CN107037726B - It is a kind of non-with first new fractional-order system synovial membrane interference observer design method - Google Patents
It is a kind of non-with first new fractional-order system synovial membrane interference observer design method Download PDFInfo
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Abstract
本发明公开了一种非同元分数阶系统滑膜干扰观测器设计方法，其具体步骤包括：激励系统获取相关数据，离线辨识系统非同元分数阶模型，然后根据系统的非同元分数阶阶次，设计滑膜干扰观测器在线观测当前时刻的扰动。本发明还公开了应用相应方法设计的产品。本发明还公开了应用相应产品对非同元分数阶系统进行控制的方法。相对于传统的抗扰动方法，本发明方法利用非同元分数阶系统模型进行滑膜干扰观测器的设计，提高了扰动观测的准确度；能够在有限时间内收敛，保证观测结果的实时性；在遇到强干扰的作用下，增强非同元分数阶系统的鲁棒性与稳定性，此外，本发明提出的方法也适用于同元分数阶系统。
The invention discloses a method for designing a synovium disturbance observer of a non-homogeneous fractional order system. The specific steps include: stimulating the system to obtain relevant data, off-line identification of the non-homogeneous fractional order model of the system, and then according to the non-homogeneous fractional order of the system order, the synovium disturbance observer is designed to observe the disturbance at the current moment online. The invention also discloses a product designed by applying the corresponding method. The invention also discloses a method for controlling non-homogeneous fractional order systems by using corresponding products. Compared with the traditional anti-disturbance method, the method of the present invention uses the non-homogeneous fractional system model to design the synovial disturbance observer, which improves the accuracy of disturbance observation; it can converge within a limited time and ensure the real-time performance of the observation results; Under the action of strong interference, the robustness and stability of non-homogeneous fractional-order systems are enhanced. In addition, the method proposed by the invention is also applicable to homogeneous fractional-order systems.
Description
Technical Field
The invention belongs to the technical field of control, and particularly relates to a design method of a non-homonymous fractional order system sliding film disturbance observer.
Background
Fractional calculus provides a more sophisticated mathematical model for complex dynamic systems. At present, the fractional order model can achieve the purpose of more accurate and concise depiction of the fractional order object. The fractional order model is an extension of the traditional integer order model, and has good fitting effect on some systems which are difficult to accurately describe by the order of the integer model. Compared with the integral order modeling, the fractional order model has the advantages of lower order, fewer parameters and higher modeling precision.
In the actual research process, the system can be divided into non-homogeneous (different orders) and homogeneous (same order) fractional order systems according to whether the fractional order of the fractional order system is the same or not. At present, for many physical systems of drivers, the fractional order characteristics of the systems have non-homogeneous characteristics, for example, the fractional order of electromagnetic energy storage and mechanical energy storage links of a flexible swing arm system in electronic manufacturing equipment are inconsistent, so the fractional order characteristics of the systems have non-homogeneous characteristics. On the other hand, the control performance of the system is seriously affected by external disturbance, perturbation of system model parameters, disturbance of friction, ripple thrust and other diversity. The requirement of high-speed and high-precision movement requires that a controlled system has strong robustness to respond to various disturbances.
In the actual anti-perturbation strategy, the existing methods can be summarized as the following two methods: passive suppression strategies and active suppression methods. The passive suppression strategy ensures the dynamic tracking performance of the system by using advanced control algorithms, such as iterative learning control, fuzzy control, adaptive control and the like. The method utilizes a relatively slow feedback adjustment mode to eliminate the influence of the disturbance on the system performance according to the system following error. These methods therefore lead to poor dynamic response performance and, in the case of strong disturbances, are more likely to lead to instability of the controlled system. The active suppression method is that firstly an observer accurately estimates the size of the disturbance and then directly eliminates the influence of the disturbance through feed-forward compensation. The method has better anti-disturbance performance and is widely applied in the control field.
The method for actively suppressing the disturbance, which is widely applied at present, is only suitable for the system with the homologism fractional order characteristic, and therefore, is not suitable for the system with the non-homologism fractional order characteristic. On the other hand, for a system with a fractional order characteristic, the current disturbance suppression method cannot guarantee the finite time convergence of disturbance observation errors.
Such as the "Second-order slipping mode to interference and fault detection in fractional-order systems" (Pisano a,m, Usai E, et, IFAC Proceedings Volumes,2011,44(1):2436-2441.), discloses a sliding mode interference observation design method for a fractional order system. However, the design method disclosed in this document has the following drawbacks or disadvantages:
(1) only applicable to homogeneous fractional order systems;
(2) only infinite time convergence of disturbance observation errors is guaranteed, and finite time convergence cannot be further guaranteed.
Disclosure of Invention
Aiming at the defects or the improvement requirements of the prior art, the invention provides a design method of a sliding mode disturbance observer of a non-homonymous fractional order system, which aims to accurately measure the actual disturbance of the system by identifying a non-homonymous fractional order state space model of the system and designing the sliding mode observer by using the non-homonymous fractional order, and then superimpose the observed measured value on the input control quantity of the system through a feedforward channel, thereby directly eliminating the influence of the disturbance on the system performance and realizing the control of the fractional order system.
To achieve the above object, according to one aspect of the present invention, there is provided a design method of a non-isobaric fractional order system synovial disturbance observer, comprising the steps of:
s1: selecting an excitation signal to excite a non-homonymous fractional order system, and acquiring a required input signal and an output signal;
s2: and carrying out state space model identification on the non-homonymous fractional order system by utilizing an optimization algorithm, wherein the state space model comprises the following steps:
wherein,(t) } is the state variable of the system, A1,A2K is a parameter of the system; d (t) ═ Δ ax (t) + de(t) represents the actual integrated disturbance of the system, u (t) is the input signal of the system; y (t) is the output signal of the system, Δ A is the uncertainty component of the system model parameters, de(t) representing unknown disturbance existing in the system, h (x (t)) is a nonlinear factor existing in the system model, ξ and upsilon are fractional orders of the system model, and D is a fractional order differential operator;
s3: designing a synovial membrane disturbance observer according to the state space model in step S2, comprising the following steps:
s11: calculating the speed following error e (t):
e(t)＝y(t)-r(t)
wherein r (t) is a given reference speed;
s12, according to the non-homonym fractional order ξ and upsilon of the system, performing fractional differentiation on the following error e (t) of the system:
wherein C is a preset real number;
s13: -integer order differentiation of the following error e (t):
s14, selecting a virtual synovial membrane observation surface according to the non-homonym fractional order ξ and upsilon:
wherein z is1Sigma is the designed virtual slide membrane surface for the selected intermediate variable,actual disturbance observed for a designed sliding mode observer;
s15: differentiating the virtual synovial observation plane σ:
wherein,-d (t) is the observed error of the system disturbance;
s16: selecting a first-order synovial differential equation to estimateThe value of (c):
where ρ isσ1,ρσ2Is the state variable of the synovial differential equation,sign is a sign function for a predefined positive real number;
can be represented by the following formula
In the formula, lambda is the error of differential equation estimation;
s17: defining dependent variablesThe following synovial membrane disturbance observer is designed, and the variation of the actual disturbance of the observation system can be obtained as follows:
wherein, TsIs the sampling time, and P (σ ') is a function of σ'.
Further, the fractional order differential operator is defined as:
α is a fractional order, n is the smallest positive integer larger than α fractional order, and I is a fractional order integral operator;
further, the fractional order integral operator is:
where α is a fractional order, t is an integration time, x (t) represents an integrated function, and Γ represents a gamma function.
Further, the optimization algorithm is a particle swarm algorithm:
wherein k' represents the current iteration number; i is 1, 2, 3 … is the number of the particle; a represents the system parameter to be identified,indicating the position currently located in the solution space;indicating the current speed of movement, c1,c2Is the acceleration constant; rand1,rand2Random number of 0 to 1, pbestijAnd gbestijRespectively representing a locally optimal solution and a globally optimal solution.
Further, the input instruction of the non-isomorphic fractional order system is as follows:
further, the variation of the actual disturbance of the observation system satisfies the following perturbation nonlinear differential equation theorem:
wherein,x is the state of the system, μ1And mu2Are the parameters of the differential equation and,μ2≥1.1M，Tsis at the time of samplingIn between, ξ (t) is the unknown system disturbance, and M represents the upper bound of the disturbance variance.
Further, the synovial membrane disturbance observer satisfies the following relationship:
wherein: mu's'1＝CKA1μ1,μ′2＝CKA1μ2,
According to the perturbation nonlinear differential equation theorem, the variables sigma' andconvergence to zero in a finite time yields:
therefore, the sliding mode disturbance observer can accurately identify the disturbance of the non-homonymous fractional order system in a limited time.
Further, the excitation signal is gaussian white noise whose amplitude distribution follows gaussian distribution.
According to another aspect of the invention, a product designed by applying the design method of the non-homonymous fractional order system slip film disturbance observer is provided.
According to another aspect of the present invention, there is provided a non-homogeneous fractional order system control method for a product, comprising: and superposing the disturbance observed by the synovial membrane disturbance observer to a control speed end, and realizing the control of the non-homonymous fractional order system through a fractional order PI controller, wherein the transfer function can be expressed as:
in the formula, KpIs the proportional coefficient of the PI controller, KiIs the integral coefficient of the PI controller, λ is the fractional order, and s is a complex variable.
In general, compared with the prior art, the above technical solution contemplated by the present invention can achieve the following beneficial effects:
(1) according to the method, a fractional order system is described through a non-homomorphic fractional order state space model, so that the effect which is more accurate than that of an integer order model can be achieved, a controlled object can be better represented, and therefore a synovial membrane interference observer designed based on the non-homomorphic fractional order can achieve a higher observation effect.
(2) The method for actively suppressing the disturbance comprehensively considers comprehensive disturbances such as system model parameter drift, external disturbance and the like, can observe the disturbance of the system within limited time, has high real-time performance, and can ensure the stability of the system when encountering strong disturbance.
(3) The synovium disturbance observer provided by the invention is not only suitable for a non-isomorphic fractional order system, but also suitable for an isomorphic fractional order system.
Drawings
FIG. 1 is a control flow chart of a flexible swing arm system involved in a design method of a non-homonymous fractional order system synovial disturbance observer according to an embodiment of the present invention;
FIG. 2 is a schematic structural diagram of a flexible swing arm system involved in a design method of a non-homonymous fractional order system synovial disturbance observer according to an embodiment of the present invention;
FIG. 3 is a control block diagram of a flexible swing arm system involved in a design method of a non-homonymous fractional order system synovial disturbance observer according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of a design step of a flexible swing arm system synovial disturbance observer related to a method for designing a non-homonymous fractional order system synovial disturbance observer according to an embodiment of the present invention;
fig. 5 is a schematic diagram of an observation effect of a flexible swing arm system synovial disturbance observer related to a non-homonymous fractional order system synovial disturbance observer design method according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
For a key component flexible swing arm system in electronic manufacturing equipment, an inductance circuit of the system can better describe the characteristics by utilizing a fractional order model due to inherent nonlinearity of charge and discharge; in addition, the system can generate fractional order dynamic characteristics due to the characteristics of uneven mass distribution of the flexible load, nonlinear elastic deformation, incapability of strictly proportional friction to the speed and the like.
Fig. 2 is a schematic structural diagram of a flexible swing arm system related to a design method of a non-homonymous fractional order system synovial disturbance observer according to an embodiment of the present invention. The flexible swing arm system in the chip sorter mainly realizes the picking and sorting of the chips, and the main structural schematic diagram is shown in fig. 2. The flexible swing arm system mainly comprises a permanent magnet synchronous motor 1 and a flexible structure, wherein the flexible structure comprises a coupler 2, a transmission device 3, a swing arm structure 4 and the like.
Along with the development of electronic manufacturing equipment technology, its work efficiency requirement is higher and higher, and the frequency is faster and faster, and the swing arm is longer and longer, and the requirement of positioning accuracy is also higher and higher simultaneously. And in the motion control of the swing arm, the angular displacement of the swing arm motor is converted into the tail end displacement of the swing arm through the amplification of the swing arm mechanism, so that the higher the motion frequency is, the longer the swing arm is, and the harder the end positioning precision is. The swing arm system of the electronic manufacturing equipment is generally divided into a single swing arm structure and a double swing arm structure, and the swing arm is driven to move by a servo motor through a transmission link; the structure has the advantages of multiple transmission links, large inertia and strong flexibility, thereby having large control difficulty.
When the flexible swing arm system is controlled, the structure of the flexible swing arm system can be considered as the combination of a permanent magnet synchronous motor and a time-varying complex load. Control of the flexible swing arm system can be resolved into a control problem for a complex servo drive system. The invention aims at controlling the speed loop of the flexible swing arm system, and aims to provide stable speed output without the flexible swing arm system, generate a smooth track in a complex and changeable environment and optimize the control performance of the system.
FIG. 3 is a control block diagram of a flexible swing arm system involved in a design method of a non-homonymous fractional order system synovial disturbance observer according to an embodiment of the present invention; fig. 4 is a schematic diagram of a design step of a flexible swing arm system slip film disturbance observer related to a non-isobaric fractional order system slip film disturbance observer design method according to an embodiment of the present invention. As shown in fig. 3 and fig. 4, the design method of the non-homonymous fractional order flexible swing arm system synovial disturbance observer provided by the present invention mainly includes three parts: identifying a non-isomorphic fractional order state space model of the flexible swing arm system; designing a fractional order synovial membrane disturbance observer to observe actual disturbance existing in the system; and introducing the observed disturbance into an input current command of the system to eliminate the influence of the disturbance on the system performance. The main design steps are summarized as follows:
the first step is as follows: gaussian white noise excites an object and collects signals
Firstly, motor speed feedback y (t) and an input current value signal u (t) in a servo system speed loop need to be acquired. Therefore, an excitation signal with a proper form is selected to excite the controlled object of the speed ring, so that the required signal is acquired, and an offline database is established. The excitation signal is selected to sufficiently excite the servo system to cover the frequency bands in which the servo system operates. In the present embodiment, gaussian white noise whose amplitude distribution follows gaussian distribution is preferably selected, whose average power is close to uniform distribution, and which has no memory. The discrete form of the white noise signal is called as a white noise sequence, the net input disturbance to the system is small, the amplitude, the period and the clock beat are easy to control, and the requirement of the optimal input signal can be met. And exciting the alternating current servo driving system by using Gaussian white noise to obtain sampling values of the rotating speed and the current of the flexible swing arm system.
Aiming at various disturbances existing in the flexible swing arm system, selecting the disturbance with the following expression to be added to the non-homomorphic fractional order state space model of the flexible swing arm system:
wherein: omega1＝1,ω2＝0.2。
In this embodiment, a particle swarm algorithm is used to complete identification of system fractional order model parameters, each iteration of a particle needs to update the state of the particle through a position update formula and a velocity update formula, for this embodiment, a represents a state space model parameter to be solved by optimization, and the formula is as follows:
wherein k' represents the current iteration number; i is 1, 2, 3 … is the number of the particle; a represents the system parameter to be identified,indicating the position currently located in the solution space;indicating the current speed of movement, c1,c2Is the acceleration constant; rand1,rand2Random number of 0 to 1, pbestijAnd gbestijRespectively representing a locally optimal solution and a globally optimal solution.
The related model parameters of the non-isomorphic fractional order state space model expression of the system can be identified and obtained:
can obtain A1＝1,A2＝-0.2xξ(t)-0.5xυ(t),h(x(t))＝0，υ＝0.2928，ξ＝0.9752，K＝143.7275。
The second step is that: the design of a sliding mode disturbance observer based on fractional order, the design of a virtual sliding mode surface and the fractional order calculus operation of a following error signal enable the sliding mode disturbance observer to observe a real-time value of actual disturbance in a limited time, and the specific design steps can be divided into the following points:
s11: calculating the speed following error e (t):
e(t)＝y(t)-r(t) (6)
wherein r (t) is a given reference speed;
s2: the following error signal of the system is differentiated in fractional order:
wherein C is set to be 1, D is a fractional order differential operator, and the order is 0 < ξ + upsilon-1 < 1;
the fractional order differential operator is defined as:
wherein n is the smallest positive integer larger than α fractional order, and I is a fractional order integral operator;
the fractional order integral operator is as follows:
where α is a fractional order, t is an integration time, x (t) represents an integrated function, and Γ represents a gamma function.
S3: performing integer order differentiation on the error signal:
s4: selecting a virtual synovial membrane observation surface:
wherein z is1Sigma is the designed virtual slide membrane surface for the selected intermediate variable,actual disturbance observed for a designed sliding mode observer;
differentiating the virtual synovial observation plane σ:
wherein,is the observation error of the system disturbance.
S5: the following first order synovial differential equation was chosen to estimate the unknown variable
Wherein: rhoσ1,ρσ2In order to be the state of the slip film equation,are set at 20 and 10.
Thus, it can be represented by the following formula
In the formula: λ is the error of the differential equation estimate.
S6: defining dependent variablesDesigning a response synovium disturbance observer, and obtaining the variation of the actual disturbance of the observation system as follows:
the third step: the most common PI controller in engineering is used for controlling the flexible swing arm system, and the transfer function can be expressed as:
in the formula, KpIs the proportional coefficient of the PI controller, KiIs the integral coefficient of the PI controller, and λ is the fractional order.
Considering the feed forward compensation of the disturbance, the final input current command of the system can be obtained as follows:
for finite time convergence of the system, the following perturbed nonlinear differential equation was used:
wherein:x is the state of the system; mu.s1And mu2Is a parameter of a differential equation, TsIs the sample time, ξ (t) is the unknown system disturbance.
For the above equation, if it can satisfyμ21.1M, then the derivative of the system state variableWill converge to zero in a finite time.
Thus for the synovial disturbance observer, the second derivative is definedIt can be deduced that:
according to the relevant expression of the disturbance observed by the sliding mode disturber, the following can be known:
in the formula: mu's'1＝CKA1μ1,μ′2＝CKA1μ2,
Further derivable is:
according to the perturbation nonlinear differential equation theorem, the variables sigma' andwill converge to zero in a finite time, so we derive:
therefore, the sliding mode disturbance observer based on the non-homonymous fractional order design can accurately identify the disturbance of the system in a limited time.
Fig. 5 is a schematic diagram of an observation effect of a flexible swing arm system synovial disturbance observer related to a non-homonymous fractional order system synovial disturbance observer design method according to an embodiment of the present invention. As shown in fig. 5, it can be seen that designing the synovial membrane disturbance observer works well.
The control rate of the system at the current moment is finally obtained through the steps, namely the current command is input, and the speed control of the flexible swing arm system is realized.
The controller parameters set by the method can obtain good position tracking performance and transient performance. The present invention may be carried out with appropriate modification without departing from the essential spirit thereof and without exceeding the scope of the subject matter to which the invention pertains.
Compared with the traditional disturbance resisting method, the method provided by the invention has the advantages that the design of the synovial membrane disturbance observer is carried out by utilizing the non-homonymous fractional order of the system, the accuracy of disturbance observation is improved, the convergence can be carried out within a limited time, the real-time property of an observation result is ensured, and the method is also suitable for the homonymous fractional order system. Therefore, the problem of disturbance suppression in a non-homonymous fractional order flexible swing arm system is solved.
The method utilizes the non-isomorphism of the fractional order model order of the flexible swing arm system to design the synovial membrane disturbance observer, can observe and compensate disturbances including external disturbance, sensor noise, parameter drift and the like in the flexible swing arm system in real time, and inhibits the influence of the system disturbance on the system performance.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (10)
1. A design method of a non-homonymous fractional order system sliding film disturbance observer is characterized by comprising the following steps: the method comprises the following steps:
s1: selecting an excitation signal to excite a non-homonymous fractional order system, and acquiring a required input signal and an output signal;
s2: and carrying out state space model identification on the non-homonymous fractional order system by utilizing an optimization algorithm, wherein the state space model comprises the following steps:
wherein x (t) { x ═ xξ(t),xυ(t) } is the state variable of the system, A1,A2K is a parameter of the system; d (t) ═ Δ ax (t) + de(t) represents the actual integrated disturbance of the system, u (t) is the input signal of the system; y (t) is the output signal of the system, Δ A is the uncertainty component of the system model parameters, de(t) representing unknown disturbance existing in the system, h (x (t)) is a nonlinear factor existing in the system model, ξ and upsilon are fractional orders of the system model, and D is a fractional order differential operator;
s3: designing a synovial membrane disturbance observer according to the state space model in step S2, comprising the following steps:
s11: calculating the speed following error e (t):
e(t)＝y(t)-r(t)
wherein r (t) is a given reference speed;
s12, according to the non-homonymous fractional order ξ and upsilon of the system, carrying out fractional order differentiation on the velocity following error e (t) of the system:
wherein C is a preset real number;
s13: integer order differentiation of the velocity following error e (t):
s14, selecting a virtual synovial membrane observation surface according to the non-homonym fractional order ξ and upsilon:
wherein z is1Sigma is the designed virtual slide membrane surface for the selected intermediate variable,actual disturbance observed for a designed sliding mode observer;
s15: differentiating the virtual synovial observation plane σ:
wherein,an observation error that is a system disturbance;
s16: selecting a first-order synovial differential equation to estimateThe value of (c):
where ρ isσ1,ρσ2Is the state variable of the synovial differential equation,sign is a sign function for a predefined positive real number;
can be represented by the following formula
In the formula, lambda is the error of differential equation estimation;
s17: defining dependent variablesThe following synovial membrane disturbance observer is designed, and the variation of the actual disturbance of the observation system can be obtained as follows:
wherein, TsIs the sampling time, P (σ ') is a function on σ', μ1And mu2Are parameters of differential equations.
2. The method for designing a non-homonymous fractional order system synovial interference observer of claim 1, wherein the method comprises the following steps: the fractional order differential operator is defined as:
wherein α is a fractional order, n is the smallest positive integer greater than α fractional order, and I is a fractional order integrator.
3. The method for designing a non-homonymous fractional order system synovial interference observer of claim 2, wherein the method comprises the following steps: the fractional order integral operator is as follows:
where α is a fractional order, t is an integration time, x (t) represents an integrated function, and Γ represents a gamma function.
4. A method for designing a non-isobaric fractional order synovial disturbance observer according to claim 3, wherein the method comprises the following steps: the optimization algorithm is a particle swarm algorithm:
wherein k' represents the current iteration number; i is 1, 2, 3 … is the number of the particle; a represents the system parameter to be identified,representing the position of the parameter set to be optimized in the solution space when the iteration number is k';represents the velocity of the particle set at the number of iterations k' +1, c1,c2Is the acceleration constant; r is1,r2Random number of 0 to 1, pbestijAnd gbestijRespectively representing a locally optimal solution and a globally optimal solution.
5. The method for designing a non-homonymous fractional order system synovial interference observer of claim 4, wherein the method comprises the following steps: the input instruction of the non-isomorphic fractional order system is as follows:
where C(s) is a transfer function and s is a complex variable.
6. The method for designing a non-homonymous fractional order system synovial interference observer of claim 5, wherein the method comprises the following steps: the variation of the actual disturbance of the observation system satisfies the following perturbation nonlinear differential equation theorem:
wherein,x is the state of the system, μ1And mu2Are the parameters of the differential equation and,μ2≥1.1M，Tsis the sampling time, ξ (t) is the unknown system disturbance, and M represents the upper bound of the disturbance variance.
7. The method for designing a non-homonymous fractional order system synovial interference observer of claim 6, wherein the method comprises the following steps: the synovial membrane disturbance observer satisfies the following relationship:
wherein, mu'1＝CKA1μ1,μ′2＝CKA1μ2,
According to the perturbation nonlinear differential equation theorem, the variables sigma' andconvergence to zero in a finite time yields:
therefore, the sliding mode disturbance observer can accurately identify the disturbance of the non-homonymous fractional order system in a limited time.
8. The method for designing a non-homonymous fractional order system synovial interference observer of claim 1, wherein the method comprises the following steps: the excitation signal is Gaussian white noise with amplitude distribution obeying Gaussian distribution.
9. A product designed using the non-isobaric fractional order system synovial interference observer design method of claim 8.
10. A non-homogeneous fractional order system control method using the product of claim 9, wherein: and superposing the disturbance observed by the synovial membrane disturbance observer to a control speed end, and realizing the control of the non-homonymous fractional order system through a fractional order PI controller, wherein the transfer function can be expressed as:
in the formula, KpIs the proportional coefficient of the PI controller, KiIs the integral coefficient of the PI controller, λ is the fractional order, c(s) is the transfer function, and s is a complex variable.
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