# continuous vs discrete systems in control theory

What are the advantages and disadvantages of using continuous or discrete systems for control? I'm going to list some that occurred to me, but would appreciate if people help complement the list. (If this question doesn't belong here, please let me know, I'll delete it).

• Data from the real world is discrete.
• Computers are discrete, hence they can more easily deal with discrete control systems.

• Sampling is not an issue.
• Hopefully this isn't your homework. They're just different. The usual advantage of "discrete" control systems are the advantages of digital systems in general -- they provide tremendous flexibility, easier engineering, and often lower parts cost as well due to more opportunities for integration. However, there are certainly times when the reverse is true. – Pete W Jan 26 at 3:43
• This isn't my homework :). It just seems to me that everybody knows why discrete systems are better, but it's not obvious to me. How do they provide flexibility? what kind of flexibility? – Schach21 Jan 26 at 4:00
• Is data from the real world really discrete though, or is it just that we are able to reduce it to discrete form in an adequate way to process? – K H Jan 26 at 5:05
• Why do you say that data from the real world is discrete? Temperature, velocity, position, rotation, time, voltage, current ... All continuous variables, as far as we can tell. – John Doty Jan 26 at 14:35
• @Schach21, that's true if we're sampling with digital circuits, but there are also ways to use continuous signals directly -- analog processing of a voltage signal is a classic example. Data from the real world, along with our use of that data, can be continuous. – Nate S. Jan 26 at 17:18

It is difficult to list all the pros and cons of analogue versus digital control but below are a few things I can think of:

1. an analogue compensation strategy, e.g. poles/zeroes placement, is fixed during the design phase and cannot be easily changed on the fly. With discrete control, you can alter the position of these frequency points while operating conditions change.

2. we all know that compensation components may be affected by tolerance and aging, naturally affecting the compensation strategy. Temperature and humidity also affect the components values in harsh environments: DSP-based compensators shield you against these effects.

3. a DSP or a µP brings intelligence capability with self-diagnosis, compensation strategy optimization to improve transient response in certain conditions. Some systems allow you to run a frequency analysis at power on and dynamically compensate the converter for steady-state operation. Some even compensate cycle-by-cycle by permanently keeping a constant phase margin regardless of the load nature (suddenly highly capacitive for instance). You can also easily modify operating parameters like the switching frequency to optimize efficiency with different loading profiles.

4. A DSP-based control usually uses less components around the chip than a classical op-amp-based compensator to which you add a voltage reference.

5. there is communication capability with a discrete-time system to signal problems or fault and report performance results.

6. the control bandwidth is naturally limited with discrete control: I know some dc-dc converters crossing over at 300-400 kHz with analogue control, something you could not easily do with a digital control.

7. delays, as pointed out by Neil_UK, can be a real plague and distort the phase margin while severely limiting your crossover selection.

8. the sensitivity of digital systems to noise when operated in proximity of power stages can be an issue in some cases, hanging the processor or affecting operation reliability.

9. and finally, the majority of power supply designers are well versed into analogue control and the Laplace transform while discrete-time engineers must be fluent in difference equations and $$\z\$$-transforms.

I remember attending a power supply manufacturers association (PSMA) conference 20 years ago where they predicted that digital control would soon be in all power supplies, including consumer applications. I can tell you that for high-volume applications like ac-dc adapters for notebook or power supplies for TVs, digital control (in the sense of what I described above) is not yet a reality and will have difficulty to be considering the low selling-price of complicated PWM controllers which became a commodity. However, it is already a reality for a long time in high-power converters for servers where one or several DSPs control the main converter and include the algorithm for power factor correction also.

There may be a false dichotomy in your question. I feel the literature you've been exposed to so far equates discrete with a digital implementation, and continuous with an analogue implementation. While that's generally true, it masks a more important difference.

In the 'bad old days' you only had analogue control systems and computers. Parameters were difficult to tune, usually by potentiometers and plug boards. Input and output data were continuous. Internally signals were usually handled continuously. They could be sampled, but it was rare.

Now, it's far cheaper to reach for an MCU, often with built-in ADCs and DACs, to do the same job. Parameters can be easily modified by changing a word in memory.

Although input and output data are still continuous, at least on the real-world side of the ADC and DAC anti-alias filters, the process of acquisition, control and output creates a latency. This latency is at least one period of the sample rate of the control system, and can often be much more.

This latency contributes a phase shift in the system feedback.

If the sampling period and the latency as much less than the period of the highest frequency significant time constant of the system, then the control is 'nearly continuous', and we can design and analyse the system with continuous theory with little error, representing the latency as merely an additional phase shift term.

If the sampling period or latency are significant compared to the system time constants, then continuous approximations are generally not good enough, and we have to use proper time domain theory and tools to design and analyse.

• Excellent point about the latency; as I like to remind the less experienced (who sometimes think digital == the only way), it doesn't matter whether you close the loop in hardware or software - it is still a closed loop that requires some form of compensation. – Peter Smith Jan 26 at 15:08

Pete W already provided good insight, let me add a few important:

The use of MCU is usually cheap, easy to implement, easy to modify. They are often the way to go except in some situations.

Speed: The big drawback of MCU based system is speed. If the control system needs to operate with a low response or high frequency, programmable systems quickly show their limits.

FPGA addresses that to some extent with programmable logic, to some extent, but they are complex to program and you are still limited by the ADC / DAC conversions.

I used to work for a company that was regulating the current on a Xenon flashlamp using a light detector and a PID system. This requires ns response time on the complete control loop and analog was the only way to achieve that.

Reliability: Another situation is where you want to have a highly reliable system, where MCU can sometimes bug or behave unpredictably.

This can happen either by software bugs, compiler bugs, programming that didn't go right, or processing errors (it does happen).

For instance, one of my designs has an analog timing control on one of the outputs of an MCU to protect the system in case of an MCU malfunction.

You can think of military, aircraft, medical, sometimes it's perhaps easier to pass certification with an analog system (you just measure it) than a software where each instruction of assembly code has to be deeply studied as well as the MCU itself being certified.

Simple System: Sometimes you need a control system that is fairly simple and it is better to use a discrete system as you do not need to program it, removing one step in the production and a software stack to maintain.

Design / Cost: If you need a fairly precise control system, handling small signals, designing a whole ADC -> MCU -> DAC chain can be tricky because of noise consideration, correct ADC drive, and so forth. High-performance ADC is expensive and you need a good front-end analog, which altogether can quickly be more expensive and complex than an analog solution.

There is certainly plenty more situation that can be described and it really depends on the specific case.

Probably the best answer so far. In both domains you're designing for stability and the placement of poles and zeroes. The criterion is mathematically equivalent with the Nyquist criterion in the complex plane of the Laplace transform being replaced by the unit circle in the Z plane of the discrete domain. The consequence of this is the following providing your analogue filtering for control in the discrete domain ensures that you never see aliasing and events are not so slow that they are outwith the sampling buffer you're using discrete systems offer benefits as noted above. For example traditional anything more complex than a three pole single roll off filter would have been nearly impossible to realise without manually adjusting or selecting components and even then stability would have been an issue. Today 15 or more pole filters are readily realisable in the discrete domain and we're only limited by the sampling rate of the of our ADCs and data rates of our DAC's. In the extreme case you could argue that digital analysis of stock market or volcanism data represents an example of discrete time analysis as it is possible to only look at pricing information on a daily basis. However note we still often reliant on continuous sensors i.e. a MEMs accelerometer or pressure sensor but we choose to sample that signal periodically because of the advances already noted. i.e Flexibility in the design, reliability component stability and accuracy and price.