Control Law Partitioning

Introduction

Control law partitioning is a non-linear control method. The controller has two parts:

• a model based portion, which cancels the non-linear terms of the system
• a servo portion, which controls the resulting linear system

After the cancellation, the closed loop system is linear.

Model based portion

For the mass spring system $m\ddot{x}+b\dot{x}+k{x}^3=u$, use the control law:

$$u=\alpha v+\beta$$

$\alpha$ and $\beta$ are functions that you choose so that, with $v$ as the new input, the system looks like a unit mass. Choose:

$$\alpha =m\beta =b\dot{x}+k{x}^3$$

Then:

$$m\ddot{x}+b\dot{x}+k{x}^3=mv+b\dot{x}+k{x}^3\Rightarrow \boxed{\ddot{x}=v}$$

Servo portion

Now control the unit mass $\ddot{x}=v$ with:

$$v={\ddot{x}}_d+k_v\dot{e}+k_{p}e$$

where $e=x_d-x$ and $x_d$ is the desired position. Combine with $\ddot{x}=v$:

$$\boxed{\ddot{e}+k_v\dot{e}+k_{p}e=0}$$

Choosing the gains

The characteristic equation is:

$$s^2+k_{v}s+k_p=0$$

• stability needs $k_v>0$ and $k_p>0$
• compare with $s^2+2\zeta {\omega }_{n}s+{\omega }_n^2=0$, so $k_v=2\zeta {\omega }_n$ and $k_p={\omega }_n^2$
• choose $\zeta \ge 1$ and ${\omega }_n\le 0.5{\omega }_{res}$

The gains do not depend on the system parameters $m$, $b$ and $k$. The same gains give the same response for any system.

Example

Take $k_p=4$, so ${\omega }_n=2$.

$k_v$	$\zeta$	Response
2	0.5	Underdamped, overshoot
8	2	Overdamped, slow
4	1	Critically damped, fastest with no overshoot