Lead Compensators

Definition

Consider a unity feedback system with reference signal $r(t)$$\large r(t)$, output signal $y(t)$$\large y(t)$, plant $P(s)$$\large P(s)$ and controller $C(s)$$\large C(s)$.

A lead compensator controller $C(s)$$\large C(s)$ is of the form:

where $p>z$$\large p > z$.

By setting $z=\alpha p$$\large z = \alpha p$ and $K=K_c\alpha$$\large K = K_c \alpha$ where $0<\alpha <1$$\large 0 \lt \alpha \lt 1$, we can rewrite it in Bode form:

Lead compensation approximates a PD controller, is used to stabilize a system and improve the transient response (i.e improve phase margin).

Design

Steps to design a lead compensator under given specifications:

1. Choose $K$$\large K$ to meet a steady state error specification
2. Find phase margin of $KP(s)$$\large KP(s)$ (either analytically or from the Bode plot)
3. Find how much extra phase is required to meet the phase margin specification. Set ${\varphi }_{max}$$\large \phi_{max}$ to be the required phase + an additional ${10}^{\circ }$$\large 10^\circ$
4. Find $\alpha =\frac{1-sin({\varphi }_{max})}{1+sin({\varphi }_{max})}$$\large \alpha = \frac{1 - sin(\phi_{max})}{1 + sin(\phi_{max})}$
5. Set ${\omega }_{max}$$\large \omega_{max}$ to be ${\omega }_{cg}$$\large \omega_{cg}$ of $KP(s)$$\large KP(s)$ and calculate $p=\frac{{\omega }_{max}}{\sqrt{\alpha }}$$\large p = \frac{\omega_{max}}{\sqrt{\alpha}}$ and $z=\sqrt{\alpha }{\omega }_{max}$$\large z = \sqrt{\alpha} \omega_{max}$
6. Check if the compensator achieves the specifications. If not, iterate.

Lag Compensators

Definition

Consider a unity feedback system with reference signal $r(t)$$\large r(t)$, output signal $y(t)$$\large y(t)$, plant $P(s)$$\large P(s)$ and controller $C(s)$$\large C(s)$.

A lag compensator controller $C(s)$$\large C(s)$ is of the form:

where $z>p$$\large z > p$.

By setting $z=\beta p$$\large z = \beta p$ where $\beta >1$$\large \beta \gt 1$, we can rewrite it in Bode form:

Lag compensation approximates a PI controller, is used to boost the DC gain.

Design

Steps to design a lead compensator under given specifications:

1. Choose $K_c$$\large K_c$ to meet the phase margin specification (with ${10}^{\circ }$$10^\circ$ buffer) by moving ${\omega }_{cg}$$\large \omega_{cg}$ to the left
2. Find the low frequency gain of $K_{c}P(s)$$\large K_cP(s)$, and determine how much extra gain $\beta$$\large \beta$ is needed for tracking specification
3. Choose $z=\beta p$$\large z = \beta p$ to be one decade below ${\omega }_{cg}$$\large \omega_{cg}$ of $K_{c}P(s)$$\large K_cP(s)$
4. Choose $p=\frac{z}{\beta }$$\large p = \frac{z}{\beta}$
5. Check if the compensator achieves the specifications. If not, iterate.