CN109901398A - A kind of peak value upper limit evaluation method of nonlinear system impulse response - Google Patents
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Abstract
The invention discloses a kind of peak value upper limit evaluation methods of nonlinear system impulse response, comprising: establishes the Liapunov multinomial level set of description system mode track；Liapunov function is projected again and state position is assessed；Effective estimated value is obtained using binary search and convex optimization method；It is applied to situation when nonlinear system equilibrium-like state value non-zero using equalization point transfer method, current invention assumes that being c ∈ (0 to the peak value upper limit estimated value of nonlinear system impulse response, ∞), establish the condition that constant c is the nonlinear system impulse response peak value upper limit, the condition is finally implemented to the convex optimization problem of several linear matrix inequality obtain after linear transformation system Liapunov multinomial, and the conservative of system impulse peak value of response upper limit estimated value can be reduced by the increase of the system Liapunov degree of polynomial.
Description
Technical field
The invention belongs to Process Control System and calculate control theory field more particularly to a kind of nonlinear system impulse sound
The peak value upper limit evaluation method answered.
Background technique
The input/output relation of system can be characterized with various indexs, and especially H- ∞ norm is (for example, system frequency is rung
The amplitude gain maximum value answered) and H-2 norm (such as root sum square of impulse response energy).These indexs are in network analysis
Played a crucial role in synthesis, thus for a long time researchers propose it is a large amount of calculate determine these index values with
And the method for controlling these index values.
The impulse response of system reflects some inherent characteristics that system itself has；And the impulse response peak value of system,
As an important indicator, the Research Literature and its contribution of relevant calculation and control method are relatively fewer.The index in response to
The instantaneous infinitely great pulse being applied on system input channel, provides the amplitude peak of system response output valve, can be used for verifying
With application to the amplitude constraint of system response output valve.Although impulse response peak index has quite in network analysis and synthesis
Importance, but how accurately to estimate and accurately determine that this index is still an outstanding question.Some classics refer to
Mark evaluation method, such as collection invariance method based on secondary liapunov function function, for determining system frequency response
When the indexs such as the square root of the sum of amplitude gain maximum value and impulse response energy, conservative as a result is lower, as a result compared with
For ideal；But the estimation result conservative for determining that impulse response peak value obtains is usually higher, is applied to H- ∞ model with it
Several and H-2 norm estimation result non-conservation forms sharp contrast.
Summary of the invention
The present invention overcomes the shortcomings of the prior art, and the side of the estimating system impulse response peak value upper limit is capable of by one kind
Method reduces the conservative of estimated value as far as possible.
In order to achieve the above objectives, the technical solution adopted by the present invention is that: on a kind of peak value of nonlinear system impulse response
Evaluation method is limited, consists of assuming that the peak value upper limit estimated value to nonlinear system impulse response is c ∈ (0, ∞), establishes constant c
∈ (0, ∞) can be asserted the condition of the nonlinear system impulse response peak value upper limit, which can be obtained by following steps:
S101 establishes the Liapunov multinomial level set of description system mode track；
S102 projects liapunov function again, and is assessed state position；
S103 obtains effective estimated value using binary search and convex optimization method；
S104 is applied to situation when system balancing state value non-zero using equalization point transfer method.
Further, it is assumed that the peak value upper limit estimated value to nonlinear system impulse response is c ∈ (0, ∞), including as follows
Step:
S201, setting natural number (including zero) integrate as N and set of real numbers as R, euclideam norm and infinity norm difference
It is expressed as | | | |2With | | | |∞, A ' is the transposition of matrix A, and A > 0 (A >=0) indicates Hermite positive definite (positive semidefinite), ∑
For multinomial square sum aggregate；
S202, with state equation description it needs to be determined that the non-linear time-invariant system of impulse response peak value:
Wherein, t ∈ R indicates time, x (t) ∈ RnIndicate system mode, u (t) ∈ R indicates input, y (t) ∈ RpIndicate system
System output,Indicate that the system mode nonlinear function matrix of suitable size, brief note are made
S203 defines the impulse response y of systemIR(t), i.e., the described non-linear time-invariant system is for impulse function input
It is x (0 that zero state response, which is system primary condition,-System when)=0 and input are u (t)=δ (t) exports y (t), wherein δ
It (t) is dirac unit impulse function；
S204, in zero initial condition x (0-Impulse function is inputted to system under)=0, is equivalent to and initial state value is arranged
ForInput corresponding system output when u (t)=0:Determine constantSo that all input channels relative to system have the infinity norm of single-pass channel shock response small
In constant c:
Establish the upper limit that constant c ∈ (0, ∞) is system impulse peak value of response.
Further, S101 includes finding a number no more than 2d, and liapunov function v (x): R of d ∈ Nn→ R,
By the definition of liapunov function it is found that v (x) is a multinomial of system mode x, to the derivative of time
It is negative, so that:
Show that the state trajectory of system originates from multinomial level set In, because of condition f (x) ∈ ∑, show byThe system mode track set out is located at
Level setIn.
Further, S102 includes being projected liapunov function system v (x) to appreciable levels collection againIt is
It is no to be located at setIn, it enables
Wherein, sk(x) it is multinomial coefficient, searches out a suitable scalar, ε > 0, ε ∈ R, so that through Li Yapunuo
The multinomial h that husband function v (x) is projected againk(x) belong to multinomial square sum aggregate, i.e. hk(x)∈∑；
Multinomial hk(x) coefficient of coefficient and multinomial v (x) is linear, hk(x) belong to multinomial square sum aggregate
Condition so that the level set of system mode track Set can not be located atAmong.If state trajectory is located at above-mentioned setIn, then by hk(x)
System mode track known to ∈ ∑ is likely located in level setAmong, it is established with S101
The description of state trajectory level setIt is conflicting.
Further, S103 includes that liapunov function v (x) is represented by following matrix forms: v (x)=b (x) ' (V
+ L (α)) b (x), wherein V is symmetrical matrix, and α is vector variable, and b (x) is a series of polynomial basis institute for being not more than d by numbers
The vector of composition, judges whether a multinomial belongs to a square sum aggregate, is equivalent to judge the corresponding linear matrix of the multinomial not
Whether equation V+L (α) >=0 is true.
Further, for input dirac unit impulse function situation, can find a constant c ∈ (0, ∞) and
Scalar ε > 0 sets up following constraint simultaneously with multiple inequality:
Constant c is the estimated value of the nonlinear system impulse response peak value upper limit.
Further, S104 is specifically included: when system has other equalization point xeMakeWhen, it can be by system mode
Equation is from equalization point xeAfter being displaced to zero balancing point situation, estimation is with zero on the nonlinear system impulse response peak value of equalization point
Limit.
Compared with prior art, the present invention has the following beneficial effects:
A kind of method for capableing of the estimating system impulse response peak value upper limit is provided, reduces the conservative of estimated value as far as possible,
It is assumed that being c ∈ (0, ∞) to the peak value upper limit estimated value of nonlinear system impulse response, establishing constant c is nonlinear system impulse
The condition of the peak value of response upper limit is implemented the condition finally will to obtain after system Liapunov multinomial progress linear transformation
Several linear matrix inequality convex optimization problem, and the conservative of system impulse peak value of response upper limit estimated value can lead to
It crosses the increase of the system Liapunov degree of polynomial and reduces.
Detailed description of the invention
Fig. 1 is flow diagram of the invention.
Fig. 2 is that the process based on binary search and the convex optimization method estimating system impulse response peak value upper limit is shown in the present invention
It is intended to.
Specific embodiment
Following is a specific embodiment of the present invention in conjunction with the accompanying drawings, technical scheme of the present invention will be further described,
However, the present invention is not limited to these examples.
As shown in Fig. 1 to 2, a kind of peak value upper limit evaluation method of nonlinear system impulse response is consisted of assuming that non-thread
Property system impulse response peak value upper limit estimated value be c ∈ (0, ∞), nonlinear system can be asserted by establishing constant c ∈ (0, ∞)
The condition of the impulse response peak value upper limit, the condition can be obtained by following steps:
S101 establishes the Liapunov multinomial level set of description system mode track；
S102 projects liapunov function again, and is assessed state position；
S103 obtains effective estimated value using binary search and convex optimization method；
S104 is applied to situation when system balancing state value non-zero using equalization point transfer method.
It is assumed that the peak value upper limit estimated value to nonlinear system impulse response is c ∈ (0, ∞), include the following steps:
S201, setting natural number (including zero) integrate as N and set of real numbers as R, euclideam norm and infinity norm difference
It is expressed as | | | |2With | | | |∞, A ' is the transposition of matrix A, and A > 0 (A >=0) indicates Hermite positive definite (positive semidefinite), ∑
For multinomial square sum aggregate；
S202, with state equation description it needs to be determined that the non-linear time-invariant system of impulse response peak value:
Wherein, t ∈ R indicates time, x (t) ∈ RnIndicate system mode, u (t) ∈ RmIndicate m dimension input, y (t) ∈ RpTable
Show that system exports,It indicates the system mode nonlinear function matrix of suitable size, is abbreviated
Make
S203 defines the single-pass channel shock response of systemInvariant system when i.e. described non-linear
It is x (0 that the impulse response united relative to i-th of input channel, which is system primary condition,-)=0 and input are u (t)=δ (t) Em
(i) system when exports y (t), wherein δ (t) is dirac unit impulse function, Em(i) be m × m unit matrix i-th column
Vector；
S204, in zero initial condition x (0-Impulse response is inputted to i-th of channel of system under)=0, is equivalent to initial shape
State value is set asInput corresponding system output when u (t)=0:Determine constant So that relative to system
All input channels have the infinity norm of single-pass channel shock response to be less than constant c:
Establish the upper limit that constant c ∈ (0, ∞) is system impulse peak value of response.
S101 includes finding a number no more than 2d, and liapunov function v (x): R of d ∈ Nn→ R, by Li Yapu
The definition of promise husband's function is it is found that v (x) is a multinomial of system mode x, to the derivative of timeIt is negative, makes
:
Show that the state trajectory of system originates from multinomial level setIn, because of condition f (x) ∈ ∑, show by The system shape to set out
State track is located at level setIn.
S102 includes being projected liapunov function system v (x) to appreciable levels collection againWhether Ji Tai is located atIn, it enables
Wherein, sk(x) it is multinomial coefficient, searches out a suitable scalar, ε > 0, ε ∈ R, so that through Li Yapunuo
The multinomial h that husband function v (x) is projected againk(x) belong to multinomial square sum aggregate, i.e. hk(x)∈∑；
Multinomial hk(x) coefficient of coefficient and multinomial v (x) is linear, hk(x) belong to multinomial square sum aggregate
Condition so that the level set of system mode track Set can not be located atAmong.If state trajectory is located at above-mentioned setIn, then by hk(x)
System mode track known to ∈ ∑ is likely located in level setAmong, it is established with S101
The description of state trajectory level setIt is conflicting.
S103 includes that liapunov function v (x) is represented by following matrix forms: v (x)=b (x) ' (V+L (α)) b
(x), wherein V is symmetrical matrix, and α is vector variable, and b (x) is as composed by a series of polynomial basis of numbers no more than d
Vector, judges whether a multinomial belongs to a square sum aggregate, is equivalent to judge the corresponding linear matrix inequality V+L of the multinomial
Whether (α) >=0 be true.
A constant c ∈ (0, ∞) and scalar ε can be found for all input channelsi> 0 sets up following formula:
Constant c is the estimated value of the nonlinear system impulse response peak value upper limit.
This method is suitable in the situation that system balancing point is zero namely S202When situation；When system has
Other equalization point xe makeWhen, it can be by system state equation from equalization point xeAfter being displaced to zero balancing point situation, estimation
With zero for equalization point the nonlinear system impulse response peak value upper limit.
Current invention assumes that the peak value upper limit estimated value to nonlinear system impulse response is c ∈ (0, ∞), establishing constant c is
The condition of the nonlinear system impulse response peak value upper limit is implemented the condition finally to carry out system Liapunov multinomial
The convex optimization problem of the several linear matrix inequality obtained after linear transformation, and system impulse peak value of response upper limit estimated value
Conservative can be reduced by the increase of the system Liapunov degree of polynomial.
Specific embodiment described herein is only an example for the spirit of the invention.The neck of technology belonging to the present invention
The technical staff in domain can make various modifications or additions to the described embodiments or replace by a similar method
In generation, however, it does not deviate from the spirit of the invention or beyond the scope of the appended claims.
Claims (7)
1. a kind of peak value upper limit evaluation method of nonlinear system impulse response, which is characterized in that consist of assuming that nonlinear system
The peak value upper limit estimated value of system impulse response is c ∈ (0, ∞), and nonlinear system impulse can be asserted by establishing constant c ∈ (0, ∞)
The condition of the peak value of response upper limit, the condition can be obtained by following steps:
S101 establishes the Liapunov multinomial level set of description system mode track；
S102 projects liapunov function again, and is assessed state position；
S103 obtains effective estimated value using binary search and convex optimization method；
S104 is applied to situation when system balancing state value non-zero using equalization point transfer method.
2. the peak value upper limit evaluation method of nonlinear system impulse response according to claim 1, which is characterized in that it is assumed that
Peak value upper limit estimated value to nonlinear system impulse response is c ∈ (0, ∞), is included the following steps:
S201, setting natural number (including zero) integrate as N and set of real numbers as R, and euclideam norm and infinity norm respectively indicate
For | | | |2With | | | |∞, A ' is the transposition of matrix A, and A > 0 (A >=0) indicates Hermite positive definite (positive semidefinite), and ∑ is more
Item formula square sum aggregate；
S202, with state equation description it needs to be determined that the non-linear time-invariant system of impulse response peak value:
Wherein, t ∈ R indicates time, x (t) ∈ RnIndicate system mode, u (t) ∈ R indicates input, y (t) ∈ RpExpression system is defeated
Out,Indicate that the system mode nonlinear function matrix of suitable size, brief note are made
S203 defines the impulse response y of systemIR(t), i.e., the described non-linear time-invariant system is directed to zero shape of impulse function input
It is that the system of x (0-)=0 and input when being u (t)=δ (t) exports y (t) that state response, which be system primary condition, wherein δ (t) is
Dirac unit impulse function；
S204, in zero initial condition x (0-Impulse function is inputted to system under)=0, is equivalent to and sets initial state value toInput corresponding system output when u (t)=0:Determine constantSo that all input channels relative to system have the infinity norm of single-pass channel shock response small
In constant c:
Establish the upper limit that constant c ∈ (0, ∞) is system impulse peak value of response.
3. the peak value upper limit evaluation method of nonlinear system impulse response according to claim 2, which is characterized in that S101
Including finding a number no more than 2d, liapunov function v (x): R of d ∈ Nn→ R, by determining for liapunov function
Justice is it is found that v (x) is a multinomial of system mode x, to the derivative of timeIt is negative, so that:
Show that the state trajectory of system originates from multinomial level set
In, because of condition f (x) ∈ ∑, show byThe system mode track set out is located at level setIn.
4. the peak value upper limit evaluation method of nonlinear system impulse response according to claim 3, which is characterized in that S102
Including being projected liapunov function system v (x) to appreciable levels collection againWhether set is located atIn, it enables
Wherein, sk(x) it is multinomial coefficient, searches out a suitable scalar, ε > 0, ε ∈ R, so that through liapunov function
The multinomial h that v (x) is projected againk(x) belong to multinomial square sum aggregate, i.e. hk(x)∈∑；
Multinomial hk(x) coefficient of coefficient and multinomial v (x) is linear, hk(x) belong to the item of multinomial square sum aggregate
Part if so, then system mode track is located at set In.
5. the peak value upper limit evaluation method of nonlinear system impulse response according to claim 4, which is characterized in that S103
Following matrix forms: v (x)=b (x) ' (V+L (α)) b (x), wherein V is couple are represented by including liapunov function v (x)
Claim matrix, α is vector variable, and b (x) is the vector as composed by a series of polynomial basis of numbers no more than d, is judged more than one
Whether item formula belongs to a square sum aggregate, is equivalent to judge whether the corresponding linear matrix inequality V+L (α) >=0 of the multinomial is true.
6. the peak value upper limit evaluation method of nonlinear system impulse response according to claim 5, which is characterized in that can be with
Finding a constant c ∈ (0, ∞) and scalar ε > 0 sets up following formula:
Constant c is the estimated value of the nonlinear system impulse response peak value upper limit.
7. the peak value upper limit evaluation method of nonlinear system impulse response according to claim 6, which is characterized in that S104
It specifically includes: when system has other equalization point xe to makeWhen, it can be by system state equation from equalization point xeIt is displaced to zero
After equalization point situation, the nonlinear system impulse response peak value upper limit with zero for equalization point is estimated.
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