CN-118331027-B - PID controller sequential reliable design method considering random and interval mixing uncertainty

CN118331027BCN 118331027 BCN118331027 BCN 118331027BCN-118331027-B

Abstract

The invention discloses a sequential reliable design method of a PID controller considering random and interval mixing uncertainty. Aiming at a PID (proportion-integration-differentiation) closed-loop control system simultaneously comprising random good interval uncertainty parameters, the method calculates the mixing reliability through a probability density evolution equation and a self-adaptive dichotomy, then builds a PID gain optimization column with the minimum control force index by taking the mixing reliability as constraint, and solves the optimization column by utilizing a sequential optimization strategy. According to the method, random and interval description is carried out on uncertain variables, a probability density evolution equation and a self-adaptive dichotomy are combined to calculate mixing reliability, and finally the PID controller with the minimum control force index is designed through a sequential optimization strategy by taking the mixing reliability as a constraint. The present invention can be used in the controller design of a PID closed loop system where there are multiple mixing uncertainty parameters and where probability information for some of the uncertainty parameters is less.

Inventors

WANG LEI
LIU JIAXIANG
Li Zeshang
Mu Yongxiang
WANG XIAOJUN

Assignees

北京航空航天大学

Dates

Publication Date: 20260505
Application Date: 20240318

Claims (7)

1. A PID controller sequential reliable design method considering random and interval mixing uncertainty is characterized in that for a PID closed-loop control system containing random and interval uncertainty variables, mixing reliability is calculated and controller optimal design is carried out by carrying out probability and non-probability interval description on the uncertainty variables, and the method comprises the following steps: the method comprises the steps of establishing a corresponding state space expression and a deterministic system according to an actual engineering system, wherein the deterministic system is a system taking all parameters into account uncertainty; Second step, setting random uncertain parameter as The interval uncertainty parameter is Wherein the probability uncertainty parameter is passed through a probability density function Carrying out quantitative description, wherein the interval uncertain parameters are quantitatively described in an interval mathematical form; the third step, calculating the mixing reliability by combining the self-adaptive dichotomy with a probability density evolution equation, wherein the third step comprises the following steps: the mixture is expressed as a probability reliability curve and a section variable abscissa axis in the section The area surrounded by the two parts is then divided into intervals by an adaptive dichotomy Selecting proper sample points, approximately dispersing integral operation into a plurality of trapezoid areas and summing; Set any interval Is divided into two points , Representation of The approximately trapezoidal area of the inner part, Representation of The approximately trapezoidal area of the inner part, Representation of Approximately trapezoidal area within: ; When the following inequality is satisfied, it is determined that the bipartite converges, then no new sample point need be added, where For a preset convergence threshold: ; If the inequality is not satisfied, continuing to add dichotomy points to the unsatisfied intervals, and repeating the third step until all the intervals meet the convergence criterion, thereby obtaining a series of sample points of the interval uncertain parameters; Let the obtained sample point disperse the original integral area into Each trapezoid has a mixed reliability R formed by the area of each trapezoid The summation performs approximate calculation: ; And fourthly, because the mixed reliability reflects the safety of the uncertainty system, replacing the limit state function in the original deterministic optimization list with reliability constraint, thereby reflecting the influence of the uncertainty on the safety of the system.
2. The method for sequential reliability design of PID controllers taking into account random and interval mixing uncertainties according to claim 1, wherein said first step comprises: Let the state space obtained by the n-degree-of-freedom control system be: ; Wherein t is the time of the time, In order to exert a control force on the surface of the body, In order to disturb the external load, As a state vector of the state vector, As the derivative of the state vector with respect to time, The system is characterized in that the system is an output vector, A is a state transfer matrix, B and E are respectively input matrixes of control force and disturbance external load, and C is an output matrix; the PID gain optimal design expression for deterministic systems is written as: ; Wherein, the 、、、、 And Respectively is 、、 Is a design domain upper and lower bound of (a), As a function of the limit state of the device, Taking an absolute value in response to the allowable value; for the time period of interest, the maximum peak value of the response is Taking absolute value, maximum valley value of response is Taking absolute value, the control force index J is controlled by the applied control force Peak of (2) Sum of the valley values The absolute values are indicated.
3. A method for sequential reliability design of PID controllers taking into account random and interval mixing uncertainties according to claim 2, characterized in that the quantification in the second step is described as: ; Wherein, the And The lower and upper bounds of the uncertainty interval parameter respectively, And For a single output system, the mixed uncertainty is considered in combination with the state space expression obtained by the n-degree-of-freedom system The system of (2) is expressed as: ; Wherein t is the time of the time, In order to exert a control force on the surface of the body, In order to disturb the external load, As a state vector of the state vector, As the derivative of the state vector with respect to time, For outputting vectors, namely responses to be observed; In order to be a state transfer matrix, And Input matrices for controlling force and disturbance external load respectively, Is an output matrix; Uncertainty parameters for a given interval Is the j-th sample point of (2) Wherein And Respectively is The original uncertainty system is simplified into a system with random uncertainty parameters only, and a probability density evolution equation considering failure is adopted for solving: ; Wherein, the In response to And random uncertainty parameters A joint probability density function under the sample points, Is the corresponding velocity vector; Is a step function, namely: ; The system fails when the system response exceeds the allowable value, and the response probability distribution is calculated by the method At random uncertainty parameters Is a distributed domain of (2) Integration over The method comprises the following steps: ; at any time at this time Probability reliability of (2) Expressed as: ; Wherein, the Indicating the probability of an event occurring, Representing a security domain of the device, In order to have an initial moment of reliability requirements, For a time period of Response within.
4. A method for sequential reliability design of PID controllers taking into account random and interval mixing uncertainties according to claim 3, characterized in that said fourth step comprises: ; Wherein, the Is an allowable value required by reliability; the sequential optimization solving strategy is adopted for solving, comprising the following steps of giving allowable values at the ith Under the condition of (1), repeating the first step to the third step to obtain the mixing reliability R under the PID gain obtained by deterministic optimization; Judging the calculated mixed reliability Whether the following convergence criteria are met: ; Wherein, the A preset convergence threshold value is set; If the above formula meets the convergence criterion, the optimization convergence is described, otherwise, the next step, i.e. the response allowable value of the deterministic optimization in the (i+1) th step, is iterated according to the reliability: ; Wherein, the For the two-step deterministic optimization, the value is obtained by responding to the variation of the allowable value at the moment before the allowable value is exceeded, namely, the reliability is equal to the static reliability of 1 at the moment when the reliability is smaller than 1 or the maximum value of the absolute value is responded Adjusting the static reliability The calculation method of (1) is as follows: the absolute value of the peak is greater than the absolute value of the valley: ; The absolute value of the peak value is smaller than the absolute value of the valley value: ; Wherein, the And Probability density functions of the responses, respectively The total area and the shadow area surrounded by the horizontal axis are solved by integrating the probability density function; Is that The time of day is responsive to the upper bound, Is that A moment response lower bound; repeating the second step to the fourth step until the mixing reliability is reached The convergence criterion or the maximum iteration step number of the system is adopted, so that the optimal PID gain which has the minimum energy consumption and meets the safety requirement is obtained.
5. The method for sequentially and reliably designing the PID controller taking into account the random and interval mixing uncertainties according to claim 1, wherein in the second step, given an interval uncertainty parameter sample point, the joint probability density function of the response and the random uncertainty parameter is solved through a probability density evolution equation, and the probability reliability is solved through integration under the condition of taking into account failure, so that the real-time evaluation of the probability reliability is realized.
6. The method for sequential reliability design of PID controller taking into account random and interval mixing uncertainties as recited in claim 1, wherein said third step uses the probability reliability calculated in the second step And selecting proper sample points in the interval uncertain parameters by adopting a self-adaptive dichotomy, and defining the mixed reliability as the mean value of the probability reliability on the interval so as to realize the solution of the mixed reliability with both efficiency and high precision.
7. The method for sequentially and reliably designing the PID controller taking random and interval mixing uncertainties into consideration according to claim 1, wherein in the fourth step, an optimized column is constructed by defining the control force index Jmin in the first step and the mixing reliability obtained in the third step as constraints, and the optimized column solving efficiency is improved by a sequential optimizing strategy to obtain the PID parameters meeting the requirements.

Description

PID controller sequential reliable design method considering random and interval mixing uncertainty Technical Field The invention relates to the technical field of vibration control and reliability evaluation, in particular to a sequential reliable design method of a PID controller considering random and interval mixing uncertainty. Background In practical engineering, active vibration control is an important problem in practical engineering. Unexpected vibrations may reduce the performance of the actual system and even cause system damage. Compared with passive vibration control, the active vibration control can effectively inhibit vibration and noise under the condition of not obviously increasing the weight of the structure, and meets the light-weight design requirements of practical engineering, particularly the aerospace field. There are various well-established empirical or analytical methods for designing active controllers, such as Proportional Integral Derivative (PID) control, pole configuration control, and optimal control. Among the above methods, optimal control, in which a controller is designed to minimize or maximize a target index while satisfying a response extremum or a control force extremum or other constraint, is a common method of guiding the design of the controller. However, there are various uncertainties in actual engineering, such as noise signals in sensor signals, dispersion of material characteristics, and measurement errors, and thus there are errors in the actual system compared to an ideal theoretical model. Because of these uncertainties, the optimal controller designed according to a deterministic system is likely to fail the constraint. In practical engineering, uncertainty should be considered when designing a controller for a system. The traditional robust control can ensure the safety of an uncertain system, but a designed closed-loop system is often too conservative and has high energy consumption. Therefore, a method of designing based on reliability (i.e., "reliable design") is proposed to balance the safety and power consumption of an uncertain system. For certain uncertainty parameters, their statistics are sufficient. In this case, these uncertainties may be quantified with a probability density function, and accordingly, reliability may be assessed with probability theory. Monte Carlo simulation defines the likelihood of failure as the ratio of samples to total samples. However, the monte carlo simulation requires a large amount of sample data, which limits its application in practical engineering. Cornell proposes a first order second moment method in which the reliability index is established under normal distribution and linear limit state functions. However, the first order second moment method lacks invariance, and thus advanced first order second moment is proposed in which a nonlinear limit state function is developed at a design point. Rice first proposes a first pass theory to calculate the probability reliability of the random process response. Davenport solves the probability density function of the steady state response stochastic process extremum under the assumption that the number of passes is a poisson distribution. VANMARCKE solve for the probability density distribution of the extremum under the assumption of a two-state markov process. Li and Chen propose probability density evolution methods based on the principle of probability conservation to obtain a decoupled probability density evolution equation. In this case, the dynamic reliability can be solved by introducing the absorption boundary conditions and integrating the joint probability density function, or a pseudo-random process can be constructed and the calculation of reliability can be converted into a one-dimensional integration problem. The Meng et al propose a semi-analytical extremum method that simulates the correlation between different moments by expanding the optimal linear estimation method, thereby improving the computational efficiency of the extremum method. However, for some uncertainties of dynamic systems in actual engineering, the samples are insufficient and their probability density functions cannot be accurately obtained, so the probability reliability is limited in this case. Their boundaries are more readily available and therefore these uncertainties are quantified in terms of non-probability intervals. Ben Haim and Elishakoff first proposed non-probability interval quantization of uncertainty, and accordingly, reliability in consideration of interval uncertainty has been developed in recent years. Mahadevan and Dey calculated the time dependent reliability of fragile structures with adaptive monte carlo methods. Wang et al performed a time-varying reliability study of the interval uncertainty vibration control system, taking into account the correlation of the first pass theory and the time moment. Chang et al propose a time dependent model to estimate the safe