System Performance

Definition 14

First Order Systems

Definition 15

A first order system is one with the transfer function of the form

After applying partial fraction decomposition to them, their step response is of the form

Second Order Systems

Definition 16

Second order systems are those with the transfer function in the form

Notice that the poles of the second order system are

  1. Undamped

  2. Underdamped

  3. Critically Damped

  4. Overdamped

The Underdamped Case

If we analyze the underdamped case further, we can first look at its derivative.

Definition 17

Definition 18

Definition 19

Since our poles are complex, we can represent them in their polar form.

Additional Poles and Zeros of a Second Order System

Suppose we added an additional pole to the second order system so its transfer function was instead

Then its step response will be

Notice that

If we instead add an additional zero to the second order system so its transfer function looks like

and its step response will look like

Stability

Recall Equation 7 which told us the time-domain solution to state-space equations was

Definition 20

Following from Definition 20, Equation 7, this means that

Theorem 3

If all poles are in the left half plane and the number of zeros is less than or equal to the number of poles, then the system is BIBO stable.

Definition 21

Theorem 4

In all other cases, the system will be unstable.

Steady State Error

We want to understand what its steady state error will be in response to different inputs.

Theorem 5

The final value theorem says that for a function whose unilateral laplace transform has all poles in the left half plane,

Using this fact, we see that for the unity feedback system,

Using these, we can define the static error constants.

Definition 22

The position constant determines how well a system can track a unit step.

Definition 23

The velocity constant determines how well a system can track a ramp.

Definition 24

The acceleration constant determines how well a system can track a parabola.

Definition 25

The system type is the number of poles at 0.

This also brings another observation.

Definition 26

The internal model principle is that if the system in the feedback loop has a model of the input we want to track, then it can track it exactly.

If instead we have a state-space system, then assuming the system is stable,

Applying this to the state space equations for a step input,

Looking at the error between the reference and the output in the 1D input case,

Margins

Definition 27

The frequency response of the system determines how it scales pure frequencies. It is equivalent to the Laplace transform evaluated on the imaginary axis.

Definition 28

Definition 29

We can imagine the gain and phase margin like placing a “virtual box” before the plant as shown in Figure 6.

The characteristic polynomial of the closed loop transfer function is

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