CN109901398B - Peak upper limit estimation method of nonlinear system impulse response - Google Patents
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Abstract
The invention discloses a peak value upper limit estimation method of nonlinear system impulse response, which comprises the following steps: establishing a Lyapunov polynomial level set describing a system state track; re-projecting the Lyapunov function and evaluating the state position; obtaining effective estimated values by using a binary search and convex optimization method; the invention adopts a balance point transfer method to be applied to the situation that the balance state value of the nonlinear system is nonzero, the invention assumes that the estimation value of the peak value upper limit of the impulse response of the nonlinear system is c epsilon (0 and infinity), establishes a condition that constant c is the peak value upper limit of the impulse response of the nonlinear system, and finally implements the condition into the convex optimization problem of a plurality of linear matrix inequalities obtained after linear conversion is carried out on the system Lyapunov polynomial, and the conservative degree of the estimation value of the peak value upper limit of the impulse response of the system can be reduced by increasing the times of the system Lyapunov polynomial.
Description
Technical Field
The invention belongs to the field of process control systems and computational control theories, and particularly relates to a method for estimating the peak upper limit of impulse response of a nonlinear system.
Background
The input-output relationships of the system can be characterized by various metrics, particularly the H-infinity norm (e.g., the maximum magnitude gain of the system's frequency response) and the H-2 norm (e.g., the square root of the sum of the impulse response energies). Since these indices play a crucial role in system analysis and synthesis, researchers have proposed a large number of methods for calculating and determining these indices and controlling them.
The impulse response of the system reflects some inherent characteristics of the system; the impulse response peak value of the system, as an important index, is relatively few in research literature and contribution of the correlation calculation and control method. The index provides a maximum amplitude of the system response output value in response to a transient infinite pulse applied to the system input channel, which can be used to verify and impose amplitude constraints on the system response output value. Although the impulse response peak index has considerable importance in system analysis and synthesis, how to accurately estimate and accurately determine the index is still a pending problem. When some classical index estimation methods, such as a set invariance method based on a quadratic Lyapunov function, are used for determining indexes such as the maximum amplitude gain value of system frequency response and the square root value of the sum of impulse response energy, the conservative degree of the result is low, and the result is ideal; however, the estimates obtained for determining the peak impulse response are generally more conservative, in contrast to their non-conservative nature as applied to the H-infinity and H-2 norm estimates.
Disclosure of Invention
The invention overcomes the defects of the prior art, and reduces the conservatism of the estimated value as much as possible by a method capable of estimating the peak value upper limit of the system impulse response.
In order to achieve the purpose, the invention adopts the technical scheme that: a peak upper limit estimation method for an impulse response of a nonlinear system includes assuming that an upper limit estimation value for a peak of an impulse response of the nonlinear system is c ∈ (0, ∞), establishing a condition under which a constant c ∈ (0, ∞) can be established as an upper limit of the peak of the impulse response of the nonlinear system, the condition being obtainable by:
s101, establishing a Lyapunov polynomial level set describing a system state track;
s102, re-projecting the Lyapunov function, and evaluating the state position;
s103, obtaining effective estimated values by utilizing a binary search and convex optimization method;
and S104, applying a balance point transfer method to the situation that the system balance state value is not zero.
Further, assuming that the peak upper limit estimation value of the impulse response of the nonlinear system is c ∈ (0, ∞), the method comprises the following steps:
s201, setting a natural number (including zero) set as N and a real number set as R, and respectively representing Euclidean norm and infinite norm as | | |. G2And | · | non-counting∞A' is the transposition of matrix A, A > 0(A ≧ 0) represents Hermite specific positive (half positive), and Σ is the sum of the squares of the polynomials;
s202, describing a nonlinear time-invariant system needing to determine an impulse response peak value by using a state equation:
wherein t ∈ R denotes time, x (t) ∈ RnRepresents the system state, u (t) e R represents the input, y (t) e RpThe output of the system is represented and,
representing appropriately sized system state nonlinearitiesFunction matrix, abbreviated as
S203, defining the impulse response y of the systemIR(t), namely the zero state response of the nonlinear time-invariant system to the impulse function input, is that the initial condition of the system is x (0)-) 0 and the system output y (t) when the input is u (t) δ (t), where δ (t) is a dirac unit impulse function;
s204, under the zero initial condition x (0)-) Inputting an impulse function to the system at 0, which is equivalent to setting the initial state value to
The corresponding system output when input u (t) is 0:
determining a constant
Such that the infinite norm with a single-channel impulse response for all input channels of the system is less than a constant c:
the constant c ∈ (0, ∞) is established as the upper limit of the system impulse response peak.
Further, S101 includes finding a Lyapunov function v (x) with a degree not greater than 2d, d ∈ N: rn→ R, defined by the Lyapunov function, v (x) is a polynomial, time-derivative of the system state x
Is negative, so that:
Further, S102 includes re-projecting the Lyapunov function system v (x) to evaluate the level set
Whether or not to be in the collection
In the middle, let
Wherein s isk(x) Finding a suitable scalar quantity epsilon > 0 epsilon R for the polynomial coefficient, so that the polynomial h obtained by re-projection of the Lyapunov function v (x)k(x) Belonging to the polynomial square sum, i.e. hk(x)∈∑；
Polynomial hk(x) Is linear with the coefficients of the polynomial v (x), hk(x) Conditional on the sum of squares of polynomials such that the level set of the trajectory of the state of the system is set
Cannot be located in a collection
Among them. If the state track is in the above set
In, then h isk(x) E sigma can know that the system state track is possibly positioned in the level set
In (2) state trace level set description established with S101
Contradict each other.
Further, S103 includes a lyapunov function v (x) which can be expressed in the form of a matrix as follows: v (x) ((V + L (α)) b (x)) where V is a symmetric matrix and α is a vector variable and b (x) is a vector composed of a series of polynomial bases with degree no greater than d, and whether a polynomial belongs to the sum of squares is judged, which is equivalent to judging whether the linear matrix inequality V + L (α) ≧ 0 corresponding to the polynomial holds.
Further, for the case of inputting a dirac unit impulse function, one constant c ∈ (0, ∞) and a scalar ε > 0 can be found such that the following constraints hold with multiple inequalities:
the constant c is an estimation value of the impulse response peak upper limit of the nonlinear system.
Further, S104 specifically includes: when the system hasOther balance points xeMake it
Then, the system state equation can be driven from the equilibrium point xeAfter shifting to the zero balance point, estimating the impulse response peak upper limit of the nonlinear system with zero as the balance point.
Compared with the prior art, the invention has the following beneficial effects:
the method can estimate the peak upper limit of the impulse response of the system, reduce the conservatism of the estimated value as much as possible, assume that the estimated value of the peak upper limit of the impulse response of the nonlinear system is c epsilon (0, infinity), establish the condition that the constant c is the peak upper limit of the impulse response of the nonlinear system, finally implement the condition as a convex optimization problem of a plurality of linear matrix inequalities obtained after linear conversion is carried out on the system Lyapunov polynomial, and the conservatism of the estimated value of the peak upper limit of the impulse response of the system can be reduced by increasing the degree of the system Lyapunov polynomial.
Drawings
FIG. 1 is a schematic flow chart of the present invention.
FIG. 2 is a schematic flow chart of estimating the peak upper limit of the impulse response of the system based on the binary search and convex optimization method in the present invention.
Detailed Description
The following are specific embodiments of the present invention and are further described with reference to the drawings, but the present invention is not limited to these embodiments.
As shown in fig. 1 to 2, a method for estimating the peak upper limit of the impulse response of a nonlinear system includes assuming that the estimation value of the peak upper limit of the impulse response of the nonlinear system is c ∈ (0, ∞), establishing a condition under which a constant c ∈ (0, ∞) can be established as the peak upper limit of the impulse response of the nonlinear system, the condition being obtained by:
s101, establishing a Lyapunov polynomial level set describing a system state track;
s102, re-projecting the Lyapunov function, and evaluating the state position;
s103, obtaining effective estimated values by using a binary search and convex optimization method;
and S104, applying a balance point transfer method to the situation that the system balance state value is not zero.
Assuming that the peak upper limit estimation value of the impulse response of the nonlinear system is c e (0, ∞), the method comprises the following steps:
s201, setting a natural number (including zero) set as N and a real number set as R, and respectively representing Euclidean norm and infinite norm as | | |. G2And | · | non-conducting phosphor∞A' is the transposition of matrix A, A > 0(A ≧ 0) represents Hermite specific positive (half positive), and Σ is the sum of the squares of the polynomials;
s202, describing a nonlinear time-invariant system needing to determine an impulse response peak value by using a state equation:
wherein t ∈ R denotes time, x (t) ∈ RnRepresents the system state, u (t) e RmRepresenting an m-dimensional input, y (t) e RpThe output of the system is represented and,
expressing a suitably sized nonlinear function matrix of the system state, abbreviated as
S203, defining single-channel impulse response of the system
Namely, the impulse response of the nonlinear time-invariant system relative to the ith input channel is the system initial condition x (0)-) 0 and input u (t) δ (t) Em(i) The system output y (t) of time, where δ (t) is the dirac unit impulse function, Em(i) Is the ith column vector of the m × m identity matrix;
s204, under the zero initial condition x (0)-) Not greater than 0Inputting impulse response to ith channel of system, which is equivalent to setting initial state value to be
The corresponding system output when input u (t) is 0:
determining a constant
Such that the infinite norm with a single-channel impulse response for all input channels of the system is less than a constant c:
the constant c ∈ (0, ∞) is established as the upper limit of the system impulse response peak.
S101 includes finding a Lyapunov function v (x) with a degree not greater than 2d, d ∈ N: rn→ R, defined by the Lyapunov function, v (x) is a polynomial, time-derivative of the system state x
Is negative, so that:
S102 includes re-projecting the Lyapunov function system v (x) to evaluate the level set
Whether or not to be located on the collecting table
In the middle, let
Wherein s isk(x) Finding a suitable scalar quantity epsilon > 0 epsilon R for the polynomial coefficient, so that the polynomial h obtained by re-projection of the Lyapunov function v (x)k(x) Belonging to the polynomial square sum, i.e. hk(x)∈∑；
Polynomial hk(x) Is linear with the coefficients of the polynomial v (x), hk(x) Conditions pertaining to polynomial squares and collections, such that the system state trace is a set of levels
Can not be located in the collection
Among them. If the state track is in the above set
In, then h isk(x) E sigma can know that the system state track is possibly positioned in the level set
In (2) state trace level set description established with S101
Contradict each other.
S103 includes a Lyapunov function v (x) that can be expressed in the form of a matrix: v (x)' (V + L (α)) b (x), where V is a symmetric matrix, α is a vector variable, and b (x) is a vector composed of a series of polynomial bases of degree no greater than d, and whether a polynomial belongs to a square set or not is determined, which is equivalent to determining whether a linear matrix inequality V + L (α) ≧ 0 corresponding to the polynomial holds or not.
One constant c ∈ (0, ∞) and scalar ε can be found for all input channelsi> 0 holds the following equation:
the constant c is an estimation value of the impulse response peak upper limit of the nonlinear system.
The method is suitable for the case that the system balance point is zero, that is, in S202
The situation of time; when the system has other balance points xe
Then, the system state equation can be driven from the equilibrium point xeAfter shifting to the zero balance point, estimating the impulse response peak upper limit of the nonlinear system with zero as the balance point.
The invention assumes that the estimation value of the peak upper limit of the impulse response of the nonlinear system is c epsilon (0, infinity), establishes the condition that the constant c is the impulse response peak upper limit of the nonlinear system, and finally implements the condition into the convex optimization problem of a plurality of linear matrix inequalities obtained after the linear conversion is carried out on the system Lyapunov polynomial, and the conservative degree of the estimation value of the peak upper limit of the impulse response of the system can be reduced by increasing the times of the system Lyapunov polynomial.
The specific embodiments described herein are merely illustrative of the spirit of the invention. Various modifications or additions may be made to the described embodiments or alternatives may be employed by those skilled in the art without departing from the spirit or ambit of the invention as defined in the appended claims.
Claims (1)
1. A peak upper limit estimation method of impulse response of a nonlinear system, comprising assuming that an estimation value of a peak upper limit for impulse response of the nonlinear system is c e (0, ∞), establishing a condition under which a constant c e (0, ∞) can be established as the peak upper limit of impulse response of the nonlinear system, the condition being obtainable by:
s101, establishing a Lyapunov polynomial level set describing a system state track;
s102, re-projecting the Lyapunov function, and evaluating the state position;
s103, obtaining effective estimated values by utilizing a binary search and convex optimization method;
s104, applying a balance point transfer method to the situation that the system balance state value is not zero; s104 specifically comprises the following steps: when the system has other balance points xeMake it
Then, the system state equation can be driven from the equilibrium point xeAfter shifting to the zero balance point, estimating the impulse response peak upper limit of the nonlinear system with zero as the balance point;
assuming that the peak upper limit estimation value of the impulse response of the nonlinear system is c e (0, ∞), the method comprises the following steps:
s201, setting a natural number (including zero) set as N and a real number set as R, and respectively representing Euclidean norm and infinite norm as | | |. G2And | · | non-conducting phosphor∞A' is the transposition of the matrix A, A is more than 0(A is more than or equal to 0) and represents hermitian (half positive) and sigma is the square sum of the polynomial;
s202, describing a nonlinear time-invariant system needing to determine an impulse response peak value by using a state equation:
wherein t ∈ R denotes time, x (t) ∈ RnRepresents the system state, u (t) e R represents the input, y (t) e RpThe output of the system is represented and,
expressing a suitably sized nonlinear function matrix of the system state, abbreviated as
S203, defining the impulse response y of the systemIR(t), namely the zero state response of the nonlinear time-invariant system to the impulse function input, is that the initial condition of the system is x (0)-) 0 and the system output y (t) when the input is u (t) δ (t), where δ (t) is a dirac unit impulse function;
s204, under the zero initial condition x (0)-) Inputting an impulse function to the system at 0, which is equivalent to setting the initial state value to
The corresponding system output when input u (t) is 0:
determining a constant
Such that the infinite norm with a single-channel impulse response for all input channels of the system is less than a constant c:
establishing a constant c epsilon (0, infinity) as the upper limit of the system impulse response peak value;
s101 includes finding a Lyapunov function v (x) with a degree not greater than 2d, d ∈ N: rn→ R, known from the definition of the Lyapunov function, v (x) being a polynomial of the system state x, the derivative with respect to time
Is negative, so that:
S102 includes re-projecting the Lyapunov function system v (x) to evaluate the level set
Whether or not to be in the collection
In the middle, let
Wherein s isk(x) Finding a suitable scalar quantity epsilon > 0 epsilon R for the polynomial coefficient, so that the polynomial h obtained by re-projection of the Lyapunov function v (x)k(x) Belonging to the polynomial square sum, i.e. hk(x)∈∑；
Polynomial hk(x) Is linear with the coefficients of the polynomial v (x), hk(x) If the condition of belonging to the polynomial square set is satisfied, the system state track is in the set
Performing the following steps;
s103 includes a Lyapunov function v (x) that can be expressed in the form of a matrix: v (x) ((V + L (α)) b (x)) where V is a symmetric matrix and α is a vector variable and b (x) is a vector composed of a series of polynomial bases with degree no greater than d, and whether a polynomial belongs to the sum of squares is judged, which is equivalent to judging whether the linear matrix inequality V + L (α) ≧ 0 corresponding to the polynomial holds.
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