# How do I get a faster system response?

I have this model in simulink (the graph is my output): The step input has amplitude 0.5 m/s, and it steps up after 0.1 seconds. The gain $$K_p=5$$.

The saturation block is to keep the voltage between -3 and 3 volts.

The transfer function is the system, and it is given by $$H(s) = \frac{16.94s + 579.5}{s^2 + 507.2s + 1224}$$

The integration block is to convert velocity (in m/s) to position (in m). So the output is position.

I want my system response to be as fast as possible, and with no overshoot. On the graph above, an overshoot can be seen. To fix this, I added a $$C_d=\frac{0.4s+1}{0.04s+1}$$ term in the feedback loop. As you can see there is no longer any overshoot. But it still takes about 3 seconds for the output to stabilize around the desired value.

How do I make the system response faster, such that the desired output value is reached sooner? Can I add a block in my simulink model perhaps?

## Edit

Putting the $$C_d$$ block in the forward path sped up the response, but not by much. About 1 sec. • If Simulink has a model for the Holy Graal, add it, otherwise, there's always a trade-off between the system's response and its bandwidth: too fast a response means higher bw which invites instability, and low bw means slow response. No free lunch. Mar 9 at 17:55
• You don't show a differentiation block -- there's an integrator in there (simulating the motor?) but no differentiator. When you say "The transfer function is the system" -- well, no, the whole system is described by the whole block diagram. What does that transfer function actually represent? Is it your motor and driver, just the motor, or what? Mar 9 at 17:59
• @TimWescott Oops, my bad I got the input and output mixed up. It's actually velocity that is the input and position that is the output, I edited the question. As for the transfer function, it represents the relation between input voltage and the wheel velocity of my robot. So $H(s) = \frac{\text{output}}{\text{input}} = \frac{v_{velocity}}{V_{voltage}}$. I hope what I have written makes sense, if not, I can elaborate.
– Carl
Mar 9 at 18:05
• why is your compensator in the feedback?
– Ben
Mar 9 at 18:48
• Yes, putting it in the forward path should speed up the response. Mar 9 at 19:11

The transfer function is $$H(s) = \frac{16.94s + 579.5}{s^2 + 507.2s + 1224}$$

This transfer function has 2 poles, one slow pole at -2.4248 and a fast pole at -504.7752. The function has a slowish zero at -34.2. Good news, your poles and zero are all in the left-half plane. It is much easier to control a system with zeroes and poles in the left-half plane than in the right-half plane. Your slow pole at -2.4248 will limit your performance. Your slowish zero at -34.2 will probably limit your performance by creating some overshoot, or maybe not. Your fast pole will have a limited impact on the closed-loop performance, so you can simply ignore it for now.

You could design your controller to cancel the slow pole at -2.4248 and the slowish zero at -34.2. This would be a lead lag compensator looking like this

$$H_{LeadLag} = K\frac{s + 2.4248}{s+34.2}$$

Then you could set a PI in series with the lead-lag compensator and tune the P and I gain to get the closed-loop performance you want.

$$H_{c} = H_{LeadLag} \frac{K_ps + K_i}{s}$$ That being said, I'm not sure my solution is particularly robust. You should check the Gang of four stability functions https://www.cds.caltech.edu/~murray/courses/cds101/fa02/caltech/astrom-ch5.pdf.

Edit 2 :

Another solution, simply use a PI controller and set the ratio of the P and I gain to 2.4248 to cancel your slow pole. This controller will not cancel your slowish zero, but maybe you don't need to cancel it to get the performance you want.

$$H_{PI} = K\frac{s+2.4248}{s}$$

• PID will be a good solution when speed is desired and limited by a low frequency pile. With PID you can virtually cancel the lowest frequency pole and the achievable response time is limited by the higher frequency pole. Mar 9 at 22:44
• (Low frequency pole, not pile!) Mar 9 at 23:17
• @Ben , thank you, this helped me out.
– Carl
Mar 10 at 10:02