We propose a robust data-driven model predictive control (MPC) scheme to control linear time-invariant systems. The scheme uses an implicit model description based on behavioral systems theory and ...past measured trajectories. In particular, it does not require any prior identification step, but only an initially measured input-output trajectory as well as an upper bound on the order of the unknown system. First, we prove exponential stability of a nominal data-driven MPC scheme with terminal equality constraints in the case of no measurement noise. For bounded additive output measurement noise, we propose a robust modification of the scheme, including a slack variable with regularization in the cost. We prove that the application of this robust MPC scheme in a multistep fashion leads to practical exponential stability of the closed loop w.r.t. the noise level. The presented results provide the first (theoretical) analysis of closed-loop properties, resulting from a simple, purely data-driven MPC scheme.
In this article, we present a novel data-driven model predictive control (MPC) approach to control unknown nonlinear systems using only measured input-output data with closed-loop stability ...guarantees. Our scheme relies on the data-driven system parameterization provided by the fundamental lemma of Willems et al. We use new input-output measurements online to update the data, exploiting local linear approximations of the underlying system. We prove that our MPC scheme, which only requires solving strictly convex quadratic programs online, ensures that the closed loop (practically) converges to the (unknown) optimal reachable equilibrium that tracks a desired output reference while satisfying polytopic input constraints. As intermediate results of independent interest, we extend the fundamental lemma to affine systems and we derive novel robustness bounds w.r.t. noisy data for the open-loop optimal control problem, which are directly transferable to other data-driven MPC schemes in the literature. The applicability of our approach is illustrated with a numerical application to a continuous stirred tank reactor.
In this article, we present a nonlinear robust model predictive control (MPC) framework for general (state and input dependent) disturbances. This approach uses an online constructed tube in order to ...tighten the nominal (state and input) constraints. To facilitate an efficient online implementation, the shape of the tube is based on an offline computed incremental Lyapunov function with a corresponding (nonlinear) incrementally stabilizing feedback. Crucially, the online optimization only implicitly includes these nonlinear functions in terms of scalar bounds, which enables an efficient implementation. Furthermore, to account for an efficient evaluation of the worst case disturbance, a simple function is constructed offline that upper bounds the possible disturbance realizations in a neighborhood of a given point of the open-loop trajectory. The resulting MPC scheme ensures robust constraint satisfaction and practical asymptotic stability with a moderate increase in the online computational demand compared to a nominal MPC. We demonstrate the applicability of the proposed framework in comparison to state-of-the-art robust MPC approaches with a nonlinear benchmark example.
In this paper, the concepts of input/output-to-state stability (IOSS) and state-norm estimators are considered for switched nonlinear systems under average dwell-time switching signals. We show that ...when the average dwell-time is large enough, a switched system is IOSS if all of its constituent subsystems are IOSS. Moreover, under the same conditions, a non-switched state-norm estimator exists for the switched system. Furthermore, if some of the constituent subsystems are not IOSS, we show that still IOSS can be established for the switched system, if the activation time of the non-IOSS subsystems is not too big. Again, under the same conditions, a state-norm estimator exists for the switched system. However, in this case, the state-norm estimator is a switched system itself, consisting of two subsystems. We show that this state-norm estimator can be constructed such that its switching times are independent of the switching times of the switched system it is designed for.
In this article, we propose a suboptimal moving horizon estimator for a general class of nonlinear systems. For the stability analysis, we transfer the "feasibility-implies-stability/robustness" ...paradigm from model predictive control to the context of moving horizon estimation in the following sense. Using a suitably defined, feasible candidate solution based on an auxiliary observer, robust stability of the proposed suboptimal estimator is inherited independently of the horizon length and even if no optimization is performed. Moreover, the proposed design allows for the choice between two cost functions different in structure: the former in the manner of a standard least squares approach, which is typically used in practice, and the latter following a time-discounted modification, resulting in better theoretical guarantees. We apply the proposed suboptimal estimator to a nonlinear chemical reactor process, verify the theoretical assumptions, and show that even a few iterations of the optimizer are sufficient to significantly improve the estimation results of the auxiliary observer. Furthermore, we illustrate the flexibility of the proposed design by employing different solvers and compare the performance with two state-of-the-art fast moving horizon estimation schemes from the literature.
This paper introduces and analyzes an improved Q-learning algorithm for discrete-time linear time-invariant systems. The proposed method does not require any knowledge of the system dynamics, and it ...enjoys significant efficiency advantages over other data-based optimal control methods in the literature. This algorithm can be fully executed off-line, as it does not require to apply the current estimate of the optimal input to the system as in on-policy algorithms. It is shown that a persistently exciting input, defined from an easily tested matrix rank condition, guarantees the convergence of the algorithm. A data-based method is proposed to design the initial stabilizing feedback gain that the algorithm requires. Robustness of the algorithm in the presence of noisy measurements is analyzed. We compare the proposed algorithm in simulation to different direct and indirect data-based control design methods.
In this paper, we study a dissipativity property which was recently used in several results on economic model predictive control to ensure optimal operation of a system at steady-state as well as ...stability. In particular, we first investigate whether this dissipativity property is not only sufficient, but also necessary for optimal steady-state operation. In the most general case, this is not true; nevertheless, under an additional controllability assumption, we show that dissipativity is in fact necessary. Second, we provide a robustness analysis of the dissipativity property with respect to changes in the constraint set, which can result in a change in the considered supply rate.
Recently, many reactive trajectory planning approaches were suggested in the literature because of their inherent immediate adaption in the ever more demanding cluttered and unpredictable ...environments of robotic systems. However, typically those approaches are only locally reactive without considering global path planning and no guarantees for simultaneous collision avoidance and goal convergence can be given. In this article, we study a recently developed circular field (CF)-based motion planner that combines local reactive control with global trajectory generation by adapting an artificial magnetic field such that multiple trajectories around obstacles can be evaluated (cf., Becker et al., 2021). In particular, we provide a mathematically rigorous analysis of this planner for static environments in the horizontal plane to ensure safe motion of the controlled robot. Contrary to existing results, the derived collision avoidance analysis covers the entire CF motion planning algorithm including attractive forces for goal convergence and is not limited to a specific choice of the rotation field, i.e., our guarantees are not limited to a specific potentially suboptimal trajectory. Our Lyapunov-type collision avoidance analysis is based on the definition of an (equivalent) 2-D auxiliary system, which enables us to provide tight, if and only if conditions for the case of a collision with point obstacles. Furthermore, we show how this analysis naturally extends to multiple obstacles and we specify sufficient conditions for goal convergence. Finally, we provide challenging simulation scenarios with multiple nonconvex point cloud obstacles and demonstrate collision avoidance and goal convergence.
In this article, we present a quasi-infinite horizon nonlinear model predictive control (MPC) scheme for tracking of generic reference trajectories. This scheme is applicable to nonlinear systems, ...which are locally incrementally stabilizable. For such systems, we provide a reference generic offline procedure to compute an incrementally stabilizing feedback with a continuously parameterized quadratic quasi-infinite horizon terminal cost. As a result, we get a nonlinear reference tracking MPC scheme with a valid terminal cost for general reachable reference trajectories without increasing the online computational complexity. The practicality of this approach is demonstrated with a benchmark example.
We present an extension of Willems' Fundamental Lemma to the class of multi-input multi-output discrete-time feedback linearizable nonlinear systems, thus providing a data-based representation of ...their input-output trajectories. Two sources of uncertainty are considered. First, the unknown linearizing input is inexactly approximated by a set of basis functions. Second, the measured output data is contaminated by additive noise. Further, we propose an approach to approximate the solution of the data-based simulation and output matching problems, and show that the difference from the true solution is bounded. Finally, the results are illustrated on an example of a fully-actuated double inverted pendulum.
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