Simultaneous state and input estimation with applications in vehicle problems
Engineering is about design of materials, structures, systems and interconnections to obtain a desired behaviour. In control engineering the focus is often on systems equipped with sensors whose output is used to provide feedback control and achieve the desired behaviour. A central paradigm in control is the separation principle, that is the optimal control action is achieved by applying the control law that is optimal under an assumption of full information to an “optimal estimate” of the system state. While in the context of linear-quadratic problems there is a well developed theory of optimal estimation and control, research on systems with inputs that are not measured directly is still ongoing. Motivated by automotive applications where it is not always feasible or practical to have sensors that measure all vehicle inputs, we aim to advance the theory on simultaneous state and input estimation and apply it to commercially important automotive examples. In particular, we formulate a deterministic estimation problem to find the input and state of a linear continuous time dynamical system which minimises a weighted integral squared error between the resulting output and the measured output. A completion of squares approach is used to find the unique optimum in terms of the solution of a Riccati differential equation. The optimal estimate is obtained from a two-stage procedure that is reminiscent of the Kalman filter. The first stage is an end-of-interval estimator for the finite horizon which may be solved in real time as the horizon length increases. The second stage computes the unique optimum over a fixed horizon by a backwards integration over the horizon. A related tracking problem is solved in an analogous manner. Making use of the solution to both the estimation and tracking problems a constrained estimation problem is solved which shows that the Riccati equation solution has a least squares interpretation that is analogous to the meaning of the covariance matrix in stochastic filtering. We show that the estimation and tracking problems considered here include the Kalman filter and the linear quadratic regulator as special cases. The infinite horizon case is also considered for both the estimation and tracking problems. Stability and convergence conditions are provided and the optimal solutions are shown to take the form of left inverses of the original system. Motivated by the intrinsically discrete nature of operation of modern computers and sensors, we then focus on systems in which the output is measured only at a discrete sequence of times. We derive two forms for the zero informational input limit for the discrete time Kalman filter in the case that there is direct feedthrough (of full column rank) of the input to the measurements. The first form is complementary to a zero informational limit filter derived recently by Bitmead, Hovd, and Abooshahab for the case where the first Markov parameter has full column rank and there is no direct feedthrough of the process noise. This form of the limit filter is closely related to a filter proposed by Gillijns and De Moor who used a constrained optimisation problem to estimate an unknown input in a standard Kalman filter with feedthrough of the unknown disturbance; more precisely, the filters coincide if the process noise covariance in Gillijns and De Moor is set to zero. A second form of the limit filter is derived from the first which takes the form of a standard Kalman filter without unknown inputs. This form is used to derive necessary and sufficient conditions for convergence and stability of the filter. These consist of a controllability condition and a minimum phase condition. We consider a deterministic estimation problem to find the input and state of a linear continuous time dynamical system with discrete time measurements which minimises a weighted sum squared error between the resulting output and the measured output. Similarly to the estimation problem with continuous time measurements we use a completion of squares approach to find the unique optimum. The optimal estimate is obtained in terms of the solution of a Riccati difference equation from a two-stage procedure that is reminiscent of the discrete time Kalman filter. The first stage is an end-of-interval estimator for the finite horizon which may be solved in real time as the horizon length increases. The second stage computes the unique optimum over a fixed horizon by a backwards recursion over the horizon. The infinite horizon case is also considered and stability conditions are provided. We apply the algorithm to two automotive examples, the first is on (offline) road elevation mapping and the second is on (online) slip estimation. In both examples we assume that the vehicle is equipped with basic sensors (e.g. accelerometers and gyroscopes (IMU), global positioning system (GPS)) and use very simple vehicle models.
estimation, least squares optimisation, optimal control
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