Dead-beat control

Dead beat controllers are often used in process control due to their good dynamic properties. They are a classical feedback controller where the control gains are set using a table based on the plant system order and normalized natural frequency.

The deadbeat response has the following characteristics:

1. Zero steady-state error
2. Minimum rise time
3. Minimum settling time
4. Less than 2% overshoot/undershoot
5. Very high control signal output

Consider that a plant has the transfer function

${\displaystyle \mathbf {G} (z)={\frac {B(z)}{A(z)}}}$

where

${\displaystyle A(z)=a_{0}+a_{1}z^{-1}+a_{2}z^{-2}+\cdots a_{m}z^{-m},}$

${\displaystyle B(z)=b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots b_{n}z^{-n}.}$

The transfer function of the corresponding dead-beat controller is[2]

${\displaystyle \mathbf {C} (z)={\frac {A(z)/B(1)}{z^{d}-B(z)/B(1)}},}$

where d is the minimum necessary system delay for controller to be realizable. For example, systems with two poles must have at minimum 2 step delay from controller to output, so d = 2.

The closed-loop transfer function is

${\displaystyle \mathbf {L} (z)={\frac {B(z)/B(1)}{z^{d}}},}$

and has all poles at the origin. In general, a closed loop transfer function which has all of its poles at the origin is called a dead beat transfer function.

• Kailath, Thomas: Linear Systems, Prentice Hall, 1980, ISBN 9780135369616
• Warwick, Kevin: Adaptive dead beat control of stochastic systems, International Journal of Control, 44(3), 651-663, 1986.
• Dorf, Richard C.; Bishop, Robert H. (2005). Modern Control Systems. Upper Saddle River, NJ 07458: Pearson Prentice Hall. pp. 617–619.{{cite book}}: CS1 maint: location (link)