An adaptive sliding mode control method based on CPL three-state boost converter

Abstract

Based on the CPL three-state Boost converter, an adaptive sliding mode control method is proposed to solve the system instability problem caused by CPL negative impedance characteristics. Firstly, the nonlinear mathematical model of the CPL three-state Boost converter is established by using the state-space averaging method, and then the linearization of the original nonlinear system is realized by using the exact feedback linearization technology. By selecting the linear sliding surface and the exponential reaching law, and introducing the adaptive mechanism to update the sliding mode switching gain in real time, an adaptive sliding mode controller is designed for the linear system. The stability and robustness of the control system are analyzed based on Lyapunov theory. Finally, the correctness and effectiveness of the proposed control strategy are verified by simulation experiments.

Description

Technical Field

[0001] This invention relates to the field of nonlinear control technology, specifically an adaptive sliding mode control method based on a CPL three-state Boost converter. Background Technology

[0002] The three-state Boost converter innovates its topology by introducing a freewheeling loop with an additional inductor current. Its core principle is to extend the discharge time while ensuring charging time, thereby effectively eliminating the system's non-minimum phase characteristics. Compared to traditional Boost converters, the improved three-state structure avoids the negative modulation phenomenon caused by non-minimum phase characteristics during duty cycle adjustment, significantly improving system stability.

[0003] However, the addition of a new mode makes system modeling increasingly difficult. When a constant power load is added to the converter, the negative impedance characteristic of the constant power load reduces the system damping coefficient, severely deteriorates the stability of the converter system, and makes the design of its controller more complex.

[0004] Therefore, in order to fully utilize the potential of the CPL three-state Boost converter and further improve its performance, this invention proposes an adaptive sliding mode control method based on differential geometry theory, and verifies the effectiveness of the proposed control method through simulation experiments. Summary of the Invention

[0005] This invention aims to overcome the shortcomings of existing technologies and provide an adaptive sliding mode control method based on a CPL three-state Boost converter.

[0006] To achieve the above objectives, the adaptive sliding mode control method based on a CPL three-state Boost converter includes the following steps: S1. Using the state-space averaging method, inductor current and output voltage are selected as state variables to establish a mathematical model of a CPL three-state Boost converter and obtain the system state-space equation. S2. Based on differential geometry theory, the nonlinear mathematical model is transformed into the Brunofsky canonical form through the precise feedback linearization method, which is equivalent to decoupling into a linear integral system. S3. Design the sliding surface and the exponential reaching law, and introduce an adaptive law to update the switching gain, and construct an adaptive sliding controller. S4. Conduct robust stability analysis under the condition of system disturbance to achieve high-precision stable control of closed-loop system.

[0007] Preferably, step S1 specifically includes: Take the inductor current of the CPL tri-state Boost converter i L and output voltagev o As a state variable, assuming the converter operates in continuous inductor current mode, based on the different on / off states of the two switches, the CPL-equipped three-state boost converter has three operating states, each with a different duty cycle. d c This refers to the duty cycle during the capacitor charging phase. d a This represents the duty cycle during the capacitor discharge phase. d f This refers to the duty cycle during the continuous flow phase.

[0008] The duty cycle in the three states satisfies the following constraints: d c + d a + d f = 1 Under these three modes, the state equations can be obtained using the KCL and KVL laws: in V in This is the power supply voltage. C For capacitors, L It's an inductor. R For load, the duty cycle of the switching transistor d c , d a 、d f ∈(0,1).

[0009] Preferably, step S2 specifically includes: Let's consider an initial state as... x ( t 0) = x For a nonlinear system with zero, its state equation can be expressed as: For the affine nonlinear system above, let the state variables be... x = ( x 1, x 2, … , x n ) T ∈ R n Control input u ∈ R Control output y ∈ R ,in f( x ), g ( x Both ) represent vector fields. When there exists x 0's neighborhood F ∈ R n This ensures that the output function satisfies the following recurrence relation: L g L f k h ( x )=0, 0≤ k ≤ r -1 L g L f r-1 h ( x )≠0 This system is said to have a relative order within the specified domain. r .

[0010] In the above formula, L f k h ( x )= ( L f k-1 h ( x )) / f ( x )= express h ( x Along the vector field f ( x )of k The first-order Lie derivative is: L f h ( x )= h ( x ) / f ( x Similarly, L g L f k h ( x )= (L f k h ( x )) / g ( x )= express L f k h ( x Along the vector field g ( x The first-order Lie derivative of ).

[0011] For a single-input single-output system, the necessary and sufficient condition for achieving exact feedback linearization is that the relative order of the system is equal to the dimension of the state space. n At this point, the following two constraints must be met: (1) In x All near 0 x ,matrix[ g ( x ) ad f g ( x ... ad f n-2 g ( x ) ad f n-1 g ( x Maintain full rank and keep the rank value constant. n ; (2) Vector field set B ={ g ( x ), ad f g ( x ),…, ad f n-2 g ( x )}, at point x The involution condition is satisfied at point 0. The Lie bracket operation between vector fields is defined as follows: ad f g ( x )= f ( x )- g ( x ).

[0012] When the above conditions are met, there exists a function or ( x ), so that at point x The relative order of the system at point 0 is equal to n At this point, coordinate transformation can be used: z = f ( x )= = have to: = In the formula, α ( x )= L f n or ( x ), β ( x )= L g L f n-1 or ( x All of them are x , a nonlinear scalar function. If we let: v = α ( x )+ β ( x ) u The original system can be precisely linearized to the Brunovsky standard form: = Az + Bv In the formula, A = , B = .

[0013] Based on this standard model, design a feedback control law: u =- + v In the formula v These are the control variables for a linear system. Based on this, we can target... v Design an effective linear controller.

[0014] To verify whether the mathematical model with the CPL three-state Boost converter satisfies the exact feedback linearization condition, the state variables are taken as follows: x =[ x 1, x 2] T =[ i L , v o ] T The mathematical model is written as an affine nonlinear mathematical model. = f ( x ) +g ( x ) d = + Among them, matrix f ( x )=[0, - x 2 / RC-P / x 2 C ] T ,matrix g ( x )=[ g 1( x ), g 2( x )] T ,and g 1( x )=[ V in / L ,0] T , g 2( x )=[ ( V in - x 2) / L , x 1 / C ] T The system output is taken as y =[ y 1( x ) , y 2( x )] T =[ x 1, x 2] T .

[0015] right y 1( x Finding the Lie derivative, we get: L g1L f 0 y 1( x )= g 1= = V in / L ≠0 L g2 L f 0 y 1( x )= g 2= =( V in - x 2) / L ≠0 It can be seen that, y 1( x relative order r 1 = 1.

[0016] right y 2( x Finding the Lie derivative, we get: L g1 L f 0 y 2( x )= g 1= =0 L g2 L f 0 y 2( x )= g 2= = x 1 / C ≠0 It can be seen that, y 2( x relative order r 2 = 1. Because r 1+ r 2=2, which equals the system dimension, indicating that the three-state Boost converter model with CPL satisfies the requirement of exact feedback linearization. Define a new matrix. β ( x )for: β ( x )= = Design control law f =[ f 1, f 2] T for: =- β -1 ( x ) + β -1 ( x ) = β -1 ( x ) + β -1 ( x ) Right now: Preferably, step S3 specifically includes: As can be seen from the above analysis, the three-state Boost converter with CPL has been completely linearized into a first-order pure integral linear system. Now, we will design an adaptive sliding mode controller for the system with the three-state Boost converter with CPL.

[0017] The error between two first-order linear subsystems is defined as: e 1= i L - i Lref ; e 2= v o - v oref ; in, i Lref and v oref These are the reference values ​​for the inductor current and the output voltage, respectively. Differentiating the above equation, we get: = ; = According to sliding mode control theory, the exponential reaching law can effectively reduce sliding mode chattering. Therefore, this invention designs the sliding surface and the exponential reaching law as follows: s 1= e1; s 2= e 2; = -c 1· sgn( s 1)- c 11 s 1; = -c 2· sgn( s 2)- c 22 s 2; In the above formula, c 1. c 2>0 represents the switching gain, which determines the rate at which the control point converges to the sliding surface. c 11 , c 22 >0 is a constant.

[0018] The control laws are designed as follows: -c 1· sgn( s 1)- c 11 s 1; = -c 2· sgn( s 2)- c 22 s 2; Since converters are often affected by uncertain interferences such as changes in circuit parameters and external disturbances during operation, it is assumed that the system has interference. F j ( t ), j =1,2. Then: = F 1( t ); = In the formula, | F j ( t |>0 is bounded, that is | F j ( t )|< Nj Let the Lyapunov function be: V 1 = 0.5 s 1 2 +0.5 s 2 2 ; Taking the first derivative of the above equation, we get: = s 1 + s 2 ; = s 1[- c 1· sgn( s 1)- c 11 s 1+ F 1( t )]+ s 2[- c 2· sgn( s 2)- c 22 s 2+ F 2( t )]; <| s 1| · F 1( t )- c 1| s 1|- c 11 s 1 2 +| s 2| · F 2( t )- c 2| s 2|- c 22 s 2 2 ; <-( c 1- N 1)| s 1|- c 11 s 1 2 -( c 2- N 2)| s 2|- c 22s 2 2 ; If <0, then the following must be satisfied: N 1< c 1. N 2< c 2 From the above equation, we can see that: switching gain c 1. c 2 depends on | F j ( t The upper bound of |) N j ,like c 1. c Choosing an excessively large value to ensure system stability may lead to excessively high speeds when the moving point approaches the sliding surface, resulting in severe chattering. Therefore, this invention introduces an adaptive law to update... c 1 and c 2.

[0019] Assumption = r 1| s 1|、 = r 2| s 2|, r 1. r 2>0; Preferably, step S4 specifically includes the following process: To verify that the closed-loop system is gradually stable, we take the Lyapunov function as: V 2= V 1+( - N 1) 2 / 2 r 1 + ( - N 2) 2 / 2 r 2; beg V The first derivative of 2 is: = s 1[- · sgn( s 1)- c 11 s 1+ F 1( t )]+ s 2[- · sgn(s 2)- c 22 s 2+ F 2( t )]+ ( - N 1) / r 1+ ( - N 2) / r 2; = s 1· F 1( t )- N 1| s 1|- c 11 s 1 2 + s 2· F 2( t )- N 2| s 2|- c 22 s 2 2 ; <-[ N 1- F 1( t )]| s 1|- c 11 s 1 2 -[ N 2- F 2( t )]| s 2|- c 22 s 2 2 ; <0; Therefore, the coefficients gradually stabilize, and the control law can be designed as follows: - · sgn( s 1)- c 11 s 1; - · sgn(s 2)- c 22 s 2; Attached Figure Description Figure 1 The topology diagram is shown below for a CPL tri-state Boost converter. Figure 2 The topology diagram of a three-state Boost converter with CPL during the capacitor charging stage; Figure 3 The topology diagram of a three-state Boost converter with CPL during the capacitor discharge stage is shown. Figure 4 The topology diagram of a CPL-equipped tri-state Boost converter in the freewheeling phase is shown. Figure 5 This is a system control block diagram with a CPL three-state Boost converter; Figure 6 This is a simulation model with a CPL three-state Boost converter; Figure 7 For loads with CPL tri-state Boost converters R Simulated waveform during the change.

[0020] Figure 8 Input voltage of a CPL three-state Boost converter V in Simulated waveform during the change. Detailed Implementation

[0021] The embodiments of the present invention will be described in further detail and clearly below with reference to the accompanying drawings and specific implementation methods.

[0022] The specific implementation scheme of the present invention to solve the above-mentioned technical problems is as follows: The first step is to use the state-space averaging method, selecting inductor current and output voltage as state variables, to establish a mathematical model of a CPL three-state Boost converter. Topology with CPL tri-state Boost converter as follows Figure 1 As shown in the figure V in Indicates the input voltage. v o For output voltage, L This is the value of the energy storage inductance. C For output capacitor, R For load resistance, Q e and Q t It forms a dual-switch structure. D and Dt These are the main diode and the freewheeling diode, respectively.

[0023] Assuming the converter operates in continuous inductor current mode, according to the switching transistor Q e and Q t With different on / off conditions, the CPL tri-state Boost converter has three operating states, and their topology diagrams are as follows: Figure 2 , Figure 3 and Figure 4 As shown. The duty cycle is different in each state, assuming... d c This refers to the duty cycle during the capacitor charging phase. d a This refers to the duty cycle during the capacitor discharge phase. d f This refers to the duty cycle during the continuous flow phase.

[0024] The duty cycle in the three states satisfies the following constraints: d c + d a + d f = 1 Under these three modes, the state equations can be obtained using the KCL and KVL laws: The second step is to transform the nonlinear mathematical model into the Brunofsky canonical form by using the precise feedback linearization method based on differential geometry theory, which is equivalent to decoupling into a linear integral system. Let's consider an initial state as... x ( t 0) = x For a nonlinear system with zero, its state equation can be expressed as: For the affine nonlinear system above, let the state variables be... x = ( x 1, x 2, … , x n ) T ∈ R n Control input u ∈ R Control output y ∈ R ,in f ( x ), g (x Both ) represent vector fields. When there exists x 0's neighborhood F ∈ R n This ensures that the output function satisfies the following recurrence relation: L g L f k h ( x )=0, 0≤ k ≤ r -1 L g L f r-1 h ( x )≠0 This system is said to have a relative order within the specified domain. r .

[0025] In the above formula, L f k h ( x )= ( L f k-1 h ( x )) / f ( x )= express h ( x Along the vector field f ( x )of k The first-order Lie derivative is: L f h ( x )= h ( x ) / f ( x Similarly, L g L f k h ( x )= ( L fk h ( x )) / g ( x )= express L f k h ( x Along the vector field g ( x The first-order Lie derivative of ).

[0026] For a single-input single-output system, the necessary and sufficient condition for achieving exact feedback linearization is that the relative order of the system is equal to the dimension of the state space. n At this point, the following two constraints must be met: (1) In x All near 0 x ,matrix[ g ( x ) ad f g ( x ... ad f n-2 g ( x ) ad f n-1 g ( x Maintain full rank and keep the rank value constant. n ; (2) Vector field set B ={ g ( x ), ad f g ( x ),…, ad f n-2 g ( x )}, at point x The involution condition is satisfied at point 0. The Lie bracket operation between vector fields is defined as follows: ad f g ( x )= f ( x )- g ( x ).

[0027] When the above conditions are met, there exists a function or ( x ), so that at point x The relative order of the system at point 0 is equal to n At this point, coordinate transformation can be used: z = f ( x )= = have to: = In the formula, α ( x )= L f n or ( x ), β ( x )= L g L f n-1 or ( x All of them are x , a nonlinear scalar function. If we let: v = α ( x )+ β ( x ) u The original system can be precisely linearized to the Brunovsky standard form: = Az + Bv In the formula, A = , B = .

[0028] Based on this standard model, design a feedback control law: u =- + v In the formula v These are the control variables for a linear system. Based on this, we can target... v Design an effective linear controller.

[0029] To verify whether the mathematical model with the CPL three-state Boost converter satisfies the exact feedback linearization condition, the state variables are taken as follows: x=[ x 1, x 2] T =[ i L , v o ] T The mathematical model can be written as an affine nonlinear mathematical model as follows: = f ( x ) +g ( x ) d = + Among them, matrix f ( x )=[0, - x 2 / RC-P / x 2 C ] T ,matrix g ( x )=[ g 1( x ), g 2( x )] T ,and g 1( x )=[ V in / L ,0] T , g 2( x )=[ ( V in - x 2) / L , x 1 / C ] T The system output is taken as y =[ y 1( x ) , y 2( x )] T =[ x 1, x 2] T .

[0030] right y 1( x Finding the Lie derivative, we get: L g1 L f 0 y 1(x )= g 1= = V in / L ≠0 L g2 L f 0 y 1( x )= g 2= =( V in - x 2) / L ≠0 It can be seen that, y 1( x relative order r 1 = 1.

[0031] right y 2( x Finding the Lie derivative, we get: L g1 L f 0 y 2( x )= g 1= =0 L g2 L f 0 y 2( x )= g 2= = x 1 / C ≠0 It can be seen that, y 2( x relative order r 2 = 1. Because r 1+ r 2=2, which equals the system dimension, indicating that the three-state Boost converter model with CPL satisfies the requirement of exact feedback linearization. Define a new matrix. β ( x )for: β ( x )= = Design control law f =[ f 1, f 2] T for: =- β -1 ( x ) + β -1 ( x ) = β -1 ( x ) + β -1 ( x ) Right now: The third step is to design the sliding surface and the exponential reaching law, and introduce an adaptive law to update the switching gain, thus constructing an adaptive sliding controller. As can be seen from the above analysis, the three-state Boost converter with CPL has been completely linearized into a first-order pure integral linear system. Now, we will design an adaptive sliding mode controller for the system with the three-state Boost converter with CPL.

[0032] The error between two first-order linear subsystems is defined as: e 1= i L - i Lref ; e 2= v o - v oref ; in, i Lref and v oref These are the reference values ​​for the inductor current and the output voltage, respectively. Differentiating the above equation, we get: = ; = According to sliding mode control theory, the exponential reaching law can effectively reduce sliding mode chattering. Therefore, this invention designs the sliding surface and the exponential reaching law as follows: s 1= e 1; s 2= e2; = -c 1· sgn( s 1)- c 11 s 1; = -c 2· sgn( s 2)- c 22 s 2; In the above formula, c 1. c 2>0 represents the switching gain, which determines the rate at which the control point converges to the sliding surface. c 11 , c 22 >0 is a constant.

[0033] The control laws are designed as follows: -c 1· sgn( s 1)- c 11 s 1; = -c 2· sgn( s 2)- c 22 s 2; Since converters are often affected by uncertain interferences such as changes in circuit parameters and external disturbances during operation, it is assumed that the system has interference. F j ( t ), j =1,2. Then: = F 1( t ); = In the formula, | F j ( t |>0 is bounded, that is | F j ( t )|< N j Let the Lyapunov function be: V 1 = 0.5 s 1 2 +0.5 s 2 2 ; Taking the first derivative of the above equation, we get: = s 1 + s 2 ; = s 1[- c 1· sgn( s 1)- c 11 s 1+ F 1( t )]+ s 2[- c 2· sgn( s 2)- c 22 s 2+ F 2( t )]; <| s 1| · F 1( t )- c 1| s 1|- c 11 s 1 2 +| s 2| · F 2( t )- c 2| s 2|- c 22 s 2 2 ; <-( c 1- N 1)| s 1|- c 11 s 1 2 -( c 2- N 2)| s 2|- c 22 s 2 2 ; If <0, then the following must be satisfied: N 1< c 1. N 2< c 2 From the above equation, we can see that: switching gain c 1. c 2 depends on | F j ( t The upper bound of |) N j ,like c 1. c Choosing an excessively large value to ensure system stability may lead to excessively high speeds when the moving point approaches the sliding surface, resulting in severe chattering. Therefore, this invention introduces an adaptive law to update... c 1 and c 2.

[0034] Assumption = r 1| s 1|、 = r 2| s 2|, r 1. r 2>0; The fourth step is to conduct robust stability analysis under the condition of system disturbance to achieve high-precision stable control of the closed-loop system.

[0035] To verify that the closed-loop system is gradually stable, we take the Lyapunov function as: V 2= V 1+( - N 1) 2 / 2 r 1 + ( - N 2) 2 / 2 r 2; beg V The first derivative of 2 is: = s 1[- · sgn( s 1)- c 11 s 1+ F 1( t )]+ s 2[- ·sgn( s 2)- c 22 s 2+ F 2( t )]+ ( - N 1) / r 1+ ( - N 2) / r 2; = s 1· F 1( t )- N 1| s 1|- c 11 s 1 2 + s 2· F 2( t )- N 2| s 2|- c 22 s 2 2 ; <-[ N 1- F 1( t )]| s 1|- c 11 s 1 2 -[ N 2- F 2( t )] | s 2|- c 22 s 2 2 ; <0; Therefore, the coefficients gradually stabilize, and the control law can be designed as follows: - · sgn( s 1)- c 11 s 1; - · sgn(s 2)- c 22 s 2; To verify the effectiveness of the control method proposed in this invention, this embodiment uses the Matlab / Simulink numerical simulation platform to construct a simulation model with a CPL three-state Boost converter for research. The specific circuit is as follows: Figure 5 As shown in the table below. The parameters of the CPL-equipped tri-state Boost converter used in this embodiment are as follows: Figure 7 Given R Simulated waveforms of the system when the Ω jumps from 53Ω to 103Ω at 0.3s and 0.35s, and when it drops back to 53Ω. R When a disturbance occurs, the output current i o Capable of rapid tracking R Transformation, while the adaptive sliding mode control strategy can enable v o Stable at 15V, which is the reference value. v oref Simulation data show that the proposed adaptive sliding mode control strategy has a strong ability to suppress load disturbances.

[0036] Figure 8 Given V in Simulated waveforms of the system when the voltage jumps from 10V to 12V at 0.15s and 0.2s, respectively, and when the voltage jumps from 12V to 10V. V in During transitions, the adaptive sliding mode control strategy can enable... v o It stabilizes at the reference value, enabling the output voltage to track the reference value quickly and accurately, demonstrating stronger robustness. In summary, this invention proposes an adaptive sliding mode control method for a CPL-equipped three-state Boost converter. First, a state-space averaging method is used to establish the system's mathematical model. Then, precise feedback linearization technology is employed to linearize the original nonlinear system. By selecting a linear sliding surface and an exponential reaching law, and introducing an adaptive mechanism to update the sliding mode switching gain in real time, an adaptive sliding mode controller is designed for the linear system. The stability and robustness of the control system are theoretically analyzed based on Lyapunov theory. Finally, simulation experiments verify that the proposed control method has advantages in resisting disturbances and improving dynamic and static performance, demonstrating good engineering application value.

[0037] The embodiments described above should be understood only as specific illustrations of this invention and are not intended to limit the specific scope of protection of this invention. After reading the description of this invention, those skilled in the art will understand that this invention can have various changes and modifications. Any changes, modifications, substitutions, combinations, simplifications, improvements, etc., made within the spirit and principles of this application should be considered equivalent substitutions and are included within the scope of protection of this invention.

Claims

1. An adaptive sliding mode control method based on a CPL three-state Boost converter, characterized in that: Includes the following steps: S1. Using the state-space averaging method, select inductor current and output voltage as state variables to establish a mathematical model of a three-state Boost converter with CPL. S2. Based on differential geometry theory, the nonlinear mathematical model is transformed into the Brunofsky canonical form through the precise feedback linearization method, which is equivalent to decoupling into a linear integral system. S3. Design the sliding surface and the exponential reaching law, and introduce an adaptive law to update the switching gain, and construct an adaptive sliding controller. S4. Conduct robust stability analysis under the condition of system disturbance to achieve high-precision stable control of closed-loop system.

2. The adaptive sliding mode control method with CPL three-state Boost converter according to claim 1, characterized in that: In step S1, the CPL three-state Boost converter operates in three modes: capacitor charging, capacitor discharging, and freewheeling, with corresponding duty cycles. d c , d a , d f satisfy: d c + d a + d f = 1. The duty cycle ranges from (0,1). Under these three modes, the state-space average equations are obtained using the KCL and KVL laws.

3. The adaptive sliding mode control method with CPL three-state Boost converter according to claim 1, characterized in that: In step S2, the mathematical model is transformed into an affine nonlinear form. The sum of the relative orders is verified to be equal to the system dimension by calculating the Lie derivative, thus satisfying the exact feedback linearization condition. The linearized matrix is ​​constructed and the control law is solved to complete the linearization and decoupling of the nonlinear system.

4. The adaptive sliding mode control method with CPL three-state Boost converter according to claim 1, characterized in that: In step S3, a sliding mode surface and an exponential reaching law are designed to effectively reduce sliding mode chattering; among them, the gain is switched. c 1. c 2>0 determines the rate at which the moving point converges to the sliding surface; at the same time, an adaptive law is introduced to update the switching gain and control the convergence rate, and an adaptive sliding mode controller is constructed based on this.

5. The adaptive sliding mode control method with CPL three-state Boost converter according to claim 1, characterized in that: In step S4, under the condition that the system has uncertain disturbances, a positive definite Lyapunov function is constructed. V ( x After differentiating with respect to time ( x If ) < 0 always holds true, according to Lyapunov's stability theorem, the converter closed-loop system satisfies global gradual stability at the equilibrium point, which can suppress the oscillation caused by the negative impedance of CPL.

6. The adaptive sliding mode control method with CPL three-state Boost converter according to claim 1, characterized in that: This adaptive sliding mode control method can be extended to various DC-DC converter topologies with CPL.

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