Parameter measurement method of static closed loop and same-time dynamic closed loop feedback system

Abstract

The present application relates to a kind of parameter measurement method of static closed loop and same time dynamic closed loop feedback system, especially the measurement method of micro inertia link, belong to system characteristic parameter measurement method and modeling method.Micro inertia link: inertia link with extremely short time constant, dynamic: the process that static system is inserted into micro inertia link and becomes quasi-dynamic system, conventional: the process that same time dynamic system is inserted into micro inertia link and becomes conventional dynamic system.Micro inertia link is usually inserted in forward channel in the loop channel of static system or same time dynamic system, the transfer function of loop channel, auxiliary function and closed loop transfer function are measured and calculated, and the stability of closed loop feedback system can be judged using Routh criterion or zero-pole point.It is suitable for static closed loop system, same time dynamic closed loop system and multiple closed loop system containing static loop and same time dynamic loop, the present application provides a kind of parameter measurement and stability analysis method, simple and direct, physical meaning is clear, and will be widely used.

Description

Technical Field

[0001] The present invention relates to a parameter calculation method for a static closed-loop and a synchronous dynamic closed-loop feedback system, in particular to a calculation method using a micro-inertia link to process a static system or a synchronous dynamic system, belonging to a system characteristic parameter calculation method and a modeling method. Background Art

[0002] With the development of society and technology, the application of automatic control systems has become increasingly widespread, and these systems have also become increasingly complex. The most common and complex structure of automatic control systems is the closed-loop feedback control system. Systems are divided into static systems and dynamic systems. The transfer function of a static system is a constant and usually a real number, while the transfer function of a dynamic system is a polynomial fraction of the complex frequency s. The numerator order of the transfer function of an actual dynamic system is not higher than the denominator order. A system with a transfer function numerator order lower than the denominator order is called a conventional dynamic system, while a system with a transfer function numerator order equal to the denominator order is called a homodynamic system. To simplify the representation of feedback control systems, some real systems are often idealized as static systems, such as operational amplifier circuits. Static systems are also the most basic systems.

[0003] There are two forms of closed-loop feedback system: negative feedback and positive feedback. According to the current theoretical closed-loop transfer function or closed-loop amplification factor calculation, they are as follows: Figure 1a and Figure 1b shown.

[0004] The transfer function of the forward channel of the static closed-loop feedback system is G(s)=K G , the transfer function of the feedback channel is G(s)=K H In the negative feedback form, the loop transfer function is Θ(s) = Θ = -K G K H , in the positive feedback form, the loop transfer function is Θ(s)=Θ=K G K H In the negative feedback form, the auxiliary function is F(s)=1-Θ(s)=1+K G K H , in the positive feedback form, the auxiliary function is F(s)=1-Θ(s)=1-K G K H , the unified form is F(s) = F = 1-Θ. In the negative feedback form, the closed-loop transfer function is In the positive feedback form, the closed-loop transfer function is The unified form is

[0005] The loop transfer function Θ(s), auxiliary function F(s) and closed-loop transfer function Ф(s) of the static closed-loop feedback system are all constants. According to the different loop magnifications Θ of the closed-loop system, the static closed-loop feedback system is divided into different feedback types or feedback states: Θ<0 negative feedback, Θ=0 no feedback, Θ>0 positive feedback; Θ>0 can be further divided into: 0<Θ<1 weak positive feedback, Θ=1 critical positive feedback, Θ>1 strong positive feedback.

[0006] The system has different stability conditions under different feedback states. When the system is deadlocked, self-oscillated or divergent, the closed-loop transfer function cannot be a constant. Therefore, the calculation formula of the closed-loop transfer function is It is only valid when the static closed-loop feedback system is stable.

[0007] If we force the calculation formula of the closed-loop transfer function to be used This can lead to contradictory results, such as Figure 2 As shown, the system is a double-closed-loop static system with negative feedback. There are two different approaches to equivalent simplification: Approach 1, which synthesizes the inner loop first and treats it as a single link, yields a loop transfer function of Θ = -4, making the system stable. Approach 2, which combines the two feedback channels into a single channel, yields a loop transfer function of Θ = 6, making the system divergent. Clearly, the results of these two approaches contradict each other.

[0008] During the design phase, the system parameters are not yet determined. Parameters are usually expressed in symbols. At this time, it is not certain what feedback state the system is in and whether the above closed-loop transfer function calculation formula can be used. The calculation loses its basis. Furthermore, in nested, compound, multi-loop feedback control systems, when the inner loop is a static closed-loop feedback system, system analysis must first calculate the closed-loop transfer function of the inner loop. Obviously, the calculation of the closed-loop transfer function of a static closed-loop feedback system is unavoidable.

[0009] In static systems, the Routh criterion, root locus method and Nyquist criterion for judging system stability are all inapplicable.

[0010] Although stability criteria such as the Routh criterion, root locus method and Nyquist criterion for dynamic systems can all be used, dynamic systems of the same order with the same numerator and denominator of the transfer function may also have the same or similar problems as static systems.

[0011] The dynamic closed-loop feedback system is usually expressed as a rational fraction of the complex frequency s, so the forward channel transfer function is expressed as The feedback channel transfer function is expressed as The loop channel transfer function of negative feedback and positive feedback is expressed as The auxiliary functions in negative and positive feedback form are expressed as The closed-loop transfer function of the system in negative and positive feedback form is expressed as The system characteristic function of negative and positive feedback is expressed as Ψ(s) = D Θ (s)-M Θ (s)=D G (s)D H (s)±M G (s)M H (s).

[0012] For example, the closed-loop channel transfer function of the system is: The auxiliary function is The closed-loop transfer function is Since the characteristic function Ψ(s) = -2 is a constant, it is impossible to determine the stability of the system.

[0013] Therefore, solving the problem of parameter measurement of loop transfer function, auxiliary function and closed-loop transfer function of static closed-loop and dynamic closed-loop systems and stability judgment of static closed-loop and dynamic closed-loop systems is of great significance for system analysis and modeling. Summary of the Invention

[0014] The technical problem to be solved by the present invention is: the measurement and calculation of parameters such as loop transfer function, auxiliary function and closed-loop transfer function of a static closed-loop feedback system and a dynamic closed-loop feedback system.

[0015] The present invention provides a parameter calculation method for a static closed-loop and a dynamic closed-loop feedback system.

[0016] The technical problem to be solved by the present invention is achieved through the following technical solutions.

[0017] A parameter calculation method for a static closed-loop feedback system is proposed. The parameters of a single closed-loop feedback system are determined by measuring the step responses of the forward channel and the feedback channel. The method is applicable to linear closed-loop systems.

[0018] Step 1: Preprocessing: disconnect the closed-loop feedback system and decompose the closed-loop into a forward channel and a feedback channel;

[0019] Step 2: Measurement: Add a step excitation signal to the input of the forward channel and the feedback channel, and measure the step response signal at the output of the forward channel and the feedback channel to obtain the step characteristics of the forward channel and the feedback channel.

[0020] Step 3: Calculation: Based on the measurement results, the forward channel transfer function G(s) = K is calculated. G The transfer function of the feedback channel H(s) = K H ;

[0021] Step 4: Dynamicize the static system: Insert a micro-inertia link with a very short time constant into the forward channel The transfer function of the forward channel becomes: The differential coefficient ε>0 and is extremely small;

[0022] Step 5: Calculation of the system loop transfer function: In the negative feedback form, the system loop transfer function is In the positive feedback form, the system loop transfer function is

[0023] Step 6: Calculation of system auxiliary function: In negative feedback form, the system auxiliary function is In the positive feedback form, the auxiliary function of the system is

[0024] Step 7: Calculation of the system closed-loop transfer function: In the negative feedback form, the closed-loop transfer function of the system is: In the positive feedback form, the closed-loop transfer function of the system is:

[0025] Step 8: System stability judgment: In negative feedback form, 1+K G K H >0 system is stable, 1+K G K H <0 system divergence, 1+K G K H =0 The system is deadlocked when there is no input and diverges when there is input; in the positive feedback form, 1-K G K H >0 system is stable, 1-K G K H <0 system divergence, 1-K G K H =0 The system is deadlocked if there is no input and diverges if there is input;

[0026] Step 9: Determine the parameter value: If the system is stable, the differential coefficient ε is treated as ε = 0 to calculate the parameters.

[0027] A parameter calculation method for a static closed-loop feedback system is proposed. The parameters of a single closed-loop feedback system are determined by measuring the step responses of the forward channel and the loop channel. The method is applicable to linear closed-loop systems.

[0028] Step 1: Preprocessing: Disconnect the feedback channel of the closed-loop feedback system. One end of the feedback channel break is the input end, and the other end is the output end.

[0029] Step 2: Loop channel measurement: Add a step excitation signal to the input end of the feedback channel break, measure the step response signal at the output end of the feedback channel break, and thus measure the step characteristics of the system loop channel;

[0030] Step 3: Forward channel measurement: Keep the feedback channel disconnected, add a step excitation signal to the input of the closed-loop system, and measure the step response signal at the output of the closed-loop system to obtain the step characteristics of the forward channel.

[0031] Step 4: Calculation: Based on the measurement results, the forward channel transfer function G(s) = K is calculated. G and the transfer function of the loop channel Θ(s) = Θ;

[0032] Step 5: Dynamicize the static system: Insert a micro-inertia link with a very short time constant into the forward channel The transfer function of the forward channel becomes: The transfer function of the loop channel then becomes: The differential coefficient ε>0 and is extremely small;

[0033] Step 6: Calculation of system auxiliary function: The system auxiliary function is

[0034] Step 7: Calculation of the system closed-loop transfer function: The closed-loop transfer function of the system is

[0035] Step 8: System stability judgment: 1-Θ>0 means the system is stable, 1-Θ<0 means the system diverges, 1-Θ=0 means the system is deadlocked if there is no input and diverges if there is input;

[0036] Step 9: Determine the parameter value: If the system is stable, the differential coefficient ε is treated as ε = 0 to calculate the parameters.

[0037] A method for calculating parameters of a same-order dynamic closed-loop feedback system is proposed. The method measures the step response or frequency response of the forward channel and the feedback channel and determines the parameters of the single closed-loop feedback system. The method is applicable to linear closed-loop systems.

[0038] Step 1: Preprocessing: disconnect the closed-loop feedback system and decompose the closed-loop into a forward channel and a feedback channel;

[0039] Step 2: Measurement: Add a step excitation signal to the input of the forward channel and the feedback channel, and measure the step response signals at the output of the forward channel and the feedback channel, thereby measuring the step characteristics of the forward channel and the feedback channel; or add a sinusoidal excitation signal with a slowly varying frequency to the input of the forward channel and the feedback channel, and measure the sinusoidal response signals at the output of the forward channel and the feedback channel, thereby measuring the frequency characteristics of the forward channel and the feedback channel;

[0040] Step 3: Calculation: Based on the measurement results, the same-order transfer function of the forward channel is calculated. and the same-order transfer function of the feedback channel

[0041] Step 4: Normalize the same system: insert a micro-inertia link with a very short time constant into the forward channel The transfer function of the forward channel becomes: The differential coefficient ε>0 and is extremely small;

[0042] Step 5: Calculation of the system loop transfer function: In the negative feedback form, the system loop transfer function is In the positive feedback form, the system loop transfer function is:

[0043] Step 6: Calculation of system auxiliary function: In negative feedback form, the system auxiliary function is In the positive feedback form, the auxiliary function of the system is

[0044] Step 7: Calculation of the system closed-loop transfer function: In the negative feedback form, the closed-loop transfer function of the system is: In the positive feedback form, the closed-loop transfer function of the system is:

[0045] Step 8: System stability judgment: system characteristic function Ψ in negative feedback form ε (s)=(εs+1)D G (s)D H (s)+M G (s)M H (s), the system characteristic function Ψ in the positive feedback form ε (s)=(εs+1)D G (s)D H (s)-M G (s)M H (s), Routh criterion is used to judge the stability of the system;

[0046] Step 9: Determine the parameter value: If the system is stable, the differential coefficient ε is treated as ε = 0 to calculate the parameters.

[0047] A method for calculating parameters of a same-order dynamic closed-loop feedback system is proposed. The method measures the step response or frequency response of the forward channel and the loop channel and determines the parameters of the single closed-loop feedback system. The method is applicable to linear closed-loop systems.

[0048] Step 1: Preprocessing: Disconnect the feedback channel of the closed-loop feedback system. One end of the feedback channel break is the input end, and the other end is the output end.

[0049] Step 2: Loop channel measurement: Add a step excitation signal to the input end of the feedback channel break, measure the step response signal at the output end of the feedback channel break, and thus measure the step characteristics of the system loop channel; or add a sinusoidal excitation signal with a slowly varying frequency to the input end of the feedback channel break, measure the sinusoidal response signal at the output end of the feedback channel break, and thus measure the frequency characteristics of the system loop channel;

[0050] Step 3: Forward channel measurement: Keep the feedback channel disconnected, add a step excitation signal to the input of the closed-loop system, and measure the step response signal at the output of the closed-loop system to measure the step characteristics of the forward channel; or add a sinusoidal excitation signal with a slowly varying frequency to the input of the closed-loop system, and measure the sinusoidal response signal at the output of the closed-loop system to measure the frequency characteristics of the forward channel;

[0051] Step 4: Calculation: Based on the measurement results, the same-order transfer function of the forward channel is calculated. and the same transfer function of the loop channel

[0052] Step 5: Normalize the same system: insert a micro-inertia link with a very short time constant in the forward channel The transfer function of the forward channel becomes: The transfer function of the loop channel then becomes: The differential coefficient ε>0 and is extremely small;

[0053] Step 6: Calculation of system auxiliary function: The system auxiliary function is

[0054] Step 7: Calculation of the system closed-loop transfer function: The closed-loop transfer function of the system is

[0055] Step 8: System stability judgment: system characteristic function Ψ ε (s)=(εs+1)D Θ (s)-M Θ (s), Routh criterion is used to judge the stability of the system;

[0056] Step 9: Determine the parameter value: If the system is stable, the differential coefficient ε is treated as ε = 0 to calculate the parameters.

[0057] A parameter calculation method for a static closed-loop or sub-dynamic closed-loop feedback system, which determines the parameters of a multi-closed-loop feedback system containing static loops or sub-loops by measuring the step response or frequency response of a channel or link, is applicable to linear multi-closed-loop systems.

[0058] Step 1: If there are cross-loops in the system closed loop, first use the equivalent migration method of nodes or links to remove the cross-loops;

[0059] Step 2: Insert a micro-inertia link into the static loop or the same dynamic loop Where: differential coefficient ε=0 + ;

[0060] Step 3: Use conventional analysis and calculation methods of dynamic systems to calculate and determine various parameters of the multi-closed-loop feedback system;

[0061] Step 4: Use Routh criterion to judge the stability of the system;

[0062] Step 5: If the system is stable, the differential coefficient ε is treated as ε = 0 to calculate the parameters.

[0063] Control systems are structurally divided into two types: open-loop systems and closed-loop systems. A single-input, single-output open-loop system is called a channel. Channels are composed of links connected in a forward direction and are categorized into simple channels, ordinary channels, and complex channels. Ordinary channels are further divided into series channels, parallel channels, and composite channels. A simple channel is composed of a single link, a composite channel is a series-parallel composite channel, and a complex channel is a channel with intersecting paths. Closed-loop systems are composed of channel feedback connections and are categorized into simple closed-loop systems, ordinary closed-loop systems, and complex closed-loop systems. Ordinary closed-loop systems are further divided into nested closed-loop systems, coupled closed-loop systems, and composite closed-loop systems. A simple closed-loop system is a single closed-loop system, a composite closed-loop system is a nested coupled composite closed-loop system, and a complex closed-loop system is a closed-loop system with intersecting loops.

[0064] The transfer function of the system is: m is the order of the numerator, n is the order of the denominator, when m<n, it is a conventional system, when m=n, it is a same-order system, when m>n, it is a constructed system, and the transfer function G(s) of the static system is a constant.

[0065] Actual closed-loop feedback systems can be divided into three types: static closed-loop feedback system, same-order dynamic closed-loop feedback system and conventional dynamic closed-loop feedback system. The static closed-loop feedback system is a closed-loop system in which the loop transfer function is a constant. The same-order dynamic closed-loop feedback system is a closed-loop system in which the numerator and denominator of the loop transfer function have the same order. The conventional dynamic closed-loop feedback system is a closed-loop system in which the denominator of the loop transfer function has a higher order than the numerator.

[0066] A static relationship refers to a proportional relationship between the dependent variable and the independent variable, and this proportion is a constant and has nothing to do with time; a dynamic relationship refers to a linear relationship between the time integral of the dependent variable and the independent variable, and this relationship is naturally related to time.

[0067] There are two common ways to describe a closed-loop feedback system: negative feedback and positive feedback. The transfer function of the forward channel of a static closed-loop system is the amplification factor of the forward channel, the transfer function of the feedback channel is the amplification factor of the feedback channel, the transfer function of the loop channel is referred to as the loop transfer function, which is the amplification factor of the loop channel, and the system auxiliary function is the system auxiliary coefficient. The system closed-loop transfer function is the system closed-loop amplification factor. The system's loop transfer function, auxiliary function, and closed-loop transfer function are the basic parameters of a closed-loop feedback system.

[0068] The static closed-loop feedback system is referred to as: static closed-loop system. When the amplification factor of the loop channel is positive, the static closed-loop system is actually positive feedback. When the amplification factor of the loop channel is negative, the static closed-loop system is actually negative feedback. The actual feedback type or feedback state of the static closed-loop system has nothing to do with the representation of positive and negative feedback.

[0069] The static closed-loop system can also be uniformly subdivided into five cases according to the positive and negative size of the loop amplification factor Θ: strong positive feedback Θ>1, critical positive feedback Θ=1, weak positive feedback 0<Θ<1, no feedback Θ=0, and negative feedback Θ<0. Among them, no feedback means no feedback and no closed loop, which can be regarded as a special case of feedback.

[0070] The present invention stems from the following concept: a so-called static system is a system with no delay in response time. A static system is only a mathematical idealized model. A truly static system does not exist in nature, and actual systems always have some delay to a greater or lesser extent. The inertia link is a link with a delay characteristic, while the micro-inertia link is a tracking link with a delay so small that it is negligible. The inertia link itself converges stably. Because the link effects of the system are additive, inserting a micro-inertia link into the system channel does not change the system's characteristics, but rather makes the idealized static system more consistent with the conditions of the actual system. Of course, even inserting a micro-inertia link into the channel of a dynamic system is not harmful, and it will not change the properties of the system.

[0071] The inertia link has tracking properties, and its time constant determines the speed of following and the magnitude of the following error. A micro-inertia link has an extremely short time constant, meaning it has extremely high following speed and accuracy, and its output and input are nearly identical. If a micro-inertia link is inserted into a static closed-loop feedback system, typically in the forward path of the static closed-loop system, the static closed-loop system instantly becomes a quasi-dynamic closed-loop feedback system. Because the micro-inertia link follows extremely quickly, the response of the quasi-dynamic closed-loop feedback system is similar to that of the corresponding static closed-loop feedback system.

[0072] A quasi-dynamic system refers to a static system containing micro-inertia links. It can be treated as a dynamic system during analysis and calculation, but its characteristics are similar to those of a static system. The process of inserting micro-inertia links into a static system is called the dynamicization process of the static system.

[0073] Although the response of the quasi-dynamic closed-loop system with the micro-inertia link inserted is not much different from that of the corresponding static closed-loop system, the criteria for the stability of the dynamic system can be used after the static system is made dynamic. The stability properties of the closed-loop system when the closed loop is equivalent to a single link are also retained, and the working principle of the system closed-loop feedback is also clearly expressed, and the static and dynamic are dialectically unified.

[0074] The static system dynamicization method unifies two systems of different nature, static system and dynamic system, so that system stability criteria such as Routh criterion, root locus method and Nyquist criterion, which are not applicable to static systems, can be applied to the stability judgment of static systems.

[0075] The working principles and stability judgment of strong positive feedback, critical positive feedback, weak positive feedback and negative feedback closed-loop feedback:

[0076] The working principle of strong positive feedback: Figure 3 As shown in the figure, a simple example of an actual static closed-loop feedback system with strong positive feedback is shown. If the initial input is 0, the output and feedback are both 0, and the superimposed output is 0; if the input is 1, the superimposed output is 1, and the output and feedback are both 2; the superimposed output is 3, and the output and feedback are both 6; the superimposed output is 7, and the output and feedback are both 14; the superimposed output is 15, and the output and feedback are both 30; the superimposed output is 31, and the output and feedback are both 62; ...; the output value becomes larger and larger, so we can judge and understand that the strong positive feedback system is divergent and unstable.

[0077] The working principle of critical positive feedback: Figure 4 As shown in the figure, a simple example of a static closed-loop feedback system with actual critical positive feedback is shown. If the input is 0, if the output and feedback are both 0, then the superimposed output is 0, the output and feedback are both 0, and the system is deadlocked; if the input is 0, if the output and feedback are both 5, then the superimposed output is 5, and the output and feedback are both 5. In this way, we can judge and understand that the critical positive feedback system is deadlocked when there is no input; if the input is 1, if the output and feedback are both 0, then the superimposed output is 1; the output and feedback are both 1; then the superimposed output is 2, the output and feedback are both 2; then the superimposed output is 3, the output and feedback are both 3, ...; the output value becomes larger and larger, so we can judge and understand that the critical positive feedback system diverges and is unstable when there is input.

[0078] The working principle of weak positive feedback: Figure 5As shown in the figure, a simple example of an actual static closed-loop feedback system with weak positive feedback is shown. If the initial input is 0, the output and feedback are both 0, and the superimposed output is 0; if the input is 1, the superimposed output is 1, and the output and feedback are both 0.5; the superimposed output is 1.5, and the output and feedback are both 0.75; the superimposed output is 1.75; the output and feedback are both 0.875; the superimposed output is 1.875, and the output and feedback are both 0.9375; the superimposed output is 1.9375, and the output and feedback are both 0.96875, ...; the output value tends to 1, so we can judge and understand that the weak positive feedback system is convergent and stable.

[0079] How negative feedback works: Figure 6a As shown, a simple example of a static closed-loop feedback system with actual negative feedback is shown. If the initial input is 0, the output and feedback are both 0, and the superimposed output is 0; if the input is 1, the superimposed output is 1, and the output and feedback are both -3; the superimposed output is -2, and the output and feedback are both 6; the superimposed output is 7, and the output and feedback are both -21; the superimposed output is -20, and the output and feedback are both 60; the superimposed output is 61, and the output and feedback are both -183; ...; the output value shows positive and negative oscillations with increasing amplitudes. The negative feedback system also seems to be divergent and unstable, but this situation is obviously completely inconsistent with the actual situation. The dynamic method of the present invention is adopted, that is, a micro-inertia link is inserted into the forward channel of the static system, such as Figure 6b As shown, if the initial input is 0, the output and feedback are both 0, and the superimposed output is 0; if the input is 1, the superimposed output is 1, and the output and feedback gradually decrease from 0, and the superimposed output also gradually decreases; when the output and feedback both decrease to -0.75, the superimposed output is 0.25, then the output and feedback remain at -0.75, the superimposed output remains at 0.25, and the output value stabilizes at -0.75; if the output and feedback values ​​reach -1 due to interference, the superimposed output is 0; then the output and feedback gradually increase from -1, and the superimposed output also gradually increases; when the output and feedback both increase to -0.75, the superimposed output is 0.25, and the output value stabilizes at -0.75; in this way, we can judge and understand that the negative feedback system is convergent and stable, which is obviously completely consistent with the actual situation.

[0080] A static closed-loop feedback system with nested double closed-loops is as follows: Figure 7 As shown in the figure, we first insert a micro-inertia link into the nested inner loop, and then analyze and calculate it as a dynamic system. The closed-loop transfer function contains a pole with a positive real part, and the system is divergent and unstable. The conclusions drawn from the two processing processes are exactly the same. Another static closed-loop feedback system with nested double closed loops is shown in the figure. Figure 8As shown in Figure 2, a comparison of the direct and dynamic approaches shows that direct processing yields contradictory results, while dynamic processing yields a strictly consistent result for a static closed-loop feedback system: convergence and stability. This shows that the dynamic approach to static systems is applicable to both positive and negative feedback systems.

[0081] Comparison of the two methods of direct processing of the dual closed-loop system with a static inner loop and a dynamic outer loop and the dynamic processing of the static inner loop Figure 9a 、 9b As shown in , 9c, and 9d, if the result of direct processing diverges, the result after dynamic processing of the inner loop must also diverge, but vice versa is not necessarily the case.

[0082] Comparison of direct processing and dynamic processing methods for the same dynamic single closed-loop system Figure 10a 、 10b As shown, when the result of direct processing is divergent, the result after dynamic processing is also divergent, and when the result of direct processing is convergent, the result after dynamic processing is not necessarily convergent.

[0083] The frequency characteristics of the closed-loop system can also be obtained by using the transfer function of the dynamic closed-loop system:

[0084] Loop channel frequency characteristics: Θ(jω)=Θ(s)| s＝jω , then the feedback depth of the closed-loop system is: |Θ(jω)|=|G(jω)H(jω)|.

[0085] When the closed-loop system is stable and the feedback depth |Θ(jω)|>>1, it is called deep feedback. In the negative feedback form, the closed-loop transfer function of the system The closed-loop transfer function of the system in positive feedback form

[0086] The deep feedback of the static system is stable only when there is negative feedback. When the feedback depth |Θ|>>1, that is, Θ<<-1, it is deep negative feedback. At this time, the closed-loop magnification of the negative feedback system is Closed-loop amplification factor of positive feedback system

[0087] It can be seen that the closed-loop amplification factor or transfer function of deep negative feedback or deep feedback when the system is stable is only related to the feedback coefficient or feedback transfer function.

[0088] The stability strength of a closed-loop system is: |F(jω)| = |1-Θ(jω)|. When the system is stable, the larger the stability strength |F(jω)|, the more stable the system is. When the system diverges, the larger the stability strength |F(jω)|, the more likely the system is to diverge. The stability strength of a static system is: |F| = |1-Θ|. The smaller the loop gain Θ, the more stable the system is.

[0089] The static system dynamic method can be used for positive feedback closed loop system, so that the stability of the system is not lost in the process of simplifying the link, and the analysis and calculation of the complex system containing positive feedback closed loop can obtain correct results. At the same time, the static system dynamic method can also be used for negative feedback closed loop system, without changing the nature of the negative feedback closed loop system, and facilitating the understanding of the stable working principle of the negative feedback closed loop system.

[0090] The static system dynamic method can also be used to determine the feedback properties of the closed loop feedback system with symbolic parameters, providing a measurement and calculation method for unknown parameter static closed loop feedback system, and ensuring the correctness of the analysis and calculation of static closed loop feedback system.

[0091] The parameter measurement and calculation method of static closed loop and dynamic closed loop feedback system provides a practical method for parameter measurement and calculation and stability analysis of static closed loop feedback system and dynamic closed loop feedback system, which is simple, intuitive, convenient to use and clear in physical meaning, and will be widely used. BRIEF DESCRIPTION OF DRAWINGS

[0092] Figure 1a 、 1b According to the current theory, the static single closed loop system closed loop transfer function of negative feedback form and positive feedback form is analyzed and calculated regularly;

[0093] Figure 2 The two different equivalent simplification methods of static double closed loop feedback system produce different stability contradiction results;

[0094] Figure 3 Simple static strong positive feedback single closed loop system;

[0095] Figure 4 Simple static critical positive feedback single closed loop system;

[0096] Figure 5 Simple static weak positive feedback single closed loop system;

[0097] Figure 6a 、 6b Simple static negative feedback single closed loop system, dynamic transformation of static negative feedback single closed loop system;

[0098] Figure 7 The two different simplification methods of static double closed loop feedback system after dynamic transformation produce the same stability results;

[0099] Figure 8 Comparison of two methods of static double closed loop feedback system direct processing and dynamic transformation processing;

[0100] Figure 9a 、 9b9c, 9d Comparison of direct processing and dynamic transformation processing of a dual closed-loop system with a static inner loop and a dynamic outer loop;

[0101] Figure 10a 、 10b Comparison of direct processing and dynamic transformation processing methods for the same dynamic single closed-loop system;

[0102] Figure 11a 、 11b Analysis and calculation of closed-loop transfer functions of static single closed-loop systems with negative feedback and static single closed-loop systems with positive feedback after dynamic transformation;

[0103] Figure 12 Stability study of a static double closed-loop feedback system with unknown parameters in the inner loop using a dynamic transformation method;

[0104] Figure 13a 、 13b Circuit implementation of anti-phase micro-inertia link and in-phase micro-inertia link;

[0105] Figure 14 An example of studying the amplification factor and stability of a static system: static closed-loop amplifier circuit;

[0106] Figure 15 An example of studying the transfer function and system stability of the same subsystem: an inverting dynamic closed-loop operation circuit;

[0107] Figure 16 An example of studying the transfer function and system stability of a same-order system: a same-phase dynamic closed-loop operation circuit. DETAILED DESCRIPTION

[0108] The present invention is described in detail below with reference to the accompanying drawings.

[0109] Example 1

[0110] Parameter Calculation and Stability Judgment of Static Single Closed-Loop Linear Feedback System

[0111] A single closed-loop linear static feedback system in the form of negative feedback is as follows Figure 11a The single closed-loop linear static feedback system in positive feedback form is shown as Figure 11b shown.

[0112] The loop of the closed-loop feedback system is disconnected, and the closed-loop is decomposed into a forward channel and a feedback channel.

[0113] A step excitation signal is added to the input end of the forward channel, and the step response signal at the output end of the forward channel is measured, thereby measuring the step characteristics of the forward channel; a step excitation signal is added to the input end of the feedback channel, and the step response signal at the output end of the feedback channel is measured, thereby measuring the step characteristics of the feedback channel.

[0114] The step response signal at the channel output end of a static system is completely synchronized with the step excitation signal at the channel input end without any delay. Therefore, if it is confirmed that it is a static system, the DC excitation and DC response measurement method can be used to obtain the amplification factor of its channel or link, that is, the transfer function.

[0115] According to the measurement results of the step characteristic, the response output is divided by the excitation input to obtain the forward channel transfer function G(s) = K of the static closed-loop feedback system. G And the feedback channel transfer function H(s) = K H .

[0116] In the negative feedback form, the loop transfer function Θ(s)=-G(s)H(s)=-K G K H = Θ, in positive feedback form, the loop transfer function Θ(s) = G(s)H(s) = K G K H =Θ.

[0117] Insert a micro-inertia link with a very short time constant in the forward channel The transfer function of the forward channel becomes: The differential coefficient ε is greater than 0 and is extremely small.

[0118] When the system is in negative feedback form, the loop transfer function is When the system is in positive feedback form, the loop transfer function is

[0119] When the system is in negative feedback form, the auxiliary function is When the system is in positive feedback form, the auxiliary function is

[0120] In the negative feedback form, the closed-loop transfer function of the system is: In the positive feedback form, the closed-loop transfer function of the system is:

[0121] Negative feedback form when 1+K G K H >0, that is, Θ=-K G K H When <1, the closed-loop feedback system is stable, and the positive feedback form is 1-K G K H >0 means Θ=K G K H When <1, the closed-loop feedback system is stable.

[0122] Conclusion: For a static closed-loop feedback system, the system is stable when there is negative feedback, i.e. Θ < 0, and when there is no feedback, i.e. Θ = 0. The system is stable when there is weak positive feedback, i.e. 0 < Θ < 1. The system is stalemate when there is no input and diverges when there is input when there is critical positive feedback, i.e. Θ = 1. The system diverges when there is strong positive feedback, i.e. Θ > 1.

[0123] When the system is stable, the differential coefficient ε is treated as ε = 0, then: the closed-loop transfer function of the system in negative feedback form is: In the positive feedback form, the closed-loop transfer function of the system is:

[0124] If the system is unstable, the closed-loop transfer function of the system is It doesn't make sense.

[0125] The larger the absolute value of the loop gain |Θ| is, the greater the feedback depth is. When Θ=0, there is no feedback, so |Θ| is called the feedback depth.

[0126] When Θ<<-1, the system is in deep negative feedback, and the negative feedback form is K G K H >>1 Closed-loop transfer function In the positive feedback form, K G K H <<-1 Closed-loop transfer function

[0127] Example 2

[0128] Calculation of Closed-Loop Transfer Function and Stability Judgment of Static Double Closed-Loop Linear Feedback System

[0129] like Figure 12 As shown, the system is a nested double closed-loop linear static closed-loop feedback system, the amplification factor of the inner loop forward channel is K, and the amplification factors of the remaining links or channels are known.

[0130] There are links with unknown parameters in the static feedback system, and the positive and negative feedback properties of the inner loop cannot be determined.

[0131] Insert a micro-inertia link in the forward channel of the inner loop The inner loop forward channel transfer function becomes: The entire system becomes a quasi-dynamic system, where the differential coefficient ε = 0 + .

[0132] The inner loop of the quasi-dynamic system is segmented and its inner loop closed loop transfer function is: The double closed-loop quasi-dynamic system is equivalent to a single closed-loop quasi-dynamic system. Continuing to link the closed-loop transfer function of the entire system is:

[0133] According to the Routh criterion or zero-pole stability principle of dynamic systems, when 1-5K>0, that is, K<0.2, the system is stable. When the system is stable, the closed-loop transfer function of the system is

[0134] Example 3

[0135] Implementation of operational amplifier circuit for micro-inertia link

[0136] The circuit implementation of the inverting micro-inertia link is as follows Figure 13a As shown, where the resistance R1 = R2 = R, and the capacitance C is very small, the time constant ε = RC> 0 is also very small, and the transfer function of the inverting micro-inertia link is:

[0137] The circuit implementation of the same-phase micro-inertia link is as follows Figure 13b As shown, the resistor R 11 =R 12 =R, R 21 =R 22 =R, and the capacitance C is very small, the time constant ε = RC> 0 is also very small, and the transfer function of the in-phase micro-inertia link is:

[0138] Example 4

[0139] A case study on the amplification factor and stability of a static system

[0140] An example is used to study the static system amplification and system stability, such as Figure 14 As shown, the circuit can be divided into two parts. The first part is the inverting adder, which consists of the op amp A1 and the resistor R 10 、R 11 、R 12 、R 13 The input-output relationship is The second part: The operational circuit with adjustable magnification is composed of op amp A2 and resistor R 20 、R 21 、R 22 、R 23 and dual potentiometer R P0 、R P1 The input-output relationship is c(t)=kx(t), where the magnitude and positive and negative polarity of the amplification factor k can be controlled by the double potentiometer R P0 、R P1 adjust.

[0141] By adjusting the potentiometer, the amplifier circuit can be in a negative feedback state or a positive feedback state. The amplifier circuit can even be in a non-feedback state by adjusting the amplification factor of the second part of the circuit to 0 through the potentiometer.

[0142] Example 5

[0143] A case study on the transfer function and system stability of the same system

[0144] An example is used to study the transfer function and system stability of the same system, such as Figure 15 As shown, the circuit can be divided into two parts. The first part is the inverting adder, which has two inputs r(t) and c(t) and one output x(t). The input and output relationship of the inverting adder is The second part: the inverse proportional inertia operation circuit has an input x(t) and an output c(t). The input and output relationship of the inverse proportional inertia operation circuit is

[0145] The two circuits are connected together through resistor R 13 Forming an overall closed-loop feedback.

[0146] Analyze using conventional methods:

[0147] The transfer function of the forward path is: The loop transfer function is: The characteristic function is: The helper function is: The closed-loop transfer function is:

[0148] When R 11 =R 12 =0.2R 13 、R 21 =R 22 =R 23 =R, the transfer function of the forward channel is: The loop transfer function is: The characteristic function is: Ψ(s) = 0.8τs + 0.6, and the auxiliary function is: The system transfer function is: Where τ = RC.

[0149] When R 11 =R 12 =R 13 、R 21 =R 22 =R 23 =R, the transfer function of the forward channel is: The loop transfer function is: The characteristic function is: Ψ(s) = -1, and the auxiliary function is: The system transfer function is: Φ(s) = -(τs+2), where τ = RC.

[0150] When R 11 =R 12=2R 13 、R 21 =R 22 =R 23 =R, the transfer function of the forward channel is: The loop transfer function is: The characteristic function is: Ψ(s) = -(τs+3), and the auxiliary function is: The system transfer function is: Where τ = RC.

[0151] Adopt the method analysis of the present invention:

[0152] When calculating the transfer function of the same dynamic system, a micro-inertia link is inserted into the forward channel, and the transfer function of the forward channel is: The loop transfer function is: The characteristic function is: The helper function is: The closed-loop transfer function is:

[0153] When R 11 =R 12 =0.2R 13 、R 21 =R 22 =R 23 =R, the transfer function of the forward channel is: The loop transfer function is: The characteristic function is: ε (s)=ετs 2 +εs+0.8τs+0.6≈ετs 2 +0.8τs+0.6, the auxiliary function is: The system transfer function is: Where τ = RC.

[0154] When R 11 =R 12 =R 13 、R 21 =R 22 =R 23 =R, the transfer function of the forward channel is: The loop transfer function is: The characteristic function is: ε (s)=ετs 2 +εs-1, the auxiliary function is: The system transfer function is: Where τ = RC.

[0155] When R 11 =R 12 =2R 13 、R21 =R 22 =R 23 =R, the transfer function of the forward channel is: The loop transfer function is: The characteristic function is: ε (s)=ετs 2 +εs-τs-3≈ετs 2 -τs-3, the auxiliary function is: The system transfer function is: Where τ = RC.

[0156] Different processing methods can produce very different results when using the same criteria to judge the stability of the system.

[0157] When the system is stable, the differential coefficient ε is treated as ε=0, and the results of analysis using the conventional method and the analysis using the method of the present invention are completely the same.

[0158] Another example can also be used to study the transfer function and system stability of the same system, such as Figure 16 shown.

Claims

1. A method for calculating parameters of a static closed-loop feedback system, which measures the step responses of the forward channel and the loop channel and determines the parameters of a single closed-loop feedback system. The method is applicable to linear closed-loop systems. The method is characterized by: first step: Pre-processing: disconnect the feedback channel of the closed-loop feedback system, with one end of the feedback channel being the input end and the other end being the output end; Step 2: Loop channel measurement: Add a step excitation signal to the input end of the feedback channel break, measure the step response signal at the output end of the feedback channel break, and thus measure the step characteristics of the system loop channel; Step 3: Forward channel measurement: Keep the feedback channel disconnected, add a step excitation signal to the input of the closed-loop system, and measure the step response signal at the output of the closed-loop system to obtain the step characteristics of the forward channel. Step 4: Calculation: Based on the measurement results, the forward channel transfer function G(s) = K is calculated. G and the transfer function of the loop channel Θ(s) = Θ; Step 5: Dynamicize the static system: Insert a micro-inertia link with a very short time constant into the forward channel The transfer function of the forward channel becomes: The transfer function of the loop channel then becomes: The differential coefficient ε>0 and is extremely small; Step 6: Calculation of system auxiliary function: The system auxiliary function is Step 7: Calculation of the system closed-loop transfer function: The closed-loop transfer function of the system is Step 8: System stability judgment: 1-Θ>0 system is stable, 1-Θ<0 system is diverging, 1-Θ=0 system is deadlocked without input and diverges with input; Step 9: Determine the parameter value: If the system is stable, the differential coefficient ε is treated as ε = 0 to calculate the parameters.

2. A method for calculating parameters of a single closed-loop feedback system in a dynamic state, which measures the step response or frequency response of a forward channel and a loop channel and calculates and determines the parameters of the single closed-loop feedback system. The method is applicable to linear closed-loop systems and is characterized by: first step: Pre-processing: disconnect the feedback channel of the closed-loop feedback system, with one end of the feedback channel being the input end and the other end being the output end; Step 2: Loop channel measurement: Add a step excitation signal to the input end of the feedback channel break, measure the step response signal at the output end of the feedback channel break, and thus measure the step characteristics of the system loop channel; or add a sinusoidal excitation signal with a slowly varying frequency to the input end of the feedback channel break, measure the sinusoidal response signal at the output end of the feedback channel break, and thus measure the frequency characteristics of the system loop channel; Step 3: Forward channel measurement: Keep the feedback channel disconnected, add a step excitation signal to the input of the closed-loop system, and measure the step response signal at the output of the closed-loop system to measure the step characteristics of the forward channel; or add a sinusoidal excitation signal with a slowly varying frequency to the input of the closed-loop system, and measure the sinusoidal response signal at the output of the closed-loop system to measure the frequency characteristics of the forward channel; Step 4: Calculation: Based on the measurement results, the same-order transfer function of the forward channel is calculated. and the same transfer function of the loop channel Step 5: Normalize the same system: insert a micro-inertia link with a very short time constant in the forward channel The transfer function of the forward channel becomes: The transfer function of the loop channel then becomes: The differential coefficient ε>0 and is extremely small; Step 6: Calculation of system auxiliary function: The system auxiliary function is Step 7: Calculation of the system closed-loop transfer function: The closed-loop transfer function of the system is Step 8: System stability judgment: system characteristic function Ψ ε (s)=(εs+1)D Θ (s)-M Θ (s), the Routh criterion is used to judge the stability of the system; Step 9: Determine the parameter value: If the system is stable, the differential coefficient ε is treated as ε = 0 to calculate the parameters.

Images