# Modern method for teaching controls

### Background

When I took my first class in controls (2006-ish as taught by an aerospace professor for a mix of ME and EE students) it was basically all done in Laplace-transform, transfer functions etc.

More recently (2012) I took a graduate level controls class at a different school and it was almost all state-space. Really it was a bunch of abstract linear algebra proofs that happened to relate to observability and controllability. I chalked up the difference to the fact that it was for grad students who would be working on a more theoretical class of problems (almost no mention was made to relate this to any real system).

Now, from talking to undergrads at the same school, I'm given to understand that state-space is the way that control theory is currently taught. Laplace methods are briefly covered but quickly dismissed as out-dated.

I work in combustion and have no real idea what's going on.

### Questions

• Is this an accurate indication of the way that controls is being taught these days?
• Either way, does that match with the state/foreseeable future of controls?

I'm also interested to know the merits of one method over the other.

Disclaimer: I've studied control theory from a mathematics perspective, not an engineer's perspective.

Classical Control Theory is based on linear systems and is also limited to them as well. Linearization is helpful in many cases, but not entirely applicable in others. Analysis tools (laplace transform, pole placement, root locus, routh hurwitz, etc...) are easiest to conduct on linear systems.

Optimal Control Theory is the more modern approach because it can handle nonlinear systems directly without using the frequency domain. The idea is that control can be achieved by minimizing a suitably chosen cost function over time. Usually, this cost function is a function of the state space variables and is usually subject to constraints defined by the dynamics, boundary conditions, and/or feasible paths. The word "optimal" doesn't mean that its necessarily better or superior to classical control theory. Here, "optimal" means that control is obtained by finding a relative extrema (usually a minimum) of a function.

Depending on how the objective function is chosen, minimization can require either solving a linear or non-linear system of equations. Fortunately, many numerical/analytical tools from optimization and/or calculus of variations can handle such problems.

Of course, minimizing a non-linear function can be a rather costly endeavor from a computational point of view. If it is possible to analyze by some other means (i.e. classical control theory), then it may be more worthwhile to do so.

• Great answer. Minor nitpick, the singular form of the word "extremum" is appropriate in the last line of the third paragraph. One might also considering entirely replacing it for the lesser-used synonym "optimum", which would more directly highlight why optimal control theory is named what it is. Jan 17, 2022 at 16:58
• Curriculum differs between universities
• Engineering curriculum differs between countries
• Curricula are updated regularly to meet new accreditation requirements
• Sometimes there is a distinct difference between graduate and undergraduate programs from the same school
• Some professors are progressive and update courses more frequently to keep up with advances

Not too long ago I too followed a Linear Systems course in a graduate program. Based on your description it appears that I too might have had a similar experience. In this case I know for sure the Prof. had not updated the course for at least 10 years.

Recommendation: I suggest learning the modern methods to solve control systems problems. I also suggest that you become a member of respective professional society; in your case it might be ASE.

Finally, some solve engineering problems using brute force methods. Qualified, skilled professional engineers use skill acquired through university studies as well as knowledge gained from being part of professional organizations to efficiently solve engineering problems. Part of the important value an engineer with undergraduate as well as graduate schooling brings to an organization is the ability to quickly recognize and solve problems efficiently.