A low-order linear system order direct identification method for low-computational complexity of different real pole linear system

Abstract

The application discloses a low-order different real pole linear system order direct identification method with small operation amount, which comprises the following steps: step one, preparing a constant input excitation and an output quantity collection system; step two, applying the constant input excitation to a system to be measured and collecting the output; step three, calculating sample elements h(r, j); step four, calculating identification base elements g(a, b); steps five to seven, respectively calculating first, second and third identification elements and first, second and third identification factors; step eight, calculating an identification factor χ Rec (1), χ Rec (2), χ Rec (3); step nine, analyzing the order Num of the system to be measured according to the identification factor; and step ten, if Num=-1 and Rec=1, then Rec=2, returning to step four for calculation, otherwise ending. The application lays a solid foundation for safe, efficient and high-performance control and operation of a low-order different real pole linear system.

Description

Technical Field

[0001] The present invention relates to the technical field of linear system order identification, and in particular to a low-order distinct real pole linear system order direct identification method with small computational complexity. Background Art

[0002] System automation and intelligence are the foundation of today's intelligent society. They are rooted in the system model. People usually use system identification to obtain the system model. System identification includes two key links: system order identification and system parameter identification. System order identification mainly determines the order of the system, that is, the complexity of the system model, while the core task of system parameter identification is to estimate the specific parameters in the system model. The former is often the basis of the latter. Most high-order linear systems are composed of low-order sub-modules or systems with different real poles of order not exceeding third order. Therefore, studying the order identification of low-order linear systems with different real poles of order not exceeding third order has very important practical significance for promoting my country's automation and intelligentization process.

[0003] At present, the order identification methods of linear systems with different real poles include three methods: constructing a Hanger matrix to find the rank or using the ratio of the determinants of the product-moment matrix to estimate the order of the model; inferring the order of the system model based on the loss function method; and estimating the model order based on the information criterion.

[0004] These methods typically involve matrix operations, which are computationally intensive, require high hardware costs or long time overhead, and are increasingly unable to meet the hardware cost and real-time requirements of automated and intelligent systems. Identification accuracy relies on relevant criteria and is not direct. Therefore, this paper proposes a low-computation, direct identification method for the order of low-order, distinct real-pole linear systems to address the problems of the existing technology. Summary of the Invention

[0005] To address the above-mentioned issues, the present invention proposes a method for direct identification of the order of low-order, distinct real-pole linear systems with minimal computational effort. This method offers the advantages of strong anti-interference capabilities, high identification accuracy, stable and reliable identification, and is economical, simple, efficient, and direct. It paves the way for rapid, precise, reliable, and high-performance automatic and intelligent control of these low-order, distinct real-pole linear systems, ensuring their safe, efficient, and high-performance operation.

[0006] To achieve the purpose of the present invention, the present invention is implemented by the following technical solutions: First, record a n 、a n-1 、a n-2 ,…,a2,a1,a0,b are constant coefficients. For an n-order linear system with input bu(t) and output y(t), it can be expressed as:

[0007]

[0008] Among them, u(t) is the unit step signal, that is,

[0009] When n=3, the system is a third-order system,

[0010] When n=2, the system is a second-order system,

[0011] When n=1, the system is a first-order system, When n=0, the system is a proportional system or a zero-order system, and a0y(t)=bu(t).

[0012] If the initial state of the system is 0, that is, For m = 0, 1, 2, 3, and y(0) = 0, the step response y(t) of the zero-order to third-order systems can be analyzed. For low-order systems with different poles, the present invention utilizes the characteristics of the poles of each order system and its step response y(t), supplemented by digital interference reduction.

[0013] A low-computation direct identification method for the order of a low-order linear system with distinct real poles comprises the following steps:

[0014] Step 1: Set the input of the system to be tested to zero and remove the constraints to put the system in a zero-equilibrium state, that is, the system's output and state are equal to zero and remain in equilibrium without change. Then prepare a constant input excitation with an amplitude of b and an output y(t) acquisition system.

[0015] Step 2: Apply the constant input stimulus prepared in step 1 to the input of the system under test, and use the prepared acquisition system to collect the output y(t) and save it. Let f(r) = y(rT), where f(r) = y(rT) is the system output value y(t) collected in the rth period, r = 1, 2, 3...N, T is the sampling period, and N is the number of samples collected.

[0016] Step 3: Calculate the sample element h(r, j), h(r, j) = f(r) - f(j), where r = 1, 2, 3 ... N, j = 2, 3 ... N, r ≠ j, initialize the number of identification times Rec = 1, and the order Num = -1;

[0017] Step 4: Let m = BasicL × Rec, r0 = m - BasicL + 1, MaxL be an integer of 0.5N, 20 < BasicL < MaxL, calculate the recognition primitive g(a,c), a = 1, 2, 3, c = 1, 2, 3, 4, where,

[0018]

[0019] Where b = 1, 2, 4;

[0020] Step 5: Calculate the first-level identification elements D111, D112, D121, and D122, as well as the first-level identification factors D11 and D12, where D11 = D111 - D112 and D12 = D121 - D122;

[0021] Step 6: Calculate the secondary identification elements D211, D212, and D221, as well as the secondary identification factors D21 and D22, where D21 = D211 - D212, and D22 = g(2,3)g(2,4)D12 - g(2,4)g(3,3)D12;

[0022] Step 7: Calculate the third-level identification elements D31, D32, D33, D34, D35, and D36, as well as the third-level identification factor D3, where D3 = D31 + D32 + D33 - D34 - D35 - D36;

[0023] Step 8. Calculate the identification factor χ Rec (1) χ Rec (2) χ Rec (3); If D12=0, then χ Rec (1)=3, otherwise If D22=0, then χ Rec (2) = 3, χ Rec (3)=1, otherwise,

[0024] Step 9: The allowable error is recorded as Tol, according to χ Rec (1) χ Rec (2) χ Rec (3) Analyze the order Num of the system to be tested;

[0025] Step 10: If Num = -1 and Rec = 1, then Rec = 2, and go to step 4 for calculation; otherwise, end. At this time, if Num = -1, the system under test is a low-order linear system with non-distinct real poles.

[0026] A further improvement is that D111, D112, D121, D122, D211 and D212 in steps 5 and 6 are calculated by the following formula:

[0027] D111=g(1,1)g(2,3)g(3,4)+g(1,4)g(2,1)g(3,3)+g(1,3)g(2,4)g(3,1

[0028] D112=g(1,1)g(2,4)g(3,3)+g(1,3)g(2,1)g(3,4)+g(1,4)g(2,3)g(3,1

[0029] D121=g(1,2)g(2,3)g(3,4)+g(1,4)g(2,2)g(3,3)+g(1,3)g(2,4)g(3,2

[0030] D122=g(1,2)g(2,4)g(3,3)+g(1,3)g(2,2)g(2,2)+g(1,4)g(2,3)g(3,2

[0031] D211=g(2,1)g(3,4)D12+g(2,4)g(3,2)D11

[0032] D212=g(2,4)g(3,1)D12+g(2,2)g(3,4)D11

[0033] D221=g(2,3)g(3,4)-g(2,4)g(3,3).

[0034] A further improvement is that in step seven, D31, D32, D33, D34, D35 and D36 are calculated by the following formula:

[0035] D31=g(1,1)g(2,2)g(2,3)g(3,3)g(3,4)

[0036] +g(1,1)g(2,3)g(2,4)g(3,2)g(3,3)

[0037] D32=g(1,2)g(2,1)g(2,4)g(3,3)g(3,3)

[0038] +g(1,2)g(2,3)g(2,3)g(3,1)g(3,4)

[0039] D33=g(1,3)g(2,2)g(2,4)g(3,1)g(3,3)

[0040] +g(1,3)g(2,1)g(2,3)g(3,2)g(3,4)

[0041] D34=g(1,1)g(2,2)g(2,4)g(3,3)g(3,3)

[0042] +g(1,1)g(2,3)g(2,3)g(3,2)g(3,4)

[0043] D35=g(1,2)g(2,1)g(2,3)g(3,3)g(3,4)

[0044] +g(1,2)g(2,3)g(2,4)g(3,1)g(3,3)

[0045] D36=g(1,3)g(2,2)g(2,3)g(3,1)g(3,4)

[0046] +g(1,3)g(2,1)g(2,4)g(3,2)g(3,4).

[0047] A further improvement is that: in the step eight, when x Rec (1) = χ Rec (2) = 3, χ Rec (3)=1, then Num=0, the system to be measured is a zero-order system, that is, a proportional system; when Num=-1 and |χ Rec (1)-χ Rec (3)-2|≤Tol,|χ Rec (2)-2χ Rec (3) -1|≤Tol, then Num=1, the system to be measured is a first-order system; when Num=-1 and |χ Rec (1)-χ Rec (2)+χ Rec (3)-1|≤Tol, then Num=2, and the system to be measured is a second-order system; when Num=-1, Rec=2, and |χ1(1)-χ2(1)|≤Tol, |χ1(2)-χ2(2)|≤Tol, |χ1(3)-χ2(3)|≤Tol, then Num=3, and the system to be measured is a third-order system.

[0048] A further improvement is that: when Num=1, the identification error IdError=max{ Rec (1)-χ Rec (3)-2, χ Rec (2)-2χ Rec (3)-1}; when Num=2, calculate the identification error IdError=χ Rec (1)-χ Rec (2)+χ Rec (3)-1; when Num=3, the identification error IdError=max{|χ1(1)-χ2(1)|, |χ1I2)-χ2(2)|, |χ1(3)-χ2I3)|} is calculated.

[0049] The beneficial effects of the present invention are as follows: the method of the present invention identifies the order of low-order different real pole linear systems, and has the advantages of strong anti-interference ability, high identification accuracy, stable and reliable identification, economy, simplicity, efficiency and directness, which lays a solid foundation for fast, accurate, reliable and high-performance automatic and intelligent control of these low-order different real pole linear systems, allowing them to operate safely, efficiently and with high performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 Flow chart of the method of the present invention.

[0051] Figure 2 This is a diagram of the test system architecture of an embodiment of the present invention.

[0052] Figure 3 2 is a circuit model diagram of each stage in an embodiment of the present invention. DETAILED DESCRIPTION

[0053] In order to deepen the understanding of the present invention, the present invention will be further described in detail below with reference to the examples. The examples are only used to explain the present invention and do not constitute a limitation on the scope of protection of the present invention.

[0054] Example

[0055] according to Figure 1 、 Figure 2 and Figure 3 As shown, this embodiment provides an experimental simulation of a method for directly identifying the order of a low-order linear system with different real poles with a small amount of computation, as follows:

[0056] As the instruction manual Figure 2 As shown in the figure, a second-order linear lumped parameter circuit model was established in the Simulink environment on the Matlab platform, with the capacitor voltage as the output, denoted as u2(t). The voltage sensor and sampling module are used to collect u2(t) and send it to the work platform for storage. White noise simulates the interference signal in actual work. It is mixed with the step signal with an amplitude of 20V. Together with the step signal, it controls the controlled voltage source to generate a 20V step DC voltage source with interference signal to supply this circuit. In the figure, R2 = 6Ω, L2 = 1H, and C2 = 0.2F. The mathematical model of this circuit is:

[0057]

[0058] The two extreme points are -1 and -5 respectively.

[0059] The specific experimental process is:

[0060] (1) The circuit input is set to 0V, the capacitor voltage in the circuit is initialized to 0V, and the inductor current is initialized to 0A, completing the zero-state initialization of the circuit.

[0061] (2) Set the gain after white noise to 0, that is, no interference is mixed in, and the circuit and detection system will work in an ideal condition. Apply a 20V step DC voltage source to this circuit, and use the acquisition system to collect the output u2It) of this circuit with a period of T = 0.13 microseconds and save it. Stop sampling after 1s and stop the excitation. Let f(r) = u2(rT), which is the circuit output value u2(t) collected in the rth period, r = 1, 2, 3.......7.69×10 6 .

[0062] (3) Calculate the sample element h(r,j) = f(r) - f(j), where r = 1, 2, 3...N, j = 2, 3...N, and r ≠ j. Initialize the number of identifications Rec = 1, the order Num = -1, and BasicL = 100.

[0063] (4) m = BasicL × Rec, r0 = m-BasicL+1, a = 1, 2, 3, b = 1, 2, 4, calculate the recognition primitive g(a, c), c = 1, 2, 3, 4.

[0064]

[0065] (5) Calculation

[0066] D111=g(1,1)g(2,3)g(3,4)+g(1,4)g(2,1)g(3,3)+g(1,3)g(2,4)g(3,1)

[0067] D112=g(1,1)g(2,4)g(3,3)+g(1,3)g(2,1)g(3,4)+g(1,4)g(2,3)g(3,1)

[0068] D121=g(1,2)g(2,3)g(3,4)+g(1,4)g(2,2)g(3,3)+g(1,3)g(2,4)g(3,2),

[0069] D122=g(1,2)g(2,4)g(3,3)+g(1,3)g(2,2)g(3,4)+g(1,4)g(2,3)g(3,2)

[0070] D11=D111―D112=-4.0390×10 -28

[0071] D12=D121―D122=-4.0390×10 -28 .

[0072] (6) Calculation

[0073] D211=g(2,1)g(3,4)D12+g(2,4)g(3,2)D11

[0074] D212=g(2,4)g(3,1)D12+g(2,2)g(3,4)D11

[0075] D21=D211―D212=1.3364×10 -51

[0076] D22=g(2,3)g(3,4)―g(2,4)g(3,3)=1.3364×10 -51 .

[0077] (7) Calculation

[0078] D31=g(1,1)g(2,2)g(2,3)g(3,3)g(3,4)+g(1,1)g(2,3)g(2,4)g(3,2)g(3,3)D32=g(1,2)g(2,1)g(2,4)g(3,3)g(3, 3)+g(1,2)g(2,3)g(2,3)g(3,1)g(3,4)D33=g(1,3)g(2,2)g(2,4)g(3,1)g(3,3)+g(1,3)g(2,1)g(2,3)g(3,2)g(3,4) D34=g(1,1)g(2,2)g(2,4)g(3,3)g(3,3)+g(1,1)g(2,3)g(2,3)g(3,2)g(3,4)D35=g(1,2)g(2,1)g(2,3)g(3,3)g(3, 4)+g(1,2)g(2,3)g(2,4)g(3,1)g(3,3)D36=g(1,3)g(2,2)g(2,3)g(3,1)g(3,4)+g(1,3)g(2,1)g(2,4)g(3,2)g(3,3)

[0079] D3=D31+D32+D33―D34―D35―D36=1.3350×10 -51 .

[0080] (8) Calculate the identification factor χ Rec (1) χ Rec (2) χ Rec (3), D12 and D22 are small but not zero, so calculate They are equal to 1, 1, and 0.9990 respectively.

[0081] (9) The allowable error is recorded as Tol = 0.001, χ Rec (1) χ Rec (2) and χRec (3) The three conditions satisfy |χ Rec (1)-χ Rec (2)+χ Rec (3)-1|≤Tol, so Num=2, IdError=-0.0044.

[0082] (10) The final result of the experiment is: Num = 2, indicating that the circuit with output y3(t) is a second-order system.

[0083] The recognition result of step (10) shows that when there is no interference, the above method proposed in this patent can fully accurately and effectively recognize the Figure 2 However, in actual work, interference is inevitable.

[0084] In order to examine the anti-interference ability of the above method of this patent, the above experiments were also carried out many times. The difference was that the "setting the gain after white noise to 0, that is, no interference mixing" in the above step (2) was changed to "adjusting the gain after white noise to" 5, 10, 20, 40, 60.

[0085] Next, attach the instructions to Figure 3 The same identification is performed on the zero-order, first-order, third-order, and fourth-order circuit systems shown in (a), (b), (c), and (d) to comprehensively verify the method. In the figure, R0 = 100Ω, R1 = 10Ω, C1 = 0.5F, R2 = 6Ω, L2 = 1H, C2 = 0.5F, R3 = 1Ω, C3 = 1F, R 40 =1Ω, R 41 =1Ω, C4=1F. Their mathematical models are

[0086] i1(t)=0.2u(t)

[0087]

[0088] The poles are 0, -0.2, {-3.2470, -1.5550, -0.1981}, {-0.7794, -0.7013, -0.5000, -0.1076}, respectively. The identification results are shown in Tables 1 and 2. Table 1 lists the identification factors calculated based on the step responses of each circuit in the experiment, and Table 2 lists the system orders calculated based on the relationships between these identification factors.

[0089] Table 1 Specific conditions of low-order linear circuit identification factors in the experiment

[0090]

[0091]

[0092] Table 2 Specific situation of the order Num (identification error IdError) of the low-order linear circuit in the experiment

[0093]

[0094] Table 2 shows that the identification method proposed in this patent can correctly and effectively identify the order of the five circuits in the experiment, both for ideal systems without interference and for systems with interference signal amplitudes no greater than three times the signal amplitude. Tables 1 and 2 show that the identification error of this method gradually increases with increasing white noise amplitude or noise-to-signal ratio, but this does not affect the final identification results. Specifically, this method achieves low order discrimination error and high accuracy for second-order and third-order systems, and exhibits excellent robustness to interference. When the interference amplitude is 300% of the signal, the method accurately identifies their orders with errors of 0.0290 and 3.2389, respectively. For first-order systems, the method is relatively sensitive to interference but also has strong interference immunity. When the interference amplitude is 300% of the signal, it can identify the system as first-order with an identification error of -0.6061. Similar studies have also been conducted on other low-order systems with different real poles, with similar results. These situations show that the identification method proposed in this patent has strong anti-interference ability, high identification accuracy, and stable and reliable identification.

[0095] The basic principles, main features, and advantages of the present invention are shown and described above. Those skilled in the art should understand that the present invention is not limited to the foregoing embodiments. The foregoing embodiments and descriptions are merely illustrative of the principles of the present invention. Various changes and modifications may be made to the present invention without departing from the spirit and scope of the present invention. Such changes and modifications are intended to fall within the scope of the present invention. The scope of protection claimed in the present invention is defined by the appended claims and their equivalents.

Claims

1. A low-computation direct identification method for the order of low-order linear systems with distinct real poles, characterized by: The following steps are involved: Step 1: A second-order linear lumped parameter circuit model was established in the Simulink environment on the Matlab platform, with the capacitor voltage as the output, denoted as u2(t). A voltage sensor and sampling module were used to collect u2(t) and send it to the work platform for storage. White noise simulated the interference signal in actual operation. It was mixed into a step signal with an amplitude of 20V and, together with the step signal, controlled the controlled voltage source to generate a 20V step DC voltage source with the interference signal to supply this circuit. Set the input of the system to be tested to zero and remove the constraints to make the system in a zero-equilibrium state, that is, the output and state of the system are equal to zero and remain in equilibrium without change. Then prepare a constant input excitation with an amplitude of b and an output y(t) acquisition system. Step 2: Apply the constant input stimulus prepared in step 1 to the input of the system under test, and use the prepared acquisition system to collect the output y(t) and save it. Let f(r) = y(rT), where f(r) = y(rT) is the system output value y(t) collected in the rth period, r = 1, 2, 3...N, T is the sampling period, and N is the number of samples collected. Step 3: Calculate the sample element h(r, j), h(r, j) = f(r) - f(j), where r = 1, 2, 3 ... N, j = 2, 3 ... N, r ≠ j, initialize the number of identification times Rec = 1, and the order Num = -1; Step 4: Let m = BasicL × Rec, r0 = m - BasicL + 1, MaxL be an integer of 0.5N, 20 < BasicL < MaxL, calculate the recognition primitive g(a,c), a = 1, 2, 3, c = 1, 2, 3, 4, where Where b = 1, 2, 4; Step 5: Calculate the first-level identification elements D111, D112, D121, and D122, as well as the first-level identification factors D11 and D12, where D11 = D111 - D112 and D12 = D121 - D122; Step 6: Calculate the secondary identification elements D211, D212, and D221, as well as the secondary identification factors D21 and D22, where D21 = D211 - D212, and D22 = g(2,3)g(2,4)D12 - g(2,4)g(3,3)D12; Step 7: Calculate the third-level identification elements D31, D32, D33, D34, D35, and D36, as well as the third-level identification factor D3, where D3 = D31 + D32 + D33 - D34 - D35 - D36; Step 8. Calculate the identification factor χ Rec (1) χ Rec (2) χ Rec (3); If D12=0, then χ Rec (1)=3, otherwise If D22=0, then χ Rec (2) = 3, χ Rec (3)=1, otherwise, Step 9: The allowable error is recorded as Tol, according to χ Rec (1) χ Rec (2) χ Rec (3) Analyze the order Num of the system to be tested; Step 10: If Num = -1 and Rec = 1, then Rec = 2, and go to step 4 for calculation; otherwise, end. At this time, if Num = -1, the system under test is a low-order linear system with non-distinct real poles.

2. The method for direct identification of the order of a low-order linear system with distinct real poles with low computational complexity according to claim 1, characterized in that: In the steps 5 and 6, D111, D112, D121, D122, D211 and D212 are calculated by the following formula D111=g(1,1)g(2,3)g(3,4)+g(1,4)g(2,1)g(3,3)+g(1,3)g(2,4)g(3,1) D112 = g(1,1)g(2,4)g(3,3)+g(1,3)g(2,1)g(3,4)+g(1,4)g(2,3)g(3,1) D121 = g(1,2)g(2,3)g(3,4)+g(1,4)g(2,2)g(3,3)+g(1,3)g(2,4)g(3,2) D122 = g(1,2)g(2,4)g(3,3)+g(1,3)g(2,2)g(2,2)+g(1,4)g(2,3)g(3,2) D211 = g(2,1)g(3,4)D12+g(2,4)g(3,2)D11 D212 = g(2,4)g(3,1)D12+g(2,2)g(3,4)D11 D221 = g(2,3)g(3,4)-g(2,4)g(3,3).

3. The method for direct identification of the order of a low-order linear system with distinct real poles with low computational complexity according to claim 1, characterized in that: In step seven, D31, D32, D33, D34, D35 and D36 are calculated by the following formula D31 = g(1,1)g(2,2)g(2,3)g(3,3)g(3,4)+g(1,1)g(2,3)g(2,4)g(3,2)g(3,3) D32 = g(1,2)g(2,1)g(2,4)g(3,3)g(3,3)+g(1,2)g(2,3)g(2,3)g(3,1)g(३,4) D33 = g(1,3)g(2,2)g(2,4)g(3,1)g(3,3)+g(1,3)g(2,1)g(2,3)g(3,2)g(3,4) D34 = g(1,1)g(2,2)g(2,4)g(3,3)g(3,3)+g(1,1)g(2,3)g(2,3)g(3,2)g(3,4) D35 = g(1,2)g(2,1)g(2,3)g(3,3)g(3,4)+g(1,2)g(2,3)g(2,4)g(3,1)g(3,3) D36 = g(1,3)g(2,2)g(2,3)g(3,1)g(3,4)+g(1,3)g(2,1)g(2,4)g(3,2)g(3,4).

4. The method for direct identification of the order of a low-order linear system with distinct real poles with low computational complexity according to claim 1, characterized in that: In the step eight, when x Rec (1) = χ Rec (2) = 3, χ Rec (3)=1, then Num=0, the system to be measured is a zero-order system, that is, a proportional system; when Num=-1 and |χ Rec (1)-χ Rec (3)-2|≤Tol,|χ Rec (2)-2χ Rec (3) -1|≤Tol, then Num=1, the system to be measured is a first-order system; when Num=-1 and |χ Rec (1)-χ Rec (2)+χ Rec (3)-1|≤Tol, then Num=2, and the system to be measured is a second-order system; when Num=-1, Rec=2, and |χ1(1)-χ2(1)|≤Tol, |χ1(2)-χ2(2)|≤Tol, |χ1(3)-χ2(3)|≤Tol, then Num=3, and the system to be measured is a third-order system.

5. The method for direct identification of the order of a low-order linear system with distinct real poles with low computational complexity according to claim 4, characterized in that: When Num=1, the identification error IdError=max{ Rec (1)-χ Rec (3)-2, χ Rec (2)-2χ Rec (3)-1}; when Num=2, calculate the identification error IdError=χ Rec (1)-χ Rec (2)+χ Rec (3)-1; when Num=3, the identification error IdError=max{|χ1(1)-χ2(1)|, |χ1(2)-χ2(2)|, |χ1(3)-χ2(3)|} is calculated.

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