# System Control - Proportional Gain and a Delay in Series

• Logio
In summary: I'll just need to do some more research on the phase margin and gain margin.In summary, the controller has a phase margin of 75 degrees and can add up to X degrees of phase lag before instability is reached.
Logio
Hi,

Just wondering if anyone could give me a few pointers in the right direction for this... thanks!

## Homework Statement

Incredibly long problem statement so I'll summarise and ask more about the concept:
I have a Plant and Controller set up in the form of a usual control system. The controller is a proportional gain in series with a time delay.
a) What is the Laplace transform of this controller.
b) Draw a bode plot (using a computer package)
c) Estimate the phase margin
d) How much extra time delay can be added to the control loop before instability.
e) Using the bode plot, draw a Nyqiust diagram.

N/A

## The Attempt at a Solution

a) Transfer function = k exp(-iD) where D is the time delay in seconds
b) Done on a computer
c) Phase margin read off the graph as the phase between -180 and the phase for the frequency with 0db gain (fairly confident this is accurate)... ANSWER = 75 degrees (for a given k and D)
d) Not sure where to start... I suspect I could consider another series delay of duration say, P, and then use some form of analysis to find P.
e) I know the phase margin and gain margin from the bode plot... is this enough to plot the Nyq. diagram or can I get more information simply?

Many thanks for any pointers on this!

the Nyquist diagram really only needs the BODE plot.

what you are trying to do is plugging in values of s into the open loop transfer function corresponding to a large semicircle surounding the positive right half plane.

What this essentially ends up being is starting at low frequency s=0 and increasing s intil you get to very high frequencies, plot the vector represented by G(s).

that is, start at w=0. look at the magnitude of the open loop transfer function at (or close to) w=0 (call it A). now look at the phase of the open loop TF at (or close to) 0 (call it theta). now on your nyquist diagram, mark the point represented by A*exp(i*theta).

do this again for higher and higher frequency values. they should converge to a single point eventually.

that represents half the graph, the other half (representing negative values of s) is the mirror image of that across the real axis.

also, the MATLAB command nyquist(tf) is useful.

Thanks for that - it makes sense.

If I have a phase margin of say X degrees, then X degrees of phase lag can be added before instability. How can I process this to calculate the amount of extra time delay the system could have before instability?

Thanks

## 1. What is proportional gain and how does it affect system control?

Proportional gain is a measure of how much a system's output changes in response to a change in its input. It is a parameter used in control systems to adjust the strength of the control action. A higher proportional gain results in a stronger control action, meaning the system will respond more aggressively to changes in its input. Conversely, a lower proportional gain results in a weaker control action and a more gradual response.

## 2. How is proportional gain determined for a system?

The optimal proportional gain for a system depends on the specific characteristics of the system. It is usually determined through a process called tuning, where the proportional gain is adjusted and the system's response is observed. The goal is to find the proportional gain that results in a stable and efficient response to changes in the system's input.

## 3. What is a delay in series and how does it impact system control?

A delay in series refers to a time delay between the input and output of a system. This delay can be caused by various factors, such as processing time or physical limitations. A delay in series can affect system control by creating a lag in the system's response to changes in the input. This can result in instability and poor performance if not properly accounted for in the control system.

## 4. How do proportional gain and delay in series work together in a control system?

In a control system, the proportional gain and delay in series are two separate parameters that need to be considered together. The proportional gain affects the strength of the control action, while the delay in series affects the timing of the response. The optimal values for both parameters need to be determined through tuning in order to achieve stable and efficient control of the system.

## 5. Can proportional gain and delay in series be adjusted in real-time?

Yes, proportional gain and delay in series can be adjusted in real-time in some control systems. This allows for the system to adapt to changes in its environment or input, ensuring optimal performance. However, proper tuning is still necessary to determine the initial values for these parameters, and real-time adjustments should be made carefully to avoid instability in the system.

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