Discrete-time system

The continuous-time solution of the state equation is given in equation (8.47). If the time interval t — to) in this equation is T, the sampling time of a discrete-time system, then the discrete-time solution of the state equation can be written as... 

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. 

All the concepts previously presented for a continuous linear system, can be formulated vis a vis for the case of a linear discrete time system described by... 

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. 

The transfer function approach will be used where appropriate throughout the remainder of this chapter. Transfer functions of continuous systems will be expressed as functions of s, e.g. as G(j) or H(s). In the case of discrete time systems, the transfer function will be written in terms of the z-transform, e.g. as G(z) or H(z) (Section 7.17). An elementary knowledge of the Laplace transformation on the part of the reader is assumed and a table of the more useful Laplace transforms and their z-transform equivalents appears in Appendix 7.1. 

G(z) Transfer function of discrete time system, i.e. in terms of — —... 

We will speak of a reverberation algorithm, or more simply, a reverberator, as a linear discrete-time system that simulates the input-output behavior of a real or imagined room. The problem of designing a reverberator can be approached from a physical or perceptual point of view. 

Finally, since the process is modelled as a discrete time system with r as the time step, the simulation time is denoted by tn = nr where the integer n is taken from 0 to 8. 

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. 

The Kalman filter problem. Considering the relations (3.258) and (3.259)) we can write the following discrete-time system ... 

Keerthi, S. S., and Gilbert, E. G., Optimal, infinite horizon feedback laws for a general class of constrained discrete time systems Stability and moving-horizon approximations, IOTA, 57, 265-293 (1998). 

State-space models can also be developed for discrete-time systems. Let the current time be denoted as k and the next time instant where input values become available as A - - 1. The equivalents of Eqs. 4.44-4.45 in discrete time are... 

In Chapter 28 we will introduce z-transforms, which constitute the main tool for the analysis of discrete-time systems and play the same role as Laplace transforms for continuous systems. 

From Continuous-Time 27 to Discrete-Time Systems... 

Figure 29.2 (a) Discrete-time system (b) corresponding block diagram. 

Figure 29.3 Block diagrams for (a) (V discrete-time systems in series (b) discrete system with two inputs and two outputs.

In previous sections we examined the dynamic characteristics of discrete-time open- and closed-loop systems and we developed the appropriate transfer functions to describe them. Nowhere, though, did we question the stability of these systems. In this section we extend the previous analysis and derive general rules that will determine the stability characteristics of the response of discrete-time systems. 

It is interesting to point out the relationship between the stability rules for continuous systems in the. v-domain (see Section 15.1) and the rule developed in this section for discrete-time systems in the z-domain. Consider a system with a continuous transfer function G(s). This system is stable if the roots of its characteristic equation (poles) lie to the left of the imaginary axis (see Figure 29.12a). Recall the relationship between the variables s and z [eq. (28.2)] ... 

Therefore, if s lies to the left of the imaginary axis, then a <0 and z < 1, which is the rule for stability of discrete-time systems. Figure 29.12 shows the corresponding regions for stability of a system in continuous form (Figure 29.12a) and discrete-time form (Figure 29.12b). 

It should be emphasized once more that the sampling period affects the stability characteristics of a discrete-time system in a very profound manner. Example 29.6 demonstrated that clearly. 

The Laplace transforms allowed us to develop simple input-output relationships for a process and provided the framework for easy analysis and design of loops with continuous analog controllers. For discrete-time systems we need to introduce new analytical tools. These will be provided by the z-transforms. 

The development of input-output models for discrete-time systems, which constitute the basis for the dynamic analysis and design of control loops... 

Like the Laplace transforms, z-transforms possess certain properties that we will find very useful in dealing with discrete-time systems. Let us examine these properties. 

Define the bounded input, bounded output stability of a discrete-time system. What is the rule for characterizing the stability of such a system ... 

Does the sampling period affect the stability of a discrete-time system, and why What would you expect to happen a system becomes destabilized when the sampling period increases or decreases Elaborate on your answer. 

Consider a process that is modeled by a third-order discrete-time system with dead time. Assume that the dead time is an integer multiple of the sampling period (i.e., ti = tT where i = integer). Then we have... 

Design a suboptimal controller for load changes when the process can be modeled by a fourth-order discrete-time system with dead time tj = tT (g = integer). Under what conditions is the resulting controller physically realizable ... 

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