Proportional controller error doesn't approach zero

Question

I'm reading this pdf. The dynamic equation of one arm is provided which is

$$ l \ddot{\theta} + d \dot{\theta} + mgL sin(\theta) = \tau $$

where

$\theta$ : joint variable.

$\tau$ : joint torque

$m$ : mass

$L$ : distance between centre mass and joint.

$d$ : viscous friction coefficient

$l$ : inertia seen at the rotation axis.

I would like to use P (proportional) controller for now.

$$ \tau = -K_{p} (\theta - \theta_{d}) $$

My Matlab code is

clear all
clc

t = 0:0.1:5;

x0 = [0; 0];

[t, x] = ode45('ODESolver', t, x0);

e = x(:,1) - (pi/2); % Error theta1

plot(t, e);
title('Error of \theta');
xlabel('time');
ylabel('\theta(t)');
grid on

For solving the differential equation

function dx = ODESolver(t, x)

dx = zeros(2,1);
%Parameters:

m  = 2;
d  = 0.001;
L  = 1;
I = 0.0023;
g  = 9.81;

T = x(1) - (pi/2);


dx(1) = x(2);

q2dot = 1/I*T - 1/I*d*x(2) - 1/I*m*g*L*sin(x(1));

dx(2) = q2dot;

The error is

My question is why the error is not approaching zero as time goes? The problem is a regulation track, so the error must approach zero.

Answer 1

If you're only using proportional force, then at some point it will be balanced by the force of gravity -- your error will converge on that balance, not zero.

To compensate for the mass of the arm, you'll need to add an integral force term. This will increase over time to counterbalance the constant force of gravity.

See also: this answer on the integral term.

Answer 2

I would like to use P (proportional) controller for now.

Just a proportional controller will never make your error stay at 0. Your system is not damped and a proportional controller acts like an undamped spring. Look at the controller equation that you wrote:

τ=−K(θ−θd)

and compare it to a spring equation:

F=Kx or F=K(x1-x2)

Your controller is acting like a spring, putting torque on the joint proportionally to it's deviation from the reference. You must use at least a PI (proportional and integrating) controller to get rid of the oscillation.

The integrating part of the controller is what makes the oscillations go away and make the steady state error approach 0. It will make your system a bit slower, but you'll have to tune the controller to behave the way you want, faster response means typically more overshoot and vice versa.

If you really need to use a P controller, you have to include a damper on the system.

Answer 3

Controllers type

A more mathematical approach to the error.

Suppose you have a close loop system like above. The equation is:

$\hspace{2.5em}$ $Y(s) = \frac{G(s)C(s)}{1+G(s)C(s)} R(s)$

The error equation is:

$\hspace{2.5em}$ $E(s) = R(s) - Y(s)$

$\hspace{2.5em}$ $E(s) = \frac{1}{1+G(s)C(s)} R(s)$ $\hspace{2.5em}[1]$

The final value theorem states that:

$\hspace{2.5em}$ $e(\infty) = \lim_{s \rightarrow 0}sE(s)$ $\hspace{2.5em}[2]$

Using $[1]$ in $[2]$:

$\hspace{2.5em}$ $e(\infty) = \lim_{s \rightarrow 0}s\frac{1}{1+G(s)C(s)} R(s)$

Ok! Now, we can conclude some cool stuffs.

• Suppose $R(s) = \frac{1}{s}$:

$\hspace{2.5em}$ $e(\infty) = \lim_{s \rightarrow 0}\frac{1}{1+G(s)C(s)}$ $\hspace{2.5em}$ From calculus, is the same as:

$\hspace{2.5em}$ $e(\infty) = \frac{1}{1+\lim_{s \rightarrow 0} (G(s)C(s))}$

$\hspace{2.5em}$ $e(\infty) = \frac{1}{1+G(0)C(0)}$

If $C(s) = K$, where $K$ is a proportional gain, $G(0)$ have a finite DC gain, your system is going to be the responsible for the finite final term. But, however, you can apply a PD control, $C(s) = \frac{K}{s}$.

$\hspace{2.5em}$ $C(0) = \lim_{s \rightarrow 0}\frac{K}{s} = \infty$

Then:

$\hspace{2.5em}$ $e(\infty) = \frac{1}{1+G(0)(\infty)} = 0$

And that's the resume why you can't get null error only with a proportional gain.