Guide Discrete-Time Control Systems (Pie)

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Chapter 4 first presents mapping between the S plane and the z plane and then discusses stability analysis of closed-loop systems in the z plane, followed by tran- sient and steady-state response analyses, design by the root-locus and frequency- response methods, and an analytical design method. Chapter 5 gives state- space representation of discrete-time systems, the solution of discrete-time state- space equations, and the pulse transfer function matrix. Then, discretization of continuous-time state-space equations and Liapunov stability analysis are treated.

Discrete-Time Control Systems

Chapter 6 presents control systems design in the state s chapter with a detailed presentation of controllability and obs present design techniques based on pole placernent, follow full-order state observen and minimum-order state observe chapter with the design of servo systems. Chapter 7 treats the approach to the design of control systems.

Then we present the design of regulator systems and control systems using the solution of Diophantine equations. The approach here is an alternative to the pole-placement approach ombined with minimum-order observ- ers. The design of model-matching control systems is included in this chapter. Finally, Chapter 8 treats quadratic optimal control problems in detail.

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Appendix ts materiak in z tEUls- form theory not included in Chapter 2. Appendix C treats pole-placement design problems when the control is a vector quantity. In each chapter, except Chapter 1,the main text isfollowedby solvedproblems and unsolved problems. The reader should study al1 solved probl Solved problems are an integral part of the text. Appendixes A, followedby solvedproblems. The reader who studiesthese solvedproblems willhave an increased understanding of the material presented.

Just as the Laplace transformation transforrns linear time-invariant differential equations into algebraic equations in S , the z transformation transforms linear time-invariant difference equations into algebraic equations in z. The main objective of this chapter is to resent definitionsof the z transform, basic theorems associated with the z transform, and methods for finding the inverse z transform.

Solving difference equations by the z transform method is also cussed. Signak, Diserete-t signals arise if the system involves a sampling of continuous-time als. The samplled signal is x O ,x T , x 2T , Such a sequence of values arising from the sampling operation is usually written asx kT. If the system involvesan iterative process carried out by a digital computer, the signal involved is a nu x O ,x l , x 2. The sequence of numbers is usualhy written as x k , where tbe argument k indicates the order in which the number occurs in the sequence, for example, x O ,x l , Although x k is a number sequence, it can be con- sidered as a sampled signal of x t when t e sampling period T is 1sec.

The z Transform The 2 transform applies to the continuous-time signal x t , sampled signal x kT , and the number sequence x k. Section gives z transforms of elementary functions. Section presents the solution of difference equations by the z transform method.

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Finally, Section gives concluding commeats. The z transform method is an operational method that is very working with discrete-time systems. In what followswe shall define of a time function or a nurnber sequence.

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The symbol7 denotes "the z transform of. Note that z is a complexvariable. Note that, when dealing with a time sequence x kT obtained by sampling a time signal x t , the z transform X z involves T explicitly. The z transform of x t , where -. For most engineering applications the one-sided z transform venient closed-form solution in its r an infinite series in z-', converges radius of absolute convergente, in u ansform method for time problems it is not necessary each cify the values of z over whichX z ks convergent.

Notice that expansion of the right-hand side of Equation gives any continuous-timefunction x t may zmkin this series indicates the position inversisn integral method see Section for details. Then, referring to Equation 2-l , we have The z Transform Chap.

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It is not necessary to specify the region of z over which suffices to know that such a region exists. Figure depicts the signal. The magnitudes of the sam values are proportional iod T. Let us obtain the z transform of x k as where a is a constant. Referring to the definition of t e z mnsform given by Equation , we obtain m w - z-- z - a on.

Another approach is to expand X s into partial fractions and use a z transform table to find the z transforms of the expanded terms. Still other approaches will be discussed in Cec. Table is such a table. In this section we present importantproperties and useful theor the time function x is z-transformable al where a is a constant. To prove this, note thatm by definition m.

The z transform possesses an Pmportant prop- erty: linearity. The shifting theorern presented here is also ieferred to as on theorem. That is, move the funct lo the right by time nT. Find the z transforms of unit-step functions that are delayed by 1 sampling period and 4 sampling periods, respectively, as shown in Figure a and b. Obtain the z transform of y k. If x t has the z transform X z , then the z transform of e-"'x t can be given by X zeuT.

This is known as the complex trans- lation theorern. Notice that Thus,. The initial value theorem is convenient for checking z transform calculations for possible errors. Determine the initial value x 0 if the z transform of x t is given by Examplie By using the initial value theorem, we find Given the z transforms of sin wt and cos wt, obtain the z transforms of e-"' sin wt and e- "' cos wt , respectively, by using the complex translation theorem.

The z Transforrn Chap. By applying the final value theorem to the given X z , we obtain ec.

ECE320 Lecture10-2d: Discrete-Time Systems Control

For the purpose of conv nt referente, these important properties and theorems are sumrnarized in Table For the z trans- form to be useful, we must be familiar with thods for finding the inverse z transform. The notation for the inverse z transform is Z-l.

The inverse z transform of X z yields the corresponding time sequence x k. Irt should be noted that only the time sequence at the sarnpling instants is obtained frorn the inverse z transform. Thus, the inverse z transform of X z yields a unique x k , but does not yield a unique x t. That is, many different ti e functions x t can have the sarne x kT. X z , the z transform of x kT or x k , is given, the operation that onding x kT or x k is called the inverse z transformation.

Other than referring to z transform fables, four methods for obfaining the inverse z transform are commonly available: The z Transform ec. Direct division method Gomputational method. The locations of the poles and zeros of X z determine the characteristics of x k , the sequence of values or numbers. S in the case of the plane analysis of linear continuous-time control systems, we often use a graphical display in the z plane of the locations of the poles and zeros of X z.

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Note that in control engineering and signal processing X z is frequently expressed as a ratio of polynomials in z-', as follows: Sherefore, in dealing with the pol of X z ,it is preferable to express X z as a ratio of polynomials in z, polynomials in z-l. In the direet division rnethod we obtain the inverse z transform by expanding X z into an infinite power series in z-l.

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This method is useful when it is difficult to obtain the closed-form expression for the inverse z transform or it is desired to find only the first several terms of x k. The direct division rnethod stems from the fact that if X z is expanded into a power series in z-', that is, if then x kT or x k is the coefficient of the zwkterm.

If X z is givenin the form of a rational function,the expansion into an infinite power series in increasing powers of z-' can be accomplished by simply dividing the numerator by the demminator, where both the numerator and denominato are wrihten in increasing pdswers of z-l. In general, the method does not yield a closed-forrn expression for x k , except in special cases.

The r Transform Chap. Figure shows a plot of this signal. En what follows, we present two corn proaches to obtain the inverse z transform. MATLA can be used for finding t transforrn. Weferring to , the input X z is the z transform of the Kronecker delta input. Since the z transforrn of the Kronecker elta input X z is equali to unity, the response of the system to this input is ce the inverse z transform of G z is given by y O ,y l ,y 2 ,..