Tensor Formulation of Kalman Filter and Linear Quadratic Gaussian Controller for Applications on Multilinear Dynamical Systems

In this work, we generalise the popular Kalman filter and Linear Quadratic Gaussian controller for use on multi-sensor and multi-agent/-target radar systems. The state-space representation for the dynamical evolution of targets and the sensor measurements is developed here using tensors in place of vectors and matrices, producing a multilinear dynamical system. In this dynamical framework, the tensor forms of the Kalman filter and the Linear Quadratic Gaussian controller are developed, allowing the simultaneous processing of (i) the inputs of all sensors, producing the estimation of the state of all agents/targets and (ii) the determination of the optimal control actions of all agents/targets. These tools are applied to implement optimal parallel waveform design and tracking control for a multi-radar system acting on multiple agents. In the study case, examined numerically, the radars can (i) estimate the state of the agents in terms of range, angular displacement, radial and angular velocities and (ii) jointly determine the agents control inputs and the radars transmitted waveforms to minimise the control cost action and the energy of the transmitted signals.