PID pole-placement algorithms

PID controllers can be applied to a large number of industrial motion control systems including dynamic- positioning systems, autopilots for steering and diving as well as path- following control systems.

Linear mass-damper-spring system

$$\begin{array}{r}m\ddot{x}+d\dot{x}+kx=0\\ \ddot{x}+2\zeta {\omega }_n\dot{x}+{\omega }_n^2=0\end{array}$$

These two equations are equal. In the first equation d is equal to the damping term and k is equal to the spring constant. The second equations includes the natural damping ratio omega and relative damping factor zeta.

$$2\zeta {\omega }_n=\frac{d}{m},{\omega }_n^2=\frac{k}{m}$$ $${\omega }_n=\sqrt{\frac{k}{m}}$$ $$\zeta =\frac{d}{2m{\zeta }_n}$$ $${\lambda }_{1,2}=-\zeta {\omega }_n\pm j\omega$$

Damped oscillator

For a damped system d > 0, the frequency of the oscillations will be smaller than the natural frequency of the undamped system.

For a marine craft a reduction of 0.5% in the natural frequency is common.

$$r=1-\frac{0.5}{100}=0.995$$ $$\omega =r{\omega }_n$$ $$a^2+(r{\omega }_n{)}^2={\omega }_n^2$$ $$a=\sqrt{1-r^2}{\omega }_n$$

The term under the square root is equal to the relative damping ratio. Undamped oscillator: a = 0

Heave, roll and pitch damping

To determine the linear damping in heave, roll and pitch, we use the following formula for linear damping:

$$d=2\zeta \sqrt{km},\zeta =\sqrt{1-r^2}$$

where r is the design parameter.

Surge, sway and yaw damping

To determine the damping in these states we need to treat each of them as a pure mass-damper system. Linear damping for such a system can be found by specifying the time constant T > 0 corresponding to:

$$m\ddot{x}+\dot{x}=\tau$$

which for the design parameter T = m/d is equivalent to

$$T\ddot{x}+\dot{x}=\frac{1}{d}\tau$$

This yields the following design formula for linear damping

$$d=\frac{m}{T}$$

(For marine craft we can specify the time constant).

SISO linear PID control

PD control law applied to a mass damper spring system:

Closed loop system:

$$m\ddot{x}+(d+K_d)\dot{x}+(k+K_p)\tilde{x}=0$$

PID pole-placement formulas:

$$K_p=m{\omega }_n^2-k$$ $$K_d=2\zeta {\omega }_{n}m-d$$ $$K_i=\frac{{\omega }_n}{10}{K}_p=\frac{{\omega }_n}{10}(m{\omega }_n^2-k)$$

(integral time should be 10 times slower than the natural frequency)

Relationship between natural frequency and control bandwidth

$${\omega }_b={\omega }_n\sqrt{1-2{\zeta }^2+\sqrt{4{\zeta }^4-4{\zeta }^2+2}}$$

For a critically damped system with relative damping ratio = 1, this expression reduces to:

$${\omega }_b={\omega }_n\sqrt{\sqrt{2}-1}\approx 0.64{\omega }_n$$

Control bandwidth gives us information about how fast of a reference we can follow.

Main result (PID controller tuning rule)