Adaptive pi type nonlinear control method, device and medium suitable for MIMO system

Abstract

The application provides a kind of adaptive PI type nonlinear control method, equipment and medium suitable for MIMO system, method includes: establishing MIMO system state equation;According to MIMO system state equation, establish MIMO system error equation;According to MIMO system error equation, design system error function S;According to system error function S, establish performance index function J z ;Select Liapunov function V1;Design system controller u d ;The stability of MIMO system controlled by system controller u d Is proved by using Liapunov function V1.The application solves the problem of being unable to effectively observe the entire operation process by using performance index function, solves the problem of being limited by square matrix and non-square matrix of MIMO system matrix by designing the controller, solves the problem of being difficult to select parameters of PID control method, and solves the problem of being unable to handle nonlinear terms by PID control method in MIMO system.

Description

Technical Field

[0001] This document relates to the field of adaptive control technology, and in particular to an adaptive PI nonlinear control method, device and medium suitable for MIMO systems. Background Technology

[0002] In industrial process control, a control system that uses the proportional, integral, and derivative components of the error generated by comparing real-time data of the controlled object with the setpoint is called a PID controller. PID control is particularly significant due to its simple structure and intuitive concept, and has been widely used in practice. Currently, the traditional PID control method compares the output value with the ideal value and re-inputs the deviation into the proportional-integral-derivative (PID) control element. Through continuous feedback adjustment, the system output gradually approaches the ideal setpoint. However, this method typically uses trial-and-error tuning to determine the PID parameters. This parameter selection method is time-consuming and makes it difficult to select optimal parameters, and the control system cannot adapt to changes in the system.

[0003] Building upon traditional PID control, combining it with modern control methods such as fuzzy control and neural network control can achieve parameter self-tuning of the PID controller. For example, BP neural networks, Mamdani fuzzy neural networks, and RBF neural networks are applied to PID control system design. Through repeated autonomous learning and training of the neural network, the difference between the control system's output value and the ideal value approaches zero infinitely. However, it is essentially still a linear control method. When considering nonlinear terms in the actual system, it uses error feedback to adjust the P and I parameters, thereby designing the control system. This error feedback form, which subtracts the expected value from the actual value, is not effective in handling nonlinear terms and unknown disturbances, resulting in a decrease in the system's accuracy and adaptability.

[0004] Nonlinear PID improves upon traditional PID by introducing nonlinear factors. The error feedback is no longer simply the error between the output and the desired value, but rather the error after nonlinear transformation, thus enhancing the system's ability to handle nonlinear factors. However, its design process still follows the traditional PID design process and does not incorporate the design process for nonlinear systems. This results in less than ideal robustness and adaptability when handling complex nonlinear systems.

[0005] Significant progress has been made in the research of control methods for multiple-input multiple-output (MIMO) systems, which exhibit nonlinearity and modeling uncertainty, resulting in a variety of advanced control methods. MIMO systems are characterized by strong nonlinearity, multi-element coupling, and system modeling uncertainty, making stability control of MIMO systems extremely difficult and challenging. Existing MIMO system control methods only address the case where the system matrix is ​​a square matrix; research on non-square matrix systems is lacking. Furthermore, existing MIMO system control methods generally employ design methods for nonlinear systems, which are difficult to apply to practical systems.

[0006] Therefore, there is an urgent need for a control method suitable for MIMO systems to solve the problem that current MIMO system control methods cannot effectively observe the entire operation process during system operation, solve the problem of the MIMO system matrix being restricted to square and non-square matrices, solve the problem of the difficulty in selecting parameters for PID control methods, and solve the problem that PID control methods in MIMO systems cannot handle nonlinear terms. Summary of the Invention

[0007] This invention provides an adaptive PI-type nonlinear control method, device, and medium suitable for MIMO systems, aiming to solve the above-mentioned problems.

[0008] This invention provides an adaptive PI-type nonlinear control method suitable for MIMO systems, comprising:

[0009] S1. Establish the state equations for the MIMO system;

[0010] S2. Based on the MIMO system state equation, establish the MIMO system error equation;

[0011] S3. Design the system error function S based on the MIMO system error equation;

[0012] S4. Establish the performance index function J based on the system error function S. z ;

[0013] S5. Select the Lyapunov function V1;

[0014] S6. Design an adaptive PI nonlinear control method. The control method includes designing a system controller u. d Automatic updates are achieved by combining the adaptive algorithm parameters in the controller with a radial neural network.

[0015] S7. Using the Lyapunov function V1, prove that the system controller u... d Stability of the controlled MIMO system;

[0016] Among them, design system controller u d By introducing a matrix, the controller can adapt to both square and non-square system matrices.

[0017] This invention provides an electronic device, comprising:

[0018] Processor; and,

[0019] A memory is configured to store computer-executable instructions, which, when executed, cause the processor to perform the steps of the adaptive PI nonlinear control method applicable to MIMO systems described above.

[0020] This invention provides a storage medium for storing computer-executable instructions, which, when executed, implement the steps of the adaptive PI nonlinear control method applicable to MIMO systems described above.

[0021] By employing the embodiments of the present invention, a performance index function J is established. z To address the problem that current MIMO system control methods cannot effectively observe the entire operation process, a system controller u is designed. d This invention addresses the limitations of MIMO systems on square and non-square matrices; it also solves the problem of parameter selection difficulties in PID control methods. The design process employs a nonlinear system design approach, and the system error feedback used is not merely the difference between the output and expected values, but a new error function is designed and superimposed, thus enabling nonlinear processing of system errors. Compared to traditional PID control methods, this adaptive PI nonlinear control method exhibits superior nonlinear processing capabilities. Attached Figure Description

[0022] To more clearly illustrate the technical solutions in one or more embodiments of this specification or in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in this specification. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0023] Figure 1 This is a flowchart of an adaptive PI nonlinear control method for MIMO systems according to an embodiment of the present invention;

[0024] Figure 2 For the embodiment of the present invention y 11 A diagram illustrating the tracking effect on the expected value;

[0025] Figure 3 For the embodiment of the present invention y 12 A diagram illustrating the tracking effect on the expected value;

[0026] Figure 4 For the embodiment of the present invention y 11 and y 12 A schematic diagram of the tracking error;

[0027] Figure 5 u, as an embodiment of the present invention 11 and u 12 Control force diagram;

[0028] Figure 6 The performance index function J in this embodiment of the invention z Schematic diagram;

[0029] Figure 7 This is a flowchart of the adaptive PI nonlinear control system according to an embodiment of the present invention;

[0030] Figure 8 This is a schematic diagram illustrating the specific implementation steps of an embodiment of the present invention. Detailed Implementation

[0031] To enable those skilled in the art to better understand the technical solutions in one or more embodiments of this specification, the technical solutions in one or more embodiments of this specification will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this specification, and not all of the embodiments. Based on one or more embodiments of this specification, all other embodiments obtained by those skilled in the art without creative effort should fall within the protection scope of this document.

[0032] Method Implementation Examples

[0033] Figure 1 This is a flowchart of an adaptive PI nonlinear control method for MIMO systems according to an embodiment of the present invention. Figure 1 As shown, the adaptive PI nonlinear control method for MIMO systems according to embodiments of the present invention specifically includes:

[0034] include:

[0035] Step S1: Establish the state equations of the MIMO system; Step S1 specifically includes:

[0036] Consider a nonlinear system with multiple inputs and multiple outputs. The system state equations are as follows:

[0037]

[0038] x k =[x k1 ,x k2 ,…,x km ] T ∈R m , u is the system control input matrix, u∈R r (r is not necessarily equal to m); y is the system output matrix, y∈R m ; It is the system control gain matrix; It is an unknown uncertainty interference in the system.

[0039] In actual operation, the controller may fail; therefore, u is redesigned, with the following settings:

[0040] u = ρu d +u p (2)

[0041] ρ=diag{ρ1(t),…,ρ r (t)}∈R r×r , 0 < ρ i (t)≤1,i=1,2,...,r;u d It is the designable part of the controller u, u d ∈R r ;u p It is the uncertain part of the controller u, u p ∈R r .

[0042] Therefore, the system state equation can be rewritten as:

[0043]

[0044] Step S2: Based on the MIMO system state equation, establish the MIMO system error equation; Step S2 specifically includes:

[0045] First, let the error be e = x1 - y. d Then the systematic error equation can be obtained:

[0046]

[0047] in, To represent the system's centralized uncertainty, for the sake of convenience in formula writing, G will be used uniformly thereafter. f d replace M replaces

[0048] Next, based on the systematic error equation, a new systematic error matrix z is designed:

[0049]

[0050] Where, λ i (i = 0, 1, ..., n-1) are given positive constants, and we take λ. n-1 =1, thus making the polynomial It is a Herwitz polynomial.

[0051] Step S3: Design the system error function S based on the MIMO system error equation; Step S3 specifically includes:

[0052] Based on S2, we further define the system's generalized error function matrix s in the form of PI, as follows:

[0053]

[0054] Where ξ is a designable constant, ξ>0; z is the error matrix defined in equation (5).

[0055] Step S4: Establish the performance index function J based on the system error function S. z Step S4 specifically includes:

[0056] Based on the MIMO system error equation, and to more conveniently evaluate the system's performance throughout the tracking process, this invention designs a performance index function J. z .

[0057]

[0058] in, ξ is a designable constant, ξ>0; Q is a weight constant matrix, Q=[λ0I m ,λ1I m ,…,λ n-2 I m ,I m ]∈R m×nm ;I m It is the identity matrix, I m ∈R m×m ;λ i (i = 0, 1, ..., n-2) has already been selected as the Hurwitz matrix in equation (5). Through design, the performance index function J is made so that... z It can be expressed as:

[0059]

[0060] From the definition of z, we can obtain:

[0061]

[0062] From the definition of s, we can further obtain:

[0063] This performance index function comprehensively reflects the overall system operation process, demonstrating the system's performance in the initial state, transient to intermediate state, and steady state. The convergence of this comprehensive performance index function is guaranteed by the subsequently designed adaptive PI-type nonlinear control method. This performance index function is obtained by superimposing the system's tracking errors to obtain the system's generalized error. Then, based on this generalized error, a PI-based approach is used to design the performance index function, giving it a PI form as well.

[0064] Step S5: Select the Lyapunov function V1; Step S5 specifically includes:

[0065] First, the Lyapunov function V1 is selected as follows:

[0066]

[0067] In the formula,

[0068] Differentiating with respect to V1, we get:

[0069]

[0070] Substitution We can obtain:

[0071]

[0072] in, Further, we can obtain Where, a = max{X m ,X d}, Ψ1 is the uncertainty function of the system's lumped synthesis; therefore, by approximating it using the RBFNN neural network function, we can obtain:

[0073]

[0074] Where Z is the input to the neural network, W1∈R η The ideal (unknown) weights; φ∈R η The basis functions are known; δ1∈R is the reconstruction error, and satisfies |δ1|<δ 1m δ 1m These are some unknown constants. Therefore, the equation can be expressed as: in, ψ(Z)=||φ||+1, therefore we can obtain:

[0075]

[0076] Step S6: Design an adaptive PI nonlinear control method. The control method includes designing a system controller u. d Automatic updates are achieved by combining the adaptive algorithm parameters in the controller with a radial neural network.

[0077] For a MIMO system where the system matrix G is a square matrix, we design a controller u. d The following has the form PI:

[0078]

[0079] Among them, K p and K I These are two designable parameters; ΔK p and ΔK I These are two parameters updated by an adaptive algorithm. To reduce the controller u... d The design difficulty, here we will discuss K p With K I ΔK p With ΔK I The relationship can be summarized as follows: K I =ξK p ΔK I =ξΔK p Where ξ is a designable constant, ξ>0. Substituting and simplifying, we get:

[0080]

[0081] For MIMO systems where the system matrix G is not a square matrix, we design the controller as follows, based on the square matrix system:

[0082]

[0083] In the controller design of this embodiment, a matrix Λ is introduced. By designing the matrix Λ, the controller can adapt to both square and non-square system matrices.

[0084] This application also solves the problem of difficulty in selecting traditional PID control parameters by adjusting the parameter ΔK. p When combined with a radial basis function neural network, the parameter ΔK is improved. p It can achieve automatic online updates, and during system operation, only one parameter needs to be updated, as shown below:

[0085] For u d Regarding the relevant parameters, we will design as follows:

[0086] (1) Choose K p >0

[0087] (2) Let

[0088] (3) Design In the formula It is an estimate of θ. The following equation can be updated to obtain:

[0089]

[0090] In the formula, ψ=||φ||+1 is a scalar function, φ can be obtained from the RBFNN neural network function; σ1 is a designable constant, σ1>0; σ0 and It is a function that satisfies the following conditions:

[0091]

[0092]

[0093] Substitute u d Then, we get:

[0094]

[0095] Substituting G and Λ, we can further obtain:

[0096]

[0097] Decomposition yields:

[0098]

[0099] because It is obliquely symmetrical, therefore we can conclude: Therefore, we can conclude that:

[0100]

[0101]

[0102] Lianlide:

[0103]

[0104] Simplifying, we get:

[0105]

[0106] Substituting equation (18) into equation (27), we get:

[0107]

[0108] It is easy to prove that: Therefore, we can conclude that:

[0109]

[0110] Depend on We can obtain:

[0111] Substitution From this, we can obtain:

[0112]

[0113] From the inequality relations, we get:

[0114]

[0115] Therefore:

[0116]

[0117] Therefore, we can conclude that:

[0118]

[0119] Where, γ1=min{2ω b K p ,σ0}>0,

[0120] Integrating both sides of equation (33), we get:

[0121]

[0122] From equations (19) and (20), we can obtain:

[0123]

[0124] Therefore, equation (34) can be expressed as:

[0125]

[0126] It is easy to see that V1(t) > 0 and Therefore, we can conclude that:

[0127] V1≤V1(0)+q (37)

[0128] Therefore, we can obtain V1∈L ∞ From the function expression (10) of V1, it can be shown that s∈L ∞ , θ∈L∞ Furthermore, from the expression for s (6), we can obtain z∈L ∞ e (i) ∈L ∞ (i = 1, ..., n-1), u d It is bounded. And because G and ρ are bounded, it can be deduced from... The expression is Since it is bounded, we can conclude that s is uniformly continuous, and further, we obtain ||s||. 2 It is consistent and continuous.

[0129] Since Γ is symmetric and positive definite, when t≥0, there exists λ(Γ)>0, which is the smallest eigenvalue of the positive definite symmetric matrix Γ. Therefore:

[0130]

[0131] From equation (36), we can derive:

[0132]

[0133] Then the performance index function J can be obtained. z :

[0134]

[0135] Therefore, the performance index function J can be obtained. z It is bounded, and at the same time, we can obtain s∈L2.

[0136] In summary, the following has been derived through proof:

[0137] 1)J z The derivative of ||s|| 2 It is consistent and continuous;

[0138] 2) When t→∞, J z It is bounded. According to Barbalat's lemma, as t→∞, the differentiable function J... z It is bounded, and It is uniformly continuous, so we can conclude that as t→∞, ||s|| 2 →0, that is, ||s||→0. Based on this, we can obtain z→0 from the expression of s. Ultimately obtainable All of them gradually approach 0.

[0139] Step S7: Use the Lyapunov function V1 to prove that the system controller u d Stability of controlled MIMO systems.

[0140] The design of the above adaptive PI nonlinear control method is used when ud Pick: This allows the system to operate stably under the control of the designed controller.

[0141] Experimental results of the adaptive PI nonlinear control method of this invention are as follows: Figures 2-6 As shown, Figure 2 Indicates the output signal y 11 For the expected value y d11 The tracking effect Figure 3 Indicates the output signal y 12 For the expected value y d12 The tracking effect Figure 4 Indicates y 11 and y 12 Tracking error, Figure 5 It is the controller u 11 and controller u 12 Control Figure 6 It describes the changes in the performance index function.

[0142] Figure 7 The flowchart of the adaptive PI nonlinear control system according to an embodiment of the present invention is as follows: Figure 7 As shown, the system tracking error is obtained by subtracting the system output value from the expected value, and an error function is established. The value of the error function is input to the PI controller, where the parameters of P and I of the PI controller are automatically updated online by the radial basis neural network. Then, through the action of the PI controller, the system control quantity u is obtained, and the control quantity is applied to the actuator to drive the entire system to operate stably.

[0143] Figure 8 This is a schematic diagram illustrating the specific implementation steps of an embodiment of the present invention. A specific embodiment of the adaptive PI-type nonlinear control method for MIMO systems according to the present invention includes the following steps:

[0144] Step 1: Consider a nonlinear system with multiple inputs and multiple outputs, and choose the system state equations x1, x2...x n Establish the state equations for the MIMO system;

[0145] Step 2: Determine the systematic error and select the systematic error equations e1, e2...e n The system error is obtained by subtracting the system output value from the expected value.

[0146] Step 3: Based on the system error equation, design the system error function s. The error function is obtained by superimposing the system errors and can be processed nonlinearly.

[0147] Step 4: Design the system performance index function J zThe performance index function is composed of the system error function, which can show the state of the overall system operation process, and is one of the core points of this invention;

[0148] Step 5: Select the Lyapunov function V1. Choosing a suitable Lyapunov function will provide a foundation for the subsequent proof of system stability.

[0149] Step 6: Design the system controller u d Design a suitable system controller so that the system can operate stably under the action of the controller;

[0150] Step 7: Prove the stability of the system. Using Lyapunov stability theory, prove that the designed control method is stable and can enable the system to operate stably.

[0151] By employing the embodiments of the present invention, the following beneficial effects are specifically achieved:

[0152] 1. The performance index function designed in this invention can comprehensively reflect the system behavior when characterizing the control performance during the overall operation of the system. It can reflect the initial, transient to intermediate and steady-state performance of the system, so that operators can observe the system's operating status in a timely manner.

[0153] 2. The adaptive PI-type nonlinear control method proposed in this invention considers both the case where the system gain matrix is ​​a square matrix and the case where the system gain matrix is ​​not a square matrix. Therefore, compared with other control methods, this control method is applicable to both square matrix and non-square matrix MIMO systems. Compared with existing control methods that only consider the case where the system matrix is ​​square, the control method of this invention has wider applicability.

[0154] 3. The adaptive PI nonlinear control method of this invention has only one online update parameter. It adopts a PI structure, which is simple in structure, has low computational cost, and is convenient for operators. Furthermore, the PI structure controller is robust to uncertain disturbances and can withstand sudden actuator failures without requiring fault detection or diagnosis. Compared to the PID control commonly used in current industrial production, the adaptive PI nonlinear control method proposed in this invention has a simpler and more convenient parameter selection capability and is better adapted to nonlinear systems, making this invention easier to apply in actual production situations.

[0155] 4. Although the system controller designed in this invention is in the form of PI control, the design process employs a nonlinear system design method. Furthermore, the system error feedback used in the design is not merely the difference between the output value and the expected value, but a new error function is designed and superimposed, thus enabling nonlinear processing of the system error. Compared with the traditional PID control method, the adaptive PI nonlinear control method of this invention has better nonlinear processing capabilities.

[0156] Device Example 1

[0157] An electronic device, comprising:

[0158] Processor; and,

[0159] A memory is configured to store computer-executable instructions, which, when executed, cause the processor to perform the steps of the method embodiments described above.

[0160] Device Example 2

[0161] A storage medium for storing computer-executable instructions, which, when executed, perform the steps of the method embodiments described above.

[0162] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. An adaptive PI-type nonlinear control method suitable for MIMO systems, characterized in that, include: S1. Establish the state equations for the MIMO system; S2. Based on the MIMO system state equation, establish the MIMO system error equation; S3. Design the system error function S based on the MIMO system error equation; S4. Establish the performance index function based on the system error function S. ; S5. Selecting Lyapunov functions ; S6. Design an adaptive PI nonlinear control method, the control method including designing a system controller. Automatic updates are achieved by combining the adaptive algorithm parameters in the controller with a radial neural network. S7. Using Lyapunov functions Proof through system controller Stability of the controlled MIMO system; Among them, design system controller By introducing a matrix, the controller can adapt to both square and non-square system matrices. S1 specifically includes: The state equations of a MIMO system are as follows: Official 1; in, , , It is the system control input matrix. , It is the system output matrix. , It is the system control gain matrix. It is an unknown uncertainty interference in the system; For the input matrix Redesign: Official 2; , , It is a controller The part that can be designed in the middle, , It is a controller The uncertain part ; The state equation of the MIMO system is rewritten as: Official 3; S2 specifically includes: Setting error The systematic error equation is obtained as follows: Official 4; in, This indicates centralized uncertainty in the system; S3 specifically includes: Design a new system error matrix based on the system error equation. : Official 5; in, Given a positive constant, take... This makes the polynomial It is a Herwitz polynomial; Define the system's generalized error function matrix in the form of PI. , Official 6; in, It is a designable constant. , It is the error matrix defined in Formula 5; S4 specifically includes: Based on the error equation of the MIMO system, a performance index function is designed. ; Official 7; in, ; It is a designable constant. ; It is a weight constant matrix. ; It is the identity matrix. ; In Equation 5, the Hurwitz polynomial is selected as the performance index function. It is expressed as: Official 8; According to the definitions of z and s, we can obtain: Official 9.

2. The method according to claim 1, characterized in that, S5 specifically includes: Choose the Lyapunov function as follows: Official 10; In the formula, ; right Differentiating, we get: Official 11; Substitution We can obtain: Official 12; in, Further, we can obtain ,in, , , It is an uncertainty function synthesized from a system set, therefore, by approximating it using an RBFNN neural network function, we can obtain: Official 13; in It is the input to the neural network. ; It is the ideal weight; These are known basis functions; It is a reconstruction error, and satisfies , They are some unknown constants. Official 14; in, , , Official 15.

3. The method according to claim 1, characterized in that, S6 specifically includes: For the system matrix Design a controller for a square matrix MIMO system. The following has the form PI: Official 16; in, and These are two designable parameters; and These are two parameters updated by an adaptive algorithm. and , and The relationships are as follows: , ,in It is a designable constant. Substituting and simplifying, we get: Official 17; For the system matrix This is a non-square matrix MIMO system. Based on the square matrix system, the controller is designed as follows: Official 18; By parameters Combining with radial basis function neural networks allows for the parameterization of parameters. It enables online automatic updates, and only one parameter needs to be updated during system operation.

4. The method according to claim 3, characterized in that, The method further includes: Will The values ​​are set as shown in the following formula, so that the system can achieve stable operation under the action of the controller; Official 19.

5. An electronic device, comprising: processor; as well as, A memory is configured to store computer-executable instructions, which, when executed, cause the processor to perform the steps of the adaptive PI nonlinear control method for MIMO systems as described in any one of claims 1-4.

6. A storage medium for storing computer-executable instructions, which, when executed, implement the steps of the adaptive PI nonlinear control method for MIMO systems as described in any one of claims 1-4.

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