A sliding mode control method for boost converter

Abstract

The application discloses a sliding mode control method suitable for a Boost converter, which comprises the following steps: introducing a nonlinear limited function to construct a mathematical model of the Boost converter, combining the sliding mode control to design a controller meeting full state constraints for the constructed Boost converter model system, and modulating a pulse for controlling on-off of a switch tube in the Boost converter by using the designed controller. The application converts a state variable (input current / output voltage) constraint problem of the Boost converter into a boundedness problem of the nonlinear limited function, and the controller designed in combination with the sliding mode control meets the full state constraints, so that the input current and output voltage constraint problems of the system are solved. Moreover, the sliding mode control method suitable for the Boost converter can realize fast response for resistive load and constant power load (CPL), and ensures global stability and robustness of the system.

Description

Technical Field

[0001] This invention belongs to the field of power electronics technology, and specifically relates to a sliding mode control method suitable for Boost converters. Background Technology

[0002] In recent years, DC microgrids, as small-scale power grids that can integrate DC renewable resources, energy storage units, and DC loads, have attracted considerable attention. Among them, DC-DC converters, with their low power consumption, small size, high precision, and wide application range, are widely used in various energy conversion scenarios. For example, DC-DC converters are often connected to batteries, supercapacitors, motors, and photovoltaic panels to achieve energy transmission and consumption. On the one hand, as more and more electrical loads are converted to power electronic converter-powered loads and servo motor drive systems, their constant power load characteristics result in negative input impedance for load increments, which easily weakens system damping, causes source-load interaction, and induces severe oscillation problems. On the other hand, with a high proportion of new energy sources, new energy storage, and new electrical loads being connected to DC microgrids, the characteristics of diversified system entities, complex grid configurations, and diverse operating modes are becoming increasingly apparent. This places more constraints on DC-DC conversion circuits, including electrical constraints, physical constraints, and subjective requirements of designers. These constraints are often nonlinear. Therefore, linear control strategies based on steady-state operating points are difficult to guarantee the global stability of the system. There is an urgent need for a nonlinear control method suitable for DC-DC converters to achieve global stability and state constraints of the system.

[0003] Currently, in order to solve the above problems, many academic papers and patents have been published and proposed corresponding solutions, such as:

[0004] 1. In their patent application No. 202211185777.0, entitled "A Sliding Mode Control Output System for a DC Microgrid Based on a Constant Power Load," Ma Wenliang et al. achieved stable voltage regulation control of a DC microgrid, greatly reducing errors in the voltage regulation modeling process. However, this method can only achieve stable voltage regulation control of the DC microgrid and cannot guarantee state constraints.

[0005] 2. In the article entitled "Research on PI-Model Predictive Control of Quasi-Z Source Bidirectional Full-Bridge DC-DC Converter", the author Qiao Lingyu proposed a control algorithm that combines PI and current MPC, and used the particle swarm optimization (PSO) algorithm to optimize the objective function of the MPC controller to achieve output stability. However, this method can only guarantee local stability and cannot guarantee global stability.

[0006] 3. Karanakos P et al. published "Direct Voltage Control of DC–DC Boost Converters Using Enumeration-Based Model Predictive Control" in IEEE Transactions on Power Electronics, 2014, 29(2):968-978. This article derives a discrete-time switching nonlinear (hybrid) model for Boost converters. This model can simultaneously capture continuous and discontinuous conduction modes, achieving optimal control by minimizing the objective function under the dynamic constraints of the model. However, in general, this method uses model predictive control, which has drawbacks such as high computational cost and sensitivity to model parameters. Summary of the Invention

[0007] In view of this, the purpose of this invention is to provide a sliding mode control method suitable for Boost converters to solve the technical problems of strong nonlinearity of DC-DC systems and the negative damping characteristics of constant power loads (CPL) in DC microgrids, which easily induce severe system oscillations, as well as the difficulty of existing control algorithms in achieving global system stability and state constraints.

[0008] The sliding mode control method of the present invention applicable to Boost converters includes the following steps:

[0009] 1) Define the following open interval:

[0010] D1:={x1∈R|I min <x1<I max}

[0011] D2:={x2∈R|V min <x2<V max}

[0012] Among them I min and I max The inductor current i in the Boost converter is respectively L The minimum and maximum allowable values, V max and V min The output voltage v in the Boost converter is respectively o The maximum and minimum allowable values, where x1 and x2 are constrained state variables;

[0013] Introduce the state constraint transition vector η = [η1, η2] T Establish a mathematical model for the Boost converter, where η1 is the constrained transformation variable corresponding to x1, η2 is the constrained transformation variable corresponding to x2, and:

[0014]

[0015] The established mathematical model of the Boost converter is as follows:

[0016]

[0017] Where v is the input voltage of the Boost converter, i o Let C be the output current of the Boost converter, C be the capacitor in the Boost converter, L be the inductor in the Boost converter, and u be the duty cycle of the switching transistor in the Boost converter. and These are the reconstructed state variables;

[0018] 2) The designed duty cycle u is:

[0019]

[0020] Where, m1=LI1(1-tanh 2 η1), I1=I max -I min ,

[0021] m2=CV1(1-tanh 2 η2), V1=V max -V min ,

[0022] α1=-ks-εsgn(s), where s is the nonlinear sliding surface, s=e1+ce2, c is the design constant, e1 and e2 are error variables, k is the design constant, and ε is the approach rate;

[0023] x2 * It is the reference value for the constrained state variable x2. is i o The second derivative, It is the first derivative of v;

[0024] α3=2vm2-2cm1i o ;

[0025] α4=-m2(V1 tanhη2+V2)+cm1(I1 tanhη1+I2), V2=V max +V min ;

[0026] 3) The pulses controlling the switching of the transistors in the Boost converter are modulated according to the duty cycle u.

[0027] The beneficial effects of this invention are:

[0028] This invention relates to a sliding mode control method for Boost converters. It models the Boost converter by designing nonlinear restricted functions, transforming the state variable (input current / output voltage) constraint problem of the Boost converter into a boundedness problem of nonlinear restricted functions. Combined with sliding mode control, a controller that satisfies all state constraints is designed, solving the input current and output voltage constraint problems of the system. Furthermore, the sliding mode control method of this invention for Boost converters can achieve fast response for resistive loads and constant power loads (CPL), ensuring the global stability and robustness of the system. Attached Figure Description

[0029] Figure 1 This is a flowchart of a sliding mode control method applicable to Boost converters. The figure shows the structure of a common Boost converter, which includes an inductor L, a switching transistor S, a diode T, and a capacitor C.

[0030] Figure 2 To connect a resistive load R to the Boost converter port, the resistance values ​​R and output voltage V are varied sequentially at 243Ω, 48Ω, 26Ω, 48Ω, and 243Ω with an interval of 0.05s. o Input current i L Waveform diagram.

[0031] Figure 3 To enable CPL at the Boost converter port, the input voltage vo and output voltage vo are measured when the input voltage changes sequentially at intervals of 0.05s: 120V, 140V, 160V, 140V, 120V. o Input current i L Waveform diagram.

[0032] Figure 4 To enable a step change CPL at the Boost converter port, the input rated voltage is v = 120V, and the total load power is P. CPL The total load power P varies sequentially with power ratings of 300W, 1200W, 2100W, 3000W, 2100W, 1200W, and 300W, at 0.05s intervals. CPL Output voltage v o and input current i L Waveform diagram.

[0033] Figure 5 To enable the Boost converter port to have CPL and power P CPL Light load and full load switching, total load power P CPL The total load power P varies sequentially from 300W to 3000W and then back to 300W, with intervals of 0.05s. CPL Output voltage vo and input current i L Waveform diagram.

[0034] Figure 6 To bring random CPLP to the Boost converter port CPL Total load power P at that time CPL Output voltage v o and input current i L Waveform diagram. Detailed Implementation

[0035] The present invention will be further described below with reference to the accompanying drawings and embodiments, but the present invention is not limited to the embodiments.

[0036] The sliding mode control method applicable to Boost converters in this embodiment includes the following steps:

[0037] 1) Define the following open interval:

[0038] D1:={x1∈R|I min <x1<I max}

[0039] D2:={x2∈R|V min <x2<V max}

[0040] Among them I min and I max The inductor current i in the Boost converter is respectively L The minimum and maximum allowable values, V max and V min The output voltage v in the Boost converter is respectively o The maximum and minimum allowable values ​​are given, and x1 and x2 are constrained state variables.

[0041] Introduce the state constraint transition vector η = [η1, η2] T Establish a mathematical model for the Boost converter, where η1 is the constrained transformation variable corresponding to x1 and η2 is the constrained transformation variable corresponding to x2.

[0042] Based on:

[0043]

[0044] And from the above interval expression, we get:

[0045]

[0046] The expressions for η1 and η2 are nonlinear restricted functions. As can be seen from the above equation, the constrained state variables x1 and x2 have been transformed into control problems of variables η1 and η2, respectively.

[0047] Let the state constraint transition vector be η = [η1, η2]. T The reference vector is η ref =[η1 * ,η2 * ] T The tracking error vector is e = η - η ref =[e1,e2] T Combining the expressions for η1 and η2 with the following formula:

[0048]

[0049] Obtain reference values ​​for variables η1 and η2:

[0050]

[0051] Where x2 * It is the reference value of the constrained state variable x2, i om1 Output current i o The measured value, v m1 The measured value of the input voltage v.

[0052] Then, according to the formula:

[0053]

[0054] By combining the expressions for η1 and η2, we obtain the mathematical model of the Boost converter:

[0055]

[0056] Where v is the input voltage of the Boost converter, i o Let C be the output current of the Boost converter, C be the capacitor in the Boost converter, L be the inductor in the Boost converter, and u be the duty cycle of the switching transistor in the Boost converter. and These are the reconstructed state variables. Clearly, the control objective of the constructed Boost converter mathematical model is transformed into: designing a controller to ensure that η can track η. ref And achieve asymptotic stability of the system.

[0057] 2) Design the controller, i.e., the duty cycle u.

[0058] Step S1: Select the sliding surface:

[0059] Since the entire state of the system is constrained and the controlled state variables are measurable, the selection of the sliding surface is based on the premise that the error between the predetermined input and the measured input is zero. A linear sliding surface is designed, and the nonlinear sliding surface design for the state-space model of the Boost system is as follows:

[0060] s = e1 + ce2, c > 0

[0061] Where s is the sliding surface, c is the design constant, and e1 and e2 are error variables.

[0062] Because x2 * It is a preset reference value, a constant positive number, and the corresponding restricted state variable η2 * Also a constant positive constant, η² * The derivative is 0, so differentiating the nonlinear sliding surface design formula yields:

[0063]

[0064] Step S2: To ensure system stability, the system will run according to a preset trajectory. The Lyapunov function is constructed as follows:

[0065]

[0066] Differentiating with respect to V, we get:

[0067]

[0068] Step S3: Let the tendency law of s be variable α1:

[0069] α1=-ks-εsgn(s),k>0

[0070] Where k is a positive constant of the design and ε is the approach rate.

[0071] Step S4: To ensure that the Lyapunov function after differentiation in step S2 is negative positive definite, assume:

[0072]

[0073] Combining the mathematical model of the Boost converter, steps S1-S3, and the above equation, we obtain:

[0074]

[0075] Where, m1=LI1(1-tanh 2 η1), I1=I max -I min ,

[0076] m2=CV1(1-tanh 2η2), V1=V max -V min ,

[0077] α1=-ks-εsgn(s), where s is the nonlinear sliding surface, s=e1+ce2, c is the design constant, e1 and e2 are error variables, k is the design constant, and ε is the approach rate;

[0078] x2 * It is the reference value for the constrained state variable x2. is i o The second derivative, It is the first derivative of v;

[0079] α3=2vm2-2cm1i o ;

[0080] α4=-m2(V1 tanhη2+V2)+cm1(I1tanhη1+I2), V2=V max +V min .

[0081] 3) The pulses controlling the switching of the Boost converter are modulated according to the designed duty cycle u.

[0082] The feasibility and stability of the sliding mode control method applicable to Boost converters described above are demonstrated below:

[0083] Substituting the formulas from steps S3 and S4 into the Lyapunov function obtained after differentiation in step S2, we get:

[0084]

[0085] According to Lyapunov's stability theorem, combined with the above equation, we can conclude that the Boost converter closed-loop system is asymptotically stable, the sliding surface s can run along the expected trajectory s→0, and the state point reaches the sliding surface. It is also necessary to prove that the system state vector can track the system reference value, i.e., e = 0. We can prove by contradiction that the error vector e = 0.

[0086] When s = 0, the sliding surface design formula in step S1 is:

[0087] s = e1 + ce2 = 0

[0088] Assuming e = [e1, e2] ≠ 0, then from the above equation, we can obtain:

[0089] e1 = -ce2

[0090] Since η1 and η2 are monotonically increasing functions of x1 and x2 respectively, therefore This makes the following equation true:

[0091]

[0092] When the system is stable, the following relationship holds:

[0093]

[0094] Simplifying the above equation, we get:

[0095]

[0096] Since io > 0, v > 0, and -μ < 0, the above equation does not hold, and the assumption also does not hold, that is:

[0097] e = [e1, e2] = 0

[0098] As can be seen from the above formula, the designed controller can achieve a tracking error of 0.

[0099] Sliding mode approach velocity analysis:

[0100] In the formula described in step S3, the reaching law of the sliding mode controller is an exponential reaching term (i.e., ) and the isodynamic approaching term (i.e. The combination of ) gives the following formula:

[0101]

[0102] Assumption:

[0103]

[0104] Combining the above two equations, we can obtain w(t)≤0, and:

[0105]

[0106] Solving the differential equation yields:

[0107]

[0108] Where t is the time variable and τ is the time constant, since w(t)≤0. but:

[0109]

[0110] From the above equation, we can see that as s deviates further from the sliding surface, V(t) will converge exponentially to 0, and its convergence rate depends on k. When s is infinitely close to the sliding surface, V(t) approaches the sliding surface at a rate ε, ensuring that it arrives in a finite time.

[0111] η(x) has the following characteristics: the absolute value of the slope of the state variable, |Δη(x) / Δx|, is proportional to its distance from the central axis, |x|. This means that the adjustment is more precise when the state variable is close to the reference value, while the error is amplified and the adjustment is accelerated when the state variable is far from the reference value. In particular, when the state variable is close to the constraint boundary, the error becomes infinitely large, and the excitation control quantity is adjusted. From the above analysis, it can be seen that the sliding mode control strategy based on nonlinear restricted functions has stronger robustness and dynamic response performance.

[0112] The sliding mode control method for Boost converters described in this embodiment is then verified through simulation:

[0113] Simulation Verification 1: Boost converter under ideal, perturbation-free conditions, set... Figure 1 The DC microgrid system with a Boost converter shown has a full-load power of 3000W and a power margin of 3.8% (maximum power is 3120W). The initial state is iL(0) = 0.001A, v o (0) = 270V, which satisfies the initial conditions for restricted control. Output voltage reference value v oref The rated input voltage is 120V, and the input voltage range is 270V. a 120-160V, switching frequency f s The frequency is 20kHz, the inductance L is 1mH, the capacitance C is 470μF, and the adaptive sliding mode coefficients 1, 2, and 3 (c,k,ε) are 5, 1×10⁻⁶ ... 4 0.1, Input current lower limit and upper limit threshold I min ,I max 0A, 26A, lower and upper threshold voltages V respectively min V max They are 255V and 285V respectively. (See diagram i) om i Lm v m and v om They are i o i L v and v o The measured value. Figure 2 To connect a resistive load R to the Boost converter port, the resistance values ​​R and output voltage V are varied sequentially at 243Ω, 48Ω, 26Ω, 48Ω, and 243Ω with an interval of 0.05s. o Input current i L Waveform diagram.

[0114] from Figure 2 The output voltage v can be seen in the image. o and input current i L Dynamic adjustment t sThe time is about 5 ms, the maximum surge voltage is 272.5 V, and the overshoot σ = 0.926%; the minimum instantaneous voltage drop is 267.8 V, and the drop rate is 0.815%. The output voltage v o and the input current i L are both maintained within the preset range, quickly track the reference value, have a fast dynamic response, and good regulation effect.

[0115] Simulation verification 2: The difference from simulation verification 1 is that the Boost converter port with a step-changing resistive load R is changed to a port with a CPL (P CPL = 2000 W), and the change in the resistance value is changed to the input voltage changing in sequence as 120 V, 140 V, 160 V, 140 V, 120 V at an interval of 0.05 s. Figure 3 For the Boost converter port with a CPL, when the input voltage changes in sequence as 120 V, 140 V, 160 V, 140 V, 120 V at an interval of 0.05 s, the input voltage v, the output voltage v o , and the input current i L waveform diagram.

[0116] From Figure 3 it can be seen that when the port has a CPL, the controller can achieve a good regulation effect throughout the process. v o and i L The dynamic regulation time t s is extremely small. The maximum surge voltage is 270.55 V, and the overshoot σ = 0.2037%; the minimum instantaneous voltage drop is 269.37 V, and the drop rate is 23.33%. v o and i L are not only maintained within the preset range, but also have extremely small ripples, a short dynamic process time, a fast response, and a good regulation effect, indicating that the system can achieve fast and reliable regulation for different input voltage levels.

[0117] Simulation verification 3: The difference from simulation verification 1 is that the Boost converter port with a step-changing resistive load R is changed to a port with a step-changing CPL, the input rated voltage v = 120 V, and the total load power P CPL changes in sequence as 300 W, 1200 W, 2100 W, 3000 W, 2'100 W, 1200 W, 300 W at an interval of 0.05 s. Figure 4 For the Boost converter port with a step-changing CPL, when the input rated voltage v = 120 V and the total load power P CPL changes in sequence as 300 W, 1200 W, 2100 W, 3000 W, 2100 W, 1200 W, 300 W at an interval of 0.05 s, the total load power P CPL , the output voltage v<00,00113>and the input current i L waveform diagram.

[0118] From Figure 4 It can be seen that the adjustment effect of the whole process is good, and the output voltage v o operates within the range of [268.2V, 271.93V], the overshoot σ is 0.715%, the maximum drop rate is 0.667%, and the input current i L operates within the range of [1.72A, 25.76A], and the output voltage v o and the input current i L always operate within the predetermined range. The system responds quickly, the dynamic process time is short, and the longest adjustment time is when the CPL jumps from 2100W to 3000W step by step, and the longest adjustment time is t s = 6ms. [[ID=**15**]]<00004**31**>[[ID=**16**]]<00004**32**>[[ID=**17**]]Simulation verification four: The difference from simulation verification one is that the Boost converter port with a step-changing resistive load R is changed to a port with a CPL and the power P[[ID=**18**]] CPL [[ID=**19**]]switches between light load and full load, and the total load power P[[ID=**20**]] CPL [[ID=**21**]]changes sequentially according to 300W, 3000W, 300W, at an interval of 0.05s. [[ID=**22**]]<00004**33**>[[ID=**23**]]For the Boost converter port with a CPL and the power P[[ID=**24**]] CPL [[ID=**25**]]switches between light load and full load, and the total load power P[[ID=**26**]] CPL [[ID=**27**]]when changing sequentially according to 300W, 3000W, 300W, at an interval of 0.05s, the total load power P[[ID=**28**]] CPL [[ID=**29**]]、output voltage v[[ID=**30**]] o [[ID=**31**]]and input current i[[ID=**32**]] L [[ID=**33**]]waveform diagram. [[ID=**34**]]<00004**34**>[[ID=**35**]]<00004**35**>[[ID=**36**]]From[[ID=**37**]]<00004**36**>[[ID=**38**]]It can be seen that the output voltage v[[ID=**39**]] o [[ID=**40**]]operates within the range of [269.4, 271.7], the maximum overshoot σ is 0.63%, the maximum drop rate is 0.22%, and the input current i[[ID=**41**]] L [[ID=**42**]]operates within the range of [2.36, 25.35], the longest adjustment time is t[[ID=**43**]] s [[ID=**44**]]= 8ms, v[[ID=**45**]] o [[ID=**46**]]and i[[ID=**47**]] L [[ID=**48**]]always operate within the predetermined range, adjust quickly, and the control effect is good. [[ID=**49**]]<00004**37**>[[ID=**50**]]<00004**38**>[[ID=**51**]]Simulation verification five: The difference from simulation verification one is that the Boost converter port with a step-changing resistive load R is changed to a port with a random CPLP[[ID=**52**]] CPL [[ID=**53**]], and let the CPLP[[ID=**54**]] CPL [[ID=**55**]]randomly change within [1728W, 2310W]. [[ID=**56**]]<00004**39**>[[ID=**57**]]For the total load power P when the Boost converter port has a random CPLP[[ID=**58**]] CPL [[ID=**59**]] [[ID=**60**]] CPL, the output voltage v o and the input current i L waveform diagram. <

[0122] From Figure 6 it can be seen that the output voltage v o can be stabilized at 270V with a small ripple, and the ripple voltage is controlled within the range [-0.1V, +0.1V]. The input current i L varies continuously within the range [14.3A, .5A] with the random CPLP CPL changing. Both the current and voltage operate within the predetermined range, with a fast response speed and high stability.

[0123] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to be limiting. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced, but any technical solutions that do not depart from the spirit and scope of the present invention should be covered within the scope of the claims of the present invention.

Claims

1. A sliding mode control method suitable for Boost converters, comprising the following steps: 1) Define the following open interval: D1:={x1∈R|I min <x1<I max } D2:={x2∈R|V min <x2<V max } Among them I min and I max The inductor current i in the Boost converter is respectively L The minimum and maximum allowable values, V max and V min The output voltage v in the Boost converter is respectively o The maximum and minimum allowable values, where x1 and x2 are constrained state variables; Introduce the state constraint transition vector η = [η1, η2] T Establish a mathematical model for the Boost converter, where η1 is the constrained transformation variable corresponding to x1, η2 is the constrained transformation variable corresponding to x2, and: The established mathematical model of the Boost converter is as follows: in, v is the input voltage of the Boost converter, i o Let C be the output current of the Boost converter, C be the capacitor in the Boost converter, L be the inductor in the Boost converter, and u be the duty cycle of the switching transistor in the Boost converter. and These are the reconstructed state variables; 2) The designed duty cycle u is: Where, m1=LI1(1-tanh 2 η1), I1=I max -I min , m2=CV1(1-tanh 2 η2),V1=V max -V min , α1=-ks-εsgn(s), where s is the nonlinear sliding surface, s=e1+ce2, c is the design constant, e1 and e2 are error variables, k is the design constant, and ε is the approach rate; x2 * It is the reference value for the constrained state variable x2. is i o The second derivative, It is the first derivative of v; α3=2vm2-2cm1i o ; α4=-m2(V1 tanhη2+V2)+cm1(I1 tanhη1+I2),V2=V max +V min ; 3) The pulses controlling the switching of the transistors in the Boost converter are modulated according to the duty cycle u.

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