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

在模型预测控制（MPC）中，通过在线求解优化问题获得控制输入。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，在MPC问题中引入了约束条件。在标准MPC中，约束是困难的，因为扰动被假定为有界的，即它们被包含在某个闭集中。因此，在最坏的情况下，可以保证约束满足，而忽略干扰可能可用的统计特性。为了获得更好的性能而结合干扰的统计特性的愿望，以及考虑作为随机过程建模的无界干扰的额外愿望，导致了机会约束MPC的发展。在机会约束的MPC中，硬确定性约束被软约束取代，软约束要求约束以指定的概率得到满足。机会约束MPC可以应用于化工、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溶液过程的可追溯性。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计算。为此，为了检查每个迭代步骤的约束，必须计算概率密度的多元积分。因此，最先进的方法可以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。保守确定性近似计算量小，但由于其保守性，不能找到最优控制策略。另一方面，基于样本的近似方法的保守性要小得多，但由于计算量大而不适用于实时应用。在本研究中，我们研究了一种新的MPC方法，它结合了保守逼近方法和随机抽样方法的优点。为此，我们使用基于确定性样本的方法来近似发生密度。这种近似方法允许通过调整样本数量来调整保守性水平。为了解决MPC优化问题，我们将采用同伦延拓方法。这些方法首先解决了当过程和测量动态为线性时无约束控制问题的解析解。然后，将约束逐步引入到控制问题中，并应用搜索算法。我们期望所提出的方法在计算上比基于随机抽样的最先进方法更快，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.