What are Transfer Functions? | Control Systems in Practice - MATLAB
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    What are Transfer Functions? | Control Systems in Practice

    From the series: Control Systems in Practice

    Brian Douglas

    This video introduces transfer functions - a compact way of representing the relationship between the input into a system and its output. It covers why transfer functions are so popular and what they are used for.

    Published: 30 Aug 2022

    A transfer function is a compact way of representing the relationship between the input into a system, and its output. It's a mathematical model that captures the linear time invariant dynamics of a system. And in this video, I want to introduce the transfer function and talk about why it's so popular and what it's used for. So I hope you stick around for it. I'm Brian, and welcome to a Matlab Tech Talk.

    It's often helpful to represent real physical systems as mathematical models because models give us a way to systematically design, analyze, and simulate systems. And we have many different types of mathematical models to choose from and each capture information in ways that are useful for different situations.

    And as an example, let's look at the suspension system for a car. As the car drives it encounters bumps in the road which the suspension will absorb and damp out over time. And these bumps are the inputs, U of t, into the system, and then the vehicle's vertical motion is the output y of t. And even though this relationship between the bump and the motion of the car is driven by complex dynamics from the springs and the shocks and the tires and all of that.

    We can model the behavior with a differential equation. In fact, this fourth order differential equation models the linear time in varying dynamics for an arbitrary suspension system. And given an impulse input, this plot shows the solution of the differential equation, or the motion of the car over time. And with this data, we can ask questions like how does the system behave over time? We can easily determine how high the car travels, and how quickly it settles back to nominal. These are time based phenomenon.

    However, instead of keeping the model in this differential equation form, we can convert it to a transfer function using the Laplace transform. And I'm not going to cover the Laplace transform in this video, but if you'd like to learn about it, I've left some links below. But for linear differential equations like this, it's enough to know that derivatives with respect to time turn into exponents of the complex variable S. So y of t, which isn't a derivative, becomes s to the 0 times y of S and y dot of t would become s to the times Y of s, since it's a single derivative. And using this logic, we can transform the differential equation in the time domain into an equivalent transfer function in the complex plane, or the S domain.

    These two models capture the same system dynamics. However, with a transfer function, we describe the system as a function of s, instead of a function of time. And s is a complex variable, sigma plus j omega, where omega relates to the frequency of a signal and sigma relates to the exponential growth or decay of a signal. So if we have the input as a function of s, or we describe the bump in the road in terms of frequency and exponentials, then we can transfer it through the system by multiplying it with the transfer function. In this way, the transfer function is the ratio of the output over the input in the S domain.

    So this is good and all, but why would we choose to model a system in this strange form instead of a differential equation? Well, for one differential equations don't do well in series. Imagine this, instead of an end to end model of the suspension system with a single differential equation, what if we had one differential equation that modeled just the tires and then another differential equation to model the shocks in the springs?

    So the output v of t of the first equation is the input to the second. And if we wanted to see how an input u affects the motion of the car y, we would have to solve for the response of the first differential equation v and then can convolve it with the impulse response of the second differential equation, which I'm calling G. And the convolution integral is a rather math heavy operation.

    On the other hand, convolution in the time domain becomes multiplication in the S domain. Therefore to combine the equivalent two transfer functions in series, we just have to multiply them. And I think we can agree that polynomial multiplication is easier than convolution. So in this way, transfer functions are nicely suited for block diagram representations since combining blocks is as easy as multiplying them.

    But there are other good reasons to choose a transfer function to model your system, and one is that we get different insights into the system by looking at the S domain instead of just the time domain. Remember, S is a complex variable that consists of the real component sigma and the imaginary component omega. So all of the values of S can be displayed on 2D plane. So, for example, as equals minus 0.1 plus 0.2j would be right here.

    Now a transfer function is a function of s So we can calculate the value of this function s over s plus 0.3 for this value of s. This is analogous to plugging in different values of t in a time domain function. So for s equals minus 0.1 plus 0.j This transfer function has the result 0.25 plus 0.75j.

    Now to show you what the entire S plane looks like for this transfer function, let me rotate the axis so that I can plot the value along the vertical. And now I'm just going to cycle through lots of different s values across the entire plane. Here the red values are the real component of the result and blue are the imaginary components. And check this out, we're left with this kind of interesting landscape of peaks and valleys. And just like how we looked at information in the time domain to understand how the system behaves over time, we can now look at this information to understand how the system behaves over frequency and exponential motion.

    Of course, there's a lot of information here, and I'll be honest, looking at it this way is cumbersome and difficult to interpret. But luckily, we don't have to concern ourselves with most of these values because all of the information in this complex plane is captured by the gain of the transfer function and the locations of the poles and zeros.

    In a transfer function poles are the values of s that cause the function to go to infinity, and zeros are values of s that cause the function to go to zero. And gain is the ratio of the output and input under steady state conditions, or more easily, it's the value of the transfer function when you set s to 0. And so we really only need to know these three things, poles, zeros, and gain, and we can understand the entire s domain, which is really powerful for analyzing system dynamics.

    For example, we can use the s domain to determine if a system is stable by looking at the location of the poles. If any pole is in the right half of the complex plane, that is that sigma has a positive value, then the system is unstable. So for example, this transfer function has a pole at -0.3, and so it's stable. And this becomes really powerful when we have transfer functions in series, since we combine them through multiplication. If any transfer function is unstable, then the end to end system is unstable. We can also analyze stability in feedback systems by solving for the closed loop transfer function and checking the location of the poles.

    But analyzing a system is just one perk of transfer functions. We can also use them to design systems specifically, when we're concerned with frequency domain characteristics like we might be with filters and feedback controllers. We can add gain, and poles, and zeros to our system in such a way as to achieve a desired behavior.

    And something that I find really amazing is that we don't even need to find the closed loop transfer function to determine closed loop stability. We actually know a lot about the closed loop system just by looking at the poles and zeros of the open loop transfer function. This means we can design a closed loop controller simply by shaping the s domain characteristics of the open loop system. And this can be accomplished with different methods, like root locus, bode plots, Nichols, and Nyquist diagrams. And, once again, I've left a lot of great information below on all of these different types of plots and how to create them and use them for analysis and design in Matlab. So I encourage you to check them out.

    Now to close out here transfer functions at first might seem like an unnecessary deviation from time domain differential equations, but hopefully you can start to see how valuable they can be for control system analysis and design. They open up many different tools and techniques that we don't have access to with differential equations. They allow for easy manipulation in block diagram form, and they can provide some intuition about stability and system response just by knowing the locations of poles and zeros. And all of this makes learning about transfer functions and using them a worthwhile effort.

    All right, that's where I'm going to leave this video. If you don't want to miss any other future Tech Talk videos, don't forget to subscribe to this channel. And if you want to check out my channel, Control System Lectures. I cover more control theory topics there as well. Thanks for watching, and I'll see you next time.