Mean-square Hâ filtering for stochastic systems ... - Semantic Scholar
Signal Processing 92 (2012) 801–806
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Mean-square HN filtering for stochastic systems: Application to a 2DOF helicopter Michael Basin a,n, Santiago Elvira-Ceja b, Edgar N. Sanchez b a b
Department of Physical and Mathematical Sciences, Autonomous University of Nuevo Leon, Mexico Center for Research and Graduate Studies (CINVESTAV), Campus Guadalajara, Mexico
a r t i c l e i n f o
abstract
Article history: Received 29 March 2011 Received in revised form 20 June 2011 Accepted 27 September 2011 Available online 5 October 2011
This paper designs the central finite-dimensional H1 filter for linear stochastic systems with integral-quadratically bounded deterministic disturbances, that is suboptimal for a given threshold g with respect to a modified Bolza–Meyer quadratic criterion including the attenuation control term with the opposite sign. The original H1 filtering problem for a linear stochastic system is reduced to the corresponding mean-square H2 filtering problem, using the technique proposed in Doyle (1989) [1]. In the example, the designed filter is applied to estimation of the pitch and yaw angles of a two degrees of freedom (2DOF) helicopter. & 2011 Elsevier B.V. All rights reserved.
Keywords: Filtering Wiener processes Linear stochastic system
1. Introduction Over the past two decades, considerable attention has been paid to the H1 estimation problem for deterministic and stochastic systems. The seminal papers on H1 control [1] and estimation [2–4] established a background for consistent treatment of controller/filtering problems in the H1 framework. The H1 filter design implies that the resulting closed-loop filtering system is robustly stable and achieves a prescribed level of attenuation from the disturbance input to the output estimation error in L2 =l2 -norm. A large number of results on this subject have been reported for systems in the general situation (see, for example [5–43] and references therein). Sufficient conditions for existence of an H1 filter, where the filter gain matrices satisfy Riccati equations, were obtained for linear deterministic systems in [4] and linear systems with state delay in [44] or with measurement delay in [45].
n
Corresponding author. E-mail addresses: [email protected], [email protected], [email protected] (M. Basin), [email protected] (S. Elvira-Ceja), [email protected] (E.N. Sanchez). 0165-1684/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2011.09.026
It should be however noted that the criteria of existence and suboptimality of solution for the central H1 filtering problems based on the reduction of the original H1 problem to the induced H2 one, similar to those obtained in [1,4] for linear systems, remain yet undeveloped for linear stochastic systems with integral-quadratically bounded deterministic disturbances. To close the gap, this paper presents the central (see [1] for definition) finite-dimensional mean-square H1 filter for linear stochastic systems, that is suboptimal for a given threshold g with respect to a modified Bolza–Meyer quadratic criterion including the attenuation control term with the opposite sign. Designing the central suboptimal meansquare H1 filter for linear stochastic systems presents a significant advantage in filtering theory and practice, since it enables one to design filtering algorithms, reducing the noise from input to output according to a prescribed attenuation level, for linear systems with both stochastic and deterministic noises. In contrast to results previously obtained for linear systems [4,44,45], this paper reduces the original H1 filtering problem to the corresponding mean-square H2 filtering problem, using the technique proposed in [1]. The obtained filter also has conventional advantages resulting from the employed
802
M. Basin et al. / Signal Processing 92 (2012) 801–806
methodology: (1) it enables one to address filtering problems for linear stochastic time-varying systems, where the linear matrix inequality technique is hardly applicable and the Hamilton–Jacobi–Bellman equationbased methods fail to provide a closed-form solution, (2) the obtained mean-square H1 filter is suboptimal, that is, optimal for any fixed g with respect to the H1 noise attenuation criterion, and (3) the obtained mean-square H1 filter is finite-dimensional. It should be commented that the proposed design of the central suboptimal mean-square H1 filters for linear stochastic systems with integral-quadratically bounded disturbances naturally carries over from the design of the optimal mean-square H2 filters for linear stochastic systems with unbounded disturbances (white noises). The entire design approach creates a complete filtering algorithm for handling the linear stochastic time-varying systems with unbounded or integral-quadratically bounded disturbances optimally for all thresholds g uniformly or for any fixed g separately. The corresponding algorithm for linear deterministic systems was developed in [4]. Note that similar H1 control and filtering problems for stochastic systems have been recently considered in [46,47], which resulted in some aerospace and mechanical applications [48,49]. The designed filter is applied to estimation of the pitch and yaw angles of a two degrees of freedom (2DOF) helicopter. The simulation results show a reliable performance of the filter, in particular, the obtained attenuation level is five times less than a given threshold. The paper is organized as follows. Section 2 presents the mean-square H1 filter problem statement for linear stochastic time-varying systems. The central suboptimal mean-square H1 filter is designed in Section 3. In Section 4, the designed filter is applied to estimation of the pitch and yaw angles of a two degrees of freedom (2DOF) helicopter. Conclusions are given in Section 5.
T
W 2 ðtÞ 2 Rp2 are independent. It is assumed that hðtÞh ðtÞ is a positive definite matrix. For the system given by (1)–(4), the following assumptions are made over the time interval ½t 0 ,t 1 :
ðAðtÞ,bðtÞÞ is stabilizable and ðC 1 ðtÞ,AðtÞÞ is detectable ðC1 Þ ðAðtÞ,GðtÞÞ is stabilizable and ðC 2 ðtÞ,AðtÞÞ is detectable ðC2 Þ ðAðtÞ,BðtÞÞ is stabilizable and ðLðtÞ,AðtÞÞ is detectable, and ðC3 Þ
HðtÞGT ðtÞ ¼ 0 and HðtÞHT ðtÞ is a positive definite matrix ðC4 Þ As usual, the first two assumptions ensure that the estimation error, provided by the designed filter, converge to zero [50]. The noise orthogonality condition HðtÞGT ðtÞ ¼ 0 is technical and represents the independence between the state and measurement deterministic disturbances. Extensive comments on the assumption ðC4 Þ can be found in [1]. The filtering problem to be addressed is as follows: develop a central suboptimal mean-square H1 filter for the linear stochastic system (S 1 ) as a linear filter based on the observations fy1 ðsÞ,t 0 rs r tg and fy2 ðsÞ,t 0 rs r tg such that the following three requirements are satisfied. (1) The resulting dynamics of the estimation error EðxðtÞÞmðtÞ, where x(t) is the state of (S 1 ) and m(t) is the mean-square H1 estimate produced by the designed filter, is asymptotically stable in the absence of disturbances, oðtÞ 0. Here, EðxðtÞÞ denotes the expectation of stochastic process x(t). (2) The variance of the mean-square H1 estimate m(t) of the system state x(t), based on the observation process YðtÞ ¼ fy1 ðsÞ,0 r s r tg, is equal to the minimum estimation error variance [51] E½ðxðtÞEðxðtÞ9F Yt ÞÞðxðtÞEðxðtÞ9F Yt ÞÞT 9F Yt
ð5Þ
E½xðtÞ9F Yt
2. Mean-square H 1 filtering problem statement Let ðO,F,PÞ be a complete probability space with an increasing right-continuous family of s-algebras F t ,t Z t 0 , and let ðW 1 ðtÞ,F t ,t Zt 0 Þ and ðW 2 ðtÞ,F t ,t Z t 0 Þ be independent Wiener processes. Consider the following linear stochastic time-varying system S 1 : dxðtÞ ¼ ðAðtÞxðtÞ þBðtÞuðtÞ þ GðtÞoðtÞÞ dt þbðtÞ dW 1 ðtÞ, xðt 0 Þ ¼ x0 ,
ð1Þ
dy1 ðtÞ ¼ C 1 ðtÞxðtÞ dt þhðtÞ dW 2 ðtÞ,
ð2Þ
y2 ðtÞ ¼ C 2 ðtÞxðtÞ þHðtÞoðtÞ,
ð3Þ
zðtÞ ¼ LðtÞxðtÞ,
means the at every time moment t. Here, conditional expectation of a stochastic matrix process xðtÞ ¼ ðxðtÞEðxðtÞ9F Yt ÞÞðxðtÞEðxðtÞ9F Yt ÞÞT with respect to the s-algebra FYt generated by the observation process Y(t) in the interval ½t 0 ,t. (3) Given a noise attenuation level g, the H1 noise attenuation condition (6) is ensured. More specifically, for any nonzero disturbance input oðtÞ 2 Ls2 ½0,1Þ, the inequality ð6Þ JzðtÞLðtÞmðtÞJ22 o g2 fJoðtÞJ22 þ EðxT ðt 0 ÞÞREðxðt 0 ÞÞg Rt T holds, where Jf ðtÞJ22 :¼ t01 f ðtÞf ðtÞ dt, t 1 is the selected filter horizon, R is a symmetric positive definite matrix, and g is a given real positive scalar.
ð4Þ n
l
where xðtÞ 2 R is the unmeasured state, uðtÞ 2 R is a known input signal, y1 ðtÞ 2 Rm1 and y2 ðtÞ 2 Rm2 are the measured observations, zðtÞ 2 Rq is the output to be estimated, oðtÞ 2 Ls2 ½0,1Þ is the deterministic disturbance input, A(t), B(t), G(t), b(t), C 1 ðtÞ, h(t), C 2 ðtÞ, H(t), and L(t) are known deterministic continuous time-varying functions of appropriate dimension. The initial condition x0 2 Rn is a Gaussian random variable such that x0, W 1 ðtÞ 2 Rp1 , and
3. Central suboptimal mean-square H 1 filter design The proposed design of the suboptimal mean-square H1 filter for linear stochastic systems is based on the general result (see Theorem 3 in [1]) reducing the H1 controller problem to the corresponding optimal H2 controller problem. In this paper, only the filtering part of this result, valid for the entire controller problem, is used.
M. Basin et al. / Signal Processing 92 (2012) 801–806
803
Then, the optimal mean-square Kalman–Bucy filter for linear stochastic systems [52] and the H1 filter for linear systems (Theorem 4 in [4]) are employed to obtain the desired result, which is given by the following theorem.
Since EðxðtÞÞ ¼ EðxðtÞÞ and the variances of the estimated errors produced by the estimates xðtÞ and m(t) are equal, the conditions 1–3 of Section 2 hold. The theorem is proved. &
Theorem 1. The central suboptimal mean-square H1 filter for the linear stochastic system (1)–(4), ensuring the minimum of the mean-square criterion (5) and the H1 noise attenuation condition (6), is given by the equation for the mean-square H1 estimate m(t)
Remark 1. The convergence of the designed mean-square H1 state estimate m(t) to the real state value x(t) is assured by the conditions ðC1 Þ and ðC2 Þ in view of the results of Theorem 7.4 and Section 7.7 in [50]. Note that boundedness of the noise-output H1 norm for the system (S 1 ), controlled by filter (7)–(9), i.e., admissibility of the mean-square H1 filter (7)–(9), is determined by the conditions I–III of Theorem 3 in [1].
dmðtÞ ¼ ðAðtÞmðtÞ þ BðtÞuðtÞÞ dt T
þPðtÞC T1 ðtÞðhðtÞh ðtÞÞ1 ½dy1 ðtÞC 1 ðtÞmðtÞ dt þSðtÞC T2 ðtÞðHðtÞHT ðtÞÞ1 ½y2 ðtÞC 2 ðtÞmðtÞ dt,
ð7Þ
with initial condition mðt 0 Þ ¼ Eðxðt 0 Þ9F Yt0 Þ, where the matrix function P(t) (minimum estimation error variance) is the solution of the differential Riccati equation _ ¼ AðtÞPðtÞ þ PðtÞAT ðtÞ PðtÞ T
T
þ bðtÞb ðtÞPðtÞC T1 ðtÞðhðtÞh ðtÞÞ1 C 1 ðtÞPðtÞ,
ð8Þ
with initial condition Pðt 0 Þ ¼ E½ðxðt 0 Þmðt 0 ÞÞðxðt 0 Þmðt 0 ÞÞT 9 F Yt0 , and the symmetric matrix function S (t) is the solution of the differential Riccati equation _ ¼ AðtÞSðtÞ þ SðtÞAT ðtÞ þ GðtÞGT ðtÞ SðtÞ SðtÞ½C T2 ðtÞðHðtÞHT ðtÞÞ1 C 2 ðtÞg2 LT ðtÞLðtÞSðtÞ, with initial condition Sðt 0 Þ ¼ R
1
ð9Þ
.
Proof. First, let us design the estimate xðtÞ satisfying the minimum variance condition (5) of Section 2. As known [51], this mean-square estimate is given by the conditional expectation xðtÞ ¼ EðxðtÞ9F Yt Þ of the system state x(t) with respect to the s-algebra FYt, generated by the observations (2) in the interval ½t 0 ,t, and is produced by the Kalman–Bucy filter [52] applied to the linear stochastic system (1) over the linear observations (2) in the presence of Gaussian disturbances (Wiener processes) W 1 ðtÞ and W 2 ðtÞ. The corresponding filtering equations for the estimate xðtÞ and the estimation error variance PðtÞ take the form dxðtÞ ¼ ðAðtÞxðtÞ þ BðtÞuðtÞ þGðtÞoðtÞÞ dt T
þPðtÞC T1 ðtÞðhðtÞh ðtÞÞ1 ½dy1 ðtÞC 1 ðtÞxðtÞ dt, with the initial condition
xðt 0 Þ ¼ Eðxðt 0 Þ9F Yt0 Þ,
ð10Þ
and
T P_ ðtÞ ¼ AðtÞPðtÞ þPðtÞAT ðtÞ þ bðtÞb ðtÞ T
PðtÞC T1 ðtÞðhðtÞh ðtÞÞ1 C 1 ðtÞPðtÞ,
Remark 2. According to the comments in subsection V.G in [1], the obtained central mean-square H1 filter (7)–(9) presents a natural choice for H1 filter design among all admissible H1 filters satisfying the inequality (6) for a given threshold g, since it does not involve any additional actuator loop (i.e., any additional external state variable) in constructing the filter gain matrix. Moreover, the obtained central mean-square H1 filter has the suboptimality property, i.e., it minimizes the criterion J ¼ JzðtÞLðtÞmðtÞJ22 g2 ðJoðtÞJ22 þ EðxT0 ÞREðx0 ÞÞ: Remark 3. Following the discussion in subsection V.G in [1], note that the complementarity condition always holds for the obtained filter (7)–(9), since the positive definiteness of the initial condition matrix R implies the positive definiteness of the filter gain matrix S(t) as the solution of (9). 4. Example This section presents the design of the central suboptimal mean-square H1 filter to estimate the pitch and yaw angles for a 2DOF helicopter, ensuring the minimum of the mean-square criterion (5) and the H1 noise attenuation condition (6) holds for g ¼ 1:1. Let the 2DOF helicopter system with the state space representation be _ ¼ AxðtÞ þ BðtÞuðtÞ þ GoðtÞ þ bc1 ðtÞ, xðtÞ
xðt 0 Þ ¼ x0 ,
ð12Þ
yðtÞ ¼ C 1 xðtÞ þ hc2 ðtÞ,
ð13Þ
y2 ðtÞ ¼ C 2 xðtÞ þHoðtÞ,
ð14Þ
zðtÞ ¼ LxðtÞ,
ð15Þ
ð11Þ
with the initial condition Pðt 0 Þ ¼ Eððxðt 0 Þxðt 0 ÞÞðxðt 0 Þxðt 0 ÞÞT 9F Yt0 Þ: Note that the latter equation coincides with (8). Now, applying the central suboptimal H1 filter for linear systems [4] to the estimate xðtÞ governed by Eqs. (10) and (11) yields the central suboptimal mean-square H1 estimate Eq. (7), where the matrix function PðtÞ satisfies Eq. (8), and the matrix S(t) in (7) satisfies Eq. (9). Note that filter (7)–(9) yields, in view of Theorems 3 and 4 in [1], the asymptotic stability of the mean value EðxðtÞÞ of the estimate (10) in the absence of disturbances and the prescribed attenuation level g for this variable: JEðxðtÞÞJ22 o g2 JoðtÞJ22 þ EðxT ðt 0 ÞÞREðxðt 0 ÞÞ.
_ T , in which y and c where the state vector is: x ¼ ½y, c, y_ , c _ are pitch are pitch and yaw angles respectively, y_ and c and yaw angular velocities, respectively. The matrices are 2 3 0 0 1 0 60 0 7 0 1 6 7 A¼6 7, 4 0 0 9:2751 5 0 0 2
0 0
6 0 6 B¼6 4 2:3667 0:2410
0
3:4955 0
3
7 0 7 7, 0:0790 5 0:7913
2
0
6 0 6 b¼6 4 0:9024 0:0919
0
3
7 0 7 7, 0:0876 5 0:8772
804
M. Basin et al. / Signal Processing 92 (2012) 801–806
0 6 0 6 G¼6 4 0:9024 0:0919
C1 ¼ C2 ¼ L ¼
H¼
0 0
0 0
0
0
0
0
0:0876
0
07 7 7, 05
0:8772
0
0
1
0
0
0
0
1
0
0
2
3
0
,
h¼
1
0
0
1
Here, u(t) is the motor voltage input, oðtÞ is an L22 disturbance input, c1 ðtÞ and c2 ðtÞ are Gaussian white noises, which are the weak mean square derivatives of standard Wiener processes W 1 ðtÞ and W 2 ðtÞ (see [51]), respectively. The Wiener processes are considered independent of each other and of a Gaussian random variable x0 serving as the initial condition in (12). Eqs. (12) and (13) present the conventional form for Eqs. (1) and (2), which is actually used in practice [53]. It can be easily verified that the noise orthogonality condition holds for the system (12)–(15). The filtering problem to be addressed is the same as described at Section 2. The filtering horizon is set from t 0 ¼ 0 to t 1 ¼ 80 s. The central suboptimal mean-square H1 filter takes the following form for the system (12)–(15): 2 3 0 0 1 0 60 0 7 0 1 6 7 _ mðtÞ ¼6 7mðtÞ 4 0 0 9:2751 5 0 0
0
0
2
3
1 60 6 þ PðtÞ6 40 0 2
1 60 6 þ SðtÞ6 40 0
3:4955
0 17 1 7 7 yðtÞ 05 0
0
0
0
1
0
0
mðtÞ
0 3 1 17 7 7 y2 ðtÞ 0 05 0
0
0
0
1
0
0
mðtÞ ,
ð16Þ
0
with mð0Þ ¼ m0 ¼ Eðx0 9F Y0 Þ, where S(t) and P(t) are the solutions to the differential Riccati equations 2 3 0 0 1 0 60 0 7 0 1 7 _ ¼6 SðtÞ 6 7SðtÞ 4 0 0 9:2751 5 0 0
0
0
2
3:4955
0
0
60 6 þ SðtÞ6 41
0
0
0 1
9:2751 0
0 2 6 6 SðtÞ6 4
0
3
0
7 0 7 7 5 0 3:4955 3
0:1736
0
0
0
0
0:1736
0
0
0
0
07 7 7SðtÞ 05
0
0
0
0
0
1
0
0
0
0:8220
7 7 7, 0:1597 5
0
0
0:1597
0:7779
2
0 : 1
1 0
3
0 60 6 þ6 40
0 1
Sð0Þ ¼ S0 ¼ R1
ð17Þ
and
,
0
0
1
0
3
60 _ ¼6 PðtÞ 6 40
0 0
0 9:2751
1 0
7 7 7PðtÞ 5
0
3:4955
0 0 2 0 0 60 0 6 þPðtÞ6 41 0
0
0
0
0
9:2751
0
3 7 7 7 5
0
1
1 60 6 PðtÞ6 40
0
0
1
0
0
0
0 3:4955 3 2 0 0 1 0 7 60 0 0 07 6 7PðtÞ þ 6 4 0 0 0:8220 05
0
0
0
0
2
0
0
0:1597
0
3
7 7 7, 0:1597 5 1
0:7779 ð18Þ
Pð0Þ ¼ E½ðx0 m0 Þðx0 m0 ÞT 9F Y0 ¼ P0 , respectively. Numerical simulations results are obtained by solving the system (12)–(15) and the filtering equations (16)–(18), with the following initial values: x0 ¼ ½0:7069,0; 0,0T , P 0 ¼ diagð10; 10,15; 5Þ, R ¼ ½0:8,0:5,0:2,0:1; 0:5,0:8,0:7,0:2; 0:2,0:7,0:9,0:4; 0:1,0:2,0:4,0:9, and m0 ¼ ½0:1745,0:5236, 0:15,0:4T . The motor voltage uðtÞ ¼ ½12:495,3:835T , the L2 disturbance oðtÞ ¼ ½o1 ðtÞ, o2 ðtÞ, o3 ðtÞ, o4 ðtÞT is realized as o1 ðtÞ ¼ 1=ð1 þtÞ, o2 ðtÞ ¼ 0:1ð1e0:3t Þ, o3 ðtÞ ¼ 0:1745=ð1 þtÞ, and o4 ðtÞ ¼ 0:0873ð1e0:3t Þ. The attenuation level value is set to g ¼ 1:1. The disturbances c1 ðtÞ and c2 ðtÞ in (12) and (13) are realized by using the built-in MatLab white noise function. As a result of the numerical simulation, the following graphs are presented: graph of the noise-output H1 norm (Fig. 1); graphs of the pitch angle and the corresponding estimation error (Fig. 2); graph of the yaw angle and the corresponding estimation error (Fig. 3). Note that the maximum value of the noise-output H1 norm T ¼ JzðtÞLðtÞmðtÞJ=ðJoðtÞJ22 þ Eðx0 ÞREðx0 ÞÞ1=2 is equal to 0.208 in the considered simulation interval, which is 0.25 0.2 H−infinity norm
2
0.15 0.1 0.05 0 0
10
20
30
40 50 Time [seg]
Fig. 1. Noise-output H1 norm T.
60
70
80
Estimation error
Pitch angle [deg]
M. Basin et al. / Signal Processing 92 (2012) 801–806
filters for nonlinear polynomial stochastic systems or linear stochastic systems with time delays.
50 true estimated
0 Acknowledgments
−50 0
10
20
30
40
50
60
70
80
0
10
20
30 40 50 Time [seg]
60
70
80
0.5 0 −0.5
Yaw angle [deg]
The authors thank the Mexican National Science and Technology Council (CONACyT) for financial support under Grants 129081 and 55584 and Sabbatical Fellowship for the first author. References
−1
Fig. 2. Above. Pitch angle. Below. Estimation error.
Estimation error
805
100 true estimated
50 0 −50 0
10
20
30
40
50
60
70
80
0
10
20
30 40 50 Time [seg]
60
70
80
0.6 0.4 0.2 0 −0.2
Fig. 3. Above. Yaw angle. Below. Estimation error.
five times less than the given H1 attenuation level, g ¼ 1:1. In addition, the estimation errors robustly converge to zero. The signal-to-noise ratio decreases virtually to zero for both, pitch and yaw, angles by t¼ 80 s, despite a sharp change in the pitch velocity at t ¼55 s.
5. Conclusions This paper designs the central finite-dimensional H1 filter for linear stochastic systems with integral-quadratically bounded deterministic disturbances, that is suboptimal for a given threshold g with respect to a modified Bolza– Meyer quadratic criterion including the attenuation control term with opposite sign. The designed filter is applied to estimation of the pitch and yaw angles of a two degrees of freedom (2DOF) helicopter. The simulation results show a reliable performance of the filter, in particular, the obtained attenuation level is five times less than a given threshold. This significant improvement is obtained due to the more reasonable selection of the filter gain matrix in the designed filter. Although this conclusion follows from the developed theory, the numerical simulation serves as a convincing illustration. The presented approach would be applied in the future to obtain the central suboptimal mean-square H1
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[23] Z.S. Duan, J.X. Zhang, C.S. Zhang, E. Mosca, A simple design method of reduced-order filters and its applications to multirate filter bank design, Signal Processing 86 (2006) 1061–1075. [24] Z.S. Duan, J.X. Zhang, C.S. Zhang, E. Mosca, Robust H2 and H1 filtering for uncertain linear systems, Automatica 42 (2006) 1919–1926. [25] Z.D. Wang, F. Yang, D.W.C. Ho, X. Liu, Robust H1 filtering for stochastic time-delay systems with missing measurements, IEEE Transactions on Signal Processing 54 (2006) 2579–2587. [26] M.S. Mahmoud, Resilient L2 =L1 filtering of polytopic systems with state delays, IET Control Theory and Applications 1 (2007) 141–154. [27] L. Wu, Z.D. Wang, H. Gao, C. Wang, Filtering for uncertain 2-D discrete systems with state delays, Signal Processing 87 (2007) 2213–2230. [28] H. Gao, T.W. Chen, H1 estimation for uncertain systems with limited communication capacity, IEEE Transactions on Automatic Control 52 (2007) 2070–2084. [29] Z. Wang, Y. Liu, X. Liu, H1 filtering for uncertain stochastic timedelay systems with sector-bounded nonlinearities, Automatica 44 (2008) 1268–1277. [30] L. Wu, D.W.C. Ho, Fuzzy filter design for Ito stochastic systems with application to sensor detection, IEEE Transactions on Fuzzy Systems 17 (2009) 233–242. [31] Z. Wang, D.W.C. Ho, Y. Liu, X. Liu, Robust H1 control for a class of nonlinear discrete time-delay stochastic systems with missing measurements, Automatica 45 (2009) 684–691. [32] G. Wei, Z. Wang, H. Shu, Robust filtering with stochastic nonlinearities and multiple missing measurements, Automatica 45 (2009) 836–841. [33] H. Gao, X. Liu, J. Lam, Stability analysis and stabilization for discrete fuzzy systems with time-varying delay, IEEE Transactions on Systems, Man and Cybernetics, Part B 39 (2009) 306–317. [34] Y. Liu, W. Wang, Fuzzy H1 filtering for nonlinear stochastic systems with missing measurements, ICIC Express Letters 3 (2009) 739–744. [35] B. Song, S. Xu, Y. Zou, Non-fragile H1 filtering for uncertain stochastic time-delay systems, International Journal of Innovative Computing Information and Control 5 (2009) 2257–2265. [36] B. Shen, Z.D. Wang, H. Shu, G. Wei, H1 filtering for nonlinear discrete-time stochastic systems with randomly varying sensor delays, Automatica 45 (2009) 1032–1037. [37] L. Wu, Z.D. Wang, Fuzzy filtering of nonlinear fuzzy stochastic systems with time-varying delay, Signal Processing 89 (2009) 1739–1753. [38] M.V. Basin, P. Shi, D. Calderon-Alvarez, Approximate finite-dimensional filtering for polynomial states over polynomial observations, International Journal of Control 83 (2010) 724–730.
[39] M.V. Basin, P. Shi, P. Soto, Central suboptimal H1 filtering for nonlinear polynomial systems with multiplicative noise, Journal of Franklin Institute 347 (2010) 1740–1754. [40] Z. Deng, P. Shi, H. Yang, Y. Xia, Robust H1 filtering for nonlinear systems with interval time-varying delays, International Journal of Innovative Computing Information and Control 6 (2010) 5527–5538. [41] R. Yang, H. Gao, P. Shi, Delay-dependent L2–L1 filter design for stochastic time-delay systems, IET Control Theory and Applications 5 (2011) 1–8. [42] M. Basin, P. Shi, D. Calderon-Alvarez, Joint state filtering and parameter estimation for linear time-delay systems, Signal Processing 91 (2011) 782–792. [43] H. Wu, J. Wang, P. Shi, A delay decomposition approach to L2–L1 filter design for stochastic systems with time-varying delay, Automatica 47 (2011) 1482–1488. [44] A. Fattou, O. Sename, J. Dion, H1 observer design for time-delay systems, in: Proceedings of the 37th IEEE Conference on Decision Control, Tampa, FL, 1998, pp. 4545–4546. [45] A. Pila, U. Shaked, C.E. de Souza, H1 filtering for continuous-time linear systems with delay, IEEE Transactions on Automatic Control 44 (1999) 1412–1417. [46] H. Dong, Z.D. Wang, H. Gao, Robust H1 filtering for a class of nonlinear net-worked systems with multiple stochastic communication delays and packet dropouts, IEEE Transactions on Signal Processing 58 (2010) 1957–1966. [47] H. Dong, Z.D. Wang, D.W.C. Ho, H. Gao, Variance-constrained H1 filtering for a class of nonlinear time-varying stochastic systems with multiple missing measurements: the finite-horizon case, IEEE Transactions on Signal Processing 58 (2010) 2534–2543. [48] H. Gao, X. Yang, P. Shi, Multi-objective robust H1 control of spacecraft rendezvous, IEEE Transactions on Control Systems Technology 17 (2009) 794–802. [49] X. Yang, H. Gao, P. Shi, G. Duan, Robust H1 control for a class of uncertain mechanical systems, International Journal of Control 73 (2010) 1303–1324. [50] A.H. Jazwinski, Stochastic Processes and Filtering Theory, Academic Press, New York, 1970. [51] V.S. Pugachev, I.N. Sinitsyn, Stochastic Systems: Theory and Applications, World Scientific, Singapore, 2001. [52] R.E. Kalman, R.S. Bucy, New results in linear filtering and prediction theory, ASME Transactions, Part D (Journal of Basic Engineering) 83 (1961) 95–108. ˚ ¨ [53] K.J. Astr om, Introduction to Stochastic Control Theory, Academic Press, New York, 1970.
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Mean-square HN filtering for stochastic systems: Application to a 2DOF helicopter Michael Basin a,n, Santiago Elvira-Ceja b, Edgar N. Sanchez b a b
Department of Physical and Mathematical Sciences, Autonomous University of Nuevo Leon, Mexico Center for Research and Graduate Studies (CINVESTAV), Campus Guadalajara, Mexico
a r t i c l e i n f o
abstract
Article history: Received 29 March 2011 Received in revised form 20 June 2011 Accepted 27 September 2011 Available online 5 October 2011
This paper designs the central finite-dimensional H1 filter for linear stochastic systems with integral-quadratically bounded deterministic disturbances, that is suboptimal for a given threshold g with respect to a modified Bolza–Meyer quadratic criterion including the attenuation control term with the opposite sign. The original H1 filtering problem for a linear stochastic system is reduced to the corresponding mean-square H2 filtering problem, using the technique proposed in Doyle (1989) [1]. In the example, the designed filter is applied to estimation of the pitch and yaw angles of a two degrees of freedom (2DOF) helicopter. & 2011 Elsevier B.V. All rights reserved.
Keywords: Filtering Wiener processes Linear stochastic system
1. Introduction Over the past two decades, considerable attention has been paid to the H1 estimation problem for deterministic and stochastic systems. The seminal papers on H1 control [1] and estimation [2–4] established a background for consistent treatment of controller/filtering problems in the H1 framework. The H1 filter design implies that the resulting closed-loop filtering system is robustly stable and achieves a prescribed level of attenuation from the disturbance input to the output estimation error in L2 =l2 -norm. A large number of results on this subject have been reported for systems in the general situation (see, for example [5–43] and references therein). Sufficient conditions for existence of an H1 filter, where the filter gain matrices satisfy Riccati equations, were obtained for linear deterministic systems in [4] and linear systems with state delay in [44] or with measurement delay in [45].
n
Corresponding author. E-mail addresses: [email protected], [email protected], [email protected] (M. Basin), [email protected] (S. Elvira-Ceja), [email protected] (E.N. Sanchez). 0165-1684/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2011.09.026
It should be however noted that the criteria of existence and suboptimality of solution for the central H1 filtering problems based on the reduction of the original H1 problem to the induced H2 one, similar to those obtained in [1,4] for linear systems, remain yet undeveloped for linear stochastic systems with integral-quadratically bounded deterministic disturbances. To close the gap, this paper presents the central (see [1] for definition) finite-dimensional mean-square H1 filter for linear stochastic systems, that is suboptimal for a given threshold g with respect to a modified Bolza–Meyer quadratic criterion including the attenuation control term with the opposite sign. Designing the central suboptimal meansquare H1 filter for linear stochastic systems presents a significant advantage in filtering theory and practice, since it enables one to design filtering algorithms, reducing the noise from input to output according to a prescribed attenuation level, for linear systems with both stochastic and deterministic noises. In contrast to results previously obtained for linear systems [4,44,45], this paper reduces the original H1 filtering problem to the corresponding mean-square H2 filtering problem, using the technique proposed in [1]. The obtained filter also has conventional advantages resulting from the employed
802
M. Basin et al. / Signal Processing 92 (2012) 801–806
methodology: (1) it enables one to address filtering problems for linear stochastic time-varying systems, where the linear matrix inequality technique is hardly applicable and the Hamilton–Jacobi–Bellman equationbased methods fail to provide a closed-form solution, (2) the obtained mean-square H1 filter is suboptimal, that is, optimal for any fixed g with respect to the H1 noise attenuation criterion, and (3) the obtained mean-square H1 filter is finite-dimensional. It should be commented that the proposed design of the central suboptimal mean-square H1 filters for linear stochastic systems with integral-quadratically bounded disturbances naturally carries over from the design of the optimal mean-square H2 filters for linear stochastic systems with unbounded disturbances (white noises). The entire design approach creates a complete filtering algorithm for handling the linear stochastic time-varying systems with unbounded or integral-quadratically bounded disturbances optimally for all thresholds g uniformly or for any fixed g separately. The corresponding algorithm for linear deterministic systems was developed in [4]. Note that similar H1 control and filtering problems for stochastic systems have been recently considered in [46,47], which resulted in some aerospace and mechanical applications [48,49]. The designed filter is applied to estimation of the pitch and yaw angles of a two degrees of freedom (2DOF) helicopter. The simulation results show a reliable performance of the filter, in particular, the obtained attenuation level is five times less than a given threshold. The paper is organized as follows. Section 2 presents the mean-square H1 filter problem statement for linear stochastic time-varying systems. The central suboptimal mean-square H1 filter is designed in Section 3. In Section 4, the designed filter is applied to estimation of the pitch and yaw angles of a two degrees of freedom (2DOF) helicopter. Conclusions are given in Section 5.
T
W 2 ðtÞ 2 Rp2 are independent. It is assumed that hðtÞh ðtÞ is a positive definite matrix. For the system given by (1)–(4), the following assumptions are made over the time interval ½t 0 ,t 1 :
ðAðtÞ,bðtÞÞ is stabilizable and ðC 1 ðtÞ,AðtÞÞ is detectable ðC1 Þ ðAðtÞ,GðtÞÞ is stabilizable and ðC 2 ðtÞ,AðtÞÞ is detectable ðC2 Þ ðAðtÞ,BðtÞÞ is stabilizable and ðLðtÞ,AðtÞÞ is detectable, and ðC3 Þ
HðtÞGT ðtÞ ¼ 0 and HðtÞHT ðtÞ is a positive definite matrix ðC4 Þ As usual, the first two assumptions ensure that the estimation error, provided by the designed filter, converge to zero [50]. The noise orthogonality condition HðtÞGT ðtÞ ¼ 0 is technical and represents the independence between the state and measurement deterministic disturbances. Extensive comments on the assumption ðC4 Þ can be found in [1]. The filtering problem to be addressed is as follows: develop a central suboptimal mean-square H1 filter for the linear stochastic system (S 1 ) as a linear filter based on the observations fy1 ðsÞ,t 0 rs r tg and fy2 ðsÞ,t 0 rs r tg such that the following three requirements are satisfied. (1) The resulting dynamics of the estimation error EðxðtÞÞmðtÞ, where x(t) is the state of (S 1 ) and m(t) is the mean-square H1 estimate produced by the designed filter, is asymptotically stable in the absence of disturbances, oðtÞ 0. Here, EðxðtÞÞ denotes the expectation of stochastic process x(t). (2) The variance of the mean-square H1 estimate m(t) of the system state x(t), based on the observation process YðtÞ ¼ fy1 ðsÞ,0 r s r tg, is equal to the minimum estimation error variance [51] E½ðxðtÞEðxðtÞ9F Yt ÞÞðxðtÞEðxðtÞ9F Yt ÞÞT 9F Yt
ð5Þ
E½xðtÞ9F Yt
2. Mean-square H 1 filtering problem statement Let ðO,F,PÞ be a complete probability space with an increasing right-continuous family of s-algebras F t ,t Z t 0 , and let ðW 1 ðtÞ,F t ,t Zt 0 Þ and ðW 2 ðtÞ,F t ,t Z t 0 Þ be independent Wiener processes. Consider the following linear stochastic time-varying system S 1 : dxðtÞ ¼ ðAðtÞxðtÞ þBðtÞuðtÞ þ GðtÞoðtÞÞ dt þbðtÞ dW 1 ðtÞ, xðt 0 Þ ¼ x0 ,
ð1Þ
dy1 ðtÞ ¼ C 1 ðtÞxðtÞ dt þhðtÞ dW 2 ðtÞ,
ð2Þ
y2 ðtÞ ¼ C 2 ðtÞxðtÞ þHðtÞoðtÞ,
ð3Þ
zðtÞ ¼ LðtÞxðtÞ,
means the at every time moment t. Here, conditional expectation of a stochastic matrix process xðtÞ ¼ ðxðtÞEðxðtÞ9F Yt ÞÞðxðtÞEðxðtÞ9F Yt ÞÞT with respect to the s-algebra FYt generated by the observation process Y(t) in the interval ½t 0 ,t. (3) Given a noise attenuation level g, the H1 noise attenuation condition (6) is ensured. More specifically, for any nonzero disturbance input oðtÞ 2 Ls2 ½0,1Þ, the inequality ð6Þ JzðtÞLðtÞmðtÞJ22 o g2 fJoðtÞJ22 þ EðxT ðt 0 ÞÞREðxðt 0 ÞÞg Rt T holds, where Jf ðtÞJ22 :¼ t01 f ðtÞf ðtÞ dt, t 1 is the selected filter horizon, R is a symmetric positive definite matrix, and g is a given real positive scalar.
ð4Þ n
l
where xðtÞ 2 R is the unmeasured state, uðtÞ 2 R is a known input signal, y1 ðtÞ 2 Rm1 and y2 ðtÞ 2 Rm2 are the measured observations, zðtÞ 2 Rq is the output to be estimated, oðtÞ 2 Ls2 ½0,1Þ is the deterministic disturbance input, A(t), B(t), G(t), b(t), C 1 ðtÞ, h(t), C 2 ðtÞ, H(t), and L(t) are known deterministic continuous time-varying functions of appropriate dimension. The initial condition x0 2 Rn is a Gaussian random variable such that x0, W 1 ðtÞ 2 Rp1 , and
3. Central suboptimal mean-square H 1 filter design The proposed design of the suboptimal mean-square H1 filter for linear stochastic systems is based on the general result (see Theorem 3 in [1]) reducing the H1 controller problem to the corresponding optimal H2 controller problem. In this paper, only the filtering part of this result, valid for the entire controller problem, is used.
M. Basin et al. / Signal Processing 92 (2012) 801–806
803
Then, the optimal mean-square Kalman–Bucy filter for linear stochastic systems [52] and the H1 filter for linear systems (Theorem 4 in [4]) are employed to obtain the desired result, which is given by the following theorem.
Since EðxðtÞÞ ¼ EðxðtÞÞ and the variances of the estimated errors produced by the estimates xðtÞ and m(t) are equal, the conditions 1–3 of Section 2 hold. The theorem is proved. &
Theorem 1. The central suboptimal mean-square H1 filter for the linear stochastic system (1)–(4), ensuring the minimum of the mean-square criterion (5) and the H1 noise attenuation condition (6), is given by the equation for the mean-square H1 estimate m(t)
Remark 1. The convergence of the designed mean-square H1 state estimate m(t) to the real state value x(t) is assured by the conditions ðC1 Þ and ðC2 Þ in view of the results of Theorem 7.4 and Section 7.7 in [50]. Note that boundedness of the noise-output H1 norm for the system (S 1 ), controlled by filter (7)–(9), i.e., admissibility of the mean-square H1 filter (7)–(9), is determined by the conditions I–III of Theorem 3 in [1].
dmðtÞ ¼ ðAðtÞmðtÞ þ BðtÞuðtÞÞ dt T
þPðtÞC T1 ðtÞðhðtÞh ðtÞÞ1 ½dy1 ðtÞC 1 ðtÞmðtÞ dt þSðtÞC T2 ðtÞðHðtÞHT ðtÞÞ1 ½y2 ðtÞC 2 ðtÞmðtÞ dt,
ð7Þ
with initial condition mðt 0 Þ ¼ Eðxðt 0 Þ9F Yt0 Þ, where the matrix function P(t) (minimum estimation error variance) is the solution of the differential Riccati equation _ ¼ AðtÞPðtÞ þ PðtÞAT ðtÞ PðtÞ T
T
þ bðtÞb ðtÞPðtÞC T1 ðtÞðhðtÞh ðtÞÞ1 C 1 ðtÞPðtÞ,
ð8Þ
with initial condition Pðt 0 Þ ¼ E½ðxðt 0 Þmðt 0 ÞÞðxðt 0 Þmðt 0 ÞÞT 9 F Yt0 , and the symmetric matrix function S (t) is the solution of the differential Riccati equation _ ¼ AðtÞSðtÞ þ SðtÞAT ðtÞ þ GðtÞGT ðtÞ SðtÞ SðtÞ½C T2 ðtÞðHðtÞHT ðtÞÞ1 C 2 ðtÞg2 LT ðtÞLðtÞSðtÞ, with initial condition Sðt 0 Þ ¼ R
1
ð9Þ
.
Proof. First, let us design the estimate xðtÞ satisfying the minimum variance condition (5) of Section 2. As known [51], this mean-square estimate is given by the conditional expectation xðtÞ ¼ EðxðtÞ9F Yt Þ of the system state x(t) with respect to the s-algebra FYt, generated by the observations (2) in the interval ½t 0 ,t, and is produced by the Kalman–Bucy filter [52] applied to the linear stochastic system (1) over the linear observations (2) in the presence of Gaussian disturbances (Wiener processes) W 1 ðtÞ and W 2 ðtÞ. The corresponding filtering equations for the estimate xðtÞ and the estimation error variance PðtÞ take the form dxðtÞ ¼ ðAðtÞxðtÞ þ BðtÞuðtÞ þGðtÞoðtÞÞ dt T
þPðtÞC T1 ðtÞðhðtÞh ðtÞÞ1 ½dy1 ðtÞC 1 ðtÞxðtÞ dt, with the initial condition
xðt 0 Þ ¼ Eðxðt 0 Þ9F Yt0 Þ,
ð10Þ
and
T P_ ðtÞ ¼ AðtÞPðtÞ þPðtÞAT ðtÞ þ bðtÞb ðtÞ T
PðtÞC T1 ðtÞðhðtÞh ðtÞÞ1 C 1 ðtÞPðtÞ,
Remark 2. According to the comments in subsection V.G in [1], the obtained central mean-square H1 filter (7)–(9) presents a natural choice for H1 filter design among all admissible H1 filters satisfying the inequality (6) for a given threshold g, since it does not involve any additional actuator loop (i.e., any additional external state variable) in constructing the filter gain matrix. Moreover, the obtained central mean-square H1 filter has the suboptimality property, i.e., it minimizes the criterion J ¼ JzðtÞLðtÞmðtÞJ22 g2 ðJoðtÞJ22 þ EðxT0 ÞREðx0 ÞÞ: Remark 3. Following the discussion in subsection V.G in [1], note that the complementarity condition always holds for the obtained filter (7)–(9), since the positive definiteness of the initial condition matrix R implies the positive definiteness of the filter gain matrix S(t) as the solution of (9). 4. Example This section presents the design of the central suboptimal mean-square H1 filter to estimate the pitch and yaw angles for a 2DOF helicopter, ensuring the minimum of the mean-square criterion (5) and the H1 noise attenuation condition (6) holds for g ¼ 1:1. Let the 2DOF helicopter system with the state space representation be _ ¼ AxðtÞ þ BðtÞuðtÞ þ GoðtÞ þ bc1 ðtÞ, xðtÞ
xðt 0 Þ ¼ x0 ,
ð12Þ
yðtÞ ¼ C 1 xðtÞ þ hc2 ðtÞ,
ð13Þ
y2 ðtÞ ¼ C 2 xðtÞ þHoðtÞ,
ð14Þ
zðtÞ ¼ LxðtÞ,
ð15Þ
ð11Þ
with the initial condition Pðt 0 Þ ¼ Eððxðt 0 Þxðt 0 ÞÞðxðt 0 Þxðt 0 ÞÞT 9F Yt0 Þ: Note that the latter equation coincides with (8). Now, applying the central suboptimal H1 filter for linear systems [4] to the estimate xðtÞ governed by Eqs. (10) and (11) yields the central suboptimal mean-square H1 estimate Eq. (7), where the matrix function PðtÞ satisfies Eq. (8), and the matrix S(t) in (7) satisfies Eq. (9). Note that filter (7)–(9) yields, in view of Theorems 3 and 4 in [1], the asymptotic stability of the mean value EðxðtÞÞ of the estimate (10) in the absence of disturbances and the prescribed attenuation level g for this variable: JEðxðtÞÞJ22 o g2 JoðtÞJ22 þ EðxT ðt 0 ÞÞREðxðt 0 ÞÞ.
_ T , in which y and c where the state vector is: x ¼ ½y, c, y_ , c _ are pitch are pitch and yaw angles respectively, y_ and c and yaw angular velocities, respectively. The matrices are 2 3 0 0 1 0 60 0 7 0 1 6 7 A¼6 7, 4 0 0 9:2751 5 0 0 2
0 0
6 0 6 B¼6 4 2:3667 0:2410
0
3:4955 0
3
7 0 7 7, 0:0790 5 0:7913
2
0
6 0 6 b¼6 4 0:9024 0:0919
0
3
7 0 7 7, 0:0876 5 0:8772
804
M. Basin et al. / Signal Processing 92 (2012) 801–806
0 6 0 6 G¼6 4 0:9024 0:0919
C1 ¼ C2 ¼ L ¼
H¼
0 0
0 0
0
0
0
0
0:0876
0
07 7 7, 05
0:8772
0
0
1
0
0
0
0
1
0
0
2
3
0
,
h¼
1
0
0
1
Here, u(t) is the motor voltage input, oðtÞ is an L22 disturbance input, c1 ðtÞ and c2 ðtÞ are Gaussian white noises, which are the weak mean square derivatives of standard Wiener processes W 1 ðtÞ and W 2 ðtÞ (see [51]), respectively. The Wiener processes are considered independent of each other and of a Gaussian random variable x0 serving as the initial condition in (12). Eqs. (12) and (13) present the conventional form for Eqs. (1) and (2), which is actually used in practice [53]. It can be easily verified that the noise orthogonality condition holds for the system (12)–(15). The filtering problem to be addressed is the same as described at Section 2. The filtering horizon is set from t 0 ¼ 0 to t 1 ¼ 80 s. The central suboptimal mean-square H1 filter takes the following form for the system (12)–(15): 2 3 0 0 1 0 60 0 7 0 1 6 7 _ mðtÞ ¼6 7mðtÞ 4 0 0 9:2751 5 0 0
0
0
2
3
1 60 6 þ PðtÞ6 40 0 2
1 60 6 þ SðtÞ6 40 0
3:4955
0 17 1 7 7 yðtÞ 05 0
0
0
0
1
0
0
mðtÞ
0 3 1 17 7 7 y2 ðtÞ 0 05 0
0
0
0
1
0
0
mðtÞ ,
ð16Þ
0
with mð0Þ ¼ m0 ¼ Eðx0 9F Y0 Þ, where S(t) and P(t) are the solutions to the differential Riccati equations 2 3 0 0 1 0 60 0 7 0 1 7 _ ¼6 SðtÞ 6 7SðtÞ 4 0 0 9:2751 5 0 0
0
0
2
3:4955
0
0
60 6 þ SðtÞ6 41
0
0
0 1
9:2751 0
0 2 6 6 SðtÞ6 4
0
3
0
7 0 7 7 5 0 3:4955 3
0:1736
0
0
0
0
0:1736
0
0
0
0
07 7 7SðtÞ 05
0
0
0
0
0
1
0
0
0
0:8220
7 7 7, 0:1597 5
0
0
0:1597
0:7779
2
0 : 1
1 0
3
0 60 6 þ6 40
0 1
Sð0Þ ¼ S0 ¼ R1
ð17Þ
and
,
0
0
1
0
3
60 _ ¼6 PðtÞ 6 40
0 0
0 9:2751
1 0
7 7 7PðtÞ 5
0
3:4955
0 0 2 0 0 60 0 6 þPðtÞ6 41 0
0
0
0
0
9:2751
0
3 7 7 7 5
0
1
1 60 6 PðtÞ6 40
0
0
1
0
0
0
0 3:4955 3 2 0 0 1 0 7 60 0 0 07 6 7PðtÞ þ 6 4 0 0 0:8220 05
0
0
0
0
2
0
0
0:1597
0
3
7 7 7, 0:1597 5 1
0:7779 ð18Þ
Pð0Þ ¼ E½ðx0 m0 Þðx0 m0 ÞT 9F Y0 ¼ P0 , respectively. Numerical simulations results are obtained by solving the system (12)–(15) and the filtering equations (16)–(18), with the following initial values: x0 ¼ ½0:7069,0; 0,0T , P 0 ¼ diagð10; 10,15; 5Þ, R ¼ ½0:8,0:5,0:2,0:1; 0:5,0:8,0:7,0:2; 0:2,0:7,0:9,0:4; 0:1,0:2,0:4,0:9, and m0 ¼ ½0:1745,0:5236, 0:15,0:4T . The motor voltage uðtÞ ¼ ½12:495,3:835T , the L2 disturbance oðtÞ ¼ ½o1 ðtÞ, o2 ðtÞ, o3 ðtÞ, o4 ðtÞT is realized as o1 ðtÞ ¼ 1=ð1 þtÞ, o2 ðtÞ ¼ 0:1ð1e0:3t Þ, o3 ðtÞ ¼ 0:1745=ð1 þtÞ, and o4 ðtÞ ¼ 0:0873ð1e0:3t Þ. The attenuation level value is set to g ¼ 1:1. The disturbances c1 ðtÞ and c2 ðtÞ in (12) and (13) are realized by using the built-in MatLab white noise function. As a result of the numerical simulation, the following graphs are presented: graph of the noise-output H1 norm (Fig. 1); graphs of the pitch angle and the corresponding estimation error (Fig. 2); graph of the yaw angle and the corresponding estimation error (Fig. 3). Note that the maximum value of the noise-output H1 norm T ¼ JzðtÞLðtÞmðtÞJ=ðJoðtÞJ22 þ Eðx0 ÞREðx0 ÞÞ1=2 is equal to 0.208 in the considered simulation interval, which is 0.25 0.2 H−infinity norm
2
0.15 0.1 0.05 0 0
10
20
30
40 50 Time [seg]
Fig. 1. Noise-output H1 norm T.
60
70
80
Estimation error
Pitch angle [deg]
M. Basin et al. / Signal Processing 92 (2012) 801–806
filters for nonlinear polynomial stochastic systems or linear stochastic systems with time delays.
50 true estimated
0 Acknowledgments
−50 0
10
20
30
40
50
60
70
80
0
10
20
30 40 50 Time [seg]
60
70
80
0.5 0 −0.5
Yaw angle [deg]
The authors thank the Mexican National Science and Technology Council (CONACyT) for financial support under Grants 129081 and 55584 and Sabbatical Fellowship for the first author. References
−1
Fig. 2. Above. Pitch angle. Below. Estimation error.
Estimation error
805
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Fig. 3. Above. Yaw angle. Below. Estimation error.
five times less than the given H1 attenuation level, g ¼ 1:1. In addition, the estimation errors robustly converge to zero. The signal-to-noise ratio decreases virtually to zero for both, pitch and yaw, angles by t¼ 80 s, despite a sharp change in the pitch velocity at t ¼55 s.
5. Conclusions This paper designs the central finite-dimensional H1 filter for linear stochastic systems with integral-quadratically bounded deterministic disturbances, that is suboptimal for a given threshold g with respect to a modified Bolza– Meyer quadratic criterion including the attenuation control term with opposite sign. The designed filter is applied to estimation of the pitch and yaw angles of a two degrees of freedom (2DOF) helicopter. The simulation results show a reliable performance of the filter, in particular, the obtained attenuation level is five times less than a given threshold. This significant improvement is obtained due to the more reasonable selection of the filter gain matrix in the designed filter. Although this conclusion follows from the developed theory, the numerical simulation serves as a convincing illustration. The presented approach would be applied in the future to obtain the central suboptimal mean-square H1
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