shown by MacGregor (51). 

Other recent developments in the field of adaptive control of interest to the processing industries include the use of pattern recognition in lieu of explicit models (Bristol (66)), parameter estimation with closed-loop operating data (67), model algorithmic control (68), and dynamic matrix control (69). It is clear that discrete-time adaptive control (vs. continuous time systems) offers many exciting possibilities for new theoretical and practical contributions to system identification and control. 

Analysis of full sheet data is useful for process performance evaluations and product value calculations. For feedback control or any other on-line application, it is necessary to continuously convert scanner data into a useful form. Consider the data vector Y ,k) for scan number k. It is separated into its MD and CD components as Y( , A ) = yM )( )+Yc )( , k) where Ymd ) s the mean of Y ,k) as a scalar and YcD -,k) is the instantaneous CD profile vector. MD and CD controllers correspondingly use these calculated measurements as feedback data for discrete time k. Univariate MD controllers are traditional in nature with only measurement delay as a potential design concern. On the other hand, CD controllers are multivariate in form and must address the challenges of controller design for large dimensional correlated systems. 

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