A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding

Set-membership estimation is usually formulated in the context of set-valued calculus and no probabilistic calculations are necessary. In this paper, we show that set-membership estimation can be equivalently formulated in the probabilistic setting by employing sets of probability measures. Inference in set-membership estimation is thus carried out by computing expectations with respect to the updated set of probability measures P as in the probabilistic case. In particular, it is shown that inference can be performed by solving a particular semi-infinite linear programming problem, which is a special case of the truncated moment problem in which only the zero-th order moment is known (i.e., the support). By writing the dual of the above semi-infinite linear programming problem, it is shown that, if the nonlinearities in the measurement and process equations are polynomial and if the bounding sets for initial state, process and measurement noises are described by polynomial inequalities, then an approximation of this semi-infinite linear programming problem can efficiently be obtained by using the theory of sum-of-squares polynomial optimization. We then derive a smart greedy procedure to compute a polytopic outer-approximation of the true membership-set, by computing the minimum-volume polytope that outer-bounds the set that includes all the means computed with respect to P

Joint waveform and guidance control optimisation for target rendezvous

The algorithm developed in this paper jointly selects the optimal transmitted waveform and the control input so that a radar sensor on a moving platform with linear dynamics can reach a target by minimising a predefined cost. The cost proposed in this paper accounts for the energy of the transmitted radar signal, the energy of the platform control input and the relative position error between the platform and the target, which is a function of the waveform design and control input. Similarly to the Linear Quadratic Gaussian (LQG) control problem, we demonstrate that the optimal solution satisfies the separation principle between filtering and optimisation and, therefore, the optimum can be found analytically. The performance of the proposed solution is assessed with a set of simulations for a pulsed Doppler radar transmitting linearly frequency modulated chirps. Results show the effectiveness of the proposed approach for optimal waveform design and optimal guidance control

Coordination of optimal guidance law and adaptive radiated waveform for interception and rendezvous problems

The authors present an algorithm that allows an interceptor aircraft equipped with an airborne radar to meet another air target (the intercepted) by developing a guidance law and automatically adapting and optimising the transmitted waveform on a pulse-to-pulse basis. The algorithm uses a Kalman filter to predict the relative position and speed of the interceptor with respect to the target. The transmitted waveform is automatically selected based on its ambiguity function and accuracy properties along the approaching path. For each pulse, the interceptor predicts its position and velocity with respect to the target, takes a measurement of range and radial velocity and, with the Kalman filter, refines the relative range and range rate estimates. These are fed into a linear quadratic Gaussian controller that ensures the interceptor reaches the target automatically and successfully with minimum error and with the minimum guidance energy consumption