A sequential reliable design method of PID controller considering random and interval mixed uncertainty

Abstract

This invention discloses a sequential reliability design method for PID controllers considering stochastic and interval-specific uncertainties. This method targets PID (proportional-integral-derivative) closed-loop control systems containing both stochastic and interval-specific uncertainties. It calculates the mixed reliability using probability density evolution equations and an adaptive bisection method, then constructs an optimization formula for the PID gain that minimizes the control force index using the mixed reliability as a constraint, and solves the optimization formula using a sequential optimization strategy. This method describes the uncertain variables stochastically and in intervals, calculates the mixed reliability using probability density evolution equations and an adaptive bisection method, and finally designs a PID controller with the minimum control force index using the mixed reliability as a constraint through a sequential optimization strategy. This invention can be used for controller design in PID closed-loop systems with multiple mixed uncertainties, some of which have limited probabilistic information.

Description

Technical Field

[0001] This invention relates to the field of vibration control and reliability assessment technology, specifically to a sequential reliability design method for PID controllers that considers random and interval mixed uncertainties. Background Technology

[0002] In practical engineering, active vibration control is a crucial issue. Unexpected vibrations can degrade the performance of a system or even cause damage. Compared to passive vibration control, active vibration control can effectively suppress vibration and noise without significantly increasing structural weight, meeting the lightweight design requirements of practical engineering, especially in aerospace. Various mature empirical and analytical methods exist for designing active controllers, such as proportional-integral-derivative (PID) control, pole placement control, and optimal control. Among these methods, optimal control is a commonly used approach to guide controller design. In optimal control, the controller is designed to minimize or maximize the target exponent while satisfying extreme response values, extreme control forces, or other constraints. However, various uncertainties exist in practical engineering, such as noise signals in sensor signals, dispersion of material properties, and measurement errors. Therefore, the actual system differs from the ideal theoretical model. Due to these uncertainties, the optimal controller designed based on deterministic systems may not satisfy the constraints. In practical engineering, uncertainties should be considered when designing controllers for systems. Traditional robust control can guarantee the safety of uncertain systems, but the designed closed-loop systems are often overly conservative and energy-intensive. Therefore, a design approach based on reliability (i.e., "reliable design") has been proposed to balance the safety and energy consumption of uncertain systems.

[0003] For certain uncertain parameters, statistical data is sufficient. In this case, these uncertainties can be quantified using probability density functions, and correspondingly, reliability can be assessed using probability theory. Monte Carlo simulation defines the failure probability as the ratio of the sample to the total sample. However, Monte Carlo simulation requires a large amount of sample data, which limits its application in practical engineering. Cornell proposed a first-order second-moment method, where the reliability index is established under a normal distribution and a linear limit state function. However, the first-order second-moment method lacks invariance, so an advanced first-order second-moment method is proposed, where the nonlinear limit state function is expanded at the design point. Rice first proposed the first-crossing theory to calculate the probabilistic reliability of the stochastic process response. Davenport solved the probability density function of the stochastic process extrema under the assumption that the number of crossings follows a Poisson distribution. Vanmarcke solved the probability density distribution of the extrema under the assumption of a two-state Markov process. Li and Chen proposed a probability density evolution method based on the principle of probability conservation, obtaining a decoupled probability density evolution equation. In this case, dynamic reliability can be solved by introducing absorbing boundary conditions and integrating the joint probability density function, or by constructing a virtual stochastic process and transforming the reliability calculation into a one-dimensional integral problem. Meng et al. proposed a semi-analytical extremum method that improves the computational efficiency of the extremum method by extending the optimal linear estimation method to simulate the correlation between different time points.

[0004] However, for some uncertainties in dynamic systems in practical engineering, the sample size is insufficient, making it impossible to accurately obtain their probability density functions, thus limiting the application of probabilistic reliability in such cases. But their boundaries are easier to obtain, so these uncertainties are quantified in non-probabilistic interval form. Ben Haim and Elishakoff first proposed the non-probabilistic interval quantification of uncertainty, and correspondingly, reliability considering interval uncertainties has been developed in recent years. Mahadevan and Dey calculated the time-dependent reliability of brittle structures using an adaptive Monte Carlo method. Wang et al. conducted a time-varying reliability study on interval-uncertain vibration control systems, considering the correlation between the first-pass theory and the time moment. Chang et al. proposed a time-dependent model to estimate the safety of mechanical structures with interval errors, where the time-dependent limit state function is transformed into a time-independent limit state function. Yang et al. considered the performance degradation of structural resistance over time and the time effect of loads, using time-dependent reliability to evaluate the safety of beam structures.

[0005] In practical engineering, various uncertainties exist. Due to the influence of factors such as sample data and correlation, the appropriate quantification methods also differ, resulting in mixed uncertainties. For these uncertain systems with mixed uncertainties, their reliability is also mixed reliability. Wen et al. proposed a mixed reliability index that simultaneously considers probabilistic and fuzzy uncertainties. Ni and Qiu proposed a mixed reliability model that considers probabilistic and fuzzy uncertainties, which can handle linear or nonlinear state functions. Shi et al. considered probabilistic and fuzzy inputs and proposed two indices to measure the safety of time-varying structures from different perspectives.

[0006] However, the current method still has two main problems: (1) the hybrid reliability index is only used as a measure of system safety and does not participate in guiding the design of PID controller; (2) in the traditional dual-loop solution strategy based on reliability optimization problem, reliability analysis and parameter optimization iteration are coupled, resulting in seriously low optimization efficiency. Summary of the Invention

[0007] To address the aforementioned technical problems, this invention provides a sequential reliability design method for PID controllers that considers both stochastic and interval uncertainties. For PID closed-loop control systems with uncertain variables, this method simultaneously considers stochastic and interval uncertain parameters, performs a mixed reliability assessment of the uncertain system, and uses the mixed reliability as a reference to conduct optimal controller design using a sequential optimization strategy. This method can be used for controller design in PID closed-loop systems with multiple mixed uncertain parameters, some of which have limited probability information. The resulting controller meets the mixed reliability requirements of the system and has high solution efficiency.

[0008] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0009] A sequential reliability design method for a PID controller considering mixed random and interval uncertainties is proposed. For a PID closed-loop control system containing random and interval uncertainties, the method calculates the mixed reliability by describing the uncertainties in probabilistic and non-probabilistic interval forms, and then uses this reliability to perform optimal controller design. The method includes the following steps:

[0010] Step 1: Establish the corresponding state-space expression and deterministic system based on the actual engineering system; the deterministic system is a system that does not consider uncertainty and whose parameters all take nominal values;

[0011] Step 2: Let the random uncertainty parameter be Θ, and the interval uncertainty parameter be b, where the probability uncertainty parameter passes through the probability density function p. Θ (θ) is quantitatively described, and the interval uncertain parameters are quantitatively described through interval mathematical forms;

[0012] Step 3: Calculate the hybrid reliability using the adaptive dichotomy method combined with the probability density evolution equation;

[0013] Step 4: Since hybrid reliability reflects the safety of an uncertain system, the limit state function in the original deterministic optimization formula is replaced with a reliability constraint, thereby reflecting the impact of uncertainty on the system's safety.

[0014] Furthermore, the first step includes:

[0015] Let the state space of the n-degree-of-freedom control system be:

[0016]

[0017] y(t)=CZ(t)

[0018] Where t is time, u(t) is the applied control force, f(t) is the external disturbance load, and z(t) is the state vector. Let y(t) be the derivative of the state vector with respect to time, y(t) be the output vector, A be the state transfer matrix, B and E be the input matrices of the control force and the external disturbance load, respectively, and C be the output matrix.

[0019] The optimal design expression for the PID gain of a deterministic system is written as:

[0020] Searching for K P ,K I ,K D

[0021] min J = max(|u max |,|u min |)

[0022] st g(t)=y cr -max(|y max |,|y min |)≥0

[0023] K Pmin ≤K P ≤K Pmax

[0024] K Imin ≤K I ≤K Imax

[0025] K Dmin ≤K D ≤K Dmax

[0026] Among them, K Pmax K Imax K Dmax KPmin K Imin and K Dmin K P K I K D The design domain has upper and lower bounds, g(t) is the limit state function, and y cr To respond to the allowed value, take the absolute value; [t0 t f [ ] represents the time period of interest; the maximum peak value of the response is y. max Take the absolute value; the maximum valley value of the response is y. min Take the absolute value; the control force index J is the peak value of the applied control force u(t). max Valley value u min This indicates that all values ​​are absolute.

[0027] Furthermore, the quantification in the second step is described as follows:

[0028] b = [b L b U ] = [b c -b r b c +b r ]

[0029] Among them, b L and b U These are the lower and upper bounds of the parameter in the uncertain interval, respectively, b c and b r Let θ and b be the center value and radius of the uncertain parameter, respectively. For a single-output system, combined with the state-space expression of an n-degree-of-freedom system, the system representing the mixed uncertainty (θ, b) is expressed as follows:

[0030]

[0031] y(θ,b,t)=C(θ,b)Z(θ,b,t)

[0032] Given the j-th sample point b of the interval uncertainty parameter b. j ∈[b L b U ], where b L and b U b j The lower and upper bounds are given, and the original uncertain system is simplified to a system with only random uncertain parameters. The probability density evolution equation considering failure is used to solve the problem.

[0033]

[0034] Where, p YΘ (y,θ,b jLet y(t) be the joint probability density function of the response y(t) and the random uncertainty parameter Θ at the sample points. H[·] is the corresponding velocity vector; H[·] is the step function, i.e.:

[0035]

[0036] The system fails when the system response exceeds the allowable value; the response probability distribution is determined by p. YΘ (y,θ,b j ,t) in the distribution domain Ω of the random uncertain parameter Θ θ The integral over gives the probability density function p. Y (y,b j ,t) is:

[0037]

[0038] At this point, the probability reliability R at any time t p (b j ,t) is represented as:

[0039]

[0040] Where Pr(·) represents the probability of the event occurring, Ω s Let t0 represent the security domain, t0 be the initial time when reliability requirements exist, and y(τ) be the response within the time period t0≤τ≤t.

[0041] Furthermore, the third step includes:

[0042] The hybrid representation is the probability reliability curve and the x-axis of the interval variable in the interval [b]. L b U The area enclosed between [b]; then, using an adaptive bisection method, the area within the interval [b] is determined. L b U Select appropriate sample points within the range, and approximate the integral operation by discretizing it into the sum of the areas of several trapezoids;

[0043] Let the bisection point of any interval [b1 b2] be... S represents the approximate trapezoidal area within [b1 b2], S1 represents the approximate trapezoidal area within [b1 b3], and S2 represents the approximate trapezoidal area within [b3 b2].

[0044] S1 = 0.5(R) p (b1)+R p (b3))(b3-b1)

[0045] S2 = 0.5(R) p (b2)+R p (b3))(b2-b3)

[0046] S = 0.5(R) p (b1)+R p (b2))(b2-b1)

[0047] The binary search is considered convergent when the following inequality is satisfied, in which case no new sample points need to be added, where ε s The pre-set convergence threshold:

[0048] |S-(S1+S2)|<ε s

[0049] If the above inequality is not satisfied, add bisection points to the intervals that are not satisfied, and repeat the third step until all intervals meet the convergence criterion, thereby obtaining a series of sample points for interval uncertain parameters.

[0050] Suppose that the obtained sample points discretize the original integration region into N. s The reliability R of the approximate trapezoid is determined by the area S of each trapezoid. i Summation for approximate calculation:

[0051]

[0052] Furthermore, the fourth step includes:

[0053] Searching for K P ,K I ,K D

[0054] min J = max(|u max |,|u min |)

[0055] st R≥R cr

[0056] K Pmin ≤K P ≤K Pmax

[0057] K Imin ≤K I ≤K Imax

[0058] K Dmin ≤K D ≤K Dmax

[0059] Among them, R cr This is the allowable value for reliability requirements;

[0060] A sequential optimization strategy is employed to solve the problem, including: at the i-th given allowable value In this case, repeat steps one through three to obtain the hybrid reliability R under the deterministic optimization of the PID gain;

[0061] Determine whether the calculated hybrid reliability R meets the following convergence criterion:

[0062] 0≤RR cr ≤ε R

[0063] Where, ε R The pre-set convergence threshold;

[0064] If the above formula meets the convergence criterion, it indicates that the optimization has converged; otherwise, iterate to the next step based on the reliability, i.e., the allowable response value for the (i+1)th step of deterministic optimization:

[0065]

[0066] in, In the two-step deterministic optimization, the change in the allowable value of the response is adjusted by the static reliability η, which is the time before the response exceeds the allowable value (i.e., when the reliability is less than 1) or when the absolute value of the response reaches its maximum (i.e., when the reliability is equal to 1). The static reliability η is calculated as follows:

[0067] The absolute value of the peak is greater than the absolute value of the trough:

[0068]

[0069] The absolute value of the peak is less than the absolute value of the trough:

[0070]

[0071] Among them, S total and S shadow These are the probability density functions p of the response. Y (y,b j The total area enclosed by the x-axis and the shaded area (t) are solved by integrating the probability density function.

[0072] Repeat steps two through four until the convergence criterion of the hybrid reliability R or the maximum number of iterations is reached, thereby obtaining the optimal PID gain with the minimum energy consumption and in compliance with safety requirements.

[0073] Furthermore, in the second step, given a sample point of uncertain parameters in an interval, the joint probability density function of the response and the random uncertain parameters is solved by the probability density evolution equation. Considering failure, the probabilistic reliability is solved by integration to achieve real-time evaluation of probabilistic reliability.

[0074] Furthermore, the third step uses the probabilistic reliability R calculated in the second step.p (b j The adaptive bisection method is used to select appropriate sample points within the interval of uncertain parameters. The mixed reliability is defined as the mean of the probabilistic reliability over the interval, so as to achieve a solution of mixed reliability that takes into account both efficiency and high accuracy.

[0075] Furthermore, in the fourth step, the minimum control force index J in the first step and the hybrid reliability obtained in the third step are defined as constraints to construct an optimization formula. The efficiency of solving the optimization formula is improved by a sequential optimization strategy, and the required PID parameters are obtained.

[0076] Beneficial effects:

[0077] (1) The hybrid reliability analysis performed in the first three steps of this invention can effectively evaluate the safety of uncertain systems, and in the fourth step, by introducing hybrid reliability constraints in the optimization formula, the hybrid reliability is used to guide the design of PID controllers.

[0078] (2) In the fourth step, the present invention adopts a sequential optimization strategy, which decouples the hybrid reliability assessment from the PID controller optimization design of the deterministic system by iterating the response allowable value, thereby achieving efficient optimization solution. Attached Figure Description

[0079] Figure 1 This is a flowchart of a sequential reliable design method for a PID controller that considers random and interval mixed uncertainties according to the present invention;

[0080] Figure 2(a) and Figure 2(b) are schematic diagrams of the areas involved in the reliability calculation process of the adaptive bisection method in this invention; wherein, Figure 2(a) is a schematic diagram of the reliability curve of the original interval, and Figure 2(b) is a schematic diagram of the reliability curve of the interval after being divided by the adaptive bisection method.

[0081] Figures 3(a), 3(b), 3(c), and 3(d) are schematic diagrams illustrating the selection of the static reliability analysis time in the sequential optimization method of this invention; wherein, Figure 3(a) is R s <1 and |y max |>|y min | The situation; Figure 3(b) shows R s =1 and |y max |>|y min | The situation; Figure 3(c) shows R s <1 and |y max |<|y min | The situation; Figure 3(d) shows R s =1 and |y max |<|y min | situation;

[0082] Figures 4(a), 4(b), 4(c), 4(d), 4(e), and 4(f) are schematic diagrams of the static reliability solution of this invention; among them, Figure 4(a) is... The situation is shown in Figure 4(b). The situation is shown in Figure 4(c). The situation is shown in Figure 4(d); cr = y (t s The case of ); Figure 4(e) is -y cr > y (t s The case of ); Figure 4(f) is -y cr < y (t s ) situation;

[0083] Figure 5 This is a schematic diagram of a 5-DOF mass spring damping system according to Embodiment 1 of the present invention;

[0084] Figures 6(a), 6(b), 6(c), and 6(d) show the specific results under the optimal PID gain in Embodiment 1 of the present invention; wherein, Figure 6(a) is the iterative process curve of sequential optimization; Figure 6(b) is the response probability density function when the interval uncertainty parameter takes the nominal value; Figure 6(c) is the probability reliability result for each sample point; and Figure 6(d) is the relative error of the probability reliability results obtained by the two methods.

[0085] Figure 7 This is the result of the dual-loop optimization solution strategy in Example 1;

[0086] Figure 8(a) and Figure 8(b) are schematic diagrams of a stiffened plate with mixed uncertainties in density and elastic modulus according to Embodiment 2 of the present invention; wherein, Figure 8(a) is a schematic diagram of loading and PID controller position; and Figure 8(b) is a schematic diagram of uncertain parameters.

[0087] Figures 9(a), 9(b), and 9(c) show different allowable reliability values ​​R. cr The reliability curve, optimization iteration history, and detailed results of the response probability density distribution under the central values ​​of the interval parameters corresponding to the optimal PID controller are shown in Figure 9(a). cr = 0.9; Figure 9(b) shows the result for R. cr = 0.95; Figure 9(c) shows the result for R. cr = 0.97. Detailed Implementation

[0088] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0089] like Figure 1As shown, this invention provides a sequential reliable design method for a PID controller that considers random and interval mixed uncertainties, comprising the following steps:

[0090] Step 1: Establish the corresponding state-space expression or transfer function based on the actual engineering system. Taking the state-space expression form as an example, let the state-space expression of an n-degree-of-freedom system be:

[0091]

[0092] y(t)=CZ(t)

[0093] Where t is time, u(t) is the applied control force, f(t) is the external disturbance load, and Z(t) is the state vector. Let y(t) be the derivative of the state vector with respect to time, and y(t) be the output vector, i.e., the response to be observed. A is the state transfer matrix, B and E are the input matrices for the control force and the external disturbance load, respectively, and C is the output matrix.

[0094] Let the PID gain of the output feedback be K. C =[K P K I K D ], where K P K I K D These are the proportional gain, integral gain, and derivative gain, respectively. Taking a single control force system as an example, the control force u(t) is:

[0095]

[0096] Where e(t) is the error signal between the response and the reference signal. Let τ be the integral term of the error signal from 0 to time t, τ be the integral variable, and e(τ) be the error signal containing the integral variable.

[0097] For PID gain design, on the one hand, the response safety requirements should be met, namely:

[0098] g(t) = y cr -|y(t)|t0≤t≤t f

[0099] Where g(t) is the limit state function, y(t) is the response, and y cr In response to the allowed value (absolute value, greater than 0), [t0t f [The time period of interest], the limit state function can be used as a constraint condition for optimization design. g(t) ≥ 0 indicates that the response does not exceed the allowable value, and the closed-loop system is safe. Conversely, g(t) < 0 indicates system failure. Let the maximum peak value (absolute value) of the response be y. maxThe maximum valley value (absolute value) of the response is y. min At this point, the limit state function can be further written as:

[0100] g(t) = y cr -max(|y max |,|y min |)t0≤t≤t f

[0101] For PID gain design, on the other hand, the goal of minimizing energy consumption should be met. The control force index J can be expressed as the peak value (absolute value) of the applied control force u(t). max Valley value (absolute value) u min This can be expressed as:

[0102] J = max(|u max |,|u min |)

[0103] In summary, the optimal design expression for PID gain in a deterministic system can be written as:

[0104] Searching for K P ,K I ,K D

[0105] min J = max(|u max |,|u min |)

[0106] st g(t)=y cr -max(|y max |,|y min |)≥0

[0107] K Pmin ≤K P ≤K Pmax

[0108] K Imin ≤K I ≤K Imax

[0109] K Dmin ≤K D ≤K Dmax

[0110] Among them, K Pmax K Imax K Dmax K Pmin K Imin and K Dmin K P K I K DThe upper and lower bounds of the design domain, when set appropriately, can improve optimization efficiency.

[0111] Step 2: Let the random uncertainty parameter be Θ, and the interval uncertainty parameter be b, where the probability uncertainty parameter passes through the probability density function p. Θ (θ) is quantitatively described, and the interval uncertainty parameter is quantitatively described using the following interval mathematical form:

[0112] b = [b L b U ] = [b c -b r b c +b r ]

[0113] Among them, b L and b U These are the lower and upper bounds of the parameter in the uncertain interval, respectively, b c and b r Here, θ and b represent the center value and radius of the uncertain parameter, respectively. Taking a single-output system as an example, and combining the state-space expression obtained from an n-degree-of-freedom system, the system considering mixed uncertainties (θ, b) can be expressed as:

[0114]

[0115] y(θ,b,t)=C(θ,b)Z(θ,b,t)

[0116] At the j-th sample point b of the given interval uncertainty parameter b. j ∈[b L b U (where b) L and b U b j Given the lower and upper bounds of the system, the original uncertain system is simplified to a system with only random uncertain parameters, which can then be solved using the probability density evolution equation:

[0117]

[0118] Where, p YΘ (y,θ,b j Let y(t) be the joint probability density function of the response y(t) and the random uncertain parameters at the sample point Θ. This is the corresponding velocity vector. The initial conditions for the above equations are:

[0119] p YΘ (y,θ,b j ,t)| t＝0 =δ(y-y0)p Θ (θ)

[0120] Where δ(·) is the Dirac function, p Θ y(t) is the probability density function of the random uncertain parameter θ, and y0 is the initial state of y(t). The boundary conditions of the probability density evolution equation are:

[0121] p YΘ (y,θ,b j ,t)| y→±∞ =0

[0122] After solving for the joint probability density function p YΘ (y,θ,b j After ,t), by considering the joint probability density function in the distribution domain Ω of the random uncertain parameters. θ Integrating over the given information, the probability density function p of the response is obtained. Y (y,b j ,t) can be calculated from this:

[0123]

[0124] The system fails when its response exceeds the allowable value. The probabilistic reliability R at any time t is given by this condition. p (b j ,t) can be represented as:

[0125]

[0126] Where Pr(·) represents the probability of the event occurring, Ω s Let represent the safety region, t0 be the initial time at which the reliability requirement exists, and y(τ) be the response during the time interval t0 ≤ τ ≤ t. Considering system failure, the probability density evolution equation can be rewritten as:

[0127]

[0128] Where H[·] is the step function, i.e.:

[0129]

[0130] That is, for the portion of the response exceeding the allowable value, its probability density becomes 0, corresponding to a decrease in reliability. Therefore, the reliability remains constant or decreases over time. The specific solution to the above equation is obtained through numerical calculation using the finite difference method (a well-known method).

[0131] Step 3: Calculate the hybrid reliability using the adaptive dichotomy method combined with the probability density evolution equation.

[0132] For the probabilistic reliability of a single sample point, at the final time t of the analysis... f It must be the minimum value, which can be simplified as After introducing the interval parameter, the hybrid reliability R can be expressed as the reliability over the interval [b]. L b U The average probability reliability within ], i.e.:

[0133]

[0134] The integral part can be represented as the probability reliability curve intersecting the x-axis of the interval variable within the interval [b]. L b U The area enclosed between [b] is shown in Figure 2(a). Then, an adaptive bisection method is used to define the area within the interval [b]. L b U Select suitable sample points within the interval [b1, b2], and approximate the integral operation by discretizing it into the sum of the areas of several trapezoids. Let the bisection point of any interval [b1, b2] be... S represents the approximate trapezoidal area within [b1 b2], S1 represents the approximate trapezoidal area within [b1 b3], and S2 represents the approximate trapezoidal area within [b3 b2].

[0135] S1 = 0.5(R) p (b1)+R p (b3))(b3-b1)

[0136] S2 = 0.5(R) p (b2)+R p (b3))(b2-b3)

[0137] S = 0.5(R) p (b1)+R p (b2))(b2-b1)

[0138] The binary search is considered convergent when the following inequality is satisfied, in which case no new sample points need to be added, where ε s The pre-set convergence threshold:

[0139] |S-(S1+S2)|<ε s

[0140] If the above inequality is not satisfied, then add bisection points to the unsatisfied intervals. Taking the interval [b1 b3] as an example, as shown in Figure 2(b), the newly added bisection points are: The area of ​​the newly divided trapezoid is (where S) 1,1 and S 1,2 To add a bisection point b 1,3 (Areas of the two new trapezoidal regions resulting from the division):

[0141] S 1,1 =0.5(R) p (b1)+R p (b1,3 ))(b 1,3 -b1)

[0142] S 1,2 =0.5(R) p (b 1,3 )+R p (b3))(b3-b 1,3 )

[0143] S1 = 0.5(R) p (b3)+R p (b1))(b3-b1)

[0144] Similar convergence criteria are:

[0145] |S1-(S 1,1 +S 1,2 )|<ε s

[0146] Repeat step three for any interval that does not meet the convergence criterion, until all intervals meet the convergence criterion. This yields a series of sample points for the interval's uncertain parameters. Let these sample points discretize the original integration region into N. s The reliability of a mixture of trapezoids can be calculated from the area S of each approximate trapezoid. i Summation for approximate calculation:

[0147]

[0148] Step 4: Hybrid reliability can reflect the safety of an uncertain system. Therefore, the limit state function in the original deterministic optimization formula can be replaced with a reliability constraint, thereby demonstrating the impact of uncertainty on system safety.

[0149] Searching for K P ,K I ,K D

[0150] min J = max(|u max |,|u min |)

[0151] st R≥R cr

[0152] K Pmin ≤K P ≤K Pmax

[0153] K Imin ≤K I ≤K Imax

[0154] K Dmin ≤KD ≤K Dmax

[0155] Among them, R cr This represents the allowable value for reliability requirements. For solving this optimization formula, the traditional double-loop optimization strategy is computationally inefficient; therefore, a sequential optimization strategy is adopted: at the i-th given allowable value... In this case, repeating steps one through three yields the hybrid reliability under the deterministically optimized PID gain.

[0156] If the calculated hybrid reliability R meets the following convergence criterion:

[0157] 0≤RR cr ≤ε R

[0158] Where, ε R If the above formula is satisfied, the optimization has converged; otherwise, the allowable response value for the next deterministic optimization step (step i+1) is determined based on the reliability threshold.

[0159]

[0160] in, The change in the allowable value of the response in the two-step deterministic optimization is solved through the following process:

[0161] The response probability density distribution is selected when the interval uncertainty parameter takes a nominal value. Let the upper and lower boundaries of the probability density distribution be respectively... and y (t), based on the relationship between the calculated mixed reliability result and 1, as shown in Figures 3(a), 3(b), 3(c), and 3(d), we consider different cases: at this time, the allowable response values ​​in the figures are all the allowable response values ​​of the i-th step deterministic optimization, that is...

[0162] Case 1: R s <1 and |y max |>|y min The system may fail at the peak, as shown in Figure 3(a):

[0163] Static reliability analysis is performed at the moment before the response exceeds the allowable value; let this moment be t. s According to the allowable value and t s Always respond to the upper bound The magnitude relationship, the static reliability η is (at this time) ):

[0164]

[0165] Figure 4(a) corresponds to the above formula. In the case shown in Figure 4(c), the above equation is correct. In the case shown in Figure 4(b), the above equation is correct. In this situation, S total and S shadow These are the probability density functions p of the response in Figure 4(b). Y (y,b j The total area enclosed by the x-axis and the y-axis, and the area of ​​the shaded region. Specifically, this can be solved by integrating the probability density function:

[0166]

[0167]

[0168] In t s At any time To make the reliability of the next iteration step close to R cr The static reliability at this point should be adjusted to η. i =1+ΔR, using the static reliability calculation expression in this case, Replace y in the expression cr It can be obtained If the value is specified, then the allowable offset for this situation is:

[0169]

[0170] Scenario 2: R s =1 and |y max |>|y min |: The system has not failed and the peak is closer to the allowable value, as shown in Figure 3(b):

[0171] Select the maximum peak y max The corresponding time is t s Then the allowable offset of the response value is:

[0172]

[0173] Here The specific solution method is the same as that for case one.

[0174] Scenario 3: R s <1 and |y max |<|y min The system may fail at a trough, as shown in Figure 3(c):

[0175] Static reliability analysis is performed at the moment before the response exceeds the allowable value; let this moment be t. s According to the allowable value and t s Always respond to the upper bound The magnitude relationship, the static reliability is (at this time) ):

[0176]

[0177] Figure 4(d) corresponds to the above equation -y cr = y (t s In the case of -y, Figure 4(f) corresponds to the above equation. cr < y (t s In the case of -y, Figure 4(e) corresponds to the above equation. cr > y (t s In the case of S, total and S shadow These are the probability density functions p of the response in Figure 4(e). Y (y,b j The total area enclosed by the x-axis and the y-axis, and the area of ​​the shaded region. Specifically, this can be solved by integrating the probability density function:

[0178]

[0179]

[0180] In t s At any time To make the reliability of the next iteration step close to R cr The static reliability at this point should be adjusted to η. i =1+ΔR, using the static reliability calculation expression in this case, Replace y in the expression cr It can be obtained If the value is specified, then the allowable offset for this situation is:

[0181]

[0182] Case 4: R s =1 and |y max |<|y min |: The system has not failed and the trough is closer to the allowable value, as shown in Figure 3(d):

[0183] Select the maximum trough (take the absolute value) y min The corresponding time is t s Then the allowable offset of the response value is:

[0184]

[0185] Here The specific solution method is the same as that for case two.

[0186] Repeat the above steps until the reliability convergence criterion or the maximum number of iterations is reached. This yields the optimal PID gain that minimizes energy consumption and meets safety requirements.

[0187] Example 1:

[0188] like Figure 5 As shown, the 10-DOF mass-spring-damped system includes a first mass block 1, a second mass block 2, a third mass block 3, a fourth mass block 4, and a fifth mass block 5. Adjacent mass blocks are connected by springs and dampers. All mass blocks have equal mass, all springs have equal stiffness, and all dampers have equal damping. Translational degree of freedom ux exists only in the horizontal direction. The mass, spring stiffness, and damping of each mass block are identical and are all uncertain parameters, as shown in Table 1 (i.e., mass m is a uniformly distributed random parameter between 0.95 and 1.05, damping c is a parameter in the interval [4.75 5.25], and stiffness k is a normally distributed random parameter with a mean of 1000 and a standard deviation of 33.33). An application of... Figure 5 The diagram shows a horizontal pulse load f(t) with an amplitude of 10 kN, and a control force u(t) is applied to the fourth mass block 4. Let the horizontal displacement response of the fifth mass block 5 be x(t). The PID controller design requirement is that the displacement response after t = 1.25 seconds does not exceed the allowable value x. cr The reliability of 0.03 meters is R. cr =95%. The PID gain design range and optimal PID parameters are shown in Table 2. To verify the accuracy of the adaptive bisection method in calculating the mixed reliability, 50 sample points were selected on average within the interval uncertainty parameter to calculate the probabilistic reliability. The reliability results of these 50 sample points were calculated on the curve obtained by the adaptive bisection method, and the relative error of the probabilistic reliability obtained by the two methods was calculated. The specific results are shown in Figures 6(a), 6(b), 6(c), and 6(d), where Figure 6(a) is the iterative history curve of the sequential optimization; Figure 6(b) is the response probability density function when the interval uncertainty parameter takes a nominal value; Figure 6(c) is the probabilistic reliability result for each sample point; and Figure 6(d) shows the relative error of the probabilistic reliability results obtained by the two methods. At the same time, the traditional double-loop optimization solution strategy was used to solve the same optimization problem to verify the efficiency and accuracy of the sequential solution strategy. The specific results are shown in Table 3 and Figure 7 As shown in Table 3, the PID gain results, optimization index J results, overall computation time, probability density evolution, and number of calls for hybrid reliability assessment obtained by the sequential optimization strategy and the dual-loop optimization strategy are listed.

[0189] Table 1

[0190]

[0191] Table 2

[0192]

[0193]

[0194] Table 3

[0195]

[0196] Table 2 shows that after considering the mixed uncertainties, the optimization index under the optimal PID gain is larger than that obtained by the initial deterministic optimization, which reflects the contradiction between system safety and energy consumption, and illustrates the importance of reliable design. Figures 6(c) and 6(d) show that the relative errors of the probabilistic reliability distributions obtained by the two methods are very small, and the adaptive bisection method can achieve sufficient accuracy with very few sample points. This demonstrates the advantages of the adaptive bisection method.

[0197] from Figure 7 As shown in Table 3, the results obtained by the two solution strategies are very similar. Since the sequential optimization strategy decouples PID gain optimization and hybrid reliability assessment, the number of reliability assessments is significantly reduced compared to the dual-loop optimization strategy, thus greatly improving its optimization efficiency.

[0198] Example 2:

[0199] Figures 8(a) and 8(b) show schematic diagrams of stiffened plates with four-sided constraints and six degrees of freedom. The elastic modulus and density of the stiffened plate material are uncertain parameters, and the density is an interval parameter ρ. I = [7410 8190] kg / m 3 The elastic modulus is a normally distributed random uncertainty parameter E ~ N(190, 12.667). 2 The finite element model was created in AYSYS using SHELL181 elements. The system damping is Rayleigh damping, P = p α M+p β K, damping coefficient p α =0.05 and p β =0.001. Two peak 10 N disturbance loads, as shown by the arrows in Figure 8(a), are applied at the center of the stiffened plate block. The PID controller sensor is applied at the square point, and the actuator is applied at the circular point. The controller design requirement is that the absolute value of the response |z(t)| does not exceed the allowable value z after 0.25 seconds. cr The reliability of 0.6mm is R. cr R crThe optimal PID gain results for different values ​​are shown in Table 4, and the closed-loop response results for the corresponding PID gain are shown in Figures 9(a), 9(b), and 9(c). For Figures 9(a), 9(b), and 9(c), the main line is the reliability curve for sample points in different intervals, the upper left corner is the response probability density distribution when the interval parameter takes the nominal value, and the lower right corner is the sequential iteration history curve.

[0200] Table 4

[0201]

[0202] As can be seen from the results in Figures 9(a), 9(b), and 9(c), higher reliability results depend on a smaller z. cr The deterministic optimization result is given below. If the reliability result in one step is less than R... cr This will result in a smaller z cr This influences the next deterministic optimization step, meaning the response constraints in the next deterministic optimization step tend to be safer, and vice versa. This demonstrates the rationality of adjusting the allowable response value in the sequential optimization strategy. Table 4 shows a larger R... cr A larger J corresponds to a requirement for greater energy consumption to achieve a more secure system; therefore, in reliable design, z... cr The settings need to balance system security and energy consumption.

Claims

1. A sequential reliable design method for a PID controller considering random and interval mixed uncertainties, characterized in that, For PID closed-loop control systems containing random and interval uncertainties, the following steps are taken to calculate the hybrid reliability and use it for optimal controller design by describing the uncertainties in terms of probabilistic and non-probabilistic intervals: Step 1: Establish the corresponding state-space expression and deterministic system based on the actual engineering system; the deterministic system is a system that does not consider uncertainty and whose parameters all take nominal values; Step 2: Let the random uncertainty parameter be... The interval uncertainty parameter is The probability uncertainty parameter is obtained through the probability density function. Quantitative description is performed, and interval uncertain parameters are quantitatively described using interval mathematical forms; Step 3: Calculate the hybrid reliability using the adaptive bisection method combined with the probability density evolution equation, including: The hybrid representation is the probability reliability curve and the x-axis of the interval variable in the interval. The area enclosed between the two points; then, the adaptive bisection method is used to find the area within the interval. By selecting appropriate sample points, the integral operation is approximately discretized into the sum of the areas of several trapezoids; Let any interval The bisection point is , express The approximate trapezoidal area inside, express The approximate trapezoidal area inside, express Approximate trapezoidal area within: ； The binary search is considered convergent when the following inequality is satisfied, in which case no new sample points need to be added: The pre-set convergence threshold: ； If the above inequality is not satisfied, add bisection points to the intervals that are not satisfied, and repeat the third step until all intervals meet the convergence criterion, thereby obtaining a series of sample points for interval uncertain parameters. Suppose that the obtained sample points discretize the original integration region as follows: The reliability R of the approximate trapezoids is determined by the area of ​​each trapezoid. Summation for approximate calculation: ； Step 4: Since hybrid reliability reflects the safety of an uncertain system, the limit state function in the original deterministic optimization formula is replaced with a reliability constraint, thereby reflecting the impact of uncertainty on the system's safety.

2. The sequential reliable design method for a PID controller considering random and interval mixed uncertainties according to claim 1, characterized in that, The first step includes: Let the state space of the n-degree-of-freedom control system be: ； Where t is time, To exert control, For the disturbance external load, For state vectors, Let be the derivative of the state vector with respect to time. Let A be the output vector, B be the state transfer matrix, E be the input matrices of the control force and the disturbance load, respectively, and C be the output matrix. The optimal design expression for the PID gain of a deterministic system is written as: ； in, , , , , and They are respectively , , The upper and lower boundaries of the design domain, Let be the limit state function. In response to the allowed value, take the absolute value; The time period of interest; the maximum peak value of the response is Take the absolute value; the maximum valley value of the response is Take the absolute value; the control force index J is determined by the applied control force. peak Valley value This indicates that all values ​​are absolute.

3. The sequential reliable design method for a PID controller considering random and interval mixed uncertainties according to claim 2, characterized in that, The quantitative description in the second step is as follows: ； in, and These are the lower and upper bounds of the parameter in the uncertain interval, respectively. and These represent the center value and radius of the uncertain parameter, respectively; for a single-output system, considering the mixed uncertainties, the state-space expression obtained from the n-degree-of-freedom system is combined. The system is represented as: ； Where t is time, To exert control, For the disturbance external load, For state vectors, Let be the derivative of the state vector with respect to time. This is the output vector, i.e., the response that needs to be observed; For the state transfer matrix, and These are the input matrices for the control force and the external disturbance load, respectively. This is the output matrix; Given interval uncertain parameters The j-th sample point ,in and They are respectively The lower and upper bounds are given, and the original uncertain system is simplified to a system with only random uncertain parameters. The probability density evolution equation considering failure is used to solve the problem. ； in, In response and random uncertain parameters The joint probability density function at the sample points. It is the corresponding velocity vector; It is a step function, that is: ； The system fails when the system response exceeds the allowable value; the response probability distribution is determined by... In random uncertain parameters Distribution area The probability density function is obtained by integrating over the integral. for: ； At any given moment Probabilistic reliability Represented as: ； in, Indicates the probability of an event occurring. Indicates a security domain. For the initial moment where reliability requirements exist, Time period The response within.

4. The sequential reliable design method for a PID controller considering random and interval mixed uncertainties according to claim 3, characterized in that, The fourth step includes: ； in, This is the allowable value for reliability requirements; A sequential optimization strategy is employed to solve the problem, including: at the i-th given allowable value In this case, repeat steps one through three to obtain the hybrid reliability R under the deterministic optimization of the PID gain; Determine the calculated hybrid reliability Does it meet the following convergence criteria: ； in, The pre-set convergence threshold; If the above formula meets the convergence criterion, it indicates that the optimization has converged; otherwise, iterate to the next step based on the reliability, i.e., the allowable response value for the (i+1)th step of deterministic optimization: ； in, In a two-step deterministic optimization, the change in the allowable value of the response is expressed as the static reliability at the moment before the response exceeds the allowable value (i.e., when the reliability is less than 1) or at the moment when the absolute value of the response reaches its maximum (i.e., when the reliability equals 1). Adjustments were made to the static reliability. The calculation method is as follows: The absolute value of the peak is greater than the absolute value of the trough: ； The absolute value of the peak is less than the absolute value of the trough: ； in, and These are the probability density functions of the response. The total area enclosed by the horizontal axis and the area of ​​the shaded region are solved by integrating the probability density function; for Always respond to the upper bound, for Always respond to the lower bound; Repeat steps two through four until hybrid reliability is achieved. The convergence criterion or maximum number of iterations is determined, thereby obtaining the optimal PID gain that minimizes energy consumption and meets safety requirements.

5. The sequential reliable design method for a PID controller considering random and interval mixed uncertainties according to claim 1, characterized in that, In the second step, given sample points of uncertain parameters in an interval, the joint probability density function of the response and random uncertain parameters is solved by the probability density evolution equation. Considering failure, the probabilistic reliability is solved by integration to achieve real-time evaluation of probabilistic reliability.

6. The sequential reliable design method for a PID controller considering random and interval mixed uncertainties according to claim 1, characterized in that, The third step uses the probabilistic reliability calculated in the second step. An adaptive bisection method is used to select appropriate sample points within the interval of uncertain parameters. The mixed reliability is defined as the mean of the probabilistic reliability over the interval, thus achieving a solution for mixed reliability that balances efficiency and high accuracy.

7. The sequential reliable design method for a PID controller considering random and interval mixed uncertainties according to claim 1, characterized in that, In the fourth step, the minimum control force index J from the first step and the hybrid reliability obtained in the third step are defined as constraints to construct an optimization formula. The efficiency of solving the optimization formula is improved by using a sequential optimization strategy to obtain the required PID parameters.

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