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7.39 This problem is intended to give you more insight into controllability and observability. Consider the circuit in Fig. 7.92, with an input voltage u(t) and an output current y(t). Source (a) Using the capacitor voltage and inductor current as state-variables, write state and output equations for the system Figure 7.92 y(t) Electric circuit for Problem 7.39 + u(t) R2 (b) Find the conditions relating R1, R2, C, and L that render the system uncon- trollable. Find a similar set of conditions that result in an unobservable system (c) Interpret the conditions found in part (b) physically in terms of the time constants of the system (d) Find the transfer function of the system. Show that there is a pole-zero cancellation for the conditions derived in part (b) (that is, when the system is uncontrollable or unobservable)

7.39 This problem is intended to give you more insight into controllability and observability. Consider the circuit in Fig. 7.92, with an input voltage u(t) and an output current y(t). Source (a) Using the capacitor voltage and inductor current as state-variables, write state and output equations for the system Figure 7.92 y(t) Electric circuit for Problem 7.39 + u(t) R2 (b) Find the conditions relating R1, R2, C, and L that render the system uncon- trollable. Find a similar set of conditions that result in an unobservable system (c) Interpret the conditions found in part (b) physically in terms of the time constants of the system (d) Find the transfer function of the system. Show that there is a pole-zero cancellation for the conditions derived in part (b) (that is, when the system is uncontrollable or unobservable)

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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Feedback Control of Dynamic Systems (7th edition) - Problem 7.39

7.39 This problem is intended to give you more insight into controllability and observability. Consider the circuit in Fig. 7.92, with an input voltage u(t) and an output current y(t). Source (a) Using the capacitor voltage and inductor current as state-variables, write state and output equations for the system Figure 7.92 y(t) Electric circuit for Problem 7.39 + u(t) R2 (b) Find the conditions relating R1, R2, C, and L that render the system uncon- trollable. Find a similar set of conditions that result in an unobservable system (c) Interpret the conditions found in part (b) physically in terms of the time constants of the system (d) Find the transfer function of the system. Show that there is a pole-zero cancellation for the conditions derived in part (b) (that is, when the system is uncontrollable or unobservable)
Transcribed Image Text:7.39 This problem is intended to give you more insight into controllability and observability. Consider the circuit in Fig. 7.92, with an input voltage u(t) and an output current y(t). Source (a) Using the capacitor voltage and inductor current as state-variables, write state and output equations for the system Figure 7.92 y(t) Electric circuit for Problem 7.39 + u(t) R2 (b) Find the conditions relating R1, R2, C, and L that render the system uncon- trollable. Find a similar set of conditions that result in an unobservable system (c) Interpret the conditions found in part (b) physically in terms of the time constants of the system (d) Find the transfer function of the system. Show that there is a pole-zero cancellation for the conditions derived in part (b) (that is, when the system is uncontrollable or unobservable)
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