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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 4, APRIL 2009
A Discrete-Time Robust Extended Kalman Filter for Uncertain Systems With Sum Quadratic Constraints Abhijit G. Kallapur, Student Member, IEEE, Ian R. Petersen, Fellow, IEEE, and Sreenatha G. Anavatti
Abstract—This technical note outlines the formulation of a novel discrete-time robust extended Kalman filter for uncertain systems with uncertainties described in terms of Sum Quadratic Constraints. The robust filter is an approximate set-valued state estimator which is robust in the sense that it can handle modeling uncertainties in addition to exogenous noise. Riccati and filter difference equations are obtained as an approximate solution to a reverse-time optimal control problem defining the set-valued state estimator. In order to obtain a solution to the set-valued state estimation problem, the discrete-time system dynamics are modeled backwards in time. Index Terms—Extended Kalman filter (EKF), integral quadratic constant (IQC), sum quadratic constant (SQC).
I. INTRODUCTION Precise state and parameter estimation is critical for many control and filtering applications. A number of state estimation techniques are available for both linear as well as nonlinear systems, of which the Kalman filter and its variants such as the extended Kalman filter (EKF) have found wide areas of application; e.g., see [1]. In spite of their ease of implementation, Kalman filters are known to diverge in some cases under the influence of nonlinearities and uncertainties [2]. In order to overcome this problem, robust versions of the Kalman filter have been derived for various classes of uncertainty description [3]–[9]. Among the many approaches developed for state estimation, the set membership state estimation approach of [10] provides a deterministic interpretation of the Kalman filter in terms of a set-valued state estimator; see also [11]. The set-valued state estimation problem involves finding the set of all states consistent with given output measurements for a time-varying system with norm bounded noise input. In [4], [12], Savkin and Petersen extended the set membership state estimation approach of [10] in order to accommodate linear uncertain continuous systems with an integral quadratic constraint (IQC) uncertainty description. These results were extended to the discrete-time case with a sum quadratic constraint (SQC) uncertainty description in [12], [13]. Later, James and Petersen [6] extended the approach in [4] to solve a robust filtering problem for a class of uncertain nonlinear systems. The solution to the set-valued state estimation problem in this case leads to a robust version of the continuous-time extended Kalman filter. A majority of the state estimation methods used in practical engineering problems are based on the use of discrete-time processes and measurements owing to the use of digital sensors and digital signal processing. This has motivated continued efforts in the literature directed towards developing suitable linear as well as nonlinear filters
Manuscript received July 21, 2008; revised November 19, 2008. Current version published April 08, 2009. This work was supported by the Australian Research Council. Recommended by Associate Editor Z. Wang. A. G. Kallapur and S. G. Anavatti are with the School of Aerospace, Civil and Mechanical Engineering, University of New South Wales at the Australian Defence Force Academy, Canberra ACT 2600, Australia (e-mail:
[email protected];
[email protected]). I. R. Petersen is with the School of Information Technology and Electrical Engineering, University of New South Wales at the Australian Defence Force Academy, Canberra ACT 2600, Australia (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAC.2008.2010962
for discrete time systems; see [12]–[15]. In particular, this motivates us to extend to the discrete-time case, the approach to robust estimation for a class of uncertain nonlinear systems given in [6]. Underlying the continuous-time set-valued state estimation approaches of [4], [6], [10], is a solution or approximate solution to a reverse-time optimal control problem. In the continuous-time case, this reverse-time optimal control problem can be solved using dynamic programming methods in terms of the forward-time system dynamics. However, this is not straightforward in the discrete-time nonlinear case, since it would require inverting the nonlinear functions describing the nonlinear discrete-time system, which may not be possible. Instead, we consider the reverse-time discrete-time system dynamics. This is the main innovation in the technical note. The set-valued state estimation problem will be solved by obtaining an approximate solution to the optimal control problem in terms of a forward-time dynamic programming equation for a class of reverse-time discrete-time uncertain nonlinear systems with an SQC uncertainty description. The remainder of this technical note is organized as follows: Section II describes the formulation of the reverse-time discrete-time nonlinear uncertain system and introduces the concept of set-valued state estimation. This set-valued state estimation problem is then expressed in terms of a corresponding optimal control problem. Section III provides an approximate solution to this optimal control problem which leads to the Riccati and filter equations that define the discrete-time REKF. Section IV considers this discrete-time REKF for an uncertain nonlinear system with a norm-bound uncertainty description and Section V presents an illustrative example which involves applying the discrete-time REKF to an estimation problem arising from FM demodulation. II. PROBLEM FORMULATION In this section, we consider a reverse-time discrete-time uncertain nonlinear system which is derived from a forward-time continuoustime uncertain nonlinear system. The uncertainties in the discrete-time uncertain system are described by an SQC, which is derived from the corresponding continuous-time uncertainty description in the form of an IQC. Furthermore, the concept of set-valued state estimation is introduced and related to a corresponding optimal control problem. A. Reverse-Time Discrete-Time Uncertain Nonlinear System We begin with a forward-time continuous-time uncertain nonlinear system of the form
x_ (t) = ac (t; x(t); u(t)) + Dc (t)w(t) z (t) = kc (t; x(t); u(t)) y(t) = cc (t; x(t)) + v(t) n
(1)
m
where x(1) 2 is the state, u(1) 2 is the known control input, w(1) 2 p and v(1) 2 l are the process and measurement uncertainty inputs, respectively, z (1) 2 q is the uncertainty output and y (1) 2 l is the measured output. ac (1), kc (1) and cc (1) are given nonlinear functions and Dc (1) is a given matrix function of time. The uncertainty associated with system (1) can be described in terms of an IQC as in [6], [12]
x(0) 0 x0 )T N (x(0) 0 x0 )
(
0018-9286/$25.00 © 2009 IEEE
s
w(t)T Qc (t)w(t) + v(t)T Rc (t)v(t)
+
0
s
d+
zt
2
k ( )k
0
dt
(2)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 4, APRIL 2009
where k1k denotes the Euclidean norm; x(0) is the initial state value and x0 is the nominal initial state. A finite difference (x(0)0 x0 ) is allowed by a non-zero value of the output uncertainty z (t) or the constant d. If z (t) = 0 and d = 0, then x(0) = x0 . Also, w(1) and v(1) represent admissible uncertainties described by,
w(t) v(t)
=
(t; x(1))
(3)
where (1) is a nonlinear time-varying dynamic uncertainty function and x(1) represents the function x(t) for all instants of time t 2 [0; T ]. The fact that w(t) and v (t) depend on x(1) in (3) indicates that they are allowed to depend dynamically on the system state. Also, N = N T > 0 is a given matrix, x0 2 n is a given state vector, d > 0 is a given constant and Qc (1), Rc (1) are given positive-definite, symmetric matrix functions of time. In order to derive a discrete-time robust set-valued state estimator, it is necessary to discretize this continuous-time uncertain system. However, as mentioned in the introduction, the discrete-time set-valued state estimator is most straightforward to derive if this system is discretized in reverse-time rather than in forward-time. In order to discretize the nonlinear uncertain system (1), standard techniques such as the Euler or Runge-Kutta methods can be applied. This will lead to a nonlinear reverse-time discrete-time uncertain system described by the state equations
x(k) = A (k; x(k + 1); u(k)) 0 B (k)w(k) z (k + 1) = K (k; x(k + 1); u(k)) y(k + 1) = C (k; x(k + 1)) + v(k + 1)
T N (x(0) 0 x ) + 0
T 01 k=0
(4)
w(k)T Q(k)w(k)+
v(k + 1)T R(k + 1)v(k + 1) d +
T 01 k=0
kz (k + 1)k2
(5)
as =
xT 2 ZT x0 ; u0 (1)j1T ; y0 (:)jT1 ; d
(7)
if and only if there exists an uncertain input sequence w(1) such that, VT [xT ; w(1)] d where the cost functional VT [xT ; w(1)] is derived from the SQC (5) as 1 T VT [xT ; w(1)] = (x(0) 0 x0 ) N (x(0) 0 x0 )
T 01
+
w(k)T Q(k)w(k)
k=0
d +
T 01 k=0
T R(k + 1)v (k + 1)
+v (k + 1)
kz (k + 1)k2
(8)
with v (k +1) = [y0 (k +1) 0 C (k; x(k +1))], z (k +1) = K (k; x(k + 1); u0 (k )). Here, the vector x(1) is the solution to the reverse-time discrete-time system (4), with input uncertainty w(1) and terminal condition x(T ) = xT . Hence
ZT x0 ; u0 (1)j1T ; y0 (1)jT1 ; d xT 2 n :
inf
w(1)
VT [xT ; w(1)] d : (9)
The optimization problem inf
w(1)
VT [xT ; w(1)]
(10)
for the system (4), defines a nonlinear optimal control problem with a sign indefinite quadratic cost function. The discrete-time REKF will be derived by finding an approximate solution to this optimal control problem. III. DISCRETE-TIME ROBUST EXTENDED KALMAN FILTER
where v (k +1) = [y (k +1) 0 C (k; x(k +1))], z (k +1) = K (k; x(k + 1); u(k )), and the admissible uncertainties w (1) and v (1) are described
w(k) v(k + 1)
y0 (1) and input sequence u0 (1). Given an output sequence y0 (1), it follows from the definition of ZT [x0 ; u0 (1)j1T ; y0 (1)jT1 ; d], that
=
where A(1), K (1) and C (1) represent discrete-time nonlinear functions and B (1) is a given time-varying matrix. The uncertainty associated with the reverse-time discrete-time system (4) can be obtained by discretizing the IQC (2) to obtain an SQC (x(0) 0 x0 )
851
(k; x(1))
(6)
and (1) is a nonlinear time-varying dynamic uncertainty function. The uncertain system (4), with the corresponding SQC uncertainty description (5) will be used to derive the robust filter and Riccati equations, which define the discrete-time REKF. B. Set-Valued State Estimation and the Corresponding Optimal Control Problem Consider y0 (k) = y (k) to be a fixed measured output and u0 (k) = u(k) a known control input for the uncertain system (4), (5), for k = 1; 2; . . . ; T . The set-valued state estimation problem involves finding the set ZT [x0 ; u0 (1)jT1 ; y0 (1)jT1 ; d] of all possible states x(T ) at time step T for the system in (4) with initial conditions and uncertainty constraints defined in (5), consistent with the measured output sequence
The optimal control problem (10) for the reverse-time discrete-time system (4) can be solved via dynamic programming. Indeed, the corresponding discrete-time (forward-time) Hamilton–Jacobi–Bellman (HJB) equation for this optimal control problem is given by
Vk3+1 (x(k + 1)) = min w(k)
Vk3 [A (k; x(k + 1); u0 (k)) 0 B (k)w(k)]
+ w (k )
T Q(k)w(k)+ v (k + 1)T R(k +1)v (k +1)
0z (k + 1)T z (k + 1)
(11)
with initial condition
V03 (x(0)) = (x(0) 0 x0 )T N (x(0) 0 x0 ) :
(12)
As a first step towards obtaining an approximate solution to the nonlinear optimal control problem (10), (4) we approximate Vk3 (1) as
Vk3 (x(k)) (x(k) 0 x^(k))T X (k) (x(k) 0 x^(k)) + 8(k) (13) ^(k ) is a vector representing the robust state estimate and X (k ) where x is a symmetric matrix. Furthermore, comparing (12) and (13), the initial ^(0) = x0 , X (0) = N and 8(0) = 0. conditions become x
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 4, APRIL 2009
(x(k + 1)
0
x^(k + 1))T X (k + 1) (x(k + 1)
+ 8(k + 1) = min
w (k )
[A (k; x(k + 1); u0 (k ))
=
0 x^(k + 1))
T X (k +1)^ x(k +1)+8(k +1)
+x ^(k +1)
Applying the approximate solution (13) to the HJB (11), we obtain
x(k +1)T
0
0
0 B(k)w(k) 0 x^(k)]T
0
2 X (k) [A (k; x(k + 1); u (k)) 0B(k)w(k) 0 x^(k)] 0
0
2 r 0r r 0r r
T ^(k ); u0 (k )) 4(k )^ x(k)+ X (k +1)^ x(k) x A (k; x T ^(k ); u0 (k )) 4(k )A (k; x ^(k ); u0 (k )) x A (k; x T ^(k )) R(k +1)y0 (k +1) + x C (k; x T C ( k; x ^(k )) R(k +1)C (k; x ^(k )) x T + x K (k; x ^(k ); u0 (k )) K (k; x ^(k ); u0 (k ))
T R(k + 1)v (k + 1) + 8(k)
+ v (k + 1)
T Q(k)w(k)
+w (k )
0 z(k + 1)T z(k + 1)
:
(14)
Solving the minimization problem in (14) using the method of completing the squares, we obtain an optimal value for w(k) (denoted by w(k)3 ), given by 01 w(k)3 = Q(k) + B (k)T X (k + 1)B (k) B (k)T X (k)
2 [A (k; x(k + 1); u(k)) 0 x^(k + 1)] :
T
+x ^(k )
(15)
0 x^(k + 1))T X (k + 1) (x(k + 1) 0 x^(k + 1))
2 rx A (k; x^(k); u (k))T 4(k)A (k; x^(k); u (k)) T + 4(k )A (k; x ^(k ); u (k )) 0rx C (k; x ^(k )) 2 R(k +1)y (k +1)+ rxC (k; x^(k))T R(k +1) 2 C (k; x^(k)) 0rx K (k; x^(k); u (k))T 2 K (k; x^(k); u (k)) 0
0
A (k; x(k +1); u0 (k))T 4(k)^ x(k) T + C (k; x(k +1)) R(k +1)C (k; x(k + 1))
0
T R(k +1)y (k + 1) 0 T 2y0 (k +1) R(k +1)C (k; x(k +1)) x^(k)T 4(k)A (k; x(k +1); u (k))+x^(k)T 4(k)^ x(k)
0
0
T
0
+ 8(k )+ A (k; x ^(k ); u0 (k )) 4(k )A (k; x ^(k ); u0 (k ))
(16)
T R(k +1)C (k; x ^(k )) T K (k; x^(k); u0 (k)) K (k; x^(k); u0 (k)) T +y (k +1) R(k +1)y (k +1)
+ C (k; x ^(k ))
0
where
X (k) 0 X (k)B (k) Q(k) + B (k)T X (k)B (k)
0
0
+ y0 (k +1)
4(k ) =
0
0
A (k; x(k +1); u0 (k))T 4(k)A (k; x(k +1); u0 (k))
0 0 0z(k +1)T z(k +1)+8(k)
0
0
+ 8(k + 1) =
rxA (k; x^(k); u (k))T 4(k) 2rx A (k; x^(k); u (k))+2rx A (k; x^(k); u (k))T 2 4(k)+4(k)+ rxC (k; x^(k))T R(k +1) 2 rx C (k; x^(k)) 0rx K (k; x^(k); u (k))T 2 rxK (k; x^(k); u (k)) x^(k) 0 2^x(k)T 0
Substituting w(k) = w(k)3 from (15) into (14) we obtain (x(k + 1)
rxA (k; x^(k); u (k))T 4(k) 2 rx A (k; x^(k); u (k))+ rx C (k; x^(k))T 2 R(k +1)rx C (k; x^(k)) 0 rx K (k; x^(k); u (k))T 2 rx K (k; x^(k); u (k)) x(k +1) 0 2x(k +1)T
#
2B(k)T X (k):
0
T 2C (k; x ^(k )) R(k +1)y0 (k +1) :
(17)
In the above equation, (1)# denotes the Moore-Penrose pseudo-inverse. If the matrix (Q(k)+ B (k)T X (k)B (k)) is positive definite, the pseudo-inverse in (17) can be replaced by a normal matrix inverse. This would hold in all cases for which a suitable solution exists for X (k); see [12]. In order to obtain an approximate solution to the nonlinear difference (16), a first order linearization of the nonlinear system (4) is performed ^(k ) about the point x
A (k; x(k +1); u0 (k)) A (k; x^(k); u0 (k))+rx A (k; x^(k); u0 (k)) 2 (x(k +1) 0x^(k)) K (k; x(k +1); u0 (k)) K (k; x^(k); u0 (k))+ rx K (k; x^(k); u0 (k)) 2 (x(k +1) 0x^(k)) C (k; x(k +1)) C (k; x^(k))+ rx C (k; x^(k)) (x(k +1) 0x^(k)) :
0
0
(19)
Comparing the left-hand-side terms with the right-hand-side terms in (19), the following recursive equations are obtained: Riccati Difference Equation X (k + 1) = rx A(k; x^(k);u0 (k))T 4(k)rx A(k; x^(k);u0 (k)) T + rx C (k; x ^(k )) R(k + 1)rx C (k; x ^(k )) T 0rx K (k; x^(k);u0 (k)) rx K (k; x^(k); u0 (k))
X (0) =N:
(20)
Filter State Equation
x^(k + 1) = x^(k) + X (k + 1)013 x^(0) = x0
(21)
where 3=
(18)
The point at which the linearization is performed corresponds to the ^(k ) as in the case of the standard EKF; e.g., see current state estimate x [1]. Substituting the linearized terms from (18) into (16), an approximate solution to this nonlinear difference equation is obtained as x(k +1)T X (k +1)x(k +1) 0 2x(k +1)T X (k +1)^ x(k +1)
rx A (k; x^(k); u (k))T 4(k)^x(k) 0 rx A (k; x^(k); u (k))T 4(k)A (k; x^(k); u (k)) T ^(k )) R(k + 1)y (k + 1) + rx C (k; x 0 rx C (k; x^(k))T R(k + 1)C (k; x^(k)) T +rx K (k; x ^(k ); u (k )) K (k; x ^(k ); u (k )) : 0
0
0
0
0
Filter Constant Equation 8(k +1) =2^ x(k)T
0
(22)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 4, APRIL 2009
2 0rx A (k; x^(k);u0 (k))T 4(k)A(k; x^(k);u0 (k)) 0 4(k)A (k; x^(k); u0 (k))+ rx C (k; x^(k))T 2 R(k +1)y0(k +1) 0rxC (k; x^(k))T R(k +1) 2C (k; x^(k))+ rxK (k; x^(k); u0 (k))T 2 K (k; x^(k); u0 (k)) + x^(k)T 2 rx A (k; x^(k); u0 (k))T 4(k) 2rx A (k; x^(k); u0 (k))+2rx A (k; x^(k); u0 (k))T 2 4(k)+4(k)+ rxC (k; x^(k))T R(k +1) 2rx C (k; x^(k)) 0rxK (k; x^(k); u0 (k))T 2 rxK (k; x^(k); u0 (k)) x^(k) T + A (k; x ^(k ); u0 (k )) 4(k )A (k; x ^(k ); u0 (k )) T +C (k; x ^(k )) R(k +1)C (k; x ^(k )) 0 K (k; x^(k); u0 (k))T K (k; x^(k); u0 (k)) T T + y0 (k +1) R(k +1)y0 (k +1) 0 2C (k; x ^(k )) 2 R(k +1)y0(k +1) T + 8(k ) 0 x ^(k +1) X (k +1)^ x(k +1) 8(0) =0
(23)
where 4(k) is defined in (17). Assuming that the recursions (20), (21), (23) have solutions x ^(k ), X (k), 8(k) such that X (k) > 0 and (Q(k)+ B (k)T X (k)B (k)) > 0 for k = 1; 2; . . . ; T , the corresponding approximate set-valued state estimate is given by
xT 2 n : ZT x0 ; u0 (1)jT1 ; y0 (1)jT1 ; d = T (x(T ) 0 x ^(T )) X (T ) (x(T ) 0 x ^(T )) d 0 8(T ) :
(24)
It should be noted that, if the reverse-time discrete-time uncertain system (4), (5) is in fact linear as in [12], [13], then the nonlinear Riccati difference equation (20), the filter state equation (21) and the filter constant (23) yield a discrete-time linear robust Kalman filter that agrees with the solution presented in [12], [13]. IV. SPECIAL CASE: NORM BOUND UNCERTAINTY In order to apply the discrete-time REKF described in Section III, a standard norm bounded uncertainty description can be considered as a special case of the SQC uncertainty description. Here, the nonlinear discrete-time reverse-time uncertain system is modeled as
x(k) = An (k; x(k + 1); u(k)) + D1 (k)11 (k) 2 Kn (k; x(k + 1); u(k)) + D2 (k)w0 (k) y(k + 1) = Cn (k; x(k + 1)) + 12 (k)Kn (k; x(k + 1); u(k)) (25) + v0 (k + 1) where the uncertain matrices 11 (1); 12 (1) satisfy the norm bound
pQ1 11 (k) 1: (26) 2R 12 (k ) Also the exogenous noise signals w0 (k) and v0 (k + 1) satisfy the en-
ergy bound
T 01 8(0) +
k=0
w0 (k)T Q2 w0 (k) + 2v0 (k + 1)T Rv0 (k + 1) d: (27)
853
Fig. 1. FM demodulation estimation using discrete-time REKF.
Here, Q1 > 0, Q2 > 0, R > 0 are given weighting matrices. The fact that the uncertain system (25), (27) is a special case of the uncertain system (4), (5) can be established by setting
A (k; (k + 1); u(k)) 0 B (k)w(k) = An (k; x(k + 1); u(k)) + [D1 (k ); D2 (k )] w (k ) (28) where w(k) = [11 (k)Kn (k; x(k + 1); u(k)) 0 w0 (k)] Q = Q01 Q0 2 v(k + 1) = 12 (k)Kn (k; x(k + 1); u(k)) + v0 (k + 1)(29) and noting that the conditions (26) and (27) will imply the SQC (5) is satisfied. When K (1) = 0, that is in the absence of uncertainty, it is clear from the above formulation that the only error in the system is contributed by exogenous noise sources. Furthermore, this reduces the estimation procedure to a set-valued state estimation version of the standard discrete-time EKF. V. ILLUSTRATIVE EXAMPLE As an example, the discrete-time REKF derived in Section III is applied to an FM demodulation problem, a continuous form of which is presented in [6]. The continuous-time equations presented in [6] are reproduced here for completeness followed by a discretization process. The continuous-time FM demodulation system described in [6] is described by the state equations
x_ (t) = Ac x(t) + Bc w(t) y(t) = Cc (t; x(t)) + v0 (t) with x(t) = ((tt)) ; Ac = 011 00 Bc = 10 10 ; w(t) = 1(wt)(t()t) 0 p Cc (t; x(t)) = 2 sin (t + (t)) where j1(t)j 1 and the noise satisfies the bound 100
w t
v t dt 10:
2 2 50 0 ( ) + 100 0 ( )
(30)
(31)
(32)
0
The above continuous-time system was discretized in order to provide a discrete-time system
x(k + 1) = Ad x(k) + Bd w(k) y(k + 1) = Cd (k; x(k + 1)) + v0 (k + 1)
(33)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 4, APRIL 2009
as shown in Fig. 1. As can be seen in this figure, the estimated signal follows the true signal with good fidelity in spite of being noisy. Fig. 2 represents the simulation results obtained by applying the standard discrete-time EKF estimator to the FM demodulation problem. In this case, the EKF was designed for the case of 1 = 0 but the simulation was carried out for the case of 1 = 1 in (33), (34). It can be seen that in this case, the estimated signal does not track the true signal but rather exhibits divergence. VI. CONCLUSION In this technical note, we have presented a novel discrete-time robust extended Kalman filter based on the reverse-time discretization of a continuous-time nonlinear uncertain system. The filter and Riccati difference equations were derived as an approximate set-valued state estimator solution obtained from a corresponding optimal control problem. The superior performance of the new filter was illustrated by comparing its estimation results with that of a standard EKF for an FM demodulation problem involving nonlinearity in the output equation. Also, future research could be directed towards applying this method to systems with more complicated nonlinearities occurring in both the state and output equations, such as the examples in [15].
Fig. 2. FM demodulation estimation using discrete-time EKF.
with
(k) ; Ad = e[A (k)
x(k) =
h
Bd = 0
w(k) = Cd (k; x(k + 1)) =
p
e[A
h
1 ]
h
1 ]
d 1 Bc
=
=
0:9048 0:0952
0:0952
0 1
0:0952
0:0048
0:0048
1(k )(k )
w0 (k) 2 sin (k 3 h + (k + 1))
REFERENCES (34)
where k = t=h, and the sample time h is set to 100 ms. The system (33) and (34) were used to simulate the discrete-time FM demodulator with initial conditions (k) = 0, (k) = 0. Since the discrete-time REKF estimation procedure considers a reverse-time uncertain system (see Section III), such a system was constructed by discretizing the system matrices (30) in reverse-time to obtain a reverse-time discrete-time uncertain system
x(k) = Ak x(k + 1) + Bk w(k) y(k + 1) = Ck (k; x(k + 1)) + v0 (k + 1)
(35)
where
(k) (k)
x(k) = Ak
=e
A 1h]
[0
h
Bk
e[0A
= 0
=
w(k) =
1:1052
=
:1052
00
h
1 ]
:1052
00
0:0052
1(k )(k )
0 1
d 1 Bc :1052
00
0:0052
w0 (k) Ck (k; x(k + 1)) = Cd (k; x(k + 1))
(36)
with a sum quadratic constraint defined by =h
100
50w0 (k )
2
k=0
h + 100v0 (k + 1)2 h
d + kz (k + 1)k2 h: (37)
The FM demodulation problem was solved using the discrete-time REKF Riccati (20) and filter (21) for the reverse-time discrete-time uncertain system (35), (36), (37). The discrete-time system (33), (34) together with the discrete-time REKF was simulated with initial conditions (0) = 0, (0) = 0 and setting 1(k) = 1 = 1. The results are
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