Unknown state and non-matching disturbance estimations with hybrid sliding mode observers - Zhou, Yong (2013)
This thesis presents a new perspective on the observer design and analysis for a class of nonlinear uncertain systems in which the uncertainties enter the systems through unknown-state-dependent distribution vectors, i.e., the systems have non-matching uncertainties in the observer sense. The main idea lies with exploiting the observabilities of the unknown states and uncertainties from the measurable outputs. This allows us to design appropriate robust terms to asymptotically track the uncertainties and thereby estimating the unmeasurable states so that precise control and monitoring of control systems can be achieved. The first part of the thesis addresses the estimation problem of a single input single output (SISO) nonlinear Lipschitz system with the unknown input being non-matching in the observer sense. A hybrid observer that combines the high gain observer with a higher order sliding mode related nonlinear feedback term is proposed. For such a hybrid observer, the high gain feedback works to constrain the estimation error to within an invariant set regardless of the initial conditions, in which the sliding mode condition is satisfied. Then, the sliding mode feedback ensures that the sliding mode surface is reached in finite time and remained thereafter. As a result, the unknown input can be recovered from the sliding mode term after all states have converged to their true values. However, the identifiability of the unknown input is strictly related to the stability of the estimation dynamics on the sliding surface, which is only dependent on the structure of the nonlinear system and is difficult to be verified. As an application example, the application of the proposed results to the series DC motor which is widely used due to its high ratio of torque per ampere of current, especially in the industrial applications that require high starting torque, is studied. The non-matching flux related motor parameter of the series DC motor is time-varying because of magnetic saturation or imperfect manufacturing. Together with the effect of an unknown external load disturbance, they often limit the corresponding control system's performance. In order to overcome such limitations, the proposed robust hybrid sliding mode observer that is developed via the Lie derivatives transformation is applied. With the measurable current and input voltage, the non-matching parameter can be exactly estimated without filtering effect, and the identified flux related parameter is then used to enhance the speed estimation performance in the presence of the external disturbance. The expected estimation performance is demonstrated through a series of experimental results. We explore how the proposed sliding mode observer design can also be applied to the rotor speed and position estimations of a surface-mounted permanent magnet synchronous motor (PMSM), in which the asymptotical stability property of the reduced order system is not satisfied. Unlike the conventional sliding mode observers, we treat the position related dynamics as new unknown system states instead of as a part of the system uncertainties, then the filtering/chattering effect on the position estimation can be completely avoided. Such methodology can be used to improve the accuracy of position estimation at low-speed situations when a one time calibration of the rotor position is available. The results are demonstrated through simulation studies. The second part of the thesis focuses on the identifiability of a class of multi-input-multi-output (MIMO) nonlinear systems with non-matching unknown inputs, i.e., without satisfying the involutive condition, but the number of the measurement outputs is assumed to be one more than the number of the unknown inputs. We shall establish conditions for uniform observability of these uncertain systems as well as the identifiability of the unknown inputs. For this class of uncertain systems the original nonlinear system can be divided into two subsystems, of which one is a square subsystem with the matching unknown inputs appearing in the corresponding last equation, and the other subsystem has non-matching unknown inputs. To handle such a nonlinear uncertain system, a high gain observer appended with multiple higher order sliding mode terms is proposed, where the nonlinear sliding mode feedbacks are designed to track the unknown inputs individually, and replace them with some nominal dynamics on the sliding mode surfaces. As a result, the uniform observability for the subsystem with the non-matching inputs can be guaranteed, and the high gain observer works to ensure that the remaining dynamics on the sliding mode surfaces is asymptotically stable. Therefore, the unknown inputs can be reconstructed after all the states have converged to their true values. A class of more general MIMO nonlinear uncertain systems in which the unknown inputs or disturbances appear in both the state dynamics and the measurement outputs, is then considered. With one more output than the number of unknown inputs, it guarantees that at least one clean output signal can be achieved in the initial stage. Then, a recursive sliding mode observer with high gain feedback is developed, in which the sliding mode feedbacks with recursive structures are designed to ensure that the sliding mode surfaces are reached sequentially, and that the valuable signals on the measurement outputs are gradually extracted by cancelling the unknown inputs in sequence. Then, the high gain feedback works to guarantee the unknown inputs and the states can be identified asymptotically.