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Roots (20 points). Consider the loop-gain transfer function L(S) = TS-a)n-m where n and m are...
Q5) Consider the following closed-loop control system Error к G(s) Let G(s) = -, a >...
Q5) Consider the following closed-loop control system Error к G(s) Let G(s) = -, a > 0. Find the range for K, in terms of a, such that: • 5/95 rise time does not exceed 2 seconds • The 3% settling time does not exceed 5 seconds
2. (15p) We shall consider a function A, defined by the recurrences A(0,n n+1 for n...
2. (15p) We shall consider a function A, defined by the recurrences A(0,n n+1 for n 20 for m>0 for m, n > 0 A(m, n) A(m-1, A(m, n-1)) = Observe that A(1,1) = A(0,A(1,0))=A(0,2) = 3 A(1,2 A(0, A(1, 1)) A(0,34 and it is now not hard to see (as can be proved by an easy induction) that A(1,n)n 2 for all n 20 1. (5p) Calculate A(2,0), A(2,1), A(2,2), and A(2,3) Then state (you are not required to...
Consider the plant P(s) = (s3)(s22s + 17) in a feedback loop with a gain K...
Consider the plant P(s) = (s3)(s22s + 17) in a feedback loop with a gain K > 0. Sketch the root locus. By applying Routh's criterion to the system in Problem 1, find the range of K > 0 such that the system is asymptotically stable. Sketch the approximate Bode plot for the plant in Problem 1. (Please exaggerate the features so that it is clear if you have understood the procedure.) Sketch the Nyquist plot for the plant in...
Problem 3. (15 points) Consider the feedback system in Figure 3, where G(s)1 (s -1)3 Ge(s)...
Problem 3. (15 points) Consider the feedback system in Figure 3, where G(s)1 (s -1)3 Ge(s) G(s) Figure 3: Problem 3 1. Let the compensator be given by a pure gain, ie, Ge(s)-K, K 0 (a) Draw the root locus of the compensated system (b) Is it possible to stabilize the systems by selecting K appropriately? If so, find the range for K such that the closed-loop system is BIBO stable. If not, explain. 2. This time, let the compensator...
Let us consider the biased random walk S,-X1 +X2+···+X,, with S: is a sequence of independent...
Let us consider the biased random walk S,-X1 +X2+···+X,, with S: is a sequence of independent randon variables with P(X,--1)-g,P(X.-1) p+q=1 and Mt Show that M.-Sn-b- calculate Elr), where τ = inf{n : S" = a or-6) with a, b > 0. 0. where Xi, X2 p. where g)n is a martingale. Use this martingale to
Consider a random sample of size n from a distribution with pdf (In O* S(x; 6)...
Consider a random sample of size n from a distribution with pdf (In O* S(x; 6) = Ox! x = 0, 1, ...;0 > 1 10 otherwise (a) Find a complete sufficient statistic for 8. (b) Find the MLE of O. (c) Find the CRLB for 6 (d) Find the UMVUE of In e. (e) Find the UMVUE of (In )? (1) Find the CRLB for (In 02
Give a context free grammar for the language L where L = {a"bam I n>:O and...
Give a context free grammar for the language L where L = {a"bam I n>:O and there exists k>-o such that m=2"k+n) 3. Give a nondeterministic pushdown automata that recognizes the set of strings in L from question 3 above. Acceptance should be by accept state. 4. 5 Give a context-free grammar for the set (abc il j or j -k) ie, the set of strings of a's followed by b's followed by c's, such that there are either a...
4. (20 points) Consider the demand function D(p;m) mP, where > 0 is consumers' average income....
4. (20 points) Consider the demand function D(p;m) mP, where > 0 is consumers' average income. The supply consists of a monopoly, whose revenue from sales is given by R(p;mpD(p;m) (a) (5 points) Compute the elasticity function, E(p;m)-D'(p;m) b) (5 points) Find the value of p such that E(p; (c) (5 points) Compute the marginal revenue function, MR(p; m) R'(p; (d) (5 points) What is the solution to the equation MR (p 0? Compare 1 your answer to your answer...
s. (15 pts)Consider all possible funtions B be a function where |4 - m and lB-n...
s. (15 pts)Consider all possible funtions B be a function where |4 - m and lB-n (a) How many such functions are possible? (b) If men, how many different bijections (1-1 and onto functions) are possible? If men, how many 1-1 functions are possible? m: [0,1,小 (c) η= [0, 1, 2, 3 J 6. (5 pts) Does |(0, 1)-(1,00)| ? If yes, give an explicit bijection. If no, demonstrate why not N because the y
8. [10 points) Consider the following algorithm procedure Algorithm(: integer, n: positive integer; 81,...a s integers with vhilei<r print (l, r, mı, arn, 》 if z > am then 1:= m + 1 if za...
8. [10 points) Consider the following algorithm procedure Algorithm(: integer, n: positive integer; 81,...a s integers with vhilei<r print (l, r, mı, arn, 》 if z > am then 1:= m + 1 if za then anstwer-1 return answer 18 and the (a) Assume that this algorithm receives as input the numbersz-32 and corresponding sequence of integers 2 | 3 1 1 4151617| 8| 9 | 10 İ 11 İ 12 | 13 | 14|15 | 16 | 17 |...
Q5) Consider the following closed-loop control system Error к G(s) Let G(s) = -, a > 0. Find the range for K, in terms of a, such that: • 5/95 rise time does not exceed 2 seconds • The 3% settling time does not exceed 5 seconds
2. (15p) We shall consider a function A, defined by the recurrences A(0,n n+1 for n 20 for m>0 for m, n > 0 A(m, n) A(m-1, A(m, n-1)) = Observe that A(1,1) = A(0,A(1,0))=A(0,2) = 3 A(1,2 A(0, A(1, 1)) A(0,34 and it is now not hard to see (as can be proved by an easy induction) that A(1,n)n 2 for all n 20 1. (5p) Calculate A(2,0), A(2,1), A(2,2), and A(2,3) Then state (you are not required to...
Consider the plant P(s) = (s3)(s22s + 17) in a feedback loop with a gain K > 0. Sketch the root locus. By applying Routh's criterion to the system in Problem 1, find the range of K > 0 such that the system is asymptotically stable. Sketch the approximate Bode plot for the plant in Problem 1. (Please exaggerate the features so that it is clear if you have understood the procedure.) Sketch the Nyquist plot for the plant in...
Problem 3. (15 points) Consider the feedback system in Figure 3, where G(s)1 (s -1)3 Ge(s) G(s) Figure 3: Problem 3 1. Let the compensator be given by a pure gain, ie, Ge(s)-K, K 0 (a) Draw the root locus of the compensated system (b) Is it possible to stabilize the systems by selecting K appropriately? If so, find the range for K such that the closed-loop system is BIBO stable. If not, explain. 2. This time, let the compensator...
Let us consider the biased random walk S,-X1 +X2+···+X,, with S: is a sequence of independent randon variables with P(X,--1)-g,P(X.-1) p+q=1 and Mt Show that M.-Sn-b- calculate Elr), where τ = inf{n : S" = a or-6) with a, b > 0. 0. where Xi, X2 p. where g)n is a martingale. Use this martingale to
Consider a random sample of size n from a distribution with pdf (In O* S(x; 6) = Ox! x = 0, 1, ...;0 > 1 10 otherwise (a) Find a complete sufficient statistic for 8. (b) Find the MLE of O. (c) Find the CRLB for 6 (d) Find the UMVUE of In e. (e) Find the UMVUE of (In )? (1) Find the CRLB for (In 02
Give a context free grammar for the language L where L = {a"bam I n>:O and there exists k>-o such that m=2"k+n) 3. Give a nondeterministic pushdown automata that recognizes the set of strings in L from question 3 above. Acceptance should be by accept state. 4. 5 Give a context-free grammar for the set (abc il j or j -k) ie, the set of strings of a's followed by b's followed by c's, such that there are either a...
4. (20 points) Consider the demand function D(p;m) mP, where > 0 is consumers' average income. The supply consists of a monopoly, whose revenue from sales is given by R(p;mpD(p;m) (a) (5 points) Compute the elasticity function, E(p;m)-D'(p;m) b) (5 points) Find the value of p such that E(p; (c) (5 points) Compute the marginal revenue function, MR(p; m) R'(p; (d) (5 points) What is the solution to the equation MR (p 0? Compare 1 your answer to your answer...
s. (15 pts)Consider all possible funtions B be a function where |4 - m and lB-n (a) How many such functions are possible? (b) If men, how many different bijections (1-1 and onto functions) are possible? If men, how many 1-1 functions are possible? m: [0,1,小 (c) η= [0, 1, 2, 3 J 6. (5 pts) Does |(0, 1)-(1,00)| ? If yes, give an explicit bijection. If no, demonstrate why not N because the y
8. [10 points) Consider the following algorithm procedure Algorithm(: integer, n: positive integer; 81,...a s integers with vhilei<r print (l, r, mı, arn, 》 if z > am then 1:= m + 1 if za then anstwer-1 return answer 18 and the (a) Assume that this algorithm receives as input the numbersz-32 and corresponding sequence of integers 2 | 3 1 1 4151617| 8| 9 | 10 İ 11 İ 12 | 13 | 14|15 | 16 | 17 |...
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