CN107356282A - Bullet train robust interval Transducer-fault Detecting Method in the case of resolution limitations - Google Patents

Bullet train robust interval Transducer-fault Detecting Method in the case of resolution limitations Download PDF

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CN107356282A
CN107356282A CN201710483450.4A CN201710483450A CN107356282A CN 107356282 A CN107356282 A CN 107356282A CN 201710483450 A CN201710483450 A CN 201710483450A CN 107356282 A CN107356282 A CN 107356282A
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周东华
张峻峰
何潇
卢晓
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Shandong University of Science and Technology
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Shandong University of Science and Technology
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    • G01—MEASURING; TESTING
    • G01D—MEASURING NOT SPECIALLY ADAPTED FOR A SPECIFIC VARIABLE; ARRANGEMENTS FOR MEASURING TWO OR MORE VARIABLES NOT COVERED IN A SINGLE OTHER SUBCLASS; TARIFF METERING APPARATUS; MEASURING OR TESTING NOT OTHERWISE PROVIDED FOR
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Abstract

The invention discloses bullet train robust interval Transducer-fault Detecting Method in the case of a kind of resolution limitations, belongs to field of signal processing, and this method includes establishing soft-sensing model step, designs robust Residual Generation device step, design error failure inspection policies step.The present invention can real time on-line monitoring sensor states, ensure the steady safe operation of bullet train, meet the practical application request of interval Transducer fault detection.

Description

Bullet train robust interval Transducer-fault Detecting Method in the case of resolution limitations
Technical field
The invention belongs to signal processing technology field, specific one kind is related in the case of resolution limitations between bullet train robust Have a rest Transducer-fault Detecting Method.
Background technology
Numerous sensors are included in bullet train, and working environment residing for these sensors is severe, intermittent fault occurs Situation is more universal, and huge potential safety hazard is constituted to bullet train safe operation.
Technology for Transducer Fault Detection can be monitored using various redundancies to sensor, be that raising sensor can By the effective ways of property, extensive concern is obtained.But existing robust sensor fault detection method heavy dependence model is not true Qualitative framework, and need to obtain the prior informations such as corresponding uncertainty structure and structural parameters.In addition, existing method is generally only Permanent sensors failure can be detected, the detection of interval sensor fault can not be carried out in the case of resolution limitations.
The content of the invention
For above-mentioned technical problem present in prior art, the present invention proposes high speed in the case of a kind of resolution limitations Train robust interval Transducer-fault Detecting Method, real time on-line monitoring sensor states, ensure that bullet train is steadily transported safely OK, it is reasonable in design, the deficiencies in the prior art are overcome, there is good effect.
To achieve these goals, the present invention adopts the following technical scheme that：
Bullet train robust interval Transducer-fault Detecting Method, comprises the following steps in the case of a kind of resolution limitations：
Step 1：Establish soft-sensing model
Shown in the state equation of bullet train critical system such as formula (1)：
X (k+1)=(Ac(k)+Aδ(k))x(k)+(Bc(k)+Bδ(k))u(k)+w(k), (1)；
Wherein,For system mode；For control input；For process noise；For procedure parameter；It is not true for procedure parameter It is qualitative；
Resolution ratio isSensor actual measurement equation such as formula (2) shown in：
Y (k)=(Cc(k)+Cδ(k))x(k)+v(k)+Δ(y(k))+f(k), (2)；
Wherein,(meet for actual measurement output)；Lured for resolution ratio Lead uncertainty；For measurement noise；For measurement parameter；For measurement parameter not Certainty；
Shown in interval sensor fault equation such as formula (3)：
Wherein,For interval sensor fault；For the pattern of i-th of intermittent fault；For The time of origin of i intermittent fault；For the extinction time of i-th of intermittent fault；Γ () is unit jump function.
According to above-mentioned state equation and actual measurement establishing equation soft-sensing model such as formula (4)：
Y (k)=(Cc(k)+Cδ(k))x(k)+v(k)+Δ(Y(k))+f(k), (4)；
Wherein,Exported for hard measurement；Induced for hard measurement uncertain.
Step 2：Design robust Residual Generation device
Step 2.1：State estimator initial value is set according to formula (5)：
Step 2.2：According to formula (6) computing system state average：
μx(k)=Ac(k-1)μx(k-1)+Bc(k-1)u(k-1), (6)；
Step 2.3：According to formula (7) computing system state second moment：
Step 2.4：Intermediate variable is calculated according to formula (8), formula (9)：
Wherein αl, l ∈ { 1,2,3,4,5 } are positive count；For on k-1 moment state estimation error covariances Boundary；
Step 2.5：State estimator gain is calculated according to formula (10)：
Kx(k)=H (k) Cc(k)TQ(k)-1, (10)；
Step 2.6：The state estimation error covariance upper bound is calculated according to formula (11)：
Step 2.7：State estimation is calculated according to formula (12), (13)：
Step 2.8：Residual error is calculated according to formula (14)：
Step 3：Design error failure inspection policies
Step 3.1：Fault detect statistic is calculated according to formula (15)
TD(k)=r (k)Tr(k), (15)；
Step 3.2：Failure determination threshold value is calculated according to formula (16)
Step 3.3：Set fault detection logic
If TD(k-1)≤JD(k-1),TD(k) ＞ JD(k), then the k moment breaks down, fault warning indicatrix Ia=1,
If TD(k-1) ＞ JD(k-1),TD(k)≤JD(k), then k moment failure vanishes, trouble shooting indicatrix Ir=1.It is excellent Selection of land, in step 1, specifically comprise the following steps：
Step 1.1：Calculate preferable measurement output average
Step 1.2：Calculate preferable measurement output second moment
Step 1.3：Calculate hard measurement output
Wherein,
Step 1.4：Calculate hard measurement and induce uncertain second moment
Wherein,
Advantageous effects caused by the present invention：
The present invention can real time on-line monitoring sensor states, ensure the steady safe operation of bullet train, meet interval pass The practical application request of sensor fault detect.
Brief description of the drawings
Fig. 1 is the flow of bullet train robust interval Transducer-fault Detecting Method in the case of resolution limitations of the present invention Figure.
Fig. 2 is the structure of bullet train robust interval Transducer-fault Detecting Method in the case of resolution limitations of the present invention Figure.
Fig. 3 is No. 1, and sensor resolution induction is uncertain and hard measurement induces probabilistic square curve map.
Fig. 4 is No. 2, and sensor resolution induction is uncertain and hard measurement induces probabilistic square curve map.
Fig. 5 is the testing result schematic diagram that intermittent fault occurs for No. 1 sensor in the present invention.
Fig. 6 is the testing result schematic diagram that intermittent fault occurs for No. 2 sensors in the present invention.
Fig. 7 is No. 1 and No. 2 sensors while the testing result schematic diagram that intermittent fault occurs in the present invention.
Embodiment
Below in conjunction with the accompanying drawings and embodiment is described in further detail to the present invention：
Bullet train robust interval Transducer-fault Detecting Method in the case of a kind of resolution limitations, its flow such as Fig. 1 institutes Show, its structure is as shown in Fig. 2 specifically comprise the following steps：
Step 1：Establish soft-sensing model
Shown in the state equation of bullet train critical system such as formula (1)：
X (k+1)=(Ac(k)+Aδ(k))x(k)+(Bc(k)+Bδ(k))u(k)+w(k), (1)；
Wherein,For system mode；For control input；For process noise；For procedure parameter；It is not true for procedure parameter It is qualitative.
Resolution ratio isSensor actual measurement equation such as formula (2) shown in：
Y (k)=(Cc(k)+Cδ(k))x(k)+v(k)+Δ(y(k))+f(k), (2)；
Wherein,(meet for actual measurement output)；Lured for resolution ratio Uncertainty is led, its square curve is as shown in Figure 3, Figure 4；For measurement noise；For measurement parameter；It is uncertain for measurement parameter.
Shown in interval sensor fault equation such as formula (3)：
Wherein,For interval sensor fault；For the pattern of i-th of intermittent fault；For The time of origin of i intermittent fault；For the extinction time of i-th of intermittent fault；Γ () is unit jump function.
Above-mentioned stochastic variable meets following condition：
Initial system state x (0) average isCovariance is P0, second moment Σ0；Noise w (k), v (k) average It is zero, covariance matrix is respectively Σw(k), Σv(k)；Parameter uncertainty Aδ(k),Bδ(k),Cδ(k) average is zero, covariance Matrix is respectively
Subscript (j) represents the jth dimension component of vector in the present invention.
According to above-mentioned state equation and actual measurement establishing equation soft-sensing model such as formula (4)：
Y (k)=(Cc(k)+Cδ(k))x(k)+v(k)+Δ(Y(k))+f(k), (4)；
Wherein,Exported for hard measurement；Uncertain, its equal Fang Qu is induced for hard measurement Line is as shown in Figure 3, Figure 4.
Circular is：
Step 1.1：Calculate preferable measurement output average
Step 1.2：Calculate preferable measurement output second moment
Step 1.3：Calculate hard measurement output
Wherein,
Step 1.4：Calculate hard measurement and induce uncertain second moment
Wherein,
Step 2：Design robust Residual Generation device
Step 2.1：State estimator initial value is set according to formula (5)：
Step 2.2：According to formula (6) computing system state average：
μx(k)=Ac(k-1)μx(k-1)+Bc(k-1)u(k-1), (6)；
Step 2.3：According to formula (7) computing system state second moment：
Step 2.4：Intermediate variable is calculated according to formula (8), formula (9)：
Wherein αl, l ∈ { 1,2,3,4,5 } are positive count.
Step 2.5：State estimator gain is calculated according to formula (10)：
Kx(k)=H (k) Cc(k)TQ(k)-1, (10)；
Step 2.6：The state estimation error covariance upper bound is calculated according to formula (11)：
Step 2.7：State estimation is calculated according to formula (12), (13)：
Step 2.8：Residual error is calculated according to formula (14)：
Step 3：Design error failure inspection policies
Step 3.1：Fault detect statistic is calculated according to formula (15)
TD(k)=r (k)Tr(k), (15)；
Step 3.2：Failure determination threshold value is calculated according to formula (16)
Step 3.3：Set fault detection logic
If TD(k-1)≤JD(k-1),TD(k) ＞ JD(k), then the k moment breaks down, fault warning indicatrix Ia=1,
If TD(k-1) ＞ JD(k-1),TD(k)≤JD(k), then k moment failure vanishes, trouble shooting indicatrix Ir=1.
Testing result is as shown in Fig. 5, Fig. 6, Fig. 7.
Certainly, described above is not limitation of the present invention, and the present invention is also not limited to the example above, this technology neck The variations, modifications, additions or substitutions that the technical staff in domain is made in the essential scope of the present invention, it should also belong to the present invention's Protection domain.

Claims (2)

1. bullet train robust interval Transducer-fault Detecting Method in the case of resolution limitations, it is characterised in that：Including as follows Step：
Step 1：Establish soft-sensing model
Shown in the state equation of bullet train critical system such as formula (1)：
X (k+1)=(Ac(k)+Aδ(k))x(k)+(Bc(k)+Bδ(k))u(k)+w(k), (1)；
Wherein,For system mode；For control input；For process noise；For procedure parameter；It is not true for procedure parameter It is qualitative；
Resolution ratio isSensor actual measurement equation such as formula (2) shown in：
Y (k)=(Cc(k)+Cδ(k))x(k)+v(k)+Δ(y(k))+f(k), (2)；
Wherein,(meet for actual measurement output)；For resolution ratio induction not Certainty；For measurement noise；For measurement parameter；Do not known for measurement parameter Property；
Shown in interval sensor fault equation such as formula (3)：
<mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>&amp;infin;</mi> </munderover> <msub> <mi>f</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>&amp;Gamma;</mi> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <msub> <mi>k</mi> <mrow> <mi>o</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> <mo>-</mo> <mi>&amp;Gamma;</mi> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <msub> <mi>k</mi> <mrow> <mi>d</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
Wherein,For interval sensor fault；For the pattern of i-th of intermittent fault；For i-th The time of origin of intermittent fault；For the extinction time of i-th of intermittent fault；Γ () is unit jump function；
According to above-mentioned state equation and actual measurement establishing equation soft-sensing model such as formula (4)：
Y (k)=(Cc(k)+Cδ(k))x(k)+v(k)+Δ(Y(k))+f(k), (4)；
Wherein,Exported for hard measurement；Induced for hard measurement uncertain；
Step 2：Design robust Residual Generation device
Step 2.1：State estimator initial value is set according to formula (5)：
Step 2.2：According to formula (6) computing system state average：
μx(k)=Ac(k-1)μx(k-1)+Bc(k-1)u(k-1), (6)；
Step 2.3：According to formula (7) computing system state second moment：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mo>&amp;Sigma;</mo> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mo>&amp;Sigma;</mo> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <msub> <mi>A</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>A</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>B</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mo>&amp;Sigma;</mo> <mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <msub> <mi>B</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>B</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mo>&amp;Sigma;</mo> <mrow> <mi>w</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>A</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>&amp;mu;</mi> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mi>u</mi> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msub> <mi>B</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>B</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msup> <msub> <mi>&amp;mu;</mi> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mi>T</mi> </msup> <msub> <mi>A</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
Step 2.4：Intermediate variable is calculated according to formula (8), formula (9)：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>Q</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>C</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>H</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>C</mi> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>C</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msub> <mi>A</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>C</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msub> <mi>A</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>&amp;mu;</mi> <mrow> <msub> <mi>C</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>c</mi> </msub> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <msub> <mi>&amp;mu;</mi> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mi>u</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msub> <mi>B</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>)</mo> </mrow> <msub> <mi>C</mi> <mi>&amp;delta;</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mrow> </msub> <mo>+</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;mu;</mi> <mrow> <msub> <mi>C</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>c</mi> </msub> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <msub> <mi>&amp;mu;</mi> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mi>u</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msub> <mi>B</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>)</mo> </mrow> <msub> <mi>C</mi> <mi>&amp;delta;</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <msub> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>C</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <msub> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>C</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msub> <mi>B</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mn>4</mn> </msub> <mo>)</mo> </mrow> <msub> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>C</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>w</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;alpha;</mi> <mn>5</mn> </msub> <mo>)</mo> </mrow> <msub> <mo>&amp;Sigma;</mo> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>5</mn> </munderover> <mrow> <mo>(</mo> <msup> <msub> <mi>&amp;alpha;</mi> <mi>l</mi> </msub> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>t</mi> <mi>r</mi> <mo>(</mo> <msub> <mo>&amp;Sigma;</mo> <mrow> <mi>&amp;Delta;</mi> <mrow> <mo>(</mo> <mi>Y</mi> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> <msub> <mi>I</mi> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mo>&amp;Sigma;</mo> <mrow> <mi>&amp;Delta;</mi> <mrow> <mo>(</mo> <mi>Y</mi> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>I</mi> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
Wherein αl, l ∈ { 1,2,3,4,5 } are positive count；For the k-1 moment state estimation error covariances upper bound；
Step 2.5：State estimator gain is calculated according to formula (10)：
Kx(k)=H (k) Cc(k)TQ(k)-1, (10)；
Step 2.6：The state estimation error covariance upper bound is calculated according to formula (11)：
Step 2.7：State estimation is calculated according to formula (12), (13)：
<mrow> <mi>r</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>Y</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msub> <mi>A</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
<mrow> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>K</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>r</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
Step 2.8：Residual error is calculated according to formula (14)：
<mrow> <mi>r</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>Y</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
Step 3：Design error failure inspection policies
Step 3.1：Fault detect statistic is calculated according to formula (15)
TD(k)=r (k)Tr(k), (15)；
Step 3.2：Failure determination threshold value is calculated according to formula (16)
Step 3.3：Set fault detection logic
If TD(k-1)≤JD(k-1),TD(k) ＞ JD(k), then the k moment breaks down, fault warning indicatrix Ia=1,
If TD(k-1) ＞ JD(k-1),TD(k)≤JD(k), then k moment failure vanishes, trouble shooting indicatrix Ir=1.
2. bullet train robust interval Transducer-fault Detecting Method in the case of resolution limitations according to claim 1, It is characterized in that：In step 1, specifically comprise the following steps：
Step 1.1：Calculate preferable measurement output average
Step 1.2：Calculate preferable measurement output second moment
Step 1.3：Calculate hard measurement output
Wherein,
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mo>(</mo> <msqrt> <mfrac> <mrow> <mi>y</mi> <mo>-</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> </msqrt> <mo>(</mo> <mrow> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mfrac> <msup> <mrow> <mo>(</mo> <mi>v</mi> <mo>-</mo> <mi>x</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mi>y</mi> <mo>-</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>}</mo> <mo>-</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mfrac> <msup> <mrow> <mo>(</mo> <mi>w</mi> <mo>-</mo> <mi>x</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mi>y</mi> <mo>-</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>}</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;times;</mo> <msup> <mrow> <mo>(</mo> <mi>&amp;Phi;</mi> <mo>(</mo> <mfrac> <mrow> <mi>w</mi> <mo>-</mo> <mi>x</mi> </mrow> <msqrt> <mrow> <mi>y</mi> <mo>-</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>)</mo> <mo>-</mo> <mi>&amp;Phi;</mi> <mo>(</mo> <mfrac> <mrow> <mi>v</mi> <mo>-</mo> <mi>x</mi> </mrow> <msqrt> <mrow> <mi>y</mi> <mo>-</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>)</mo> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
Step 1.4：Calculate hard measurement and induce uncertain second moment
Wherein,
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>h</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>y</mi> <mo>-</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>-</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mo>(</mo> <mi>x</mi> <mo>-</mo> <mn>2</mn> <mi>z</mi> <mo>+</mo> <mi>v</mi> <mo>)</mo> </mrow> <msqrt> <mfrac> <mrow> <mi>y</mi> <mo>-</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> </msqrt> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mfrac> <msup> <mrow> <mo>(</mo> <mi>v</mi> <mo>-</mo> <mi>x</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mi>y</mi> <mo>-</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>-</mo> <mn>2</mn> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mo>)</mo> </mrow> <msqrt> <mfrac> <mrow> <mi>y</mi> <mo>-</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> </msqrt> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mfrac> <msup> <mrow> <mo>(</mo> <mi>w</mi> <mo>-</mo> <mi>x</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mi>y</mi> <mo>-</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>}</mo> <mo>)</mo> <mo>&amp;times;</mo> <msup> <mrow> <mo>(</mo> <mi>&amp;Phi;</mi> <mo>(</mo> <mfrac> <mrow> <mi>w</mi> <mo>-</mo> <mi>x</mi> </mrow> <msqrt> <mrow> <mi>y</mi> <mo>-</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>)</mo> <mo>-</mo> <mi>&amp;Phi;</mi> <mo>(</mo> <mfrac> <mrow> <mi>v</mi> <mo>-</mo> <mi>x</mi> </mrow> <msqrt> <mrow> <mi>y</mi> <mo>-</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>)</mo> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow> 3
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