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1、Decompositionbased least squares iterative estimation algorithm for output error moving average systems May 14, 2013 Outline 1. Background 2. Idea 3. Problem formulation 4. Identifi cation algorithms 5. Algorithm description 6. Example 7. Conclusions 1 1 Background Large-scale system: The number of
2、the system parameters is large. Online identifi cation: For some practical application and the adaptive control, we need estimate the system parameter real time. Output error moving average systems(OEMA): The output error model disturb by a colored noise with moving average model. Traditional iterat
3、ive algorithm: The estimate accuracy is higher than the recursive algorithm, but it has a heavier computational burden. 2 2 Idea This paper presents a decomposition based least squares iterative identifi cation algorithm for output error moving average (OEMA) systems. The basic idea is to decompose
4、an OEMA system into three subsystems and to identify each subsystem, respectively. Compared with the least squares based iterative algorithm, the proposed algorithm has a less computational burden.The simulation results show the performance of the proposed algorithm. 3 3 Problem formulation OEMA sys
5、tem : Let us introduce some notations fi rst. “A =: X” or “X := A” stands for “A is defi ned as X”; the superscript T denotes the matrix transpose; the norm of a matrix X is defi ned by kXk2:= trXX T; 1n represents an n-dimensional column vector whose elements are all 1. Consider the following OEMA
6、system, y(t) = B(z) A(z) u(t) + D(z)v(t),(1) where u(t) is the input sequence of the system, y(t) is the output sequence of the system, v(t) is a white noise sequence with zero mean and variance 2, and A(z), B(z) and D(z) are the polynomials in the 4 unit backward shift operator z1i.e., z1 y(t) = y(
7、t 1), and defi ned by A(z):=1 + a1z1+ a2z2+ + anazna, B(z):=b1z1+ b2z2+ + bnbznb, D(z):=1 + d1z1+ d2z2+ + dndznd, The orders na, nband ndare assumed to be known. Without loss of generality, and that y(t) = 0, u(t) = 0 and v(t) = 0 for t 6 0. The goal of this paper is to apply the decomposition techn
8、ique and to develop a new least squares based iterative identifi cation algorithm for estimating the system parameters ai, biand di, and transform the original identifi cation problem into three subproblems with small sizes. 5 Defi ne the true output of the system, x(t) := B(z) A(z) u(t).(2) Defi ne
9、 the parameter vectors: := 1 2 3 Rn, n := na+ nb+ nd, 1:=a1,a2,anaT Rna, 2:=b1,b2,bnbT Rnb, 3:=d1,d2,andT Rnd, 6 and the information vectors: (t):= (t) (t) (t) Rn, (t):=x(t 1),x(t 2),x(t na)T Rna,(3) (t):=u(t 1),u(t 2),u(t nb)T Rnb, (t):=v(t 1),v(t 2),v(t nd)T Rnd.(4) Then (2) can be written as x(t)
10、 = T(t)1+ T(t)2.(5) Using (2) and (5), we can obtain the following identifi cation model from (1): y(t)=x(t) + D(z)v(t) 7 =T(t)1+ T(t)2+ T(t)3+ v(t)(6) =T(t) + v(t).(7) Defi ne three intermediate variables: y1(t):=y(t) T(t)2 T(t)3,(8) y2(t):=y(t) T(t)1 T(t)3,(9) y3(t):=y(t) T(t)1 T(t)2.(10) From (8)
11、(10), the system in (6) can be decomposed into the following three fi ctitious subsystems: y1(t)=T(t)1+ v(t),(11) y2(t)=T(t)2+ v(t),(12) y3(t)=T(t)3+ v(t).(13) 8 From (8)(10), we have Y 1=Y 22 33, (14) Y 2=Y 11 33, (15) Y 3=Y 11 22. (16) From (11)(13), we have Y i = ii+ V , i = 1,2,3. Defi ne three
12、quadratic criterion functions: Ji(i) := kY i iik2, i = 1,2,3. Letting the partial derivatives of Ji(i), with respect to ibe zero gives Ji(i) i = 2T iYi ii = 0, i = 1,2,3. 9 Assume that the information vectors (t), (t) and (t) are persistently exciting, that is, T 11, T 22 and T 33 are non-singular.
13、From the above equations, we can obtain the following least squares estimates of the parameter vectors i: i= T ii 1T iYi, i = 1,2,3. Substituting (14)(16) into the above equations, respectively, gives 1=T 11 1T 1Y 22 33,(17) 2=T 22 1T 2Y 11 33,(18) 3=T 33 1T 3Y 11 22.(19) 10 4 Identifi cation algori
14、thms Applying the Gauss-Seidel method to coordinate the solutions in the diff erent subsystems, the decomposition least squares based iterative (D- LSI) identifi cation algorithm for estimating 1, 2and 3of the OEMA systems can be summarized as follows: 1(k)= T 1(k) 1(k)1 T 1(k)Y 2 2(k 1) 3(k)3(k 1), (20) 2(k)=T 22 1T 2Y 1(k)1(k) 3(k)3(k 1),(21) 3(k)= T 3(k) 3(k)1 T 3(k)Y 1(k)1(k) 22(k),(22) Y =y(1),y(2),y(L)T,(23) 1(k)= k(1), k(2), k(L)T,(24) 2=(1),(2),(L)T,(25) 3(k)=k(1), k(2), k(L)T,(26) k(t)= xk1(t 1), xk1(t 2), xk1(t na) T, (27) 11 (t)=u(t 1),u(t 2),u(t nb)T(28) k(t)= vk1(t 1), vk1(t 2)
