Performance of finite horizon NMPC Formulations

Ideally one would like to use an infinite horizon NMPC formulation, since in the nominal case the closed-loop trajectories do coincide with the open-loop predicted ones (principle of optimality). The main problem is that infinite horizon schemes can often not be applied in practice, since the open-loop optimal control problem cannot be solved sufficiently fast. Using finite horizon however it is by no means true that a repeated minimization over a finite same stage cost F). in fact the two solution will in general differ significantly if a short horizon is chosen.

The QIH-NMPC strategy outlined in the provision section allows in the principle to recover the performance of the infinite horizons scheme without jeopardizing the closed-loop stability . the value function resulting from problem 2 can be seen as an upper bound of the infinite horizon cost, to be more precise if the terminal penalty function E is chosen such that a corresponding local control law is a good approximation of the control resulting from infinite horizon control law in a neighborhood of the origin, the performance corresponding to problem 2 can recover the performance of the infinite horizon cost even for short horizon (assuming the terminal region constraint can be satisfied ).

Robustness

So far only the nominal control problem was considered , the NMPC scheme discussed before do require that the actual system is identical to the model used for predication, i ,e. that no mode/plant mismatch or unknown disturbances are present . clearly this is a very unrealistic assumption for practical and the development of a NMPVC framework to address robustness issues is of paramount importance, in this note the nonlinear uncertain system is assumed to be given by:
Where the uncertainty d(.) satisfies is assumed to be compact like in the nominal stability and performance case , the resulting difference between the predicted open-loop and actual closed �loop trajectory is the main obstacle, as additional problem the uncertainty d hitting the system now leads not only to one single future trajectory in the prediction instead a whole tree of possible solution must analyzed .
Even thought the analysis of robustness properties in nonlinear NMPC must still be considered as an unsolved problem in general, some preliminary result are available . in principle one must distinguish between two approaches to consider the robustness question . firstly one can examine the robustness of the NMPC scheme designed for nominal stability and by this take the uncertainty/disturbance only indirectly into account [40,47] second secondly one k can consider to design NMPC scheme that directly take into account the uncertainty/ disturbances.

inherent robustness of NMPC

As mentioned above, inherent robustness corresponds to the fact that nominal NMPC can cope with input model uncertainties without taking them directly into account. This fact stems from the close relation of NMPC to optimal control. Assuming that the system under consideration is of the following (input affine ) form

Where q is positive definite, that there are no constraints on the state and the input and the resulting control law and the value function satisfies future assumptions (u being continuously differentiable and the value function being twice continuously differentiable ). Then one can show [47] that the NMPC control law is inverse optimal, I, e. it is also optimal for a modified optimal control problem spanning over an infinite horizon. due to this inverse optimality the NMPC control law inherits the same robustness properties as infinite horizon optimal control assuming that the sampling time goes to zero. In particular, the closed-loop is robust with respect to sector bounded input uncertainties; The nominal NMPC controller also stabilizes system of the form.

Robust NMPC Schemes

At least three different robust NMPC formulation exits:
� Robust NMPC solving an open-max problem [18,45]:
In this formulation the standard NMPC setup kept. However now the cost function optimized is given by the worst case disturbance �sequence� occurring i.e.

The resulting open-loop optimization is a min-max problem. The key problem is that adding stability constraints like in the nominal case, might lead to the fact that no feasible solution can be found at all. These mainly stems from the fact, that input signal must �reject� all possible disturbances and guarantee the satisfaction of the stability constraints.

• H- NMPC [11,18,46]: Another possibility is to consider the standard H problem in a receding horizon framework. The key obstacle is, that an infinite horizon min-max problem must be solved (solution of the nonlinear Hamilton-Jacobin �Isaacs equation). Modifying the NMPC function similar to the H problem and optimizing over a sequence of control laws robustly stabilizing finite horizon H-NMPC formulation can connection to the first approach.
• Robust NMPC optimizing a feedback controller used during the sampling times [45]: The open- loop formulation of the robust stabilization problem can be seen as very conservative, since only open-loop control is used during the sampling times, i.e. the disturbance are not directly rejected in between the sampling instances. To overcome this problem it has been proposed not to optimized over the input signal. Instead of optimizing the open-loop input signal directly ,a feedback controller is optimized, i,e, the decision variable U is not considered as optimized variable instead a �sequence� of control laws u=kx) applied during the sampling times is optimized. Now the optimization problem has as optimization variables the parameterization of the feedback controllers (K1��Kn). While this formulation is very attractive since the conservation is reduced, the solution is often prohibitively complex.