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, [8] has discussed [H.sub.2] or Linear Quadratic Gaussian (LQG) control, while [9] 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].