CN107168925A - A kind of state of linear system and Unknown inputs method - Google Patents
A kind of state of linear system and Unknown inputs method Download PDFInfo
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
Abstract
The present invention relates to a kind of state of linear system and Unknown inputs method, comprise the following steps：S1, obtains the input vector and output vector of the linear time varying system with Unknown worm, introduces auxiliary output vector, system is met observer matching condition, then designs High-Order Sliding Mode observer, obtains aiding in the estimation of output vector；S2, it regard the estimation for aiding in output vector as output vector, it regard the input vector of original system as input vector, construction new linear time varying system and its reduced dimension observer, new system mode is estimated, and according to the estimation of the obtained auxiliary output vector derivative of state estimation result and step S1, reconstruct Unknown worm compared with prior art, the present invention can eliminate the state of the influence of Unknown worm and accurate estimating system, the derivative of auxiliary output will be used to reconstruct Unknown worm, with certain scientific theory meaning and actual application background.
Description
Technical field
The present invention relates to a kind of method for estimating state of linear system, more particularly, to a kind of state of linear system and not
Know input method of estimation.
Background technology
With modern industry produce it is growing, by engineering, actually the abstract mathematical modeling system drawn is also increasingly multiple
Miscellaneous, to meet the higher and higher requirement for controlling it to propose, computer has become indispensable effective in control field
Instrument.Due to discrete data signal can only be handled when computer is in data storage, calculating, when a system is real with computer
It is discrete variable to be needed when now controlling variables transformations, and the system of research is also accordingly discrete-time system, therefore discrete system
The research of system is of increased attention.Although the part conclusion of continuous system can be generalized to discrete system, from
The self structure for the system of dissipating and the control increasingly improved require to determine that it should also have the research method different from continuous system.
Research for Unknown Input Observer (UIO) can be traced back in the 1970s, most causing in the late three decades
The extensive attention of many researchers, in early stage, how main discussion avoids the influence of unknown disturbances, later state estimation and system
The method of reconstruct turns into the focus discussed.
Particularly, State Observer that is, state reconstruction problem, utilizing can directly observe in original system
Information (as inputted and exporting) constructs a new system as input signal, and make it that the output signal of new system is asymptotic or refers to
Number converges on the state vector of original system, and this realizes that the new system of state reconstruction is exactly observer.According to the function of observer,
Two classes can be divided into；One is the state observer for reconstructing original system state；Two be reconstruct original system function of state (such as feedback linearization
Function) Function Observer.According to the structure of observer, state observer can be divided into two classes：One is with being observed system
The consistent full micr oprocessorism of dimension, two be the reduced dimension observer that dimension is less than the system that is observed.
For linear system, the concept of optimal linear filtering device is introduced in nineteen sixty by Kalman so that mean square deviation is estimated
The error of meter is minimized, normally referred to as Kalman filter.Luenberger proposed that linear system observer was set in 1964
Meter method, Linear Time-Invariant System is transformed to standard type to design observer by he by equivalence transformation, it is proposed that Luenberger
The design method of type observer.This two classes observer provides more perfect theoretical base to subsequent linear system observer research
Plinth.For the Design of Observer of nonlinear system, there is presently no a total method, so for linear system more
For complexity, but different design methods can be found for different nonlinear systems.The method being applied primarily to can be concluded
It is as follows：One is class Lyapunov methods, and this method is using more ripe Lyapunov Theory of Stability in Design of Observer
Error system stability analyzed, whether the system of being observed can be reconstructed with the observer of test design, based on the method portion
Single cent offers the design problem for the exponential type observer for discussing nonlinear system；Two be coordinate transformation method, and this method utilizes coordinate
General nonlinear system is converted to nonlinear system standard type by conversion, may then pass through POLE PLACEMENT USING to design observation
Device.Bestle and Zeitz propose the standard type of nonlinear system earliest, and then many documents are carried out to this kind of design method
Inquire into.
The content of the invention
It is an object of the present invention to overcome the above-mentioned drawbacks of the prior art and provide a kind of shape of linear system
State and Unknown inputs method, solve the state estimation containing Unknown worm in the case of observer matching condition is unsatisfactory for and ask
Topic.
The purpose of the present invention can be achieved through the following technical solutions：
A kind of state of linear system and Unknown inputs method, comprise the following steps：
S1, obtain with Unknown worm linear time varying system input vector and output vector, introduce auxiliary output to
Amount, system is met observer matching condition, then design High-Order Sliding Mode observer, obtain aid in output vector estimation and its
The estimation of derivative；
S2, will aid in the estimation of output vector and the input vector of original system as input vector, constructs reduced dimension observer,
New system mode is estimated, and estimating according to the obtained auxiliary output vector derivative of state estimation result and step S1
Meter, reconstructs Unknown worm, obtains the estimation of Unknown worm.
Described step S1 comprises the following steps：
S11, the relational expression for setting up linear time varying system is as follows：
Wherein, x ∈ Rn,y∈Rp,u∈Rm, η ∈ Rq, x is state vector,For derivative of the state to the time, y for output to
Amount, u is input vector, and η is Unknown worm, coefficient matrices A ∈ Rn×n, B ∈ Rn×m, D ∈ Rn×q, C ∈ Rr×nFor known constant square
Battle array, and rank (C)=p, CT=[c1 T … ci T … cp T], rank (D)=q, in formula 1, only y is measurable, and u is known
's；
S12, construction auxiliary output such as following formula：
Wherein, CaFor auxiliary output matrix, yaTo aid in output vector, γ=γ1+γ2+…+γp；
S13, according to formula (1), (2), constructs High-Order Sliding Mode observer, obtains auxiliary output vector yaEstimationAnd its lead
Several estimationsSuch as following formula：
Wherein,It can be tried to achieve with the differential equation computational methods such as Runge Kutta,λi,j(i=1,2 ..., p；J=1,2 ... γi+ 1) for just
Number.
In described step S12, γiFor integer and 1≤γi≤ri, (i=1,2 ..., p), riIt is to meetSmallest positive integral.
Described step S2 comprises the following steps：
S21, matrix S is obtained by Schimidt orthogonalizationa∈Rγ×γ, make SaMeetWherein expand matrixAndWillN × n orthogonal matrix is extended for,Wherein Ma∈R(n -γ)×n, utilize equivalent variationsFormula (2) is set to be equivalent to following formula：
Wherein
S22, according to formula (3), (4), constructs reduced dimension observer such as following formula：
Wherein,For reduced dimension observer state,For derivative of the observer state to the time, For coefficient matrix,For the estimation of state vector,
S23, reconstructs Unknown worm, such as following formula：
Wherein,For the estimation of Unknown worm,
It is defeated to aid in
Go out derivativeEstimation,
In described step S22, coefficient matrixSolution procedure include：Solve LMIObtain matrixWhereinFor positive definite matrix, order Wherein Take
In described step S22, reduced dimension observer existence condition includes：
1) system of formula (1) meets minimum phase condition；
2) state vector x, Unknown worm η and its derivativeAll it is bounded, and η is the continuous function on the time；
3) there is γiSo that CaFull rank simultaneously meets observer matching condition rank (D)=rank (CaD)=q.
Described minimum phase condition is：For all plural s (Re (s) >=0), meet
Wherein I is unit matrix.
Compared with prior art, the present invention considers estimating for state when observer matching condition is unsatisfactory for and Unknown worm
Meter problem, by designing High-Order Sliding Mode observer, can not only estimate that auxiliary is exported can also estimate the derivative of auxiliary output, then
The estimation of auxiliary output is used to construct reduced dimension observer, and this reduced dimension observer can eliminate the influence of Unknown worm and accurately estimate
The state of system, the derivative of auxiliary output will be used to reconstruct Unknown worm.Therefore, the present invention has certain scientific theory meaning
With actual application background.
Failure can be not only detected, the influence of failure can also be eliminated：The error of observer isIts
Error equation can be expressed as Derivative for error on the time, due toSo thatSo observer completely eliminates the influence of failure；
Failure reconfiguration need not assume that fault-signal is constant or slowly varying, partly breach observer matching condition
Rank (CD)=rank (D) limitation.
Brief description of the drawings
Fig. 1 is the flow chart of the inventive method.
Embodiment
The present invention is described in detail with specific embodiment below in conjunction with the accompanying drawings.The present embodiment is with technical solution of the present invention
Premised on implemented, give detailed embodiment and specific operating process, but protection scope of the present invention is not limited to
Following embodiments.
Embodiment
As shown in figure 1, the state estimation based on dimensionality reduction reduced dimension observer and High-Order Sliding Mode observer is reconstructed with Unknown worm
Method, comprises the following steps：
Step 1：Introducing auxiliary output vector causes observer matching condition to be satisfied, and can survey output design based on system exists
Both the High-Order Sliding Mode observer of the accurate estimation that can also obtain its derivative of output can have been aided in finite time.
Step 2:Obtain aiding in the estimation exported as a kind of new reduced dimension observer of system output construction by the use of step 1,
The observer can eliminate the influence asymptotic estimates system mode of Unknown worm, while the derivative based on state and auxiliary output
Estimation provides a kind of building method of Unknown worm.
Step 1 includes the steps：
(1) shown in the linear time varying system such as formula (1) with Unknown worm：
Wherein, x ∈ Rn,y∈Rp,u∈RmWith η ∈ RqRespectively state vector, output vector, input vector and unknown defeated
Enter, rank (C)=p and CT=[c1 T … ci T … cp T], rank (D)=q.
(2) output of construction auxiliary is as shown in formula (2)：
Whereinγ=γ1+γ2+…+
γp, γiFor integer and 1≤γi≤ri, riIt is to meet(i=1,2 ..., minimum p)
Integer.
(3) High-Order Sliding Mode observer is constructed, according to formula (1), (2) construction High-Order Sliding Mode observer is as follows：
WhereinAnd λi,j(i=1,
2,…,p；J=1,2 ... γi+ 1) it is positive number.
Step 2 comprises the following steps：
(1) matrix S is obtained by Schimidt orthogonalizationa∈Rγ×γIt meetsWhereinAndExpand matrixFor orthogonal matrixWherein Ma∈R(n-γ)×n.Utilize equivalent variationsSystem (2) is equivalent to
WhereinAnd
(2) reduced dimension observer is constructed
Solve LMI
Obtain matrixWhereinFor positive definite matrix.
Decomposing system vectorWhereinWhile decomposition coefficient matrix
WhereinAnd
Take
According to formula (3) and (4), design reduced dimension observer is as follows：
WhereinIt is that the auxiliary obtained by High-Order Sliding Mode observer (3) exports yaEstimation.
(3) reconstruct of Unknown worm can be expressed as：
Wherein,
(4) existence condition of fixed pattern (5) observer is given：
The system of condition 1 (1) meets minimum phase condition, i.e., for all plural s (Re (s) >=0), meets
Condition 2 state x (t), Unknown worm η (t) and its derivativeAll it is bounded, meanwhile, η (t) is on the time
Continuous function；
There is γ in condition 3iSo that CaFull rank simultaneously meets observer matching condition rank (D)=rank (CaD)=q.
The present invention considers the state when observer matching condition is unsatisfactory for and the estimation problem of Unknown worm, passes through design
High-Order Sliding Mode observer can not only estimate that auxiliary output can also estimate the derivative of auxiliary output, then the estimation of auxiliary output is used
In construction reduced dimension observer, this reduced dimension observer can eliminate the state of the influence of Unknown worm and accurate estimating system, auxiliary
The derivative of output will be used to reconstruct Unknown worm.Therefore, the present invention has certain scientific theory meaning and actual application background.
Claims (7)
1. a kind of state of linear system and Unknown inputs method, it is characterised in that comprise the following steps：
S1, obtains the input vector and output vector of the linear time varying system with Unknown worm, introduces auxiliary output vector, makes
System meets observer matching condition, then designs High-Order Sliding Mode observer, obtains aiding in estimation and its derivative of output vector
Estimation；
S2, will aid in the estimation of output vector and the input vector of original system as input vector, reduced dimension observer is constructed, to new
System mode estimated, and according to the estimation of the obtained auxiliary output vector derivative of state estimation result and step S1, weight
Structure goes out Unknown worm, obtains the estimation of Unknown worm.
2. a kind of state of linear system according to claim 1 and Unknown inputs method, it is characterised in that described
Step S1 comprise the following steps：
S11, the relational expression for setting up linear time varying system is as follows：
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Rank (C)=p, CT=[c1 T … ci T … cp T], rank (D)=q；
S12, construction auxiliary output such as following formula：
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Wherein, CaFor auxiliary output matrix, yaTo aid in output vector, γ=γ1+γ2+…+γp；
S13, according to formula (1), (2), constructs High-Order Sliding Mode observer, obtains auxiliary output vector yaEstimationAnd its derivative
EstimationSuch as following formula：
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Wherein,λi,j(i=1,2 ..., p；J=1,2 ...
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3. a kind of state of linear system according to claim 2 and Unknown inputs method, it is characterised in that described
Step S12 in, γiFor integer and 1≤γi≤ri, (i=1,2 ..., p), riIt is to meetSmallest positive integral.
4. a kind of state of linear system according to claim 2 and Unknown inputs method, it is characterised in that described
Step S2 comprise the following steps：
S21, matrix S is obtained by Schimidt orthogonalizationa∈Rγ×γ, make SaMeetWherein expand matrix
AndWherein Ma∈R(n-γ)×n, utilize equivalent variationsMake formula (2) etc.
Valency is in following formula：
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S22, according to formula (3), (4), constructs reduced dimension observer such as following formula：
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<msub>
<mover>
<mi>K</mi>
<mo>&OverBar;</mo>
</mover>
<mi>a</mi>
</msub>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<msubsup>
<mi>S</mi>
<mi>a</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<msub>
<mover>
<mi>y</mi>
<mo>^</mo>
</mover>
<mi>a</mi>
</msub>
<mo>+</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mover>
<mi>K</mi>
<mo>&OverBar;</mo>
</mover>
<mi>a</mi>
</msub>
</mtd>
<mtd>
<msub>
<mi>I</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<mi>r</mi>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mover>
<mi>B</mi>
<mo>&OverBar;</mo>
</mover>
<mi>u</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mo>=</mo>
<msubsup>
<mi>W</mi>
<mi>a</mi>
<mi>T</mi>
</msubsup>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<msubsup>
<mi>S</mi>
<mi>a</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<msub>
<mover>
<mi>y</mi>
<mo>^</mo>
</mover>
<mi>a</mi>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>z</mi>
<mo>^</mo>
</mover>
<mn>2</mn>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>K</mi>
<mo>&OverBar;</mo>
</mover>
<mi>a</mi>
</msub>
<msubsup>
<mi>S</mi>
<mi>a</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<msub>
<mover>
<mi>y</mi>
<mo>^</mo>
</mover>
<mi>a</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein,For reduced dimension observer state,For derivative of the observer state to the time, For coefficient matrix,For the estimation of state vector,
S23, reconstructs Unknown worm, such as following formula：
<mrow>
<mover>
<mi>&eta;</mi>
<mo>^</mo>
</mover>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<msup>
<mi>G</mi>
<mi>T</mi>
</msup>
<mi>G</mi>
<mo>)</mo>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<msup>
<mi>G</mi>
<mi>T</mi>
</msup>
<mo>&lsqb;</mo>
<msub>
<mover>
<mi>&xi;</mi>
<mo>^</mo>
</mover>
<mi>a</mi>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>C</mi>
<mo>~</mo>
</mover>
<mi>a</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mo>+</mo>
<mi>B</mi>
<mi>u</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>,</mo>
</mrow>
Wherein,For the estimation of Unknown worm,
It is defeated to aid in
Go out derivativeEstimation,
<mrow>
<msub>
<mover>
<mi>C</mi>
<mo>~</mo>
</mover>
<mi>a</mi>
</msub>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msup>
<mrow>
<mo>(</mo>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
<msup>
<mi>A</mi>
<mrow>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
</mtd>
<mtd>
<msup>
<mrow>
<mo>(</mo>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
<msup>
<mi>A</mi>
<mrow>
<msub>
<mi>&gamma;</mi>
<mn>2</mn>
</msub>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
</mtd>
<mtd>
<mo>...</mo>
</mtd>
<mtd>
<msup>
<mrow>
<mo>(</mo>
<msub>
<mi>c</mi>
<mi>p</mi>
</msub>
<msup>
<mi>A</mi>
<mrow>
<msub>
<mi>&gamma;</mi>
<mi>p</mi>
</msub>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
<mi>G</mi>
<mo>=</mo>
<msub>
<mover>
<mi>C</mi>
<mo>~</mo>
</mover>
<mi>a</mi>
</msub>
<mi>D</mi>
<mo>.</mo>
</mrow>
5. a kind of state of linear system according to claim 4 and Unknown inputs method, it is characterised in that described
In step S22, coefficient matrixSolution procedure include：Solve LMI
Obtain matrixWhereinFor positive definite matrix, order
Wherein Take
6. a kind of state of linear system according to claim 4 and Unknown inputs method, it is characterised in that described
Step S22 in, reduced dimension observer existence condition includes：
1) system of formula (1) meets minimum phase condition；
2) state vector x, Unknown worm η and its derivativeAll it is bounded, and η is the continuous function on the time；
3) there is γiSo that CaFull rank simultaneously meets observer matching condition rank (D)=rank (CaD)=q.
7. a kind of state of linear system according to claim 6 and Unknown inputs method, it is characterised in that described
Minimum phase condition be：For all plural s (Re (s) >=0), meetWherein I is single
Bit matrix.
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| CN111880411A (en) * | 2020-08-12 | 2020-11-03 | 深圳职业技术学院 | Dynamic extended regression and interactive estimation method of linear system |
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| CN113110571A (en) * | 2021-03-31 | 2021-07-13 | 北京航天飞腾装备技术有限责任公司 | Method for estimating flight attack angle based on dimension reduction state observer |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN112041699A (en) * | 2018-02-23 | 2020-12-04 | 卡帕奇欧特私人有限公司 | Reconstruction system and method |
| CN108749915A (en) * | 2018-06-08 | 2018-11-06 | 扬州大学 | A kind of electric power-assisted steering apparatus design method based on sliding mode observer |
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| CN113110571A (en) * | 2021-03-31 | 2021-07-13 | 北京航天飞腾装备技术有限责任公司 | Method for estimating flight attack angle based on dimension reduction state observer |
| CN113110571B (en) * | 2021-03-31 | 2023-03-03 | 北京航天飞腾装备技术有限责任公司 | Method for estimating flight attack angle based on dimension reduction state observer |
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