Systems & Control Letters 14 (1990) 409-410 409 North-Holland

On a fallacious conjecture about the stabilizability properties of solutions of the Riccati difference equation

G i u s e p p e D e N I C O L A O

Centro Teoria dei Sistemi-C.N.R., c/o Dipartimento di Elet- tronica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Received 3 January 1990 Revised 17 February 1990

Abstract: In [1], among other results, some conjectures concern- ing the monotonicity and stabilizing properties of solutions of the difference Riccati equation were proven to be fallacious by means of suitable counterexamples. Moreover, a 'possibly fal- lacious conjecture' was formulated for which no counterexam- pie had been found. In this letter, such a counterexample is provided together with an interpretation of the somewhat counterintuitive behaviour of the Riccati equation.

Keywords: Riccati difference equation; Kalman filtering; stabilizability; fake Riccati equation.

Consider the discrete-t ime stochastic l inear sys- tem

x ( t + 1) = F x ( t ) + w ( t ) ,

y ( t ) = H x ( t ) + v ( t ) ,

where w( t ) and v( t ) are independent white noises with variance Q > 0 and R > 0, respectively. The associated K a l m a n filter is

x ( t + l i t ) = F x ( t l t - 1)

+ K ( t ) [ y ( t ) - H x ( t [ t - 1)] ,

K ( t ) = F P ( t ) H T [ H P ( t ) H T + R] -1

where P( t ) satisfies the Riccati Difference Equa- t ion (RDE)

P ( t + 1) = F P ( t ) F T

- F P ( t ) H T [ H P ( t ) H T + R] -1

" H P ( t ) F T + Q ,

with initial condi t ion P0. The cons tant solutions

.5

°:[i Then

P( t ) = P of the R D E satisfy the Algebraic Riccati Equat ion (ARE)

P = F P F T - F P H T ( H P H T + R ) - a H P F T + Q.

(1)

In the sequel, P + will denote the stabilizing solu- t ion of the ARE, which we assume to exist, i.e. P+ is a real symmet r ic normegat ive solution of (1) such that all the eigenvalues of the associated closed loop mat r ix L + = F - K + H belong to the open unit disk. In the context of the so-called Fake Riccat i Equa t ion [2] analysis, the ' f rozen ' predic tor associated with the gain K( t ) is consid- ered. Precisely, P ( t ) is said to be stabilizing (for the frozen predic tor at t) if the eigenvalues of L ( t ) = F - K ( t ) H belong to the open unit disk.

In [1] the following conjecture was stated as a possibly fallacious one, but no counterexample was provided.

Conjecture. Suppose P0 > P+ and P0 and P+ are bo th stabilizing. Then { P ( t ) } is stabilizing.

Counterexample. Let R = 1 and

0.5 0 , 0 2

0 , H T = •

0

o - ~ ] P + = 0 0 0 ] ,

-~ 0 ~ + 2 ¢ ~ is the stabilizing solution. Taking

[o o 1 3 0 0

Po = 20 0 > P + . 0 12

016%6911/90/$3.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

410 D. De Nicolao / A fallacious conjecture on solutions of RDE

it turns out that

[0.0938]

=[ 0. 06 1 _0.093s 1 0 , Lo 0 0.5

- 1 . 5 0 0.5

and L o is asymptotically stable. At the next step we have

• 21.6094 10 -2 .25 ] P ( 1 ) = 10 5 0 1,

- 2 . 25 0 13

0.6326 ] K ( 1 ) = 0.1607 ,]

0.6911

and, even though P ( 1 ) > P+, L(1) is no longer stable since it has an eigenvalue equal to 1.5961.

Remark. The 'strange' behaviour of the RDE in the above example can be roughly interpreted as follows. Let

e i ( t ) = x i ( t ) - x i ( t l t - 1)

be the prediction error relative to the i-th state variable. When predicting x3(t + 1), the Kalman filter uses the output

y ( t ) = x l ( t ) + x3(t ) + o ( t )

= x a ( t l t - 1) + e l ( t ) + x 3 ( t l t - 1)

+ e 3 ( t ) + v ( t )

to correct the open-loop prediction based on x 3 ( t l t - 1). Therefore, as long as the correlation between the prediction errors e l ( t ) and e3(t ) is

negligible, e~(t) acts as a further disturbance, whose variance is kept into account by suitably changing the value of K3(t ). More precisely, a higher variance ratio between el( t ) and e3(t ) leads to smaller values of K3(t ) to reflect the di- minished reliability of the measurement. Now, because of the high initial uncertainty on x2(0), which propagates to e1(1), the variance ratio is much greater at time 1 than at time 0. As a consequence, K3(1 ) is smaller than K3(0 ) in such a way that the Kalman gain K(1) is no longer able to stabilize the unstable eigenvalue of F. In other words, as far as the prediction of x3(2 ) is con- cerned, the Kalman filter does not trust enough the measurement at time t = 1 and cannot do better than operating in a 'quasi-open-loop' mode while waiting for better information.

Acknowledgement

Research for this paper has been supported by the Centro di Teoria dei Sistemi of the Italian national Research Council (CNR) and by the Ministry of University and Scientific Research (MURS).

References

[1] R.R. Bitmead, M.R. Gevers, I.R. Petersen and R.J. Kaye, Monotonicity and stabilizability properties of solutions of the Riccati difference equation: Propositions, lemmas, the- orems, fallacious conjectures and counterexamples, Sys- tems Control Lett. 5 (1985) 309-315.

[2] M.A. Poubelle, R.R. Bitmead and M.R. Gevers, Fake alge- braic Riccati techniques and stability, IEEE Trans. Auto- mat. Control 33 (1988) 379-381.