Contributions to Sliding Mode Control and Observation of Nonlinear Uncertain Systems

Most control approaches are model-based methods and require a precise mathematical description of the considered dynamical system. System modeling offers the advantage that the controllable system dynamics can be affected by control in such a way that some given desired dynamics is achieved. However, most technical systems are complex and it is difficult and time-consuming to describe them exactly or it may even be impossible. Challenges that arise in system modeling can be for instance: parameters that are not precisely known or that vary in time, dynamics that exist but are unmodeled, or especially for modern control systems interactions with the environment that lead to unknown exogenous inputs. Consequently, it is common that a discrepancy between the modeled dynamics and the true dynamics exists. The applied controller is required to be robust in the sense that the control goals are also guaranteed to be achieved in presence of the model discrepancy. Sliding mode control (SMC) is such a robust control method. It can make the system dynamics invariant to disturbances that appear in the input channel so that the desired dynamics are still achieved. However, besides its strong robustness properties sliding mode control also has some disadvantages:

  • Conventional sliding mode control leads to a high frequent switching effect in the input signal denoted as chattering making the controller inapplicable in practice. Higher order SMC approaches may effectively mitigate the chattering but typically require higher order time derivatives of the measured signal. As a consequence, the whole approach becomes more sensitive to noise. Adaptive SMC approaches have been developed to reduce the chattering as well. However, chattering reduction can only be achieved to a certain extend as otherwise the control goals may not be achieved anymore.
  • The handling of constraints in the context of sliding mode control is not straight forward as due to the model uncertainty standard add-on control approaches like the invariance control method or the reference governor approach can not be applied directly. Constrained SMC approaches exist but they typically only consider box constraints with time-invariant bounds which limits the field of applications. Another problem is that most of the existing constrained SMC approaches only solve the constrained control problem in theory but are not applicable in practice due to chattering. Approximation techniques are required to be applied to mitigate the chattering. However, as the approximation techniques modify the original control laws it can not be guaranteed anymore that the constraints remain satisfied in general.

State estimation approaches are widely applied in the field of control and system monitoring. Typically, the estimation approaches are model-based methods and require a precise system description to be known. Design principles of sliding mode control can also be applied to design state estimation approaches leading to the so-called sliding mode observers. However, in contrast to the SMCs the robustness of the sliding mode observers (SMOs) is much more restricted. General design concepts for SMO approaches that can handle model uncertainty only exist for specific classes of linear systems with unknown inputs and for nonlinear systems in companion form. There exist SMO approaches for other classes of systems as well, but most of them require a precise model description.

Based on the aforementioned limitations and problems of sliding mode control and observation the following topics are discussed within this thesis.
A nonlinear state estimation approach denoted as smooth variable structure filter (SVSF) is considered. The SVSF combines the predictor corrector scheme of the Kalman filter with design elements known from sliding mode control and observation. It is applicable to nonlinear systems and can handle model uncertainty. However, the estimation performance of the SVSF highly depends on the choice of some tuning parameters. In this thesis a reformulation of the SVSF is stated which gives an easy interpretation on how the tuning parameters affect the behavior of the filter. An equation to determine the error covariance of the reformulated filter is derived. Based on the error covariance a new filter gain is determined which minimizes the mean squared estimation error. If the new gain is applied to the reformulated filter it can be shown that the obtained estimation algorithm equals the one of the extended Kalman filter (EKF). To further investigate the robustness of the SVSF a combined estimation approach is formulated. The combined approach is equal to the reformulated SVSF but has a combined gain  consisting of a weighted sum of the SVSF and the EKF gains. To optimize the filter parameters and weighting factors a parameter optimization scheme is proposed. The scheme neither requires any experiments on the real system to be conducted nor does it require the true system description to be known. The optimization scheme is generic in the sense that it can be applied to optimize any model-based state estimation approach that has to deal with an imprecise system description. The performance of the proposed combined estimation approach is compared with the one of the SVSF. A nonlinear system is simulated and the system description is assumed to be unknown. Both, the parameters of the combined approach and the ones of SVSF are optimized using the proposed scheme. For the considered system the combined approach outperforms the SVSF. Further, from the optimized parameters it can be seen that the robustness of the combined approach is mainly achieved by the EKF gain. That means that the combined approach behaves structural similar to the EKF but with a parameterized gain that is optimized by the proposed optimization scheme to handle model uncertainty.

Further, adaptive sliding mode control is considered in this thesis. The ability of adaptive sliding mode control to mitigate the chattering effect is improved. Therefore, a new data-driven adaptive SMC for nonlinear systems is designed. The system description is not necessarily required to be explicitly known but the dynamics are assumed to be sufficiently slow. Input- and output-data of the system is applied to train a linear local model that predicts the future system behavior. The local model is trained by a Kalman filter which allows to update the model at each time step based on the incoming data of the system. Using the prediction capabilities of the local model an optimization problem is formulated to minimize the squared control error and the input energy. The optimal control input is determined and combined with a first order adaptive sliding mode controller. Weighting functions are introduced to guarantee reaching of a subspace around the sliding manifold from which follows that the tracking error is bounded. In the vicinity of the sliding surface the gain of the sliding mode controller is scaled down. As a consequence, the control law is dominated by the optimal control input and the chattering effect can be effectively mitigated. Set point tracking of a nonlinear system is considered to compare the proposed data-driven approach with a conventional first order adaptive SMC. In contrast to the conventional adaptive SMC the developed approach achieves stationary accurate tracking without noticeable chattering. For the considered specific system the control results of the proposed method are achieved even without concrete knowledge of the system parameters.

Finally, the development of a constrained sliding mode controller is considered in this thesis. The approach is applicable to nonlinear relative degree two systems and assumes the first time derivative of the control variable to be constrained. The bounds of the constraints may explicitly depend on time. The controller is designed based on a combination of two SMC sub-controllers. In contrast to the existing approaches a smooth transition between the sub-controllers is achieved and chattering mitigation is considered in such a way that it does not lead to constraint violation. As the developed method is based on sliding mode control design it can handle model uncertainty. The proposed approach guarantees convergence of the tracking error with respect to a domain that can be specified based on the controller parameters. In addition, a maximum time period can be stated after which convergence of the tracking error is guaranteed to be achieved. As the proposed controller allows to update the constraints online it is tested on a robotic system with time-dependent velocity constraints. The angular velocities change online dependent on the distance between the end effector and the waypoints. Moreover, the robot has to accomplish a pick and place problem in which an unknown payload is considered. The payload servers as a disturbance and is required to be rejected by the controller. Based on the developed theory about the proposed controller the tracking error bounds can be determined beforehand. The maximum time period that is required to accomplish the pick and place problem can also be stated in advance. The results have been successfully confirmed by simulation. As the proposed controller is well suited for velocity constraint control of robotic systems a safety concept for human-robot collaboration tasks is developed in addition. The proposed concept guarantees that the robot velocity is restricted to a desired value in the moment when the human and the robot are in contact with each other. A scenario of human robot interaction is considered to test the developed concept. The theoretical controller performance is confirmed by the simulation results.


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