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Problem 5 Consider the system of transfer function H(z) 1-2¹ +az-2 1+2-¹+az-²2* where a is a real parameter (with sign). Give the exact range of values of a for which the system is stable. Warning: For certain values of a, the poles of the system are real and for other values of a, they are complex. Hint: When the poles are real, you may wish to use the inequality |x ±y| ≤ |x| + |y|.

Problem 5 Consider the system of transfer function H(z) 1-2¹ +az-2 1+2-¹+az-²2* where a is a real parameter (with sign). Give the exact range of values of a for which the system is stable. Warning: For certain values of a, the poles of the system are real and for other values of a, they are complex. Hint: When the poles are real, you may wish to use the inequality |x ±y| ≤ |x| + |y|.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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Problem 5: The poles A = 1-4a Case 1: Azo poles: ?= -1± √I-4a' 2 |d₂| = 2/² | -1 ± √√₁-4a| ≤ = (1 + √1-4a) = |d-1 + So the system is stable IFF Id_|<1 1λ=1 <1 1+√T-49 <2 Case 2. A<O d+ = The system are O 0 Poles : -1±j√=A 2 | d + 1² = ²/₁ (1 + 4a − 1) = a Conclusion: real poles. S V/ stable ^ -15 the roots of H='+az? -15 4 |d₂| = 2/²2 | -1 ± √₁-4a|≤ 1/2 | = + is stable IFF √1-4a < 1 + (−1+ j√42-1) So the system is stable IFF lakı complex poles stable (1+√1-4a) 0<a<1 1-49 <1 a> o
Transcribed Image Text:Problem 5: The poles A = 1-4a Case 1: Azo poles: ?= -1± √I-4a' 2 |d₂| = 2/² | -1 ± √√₁-4a| ≤ = (1 + √1-4a) = |d-1 + So the system is stable IFF Id_|<1 1λ=1 <1 1+√T-49 <2 Case 2. A<O d+ = The system are O 0 Poles : -1±j√=A 2 | d + 1² = ²/₁ (1 + 4a − 1) = a Conclusion: real poles. S V/ stable ^ -15 the roots of H='+az? -15 4 |d₂| = 2/²2 | -1 ± √₁-4a|≤ 1/2 | = + is stable IFF √1-4a < 1 + (−1+ j√42-1) So the system is stable IFF lakı complex poles stable (1+√1-4a) 0<a<1 1-49 <1 a> o
Problem 5 Consider the system of transfer function H(z) = 1-z-¹+az-2 1+2-¹+az-2' where a is a real parameter (with sign). Give the exact range of values of a for which the system is stable. Warning: For certain values of a, the poles of the system are real and for other values of a, they are complex. Hint: When the poles are real, you may wish to use the inequality |xy| ≤ |x| + |y|.
Transcribed Image Text:Problem 5 Consider the system of transfer function H(z) = 1-z-¹+az-2 1+2-¹+az-2' where a is a real parameter (with sign). Give the exact range of values of a for which the system is stable. Warning: For certain values of a, the poles of the system are real and for other values of a, they are complex. Hint: When the poles are real, you may wish to use the inequality |xy| ≤ |x| + |y|.
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