CN108897219B - Chemical uncertain industrial process constraint prediction control method - Google Patents

Chemical uncertain industrial process constraint prediction control method Download PDF

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CN108897219B
CN108897219B CN201810760493.7A CN201810760493A CN108897219B CN 108897219 B CN108897219 B CN 108897219B CN 201810760493 A CN201810760493 A CN 201810760493A CN 108897219 B CN108897219 B CN 108897219B
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胡晓敏
余哲
邹洪波
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Hangzhou Dianzi University
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    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
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Abstract

The invention discloses a constraint prediction control method for an uncertain chemical industrial process, which comprises the following steps: step 1, establishing an equivalent two-dimensional model; and 2, designing a predictive controller. The method comprises the steps of firstly, analyzing a state space model of an uncertain system with disturbance, determining a system state error and an output tracking error according to an iterative learning control strategy, and establishing an equivalent two-dimensional model; and then establishing a Lyapunov function based on a closed-loop prediction model, solving the update law of the system according to the constraint relation between the target function of the prediction control and the Lyapunov function, and further obtaining the controlled variable acting on the controlled object. The chemical uncertain batch process constraint prediction control method not only solves the problems of uncertainty and disturbance, but also ensures the stability of the system.

Description

Chemical uncertain industrial process constraint prediction control method
Technical Field
The invention belongs to the technical field of automation, and relates to a chemical uncertain batch process constraint prediction control method.
Background
The problems of uncertainty and disturbance exist in the actual production process, the problems can cause the uncertain change of the system, the control performance of the system is deteriorated, and the produced product can not meet the product quality requirement. The batch process fault-tolerant control technology in the one-dimensional system model is only researched in the time direction or the batch direction, and obviously, the control precision cannot meet the requirement. Although the batch process robust control technology in the two-dimensional system model can effectively deal with the uncertainty problem, the state deviation problem in the system cannot be solved, and the deviation problems will have adverse effects on the continuous and stable operation and the control performance of the system, and even influence the product quality. It is therefore desirable to provide a more efficient control method that addresses the uncertainty and disturbance of the system.
Disclosure of Invention
The invention aims to better solve the problems of uncertainty and disturbance in actual production, and further provides a chemical uncertain batch process constraint prediction control method. The method comprises the steps of firstly, analyzing a state space model of an uncertain system with disturbance, determining a system state error and an output tracking error according to an iterative learning control strategy, and establishing an equivalent two-dimensional model; and then establishing a Lyapunov function based on a closed-loop prediction model, solving the update law of the system according to the constraint relation between the target function of the prediction control and the Lyapunov function, and further obtaining the controlled variable acting on the controlled object. The chemical uncertain batch process constraint prediction control method not only solves the problems of uncertainty and disturbance, but also ensures the stability of the system. The specific technical scheme is as follows:
the method comprises the following steps:
step 1, establishing an equivalent two-dimensional model, which comprises the following specific steps:
1.1 the state space model of an uncertain system with perturbations is as follows:
Figure BDA0001727465690000011
wherein the content of the first and second substances,
Figure BDA0001727465690000012
t is time, k is batch, x (t +1, k), x (t, k) u (t, k) w (t, k) y (t, k) respectively represent the state at time t of the k +1 th batch, the state, input, unknown disturbance and output at time t of the k-th batch,
Figure BDA0001727465690000013
A,B2,C2all represent a system matrix of appropriate dimensions and Δ a (t, k) represents the perturbation matrix at time t for the kth lot.
1.2 consider fault gain, the system model is as follows:
Figure BDA0001727465690000021
where α represents a constant between 0 and 1.
1.3 the iterative learning control law is as follows:
u(t,k)＝u(t,k-1)+r(t,k)
u(0,k)＝0
wherein u (t, k-1) represents the input of the k-1 th batch at the time t, r (t, k) represents the updating law of the k-1 th batch at the time t, and u (0, k) represents the initial iteration value.
1.4 the system state error and output tracking error model is as follows:
Figure BDA0001727465690000022
wherein the content of the first and second substances,
Figure BDA0001727465690000023
Figure BDA0001727465690000024
Figure BDA00017274656900000210
A1，B，C1a matrix of the system is represented,
Figure BDA0001727465690000025
representing a perturbation matrix, I representing an identity matrix, 0 representing a zero matrix, δ (x (t, k)) representing a system state error at time t of the kth batch, δk(x (t +1, k)) represents the system state error at time t +1 for the kth lot, e (t +1, k-1) represents the output tracking error at time t +1 for the kth-1 lot, and e (t, k +1) represents the output tracking error at time t for the kth +1 lot.
1.5 equivalent two-dimensional models are as follows:
Figure BDA0001727465690000026
wherein the content of the first and second substances,
Figure BDA0001727465690000027
the designation @ @ is for the definition of,
Figure BDA0001727465690000028
respectively expressed as different expansion states of the system at the time t of the kth batch,
Figure BDA0001727465690000029
represents the spread of the system interference at time t for the kth lot, e (t, k-1) represents the output tracking error at time t for the kth lot, e (t, k) represents the output tracking error at time t for the kth lot, and Z (t, k) represents the output of the controller at time t for the kth lot.
Step 2, designing a predictive controller, which comprises the following specific steps:
2.1 the objective function of the system is as follows:
Figure BDA0001727465690000031
wherein i is 0, 1,2 … ∞, J∞(T, k) represents the objective function of the system at time T of the kth batch, T represents the transposed symbol of the matrix,
Figure BDA0001727465690000032
the expansion state of the system at the t + i moment predicted by the t moment of the kth batch is shown, R (t + i | t, k) represents the updating law of the t + i moment predicted by the t moment of the kth batch, and Q and R are weight matrixes with proper dimensions.
2.2 closed-loop prediction model as follows:
Figure BDA0001727465690000033
wherein the content of the first and second substances,
Figure BDA0001727465690000034
represents the expansion state of the system at the t + i moment predicted by the t moment of the kth batch,
Figure BDA0001727465690000035
represents the spread of the system disturbance at time t + i predicted at time t for the kth lot, y (t + i | t, k) represents the output of the system at time t + i predicted at time t for the kth lot, and Z (t + i | t, k) represents the output of the controller at time t + i predicted at time t for the kth lot.
2.3 according to step 2.2, the Lyapunov function is as follows:
Figure BDA0001727465690000036
where V (t, k) represents the lyapunov function of the kth lot at time t, and M represents a set matrix.
2.4 to meet the system stability requirements, it is necessary to meet:
maxJ∞(t,k)≤Vh(0,k)+Vv(t,0)
wherein, maxJ∞(t, k) represents the maximum value of the objective function of the system at time t of the kth batch, Vh(0,k)，Vv(t,0) are expressed as the initial value of the Lyapunov function at time 0 of the kth lot and the initial value of the Lyapunov function at time t of the 0 th lot, respectively.
2.5 according to step 2.3 and step 2.4, obtaining a system control law:
Figure BDA0001727465690000037
where K (t, K) represents the gain of the system at time t for the kth lot.
2.6 according to step 1.3 and step 2.5, a robust predictive controller is obtained:
Figure BDA0001727465690000041
u(0,k)＝0
and 2.7, according to the steps 2.1 to 2.6, sequentially solving the control quantity u (t, k) in a circulating mode, and then acting the control quantity on the controlled object.
Detailed Description
Taking an injection molding process in an actual process as an example:
step 1, establishing a two-dimensional model equivalent to an injection molding process, which comprises the following specific steps:
1.1 the state space model of the injection molding process is as follows:
Figure BDA0001727465690000042
wherein the content of the first and second substances,
Figure BDA0001727465690000043
t is time, k is batch, x (t +1, k), x (t, k) u (t, k) w (t, k) y (t, k) respectively represent the state at time t of the (k +1) th batch, the state at time t of the kth batch, the valve opening, the unknown disturbance and the cavity pressure,
Figure BDA0001727465690000044
A,B2,C2all represent a system matrix of appropriate dimensions and Δ a (t, k) represents the perturbation matrix at time t for the kth lot.
1.2 consider the failure gain, the model of the injection molding process is as follows:
Figure BDA0001727465690000045
where α represents a constant between 0 and 1.
1.3 the iterative learning control law of the injection molding process is as follows:
u(t,k)＝u(t,k-1)+r(t,k)
u(0,k)＝0
wherein u (t, k-1) represents the valve opening at the time t of the k-1 th batch, r (t, k) represents the update law at the time t of the k-1 th batch, and u (0, k) represents the initial valve opening.
1.4 the system state error and output tracking error model is as follows:
Figure BDA0001727465690000046
wherein the content of the first and second substances,
Figure BDA0001727465690000047
Figure BDA0001727465690000051
Figure BDA0001727465690000052
A1，B，C1a matrix of the system is represented,
Figure BDA0001727465690000053
representing a perturbation matrix, I representing an identity matrix, 0 representing a zero matrix, δ (x (t, k)) representing a system state error at time t of the kth batch, δk(x (t +1, k)) represents the system state error at time t +1 for the kth lot, e (t +1, k-1) represents the cavity pressure tracking error at time t +1 for the kth-1 lot, and e (t, k +1) represents the cavity pressure tracking error at time t for the kth +1 lot.
1.5 the two-dimensional model equivalent to the injection molding process is as follows:
Figure BDA0001727465690000054
wherein the content of the first and second substances,
Figure BDA0001727465690000055
the designation @ @ is for the definition of,
Figure BDA0001727465690000056
respectively expressed as different expansion states of the system at the time t of the kth batch,
Figure BDA0001727465690000057
indicating the spread of system disturbances at time t for the kth batch, e (t, k-1) indicating the cavity pressure tracking error at time t for the kth batch, e (t, k) indicating the cavity pressure tracking error at time t for the kth batch, and Z (t, k) indicating the cavity pressure of the controller at time t for the kth batch.
Step 2, designing a predictive controller, which comprises the following specific steps:
2.1 the objective function of the injection molding process is as follows:
Figure BDA0001727465690000058
wherein i is 0, 1,2 … ∞, J∞(T, k) represents the objective function of the system at time T of the kth batch, T represents the transposed symbol of the matrix,
Figure BDA0001727465690000059
the expansion state of the system at the t + i moment predicted by the t moment of the kth batch is shown, R (t + i | t, k) represents the updating law of the t + i moment predicted by the t moment of the kth batch, and Q and R are weight matrixes with proper dimensions.
2.2 closed-loop prediction model of injection molding process as follows:
Figure BDA00017274656900000510
wherein the content of the first and second substances,
Figure BDA0001727465690000061
represents the expansion state of the system at the t + i moment predicted by the t moment of the kth batch,
Figure BDA0001727465690000062
represents the spread of system disturbances at time t + i predicted at time t for the kth lot, y (t + i | t, k) represents the cavity pressure of the system at time t + i predicted at time t for the kth lot, and Z (t + i | t, k) represents the cavity pressure of the controller at time t + i predicted at time t for the kth lot.
2.3 according to step 2.2, the Lyapunov function is as follows:
Figure BDA0001727465690000063
where V (t, k) represents the lyapunov function of the kth lot at time t, and M represents a set matrix.
2.4 to meet the system stability requirements, it is necessary to meet:
maxJ∞(t,k)≤Vh(0,k)+Vv(t,0)
wherein, maxJ∞(t, k) represents the maximum value of the objective function of the system at time t of the kth batch, Vh(0,k)，Vv(t,0) are expressed as the initial value of the Lyapunov function at time 0 of the kth lot and the initial value of the Lyapunov function at time t of the 0 th lot, respectively.
2.5 according to step 2.3 and step 2.4, obtaining a system control law:
Figure BDA0001727465690000064
where K (t, K) represents the gain of the system at time t for the kth lot.
2.6 according to step 1.3 and step 2.5, a robust predictive controller is obtained:
Figure BDA0001727465690000065
u(0,k)＝0
and 2.7, according to the steps 2.1 to 2.6, sequentially and circularly solving the valve opening u (t, k), and then acting the valve opening u (t, k) on the injection molding process.

Claims (1)

1. A chemical uncertain industrial process constraint prediction control method comprises the following steps:
step 1, establishing an equivalent two-dimensional model;
step 2, designing a prediction controller;
the step 1 is as follows:
1.1 the state space model of an uncertain system with perturbations is as follows:
Figure FDA0002818115080000011
wherein the content of the first and second substances,
Figure FDA0002818115080000012
t is time, k is batch, x (t +1, k), x (t, k), u (t, k), w (t, k), y (t, k) respectively represent the state at time t of the k +1 th batch, the state, input, unknown disturbance and output at time t of the k-th batch,
Figure FDA0002818115080000013
A,B2,C2all represent system matrices of appropriate dimensions, Δ a (t, k) represents the perturbation matrix at time t of the kth lot;
1.2 consider fault gain, the system model is as follows:
Figure FDA0002818115080000014
wherein α represents a constant between 0 and 1;
1.3 the iterative learning control law is as follows:
u(t,k)＝u(t,k-1)+r(t,k)
u(0,k)＝0
wherein u (t, k-1) represents the input of the kth-1 batch at the time of t, r (t, k) represents the updating law of the kth batch at the time of t, and u (0, k) represents the initial iteration value;
1.4 the system state error and output tracking error model is as follows:
Figure FDA0002818115080000015
wherein the content of the first and second substances,
Figure FDA0002818115080000016
Figure FDA0002818115080000017
Figure FDA0002818115080000018
A1，B，C1a matrix of the system is represented,
Figure FDA0002818115080000019
representing a perturbation matrix, I representing an identity matrix, 0 representing a zero matrix, δ (x (t, k)) representing a system state error at time t of the kth batch, δk(x (t +1, k)) representsThe system state error at the t +1 th batch, the output tracking error at the t +1 th batch is represented by e (t +1, k-1), and the output tracking error at the t +1 th batch is represented by e (t, k + 1);
1.5 equivalent two-dimensional models are as follows:
Figure FDA0002818115080000021
wherein the content of the first and second substances,
Figure FDA0002818115080000022
the designation @ @ is for the definition of,
Figure FDA0002818115080000023
respectively expressed as different expansion states of the system at the time t of the kth batch,
Figure FDA0002818115080000024
representing the spread of the system interference at the time t of the kth batch, e (t, k-1) representing the output tracking error at the time t of the kth batch, e (t, k) representing the output tracking error at the time t of the kth batch, and Z (t, k) representing the output of the controller at the time t of the kth batch;
the step 2 is as follows:
2.1 the objective function of the system is as follows:
Figure FDA0002818115080000025
wherein i is 0, 1,2 … ∞, J∞(T, k) represents the objective function of the system at time T of the kth batch, T represents the transposed symbol of the matrix,
Figure FDA0002818115080000026
represents the expansion state of the system at the t + i moment predicted at the t moment of the kth batch, R (t + i | t, k) represents the update law of the t + i moment predicted at the t moment of the kth batch, Q, R represents the weight moment of a proper dimensionArraying;
2.2 closed-loop prediction model as follows:
Figure FDA0002818115080000027
wherein the content of the first and second substances,
Figure FDA0002818115080000028
represents the expansion state of the system at the t + i moment predicted by the t moment of the kth batch,
Figure FDA0002818115080000029
represents the spread of the system interference at the t + i moment predicted at the t moment of the kth batch, y (t + i | t, k) represents the output of the system at the t + i moment predicted at the t moment of the kth batch, and Z (t + i | t, k) represents the output of the controller at the t + i moment predicted at the t moment of the kth batch;
2.3 according to step 2.2, the Lyapunov function is as follows:
Figure FDA0002818115080000031
v (t, k) represents a Lyapunov function of the kth batch at the time t, and M represents a set matrix;
2.4 to meet the system stability requirements, it is necessary to meet:
maxJ∞(t,k)≤Vh(0,k)+Vv(t,0)
wherein, maxJ∞(t, k) represents the maximum value of the objective function of the system at time t of the kth batch, Vh(0,k)，Vv(t,0) are respectively expressed as the initial value of the lyapunov function at the time of 0 of the kth batch and the initial value of the lyapunov function at the time of t of the 0 th batch;
2.5 according to step 2.3 and step 2.4, obtaining a system control law:
Figure FDA0002818115080000032
wherein K (t, K) represents the gain of the system at the time t of the kth batch;
2.6 according to step 1.3 and step 2.5, a robust predictive controller is obtained:
Figure FDA0002818115080000033
and 2.7, according to the steps 2.1 to 2.6, sequentially solving the control quantity u (t, k) in a circulating mode, and then acting the control quantity on the controlled object.
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