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 ( Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru -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 Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru -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 Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru roots; for this case, the output will become unbounded for any input.

Answer: this control system is marginally stable ( Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru ).

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Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

7.2 Define the equivalent transfer function of feedback loop:

Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

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

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Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

8.2 Define gain-phase frequency characteristic of the link with the following transfer function Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru .

Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

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

Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

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Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

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

M(t)

Z₁(t)

Z₂(t)

Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

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

Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

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Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

10.2 Define the transfer function Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

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

Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

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

z(t)

x(t)

Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

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

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Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

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

Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

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

Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru

The control system with Fig.18.5 Step response functions of control systems (a – stable, b – unstable) - student2.ru characteristic equation is unstable since one root is located in the right half plane of s-plane.