Chapter 2 The z Transform

Introduction A mathematical tool used to the analysis and synthesis of discrete-time control systems: the z transform. The Laplace transform in continuous-time systems. With the z transform method, the solutions to linear difference equations become algebraic in nature. Discrete-Time Signals The sampled signal: x(0), x(T), x(2T),…, where T is the sampling period. If the system involves an iterative process carried out by a digital computer, the signal involved is a number sequence: x(0), x(1), x(2),…. – it can be considered as a sampled signal of x(t) when the sampling period T is 1 sec. Chap. 2 The z Transform 41-142

The z Transform One-sided z Transform For most engineering applications the one-sided z transform will have a convenient close-form solution in its region of convergence. The z transform of any continuous-time function may be written in the series form by inspection. The inverse transform can be obtained by inspection if X(z) is given in the series form. Chap. 2 The z Transform 41-142

z Transforms of Elementary Functions Unit-Step Function The series converges if Unit step sequence Chap. 2 The z Transform 41-142

z Transforms of Elementary Functions Unit-Ramp Function Chap. 2 The z Transform 41-142

z Transforms of Elementary Functions Polynomial Function ak Exponential Function Chap. 2 The z Transform 41-142

z Transforms of Elementary Functions Sinusoidal Function See Table 2-1 on pages 29-30 Chap. 2 The z Transform 41-142

Important Properties and Theorems of the z Transform Multiplication by a Constant Linearity of the z Transform Chap. 2 The z Transform 41-142

Important Properties and Theorems of the z Transform Multiplication by ak Shifting Theorem Chap. 2 The z Transform 41-142

Important Properties and Theorems of the z Transform Shifting Theorem (cont.) Chap. 2 The z Transform 41-142

Important Properties and Theorems of the z Transform Example 2-3 Note that z- 1 represents a delay of 1 sampling period T, regardless of the value of T. Chap. 2 The z Transform 41-142

Important Properties and Theorems of the z Transform Complex Translation Theorem Initial Value Theorem Chap. 2 The z Transform 41-142

Important Properties and Theorems of the z Transform Final Value Theorem Chap. 2 The z Transform 41-142

Important Properties and Theorems of the z Transform Example 2-9 Determine the final value x() of Chap. 2 The z Transform 41-142

The Inverse z Transform The notation for the inverse z transform is Z-1. The inverse z transform of X(z) yields the corresponding time sequence x(k). Only the time sequence at the sampling instants is obtained from the inverse z transform. The inverse z transform yields a time sequence that specifies the values of x(t) only at discrete instants of time. Many different time functions x(t) can have the same x(kT). Chap. 2 The z Transform 41-142

The Inverse z Transform A z transform table: an obvious method for finding the inverse z transform. Table 2-2 Four methods for obtaining the inverse z transform: Direct division method Computational method Partial-fraction-expansion method Inversion integral method Chap. 2 The z Transform 41-142

The Inverse z Transform Poles and Zeros in the z Plane The locations of the poles and zeros of X(z) determine the characteristics of x(k), the sequence of values or numbers. We often use a graphical display in the z plane of the locations of the poles and zeros of X(z). poles at z=-1, z=-2 zeros at z=0, z=-0.5 Chap. 2 The z Transform 41-142

The Inverse z Transform Direct Division Method Example 2-10 Chap. 2 The z Transform 41-142

The Inverse z Transform Computational Method Consider a system For the Kronecker delta input, X(z) =1 The inverse z transform of G(z) is given by y(0),y(1),y(2),… Two approaches to obtain the inverse transform: MATLAB approach Difference equation approach Chap. 2 The z Transform 41-142

The Inverse z Transform Computational Method (cont.) MATLAB approach num=[0 0.4673 -0.3393] den=[1 -1.5327 0.6607] x=[1 zeros(1,40)] y=filter(num,den,x) y = 0 0.4673 0.3769 0.2690 0.1632 0.0725 0.0032 -0.0429 -0.0679 -0.0758 -0.0712 -0.0591 -0.0436 -0.0277 -0.0137 -0.0027 0.0050 0.0094 0.0111 0.0108 0.0092 0.0070 0.0046 0.0025 0.0007 -0.0005 -0.0013 -0.0016 -0.0016 -0.0014 -0.0011 -0.0008 -0.0004 -0.0002 0.0000 0.0002 0.0002 0.0002 0.0002 0.0002 0.0001 Chap. 2 The z Transform 41-142

The Inverse z Transform Computational Method (cont.) MATLAB approach Chap. 2 The z Transform 41-142

The Inverse z Transform Computational Method (cont.) Difference Equation Approach for k = -2 for k = -1 Chap. 2 The z Transform 41-142

The Inverse z Transform Partial-Fraction-Expansion Method Each expanded term has a form that may easily be found from commonly available z transform tables. for simple pole for multiple pole Chap. 2 The z Transform 41-142

The Inverse z Transform Partial-Fraction-Expansion Method (cont.) Example 2-14 Chap. 2 The z Transform 41-142

The Inverse z Transform Inversion Integral Method for a simple pole for a multiple pole Chap. 2 The z Transform 41-142

The Inverse z Transform Inversion Integral Method (cont.) Example 2-16 Two simple poles: z=z1=1 and z=z2=e-aT Chap. 2 The z Transform 41-142

z Transform Method for Solving Difference Equations The linear time-invariant discrete-time system characterized by the following linear difference equation: Discrete function z Transform Chap. 2 The z Transform 41-142

z Transform Method for Solving Difference Equations Example 2-18 Chap. 2 The z Transform 41-142

Concluding Comments The basic theory of the z transform method has been presented. z transform : linear time invariant discrete-time systems Laplace transform: linear time-invariant continuous-time systems With the z transform method, linear time-invariant difference equations can be transformed into algebraic equations. The z transform method allows us to use conventional analysis and design techniques available to analog control systems. Chap. 2 The z Transform 41-142