IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 9, SEPTEMBER 2001 2103 Finite-Horizon Robust Kalman Filter Design Minyue Fu, Senior Member, IEEE, Carlos E. de Souza, Senior Member, IEEE, and Zhi-Quan (Tom) Luo, Member, IEEE Abstract—In this paper, we study the problem of finite-horizon Kalman filtering for systems involving a norm-bounded uncertain block. A new technique is presented for robust Kalman filter de- sign. This technique involves using multiple scaling parameters that can be optimized by solving a semidefinite program. The use of optimized scaling parameters leads to an improved design. A re- cursive design method that can be applied to real-time applications is also proposed. Index Terms—Adaptive filtering, Kalman filtering, robust filtering, robust signal processing. I. INTRODUCTION F INITE-horizon Kalman filters, including recursive least-squares filters as a special case, are widely used in signal processing applications. Compared with infinite-horizon Kalman filters, the finite-horizon ones can offer a better transient performance, which is an important property for applications where signals are nonstationary. One of the problems with Kalman filters, which has been well recognized now, is that they can be sensitive to system data, or in other words, they may lack robustness. A typical phenomenon is that the performance of the filter, although being optimal for a “nominal” system, may deteriorate very quickly as the system data drift; see, e.g., [4]. This is, of course, not acceptable for applications where a good system model is hard to obtain or the system drifts. Motivated by this problem, a number of pa- pers have attempted to generalize the classical Kalman filter to systems involving a norm-bounded uncertain block; see [2]–[6]. Note that norm-bounded blocks are used to represent inaccura- cies in the system model. The resulting filters are often called robust Kalman filters. The design of robust Kalman filters faces a major obstacle in comparison with the classical Kalman filters. There are two pre- vailing properties possessed by classical finite-horizon Kalman filters. First, an optimal filter at time leads to an optimal filter at . That is, an optimal filter at produces a minimum state estimation error at (in the variance sense), which is the best initial condition for the filter design at . Second, the op- timal filter for state estimation is also optimal for estimation of any other signal, provided it is a linear function of the state. Un- fortunately, neither of the two properties carries through when Manuscript received May 4, 1999; revised May 14, 2001. The associate editor coordinating the review of this paper and approving it for publication was Dr. Joseph M. Francos. M. Fu is with the Department of Electrical and Computer Engineering, the University of Newcastle, Callaghan, Australia (e-mail: eemf@ee.new- castle.edu.au). C. E. de Souza is with the Department of Systems and Control, Laboratório Nacional de Computação Científica—LNCC, Petrópolis, Brazil. Z.-Q. Luo is with the Department of Electrical and Computer Engineering, MacMaster University, Hamilton, ON L8S 4K1 Canada. Publisher Item Identifier S 1053-587X(01)07068-4. the system involves uncertainties. More precisely, a filter that produces a small state estimation error at time may worsen the state estimation at time . Similarly, a filter that minimizes the state estimation error may not be optimal for estimation of the signal of interest, even when it is a linear combination of the state. A commonly used technique for robust Kalman filter design is to apply the so-called S-Procedure, which replaces the un- certainty block with a scaling parameter. This yields an upper bound for the covariance of the estimation error. Two types of scaling parameters have been used: constant and time-varying. A constant scaling parameter ( ) is used in [3], [4], and [6] and is most suitable for infinite-horizon or stationary filtering prob- lems. One serious problem with using a constant scaling pa- rameter is that the entailed conservatism can aggregate quickly as time evolves and may lead to a very poor estimator. Time- varying scaling parameters ( ) are more flexible, and if they are carefully chosen, the amount of conservatism can be reduced. Two papers have used time-varying scaling parameters. In [5], a simple formula is given, but the scaling parameter is not opti- mized in any way. In [2], the scaling parameter is chosen using a semidefinite program. However, as we will reveal later, the scaling parameter obtained at time using [2] may lead to a poor estimation at future times. In addition, the semidefinite program to be solved in [2] is quite cumbersome. In this paper, we intend to carry out some deeper study on finite-horizon Kalman filtering for systems involving a norm- bounded uncertain block. Our focus will be on how to choose scaling parameters. A summary of our results is given as fol- lows. • We show that optimal scaling parameters for time may lead to poor estimation at future times. Subsequently, two types of scaling parameters are suggested: one optimal for time and one used for the future. In fact, at each time , all the scaling parameters need to be reoptimized. • The design of the estimator has the following separation properties. — The covariance of the estimation error at de- pends only on the scaling parameters and the system data and not on other parameters in the filter. Thus, the scaling parameters can be optimized first. In particular, we note that they depend on the signal to be estimated. — Once the scaling parameters are determined, an op- timal filter can be generated using an algebraic for- mula. In particular, we note that the optimal filter does not explicitly depend on the signal to be estimated. Im- plicit dependence happens only through the scaling pa- rameters. 1053–587X/01$10.00 © 2001 IEEE