Design of Input Signals and Initial States. V.I. Denisov, V.M. Chubich, O.S. Chernikova. Novosibirsk State Technical University, Novosibirsk, Russia. Abstract ...
АПЭП – 2014 XII МЕЖДУНАРОДНАЯ КОНФЕРЕНЦИЯ
Active Parametric Identification of Stochastic Linear Discrete System Based on Optimal Design of Input Signals and Initial States V.I. Denisov, V.M. Chubich, O.S. Chernikova Novosibirsk State Technical University, Novosibirsk, Russia Abstract – Under consideration are some theoretical and applied aspects of the active parametric identification of the multidimensional stochastic linear discrete systems described by state space models. Original results are obtained in the case that the parameters of mathematical models to be estimated appear in various combinations in the state and measurement equations, as well as the covariance matrices of the process noise and measurement errors. By an example of one model structure, the efficiency and sensibility is shown of applying the active identification procedure based on optimal design of input signals and initial states.
tried to generalize the results obtained in these papers and constructed new algorithms connected with the simultaneous design inputs and initial states.
Index Terms – Parameter estimation, maximum likelihood method, (Fisher) information matrix, experimental design.
k = 0,1,..., N − 1 . is the state n - vector; u ( tk ) is deter-
N THE TERMS OF THE WAY OF THE behavior present methods of identification can be divided into passive and active. In the time of passive identification of dynamic systems only real operating signals and initial state are used and modes of operation are not disturbed. Fully enough description of passive identification methods is given in [1]. On the contrary, active identification assumes violation of technological mode and uses some special synthesized designs [2]. Mentioned designs are results of extreme problem solution for some preliminary chosen functionals of information (or covariance) matrices for vector of parameters to be estimated. The difficulties connected with necessity of violation of a technological mode, are provide by increase of efficiency and correctness of researches. Methods of active identification give to the experimenter considerably greater possibilities in design of qualitative model in comparison with methods of passive identification. These possibilities are caused by the ideology of the active identification based on a combination of traditional parameter estimation techniques with the concept of experiment design. The procedures of active parametric identification of stochastic linear discrete systems based on optimal design of input signals or initial states were developed in [3] and [4] respectively. In this paper, the authors 978-1-4799-/14/$31.00 © 2014 IEEE
Consider the following model of controllable, observable and identifiable stochastic linear discrete system in the state space: x tk +1 = a u(tk ), tk + F tk x tk + Γ tk w tk ; (1)
( ) ( ) ( ) ( ) ( ) ( ) y ( tk +1 ) = A ( tk +1 ) + H ( tk +1 ) x ( tk +1 ) + v ( tk +1 ) ,
Here x ( tk )
(2)
ministic control (input) r - vector; w ( tk ) is the pro-
I. INTRODUCTION
I
II. PROBLEM DEFENITION
cess noise p - vector; y ( tk +1 ) is the measurement (output) m - vector; v ( tk +1 ) is the measurement error m - vector. Suppose that • the random vectors w ( tk ) and v ( tk +1 ) form stationary white Gaussian sequences with E ⎡⎣ w(tk )⎤⎦ = 0 ,
E ⎡⎣ w(tk )wT (ti ) ⎤⎦ = Qδ ki , E ⎣⎡v(tk ) ⎦⎤ = 0 , E ⎡⎣v(tk +1 )vT (ti +1 ) ⎤⎦ = Rδ ki , E ⎡⎣v(tk )wT (ti )⎤⎦ = 0 , k , i = 0,1, , N − 1
( Ε [⋅] denotes the mathematical expectation, •
574
δ ki - Kronecker symbol); the initial state x ( t0 ) is normally distributed with parameters
Ε ⎡⎣ x ( t0 )⎤⎦ = x ( t0 ) ,
{
Ε ⎡⎣ x( t0 ) − x ( t0 )⎤⎦ ⎡⎣ x ( t0 ) − x ( t0 )⎤⎦
T
} = Ρ (t ) 0
and is uncorrelated with w ( tk ) and v ( tk +1 ) for all values of k ;
2014 12TH INTERNATIONAL CONFERENCE APEIE – 34006 •
the vectors a ( u (tk ), tk ) , A ( tk +1 ) and matrices
initial state x i (t0 ) . Then as a result of carrying out an
F ( tk ) , Γ ( tk ) , H ( tk +1 ) , Q , R depend on unknown parameters Θ = (θ 1 ,θ 2 , ,θ s ) ∈ ΩΘ .
identification experiment for design ξν the following set will be generated
For the mathematical model (1), (2), taking the listed a priori assumptions into account, we need to develop an active parametric identification procedure based on optimal design of the input signals and initial states and study its efficiency and reasonability of application. In such mathematical statement this task is considered and solved for the first time. III. THEORY
{(
}
)
Ξ = α i , Yi , j , j = 1,2,..., ki , i = 1,2,..., d ,
d
∑ ki = ν i =1
The apriori assumptions stated in section II allow to use maximum likelihood (ML) method for parameter estimation. According to this method it is necessary to find such values of parameters Θˆ for which
Θˆ = arg min ⎡⎣ χ (Θ ; Ξ ) ⎤⎦ = arg min ⎡⎣ − ln L (Θ ; Ξ ) ⎤⎦ Θ ∈ΩΘ
(4)
Θ ∈ΩΘ
Procedure of active parametric identification of systems with preliminary chosen model structure assumes performance of following stages: Stage 1. Calculation of parameters estimates with use of measuring data corresponding to some experiment design. Estimation of unknown parameters of mathematical model it is carried out according to observation data Ξ according to criterion of identification χ . Gathering of numerical data occurs during carrying out of identification experiments which are carried out under some design ξν . Assume that experimenter can make ν system starts. He gives signal U1 on the input of the system
and (see, for example, [5])
with initial state x 1 (t0 ) k1 times, signal U 2 with ini-
Ρ ( tk +1 | tk ) = F ( tk ) Ρ ( tk | tk ) F T ( tk ) + Γ ( tk ) QΓ T ( tk ) ;
χ (Θ ; Ξ ) =
ν N −1 Nmν ln 2π + ∑ ln det Β ( tk +1 ) + 2 k =0 2
T −1 1 d ki N −1 i , j ⎡ε ( tk +1 )⎤ ⎡⎣Β ( tk +1 )⎤⎦ ⎡ε i , j ( tk +1 )⎤ , ∑∑ ∑ ⎣ ⎦ ⎣ ⎦ 2 i =1 j =1 k = 0
where
ε i , j ( tk +1 ) and Β ( tk +1 ) are defined according
to corresponding recursive equations of discrete Kalman filter [6]:
(
)
xˆ i, j ( tk +1 | tk ) = a ui (tk ), tk + F ( tk ) xˆ i, j (tk | tk ) ; −1
K ( tk +1 ) = Ρ ( tk +1 | tk ) Η T ( tk +1 ) ⎡⎣Β ( tk +1 ) ⎤⎦ ;
tial state x 2 (t0 ) - k2 times etc., at last, signal U d
yˆ i , j ( tk +1 | tk ) = A ( tk +1 ) + Η ( tk +1 ) xˆ i , j ( tk +1 | tk ) ;
with initial state x d (t0 ) - kd times. In this case dis-
ε i , j ( tk +1 ) = y i , j ( tk +1 ) − yˆ i , j ( tk +1 | tk ) ;
crete normalized experiment design ξν is 1
2
⎧ α ,α , ⎪ ξν = ⎨ k 1 k 2 ⎪⎩ ν , ν ,
, α ⎫⎪ k d ⎬ , α i ∈ Ω α , i = 1,2,..., d , (3) , ν ⎪⎭ where the point of design spectrum α i has the fol-
lowing structure
T
T ⎡ T i ⎤ ⎡ i⎤ ⎣α ⎦ = ⎢U i ; x ( t0 ) ⎥ = ⎣ ⎦
)
(
) (
periment
are
)
(
)
defined
Ωα = ΩU × Ω x (t0 )
by ⊂ RNr + n .
the
)
(
bounded
T
⎤ ⎥ ex⎦ set
T ,..., yi , j (t N ) ⎤ ⎥⎦ output realization with number j ( j = 1, 2, , ki ) corresponding to input signal U i ( i = 1, 2,..., d ) with
(
T
) ,( y
i, j
(t2 )
T
)
(
Ρ ( tk +1 | tk +1 ) = ⎣⎡Ι − Κ ( tk +1 ) Η ( tk +1 )⎦⎤ Ρ ( tk +1 | tk ) , for
k = 0,1,..., N − 1 , j = 1, 2,..., ki , i = 1, 2,..., d initial
conditions
xˆ i , j ( t0 | t0 ) = x i ( t0 ) ,
Ρ ( t0 | t0 ) = Ρ ( t0 ) .
Designate through
YiT, j = ⎡ yi , j (t1 ) ⎢⎣
xˆ i , j ( tk +1 | tk +1 ) = xˆ i , j ( tk +1 | tk ) + Κ ( tk +1 ) ε i , j ( tk +1 ) ;
with
T T T i ⎡ = ⎢ u i ( t0 ) , u i ( t1 ) ,..., u i ( t N −1 ) ; x ( t0 ) and⎣ restrictions on the conditions of carrying out
(
Β ( tk +1 ) = Η ( tk +1 ) Ρ ( tk +1 | tk ) Η T ( tk +1 ) + R ;
d
)
We use Sequential Quadratic Programming (SQP) method for calculation of the conditional minimum χ (Θ ; Ξ ) . Stage 2. Synthesis of experiment design based on the received estimates and chosen criterion optimality. Optimal input signals and initial states choice allows to the experimenter to prepare the most informative observation data with given number of system starts. These data are used for unknown parameters estimation from stage 1. Under continuous normalized design ξ we will understand the collection
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АПЭП – 2014 XII МЕЖДУНАРОДНАЯ КОНФЕРЕНЦИЯ
⎧ 1 2 ,α q ⎫ ξ = ⎨αp ,,αp ,, ⎬, ⎩ 1 2 , pq ⎭
(5)
q
pi ≥ 0,
∑ p i = 1, α i ∈ Ω α , i = 1,2,..., q. i =1
For design (5) normalized information matrix Μ
(ξ )
are defined by q
Μ (ξ ) = ∑ p i Μ (α i ),
(6)
i =1
where (Fisher) information matrices for one-point designs
(
⎡ ∂ 2 ln L α i ,Θ Μ α ,Θ = − Ε ⎢ Y ⎢ ∂ Θ ∂Θ T ⎣ depend on unknown parameters Θ (this
(
i
)
) ⎤⎥ , ⎥ ⎦
fact allows us to speak further only about locally optimal design). In [7] an algorithm of calculation of (Fisher) information matrices for model (1), (2) is given. The quality of parameter estimation can be improved by construction of experiment design which optimizes some convex functional X of information matrix Μ (ξ ) , i.e. we should solve extreme task *
ξ = arg min X [ M (ξ )] .
Another approach (it is called dual) to solving the optimal problem (7) is based on the equivalence theorem. In this case the problem under the study is no longer a convex programming problem, but the size of the parameter vector being varied can turn out substantially smaller than in the direct approach [2]. Applying of the gradients in the direct or dual procedures leads on the significantly improve the speed of problem solving. It is impossible without calculation derivatives (Fisher) information matrix by the components of input signal and initial state. In this case, the gradient optimality criterion at points of design spectrum has the following block structure: ⎡ ∂X ⎡⎣ M ( ξ ) ⎤⎦ ⎤ ⎡ ∂X ⎡⎣ M (ξ ) ⎤⎦ ⎤ ⎢ ∂u i ( t ) ⎥ ⎥ 0 ⎢ ⎥ ⎢⎢ ⎥ ∂X ⎣⎡ M (ξ ) ⎦⎤ ⎢ ∂U i ⎥ ⎢ ∂X ⎡ M ξ ⎤ ⎥ , = ( ) ⎣ ⎦⎥ ⎢ ∂X ⎡⎣ M (ξ ) ⎤⎦ ⎥ = ⎢ ∂α i i ⎢ ⎥ u t ∂ ( ) ⎢ ⎥ i N −1 ⎣⎢ ∂x ( t0 ) ⎥⎦ ⎢ ∂X ⎡⎣ M (ξ ) ⎤⎦ ⎥ ⎢ ⎥ i ⎣⎢ ∂x ( t0 ) ⎦⎥
i = 1,2,..., q . The
X ⎡⎣ M (ξ )⎤⎦ = − ln det M (ξ ) ,
and for an А – optimality criterion
X ⎡⎣ M (ξ )⎤⎦ = − SpM −1 (ξ ) .
Solving the problem of designing experiments, we act in a certain way on lower bound in the RaoCramer inequality: for a D – optimal design we minimize the volume, and for an А – optimal design minimize the sum of the squares axis lengths of the concentration ellipsoid of the estimates for the parameters. We can solve the optimization task of finding the minimum of (7) using the general numerical methods for finding extrema. The two approaches are possible. The first (direct) approach is to seek the minimum of the functional X ⎡⎣ M (ξ ) ⎤⎦ on the space inforwhere mation matrices ΩM ,
ΩM = {M (ξ ) ⊂ Rs×s | ξ ∈ Ωξ } , taking into account that design spectrum (5) consist of q =
s ( s + 1) +1 2
of
calculation
of
gradients
∂X ⎡⎣ M ( ξ ) ⎤⎦ ∂X ⎡⎣ M ( ξ ) ⎤⎦ and for D –optimality i ∂u ( t k ) ∂ x i ( t0 )
(7)
ξ ∈Ωξ
For a D –optimality criterion
algorithms
and A – optimality criterions are given in [3] and [8] respectively. Stage 3. Recalculation of estimates of unknown parameters on the measuring data corresponding to the optimal design. Practical application of the constructed continuous optimum design is complicated because weights pi* are, generally speaking, any real numbers in the range from zero to unit. It is simple to notice that in case of given number of possible starts of the system v values ki* = vpi* can be noninteger numbers. Carrying out of experiment demands a rounding off of values
ki* to integers. It is obvious that the design received as a result of such rounding off will differ from the optimum continuous design. An approach to the optimum continuous design is better when the number of possible starts is bigger. The possible algorithm of "rounding off" of the continuous design to the discrete design is given in [9]. Next, prepare a discrete design
⎧α*1 (t0 ), α*2 (t0 ), , α*q (t0 ) ⎫ ⎪ ⎪ ξν = ⎨ k1* kq* ⎬ , k2* ⎪⎩ ν , ν , , ν ⎪⎭ *
points. Since X ⎡⎣ M (ξ ) ⎦⎤ is convex functional, we have a problem of convex programming and we find the conditional minimum (7) with help of SQP method.
will conduct the identification experiment and recalculate estimation of unknown parameters.
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2014 12TH INTERNATIONAL CONFERENCE APEIE – 34006 IV. EXPERIMENTAL RESULTS
ki N −1
q
Consider the following model of stochastic linear discrete dynamic system:
⎛ θ1 1 ⎞ ⎛ 0.58 ⎞ ⎛ 1 0⎞ x ( tk +1 ) = ⎜ ⎟ x ( tk ) + ⎜ ⎟ u ( tk ) + ⎜ ⎟ w ( tk ) , ⎝ 0.57 ⎠ ⎝ 0 1⎠ ⎝ θ2 0 ⎠ where parameters to be estiy ( tkθ+11 ,) =θ 2(1,are 0 ) xthe ( tk +unknown 1 ) + v ( tk +1 ) , k = 0,1,..., N − 1; mated, i.e. Assume s = 2. 0 ≤ θ1 ≤ 2 , −1 ≤ θ 2 ≤ 1 , N = 20 and all listed a priori assumptions from section II are satisfied so that
⎛ 0.1 0 ⎞ ⎟ δ ki = Qδ ki , ⎝ 0 0.1⎠ Ε ⎡⎣v ( tk +1 ) v ( ti +1 )⎤⎦ = 0.02δ ki = Rδ ki ,
Ε ⎡⎣ w ( tk ) wT ( ti ) ⎤⎦ = ⎜
Y − Yˆ *
δY* =
Y
∑∑ ∑ ( yi, j (tk ) − yˆ*i, j (tk tk −1 ) ) =
i =1 j =1 k = 0 q ki N −1
∑∑ ∑ ( y i =1 j =1 k = 0
TABLE I RESULTS OF PROCEDURES ACTIVE IDENTIFICATION BASED ON DESIGN OF INPUT SIGNALS AND (OR) INITIAL STATE
⎝ ⎠
T ⎛ 0.1 0 ⎞ Ε ⎡⎣ x( t0 ) − x ( t0 )⎤⎦ ⎡⎣ x ( t0 ) − x ( t0 )⎤⎦ = ⎜ ⎟ = Ρ ( t0 ) . ⎝ 0 0.1⎠ Chose design area Ωα = ΩU × Ω x (t ) , where 0
Discrete experiment design
Ω U = {U ∈ RN −1 − 5 ≤ u ( tk ) ≤ 5, k = 0,1,..., N − 1} ,
T
}
Initial design
θ1* = 0.52 , θ2* = −0.01 . The quality of identification in the parameter space and response space will be judged by values of fac* Y
tors δθ , δθ and δ Y , δ accordingly. These factors are calculated on following formulas:
θ − θˆ
δθ =
δY =
θ*
Y − Yˆ Y
=
∑(
θi* − θˆi
i =1
s
2
)
∑q (kθi*N) −1 i =1
i
s
2
, δθ* =
∑∑ ∑ ( y =
ˆ*
*
θ −θ
i, j
θ*
(tk ) − yˆ i =1 j =1 k = 0 q ki N −1 i, j
∑∑ ∑ ( y i =1 j =1 k = 0
=
i, j
∑( i =1
s
∑2(θi* )
2
)
)
2
)
k =5 ⎩⎪ ⎭⎪ U corresponds to the signal №1 from Tabl. III and x (t0 ) corresponds to the initial state №1 from the Tabl. II Optimal design of initial states
⎧⎪α = ⎡U T ; x*T (t0 ) ⎤ ⎪⎫ ⎣ ⎦⎬ ξ* = ⎨ ⎩⎪
k =5
U corresponds to the signal №1 *
from Tabl. III and x (t0 ) corresponds to the initial state №2 from the Tabl. II Optimal design of input signals
,
δθ = 0.151, δY = 0.106 θˆ *1 = 0.495 , θˆ * = 0.048 2
δθ* = 0.121 , δ Y* = 0.91
( )
U * corresponds to the signal №2 from Tabl. III and x (t0 ) corre-
δθ* = 0.063 , δ Y* = 0.065
sponds to the initial state №1 from the Tabl. II Optimal design of input signals and initial states (the optimal design has turned out two point)
577
2
θˆ *1 = 0.512 , θˆ * = 0.022
*
;
θˆ1 = 0.487 , θˆ = 0.061
⎪⎭
⎧ ⎡ * T ; x T (t ) ⎤ ⎫ ⎪α = ⎢ U 0 ⎥⎪ ⎣ ⎦⎬ ξ =⎨ ⎪⎩ ⎪⎭ k =5
2
i =1
(tk tk −1 )
(tk )
θi* − θˆi*
T
⎪⎧α = ⎣⎡U ; x (t0 ) ⎦⎤ ⎪⎫ ξ =⎨ ⎬
and D – optimality criterion. In order to weaken the dependence of estimation results on measurement data we will make five independent starts of system, i.e.ν = 5 , and then we will average received estimates of unknown parameters. Realizations of output signal we will get by the computer modeling with true values of parameters
s
Values of parameters estimates θ
Ω x (t0 ) = {x (t0 )∈ R2 − 1 ≤ x j (t0 ) ≤ 1, j = 1,2}
*
)
,
initial experiment design; θˆ* are estimates of unknown parameters values based on the synthesized input signal and (or) initial states; Results of the active parametric identification procedure based on optimal design of input signals and (or) initial state are presented in Tabl. I.
⎛ 1⎞
*
(tk )
2
where θ * are true values of parameters; θˆ are estimates of unknown parameters values based on the
Ε ⎡⎣ x ( t0 ) ⎤⎦ = ⎜ ⎟ = x ( t0 ) , 1
{
i, j
2
2
θˆ *1 = 0.508 , θˆ *2 = -0.018
АПЭП – 2014 XII МЕЖДУНАРОДНАЯ КОНФЕРЕНЦИЯ
α2 ⎫ ⎧ α1 ⎬ ⎩ k1 = 3 k2 = 2 ⎭
ξ* = ⎨
T T αi = ⎡⎢(Ui* ) ; x*i (t0 ) ⎤⎥ , ⎣ ⎦
(
)
U i*
corresponds to the signals №3 (i=1) and №4 (i=2) from Tabl. III, and
x*i (t0 ) corresponds to the initial
δθ* = 0.027 , δ Y* = 0.041
3
state №1 (i=1) and №2 (i=2) from Tabl. II.
TABLE II INITIAL STATE FOR TABLE I Initial state № 1 2
Initial state x (t0 )
4
T
(1,1)
(−1, −1)T TABLE II INPUT SIGNALS FOR TABLE I
Input signal №
1
V. DISCUSSION OF RESULTS
Input signal U
The results presented in the Tabl. I, show that design of the initial conditions only conditions led to better estimation by 3% in the parameter space, and 1.5% in the space of responses, while design of the input signals only contributed to the improvement of evaluation results at 8.8% and 4.1%, respectively. The most successful results were obtained for the case of designing of the inputs and initial conditions (improvement on 12.4% and 6.5%). Thus, the authors consider that the applying of the active parametric identification procedures based on optimal design of input signals and initial states is helpful and advisable. VI. CONCLUSION
2
In this paper was shown that design of the input signals and the initial states has a defining impact on the functional dependence of state variables and model parameters. For the first time we considered and solved the problem of active parametric identification of the Gaussian linear discrete systems in the case when the unknown parameters in the state and control equations as well as the covariance matrices of the dynamic noise and measurement noise are contained. The original gradient algorithms of active identification based 578
2014 12TH INTERNATIONAL CONFERENCE APEIE – 34006 on optimal design of input signals and initial states were developed. The work is executed under the auspices of the Ministry of education and science of the Russian Federation (№2014/138, project № 1689).
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[3]
[4]
[5] [6] [7]
[8]
[9]
L. Ljung. System identification: Theory the user. - M.: Nauka, 1991. - 432 p. (in Russian). V.I. Densov, V.M. Chubich, O.S. Chernikova and D.I. Bobyleva. Active parametric identification of stochastic linear systems. Novosibirsk: Novosibirsk State Technical University, 2009. - 192 p. (in Russian). V.M. Chubich. Information technology of active parametric identification of stochastic quasilinear discrete systems / / Computer Science and Applications, 2011. V.5. №.1. P. 46-57. (in Russian). V.M. Chubich, O.S. Chernikova. Optimal parameter estimation of Gaussian models of linear discrete systems based on design the initial state. Vestnik Novosibirsk State Technical University, 2013. №4(53). P. 31 – 40. (in Russian). Astrom K.J. Maximum likelihood and prediction errors methods // Automatica, 1980. V. 16. P. 551 - 574. M.A. Ogarkov. Methods for statistical estimate of the parameters of random processes. - M.: Energoatomizdat, 1980. - 208 p. (in Russian). V.M. Chubich. The calculation of the Fisher information matrix in the problem of active parametric identification for stochastic nonlinear discrete systems. Vestnik Novosibirsk State Technical University, 2009. № 1(34). P. 23 – 40. (in Russian). V.M. Chubich. Design of the initial state in the problem of parametric identification of active Gaussian linear discrete systems. Vestnik Novosibirsk State Technical University, 2011. № 1(42). P. 39 – 46. (in Russian). S.M. Yermakov, A.A. Zhiglyavskiy. Mathematical theory of optimal experiment.- M.: Nauka, 1987. - 320 p. (in Russian).
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Denisov Vladimir I. was born in 1936, D.Sc. (Engineering), professor, assistant of Vice-Rector on scientific work, Novosibirsk State Technical University. Founder of the Novosibirsk school mathematical theory analysis and experimental design. He has authored or coauthored over 200 journal and conference papers. Chubich Vladimir M. was born in 1962, PhD, associate professor of the department of the applied mathematics, Novosibirsk State Technical University. His research has been concentrated on the areas of analysis and experiment design of stochastic dynamic system. He has authored or coauthored over 50 journal and conference papers.
Chernikova Oksana S. was born in PhD, associate professor of the department of the applied mathematics, Novosibirsk State Technical University. Her research has been concentrated on the areas of active identification of stochastic dynamic system. She has authored or coauthored over 25 journal and conference papers.
