Beyond Control: Maths of a Control System

Control systems are all around us, and understanding them is going to make you much better at hardware design. In the last article — Beyond Control: The Basics of Control Systems — we looked at an overview of what a control systems are in general with the example: "everything in between water and time is a control system". We also observed control systems in nature, where I described my keen ability to fill a glass of water without catastrophic results. That discussion involved the basic concept of a block diagram (without maths) and we expanded that a bit to see what our satellite dish example would look like (still without maths).

I promised some big ugly maths in this article, and we'll get to that in a bit, never you fear. First let's have a look at how some basic elements: resistors, inductors, and capacitors are defined in the time domain. Don't let these first few definitions turn you off. No matter how you feel about calculus, you don't necessarily need to fully understand each equation. What's more important is how the equations themselves combine to solve the circuit. Also important is that I will do everything possible to get out of doing difficult math. So stick with me through the article and you'll learn that agony-saving trick for yourself!

A quick recap on transfer functions before we get going might be beneficial. A control system is used to define electromechanical behavior. For example: our satellite dish (from the previous article) at some point will need to be moved from one position to another position and as control engineers it is our job to determine just how this action will take place. I'm not talking about setting the mood for the big emotional robotic rotation, more like: not damaging the equipment or any people that might be nearby when moving the dish. For many reasons the dish would need to be moved with extreme care and in a very precise manner. The control system is the mathematical definition of that movement. Often the maths of the definition are nasty differential equations, (remember I'm avoiding any math that can be avoided, right?) so, instead of using differential equations to define the system, the transfer function will define the system with algebra, relating the output of the system to the input.

We aren't going to be grinding out anything difficult but if you're still worried about the calculus bits you should add [Will Sweatman]s Calculus is Not Hard series to your reading list.

Now let's get familiar with some passive transients, shall we?

Defining Passive Components Mathematically

A resistor is an element that can be used to alter voltage and/or current, serving a variety of purposes that quickly become obvious and familiar when designing and building circuits. Although this is not an all encompassing definition for resistors it does allow us to communicate with the rest of the world what it is we're using "that little one with the different colored stripes" for in our circuits.

If we wanted to explain this to someone with a mathematical background then we need to alter this definition so that it would be useful to them. For resistors this is not a problem, get really close to the mic and in a deep voice say "Ohm's Law" while making eye contact (mean mugging the kids are calling it), drop the mic and walk out.

In the case that you forgot, Ohms Law states:

That is voltage is equal to the product of current and resistance.

Defining energy storing elements is not quite as simple. Transients require a little bit of fancy footwork and a careful exit strategy to keep from stepping on our own tongue while getting out of the room alive.

Lets have a look at how each element is analytically defined in terms of voltage and current in the time domain:

Resistor

I have included the symbol for a resistor that should suit you no matter which side of the pond you reside on. The calculations we use for resistors are the very essence of Ohm's Law: V=IR or voltage is equal to the product of current and resistance. Fair enough, although it doesn't seem that spectacular the first 600 times you use it, at some point you start getting creative with Ohm's Law and it comes in handier than a spare $20 in your shoe. For our purposes we can think of the value a resistor represents as a constant, if you aren't up to snuff on your calculus: constants = easy like Sunday morning.

Inductor

This isn't quite as easy to explain to someone without a little calculus under their belt. For the most part if your response to those equations is something like "right, that's bananas. I don't like it, and I won't stand for it." then we'll be okay. At this point in our controls discussion we aren't required to work through any calculus. Believe you me, I don't want to explain it any more than you want me to. The point of including the calculus equations is to remind those of us who haven't had to work through a derivative (that's the one on the left with 2 many  d's) or an integral (the one on the right with big stupid ∫ with numbers on the wrong ends of it) in a while just how much we don't remember the mechanics of it.

Capacitor

A good eye will spot the similarities in the capacitor and inductor equations.

 

Time Domain Analysis

You can see how things can get mathematically hairy if we have a few capacitors (C) and inductors (L) in a circuit that needs to be defined analytically. Just to drive the point home, have a look at a basic series RLC circuit (I'm assuming you can make the leap of faith as to what the R represents).

The series RLC circuit shown above can be modeled using the time domain definitions that we previously discussed. The voltage equation for each component is what goes into the right side of the equation. When you get all the bits together it's basically an intimidating version of Ohms Law, as you can see below.

 

If you're breaking into a cold sweat as a result of that equation take a deep breath, step back, and let's talk about this. I said it was going to be a big hairy Ohm's Law equation remember? Take a look at the term on the left side of the equation and the first term on the right side and cover the rest with your hand, now ignore the (t) in both terms and what do we have? You got it, V=IR and the following terms are the definitions of voltage for each component. Adding the voltage of each component together results in the total voltage in the circuit for any time t (aka: voltage as a function of time). The symbol tau used in the capacitor term of the equation is a time constant as well. We won't get into how to find tau at the moment but I just told you this equation is a function of time and didn't want you calling me out on tau.

In order to evaluate this equation lets get rid of the integral by taking the derivative of the entire equation,  leaving us with a 2nd order differential equation. I realize that you weren't warned about the differential equations but just hold on a sec before you run to the comments to taddle, it'll be quick like a Band-Aid.

A differential equation (DFQ) can be solved in the time domain but look at that monster we've created, it must be a joke, right? Also keep in mind that we haven't replaced the RLC variables with the component values, this has the potential to get horrific. Replacing L with 20E-6 won't make the arithmetic any different, it will however cause it to look even more intimidating and most importantly: when I have to keep track of numbers on paper rather than symbols, mistakes will be made. Conclusion: lets attack in another direction.

 

S-Domain = Algebra instead of Differential Equations

As you can see from the differential equation we decided to steer clear of, this is complicated. To simplify things we can handle these nasty time domain equations in the s-domain. It might be easier to wrap your head around this concept if you think of the s-domain as the complex frequency domain. This allows us to solve complex calculus equations using algebra. But how can this be? Why would calculus be a thing if we can just algebra our way out of it? That's a great question: because Laplace. The Laplace Transform is how we get from the time domain to the s-domain. Not surprisingly we use the Inverse Laplace Transform to return to the time domain when we're done messing about in the s-domain.

The s-domain equivalent of a derivative is the complex variable s. Remember this substitution and things will start to make sense as we go from one domain to the other.

 

Laplace Transform

The general form of the Laplace Transform can be seen above and if you desire it can be used as described. We aren't going to need it in this article but it will come up again in a subsequent article when we go from the s-domain to the time domain. So here it is in all its glory for you to refer back to at a later date, but for now, we should keep moving.

Passives S-Domain equivalents

The s-domain will allow us to replace calculus with algebra. This can be done not by magic, but careful redefinition of our mathematical statements, expressions, and equations. Resistance in the time domain is equivalent to impedance (Z) in the s-domain so our Ohm's Law equation in the s-domain is:

Below you will find the s-domain equivalent impedance for each component. Derive them on your own if you wish, this article is already mathy enough for me without anymore proofs.

Resistor s-domain Impedance

 

Inductor s-domain Impedance

 

Capacitor s-domain Impedance

 

Control Systems, was it?

Meanwhile, back to the point: I bet you thought I forgot what we were even doing here? You're right, I got all caught up in multi-domain definitions of circuit elements but don't touch that dial because I'm about to shoe-horn some controls in here. Lets see how (if at all) this relates to a control system.

Lets find the Transfer Function relating the capacitor voltage to the supply voltage using the original RLC circuit.

We'll step through the process of each approach, the DFQ and transform.

 

Differential Equation with Mesh Analysis

Starting with the resistor and working our way around the loop in a clockwise rotation we have:

Using the relationship between current and charge:

We can substitute out every i in the equation. This results in a 2nd order DFQ:

We of course remember that the charge of a capacitor is equal to the product of the voltage across the capacitor and its capacitance:

We use the relationship to replace the charge q:

Take step back to observe the beautiful differential equation we have constructed, this is an emotional moment we're sharing right now. Dry your eyes and lets move from the time domain to the s-domain by using the Laplace Transform. By substituting the complex variable s for the derivatives and factoring out the voltage across the cap (Vc(s)) we have a much cleaner equation:

Finally, we can solve the equation for the ratio of output to input for our control system, resulting in the transfer function:

 

Transform Method with Mesh Analysis

If we redraw our circuit with s-domain equivalent impedances before creating the loop equation we have this circuit:

Now we write our loop equation the same way we have each time when looking at the schematic:

Solve for current over voltage:

Using the impedance equivalent of Ohms Law for our capacitor voltage:

We can solve for current and use the result to replace the current in our "current over voltage" equation:

We can solve the equation for the ratio of output to input for our control system, resulting in the transfer function:

 

Why did we arrive at the same equation twice?

I really hope this is rhetorical because it is SO exciting that this happened! I mean we TOTALLY avoided solving a 2nd order differential equation! We just blasted right thru the defense with nothing but algebra, made a few crossovers, one behind the back pass and straight dunked a transfer function like a boss! I don't play baseball but that sounds like an easy win to me.

 

Resulting Transfer Function

We know that the transfer function is the mathematical representation of the control system, so we can take our transfer function and put it in the block diagram of the control system:

 

What does it all mean?

The fact that this is very technical is not lost on me, so maybe I can shed some light by trying to bring this full circle: We can design an RLC circuit to be a notch filter for an AC signal, which is also known as a band-stop filter. This is useful for a variety of applications such as blocking pesky 60Hz noise. One way we can do this is by taking the output of our circuit across the inductor and capacitor, like so:

This is useful for blocking only part of an input signal. However we aren't blocking in the form of clipping the signal, which would be blocking any voltage over a specified value. We are blocking parts of the signal which are defined by a range of frequencies, in this case it would be the "stop band". We can define this range of undesirable frequencies by choosing inductor and capacitor values which create resonance at the center of the desired stop-band. I think we should forego any further maths for now as I've had my fill and I think you're picking up what I'm putting down at this point.

Conclusion

We covered how to create a transfer function from a simple RLC circuit. We defined passive elements in multiple domains. We used those definitions to find a simple RLC circuits transfer function. In the next installment of Beyond Control I plan to discus how we can find the time response from a transfer function and what some of its characteristics look like. We will also look at poles and zeros as a way to determine how a system will respond to various inputs and then I'll then explain what poles and zeros are, cheers!

Filed under: Hackaday Columns