- Email: [email protected]
Robust State Estimation for Discrete-Time Uncertain Systems with a Deterministic Description of Noise and Unqrtainty
Copyright © IFAC Youth Automation. Beijing. PRC. 1995
ROBUST STATE ESTIMATION FOR DISCRETE-TIME UNCERTAIN SYSTEMS WITH A DETERMINISTIC DESCRIPTION OF NOISE AND UNCERTAINTY
A. V. SAVKIW and 1. R. PETERSEW' • Department of Electrical and Electronic Engineering and Cooperative Research Center for Sensor Signal and Information Processing, University of Melbourne, Parkville, Victoria 3052, Australia, email: [email protected]. "Department of Electrical Engineering, Australian Defence Force Academy, Campbell, 2600, Australia, email: [email protected].
Abstract. The paper presents a new approach to robust state estimation for a class of uncertain discrete-time systems with a deterministic description of noise and uncertainty. The main result is a recursive scheme for constructing an ellipsoidal state estimation set of all states consistent with the measured output and the given noise and uncertainty description. The paper also includes a result on model validation whereby it can be determined if the assumed model is consistant with measured data. Keywords. Signal Processing Algorithms; Robust Estimation; Kalman Filters; Modelling; Validation; Uncertain Dynamical Systems; Riccati Equations
1. INTRODUCTION
new uncertainty description is referred to as a Sum Quadratic Constraint. It ailows for a large class of nonlinear, dynamic uncertainties. Furthermore, our main result shows that for this uncertainty description , our robust filter can be used to determine if the assumed model is consistant with the given output measurements. Such m C) d~l validation results cannot be obtained with a stochastic description of noise and uncertainty.
The Kalman Filter has seen wide application in areas such as deconvolution, channel equalization, FM demodulation , radar tracking, inertial navigation and global positioning systems ; e.g., see (Anderson and Moore 1979. Peng and Chen 1994, Chen and Chen 1994) and the references therein. The Kalman filter gives an optimal estimate of the state of a given process based on output measurements. However, application of the standard Kalman Filter requires an accurate model of the process under consideration and if there are errors in the process model, the Kalman Filter may lead to poor performance; e.g., see (Theodor et al. 1994). In this paper, we present a new method for robust state estimation for the case in which the process model contains uncertainties and nonlinearities. The starting point for our approach is the deterministic interpretation of the discrete-time Kalman Filter given in (Bertsekas and Rhodes 1971). In (Bertsekas and Rhodes 1971), the Kalman Filter is shown to give a state estimate in the form oi an ellipsoidal set of all possible states consistent with the given process measurements and a deterministic description of the noise. We extend the approach of (Bertsekas and Rhodes 1971) to consider discrete-time uncertain process models which have a determin ist description of the noise and l1ncertainty. This
Our main result is a state estimator which gives a state estimate in the form of an ellipsoidal set of all possible states consistent with the given process measurements and our deterministic description of the noise and uncertainty. Our results give an exact characterization d this set. This is opposed to existing results on robust filtering such as (Theodor et al. 1994) which minimize only an upper bound on the estimation error for the problems they consider. As in the standard Kalman Filter, our state estimate is constructed recursively from the output measurements. However, the form of our state estimator equations are somewhat different from the standard Kalman Filter equations. In addition , we present an illustrative example which shows how our results can be applied to a robust deconvolution problem with uncertainty in the signal model.
289
Then condition (2) is satisfied.
2. PROBLEM STATEMENT Consider the time-varying uncertain discrete-time system defined for t = 0,1, ... ,T:
x(t + 1) z(t) y(t)
= = =
A(t)x(t) + B(t)w(t); K(t)x(t); C(t)x(t) + v(t)
(1)
where x(t) E Rn is the state, w(t) E RP and E RI are the uncertainty inputs, z(t) E Rq is the uncertainty output, y(t) E RI is the measured output, and A(t), B(t), K(t) and C(t) are given matrices such that A( t) is non-singular for t = 0, 1, ... ,T.
v(t)
State Estimation Problem The main result of this paper concerns the following state estimation problem. Let y(t) = Yo(t) be a fixed measured output of the uncertain system (1), (2) for t = 1,2, . .. , T. Then, find the corresponding set XT[xo,yo(-)I[,d] of all possible states x(T) at time T for the system (1) with uncertainty inputs and initial conditions satisfying the constraint (:?). Notation The system (1) , (2) is said to be robustly observable if the set XT[xo , Yo (.)1;' dj is bounded for any xo, any yo(-) and any d. Let Yo(t), t = 1,2, ... , T be a given output sequence. The output Yo (.) is said to be realizable if there e..~ ist sequences [x{·), w{ ·), v(·)] satisfying conditions (1), (2) with y{t) = Yo(t).
System Uncertainty The uncertainty in the above system is described by an equation of the form: [w(t) v(t)] = 4>(t,x(')I~) where the following Sum Quadratic Constraint is satisfied. Let Xo = X~ > 0 be a given matrix, Xo ERn be a given vector, d > 0 be a given constant, and Q( t) and R( t) be given positive-definite symmetric matrices defined for t = 0,1. . . . , T . Then we will consider the uncertainty inputs w(·) and v(·) and initial conditions x(O) such that
T-l l:)w(t)'Q(t)w(t) + v(t + I)' R(t + l)v(t + 1))
In addition to solving the state estimation problem mentioned above, the main result of this paper also solves the following problem: Given an output sequence yo(-), determine if this output is realizable for the uncertain system (1), (2). If a given measured output sequence is not realizable for the given uncertain system model, we can say that this model is invalidated by the measured data. Thus, the results of this paper are useful in the question of model validation.
t=o
<
+(x(O) - xo)'Xo(x(O) - xo) T-l d+ IIz(t + 1)112.
L
3. THE MAIN RESULT
(2) Our solution to the above state estimation ;)foblem involves the following Riccati difference equation:
t=o
Here 11 . 11 denotes the standard Euclidean norm . The uncertain system (I), (2) allows for uncertainty satisfying a standard norm bound constraint. In this case, the uncertain system would be described by the state equations
x(t + 1) y(t)
=
F(t + 1)
=
S(t + 1)
[A(t) + B(t)~l(t)K(t)]x(t) +B(t)nl(t); [C(t) + 6 2(t)K(t)]x(t) +n2(t);
=
B(t)'S(t)B(t) + Q(t) 2: 0 & N(B(t)'S(t)B(t) + Q(t)) c N(A(t)'S(t)Blt))
S
d. To verify that such uncertainty is adm issible for the uncertain system (11. (:?), let w(t) ~l (t)K(t)x(t) + ndt) for t 0.1, ... , T and v(t) = 6 2 (t)K(t)x(t) + n2(t) for t = 1.2, . .. , T where II[6 1 (t)'Q(t)! 6 2 (t)'R(t)!111 S 1 for all t.
=
1)] 1,4)
where A(t) ~ A(t)-r, B(t) ~ A,(t)-l B(t) and (.)# denotes the Moore-Penrose pseudo-inverse: e.g., see (Anderson and Moore 1979). Solutions to this Riccati equation will he required to satisfy the following condition:
where 6dt) and 6 2 (t) are uncertainty matrices, nl (t) and n2 (t) are noise sequences, and 11 . 11 denotes the standard induced matrix norm. Also, the initial conditions and noise sequences would be required to satisfy the inequality (x(O) - xo)'Xo(x(O) - xo) +
'£:=1 n2(t)'R(t)n2(t)
A(t)'S(t) [A(t) - B(t)F(t+
+C(t + I)' R(t + 1)C(t + 1) -K(t + 1)'K(t + 1), 5(0) = Xo
II[6 1 (t)'Q(t)! 6 2 (t)'R(t)t]II S 1 (3)
,£:=~l nl(t)'Q(t)nl(t) +
[B(t)'S(t)B{t) + Q(t)r B(t)'5(tU.(t)
=
.) )
=
for t 1. 2 .. .. , T . Here XI .) denotes the operation of taking the null space of a matrix. Also, we consider a set of state equations oi ~ he
290
form
77(t + 1) = [A(t) -
J[xr, w(·)]
~(t)F(t+ 1)]' 77(t)
(x(O ) - xo)' Xo(x(O) - xo)
w(t)'Q(t)w(t) ) -x(t + I)'K(t + I)' + ~ ( x K(t + l)x(t + 1) +e(t+l)'R(t+I )e(t+l ) r-l
+C(t + I)' R(t + I)yo(t + 1), 77(0) = Xoxo, g(t+ 1) = g( t) + Yo (t + 1)' R( t + 1)yo (t + 1) -77(t)' B(t) x [B(t)'S(t)B(t) g(O ) = x~Xoxo .
£
~
(8) where e(t+1) = Yo(t+1 ) -C(t+1)x(t+1) and x(·) is the solution to (1) with input w( ·) and boundary condition x(T) Xr .
+ Q(t)t B(t)'77(t) ,
=
(6) Now suppose the uncertain system (1 ), (2) is robustly observable and consider the functional (8) with Xo = and Yot-) == 0. In this case, J is a homogeneous quadratic functional with a terminal cost term . Also, consider the set Xr[O , 0, 1] corresponding to Xo = 0, Yo (·) == and d = 1. Since Xr [0, 0,1] is bounded. there exists a constant hr > such that all vectors Xr E Rn with !lxr!l = hr do not belong to the set Xr[O , O, 1]. Hence, J[xr ,w(.) ] > 1 for all Xr E Rn such that !lxr!l = hr and for all w( ·). Since, J is a homogeneous quadratic functional , we have J[axr , aw( ·)] = a2 J[xr , w(') ] and the condition, J[xr , w( ·)] > 1 for II xr " = hr , implies that m( xr) > for all Xr i= 0 where
Note that if the matrix B(t)'S(t)B(t) + Q(t) is positive-definite, then condition (5) holds automatically and the pseudo-inverse in (4) and (6) can be replaced by a normal matrix inverse. This situation will hold in almost all cases for which a suitable solution exists to Riccati equation (4).
°
°
°
Theorem 1 Consider the uncertain system (1), (2). Then the following statements hold: (i) The uncertain system (1). (2) is robustly observable if and only if there exists a solution to Riccati equation (4) satisfying condition (5) and S(T) > 0.
°
(ii ) Let Yo(t ), t = 1, 2, ... . T be a given output sequence and suppose the system (1) , (2) is robustly observable. Then. Yo (.) is realizable if and only if pr(Yo\ ') ) ~ -d where
m(xr ) £ infw(-) J[xr ,w(·) J. The optimization problem infw(- ) J[xr , w(·)] subject to the constraint defined by the system (1) is a linear quadratic optimal control problem in which time is reversed. In this linear quadratic optimal control problem, a sign indefinite quadratic cost function is being considered. Using a known result from linear quadratic optimal control theory, we conclude that the condition m( xr) > implies that there exists a solution to Riccati equation (4) satisfying condition (5) and S(T) > OJ e.g., see (Clements and Anderson 1978, Lewis 1986).
pr (Yo (' )) £ T/(T)'S (T)-iT/\T) - g(T) and
77(T ) and g(T ) are defined by the equations (6).
(iii) If the uncertain system (1). (2) is robustly observable, then
°
We have shown above that an output sequence Yo (.) is realizable if and only if there exists a vector Xr E Rn and an uncertainty input w(·) such that the condition J[xr , w(- )] ::; d holds. Now with J defined as in (8) , consider the optimization problem inf w(.) Jlxr , w(· )J. This problem is a linear quadratic optimal tracking problem in which the system operates in reverse time. In fact , the only difference between this tracking problem and the tracking problem considered in (Lewis 1986 ) is that in this paper, we have a sign indefinite quadratic cost function and time is reversed. The solution to this tracking problem is well known {e.g. ~ ce ( Lewis 1!J86)) . Indeed. if there exists il. solution t.o Riccati equation (4) satisfying (5) .1.nd S(T) > 0, t.hen the infimum inf",(.) J [XT , w(·) ]
(7)
Proof Given an output sequence YO( ') , we have by the definition of Xr[xo,Yo( · )I~.d], that Xr E Xr[xO,Yo(')I; , d] if and only if there exist sequences x(-), w(·) and v(·) satisfying equation (I ) and such that x(T) = Xr , the constraint (2) holds. and Yo(t) = C(t)x(t) + Iltt). Substitution of t his into (2) implies that Xr E Xr lxo.]o \ ·)I; , dl if a.nd only if there exists an input sequence w( · I such t.hat J[xr.lL'I ·I ] :::; d. Here FXT . :L·I·1j is defined hy
291
will be achieved for any Xo and yo(·); e.g., see (Clements and Anderson 1978, Lewis 1986). Furthermore, as in (Lewis 1986), we can write
m(XT)
....
min J[XT, w( ·)]
.
w(·)
=
x~S(T)XT
- 2x~7](T)
+ g(T)
-""8':" !
(9) where [7](' ),g(')] is the solution to state equations (6). However, since S(T) > 0, it follows that the set XT[xo,yoOli,d] = {XT : m(xT). S d} is bounded. Since, Xo, YO(-) and d were arbItrary, it follows that the uncertain system must be robustly observable. Conversely, we have proved above that if the uncertain system (1), (2) is robustly observable, then there exists a solution to Riccati equation (4) satisfying (5) and S(T) > O. Thus , we have established statement (i). Now suppose the system is robustly observable. As above, it follows that there will exist a solution to Riccati equation (4) satisfying (5) and S(T) > O. Also , it is clear that the output Yo (.) is realizable if and only if there exists a vector XT ERn such that m( XT) S d. This and (9) imply that the realizability of yo( ·) is equivalent to the condition PT (yo (.)) 2: -d . Thus we have established statement (ii). Also, we have shown above that XT[xO ,Yo( ' )li , d] is the set of all XT E Rn such that m( XT) S d. Then from (9), condition (7) follows immediately. Thus. we have proved statement (iii). 0
-Fig. 1. Robust Deconvolution System. where 10llx(0)II2+ L;=~I nI (t)2+ L;=l n2(t)2 S 1 and II[~l(t)' ~2(t)']II S 1. In this state space description, Xl (t) and X2 (t) are the state variables of the signal model. The required signal u( t) corresponds to Xl(t) . Also, X3(t) is the state variable of the channel model. We consider this system over a finite time interval of T = 100 samples. To apply our results to this deconvolution problem. we consider a corresponding uncertain system of the form (1) in which the uncertainty satisfies the sum quadratic constraint. In this case, the matrices A.. X o, B, Xo, K, C , Q and R are given by
A
=
[ 1.9~ 0.4
[ 4. ILLUSTRATIVE EXAMPLE
B
To illustrate the results of this paper, we consider a robust deconvolution problem similar to those considered in (Chen and Chen 1994). However, we have a different uncertainty description than was considered in (Chen and Chen 1994). A block diagram of the system under consideration is shown if Figure 1. In this robust deconvolution problem, the uncertain parameter ~I(t) repr~sents the uncertainty in the signal model natural frequency and the uncertain parameter ~2(t) represents an uncertain direct feed through in the channel. Combining the signal model and the channel model, we obtain the following uncertain system of the form (3) :
X(t+ 1)
=
[
+ y( t)
1.987.:.l!
~1
0.4
0
=
C K
0
~O ~O ~
o
0
~]
;
0.2
];
10
U], x·=[n ' [001];
=
Q =
[0.018 0 0]; 1 and R
= 1.
To illustrate the performance of our state estimator, we consider the uncertainties and noise signals to be such that ~1 (t) == 1, 6 2 (t) == 0, nl (0) = 0.5. nl(t) = 0 for t = 1,2, .. . ,100, and n2(t"1 = _1 sin( t/l0) for t = 1,2, ... ,100. With the ini105 . h' tial condition X ( 0 ) = 0, . It .IS stralg t Iorwara. to verify that the uncertainty input sequences U ' o t I = 6 1 Kx(t) + ndt) and v(t) = 6 zKx(t) -:- n2(t satisfy the integral quadratic constraint (21. We :lOW apply our state estimator to the linear system corresponding to this uncertainty realization. F:gure 2 shows the resulting estimate of the signal :':1 tI. upper and lower bounds on u( t) and the true \-aiue of u(t) for t = 1.2, . ... 100. The estimated ';aiue of the state vector corresponds to the center oi the
[nr 1
6 2x dt)
-~
+ I3(t) + n2(t) 292
tering. Prentice HalL Englewood Cliffs, N.J. Bertsekas, D. P. and I. B. Rhodes (1971). 'Recursive state estimation for a set-membership description of uncertainty'. IEEE Transactions on Automatic Control 16(2), 117-128. Chen, Y. and B.S. Chen (1994). 'Minimax robust deconvolution filters under stochastic parametric and noise uncertainties' . IEEE Transactions on Signal Processing 42(1), 32-45. Clements, D. J. and E. D. 0 Anderson (1978). Singular Optimal Control: The LinearQuadratic Problem. Springer-Verlag. Berlin. Lewis, F. L. (1986). Optimal Control. Wiley. New York. Peng, S. and E.S. Chen (1994). ' A deconvolution filter for multichannel nonminimum phase systems via the minimax approach '. Signal Processing 36(1),71-90. Theodor, Y., U. Shaked and C.E. de Souza (1994). 'A game theory approach to robust discretetime Hoc-estimation'. IEEE Transactions on Signal Processing 42(6), 1486-1495.
- ... - ... .. ,
'. 20
'.
, ./
"',
.'
'. '.
-'0 trw yalue 01 u(t)
-
-20
- - ~va"'oIu(t) ._ .• _ bound on u('1
.. .. . ioItIwet bound on u(t)
-300
'0
20
30
40
50
60
70
SO
lIO
'00
lime srep
Fig. 2. Estimated value of u(t) with ~1
=L
ellipsoid of possible states described by equation (7). Indeed, referring to equation (7), the state estimate at time t is given by x( t) = S( t) -11]( t). In this example, the required estimate of the signal u( t) corresponds to the first component of this estimated value of the state vector. Also, the upper and lower bounds on the signal u(t) are obtained by projecting the ellipsoidal set of possible values of the state vector onto its first component. In addition to the uncertainty realization described above we also considered another uncertainty realization in which the value of ~1 was replaced by ~1 = - L Apart from this change, which corresponds to a change in the natural frequency of the signal model, the uncertainty realization is the same as above. Figure 3 shows the corresponding simulations for this case. Note that in both cases, a good estimate is obtained for the actual signal u( t) in spite of large uncertainty in the signal modeL
-.".._010(11 - - • _ _ 01001 · _ · · _ _ ndooo(l)
10
" ' _ _ 0011(1)
, ~
.
li>
,- ,
,.
,
.
,.
'
~'
,
•
"'WV -5
m
20
30
~
~
60
M
SO
~
m
lime SlIP
Fig. 3. Estimated value of uln with
~I
=- L
REFERE~CES
Andcrson. B. and J:B. \loore I (979) . Optimal Fil·
293
ROBUST STATE ESTIMATION FOR DISCRETE-TIME UNCERTAIN SYSTEMS WITH A DETERMINISTIC DESCRIPTION OF NOISE AND UNCERTAINTY
A. V. SAVKIW and 1. R. PETERSEW' • Department of Electrical and Electronic Engineering and Cooperative Research Center for Sensor Signal and Information Processing, University of Melbourne, Parkville, Victoria 3052, Australia, email: [email protected]. "Department of Electrical Engineering, Australian Defence Force Academy, Campbell, 2600, Australia, email: [email protected].
Abstract. The paper presents a new approach to robust state estimation for a class of uncertain discrete-time systems with a deterministic description of noise and uncertainty. The main result is a recursive scheme for constructing an ellipsoidal state estimation set of all states consistent with the measured output and the given noise and uncertainty description. The paper also includes a result on model validation whereby it can be determined if the assumed model is consistant with measured data. Keywords. Signal Processing Algorithms; Robust Estimation; Kalman Filters; Modelling; Validation; Uncertain Dynamical Systems; Riccati Equations
1. INTRODUCTION
new uncertainty description is referred to as a Sum Quadratic Constraint. It ailows for a large class of nonlinear, dynamic uncertainties. Furthermore, our main result shows that for this uncertainty description , our robust filter can be used to determine if the assumed model is consistant with the given output measurements. Such m C) d~l validation results cannot be obtained with a stochastic description of noise and uncertainty.
The Kalman Filter has seen wide application in areas such as deconvolution, channel equalization, FM demodulation , radar tracking, inertial navigation and global positioning systems ; e.g., see (Anderson and Moore 1979. Peng and Chen 1994, Chen and Chen 1994) and the references therein. The Kalman filter gives an optimal estimate of the state of a given process based on output measurements. However, application of the standard Kalman Filter requires an accurate model of the process under consideration and if there are errors in the process model, the Kalman Filter may lead to poor performance; e.g., see (Theodor et al. 1994). In this paper, we present a new method for robust state estimation for the case in which the process model contains uncertainties and nonlinearities. The starting point for our approach is the deterministic interpretation of the discrete-time Kalman Filter given in (Bertsekas and Rhodes 1971). In (Bertsekas and Rhodes 1971), the Kalman Filter is shown to give a state estimate in the form oi an ellipsoidal set of all possible states consistent with the given process measurements and a deterministic description of the noise. We extend the approach of (Bertsekas and Rhodes 1971) to consider discrete-time uncertain process models which have a determin ist description of the noise and l1ncertainty. This
Our main result is a state estimator which gives a state estimate in the form of an ellipsoidal set of all possible states consistent with the given process measurements and our deterministic description of the noise and uncertainty. Our results give an exact characterization d this set. This is opposed to existing results on robust filtering such as (Theodor et al. 1994) which minimize only an upper bound on the estimation error for the problems they consider. As in the standard Kalman Filter, our state estimate is constructed recursively from the output measurements. However, the form of our state estimator equations are somewhat different from the standard Kalman Filter equations. In addition , we present an illustrative example which shows how our results can be applied to a robust deconvolution problem with uncertainty in the signal model.
289
Then condition (2) is satisfied.
2. PROBLEM STATEMENT Consider the time-varying uncertain discrete-time system defined for t = 0,1, ... ,T:
x(t + 1) z(t) y(t)
= = =
A(t)x(t) + B(t)w(t); K(t)x(t); C(t)x(t) + v(t)
(1)
where x(t) E Rn is the state, w(t) E RP and E RI are the uncertainty inputs, z(t) E Rq is the uncertainty output, y(t) E RI is the measured output, and A(t), B(t), K(t) and C(t) are given matrices such that A( t) is non-singular for t = 0, 1, ... ,T.
v(t)
State Estimation Problem The main result of this paper concerns the following state estimation problem. Let y(t) = Yo(t) be a fixed measured output of the uncertain system (1), (2) for t = 1,2, . .. , T. Then, find the corresponding set XT[xo,yo(-)I[,d] of all possible states x(T) at time T for the system (1) with uncertainty inputs and initial conditions satisfying the constraint (:?). Notation The system (1) , (2) is said to be robustly observable if the set XT[xo , Yo (.)1;' dj is bounded for any xo, any yo(-) and any d. Let Yo(t), t = 1,2, ... , T be a given output sequence. The output Yo (.) is said to be realizable if there e..~ ist sequences [x{·), w{ ·), v(·)] satisfying conditions (1), (2) with y{t) = Yo(t).
System Uncertainty The uncertainty in the above system is described by an equation of the form: [w(t) v(t)] = 4>(t,x(')I~) where the following Sum Quadratic Constraint is satisfied. Let Xo = X~ > 0 be a given matrix, Xo ERn be a given vector, d > 0 be a given constant, and Q( t) and R( t) be given positive-definite symmetric matrices defined for t = 0,1. . . . , T . Then we will consider the uncertainty inputs w(·) and v(·) and initial conditions x(O) such that
T-l l:)w(t)'Q(t)w(t) + v(t + I)' R(t + l)v(t + 1))
In addition to solving the state estimation problem mentioned above, the main result of this paper also solves the following problem: Given an output sequence yo(-), determine if this output is realizable for the uncertain system (1), (2). If a given measured output sequence is not realizable for the given uncertain system model, we can say that this model is invalidated by the measured data. Thus, the results of this paper are useful in the question of model validation.
t=o
<
+(x(O) - xo)'Xo(x(O) - xo) T-l d+ IIz(t + 1)112.
L
3. THE MAIN RESULT
(2) Our solution to the above state estimation ;)foblem involves the following Riccati difference equation:
t=o
Here 11 . 11 denotes the standard Euclidean norm . The uncertain system (I), (2) allows for uncertainty satisfying a standard norm bound constraint. In this case, the uncertain system would be described by the state equations
x(t + 1) y(t)
=
F(t + 1)
=
S(t + 1)
[A(t) + B(t)~l(t)K(t)]x(t) +B(t)nl(t); [C(t) + 6 2(t)K(t)]x(t) +n2(t);
=
B(t)'S(t)B(t) + Q(t) 2: 0 & N(B(t)'S(t)B(t) + Q(t)) c N(A(t)'S(t)Blt))
S
d. To verify that such uncertainty is adm issible for the uncertain system (11. (:?), let w(t) ~l (t)K(t)x(t) + ndt) for t 0.1, ... , T and v(t) = 6 2 (t)K(t)x(t) + n2(t) for t = 1.2, . .. , T where II[6 1 (t)'Q(t)! 6 2 (t)'R(t)!111 S 1 for all t.
=
1)] 1,4)
where A(t) ~ A(t)-r, B(t) ~ A,(t)-l B(t) and (.)# denotes the Moore-Penrose pseudo-inverse: e.g., see (Anderson and Moore 1979). Solutions to this Riccati equation will he required to satisfy the following condition:
where 6dt) and 6 2 (t) are uncertainty matrices, nl (t) and n2 (t) are noise sequences, and 11 . 11 denotes the standard induced matrix norm. Also, the initial conditions and noise sequences would be required to satisfy the inequality (x(O) - xo)'Xo(x(O) - xo) +
'£:=1 n2(t)'R(t)n2(t)
A(t)'S(t) [A(t) - B(t)F(t+
+C(t + I)' R(t + 1)C(t + 1) -K(t + 1)'K(t + 1), 5(0) = Xo
II[6 1 (t)'Q(t)! 6 2 (t)'R(t)t]II S 1 (3)
,£:=~l nl(t)'Q(t)nl(t) +
[B(t)'S(t)B{t) + Q(t)r B(t)'5(tU.(t)
=
.) )
=
for t 1. 2 .. .. , T . Here XI .) denotes the operation of taking the null space of a matrix. Also, we consider a set of state equations oi ~ he
290
form
77(t + 1) = [A(t) -
J[xr, w(·)]
~(t)F(t+ 1)]' 77(t)
(x(O ) - xo)' Xo(x(O) - xo)
w(t)'Q(t)w(t) ) -x(t + I)'K(t + I)' + ~ ( x K(t + l)x(t + 1) +e(t+l)'R(t+I )e(t+l ) r-l
+C(t + I)' R(t + I)yo(t + 1), 77(0) = Xoxo, g(t+ 1) = g( t) + Yo (t + 1)' R( t + 1)yo (t + 1) -77(t)' B(t) x [B(t)'S(t)B(t) g(O ) = x~Xoxo .
£
~
(8) where e(t+1) = Yo(t+1 ) -C(t+1)x(t+1) and x(·) is the solution to (1) with input w( ·) and boundary condition x(T) Xr .
+ Q(t)t B(t)'77(t) ,
=
(6) Now suppose the uncertain system (1 ), (2) is robustly observable and consider the functional (8) with Xo = and Yot-) == 0. In this case, J is a homogeneous quadratic functional with a terminal cost term . Also, consider the set Xr[O , 0, 1] corresponding to Xo = 0, Yo (·) == and d = 1. Since Xr [0, 0,1] is bounded. there exists a constant hr > such that all vectors Xr E Rn with !lxr!l = hr do not belong to the set Xr[O , O, 1]. Hence, J[xr ,w(.) ] > 1 for all Xr E Rn such that !lxr!l = hr and for all w( ·). Since, J is a homogeneous quadratic functional , we have J[axr , aw( ·)] = a2 J[xr , w(') ] and the condition, J[xr , w( ·)] > 1 for II xr " = hr , implies that m( xr) > for all Xr i= 0 where
Note that if the matrix B(t)'S(t)B(t) + Q(t) is positive-definite, then condition (5) holds automatically and the pseudo-inverse in (4) and (6) can be replaced by a normal matrix inverse. This situation will hold in almost all cases for which a suitable solution exists to Riccati equation (4).
°
°
°
Theorem 1 Consider the uncertain system (1), (2). Then the following statements hold: (i) The uncertain system (1). (2) is robustly observable if and only if there exists a solution to Riccati equation (4) satisfying condition (5) and S(T) > 0.
°
(ii ) Let Yo(t ), t = 1, 2, ... . T be a given output sequence and suppose the system (1) , (2) is robustly observable. Then. Yo (.) is realizable if and only if pr(Yo\ ') ) ~ -d where
m(xr ) £ infw(-) J[xr ,w(·) J. The optimization problem infw(- ) J[xr , w(·)] subject to the constraint defined by the system (1) is a linear quadratic optimal control problem in which time is reversed. In this linear quadratic optimal control problem, a sign indefinite quadratic cost function is being considered. Using a known result from linear quadratic optimal control theory, we conclude that the condition m( xr) > implies that there exists a solution to Riccati equation (4) satisfying condition (5) and S(T) > OJ e.g., see (Clements and Anderson 1978, Lewis 1986).
pr (Yo (' )) £ T/(T)'S (T)-iT/\T) - g(T) and
77(T ) and g(T ) are defined by the equations (6).
(iii) If the uncertain system (1). (2) is robustly observable, then
°
We have shown above that an output sequence Yo (.) is realizable if and only if there exists a vector Xr E Rn and an uncertainty input w(·) such that the condition J[xr , w(- )] ::; d holds. Now with J defined as in (8) , consider the optimization problem inf w(.) Jlxr , w(· )J. This problem is a linear quadratic optimal tracking problem in which the system operates in reverse time. In fact , the only difference between this tracking problem and the tracking problem considered in (Lewis 1986 ) is that in this paper, we have a sign indefinite quadratic cost function and time is reversed. The solution to this tracking problem is well known {e.g. ~ ce ( Lewis 1!J86)) . Indeed. if there exists il. solution t.o Riccati equation (4) satisfying (5) .1.nd S(T) > 0, t.hen the infimum inf",(.) J [XT , w(·) ]
(7)
Proof Given an output sequence YO( ') , we have by the definition of Xr[xo,Yo( · )I~.d], that Xr E Xr[xO,Yo(')I; , d] if and only if there exist sequences x(-), w(·) and v(·) satisfying equation (I ) and such that x(T) = Xr , the constraint (2) holds. and Yo(t) = C(t)x(t) + Iltt). Substitution of t his into (2) implies that Xr E Xr lxo.]o \ ·)I; , dl if a.nd only if there exists an input sequence w( · I such t.hat J[xr.lL'I ·I ] :::; d. Here FXT . :L·I·1j is defined hy
291
will be achieved for any Xo and yo(·); e.g., see (Clements and Anderson 1978, Lewis 1986). Furthermore, as in (Lewis 1986), we can write
m(XT)
....
min J[XT, w( ·)]
.
w(·)
=
x~S(T)XT
- 2x~7](T)
+ g(T)
-""8':" !
(9) where [7](' ),g(')] is the solution to state equations (6). However, since S(T) > 0, it follows that the set XT[xo,yoOli,d] = {XT : m(xT). S d} is bounded. Since, Xo, YO(-) and d were arbItrary, it follows that the uncertain system must be robustly observable. Conversely, we have proved above that if the uncertain system (1), (2) is robustly observable, then there exists a solution to Riccati equation (4) satisfying (5) and S(T) > O. Thus , we have established statement (i). Now suppose the system is robustly observable. As above, it follows that there will exist a solution to Riccati equation (4) satisfying (5) and S(T) > O. Also , it is clear that the output Yo (.) is realizable if and only if there exists a vector XT ERn such that m( XT) S d. This and (9) imply that the realizability of yo( ·) is equivalent to the condition PT (yo (.)) 2: -d . Thus we have established statement (ii). Also, we have shown above that XT[xO ,Yo( ' )li , d] is the set of all XT E Rn such that m( XT) S d. Then from (9), condition (7) follows immediately. Thus. we have proved statement (iii). 0
-Fig. 1. Robust Deconvolution System. where 10llx(0)II2+ L;=~I nI (t)2+ L;=l n2(t)2 S 1 and II[~l(t)' ~2(t)']II S 1. In this state space description, Xl (t) and X2 (t) are the state variables of the signal model. The required signal u( t) corresponds to Xl(t) . Also, X3(t) is the state variable of the channel model. We consider this system over a finite time interval of T = 100 samples. To apply our results to this deconvolution problem. we consider a corresponding uncertain system of the form (1) in which the uncertainty satisfies the sum quadratic constraint. In this case, the matrices A.. X o, B, Xo, K, C , Q and R are given by
A
=
[ 1.9~ 0.4
[ 4. ILLUSTRATIVE EXAMPLE
B
To illustrate the results of this paper, we consider a robust deconvolution problem similar to those considered in (Chen and Chen 1994). However, we have a different uncertainty description than was considered in (Chen and Chen 1994). A block diagram of the system under consideration is shown if Figure 1. In this robust deconvolution problem, the uncertain parameter ~I(t) repr~sents the uncertainty in the signal model natural frequency and the uncertain parameter ~2(t) represents an uncertain direct feed through in the channel. Combining the signal model and the channel model, we obtain the following uncertain system of the form (3) :
X(t+ 1)
=
[
+ y( t)
1.987.:.l!
~1
0.4
0
=
C K
0
~O ~O ~
o
0
~]
;
0.2
];
10
U], x·=[n ' [001];
=
Q =
[0.018 0 0]; 1 and R
= 1.
To illustrate the performance of our state estimator, we consider the uncertainties and noise signals to be such that ~1 (t) == 1, 6 2 (t) == 0, nl (0) = 0.5. nl(t) = 0 for t = 1,2, .. . ,100, and n2(t"1 = _1 sin( t/l0) for t = 1,2, ... ,100. With the ini105 . h' tial condition X ( 0 ) = 0, . It .IS stralg t Iorwara. to verify that the uncertainty input sequences U ' o t I = 6 1 Kx(t) + ndt) and v(t) = 6 zKx(t) -:- n2(t satisfy the integral quadratic constraint (21. We :lOW apply our state estimator to the linear system corresponding to this uncertainty realization. F:gure 2 shows the resulting estimate of the signal :':1 tI. upper and lower bounds on u( t) and the true \-aiue of u(t) for t = 1.2, . ... 100. The estimated ';aiue of the state vector corresponds to the center oi the
[nr 1
6 2x dt)
-~
+ I3(t) + n2(t) 292
tering. Prentice HalL Englewood Cliffs, N.J. Bertsekas, D. P. and I. B. Rhodes (1971). 'Recursive state estimation for a set-membership description of uncertainty'. IEEE Transactions on Automatic Control 16(2), 117-128. Chen, Y. and B.S. Chen (1994). 'Minimax robust deconvolution filters under stochastic parametric and noise uncertainties' . IEEE Transactions on Signal Processing 42(1), 32-45. Clements, D. J. and E. D. 0 Anderson (1978). Singular Optimal Control: The LinearQuadratic Problem. Springer-Verlag. Berlin. Lewis, F. L. (1986). Optimal Control. Wiley. New York. Peng, S. and E.S. Chen (1994). ' A deconvolution filter for multichannel nonminimum phase systems via the minimax approach '. Signal Processing 36(1),71-90. Theodor, Y., U. Shaked and C.E. de Souza (1994). 'A game theory approach to robust discretetime Hoc-estimation'. IEEE Transactions on Signal Processing 42(6), 1486-1495.
- ... - ... .. ,
'. 20
'.
, ./
"',
.'
'. '.
-'0 trw yalue 01 u(t)
-
-20
- - ~va"'oIu(t) ._ .• _ bound on u('1
.. .. . ioItIwet bound on u(t)
-300
'0
20
30
40
50
60
70
SO
lIO
'00
lime srep
Fig. 2. Estimated value of u(t) with ~1
=L
ellipsoid of possible states described by equation (7). Indeed, referring to equation (7), the state estimate at time t is given by x( t) = S( t) -11]( t). In this example, the required estimate of the signal u( t) corresponds to the first component of this estimated value of the state vector. Also, the upper and lower bounds on the signal u(t) are obtained by projecting the ellipsoidal set of possible values of the state vector onto its first component. In addition to the uncertainty realization described above we also considered another uncertainty realization in which the value of ~1 was replaced by ~1 = - L Apart from this change, which corresponds to a change in the natural frequency of the signal model, the uncertainty realization is the same as above. Figure 3 shows the corresponding simulations for this case. Note that in both cases, a good estimate is obtained for the actual signal u( t) in spite of large uncertainty in the signal modeL
-.".._010(11 - - • _ _ 01001 · _ · · _ _ ndooo(l)
10
" ' _ _ 0011(1)
, ~
.
li>
,- ,
,.
,
.
,.
'
~'
,
•
"'WV -5
m
20
30
~
~
60
M
SO
~
m
lime SlIP
Fig. 3. Estimated value of uln with
~I
=- L
REFERE~CES
Andcrson. B. and J:B. \loore I (979) . Optimal Fil·
293