Active disturbance rejection controller (ADRC) has poor control performance under strong disturbance system and large time delay. The main influence factor is the static parameter mechanism. Therefore, an ADRC based on joint optimization of differential evolution algorithm and particle swarm optimization (PSO) is designed. The particle swarm optimization (PSO) algorithm is used to optimize the momentum estimation coefficients of the anti-extended state observer in the disturbance rejection controller.
The servo mechanism triggered by the error threshold is used to improve the calculation speed of dynamic optimization. The mutation, crossover and selection operators of the differential evolution algorithm are used to improve the diversity of the PSO algorithm and prevent falling into trouble. The partial optimum is used to improve the convergence accuracy of the algorithm. The experimental results in thermal time-delay system show that the control effect of the improved algorithm is improved under strong disturbance system and large time-delay condition, and the anti-jamming performance and robustness are improved. Active Disturbances Rejection Control (ADRC) is widely used in robot control, motor control, power system and flight control because it can achieve high-precision control without precise mathematical model of the controlled object and has unique advantages in robustness, anti-interference ability and response speed. Extensive application, with strong industrial practical value [2]. However, due to the large number of parameters related to ADRC and the difficulty of adjustment, there are many improved optimization algorithms for ADRC at present, including ADRC optimized by chaotic particle swarm optimization, immune particle swarm optimization and analog control. These improved algorithms effectively improve the efficiency of parameter adjustment of ADRC and play a comparative role in the practical application of ADRC. The ADRC itself is a static optimization mechanism under the parameter line. It is easy to cause output overshoot and instability under strong disturbance and large time-delay environment such as thermal system, and system error is easy to lead to system disorder [6]. Aiming at this defect, this paper uses differential evolution algorithm (Differential Evolution Algorithm). The joint algorithm of ybrid algorithm, DE and Particle Swarm Optimization (PSO) is used to dynamically optimize the ADRC algorithm, and PSO is used to optimize the momentum estimation coefficients of the extended state observer in ADRC. At the same time, the diversity of the particle swarm optimization algorithm is improved by combining the mutation, crossover and selection operators of the differential evolution algorithm. Prevent PSO from falling into local optimum and improve PSO’s global search ability. PSO algorithm [7] is a population-based stochastic optimization algorithm, which simulates the swarm behavior of birds, insects, fish and so on. The algorithm is initialized as a group of stochastic solutions.
After iteration, the optimal solution is obtained. In each iteration, the particles are updated by tracking the local optimal solution [Pbest] and the global optimal solution [Gbest]. In the formula, [c1, c2] is a learning factor, also known as acceleration constant, [rand?] is a random function in the interval [0, 1]. The main principle of Differential Evolution Algorithms [8] is to perform evolutionary operations according to the operations of hybridization, mutation and selection. New individuals after hybridization and mutation participate in the competition of paternity, and select the generation with good fitness as the next generation according to the fitness of the new generation and the paternity. In the formula, [pr1, pr2, pr3] are three mutually dissimilar individuals selected arbitrarily from the population; [_] is a scaling factor, whose function is to adjust the influence of mutation operation to improve the controllability of mutation operation. In the formula, [j < 1, D,] [D] is the solution space dimension, [PCR] is the mutation probability and [PCR < 0, 1;] [rand bj] is the [j] value of the same random number generator. In the differential evolution algorithm, the principle of genetic substitution optimization is used to select the operation, and the selection substitution is carried out when the evolution index of the offspring is better than that of the paternal generation. If the offspring are not optimized, the paternal generation will continue directly to the offspring. PSO algorithm and differential evolution algorithm are both heuristic algorithms within the population, but the global search ability of PSO algorithm is insufficient, and it is easy to fall into local optimal solution [9]. The local search ability of PSO algorithm can be improved by improving the diversity of population.
Therefore, a hybrid optimization algorithm of particle swarm optimization and differential evolution algorithm is proposed, which uses differential evolution. The crossover operator and selection operator of the algorithm can improve the ability of local search and memory. Meanwhile, the mutation operator can effectively improve the correlation and retrieval speed of the algorithm while ensuring the diversity of the population. The mutation, crossover and selection operators of the differential evolution algorithm are integrated to ensure the existence of excellent individuals in the population and avoid the trapping of the algorithm. Local optimum. In the formula, [_s, _e] is the initial inertia weight and the final inertia weight respectively, [k] is the control factor, which controls the smoothness of the inertia weight curve with time. The principle of PSODE (Particle Swarm Optimization’s Differential Evolution Algorithms) is that when a particle in a group falls into a local optimal value, the following position is determined according to the direction of the particle population, and the following position is also determined according to the optimal individual information of the differential evolution algorithm. The location information of the two algorithms is optimized by combining the two algorithms. It guides the particles trapped in the local optimum to evolve to the position of the global optimum. In the formula, [Xmin, Xmax] is the preset search boundary of the algorithm, [rand 0, 1] is the random number in the random generated [0, 1] interval, through which the particles will be updated continuously. ADRC [10] consists of three main components, the corresponding functions are: non-linear tracking differentiator, which is used to achieve fast non-overshoot tracking of input signals and give good differential signals; extended state observer, which is used to estimate disturbance and state of the system; and non-linear state error feedback controller, which is used to obtain control variables. 。 Generally, the simplest ADRC only contains the nonlinear state error feedback control law, while the most complex ADRC consists of the above three parts. The typical second-order ADRC structure is shown in Figure 1, where [v(t)] is the setting signal, [v 1, thermostatic element V 2] is the tracking signal of [v(t)] and its differential, [e1, e2] is the error, [y(t)] is the output of the controlled system, [w(t)] is the system disturbance, [z1, z2] is the tracking signal of output [y(t)], [z3] is the observation value of unknown disturbance, [u0] is the input of the nonlinear state error feedback controller. The output control signal; [u] is the control signal after disturbance compensation. In the formula, [r] is the tracking speed parameter, [h0] is the filtering factor. In the formula, [beta 01,] [beta 02,] [beta 03] is the momentum estimation coefficient of the extended state observer, which determines the accuracy of the estimated disturbance of the extended state observer, [fal?] is a non-linear feedback function [11]. Due to the inadequacy of the static estimation mechanism of ADRC, the error estimation range tends to overshoot in the face of large disturbance jump or serious time delay, which will lead to system disorder. Therefore, a joint algorithm of differential evolution algorithm and particle swarm optimization is introduced to optimize ADRC dynamically in order to improve the control effectiveness of ADRC systems with strong disturbance and large time delay. Fruit. PSODE hybrid optimization algorithm is based on stochastic solution and uses differential evolution algorithm to control PSO algorithm to avoid falling into local optimal solution, so as to obtain global optimal solution quickly. It has simple implementation structure, fast calculation speed, and can use real coding. It has certain advantages in real-time online optimization algorithm. The ADRC structure of PSODE hybrid optimization algorithm is shown in Fig. 2. The PSODE hybrid optimization algorithm is placed before the extended state observer, and the three momentum estimation coefficients of the extended state observer [beta 01,] [beta 02,] [beta 03] are optimized. Because ADRC has anti-jamming ability, in order to ensure the real-time performance of PSODE hybrid optimization algorithm and improve calculation efficiency, an error threshold [T,] is set up to implement PSODE online optimization after the error exceeds the error threshold, not every operation is optimized, so that the performance of the algorithm is guaranteed at the same time. Efficiency reduces optimization time. At the same time, the complexity of PSODE hybrid optimization algorithm is high. Considering the real-time performance of the system, compatibility of PSODE algorithm must be improved. In order to determine the selection operation after differential evolution in PSODE algorithm, it is necessary to determine the fitness [12] to compare the advantages and disadvantages of parent and offspring. In the formula, [q] is the neighborhood range, [ndi, j] is the dominant value of particles, when [i] is better than [j], the value is 1, otherwise it is 0. In the formula, [et, ess, Mp] are feedback error, steady-state error and overshoot respectively, [Ke, Kss, Kmp] are corresponding coefficients. In this way, in the operation of PSODE algorithm, the particle selection operation can determine the merits of mutation individuals and parents according to fitness [F], so as to retain the elite of the whole population, and make the optimization algorithm easier to obtain the global optimal value. In order to verify the performance of the PSODE joint optimization algorithm proposed in this paper, experiments are carried out in a typical thermal time-delay system [13], and the performance is compared with that of ADRC algorithm optimized by ADRC and PSO. In the formula, [kp, Tp, _] are all system state parameters, which vary with the change of boiler operating conditions. In the experiment, [65%] load state is used, and the main related parameters are set as shown in Table 1. In order to test the robustness of various algorithms, the system load is set to jump from [65%] to [100%], corresponding to the jump perturbation of each parameter. The step response curves of the three algorithms are shown in Fig. 3. It can be seen that the output overshoots of ADRC, PSO? ADRC and PSODE? ADRC controllers are [23.75%,] [14.
53%] and [9.71%], respectively, but compared with ADRC’s system regulation time, PSO? ADRC and PSODE? ADRC’s regulation time are prolonged by 67 s and 81 s, respectively. In order to test the robustness of various algorithms, a long time disturbance signal is set up. The step response curves of the three algorithms are shown in Fig. 4. It can be seen that the output overshoots of ADRC, PSO? ADRC and PSODE? ADRC controllers are 21.87%, 9.13% and 5.26% respectively, but compared with ADRC’s system regulation time, PSO? ADRC and PSODE? ADRC’s regulation time is prolonged by 52 s and 62 s respectively. The experimental results show that the robust and anti-jamming performance of the ADRC optimized by PSODE algorithm are improved to a certain extent compared with PSO? ADRC and ADRC, but the corresponding adjustment time is also increased, which affects the real-time performance of the system to a certain extent. Aiming at the deficiency of static estimation mechanism of ADRC system, which results in large disturbance jump or poor adaptability when time delay is serious, PSO algorithm and differential evolution algorithm are used to optimize ADRC on-line after system error exceeds the threshold value, and the momentum estimation coefficient of extended state observer is adjusted to improve the estimation of extended state observer. The accuracy of calculating disturbance can improve the anti-jamming performance of ADRC in severe time delay and strong disturbance environment while ensuring real-time performance. The experimental results show that the ADRC robustness and anti-jamming performance of PSODE hybrid optimization algorithm are improved, but the real-time performance is affected by the increase of system complexity, which will be the focus of future research.