A control method for a nonlinear induction heating circuit system based on output feedback

Abstract

The present invention discloses a control method for a nonlinear induction heating circuit system based on output feedback. For the nonlinear induction heating circuit system, a low-gain output feedback controller is designed based on an adaptive output feedback method, taking into account the situation that some system information is unknown, there are disturbances in the system sensor, and there is input lag, so as to achieve the asymptotic stability of the system. First, a mathematical model of the system is established; then a suitable state is selected to transform it into a state space expression with measurement uncertainty and input lag, and then the negative influence of the system measurement uncertainty is avoided by an improved backstepping method to obtain an output feedback controller, and then the influence of the system input lag is processed by the selected Lyapunov‑Krasovskii functional. This method takes into account the influence of measurement uncertainty and input lag at the same time, and is more in line with actual engineering conditions.

Description

Technical Field

[0001] The present invention belongs to the technical field of control theory and control engineering, and relates to an adaptive control method for a nonlinear system with measurement uncertainty. Specifically, it relates to a control method for a nonlinear induction heating circuit system based on output feedback. Background Art

[0002] Linear control theory is the most mature and fundamental component branch in system and control theory, and is the cornerstone of modern control theory. In the past few decades, scholars have developed a relatively complete set of control theories for linear systems. However, in the engineering field, purely linear systems hardly exist, and most actual systems contain some nonlinear parts, such as dead zones, hysteresis, saturation, etc. Traditional control methods often ignore these nonlinear links or approximate the nonlinear parts linearly. But as systems in engineering become more and more complex and the requirements for control effects become higher and higher, many systems are no longer suitable for being modeled as linear systems for processing. At this time, more and more scholars begin to study nonlinear systems. However, due to the more diverse and complex structures of nonlinear systems, a general nonlinear control theory cannot be formed, and only different control schemes can be proposed for nonlinear systems with different structures. Therefore, studying the control methods of nonlinear systems is a very meaningful research topic both theoretically and in practical applications.

[0003] As is well known, there are many actual systems, such as communication systems, chemical engineering systems, aerospace systems, power systems, and network transmission systems, etc., whose current states are inevitably affected by past states, that is, the change rate of the current state is not only related to the state at the current moment, but also depends on the state at a certain past moment or a certain period of time - this characteristic is called time delay. When studying the motion laws of objective things in nature, due to their complexity and diversity, there is always an inevitable lag phenomenon. Therefore, time delay and time-delay systems are a practical problem commonly encountered in real life and engineering technology. The existence of time delay, on the one hand, makes the dynamic performance of the system deteriorate or even causes the system to be unstable. But on the other hand, time delay can be used to improve the control effect in some control systems. For example, repetitive control systems and finite-time stability control, etc., all need to use time delay to achieve this purpose. In order to better use time delay to solve practical problems and avoid its adverse consequences, it is very necessary to study the influence of time delay phenomena on dynamic systems.

[0004] State feedback and output feedback are two main feedback strategies in control system design. The principle is to use the observed state and output as feedback quantities respectively to form a feedback law, realizing the closed-loop control of the system to meet the desired requirements for system performance indicators. However, in actual production and life, due to economic and other reasons, all states of the system cannot be measured. At this time, it is necessary to construct an observer to estimate the system state and design a controller based on the observer and system output. At the same time, in the actual production process, due to imperfect measurement and insufficient understanding of people, for example, due to environmental factors or measurement errors of measuring instruments such as sensors or inaccurate system parameters, the measured values obtained have dispersion, that is, the results measured each time are not the same value, but many values scattered in a certain area with a certain probability. Or it can be said that the output of the system may have a significant difference from the actual value, or there is uncertain information in the system output, which is called measurement uncertainty. In biomedical equipment, circuits, electrical equipment and mechanical systems, the uncertainty of system output is often encountered. Therefore, how to handle measurement uncertainty has also become the research object of scholars in the control field. Summary of the Invention

[0005] Aiming at the deficiencies of the prior art, the present invention proposes a control method for a non-linear induction heating circuit system based on output feedback, realizing the adaptive adjustment of the closed-loop system state for a system with measurement uncertainty and input time delay.

[0006] A control method for a non-linear induction heating circuit system based on output feedback specifically includes the following steps:

[0007] Step 1: Establish the dynamic model of the non-linear induction heating circuit system

[0008] Ignoring the influence of circuit element deformation on the system, establish the following dynamic model of the non-linear induction heating circuit system:

[0009]

[0010]

[0011] y = θ(dV C -i L ) (1)

[0012] Wherein, respectively represent the voltages of two voltage sources, V C represents the voltage across the capacitor C, R 1 , R 2 represent the resistance values, L represents the inductance value of the inductor L, i L represents the current flowing through the inductor L, dV CLet \(I\) denote the current passing through the voltage - controlled current source, \(y\) denote the output signal measured by the system sensor, \(t\) denote time, \(\text{sign}(\cdot)\) represent the sign function, \(s\) represent the time delay due to state switching, \(\theta\) represent the measurement uncertainty, and they satisfy θ and are unknown constants. respectively denote the first - order derivatives of \(V\) C and \(i\) L with respect to time. \(C = 1nF\), \(R\) 1 \(=R\) 2 \(= 1k\Omega\), \(L = 1mH\), \(d = 2mA / V\).

[0013] Meanwhile, define \(x\) 1 \(= 2V\) C \(-i\) L and \(x\) 2 \(= V\) C \(-i\) L . Convert the dynamic model of the system into the state - space representation form:

[0014]

[0015]

[0016] \(y=\theta x\) 1 (2)

[0017] where respectively denote the first - order derivatives of \(x\) 1 and \(x\) 2 with respect to time.

[0018] Step 2: Design an adaptive output - feedback controller

[0019] For the system state - space expression obtained in Step 1, design an output - feedback controller:

[0020]

[0021] where is the state value of the state reconstructor, \(u\) represents the control input, respectively denote the first - order derivatives with respect to time, represents the dynamic gain initial value, \(a\) 1 and \(a\) 2 are the coefficients of the Hurwitz polynomial \(g\) 2 +\(a\) 1 \(g + a\) 2 , represents the adaptive gain, denotes The first derivative, where ρ is a constant, and k 1 , k 2 , q are constants to be solved for.

[0022] Step 3: Control effect analysis and related parameter selection

[0023] Based on the system state-space expression and output feedback controller obtained in Steps 1 and 2, define the following error state transformation:

[0024]

[0025] And construct a set of virtual control laws y * , And state transformation η 1 , η 2 As follows:

[0026]

[0027] Where k 1 = 1, k 2 = 3k 1 + 1 = 4 and

[0028] Let ε = [ε 1 , ε 2 T , and transform systems (2) and (3) into the following form:

[0029]

[0030] Where Φ = [φ 1 , 0] T , Are the first derivatives of ε 1 , ε 2 with respect to time, respectively. At the same time, there is:

[0031]

[0032] Where Is the first derivative of η 1 with respect to time.

[0033] To illustrate the correctness of the designed controller above and to handle the time delay in the input, construct the following Lyapunov-Krasovskii functional for the non-linear induction heating circuit system:

[0034]

[0035] Taking the derivative of it, we can get:

[0036] ​

[0037] where β i , i = 1, 2, 3 and b i , i = 1, 2 are some constants.

[0038] Based on formula (9), it can be obtained that is bounded, ||x|| 2 < +∞, and Furthermore, it can be obtained by Barbalat's lemma that

[0039]

[0040] Therefore, it can be concluded that all signals of the closed-loop system are ultimately convergent to zero.

[0041] The present invention has the following beneficial effects:

[0042] A low-gain output feedback controller is designed based on the adaptive output feedback control method for the measurement uncertainty and input time delay existing in the system. This method allows the bound of the measurement uncertainty to be unknown. Combining the backstepping method and design constants, it effectively solves the situation where the bound of the measurement uncertainty is unknown. The designed controller based on the low-gain observer has robustness and can effectively compensate for the measurement uncertainty of the system. In addition, the problem of input time delay is effectively solved by the Lyapunov-Krasovskii functional, thereby realizing the adaptive regulation of the state of the closed-loop system. This control method is more in line with the actual situation in engineering applications and is more universal. Description of the Drawings

[0043] Figure 1 is the circuit schematic diagram of the nonlinear induction heating circuit system;

[0044] Figure 2 is the state trajectory diagram of the closed-loop system composed of the system state space expression and the output feedback controller;

[0045] Figure 3 is the control input and adaptive gain trajectory diagram of the closed-loop system;

[0046] Figure 4 is the capacitor voltage and inductor current trajectory diagram of the nonlinear induction heating circuit system. Detailed Embodiment

[0047] The following further explains the present invention with reference to the drawings;

[0048] A control method for a nonlinear induction heating circuit system based on output feedback specifically includes the following steps:

[0049] Step 1: Establish the dynamic model of the non - linear induction heating circuit system

[0050] The schematic diagram of the non - linear induction heating circuit is as shown Figure 1 below. Ignoring the influence of the deformation of some circuit components in the system on the model, the following dynamic model is established:

[0051]

[0052]

[0053] y = θ(dV C - i L ) (1)

[0054] Wherein, respectively represent the voltages of two voltage sources, V C represents the voltage across the capacitor C, i L represents the current flowing through the inductor L, C = 1nF, R 1 = R 2 = 1kΩ, L = 1mH, dV C represents the current through the voltage - controlled current source, d = 2mA / V. y represents the output signal measured by the system sensor, t represents time, sign(·) represents the sign function, s represents the time delay caused by the state switching, θ represents the measurement uncertainty, and satisfies θ 、 are unknown constants. respectively represent the first - order derivatives of V C 、i L with respect to time.

[0055] Define x 1 = 2V C - i L , x 2 = V C - i L . Convert the dynamic model of the system into the state - space representation form:

[0056]

[0057]

[0058] y = θx 1 (2)

[0059] Wherein, respectively represent the first - order derivatives of x 1 、x 2 with respect to time.

[0060] Step 2: Design an adaptive output feedback controller:

[0061] For the system state space expression obtained in Step 1, design an output feedback controller:

[0062]

[0063] where is the state value of the state reconstructor, u represents the control input, respectively represent the first-order derivative with respect to time, represents the adaptive gain the initial value of,, a 1 a 2 are the coefficients of the Hurwitz polynomial g 2 + a 1 g + a 2 represents the first-order derivative of, ρ is a constant, k 1 k 2 q are constants to be solved.

[0064] Step 3: Analyze the control effect:

[0065] Analyze the control performance of the output feedback controller designed in Step 2.

[0066] Define the following matrices:

[0067]

[0068] Since matrices A and D are Hurwitz matrices, there exist appropriate a 1 a 2 and a positive definite matrix P = P T satisfying:

[0069] A T P + PA ≤ -I 2 , DP + PD ≥ 0 (5)

[0070] where I 2 represents the 2D identity matrix.

[0071] Define the following error state transformation:

[0072]

[0073] Then construct a set of virtual control laws y * , and state transformations η 1 η 2 :​

[0074]

[0075] where \(k\) 1 = 1, \(k\) 2 = 3\(k\) 1 + 1 = 4,

[0076] Let \(\varepsilon=[\varepsilon\) 1 , \(\varepsilon\) 2 \) T , and transform the state - space expression (2) of the system and the output - feedback controller (3) into:

[0077]

[0078] where \(\varPhi=[\varphi\) 1 , 0]\) T , respectively represent the first - order derivatives of \(\varepsilon\) 1 , \(\varepsilon\) 2 with respect to time.

[0079]

[0080] where represents the first - order derivative of \(\eta\) 1 with respect to time.

[0081] To ensure the correctness of the above - designed output - feedback controller and to handle the time - delay existing in the input, a Lyapunov - Krasovskii functional is constructed as follows:

[0082]

[0083] Taking the derivative of it, we can get:

[0084]

[0085] where \(\beta\) i , \(i = 1,2,3\) and \(b\) i , \(i = 1,2\) are some constants.

[0086] Assume that the maximum existence interval of the solution is \([0,t\) f ), and \(t\) f \in(0,+\infty]\). Based on formula (11), prove that all states of the closed - loop system tend to zero:

[0087] ① Prove that the adaptive gain is bounded by using the proof by contradiction:

[0088] Assume that Therefore, there exists a time \(t\) 1∈(0, t f ) satisfies:

[0089] such that:

[0090]

[0091] where η = [η 1 , η 2 T , σ is a constant satisfying 1 < σ < 1 - k 2 q + q, and at the same time formula (12) also shows that ||ε|| < +∞, ||η|| < +∞, so:

[0092]

[0093] Therefore, the adaptive gain is bounded.

[0094] ② Prove that is bounded:

[0095] Redefine the state transformation as follows:

[0096]

[0097] Let z = [z 1 , z 2 T , a = [a 1 , a 2 T , B = [0, 1] T , and we can get:

[0098]

[0099] Select the Lyapunov function V z = z T Pz, and take its derivative to get:

[0100]

[0101] where r is a constant, and by integrating and transforming formula (16) on (0, t), we can get:

[0102]

[0103] ③ Prove that x = [x 1 , x 2 T is bounded:

[0104] Define the new state transformation as follows: ​​​​

[0105]

[0106] Let ξ = [ξ 1 , ξ 2 T , and select the following Lyapunov - Krasovskii functional

[0107]

[0108] where c, is a constant. Select L * ≥ 32c||P||, and taking the derivative of formula (18) gives:

[0109]

[0110] where c ξ is a constant. By integrating formula (19) over (0, t), we can obtain: ||x|| 2 < +∞, In summary, we have: t f = +∞, and then by Barbalat's lemma, we get:

[0111]

[0112] Therefore, it can be concluded that all signals of the closed - loop system are ultimately convergent to zero.

[0113] Step 4: Select the controller parameters:

[0114] In this embodiment, select the observer parameter a 1 = 3, a 2 = 3 and the design parameter q = 0.01, and obtain the following output - feedback controller:

[0115]

[0116] The initial value of the system is x 1 (τ) = 1, x 2 (τ) = - 1, The initial value of the observer The initial value of the adaptive dynamic gain The system measurement uncertainty is θ = 0.8. Figure 2 、 Figure 3 are respectively the state trajectory diagram of the closed - loop system composed of the system state - space expression and the output - feedback controller, and the trajectory diagrams of the control input and the adaptive gain of the closed - loop system; Figure 4 is the trajectory diagram of the capacitor voltage and inductor current of the original system. It can be seen from the figure that whether it is the system state - space expression or the state of the output - feedback controller, they ultimately tend to zero, and the dynamic gain​ It is bounded, and its transient process does not oscillate violently and can be applied to practice. Therefore, it can be obtained that the proposed controller achieves the expected control objective - asymptotic stability.

Claims

1. A control method for a nonlinear induction heating circuit system based on output feedback, characterized in that: In this nonlinear induction heating circuit system, the values ​​of the circuit elements are C = 1nF, R1 = R2 = 1kΩ, L = 1mH, d = 2mA / V; The control method specifically includes the following steps: Step 1: Build a dynamic model of the nonlinear induction heating circuit system Ignoring the influence of circuit component deformation on the system, a dynamic model of the nonlinear induction heating circuit system is established: in, Represents the voltage of the voltage source, V C represents the voltage across the capacitor C, R1 and R2 represent the resistance values, L represents the inductance value of the inductor L, i L represents the current flowing through the inductor L, dV C represents the current through the voltage-controlled current source, y represents the output signal measured by the system sensor, t represents time, sign(·) represents the sign function, s represents the time delay caused by state switching, θ represents the measurement uncertainty, and satisfies is an unknown constant; Respectively represent V C 、i L The first derivative with respect to time t; At the same time, define x1=2V C -i L 、x2=V C -i L , Convert the dynamic model of the system into a state-space representation: in, Respectively represent the first-order derivatives of x1 and x2 with respect to time; Step 2: Design an Adaptive Output Feedback Controller According to the system state space expression obtained in step 1, the output feedback controller is designed: in, is the state value of the state reconstructor, u represents the control input, Respectively The first derivative with respect to time, Indicates dynamic gain The initial value of a1 and a2 are the Hurwitz polynomials g 2 +a1g+a2 coefficient, express The first-order derivative of , ρ is a constant, k1, k2, q are constants to be solved; Step 3: Control effect analysis and related parameter selection Based on the system state space expression and output feedback controller obtained in steps 1 and 2, the following error state transformation is defined: And construct a set of virtual control laws And the state transformation η1,η2 is as follows: where k1=1, k2=3k1+1=4 and Let ε=[ε1,ε2] T , transforming systems (2) and (3) into the following form: in, They are the first-order derivatives of ε1 and ε2 with respect to time, and we have: in is the first-order derivative of η1 with respect to time; In order to handle the time lag in the input, the following Lyapunov-Krasovskii functional is constructed for the nonlinear induction heating circuit system: P represents a positive definite matrix; taking its derivative we get: Among them, β1, β2, β3, b1, b2 are constants; Based on formula (9), we get is bounded, ||x|| 2 <+∞, as well as Then, through Barbarat's lemma, we get: Therefore, all signals in a closed-loop system will eventually converge to zero.

2. A control method for a nonlinear induction heating circuit system based on output feedback as claimed in claim 1, characterized in that: Hurwitz polynomial g 2 The coefficient of +a1g+a2 is a1=a2=3.

3. The control method of a nonlinear induction heating circuit system based on output feedback as claimed in claim 1, characterized in that: Constant q=0.01.

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