Stability and convergence analysis of intelligent control algorithm based on fuzzy set theory in dynamic environment

With the rapid development of modern science and technology, a large number of complex and precise physical control systems, chemical control systems and other nonlinear systems have been produced, and the data-driven control problem of nonlinear systems has been a hot research topic in control theory and control engineering [1-2]. Since the 1970s, the development of system model-based control theory has made significant breakthroughs and has become increasingly perfect, especially the linearization technique for nonlinear systems, whose application has been a milestone for the progress of nonlinear science [3-4]. However, with the increasing degree of system complexity, as well as people’s more and more demanding requirements for system control performance indicators, it is difficult to meet the requirements of the actual system control performance by using only the system model-based control theory and linearization techniques to design controllers for nonlinear systems [5-7]. The main reason is that in the process of linearization of nonlinear systems, certain higher-order terms in the actual system are ignored, so the system model obtained by the linearization technique is often not accurate enough, and then the design of controllers using linear system control theory will inevitably reduce the control performance of the system [8-10]. If the higher-order terms in the actual system are not ignored and the complexity of the internal structure of the actual system is taken into account, the exact mathematical model of the actual system may be too complex to be established, and the model-based control method is dependent on the exact model of the system, which undoubtedly brings great difficulties to the study of nonlinear system control problems [11-13]. In addition, even if the actual system model can be accurately established by the mathematical model, the actual system model will be very complex, and the controller designed based on such a complex system model will inevitably be complex, and the robustness and applicability of the controller cannot be guaranteed [14-16].

Since the system model-based control methods are limited in solving the control problems of nonlinear systems, the algorithms that directly utilize the online observable data of the system for the intelligent control of nonlinear systems have gradually become a hot topic of attention and research [17-19]. At present, there is no uniform definition of data-based intelligent control algorithms, which generally refers to the design of a controller using only the online or offline input and output data of the controlled object or the system information obtained through data processing, while the stability, convergence and robustness of the closed-loop system based on the controller can be proved by rigorous mathematical tools under some reasonable assumptions [20-22]. Compared with the model-based control method, the data-based intelligent control algorithm has two advantages, one is that the data-based intelligent control algorithm can not rely on the precise mathematical model of the nonlinear system, which solves the major problem of the nonlinear system control problem that is difficult to accurately model, and the data-based intelligent control algorithm is more flexible in the restriction of the performance index of the nonlinear system. Secondly, the form of its performance indicator function can have other types, such as entropy function, avoiding the incomplete description of quadratic indicators [23-26].

This paper analyzes the feasibility of combining fuzzy set theory and controller optimization tasks from the perspective of optimizing the control performance of control systems. Fuzzy algorithms and traditional immune algorithms are introduced into traditional PID controllers so that the PID controllers can be adaptively adjusted to improve the stability and convergence of intelligent control algorithms. Considering the uncertainty of the controller control parameters under different interaction environments, the stability and convergence analysis method based on the fuzzy immune control algorithm is designed. Comparative analysis is used to verify the relevant performance of the algorithm in this paper, and the performance of the algorithm is further verified in a simulation environment under a variety of local shaded environments and dynamic environments.

2

Control system construction based on fuzzy immune PID algorithm

2.1

Fuzzy set theory

In real life and work, multi-attribute decision-making problems generally have complexity. Traditional mathematics has established a complete theoretical system. It is described in detail by precise expressions and concepts, which is of great help to social production practice. But in fact, it is not possible to solve some uncertainty problems by means of traditional mathematics. Many evaluative attributes are often difficult to be expressed directly and precisely in numerical terms. For example, the good and bad things, the size is not a clear boundary, has a certain degree of ambiguity. This then becomes less reasonable and accurate when solved in the rigorous way of traditional mathematics. The theory of fuzzy mathematics has flourished precisely to characterize things more precisely.

Fuzzy language is a comprehensive evaluation based on fuzzy mathematics [27], which is used to describe the characteristics of objects with ambiguity and uncertainty as abstract representations. This requires mathematical methods to convert fuzzy representations of attributes into precise forms of expression. In fuzzy set theory, the absolute “belonging” in traditional mathematics is transformed into the relative “belonging”. At the same time, it quantifies the belonging relationship between elements and sets, and considers that the degree of belonging of different elements to the same set may be different, and gives the concept of degree of affiliation. The fuzzy evaluation is transformed into the specific mathematical form of affiliation function to effectively deal with unclear and quantitatively difficult fuzzy information.

$\tilde{A}$ is said to be a fuzzy set on U if for all elements x in the domain U, there is a number $\tilde{A} (x)$ between the interval [0, 1] that corresponds to it one-to-one. Called mapping: (1) $ϕ_{\tilde{A}} (x) : U \to [0, 1]$

$\tilde{A} \to ϕ_{\tilde{A}} (x)$ is the affiliation function of the fuzzy set $\tilde{A}$ .

The affiliation function [28] is characterized by the fact that it does not indicate absolute belonging or non-belonging, but indicates a relationship of degree of belonging with a value between (0, 1), which is the biggest difference with traditional sets. The closer the value is to 1, the higher the degree to which x belongs to $\tilde{A}$ . Conversely, the lower the degree of belonging.

Let F(R) be the whole of the fuzzy set over the real number field R if there is $\tilde{A} \in F (R)$ and there exists a closed interval [α, χ] ⊂ R, α ≤ χ such that: 1)

on [α, χ], $ϕ_{\tilde{A}} (x) \equiv 1$ ;

2)

On (−∞, α), $ϕ_{\hat{A}} (x)$ is a right continuous increasing function and $0 \leq ϕ_{\hat{A}} (x) < 1$ ;

3)

on (χ, ∞), $ϕ_{\tilde{A}} (x)$ is a left continuous decreasing function and $0 \leq ϕ_{\tilde{A}} (x) < 1$ . Then $\tilde{A}$ is said to be a fuzzy number on R.

The affiliation function of a triangular fuzzy number can be expressed as: (2) $ϕ_{\tilde{A}} (x) = {\begin{array}{l} \frac{x}{ς - ρ} - \frac{ρ}{ς - ρ}, ρ \leq x \leq ξ \\ \frac{x}{ς - τ} - \frac{τ}{ς - τ}, ξ \leq x \leq τ \\ 0, x \leq ρ or x \geq τ \end{array}$

where ρ ≤ ς ≤ τ, the triangular fuzzy number can be written as: $\tilde{A} = (ρ, ς, τ)$ .

2.2

Principles of fuzzy immune PID controller design

The traditional mathematical model of PID control [29], as shown in equation (3): (3) $u (t) = k_{p} e (t) + k_{i} \int_{0}^{t} e (t) + k_{d} \frac{d e (t)}{d t}$

Where, k_p is the proportional session weight. k_i is the integral link weight. k_d is the differential link weight. e(t) is the deviation of the actual output value from the input value. u(t) is the actual output value.

The traditional PID control model is discretized for the operation of the model, so that it can be obtained: (4) $u (k) = k_{p} e (k) + k_{i} \sum_{j = 0}^{k} e (j) + k_{d} \frac{e (k) - e (k - 1)}{T}$

Where, e(k) is the deviation between the actual value and the input value for the knd sampling. e(k − 1) is the deviation between the actual value and the input value sampled by the intelligent control at the k − 1th time. T is the period sampled by the motor encoder. u(k) is the actual output of the controlled object.

The use of fixed PID parameters to characterize an uncertain model is inaccurate and suffers from high errors. Therefore, its three control parameters need to be adjusted online to cope with the changes occurring in the system by incorporating fuzzy control, the principle of which is shown in Fig. 1.

In this chapter, the fuzzy logic controller uses a two-input three-output MIMO system. Its control principle is shown in Fig. 2, in which the operation process of the core part of the fuzzy controller can be divided into the following four parts:

The input and output values used for fuzzy control are fuzzy quantities, so it is necessary to fuzzify the exact data of the inputs. The discussion domain of the inputs is pre-set and the domain is divided into seven levels. The two input values of the fuzzy logic controller are e and Δe, which are the deviation and increment of deviation from the desired and actual values of the PID controller respectively.

The fuzzy rule base is able to define the relationship of complex nonlinear systems, which is based on a large number of experiments and long-term engineering practice.

Based on the fuzzy relationship, fuzzy rules are established to describe the relationship between model parameters.

Fuzzy reasoning is an approximate reasoning process, and the fuzzy reasoning method chosen in this paper is Mamdni method. Its general form of fuzzy decision reasoning is:

Rule: If (E is H_i) and (ΔE is G_i) then (Z is K_i), then its control expression is: (5) $T_{i} = (H_{i} \times G_{i}) \times K_{i}$

The affiliation function of T_i is: (6) $μ_{T_{i}} (E, Δ E, Z) = μ_{H_{i}} (e_{i}) \cap μ_{G_{i}} (Δ e_{i}) \cap μ_{K_{i}} (z_{i}) \forall e i \in E, \forall Δ e_{i} \in Δ E, \forall z_{i} \in Z$

The full rule correspondence can be expressed as a concatenation of all T_i’s with the rule of inference $T = \cup_{i}^{n} T i$ .

The affiliation function of T is: (7) $μ_{T} (E, Δ E, Z) = \cup_{i = 1}^{n} [μ_{T_{i}} (E, Δ E, Z)]$

The controlled object K can be denoted as K = (H × G) · R and hence the affiliation function of K can be denoted as: (8) $μ_{K} (Z) = \underset{\underset{Δ e \in Δ E}{e \in E}}{\cup} μ_{T} (E, Δ E, Z) \cap [μ_{H} (e) \cap μ_{G} (Δ e)]$

Intelligent control systems need to receive accurate values, so it is necessary to convert the fuzzy quantity output from the fuzzy controller into accurate values that can be used in engineering applications, and the process of changing from fuzzy quantity to clear quantity is defuzzification. The weighted average method is chosen for defuzzification. After fuzzy reasoning to get the set of fuzzy quantity K and weighted average of the affiliation function of K, the output quantity K is transformed into an accurate value and its weighted average expression is: (9) $z_{o u t} = \frac{\int z μ_{K} (z) d z}{\int μ_{K} (z) d z}$

It needs to be discretized in the field of engineering applications: (10) $z_{o u t} = \frac{\sum_{i = 1}^{n} z_{i} μ_{K} (z_{i})}{\sum_{i = 1}^{n} μ_{K} (z_{i})}$

2.3

Improved PID controller construction based on fuzzy immune algorithm

Immunization algorithms are often used in control systems for their good feedback performance. Combining the artificial immune algorithm with fuzzy control realizes a more precise control of the controlled object with fewer rules and ensures the efficiency of the intelligent control system operation.

The artificial immune algorithm operates on the principle that when T a cell detects an antigen invasion, it sends a signal to the enhancing cell T_H and the suppressor cell T_s, and under the joint intervention of the enhancing cell T_H and the suppressor cell T_s it generates B a cell, which B produces an antibody to inhibit the antigen, and finally completes the adaptive regulation of the immune system according to the feedback of the antigen concentration.

The corresponding value of T_H antigen concentration is: (11) $T_{H} (k) = k_{1} \cdot n (k) \cdot T_{H} (k)$

where k₁ is the excitation factor.

The output value of T_s cell is: (12) $T_{S} (k) = k_{2} f [Δ s (k)] n (k) T_{S} (k) = k_{2} f [Δ s (k)]$

where k₂ is the inhibitory factor; f(·) is a nonlinear function describing T_S, where the domain of f(·) is [0, 1]; and the total stimulus to the B cell can be expressed as: (13) $S (k) = T_{H} (k) - T_{S} (k) = k_{1} n (k) - k_{2} f [Δ s (k)] n (k) S (k)$

The discrete form of the positional PID controller is: (14) $u (k) = k_{p} e (k) + k_{i} \sum_{j = 0}^{k} e (j) + k_{d} \frac{e (k) - e (k - 1)}{T}$

The P controller in the PID controller is used to control the accuracy, and a suitable P link can make the overshoot of the system much lower and the stability of the system enhanced. Where P controller is: (15) $u (k) = k_{p} \cdot e (k) u (k)$

With the addition of the immunization mechanism: (16) $u (k) = k_{1} e (k) - k_{2} f [Δ u (k)] e (k) u (k)$ (17) $u (k) = N (1 - γ f [Δ u (k)]) e (k)$ (18) $k_{p} = N (1 - γ f [Δ u (k)])$

Where, N = k₁ is the control reaction speed, the larger the value of N, the faster the reaction speed of the system; γ = k₂/k₁ is the control reaction effect; f(·) = [u(k), Δu(k)] is the inhibition function of T_s; and u(k) is the antigen concentration input, hence the improved immune control PID: (19) $u (k) = N (1 - γ [Δ u (k)]) e (k) + k_{i} \sum_{j = 0}^{k} e (j) + k_{d} \frac{e (k) - e (k - 1)}{T}$

After completing the construction of the immune PID controller, the fuzzy immune controller for the P link is constructed by fusing it with the fuzzy control. The control refinement of the P link enables more accurate control.

The value of k_p is determined by three parameters, i.e., N, γ, f(·). The inhibition function is a nonlinear function with a domain of [0,1], and a fuzzy relationship can also be established between the excitation factor and the inhibition factor through μ, so the following design is carried out.

In order to improve the accuracy of the simulation of the controlled object, the proportional link N is optimized twice by using the fuzzy immune algorithm to obtain more accurate control, and the control process is shown in Fig. 3.

The input of the fuzzy immune processing module is N, u, du/dt, from which γ and f(·) can be obtained through the fuzzy logic relationship, and a more accurate value of the k_p parameter can be obtained, realizing the secondary optimization of the proportional parameters.

2.4

Stability and convergence analysis methods

In order to eliminate the contact force tracking error caused by environmental uncertainty, an adaptive compensation term is introduced for online correction of trajectory error, which also makes the internal structure and parameters of the controller partially changed, and the stability of the intelligent control model needs to be determined to ensure that the system can operate safely. Therefore, a detailed determination process is given for the stability of the intelligent control system, which is used to guide the design of the intelligent controller.

Substituting the adaptive compensation term into the adaptive conduction control strategy, the contact force tracking error equation can be obtained: (20) $m_{d} D^{β} \hat{e} (t) + b_{d} (D^{γ} \hat{e} (t) + Φ (t - λ) + η \frac{f_{d} (t - λ) - f_{e} (t - λ)}{b_{d}}) = f_{e} (t) - f_{d} (t)$

Reorganizing the above equations yields: (21) $m_{d} D^{β} \hat{e} (t) + b_{d} D^{γ} \hat{e} (t) + b_{d} Φ (t - λ) - η e_{f} (t - λ) = e_{f} (t)$

It can be obtained by rewriting and differentiating the pure stiffness contact environment model f_e = k_e(x_e − x) = −k_ee according to the one described earlier: (22) $\begin{array}{l} e = - \frac{f_{e}}{k_{e}} \\ D^{γ} e = - \frac{D^{γ} f_{e}}{k_{e}} \\ D^{β} e = - \frac{D^{β} f_{e}}{k_{e}} \end{array}$

Organizing is available: (23) $\begin{array}{l} m_{d} D^{β} (k_{e} δ x_{e} - f_{d} (t)) + b_{d} D^{γ} (k_{e} δ x_{e} - f_{d} (t)) \\ = m_{d} D^{β} e_{f} (t) + b_{d} D^{γ} e_{f} (t) \\ + k_{e} (e_{f} (t) - b_{d} Φ (t - λ) + η e_{f} (t - λ)) \end{array}$

Assuming that the contact force due to the environmental position estimation error is ${\hat{f}}_{e} = k_{e} δ x_{e}$ and substituting it into the above equation yields: (24) $\begin{array}{l} m_{d} D^{β} ({\hat{f}}_{e} (t) - f_{d} (t)) + b_{d} D^{γ} ({\hat{f}}_{e} (t) - f_{d} (t)) \\ = m_{d} D^{β} e_{f} (t) + b_{d} D^{γ} e_{f} (t) \\ + k_{e} (e_{f} (t) - b_{d} Φ (t - λ) + η e_{f} (t - λ)) \end{array}$

Labeling $v (t) = {\hat{f}}_{e} (t) - f_{d} (t)$ and its substitution into the reduced form gives: (25) $\begin{array}{l} m_{d} D^{β} e_{f} (t) + b_{d} D^{γ} e_{f} (t) + k_{e} (e_{f} (t) - b_{d} Φ (t - λ) + η e_{f} (t - λ)) \\ = m_{d} D^{β} v (t) + b_{d} D^{γ} v (t) \end{array}$

Rewriting the above equation gives: (26) $\begin{array}{l} m_{d} D^{β} e_{f} (t) + b_{d} D^{γ} e_{f} (t) + k_{e} e_{f} (t) + η k_{e} \sum_{i = 1}^{n} e_{f} (t - i λ) \\ = m_{d} D^{β} v (t) + b_{d} D^{γ} v (t) \end{array}$

A Laplace transform is performed to obtain: (27) $\begin{array}{l} m_{d} s^{β} e_{f} (s) + b_{d} s^{γ} e_{f} (s) + k_{e} e_{f} (s) + η k_{e} e_{f} (s) \sum_{i = 1}^{n} e^{- i λ s} \\ = m_{d} s^{β} v (s) + b_{d} s^{γ} v (s) \end{array}$

Defining the intelligent control contact force tracking error e_t(s) as the system output and v(s) as the system input, there is a system transfer function as follows: (28) $\frac{e_{f} (s)}{v (s)} = \frac{m_{d} s^{β} + b_{d} s^{γ}}{m_{d} s^{β} + b_{d} s^{γ} + k_{e} + η k_{e} \sum_{i = 1}^{n} e^{- i λ s}}$

The stability characteristic equation for the transfer function of the above intelligent control system is: (29) $m_{d} s^{β} + b_{d} s^{γ} + η k_{e} \sum_{i = 1}^{n} e^{- i λ s} + k_{e} = 0$

where the sampling period λ generally takes a very small value when 0 < λ < 1, |e⁻λs| ≠ 1, and n is a sufficiently large natural number. Therefore, where and the function can be expressed as: (30) $\sum_{i = 1}^{n} e^{- i λ s} = \frac{1}{1 - e^{- λ s}} - 1$

When the sampling rate is very fast, i.e., sampling period λ ≪ 1, this delay term can be expanded at point 0 using Taylor’s formula: (31) $e^{- λ s} = 1 - λ s + o (s^{2})$

where the cosine term o(s²) denotes the higher order infinitesimal of s², which is negligible, then there is: (32) $e^{- λ s} \approx 1 - λ s$

Substituting the above equation into the stability characteristic equation and rewriting it yields: (33) $m_{d} λ s^{1 + β} + b_{d} λ s^{1 + γ} + λ k_{e} (1 - η) s + η k_{e} = 0$

Observing the above equation, it can be found that the system can be equated to a linear time-invariant system, and the desired intelligent control conductance parameters, sampling period, and adaptive update rate are selected and substituted into the above equation, and then the stability determination of the intelligent control system is carried out.

The pole distribution can be obtained by applying the Routh criterion. From this, it can be seen whether all the poles of the intelligent control system are located within the stability region, and thus whether the system is stable or not can be determined.

For a good intelligent control force control system, it must be ensured that the contact force error tends to zero, i.e., $\lim_{i \to \infty} e_{i} (t) = 0$ . Therefore, a combination of the sum function as well as the intelligent control error transfer function and the Laplace Transform Terminal Value Theorem can be obtained: (34) $\begin{array}{rcl} \lim_{t \to \infty} e_{f} (t) & = & \lim_{s \to 0} s e_{f} (s) \\ = & \lim_{s \to 0} \frac{s (m_{d} λ s^{β + 1} + b_{d} λ s^{γ + 1}) v (s)}{m_{d} λ s^{β + 1} + b_{d} λ s^{γ + 1} + λ k_{e} (1 - η) s + η k_{e}} \\ = & 0 \end{array}$

Clearly, the above equation holds when t → ∞, regardless of whether the input signal v(s) is a pulse, step, ramp, or sinusoidal signal function, then there is f_e → f_d. In practice, this means that the contact force tracking error of the intelligent control with the environment will eventually converge to zero.

3

Control system simulation case study

Consider a nonlinear system of the following form: (35) ${\begin{array}{l} {\dot{x}}_{1} = 0.5 x_{1}^{2} + (1 + 0.1 x_{1}^{2}) x_{2} + D (u_{1}) \cos x_{1} + D (u_{2}) \sin x_{2} \\ {\dot{x}}_{2} = x_{1} x_{2} + D (u_{1}) \sin x_{1} + D (u_{2}) (2 + \cos x_{1} x_{2}) \end{array}$

The above system can be simplified as: (36) $[\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \end{matrix}] = [\begin{matrix} 0.5 x_{1} & 1 + 0.1 x_{1}^{2} \\ 0.5 x_{2} & 0.5 x_{1} \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] + [\begin{matrix} \cos x_{1} & \sin x_{2} \\ \sin x_{1} & 2 + \cos x_{1} x_{2} \end{matrix}] [\begin{matrix} D (u_{1}) \\ D (u_{2}) \end{matrix}]$

y₁ = x₁, y₂ = x₂ of them: (37) $D (u_{i}) = {\begin{array}{l} 0.1 \sin x_{i} u_{i} - 1.25 e^{- 0.0.1 t}, & u_{i} \geq 0.6 \\ 0, & - 0.6 < u_{i} < 0.6 \\ 0.1 \sin x_{i} u_{i} + 1.25 e^{- 0.0.1 t}, & u_{i} \leq - 0.6 \end{array}$

Design parameter Q = diag(10, 10) is selected and C_k = 0.8, γ_1f = γ_2f = 0.01, γ_Ig = γ_2g = 0.5, K = [1, 5], γ_ID = 2, γ_2D = 2 is taken in the bionic adaptive law.

Set the initial condition as X(0) = (0, 10)^T, add the internal disturbance to the controlled system, and replace the system state input X with (1 + δ₁)X, where δ₁ is the random disturbance value, which is taken as δ = 0.1 in this paper, and also add the external disturbance, which is taken as δ₂ = 0.5 sin(t) in this paper.

X₁ and X₂ are the controller based on fuzzy immune PID algorithm designed in this paper and the traditional controller respectively. Figs. 4 and 5 show the output trajectories of X₁ and X₂ with internal disturbance, Figs. 6 and 7 show the output trajectories of X₁ and X₂ with external disturbance, and Figs. 8 and 9 show the output trajectories of X₁ and X₂ with both internal and external disturbance when the magnitude coefficient C_K is taken to be 0, 0.5, and 0.8 with three different values. From the simulation effect in the figure, it can be seen that under the same interference, the controller based on the fuzzy immune PID algorithm designed in this paper has a better adaptive law than the traditional controller, which makes the system biologically adaptive, and the output trajectory of this paper’s controller oscillates in the range of -5 to 1 as a whole, while that of the traditional controller oscillates in the range of -5 to 10 as a whole, and the fluctuation is much larger than that of the controller based on the fuzzy immune PID algorithm based controller, which shows that the algorithm in this paper improves the convergence speed and anti-interference of the intelligent control system. It can be concluded that the controller based on the fuzzy immune PID algorithm proposed in this chapter makes the system highly resistant to interference.

It is obvious through simulation that the design of the fuzzy controller makes the system biologically adaptive, strong stability and anti-interference, which verifies the effectiveness and superiority of the control system based on the fuzzy immune PID algorithm.

4

Optimized hybrid algorithm result analysis

4.1

Analysis of results in multiple localized shading environments

In order to verify the convergence of the optimized hybrid algorithm for maximum power tracking in practical applications, four PV array local shadow distribution scenarios are therefore designed for verification. The P-U characteristic curves of the PV array output and the maximum power tracking output characteristics for the four environments are shown in Fig. 10 and Fig. 11, respectively.

The optimized hybrid algorithm tracks the maximum power value of 705W in environment 1, and the error with the actual maximum power is only 0.05 W. The maximum power value of 1040W in environment 1 is only 0.8 W. The maximum power value of 842W in environment 2 is only 1.59 W. The maximum power value of 842W in environment 3 is only 1.59 W. The maximum power value of 1040W in environment 4 is only 0.8 W. The maximum power value of 842W in environment 5 is only 1.59 W. The maximum power value of 842W in environment 6 is only 1.59 W. The maximum power value of 842W in environment 6 is only 1.59 W. The tracked maximum power value is 859W and the error with the actual maximum power is only 0.35 W. The convergence of the optimized hybrid algorithm for global maximum power tracking is demonstrated.

4.2

Comparison of results in a dynamic environment

In order to be able to explore whether the hybrid algorithm has a good maximum power tracking effect even in dynamic environments, the environment is therefore set to change as follows: the power output characteristics are five peaks in 0-0.4s, and the light intensity of PV1 is reduced from 1000W/s² to 600W/s² at the 0.4s, and the simulation running time is 1 second.

The maximum power tracking output characteristic of the immunization algorithm is shown in Fig. 12. When the external environment in 0.4s by the light from 1000W/s² down to 600W/s², the immune algorithm maximum power output from 709.55W down to 698.69W, but the tracking generated by the degree of steady-state oscillation and in the environment before the change is consistent, it can be shown that in the external environment occurs in the oscillation of the maximum power point tracking sensitive. The maximum oscillation of the system output decreases from 88.5W to 72.8W before the environment change, and it can be found that when the maximum power of the system output decreases, the error oscillations generated by the immune algorithm for the maximum power tracking will also decrease.

The fuzzy controlled maximum power tracking output characteristics are shown in Fig. 13. When the external environment is reduced from 10005/s² to 6005/s² by light in 0.4s, the maximum power output of the fuzzy control decreases from 580.34 to 546.45W, and the oscillation range is within 4W, which does not change much relative to the power output oscillation before the environment change. From this, it can be found that the fuzzy control has little effect on the steady state oscillations of its system output when the environment is changed.

The maximum power tracking output characteristics of the optimized hybrid algorithm are shown in Fig. 14. When the external environment in 0.4s by the light from 1000Ws² down to 600Ws², the power output under the optimized hybrid algorithm is almost no oscillation, and the steady state oscillation of the system output is also kept within 0.4W, which has a very good anti-interference effect.

The maximum power output characteristic curves of optimized hybrid algorithm, immune algorithm, and fuzzy control are compared in the PV system in a locally shaded environment, respectively, and the local output contrasts of the three algorithms in the MPPT in a dynamic environment are shown in Fig. 15. The tracking effect of the optimization hybrid algorithm on the system in the face of sudden changes in the external environment combines the advantages of the immune algorithm and fuzzy control, reduces the oscillations when the output power drops, and the steady state oscillations of the system output are very small. Therefore, it can be judged that the optimized hybrid algorithm has a very good maximum power tracking effect under the local shadow of the dynamic environment.

Through the above comparative analysis, it can be found that the optimized hybrid algorithm designed in this project has a very good tracking effect compared with the traditional immune algorithm and the traditional fuzzy control in the application of the maximum power tracking of the intelligent control system, not only in the maximum power tracking speed has been greatly improved, but also improves the stability of the system, and in the face of the change of the external environment, it also has a very fast and stable tracking The tracking effect is very good in the application of maximum power tracking.

5

Conclusion

Aiming at the shortcomings of the traditional control system, the article constructs a PID controller incorporating the fuzzy immune algorithm, and verifies the effect of the improved hybrid algorithm proposed in this paper on the stability and convergence of the PID controller in a dynamic environment.

The PID controller with fuzzy immune algorithm designed in this paper is subjected to the anti-interference experiments of internal disturbance, external disturbance and parallel internal and external disturbance respectively, and it is found that under the same disturbance, the controller based on fuzzy immune PID algorithm designed in this paper has a better anti-interference ability than the traditional controller, so that the oscillation range of the control system’s output trajectory is kept at -5~1, which is much smaller than that of the traditional controller’s output trajectory, which illustrates that the The PID controller incorporating the fuzzy immune algorithm in this paper improves the convergence speed and stability of the intelligent control system.

In order to further verify the application effect of the optimization algorithm designed in this paper in the actual dynamic environment, this system is built in the simulation platform, and the convergence of the maximum power tracking of the model in this paper is verified in a variety of local shaded environments, and the PID controller incorporating the fuzzy immune algorithm is simulated and analyzed, compared with the controller using the traditional immune algorithm and fuzzy control algorithm alone, respectively, in the dynamic environment, and it is found that this paper’s The optimization algorithm is found to have small steady state oscillations when the system is running under dynamic environment, and the oscillation range is within 0.4W, which is much smaller than that of the fuzzy control algorithm and the immune algorithm.

Language:: English

Publication timeframe:: 1 times per year
Journal Subjects:: Life Sciences, Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics, Physics, other

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Stability and convergence analysis of intelligent control algorithm based on fuzzy set theory in dynamic environment

Weiqiang Hu

Published Online: Sep 26, 2025

Received: Dec 29, 2024

Accepted: Apr 19, 2025

DOI: https://doi.org/10.2478/amns-2025-1065

KeywordsFuzzy control, Immune algorithm, Routh criterion, Disturbance immunity, Intelligent control algorithm

© 2025 Weiqiang Hu, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Fuzzy control, Immune algorithm, Routh criterion, Disturbance immunity, Intelligent control algorithm