### Glossary of Control Engineering Terms - L

Left half plane: In the complex plane the half with negative real numbers. For continuous systems the poles in the left half plane are stable.

Legacy Systems: Computer systems or application programs, which are outdated and incompatible with other systems, but are too costly to replace or redesign. They are often large, intimidating, and difficult to modify.

Lag System: See first order system.

Lag Time: Lag time is the amount of time after the dead time that the process variable takes to move 63.3% of its final value after a step change in valve position. Lag time is also called a capacity element or the time constant of a first order system. Very few real processes are pure lag. Almost all real processes contain some dead time.

Lambda PID Tuning: A PID tuning method that uses a low-order linear model of the plant to determine the controller gains to give a desired closed loop time constant. Also called IMC PID tuning since it is based on Internal Model Control ideas.

Laplace Transform: A transform to convert a continuous time differential equation into a transfer function from which system connections can be manipulated easier and frequency responses calculated. Sister of the Z-transform.

Linear System: A system is said to be linear if it is scalable and obeys the principle of superposition. What this means in practice is that if the input signal is doubled, the output will be doubled also, and that if an input signal is a summation of several different waveforms (e.g. a step response can be regarded as being made up of many sine waves) then the output is the summation of the individual output responses. A non-linear system is something that does not obey these rules, which is virtually every practical system, since physical limits mean there is always a limit to how far the scalability rule applies.

Load Disturbance: An upset to the process (that is not from changing the set-point) which drives the system away from its desired operating point. It is typically a low frequency disturbance and can be modelled as an additive signal on the input to the process or more commonly added onto the system output.

LQG (Linear Quadratic Gaussian): When the plant dynamics are linear and the measurement noise and disturbance signals are zero mean Gaussian stochastic processes then the optimal controller can be calculated using the separation theorem which is: solve the deterministic linear quadratic regulator (LQR) problem to get the controller state-feedback gain. The optimal estimate of the unknown state vector is given by a Kalman filter.

LQR (Linear Quadratic Regulator): An optimal state feedback controller. The state feedback gain is calculated by minimising a performance index which is the time dependent integral of the weighted sum of the squares of the states and the controller output. The weighting matrices are symmetrical and real numbers. A linear state-space model is used for the gain calculation.