Non-singular terminal sliding mode control algorithm for DC-DC converter BUCK circuit at specified time

Abstract

The invention discloses a specified time nonsingular terminal sliding mode control method for a DC buck converter. The output voltage of the DC-DC buck converter is adjusted and controlled at specified time. The flow chart is shown in the figure 1 in the abstract drawing. The method comprises the following steps: firstly, obtaining a state space mathematical model of a step-down circuit under the condition of considering the uncertainty of inductance parameters; then, aiming at the mathematical model, constructing a sliding mode variable, and proposing a novel nonsingular terminal sliding mode control algorithm, thereby realizing specified time output voltage mediation control of which convergence time does not depend on a system initial value; finally, a conclusion can be obtained through Lyapunov analysis, which indicates that the closed-loop system can complete global specified time stabilization. Compared with an existing controller method, a plurality of control methods can only ensure that output errors converge to the neighborhood of an original point within specified time. According to the controller method provided by the invention, a specified time nonsingular terminal sliding mode algorithm (PSM) is designed on the premise of no singular problem and no dependence on an initial state in a system state convergence process, and it is ensured that an output voltage error can be accurately converged to zero within specified time.

Description

Technical Field

[0001] The present invention relates to the technical fields of DC-DC buck converters, nonsingular terminal sliding mode control, and finite-time stability, and particularly relates to a finite-time nonsingular terminal sliding mode controller method for a DC-DC buck converter. Background Art

[0002] Due to advantages such as high efficiency, small size, and high stability, DC-DC converters have been widely used in DC motor drives, computer systems, communication equipment, and other industrial systems. As a basic unit circuit, DC-DC converters are widely used in various power electronic devices, and their stability plays a crucial role in the application of power electronic devices in some high-tech industries. Among them, the DC-DC buck converter is one of the most important switching converters. With the continuous development of new applications, the requirements for the dynamic response speed and stability accuracy of DC-DC buck converters are getting higher and higher. Therefore, it is particularly important to select the optimal control method with the goal of achieving precise regulation of the output voltage of the DC-DC buck converter.

[0003] Linear average mathematical models are often used in the control design problems of DC-DC converters, so PID control is widely used. However, the PID controller cannot eliminate the influence of lumped disturbances composed of uncertainties and external disturbances. Therefore, the sliding mode control strategy brings new solutions to the DC-DC buck converter with lumped disturbances. Sliding mode control (SMC) is widely used in buck control due to its strong robustness and simple physical implementation. It should be noted that traditional SMC controllers may encounter two problems: one problem is the poor adjustability of the convergence time, which greatly limits the application scenarios of the algorithm; the second problem is the singularity problem caused by time-varying functions, which limits its practical application range. Nonsingular terminal sliding mode technology has been widely used to solve these two problems. However, traditional nonsingular terminal sliding mode can only achieve finite-time convergence of the system, and its convergence time is severely restricted by the initial conditions and will increase infinitely when the initial conditions tend to infinity. To solve this limitation, the phenomenon of finite-time convergence has been continuously developed. In recent years, the application of finite-time control in DC-DC converters has gradually attracted the attention of scholars, but compared with finite-time control, the relevant achievements are not many. Summary of the Invention

[0004] In view of the above series of problems, the present invention designs a finite-time nonsingular terminal sliding mode controller method for a DC-DC buck circuit system with matched disturbances. The present invention solves the possible singularity problem in the convergence process of traditional finite-time algorithms. The technical solution of the present invention is a finite-time nonsingular terminal sliding mode controller method for a DC-DC buck converter, and the specific steps are as follows:

[0005] Step 1. The DC-DC buck converter can be expressed by the average state-space equation as follows:

[0006]

[0007] Where E is the input voltage source, S represents the semiconductor switch, D is the diode, C and L0 are the nominal values of the filter capacitor and inductor, R represents the load resistor, the output voltage of the load resistor is V0, and the inductor current is i L , and α is the control signal of the PWM.

[0008] Step 2. In actual situations, there are certain errors and uncertainties in the parameters of the DC-DC converter. In this invention, the uncertainty of the inductor parameter is taken as the error. Therefore, the average state-space equation (1) is rewritten as:

[0009]

[0010] Where L = L0 + ΔL, and ΔL is the error value of the inductor. Define the error variable x1 = V s -V eq , V eq is the expected output voltage, and the error model of the DC-DC buck converter is obtained:

[0011]

[0012] Where d(t) is the matching perturbation caused by the uncertain parameters of the inductor in the system.

[0013] Step 3. Define the sliding-mode variable s = x2 + k1μ(t)x1, where The definition of the time-varying scaling function is as follows: Where T is a specified time constant designed artificially, T ≥ 0. From the definition of the time-varying scaling function μ1(t), we can get: Where m ≥ 1.

[0014] The controller design needs to meet the following assumptions:

[0015] There exists a constant D0 ≥ 0 such that |d(t)| ≤ D0.

[0016] On the premise of meeting the assumptions, the specified-time nonsingular terminal sliding-mode controller is designed as follows:

[0017]

[0018] Where D0 is given by the assumptions; where The closed-loop system satisfies global specified-time stability and no singularity problem.

[0019] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0020] 1. Many traditional control methods of DC-DC buck circuits have limited-time convergence of errors, and their convergence time is affected by initial conditions; different from other algorithms, the method proposed by the present invention can achieve specified-time convergence, and the convergence time does not depend on the initial conditions of the system, so it has a wider range of applications;

[0021] 2. The specified-time control algorithm of the existing DC-DC buck circuit may have singularity problems at specified time points. The present invention realizes the non-singularity problem during the convergence process through the design of a novel non-singular terminal sliding mode controller. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] Figure 1 is a design flow chart of a specified-time non-singular terminal sliding mode controller method for a DC-DC buck converter according to the present invention.

[0023] Figure 2 is a schematic diagram of the DC-DC buck converter system according to the present invention.

[0024] Figure 3 is a comparison diagram of simulation results of output voltage regulation of a PSM controller and a traditional SMC controller when the expected output voltage is 2V on the premise that the specified time of the design of the present invention is 3 seconds.

[0025] Figure 4 is a comparison diagram of simulation results of output voltage regulation of a PSM controller and a traditional SMC controller when the expected output voltage is 12V on the premise that the specified time of the design of the present invention is 3 seconds.

[0026] Figure 5 is a comparison diagram of simulation results of output voltage regulation of a PSM controller and a traditional SMC controller when the expected output voltage is 20V on the premise that the specified time of the design of the present invention is 3 seconds. DETAILED DESCRIPTION OF THE INVENTION

[0027] In order to make the design idea and theory of the present invention clearer, the specified-time controller designed by the present invention will be described in detail from several aspects such as establishment, design principle, and proof. The following will describe the present invention in detail with reference to the accompanying drawings and specific design methods.

[0028] As shown in the specification Figure 1 The present invention provides a design flow of a specified-time variable-gain second-order sliding mode controller method for a DC-DC buck converter.

[0029] According to the design of the present invention, for a DC-DC buck converter system with parameter errors, the controller can accurately regulate the output voltage within a specified time. The present invention introduces in detail the design steps of the controller parameters with a specified-time non-singular output voltage regulation algorithm. The technical solution of the present invention is a specified-time non-singular terminal controller method for a DC-DC buck converter, and the specific steps are as follows:

[0030] Step 1, as shown in the accompanying drawings of the specification Figure 2 is the schematic diagram of the DC-DC buck converter system. Where E is the input voltage source, S represents the semiconductor switch, D is the diode, C and L are the filter capacitor and inductor, and R represents the load resistor. The DC-DC buck converter can be expressed by the average state space equation as:

[0031]

[0032] Where the output voltage of the load resistor is V s , the inductor current is i L , and α is the control signal of PWM.

[0033] Step 2, in actual situations, there are certain errors and uncertainties in the parameters of the converter. The present invention takes the uncertainty of the inductor parameter as the error, so the state space equation (1) is rewritten as

[0034]

[0035] Where L = L0 + ΔL, and ΔL is the error value of the inductor. Define the error variable x1 = V s -V eq , V eq is the expected output voltage, and the error model of the DC-DC buck converter is obtained:

[0036]

[0037] Where d(t) is the matching disturbance caused by the uncertain parameters of the inductor in the system.

[0038] Step 3, define the sliding mode variable s = x2 + k1μ(t)x1, where The definition of the time-varying scaling function is as follows: Where T is the specified time constant designed artificially, T≥0, and from the definition of the time-varying scaling function μ1(t), it can be obtained that: Where m≥1.

[0039] The controller design needs to meet the following assumptions:

[0040] There exists a constant D0≥0, such that |d(t)|≤D0.

[0041] On the premise of meeting the assumed conditions, the specified-time nonsingular terminal sliding mode controller is designed as follows:

[0042]

[0043] where \(D_0\) is given by the assumed conditions; where

[0044] Step 4: Conduct strict proof and derivation for the designed controller of the present invention.

[0045] Before the proof, it is necessary to explain the theoretical lemmas that will be used.

[0046] Lemma 1 If is a real variable, \(c, d>0\) are positive constants, then for any given function \(\gamma>0\) and there is

[0047]

[0048] (Qian C, Lin w. A continuous feedback approach to global strong stabilization of nonlinear systems[J]. IEEE Transactions on Automatic Control, 2001, 46(7): 1061 - 1079.)

[0049] Next, the relevant theory on specified-time stability is given. Consider the following system:

[0050]

[0051] There is \(f(x(t))\): is a nonlinear Lebesgue integrable function and satisfies \(f(0)=0\). The initial state of the system is:

[0052] Definition 1 For If the origin of system (5) is globally asymptotically stable, and any solution \(x(t, x_0)\) of system (5) reaches the equilibrium point at a certain time \(T\), that is, \(T\): Then the origin of the system (5) is said to be globally finite-time stable. (Shi s, Dai L, Min H, et al. Non-singular terminal sliding mode controller design for nonlinear systems with prescribed convergence time guarantees[J]. International Journal of Robust and Nonlinear Control, 2024, 34(4): 2597-2613.)

[0053] Definition 2 For If the nonlinear system (2.4.5) is globally finite-time stable and the convergence time T can be specified artificially, then this system is said to be globally specified-time stable.

[0054] (Shi s, Dai L, Min H, et al. Non-singular terminal sliding mode controller design for nonlinear systems with prescribed convergence time guarantees[J]. International Journal of Robust and Nonlinear Control, 2024, 34(4): 2597-2613.)

[0055] Lemma 2 Consider the system (5). If there exists a positive definite Lyapunov function V(x(t)) that is radially unbounded and satisfies:

[0056]

[0057] where c > 0, n ≥ 0, and ω(t) satisfies:

[0058]

[0059] where δ ≥ 0, T is an artificially designed time constant, then the system (5) is specified-time stable. And when t ≥ 0, x(t) is bounded.

[0060] (Shi s, DaiL, Min H, et al. Non-singular terminal sliding mode controller design for nonlinear systems with prescribed convergence time guarantees[J]. International Journal of Robust and Nonlinear Control, 2024, 34(4): 2597-2613.)

[0061] Proof: The proof method can be divided into three parts.

[0062] First step: First, prove that the sliding mode surface s converges in the specified time. Take the positive definite Lyapunov function The derivative of V1(s(t)) can be calculated as

[0063]

[0064] According to Substituting into (8), we can get:

[0065]

[0066] According to Lemma 2, it can be obtained that the sliding mode surface s converges in the specified time, and s is bounded when t≥0.

[0067] Solving (9), we get:

[0068]

[0069] Then the proof of the convergence of the sliding mode surface s in the specified time in the first step is completed.

[0070] Second step: Prove that the error signal x1 converges in the specified time T. From the error variable x1 = V s -V eq and the definitions of the sliding mode variable s = x2 + k1μ(t)x1:[[]]

[0071]

[0072] Take the positive definite Lyapunov function The derivative of V2(x1(t)) can be calculated as:

[0073]

[0074] Since μ(t)≥1, applying Lemma 2, at this time taking c = d = 1, γ = (1 - λ)k1, and substituting into (11) we get:

[0075]

[0076] Substituting into (14), we get:

[0077]

[0078] Applying Lemma 2.1, then \(x_1\) converges at the specified time \(T\), and when \(t\geq0\), \(x_1\) is bounded. Thus, the proof of the convergence of the error signal \(x_1\) at the specified time \(T\) in the second step is completed.

[0079] Step 3: Prove the boundedness of the signals in the entire algorithm convergence process. From the first and second steps, it is obtained that the sliding mode surface \(s\) and the state variable \(x_1\) are bounded. Therefore, it is only necessary to prove the boundedness of the controller \(u\) and the state variable \(x_2\). Since the singularity problem in the entire convergence process is caused by the time-varying scaling function \(\mu(t)\), when \(t\geq T\), \(\mu(t)=1\). So, it is only necessary to further discuss the boundedness of the controller \(u\) and the state variable \(x_2\) for \(0\leq t\lt T\).

[0080] When \(0\leq t\lt T\), is bounded, that is, there exists Substituting into (16), we get:

[0081]

[0082] By solving (17), we get:

[0083]

[0084] where Because By solving through (18), we get:

[0085]

[0086] It has been defined in (3) and (4) From this, we can obtain: \(4m - 2\lambda k_1T\leq0\), \(4m - 2k_2T - 1\leq0\), which indicates that: \(\mu\) 2 (t)x1(t) is bounded, that is:

[0087]

[0088] Obviously, \(2m - 2\lambda k_1T\leq0\) and \(2m - 2k_2T - 1\leq0\) hold; so \(\mu(t)x_1(t)\) is bounded. And

[0089] x2 = s - k1μ(t)x1 (22)

[0090] Then \(x_2\) is bounded. Similarly

[0091]

[0092] From the definition of the controller \(u\) in (3), all variables in the controller \(u\) are bounded for \(0\leq t < T_0\). Therefore, the controller \(u\) is bounded for \(0\leq t < T_0\). Thus, the boundedness of the entire algorithm convergence process in the third step is proven. Thus, the proof is complete.

[0093] Embodiment

[0094] To verify the effectiveness of the proposed prescribed-time nonsingular terminal sliding mode control algorithm (PSM) of the present invention, the present invention uses a DC-DC buck converter system for simulation experiments. The DC-DC buck converter system equation description is the same as formula (2). Among them, the converter parameters are selected according to Table 1 in the specification, and the controller parameters are according to Table 2 in the specification.

[0095] Table 1 DC-DC Buck converter parameter settings

[0096] Tab.1 Parameter settings of DC-DC Buck converter

[0097]

[0098] Table 2 Prescribed-time nonsingular terminal sliding mode controller constant parameter settings

[0099] Tab.2 Parameter configuration of constant terms in prescribed-time nonsingular terminal sliding mode controllers

[0100]

[0101] According to the various parameters provided in Table 1 and Table 2, the specific expression of the controller can be obtained as

[0102]

[0103] First, set the specified time T to 3 seconds. In this paper, the designed controller (PSM) is compared with the traditional sliding mode controller (SMC) under different desired output voltages, and the stability curves obtained by the three controller methods are compared. The present invention compares the control performances between the proposed specified-time non-singular terminal sliding mode controller and the traditional sliding mode controller (SMC). After the state is started, observe the transient performance of the system output voltage from the start time t = 0 s to the specified time T, which stabilizes from 0 V to the tracking reference voltage under different desired output voltage conditions, and the steady-state performance of the system after the specified time T, so as to judge the control performance of the controller. The simulation results are as shown in the appendix of the specification Figure 3 , Figure 4 , Figure 5 As shown, for the specified-time non-singular terminal sliding mode algorithm (PSM) proposed by the present invention, under different desired output voltage conditions, the system output voltage can reach the desired output voltage and stabilize before the specified time T. However, for the traditional sliding mode algorithm (SMC), when the desired output voltage is 2 V, the system output voltage can basically reach the desired output voltage at the specified time. But as the desired output voltage increases, the convergence time of the traditional sliding mode algorithm increases significantly, and the system output voltage cannot reach the desired output voltage within the specified time T.

[0104] It should be understood that the above are only the general steps of the present invention and are not used to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A specified-time nonsingular terminal sliding mode controller for a DC-DC buck converter, characterized in that The method includes the following steps: S1. Establish the average state - space equation of the DC - DC buck converter; S2. Introduce the uncertainty parameter of the inductor parameter, and determine the new state - space equation and the error state - space equation; S3. Construct the sliding - mode variable, design the non - singular terminal sliding - mode control algorithm, so as to realize the output voltage regulation control at the specified time.

2. The specified-time nonsingular terminal sliding mode controller for a DC-DC buck converter according to claim 1, characterized in that, The specific content of step S1 is as follows: The DC - DC buck converter can be expressed by the average state - space equation as: Where E is the input voltage source, T represents a semiconductor switch, D is a diode, C and L0 are the nominal values of the filter capacitor and inductor, R represents the load resistor, and the output voltage of the load resistor is V S , the inductor current is i L , and α is the control signal of PWM.

3. The specified-time nonsingular terminal sliding mode controller for a DC-DC buck converter according to claim 1, wherein The specific content of step S2 is as follows: In the actual situation, there is a certain error in the inductor parameter of the DC - DC buck converter, which has uncertainty. In the present invention, the uncertainty parameter of the inductor parameter is taken as the error, so the average state - space equation (1) is rewritten as: where \(L = L_0+\Delta L\), and \(\Delta L\) is the error value of the inductor. Define the error variable \(x_1 = V\) s -V eq , \(V\) eq is the desired output voltage, and the error model of the DC-DC buck converter is obtained as follows: where d(t) is the matching disturbance caused by the uncertain parameters of the inductor in the system.

4. The specified-time nonsingular terminal sliding mode controller of a DC-DC buck converter according to claim 1, characterized in that The specific content of step S3 is as follows: Define the sliding mode variable \(s = x_2 + k_1\mu(t)x_1\). The definition of the time-varying scaling function is as follows: where \(T\) is a specified time constant pre-designed artificially, \(T\geq0\). From the definition of the time-varying scaling function \(\mu_1(t)\), we can obtain: where \(m\geq1\). The controller design needs to meet the following assumption conditions: There exists a constant D0≥0 such that |d(t)|≤D0. On the premise of meeting the assumption conditions, the non - singular terminal sliding - mode controller at the specified time is designed as follows: where D0 is given by the hypothesis condition; where The closed - loop system satisfies global specified - time stability and no singularity problem.

Images