Proportional, PD and PID Control

These are the three common linear controllers $G_{c} (s)$ for the single joint loop in Motion Control Schemes. The plant is:

G_{p} (s) = \frac{n K _{a}}{s ( R _{a} J _{e} s + R _{a} f _{e} + K _{a} K _{b} )}

The error is $e (t) = θ_{L}^{d} (t) - θ_{L} (t)$ .

The motor voltage is proportional to the error:

V_{a} (t) = K_{p} e (t) G_{c} (s) = K_{p}

Open loop:

G (s) = \frac{n K _{a} K _{p}}{s ( R _{a} J _{e} s + R _{a} f _{e} + K _{a} K _{b} )}

Closed loop:

\frac{θ _{L} ( s )}{θ _{L}^{d} ( s )} = \frac{n K _{a} K _{p}}{R _{a} J _{e} s ^{2} + ( R _{a} f _{e} + K _{a} K _{b} ) s + n K _{a} K _{p}}

This is a second order system. It is always stable if all the parameters are positive.

Add a derivative term:

V_{a} (t) = K_{p} e (t) + K_{v} \overset{e}{˙} (t) G_{c} (s) = K_{p} + s K_{v}

Open loop:

G (s) = \frac{n K _{a} ( K _{p} + s K _{v} )}{s ( R _{a} J _{e} s + R _{a} f _{e} + K _{a} K _{b} )}

Closed loop:

\frac{θ _{L} ( s )}{θ _{L}^{d} ( s )} = \frac{n K _{a} ( K _{p} + K _{v} s )}{R _{a} J _{e} s ^{2} + ( R _{a} f _{e} + K _{a} K _{b} + n K _{a} K _{v} ) s + n K _{a} K _{p}}

The derivative gain $K_{v}$ adds to the damping term (the coefficient of $s$ ).

Add an integral term:

V_{a} = K_{p} e + K_{v} \overset{e}{˙} + K_{i} \int e d t G_{c} (s) = K_{p} + s K_{v} + \frac{K _{i}}{s}

Open loop:

G (s) = \frac{n K _{a} ( K _{p} s + K _{v} s ^{2} + K _{i} )}{s ^{2} ( R _{a} J _{e} s + R _{a} f _{e} + K _{a} K _{b} )}

Closed loop:

\frac{θ _{L} ( s )}{θ _{L}^{d} ( s )} = \frac{n K _{a} ( K _{p} s + K _{v} s ^{2} + K _{i} )}{R _{a} J _{e} s ^{3} + ( R _{a} f _{e} + K _{a} K _{b} + n K _{a} K _{v} ) s ^{2} + n K _{a} K _{p} s + n K _{a} K _{i}}

This is a third order system. The integral term removes the steady state error.

JJ