CN103048923B - For the consistency constraint method of service water tank liquor bit string level Predictive Control System - Google Patents

Abstract

The invention discloses a kind of consistency constraint method for service water tank liquor bit string level Predictive Control System, 1) set up the control structure model of external loop pfc controller and inner looping PID controller respectively; 2) judge whether external loop pfc controller retrains consistent with inner looping PID controller; 3) adopt constraint backoff algorithm, by pre-loaded in external loop PFC control law for the constraint of inner looping PID controller, and obtain the optimization of external loop pfc controller and calculate stylish constraint condition set; 4) external loop pfc controller is optimized calculating under new constraint condition set, obtains the setting value of inner looping PID controller, and this setting value is supplied to inner looping PID controller to follow the tracks of this setting value.The present invention is by and levels constraint inconsistence problems saturated to the bottom loop control unit occurred in PFC-PID serials control loop, calculated by constraint rollback, make to optimize gained setting value practical for bottom loop, improve the control performance of system, improve productive unit economic benefit.

Description

Consistency constraint method for industrial water tank liquid level cascade predictive control system

Technical Field

The invention relates to a consistency constraint method, in particular to a consistency constraint method for an industrial water tank liquid level cascade prediction control system.

Background

In industrial processes, due to physical characteristics of actuators and production devices, such as linear working range of valves, adjustment rate of valves, volume and liquid level limitations of oil tanks, etc., controllers have a certain range of operability and require process output within a certain safety range. Meanwhile, in order to maximize the economic benefit of an enterprise on the premise of ensuring the product quality, it is generally desirable to make the operating point approach the constraint limit as much as possible within a certain probability constraint range to realize the edge clamping operation. Therefore, two-level constraint strategies, hard constraint and soft constraint, are usually set according to the constraint mandatory property and optimization property. The basic requirement of the production operation is to ensure that the variables do not violate the hard constraint limits, otherwise emergency measures such as stopping and stopping production are taken to prevent accidents. Once a shutdown occurs, this can result in millions of dollars or more of economic loss to a continuous process industrial enterprise. Thus, businesses often treat parking as a production incident, while avoiding it as much as possible. For soft constraint limits, if a production process operating point is out of limits, an alarm may be issued to alert an operating engineer to make adjustments to the production conditions.

For conventional PID control schemes, the operation engineer ensures that the operating point does not violate the hard constraint limits based on production experience. The production experience of the operating engineer and the understanding of the production process have a great influence on the effectiveness of the control. For soft constraints and economic performance, due to the lack of necessary deterministic model information, it is difficult to guarantee optimization, and it is common practice to guarantee safety at the cost of lost economic benefit, so the operation process fluctuates over a large range and a large back-off value is reserved from operating soft constraints.

Predictive Control (MPC) is a typical representative of advanced control strategies, and has been widely applied to the process industries of petroleum, chemical industry, paper making, pharmacy, etc. in recent years. One of the main characteristics of predictive control is that it can be convenient to deal with constraint problems. According to a process model obtained by identifying a production system, the future process output is predicted, and the process output and the model prediction output are fed back to a controller for optimization control, so that the control effect is improved, the process output fluctuation is compressed, and further, on the premise of ensuring that the hard constraint is not violated, the soft constraint within a certain probability range is allowed to exceed the limit, an operation point is further close to the constraint limit to realize the card edge control, and the economic benefit is maximized. It can therefore be said that the improvement in economic efficiency of predictive control depends mainly on two factors: first, the control effect is improved to reduce production fluctuations; second, the backoff problem is handled reasonably to bring the operating point close to the constraint limit. Therefore, the reasonable treatment process constraint is an important problem in the process industry for the safety of production and the improvement of the economic benefit of enterprises.

Predictive Function Control (PFC) is called a third generation predictive controller. The method has the characteristics of simple algorithm, no need of processing quadratic optimization problem, small calculated amount and the like, so that the method can be conveniently embedded into a DCS (distributed control System) or a PLC (programmable logic controller), not only can process slow response processes of process industry and the like, but also can be effectively applied to fast control processes such as robots, aircrafts, weapon control systems and the like. In the flow industries of petroleum, chemical engineering and the like, compared with a general quadratic form predictive control algorithm, the PFC can avoid solving the quadratic form optimization problem, and has a simple control structure, so that the PFC is closer to a PID (proportion integration differentiation) controller and can be directly applied to the dynamic control of a bottom basic loop.

The PFC-PID cascade control loop is in accordance with the control structure of a generic MPC-PID cascade control loop commonly used in industrial processes. Because the PID control algorithm adopted by more than 95% of control schemes in the process industry has been applied for decades or even hundreds of years, the MPC (PFC) -PID cascade control structure is adopted to take compatibility and optimality into consideration, so that the MPC (PFC) -PID cascade control structure is more easily accepted by industrial enterprises and operation engineers.

The characteristic of this kind of PFC-PID cascade control loop is that the object controlled by the inner loop PID controller and the outer loop PFC controller is the same process, which is a typical control scheme in industrial process. The role of the inner loop controller is "transparent" to the outer loop controller, and thus this cascade control is referred to as "conduction control".

Liquid level control is a typical process control problem in industrial production, and accurate measurement and effective control of liquid level are important indexes of high quality, high yield, low consumption and safe production of some equipment. Aiming at a generalized object of a water tank liquid level system, the method adopts first-order inertia and pure lag to approximate equivalence, and takes the fact that the time constant and lag time are both large, so that a prediction function control strategy with strong robustness and easiness in engineering implementation is adopted. The industrial water tank liquid level control system mainly comprises a water tank control unit, a DCS control cabinet and an upper computer configuration and monitoring environment. A PFC-PID conduction control loop is designed under the system, and mainly comprises an outer loop PFC controller and an inner loop PID controller, and the control flow is shown in figure 2. The inner loop adopts proportional P control to process the dynamic characteristics of the process, and the outer loop adopts a PFC control strategy tracking set value without an explicit integrator. The outer loop provides a set point for the inner loop through optimal calculation, and the inner loop tracks the set point through dynamic control. However, due to the problem of inconsistent constraints among loops at all levels, it is difficult to ensure the correctness and feasibility of the optimization result of the outer loop. Aiming at the problem, the constraint consistency analysis is carried out on the control of the multilayer structure prediction function, and the constraint of the inner loop is transferred to the outer loop step by step through the backspacing calculation of the cascade loop, so that the optimization set value is guaranteed to be feasible for the dynamic control of the basic loop. The improvement of the control effect of the constraint backspacing calculation method is verified through simulation analysis.

Disclosure of Invention

In order to overcome the defects of the prior art, the invention provides the consistency constraint method for the industrial water tank liquid level cascade prediction control system, which aims at the problems of bottom loop controller saturation and inconsistent upper and lower layer constraints in a PFC-PID cascade control loop, ensures that an outer loop controller calculates a correct optimal set value, prevents the phenomena of fluctuation, overshoot and the like caused by error calculation, improves the control performance and improves the economic benefit of a production unit.

The technical scheme of the invention is as follows: a consistency constraint method for an industrial water tank liquid level cascade predictive control system comprises the following steps:

1) establishing a control structure model of an outer loop PFC controller and a control structure model of an inner loop PID controller for an industrial water tank liquid level cascade prediction control system;

2) judging whether the constraints of the outer loop PFC controller and the constraints of the inner loop PID controller are consistent, if so, performing the step 3), and if not, returning to the step 1) to reestablish a control structure model of the outer loop PFC controller and a control structure model of the inner loop PID controller;

3) under the condition that the constraints of the outer loop PFC controller and the inner loop PID controller are consistent, the constraint of the inner loop PID controller is pre-loaded in the calculation process of the outer loop PFC control law by adopting a constraint rollback algorithm, and a new constraint condition set when the outer loop PFC controller carries out optimization calculation is obtained;

4) and the outer loop PFC controller performs optimization calculation under a new constraint condition set to obtain a set value of the inner loop PID controller, and the set value is provided for the inner loop PID controller so that the inner loop tracks the set value.

Further setting up a control structure model of an outer loop PFC controller and a control structure model of an inner loop PID controller for the industrial water tank liquid level cascade prediction control system in the step 1),

the control structure model of the outer loop PFC controller is

$u_{ext}\left(k\right)=\frac{({\mathrm{SP}}_{ext}-y_p(k\left)\right)(1-{{\mathrm{\λ}}_{ext}}^h)-y_m\left(k\right){a_{mext}}^h+y_m\left(k\right)}{K_{mext}(1-{a_{mext}}^h)}---(1-1)$

In which the subscript p represents the process;

u_ext(k) the output of the PFC prediction function controller of the outer loop at the moment k;

SP_extpredicting a set value of a function controller for the PFC of the outer loop;

y_p(k) and y_p(k + h) respectively representing the output of the industrial water tank liquid level cascade prediction control system at the k moment and the k + h moment;

indicating an out of time k loop deviation_ext(k)＝SP_ext-y_p(k) Out-of-loop deviation to time k +1_ext(k+1)＝SP_ext-y_p(k +1) attenuation coefficient;

T_sis a sampling period;

CLTR is the closed loop response time of a first-order discrete system, namely the time of reaching 95 percent of a set value;

controlling the model attenuation coefficient for the outer loop PFC prediction function;

T_mis the model time constant;

K_mextan outer loop model gain;

the inner loop is controlled by a proportion P, and the gain of the controller is set to be K_intThen the control structure model of the inner loop PID controller is

u_int＝(SP_int-y_p(k))K_int(1-2)

In the formula, ui_ntIs the output of the inner loop controller;

SP_intis the set value of the inner loop;

y_p ^(k)the output of the industrial water tank liquid level cascade prediction control system at the moment k;

K_intis the inner loop controller gain.

Further setting that in the step 2), whether the external loop PFC controller and the internal loop PID controller are constrained to be consistent or not is judged, and the following method is adopted:

① when the inner loop is controlled by the proportion P, the set value SP 'of one inner loop is artificially set for the inner loop proportion control law represented by the control structure model (1-2) of the inner loop PID controller'_intAnd the set value SP 'is made'_int>SP_intThe corresponding inner loop control law is

u′_int＝(SP′_int-y_p(k))K_int(2-1)

In the formula u_int' taking SP for inner loop set point_int' time, the output value of the inner loop controller;

subtracting the control structure model (1-2) of the inner loop PID controller from the control law (2-1) of the inner loop to obtain

u′_int-u_int＝(SP′_int-y_p(k))K_int-(SP_int-y_p(k))K_int(2-2)

Namely, it is

u′_int-u_int＝(SP′_int-SP_int)K_int(2-3)

③ when K_int>0, since SP is assumed'_int>SP_intU's of'_int(k)-u_int(k)>0, namely when SP'_int>SP_intIn each case, there is u'_int(k)>u_int(k)；

When K is_int<0, since SP is assumed'_int>SP_intU's of'_int(k)-u_int(k)<0, namely when SP'_int>SP_intIn each case, there is u'_int(k)<u_int(k)；

Therefore, for an industrial water tank liquid level cascade predictive control system controlled by a proportion P, the output of the controller is strictly monotonous with the set value;

for PFC-PID cascade conduction control loop, the output of the outer loop controller is the set value of the inner loop controller, namely

u_ext(k)＝SP_int(2-4)

When u is strictly monotonic with the output of the inner loop controller and the output of the outer loop controller_min≤u_int≤u_maxXi (u)_min)≤u_ext≤Ξ(u_max)；

I.e., the operating constraints of the outer loop PFC controller are consistent with the operating constraints of the inner loop PID controller.

Further setting that, in step 3), the operation constraint of the inner loop PID controller is loaded in the calculation process of the outer loop PFC control law in advance, and a constraint condition set of the outer loop PFC controller during the optimization calculation is obtained, and adopting the following method:

establishing an optimization model of an outer loop PFC controller

$\begin{array}{cc}\underset{{\mathrm{\&Delta;p}}_i,{\mathrm{\&Delta;m}}_i}{\min }& J=\underset{i=1}{\overset{n}{\&Sigma;}}{({\mathrm{\&Delta;p}}_i-{\mathrm{\&Delta;m}}_i)}^2\\ s.t.& \mathrm{\&Delta;}p={\&lsqb;{\mathrm{\&Delta;p}}_1,L,{\mathrm{\&Delta;p}}_n\&rsqb;}^T\\ & \mathrm{\&Delta;}m={\&lsqb;{\mathrm{\&Delta;m}}_1,L,{\mathrm{\&Delta;m}}_n\&rsqb;}^T\\ & \mathrm{\&Delta;}p=({\mathrm{SP}}_{ext}-y_p(k\left)\right)(1-{{\mathrm{\&lambda;}}_{ext}}^h)\\ & \mathrm{\&Delta;}m=y_m\left(k\right){a_{mext}}^h+u\left(k\right)K_{mext}(1-{a_{mext}}^h)-y_m\left(k\right)\\ & u_{ext\min }\&le;u_{ext}\&le;u_{ext\max }\\ & {\mathrm{SP}}_{ext\min }\&le;{\mathrm{SP}}_{ext}\&le;{\mathrm{SP}}_{ext\max }\end{array}---(3-1)$

In the formula,n is the number of variables for optimizing the objective function;

in the optimization time domain, the deviation between the process output increment delta p and the model prediction output increment delta m is minimized, and the control law of the outer loop PFC controller can be obtained;

u_extmin≤u_ext≤u_extmaxand SP_extmin≤SP_ext≤SP_extmaxRespectively represents u_extAnd SP_extA constraint of (1), wherein u_extmin、u_extmaxRespectively represents u_extMinimum and maximum values of, SP_extmin、SP_extmaxRespectively represent SP_extMinimum and maximum values of;

since the operating constraints of the outer loop PFC controller are identical to the operating constraints of the inner loop PID controller, when u is_min≤u_int≤u_maxXi (u)_min)≤u_ext≤Ξ(u_max)；

The operation constraint of the inner loop PID controller is loaded in advance in the calculation process of the outer loop PFC control law, namely xi (u)_min)≤u_ext≤Ξ(u_max) Performing optimization calculation on the formula (3-1) for constraint conditions, and then

$\left\{\begin{array}{c}{u}_{ext\min }\&le;u_{ext}\&le;u_{ext\max }\\ \mathrm{\&Xi;}\left(u_{\min }\right)\&le;u_{ext}\&le;\mathrm{\&Xi;}\left(u_{\max }\right)\end{array}\right.---(3-2)$

Thus obtaining a new constraint max u_extmin,Ξ(u_min)}≤u_ext≤min{u_extmax,Ξ(u_max)}。

Further setting that, in step 4), the outer loop PFC controller performs optimization calculation under a new constraint condition set to obtain a set value of the inner loop PID controller, and provides the set value to the inner loop PID controller to make the inner loop track the set value, and the method is as follows:

under the new constraint condition, the optimization calculation is carried out on the external loop PFC controller, and the optimization model is

$\begin{array}{cc}\underset{{\mathrm{\&Delta;p}}_i,{\mathrm{\&Delta;m}}_i}{\min }& J=\underset{i=1}{\overset{n}{\&Sigma;}}{({\mathrm{\&Delta;p}}_i-{\mathrm{\&Delta;m}}_i)}^2\\ s.t.& \mathrm{\&Delta;}p={\&lsqb;{\mathrm{\&Delta;p}}_1,L,{\mathrm{\&Delta;p}}_n\&rsqb;}^T\\ & \mathrm{\&Delta;}m={\&lsqb;{\mathrm{\&Delta;m}}_1,L,{\mathrm{\&Delta;m}}_n\&rsqb;}^T\\ & \mathrm{\&Delta;}p=({\mathrm{SP}}_{ext}-y_p(k\left)\right)(1-{{\mathrm{\&lambda;}}_{ext}}^h)\\ & \mathrm{\&Delta;}m=y_m\left(k\right){a_{mext}}^h+u\left(k\right)K_{mext}(1-{a_{mext}}^h)-y_m\left(k\right)\\ & u_{ext\min }\&le;u_{ext}\&le;u_{ext\max }\\ & {\mathrm{SP}}_{ext\min }\&le;{\mathrm{SP}}_{ext}\&le;{\mathrm{SP}}_{ext\max }\end{array}---(3-1)$

Solving (4-1) to obtain an optimized objective functionObtaining the minimum value J_minU of time_extIs the optimal solution u of the optimization problem_extoptIt is used as the set value SP of the PID controller_intIs provided to the inner loop.

In step 1), when the outer loop controller is a PFC prediction function controller, firstly, a PFC algorithm is adopted to calculate a controller structure model;

① A first-order discrete system is constructed when the sampling period is T_sThe process gain is K_pThe process output is

CV(k)＝y_p(k)＝y_p(k-1)a_p+(1-a_p)K_pu(k-1)(1-3)

In which the subscript p represents the process;

CV (k) is the control variable at time k;

y_p ^(k)the output of the industrial water tank liquid level cascade prediction control system at the moment k;

is the process attenuation coefficient;

T_pis the process time constant;

u (k-1) is the output of the PFC prediction function controller at the k-1 moment;

modeling an output function as

y_m(k)＝y_m(k-1)a_m+(1-a_m)K_mu(k-1)(1-4)

In the formula, the subscript m represents a model;

y_m(k) is the output of the model at the moment k;

is the model attenuation coefficient;

T_mis the model time constant;

K_mis the model gain;

in the optimization time domain, the deviation between the process output increment delta p and the model prediction output increment delta m is minimized, and one-step optimization is adopted, namely, the process output increment delta p is equal to the model prediction output increment delta m

Wherein,

Δp＝(k)-(k+h)

＝(k)-(k)λ^h

＝(k)(1-λ^h)

＝(SP-y_p(k))(1-λ^h)(1-5)

in the formula, SP is a set value;

(k)＝SP-y_p(k) industrial water tank liquid level cascade prediction control system output y representing set values SP and k_p(k) A deviation of (a);

(k+h)＝SP-y_p(k+h)＝(k)λ^hindustrial water tank liquid level cascade prediction control system output y representing set values SP and k + h_p(k + h) deviation;

represents the attenuation coefficient of the deviation from the k time to the k +1 time;

λ^his the h-th power of lambda and represents the attenuation coefficient of the deviation from the time k to the time k + h;

for the first-order discrete system, taking the system time constant of the industrial water tank liquid level cascade predictive control systemCLTR is the closed loop response time of a first-order discrete system, namely the time of reaching 95 percent of a set value;

and is

Δm＝y_m(k+h)-y_m(k)(1-6)

In the formula, y_m(k + h) is the predicted output of the model at the moment k + h and is represented by the free response y_mFree(k + h) and forced response y_mForce(k + h) composition, i.e.

y_m(k+h)＝y_mFree(k+h)+y_mForce(k+h)(1-7)

Wherein,

$y_{mFree}(k+h)=y_m\left(k\right)a_m^h---(1-8)$

$y_{mForce}(k+h)=u\left(k\right)K_m(1-a_m^h)---(1-9)$

in the formula, y_mFree(k + h) denotes the free response, y_mForce(k + h) represents a forced response;

therefore, the temperature of the molten steel is controlled,

$\mathrm{\&Delta;}m=y_m\left(k\right)a_m^h+u\left(k\right)K_m(1-a_m^h)-y_m\left(k\right)---(1-10)$

from Δ p ═ Δ m

$(SP-y_p(k\left)\right)(1-{\mathrm{\&lambda;}}^h)=y_m\left(k\right)a_m^h+u\left(k\right)K_m(1-a_m^h)-y_m\left(k\right)---(1-11)$

After being sorted, the general form of the PFC controller is obtained

$u\left(k\right)=\frac{(SP-y_p(k\left)\right)(1-{\mathrm{\&lambda;}}^h)-y_m\left(k\right)a_m^h+y_m\left(k\right)}{K_m(1-a_m^h)}---(1-12)$

③ the control structure model of the outer loop PFC controller, which can be obtained from the formula (1-12)

$u_{ext}\left(k\right)=\frac{({\mathrm{SP}}_{ext}-y_p(k\left)\right)(1-{{\mathrm{\&lambda;}}_{ext}}^h)-y_m\left(k\right){a_{mext}}^h+y_m\left(k\right)}{K_{mext}(1-{a_{mext}}^h)}---(1-1)$

In the formula u_ext(k) The output of the PFC prediction function controller of the outer loop at the moment k;

SP_extpredicting a set value of a function controller for the PFC of the outer loop;

y_p(k) and y_p(k + h) respectively representing the output of the industrial water tank liquid level cascade prediction control system at the k moment and the k + h moment;

indicating an out of time k loop deviation_ext(k)＝SP_ext-y_p(k) Out-of-loop deviation to time k +1_ext(k+1)＝SP_ext-y_pThe attenuation coefficient of (k +1) is,CLTR is the closed loop response time of a first-order discrete system, namely the time of reaching 95 percent of a set value;

controlling the model attenuation coefficient for the outer loop PFC prediction function;

T_mis the model time constant;

K_mextan outer loop model gain;

④ the inner loop is controlled by proportion P, and the gain of the controller is set to K_intThereby obtaining a control structure model of the inner loop PID controller

u_int＝(SP_int-y_p(k))K_int(1-2)

In the formula, the inner loop is controlled by a proportion P;

u_intis the output of the inner loop controller;

SP_intis the set value of the inner loop;

y_p ^(k)the output of the industrial water tank liquid level cascade prediction control system at the moment k;

K_intis the inner loop controller gain.

The invention has the advantages that the problem of inconsistent upper and lower layer constraints can be avoided by controlling the saturation of the bottom layer loop controller in the PFC-PID cascade control loop, the set value obtained by dynamic optimization is feasible for the bottom layer control loop through constraint rollback calculation, and the control performance of the industrial water tank liquid level cascade predictive control system is further improved. Particularly, in the industrial water tank liquid level cascade prediction control system, when the actuator is in a saturated state, a correct inner loop model is provided for the outer loop so as to ensure that the outer loop controller can calculate a correct optimal set value, the phenomena of fluctuation, overshoot and the like caused by error calculation are prevented, the control performance of the industrial water tank liquid level cascade prediction control system is improved, and the economic benefit brought by a production unit in the industrial water tank liquid level cascade prediction control system is improved.

Drawings

Fig. 1 is a flow chart of PFC-PID cascade conduction control according to the present invention.

FIG. 2 is a schematic structural diagram of the cascade prediction function control system for industrial water tank liquid level according to the present invention.

Fig. 3 is a block diagram of a cascade conduction control loop configuration according to the present invention.

FIG. 4 is a schematic of the prediction function control of the present invention.

Detailed Description

The invention is further described below with reference to the accompanying drawings:

an industrial water tank (which can be understood as one of water tanks, namely the water tank with a tank cover) liquid level cascade control system is designed aiming at a CS4000 process control system adopted by a certain domestic company:

as shown in fig. 1 and 2, the PFC-PID conduction control loop structure of the industrial water tank level control system includes a water tank control unit, a DCS control cabinet 1, and an upper computer configuration and monitoring environment.

In FIG. 1, SP_optFor optimal setting of the PFC prediction function controller, SP_extSet-point of the prediction function controller for the outer loop PFC, SP_intMSet point, MV, for an inner loop PID controller model_extPredicting the manipulated variable, MV, of the function controller for the outer loop PFC_intFor the manipulated variable of the inner loop PID controller, dist is the system disturbance, MV_intMThe operating variable of the inner loop PID controller model, Gp is the control object, CV is the system control variable, and P (ID) is the proportion P control adopted by the PID controller. MV is the system input value, which in this embodiment refers to the valve opening value; CV is the system output value, referred to in this embodiment as tank level.

In fig. 2, a DCS control cabinet 1, a water tank 2, an electromagnetic valve 3, and a water storage tank 4.

As shown in fig. 3, the PFC-PID conduction control loop is constructed by a modular programming environment, wherein the main modules include an outer loop PFC controller module and an inner loop PID controller module. The inner loop adopts proportional P control to process the dynamic characteristics of the process, and the outer loop adopts a PFC control strategy tracking set value without an explicit integrator.

1) Establishing a control structure model of an outer loop PFC controller and a control structure model of an inner loop PID controller for an industrial water tank liquid level cascade predictive control system;

when the outer loop controller is a PFC prediction function controller, firstly, a PFC algorithm is adopted to calculate a controller structural model.

Establishing a first-order discrete system with a sampling period of T_sThe process gain is K_pThe process output is

CV(k)＝y_p(k)＝y_p(k-1)a_p+(1-a_p)K_pu(k-1)(1-3)

In the formula, the subscript p represents a process;

CV is a control variable of the system;

CV (k) is a control variable of the system at time k;

y_p ^(k)the measured value of the water tank liquid level at the moment k is obtained;

the attenuation coefficient of the liquid level of the water tank;

T_pis the process time constant;

u (k-1) is the valve opening value at the moment of k-1;

the equation (1-3) reflects the relationship between the system control variable CV (tank level) and the manipulated variable MV (valve opening, i.e., the opening of the solenoid valve 3). Process output y_p ^(k)The measured value of the water tank liquid level at the moment k can be directly detected in a water tank liquid level system.

Modeling an output function as

y_m(k)＝y_m(k-1)a_m+(1-a_m)K_mu(k-1)(1-4)

In the formula, the subscript m represents a model (model);

is the model attenuation coefficient;

T_mis the model time constant;

K_mis the model gain;

and the set value of the cascade loop, namely the input value of the PFC controller of the outer loop is determined by upper layer steady state optimization and represents the actually required water tank level value. In the parameter identification process, the change of the set value is irrelevant to steady state optimization, and the set value at this time is given manually at will, so as to observe how the output changes when the input changes. Therefore, the set value of the cascade loop is changed, the initial output and the steady-state output of the PFC controller are analyzed, the parameter identification is carried out on the loop model in the PFC closed loop in sequence, and the K can be obtained_m＝1.9，T_m＝29

As shown in fig. 4, in the optimization time domain, the deviation between the process output increment Δ p and the model prediction output increment Δ m is minimized, that is, Δ p is equal to Δ m.

Wherein,

Δp＝(k)-(k+h)

＝(k)-(k)λ^h

＝(k)(1-λ^h)

＝(SP-y_p(k))(1-λ^h)(1-5)

in the formula, the SP is a set value provided by the upper computer to the PFC controller;

(k)＝SP-y_p(k) indicating the set values SP and k moment water tank liquid level y_p(k) A deviation of (a);

(k+h)＝SP-y_p(k+h)＝(k)λ^hindicating the set value SP and k + h moment water tank liquid level y_p(k + h) deviation;

represents the attenuation coefficient of the deviation from the k time to the k +1 time;

λ^his the h-th power of λ, representing the attenuation coefficient of the deviation from time k to time k + h.

For the first-order discrete system, the system time constant is takenCLTR is the system closed loop response time, i.e. the time to reach 95% of the set value;

and is

Δm＝y_m(k+h)-y_m(k)(1-6)

In the formula, y_m(k + h) is the predicted output of the model at the moment k + h and is represented by the free response y_mFree(k + h) and forced response y_mForce(k + h) composition, i.e.

y_m(k+h)＝y_mFree(k+h)+y_mForce(k+h)(1-7)

Wherein,

$y_{mFree}(k+h)=y_m\left(k\right)a_m^h---(1-8)$

$y_{mForce}(k+h)=u\left(k\right)K_m(1-a_m^h)---(1-9)$

in the formula, y_mFree(k + h) denotes the free response, y_mForce(k + h) represents a forced response;

therefore, the temperature of the molten steel is controlled,

$\mathrm{\&Delta;}m=y_m\left(k\right)a_m^h+u\left(k\right)K_m(1-a_m^h)-y_m\left(k\right)---(1-10)$

from Δ p ═ Δ m

$(SP-y_p(k\left)\right)(1-{\mathrm{\&lambda;}}^h)=y_m\left(k\right)a_m^h+u\left(k\right)K_m(1-a_m^h)-y_m\left(k\right)---(1-11)$

After finishing to obtain

$u\left(k\right)=\frac{(SP-y_p(k\left)\right)(1-{\mathrm{\&lambda;}}^h)-y_m\left(k\right)a_m^h+y_m\left(k\right)}{K_m(1-a_m^h)}---(1-12)$

Therefore, the general form of the PFC controller is obtained;

the control structure model of the outer loop PFC controller can be obtained from the formulas (1-12)

$u_{ext}\left(k\right)=\frac{({\mathrm{SP}}_{ext}-y_p(k\left)\right)(1-{{\mathrm{\&lambda;}}_{ext}}^h)-y_m\left(k\right){a_{mext}}^h+y_m\left(k\right)}{K_{mext}(1-{a_{mext}}^h)}---(1-1)$

In the formula u_ext(k) The output of the PFC prediction function controller of the outer loop at the moment k;

SP_extpredicting a set value of a function controller for the PFC of the outer loop;

y_p(k) and y_p(k + h) represents the outputs of the system at the time k and the time k + h, respectively;

indicating an out of time k loop deviation_ext(k)＝SP_ext-y_p(k) Out-of-loop deviation to time k +1_ext(k+1)＝SP_ext-y_pThe attenuation coefficient of (k +1) is,CLTR is the system closed loop response time, i.e. the time to reach 95% of the set value;

controlling the model attenuation coefficient for the outer loop PFC prediction function;

T_mis the model time constant;

K_mextan outer loop model gain;

the inner loop is controlled by proportion P, and the gain of the controller is set to K_intThereby obtaining a control structure model of the inner loop PID controller

u_int＝(SP_int-y_p(k))K_int(1-2)

In the formula, the inner loop is controlled by proportion P, namely, the opening of the valve is in linear relation with the liquid level of the water tank;

u_intis the valve opening value;

SP_intis the set value of the liquid level of the water tank;

y_p ^(k)the measured value of the water tank liquid level at the moment k is obtained;

K_intis the inner loop PID controller gain.

2) Judging whether the external loop PFC controller and the internal loop PID controller are in accordance with the constraint:

when the inner loop adopts proportion P control, the set value SP 'of the water tank liquid level is artificially set for the inner loop proportion control law represented by the control structure model (1-2) of the inner loop PID controller'_intAnd the set value SP 'is made'_int>SP_intThe corresponding inner loop control law is

u′_int＝(SP′_int-y_p(k))K_int(2-1)

In the formula u_intTaking SP for the set value of the liquid level of the water tank_int' the opening value of the valve of the water tank 2 is equal;

subtracting the control structure model (1-2) of the inner loop PID controller from the inner loop control law (2-1) to obtain

u′_int-u_int＝(SP′_int-y_p(k))K_int-(SP_int-y_p(k))K_int(2-2)

Namely, it is

u′_int-u_int＝(SP′_int-SP_int)K_int(2-3)

When K is_int>0, since SP is assumed'_int>SP_intU's of'_int(k)-u_int(k)>0, namely when SP'_int>SP_intIn each case, there is u'_int(k)>u_int(k)；

When K is_int<0, since SP is assumed'_int>SP_intU's of'_int(k)-u_int(k)<0, namely when SP'_int>SP_intIn each case, there is u'_int(k)<u_int(k)；

Thus, for a proportional P controlled system, the controller output is strictly monotonic with the set point;

for the industrial water tank liquid level cascade prediction control system, the output of the inner loop controller is the opening value of a valve of a water tank 2, and the set value is the set value of the water tank liquid level.

For a PFC-PID cascade conduction control loop, the output of the outer loop controller is the set value of the inner loop controller, namely

u_ext(k)＝SP_int(2-4)

When u is strictly monotonic with the output of the inner loop controller and the output of the outer loop controller_min≤u_int≤u_maxXi (u)_min)≤u_ext≤Ξ(u_max)；

I.e., the operating constraints of the outer loop PFC controller are consistent with the operating constraints of the inner loop PID controller.

3) Under the condition that the constraint of an inner loop is not considered, an outer loop PFC control law (a control structure model of an outer loop PFC controller) is calculated, the controller output obtained through solving, namely the optimal water tank level value is used as a set value of a PID loop, and control operation is carried out on the DCS control cabinet 1.

The operation constraint of the inner loop PID controller is loaded in the calculation process of the outer loop PFC control law in advance, a constraint condition set of the outer loop PFC controller during optimization calculation is obtained, and the following method is adopted:

under the condition that the constraints of the outer loop PFC controller and the constraints of the inner loop PID controller are consistent, the constraints of the inner loop PID controller are loaded in the calculation process of the outer loop PFC control law in advance by adopting a constraint rollback algorithm, and a constraint condition set of the outer loop PFC controller during optimization calculation is obtained.

Establishing an optimization model of an outer loop PFC controller

$\begin{array}{cc}\underset{{\mathrm{\&Delta;p}}_i,{\mathrm{\&Delta;m}}_i}{\min }& J=\underset{i=1}{\overset{n}{\&Sigma;}}{({\mathrm{\&Delta;p}}_i-{\mathrm{\&Delta;m}}_i)}^2\\ s.t.& \mathrm{\&Delta;}p={\&lsqb;{\mathrm{\&Delta;p}}_1,L,{\mathrm{\&Delta;p}}_n\&rsqb;}^T\\ & \mathrm{\&Delta;}m={\&lsqb;{\mathrm{\&Delta;m}}_1,L,{\mathrm{\&Delta;m}}_n\&rsqb;}^T\\ & \mathrm{\&Delta;}p=({\mathrm{SP}}_{ext}-y_p(k\left)\right)(1-{{\mathrm{\&lambda;}}_{ext}}^h)\\ & \mathrm{\&Delta;}m=y_m\left(k\right){a_{mext}}^h+u\left(k\right)K_{mext}(1-{a_{mext}}^h)-y_m\left(k\right)\\ & u_{ext\min }\&le;u_{ext}\&le;u_{ext\max }\\ & {\mathrm{SP}}_{ext\min }\&le;{\mathrm{SP}}_{ext}\&le;{\mathrm{SP}}_{ext\max }\end{array}---(3-1)$

In the formula,for optimizing the objective function, n is the number of variables. In the optimization time domain, the deviation between the process output increment delta p and the model prediction output increment delta m is minimized, and the control law of the outer loop PFC controller can be obtained.

Therefore, u_extmin≤u_ext≤u_extmaxAnd SP_extmin≤SP_ext≤SP_extmaxRespectively represents u_extAnd SP_extA constraint of (1), wherein u_extmin、u_extmaxRespectively represents u_extMinimum and maximum values of, SP_extmin、SP_extmaxRespectively represent SP_extMinimum and maximum values of;

since the operating constraints of the outer loop PFC controller are identical to the operating constraints of the inner loop PID controller, when u is_min≤u_int≤u_maxXi (u)_min)≤u_ext≤Ξ(u_max)；

PID control of inner loopThe operation constraint of the device is loaded in advance in the calculation process of the outer loop PFC control law, namely xi (u)_min)≤u_ext≤Ξ(u_max) Performing optimization calculation on the formula (3-1) for constraint conditions, and then

$\left\{\begin{array}{c}{u}_{ext\min }\&le;u_{ext}\&le;u_{ext\max }\\ \mathrm{\&Xi;}\left(u_{\min }\right)\&le;u_{ext}\&le;\mathrm{\&Xi;}\left(u_{\max }\right)\end{array}\right.---(3-2)$

Thus obtaining a new constraint max u_extmin,Ξ(u_min)}≤u_ext≤min{u_extmax,Ξ(u_max)}。

4) And the outer loop PFC controller performs optimization calculation under a new constraint condition set to obtain a set value of the inner loop PID controller, and the set value is provided for the inner loop PID controller so that the inner loop tracks the set value.

Under the new constraint condition, the optimization calculation is carried out on the external loop PFC controller, and the optimization model is

$\begin{array}{cc}\underset{{\mathrm{\&Delta;p}}_i,{\mathrm{\&Delta;m}}_i}{\min }& J=\underset{i=1}{\overset{n}{\&Sigma;}}{({\mathrm{\&Delta;p}}_i-{\mathrm{\&Delta;m}}_i)}^2\\ s.t.& \mathrm{\&Delta;}p={\&lsqb;{\mathrm{\&Delta;p}}_1,L,{\mathrm{\&Delta;p}}_n\&rsqb;}^T\\ & \mathrm{\&Delta;}m={\&lsqb;{\mathrm{\&Delta;m}}_1,L,{\mathrm{\&Delta;m}}_n\&rsqb;}^T\\ & \mathrm{\&Delta;}p=({\mathrm{SP}}_{ext}-y_p(k\left)\right)(1-{{\mathrm{\&lambda;}}_{ext}}^h)\\ & \mathrm{\&Delta;}m=y_m\left(k\right){a_{mext}}^h+u\left(k\right)K_{mext}(1-{a_{mext}}^h)-y_m\left(k\right)\\ & \max \{u_{ext\min },\mathrm{\&Xi;}\left(u_{\min }\right)\}\&le;u_{ext}\&le;\min \{u_{ext\max },\mathrm{\&Xi;}\left(u_{\max }\right)\}\\ & {\mathrm{SP}}_{ext\min }\&le;{\mathrm{SP}}_{ext}\&le;{\mathrm{SP}}_{ext\max }\end{array}---(4-1)$

Solving (4-1) to obtain an optimized objective functionObtaining the minimum value J_minU of time_extIs the optimal solution u of the optimization problem_extoptIt is used as the set value SP of the PID controller_intIs provided to the inner loop.

Under the new constraint condition, the PFC control law of the outer loop is calculated, and constraint consistency backspacing calculation enables constraint information to be transmitted to the PFC controller under the condition that the PID controller is saturated, so that the set value obtained by optimization and solution of the PFC controller is guaranteed to be correct and feasible, and the control performance is improved.

Claims (4)

1. A consistency constraint method for an industrial water tank liquid level cascade predictive control system is characterized by comprising the following steps:

1) establishing a control structure model of an outer loop PFC controller and a control structure model of an inner loop PID controller for an industrial water tank liquid level cascade prediction control system;

the control structure model of the outer loop PFC controller is

$u_{ext}\left(k\right)=\frac{({\mathrm{SP}}_{ext}-y_p(k\left)\right)(1-{{\mathrm{\&lambda;}}_{ext}}^h)-y_m\left(k\right){a_{mext}}^h+y_m\left(k\right)}{K_{mext}(1-{a_{mext}}^h)}---(1-1)$

In which the subscript p represents the process;

u_ext(k) the output of the PFC prediction function controller of the outer loop at the moment k;

SP_extpredicting a set value of a function controller for the PFC of the outer loop;

y_p(k) and y_p(k + h) respectively representing the output of the industrial water tank liquid level cascade prediction control system at the k moment and the k + h moment;

indicating an out of time k loop deviation_ext(k)＝SP_ext-y_p(k) Out-of-loop deviation to time k +1_ext(k+1)＝SP_ext-y_p(k +1) attenuation coefficient;

T_sis a sampling period;

CLTR is the closed loop response time of a first-order discrete system, namely the time of reaching 95 percent of a set value;

controlling the model attenuation coefficient for the outer loop PFC prediction function;

T_mis the model time constant;

K_mextan outer loop model gain;

when the outer loop controller is a PFC prediction function controller, firstly, a PFC algorithm is adopted to calculate a controller structural model,

① A first-order discrete system is constructed when the sampling period is T_sThe process gain is K_pThe process output is

CV(k)＝y_p(k)＝y_p(k-1)a_p+(1-a_p)K_pu(k-1)(1-3)

In which the subscript p represents the process;

CV (k) is the control variable at time k;

y_p(k) the output of the industrial water tank liquid level cascade prediction control system at the moment k;

is the process attenuation coefficient;

T_pis the process time constant;

u (k-1) is the output of the PFC prediction function controller at the k-1 moment;

modeling an output function as

y_m(k)＝y_m(k-1)a_m+(1-a_m)K_mu(k-1)(1-4)

In the formula, the subscript m represents a model;

y_m(k) is the output of the model at the moment k;

is the model attenuation coefficient;

T_mis the model time constant;

K_mis the model gain;

in the optimization time domain, the deviation between the process output increment delta p and the model prediction output increment delta m is minimized, and one-step optimization is adopted, namely the process output increment delta p is equal to the model prediction output increment delta m,

wherein,

Δp＝(k)-(k+h)

＝(k)-(k)λ^h

＝(k)(1-λ^h)

＝(SP-y_p(k))(1-λ^h)(1-5)

in the formula, SP is a set value;

(k)＝SP-y_p(k) industrial water tank liquid level cascade prediction control system output y representing set values SP and k_p(k) A deviation of (a);

(k+h)＝SP-y_p(k+h)＝(k)λ^hindicates the set values SP and k + hOutput y of industrial water tank liquid level cascade predictive control system_p(k + h) deviation;

represents the attenuation coefficient of the deviation from the k time to the k +1 time;

λ^his the h-th power of lambda and represents the attenuation coefficient of the deviation from the time k to the time k + h;

for the first-order discrete system, taking the system time constant of the industrial water tank liquid level cascade predictive control systemCLTR is the closed loop response time of a first-order discrete system, namely the time of reaching 95 percent of a set value;

and is

Δm＝y_m(k+h)-y_m(k)(1-6)

In the formula, y_m(k + h) is the predicted output of the model at the moment k + h and is represented by the free response y_mFree(k + h) and forced response y_mForce(k + h) composition, i.e.

y_m(k+h)＝y_mFree(k+h)+y_mForce(k+h)(1-7)

Wherein,

$y_{mFree}(k+h)=y_m\left(k\right)a_m^h---(1-8)$

$y_{mForce}(k+h)=u\left(k\right)K_m(1-a_m^h)---(1-9)$

in the formula, y_mFree(k + h) denotes the free response, y_mForce(k + h) represents a forced response;

therefore, the temperature of the molten steel is controlled,

$\mathrm{\&Delta;}m=y_m\left(k\right)a_m^h+u\left(k\right)K_m(1-a_m^h)-y_m\left(k\right)---(1-10)$

from Δ p ═ Δ m

$(SP-y_p(k\left)\right)(1-{\mathrm{\&lambda;}}^h)=y_m\left(k\right)a_m^h+u\left(k\right)K_m(1-a_m^h)-y_m\left(k\right)---(1-11)$

After being sorted, the general form of the PFC controller is obtained

$u\left(k\right)=\frac{(SP-y_p(k\left)\right)(1-{\mathrm{\&lambda;}}^h)-y_m\left(k\right)a_m^h+y_m\left(k\right)}{K_m(1-a_m^h)}---\left(1-12\right)$

③ the control structure model of the outer loop PFC controller, which can be obtained from the formula (1-12)

$u_{ext}\left(k\right)=\frac{({\mathrm{SP}}_{ext}-y_p(k\left)\right)(1-{{\mathrm{\&lambda;}}_{ext}}^h)-y_m\left(k\right){a_{mext}}^h+y_m\left(k\right)}{K_{mext}(1-{a_{mext}}^h)}---(1-1)$

In the formula u_ext(k) The output of the PFC prediction function controller of the outer loop at the moment k;

SP_extpredicting a set value of a function controller for the PFC of the outer loop;

y_p(k) and y_p(k + h) respectively representing the output of the industrial water tank liquid level cascade prediction control system at the k moment and the k + h moment;

indicating an out of time k loop deviation_ext(k)＝SP_ext-y_p(k) Out-of-loop deviation to time k +1_ext(k+1)＝SP_ext-y_pThe attenuation coefficient of (k +1) is,CLTR is the closed loop response time of a first-order discrete system, namely the time of reaching 95 percent of a set value;

controlling the model attenuation coefficient for the outer loop PFC prediction function;

T_mis the model time constant;

K_mextan outer loop model gain;

④ the inner loop is controlled by proportion P, and the gain of the controller is set to K_intThereby obtaining a control structure model of the inner loop PID controller

u_int＝(SP_int-y_p(k))K_int(1-2)

In the formula, the inner loop is controlled by a proportion P;

u_intis the output of the inner loop controller;

SP_intis the set value of the inner loop;

y_p(k) the output of the industrial water tank liquid level cascade prediction control system at the moment k;

K_intan inner loop controller gain;

2) judging whether the constraints of the outer loop PFC controller and the constraints of the inner loop PID controller are consistent, if so, performing the step 3), and if not, returning to the step 1) to reestablish a control structure model of the outer loop PFC controller and a control structure model of the inner loop PID controller;

3) under the condition that the constraints of the outer loop PFC controller and the inner loop PID controller are consistent, the constraint of the inner loop PID controller is pre-loaded in the calculation process of the outer loop PFC control law by adopting a constraint rollback algorithm, and a new constraint condition set when the outer loop PFC controller carries out optimization calculation is obtained;

4) and the outer loop PFC controller performs optimization calculation under a new constraint condition set to obtain a set value of the inner loop PID controller, and the set value is provided for the inner loop PID controller so that the inner loop tracks the set value.

2. The method of claim 1, wherein the method comprises: in step 2), whether the external loop PFC controller and the internal loop PID controller are constrained to be consistent or not is judged, the following method is adopted,

① when the inner loop is controlled by the proportion P, the set value SP 'of one inner loop is artificially set for the inner loop proportion control law represented by the control structure model (1-2) of the inner loop PID controller'_intAnd the set value SP 'is made'_int＞SP_intThe corresponding inner loop control law is

u′_int＝(SP′_int-y_p(k))K_int(2-1)

In the formula u_int' taking SP for inner loop set point_int' time, the output value of the inner loop controller;

subtracting the control structure model (1-2) of the inner loop PID controller from the control law (2-1) of the inner loop to obtain

u′_int-u_int＝(SP′_int-y_p(k))K_int-(SP_int-y_p(k))K_int(2-2)

Namely, it is

u′_int-u_int＝(SP′_int-SP_int)K_int(2-3)

③ when K_int> 0, since SP is assumed'_int＞SP_intU's of'_int(k)-u_int(k) > 0, i.e. when SP'_int＞SP_intIn each case, there is u'_int(k)＞u_int(k)；

When K is_int< 0, since SP 'has been assumed'_int＞SP_intU's of'_int(k)-u_int(k) < 0, i.e. when SP'_int＞SP_intIn each case, there is u'_int(k)＜u_int(k)；

Therefore, for an industrial water tank liquid level cascade predictive control system controlled by a proportion P, the output of the controller is strictly monotonous with the set value;

for PFC-PID cascade conduction control loop, the output of the outer loop controller is the set value of the inner loop controller, namely

u_ext(k)＝SP_int(2-4)

When u is strictly monotonic with the output of the inner loop controller and the output of the outer loop controller_min≤u_int≤u_maxXi (u)_min)≤u_ext≤Ξ(u_max)；

I.e., the operating constraints of the outer loop PFC controller are consistent with the operating constraints of the inner loop PID controller.

3. The method of claim 2, wherein the method comprises: in step 3), the operation constraint of the inner loop PID controller is loaded in the calculation process of the outer loop PFC control law in advance, and a constraint condition set of the outer loop PFC controller during optimization calculation is obtained, by adopting the following method,

establishing an optimization model of an outer loop PFC controller

$\underset{{\mathrm{\&Delta;p}}_i,{\mathrm{\&Delta;m}}_i}{min}J=\underset{i=1}{\overset{n}{\&Sigma;}}{({\mathrm{\&Delta;p}}_i-{\mathrm{\&Delta;m}}_i)}^2$

s.t.Δp＝[Δp₁,…,Δp_n]^T

Δm＝[Δm₁,…,Δm_n]^T(3-1)

Δp＝(SP_ext-y_p(k))(1-λ_ext ^h)

Δm＝y_m(k)a_mext ^h+u(k)K_mext(1-a_mext ^h)-y_m(k)

u_extmin≤u_ext≤u_extmax

SP_extmin≤SP_ext≤SP_extmax

In the formula,n is the number of variables for optimizing the objective function;

in the optimization time domain, the deviation between the process output increment delta p and the model prediction output increment delta m is minimized, and the control law of the outer loop PFC controller can be obtained;

u_extmin≤u_ext≤u_extmaxand SP_extmin≤SP_ext≤SP_extmaxRespectively represents u_extAnd SP_extA constraint of (1), wherein u_extmin、u_extmaxRespectively represents u_extMinimum and maximum values of, SP_extmin、SP_extmaxRespectively represent SP_extMinimum and maximum values of;

since the operating constraints of the outer loop PFC controller are identical to the operating constraints of the inner loop PID controller, when u is_min≤u_int≤u_maxXi (u)_min)≤u_ext≤Ξ(u_max)；

The operation constraint of the inner loop PID controller is loaded in advance in the calculation process of the outer loop PFC control law, namely xi (u)_min)≤u_ext≤Ξ(u_max) Performing optimization calculation on the formula (3-1) for constraint conditions, and then

$\left\{\begin{array}{c}{u}_{ext\min }\&le;u_{ext}\&le;u_{ext\max }\\ \mathrm{\&Xi;}\left(u_{\min }\right)\&le;u_{ext}\&le;\mathrm{\&Xi;}\left(u_{\max }\right)\end{array}\right.---\left(3-2\right)$

Thus obtaining a new constraint max u_extmin,Ξ(u_min)}≤u_ext≤min{u_extmax,Ξ(u_max)}。

4. The method of claim 3, wherein the method comprises: in step 4), the outer loop PFC controller performs optimization calculation under a new constraint condition set to obtain a set value of the inner loop PID controller, and provides the set value to the inner loop PID controller to enable the inner loop to track the set value, the method is as follows,

under the new constraint condition, the optimization calculation is carried out on the external loop PFC controller, and the optimization model is

$\underset{{\mathrm{\&Delta;p}}_i,{\mathrm{\&Delta;m}}_i}{min}J=\underset{i=1}{\overset{n}{\&Sigma;}}{({\mathrm{\&Delta;p}}_i-{\mathrm{\&Delta;m}}_i)}^2$

s.t.Δp＝[Δp₁,…,Δp_n]^T

Δm＝[Δm₁,…,Δm_n]^T(4-1)

Δp＝(SP_ext-y_p(k))(1-λ_ext ^h)

Δm＝y_m(k)a_mext ^h+u(k)K_mext(1-a_mext ^h)-y_m(k)

max{u_extmin,Ξ(u_min)}≤u_ext≤min{u_extmax,Ξ(u_max)}

SP_extmin≤SP_ext≤SP_extmax

Solving (4-1) to obtain an optimized objective functionObtaining the minimum value J_minU of time_extIs the optimal solution u_extoptIt is used as the set value SP of the PID controller_intIs provided to the inner loop.