Linear and Time Invariant Systems

08/13/17

“What is the most ideal form for controls systems?”

There is a motley of types of control systems out there. So before we begin any sort of analysis, let’s start with the most simple form, known as Linear and Time Invariant Systems. LTI systems have three properties.

• Homogeneity If an input signal is scaled by a constant then the output will be scaled by the same constant

• Superposition If two unique inputs are summed together, then the sum of their outputs will be produced.

• Time Invariance The system will perform the same way no matter what the time is.

Unfortunately, Most controls systems are not LTI systems, but they are still important to study due to their easy to solve structure.

PID Control

08/12/17

“What is one model for a closed loop controller?”
Imagine a robot moving from one spot to another. If it was operating under a closed loop controller system, it would work by sensing the target location, comparing it to the current location and performing an error estimation. However, what is one way that we can implement this? Well, let’s begin with one idea; for every second we are not at our setpoint (destination) let’s take how far we are, take it as an error value, and put it on a graph. Then, let’s take the proportion (or magnitude of the error), integral (area under the graph) and derivative (current rate of change) and combine these values to estimate how far we are from our desired value. This type of control is known as proportional-integral-derivative control (or PID) and is implemented in control systems worldwide.

Bode plots

08/11/17

“How can we plot the gain and phase shift for a transfer function?”

A transfer function will change the magnitude and phase of a sinusoid in some way. So wouldn’t it be logical if we could plot this out on a graph? Well, let’s think about how we could do this ourselves. First, since we have to plot two different outputs (gain and phase shift) let’s put make two separate graphs side by side. Then, let’s put the input (frequency) on the x axis and the output (gain or phase shift) on the y axis. Now, since our input variable will cover an extremely large range, let’s make it on a logarithmic scale. Specifically, let’s take a frequency as an input, plug it into the formula 20log10(omega)), and then graph. Since the units on the x axis are not normal numbers but rather ratios, let’s give them the unit decibels. This type of plot is called a bode plot and is used for analyzing control systems worldwide.

Setpoints

08/11/17

“What is the desired output for an objective function?”
Objective functions in control systems work by trying to compare a measured value to a setpoint, or ideal value. If there is a discrepancy between these two values, then the system will give back feedback to rectify the error.

Impulse response

08/11/17

“How does a dynamic system respond to an external response?”

Dynamic systems are not isolated from the external world and are therefore subject to external inputs. The response of the system is known as an impulse response and can consist of any form of sinusoidal function.

Transfer functions

08/10/17

“What causes controls systems to change input to output?”
All controls systems have an input (usually some sinusoidal wave) and an output (the wave modified either through addition, multiplication, differentiation, or integration). However, what exactly causes this mathematical change? Well, the key behind this is something known as the transfer function. The transfer function H(s) is defined as the ratio of the output function Y(s) over the input function X(s) (H(s) = Y(s)/X(s)). If we rearrange this formula, then we can see that the input function multiplied by the transfer function is equal to the output function (H(s)*X(s) = Y(s)).

Poles and Zeros

08/10/17

“When does a transfer function go to zero or infinity?”

Transfer functions are usually made up of two polynomials, one in the numerator and one in the denominator. When the polynomial in the denominator (known as the pole) goes to zero, the transfer function will become infinitely large, while when the ones in the numerator go to zero, the function becomes a zero (hence the term zero for such functions). If a transfer function has more poles, then it becomes more unstable, while more zeros will make it more stable. Because of this, controls engineers try to maximize the pole-to-zero ratio.