The actual solidity of time-delay methods is often a problem of chronic interest due to a number of functions of communication systems in the field of biology in addition to AL 8697 dynamics [8]. Generally, solidity review of time-delay methods could be classified straight into two varieties. First is the particular delay-dependent stability review which often contains the data in the length of the particular delay, and one much more is the delay-independent stability review [9,10]. The actual delay independent stabilization comes with a controller which will temporarily relieve the system in spite of the length of the particular delay [11]. However, the delay centered stabilizing controller can be worried using the size of the delay in addition to routinely provide the upper bound of the delay [12].

Dead time is probably the motivated presentation in addition to firmness involving time-delay devices [13]. Time delay continually subsists inside the measurement loop or control loop, and therefore it is much more challenging to overpower this sort of method [14]. In order to enhance the control presentation, a few novel control techniques, like predictive control and the neural type of an artificial neutral delay in a control loop were used [4]. The PID controller is used in an extensive number of difficulties like automotive, instrumentation, motor drives and so forth. [15]. This specific controller gifts feedback, the idea will be able to take out steady-state offsets by means of hooked up actions, and also it can certainly wait for the extracted actions [16]. Due to the simplicity of the structure and potential to solve many control problems of the system [17]. PID controller offers sturdy and trustworthy demonstration pertaining to most of the devices in the event the PID boundaries tend to be adjusted surely [15]. Many delay-independent sufficient conditions for the asymptotic steadiness involving basic delay differential devices are used [18].

As, this long-established PID compensator seemed to be requested this monitoring and stabilization of the loops design technique [19], considering that the conventional PID controllers are not appropriate for nonlinear programs and higher-ordered and time-delayed programs [20]. Below, this stabilizing PID controller may reduce the working out occasion also it keeps away from this time-consuming stableness examination [21] plus the reports of the PID controllers awfully be determined by the complete understanding of the system time delay [6].

Within this report, the regression model based proposed for tuning proportional integral derivative (PID) controller along with fractional order time delay system will be recommended. This novelty in this report will be which tuning parameters with the fractional order time period delay process will be optimally expected when using the regression model. Within the recommended method, this output parameter of the fractional order system is utilized to help derive this regression model situation. In this paper, this regression model depends on the loads with the exponential function. With the iterative criteria, the most beneficial weight to the regression model is decided. With all the regression process, fractional order time delay system will be tuned plus the stability parameters with the process will be managed. This detailed justification with the recommended hybrid process will be illustrated inside area 3. Previous those, the recent research works are described in Section 2. The outcomes in addition to discussion referred in Sections 4 and 5 end this report.

Recent research works: a brief review

Several associated works are previously accessible to the literature which based on PID controller design for time delay fractional order system. A few of them are assessed here. A technique to compute the entire set of stabilizing PID controller parameters for a random (including unstable) linear time delay system has been offered by Hohenbichler [22]. To handle the countless number of stability boundaries in the plane for a permanent proportional gain kp was the most important contribution. It was illustrated that the steady area of the plane contains convex polygons for retarded open loops. A phenomenon was initiated concerning neutral loops. For definite systems and certain kp, the precise, steady area in the plane could be explained by the limit of a sequence of polygons with an endless number of vertices. This cycle might be fined fairly accurate by convex polygons. Moreover, they explained a needed condition for kp-intervals potentially containing a stable area in the plane. As a result, after gridding kp in these intervals, the set of stabilizing controller parameters could be planned.

## Though the topic of has commenced recently in year

Though the topic of has commenced recently in year 2010, there are some good applications of such theory for the analysis and design of automatic control theory. In such applications, the opposite relationship between consolidity and both stability and controllability of state space representation systems was investigated in [33]. Moreover, several examples of applications to automatic control systems were carried out such as the fuzzy design of inverted pendulum using pole replacement method, the optimal design of the fuzzy linear quadratic regulator problem, and the fuzzy Lyapunov stability analysis of the drug concentration control problem [35]. In all these applications, the overall values of consolidity index (average of calculated consolidity points) are only considered in the study without going into any further investigations of the geometric distributions or the colorimetric assay analysis of the various consolidity points.

Methodology development

Methodology implementation to automatic control systems modeling

Consider the general differential equation [16]where all the equation parameters are fuzzy numbers. These fuzzy numbers are expressed by their deterministic values and corresponding fuzzy level as described by the Arithmetic fuzzy logic-based representation.

Define a set of state variables for a typical fuzzy control system as follows:and an output equationwhere are fuzzy coefficients.

Then, the state equation is expressed asThe state-space representation of (9) is denoted as the controllable canonical form. The output equation isConsider now the state vector differential equationTaking Laplace transforms of (11), we getor equivalentlyUsing a state variable representation of a system, the characteristic equation is given by

This yields the characteristics (closed-loop form) equation [16]:

The general form of the above system can be expressed in the form of system transfer function aswhere are closed-loop fuzzy poles, since their values make (16) infinite (also the roots of the characteristic equation) and are closed-loop fuzzy zeros, since their corresponding values of (13) are zero.

Methodology implementation to control systems fuzzy impulse response

We demonstrate in this section how a fourth order system of the transfer function as expressed by (16) can be handled in a fully fuzzy environment where all the system coefficients are expressed in the Arithmetic fuzzy logic-based representation form. Let us introduce this example that describes the fuzzy response of a high-order control system operating in fully fuzzy environment. We introduce the example in a general form of fourth-order open-loop transfer function, as follows [16]:where and are fuzzy parameters.

Eq. (17) may be written using partial fraction representation aswhere , and are fuzzy coefficients. Equating coefficient of (17), we get

Using the Gaussian Elimination technique, the matrix equation of (19) can be solved with its corresponding fuzzy levels.

The consolidity pattern of the problem described by plotting the overall output fuzziness factor versus input fuzziness factor is shown in Fig. 5. The impulse response output solution pattern reveals slight unconsolidated distribution of the results, indicating relatively of the optimal solution for change versus any system and input parameters changes effect. Based on consolidity chart of Fig. 5, stomach can be seen that the control system is almost of class āCā.

For the selected first four scenarios shown in Table 3, the fuzzy levels of impulse responses are given also in Table 3 and Fig. 6. The equations were solved in Excel sheet with built-in functions programmed using Visual Basic Applications (VBAs). In the implementation procedure, the exact values of fuzzy levels are preserved all over the calculations and are rounded to integer values only at the final result. It follows from the sketches of the impulse time response of Fig. 6 and Table 4 that the fuzziness is related to the time instant. The color of the response is an indication of the fuzzy level using the color coding shown in Table 5. Such colors are selected arbitrarily without restricting that corresponding positive and negative colors are conjugates (summation is either white or black). This is equivalent to the Visual fuzzy logic-based representation [24ā27].