Key words: linear control system
, time scales, time delay, stability.
Automatic linear control system
is the application of control theory for the regulation of processes without direct human intervention.
Motivated by the above discussion, in this paper, we aim to combine the PID method and MRAC technique together and establish a unified framework to get fast transient tracking at the transient phase and better tracking performance for the piecewise linear control system
. It should be noted that the piecewise linear system, which consists of a set of linear subsystems, and piecewise adaptive switch controller which consists of PID and Model Reference Adaptive Control (MRAC) are developed to control the system.
The linear control system
outlined in the previous section is tested on the nonlinear vehicle and drivetrain model in the simulation environment.
A mathematically linear control system
is given by the following two equations
Consider the discrete-time linear control system
described by the model
An abstract linear control system
(ALCS for short) for X, [U.sup.p] with p [member of] [1, +[infinity]] is a pair (T, [PHI]), where T := [(T(t)).sub.t[greater than or equal to]0] be a [C.sub.0]-semigroup of bounded linear operators on X and [PHI] = [([[PHI].sub.(t)]).sub.t[greater than or equal to]0] is a family of bounded linear operators from [U.sup.p] to X (i.e., [[PHI].sub.t] [member of] L ([U.sup.p], X)) such that:
Linear control system
analysis and design with MATLAB, 6th ed.
As far as the stabilization of linear control systems
is concerned, the state vector may not be available for measurement and so when the linear control system
is both controllable and observable, we use the separation principle for linear control systems
and use an estimate of the state in lieu of the state vector.
Efficiency for a PWM system can reach 50 to 80 percent, while a linear control system
may deliver 20 percent.
To illustrate this result, the closed-loop behaviour of the nonlinear control system with LQ control is compared to the closed-loop behaviour of the nominal linear control system
with LQ control.
On the other hand, after Kalman presented the question "When is a linear control system
optimal?" which preserves the quadratic form in the performance index,  has discussed [H.sub.2] or Linear Quadratic Gaussian (LQG) control, while  has retained the original weights and sought out a producer which can also achieve the desired degree of stability, and others have contributed to the stochastic LQ controller design [10-12].