# Meaning of time parameters in control theory

I know in control theory there are different time parameters, but I'm not sure why they are used and defined that way, with my main doubts being about rise time; definitions from here:

• Delay time (td) is the time required to reach at 50% of its final value by a time response signal during its first cycle of oscillation.

• Rise time (tr) is the time required to reach at final value by a under damped time response signal during its first cycle of oscillation. If the signal is over damped, then rise time is counted as the time required by the response to rise from 10% to 90% of its final value.

• Peak time (tp) is simply the time required by response to reach its first peak i.e. the peak of first cycle of oscillation, or first overshoot.

• Settling time (ts) is the time required for a response to become steady. It is defined as the time required by the response to reach and steady within specified range of 2 % to 5 % of its final value.

As far as I can understand:

• delay time gives an idea on how fast is the system to start.

• rise time I guess gives an idea on how fast is the system to get to the final output value, so you know it's at least oscillating around (or it's near for under damped systems) to the final value, but I'm not clear why a different definition for under and over damped systems; and why in the latter case is from 10% to 90% instead of 0%-90% or 0%-100%. I read somewhere, but unfortunately I don't remember where, that it starts from 10% to ignore initial transients, but also if that's is true I'm not sure why stopping at 90%.

• peak time is quite straightforward, but I'm not sure why it's useful to know how long it takes to have the first peak, given all the other defined times.

• settling time I think is how long the systems takes to - well - settles, to stop its transient (and I guess this is useful in addition to rise time because a system could be fast to reach 90% of the exit, but slow to settle?).

Am I missing something? Why that unusual definition for the rise time ("unusual" as in, there are two definition, and while all the other starts at 0% one of those starts at 10%).

The definitions for timings can vary among various sources but, as I know them:

1. Correct, the lower the td the faster the system responds;
2. It's considered from 10%-90% for consistency; you may also see 20%-80% and the reasons may include noise, or very high order systems with complex poles/zeroes that make the rise time "wobbly" but, still, consistent for all cases. It doesn't make sense to discriminate for the underdamped response since it can be difficult to know where the final settling time is without measuring it first. And this tr tells about the bandwidth of the system: the faster the rise time the more bandwidth it has;
3. The peak tells about the damping, for 2nd order systems, at least (3rd orders and higher have various dampings for each of their consituent 2nd/1st order stages so, it's a bit more complicated);
4. The settling time may even be defined as low as 1% of the total time and it tells about the real part of the pole, for 2nd order systems (for 3rd+, it's more complicated, like in point 2., above), since for the 2nd order the exponential curve defining the envelope of the response (impulse or step) is directly related to $$\\text{e}^{\Re(p)t/2}\$$. It also tells you about the time when the system can be considered to be in steady-state, or how long you have to wait until it can be considered so.

[edit, given the comments below]

since rise time is measured 10%-90% to avoid initial noise, why are the other times from 0% (like delay time, 0%-50%)?

For td, it's not so much the 0%-50% part as it is the time when the output reaches half the value. E.g. for a step input (0...1 V), when does the output reach 0.5 V? If the system was an inverter then it would have been from 100%-50%. If it were as you say, from 10%, the final value would have still been 50% since that is the half value. So it doesn't matter where you start off, 0%, 10%, 27.4%, etc, it's the half-value that matters.

For tp, noise or not, it has a fixed value given by the transfer function, therefore it doesn't make sense to talk about this as measuring the time when it goes from 0% to max, just the maximum value, for which either some derivative is needed, or some peak detection.

ts is similar to tp, in that you need to wait until the output has settled, you don't know when, but when it does that's the time for that particular value. Don't forget that, in this case, you can't just settle for the first value that falls under the threshold, at least not for the under-damped case; for the over- and critically-damped, sure, but under-damped will oscillate and you need to measure the two, consecutive oscillations that don't rise above the threshold, then take the earlier measurement.

For all the three cases above, the values that were required to be measured had nothing to do with the slope, which is the case for tr. That's why tr needs two values between which the measurement needs to be considered, and they need to be sensibly chosen. 0%-100% can't even be theoretical since the outputs are asymptotic in all three cases. You could argue with limits but, a limit is until infinity, and you're not Chuck Norris (are you?).

[edit 2]

Let me try again: no measuremetns can be known a priori, it's why you're measuring. This means that your only reference is the input and, since for this case it's a step input, you know for sure that t=0 is the moment when you applied the step. Or the 0%, if you really insist in using percentages. And all the measurements are referenced to this initial time point since you're measuring timings.

Now, for tp you need to wait for the moment when the derivative (or the impulse response) crosses zero the first time and is negative. This can be done since it only applies to the under-damped case and the measuring is done without problems.

For ts you need to wait until the output goes below the threshold and stays there. This applies to all the X-damped cases since all of them are dependent on the $$\\mathrm{e}^{-t}\$$ envelope, even if the under-damped case wobbles (see my initial answer).

For td you need to measure the output when it reaches half the maximum. E.g. if the input is a step of 1 V and the gain of the system is 3 then you're measuring the 1.5 V point. Or 50%, if you wish. Again, this can be done since there aren't any special considerents for any of the three cases.

But for tr, ideally, by using the 0% reference, this time, because you're talking about a slope, the end point has to be 100%. You can't use 0%-90% (or 0-50, or anything that's not 100%) because the response is nonlinear so there will be errors in the readings. Note that the symmetry is around the 50% point. But no system ever reaches 100%. The under-damped does but only because it oscillates: it reaches 100% then goes above, then comes back, reaches 100% again, goes below, returns, etc, but the 100% must be reached and the system must stay there, which doesn't happen. Even if some would, but others wouldn't, it wouldn't make sense to use different trip points for some but not for others. Therefore the convention was chosen to have 10%-90% (or whatever other limits), to be able to measure all three systems equally, while preserving the symmetry around the 50% point. And it's around this half-point because it's a slope: imagine a pin in the middle point, around which the slope can be rotated.

I don't know what else to say. It really looks like you're stuck on percentages, it would do you good to avoid being too rigid and try different points of view.

• Thanks! Why measuring starts from 10% only for rise time? I'd think that if the noise is an issue in identifying when the system leaves 0%, the same issue holds true for all the defined time, not just for the rise time. Sep 3, 2022 at 11:16
• @Mauro Read point 2 again. I never said that it's one way for one case and another way for some other case. It's consistent for all to be considered 10%-90%. Can you post a reference for where you read that? For my part, I haven't seen it described from 10%-100%. Maybe there was an additional explanation in there that acted as an exception (which only strengthens the rule)? Sep 3, 2022 at 19:02
• Your reference does not say 10%-100%, anywhere. It says: "is the time required to reach at final value by a under damped time response signal during its first cycle of oscillation". The translation is a bit poor, but nowhere does it say 100%, it says to reach at [a?] final value. That value is chosen to be 90% (or 20%-80%, it's not a rule, it's a guideline). If they did mean 100% then they are not very good at this. The derivations in there are very nice, though. Sep 4, 2022 at 7:16
• @Mauro I've updated my answer again. I'm not sure what else to say. Sep 5, 2022 at 8:29
• @Mauro I'm glad I could help, there's nothing to be sorry about. And thank you for the extra points but, even if it's late now, maybe it would have been better to have waited to see maybe there are other answers. At any rate, if the purpose is achieved then confetti. Sep 5, 2022 at 16:27

The 10%-90% definition of rise time is for measurement practicality.

If you are going to make a well-defined time measurement, you need a signal changing fairly quickly. If you try to make a 0%-100% measurement, you can never be sure when the trace has left the 0% value, or (in an over-damped system) arrived at the 100% value. An oscilloscope trace may have several percent of full scale noise on it. Choosing 10% away from the start and end points allows for noise, and gives time for the waveform to get slewing.

To me, the oddity is the 10%-100% choice in an under-damped system. It meets the waveform slew-rate test for a well underdamped system, but I'm not sure how you'd make the decision in the boundary case between a critically damped system, and one with only a few percent overshoot. The measurement limit would jump between 90% and 100% for barely any change in the system behaviour. If you were making measurements on these boundary cases, you would need to stick with 90% for consistency, and state this when presenting the results.

• Make sense, thanks; then why the other times and rise time for under damped systems are defined from 0%? Aug 29, 2022 at 14:46

Did you miss anything? Briefly;

1. Time delay ( from a step pulse )

• where do you measure?

• correct, usually at 50% amplitude of nominal levels? unless the detector uses a different threshold like TTL or has hysteresis. (; But you don't always know where 0% starts with one waveform, as you said, so there must be an accurate start reference.
• why? - because time delay can be from propagation delay or risetime from stored energy or an unknown order of exponential delays - depending on the linearity or dynamic range, delay might be defined to any threshold that is important to error correction.

• what about sine waves?

• the rate of change of phase is called group delay $$\\tau_g(\omega)=d \phi/d\omega\$$. Some control systems need to know this property from different distortions caused by filtered spectrum at the band-edge.
• what about baseband data?

• There is a common property of jitter called Inter-Symbol-Interference that degrades SNR, eye pattern and Bit Error Rate. In modern differential lines like 1Gbps, the control system equalizes the frequency amplitude and phase response with test patterns in order to pre-compensate for the timing errors in order to maximize bandwidth of data to make it error free.
• If this is new to you , imagine a transition has not settled to target voltage before the next transition occurs. That residual voltage will affect the expected delay at 50% or even 10 to 90%. Why can it change? When one is using most of the available bandwidth unlike simple logic signals, delay and jitter becomes critical to control the data recovery. If the channel delay is different for different frequencies or pulse widths e.g. T/4, T/2, 3T/4, T or 1f 1.5f 2f or 1T, 1.5T, 2T then ISI occurs. Choosing the right filter such as raised cosine or using digital equalization with CM chokes are some ways of minimizing error in the recovered clock to sample the data in a PLL with skew and jitter-induced errors on the control clock error signal.

• also the time delay to the 1st oscillation cycle is usually longer than the stable cycles, unless it has exactly the same initial condition on power up or enable. This is always true for Schmitt Trigger oscillators from 50% hysteresis.

• also keep in mind logic chips are simply analog amplifiers that limit with logical inputs and loosely-controlled switching thresholds with some small shoot-through currents from cross-conduction of complementary drivers. So when analyzing ringing or glitches, in a control system, keep in mind mis-matched driver impedances (too low) and too long inductive scope ground clips feeding a coaxial capacitance.

2. Rise time

The frequency response for a 1st order filter, or scope bandwidth relates directly to the 10 to 90% time delay $$\\tau _d= 0.35/f_{-3dB}\$$.

You can correlate this with the asymptotic time delay of 0 to 63% if you have the step input to compare to find the start, but it's a shorter delay time than the 80% delta from 10~90%.

3. peak time This is greatly affected by the phase shift of harmonics, so it's not a common parameter.

4. settling time

The most critical settling time used every day is on your hard disk drive. The % tolerance for settling time is measured for a fixed % percentage of track space position error signal or PES. At one time the threshold was PES = 5% or 10% But this depends on the degradation of timing margin from adjacent track interference (ATI). The strength of the magnet can vary in direction at the ends and temperature so usually this is calibrated on startup. The worst thing that can happen is with insufficient damping overshoot the track after detecting on track (Seek End) then overshoot > 10% during a coincidental write sector. These are rare but must be avoided by a perfect servo control system under a wide range of magnet strengths, gap, and temperature in the fastest possible speed with no overshoot.

Settling time increases greatly with propagation time with stiction or hysteresis or time delay in a heating control system. So predictor correction filtering is common by differentiation or sensing velocity and acceleration when regulating position. This converts 3 order delays into 2nd or 1st order control feedback for an easier feedback filter to avoid the group delays from integral delays.

These are just some random thoughts that may be of interest in how to specify a system with signal error specs.

p.s. your link is an excellent resource for the theory of derivation of transfer functions with Q or $$\1/2\zeta\$$ and all EE subjects with clear examples.

## Exec. Summary

• use 50% for most delays

• Use 0 to 63% to measure rise time $$\\tau\$$ = RC or = L/R

• for most signal risetime, use 10% to 90% T=0.35/ f(-3dB)

• exception: Use Slew Rate for current limited risetime into a given load pF