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Fig.18.5 Step response functions of control systems (a – stable, b – unstable)




A system is stable if all the poles of the transfer function are in the left-hand s-plane.

A system is not stable if not all the roots are in the left-hand plane.

If the characteristic equation has simple roots on the imaginary axis ( -axis) with all other roots in the left half-plane, the steady-state output will be sustained oscillations for a bounded input, unless the input is a sinusoid (which is bounded) whose frequency is equal to the magnitude of the -axis roots. For this case, the output becomes unbounded. Such a system is called marginally stable, since only certain bounded inputs (sinusoids of the frequency of the poles) will cause the output to become unbounded.

For an unstable system, the characteristic equation has at least one root in the right half of the S-plane or repeated roots; for this case, the output will become unbounded for any input.

Answer: this control system is marginally stable ().

№7

7.2 Define the equivalent transfer function of feedback loop:

To increase your grade you have to explain how to obtain equivalent transfer functions using Matlab functions.

№8

8.2 Define gain-phase frequency characteristic of the link with the following transfer function .

This link corresponds to a differentiating link (Lecture 1.11).

№9

9.1 Define the value of M(t) signal after Z2(t) disturbance input signal is applied.

M(t)
Z1(t)  
Z2(t)

We take into account the superposition principle.

Z2(s) is an input signal, Y(s) is an output signal.

On condition that X(s) = 0 and Z1(s) = 0

№10

10.2 Define the transfer function

M(s)
Z1(s)  
Z2(s)

We take into account the superposition principle.

Z1(s) is an input signal, Y(s) is an output signal.

On condition that X(s) = 0 and Z2(s) = 0

10.3 Compare the characteristics of the 2nd order link before and after the introduction of the negative differentiating feedback loop:

z(t)
x(t)

Thus, we have the change only in dynamic characteristics (transient characteristics).

№11

11.2 Define the value of M(t) signal after Z1(t) disturbance input signal is applied.

M(t)
Z1(t)  
Z2(t)

We take into account the superposition principle.

Z1(s) is an input signal, Y(s) is an output signal.

On condition that X(s) = 0 and Z2(s) = 0

11.3 Compare the stability indicators for 2 control systems with the following characteristic equations

The control system with characteristic equation is unstable since one root is located in the right half plane of s-plane.

№12

12.2 Explain the construction of static load characteristic.





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