^{2} University of Science and Technology Liaoning, National Financial Security and System Equipment Engineering Research Center, Anshan 114044, China
Polyvinyl chloride is one of the five thermoplastic synthetic resins, widely used in all fields such as industry, agriculture, construction public utilities etc.^{[1]} The temperature of polymerizing reaction is the most important controlled parameter in the production process. At present, the automatic control of temperature at the bottom of the polymerizer still givens priority to PID control^{[2]}, and the application of some advanced controlling technology has also achieved good controlling effect. In 1995, a design scheme of fuzzy logic and artificial neural network feedback control was developed in a batch polymerizing reactor temperature control system, to improve the stability of the system^{[3]}. In 2003, on the basis of the conventional PID control of the polymerization kettle and in view of Laguerre function identification and predictive control, an indirect adaptive predictive control strategy was proposed, so as to shorten the producting cycle of the polymerizer and improve the equipment
Modelfree adaptive control is a control strategy that lets the controlled system get rid of relying any model, uses only input and output data of the controlled system to design the designer, so essentially overcomes the "unmodeled dynamics and robustness problem". In 1994, cosponsored by the Hou and Han, the model free adaptive control (MFAC) theory, marked a major breakthrough of the model free control theory^{[8]}, whose thinking is that the controller is designed using only the controlled system I/O data and the controller does not contain any information of the mathematical model of the controlled process. In 1999, an adaptive PI control algorithm is developed for a class of SISO nonlinear discretetime systems based on a generalized predictive control (GPC) approach, the simulation and realtime experiment are provided for real nonlinear systems which are known to be difficult to model and control^{[9]}. In 2005, aimed at the characteristic of the large of lag timevarying nonlinear complex systems is difficult to control, the tin thickness modelfree adaptive control system is proposed, the system is successfully put into operation and achieved good control effect^{[10]}. In 2006, Shang Chuankou, Li Wei and others used the modelfree adaptive control combined with the traditional PID control, and applied it to the main steam temperature control system, simulation results are given to show that it has achieved good control effect to overcome the time delay and large disturbance^{[11]}. In 2009, aimed at the problem of multimotor synchronization in multiwire saw, the algorithm of MFAC is used in motor control scheme, and the feasibility and effectiveness of the method are proved by the prototype test method^{[12]}. In 2011, aimed at the outlet pressure of boiler with the characteristics of typical nonlinear and coupled, a model free adaptive control method is used to maintain it at the respected values, compared with the traditional PID control algorithm, the simulation verifies the feasibility and robustness of the control system^{[13]}. In 2016, Hu and Tang^{[14]} proposed a modelfree adaptive datadriven SMC algorithm, this new discretetime nonlinear systems modelfree control algorithm obtained better control performance through the simulations for the linear motor position and the information tracking speed, which also achieved robust and accurate traceability.
Because the polymerizing temperature possess features nonlinearity, timevarying and multivariables, this paper proposes a modelfree adaptive control algorithm based on particle swarm optimization. Firstly, this paper designs a modelfree adaptive controller. Secondly, the particle swarm optimization algorithm is used to optimize key parameters of the controller. Then, the identified polymerization temperature model is optimized; finally, the simulation results are given to verify the effectiveness of the proposed control algorithm.
The rest of the paper is organized as follows. PVC polymerization process is presented in Section 2. PVC polymerization process temperature control based on modelfree adaptive is presented in Section 3. Standard algorithm of PSO is presented in Section 4. Modelfree adaptive control algorithm based on PSO optimization is presented in Section 5. Simulation study is made in Section 6, and compared with other methods. Finally, conclusions and some remarks are given in Section 7.
2 PVC polymerization processThe PVC resin is produced by suspension method production technology method. Polymerization process production device is shown in Fig. 1. Liquid vinyl chloride (VCM) in the presence of dispersants, by mixing effect dispersed droplet, suspended in water, soluble in VCM of initiator on the polymerization temperature decomposition into radicals, VCM polymerization reactions. According to the free radical polymerization mechanism of VCM suspension polymerization, including chain initiation, chain growth, chain transfer and chain termination reactions.
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Fig. 1. Flowchart of polymerizing production process 
In the process of polyvinyl chloride (PVC) polymerization, firstly, the various raw materials and additives are added to the reaction kettle, which can be fully dispersed homogeneously by mixing blade's agitation, and then the reaction is started by adding a proper proportion of the initiator. At the same time, the cooling water is injected into the external jacket of the reaction kettle and the inner baffle, whose main purpose is to remove the reaction heat. When the reaction is terminated, the termination agent is joined. At this time the conversion rate of polyvinyl chloride (VCM) is about
On the mechanism, PVC polymerization degree only lies on the temperature, so the temperature control is of great importance. The temperature deviation is controlled between
The initiator is decomposed, to primary radicals:
$ I\xrightarrow {{K_d }}2R^ \bullet $ 
which
The primary radicals with monomer addition monomer free radical is
$ R^ \bullet + M\xrightarrow{{K_i }}RM^ \bullet $ 
which
$ \begin{equation} \frac{{{\rm d}[R^ \bullet ]}}{{{\rm d}t }} = 2K_d [I]. \end{equation} $  (1) 
Due to the rate of monomer free radical formation is much greater than the initiator decomposition rate, and because of the influence of the side reactions, the free radicals cannot be completely used to induce VCM. So
$ \begin{equation} R_i = 2fK_d [I]. \end{equation} $  (2) 
In the formula,
$ RM^ \bullet \mathop \to \limits_{K_{p1} }^{ + M} RM_1 ^ \bullet \mathop \to \limits_{K_{p2} }^{ + M} RM_2 ^ \bullet \mathop \to \limits_{K_{p3} }^{ + M} \cdots RM_n ^ \bullet $ 
$ K_{p1} = K_{p2} = K_{p3} = \cdots = K_p. $ 
With
$ \begin{equation} R_p =  \left[ {\frac{{{\rm d}\left[ M \right]}} {{{\rm d}t}}} \right]_p = K_p \left[ M \right]\left[ {M^ \bullet } \right]. \end{equation} $  (3) 
In the formula,
When the free radical molecules up to a certain length to happen chain termination reactions. Chain termination reaction is respectively divided into the coupling termination and disproportionation termination, expressed as follows:
$ M_x ^ \bullet + M_y ^ \bullet \xrightarrow{{K_{tc} }}M_{X + Y}. $ 
Coupled termination: the degree of polymerization is the sum of two chain radicals on the number of monomer units. The generated macromolecular as initiator residues on both ends.
$ \begin{equation} R_{tc} = 2K_{tc} \left[ {M^ \bullet } \right]^2\\ M_x ^ \bullet + M_y ^ \bullet \xrightarrow{{K_{tc} }}M_X + M_Y. \end{equation} $  (4) 
Disproportionation termination: the degree of polymerization is the original chain radicals contained in the number of monomer units, each contains a initiator residues end groups (one is saturated end group, the other one is unsaturated end group).
$ \begin{equation} R_{td} = 2K_{td} \left[ {M^ \bullet } \right]^2. \end{equation} $  (5) 
The way of termination reaction depends on the structure of the monomer and polymerization temperature. Termination of the total rate is
$ \begin{equation} R_t = \frac{{{\rm d}\left[ {M^ \bullet } \right]}} {{{\rm d}t}} = 2K_t \left[ {M^ \bullet } \right]^2. \end{equation} $  (6) 
In the formula,
In polymerization of VCM, macromolecular free radical can capture a chlorine atom or atomic hydrogen and termination from monomer, fluxing agent, initiator or macromolecules, loss of atoms in molecules will become free radical, continues to grow a new chain reaction.
When the VCM polymerization, macromolecular chain transfer to monomer free radical and significantly became the primitive response to determine the molecular weight of PVC resin.
The primary control variables of the polymerization process variables are the polymerizer temperature, pressure, jacket water flow baffle flow of water injected into the water flow, the seal water flow, the jacket outlet temperature, the shutter outlet temperature, etc. Among them, the polymerization temperature is a very important parameter, which determines the molecular weight of PVC, which corresponds to the different types of PVC products. The polymerization reaction is intermittent operation of the exothermic reaction, the use of largescale polymerizer and higher response speed can increase its productivity, it is advantageous from economic considerations. However, the largescale polymerizer, poor heat transfer, uneven mixing and the reaction speed is fast, resulting in that the temperature control becomes very difficult. So we should control the polymerization kettle temperature in order to reduce the production costs and improve PVC product quality.
2.4 Relationship between polymerizing temperature and conversing rateThe factors influencing conversion velocity are initiator concentration, reaction temperature and polymerize degree. Polymerization velocity is given by the following equation.
$ \begin{equation} {R_c} = {K_p}[ M ]{\left( {\frac{{{R_i}}}{{2{K_t}}}}\right)}^{ \frac {1} {2}} \end{equation} $  (7) 
where
$ \begin{equation} {K_p} = A{\rm e}^ {\frac {E} {RT}} \end{equation} $  (8) 
where
It can be seen from (7) that the reaction velocity has relationship with initiator concentration and reaction temperature of constant kind. From (8), when
$ \begin{equation} {R_c} = A{{\rm e}^ { \frac { E} {RT}}[ M ]{\left( {\frac{{{R_i}}}{{2{K_t}}}} \right)}^ {\frac {1}{2}}} \end{equation} $  (9) 
where
It is known from (9) that the polymerizing rate of VCM (conversion rate) is the function of temperature. The polymerization degree decreases with an increase in temperature. Therefore, polymerization degree only has a relationship with temperature for the VCM polymerization reaction^{[15]}.
3 PVC polymerization process temperature control based on modelfree adaptive 3.1 Mechanism modeling of polymerizer temperaturePVC polymerizing process is a typically batch exothermic reaction process. According to the heat balance principle, the total calories instantaneously released in polymerizing reaction is equal to the heat amount removed by jacket water and baffle plate water, the absorbed heat by injection water and sealing water, and the heat making the polymerizer temperature change. Thus the following equations are obtained.
$ \begin{equation} \Delta {H_R}{M_0}{R_C} = {Q_j} + {Q_b} + {Q_i} + {Q_s} + {Q_f} \end{equation} $  (10) 
where
$ \begin{equation} {Q_j} = {C_w}{u_j}\left( {{T_j}  {T_{j0}}} \right) \end{equation} $  (11) 
$ \begin{equation} {Q_b} = {C_w}{u_b}\left( {{T_b}  {T_{b0}}} \right) \end{equation} $  (12) 
$ \begin{equation} {Q_i} = {C_w}{u_i}\left( {T  {T_{i0}}} \right) \end{equation} $  (13) 
$ \begin{equation} {Q_s} = {C_w}{u_s}\left( {T  {T_{s0}}} \right) \end{equation} $  (14) 
$ \begin{equation} {Q_f} = {C_v}{m_v}T + {C_p}{m_p}T + {C_w}{\rho _v}VT. \end{equation} $  (15) 
The outlet temperature of the baffle plate water and jacket water is defined as follows:
$ \begin{equation} {T_j} = \frac{{T + {\alpha _j}{T_{j0}}}}{{1 + {\alpha _j}}}, \quad {\alpha _j} = \frac{{{C_W}{u_j}}}{{{K_j}{A_j}}} \end{equation} $  (16) 
$ \begin{equation} {T_b} = \frac{{T + {\alpha _b}{T_{b0}}}}{{1 + {\alpha _b}}}, \quad {\alpha _b} = \frac{{{C_W}{u_b}}}{{{K_b}{A_b}}} \end{equation} $  (17) 
Based on the (10)(17), the differential equation of the polymerizer temperature is deduced as follows:
$ \begin{equation} \begin{gathered} \dot T = \frac{{\Delta {H_R}{m_O}{{\dot R}_C}  {C_W}{u_j}\left( T  T_{j0} {\frac {T_{j0}} {1}} + {\alpha _j} \right)}} { {m_0}\left( {C_v}\left( {1  {R_C}} \right) + {C_P}{R_C} + {C_w}{\rho _w}\left( {\frac{{R_C} \left({ {\rho _P}  {\rho _v}} \right)} {{\rho _P}{\rho _v}}} \right)\right)}  \\ \frac{ {C_W}{u_b}\left( {T  {\frac{T_{b0}}{ 1}} + {\alpha _b}} \right)} { {m_0}\left( {C_v}\left( {1  {R_C}} \right) + {C_P}{R_C} + {C_w}{\rho _w}\left( {\frac{{R_C} \left( {\rho _P}  {\rho _v} \right)} {{\rho _P}{\rho _v}}} \right) \right) }  \\ \frac{ {C_W}\left( {\frac{ {m_0}{\rho _w}\left( {{\rho _P}  {\rho _v}} \right)} {{\rho _P}{\rho _v}}} \right){{\dot R}_C}\left( {T  {T_{i0}}} \right) } { {m_0}\left( {{C_v}\left( {1  {R_C}} \right) + {C_P}{R_C} + {C_w}{\rho _w}\left( {\frac{{R_C} \left( {{\rho _P}  {\rho _v}} \right)} {{\rho _P}{\rho _v}}} \right) } \right)}. \\ \end{gathered} \end{equation} $  (18) 
On the other hand, it is known from (9) that the polymerizing rate of VCM (conversion rate) is the nonlinear function of temperature, while the temperature is an important factor affecting PVC production quality. To make sure the quality of PVC production, the polymerizer temperature has to be controlled to a certain constant value. So the modelfree adaptive model is set up based on the conversion rate of VCM.
3.2 SISO discretetime nonlinear systemA general nonlinear discretetime system can be expressed as
$ \begin{equation} y(k + 1) = f[Y_k^{k  n}, u(k), U_{k  1}^{k  m}, k] \end{equation} $  (19) 
where
$ Y_k^{k  n} = \{ y(k), y(k  1), \cdots , y(k  n)\} $ 
$ U_{k  1}^{k  m} = \{ u(k  1), u(k  2), \cdots , u(k  m)\} $ 
and
Assumption 1. The output and input of system (19) are observable and controllable.
Assumption 2. The controlled input of current system
Assumption 3. System (19) satisfies the generalized Lipschitz theorem. That is for any
$ \Delta y(k + 1) \le b\Delta u(k) $ 
$ \Delta y(k + 1) = y(k + 1)  y(k) $ 
where
Lemma 1. For nonlinear system (19), if it meets the Assumptions 13, then when
$ \begin{equation} \Delta y(k + 1) = \phi (k)\Delta u(k). \end{equation} $  (20) 
The SISO systems (19) can be written into the dynamic linear form of (20):
$ \Delta y(k + 1) = \phi (k)\Delta u(k). $ 
1) The design of control law algorithm
Considering the control input criterion function:
$ J(u(k)) = {y^*}(k + 1)  y(k + 1){^2} + \lambda u(k)  u(k  1){^2} $ 
where
Put (20) into the criterion function for
$ u(k) = u(k  1) + \frac{{{\rho _k}\phi (k)}}{{\lambda + \phi (k){^2}}}[{y^*}(k + 1)  y(k)] $ 
where
By online estimation,
2) Pseudo partial derivative estimation algorithm
Take into account that the symmetry similar structural system has a certain superiority^{[16]}.
Here is a symmetry similar estimation algorithm with control law.
$ \begin{array}{l} J(\phi (k)) = {\left( {{y^0}(k)  y(k  1)  \phi (k)\Delta u(k  1)} \right)^2}+\\ \qquad \qquad\;\; \mu {\left( {\phi (k)  \hat \phi (k  1)} \right)^2} \end{array} $ 
where
Then the pseudo partial derivative estimation algorithm is:
$ \begin{array}{l} \hat \phi (k) = \hat \phi (k  1) +\displaystyle \frac{{{\eta _k}\Delta u(k  1)}}{{\mu + \Delta u(k  1){^2}}}\times\\ \qquad \quad \left( {\Delta y(k)  \hat \phi (k  1)\Delta u(k  1)} \right) \end{array} $ 
where
3) The design of modelfree adaptive controller
Based on previously obtained control law algorithms and parameter estimation algorithm, MFAC algorithm can be written as
$ \begin{equation} \begin{array}{l} \hat \phi (k) = \hat \phi (k  1) + \frac{{{\eta _k}\Delta u(k  1)}}{{\mu + \Delta u(k  1){^2}}}\times \\ \qquad \quad \left( {\Delta y(k)  \hat \phi (k  1)\Delta u(k  1)} \right) \end{array} \end{equation} $  (21) 
$ \begin{equation} \hat \phi (k) = \hat \phi (1), \quad \hat \phi (k) \le \varepsilon or\Delta u(k  1) \le \varepsilon \end{equation} $  (22) 
$ \begin{equation} u(k) = u(k  1) + \frac{{{\rho _k}\hat \phi (k)}}{{\lambda + \hat \phi (k){^2}}}[{y^*}(k + 1)  y(k)] \end{equation} $  (23) 
where
The PSO is a population based optimization technique, where the population is called as warm. A simple explanation of the PSO's operation is as follows. Each particle represents a possible solution to the optimization task at hand. For the remainder of this paper, reference will be made to unconstrained minimization problems. During each iteration each particle accelerates in the direction of its own personal best solution found so far, as well as in the direction of the global best position discovered sofar by any of the particles in the swarm. This means that if a particle discovers a promising new solution, all the other particles will move closer to it, exploring the region more thoroughly in the process.
Assume a
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Fig. 2. Detailed flowchart of the PSO 
So far, the position and velocity of
$ \begin{equation} \begin{array}{l} v_{id}^{k + 1} = v_{id}^k + {c_1}{r_1}(p_{id}^k  x_{id}^k)+ \\ \qquad \quad\; {c_2}{r_2}(p_{gd}^k  x_{id}^k) \end{array} \end{equation} $  (24) 
$ \begin{equation} x_{id}^{k + 1} = x_{id}^k + v_{id}^{k + 1} \end{equation} $  (25) 
where
Bring the inertia weight factor into (12) type to correct it, namely
$ \begin{equation} v_{id}^{k + 1} = wv_{id}^k + {c_1}{r_1}(p_{id}^k  x_{id}^k) + {c_2}{r_2}(p_{gd}^k  x_{id}^k) \end{equation} $  (26) 
where
Step 1. Initialization.
Because two parameters of modelfree adaptive controller requires optimization and tuning, so the search space dimension is identified (
Step 2. Select the fitness function.
Goal of optimization is to ensure the system output can follow a given reference trajectory in the input
$ Q = \sum\limits_{k = 0}^\infty {\left\{ {{{[{y^*}(k + 1)  y(k + 1)]}^2} + {{[u(k)  u(k  1)]}^2}} \right\}}. $ 
The fitness function ensures that the system output can track the given reference trajectory, it also ensures the control energy as small as possible.
Step 3. Evaluate fitness of each particle.
According to the given fitness function by (2), the fitness value of the particle is calculated and determined.
Step 4. Update individual extreme.
For each particle, its fitness value is compared with the individual extreme. If the particle is better than the current individual extreme, then set
Step 5. Update the global extremum.
If the individual extreme of all the particles are best to the current global extremum, then set
Step 6. Update state of particles.
Using (25) and (26) to update each particle
Step 7. To test whether meets of end condition.
If the current number of iterations reaches a preset maximum number
Finally, output optimal value is used to optimize key parameters of the controller.
6 Research on simulation of polymerizing process 6.1 Construct model for polymerization temperatureThe polymerization process is always accompanied by volume shrinkage. The volume of organic phase in the polymerizer is
$ \begin{equation} V = \frac{M}{\rho} \end{equation} $  (27) 
where
During the reaction, the density of organic phase in polymerizer at any time is
$ \begin{equation} \rho = \frac{M}{{\frac{{{M_0}(1  R)}}{{{\rho _v}}} + \frac{{{M_0}R}}{{{\rho _p}}}}}. \end{equation} $  (28) 
Ignore the heat exchange between the reactor and the outside, and at the same time, ignore other heat loss in heat exchange process. Then the exothermic heat of reaction in the unit time is
$ \begin{equation} Q = \sum\limits_{i = 1}^n {{u_i}{c_w}({T_o}  {T_i})}. \end{equation} $  (29) 
where
Ignore the temperature of polymerization reaction kettle affected by conversion rate, thereby the function relationship between the heat released and the temperature inside the reactor can be expressed as
$ \begin{equation} \frac{{{\rm d}Q}}{{{\rm d}t}} = \frac{{{{\rm d}^2}T}}{{{\rm d}{t^2}}} + {c_1}\frac{{{\rm d}T}}{{{\rm d}t}} + {c_2}T. \end{equation} $  (30) 
The removed heat by cooling water and injected water in the unit time is expressed as
$ \begin{equation} {u_i}{c_w}({T_o}  {T_i}) = {c_w}U(T  \tau ) \end{equation} $  (31) 
where
Simultaneous (30) and (31), the relationship between the temperature of the kettle and the valve opening is obtained.
$ \begin{equation} \frac{{{{\rm d}^2}T}}{{{\rm d}{t^2}}} + {c_1}\frac{{{\rm d}T}}{{{\rm d}t}} + {c_2}T = {c_w}U(t  \tau ). \end{equation} $  (32) 
So, the equation (32) is the model of the polymerizer temperature
The equation of the valve opening degree for the cooling water and the injected water in polymerizing process can be expressed as
$ \begin{equation} y''(t) + ay'(t) = bu(t  \tau ) \end{equation} $  (33) 
where
On the discrete disposal of (33), select 300 groups of collected data, and use the least squares method to identify the data. Select
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Fig. 3. Leastsquares fitting results 
As can be seen in the simulation results, using the least squares method to identify the parameters of the controlled object, the ultimate goodness of fit runs up to
Aiming at the control of the reacting temperature of polymerizing process, under the environment of MATLAB language, the MFAC method adopted in this paper is compared with the controlling effect of traditional PID. According to the actual reacting temperature required by a certain type of PVC products from a certain chemical group, the polymerization temperature in simulation is set to 56.4℃, and initial temperature of polymerization reactor is 40℃.
In view of the actually operating conditions during polymerizing process, considering two cases of normal operation and operation suffering from sudden disturbance, the output responses between MFAC and traditional PID control system are compared.
1) When the polymerization kettle run in good condition, parameters of the modelfree adaptive controller are set: step sequence
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Fig. 4. Polymerizer normal operation of the kettle temperature control result 
From Fig. 4, the modelfree adaptive control method has faster response, the ability to adapt is better than other control method.
The analysis of simulation results is given in Table 1.
It can be seen from Table 1, compared with PID control algorithm, the modelfree adaptive control method has the smaller overshoot, the output of the system can stably track the setting value.
2) It is supposed that polymerization kettle suffered from a sudden disturbance during operating process, for example, factors like reacting material suddenly heated unequally, which led to temperature of polymerization kettle changing in a sudden. In the 100th minute, step disturbance with amplitude of 2℃ is added to the simulation, and the simulation results are shown in Fig. 5. Compared with PID control strategy, in the face of random interference, using MFAC method can be quicker to overcome the disturbance, which restores the system to the set value as soon as possible, at the same time, ensuring less bias and stronger robustness to the system.
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Fig. 5. Polymerization kettle temperature control results by step disturbance 
From Fig. 5, the modelfree adaptive control method can be quicker to overcome the disturbance, the ability to antiinterference is better than other control method.
The analysis of simulation results is given in Table 2.
It can be seen from Table 2, compared with PID control algorithm, the modelfree adaptive control method has the smaller deviation and stronger robustness.
3) Using PSO algorithm to optimize and tune parameters of model free adaptive controller, parameters of the modelfree adaptive controller are set: step sequence
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Fig. 6. PSOMFAC, MFAC and PID temperature control curve comparison 
From Fig. 6, after using PSO algorithm to optimize and tune parameters of model free adaptive controller, the method of PSOMFAC can be faster to achieve the stable time.
The analysis of simulation results is given in Table 3.
It can be seen from Table 3, compared with the controlling effects of MFAC and PID, the overshoot of PSOMFAC control method is the smallest, and the settling time is the shortest.
7 ConclusionsAccording to the theory of modelfree adaptive control, a model free adaptive control algorithm for the polymerization temperature is proposed. On this basis, the particle swarm optimization algorithm is used to optimize and tune the key parameters of the controller. Using real operating data exist in the polymerizing process, the polymerization temperature model is established. This modelfree adaptive control algorithm is applied to PVC polymerization, and compared the traditional PID in terms of controlling effect. The simulation shows the effectiveness of the MFAC algorithm. Moreover under the same conditions, the modelfree adaptive control strategy optimized by the particle swarm can obtain better tracking effect. In future research, the model free adaptive control algorithm will be applied to the actual operation of polymerization temperature of the PVC production. In addition, this control algorithm provides an excellent reference for other complex industrial temperature control.
NotationsThis work was supported by University of Science and Technology Liaoning, National Financial Security and System Equipment Engineering Research Center (No. USTLKFGJ201502).
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