text: Modeling error or external disturbances can severely degrade the performance of Model Predictive Control (MPC) in real-world scenarios. Robust MPC (RMPC) addresses this limitation by optimizing over feedback policies but at the expense of increased computational complexity. Tube MPC is an approximate solution strategy in which a robust controller, designed offline, keeps the system in an invariant tube around a desired nominal trajectory, generated online. Naturally, this decomposition is suboptimal, especially for systems with changing objectives or operating conditions. In addition, many tube MPC approaches are unable to capture state-dependent uncertainty due to the complexity of calculating invariant tubes, resulting in overly-conservative approximations. This work presents the Dynamic Tube MPC (DTMPC) framework for nonlinear systems where both the tube geometry and open-loop trajectory are optimized simultaneously. By using boundary layer sliding control, the tube geometry can be expressed as a simple relation between control parameters and uncertainty bound; enabling the tube geometry dynamics to be added to the nominal MPC optimization with minimal increase in computational complexity. In addition, DTMPC is able to leverage state-dependent uncertainty to reduce conservativeness and improve optimization feasibility. DTMPC is demonstrated to robustly perform obstacle avoidance and modify the tube geometry in response to obstacle proximity.
  @cite_8: Abstract Motivated by requirements in the process industries, the largest user of model predictive control, we re-examine some features of recent research on this topic. We suggest that some proposals are too complex and computationally demanding for application in this area and make some tentative proposals for research on robust and stochastic model predictive control to aid applicability
  @cite_10: This paper recalls a few past achievements in Model Predictive Control, gives an overview of some current developments and suggests a few avenues for future research.
  @cite_13: 
  @cite_12: This paper introduces elastic tube model predictive control (MPC) synthesis. The proposed framework is a natural generalization of the rigid and homothetic tube MPC design methods. The cross-sections of the employed state and control tubes are allowed to change more elastically, while the local component of the tubes control policy is permitted to take a more general form. The related stabilizing terminal conditions are also adequately generalized in order to take advantage of more flexible tubes and tubes control policy parameterizations. These novel features result in an improved tube MPC at the cost of a manageable increase in computational complexity.
  @cite_17: The robust model predictive control for constrained linear discrete time systems is solved through the development of a homothetic tube model predictive control synthesis method. The method employs several novel features including a more general parameterization of the state and control tubes based on homothety and invariance, a more flexible form of the terminal constraint set and a relaxation of the controlled dynamics of the sets that define the state and control tubes. Under natural assumptions, the proposed method is computationally efficient and it induces strong system theoretic properties.
summary: A number of works have been published on the stability, feasibility, and performance of linear tube MPC @cite_10 @cite_13 @cite_8 . While this is an effective strategy to achieve robustness, decoupling the nominal MPC problem and controller design is suboptimal. Rakovi @cite_17 showed that the region of attraction can be enlarged by parameterizing the problem with the open-loop trajectory and tube size. The authors presented the homothetic tube MPC (HTMPC) algorithm that treated the state and control tubes as homothetic copies of a fixed cross-section shape, enabling the problem to be parameterized by the tube’s centers (i.e., open-loop trajectory) and a cross-section scaling factor. The work was extended to tubes with varying shapes, known as elastic tube MPC (ETMPC), but at the expense of computational complexity @cite_12 . Both HTMPC and ETMPC possess strong theoretical properties and have the potential to significantly improve performance but a nonlinear extension has yet to be developed.