Wide and thick plate hot working guaranteed performance control method based on electric furnace power supply switching

Abstract

The invention discloses a wide and thick plate hot working furnace temperature guarantee performance control method based on an electric furnace power supply switching device. The method comprises the following steps: firstly, establishing a partial differential equation mathematical model and corresponding performance indexes of the wide and thick plate hot working system of the electric furnace power supply switching device; and then, designing a furnace temperature state feedback controller of the wide and thick plate hot working system of the electric furnace power supply switching device. And then, a closed-loop control system is constructed according to the wide and thick plate hot working mathematical model of the electric furnace power supply switching device and a furnace temperature state feedback controller. And the Lyapunov stability theorem, the Poincare inequality and the linear matrix inequality theory are utilized to obtain sufficient conditions for the existence of the hot working guaranteed performance control of the wide and thick plate of the electric furnace power supply switching device. A state feedback controller and a switching rule are designed, so that the temperature stability of the wide and thick plate hot working system of the electric furnace power supply switching device can be ensured, and the performance index of the wide and thick plate hot working system of the switching device can meet a certain requirement. According to the method, the influence of the Laplace operator coefficient in the electric furnace power switching device wide and thick plate hot working mathematical model on the system is fully considered, and the design is more scientific and reasonable. The method has the advantages that feedback control is carried out on the furnace temperature state of wide and thick plate hot working through the furnace temperature power supply cut-off device, engineering implementation is easy, and the purposes that the wide and thick plate hot working furnace temperature is stable and certain performance indexes are met under switching and external nonlinear disturbance of an electric furnace power supply switching control system can be achieved.

Description

Technical Field

[0001] The invention relates to the technical field of industrial furnace temperature, and in particular to a method for controlling the temperature preservation performance of a wide and thick plate hot processing furnace of an electric furnace power supply switching device. Background Art

[0002] The electric furnace power switching device is essentially a switching system device. The switching system is a very important type of hybrid system, consisting of a set of subsystems and switching rules. Since the switching system has a wide range of applications in heat exchange systems, robotic systems, power systems, transportation systems and other fields, the switching system has been widely studied. When the subsystem of the switching system is modeled by partial differential equations, it is called a distributed parameter switching system. Compared with the concentrated parameter switching system modeled by ordinary differential equations, the distributed parameter switching system can effectively improve the control accuracy. Since the distributed parameter switching system has infinite degrees of freedom, the study of switching distributed parameter systems is difficult. The study of the control problem of distributed parameter switching systems has always been a difficult topic in the field of control research.

[0003] For control systems, stability is a prerequisite for the existence of the system. In addition to stability, the system is generally required to meet certain performance indicators, such as the system's performance indicators do not exceed an integral quadratic form, that is, guaranteed performance control. How to design a reasonable guaranteed performance controller so that the system's performance indicators do not exceed a certain upper limit under stability conditions is a challenging topic. Summary of the invention

[0004] For the wide and thick plate hot processing system of the power switching device, switching and external nonlinear disturbance factors can easily lead to the instability of the furnace temperature of the system. In order to solve the technical problems of furnace temperature state stability and performance-guaranteed control of the wide and thick plate hot processing system of the electric furnace power switching device, the present invention provides a performance-guaranteed control method for the furnace temperature of the wide and thick plate hot processing system based on the electric furnace power switching device. The present invention models the wide and thick plate hot processing system of the electric furnace power switching device into a partial differential equation mathematical model (i.e., a distributed parameter switching system mathematical model). And for the distributed parameter switching system, a state feedback controller and switching rules are designed, and the Lyapunov stability theory and linear matrix inequality theory are used to derive the sufficient conditions for performance-guaranteed control of the distributed parameter switching system for hot processing of wide and thick plates of the electric furnace power switching device. It not only ensures the stability of the furnace temperature of the distributed parameter switching system for hot processing of wide and thick plates of the electric furnace power switching device, but also enables the performance indicators of the furnace temperature to meet certain requirements.

[0005] The purpose of the present invention is achieved by the following technical solution. It comprises the following steps:

[0006] Step S1: Establish a partial differential equation mathematical model of the wide and thick plate hot processing system of the electric furnace power switching device and the corresponding performance indicators. The partial differential equation mathematical model is used to describe the temperature state change process of the wide and thick plate hot processing of the electric furnace power switching device;

[0007] Step S2: designing a furnace temperature state feedback controller for hot processing of wide and thick plates in an electric furnace power switching device;

[0008] Step S3: constructing a closed-loop control system based on the partial differential equation mathematical model of wide and thick plate hot processing of the electric furnace power supply switching device and the furnace temperature state feedback controller;

[0009] Step S4: design the parameter adjustment law and switching rule of the wide and thick plate hot processing closed-loop control system of the electric furnace power switching device, and determine the guaranteed performance control conditions of the closed-loop control system;

[0010] Step S5: realizing the temperature preservation performance control of the wide and thick plate hot processing furnace of the electric furnace power switching device.

[0011] Furthermore, the partial differential equation mathematical model of the wide and thick plate hot processing system of the electric furnace power supply switching device constructed in step S1 is specifically:

[0012]

[0013] Where (x, t)∈Ω×(0,+∞), x represents the coordinate, t represents the time, Represents a smooth boundary The bounded region of (i,j∈Γ),Γ={1,2,3,···,n},W(x,t)∈R m represents temperature, U(x,t)∈R s represents the control input, is the Laplace operator, the dissipation coefficient is a constant, For a given matrix of appropriate dimensions, f i (W(x,t)) is a nonlinear perturbation, ||f i (W(x,t))||≤G i ||W(x,t)||.

[0014] Its boundary conditions are:

[0015]

[0016] or

[0017]

[0018] Where N is the boundary The unit external normal vector of .

[0019] Furthermore, the performance indicators of the constructed partial differential equation mathematical model of the wide and thick plate hot processing system of the electric furnace power switching device (i.e., the distributed parameter switching system) are specifically as follows:

[0020]

[0021] Where Q and R are positive definite weight matrices.

[0022] Furthermore, the control input of step S2 is the feedback of all temperature state variables of the wide and thick plate hot processing control system of the electric furnace power switching device. The controller is:

[0023] U(x,t)=K i W(x,t)

[0024] Where K i is an adjustable control gain.

[0025] Further, the method for controlling the temperature preservation performance of a wide and thick plate hot processing furnace of an electric furnace power supply switching device is characterized in that the closed-loop electric furnace power supply switching device wide and thick plate hot processing control system of step S3 is:

[0026]

[0027] Furthermore, the sufficient conditions for the parameter adjustment law and switching rule of the wide and thick plate hot processing control system designed in the step S5 are:

[0028] Sufficient condition 1 When there exists a matrix P i >0,K i (i=1,2,···,n), so that the matrix inequality

[0029] Condition 1.1

[0030]

[0031] in

[0032] Condition 1.2 Switching rules

[0033]

[0034] Sufficient condition 2: When there exists a matrix X i >0,S i , (i=1,2,···,n), so that

[0035] Condition 2.1

[0036] in

[0037]

[0038] Condition 2.2 Switching rules

[0039]

[0040] Further, the step S4 designs the electric furnace power supply switching device wide and thick plate hot processing closed-loop control system, under the condition of the step S5, to achieve furnace temperature stability and furnace temperature performance indicators meet certain requirements, and the performance upper limit is:

[0041] J * =max{∫ Ω W T (x,t0)P1W(x,t0)dx,…,∫ Ω W T (x,t0)P n W(x,t0)dx}(9)

[0042] Thereby realizing the temperature preservation performance control of the wide and thick plate hot processing furnace of the electric furnace power switching device. DETAILED DESCRIPTION

[0043] The following is combined with Figure 1 The implementation of the present invention is described in detail.

[0044] The embodiment of the present invention is a wide and thick plate hot processing of an electric furnace power switching device. The provided design method requires that the wide and thick plate hot processing temperature of the electric furnace power switching device can be stabilized under the switching and external nonlinear disturbance, and the performance indicators of the wide and thick plate hot processing system of the device can meet certain requirements.

[0045] In order to achieve the above objectives, this example first establishes the partial differential equation mathematical model of the wide and thick plate hot processing system of the electric furnace power switching device and the corresponding performance indicators. Then, the furnace temperature state feedback controller and switching signal of the wide and thick plate hot processing system of the electric furnace power switching device are designed. Then, a closed-loop control system is constructed based on the mathematical model of the wide and thick plate hot processing system of the electric furnace power switching device and the furnace temperature state feedback controller. Using the Lyapunov stability theorem, Poincare inequality and linear matrix inequality theory, sufficient conditions for the existence of guaranteed performance control of the wide and thick plate hot processing system of the electric furnace power switching device are obtained.

[0046] like Figure 1 As shown, a method for controlling the temperature preservation performance of a wide and thick plate hot processing furnace based on an electric furnace power supply switching device comprises the following steps:

[0047] Step S1: Establish a partial differential equation mathematical model of the wide and thick plate hot processing system of the electric furnace power switching device and the corresponding performance indicators. The partial differential equation mathematical model is used to describe the temperature state change process of the wide and thick plate hot processing of the electric furnace power switching device;

[0048] The partial differential equation mathematical model of the wide and thick plate hot processing system of the electric furnace power switching device is established (that is, the mathematical model of the distributed parameter switching system):

[0049]

[0050] Where (x, t)∈Ω×(0,+∞), x represents the coordinate, t represents the time, Represents a smooth boundary The bounded region of (i,j∈Γ),Γ={1,2,3,···,n},W(x,t)∈R m represents temperature, U(x,t)∈R s represents the control input, is the Laplace operator, the dissipation coefficient is a constant, For a given matrix of appropriate dimensions, f i (W(x,t)) is a nonlinear perturbation, ||f i (W(x,t))||≤G i ||W(x,t)||.

[0051] Its boundary conditions are:

[0052]

[0053] or

[0054]

[0055] Where N is the boundary The unit external normal vector of .

[0056] The performance indicators of the partial differential equation mathematical model of the wide and thick plate hot processing system of the electric furnace power supply switching device constructed in step S1 (i.e., the distributed parameter switching system) are specifically:

[0057]

[0058] Where Q and R are positive definite weight matrices.

[0059] Step S2: designing a furnace temperature state feedback controller for a wide and thick plate hot processing system of an electric furnace power supply switching device;

[0060] Its state feedback control is:

[0061] U(x,t)=K i W(x,t) (5)

[0062] Step S3: Construct a closed-loop control system based on the partial differential equation mathematical model of the wide and thick plate hot processing system of the electric furnace power switching device and the furnace temperature state feedback controller

[0063]

[0064] Step S4: Design the parameter adjustment law and switching rules of the closed-loop control system to determine the guaranteed performance control conditions of the closed-loop system.

[0065] In step S4, the Lyapunov function is constructed as

[0066]

[0067] Where P i is a positive definite matrix.

[0068] According to Gauss's convergence theorem, Poincare's inequality and boundary conditions, we have

[0069]

[0070] V i Derivative of the state trajectory of the closed-loop system with respect to time t

[0071]

[0072] further

[0073]

[0074] when

[0075] Take switching rules

[0076] Then there must exist i∈Ω i , so that

[0077] That is, when i∈Ω i region, for each subsystem i, there is a corresponding

[0078] Then, the system (1) is asymptotically stable under the action of the controller (5) and the switching rule (11). And

[0079] J * =max{∫ Ω W T (x,t0)P1W(x,t0)dx,…,∫ Ω WT (x,t0)P n W(x,t0)dx}(12)

[0080] is a performance upper bound of the system. Therefore, controller (5) is a guaranteed cost control law for system (1).

[0081] Furthermore, according to Shur's complement lemma, we get

[0082]

[0083] and then

[0084]

[0085] Formula (13) and formula (14) are equivalent.

[0086] when Take the switching rule (11)

[0087] Step S5: realizing the furnace temperature maintenance performance control of the wide and thick plate hot processing system of the electric furnace power supply switching device.

[0088] According to step S4, there must exist i∈Ω i , so that equation (14) holds, that is, equation (13) holds, then the system (1) is asymptotically stable under the action of controller (2) and switching rule (11); controller (5) is a guaranteed cost control law of system (1), and a performance upper bound of the system is

[0089] J * =max{∫ Ω W T (x,t0)P1W(x,t0)dx,…,∫ Ω W T (x,t0)P n W(x,t0)dx}.

[0090] Furthermore, assuming there are two electric furnace power switching devices, the specific parameters are given as follows:

[0091] A1=[1 2 1; 2 1 1; 2 3 1], B1=[5 2 3; 2 4 6; 3 2 4], G1=[1 1 3; 1 1 2; 1 21],

[0092] D1 = 0.6;

[0093] A2=[2 2 1; 1 0.7 1; 3 1 2], B2=[2 5 3; 3 2 4; 4 2 4], G2=[1 2 1; 1 0.3 1; 23 1],

[0094] D2=0.8.

[0095] The positive definite weight matrix is ​​Q = [100; 010; 001]; R = [100; 010; 001];

[0096] The switching rule is (11), and the linear matrix inequality (14) is solved using the MATLAB toolbox.

[0097] The state feedback gain matrix of the guaranteed cost control of the wide and thick plate hot processing system of the electric furnace power switching device is obtained as follows:

[0098] K1=[9.587 1.400 1.232; 1.400 9.651 0.241; 1.232 0.241 12.14];

[0099] K2=[13.84 1.151 3.332; 1.151 11.96 1.455; 3.332 1.455 13.13].

[0100] Furthermore, it can be obtained that the furnace temperature of the wide and thick plate hot processing system of the electric furnace power switching device is stable and an upper bound of the performance index is J * =10.46.

[0101] The beneficial effects of the present invention are: the furnace temperature state of the wide and thick plate hot processing system of the electric furnace power switching device is used for feedback control, the engineering implementation is simple, and the problem of furnace temperature instability of the wide and thick plate hot processing control system of the electric furnace power switching device caused by switching and external nonlinear disturbance factors can be solved. The present invention fully considers the influence of the Laplace operator coefficient in the mathematical model of the partial differential equation of the wide and thick plate hot processing system of the electric furnace power switching device on the system, and such a design is more scientific and reasonable. BRIEF DESCRIPTION OF THE DRAWINGS

[0102] Figure 1 It is a principle block diagram of the present invention.

Claims

1. A wide and thick plate hot processing furnace temperature maintenance performance control based on an electric furnace power supply switching device, characterized in that: The method comprises: Step S1: Establish a partial differential equation mathematical model of the wide and thick plate hot processing system of the electric furnace power switching device and the corresponding performance indicators. The partial differential equation mathematical model is used to describe the temperature state change process of the wide and thick plate hot processing of the electric furnace power switching device; Step S2: Designing a furnace temperature state feedback controller for hot processing of wide and thick plates in an electric furnace power switching device; Step S3: constructing a closed-loop control system based on the partial differential equation mathematical model of the wide and thick plate hot processing system of the electric furnace power switching device and the furnace temperature state feedback controller; Step S4: design the parameter adjustment law and switching rule of the wide and thick plate hot processing closed-loop control system of the electric furnace power switching device, and determine the guaranteed performance control conditions of the closed-loop control system; Step S5: realizing the temperature preservation performance control of the wide and thick plate hot processing furnace of the electric furnace power switching device.

2. The method for controlling the temperature preservation performance of a wide and thick plate hot processing furnace of an electric furnace power supply switching device according to claim 1 is characterized in that: The partial differential equation mathematical model of the wide and thick plate hot processing system of the electric furnace power supply switching device constructed in step S1 is specifically: Where (x, t)∈Ω×(0,+∞), x represents the coordinate, t represents the time, Represents a smooth boundary The bounded region of (i,j∈Γ),Γ={1,2,3,…,n},W(x,t)∈R m represents temperature, U(x,t)∈R s represents the control input, is the Laplace operator, the dissipation coefficient is a constant, For a given matrix of appropriate dimensions, f i (W(x,t)) is a nonlinear perturbation, ||f i (W(x,t))||≤G i ||W(x,t)||. Its boundary conditions are: or Where N is the boundary The unit external normal vector of ; The performance indicators of the partial differential equation mathematical model of the wide and thick plate hot processing system of the electric furnace power supply switching device constructed in step S1 (i.e., the distributed parameter switching system) are specifically: Where Q and R are positive definite weight matrices.

3. The method for controlling the temperature preservation performance of a wide and thick plate hot processing furnace of an electric furnace power supply switching device according to claim 1, characterized in that: The control input of step S2 is the feedback of all temperature state variables of the wide and thick plate hot processing control system of the electric furnace power switching device, and the controller is: U(x,t)=K i W(x,t) Where K i is an adjustable control gain.

4. The method for controlling the temperature preservation performance of a wide and thick plate hot processing furnace of an electric furnace power supply switching device according to claim 1, characterized in that: The control system for hot working of thick and wide plates of the closed-loop electric furnace power supply switching device in step S3 is:

5. The method for controlling the temperature preservation performance of a wide and thick plate hot processing furnace of an electric furnace power supply switching device according to claim 1, characterized in that: The step S5 designs the parameter adjustment law and switching rule of the closed-loop control system, determines the guaranteed performance control condition of the closed-loop system, and the sufficient condition to be satisfied is: Sufficient condition 1 When there exists a matrix P i >0,K i (i=1,2,…,n), so that the matrix inequality Condition 1.1 in Condition 1.2 Switching rules Sufficient condition 2: When there exists a matrix X i >0,S i , (i=1,2,…,n), so that the linear matrix inequality Condition 2.1 Established, of which Condition 2.2 Switching rules 6. The method for controlling the furnace temperature maintenance performance of a wide and thick plate hot processing system of an electric furnace power supply switching device according to claim 1 is characterized in that: The step S4 designs the electric furnace power supply switching device wide and thick plate hot processing closed-loop control system, and under the condition of the step S5, the furnace temperature is stable and the performance index of the furnace temperature meets certain requirements, and the upper limit of the performance is: J * =max{∫ Ω W T (x,t0)P1W(x,t0)dx,…,∫ Ω W T (x,t0)P n W(x,t0)dx} Thereby realizing the furnace temperature maintenance performance control of the wide and thick plate hot processing system of the electric furnace power supply switching device.

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