Design of Optimal UKF State Observer–Controller for Stochastic Dynamical Systems

"Stochastic noise processes" passed through highly nonlinear systems, always pose a significant threat to the industrial plant's stability. A novel generalized optimal “unscented Kalman filter state observer-controller” (UKFOC) algorithm is presented to control these plants effectively and efficiently. The proposed optimal UKFOC provides state “estimation and control” simultaneously, omitting the system's need for a separate controller. The “trajectory exit probability” from the desired boundaries is minimized based on the large deviation principle with the bounded instantaneous “trajectory tracking error” and the “state tracking error.” These boundaries are rationally computed from the error statistics. The convergence and robustness are realized in terms of the error energy under the influence of the noise and the small parametric uncertainties, respectively. The algorithm's superior performance is demonstrated with respect to Lyapunov control and adaptive Lyapunov control based techniques. Finally, the UKFOC is implemented and tested for the commercially available Phantom Omni robot to demonstrate the potential application on a real-time basis.

I. Introduction

Stable tracking control under the effect of noise is a prerequisite for most of the industrial robotic applications. Broadly, the tracking is implemented using both controllers and observers. The estimation by observers is widely done via filtering methods and model estimation mechanisms. For example, the Kalman filter, extended Kalman filter (EKF), unscented Kalman filter (UKF), and particle filter form filter-based methods, whereas, unknown input observer in disturbance accommodation control, disturbance observer, perturbation observer, extended state observer, equivalent input disturbance based estimator and uncertainty and disturbance estimator are some user-defined model-based observers [1]–[3]. The superiority of filter-based observers is proven because they are considered a part of the state variable of the nonlinear system, which results in high robustness compared to the other observers. Moreover, the derivative of the state in the model-based observer is always spiky, whereas, in filter-based observers, the inherent frequency smoothing is present. Although the observers have been mainly used to estimate the disturbances. After estimation, the control actions are applied to minimize the error between variables of interest. It should be noted that if the disturbance is known, it is a straightforward strategy to eliminate disturbance using the feedforward mechanism. However, it is not directly measurable and is a costly affair to measure it due to the use of various sensors for measurement. Furthermore, EKF is widely used in such applications where the disturbances are nonmeasurable. The EKF is applied to linearized dynamics of the system, assuming that the disturbance and unmodeled forces are Gaussian [27]. However, in stochastic nonlinear systems, the first-order linearization dilutes the effect of nonlinear action on the system during the modeling of the EKF recursive equations. To overcome this disadvantage of EKF, Sigma points EKF (SEKF) had been proposed, where the implementation of SEKF is far easier because there is no Jacobian calculation involved [5], [6]. However, in SEKF, weights are assigned to a large number of samples, which according to the central limit theorem (CLT), is an approximation to the normal distribution and hence deviates from the Gaussian distribution. On the other hand, in UKF, the sample points are propagated through the true nonlinear system, which in fact converges to true mean and covariance of the Gaussian variable in view of the strong law of large numbers. The linearized implementation of the UKF while retaining the true nonlinearity of the system is still underdeveloped and will be investigated.