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Signal Adapted Modulation Functions for Identification of Linear Continuous Time Systems
Copyright © IFAC 10th Triennial World Congress. Munich . FRG. 1987
14.5-2 VARIOUS IDENTIFICATION TECHNIQUES
SIGNAL ADAPTED MODULATION FUNCTIONS FOR IDENTIFICATION OF LINEAR CONTINUOUS TIME SYSTEMS F.
J.
Kraus and M. F. Senningt
Department of Automatic Control and Industrial Electronics, Swiss Federal Institute of T echnology (E TH), CH -8092 Zurich. Switzerland
Abstract. In the literature on Modulation Function Identification of linear continuous time systems, little attention is paid to the choice of the modulation function. It is shown that an a priori chosen modulation function may cause the identification to fail, even though the system is persistently excited. As remedy, an appropriate modulation function needs to be chosen in accordance with the measurement signals. Based on a Ritz parametrization, a family of modulation functions is proposed. Existence conditions are given. Among the members of this family, the optimally signal adapted modulation functions guarantee not to filter out necessary information for the parameter identification. The special case of orthogonal series is discussed. Its relationship to the classical Fourier series of the measurement signals is investigated. Keyword. Identification. Parameter estimation. Parameter fitting. Ritz parametrization. Modulation function. Template function.
I.
system usually have a physical interpretation. Further, with decreasing sampling time, the denominator polynomial coefficients of the corresponding discrete time system in the frequency domain description converge toward the binomial coefficients. The corresponding nominator coefficients converge toward zero. Hence, for the discrete time system identification with high sampling rate, the requirements on the accuracy are considerably sharpened [1]. Moreover, if the system to be identified is in a cascade of systems , the input signal is not in a sampled data form which prevents a discrete time system parameter identification.
INTRODUCTION
Consider a single input, single output plant which is modelled by the linear continuous time system (1) of order n. an_l y (n-l) + + b _ u(n-l) +
n 1
(1 )
where y(t) and u(t) denote the output and input signals with the corresponding higher order time derivatives y
section 2 reviews the basics of the classical modulation function identification technique which is commonly used for the identification of continuous time system parameters. In the literature, little attention is paid to the choice of the modulation function. The impact of this device is illustrated by an academic example. The filtering effect of the modulation function is shown to cause the identification to fail even
t Dept. Packing Machineries, Swiss Industrial Company, CH-8212 Neuhausen,
switzerland. 245
F. J. Kraus and M. F. Senning
246
though the measurement signals contain enough information to identify the parameters uniquely. In the subsequent section, based on a Ritz parametrization, a general family of modulation functions is defined. The existence properties of this family are discussed. section 4 copes with special members of this modulation function family, the so-called optimally signal adapted modulation functions, which do not filter out information necessary to identify the system parameters. In section 5 a special case is investigated: The use of orthogonal series which are closed w.r.t. derivation simplifies the computation of the optimally signal adapted modulation functions considerably. Further, an illustration of the method based on trigonometric Fourier series is given. Finally, the relationship between the optimally signal adapted modulation functions and the Fourier series of the measurement signals is discussed.
II.
PRINCIPLES OF MODULATION FUNCTION IDENTIFICATION
To introduce the notation and to enlight some severe drawbacks of the modulation function identification, its basic principles are briefly reviewed. If, in system (1), in addition to the measurable signals y(t) and u(t), their time derivatives y
The basic idea behind the modulation function identification is the premultiplying of the system (1) with a so-called modulation function. Integrating this product per partes enables the elimination of the unmeasurable time deri vati ves of the signals y(t) and U(t). To present the basics of the classical modulation function identification in Theorem 1, the modulation function needs to be defined. Definition 1. Let T e ~, T > 0, be given. A n-fold differentiable function ~: [O,T] ~ ~ is termed modulation function if it satisfies the boundary conditions (3), where the vector function]£: [0, T] ~ JlIl is defined according to (2) •
(~(t) ,~(t), ••. ,~(n-l) (t»'
(2 )
]£(0)
=Q
(3)
Theorem 1. For a given Tm, Tm > 0, let the functions u,y: [O,Tm] ~ ~ be given, where y( . ) is generated by the system (1) with the input signal u(.). For T'Tm, let ~: [O,T] ~ ~ be a modulation function, and define the vector ]£(t) according to (2). Then, for any T e [O,Tm-T], the parameter vector ~ satisfies the measurement equation (4) , where ~ is defined according to (5) •
J
x' []£(t)y(t+T) ]dt - 0 ]£(t)U(t+T)
T
J~(n)(t)y(t+T)dt
(4)
o
Proof. The proof is straightforward and is, thus, merely summarized. Premultiplying (1) with the modulation function ~(t) and integrating over the period [T,T+T] together with a time transformation and inserting the boundary conditions (3) of the modulation function yield (4) . For the ease of notation, the changes of sign due to the integration per partes are taken regard for in the parameter vector ~.
•
To identify the parameters aj' bj , i = 0, ••• , n-l, of the persistently excited system (1) with unmeasurabe higher order time deri vati ves of the measurable signals y(t) and u(t), Theorem 1 provides a linear scalar equation in ~ for each data window [T,T+T]. The measurement equations of different data windows [Tj' Tj+T] , i = 1, •.. , f , f > 2n, define a linear, possibly overdetermined, equation set for the parameter vector ~. Parameter identification amounts to fit this linear equation in a least-squares sense. Note that different modulation functions can be used to form this set of linear equations in ~, rather than different data windows. Even though the signals y(t) and U(t) in the measurement equation (4) are continuous over time, the integrals are commonly computed numerically with some degree of discretization. It is important to note that (4) can be interpreted as a convolution of the signals y(t) and u(t) with a time function ]£(T-t). Thus, the modulation function identification amounts to smooth the measurements with a filter defined according to the modulation function. Accordingly, even though the measurement signals y(t) and u(t) contain enough information to identify the parameters uniquely, the modulation function identification may fail due to loss of information caused by the filtering of the measurements.
247
Identification of Linear Continuous Time Systems
In the past, little attention was paid to this inherent phenomenon of the classical modulation function identification, which is illustrated in the subsequent example. Given the system (1') and the input signal (6) which persistently excites
Lemma 1. A function
= Q'
(11)
where
(1') ,
B
[f(0),f(O), ... ,f(n-1)(O),f(T), .•. , f(n-l) (T)]
sin(2t) + sin(3t)
u(t)
(6)
Choose the modulation function
=
=0 =0
1 - COS(t) ,
T =
2n
~(O) = ~(T)
Proof . . According to Definition 1, a modulation function satisfies the boundary condition (3). Thus, the virtue of (9), the boundary condition (13) or, equivalently (11), is obtained.
(7)
Q'
0
=0
This example enlightens the importance of the choice of the modulation function. It is crucial not to a priori choose the modulation function, without taking the measurement signals y(t) and u(t) into account.
[p'f(O), ..• ,p'f(n-1)(0),p'f(T), ... , p'f(n-1) (T)]
(3' )
By orthogonality of the input signal u(t) and the modulation function
(12)
(13)
• A trivial modulation function is the zero function over [O,T]. Lemma 2 gives the existence conditions for non-trivial modulation functions in the space Fk •
Lemma 2. A non-trivial modulation function
(14)
Proof. By Lemma 1, vector R must be within the left null-space of the boundary matrix B. For a non-trivial vector R, i.e. R ~ Q, to satisfy this condition, (14) must hold.
•
Ill. RITZ FAMILIES OF MODULATION FUNCTIONS Let Cn[O,T] be a linear space of real-valued, n-fold differentiable functions f(t) which are defined on the interval [O,T], T > O. Further, let Fk c Cn[O,T] be a k-dimensional sub-space which is spanned by the basis (8 )
with fj(t) E Cn[O,T]. with respect to this basis, an arbitrary element ~(t) E Fk is representable by ~(t)
k
I Pi fi(t) i=l
= R'f(t)
(9 )
Theorem 2 gives the description of the complete set of modulation functions in the space Fk . Theorem 2. Let f(t) be the basis (8) and B the corresponding boundary matrix (12) . Further, define v = k - rank(B) and let V be an orthogonal matrix the columns of which span the left kernel of B. Then, the set of all modulation functions in the space Fk constitute the v-dimensional linear subspace ~ (15). ~
= (g'Vf(t),
gE
~v)
(15)
Proof. To prove the Theorem constructively, we introduce the SVD of the boundary matrix B:
where (10) By the linear independence of the basis elements fj(t), i = l, ... ,k, over the interval [O,T], the representation (9) is unique. In the subsequent lemma, two existence results for modulation functions in the space Fk are presented.
(16) where L = diag( 0j)' 0j > 0 Vi, and the matrix (VI V2 ) is orthonormal. Since L E ~xr with r = rank(B), the dimension v of the basis V2 is v = k - rank(B) Due to (16) and the orthogonali ty of the matrix (VI V2 ), i.e. V~V2 = 0, we conclude that the columns of V2 span the left kernel of B, span(V2 ) = ker(B'). Thus, all parameter vectors J2 of the
F. J. Kraus and M. F. Senning
248
Ri tz series (9) which satisfy condition (11) must be of the form
the
boundary
T
f f(t)y(t+T)dt
g E Jlv
Taking V V2 and q>(t) .. R'f(t), assertion in the Theorem.
proves the
T
•
o
By Theorem 2, all modulation functions corresponding to a given Ritz basis fIt) are parametrized. Note that, since V is orthogonal, the functions 1(t) = V'f(t) constitute a basis for the space <%> of all modulation functions in Fk . Thus, contrary to the entries of fIt), the entries of 1(t) are modulation functions.
The rest of this paragraph is devoted to a special case for which the Ritz series is particularly favourable. If the space Fk is closed w.r.t. derivation, the computation of the higher order time derivatives of the modulation function in (4) is significantly simplified. In this case, in lieu of Theorem 1, the modulation function identification is stated in Theorem 3.
f f(t)u(t+T)dt
Lemma 3. If Fk is closed w.r.t. derivation, the i-th time derivative of the modulation function is expressable by (17)
By ( 2) and Lemma 3, (R'f(t), •.. ,R'Dn-1f(t))'. Extracting the time independent part of the integral expression in (4), yields the matrix Q(p) (20) on the left-hand side, and 12' Dn on the right-hand side. The remaining of the integrals is given by l!:y (T) and l!:u(T), (21) and (22), respectively.
= D fIt)
(18)
for an appropriate matrix D. The time derivative of q>(t) is accordingly ~(t) = 12' D fIt). Repeating this procedure i-fold, yields (17).
• Theorem 3. Let the assumptions of Theorem 1 hold. Then, the parameter vector ~ satisfies the measurement equation (19), where Q(P), l!:y(T) and l!:u(T) are defined according to (20), (21), and (22), respectively.
(19)
Q(P)
R'D 12' [
~'Dn-l
1
=
•
Due to Theorem 3, the computational burden of the modulation function identification is significantly reduced if the Ritz basis is closed w.r.t. deri vation. For each measurement equation (19) , 1. e. for each T, solely the 2k integral expressions (21) and (22) need to be computed. without the closedness assumption (Theorem 1), (2n+1)k integrals have to be evaluated for each measurement equation. In addition, the closedness assumption enables a simplified construction of the boundary matrix B. B = [f(0), ... ,D n- 1f(0),f(T), ... ,Dn- 1f(T)1
Obviously, with exception of the (n+1)-th column, the columns of B are generated by premultiplying their predecessors with the matrix D.
(17)
Proof. Since Fk is closed w.r.t. derivation, fIt) E Fk . Thus, fIt) is a linear combination of the basis fIt): fIt)
(22)
Proof.
~(t)
Definition 2. A linear space F is closed w.r.t. derivation if g(t) E F, V g(t) E F.
i = 1,2, ...
(21)
o
IV. OPTlMALLY SIGNAL ADAPTED MODULATION FUNCTIONS Paragraph 2 reveals that modulation function identification uses signals that are obtained by filtering the measurement signals y(t) and u(t). To avoid the information in y(t) and u(t) needed for identification to be filtered out, the modulation function must be appropriately chosen. Since commonly no modulation function exists that is simultaneously adapted to all signals y(t), ... ,y(t) = g'1(t) Fk is termed optimally adapted to the signal 7)(i) (t) if it is a solution of the unconstrained optimization problem (23),
E
(20)
Ide ntificatio n of Linea r Co ntinu ous Time Sys tems
--
min 1~(t)-g'f(i)(t)12
(23)
gellv
where fIt)
249
= V'I(t) and 1· 1 denotes the ~-norm.
Theorem 4. The modulation function ~(t) = g'f(t) is optimally adapted to the signal ~(j) (t) if and only if
.
By (21) and (22), for fj(t) being the Fourier functions over T, l!y{T) and l!u(T) are vectors with the Fourier coefficients of the time shifted measurement signals y(t+T) and U(t+T), respectively. Thus, by virtue of Theorem 3, the only measurement dependent information needed to form the measurement equation (19) for a given modulation function ~ (t) = ~'f( t) is the Fourier coefficient of the signals y(t+T) and U(t+T).
(24)
The case of optimally signal-adapted modulation functions which are based on Fourier functions is stated in Theorem 5:
where T
Mi
J f(i)(t)f(i)'(t)
dt
(25)
o T
Ii
J f(i)(t)~(t)
dt
(26)
o Proof.
The proof is standard, and, thus, omitted.
For the signals ~(t) = y(t+T), ~(t) = U(t+T) and their time derivatives, we obtain 2n different modulation functions. By (4), each of these give a linear equation for the 2n-dimensional parameter vector ~. Using different T, i.e. different measurement data windows, results in further sets of 2n linear equations for ~. This overdetermined system of linear equations forms a fitting problem for the unknown ~. Note that in contrast to the classical modulation function identification, different modulation functions result for different T. Note that only Ij depends on the signal ~(t) . Hence, the matrix Mj can be computed a priori without knowledge of the measurement signal ~(t). Since the classical modulation function is chosen a priori, the measurement noise is asymptotically filtered out. In this sense, the classical modulation function identification is an Instrumental Variable Method [5]. Contrarily, the optimally signal adapted modulation functions depend on the measurement signals and are, thus, corrupted by measurement noise. Consequently, they do not possess the IV property.
V.
FOURIER-BASED MODULATION FUNCTIONS
In this section, we cope with modulation functions based on trigonometric Fourier series. Id est, the Ritz basis functions are trigonometric Fourier functions. Subsequent to the investigation of the case of a priori fixed modulation functions, the optimally signal-adapted modulation functions are considered.
Theorem 5. Let the basis elements fj(t) be the Fourier functions over the period T. For a given T and i e [O,n-1], let ~ yj(t) and ~uj(t) be the modulation functions which are optimally adapted to the signals y{j)(t+T) and u(i) (t+T), respectively. Then, applying ~ y j(t) and ~u j (t), i = O, ... ,n-1, results in the 2n linear equations (27) for the parameter vector ~ :
[~ ~:l' [~l
(27)
=
where the (j,') -th entry of the submatrices Ayz and the j-th entry of the sub vector £y are (28)
(29) Proof. The proof is straightforward and thus merely summarized. By the closedness property of the Fourier functions, the inserting of the optimally signal adapted modulation functions according to Theorem 4 into the measurement equations (19) of Theorem 3 yields the linear equation (27).
•
Note that (27) has 2n equations for the 2n unknowns in ~. To obtain an overdetermined fitting problem for ~, different data windows [T,T+T] have to be processed and the measurement equations have to be concatenated.
To illustrate the method, the example in paragraph 2 is considered. Let the system (1') have the parameters aO = -2, a l = -3, bO = 2, b l = O. Further, let the system operate, in its steady state, driven by the input signal (6). Recall, if the modulation functions (7) are applied , the identification is impossible due to the infinite condition number of the fitting matrix. If the Ritz basis function is chosen to be the
Fourier function
F. J. Kraus and M. F. Senning
250
REFERENCES
fIt)
~[~
, sint, cost, sin2t, cos2t, sin3t, cos3t]'
the fitting problem based on the signal adapted modulation function is
.0504 .0126 -.0209 [ -.1962
.0188 -.0209 .0667 .0744 .2143 .1418 -.2077 0.0000
-.3214 -.2077 0.0000 2.0000
1 If=
optimally
[-.1989 -.0764 .0451 1.0154
1
The condition number is in this case K = 643, i.e. the parameter vector If is identifyable.
VI. CONCLUSIONS Even for persistently excited systems, choosing a modulation function without taking regard to the mesurement signals may cause a loss of accuracy in Adapted Modulation Functions guarantee the identifiability of the parameters if the system is persistently excited. The optimally signal adapted modulation functions are special members of a Ritz parametrized function family. If the basis. of this family is closed w. r . t. derivation, the computation of the optimally signal adapted modulation function is considerably simplified. Moreover, if the basis is given by the trigonometric Fourier functions, the only measurement dependent information needed to identify the parameters are the Fourier coefficients of the measurement signals y(t) and u(t). In this sense, the optimally signal adapted modulation function identification is closely related to the Fourier analysis of the measurement signals.
[1] F.J. Kraus, and M.F. Senning: Parameter Weighted Least Squares Fitting. IFAC Cont. on Digital Computer Applications to Process Control, Vienna, 1985. [2] M. Shinbrot: On the Analysis of Linear and Nonlinear Systems. Trans. ASME, 1957-2. [3] P. Eykhoff: System Identification: an Approach to a Coherent pictore through Template Functions. Electr. Letters 16(1980). [4] A.E. Pearson, F.C. Lee: Parameter Identification for a Class of Polynomial Differential Systems. 23rd IEEE Conf. on Decision and Control, Las Vegas, 1984. [5] T. SOderstrom, P.G. stoica Instrumental Variable Methods for System Identification, springer Verlag, 1983.
14.5-2 VARIOUS IDENTIFICATION TECHNIQUES
SIGNAL ADAPTED MODULATION FUNCTIONS FOR IDENTIFICATION OF LINEAR CONTINUOUS TIME SYSTEMS F.
J.
Kraus and M. F. Senningt
Department of Automatic Control and Industrial Electronics, Swiss Federal Institute of T echnology (E TH), CH -8092 Zurich. Switzerland
Abstract. In the literature on Modulation Function Identification of linear continuous time systems, little attention is paid to the choice of the modulation function. It is shown that an a priori chosen modulation function may cause the identification to fail, even though the system is persistently excited. As remedy, an appropriate modulation function needs to be chosen in accordance with the measurement signals. Based on a Ritz parametrization, a family of modulation functions is proposed. Existence conditions are given. Among the members of this family, the optimally signal adapted modulation functions guarantee not to filter out necessary information for the parameter identification. The special case of orthogonal series is discussed. Its relationship to the classical Fourier series of the measurement signals is investigated. Keyword. Identification. Parameter estimation. Parameter fitting. Ritz parametrization. Modulation function. Template function.
I.
system usually have a physical interpretation. Further, with decreasing sampling time, the denominator polynomial coefficients of the corresponding discrete time system in the frequency domain description converge toward the binomial coefficients. The corresponding nominator coefficients converge toward zero. Hence, for the discrete time system identification with high sampling rate, the requirements on the accuracy are considerably sharpened [1]. Moreover, if the system to be identified is in a cascade of systems , the input signal is not in a sampled data form which prevents a discrete time system parameter identification.
INTRODUCTION
Consider a single input, single output plant which is modelled by the linear continuous time system (1) of order n. an_l y (n-l) + + b _ u(n-l) +
n 1
(1 )
where y(t) and u(t) denote the output and input signals with the corresponding higher order time derivatives y
section 2 reviews the basics of the classical modulation function identification technique which is commonly used for the identification of continuous time system parameters. In the literature, little attention is paid to the choice of the modulation function. The impact of this device is illustrated by an academic example. The filtering effect of the modulation function is shown to cause the identification to fail even
t Dept. Packing Machineries, Swiss Industrial Company, CH-8212 Neuhausen,
switzerland. 245
F. J. Kraus and M. F. Senning
246
though the measurement signals contain enough information to identify the parameters uniquely. In the subsequent section, based on a Ritz parametrization, a general family of modulation functions is defined. The existence properties of this family are discussed. section 4 copes with special members of this modulation function family, the so-called optimally signal adapted modulation functions, which do not filter out information necessary to identify the system parameters. In section 5 a special case is investigated: The use of orthogonal series which are closed w.r.t. derivation simplifies the computation of the optimally signal adapted modulation functions considerably. Further, an illustration of the method based on trigonometric Fourier series is given. Finally, the relationship between the optimally signal adapted modulation functions and the Fourier series of the measurement signals is discussed.
II.
PRINCIPLES OF MODULATION FUNCTION IDENTIFICATION
To introduce the notation and to enlight some severe drawbacks of the modulation function identification, its basic principles are briefly reviewed. If, in system (1), in addition to the measurable signals y(t) and u(t), their time derivatives y
The basic idea behind the modulation function identification is the premultiplying of the system (1) with a so-called modulation function. Integrating this product per partes enables the elimination of the unmeasurable time deri vati ves of the signals y(t) and U(t). To present the basics of the classical modulation function identification in Theorem 1, the modulation function needs to be defined. Definition 1. Let T e ~, T > 0, be given. A n-fold differentiable function ~: [O,T] ~ ~ is termed modulation function if it satisfies the boundary conditions (3), where the vector function]£: [0, T] ~ JlIl is defined according to (2) •
(~(t) ,~(t), ••. ,~(n-l) (t»'
(2 )
]£(0)
=Q
(3)
Theorem 1. For a given Tm, Tm > 0, let the functions u,y: [O,Tm] ~ ~ be given, where y( . ) is generated by the system (1) with the input signal u(.). For T'Tm, let ~: [O,T] ~ ~ be a modulation function, and define the vector ]£(t) according to (2). Then, for any T e [O,Tm-T], the parameter vector ~ satisfies the measurement equation (4) , where ~ is defined according to (5) •
J
x' []£(t)y(t+T) ]dt - 0 ]£(t)U(t+T)
T
J~(n)(t)y(t+T)dt
(4)
o
Proof. The proof is straightforward and is, thus, merely summarized. Premultiplying (1) with the modulation function ~(t) and integrating over the period [T,T+T] together with a time transformation and inserting the boundary conditions (3) of the modulation function yield (4) . For the ease of notation, the changes of sign due to the integration per partes are taken regard for in the parameter vector ~.
•
To identify the parameters aj' bj , i = 0, ••• , n-l, of the persistently excited system (1) with unmeasurabe higher order time deri vati ves of the measurable signals y(t) and u(t), Theorem 1 provides a linear scalar equation in ~ for each data window [T,T+T]. The measurement equations of different data windows [Tj' Tj+T] , i = 1, •.. , f , f > 2n, define a linear, possibly overdetermined, equation set for the parameter vector ~. Parameter identification amounts to fit this linear equation in a least-squares sense. Note that different modulation functions can be used to form this set of linear equations in ~, rather than different data windows. Even though the signals y(t) and U(t) in the measurement equation (4) are continuous over time, the integrals are commonly computed numerically with some degree of discretization. It is important to note that (4) can be interpreted as a convolution of the signals y(t) and u(t) with a time function ]£(T-t). Thus, the modulation function identification amounts to smooth the measurements with a filter defined according to the modulation function. Accordingly, even though the measurement signals y(t) and u(t) contain enough information to identify the parameters uniquely, the modulation function identification may fail due to loss of information caused by the filtering of the measurements.
247
Identification of Linear Continuous Time Systems
In the past, little attention was paid to this inherent phenomenon of the classical modulation function identification, which is illustrated in the subsequent example. Given the system (1') and the input signal (6) which persistently excites
Lemma 1. A function
= Q'
(11)
where
(1') ,
B
[f(0),f(O), ... ,f(n-1)(O),f(T), .•. , f(n-l) (T)]
sin(2t) + sin(3t)
u(t)
(6)
Choose the modulation function
=
=0 =0
1 - COS(t) ,
T =
2n
~(O) = ~(T)
Proof . . According to Definition 1, a modulation function satisfies the boundary condition (3). Thus, the virtue of (9), the boundary condition (13) or, equivalently (11), is obtained.
(7)
Q'
0
=0
This example enlightens the importance of the choice of the modulation function. It is crucial not to a priori choose the modulation function, without taking the measurement signals y(t) and u(t) into account.
[p'f(O), ..• ,p'f(n-1)(0),p'f(T), ... , p'f(n-1) (T)]
(3' )
By orthogonality of the input signal u(t) and the modulation function
(12)
(13)
• A trivial modulation function is the zero function over [O,T]. Lemma 2 gives the existence conditions for non-trivial modulation functions in the space Fk •
Lemma 2. A non-trivial modulation function
(14)
Proof. By Lemma 1, vector R must be within the left null-space of the boundary matrix B. For a non-trivial vector R, i.e. R ~ Q, to satisfy this condition, (14) must hold.
•
Ill. RITZ FAMILIES OF MODULATION FUNCTIONS Let Cn[O,T] be a linear space of real-valued, n-fold differentiable functions f(t) which are defined on the interval [O,T], T > O. Further, let Fk c Cn[O,T] be a k-dimensional sub-space which is spanned by the basis (8 )
with fj(t) E Cn[O,T]. with respect to this basis, an arbitrary element ~(t) E Fk is representable by ~(t)
k
I Pi fi(t) i=l
= R'f(t)
(9 )
Theorem 2 gives the description of the complete set of modulation functions in the space Fk . Theorem 2. Let f(t) be the basis (8) and B the corresponding boundary matrix (12) . Further, define v = k - rank(B) and let V be an orthogonal matrix the columns of which span the left kernel of B. Then, the set of all modulation functions in the space Fk constitute the v-dimensional linear subspace ~ (15). ~
= (g'Vf(t),
gE
~v)
(15)
Proof. To prove the Theorem constructively, we introduce the SVD of the boundary matrix B:
where (10) By the linear independence of the basis elements fj(t), i = l, ... ,k, over the interval [O,T], the representation (9) is unique. In the subsequent lemma, two existence results for modulation functions in the space Fk are presented.
(16) where L = diag( 0j)' 0j > 0 Vi, and the matrix (VI V2 ) is orthonormal. Since L E ~xr with r = rank(B), the dimension v of the basis V2 is v = k - rank(B) Due to (16) and the orthogonali ty of the matrix (VI V2 ), i.e. V~V2 = 0, we conclude that the columns of V2 span the left kernel of B, span(V2 ) = ker(B'). Thus, all parameter vectors J2 of the
F. J. Kraus and M. F. Senning
248
Ri tz series (9) which satisfy condition (11) must be of the form
the
boundary
T
f f(t)y(t+T)dt
g E Jlv
Taking V V2 and q>(t) .. R'f(t), assertion in the Theorem.
proves the
T
•
o
By Theorem 2, all modulation functions corresponding to a given Ritz basis fIt) are parametrized. Note that, since V is orthogonal, the functions 1(t) = V'f(t) constitute a basis for the space <%> of all modulation functions in Fk . Thus, contrary to the entries of fIt), the entries of 1(t) are modulation functions.
The rest of this paragraph is devoted to a special case for which the Ritz series is particularly favourable. If the space Fk is closed w.r.t. derivation, the computation of the higher order time derivatives of the modulation function in (4) is significantly simplified. In this case, in lieu of Theorem 1, the modulation function identification is stated in Theorem 3.
f f(t)u(t+T)dt
Lemma 3. If Fk is closed w.r.t. derivation, the i-th time derivative of the modulation function is expressable by (17)
By ( 2) and Lemma 3, (R'f(t), •.. ,R'Dn-1f(t))'. Extracting the time independent part of the integral expression in (4), yields the matrix Q(p) (20) on the left-hand side, and 12' Dn on the right-hand side. The remaining of the integrals is given by l!:y (T) and l!:u(T), (21) and (22), respectively.
= D fIt)
(18)
for an appropriate matrix D. The time derivative of q>(t) is accordingly ~(t) = 12' D fIt). Repeating this procedure i-fold, yields (17).
• Theorem 3. Let the assumptions of Theorem 1 hold. Then, the parameter vector ~ satisfies the measurement equation (19), where Q(P), l!:y(T) and l!:u(T) are defined according to (20), (21), and (22), respectively.
(19)
Q(P)
R'D 12' [
~'Dn-l
1
=
•
Due to Theorem 3, the computational burden of the modulation function identification is significantly reduced if the Ritz basis is closed w.r.t. deri vation. For each measurement equation (19) , 1. e. for each T, solely the 2k integral expressions (21) and (22) need to be computed. without the closedness assumption (Theorem 1), (2n+1)k integrals have to be evaluated for each measurement equation. In addition, the closedness assumption enables a simplified construction of the boundary matrix B. B = [f(0), ... ,D n- 1f(0),f(T), ... ,Dn- 1f(T)1
Obviously, with exception of the (n+1)-th column, the columns of B are generated by premultiplying their predecessors with the matrix D.
(17)
Proof. Since Fk is closed w.r.t. derivation, fIt) E Fk . Thus, fIt) is a linear combination of the basis fIt): fIt)
(22)
Proof.
~(t)
Definition 2. A linear space F is closed w.r.t. derivation if g(t) E F, V g(t) E F.
i = 1,2, ...
(21)
o
IV. OPTlMALLY SIGNAL ADAPTED MODULATION FUNCTIONS Paragraph 2 reveals that modulation function identification uses signals that are obtained by filtering the measurement signals y(t) and u(t). To avoid the information in y(t) and u(t) needed for identification to be filtered out, the modulation function must be appropriately chosen. Since commonly no modulation function exists that is simultaneously adapted to all signals y(t), ... ,y
E
(20)
Ide ntificatio n of Linea r Co ntinu ous Time Sys tems
--
min 1~(t)-g'f(i)(t)12
(23)
gellv
where fIt)
249
= V'I(t) and 1· 1 denotes the ~-norm.
Theorem 4. The modulation function ~(t) = g'f(t) is optimally adapted to the signal ~(j) (t) if and only if
.
By (21) and (22), for fj(t) being the Fourier functions over T, l!y{T) and l!u(T) are vectors with the Fourier coefficients of the time shifted measurement signals y(t+T) and U(t+T), respectively. Thus, by virtue of Theorem 3, the only measurement dependent information needed to form the measurement equation (19) for a given modulation function ~ (t) = ~'f( t) is the Fourier coefficient of the signals y(t+T) and U(t+T).
(24)
The case of optimally signal-adapted modulation functions which are based on Fourier functions is stated in Theorem 5:
where T
Mi
J f(i)(t)f(i)'(t)
dt
(25)
o T
Ii
J f(i)(t)~(t)
dt
(26)
o Proof.
The proof is standard, and, thus, omitted.
For the signals ~(t) = y(t+T), ~(t) = U(t+T) and their time derivatives, we obtain 2n different modulation functions. By (4), each of these give a linear equation for the 2n-dimensional parameter vector ~. Using different T, i.e. different measurement data windows, results in further sets of 2n linear equations for ~. This overdetermined system of linear equations forms a fitting problem for the unknown ~. Note that in contrast to the classical modulation function identification, different modulation functions result for different T. Note that only Ij depends on the signal ~(t) . Hence, the matrix Mj can be computed a priori without knowledge of the measurement signal ~(t). Since the classical modulation function is chosen a priori, the measurement noise is asymptotically filtered out. In this sense, the classical modulation function identification is an Instrumental Variable Method [5]. Contrarily, the optimally signal adapted modulation functions depend on the measurement signals and are, thus, corrupted by measurement noise. Consequently, they do not possess the IV property.
V.
FOURIER-BASED MODULATION FUNCTIONS
In this section, we cope with modulation functions based on trigonometric Fourier series. Id est, the Ritz basis functions are trigonometric Fourier functions. Subsequent to the investigation of the case of a priori fixed modulation functions, the optimally signal-adapted modulation functions are considered.
Theorem 5. Let the basis elements fj(t) be the Fourier functions over the period T. For a given T and i e [O,n-1], let ~ yj(t) and ~uj(t) be the modulation functions which are optimally adapted to the signals y{j)(t+T) and u(i) (t+T), respectively. Then, applying ~ y j(t) and ~u j (t), i = O, ... ,n-1, results in the 2n linear equations (27) for the parameter vector ~ :
[~ ~:l' [~l
(27)
=
where the (j,') -th entry of the submatrices Ayz and the j-th entry of the sub vector £y are (28)
(29) Proof. The proof is straightforward and thus merely summarized. By the closedness property of the Fourier functions, the inserting of the optimally signal adapted modulation functions according to Theorem 4 into the measurement equations (19) of Theorem 3 yields the linear equation (27).
•
Note that (27) has 2n equations for the 2n unknowns in ~. To obtain an overdetermined fitting problem for ~, different data windows [T,T+T] have to be processed and the measurement equations have to be concatenated.
To illustrate the method, the example in paragraph 2 is considered. Let the system (1') have the parameters aO = -2, a l = -3, bO = 2, b l = O. Further, let the system operate, in its steady state, driven by the input signal (6). Recall, if the modulation functions (7) are applied , the identification is impossible due to the infinite condition number of the fitting matrix. If the Ritz basis function is chosen to be the
Fourier function
F. J. Kraus and M. F. Senning
250
REFERENCES
fIt)
~[~
, sint, cost, sin2t, cos2t, sin3t, cos3t]'
the fitting problem based on the signal adapted modulation function is
.0504 .0126 -.0209 [ -.1962
.0188 -.0209 .0667 .0744 .2143 .1418 -.2077 0.0000
-.3214 -.2077 0.0000 2.0000
1 If=
optimally
[-.1989 -.0764 .0451 1.0154
1
The condition number is in this case K = 643, i.e. the parameter vector If is identifyable.
VI. CONCLUSIONS Even for persistently excited systems, choosing a modulation function without taking regard to the mesurement signals may cause a loss of accuracy in Adapted Modulation Functions guarantee the identifiability of the parameters if the system is persistently excited. The optimally signal adapted modulation functions are special members of a Ritz parametrized function family. If the basis. of this family is closed w. r . t. derivation, the computation of the optimally signal adapted modulation function is considerably simplified. Moreover, if the basis is given by the trigonometric Fourier functions, the only measurement dependent information needed to identify the parameters are the Fourier coefficients of the measurement signals y(t) and u(t). In this sense, the optimally signal adapted modulation function identification is closely related to the Fourier analysis of the measurement signals.
[1] F.J. Kraus, and M.F. Senning: Parameter Weighted Least Squares Fitting. IFAC Cont. on Digital Computer Applications to Process Control, Vienna, 1985. [2] M. Shinbrot: On the Analysis of Linear and Nonlinear Systems. Trans. ASME, 1957-2. [3] P. Eykhoff: System Identification: an Approach to a Coherent pictore through Template Functions. Electr. Letters 16(1980). [4] A.E. Pearson, F.C. Lee: Parameter Identification for a Class of Polynomial Differential Systems. 23rd IEEE Conf. on Decision and Control, Las Vegas, 1984. [5] T. SOderstrom, P.G. stoica Instrumental Variable Methods for System Identification, springer Verlag, 1983.