In order to solve the problem of parameter tuning of proportional integral differential (PID) controller in traditional industrial control, an algorithm for parameter determination of controller based on internal model method (IMC) and system identification is proposed.Firstly, the controlled process is identified as a first-order plus-delay (FOPDT) or second-order plus-delay (SOPDT) model by using the corresponding relationship between input and transient output under the excitation of open-loop step signals, and then the parameters of the controller are determined by IMC algorithm.For the determination of the filter parameter lambda introduced in the internal model method, a method of determining the filter parameter lambda by introducing two parameters, gamma and_, and establishing a relationship with the square of the output error is proposed.The simulation results show that compared with the traditional IMC-based PID control algorithm, the proposed algorithm can increase about 20% in the absence of input disturbance and about 10% in the presence of input disturbance for the index of the sum of absolute values of output errors (IAE).The simulation results show that the proposed tuning method improves the transient response speed of the system while guaranteeing the robustness of the system, and effectively suppresses the overshoot of the system output when the unit step signal is used to excite the system.Controller plays an important role in industrial production process. The traditional Proportional Integral Derivative (PID) controller is widely used because of its easy design, adjustment and application.However, with the modernization of industrial process, the combination of traditional PID controller and intelligent control has been favored by researchers. Intelligent PID control includes fuzzy PID controller, neural network PID controller, genetic algorithm PID controller  and IMCPID controller based on internal model control (IMC).IMCPID controller has good robustness and tracking control performance for large time-delay systems, and only one parameter of the controller needs to be adjusted to achieve the desired closed-loop response.For the design of PID controller based on IMC, many literatures have proposed some new control methods or tuning strategies for specific problems.For example, when there are pure integral links in the controlled object model, reference  proposes a parameter design method of PID controller based on frequency response matching, and reference  proposes that pure integral links be approximated as first-order lag links with large time constants.For the problem of ADRC, the concept of Active Disturbance Rejection Control (ADRC) is introduced into the internal model control by analyzing the relationship between the internal model control and the disturbance observation control.In addition, the IMCPID control is extended to fractional order [6-7].However, the filter parameters of the traditional IMC-based PID controller are usually fixed, which makes the response speed of the system slow and the gain of the PID control limited.In order to solve this problem, the method of increasing the response speed by introducing parameter alpha without changing lambda is proposed in reference . In the simulation process, it has the advantages of fast response and small overshoot.
However, its disadvantage is that the control output of the controller structure is close to the ideal impulse response, and the control effect can not be achieved when the executing mechanism is limited. Therefore, the index of the sum of the absolute values of the output error (IAE) is taken as the benchmark for comparison between the proposed algorithm and the classical IMCPID algorithm.In reference , the simple hysteresis tuning method of single input single output system in reference  is extended to Smith predictor controller.However, this method is not applicable to the first-order lag process model, which may cause saturation problems in practical applications and is sensitive to model mismatch .In reference , a self-tuning method of parameter lambda in internal model control principle is proposed by using fuzzy tuning rules, but the tuning of other control parameters in PID controller is mentioned.Based on this, this paper adopts the algorithm of designing PID controller based on IMC, and proposes to determine the value of lambda according to the square of the system output and reference input error. It can improve the system’s transient response speed and effectively suppress the system overshoot while ensuring the system’s robustness.In addition, when the model is seriously mismatched or the model obtained from experience is either non-typical first order Plus Dead Time (FOPDT) or second order Plus Dead Time (SOPDT), the open-loop identification method mentioned in reference  is proposed, that is, the process is stimulated by unit step signal, according to the system output, and the traditional least square method is used. The process parameters are re-identified, and the identified parameters are re-applied to the tuning process of the controller parameters.The paper adopts the following organizational structure.In the first part, the basic principle of IMC is introduced. In the second part, the self-tuning control strategy of PID controller based on IMC is introduced. The identification methods of FOPDT model and SOPDT model are introduced in detail, and the tuning strategy of filter parameter lambda is emphasized.The third part carries on the simulation experiment and carries on the comparative analysis with the algorithm in reference  and the classical IMCPID control algorithm.The fourth part is a summary.Internal model control (IMC) is developed from Smith predictive controller in 1950s.
It overcomes the shortcomings of Smith predictive controller which requires accurate model and has poor robustness. It has good control effect on systems with large time delay and model mismatch, and has the ability of restraining process disturbance in structure.The basic principle of the internal model method is shown in Figure 1.In Figure 1: GIMC (S) represents internal model controller, GP (S) represents real model of system, G’P (S) represents model of identification system, D (t) represents external disturbance, R (t) represents external reference input (expected output), y (t) represents actual output of system, e (t) represents output error of system.The first step of IMC controller design is to decompose and identify the model G’P (S), G’P (S) = G’P (S) G’P – (S), and G’P (S) as the irreversible part of the model, including the zero or pole in the right half plane of the model and the time-delay term of the system.When GP-‘-1 (S) is irregular, the order n of the filter can be selected to make the controller realizable and the controlled system stable.P2 (s) = Km (T1ms 1) (T2ms 1) e-theta s (4) represents the FOPDT model and formula (4) represents the SOPDT model.Km is process gain, Tm, T1m, T2m and theta are time constants and positive values.In reference , the basic design method of IMC controller for these two different processes is introduced in detail, which is not discussed here.The classical IMCPID controller chooses lambda=theta, which is the object of comparison based on the algorithm in reference .When the process model is second order, it can be found that the traditional PID controller is irregular, so in this paper, a low-pass filter 1 / (tfs1) is added to the differential link. In the subsequent simulation, TF is selected as 1% * TD, TD is the time constant of the differential link of the controller.In Chapter 1, it can be seen from Formulas (5) and (6) that there is only one uncertain parameter of PI/PID controller, and the other parameters are obtained from the process model.The ideal output response curve of the system should have fast response and small overshoot on the premise of ensuring the stability of the system. It is obvious that adjusting only one parameter can not achieve this effect.When a faster transient response speed is required, a smaller lambda is needed, but this will inevitably lead to an increase in overshoot; when a smaller output overshoot is required, a larger lambda is needed, but the response speed will inevitably decrease.Therefore, a control system with only one parameter adjustment will inevitably lose sight of one another between response speed and overshoot.Where R is the reference input of the system, theta is the pure time delay of the process model, e = y-r, gamma and_are the parameters introduced in this paper.By substituting Formula (7) into the proportional gain Kp of the controller, it can be concluded that Kp is positively correlated with e2.For PI controller, in the initial stage of system response, E2 achieves the maximum, so Kp also achieves the maximum, and the proportional part of the controller plays a major role. At this time, the response of the system is faster than that of other methods.
As the system begins to have output, E2 decreases slowly, and the corresponding Kp decreases gradually, and the integral part begins to play a major role. When the integral role is strong enough, the output of the system will overshoot.The gamma introduced here can adjust the overshoot of the system while guaranteeing the fast response of the system. Obviously, gamma is negatively correlated with the gain Kp of the controller. When gamma increases, the Kp decreases, and the overshoot of the corresponding output can even achieve the effect of no overshoot. On the contrary, the overshoot of the output increases.After a lot of simulation verification, gamma is generally taken as [0.30.8].When the system output is near the set value, e2/r is very small. In order to ensure that the system response is not too slow when the system approaches the set value, the parameter_is introduced to limit the minimum value of e2/r. The larger the value of_, the faster the response is near the set value of the system. However, if the value is too large, the system will oscillate near the set value (at this time, the oscillation convergence is determined by the structure of the IMCPID controller). Generally, the parameter_ <[0.3.0.5].In reference , it is proposed that the absolute sum of system output errors (IAE) index is related to filter parameter lambda and system pure delay parameter theta, and when lambda = theta, the IAE index based on IMCPID is the smallest, so lambda = f (theta) is chosen here.When e2_r, gamma=1, the tuning method in this paper is the traditional IMCPID control method.This section mainly introduces the knowledge of system identification used in the tuning process of this paper. For more methods of system identification, please refer to reference .In most cases, the mathematical model of the controlled object is too complex or can not be established by theoretical modeling method. Furthermore, the characteristics of the controlled object have changed during operation. At this time, online modeling is needed, that is, the mathematical model of the controlled object is established by system identification method.Generally speaking, system identification can be divided into classical methods, modern methods and methods combined with neural network and genetic algorithm.After comparing various identification methods, the step response method of classical identification method is used to identify the process model.Because it belongs to open-loop identification, and applies the transient response data of the system, it does not need the condition that the system changes in the stable range as relay identification does.In this paper, the controlled object is identified as FOPDT model and FOPDT model. The STEP-based identification of controlled object in reference  has the advantages of less computation, simplicity and convenience.In the subsequent simulation process, this paper focuses on the tracking of the step response of the controlled object, which is consistent with the idea of identification method.The tuning control system of PID controller based on internal model method is shown in Fig. 2.In Figure 2, K1 and K2 are bidirectional switches and K3 are normally closed contacts.The normal running system, K1 connected.When the controlled object model needs to be identified as FOPDT or SOPDT, it needs to disconnect K3 and turn on K2. Assuming that the controlled object is in zero initial state, the process is stimulated by a step signal whose amplitude is Mu at t = 0, and the output is y (t).Then take the natural index of the value obtained from equation (18) to obtain km and θ.The identification principle of SOPDT model is the same as that of FOPDT. There is no further discussion here, but a brief description of the results is given.Note: In reference , when the above method is used for process identification, if the measurement of Y is noisy, the least squares estimates are inconsistent, so the auxiliary variable matrix Z=[z1z2]is chosen. ZN] T, where zi= [1/t2i1/ti1tit2i], then_= (ZT) – 1Z.The specific proof is shown in document .In order to illustrate the effectiveness of the proposed method, three examples are given to verify the effectiveness of the proposed method.The first example compares the results of the proposed method with those of the other two methods. The effectiveness of the proposed method is illustrated by two performance indices: overshoot and steady-state time.The second example shows that when the model does not match the real process, the proposed algorithm is also robust and has the ability to suppress disturbances.The third example shows the effectiveness of the identification method used in this paper when the model is unknown.In the fourth example, when the output of the controller is limited, the IAE performance of the proposed method is compared with that of reference . The superiority of the proposed method is illustrated from the point of view of control energy.The simulation results are shown in Fig. 3 (a).Secondly, fixed gamma to observe the effect of different_on the system response speed and system overshoot when approaching the steady-state value.The simulation results are shown in Fig. 3 (b) and Fig. 3 (c).It can be seen from Fig. 3 (a) that when_is fixed, the smaller the lambda is, the larger the overshoot of the system is.Fig. 3 (b), (c) show that the effect of fixed lambda on overshoot is approximately 2.48% when the system has overshoot, and 24.07% when the system has overshoot. When the system has no overshoot, _=0.5 is nearly 40 s faster than_=0.3, and when_=0, it can not reach steady state value in simulation time.In this paper, the control parameters gamma = 0.6, _ = 0.4 are selected in the algorithm, and the algorithm in reference  (KIMC for short) is selected a = 0.5 (the same value is used in subsequent simulation). The differential constants Ti and Td of the three methods are the same, respectively: Ti = 23, Td = 102/23.The simulation results are shown in Fig. 4.From Figure 4, it can be seen that the controller in this paper can effectively control overshoot and improve the system response speed after the reasonable selection of lambda and_.On IAE performance index, the IAE index value of KIMC algorithm as benchmark is 14.40, the IAE value of this algorithm is 19.36, and that of classical IMCPID algorithm is 22.69. The IAE index of this algorithm is 23.13% higher than that of classical IMCPID.The simulation is 300s. Among the three methods, the controller parameter Ti = 2 and Td = 0.5.In this algorithm, gamma = 0.6 and_ = 0.5 are selected. When t = 100s, step disturbance with amplitude of 0.3 is added. When t = 200s, the model mismatches and the proportional gain of the process model changes from 1 to 1.1.The simulation results are shown in Fig. 5.Fig. 5 shows that the control algorithm can effectively suppress disturbance and model mismatch, but the speed of adjustment is less than KIMC and IMCPID controllers.In IAE performance index, the IAE index value of KIMC algorithm as benchmark is 20.48, the IAE value of this algorithm is 27.98, and that of classical IMCPID algorithm is 29.65. The IAE index of this algorithm is 8.15% higher than that of classical IMCPID.The identification result is to illustrate the effectiveness of the controller designed with the parameters of the model in the process of closed-loop control. Therefore, in the simulation process, thermostatic element the real process and the identified process model are respectively used as the controlled object for closed-loop control.In this paper, the principle of “single variable” is adopted, that is, only considering the identification effect, the classical IMCPID algorithm is used to design the controller.The simulation results are shown in Figure 6.From Figure 6, we can see that the curve of “real process of PID control” coincides with “SOPDT model of PID control”, which shows the validity of the identification method adopted in this paper, and that the second-order identification of the system can better approximate the higher-order controlled object.The algorithm is applied to the control of the high-order controlled object. Because the second-order model can better approximate the original process, the parameters of SOPDT model are used to design the PID controller when choosing the parameters of the controller.The parameters of this algorithm are chosen as follows: gamma = 0.6, _ = 0.
5. The results are shown in Figure 7.In IAE performance index, the IAE index value of KIMC algorithm as benchmark is 31.53, the IAE value of this algorithm is 41.61, and that of classical IMCPID algorithm is 45.80. Compared with the IAE index of classical IMCPID, the IAE index of this algorithm is increased by 13.29%.In this algorithm and KIMC and IMCPID algorithms, the controller parameters Ti = 3, Td = 2/3.In this algorithm, gamma = 0.
6 and_ = 0.5 are selected, and the output amplitude of the controller is limited to 3.
The simulation results are shown in Fig. 8.From Figure 8, it can be seen that the response of the system to step signals is not greatly affected when the output of the controller is limited, while the control effect of the algorithm in the literature  varies greatly and even overshoots occur. Therefore, it can be explained that the control energy requirement of the algorithm in this paper is not high, which is in line with the practical application.When the control output is unrestricted, the IAE values of the proposed algorithm and KIMC algorithm are 10.29 and 7.5, respectively. When the control output is limited, the IAE values are 10.31 and 8.42, respectively. It can be seen that the IAE indexes of the proposed algorithm do not change much.In this paper, an improved tuning algorithm of PID controller based on IMC is proposed.
By optimizing the filter parameter lambda, two parameters gamma and_are introduced.Gamma not only improves the system’s rapidity, but also restrains overshoot. _is introduced to overcome the shortcoming of slow regulation speed when the output error is small, which is the need of this algorithm.
Compared with the classical IMCPID algorithm, the effectiveness of this algorithm is verified.When the controlled object needs model identification, the open-loop step excitation method is used to identify the controlled process, and its effectiveness is verified by an example.The disadvantage is that the selection of gamma and_depends on experience and has strong subjectivity.But on the other hand, users can improve IAE indicators and have a certain range of choices for fast response and overshoot indicators.