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On State Estimation of Unknown Systems Based on Subspace State-Space System Identification
suppose we can select the correct order, that. is, the same order as a system to be identified. The system and an identified model are related by a similarity transformation. On considering recursive identification, a similarity transformation by which an identified model is related to the system at an update step may not be identical to onc at another update step.
is Ilonsingular, where the notation as follows:
Unfortunately, conventional state estimation methods, e.g., Kalman filter or so-called obser'ver in control theory, are not available directly for the situation which we mentioned above because they need a fixed estimate of the order of a system and a fixed similar transformation commonly over all update steps.
If the output measurement of the system (1) is contaminated by the noise term VI.;, then the contaminated measurement fh is described as
N
L uv(j)'uv(j)T
(2)
j=1
U v (·)
is defined
(3)
(4)
Let the noise VI.; be a zero-mean Gaussian white noise with covariance
To overcome these difficulties, we introduce a new kind of "state" which maintains its meaning as a state over different update steps. This "state" must reflect some intrinsic properties of the system which are invariant with respect to choices of the order of models and the degree of freedom of similarity transformations. We identify such a "state" which possesses such properties.
V"
It is assumed that is independent of Ui for any i and the initial state Xo.
Remark 1. If u is a quasi-stationary signa\(Ljung, 1999), replace the expectation E by E where
This paper is organized as follows . In Section 2, some assumptions required in the paper are introduced. Section 3 gives a brief review of the basic 4SID method. In Section 4, we introduce a recursive algorithm of the basic 4SID which is based on the matrix inversion lemma (Oku and Kimura, 1999). In Section 5, we consider a state estimation problem of a MIMO model derived by the 4SID method . A vector made by premultiplying the state of a system to be identified by the extended observability matrix of it is employed as the "stale" mentioned above. We also present estimates of such a new" state" at the past, the present and the one-step ahead steps, respectively, in this section . Finally, an illustrative example is shown in Section 6.
_
1
Eu(k) :=
J~oo!vI
M
L Eu(k). "=1
3, BRIEF REVIEW OF BASIC 4SID
In this section, we give a brief review of basic 4SID for noise-free systems. For details, see Verhaegen and Dewilde(1992a). For a finite sequence of input-output (for short, "I/O") data {(Uj, Yj)}, j = 1,," ,N, uv(i) E JR"v, Yv (i) E JRl1tv are defined as follows:
Ui-V+I] uv(i)
:=
[
:
Yi - V+l] '
Yv(i)
1ti
:=
:
[
Yi (5)
2. PRELIMINARIES
Then, from (1), it follows that
In this section, some assumptions required in this paper are introduced.
Yv(i) =
= Axl.; + Bu",
y"
= Cx" + Du",
(6)
where
A discrete-time LTI system to be identified is described as follows: X"+l
r vXi-v+1 + Hvuv(i),
r v :=
(la) (lb)
u"
where XI.; E JRn, E JRr and Yk E JR7n. Let the matrix A be stable and the pair (A, C) be observable. The initial state of the system Xo is assumed to be O. A sequence of input data Ui is assumed to be a realization of a bounded stochastic process which is independent of the initial state Xo. We assume that there exists an integer No > 0 such that for a given integer v satisfying v 2 n + 1, a given input sequence {ud and 'iN 2 No
H"=
[C T (CA)T .. , (CAv-l)T
f',
[c5~'B CAJ-'B. J
(7)
(8)
The equation (6) yields the following matrix equation with respect to the input-state-output relations:
where
242
proposed in this paper is inferior to the one due to Verhaegen and Deprettere.
(11)
On the other hand, from a theoretical point of view, the algorithm can clarify the fundamental structure of the recursive 4SID algorithm, specially the relations among input, output and state vectors of a system to he identified. This advantage led us to recusive POMOESP (or PI-MOESP) algorithms (Oku and Kimura, 1999). It is also useful to analyze some properties of the 4SID schemes which will be stated in the following sections. For the details of MOESP(MIMO Output-Error State sPace model identification) schemes, see Verhaegen and Dewilde(1992a; 1992b) and Verhaegen(1993; 1994).
Note that UN and YN are rvx (N -v+l) and rnvx (N - v + 1) matrices which consist of N I/O data, respectively. r v E R mv x nand H v E R mv x ,·v are said to he an extended observability matrix and a
lower· triangular block Toeplitz matrix containing the Markov pammeters, respectively. Notice that the matrix U NUJr is invertible because (3) is invertible. We define the following two projection matrices l
lIu N := UJr (UNUJrr UN,
( 12)
IIU N := I - lI uN •
(13)
From (15), postmultiplication of YNIIU N by its transpose yields
Then, postmultiplication of the equation (9) by 11.1 UN yields the relation YNIIU N
= rvXNIIU
N
•
-= .-N:
(14)
= [EN
E~ ] [S 0] [(:~T ],
n.l
FiN 1[(fll:l N
(y 11.1) T - Y 11.1 yT
[~N u;~ I
0
(15)
(19)
it is known that for a sufficient large N there exists a nonsingular matrix T such that
Algorithm 1. (Oku and Kimura, 1999) Suppose =-N-l, PN- 1 and YN- I UJr_1 have already been obtained. When a pair of the N-th I/O data
( 16) From EN and E~ a state-space representation of the resulting model (AT, B T , CT, DT) can be calculated such that AT CT
= TAT - I, = CT- ' ,
BT DT
= TB,
= D.
(\8)
The relation (18) means that both EN and E~ can be derived from computation of eigenvalues of the real symmetric matrix =-N := YNIIUN YJ, which is called compressed I/O data matrix in this paper. A recursive algorithm of updating this compressed I/O data matrix is derived in the sequel. Let
Note that YNIIU N in the left hand side of (14) can be made of the input and output HankeJ matrices, (10) and (ll). If the singular value decomposition of the left side of (14) is obtained as YNIIU N
Y
(UN,YN) is obtained, we can construct Uv(N) and yv(N) from (5). Then, the N-th compressed I/O data matrix =-N can be updated recursively by the following procedure:
( 17)
=-N
A detailed procedure of the derivation of such matrices has been shown by Verhaegen and Dewilde(1992a).
= =-N - I + QNeNej;",
(20)
where
PN
= PN - I -
QNPN - 1U,/(N)Uv(N{ PN - I , (21)
The recursive scheme of identification is reduced to obtaining the recursive computation of Y N N •
IIU
(22)
4. A RECURSIVE ALGORITHM OF THE BASIC 4SID
(23)
A recursive 4SID algorithm based on the RQ factorization has already been proposed by Verhaegen and Deprettere( 1991). In this paper, we introduce a recursive algorithm which is based OIl the matrix inversion lemma (Oku and Kimura, 1999). It is well known that algorithms based on the matrix inversion lemma sometimes exhibit numerically poor performance with respect to the round-off error. Algorithms based on matrix factorization (e.g. QR or UD factorization methods) were devised as a remedy for this problem initially in state estimation (Bierman, 1977) . Therefore, from the numerical point of view, the algorithm
Proof. See Oku and Kimura(1999).
0
Remark 2. This algorithm works well also for the case where the contaminated measurement fh is observed because
.
hm N N-tCXJ
-
1 v
T
+ 1 VNUN = 0,
(25)
where VN is a matrix composed of the noise VI" in the same manner as (10). In this case, this algorithm is modified by replacing Yv(N), Yi , =-i
243
(i = N - I
or N) and eN by
tlj. v
(N) , I,i-t,
estimates of r", H" and (A, E, C, D) at the step N be denoted as
-::: ...... 1...-- ,i-rr1.. I1 U }i-'1' i
and
. (26)
EN, RN, (AN, EN, eN, DN), respectively. These matrices are obtained from the Algorithm 1, the eigenvalue decomposition of (20).
5. ON STATE ESTIMATION OF MIMO MODELS A principal advantage of 4SID schemes is that the order of a model, denoted by n, can be chosen to be the number of dominant singular values in (15). This means that, when we apply recursive 4SID for the system, we may take different choices of the order of models at different update steps.
Now, based on (6), we consider that an estimate ZN-v+!lN of the quasi-state ZN-v+l is given by the following equation:
If we can select the correct order, that is , the same order as the system, from (17) the system and an identified model are related by a similarity transformation matrix T. On considering recursive identification, a similarity transformation matrix TI by which an identified model are related to the system at an update step NI may not be identical to one T2 at another update step N 2 , i.e., TI #- T2 for NI #- N 2 · In this section we consider a state estimation problem of unknown multivariable systems utilizing the recursive 4SID. The problem is very important because the knowledge of the present state of the identified models can be a clue to design an adaptive control law for the system. Unfortunately, conventional state estimation methods, e.g. Kalman filter or observer, are not available directly for our case because they assume that the order of the system is fixed and similarity transformations are common over all update steps.
The third term of the right hand side of (29) plays a role of cancelling the noise effect. We shall explain the reason. From (1) the estimates zNIN and ZN+1IN of the quasi-states ZN and ZN+l, respectively, are calculated respectively as
ZNIN:= E N (AA N )v-1EtN ZN- ,,+lIN v- I + EN :L(A N )i - l EN'UN - i, A
From (28), the states of the estimated model (AN, BN , eN, D N ) are calculated as
(32) (33) (34) The rest of this section is devoted to the explanation of why the third term of the right hand side of (29) eliminates the noise effect. From (4) and (9), we obtain
(27)
Zi E ]Rmv.
We take Zi as a state at step i. The term Zi is named "the quasi-state at step i" in this paper. Note that the dimension of Zi is independent of the order of the system n and of the order of a model n. From the assumption that r v is of full column rank, the state of the system is denoted by
YN
rt
is the Moore-Penrose inverse of
r
(E~)'rYNUJ,
= y,,(i+v-I) -
(35)
= (E!vf HvUNU~ + (E~fvNU~. (36)
v'
Because of (25), we can neglect the second term of the right hand side of (36) for sufficient large N, i.e.,
If it were not for the measurement noise and if the lower triangular block Toeplitz matrix containing the Markov parameters Hv in (8) were known, we could easily obtain the quasi-state Zi from (6), i.e.,
Zi
= r"X N- v+1 + H"UN + VN.
It is assumed that there exists a nonsingular matrix T such that r = ENT. The premultiplication of (35) by (EM T and the postmultiplication by U~ yield
(28) where
(30)
i=1
To overcome this difficulty, we introduce a new kind of state which is still meaningful over different update steps. Let
Zi := r vXi,
A
1..'1'T ( EN) YNU N ~
Hv'Uv(i+v-I).
1..T
T
(EN) HvUNU N ·
From (19),
Since the matrix Hv is unknown but can be estimated by the recursive identification, the quasistate Zi can also be estimated recursively.
(E~f (Yv(N) - H,,'Uv(N))
~ (E~f (iiv(N) - frNU~PN'Uv(N)) .
When a pair of data (UN, Y N) is obtained at step N , Uv(N) and Yv(N) are composed of (5). Let
Note that, frolll (22),
244
• The one-step-ahead predictors of the model zN+lIN and xN+lIN are ohtained from the following equations:
6. A NUMERICAL EXAMPLE
(37)
In order to illustrate the usefulness of our idea presented in the previous section, we consider a tracking control problem.
(38)
Consider an unknown system described as (1) with
Therefore, we have (Elvf' (y1/(N) - HI/UI/(N))
= (EIvl
T
ONCN.
On the other hand, without noise we could derive
A
=
(EIv)T (YI/(Nl - HI/UI/(Nl)
.IT( T ) =(E N ) YI/(N) - YNUNPNUI/(N)
=(Elvf ON (YI/(N) =(Elvf 0NCN· Since f v
= EnT,
-0.18
(Elvf ONCN =(EIv)T ON (fvXN-I/+I (39)
Hence, in comparison with (38) to (39) we can regard the term E;(E;)1' ONCN as estimation error of ZN-v+1 due to the meSllrement noise. Now, we summarize the estimation scheme of the state.
Define ~k+1 - ~k := r -
EN, fIN, (AN, B N , CN , DN )
ZN-v+IIN =Yv(N) - HNul/(N)
- EIv(Elvf oNeN, Et '
NZN-v+lIN '
• The present states of the model zNIN and xNIN are obtained from the following equations: , E A' v-IE t '
=
N
N
ih (=: Ed·
Since the parameters of the system (40) A, B and C are assumed to be unknown , we use parameters of an estimated model AN, B Nand CN for estimating the state. The model is obtained by the recursive 4SID identification with taking v = 5. Let the order of the model be described as nN. Then, we obtain the following extended state-space representation:
• Past states of the model zN-v+lIN and XN-v+1IN are obtained from the following equations:
zNIN
(40)
-0.3
Suppose we have N pairs of I/O data (Ui ' Y;), i = 1"" , N. To solve this problem, one-step-ahead control input UN+I is decided according to the following procedure.
• A model of the system at step N results from recursive 4SID.
=
0.3
The measurement y is contaminated by noise v which is a zero mean white noise with covariance 0.25 * eye(2), where eye(j) is a MATLAB command which makes the j x j identity matrix.
T _ IPN - IUv(N) ) -fvXN - vU N
xN - v+lIN
[-0.45 0.3] 0.15 0.6 , 0.3 0.2
and D = O. Our control objective is to drive the tracking output Y to a desired constant reference signal r·. Here, r = [8, -5]1'. Our control strategy is shown in Figure 1.
we obtain
,
=
C = [-0.18 -0.135 0.18] ,
YN - I UJr _ IPN-IUI/(N))
=0.
0.8 0.1 0.3] 0.1 0.55 0.2 ,B [ -0.1 0.3 0.7
[
~::: ]
= [ _A6N
~] [~:] + [ BoN ] Uk + [~] r'. (41)
We apply the linear-quadratic regulator design for the extended model with weighting matrices Q = eye(nN) and R = eye(2). Let the resulting controller be described as
NZN-v+IIN
v- I
+ EN L A~I BN 'UN - i, i= 1
(42)
245
is Ilonsingular, where the notation as follows:
Unfortunately, conventional state estimation methods, e.g., Kalman filter or so-called obser'ver in control theory, are not available directly for the situation which we mentioned above because they need a fixed estimate of the order of a system and a fixed similar transformation commonly over all update steps.
If the output measurement of the system (1) is contaminated by the noise term VI.;, then the contaminated measurement fh is described as
N
L uv(j)'uv(j)T
(2)
j=1
U v (·)
is defined
(3)
(4)
Let the noise VI.; be a zero-mean Gaussian white noise with covariance
To overcome these difficulties, we introduce a new kind of "state" which maintains its meaning as a state over different update steps. This "state" must reflect some intrinsic properties of the system which are invariant with respect to choices of the order of models and the degree of freedom of similarity transformations. We identify such a "state" which possesses such properties.
V"
It is assumed that is independent of Ui for any i and the initial state Xo.
Remark 1. If u is a quasi-stationary signa\(Ljung, 1999), replace the expectation E by E where
This paper is organized as follows . In Section 2, some assumptions required in the paper are introduced. Section 3 gives a brief review of the basic 4SID method. In Section 4, we introduce a recursive algorithm of the basic 4SID which is based on the matrix inversion lemma (Oku and Kimura, 1999). In Section 5, we consider a state estimation problem of a MIMO model derived by the 4SID method . A vector made by premultiplying the state of a system to be identified by the extended observability matrix of it is employed as the "stale" mentioned above. We also present estimates of such a new" state" at the past, the present and the one-step ahead steps, respectively, in this section . Finally, an illustrative example is shown in Section 6.
_
1
Eu(k) :=
J~oo!vI
M
L Eu(k). "=1
3, BRIEF REVIEW OF BASIC 4SID
In this section, we give a brief review of basic 4SID for noise-free systems. For details, see Verhaegen and Dewilde(1992a). For a finite sequence of input-output (for short, "I/O") data {(Uj, Yj)}, j = 1,," ,N, uv(i) E JR"v, Yv (i) E JRl1tv are defined as follows:
Ui-V+I] uv(i)
:=
[
:
Yi - V+l] '
Yv(i)
1ti
:=
:
[
Yi (5)
2. PRELIMINARIES
Then, from (1), it follows that
In this section, some assumptions required in this paper are introduced.
Yv(i) =
= Axl.; + Bu",
y"
= Cx" + Du",
(6)
where
A discrete-time LTI system to be identified is described as follows: X"+l
r vXi-v+1 + Hvuv(i),
r v :=
(la) (lb)
u"
where XI.; E JRn, E JRr and Yk E JR7n. Let the matrix A be stable and the pair (A, C) be observable. The initial state of the system Xo is assumed to be O. A sequence of input data Ui is assumed to be a realization of a bounded stochastic process which is independent of the initial state Xo. We assume that there exists an integer No > 0 such that for a given integer v satisfying v 2 n + 1, a given input sequence {ud and 'iN 2 No
H"=
[C T (CA)T .. , (CAv-l)T
f',
[c5~'B CAJ-'B. J
(7)
(8)
The equation (6) yields the following matrix equation with respect to the input-state-output relations:
where
242
proposed in this paper is inferior to the one due to Verhaegen and Deprettere.
(11)
On the other hand, from a theoretical point of view, the algorithm can clarify the fundamental structure of the recursive 4SID algorithm, specially the relations among input, output and state vectors of a system to he identified. This advantage led us to recusive POMOESP (or PI-MOESP) algorithms (Oku and Kimura, 1999). It is also useful to analyze some properties of the 4SID schemes which will be stated in the following sections. For the details of MOESP(MIMO Output-Error State sPace model identification) schemes, see Verhaegen and Dewilde(1992a; 1992b) and Verhaegen(1993; 1994).
Note that UN and YN are rvx (N -v+l) and rnvx (N - v + 1) matrices which consist of N I/O data, respectively. r v E R mv x nand H v E R mv x ,·v are said to he an extended observability matrix and a
lower· triangular block Toeplitz matrix containing the Markov pammeters, respectively. Notice that the matrix U NUJr is invertible because (3) is invertible. We define the following two projection matrices l
lIu N := UJr (UNUJrr UN,
( 12)
IIU N := I - lI uN •
(13)
From (15), postmultiplication of YNIIU N by its transpose yields
Then, postmultiplication of the equation (9) by 11.1 UN yields the relation YNIIU N
= rvXNIIU
N
•
-= .-N:
(14)
= [EN
E~ ] [S 0] [(:~T ],
n.l
FiN 1[(fll:l N
(y 11.1) T - Y 11.1 yT
[~N u;~ I
0
(15)
(19)
it is known that for a sufficient large N there exists a nonsingular matrix T such that
Algorithm 1. (Oku and Kimura, 1999) Suppose =-N-l, PN- 1 and YN- I UJr_1 have already been obtained. When a pair of the N-th I/O data
( 16) From EN and E~ a state-space representation of the resulting model (AT, B T , CT, DT) can be calculated such that AT CT
= TAT - I, = CT- ' ,
BT DT
= TB,
= D.
(\8)
The relation (18) means that both EN and E~ can be derived from computation of eigenvalues of the real symmetric matrix =-N := YNIIUN YJ, which is called compressed I/O data matrix in this paper. A recursive algorithm of updating this compressed I/O data matrix is derived in the sequel. Let
Note that YNIIU N in the left hand side of (14) can be made of the input and output HankeJ matrices, (10) and (ll). If the singular value decomposition of the left side of (14) is obtained as YNIIU N
Y
(UN,YN) is obtained, we can construct Uv(N) and yv(N) from (5). Then, the N-th compressed I/O data matrix =-N can be updated recursively by the following procedure:
( 17)
=-N
A detailed procedure of the derivation of such matrices has been shown by Verhaegen and Dewilde(1992a).
= =-N - I + QNeNej;",
(20)
where
PN
= PN - I -
QNPN - 1U,/(N)Uv(N{ PN - I , (21)
The recursive scheme of identification is reduced to obtaining the recursive computation of Y N N •
IIU
(22)
4. A RECURSIVE ALGORITHM OF THE BASIC 4SID
(23)
A recursive 4SID algorithm based on the RQ factorization has already been proposed by Verhaegen and Deprettere( 1991). In this paper, we introduce a recursive algorithm which is based OIl the matrix inversion lemma (Oku and Kimura, 1999). It is well known that algorithms based on the matrix inversion lemma sometimes exhibit numerically poor performance with respect to the round-off error. Algorithms based on matrix factorization (e.g. QR or UD factorization methods) were devised as a remedy for this problem initially in state estimation (Bierman, 1977) . Therefore, from the numerical point of view, the algorithm
Proof. See Oku and Kimura(1999).
0
Remark 2. This algorithm works well also for the case where the contaminated measurement fh is observed because
.
hm N N-tCXJ
-
1 v
T
+ 1 VNUN = 0,
(25)
where VN is a matrix composed of the noise VI" in the same manner as (10). In this case, this algorithm is modified by replacing Yv(N), Yi , =-i
243
(i = N - I
or N) and eN by
tlj. v
(N) , I,i-t,
estimates of r", H" and (A, E, C, D) at the step N be denoted as
-::: ...... 1...-- ,i-rr1.. I1 U }i-'1' i
and
. (26)
EN, RN, (AN, EN, eN, DN), respectively. These matrices are obtained from the Algorithm 1, the eigenvalue decomposition of (20).
5. ON STATE ESTIMATION OF MIMO MODELS A principal advantage of 4SID schemes is that the order of a model, denoted by n, can be chosen to be the number of dominant singular values in (15). This means that, when we apply recursive 4SID for the system, we may take different choices of the order of models at different update steps.
Now, based on (6), we consider that an estimate ZN-v+!lN of the quasi-state ZN-v+l is given by the following equation:
If we can select the correct order, that is , the same order as the system, from (17) the system and an identified model are related by a similarity transformation matrix T. On considering recursive identification, a similarity transformation matrix TI by which an identified model are related to the system at an update step NI may not be identical to one T2 at another update step N 2 , i.e., TI #- T2 for NI #- N 2 · In this section we consider a state estimation problem of unknown multivariable systems utilizing the recursive 4SID. The problem is very important because the knowledge of the present state of the identified models can be a clue to design an adaptive control law for the system. Unfortunately, conventional state estimation methods, e.g. Kalman filter or observer, are not available directly for our case because they assume that the order of the system is fixed and similarity transformations are common over all update steps.
The third term of the right hand side of (29) plays a role of cancelling the noise effect. We shall explain the reason. From (1) the estimates zNIN and ZN+1IN of the quasi-states ZN and ZN+l, respectively, are calculated respectively as
ZNIN:= E N (AA N )v-1EtN ZN- ,,+lIN v- I + EN :L(A N )i - l EN'UN - i, A
From (28), the states of the estimated model (AN, BN , eN, D N ) are calculated as
(32) (33) (34) The rest of this section is devoted to the explanation of why the third term of the right hand side of (29) eliminates the noise effect. From (4) and (9), we obtain
(27)
Zi E ]Rmv.
We take Zi as a state at step i. The term Zi is named "the quasi-state at step i" in this paper. Note that the dimension of Zi is independent of the order of the system n and of the order of a model n. From the assumption that r v is of full column rank, the state of the system is denoted by
YN
rt
is the Moore-Penrose inverse of
r
(E~)'rYNUJ,
= y,,(i+v-I) -
(35)
= (E!vf HvUNU~ + (E~fvNU~. (36)
v'
Because of (25), we can neglect the second term of the right hand side of (36) for sufficient large N, i.e.,
If it were not for the measurement noise and if the lower triangular block Toeplitz matrix containing the Markov parameters Hv in (8) were known, we could easily obtain the quasi-state Zi from (6), i.e.,
Zi
= r"X N- v+1 + H"UN + VN.
It is assumed that there exists a nonsingular matrix T such that r = ENT. The premultiplication of (35) by (EM T and the postmultiplication by U~ yield
(28) where
(30)
i=1
To overcome this difficulty, we introduce a new kind of state which is still meaningful over different update steps. Let
Zi := r vXi,
A
1..'1'T ( EN) YNU N ~
Hv'Uv(i+v-I).
1..T
T
(EN) HvUNU N ·
From (19),
Since the matrix Hv is unknown but can be estimated by the recursive identification, the quasistate Zi can also be estimated recursively.
(E~f (Yv(N) - H,,'Uv(N))
~ (E~f (iiv(N) - frNU~PN'Uv(N)) .
When a pair of data (UN, Y N) is obtained at step N , Uv(N) and Yv(N) are composed of (5). Let
Note that, frolll (22),
244
• The one-step-ahead predictors of the model zN+lIN and xN+lIN are ohtained from the following equations:
6. A NUMERICAL EXAMPLE
(37)
In order to illustrate the usefulness of our idea presented in the previous section, we consider a tracking control problem.
(38)
Consider an unknown system described as (1) with
Therefore, we have (Elvf' (y1/(N) - HI/UI/(N))
= (EIvl
T
ONCN.
On the other hand, without noise we could derive
A
=
(EIv)T (YI/(Nl - HI/UI/(Nl)
.IT( T ) =(E N ) YI/(N) - YNUNPNUI/(N)
=(Elvf ON (YI/(N) =(Elvf 0NCN· Since f v
= EnT,
-0.18
(Elvf ONCN =(EIv)T ON (fvXN-I/+I (39)
Hence, in comparison with (38) to (39) we can regard the term E;(E;)1' ONCN as estimation error of ZN-v+1 due to the meSllrement noise. Now, we summarize the estimation scheme of the state.
Define ~k+1 - ~k := r -
EN, fIN, (AN, B N , CN , DN )
ZN-v+IIN =Yv(N) - HNul/(N)
- EIv(Elvf oNeN, Et '
NZN-v+lIN '
• The present states of the model zNIN and xNIN are obtained from the following equations: , E A' v-IE t '
=
N
N
ih (=: Ed·
Since the parameters of the system (40) A, B and C are assumed to be unknown , we use parameters of an estimated model AN, B Nand CN for estimating the state. The model is obtained by the recursive 4SID identification with taking v = 5. Let the order of the model be described as nN. Then, we obtain the following extended state-space representation:
• Past states of the model zN-v+lIN and XN-v+1IN are obtained from the following equations:
zNIN
(40)
-0.3
Suppose we have N pairs of I/O data (Ui ' Y;), i = 1"" , N. To solve this problem, one-step-ahead control input UN+I is decided according to the following procedure.
• A model of the system at step N results from recursive 4SID.
=
0.3
The measurement y is contaminated by noise v which is a zero mean white noise with covariance 0.25 * eye(2), where eye(j) is a MATLAB command which makes the j x j identity matrix.
T _ IPN - IUv(N) ) -fvXN - vU N
xN - v+lIN
[-0.45 0.3] 0.15 0.6 , 0.3 0.2
and D = O. Our control objective is to drive the tracking output Y to a desired constant reference signal r·. Here, r = [8, -5]1'. Our control strategy is shown in Figure 1.
we obtain
,
=
C = [-0.18 -0.135 0.18] ,
YN - I UJr _ IPN-IUI/(N))
=0.
0.8 0.1 0.3] 0.1 0.55 0.2 ,B [ -0.1 0.3 0.7
[
~::: ]
= [ _A6N
~] [~:] + [ BoN ] Uk + [~] r'. (41)
We apply the linear-quadratic regulator design for the extended model with weighting matrices Q = eye(nN) and R = eye(2). Let the resulting controller be described as
NZN-v+IIN
v- I
+ EN L A~I BN 'UN - i, i= 1
(42)
245