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 colorimetric assay 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 ā€œCā€.
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 [24ā€“27].