y(t)= h(t)*x(t) where h(t) is a decaying exponential and x(t)= sin(5t) u(t). Find y(t) using convolution theorem. I'm confused about the sine wave. If i write sinusoid in exponential form then I get imaginary parts as well. can someone please help
Electronic – Convolution with sinusoids using convolution theorem
convolution
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I wouldn't go so far as to call PID outdated. But there certainly is room for improvement. One way in which I have auto-tuned PID control loops is to use the Nelder-Mead method which is a form of hill climbing simplex algorithm. It has the benefit of being able to converge and reconverge on a target parameter that moves over time.
From this paper:
For example in our case of PID parameters tuning {KP, KI, KD} a simplex is tetrahedron. Nelder–Mead generates a new test position of simplex by extrapolating the behavior of the objective function measured at each test point arranged as the simplex. The algorithm then chooses to replace one of these test points with the new test point and so the technique progresses.
My particular application was for motor control. We had two loops, a PID current control loop and a PI velocity control loop. We set our vertices to P, I, and D respectively and ran statistics on the output of the loop. We then ran the reflection, expansion, contraction, and reduction over and over again until the current or velocity control targets generated were within a few standard deviations.
With our product, the VP was very concerned with how the motor "sounded". And as it turned out, it "sounded" better when the current target bounced a bit more than was mathematically optimal. So, our tuning was done "live" in that we let the algorithm seek while the motor was running so that user's perception of the motor sound was also taken into account. After we found parameters that we liked, they were hard-coded and not changed.
This probably would not be ideal for you since you state, "putting the system in oscillation even as a part of auto-tuning is not acceptable to the users". Our system would most certainly oscillate and do other horrible things while it was auto-tuning.
However, you could run two copies of the PID controller. One that was "live" and actually controlling the process. And a second that was constantly being auto-tuned while being fed the same inputs as the "live" controller. When the output of the auto-tuned controller became "better" or more stable, you could swap the coefficients into the "live" controller. The controller would then perform corrections to the process until the desired performance was achieved. This would prevent oscillations that can be perceived by the user during auto-tuning. But if the inputs change drastically and the PID controller is no longer optimal, the auto-tuning can swap in new coefficients as they become available.
Convolution basically tells you how similar two signals are as one of them is shifted in the x-axis and reflected. So consider taking your signal \$g\$ and shifting it by various amounts. The convolution will have a peak when the two signals are mirror images across the y-axis. In this case, they already are, so the peak convolution occurs with a shift of 0.
The most important thing to understand is that the meaning of the x and y axes have changed from (time, value) to (shift, alignment).
I've ignored the meaning of the y-axis magnitude. When I've used the convolution before it's been with normalized signals so that the values ranged from 0 to 1, but that is certainly not always the case.
Best Answer
Hint: You have to combine the resulting complex exponentials into sine and cosine terms:
$$\sin x=\frac{e^{jx}-e^{-jx}}{2j}\\ \cos x=\frac{e^{jx}+e^{-jx}}{2}$$
If you use \$h(t)=e^{-at}u(t)\$, you should end up with the expression
$$y(t)=\frac{1}{a^2+25}\left[a\sin(5t)-5\cos(5t)+5e^{-at}\right]u(t)$$