Design method and system for fractional order [PI] controller of inertia and delay fractional order object

Abstract

The invention provides a method and a system for designing a fractional order [PI] controller of an inertia and delay fractional order object. The method comprises the following steps of: describing a controlled object by adopting an inertia and delay fractional order transfer function; selecting a fractional order [PI] controller as a feedback controller; the closed-loop system is composed of a controlled object and a fractional order [PI] controller. Obtaining a characteristic equation of a closed-loop system; traversing the order and frequency of the controller, solving to obtain a complex root stability domain, and combining with a real root stability domain to obtain a fractional order [PI] controller stability domain; the proportional gain, the integral gain and the order of the controller are uniquely solved based on three constraint equations of a given crossing frequency, a phase margin and a condition that the slope of an open-loop phase of a closed-loop system at the crossing frequency is zero (also known as phase flatness); if no solution exists, the crossing frequency and the phase margin are given again for solution. The fractional order [PI] controller obtained through the method has high robustness and has practical application value.

Description

Technical Field

[0001] The present invention relates to a fractional order [PI] controller design method, and in particular to a fractional order [PI] controller design method and system for an inertia plus delay fractional order object. Background Art

[0002] Shape (k p is the controller proportional gain, k i The fractional-order [PI] controller (where π is the integral gain of the controller and r is the controller order) is the product of the development of fractional-order calculus. It can provide more degrees of freedom and parameter adjustment space than integer-order controllers, achieve more flexible and precise control performance, and have better robustness and adaptability to complex systems. The fractional-order object describes the dynamic characteristics of the controlled object through fractional-order differential equations. It is a generalized form of the integer-order object and can more accurately describe the dynamic characteristics of the actual system.

[0003] For fractional-order objects, the form There is a lack of research on the calculation method of the stable domain of the parameters of the fractional-order [PI] controller. There is little research on the current fractional-order [PI] controller tuning technology. The existing tuning methods are designed for integer-order first-order inertia plus delay objects and lack generality. In addition, there are also methods based on evolutionary algorithms to optimize the parameters of fractional-order [PI] controllers. Based on robustness constraints, the minimum control performance index can be solved. This method has the disadvantages of slow solution speed and easy convergence of parameters to local optimality. With the development of fractional-order calculus theory and technology, more and more controlled objects are described by fractional-order transfer functions. Current research lacks direct calculation methods for fractional-order [PI] controller parameters under typical robust indicators (such as maximum sensitivity function, phase margin, amplitude margin, phase flatness, crossover frequency, etc.). Robust tuning of fractional-order [PI] controller parameters for fractional-order objects is very necessary. Summary of the invention

[0004] The purpose of the present invention is to solve the problem of robust design of a fractional-order [PI] controller for a class of fractional-order objects, and to provide a design method and system for a fractional-order [PI] controller for an inertia-plus-delay fractional-order object.

[0005] In a first aspect, the present invention provides a method for designing a fractional-order [PI] controller for an inertia-plus-delay fractional-order plant, comprising the following steps:

[0006] 1) The fractional-order [PI] controller of the inertia-delay fractional-order object is established as a closed-loop system consisting of the controlled object and the feedback controller;

[0007] The controlled object adopts the inertia plus delay fractional order transfer function G p(s) Description, as follows:

[0008]

[0009] In the formula, G p (s) is the inertia plus delay fractional order transfer function, s is the differential operator, K is the gain of the controlled object, and K∈[-10 10 ,0∪0,10 10 ], T is the time constant of the controlled object, T∈(0,10 10 ], α is the order of the controlled object, α∈(0,2), L is the delay time constant of the controlled object, L∈(0,10 10 ], e is a natural constant;

[0010] The feedback controller adopts a fractional order [PI] controller, whose transfer function G c (s) are as follows:

[0011]

[0012] In the formula, k p is the controller proportional gain, with k p ∈[-10 10 ,10 10 ],k i is the controller integral gain, with k i ∈[-10 10 ,10 10 ], r is the controller order, r∈(0,2); k p , k i and r are the fractional-order [PI] controller parameters to be tuned;

[0013] The characteristic equation of the closed-loop system G ce (k p ,k i ,r,s) is:

[0014] G ce (k p ,k i ,r,s)=s r (Ts a +1)+Ke -Ls (k p s+k i ) r =0 (3)

[0015] 2) Based on the given controller order r, traverse the system frequency ω∈0+∞, calculate the complex root stability values ​​of the fractional-order [PI] controller parameters respectively, and use the final complex root stability values ​​as the complex root stability domain of the fractional-order [PI] controller parameters; where,

[0016] Given the controller order r and the system frequency ω, the complex root stable value of the fractional-order [PI] controller parameters at the system frequency ω is calculated by the following formula:

[0017]

[0018] and

[0019]

[0020] In the formula, R2 is the intermediate variable, θ is the intermediate variable, ω is the system frequency;

[0021] Assume that the real root stability region of the fractional-order [PI] controller parameters is:

[0022]

[0023] 3) Combined with the real root stability domain of the fractional-order [PI] controller parameters, the parameter stability domain of the fractional-order [PI] controller under the controller order r is obtained;

[0024] 4) Given the crossover frequency ω gc and phase margin And based on a given crossover frequency ω gc and phase margin Calculate the fractional order [PI] controller parameters by solving equations 1, 2 and 3 to solve for the proportional gain k of the fractional order [PI] controller p , integral gain k i and controller order r;

[0025] If the solution is an empty set, then re-set the crossing frequency ω gc and phase margin Recalculate and solve equation 1, equation 2, and equation 3;

[0026] If the solution is a non-empty set, the proportional gain k is p , integral gain k i and controller order r are the fractional-order [PI] controller parameters that satisfy the constraints;

[0027] Given the crossover frequency ω of the closed-loop system gc and phase margin Solving equation 1 for fractional-order [PI] controller parameters:

[0028]

[0029] Given the crossover frequency ω of the closed-loop system gcand phase margin Solving equation 2 for fractional-order [PI] controller parameters:

[0030]

[0031] On the basis that the open-loop phase of the closed-loop system has a slope of zero at the crossover frequency, the fractional-order [PI] controller parameters are solved in equation 3:

[0032]

[0033] In formula (7) to formula (9), R 2gc is the intermediate variable, θ gc is the intermediate variable,

[0034] In a second aspect, the present invention provides a design system for a fractional-order [PI] controller of an inertia-plus-delay fractional-order plant, comprising:

[0035] The closed-loop system establishment module is used to establish the fractional-order [PI] controller of the inertia plus delay fractional-order object into a closed-loop system consisting of a controlled object and a feedback controller; wherein,

[0036] The controlled object adopts the inertia plus delay fractional order transfer function G p (s) Description, as follows:

[0037]

[0038] In the formula, G p (s) is the inertia plus delay fractional order transfer function, s is the differential operator, K is the gain of the controlled object, and K∈[-10 10 ,0∪0,10 10 ], T is the time constant of the controlled object, T∈(0,10 10 ], α is the order of the controlled object, α∈(0,2), L is the delay time constant of the controlled object, L∈(0,10 10 ], e is a natural constant;

[0039] The feedback controller adopts a fractional order [PI] controller, whose transfer function G c (s) are as follows:

[0040]

[0041] In the formula, k p is the controller proportional gain, with k p ∈[-10 10 ,10 10 ],k i is the controller integral gain, with ki ∈[-10 10 ,10 10 ], r is the controller order, r∈(0,2); k p , k i and r are the fractional-order [PI] controller parameters to be tuned;

[0042] The characteristic equation of the closed-loop system G ce (k p ,k i ,r,s) is:

[0043] G ce (k p ,k i ,r,s)=s r (Ts a +1)+Ke -Ls (k p s+k i ) r =0 (3)

[0044] The complex root stability domain calculation module is connected to the closed-loop system establishment module, and is used to traverse the system frequency ω∈0+∞ based on a given controller order r, and calculate the complex root stability values ​​of the fractional-order [PI] controller parameters respectively, and use the final complex root stability values ​​as the complex root stability domain of the fractional-order [PI] controller parameters; wherein,

[0045] Given the controller order r and the system frequency ω, the complex root stable value of the fractional-order [PI] controller parameters at the system frequency ω is calculated by the following formula:

[0046]

[0047] and

[0048]

[0049] In the formula, R2 is the intermediate variable, θ is the intermediate variable, ω is the system frequency;

[0050] A parameter stability domain calculation module is connected with the closed-loop system establishment module and the complex root stability domain calculation module, and is used to combine the real root stability domain of the fractional order [PI] controller parameters to obtain the parameter stability domain of the fractional order [PI] controller under the controller order r;

[0051] Assume that the real root stability region of the fractional-order [PI] controller parameters is:

[0052]

[0053] The parameter solving module is connected with the parameter stable domain calculation module and the complex root stable domain calculation module to give the crossing frequency ω gc and phase margin And based on a given crossover frequency ω gc and phase margin Calculate the fractional order [PI] controller parameters by solving equations 1, 2 and 3 to solve for the proportional gain k of the fractional order [PI] controller p , integral gain k i and controller order r;

[0054] If the solution is an empty set, then re-set the crossing frequency ω gc and phase margin Recalculate and solve equation 1, equation 2, and equation 3;

[0055] If the solution is a non-empty set, the proportional gain k is p , integral gain k i and controller order r are the fractional-order [PI] controller parameters that satisfy the constraints;

[0056] Given the crossover frequency ω of the closed-loop system gc and phase margin Solving equation 1 for fractional-order [PI] controller parameters:

[0057]

[0058] Given the crossover frequency ω of the closed-loop system gc and phase margin Solving equation 2 for fractional-order [PI] controller parameters:

[0059]

[0060] On the basis that the open-loop phase of the closed-loop system has a slope of zero at the crossover frequency, the fractional-order [PI] controller parameters are solved in equation 3:

[0061]

[0062] In formula (7) to formula (9), R 2ggc is the intermediate variable, θ gc is the intermediate variable,

[0063] In a third aspect, the present invention provides a fractional-order [PI] controller for an inertia-plus-delay fractional-order object, which is designed using the design method of the fractional-order [PI] controller for an inertia-plus-delay fractional-order object.

[0064] In a fourth aspect, the present invention provides a fractional-order [PI] controller design device, comprising:

[0065] one or more processors;

[0066] a memory for storing one or more programs,

[0067] When the one or more programs are executed by the one or more processors, the one or more processors are caused to perform the steps of the method for designing a fractional-order [PI] controller for an inertia-plus-delay fractional-order plant.

[0068] In a fifth aspect, the present invention provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the steps of the method for designing a fractional-order [PI] controller for an inertia-plus-delay fractional-order object as described above.

[0069] Compared with the prior art, the present invention has outstanding substantive features and significant progress, specifically:

[0070] 1. The present invention solves a method for calculating the stable domain of parameters of a fractional-order [PI] controller for a class of fractional-order objects. The method obtains the stable domain by calculating the real root stable domain and the complex root stable domain, and has the advantages of being simple and easy to calculate.

[0071] 2. The present invention solves a method for robust design of parameters of a fractional-order [PI] controller for a class of fractional-order objects. The method can directly calculate the parameters of the fractional-order [PI] controller under the constraints of phase margin, crossover frequency and phase flatness. It has the advantages of clear principle and easy implementation. The fractional-order [PI] controller with calculated parameters has a strong ability to cope with the gain uncertainty of the controlled object.

[0072] 3. Compared with the existing fractional-order [PI] controller parameter tuning method, the present invention can uniquely determine the proportional gain, integral gain and controller order of the fractional-order [PI] controller through three equations, which can ensure the uniqueness of the controller parameters and avoid the local optimal solution of the fractional-order [PI] controller optimized based on the evolutionary algorithm. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1 The invention discloses a closed-loop system composed of a fractional-order [PI] controller and an inertia-plus-delay fractional-order object.

[0074] Figure 2 It is the stability region of the fractional-order [PI] controller at a given order calculated by the present invention in Example 1.

[0075] Figure 3 This is the control effect of the present invention in Example 1.

[0076] Figure 4 This is the control effect of the present invention in Example 1 when there is uncertainty in the controlled object. DETAILED DESCRIPTION

[0077] In order to make the purpose, technical solution and advantages of the present application more clearly understood, the present application is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and are not used to limit the present application.

[0078] Example 1

[0079] The present embodiment provides a method for designing a fractional-order [PI] controller for an inertia-plus-delay fractional-order plant, comprising the following steps:

[0080] 1) The fractional-order [PI] controller of the inertia-delay fractional-order object is established as a closed-loop system consisting of the controlled object and the feedback controller;

[0081] The controlled object adopts the inertia plus delay fractional order transfer function G p (s) Description, as follows:

[0082]

[0083] In the formula, G p (s) is the inertia plus delay fractional order transfer function, s is the differential operator, K is the gain of the controlled object, and K∈[-10 10 ,0∪0,10 10 ], T is the time constant of the controlled object, T∈(0,10 10 ], α is the order of the controlled object, α∈(0,2), L is the delay time constant of the controlled object, L∈(0,10 10 ], e is a natural constant;

[0084] In this embodiment, K=1, T=1, α=1, and L=0.5.

[0085] The feedback controller adopts a fractional order [PI] controller, whose transfer function G c (s) are as follows:

[0086]

[0087] In the formula, k p is the controller proportional gain, with k p ∈[-10 10 ,10 10 ],k i is the controller integral gain, with k i ∈[-10 10 ,10 10], r is the controller order, r∈(0,2); k p , k i and r are the fractional-order [PI] controller parameters to be tuned.

[0088] The characteristic equation of the closed-loop system G ce (k p ,k i ,r,s) is:

[0089] G ce (k p ,k i ,r,s)=s r (Ts a +1)+Ke -Ls (k p s+k i ) r =0 (3).

[0090] 2) Based on the given controller order r, traverse the system frequency ω∈0+∞, calculate the complex root stability values ​​of the fractional-order [PI] controller parameters respectively, and use the final complex root stability values ​​as the complex root stability domain of the fractional-order [PI] controller parameters;

[0091] Given the controller order r and system frequency ω, the complex root stable values ​​of the fractional-order [PI] controller parameters at the system frequency ω are calculated by equations (4) and (5);

[0092]

[0093] and

[0094]

[0095] In the formula, R2 is the intermediate variable, θ is the intermediate variable, ω is the system frequency;

[0096] 3) Combined with the real root stability domain of the fractional-order [PI] controller parameters, the parameter stability domain of the fractional-order [PI] controller under the controller order r is obtained; let the real root stability domain of the fractional-order [PI] controller parameters be:

[0097]

[0098] In this embodiment, r = 1, we can get Figure 2 The parameter stability domain of the fractional-order [PI] controller under the controller order r=1; illustrating the effectiveness of the method.

[0099] 4) Given the crossover frequency ω gc and phase margin And based on a given crossover frequency ω gc and phase margin Calculate the fractional order [PI] controller parameters by solving equations 1, 2 and 3 to solve for the proportional gain k of the fractional order [PI] controller p , integral gain k i and controller order r;

[0100] If the solution is an empty set, then re-set the crossing frequency ω gc and phase margin Recalculate and solve equation 1, equation 2, and equation 3;

[0101] If the solution is a non-empty set, the proportional gain k is p , integral gain k i and controller order r are the fractional-order [PI] controller parameters that satisfy the constraints;

[0102] Given the crossover frequency ω of the closed-loop system gc and phase margin Solving equation 1 for fractional-order [PI] controller parameters:

[0103]

[0104] Given the crossover frequency ω of the closed-loop system gc and phase margin Solving equation 2 for fractional-order [PI] controller parameters:

[0105]

[0106] On the basis that the open-loop phase of the closed-loop system has a slope of zero at the crossover frequency (also known as phase flatness), the fractional-order [PI] controller parameters are solved in equation 3:

[0107]

[0108] In formula (7) to formula (9), R 2gc is the intermediate variable, θ gc is the intermediate variable,

[0109] In this embodiment, the crossover frequency ω of the given closed-loop system gc = 0.1 and phase margin Get the proportional gain k p =0.9539, integral gain k i =0.3283, controller order r=1.9800.

[0110] In this embodiment, the crossover frequency ω can be obtained gc =0.1HZ, phase margin Parameters of fractional-order [PI] controller with phase flatness and other constraints: proportional gain k p =0.9539, integral gain k i =0.3283 and controller order r=1.9800. Combined with the controlled object, we can get Figure 1 The tracking and anti-interference performance of the closed-loop system is shown in Figure 3 As shown in the figure, the specific simulation process is:

[0111] At the beginning of the simulation, the closed-loop system is in a steady state. At 0s, the set value is changed from 0 to 1. At 20s, the closed-loop control quantity is disturbed from 0 to 1, and the simulation ends at 40s.

[0112] Through simulation, it can be seen that the closed-loop system proposed by the method of the present invention, which directly calculates the parameters of the fractional-order [PI] controller based on a given phase margin, crossover frequency and phase flatness constraint, has a relatively fast tracking capability and a strong anti-interference ability.

[0113] In order to analyze the ability of the tuned fractional-order [PI] controller to cope with the uncertainty of the controlled object gain, G is modified. p In (s), the gains K of the controlled objects are 120%K and 80%K respectively. Repeating the above simulation, we can get Figure 4 The results shown in the figure show that the adjusted fractional-order [PI] controller has a strong ability to cope with the gain uncertainty of the controlled object.

[0114] Example 2

[0115] This embodiment provides a design system for a fractional-order [PI] controller of an inertia-plus-delay fractional-order plant, including:

[0116] A closed-loop system establishment module, used to establish a fractional-order [PI] controller of an inertia-plus-delay fractional-order object into a closed-loop system consisting of a controlled object and a feedback controller;

[0117] The complex root stability domain calculation module is connected with the closed-loop system establishment module, and is used to traverse the system frequency ω∈0+∞ on the basis of a given controller order r, and respectively calculate the complex root stability values ​​of the fractional-order [PI] controller parameters, and use the final complex root stability values ​​as the complex root stability domain of the fractional-order [PI] controller parameters;

[0118] A parameter stability domain calculation module is connected with the closed-loop system establishment module and the complex root stability domain calculation module, and is used to combine the real root stability domain of the fractional order [PI] controller parameters to obtain the parameter stability domain of the fractional order [PI] controller under the controller order r;

[0119] The parameter solving module is connected with the parameter stable domain calculation module and the complex root stable domain calculation module to give the crossing frequency ω gc and phase margin And based on a given crossover frequency ω gc and phase margin Calculate the fractional order [PI] controller parameters by solving equations 1, 2 and 3 to solve for the proportional gain k of the fractional order [PI] controller p , integral gain k i and controller order r;

[0120] If the solution is an empty set, then re-set the crossing frequency ω gc and phase margin Recalculate and solve equation 1, equation 2, and equation 3;

[0121] If the solution is a non-empty set, the proportional gain k is p , integral gain k i and the controller order r are the fractional-order [PI] controller parameters that satisfy the constraints.

[0122] In this embodiment, the implementation scheme of the design system of the fractional-order [PI] controller for the inertia-plus-delay fractional-order object to solve the problem is similar to the implementation scheme described in the method of Example 1, and will not be repeated here.

[0123] Example 3

[0124] This embodiment provides a fractional-order [PI] controller for an inertia-plus-delay fractional-order object, which is designed using the design method of the fractional-order [PI] controller for an inertia-plus-delay fractional-order object described in Example 1.

[0125] Example 4

[0126] Each module in the embodiment 2 system can be implemented in whole or in part by software, hardware, or a combination thereof. Each of the above modules can be embedded in or independent of a processor in a computer device in the form of hardware, or can be stored in a memory in a computer device in the form of software, so that the processor can call and execute operations corresponding to each of the above modules.

[0127] In an exemplary embodiment, a fractional-order [PI] controller design device is provided, and the computer device may be a terminal. The computer device further includes a processor, a memory, an input / output interface, a communication interface, a display unit, and an input device. The processor, the memory, and the input / output interface are connected via a system bus, and the communication interface, the display unit, and the input device are connected to the system bus via the input / output interface. The processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system and a computer program. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The input / output interface of the computer device is used to exchange information between the processor and an external device. The communication interface of the computer device is used to communicate with an external terminal in a wired or wireless manner, and the wireless manner may be implemented through WIFI, a mobile cellular network, NFC (near field communication) or other technologies. When the computer program is executed by the processor, the steps of the design method of the fractional-order [PI] controller of the inertial plus delay fractional-order object are implemented. The display unit of the computer device is used to form a visually visible picture, which may be a display screen, a projection device, or a virtual reality imaging device. The display screen can be a liquid crystal display screen or an electronic ink display screen, and the input device of the computer device can be a touch layer covering the display screen, or a button, trackball or touchpad set on the computer device shell, or an external keyboard, touchpad or mouse.

[0128] Those skilled in the art will appreciate that the structure of the above-mentioned computer device is only a partial structure related to the solution of the present application, and does not constitute a limitation on the computer device to which the solution of the present application is applied. The specific computer device may include more or fewer components, or combine certain components, or have a different arrangement of components.

[0129] In an exemplary embodiment, a computer-readable storage medium is provided, on which a computer program is stored, and when the computer program is executed by a processor, the steps of a method for designing a fractional-order [PI] controller for an inertia-plus-delay fractional-order plant are implemented.

[0130] Those skilled in the art can understand that all or part of the processes in the above-mentioned embodiment methods can be completed by instructing the relevant hardware through a computer program, and the computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above-mentioned methods. Among them, any reference to the memory, database or other medium used in the embodiments provided in the present application can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetoresistive random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory can include random access memory (RAM) or external cache memory, etc. As an illustration and not limitation, RAM can be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM). The database involved in each embodiment provided in this application may include at least one of a relational database and a non-relational database. Non-relational databases may include distributed databases based on blockchains, etc., but are not limited to this. The processor involved in each embodiment provided in this application may be a general-purpose processor, a central processing unit, a graphics processor, a digital signal processor, a programmable logic device, a data processing logic device based on quantum computing, etc., but are not limited to this.

[0131] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0132] The above is only a specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art can easily think of changes or substitutions within the technical scope disclosed by the present invention, which should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims

1. A design method for a fractional-order [PI] controller for an inertia-plus-delay fractional-order plant, characterized in that: The following steps are involved: 1) Establish a closed-loop system consisting of the controlled object and the feedback controller; The controlled object adopts the inertia plus delay fractional order transfer function G p (s) Description, as follows: In the formula, G p (s) is the inertia plus delay fractional order transfer function, s is the differential operator, K is the gain of the controlled object, and K∈[-10 10 ,0∪0,10 10 ], T is the time constant of the controlled object, T∈(0,10 10 ], α is the order of the controlled object, α∈(0,2), L is the delay time constant of the controlled object, L∈(0,10 10 ], e is a natural constant; The feedback controller adopts a fractional order [PI] controller, whose transfer function G c (s) are as follows: In the formula, k p is the controller proportional gain, with k p ∈[-10 10 ,10 10 ],k i is the controller integral gain, with k i ∈[-10 10 ,10 10 ], r is the controller order, r∈(0,2); k p , k i and r are the fractional-order [PI] controller parameters to be tuned; The characteristic equation of the closed-loop system G ce (k p ,k i ,r,s) is: G ce (k p ,k i ,r,s)=s r (Ts a +1)+Ke -Ls (k p s+k i ) r =0 (3) 2) Based on the given controller order r, traverse the system frequency ω∈0+∞, calculate the complex root stability values ​​of the fractional-order [PI] controller parameters respectively, and use the final complex root stability values ​​as the complex root stability domain of the fractional-order [PI] controller parameters; where, Given the controller order r and the system frequency ω, the complex root stable value of the fractional-order [PI] controller parameters at the system frequency ω is calculated by the following formula: and In the formula, R2 is the intermediate variable, θ is the intermediate variable, θ∈(-π,π), ω is the system frequency; Assume that the real root stability region of the fractional-order [PI] controller parameters is: 3) Combined with the real root stability domain of the fractional-order [PI] controller parameters, the parameter stability domain of the fractional-order [PI] controller under the controller order r is obtained; 4) Given the crossover frequency ω gc and phase margin And based on a given crossover frequency ω gc and phase margin Calculate the fractional order [PI] controller parameters by solving equations 1, 2 and 3 to solve for the proportional gain k of the fractional order [PI] controller p , integral gain k i and controller order r; If the solution is an empty set, then re-set the crossing frequency ω gc and phase margin Recalculate and solve equation 1, equation 2, and equation 3; If the solution is a non-empty set, the proportional gain k is p , integral gain k i and controller order r are the fractional-order [PI] controller parameters that satisfy the constraints; Given the crossover frequency ω of the closed-loop system gc and phase margin Solving equation 1 for fractional-order [PI] controller parameters: Given the crossover frequency ω of the closed-loop system gc and phase margin Solving equation 2 for fractional-order [PI] controller parameters: On the basis that the open-loop phase of the closed-loop system has a slope of zero at the crossover frequency, the fractional-order [PI] controller parameters are solved in equation 3: In formula (7) to formula (9), R 2gc is the intermediate variable, θ gc is the intermediate variable, θ gc ∈(-π,π).

2. A design system for a fractional-order [PI] controller for an inertia-plus-delay fractional-order plant, characterized in that: include: The closed-loop system establishment module establishes a closed-loop system consisting of a controlled object and a feedback controller; The controlled object adopts the inertia plus delay fractional order transfer function G p (s) Description, as follows: In the formula, G p (s) is the inertia plus delay fractional order transfer function, s is the differential operator, K is the gain of the controlled object, and K∈[-10 10 ,0∪0,10 10 ], T is the time constant of the controlled object, T∈(0,10 10 ], α is the order of the controlled object, α∈(0,2), L is the delay time constant of the controlled object, L∈(0,10 10 ], e is a natural constant; The feedback controller adopts a fractional order [PI] controller, whose transfer function G c (s) are as follows: In the formula, k p is the controller proportional gain, with k p ∈[-10 10 ,10 10 ],k i is the controller integral gain, with k i ∈[-10 10 ,10 10 ], r is the controller order, r∈(0,2); k p , k i and r are the fractional-order [PI] controller parameters to be tuned; The characteristic equation of the closed-loop system G ce (k p ,k i ,r,s) is: G ce (k p ,k i ,r,s)=s r (Ts a +1)+Ke -Ls (k p s+k i ) r =0 (3) The complex root stability domain calculation module is connected to the closed-loop system establishment module, and is used to traverse the system frequency ω∈0+∞ based on a given controller order r, and calculate the complex root stability values ​​of the fractional-order [PI] controller parameters respectively, and use the final complex root stability values ​​as the complex root stability domain of the fractional-order [PI] controller parameters; wherein, Given the controller order r and the system frequency ω, the complex root stable value of the fractional-order [PI] controller parameters at the system frequency ω is calculated by the following formula: and In the formula, R2 is the intermediate variable, θ is the intermediate variable, θ∈(-π,π), ω is the system frequency; A parameter stability domain calculation module is connected with the closed-loop system establishment module and the complex root stability domain calculation module, and is used to combine the real root stability domain of the fractional order [PI] controller parameters to obtain the parameter stability domain of the fractional order [PI] controller under the controller order r; Assume that the real root stability region of the fractional-order [PI] controller parameters is: The parameter solving module is connected with the parameter stable domain calculation module and the complex root stable domain calculation module to give the crossing frequency ω gc and phase margin And based on a given crossover frequency ω gc and phase margin Calculate the fractional order [PI] controller parameters by solving equations 1, 2 and 3 to solve for the proportional gain k of the fractional order [PI] controller p , integral gain k i and controller order r; If the solution is an empty set, then re-set the crossing frequency ω gc and phase margin Recalculate and solve equation 1, equation 2, and equation 3; If the solution is a non-empty set, the proportional gain k is p , integral gain k i and controller order r are the fractional-order [PI] controller parameters that satisfy the constraints; Given the crossover frequency ω of the closed-loop system gc and phase margin Solving equation 1 for fractional-order [PI] controller parameters: Given the crossover frequency ω of the closed-loop system gc and phase margin Solving equation 2 for fractional-order [PI] controller parameters: On the basis that the open-loop phase of the closed-loop system has a slope of zero at the crossover frequency, the fractional-order [PI] controller parameters are solved in equation 3: In formula (7) to formula (9), R 2gc is the intermediate variable, θ gc is the intermediate variable, θ gc ∈(-π,π).

3. A fractional-order [PI] controller for an inertia-plus-delay fractional-order plant, characterized in that: The design is performed using the design method of the fractional-order [PI] controller for the inertia plus delay fractional-order object described in claim 1.

4. A fractional order [PI] controller design device, characterized in that: include: one or more processors; a memory for storing one or more programs, When the one or more programs are executed by the one or more processors, the one or more processors are caused to perform the steps of the method for designing a fractional-order [PI] controller for an inertia-plus-delay fractional-order plant as claimed in claim 1.

5. A computer-readable storage medium storing a computer program, characterized in that: When the program is executed by a processor, the steps of the method for designing a fractional-order [PI] controller for an inertia-plus-delay fractional-order plant as claimed in claim 1 are implemented.

Images