How do I design a PI controller with unity DC gain?

[glorimda]

I'm doing my homework, and encountered a problem.

The problem is the one as attached. It includes a part of solution that I don't understand.

I've known that when designing the PI controller, we start from D(s) = (Kp + Ki/S)

The condition of the PI controller is, firstly, unity DC gain.

So the way I started is setting D(0) = 1, but couldn't proceed anymore because there is 1/s

in D(s), also it is obviously different from the solution.

The second way was setting whole transfer function of the system Gcl(0) = 1,

but couldn't find the right solution.

According to solution, it approaches Kp and Ki separately.

And it uses (G(0)/(1+Kp*G(0))=1), but I can't understand this.

Could somebody help me out??

 

[X89codered89X]

The secret is in the fact that there is a pole @ $s=0$ in the plant. When you apply the feedback controller, You'll have $H(s) = \frac{G_c(s)G(s)}{1 + G_c(s)G(s)}$ where $G_c(s)$ is the controller.

You start with the criterion they gave you $\frac{G(0)}{1 + K_pG(0)} = 1$. In order to use this, you first have to plug in G(s). Notice, that of course, if we then set $s=0$, that the expression blows up, so instead we have to do the Laplace domain equivalent of L'Hopital's rule, which is multiplying the expression by a fancy form of one, or $\frac{s}{s}$. Assigning ${\hat{G}}(s) = \frac{210}{(5s+7)(s+3)}$, then you'll have $\frac{{\hat{G}}(s)}{s + K_p{\hat{G}}(s)}$. Notice now that if you plug in 0, the expression simplifies to $\frac{21}{K_p21}$ which, set equal to one, produces the result that $K_p=1$.

Unfortunately, I remember less about the phase margin controller design. Maybe I'll brush up and get back to you (I love reviewing. I seriously might.). But I will say this... The design centers around starting with the bode plot for G(s). Since the new plant will be Gc(s)G(s), the phases will add, and you have to cancel out the unwanted phase in the plant with the controller. It's also standard to add in 5 degrees of safety within the design procedure since it's known that this procedure involves some, perhaps strong, approximation. This is vague and I'm sorry I don't have more details. But I hope this helps get you started...

 

[glorimda]

Thanks for reply.
In fact, I'm still unclear about the part (G(0)/(1+Kp*G(0))=1) which is from the solution.
In the problem, they just gave me the condition that 'PI controller with a unity dc gain'
and in the solution, the equation '(G(0)/(1+Kp*G(0))=1)' just came up. The equation is not from the problem, but from solution. What I wondered was how to induce
'PI controller with a unity dc gain' ----> (G(0)/(1+Kp*G(0))=1)
The rest of the part you explained, I got it, but not this part.
I hope that you understood my question. ^^!

 

[X89codered89X]

Ohhhhhh I see what you're saying. Honestly I don't know. It seems like the textbook is asking you to take more approximations than necessary, and not giving you some of the information. It depends. Where is the controller assumed to be placed in the loop? Between the E(s) signal and the G(s) plant or in the Feedback Loop on Y(s)? The way the problem is posed makes it look like it's in the feedback loop on Y(s). That may yield something.

 

[glorimda]

I guess the controller is supposed to be placed between E(s) and plant G(s), because in the problem I attached, it says 'unity feedback' which means H(s) = 1.

All I got is identical to the one I attached.

I'm many times bothered due to lack of explanation and sources regarding exercises in

the textbook. >.<