Dead-beat Control

In discrete-time control theory, the dead-beat control problem consists of finding what input signal must be applied to a system in order to bring the output to the steady state in the smallest number of time steps.

For an ''N''th-order linear system it can be shown that this minimum number of steps will be at most ''N'' (depending on the initial condition), provided that the system is null controllable (that it can be brought to state zero by ''some'' input). The solution is to apply feedback such that all poles of the closed-loop transfer function are at the origin of the ''z''-plane. (For more information about transfer functions and the ''z''-plane see z-transform). Therefore the linear case is easy to solve. By extension, a closed loop transfer function which has all poles of the transfer function at the origin is sometimes called a dead beat transfer function.

For nonlinear systems, dead beat control is an open research problem. (See Nesic reference below).

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:
# Zero steady-state error
# Minimum rise time
# Minimum settling time
# Less than 2% overshoot/undershoot
# Very high control signal output

Transfer functions

Consider the transfer function of a plant

:$\mathbf(z) = \frac z^$

with polynomials

:$A(z) = a_ + a_ z^ + a_ z^ + \cdots a_ z^,$

:$B(z) = b_ + b_ z^ + b_ z^ + \cdots b_ z^$

an
discrete time delay
$e^$.

The corresponding dead-beat controller is noted as

:$\mathbf(z) = \frac$

and the closed-loop transfer function is calculated as

:$\mathbf(z) = \frac.$

References

*Kailath, Thomas: ''Linear Systems'', Prentice Hall, 1980,

Nesic et al.:''Output dead beat control for a class of planar polynomial systems''
* Kevin Warwick, Warwick, Kevin: ''Adaptive dead beat control of stochastic systems'', International Journal of Control, 44(3), 651-663, 1986.
*

Control theory

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