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The proportional-integral-derivative (PID) is still the most common controller and stabilizer used in industry due to its simplicity and ease of implementation. In most of the real applications, the controlled system has parameters which slowly vary or are uncertain. Thus, PID gains must be adapted to cope with such changes. In this paper, adaptive PID (APID) controller is proposed using the recursive least square (RLS) algorithm. RLS algorithm is used to update the PID gains in real time (as system operates) to force the actual system to behave like a desired reference model. Computer simulations are given to demonstrate the effectiveness of the proposed APID controller on SISO stable and unstable systems considering the presence of changes in the systems parameters.

A challenging problem in designing a PID controller is to find its appropriate gain values (i.e., proportional gain

Adaptive control has been commonly used during the past decades specially the model reference adaptive control (MRAC). Its objective is to adapt the parameters of the control system to force the actual process to behave like some given ideal model which is demonstrated in [

An adaptive PID controller is presented in [

In the case of unstable systems, few researchers study the behaviour of the adaptive PID techniques on unstable systems and examine its ability to stabilize them as verified in [

In this paper, the direct method of adaptive control is considered. RLS algorithm is used as adaptation mechanism to tune the PID gains automatically online to force the actual process to behave like the reference model. The proposed approach has also the ability to stabilize the unstable system. Adding some parameters variations in actual process during its operation time confirms the proposed controller adaptation capability and robustness against process variation in both stable and unstable cases.

The structure of this paper is as follows. In Section

Consider a system shown in Figure

Closed-loop system.

In conventional control, the PID controller can be expressed as

Note that, approximately, the transfer function of integrator

Simply, the controller transfer function [

Define

So (

In most of practical applications, the actual structure of the controlled system is unknown or varying. Therefore, the adaptive mechanism is used for self-adjustment of the PID gains to achieve the best tracking performance. The proposed technique controls the motion of both stable and unstable systems to follow the ideal trajectory provided by a designer defined reference model

The proposed controller objective is to find the coressponding controller parameters (PID gains) using RLS algorithum as adaptation mechanism such that the closed-loop transfer function is more or less equal to the reference model transfer function. In other words, the reference output

So it can be written as

Hence, (

Applying the controller transfer function

Now the modified estimation error of RLS can be defined as

This means that

Based on the RLS algorithms, we tune the parameters

On the other hand, in order to apply the classical equations of the RLS estimation algorithm used to find the parameters

To build the RLS algorithm using the 2nd-Level S-Function in Matlab, the first term in right hand side of (

RLS is an algorithm which recursively finds the optimal estimate

According to the above RLS algorithm equations, the controller parameters

In order to avoid such a problem and reduce the variation of the controller parameters

In order to illustrate the main features of the proposed APID using RLS, simulation examples are now presented. The following examples cover stable and unstable systems cases and consider the changes in the system parameters during simulation time.

Consider the plant used in [

Let the sampling time be

The reference input signal is chosen to be a delayed square wave.

The proposed technique in Section

In order to evaluate the proposed control method to plant uncertainties, we consider the case where there exists a change in one of the system poles at 155 s (i.e., the pole

The output by the proposed APID using RLS controller is shown in Figure

Output in stable system case.

Now, consider that the unstable system stated in [

And reference model can be expressed as

The reference input signal is chosen to be a delayed square wave.

The proposed technique in Section

It is shown in Figure

Output in unstable system case.

In this paper, adaptive PID (APID) controller is proposed using RLS algorithm which updates the PID gains automatically online to force the actual system to behave like a desired reference model. Numerical examples have been shown to confirm the tracking capability of the proposed controller when it is applied to both stable and unstable systems. It also proves the efficiency of the controller during the changes of system parameters during operation of system. Moreover comparisons are made between the proposed APID and the adaptive controller presented in [

The authors declare that there is no conflict of interests regarding the publication of this paper.