Best way to learn control theory for mechanical engineers

[JTC]

Hello,

I have begun to teach myself Control Theory.

I am looking for a book that is focused for mechanical engineers. I do not mind examples in electrical engineering, but they bore me (no offense).

Also, I find some books begin with Laplace Transforms. Yet I found this online lecture series that begins with Eigenvalues (which I like a lot):



Could someone please put these two different beginnings in context?
Why begin a book on Control Theory with Laplace Transforms?
Why begin a book on Control Theory with Eigenvalues?
Differences? Similarities?
And could you recommend a book (or an online youtube series) that teaches it for Mechanical Engineers?

 

[donpacino]

A few things.

Control theory is an interdisciplinary field. You NEED to be able to learn and use other disciplines. Let's say you are trying to design a control system to control the yaw of an airplane. Well you need the fluid dynamics to tell you what effect moving a fin will have on the system, you need to know what effects your feedback will have (these days sensors will be electrical), you need to know how your control system is implemented (these days digital electronics so you will have a discrete control system), and how your system is actuated (electrical/mechanical).

Laplace transforms are insanely important in control theory. They will allow you to look at problems in the frequency domain, which will give you new tools to solve control problems.

If you have learned about transfer functions, eigenvalues will help you get the poles of your system, which will determine your system dynamics.

 

[JTC]

A few things.

Control theory is an interdisciplinary field. You NEED to be able to learn and use other disciplines. Let's say you are trying to design a control system to control the yaw of an airplane. Well you need the fluid dynamics to tell you what effect moving a fin will have on the system, you need to know what effects your feedback will have (these days sensors will be electrical), you need to know how your control system is implemented (these days digital electronics so you will have a discrete control system), and how your system is actuated (electrical/mechanical).

Laplace transforms are insanely important in control theory. They will allow you to look at problems in the frequency domain, which will give you new tools to solve control problems.

If you have learned about transfer functions, eigenvalues will help you get the poles of your system, which will determine your system dynamics.

Thank you for your response, but I did not explain myself clearly...

I still don't see the "distinction" between doing control theory with eigenvalues vs. Laplace transforms. I am forced to guess... Once a system is linearized, one can use eigenvalues, and they are more useful, especially when it comes to controll-ability and observability. However, Laplace transforms can solve non-linear systems and ALSO give a view of things in phase space. Is that all it comes down to?With regard to a book, yes I understand you need circuit design but I am not interested in learning that. I am interested in learning enough about control theory to collaborate with someone. I don't want a book with circuits. Yes, the example you gave was fluids, but I consider that part of mechanical engineering. I am hoping for a tutorial style book for mechanical engineers that structures the field with regard to the solution methods above.

 

[donpacino]

Most control theory books are written from a systems perspective. They are going to have questions about chemistry, fluid dynamics, hydraulics, and mechanics. Refusing to read one because it has some circuits in it is foolish, and I don't know of any that don't have any circuity in them (not that i know every controls book). I'd go to google and type in control theory for mechanical engineers.

Laplace transforms and eigenvalues are two completely different things. Saying eigenvalues vs laplace transforms is like saying "should I drive a car to work" vs "I think I am going to eat shrimp for dinner." In fact it is likely that for a true start to finish controls problem, you will use both tools to solve the problem. where did you get the idea it was one or the other?

 

[JTC]

Most control theory books are written from a systems perspective. They are going to have questions about chemistry, fluid dynamics, hydraulics, and mechanics. Refusing to read one because it has some circuits in it is foolish, and I don't know of any that don't have any circuity in them (not that i know every controls book). I'd go to google and type in control theory for mechanical engineers.

Laplace transforms and eigenvalues are two completely different things. Saying eigenvalues vs laplace transforms is like saying "should I drive a car to work" vs "I think I am going to eat shrimp for dinner." In fact it is likely that for a true start to finish controls problem, you will use both tools to solve the problem. where did you get the idea it was one or the other?

Thanks for the first paragraph. I will try to look for ME specific, but if I have to deal with EE, so be it (no offense to EE people -- I just know what I like).

As for your second... yes, you hit the nail on the head. And I will ask you to reiterate so I can be sure.

I simply stumbled on that video series in the original post and it was exciting to learn. It was all eigenvalues, and I got that.
But the textbooks all begin with Laplace and that is fine, too.

I am just trying to understand the difference... Why are different tools are used? Why do books choose to open with one or the other?
In my own post previous to this one, did I contextualize the issue properly?

 

[donpacino]

Why are different tools are used?

For two reasons.
1. They solve different problems
2. They solve the same problem different ways, which may have advantages and disadvantages.
In this case they solve different problems

Why do books choose to open with one or the other?

It depends on what method of teaching the author wants to use.

 

[donpacino]

https://www.amazon.com/dp/0136156738/?tag=pfamazon01-20

here is the texbook I used in college. Its very readable, and has a lot of good MATLAB (a very useful program for controls engineering) examples. Matlab will help you a LOT. This textbook has a variety of examples and problems used. Yes it does have electrical stuff, but like most other books, it is in there to teach you controls engineering, not electrical stuff. Really any book that says it covers "state space" "frequency analysis" and "root locus" will be a good start.

for reference state space is considered modern design , root locus is older time domain based design, and then there frequency analysis which is older frequency domain based design.

Often they are all used to some extent.

note: you might want to buy an older edition book so it will be easier.

 

[FactChecker]

Classical control theory uses Laplace transforms to manipulate single input/single output signals and then combine them for multiple input / multiple output systems. Combining them is an art and the theory does not help in a systematic way. Simple combinations are easy: Suppose you want to obtain a good position estimate by combining GPS position signals and accelerometer signals. GPS is good for general location but does not respond instantly to small changes. Accelerometers are good for detecting instantaneous changes and you can integrate the signal to get instantaneous position changes, but the integral will drift off and errors will accumulate. By using a low pass filter on the GPS signal and a high pass filter on the integrated accelerometer (actually there is no need to integrate and then high pass, but that is a detail), you can combine the best of both signals to get a position estimate. That is called a "complimentary filter". This is all nice, but things get messy and it doesn't help much if there are a lot of interacting inputs and outputs within the system.

Modern control law theory uses the state space and eigenstructure of the linear combination of multiple inputs and outputs to analyze complicated systems. It shows all the state interactions. Some control laws actually translate the desired system behavior into a desired eigenstructure and force the system toward that overall eigenstructure. Usually a modern control system still has an eigenstructure part surrounded by classical Laplace transforms which pre-process the individual inputs and post-process the individual outputs. A lot of the inputs and outputs can be improved with simple Laplace transformations.

PS. I hope this explanation gives you a general idea. Please do not take anything here too literally. I am working from vague memory of long ago.

 

[donpacino]

sually a modern control system still has an eigenstructure part surrounded by classical Laplace transforms which pre-process the individual inputs and post-process the individual outputs.

A laplace transform just a method to convert time to frequency domain. Its used in both classical and modern control theory to help model and design a system. Its not something that's just used to pre-process inputs and outputs or combine signals or combine IOs. Any processing of real signals would likely be done with an FFT, not a laplace transform which is better for more theoretical paper based equation type calculations (the design and analysis stage, not implementation of an algorithm).

I don't think I would compare something like a kalman filter (the type of filter used to combine INS and gps systems) to a laplace transform, they're as different as eigenvalues vs laplace transforms.

 

[donpacino]

that being said overall that's a portrayal of some of the work and structure of control theory work that i'd agree with, outside of my nitpicky comments.

 

[FactChecker]

A laplace transform just a method to convert time to frequency domain. Its used in both classical and modern control theory to help model and design a system. Its not something that's just used to pre-process inputs and outputs or combine signals or combine IOs. Any processing of real signals would likely be done with an FFT, not a laplace transform which is better for more theoretical paper based equation type calculations (the design and analysis stage, not implementation of an algorithm).

I don't think I would compare something like a kalman filter (the type of filter used to combine INS and gps systems) to a laplace transform, they're as different as eigenvalues vs laplace transforms.

I have seen very complicated control systems for major programs implemented in the discrete Tustin transforms that correspond to Laplace transforms.

 

[donpacino]

a bi-linear transform (tustin) converts continuous domain to discrete domain, which is different than time to frequency domain (laplace). Although laplace can be done in the discrete time steps (with a z transform)

 

[donpacino]

note: I am guessing you were talking about a bi linear transform. I guess it is possible for a system to implement a laplace transform (I've done it in analysis scripts) but I can't think of too many ways it would be useful to implement in a control loop, unless you were doing automated system recognition and characterization of something.

 

[FactChecker]

a bi-linear transform (tustin) converts continuous domain to discrete domain, which is different than time to frequency domain.

Sorry. I think I used wrong terminology and should have said "z-transformation" when I said "Tustin transformation". The substitution of (2/T)(z-1)/(z+1) for s converts a Laplace transformation to a corresponding discrete equivalent. Those are the z-transformations that we used in discrete systems. Occasionally, the exact response to a particular frequency was required and a "prewarp" was done to match the discrete response to the required continuous response at that frequency. But that was very rarely required at the frame rates we used and the frequencies we were concerned about.

 

[donpacino]

ahhh that explains it, for op a z-transform is the descrete domain equivalent of the laplace transform. so you are right in the using it in control loop regards, although doing it with an actual laplace transform would require symbolic manipulations of a system which typically is not done. The same can be said for many things in the continuous domain though.

In general though, the Laplace transform (and z-transofrm) are used much more in the analysis and design of a system, and less in the implementation of a system. Adding the time to frequency and vice versa conversions into a control loop are rather advanced topics, even when you get into modern control systems.

 

[FactChecker]

ahhh that explains it, for op a z-transform is the descrete domain equivalent of the laplace transform. so you are right in the using it in control loop regards, although doing it with an actual laplace transform would require symbolic manipulations of a system which typically is not done. The same can be said for many things in the continuous domain though.

Once a library of common transformations is built up, it takes care of the majority of low-order transformations. If a transformation has higher orders of s and can't be factored, a symbolic manipulation can be done using available tools.

In general though, the Laplace transform (and z-transofrm) are used much more in the analysis and design of a system, and less in the implementation of a system.

I can only speak for the programs that I was involved in or reasonably familiar with.

 

[donpacino]

I can only speak for the programs that I was involved in or reasonably familiar with.

Well laplace transforms will be used in the analysis & design of most if not all programs. using them as an implementation tool is an edge case for very advanced systems

 

[donpacino]

Once a library of common transformations is built up, it takes care of the majority of low-order transformations. If a transformation has higher orders of s, a symbolic manipulation can be done using available tools.

Its not about the ability to do something. Its about the need to do it. If you are not preforming live characterization of a system you won't need to implement it as an algorithm in a control loop.

 

[FactChecker]

Well laplace transforms will be used in the analysis & design of most if not all programs. using them as an implementation tool is an edge case for very advanced systems

I've seen them converted directly to discrete z-transformations and implemented in several fighter aircraft digital flight controls.

That being said, I admit that they may be using other techniques in the most recent programs.

 

[donpacino]

I've seen them converted directly to discrete z-transformations and implemented in several fighter aircraft digital flight controls.

they are NOT used heavily in the actual control loop itself... z transforms and laplace transforms are used to derive frequency domain expressions that are then used in the design of the control system. Actually needing to implement a z transform in the control loop is rare, even with complicated systems. There is a difference between using a z transform to determine an equation, then solving a problem with a filter, and implementing a z transform in a control loop.

Using it in a loop is an edge case. If you talk about flight control systems you should know that the vast majority of controls work does not have laplace and z transforms in a controller. Even the kalman filter that you talked about earlier does not actually have a z transform in it (z transforms are used to help design the controller).

 

[FactChecker]

they are NOT used heavily in the actual control loop itself... z transforms and laplace transforms are used to derive frequency domain expressions that are then used in the design of the control system. Actually needing to implement a z transform in the control loop is rare, even with complicated systems. There is a difference between using a z transform to determine an equation, then solving a problem with a filter, and implementing a z transform in a control loop.

Using it in a loop is an edge case. If you talk about flight control systems you should know that the vast majority of controls work does not have laplace and z transforms in a controller. Even the kalman filter that you talked about earlier does not actually have a z transform in it (z transforms are used to help design the controller).

It was my job for decades up until I retired a couple of years ago, so I'll let that speak for itself. (You mentioned the Kalman filter, not I.)

 

[donpacino]

It was my job for decades up until a couple of years ago, so I'll let that speak for itself. (You mentioned the Kalman filter, not I.)

argumentum ad verecundiam...It's also my job
A kalman filter is essentially what you described in the beginning of post 8.

 

[donpacino]

I've worked on multiple aircraft programs that did NOT have any z transforms or laplace implemented in the control loop. there are way more efficient methods of extracting frequency information without the baggage a z transform or laplace will carry. need a z transform where an fft wouldn't do implies you are manipulating the underlying equations in a way that simply changing the co-efficents will not be able to handle, which is in itself an edge case, when compared to ops original question as to their use in control systems.

 

[FactChecker]

argumentum ad verecundiam...It's also my job
A kalman filter is essentially what you described in the beginning of post 8.

A complimentary filter and a Kalman filter have some similarities but are very different.

 

[donpacino]

A complimentary filter and a Kalman filter have some similarities but are very different.

100% agree with that. i really was just referring to a structure that combines multiple inputs

 

[FactChecker]

I've worked on multiple aircraft programs that did NOT have any z transforms or laplace implemented in the control loop. there are way more efficient methods of extracting frequency information without the baggage a z transform or laplace will carry. need a z transform where an fft wouldn't do implies you are manipulating the underlying equations in a way that simply changing the co-efficents will not be able to handle, which is in itself an edge case, when compared to ops original question as to their use in control systems.

As I stated before, I can only speak for the programs I was on and those I was reasonably familiar with.

 

[donpacino]

As I stated before, I can only speak for the programs I was on and those I was reasonably familiar with.

But as an experienced controls engineer, how can you not agree that it not at all common to have them in a loop. having it in the loop implies you need to on the fly do a task that an fft could not do

 

[FactChecker]

But as an experienced controls engineer, how can you not agree that it not at all common to have them in a loop. having it in the loop implies you need to on the fly do a task that an fft could not do

Because after all the design work is done, the frequency characteristics of the system at particular flight conditions (or in particular tasks) are known. It is only necessary to implement control laws that respond as desired to those frequencies. Flight control gains and other parameters are in tables that look up the required values for any particular flight condition. They give the z-transforms the desired frequency response. At that point, a FFT is not needed and would not help by replacing any part of it.

 

[FactChecker]

there are way more efficient methods of extracting frequency information without the baggage a z transform or laplace will carry.

A z or Laplace transformation in a flight control does not "extract frequency information". When given an input function of time, it naturally responds with specific gains and phase to frequencies in the input. That is what is needed in control laws to maintain adequate gain and phase margins and to give the desired system response.

Implementing a z-transform requires only a few arithmetic operations and it will apply the desired frequency response to an input signal. They are trivial. A FFT is thousands of times more computationally intensive. I guess the most extreme example is the single frame delay. For the given frame rate it has a known frequency response (approximately e^-sT, where T=time delay), which is often needed and the code requires no calculations at all:

output = prior_input
prior_input = input

Implementing general first or second order z-transformations only takes a dozen or so arithmetic operations. You can put many hundreds (thousands?) of them in a flight control to shape the behavior and still run in real time.

 

[paralleltransport]

Returning to the original post, I agree mostly with what was said. People like to say there's two related branch of control theory, one called classical control theory and the other one is "modern" control theory. Classical Control theory was developed in the 20's-30's with people like Nyquist, Bode etc... who were trying to gain intuition on simple control problems that were mostly SISO (single input, single output), like stabilizing an Op-Amp. Classical Control theory starts with the study of the transfer function of the system, which is the laplace transform of the impulse response. A SISO linear time invariant system has the property that sinusoidal inputs give sinusoidal outputs, hence why a frequency response characterisation is useful and why laplace/fourier transforms are sufficient. Modern control theory extends that study to more complicated systems, MIMO. Because of the many outputs and state variables, a good way to organize that information is in the form of systems of Linear ODE, hence why linear algebra is being used systematically (eigenvalues, and linear system theory).

Now here comes my opinion. Unless you are going to be a controls engineer (like you are tasked with developing new controls algorithm for things like underactuated robots, etc...), classical control theory is plenty to use already, and you'll likely not have to use modern control theory. Classical control theory is a bit less powerful, however, because it is simpler it can give you much more design intuition on loop gain shaping techniques. Rules like Root Locus, Nyquist Diagram, phase margin, gain margin, DC Loop gain, Bode Plots are all classical control tools I use on a daily basis to tweak control loops and improve their behavior. Rarely have to think about reachability/controllability etc.