Presented By: M Junaid Khan Associate Professor, Dept. of Electronic and Power Engineering contactjunaid 86-15251605382 National University of Science and Technology Pakistan 1 Contents nReview of last lecture nDesign by Emulation - Indirect Design Method nMethods to Discretize Continuous Controller nForward Rectangular Rule nBackward Rectangular Rule nTrapezoidal Rule nBilinear OR Tustins Transformation nZOH Equivalent Step Invariance Method nPole Zero Mapping Matched Pole Zero Mapping nBilinear Transformation with Frequency Pre-warping nAnalyzing Performance of Discrete System Numerical Integration Methods 2 Review- Two Ways to Design a Digital Controller Indirect Design: First design a continuous time controller and then discretize it using some discretization technique to obtain an equivalent digital controller. Direct Design: Discretize the plant first to obtain a discrete-time system and then apply digital control system design techniques Indirect Design Direct Design 3 Review- Strategy of Indirect Design 1. Having a continuous transfer function D(s), find the best discrete equivalent D(z) using any suitable method of conversion. 2. Judge the effectiveness of the digital design by comparing its frequency response with that of D(s) 3. Selected sampling frequency is kept higher than frequency of the input signals 4. For input signals which are at high frequency i.e. approaching the Nyquist rate (fs/2) or folding frequency, the fidelity of D(z) compared with D(s) will deteriorate. This means that: 1.if the sampling frequency fs is less than double of signal frequency, the performance of D(z) will be bad. 4 Review- Discrete Approximations Forward Rectangular Rule: It is simple to apply, but a stable system can become unstable, so it is impractical to use this approximation. Backward Rectangular Rule : A stable system will result in a stable system, but there are large distortions in dynamic response and frequency response properties Trapezoidal Rule : A stable system will remain stable, however it can cause frequency distortion or warping. Frequency pre-warping can decrease the distortion in frequency response. 5 Review- Discrete Approximations Remarks Forward Rectangular Rule is not used in practical applications. Backward Rectangular Rule always maps a stable continuous controller to a stable discrete controller. However, some unstable continuous controller can also be transformed into stable discrete controllers The bilinear transformation (trapezoidal or Tustins approximation) maps the left half s plane into the unit disc. Hence, stable continuous controllers are approximated by stable discrete controllers and unstable continuous controllers are mapped to unstable discrete controllers In practice, the Tustins approximation (bilinear transformation) is the approximation of choice for converting continuous-time controllers to discrete-time controllers. In fact, some computer-aided programs (e.g. MATLAB) dont even have the option to approximate with forward or backward difference methods 6 Indirect design method Strategies: 1. Emulation by ZOH Equivalent - Step- invariance method This method simply assumes that the signal entering the microprocessor is constant over the sampling time (the function of the ZOH DAC on the output signal ) 7 Indirect design method ZOH Equivalent or Step-invariance method Convert D(s) to (D(z) (suitable for implementation on a microprocessor) based on a sampling time of 0.1 second by ZOH method. Example 8 Indirect design method ZOH Equivalent or Step-invariance method Remarks: n1. A stable system will remain stable n2. Frequency folding phenomena may occur, but thanks to the low-pass characteristics of the ZOH, it is a little better. n3. Complex computation for large-scale systems n4. Steady-state value is invariant, i.e., G(s)|s=0=H(z)|z=1 9 Indirect design method 2. Pole and Zero Mapping Since every pole and zero of D(s) in the s-plane has its equivalent position in the z-plane through the mapping: then its seems reasonable to form D(z) from D(s) by mapping the positions of the poles and zeroes in terms s to positions in the z-plane using equations above. A simple example will demonstrate the Method. If Then the positions of the finite poles and zeroes of D(s) are: 10 Indirect design method 2. Pole and Zero Mapping Using the mapping, these map to positions in the s-plane given by: Thus D(z) is given by: The value of K is selected to ensure the gain of D(s) and D(z) are the same at some specific frequency, usually zero frequency (DC gain). The DC gain in the s-plane is determined when s = 0 and in the z-plane when z = 1 11 Indirect design method Pole and Zero Mapping Thus for equal DC gain: And thus the equivalent transfer function is given by: 12 Indirect design method This is a popular method and has a valid rational, and for transfer functions with as many zeroes as poles in D(s) it is a re