1. Ch.10 Design of Digital Filters and Controllers

2. 10.1 Discretization The sampled signal x s (t) = x(t) p(t) where p(t) is the sampling pulse signal, with T as the sampling period and ω s, the sampling frequency is equal to 2  /T. A signal reconstructor (low pass filter) can be used to regenerate the continuous signal from the samples.

3. 10.1 Discretization (cont.) Anti-aliasing filters can be used to bandlimit a signal before sampling, but they are not “ideal” in the real world and distortion can still occur. An alternative process to reconstructing with a simple low pass filter is to use a hold circuit (which has a low pass frequency response—Fig. 10.4). These hold circuits are often used in control systems.

4. 10.1.2 System Discretization If a linear time-invariant continuous-time system with transfer function H(s) has an output y(t) for a given input x(t), then consider the discretization of the input and output signals with sampling time T; this would be x[n] and y[n]. A discretization of H(s) would be H d (z) such that x[n] and y[n] are the input/output pair. Pages 540-542 show that H d (z) = H(s)| s = (1/T) ln z

5. 10.1.2 (cont.) Substitution directly does not provide a rational function of z (and so the result cannot be used without an approximation.) However, (1/T) ln z ≈ (2/T) (z-1)/(z+1) And so s = (2/T)(z-1)/(z+1) and also the inverse transformation, z={1+(T/2)s}/{1-(T/2)s}. This is known as the bilinear transformation. This transforms in the open left-hand s-plane into the open unit circle in the z-plane and vice versa.

6. 10.2 Design of IIR Filters STEP 1: The design specification is given in terms of the continuous-time frequency spectrum. STEP 2: The analog specifications are meet using well established design methods (using prototype analog filters, such as the Butterworth and Chebyshev filters.)

7. 10.2 Design of IIR Filters STEP 3: If H(s) (and the frequency response H(ω) ) meet the design requirements, the bilinear transformation can be used to find H d (z). STEP4: Check the result to see if it meets the requirements. Note: In practice, the result is an approximation, and “prewarping” techniques are used to improve the result (page 545-546).

8. 10.5 Design of Digital Controllers As computer technology has become more economic, smaller, and more powerful, digital control of continuous-time systems has become very standard. Applications include engine controllers in automobiles, flight controls on aircraft, equipment control in manufacturing systems, robotics, climate control in buildings, and process controllers in chemical plants.

9. 10.5 (cont.) Digital control started being commonplace in the 1970s and 1980s. Theory of continuous-time control was mature at this time. The first methods for digital control was discretizing a continuous time controller. Figure 10.27 illustrates the block diagrams. More recently methods that map the continuous-time plant into the discrete-time domain and then designing the controller using methods related to the root-locus technique or transfer function techniques using Bode plots.

10. 10.5 (cont.) The analog emulation method is similar to the design of digital filters, by the use of analog prototypes. The bilateral transform can be used to map an analog controller G c (s) to a digital controller with transfer function G d (z) Another method is called response matching.

11. Response Matching Consider a continuous-time system with transfer function G(s), and let y(t) be the output resulting from input x(t) with zero initial conditions. The response matching problem is to construct a discrete-time system, G d (z), such that when the input is x[n], the output is y[n]= y(nT) = y(t)| t=nT where T is the sampling interval. Hence, if X(z) and Y(z) are the transforms of the discretized version of x(t) and y(t) then we have G d (z) = Y(z)/X(z).

12. Step-Response Matching Of particular interest in digital control is step- response matching, where x(t) is a step function. Step 1: If G(s) is the system function of the given continuous-time system, then the output y(t) can be found from the inverse of Y(s)=G(s)/s. Step 2: y(t) is discretized to get y[n]. Step 3: Y(z) is found from the transform of y[n]. Step 4: G d (z) = Y(z) (z-1)/z (recall u[n]↔z/(z-1))