Lead Compensator Design: Mathematical Derivation

This document derives the complete mathematical framework for lead compensator design to maximize DC gain while maintaining phase margin.

1. System Model

Why the First-Order Slope Model is Valid

In a typical PID control loop, the open-loop gain has several regions:

Integration region (low frequency): Steep slope for zero steady-state error
Proportional region (mid frequency): Flat plateau
Differentiation region (higher frequency): Small gain boost
Rolloff region (around crossover): Descent to unity gain

The crossover frequency must be far beyond the PID region to maintain phase margin. At these frequencies:

The integrator contribution has rolled off to a simple $1/\omega$ slope
The proportional gain is just a constant (which we absorb into $A$)
The derivative term (if any) has minimal effect
The plant often provides natural first-order rolloff (motor inertia, thermal time constants, etc.)

Therefore, around crossover, we can approximate the loop as:

$$ |L(j\omega)| \approx \frac{A}{\omega} $$

This is the same reasoning used in the Control Loop Bandwidth analysis, where we showed that delay-limited loops naturally have a first-order slope at crossover.

Plant + Controller Around Crossover

The open-loop transfer function around crossover is modeled as a first-order slope:

$$ |L(j\omega)| = \frac{A}{\omega} $$

where: - $A$ is the DC gain (the slope gain coefficient) - The phase is approximately $-90°$ (first-order rolloff)

This model is accurate because: - Crossover is far from the PID corner frequencies - The $1/\omega$ behavior dominates - Higher-order dynamics are negligible at these frequencies

Delay

A pure delay $\tau$ adds phase:

$$ \angle\text{delay}(j\omega) = -\omega\tau \quad \text{[radians]} $$

2. Baseline Design (No Lead Compensation)

Phase Margin Condition

At the baseline crossover $\omega_{c1}$, we require phase margin $\phi_m$:

$$ -\frac{\pi}{2} - \omega_{c1}\tau = -\pi + \phi_m \quad \text{[radians]} $$

Solving for $\omega_{c1}$:

$$ \boxed{\omega_{c1} = \frac{\pi/2 - \phi_m}{\tau}} \quad \text{[rad/s]} $$

where $\phi_m$ is in radians. If given in degrees:

$$ \omega_{c1} = \frac{\pi}{180} \cdot \frac{90° - \phi_m°}{\tau} $$

Baseline DC Gain

At crossover, the magnitude condition is:

$$ |L(j\omega_{c1})| = \frac{A_1}{\omega_{c1}} = 1 $$

Therefore, the baseline DC gain is:

$$ \boxed{A_1 = \omega_{c1}} $$

This is the maximum gain achievable with the specified phase margin and delay, without lead compensation.

3. Lead Compensator

Transfer Function

The lead compensator has the form:

$$ F(s) = \frac{1 + s/\omega_z}{1 + s/\omega_p} $$

where $\omega_z < \omega_p$ (zero before pole).

Key properties: - Unity DC gain: $F(0) = 1$ (does not affect low-frequency PID gain) - High-frequency gain: $|F(j\omega)| \to \alpha$ as $\omega \to \infty$

Pole-Zero Ratio

Define:

$$ \alpha = \frac{\omega_p}{\omega_z} > 1 $$

This is the high-frequency gain and the primary design constraint (typically $\alpha \le 10$).

Magnitude Response

$$ |F(j\omega)| = \sqrt{\frac{1 + (\omega/\omega_z)^2}{1 + (\omega/\omega_p)^2}} $$

Phase Response

$$ \angle F(j\omega) = \arctan\left(\frac{\omega}{\omega_z}\right) - \arctan\left(\frac{\omega}{\omega_p}\right) $$

The lead compensator provides positive phase (phase lead) in the frequency range between $\omega_z$ and $\omega_p$.

4. Compensated System

Magnitude Condition

With lead compensation, the loop magnitude is:

$$ |L_{\text{comp}}(j\omega)| = \frac{A}{\omega} \cdot |F(j\omega)| $$

At the new crossover $\omega_{c2}$:

$$ \frac{A}{\omega_{c2}} \cdot |F(j\omega_{c2})| = 1 $$

Analytic Crossover Solution

Substituting the lead magnitude and squaring:

$$ \frac{A^2}{\omega_c^2} \cdot \frac{1 + (\omega_c/\omega_z)^2}{1 + (\omega_c/\omega_p)^2} = 1 $$

Let $x = \omega_c^2$ and $\omega_p = \alpha\omega_z$:

$$ \frac{A^2}{x} \cdot \frac{1 + x/\omega_z^2}{1 + x/(\alpha^2\omega_z^2)} = 1 $$

Expanding:

$$ A^2 \left(1 + \frac{x}{\omega_z^2}\right) = x \left(1 + \frac{x}{\omega_p^2}\right) $$

$$ A^2 + \frac{A^2 x}{\omega_z^2} = x + \frac{x^2}{\omega_p^2} $$

Multiply by $\omega_z^2\omega_p^2$ and rearrange:

$$ \boxed{x^2 + x\omega_p^2\left(\frac{\omega_z^2 - A^2}{\omega_z^2}\right) - A^2\omega_p^2 = 0} $$

Solve using the quadratic formula, taking the positive root:

$$ \omega_c = \sqrt{x} $$

Special case: When $\alpha = 1$ (no lead), this reduces to $\omega_c = A$.

Phase Condition

The total phase at crossover is:

$$ \angle L_{\text{comp}}(j\omega_c) = -\frac{\pi}{2} - \omega_c\tau + \arctan\left(\frac{\omega_c}{\omega_z}\right) - \arctan\left(\frac{\omega_c}{\omega_p}\right) $$

The phase margin is:

$$ \text{PM} = \pi + \angle L_{\text{comp}}(j\omega_c) $$

5. Optimization Problem

Variables (Normalized)

To make the problem tau-independent and numerically well-conditioned, we optimize normalized variables:

$$ \begin{aligned} A_{\text{ratio}} &= \frac{A}{A_1} \ge 1 \ \omega_{z,\text{ratio}} &= \frac{\omega_z}{\omega_{c1}} \ge 1 \ \alpha &= \frac{\omega_p}{\omega_z}, \quad 1 \le \alpha \le \alpha_{\max} \end{aligned} $$

All variables are $O(1)$, making COBYLA's numerical scaling work well.

Objective

Maximize $A_{\text{ratio}}$ (the DC gain improvement):

$$ \max_{A_{\text{ratio}}, \omega_{z,\text{ratio}}, \alpha} A_{\text{ratio}} $$

Constraint

Phase margin at the new crossover must meet or exceed the target:

$$ \text{PM}(A_{\text{ratio}} \cdot A_1, \omega_{z,\text{ratio}} \cdot \omega_{c1}, \alpha) \ge \phi_m $$

Bounds

$$ \begin{aligned} A_{\text{ratio}} &\in [1, 100] \quad \text{(effectively unbounded above)} \ \omega_{z,\text{ratio}} &\in [1, 100] \quad \text{(zero at or above baseline)} \ \alpha &\in [1, \alpha_{\max}] \quad \text{(typically } \alpha_{\max} = 10\text{)} \end{aligned} $$

6. Physical Interpretation

The DC Gain A

The DC gain $A$ determines: - Disturbance rejection: Higher $A$ → better rejection - Tracking accuracy: Higher $A$ → tighter tracking
- Steady-state error: Higher $A$ → lower error

In a PID controller, $A$ is proportional to the integral gain.

The Improvement Ratio

The ratio $A/A_1$ represents how much we can increase the PID gain beyond what's possible without lead compensation, while maintaining the same phase margin.

Example Results (for $\tau = 0.02$ s):

Phase Margin	$A_1$	$A$	$A/A_1$	Improvement
45°	39.3	54.5	1.39	39%
60°	26.2	43.8	1.67	67%
75°	13.1	33.8	2.58	158%

Why Alpha Matters

The parameter $\alpha$ (high-frequency gain) is constrained because: - Noise amplification: High $\alpha$ amplifies sensor noise - Unmodeled dynamics: Excites resonances and parasitics - Actuator stress: Higher bandwidth demands more actuator effort

Typical practical limit: $\alpha_{\max} = 10$.

7. Design Procedure

Specify requirements:
Delay: $\tau$
Phase margin: $\phi_m$
Maximum alpha: $\alpha_{\max}$ (default: 10)
Compute baseline:
$\omega_{c1} = (\pi/2 - \phi_m)/\tau$
$A_1 = \omega_{c1}$
Optimize:
Variables: $(A_{\text{ratio}}, \omega_{z,\text{ratio}}, \alpha)$
Objective: Maximize $A_{\text{ratio}}$
Constraint: Phase margin $\ge \phi_m$
Solver: COBYLA
Extract results:
Optimal DC gain: $A = A_{\text{ratio}} \cdot A_1$
Lead zero: $\omega_z = \omega_{z,\text{ratio}} \cdot \omega_{c1}$
Lead pole: $\omega_p = \alpha \cdot \omega_z$
New crossover: $\omega_{c2}$ (from analytic solution)

8. Key Insights

Tau Independence (Theoretical)

The normalized optimization problem is mathematically independent of $\tau$: - All variables scale with baseline values - The constraint involves only ratios - The product $\omega_{c1} \cdot \tau = \pi/2 - \phi_m$ is constant

Numerical Considerations

For very large $\tau$ (> 1 s), numerical convergence can be slow because absolute frequencies become very small. However, for all practical control systems ($\tau = 0.001$ to $0.1$ s), the optimizer converges reliably.

Zero and Pole Placement

The optimal design typically places: - Zero $\omega_z$: Near or slightly above the baseline crossover - Pole $\omega_p = \alpha\omega_z$: Much higher (factor of $\alpha$)

This provides phase lead in the critical frequency region while maintaining unity DC gain.

9. Implementation

This derivation is implemented in lead_lag_design.py:

from lead_lag_design import LeadCompensatorDesign

# Design for 20ms delay, 45° phase margin
design = LeadCompensatorDesign(tau=0.02, phi_m_deg=45)

print(f"Baseline gain: A1 = {design.A1:.2f}")
print(f"Optimized gain: A = {design.A:.2f}")
print(f"Improvement: {design.A/design.A1:.2f}x")
print(f"Lead zero: {design.wz:.2f} rad/s")
print(f"Lead pole: {design.wp:.2f} rad/s")
print(f"Alpha: {design.alpha:.2f}")

References

Franklin, Powell, Emami-Naeini: "Feedback Control of Dynamic Systems"
Åström, Murray: "Feedback Systems: An Introduction for Scientists and Engineers"