z-Plane Analysis of Discrete-Time Control Systems Chapter 3 z-Plane Analysis of Discrete-Time Control Systems

Chap. 3 z-Plane Analysis of Discrete-Time Control Systems Introduction Backgrounds necessary for the analysis and design of discrete-time control systems in the z plane are presented. The main advantage of the z transform method: it enables us to apply conventional continuous-time design methods to discrete-time systems. The chapter covers: Mathematical representation of the sampling operation The convolution integral method for obtaining the z transform The sampling theorem based on the fact that the Laplace transform of the sampled signal is periodic Mathematical modeling of digital controllers in terms of pulse transfer function Realization of digital controllers and digital filters Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Impulse Sampling and Data Hold A fictitious sampler The output of the sampler is a train of impulses. Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Impulse Sampling and Data Hold Impulse Sampling (cont.) Let’s define a train of unit impulses: The sampler may be considered a modulator with The modulating signal: the input x(t) The carrier : the train of unit impulses T(t) Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Impulse Sampling and Data Hold Impulse Sampling (cont.) The Laplace transform of x*(t) If we define or Hence we may write Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Impulse Sampling and Data Hold Impulse Sampling (cont.) Summary If the continuous-time signal x(t) is impulse sampled in a periodic manner, the sampled signal may be represented by The Laplace transform of the impulse-sampled signal x*(t) has been shown to be the same as the z transform of signal x(t) if Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Impulse Sampling and Data Hold Data-Hold Circuits Data-hold: a process of generating a continuous-time signal h(t) from a discrete-time sequence x(kT). A hold circuit approximately reproduces the signal applied to the sampler. The simplest data-hold: zero-order hold (clamper) Note that signal h(kT) must equal x(kT): n-th order hold Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Impulse Sampling and Data Hold Zero-Order Hold Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Impulse Sampling and Data Hold Zero-Order Hold (cont.) A real sampler and zero-order hold Since Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Impulse Sampling and Data Hold Zero-Order Hold (cont.) Mathematical model: an impulse sampler and transfer function From the figure Since Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Impulse Sampling and Data Hold Transfer function of First-Order Hold n-th order hold 1-st order hold By applying the condition that Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Impulse Sampling and Data Hold Transfer function of First-Order Hold (cont.) Derivation of the transfer function Taking the Laplace transform Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Impulse Sampling and Data Hold Transfer function of First-Order Hold (cont.) Derivation of the transfer function (cont.) The transfer function Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Reconstructing Original Signals from Sampled Signals Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Reconstructing Original Signals from Sampled Signals Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Reconstructing Original Signals from Sampled Signals Sampling Theorem If the sampling frequency is sufficiently high compared with the highest-frequency component involved in the continuous-time signal, the amplitude characteristics of the continuous-time signal may be preserved in the envelope of the sampled signal. To reconstruct the original signal from a sampled signal, there is a certain minimum frequency that the sampling operation must satisfy. We assume that x(t) does not contain any frequency components above 1 rad/sec. Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Reconstructing Original Signals from Sampled Signals Sampling Theorem The frequency spectrum: If s, defined as 2/T is greater than 21, where 1 is the highest-frequency component present in the continuous-time signal x(t), then the signal x(t) can be reconstructed completely from the sampled signal x*(t). Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Reconstructing Original Signals from Sampled Signals Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Reconstructing Original Signals from Sampled Signals Ideal Low-pass filter The ideal filter attenuates all complementary components to zero and will pass only the primary component. If the sampling frequency is less than twice the highest-frequency component of the original continuous-time signal, even the ideal filter cannot reconstruct the original continuous-time signal. Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Reconstructing Original Signals from Sampled Signals Ideal Low-pass filter Is NOT Physically Realizable For the ideal filter an output is required prior to the application of the input to the filter – physically not realizable. Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Reconstructing Original Signals from Sampled Signals Frequency-Response Characteristics of the ZOH Transfer function of ZOH Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Reconstructing Original Signals from Sampled Signals Frequency-Response Characteristics of the ZOH(cont.) Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Reconstructing Original Signals from Sampled Signals Frequency-Response Characteristics of the ZOH(cont.) The comparison of the ideal filter and the ZOH. ZOH is a low-pass filter, although its function is not quite good. The accuracy of the ZOH as an extrapolator depends on the sampling frequency. Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Reconstructing Original Signals from Sampled Signals Folding The phenomenon of the overlap in the frequency spectra. The folding frequency (Nyquist frequency): N In practice, signals in control systems have high-frequency components, and some folding effect will almost always exist. Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Reconstructing Original Signals from Sampled Signals Aliasing The phenomenon that the frequency component ns 2 shows up at frequency 2 when the signal x(t) is sampled. To avoid aliasing, we must either choose the sampling frequency high enough or use a prefilter ahead of the sampler to reshape the frequency spectrum of the signal before the signal is sampled. Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Reconstructing Original Signals from Sampled Signals Hidden Oscillation An oscillation existing in x(t) between the sampling periods. For example, if the signal is sampled at t=0, 2/3, 4 /3,…, then the sampled signal will not show the frequency component with =3 rad/sec. Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function Convolution Summation For the continuous time-system For the discrete-time system For a physical system a response cannot precede the input Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function Convolution Summation (cont.) The value of the output y(t) at the sampling instants t=kT are given by Since we assume that x(t)=0 for t <0 It is noted that if G(s) is a ratio of polynimials in s and if the degree of the denominator polynomial exceeds that of the numerator polynomial only by 1 the output y(t) is discontinuous. Convolution summation Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function Convolution Summation (cont.) In analyzing discrete-time control systems it is important to remember that the system response to the impulse-sampled signal may not portray the correct time-response behavior of the actual system unless the transfer function G(s) of the continuous-time part of the system has at least two more poles than zeros, so that Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function The z transform of y(kT) Pulse transfer function to the Kronecker delta input Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function Starred Laplace Transform of the Signal involving both Ordinary and Starred Laplace Transform Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function General Procedures for Obtaining Pulse Transfer Functions Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function Pulse Transfer Function of Cascaded Elements Note that Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function Pulse Transfer Function of Closed-Loop Systems Refer to Table 3-1 Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function Table 3-1: Five typical configurations for closed-loop discrete-time control systems Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function Pulse Transfer Function of a Digital Controller The input to the digital controller is e(k) and the output is m(k) The z transform of the equation Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function Closed-loop Pulse Transfer Function of a Digital Control System Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function Pulse Transfer Function of a Digital PID Controller The PID control action in analog controllers Discretization of the equation to obtain the pulse transfer function Define Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function Pulse Transfer Function of a Digital PID Controller(cont.) Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

The Pulse Transfer Function Obtaining response between consecutive sampling instants Laplace transform method Modified z transform method State-space method Laplace Transform Method Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters A computational algorithm that converts an input sequence of numbers into an output sequence in such a way that the characteristics of the signal are changed in some prescribed fashion. The block diagram realization of digital filters using delay elements, adders, and multipliers will be discussed. The general form of the pulse transfer function PID controller Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Direct Programming From the diagram Total number of delay element: m+n Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Standard Programming The number of delay elements is minimized. from n+m to n Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Standard Programming (cont.) Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Standard Programming (cont.) Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Standard Programming (cont.) It is important to have a good level of accuracy. There are three sources of errors affect the accuracy. The error due to the quantization of the input signal into a finite number of discrete levels. The error due to the accumulation of round-off errors in the arithmetic operations in the digital system. The error due to quantization of the coefficients of the pulse transfer function. For decomposing higher-order pulse transfer functions in order to avoid coefficient sensitivity problem, the following three approaches are commonly used: Series programming Parallel programming Ladder programming Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Series Programming A series connection of first-order and/or second-order pulse transfer functions. Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Parallel Programming Expanding the pulse transfer function into partial fractions. Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Parallel Programming (cont.) Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Ladder Programming Expanding the pulse transfer function into the continued-fraction form Define Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Ladder Programming (cont.) Note that Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Infinite-Impulse Response Filter and Finite-Impulse Response Filter Digital filters may be classified according to the duration of the impulse response. Infinite-Impulse Response Filter Not all ai’s are zero. An infinite number of nonzero samples. Also called recursive filter Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Infinite-Impulse Response Filter and Finite-Impulse Response Filter (cont.) Finite-Impulse Response Filter The impulse response is limited to a finite number of samples defined over a finite range of time intervals. The impulse response sequence is finite. Also called nonrecursive filter Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Realization of a Finite-Impulse Response Filter Suppose we decide to employ the N immediate past values of the input and the current input. Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142

Realization of Digital Controllers and Digital Filters Realization of a Finite-Impulse Response Filter (cont.) The characteristics of the finite-impulse response filter: The finite-impulse response filter is nonrecursive. Because of the lack of feedback, the accumulation of errors in past outputs can be avoided in the processing of the signal. Implementation of the finite-impulse response filter does not require feedback, so the direct programming and standard programming are identical. Also, implementation may be achieved by high-speed convolution using the FFT. The poles of the pulse transfer function of the FIR filter are at the origin, and therefore, it is always stable. If the input signal involves high-frequency components, then the number of delay elements needed in the finite-impulse response filter increases and the amount of time delay becomes large. Chap. 3 z-Plane Analysis of Discrete-Time Control Systems 41-142