Chance-Constrained Model Predictive Control based on Deterministic Density Approximation and Homotopy Continuation

在模型预测控制（MPC）中，控制输入通过在线求解优化问题来获得。为了适用于实际应用，MPC必须提供闭环稳定性和对内源性和外源性干扰的鲁棒性。为此目的，限制被引入到MPC问题。在标准MPC中，约束是困难的，因为假设干扰是有界的，即，它们包含在某个封闭集合中。因此，约束满足保证最坏的情况下，忽略可能可用的统计特性的干扰。为了实现更好的性能，并考虑无限的干扰建模为随机过程的愿望，将统计特性的干扰，以导致机会约束MPC的发展。在机会约束MPC中，硬确定性约束被软约束所取代，软约束要求约束以指定的概率被满足。机会约束MPC可以应用于化学工程、机器人辅助手术、跟踪控制、节能温度控制等领域。机会约束MPC的主要挑战是MPC求解过程的易处理性。这个过程包括一个优化过程，通常必须使用迭代方法进行数值评估。为此，为了检查在每个迭代步骤的约束，概率密度的多元积分必须计算。因此，最先进的方法通过保守的确定性近似或基于随机样本的近似来近似这些密度。这两种方法各有优缺点。保守的确定性近似产生低的计算负担，但他们没有找到最优的控制策略，由于其保守性。另一方面，基于样本的近似方法保守性要小得多，但由于它们的计算负担，它们不适用于实时应用。在这个建议中，我们研究了一种新的MPC方法，结合了保守近似方法和随机采样方法的优点。为了这个目的，我们近似的发生密度使用一个确定性的基于样本的方法。这种近似方法允许通过调整样本的数量来调整保守性水平。为了解决MPC优化问题，我们将应用同伦连续方法。这些方法首先解决了无约束控制问题，如果过程和测量动态是线性的，则可以获得解析解。然后，约束逐步引入到控制问题和搜索算法可以应用。我们希望所提出的方法是计算速度比国家的最先进的方法的基础上随机采样产生相同质量的优化结果。

In Model Predictive Control (MPC), control inputs are obtained by online solving an optimization problem. In order to be applicable in real-world applications, MPC has to provide closed-loop stability and robustness to endogenous and exogenous disturbances. For this purpose, constraints are introduced into the MPC problem. In standard MPC, the constraints are hard, because the disturbances are assumed to be bounded, i.e., they are contained within some closed set. Thus, constraint satisfaction is guaranteed for the worst case scenario ignoring probably available statistical properties of the disturbances. The desire to incorporate statistical properties of disturbances in order to achieve better performance and the additional desire to consider unbounded disturbances modeled as stochastic processes led to the development of chance-constrained MPC. In chance-constrained MPC, hard deterministic constraints are replaced by soft constraints that require that the constraints are satisfied with a specified probability. Chance-constrained MPC can be applied in chemical engineering, robotically-assisted surgery, tracking control, energy-efficient temperature control, etc.The main challenge of chance-constrained MPC is the tractability of the MPC solution process. This process consists of an optimization procedure that generally has to be evaluated numerically using iterative methods. For this purpose, in order to check the constraints at each iteration step, multivariate integrals of probability densities have to be computed. State-of-the-art methods therefore approximate these densities, either by means of conservative deterministic approximations or stochastic sample-based approximations. Both these approaches have their advantages and disadvantages. Conservative deterministic approximations yield low computational burden but they do not find optimal control policies due to their conservatism. Sample-based approximation methods on the other hand are far less conservative but they are not applicable in real-time applications due to their computational burden.In this proposal, we investigate a new MPC approach that combines the advantages of the conservative approximation methods and the stochastic sampling methods. For this purpose, we approximate the occurring densities using a deterministic sample-based method. This approximation method allows to tune the conservatism level by adapting the number of samples. In order to solve the MPC optimization problem, we will apply homotopy continuation methods. These methods first solve the unconstrained control problem for which analytical solutions are available if process and measurement dynamics are linear. Then, the constraints are introduced progressively into the control problem and search algorithms can be applied. We expect the proposed method to be computationally faster than state-of-the-art methods based on stochastic sampling yielding optimization results with the same quality.