Direct synthesis of a controller (plus simulink modeling)

[gfd43tg]

Homework Statement

Homework Equations

The Attempt at a Solution

Part (a)
Starting with the formula

$$ g_{c} = \frac {g_{CL}}{g_{p}(1-g_{CL})} $$

The controller transfer function is determined,

$$ g_{c} = \frac { \frac {1}{\lambda s + 1}}{\frac {k_{p}}{(\tau_{p1}s+1)(\tau_{p2}s+1)}
- \frac {k_{p}}{(\tau_{p1}s_1)(\tau_{p2}s+1)(\lambda s+1)}} $$

$$ = \frac {1}{\lambda s + 1} \times \frac
{(\tau_{p1}s+1)(\tau_{p2}s+1)(\lambda s+1)}{K_{p} \lambda s} $$

$$ = \frac {1}{K_{p} \lambda} \times \frac {\tau_{p1} \tau_{p2} s^{2} +
(\tau_{p1}+\tau_{p2})s + 1}{s} $$

$$ = \frac {\tau_{p1} \tau_{p2}}{K_{p} \lambda}s + \frac
{\tau_{p1}+\tau_{p2}}{K_{p} \lambda} + \frac {1}{K_{p} \lambda s} $$

$$ = \frac {\tau_{p1}+\tau_{p2}}{K_{p} \lambda} \bigg [ 1 + \frac
{\tau_{p1} \tau_{p2}}{\tau_{p1}+\tau_{p2}} s + \frac
{1}{\tau_{p1}+\tau_{p2}} \frac {1}{s} \bigg ] $$

Which is of the form of a PID Controller,

$$ g_{c,PID} = k_{c} \bigg ( 1 + \frac {1}{\tau_{I}s}+ \tau_{D}s \bigg )
$$

Where $k_{c}= \frac {\tau_{p1} + \tau_{p2}}{K_{p} \lambda}$, $\tau_{I} = \tau_{p1} + \tau_{p2}$, and $\tau_{D} = \frac {\tau_{p1} \tau_{p2}}{\tau_{p1}+\tau_{p2}}$

Part (b)
Here is my simulink model

However, because the numerator coefficient is second order, and the denominator is first order, I cannot run the simulation due to an error. I know that it is a big no-no to have higher order numerators, but how should I go about solving the problem?

 

[gfd43tg]

A second thought came to mind, I just used a PID block

And here is my input to the PID controller block

And here is my output

So obviously this controller is not working, or I am inputting the P, I, and D incorrectly into the controller

 

[milesyoung]

However, because the numerator coefficient is second order, and the denominator is first order, I cannot run the simulation due to an error. I know that it is a big no-no to have higher order numerators, but how should I go about solving the problem?

If the numerator order of the controller is larger than that of the denominator, then it represents a non-causal system, which is what the solver is complaining about.

To put in another way: it has infinite gain as ω → ∞. You can fix this by adding a pole at some high frequency that has no real influence on the behavior of your system. This what the PID controller block does in Simulink, which is controlled by the 'Filter coefficient (N)'.

The $g_cg_p$ system is causal, however, so you could enter that instead.

So obviously this controller is not working, or I am inputting the P, I, and D incorrectly into the controller

You did everything right, but you just mixed up the I and D parameters in the PID block. I think you also forgot parenthesis around the (tau1 + tau2) coefficient in your plant transfer function (edit: I'm not sure this bit is actually necessary, it just looks off. Simulink might interpret it correctly).

 

[gfd43tg]

Great, I see that I made the mistake about switching them. Also, I changed the transfer function to have the parenthesis, thanks for catching that! Also, I am a little confused when it asks if I successfully converted the system to a first order process. Does that just mean if the controller works, then yes? But I want to know a little deeper what exactly they mean by that.

 

[milesyoung]

Also, I am a little confused when it asks if I successfully converted the system to a first order process. Does that just mean if the controller works, then yes? But I want to know a little deeper what exactly they mean by that.

If your controller gains really do solve the assignment, then the closed-loop system response should be that of this system:
$$
g_{CL} = \frac{1}{s + 1}
$$
Your expressions for the controller gains were found as to guarantee this. What does the step response of this system look like, i.e. if you simulate the system (open loop): step input → g_CL → scope ?

Does this response match that of the closed-loop system with the PID controller? If so, then you've solved the assignment.

Before you simulate anything, try to characterize how the step response of $g_{CL} = \frac{1}{s + 1}$ should look. Tips:
- If it's a first-order system, will it have any overshoot?
- How can you tell what the time constant of a first-order system is by inspecting its step response?

And so on.