What is the effect of a complex pole on control system stability?

[geft]

Let's say I have the following transfer function:

$$G(s)=\frac { s-1 }{ { s }^{ 4 }+2{ s }^{ 3 }+{ 3 }s^{ 2 }+{ 4s }+5 }$$

Which is run through MATLAB to obtain the pole-zero format:

$$G(s)=\frac { s-1 }{ ({ s }^{ 2 }+2.576s+2.394)({ s }^{ 2 }-0.5756s+2.088) }$$

Using a quadratic solver such as this one, both poles are found to be complex.

I still can't tell the difference between a pole and a zero in terms of system stability. From my understanding of poles and zeroes, roots that are located on the left hand side make the system stable while those on the right hand side make it unstable. Therefore, am I correct to assume that since the zero is 1, the system is unstable? And since the poles are complex, the system oscillates forever without reaching a steady state?

 

[RoshanBBQ]

Stability is determined by poles, not zeros. Poles on the complex axis make the system marginally stable, which means the system is unstable since a bounded input at the frequency on the axis will result in an unbounded output.

 

[D H]

Your transfer function has two poles that lie to the right of the imaginary axis, so it's unstable per the Nyquist criterion.

 

[geft]

I see, so the system is unstable because the poles are to the right of the y axis, making the response an increasing sinusoidal? Won't it be canceled to a degree by the other two poles which work to reduce the response?

If the zero has no effect on stability, what does it affect then?

 

[DeeNos]

The presence of complex poles in a control system can have a significant effect on its stability. In order to understand this, we must first understand the concept of poles and zeros in a transfer function.

Poles and zeros are the roots of the denominator and numerator, respectively, of a transfer function. They represent the values of s for which the transfer function becomes infinite or zero. Poles are also known as the eigenvalues of the system, and they determine the dynamic behavior of the system. Zeros, on the other hand, do not have as much influence on the system's stability.

In the given transfer function, G(s), it is important to note that the poles are complex. This means that they have both a real and an imaginary component. The real component of a complex pole determines the damping of the system, while the imaginary component determines the oscillatory behavior.

In general, a system with complex poles can exhibit oscillatory behavior, leading to a response that never reaches a steady state. This can make the system unstable and difficult to control. However, the exact effect of complex poles on stability depends on the position of the poles in the complex plane.

In this case, the poles are found to be complex and located in the left half-plane, indicating that the system is stable. The presence of a complex pole does not necessarily make the system unstable. It is the location of the pole in relation to the imaginary axis that determines stability.

In summary, the effect of a complex pole on control system stability depends on its location in the complex plane. In general, complex poles can lead to oscillatory behavior, but their impact on stability can vary depending on the characteristics of the system.