Nonlinear Model Predictive Control (NMPC)

Nonlinear model predictive control (NMPC) [13] is based on minimising a cost function $J$ defined over a future window of fixed length $T_c$. For example, the quadratic difference between the predicted future observations $\mathbf{x}$ and a reference signal $\mathbf{r}$ can be used:

$$J(\mathbf{s}(t_0),\mathbf{u}(t_0),\dots,\mathbf{u}(t_0+T_c-1)) = \sum_{\tau=1}^{T_c}\left\vert\mathbf{x}(t_0+\tau)-\mathbf{r}\right\vert^2.$$ (9)

Then $J$ is minimised w.r.t. the control signals $\mathbf{u}$ and the first one $\mathbf{u}(t_0)$ is executed. In this paper, the states and observations (but not control signals) are modelled probabilistically so we actually minimise the expected cost $E_q\{J\}$. The current guess $\mathbf{u}(t_0),\dots,\mathbf{u}(t_0+T_c-1)$ defines a probability distribution over future states and observations. This inference can be done with a single forward pass, when ignoring the policy mapping, that is, the dependency of the state on future control signals. In this case, it makes sense to ignore the policy mapping anyway, since the future control signals do not have to follow the policy. Minimisation of $E_q\{J\}$ is done with a certain quasi-Newton algorithm [12]. For that, the partial derivatives $\partial \mathbf{x}(t_2) / \partial \mathbf{u}(t_1)$ for all $t_0\leq t_1 < t_2 \leq t_0+T_c$ are computed efficiently based on the chain rule and dynamic programming. Details are left for future publications due to lack of space. The use of a cost function makes NMPC very versatile. Costs for control signals and observations can be set for instance to restrict values within bounds etc. Quadratic costs such as (9) make things easy for the optimisation algorithm.

Table I: Control Scheme Summary

Scheme	Based on	Data	Speed
DC	internal MLP	task-oriented	fast
OIC	probabilistic inference	general	slow
NMPC	cost minimisation	general	slow