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 application relates to a fractional order [PI] controller design method, in particular to a fractional order [PI] controller design method and system for an inertia plus delay fractional order object. BACKGROUND

[0002] In typical industrial process control such as chemical process and wind power generation process, there is a class of inertia plus delay fractional order processes, such as wind power generation variable pitch control system, which is generally described by an inertia plus delay fractional order system , wherein G p (s) is an inertia plus delay fractional order transfer function, s is a differential operator, K is a gain of the controlled object, T is a time constant of the controlled object, a is an order of the controlled object, L is a delay time constant of the controlled object, and e is a natural constant; taking the wind power generation variable pitch control system as an example, the meanings of the parameters in the above formula are as follows: The gain K is a "control signal-pitch angle" static gain of a variable pitch actuator (such as a servo motor and a speed reducer); The time constant T is the mechanical inertia (such as the blade moment of inertia and the transmission mechanism damping) of the variable pitch system, the greater the time constant T, the stronger the inertia, and the slower the response of the pitch angle to the control signal; The fractional order a is the core nonlinearity of the variable pitch system, which is derived from the blade aerodynamic characteristics: the dynamic relationship between the lift and the drag of the blade when the wind speed changes cannot be accurately described by an integer order (such as a first order or a second order) linear model (for example, the difference in aerodynamic damping at low wind speed and high wind speed); and the fractional order model can more flexibly fit this "between integer orders" nonlinearity through a continuous adjustable order a∈(0, 2), and has higher accuracy than an integer order model (such as a first order inertia with a=1); The delay time L is a double delay of the variable pitch system: 1. Signal delay: time lag of wind speed detection (such as wind speed sensor sampling and data transmission); 2. Execution delay: mechanical response lag of the servo motor starting and the transmission mechanism overcoming the gap after the control signal is issued, which is completely consistent with the actual scene of "delay in wind speed fluctuation influence".

[0003] At present, the inertia plus delay fractional order system in typical industrial process control such as chemical process and wind power generation process widely uses a traditional feedback control mode based on output error, and the controller used is mainly a traditional proportional-integral-derivative (Proportional-Integral-Derivative, PID) controller. The traditional PID controller model is as follows, G c (s)=k p +ki s + k d s, where k p is the proportional gain, k i is the integral gain, k d is the derivative gain.

[0004] The traditional PID controller has the advantages of simple structure and easy implementation, but for an inertia plus delay fractional order object system, it cannot provide more freedom and parameter adjustment space to meet the requirements of more flexible and accurate control performance. Therefore, it needs to be improved and extended to a fractional order [PI] controller. The model of the fractional order [PI] controller is as follows, where k p is the proportional gain, k i is the integral gain, and r is the order. The fractional order [PI] controller can obtain better control performance than the traditional PID controller for an inertia plus delay fractional order system.

[0005] The fractional order [PI] controller in the form of (k p is the proportional gain of the controller, k i is the integral gain of the controller, and r is the order of the controller) is the product of the development of fractional calculus. Because it can provide more freedom and parameter adjustment space than the integer order controller, it can achieve more flexible and accurate control performance, better robustness and adaptability to complex systems. The fractional order object is described by a fractional differential equation to describe the dynamic characteristics of the controlled object, which is a generalization of the integer order object and can more accurately describe the dynamic characteristics of the actual system.

[0006] The parameter stability domain calculation method of the fractional order [PI] controller in the form of for a fractional order object lacks research. The current tuning technology of the fractional order [PI] controller is less studied. The existing tuning method is designed for an integer order first-order inertia plus delay object and lacks generality. In addition, the optimization of the fractional order [PI] controller parameters based on evolutionary algorithms can be based on robustness constraints to solve the minimum control performance index. This method has the disadvantages of slow solving speed and easy convergence to local optimum. With the development of fractional calculus theory and technology, more and more controlled objects are described by fractional transfer functions. The current research lacks direct calculation methods of the fractional order [PI] controller parameters under typical robustness indicators (such as maximum sensitivity function, phase margin, amplitude margin, phase flatness, and crossover frequency). It is necessary to robustly tune the parameters of the fractional order [PI] controller for a fractional order object. SUMMARY

[0007] The application aims to solve the robust design problem of fractional [PI] controller for a class of fractional order objects, and provides a design method and system of fractional [PI] controller for inertia plus delay fractional order objects.

[0008] In the first aspect, the application provides a design method of fractional [PI] controller for inertia plus delay fractional order objects, comprising the following steps: 1) establishing a closed-loop system composed of a controlled object and a feedback controller; wherein, A class of actual industrial systems as controlled objects adopts inertia plus delay fractional transfer function G p (s) is described as follows: (1) In the formula, G p (s) is an inertia plus delay fractional transfer function, s is a differential operator, K is the gain of the controlled object, 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 ], and e is a natural constant; The feedback controller adopts a fractional [PI] controller, and its transfer function G c (s) is as follows: (2) In the formula, k p is the proportional gain of the controller, k p ∈[-10 10 ,10 10 ], k i is the integral gain of the controller, k i ∈[-10 10 ,10 10 ], r is the order of the controller, r∈(0,2); k p , k i and r are the fractional [PI] controller parameters to be determined; The characteristic equation of the closed-loop system is: (3) 2) On the basis of a given controller order r, traverse the system frequency ω∈0+∞, respectively calculate the complex root stable value of the fractional [PI] controller parameters, and take the finally obtained complex root stable value as the complex root stable domain of the fractional [PI] controller parameters; wherein, ​Given the controller order r and the system frequency ω, the fractional [PI] controller parameters are calculated at the system frequency ω by the following formula: (4) and (5) where R2 is an intermediate variable, θ is an intermediate variable, θ∈(-π,π), and ω is the system frequency; Let the real root stability region of the fractional [PI] controller parameters be: (6) 3) Combined with the real root stability region of the fractional [PI] controller parameters, the parameter stability region of the fractional [PI] controller under the controller order r is obtained; 4) Given the crossover frequency ω gc and the phase margin , and based on the given crossover frequency ω gc and the phase margin , the solving equation one, the solving equation two and the solving equation three of the fractional [PI] controller parameters are calculated to solve the proportional gain k p , the integral gain k i and the controller order r of the fractional [PI] controller; If the solution is an empty set, the crossover frequency ω gc and the phase margin are redefined, and the solving equation one, the solving equation two and the solving equation three are recalculated; If the solution is a non-empty set, the proportional gain k p , the integral gain k i and the controller order r obtained are the fractional [PI] controller parameters that meet the constraints; Given the crossover frequency ω gc and the phase margin of the closed-loop system, the solving equation one of the fractional [PI] controller parameters is: (7) Given the crossover frequency ω gc and the phase margin of the closed-loop system, the solving equation two of the fractional [PI] controller parameters is: (8) On the basis that the open-loop phase of the closed-loop system has a slope of zero at the crossover frequency, the solving equation three of the fractional [PI] controller parameters is: (9) In formula (7) to formula (9), R 2gc is an intermediate variable, , θ gc is an intermediate variable, , θ gc ∈(-π, π); 5) set the obtained k p value, k i value and r value as the proportional gain, integral gain and order of the fractional [PI] controller satisfying the constraints into the fractional [PI] controller, that is, obtain the feedback controller satisfying the control requirements.

[0009] In a second aspect, the present application provides a design system of a fractional [PI] controller of an inertia plus delay fractional order object, comprising: a closed loop system establishment module for establishing a closed loop system composed of a controlled object and a feedback controller; wherein, a controlled actual industrial system is described by an inertia plus delay fractional transfer function G p (s) as follows: (1) In the formula, G p (s) is an inertia plus delay fractional transfer function, s is a differential operator, K is a controlled object gain, K∈[-10 10 , 0)∪(0, 10 10 ], T is a controlled object time constant, 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 ), and e is a natural constant; the feedback controller adopts a fractional [PI] controller, and the transfer function G c (s) of the controller is as follows: (2) In the formula, k p is a controller proportional gain, k p ∈[-10 10 , 10 10 ], k i is a controller integral gain, k i ∈[-10 10 , 10 10 ], and r is a controller order, r∈(0, 2); k p , k i and r are fractional [PI] controller parameters to be determined; the characteristic equation of the closed loop system is : (3) The complex root stability region calculation module, connected to the closed-loop system establishment module, is used to calculate the complex root stable values ​​of the fractional-order [PI] controller parameters by iterating through the system frequencies ω∈0+∞, given a controller order r. The final complex root stable values ​​are then used as the complex root stability region of the fractional-order [PI] controller parameters. Given the controller order r and the system frequency ω, the complex root stable values ​​of the fractional-order [PI] controller parameters at the system frequency ω are calculated using the following formula: (4) and (5) In the formula, R2 is an intermediate variable. θ is an intermediate variable. θ∈(-π,π), ω is the system frequency; The parameter stability domain calculation module, connected to the closed-loop system establishment module and the complex root stability domain calculation module, is used to combine the real root stability domain of the fractional-order [PI] controller parameters to obtain the stability domain of the fractional-order [PI] controller at the controller order. The parameter stability region is as follows; Let the real root stability region of the fractional-order [PI] controller parameters be: (6) The parameter solving module, connected to the parameter stability domain calculation module and the complex root stability domain calculation module, is used to calculate the crossover frequency ω. gc and phase margin And based on the given crossover frequency ω gc and phase margin Solve equations one, two, and three to calculate the parameters of the fractional-order [PI] controller, and then calculate the proportional gain k of the fractional-order [PI] controller. p Integral gain k i and controller order r; If the solution yields an empty set, then the crossing frequency ω is redefined. gc and phase margin Recalculate and solve equations one, two, and three. If the solution yields a non-empty set, then the proportional gain k is obtained. p Integral gain k i The controller order r is a fractional-order [PI] controller parameter that satisfies the constraints; Given the crossover frequency ω of the closed-loop system gc and phase margin Equation 1 for solving the parameters of a fractional-order [PI] controller: (7) Given the crossover frequency ω gc and phase margin , the solving equation two of fractional order [PI] controller parameters: (8) Based on the open-loop phase of the closed-loop system at the crossover frequency is out of the slope of zero, the solving equation three of fractional order [PI] controller parameters: (9) In equation (7)-equation (9), R 2gc is an intermediate variable, , θ gc is an intermediate variable, , θ gc ∈(-π,π); The design module is used to set the obtained k p value, k i value and r value as the proportional gain, integral gain and order of the fractional order [PI] controller that meets the constraints into the fractional order [PI] controller, that is, the feedback controller that meets the control requirements is obtained.

[0010] In a third aspect, the present application provides a fractional order [PI] controller of an inertia plus delay fractional order object, which is designed by using the design method of the fractional order [PI] controller of the inertia plus delay fractional order object.

[0011] In a fourth aspect, the present application provides a fractional order [PI] controller design device, which comprises: 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 execute the steps of the design method of the fractional order [PI] controller of the inertia plus delay fractional order object.

[0012] In a fifth aspect, the present application provides a computer readable storage medium storing a computer program, which is executed by a processor to realize the steps of the design method of the fractional order [PI] controller of the inertia plus delay fractional order object.

[0013] The present application has outstanding substantial features and significant progress compared with the prior art, and specifically: 1、The present application solves the fractional order [PI] controller parameter robust design method of a class of fractional order objects, which can directly adjust the parameters (proportional gain k pintegral gain k i and order r), so as to design the fractional order [PI] controller with strong ability to cope with gain uncertainty of the controlled object.

[0014] 2, the method only needs to give the phase margin, crossover frequency and phase flatness constraint of the closed-loop control system, and then the stable domain is obtained by calculating the real root stable domain and the complex root stable domain, so as to adjust the parameters (proportional gain k p integral gain k i and order r), the method is simple and easy to realize.

[0015] 3, compared with the existing fractional order [PI] controller parameter tuning method, the proportional gain, integral gain and controller order of the fractional order [PI] controller can be uniquely determined by calculating three equations, which can ensure the uniqueness of the obtained fractional order [PI] controller parameters, and avoid the local optimal solution problem when the evolutionary algorithm is used to optimize the fractional order [PI] controller. BRIEF DESCRIPTION OF DRAWINGS

[0016] Figure 1 The closed-loop system composed of the fractional order [PI] controller and the inertia plus delay fractional order object of the present application.

[0017] Figure 2 The fractional order [PI] controller stable domain of the present application under the given order calculated in example 1.

[0018] Figure 3 The control effect of the present application in example 1.

[0019] Figure 4 The control effect of the present application in example 1 when the controlled object has uncertainty. DETAILED DESCRIPTION

[0020] In order to make the purpose, technical scheme and advantages of the present application clearer, the present application will be further described in detail below in combination with the drawings and examples. It should be understood that the specific examples described herein are only used to explain the present application, and are not used to limit the present application.

[0021] Example 1 The design method of the fractional order [PI] controller of the inertia plus delay fractional order object proposed in this embodiment includes the following steps: 1) the fractional order [PI] controller of the inertia plus delay fractional order object is established as a closed-loop system composed of the controlled object and the feedback controller; wherein, A kind of actual industrial system controlled is used as the controlled object inertia plus delay fractional order transfer function G p(s) Description, as follows: (1) In the formula, G p (s) is the fractional-order transfer function with inertia and delay, s is the differential operator, and K is the gain of the controlled object, where K∈[-10]. 10 ,0)∪(0,10 10 ], T is the time constant of the controlled object, where T∈(0,10) 10 ], where α is the order of the controlled object, α∈(0,2), and L is the delay time constant of the controlled object, L∈(0,10). 10 ), where e is the natural constant; In this embodiment, K=1, T=1, α=1, and L=0.5.

[0022] The feedback controller uses a fractional-order [PI] controller with a transfer function G. c (s) are as follows: (2) In the formula, k p For the proportional gain of the controller, we have k p ∈[-10 10 10 10 ], k i For the controller integral gain, we have k i ∈[-10 10 10 10 ], r is the controller order, where r∈(0,2); k p k i r and r are the parameters of the fractional-order [PI] controller to be tuned.

[0023] Characteristic equation of a closed-loop system for: (3).

[0024] 2) Given the controller order r, iterate through the system frequencies ω∈0+∞ and calculate the complex root stable values ​​of the fractional-order [PI] controller parameters respectively. The final complex root stable value is taken as the complex root stability region of the fractional-order [PI] controller parameters. Given the controller order r and the system frequency ω, the complex root stable values ​​of the fractional-order [PI] controller parameters at the system frequency ω are calculated using equations (4) and (5). (4) and (5) In the formula, R2 is an intermediate variable. θ is an intermediate variable. , θ ∈ (-π, π), ω is the system frequency; 3) Combining the real root stability region of the fractional order [PI] controller parameters, the parameter stability region of the fractional order [PI] controller under the controller order r is obtained; the real root stability region of the fractional order [PI] controller parameters is: (6) In this embodiment, r = 1, and the parameter stability region of the fractional order [PI] controller under the controller order r = 1 is obtained; the effectiveness of the method is illustrated. Figure 2

[0025] 4) Given the crossover frequency ω gc and the phase margin , and based on the given crossover frequency ω gc and the phase margin , the solving equation one, the solving equation two and the solving equation three of the fractional order [PI] controller parameters are calculated, and the proportional gain k p , the integral gain k i and the controller order r of the fractional order [PI] controller are solved. If the solved result is an empty set, the crossover frequency ω gc and the phase margin are redefined, and the solving equation one, the solving equation two and the solving equation three are recalculated. If the solved result is a non-empty set, the proportional gain k p , the integral gain k i and the controller order r are the fractional order [PI] controller parameters satisfying the constraints. Given the crossover frequency ω gc and the phase margin of the closed-loop system, the solving equation one of the fractional order [PI] controller parameters is: (7) Given the crossover frequency ω gc and the phase margin of the closed-loop system, the solving equation two of the fractional order [PI] controller parameters is: (8) 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 solving equation three of the fractional order [PI] controller parameters is: (9) In formula (7) to formula (9), R 2gc is an intermediate variable, , θ gc is an intermediate variable, ​​θ gc ∈(-π,π); In this embodiment, the crossover frequency ω of the closed-loop system is given. gc =0.1 and phase margin =73º, thus obtaining the proportional gain k. p =0.9539, Integral gain k i =0.3283, controller order r=1.9800.

[0026] 5) Obtain k p =0.9539, k i =0.3283 and r=1.9800 are used as the proportional gain, integral gain and order of the fractional-order [PI] controller to satisfy the constraints, and thus a feedback controller that meets the control requirements is obtained.

[0027] In this embodiment, the crossover frequency ω can be obtained. gc =0.1Hz, phase margin Fractional-order [PI] controller parameters with constraints such as =73º and phase flatness, and proportional gain k. p =0.9539, Integral gain k i =0.3283 and controller order r=1.9800. Combining this with the controlled object, we can obtain... Figure 1 The tracking and anti-interference performance of the closed-loop system shown is as follows: Figure 3 As shown, the specific simulation process is as follows: At the start of the simulation, the closed-loop system is in steady state. At 0s, the setpoint is changed from 0 to 1. At 20s, the control variable of the closed-loop is disturbed, changing from 0 to 1. The simulation ends at 40s.

[0028] Simulation results show that the closed-loop system proposed in this 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 relatively fast tracking capability and strong anti-interference capability.

[0029] To analyze the ability of the tuned fractional-order [PI] controller to handle the uncertainty of the controlled object's gain, G was modified. p In (s), the controlled object gain K is 120%K and 80%K respectively. Repeating the above simulation, we can obtain... Figure 4 As shown in the figure, the tuned fractional-order [PI] controller has a strong ability to cope with the uncertainty of the controlled object's gain.

[0030] Example 2 This embodiment provides a design system for a fractional-order (PI) controller for an inertial-delayed fractional-order object, including: The closed loop system establishing module is configured to establish the fractional order [PI] controller of the inertia plus delay fractional order object as a closed loop system composed of the controlled object and a feedback controller. The complex root stability domain calculation module is connected with the closed loop system establishing module and is configured to calculate the complex root stability value of the fractional order [PI] controller parameter based on the given controller order r and traverse the system frequency ω∈0+∞, and take the finally obtained complex root stability value as the complex root stability domain of the fractional order [PI] controller parameter. The parameter stability domain calculation module is connected with the closed loop system establishing module and the complex root stability domain calculation module and is configured to obtain the parameter stability domain of the fractional order [PI] controller under the controller order r in combination with the real root stability domain of the fractional order [PI] controller parameter. The parameter solving module is connected with the parameter stability domain calculation module and the complex root stability domain calculation module and is configured to give the crossing frequency ω gc and the phase margin , and calculate the solving equation one, the solving equation two and the solving equation three of the fractional order [PI] controller parameter based on the given crossing frequency ω gc and the phase margin , and solve the proportional gain k p , the integral gain k i and the controller order r of the fractional order [PI] controller. If the solved result is an empty set, the crossing frequency ω gc and the phase margin are re-given, and the solving equation one, the solving equation two and the solving equation three are re-calculated. If the solved result is a non-empty set, the proportional gain k p , the integral gain k i and the controller order r are the fractional order [PI] controller parameters meeting the constraints. The design module is configured to set the obtained k p value, k i value and r value as the proportional gain, integral gain and order of the fractional order [PI] controller meeting the constraints to the fractional order [PI] controller, that is, obtain the feedback controller meeting the control requirements.

[0031] In this embodiment, the implementation scheme of the problem solving of the fractional order [PI] controller of the inertia plus delay fractional order object is similar to the implementation scheme described in the method of embodiment 1, and will not be described here.

[0032] Embodiment 3 This embodiment provides a fractional order [PI] controller of an inertia plus delay fractional order object, which is designed by using the design method of the fractional order [PI] controller of the inertia plus delay fractional order object described in embodiment 1.

[0033] Embodiment 4 The various modules in the system of Embodiment 2 can be implemented in whole or in part by software, hardware, and combinations thereof. The modules described above can be embedded in the processor in the computer device in hardware form or independent of the processor in the computer device, or stored in the memory in the computer device in software form, so as to be called and executed by the processor to perform the operations corresponding to the modules.

[0034] In an exemplary embodiment, a fractional [PI] controller design apparatus is provided, and the computer device can 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 through a system bus, and the communication interface, the display unit, and the input device are connected to the system bus through the input / output interface. The processor of the computer device is configured 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 operating system and the computer program in the non-volatile storage medium to run. The input / output interface of the computer device is configured to exchange information between the processor and external devices. The communication interface of the computer device is configured to perform wired or wireless communication with external terminals, and the wireless communication can be achieved through WIFI, mobile cellular network, NFC (Near Field Communication), or other technologies. The computer program is executed by the processor to implement the steps of the design method of the fractional [PI] controller of the inertia plus delay fractional order object. The display unit of the computer device is configured to form a visually visible picture, which can 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 overlaid on the display screen, or a key, trackball, or touchpad arranged on the shell of the computer device, or an external keyboard, touchpad, or mouse, etc.

[0035] Those skilled in the art can understand that the structure of the computer device described above is only part of the structure related to the scheme of the present application, and does not constitute a limitation on the computer device to which the scheme of the present application is applied. The specific computer device can include more or fewer components, or combine certain components, or have a different arrangement of components.

[0036] In an exemplary embodiment, a computer readable storage medium is provided, and the computer readable storage medium stores a computer program. The computer program is executed by the processor to implement the steps of the design method of the fractional [PI] controller of the inertia plus delay fractional order object.

[0037] 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. The computer program can be stored in a non-volatile computer readable storage medium, and when the computer program is executed, the processes of the above-mentioned embodiments of the methods can be included. Any reference to 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. The non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical storage, high-density embedded non-volatile memory, resistive memory (ReRAM), magnetoresistive random access memory (MRAM), ferroelectric memory (FRAM), phase change memory (PCM), graphene memory, etc. The volatile memory can include random access memory (RAM) or external cache memory, etc. As an illustration but not limitation, the RAM can be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM), etc. The database involved in the embodiments provided in the present application can include at least one of a relational database and a non-relational database. The non-relational database can include a distributed database based on a block chain, etc., without being limited thereto. The processor involved in the embodiments provided in the present application can be a general-purpose processor, a central processing unit, a graphics processing unit, a digital signal processor, a programmable logic device, a data processing logic device based on quantum computing, etc., without being limited thereto.

[0038] Any combination of the technical features of the above embodiments can be made. In order to make the description simple, all possible combinations of the technical features in the above embodiments are not described, however, as long as the combination of the technical features does not exist, it should be considered as the scope disclosed in the present application.

[0039] The above is only a specific implementation of the present application, but the protection scope of the present application is not limited thereto. Any person skilled in the art can easily think of changes or replacements within the technical range disclosed in the present application, which should be covered within the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A method of designing a fractional order [PI] controller for an inertial plus time- delayed fractional order plant, characterized by, The method comprises the following steps: 1) establishing a closed loop system composed of a controlled object and a feedback controller; wherein, A class of actual industrial systems as controlled object is taken as controlled object to adopt inertia plus delay fractional order transfer function G p (s) description, as follows: （1） In the formula, G p (s) is an inertial plus delay fractional order transfer function, s is a differential operator, K is a gain of the controlled object, has K ∈ [-10 10 , 0)∪(0, 10 10 ], T is a time constant of the controlled object, has T ∈ (0, 10 10 ], α is an order of the controlled object, has α ∈ (0, 2), L is a delay time constant of the controlled object, has L ∈ (0, 10 10 ], and e is a natural constant; The feedback controller employs a fractional order [PI] controller whose transfer function G c (s) is as follows: （2） where 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 ], and r is the controller order, with r ∈ (0, 2); k p , k i , and r are the tuning fractional order [PI] controller parameters; Characteristic equation of a closed loop system is: （3） 2) on the basis of a given controller order r, traversing system frequency ω∈0+∞, respectively calculating the complex root stable value of the fractional order [PI] controller parameter, and taking the finally obtained complex root stable value as the complex root stable domain of the fractional order [PI] controller parameter; wherein, on the basis of a given controller order r and system frequency ω, the complex root stable value of the fractional order [PI] controller parameter at the system frequency ω is calculated by the following formula: （4） and （5） where R2 is an intermediate variable, , θ is an intermediate variable, , θ ∈ (-π, π), ω is the system frequency; the real root stable domain of the fractional order [PI] controller parameter is set as: （6） 3) combining the real root stable domain of the fractional order [PI] controller parameter, the parameter stable domain of the fractional order [PI] controller at the controller order r is obtained; 4) given the crossover frequency ω gc and the phase margin , the solution equations one, two and three are calculated to solve the fractional order [PI] controller's proportional gain k p , integral gain k i and controller order r based on the given crossover frequency ω gc and the phase margin ; If the solution is an empty set, then re-define the crossing frequency ω gc and the phase margin and re-compute solution equation one, solution equation two and solution equation three; If the solution is a non-empty set, then the proportional gain k p , the integral gain k i and the controller order r are the fractional [PI] controller parameters that satisfy the constraints. The crossover frequency ω of the given closed-loop system gc and phase margin The fractional [PI] controller parameters solving equation one: （7） The crossover frequency ω of the given closed-loop system gc and phase margin The solving equation two of fractional order [PI] controller parameters: （8） on the basis of the open loop phase of the closed loop system crossing the frequency with a slope of zero, the solving equation three of the fractional order [PI] controller parameter is: （9） In formula (7) - formula (9), R 2gc is an intermediate variable, , θ gc is an intermediate variable, , θ gc ∈(-π,π) 5) the obtained k p value, k i value and r value are set to the fractional order [PI] controller as the proportional gain, integral gain and order of the fractional order [PI] controller satisfying the constraints, that is, a feedback controller satisfying the control requirements is obtained.

2. A system for designing a fractional order [PI] controller of an inertial plus delayed fractional order object, characterized by, comprise: a closed loop system establishing module, which establishes a closed loop system composed of a controlled object and a feedback controller; wherein, A class of actual industrial systems as controlled object is taken as controlled object to adopt inertia plus delay fractional order transfer function G p (s) description, as follows: （1） In the formula, G p (s) is an inertial plus delay fractional order transfer function, s is a differential operator, K is a gain of the controlled object, has K ∈ [-10 10 , 0)∪(0, 10 10 ], T is a time constant of the controlled object, has T ∈ (0, 10 10 ], α is an order of the controlled object, has α ∈ (0, 2), L is a delay time constant of the controlled object, has L ∈ (0, 10 10 ], and e is a natural constant; The feedback controller employs a fractional order [PI] controller whose transfer function G c (s) as follows: （2） where 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 ], and r is the controller order, with r ∈ (0, 2); k p , k i , and r are the tuning fractional order [PI] controller parameters; Characteristic equation of a closed loop system is: （3） a complex root stable domain calculating module, which is connected with the closed loop system establishing module and is used for, on the basis of a given controller order r, traversing system frequency ω∈0+∞, respectively calculating the complex root stable value of the fractional order [PI] controller parameter, and taking the finally obtained complex root stable value as the complex root stable domain of the fractional order [PI] controller parameter; wherein, on the basis of a given controller order r and system frequency ω, the complex root stable value of the fractional order [PI] controller parameter at the system frequency ω is calculated by the following formula: （4） and （5） where R2 is an intermediate variable, , θ is an intermediate variable, , θ ∈ (-π, π), ω is the system frequency; The 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 for combining the real root stability domain of the fractional order [PI] controller parameter to obtain the parameter stability domain of the fractional order [PI] controller under the controller order . the real root stable domain of the fractional order [PI] controller parameter is set as: （6） The parameter solving module, connected to the parameter stability domain calculation module and the complex root stability domain calculation module, is used to calculate the crossover frequency ω. gc and phase margin And based on the given crossover frequency ω gc and phase margin Solve equations one, two, and three to calculate the parameters of the fractional-order [PI] controller, and then calculate 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-define the crossing frequency ω gc and the phase margin and re-compute solution equation one, solution equation two and solution equation three; If the solution is a non-empty set, then the proportional gain k p , the integral gain k i and the controller order r are the fractional [PI] controller parameters that satisfy the constraints. The crossover frequency ω of the given closed-loop system gc and phase margin The fractional [PI] controller parameters solving equation one: （7） The crossover frequency ω of the given closed-loop system gc and phase margin The solving equation two of fractional order [PI] controller parameters: （8） on the basis of the open loop phase of the closed loop system crossing the frequency with a slope of zero, the solving equation three of the fractional order [PI] controller parameter is: （9） In formula (7) - formula (9), R 2gc is an intermediate variable, , θ gc is an intermediate variable, , θ gc ∈(-π,π) a design module for setting the obtained k p value, k i value and r value as the proportional gain, integral gain and order of the fractional [PI] controller satisfying the constraint into the fractional [PI] controller, that is, obtaining the feedback controller satisfying the control requirement.

3. A fractional order [PI] controller of an inertial plus delayed fractional order object characterized by: The fractional order [PI] controller of an inertial plus delay fractional order object is designed by using the design method of the fractional order [PI] controller of an inertial plus delay fractional order object according to claim 1.

4. A fractional [PI] controller design apparatus characterized by comprising: comprise: 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 execute the steps of the design method of the fractional order [PI] controller of an inertial plus delay fractional order object according to claim 1.

5. A computer readable storage medium storing a computer program, characterized in that, The programs are executed by the processors to implement the steps of the design method of the fractional order [PI] controller of an inertial plus delay fractional order object according to claim 1.

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