Digital Control Systems 1.01

Digital control systems never did get enough attention during my Master System & Control in my opinion. In the various courses which I was taught, continuous-time control systems were always examined, but almost never digital control systems. For most courses this is also perfectly fine. However, you would expect that there would be also a more in-depth course about digital control systems into the curriculum. Especially, since almost all control systems are a digital control system.

I remember a course in which each group had to program a robot to convey pizza’s from one cabinet to another. A group mentioned, during their presentation, that they used a PID controller to control the various axis of the robot. I knew all the groups had designed their control algorithms in the continuous-time domain, therefore I asked; i) what discretization method(s) they had used, ii) if they had check the stability and iii) if they had checked the robustness (phase, gain and modulus margin) of their digital control system with respect to the continuous-time domain controller. Unfortunately none of my questions were answered. This disappointed me and was an acknowledgement of my opinion that not enough attention was given to digital control systems. Since this was one of the last courses given before one started their internship and graduation.

For the above reason, I will do a couple of blogs about digital control systems. I will try to focus on a single topic each blog and keep it clear and easy to understand; “If you can’t explain it simply, you don’t understand it well enough.”. In this first blog I will briefly go into the pros and cons of digital control systems and its analog counterpart. Followed by the introduction of the various type of signals that are present in a digital control system.

Pros and cons of digital control systems

Below is a list of pros and cons for both digital as well as analog control systems. Remark that there might be many more and that some pros or cons are subject to opinion, experience or interpretation.

Analog control Digital control
Pros
  • Robust; do not (often) break down
  • Continuous processing; no inherent bandwidth limits
  • Hard to modify; time-consuming, hardware needs to be rebuilt
  • Flexible; ability to create complicated controllers, easy to modify and fast to develop, less susceptible to aging and environmental variations
  • Robust; performance or safety can be monitored, errors can be dealt with appropriately
  • Diagnostics; on-the-fly testing, parameters can be adjusted, measurement signals can be stored
  • Low cost
Cons
  • Slow development; difficult to develop accurate designs due to tolerances in components
  • Complex: difficult to create complicated controllers, also hard to modify, testing alternatives is difficult, hard to build in intelligence, difficult to do multi-input-multi-output control
  • Disruptive noise
  • High cost
  • Discrete; signal might be inaccurate due to finite word length, time delays in control loop, discrete controller, stability might be an issue
  • Complex; complicated controllers, difficult to obtain a high bandwidth

While analog control systems have considerable disadvantages the flexibility of digital control systems stands-out. The latter made sure that companies adopted digital over their counterpart because it opened multiply doors. For instance, the rapid development of a complex controllers which incorporated optimization algorithms to maximize performance and profit. All created at the tenth of the costs for the same analog control system.

Types of signals in a digital control systems

The main disadvantage of digital control systems is found in the fact that one has to deal with discrete variables instead of continuous ones. Consequently, let us define the different types of signals. A Continuous-time signal is a signal for which every point in the time domain, this can be infinitely, an amplitude is defined. Similarly, a discrete-time signal is a signal for which only a sequence of points in time an amplitude is defined. Note that these points in time do necessarily have to be equidistant. In case it is, the signal has an associated periodic interval. Likewise to the domain, signals may also be continuous or discrete with respect to their range, i.e., amplitude.

Summarized; in general signals can either be continuous or discrete with respect to their domain or range. Herein, the domain is in general time, in case of control systems and the range can be anything; Voltage, Amperes, Newtons, meter per second, et cetera. Figure 1 shows the four possible combinations.

Figure 1. The four possible combinations of signal types.
Figure 1. The four possible combinations of signal types. Matlab code.

In Figure 1(a) an analog signal is displayed, which is both continuous in time and amplitude. Figure 1(b) shows a sampled signal; continuous in amplitude and discrete in time; Figure 1(c) the quantized signal, discrete in amplitude and continuous in time. Finally Figure 1(d), the digital signal. This signal is both discrete in time and amplitude. Remark: the exact definition of a quantized signal and sampled signal are often ambiguous. Often it is implicitly assumed that a quantized signal is also sampled, e.g., discrete in time. Likewise, the sampled signal is sometimes in explicitly assumed to be quantized as well. Lastly, the digital signal can be obtained by sampling and quantization of the analog signal.

We have read about the pros and con, and defined the different signal types present in a digital control system. Conclusively, I want you to point to the books in the references below, these are the best literature references in this field. I also use them as a reference for these blogs, and will try to point or refer to a chapter or section from the book when addressing a certain topic. In the next blog we will focus on the different system components of a digital control system.

References

[1] Levine, William S., ed. “The Control Handbook: Control System Fundamentals”. CRC press, 2010.
[2] Åström, Karl J., and Björn Wittenmark. “Computer-controlled systems: theory and design”. Courier Corporation, 2013.
[3] Franklin, Gene F., et al. “Feedback control of dynamic systems”. Vol. 3. Reading, MA: Addison-Wesley, 1994.
[4] Wittenmark, Björn, Karl Johan Åström, and Karl-Erik Årzén. “Computer control: An overview”. IFAC Professional Brief, 2002.
[5] Ogata, K. “Discrete-Time Control Systems”. Pearson Education, 1995.

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