optimization to a "min-max" problem, minimizing the worst case of the objective function, maximized over all possible plants from the uncertainty set. Constraint Tightening MPC. Here the state constraints are enlarged by a given margin so that a trajectory can be guaranteed to be found under any evolution of disturbance. Tube MPC. This uses an independent nominal model of the system, and uses a feedback controller to ensure the actual state converges to the nominal state. The amount of separation required from the state constraints is determined by the robust positively invariant (RPI) set, which is the set of all possible state deviations that may be introduced by disturbance with the feedback controller. Multi-stage MPC. This uses a scenario-tree formulation by approximating the uncertainty space with a set of samples and the approach is non-conservative because it takes into account that the measurement information is available at every time stage in the prediction and the decisions at every stage can be different and can act as recourse to counteract the effects of uncertainties. The drawback of the approach however is that the size of the problem grows exponentially with the number of uncertainties and the prediction horizon. Tube-enhanced multi-stage MPC. This approach synergizes multi-stage MPC and tube-based MPC. It provides high degrees of freedom to choose the desired trade-off between optimality and simplicity by the classification of uncertainties and the choice of control laws in the predictions. Commercially available MPC software Commercial MPC packages are available and typically contain tools for model identification and analysis, controller design and tuning, as well as controller performance evaluation. A survey of commercially available packages has been provided by S.J. Qin and T.A. Badgwell in Control Engineering Practice 11 (2003) 733–764. MPC vs. LQR Model predictive control and linear-quadratic regulators are both expressions of optimal control, with different schemes of setting up optimisation costs. While a model predictive controller often looks at fixed length, often graduatingly weighted sets of error functions, the linear-quadratic regulator looks at all linear system inputs and provides the transfer function that will reduce the total error across the frequency spectrum, trading off state error against input frequency. Due to these fundamental differences, LQR has better global stability properties, but MPC often has more locally optimal[?] and complex performance. The main differences between MPC and LQR are that LQR optimizes across the entire time window (horizon) whereas MPC optimizes in