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)

Question

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)

Image Transcription

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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