Robust Estimation Algorithms      

The fundamental problem that we consider is posed in the framework of the following figure.

The plant consists of a "nominal" linear time-varying system, and is affected by multiplicative uncertainties that have a stochastic description. The aim is to devise algorithms that are implemented in the block labeled "Estimator", so as to "minimize" the estimation error. There are a number of criteria that can used to effect this minimization. Our work, focuses on two criteria, and leads to the design of two different estimation algorithms:

• Robust Adaptive and Steady-State Kalman Filters .
  Assuming that the nominal linear time-varying system parameters can be measured in real-time, and assuming that the initial covariance of the state vector is known only to lie in a polytope, we present a recursive algorithm, that at each step minimizes the mean square estimation error for white noise w. The filter that implements the algorithm has the one-step predictor-corrector structure, and at each step, the filter coefficients are obtained by solving convex optimization problem based on linear matrix inequalities.
  We also consider the behavior of the algorithm in steady-state. In particular, we show that the algorithm is converges when the system is mean square stable and the state-space matrices are time-invariant.

• Asymptotic H-infinity Optimal Filter.
  Suppose that the nominal system is time-invariant, and that the input w is a signal with energy bounded by one. We design a filter that minimizes the worst-case energy of the estimation error. (In the ideal condition that there is no uncertainty, this problem is a standard H-infinity filtering problem.)

s is the signal which is transmitted through the channel, w is the -10 dB white noise that corrupts the received signal y . The channel model is affected by time-varying uncertainties that are a combination of both deterministic and stochastic parametric uncertainties (see references for details). The following figures show the improvement obtained by our algorithm over the standard Kalman filter.

Moreover, our algorithms offer the potential to compare the steady-state performance of the robust Kalman filter with that of the the robust H-infinity filter; the respective performance measures are denoted gamma_2 and gamma_infinity in the following figure:

The following publications summarize our work:

• F. Wang and V. Balakrishnan, `` Robust Adaptive Kalman Filters for Linear Time-Varying Systems with Stochastic Parametric Uncertainties''. In Proc. IEEE American Control Conf., San Diego, CA, June 1999. Finalist, best student paper award .
• F. Wang and V. Balakrishnan, `` Robust Estimators for Systems with Both Deterministic and Stochastic Uncertainties''. In Proc. IEEE Conf. on Decision and Control, Phoenix, Arizona, December 1999.
• F. Wang and V. Balakrishnan, `` Robust Adaptive Kalman Filters for Linear Time-Varying Systems with Stochastic Parametric Uncertainties''. Submitted to the IEEE Trans. AC, September 1999.