Digital Control & Digital Filters. Lectures 1 & 2

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Kevin Russell
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1 Digital Controls & Digital Filters Lectures 1 & 2, Professor Department of Electrical and Computer Engineering Colorado State University Spring 2017

2 Digital versus Analog Control Systems Block diagrams of typical analog and digital (sampled-data) control systems are shown below. As can be seen the plant is still analog while the controller is replaced by a sampler (A/D converter), a digital controller (filter), and a hold device (D/A converter).

3 Why Digital Control? Benefits: Reduced Sensitivity and Robustness: Unlike analog controllers digital ones are not sensitive to environmental variations and aging. Better Adaptability: Parameters of digital controllers can more easily be adapted to the changes in a changing plant i.e. adaptive control strategies are more suited for digital control systems. Form Factor: Digital devices are more compact and lightweight comparing to their analog counterparts. Cost Effectiveness: Digital controllers are generally cheaper. More Reliable: Digital controllers are more reliable as their characteristics don t drift with time. Drawback: Quantization Effects: In designing digital systems one must be aware of the quantization effects (e.g., roundoff and truncation) due to finite word length of the processors.

4 Example Examples below show block diagrams of an autopilot system with position and rate feedback for: (a) analog control, (b) digital control, and digital control with multirate sampling for situations where the signals have different bandwidths.

5 Review of z-transform The role of z-transform to digital systems is similar to that of Laplace transform to continuous-time systems. Definition 1: The z-transform of a two-sided sequence {x(n)} is defined by X(z) = n= x(n)z n If {x(n)} is a right-sided sequence, i.e. x(n) = 0, X(z) = n=0 x(n)z n n < 0 then Example 1: Let x(n) = a n u s(n), where u s(n) is the unit step function defined as, { 1 n 0 u s(n) = 0 n < 0 Find X(z). Using n=0 an = 1 for a < 1, we have 1 a X(z) = n=0 an z n = n=0 ( a z )n = 1 1 a z if z > a

6 z-transform Thus the region of convergence (ROC) for this example extends from circle with radius a to. Example 2: Let x(n) = A sin(ω 0nT ), n 0. Find X(z). Using Euler formula, X(z) = n=0 Asin(Ω0nT )z n = A e jω 0 nt e jωnt n=0 z n 2j [ = A 2j [ n=0 ( ejω 0 T ) n ] z n=0 ( e jω 0 T ) n = A z 2j 2Az 1 sin(ω 0 T ) 1 2z 1 cos(ω 0 T )+z e jω 0 T z 1 ] 1 = 1 e jω 0 T z 1 exists when ejω 0 T z < 1 or z > 1 (since e jω 0T = 1) Thus, ROC extends from circle with radius 1 to similar to that of Example 1.

7 z-transform Remarks: 1 For a right-sided sequence ROC is outside a circle bounded on the inside by largest magnitude pole and on the outside by. 2 For a left-sided sequence ROC is inside a circle bounded on the outside by smallest magnitude pole and on the inside by 0. 3 For a two-sided sequence ROC is within a ring (between two circles) bounded on the inside by the pole with largest magnitude for n 0 (i.e. right-sided part) and on the outside by the pole with smallest magnitude for n < 0. 4 Note: ROC should NOT enclose a pole. a N >... > a 2 > a 1 b 1 < b 2 <... < b M

8 Properties 1. Linearity Let x 1 (n) z X 1 (z) ROC R 1 < z < R 2 x 2 (n) z X 2 (z) ROC R 3 < z < R 4 Then ax 1 (n) + bx 2 (n) Z ax 1 (z) + bx 2 (z) ROC R 5 < z < R 6 where R 5 = max(r 1, R 3 ) R 6 = min(r 2, R 4 ) Remark: If linear combination leads to pole-zero cancellation, ROC may be larger. For example, both x 1 (n) = a n u s (n) and x 2 (n) = a n u s (n 1) have ROCs z > a but x 1 (n) x 2 (n) = δ(n) has a ROC which is the entire z-plane because X 1 (z) = z z a and X 2(z) = a z a hence X 1(z) X 2 (z) = z a z a = 1, i.e. ROC is everywhere.

9 z-transform Properties-Cont. 2. Shift-in-Time Let x(n) z X(z) ROC R 1 < z < R 2 Then x(n n 0 ) z z n0 X(z) ROC R 1 < z < R 2 Thus, ROCs are the same except possibly at z = 0 or z =. To see this, consider example, x 1 (n) = δ(n) z X 1 (z) = 1 ROC is everywhere on z-plane x 2 (n) = δ(n 1) z X 2 (z) = 1 z ROC is everywhere except at z = 0 x 3 (n) = δ(n + 1) Z X 3 (z) = z ROC is everywhere except at z = Generalizations: [ x(n n 0 ) z z n0 X(z) ] 1 k= n 0 x(k)z k with IC s x( 1),.., x( n 0 ). [ x(n + n 0 ) z z n0 X(z) n 0 1 k=0 ]. x(k)z k

10 z-transform Properties-Cont. 3. Multiplication by an Exponential Sequence Let x(n) z X(z) ROC R 1 < z < R 2 Then a n x(n) z X(z/a) ROC a R 1 < z < a R 2 Thus, ROC is scaled by a. 4. Differentiation Let x(n) z X(z) ROC R 1 < z < R 2 Then nx(n) z z dx(z) dz ROC R 1 < z < R 2 i.e. ROC is unchanged. 5. Conjugate (Complex Signals) Let x(n) Z X(z) ROC R 1 < z < R 2 Then x (n) Z X (z ) ROC R 1 < z < R 2 i.e. ROC is unchanged.

11 z-transform Properties-Cont. 6. Initial Value Theorem If x(n) = 0 n < 0 (right-sided) Then x(0) = lim z X(z) 7. Final Value Theorem (FVT) If x(n) = 0 n < 0 Then lim n x(n) = lim z 1 (1 z 1 )X(z) = lim z 1 (z 1)X(z) Condition: As long as (1 z 1 )X(z) does not have a pole on or outside the unit circle. Note: FVT is very useful for steady-state error analysis in control systems. Example: For x(n) = sin(ωn) from Table 2.3, we have X(z) = Now, if we use FVT lim z 1 (1 z 1 )X(z) = lim z 1 z sin(ω) z 2 2zcos(Ω)+1. (z 1)(zsin(Ω) z(z 2 2zcos(Ω)+1) = 0.

12 z-transform Properties-Cont. This result contradicts with lim n = sin(ωn) =? due to the fact that the above condition for using FVT is not satisfied since (1 z 1 )X(z) has poles on the unit circle z 1,2 = e ±jω. 8. Linear Convolution If y(n) = x(n) h(n) where stands for linear convolution operation i.e. y(n) = x(n) h(n) = k= x(k)h(n k) = and x(n) z X(z) ROC R 1 < z < R 2 h(n) z H(z) ROC R 3 < z < R 4 k= x(n k)h(k) Then Y (z) = X(z)H(z) ROC max[r 1, R 3 ] < z < min[r 2, R 4 ] Note: If a pole that borders on the ROC of one of the z-transforms is cancelled by a zero of the other, then the ROC of Y (z) will be larger.

Your solutions for time-domain waveforms should all be expressed as real-valued functions.

Your solutions for time-domain waveforms should all be expressed as real-valued functions. ECE-486 Test 2, Feb 23, 2017 2 Hours; Closed book; Allowed calculator models: (a) Casio fx-115 models (b) HP33s and HP 35s (c) TI-30X and TI-36X models. Calculators not included in this list are not permitted.