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There is a lot of overlap but some differences in emphasis. Control Engineering is also older than DSP. If you have a traditional EE education, you don’t really make much of a distinction.

State variables are the more typical perspective in Controls. The first edition of Oppenheim and Schafer 1975, had a chapter on state variables, but they dropped it over the years. You need to understand state variables to do Kalman Filtering which is an area of overlap. Linear Estimation and Linear Controls are duals of each-other.

I would also say that hybrid continuous/discrete time systems are more common in Controls but there are many examples for DSP as well.

DSP is almost always done on uniform sampling. State Variables can work with nonuniform sampling as well.

I’ve never heard of anti causal Control System but forward backward filtering in time is common in DSP. Controls are inherently causal. The one sided Laplace transform is more common in controls.

Stability in feed back loops is important in both areas. An advanced control systems class will cover topics like Lyaponov stability. You typically don’t see that covered in DSP but there are DSP papers that use that technique.

Control Theory shows up in mechanical engineering. DSP shows up in finance. There is a-lot of both in robotics which also uses computer vision.

In RADAR, waveforms and filtering are more DSP at the front end, but the tracking systems at the back end are more Controls like.

If I had to use a single word to describe each.

Controls: feed back

Signal Processing: sensing

or maybe using a phrase

Controls: in-the-present

DSP: in-the-groove

I did my signal processing Ph.D. in a control systems department. My take is that signal processing is open loop; control systems close the loop.

Apart from that, the mathematics behind both are very similar. It's the applications that are generally very different.

Both draw on Linear System Theory (a.k.a. "Signals and Systems"). So also does Communications Systems and Linear Electric Circuits, Electronic Circuits,and Distributed Networks (a.k.a. Transmission Lines).

Both worry about system stability. Poles have to be inside the unit circle. DSP is actually broader than either Controls or Communications.

Control Systems usually is more interested in time-domain behavior; impulse response and step response. Routh-Hurwitz criterion (or its discrete-time counterpart) and Root-Locus techniques is something that Control guys worry about. I have never really worried about it.

It used to be that State-Variable systems was in the Controls purview, but ever since the Kalman Filter, I've seen State-Variable representations (with the A, B, C, D matrices) appear more often in DSP.

Many DSP problems outside of Controls are less concerned about time-domain behavior and more concerned about frequency-domain behavior.

Image Processing is more closely related to DSP than it is to Controls.

I dunno of the Controls guys worry at all about the FFT and such.

All of these disciplines have a practical end that becomes Electronics. Worrying about how DSP or CPU chips are hooked up to A/D and D/A converters and to memory and to other peripherals. I dunno how much Controls guys worry about quantization error, but they should.

There's a fairly simple distinction.

Signal processing is a set of tools that can be used for control engineering.

Control engineering is about making something move how you want it to move. Some of the tools of signal processing will help with that (and some won't; backward filtering doesn't happen in real-time without a TARDIS).

Signal processing is largely concerned with frequency response (gain), because that's most of what affects what you hear. Phase and group delay are issues, but often not the major ones.

In control engineering though, you generally want something to move to a position and then not move. In doing this, there's a fundamental principle - if you can't see it, you can't correct it. If your position measurement is filtered in ways which delay the measurement badly, the control loop doesn't know where it is (or doesn't get that information fast enough) and so can't move appropriately. Or worse, if it gets the information too late then it may even try to move in the wrong direction.

So control engineering tends to use filters like Butterworth which may not do such a good job of filtering, but which have much more benign effects on signals. Or it may not even use filters at all, because noise on signals may not affect the movement of the system if you have a slow control loop or a system with a lot of inertia.

The best textbook I know of is Modern Control Engineering by Ogata. I can thoroughly recommend that. It stops just short of state-space control, but for most control work you're rarely going to need that.

Control engineering are often taught in similar or even same courses of study, up the masters' degrees. In the general system modeling approach, where inputs ($I$) and outputs ($O$) are related through systems ($S$), I would say that, for a target $O$, they either work on $S$ or $I$:

• 
  control engineers tend to put (strong) contraints on outputs of a system, and are dedicated to find inputs that meet the contraints
  


• 
  signal processing people to tend put (strong) expectations on outputs, and strive to find systems that convert inputs appropriately.

As a consequence, their tools are very similar, and it is like they sometimes use them is a dual manner. Even if their backgrounds are very close, I have noticed some difficulties in their intercommunication. To some extend, this situation reminds me of George Bernard Shaw's:

The United States and Great Britain are two countries separated by a
common language.

Hence, signal/image processing and control engineering are two close disciplines, separated by a set of common tools.

• The requirement, for causal, real-time system implementations (where time is the independent parameter) that continuously minimize an output error with respect to a reference criterion, distinguishes the control systems discipline.




• You could search MIT Open Courseware, such as https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-30-feedback-control-systems-fall-2010/




• The free MATLAB workalike Scilab (https://scilab.org) provides access to many proven libraries supporting control systems design and analysis.




• Python's NumPy and SciPy (https://scipy.org) can substitute for Scilab, if you prefer, while SymPy (https://sympy.org) can help with symbolic (computer algebra system) manipulations. Anaconda Jupyter notebooks (https://anaconda.org) will allow you to document your development with Markdown typesetting and LaTeX expression rendering, together with interactive code and output blocks.




• To render signal flow graphs, which frequently summarize control systems, you might use Graphviz (https://graphviz.org).




• Roger Labbe explains Kalman filters very effectively: https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python The estimated system state is the object of control for a Kalman filter.