1. 1 ENGG 1015 Tutorial Systems, Control and Computer Arithmetic 11 Dec Learning Objectives  Analyse systems and control systems  Interpret computer arithmetic News  HW3 deadline (Dec 3)  Project Competition (Dec 5) Ack.: MIT OCW 6.01, 6.003

2. System Function Write an expression for the system function for this whole system, in terms of n 1, d 1, n 2, d 2, n 3, d 3 2

3. Difference Equations 3

4. Find the Pole(s) Let.. Determine the pole(s) of H 3 and the pole(s) of. 4

5. Finding Equations and Poles For k = 0.9 5

6. Conversion between Block Diagrams (1) The system that is represented by the following difference equation y[n] = x[n] + y[n − 1] + 2y[n − 2] can also be represented by the left block diagram. It is possible to choose coefficients for the right block diagram so that the systems represented by the left and right block diagrams are “equivalent”. 6

7. Conversion between Block Diagrams (2) 7

8. Feedback (1) Let H represent a system with input X and output. Assume that the system function for H can be written as a ratio of polynomials in R with constant, real-valued, coefficients. In this problem, we investigate when the system H is equivalent to the following feedback system where F is also a ratio of polynomials in R with constant, real-valued coefficients. 8

9. Feedback (2) Example 1: Systems 1 and 2 are equivalent when Example 2: Systems 1 and 2 are equivalent when Which expressions for F guarantees equivalence of Systems 1 and 2? 9

10. Feedback (3) Let E represent the output of the adder. Then 10 E

11. What’s Cooking (1) Sous vide ("under vacuum") cooking involves cooking food at a very precise, fixed temperature T (low enough to keep it moist, but high enough to kill any pathogens). In this problem, we model the behavior of the heater and water bath used for such cooking. Let I be the current going into the heater, and c be the proportionality constant such that Ic is the rate of heat input. The system is thus described by the following diagram: 11

12. What’s Cooking (2) 12

13. What’s Cooking (3) Let the system start at rest (all signals are zero). Suppose I[0]= 100 and I[n]= 0 for n>0. What is the plot when k 1 = 0.5 and k 2 = 0? What is the plot when k 1 = 1 and k 2 = 0.5 ? 13

14. Personal Savings (1) You and your friend Waverly have accounts in rival banks. Each month, your bank deposits your interest from last month into your account, leaving your new balance equal to α times your old balance. Waverly’s bank is similar but the constant is γ instead of α. Each month, you make an additional deposit (into your account) of x[n] dollars plus β times the balance in Waverly’s account from last month. Each month, Waverly withdraws (from her account) δ times the balance in your account from last month. 14

15. Personal Savings (2) We wish to describe the balances in these bank accounts as a linear system. Let y[n] and w[n] represent last month’s balances in your account and in Waverly’s account, respectively. Let x[n] represent the input to the system, and let w[n] represent the output. Determine a system function to describe the relation between the signals X and W. (The system function should not depend on Y.) 15

16. Personal Savings (3) 16

17. Personal Savings (4) Determine if Waverly’s balance oscillates and diverges  α = 0.1, β = 0.5, γ = 0.1, δ = 0.5 Oscillates over time; Magnitude converges  α = 1.1, β = 1.1, γ = 1.1, δ = 1.5 Oscillates over time; Magnitude diverges  α = 0.5, β = 0.1, γ = 1, δ = 0.1 Not oscillates over time; Magnitude converges  α = 1.5, β = 0.1, γ = 1, δ = 0 Not oscillates over time; Magnitude diverges 17

18. 18 Computer Arithmetic (1) Convert the following decimal values to binary: a) 205 b) 2133 Perform the following operations in the 2’s complement system. Use eight bits (including the sign bit) for each number. a) add +9 to +6 b) add +14 to -17c) add +19 to -24

19. 19 Computer Arithmetic (2) Convert the following decimal values to binary: a) 205 b) 2133 205 10 = 1 x 2 7 + 1 x 2 6 + 1 x 2 3 + 1 x 2 2 + 1 x 2 0 = 11001101 2 2133 10 = 1 x 2 11 + 1 x 2 6 +1 x 2 4 + 1 x 2 2 + 1 x 2 0 = 100001010101 2

20. 20 Computer Arithmetic (3) Perform the following operations in the 2’s complement system. Use eight bits (including the sign bit) for each number. a) add +9 to +6b) add +14 to -17c) add +19 to -24

21. 21 Overflow Overflow: Add two positive numbers to get a negative number or two negative numbers to get a positive number For 2’s complement, (+1)+(+6) = +7  OK (+1)+(+7) = -8  Overflow (-1)+(-8) = +7  Overflow (-6)+(+7) = -1  OK

22. 22 Addition using 2 ’ s Complement (1) Perform the following computations.  Indicate on your answer if an overflow has occurred. 01000000 + 01000001(64 + 65) 00000111 − 11111001(7 - -7)

23. 23 Addition using 2 ’ s Complement (2) 01000000 (64) + 01000001 (65) ---------------- 10000001 (-127)  Overflow 00000111 (7) + 00000111 (7) ---------------- 00001110 (14)  No Overflow 00000111 - 11111001 = 00000111 + (-11111001) = 00000111+00000111