Nonlinear model-based predictive control

The articles Nonlinear Model-Based Predictive Control and Model Predictive Control overlap thematically. Help me to better differentiate or merge the articles (→  instructions ) . To do this, take part in the relevant redundancy discussion . Please remove this module only after the redundancy has been completely processed and do not forget to include the relevant entry on the redundancy discussion page{{ Done | 1 = ~~~~}}to mark. Errordynamic ( discussion ) 10:03, 9 Mar. 2018 (CET)

The nonlinear model predictive control , in the mostly English literature as a non-linear model predictive control referred to is a method from the branch of control theory and control engineering . This type of controller is used specifically to handle non-linear processes with restrictions and without linearization.

Although most of the processes examined in control theory are non-linear , they can be linearized without major deviations and therefore controlled by linear controllers . The non-linear model predictive control offers a possibility for process modeling if these processes are not to be linearized. The implementation of a non-linear model predictive control is more complex than that of a linear control.

functionality

A non-linear model predictive control algorithm consists of three basic steps.

First, with the aid of a time-discrete or time-continuous dynamic model of the process to be regulated, a prediction of the state development is calculated as a function of the control signals for a fixed finite time horizon. The resulting trajectory and the controls used are then evaluated with the aid of a target functional. With the help of (direct or indirect) optimization methods, a control for this time horizon is now determined that minimizes the given target function. Since the resulting control was only determined for a finite time horizon, it has to be expanded in order to control the process as optimally as possible for an indefinite period of time and at the same time to observe all restrictions .

Therefore, in the second step, the first element from this control is applied to the process.

Then, in the third step, the optimization horizon is moved forward by the length of the validity of the implemented control element and the control process is started from the beginning. In the second step, the remaining control elements can either be deleted or used as a starting point for optimization in the subsequent NMPC step. In addition, this also enables new measurement data to be taken into account, which closes the control loop and results in regulation from the various consecutive controls.

Mathematical formulation

The following problem is considered here:

${\displaystyle {\text{Bestimme}}\;\mu _{N}(x(n)):=argmin_{u\in {\mathcal {U}}}J_{N}(x_{0},u)}$

under the constraints

${\displaystyle J_{N}(x_{0},u)=\sum _{k=0}^{N-1}l(x(k),u(k))+F(x(N))}$

${\displaystyle x(k)\in \mathbb {X} \qquad \forall k\in \{0,\ldots ,N\}}$

${\displaystyle u(k)\in \mathbb {U} \qquad \forall k\in \{0,\ldots ,N-1\}}$

At the same time, the dynamics of the system must be taken into account. Depending on the problem, this is done through a time-continuous or time-discrete non-linear control system of the form

${\displaystyle {\dot {x}}(t)=f(x(t),u(t))}$

or

${\displaystyle x(k+1)=f(x(k),u(k))}$

given. Here, time-continuous systems are usually understood and implemented as scanning systems , i.e. H. the set of control functions consists of the set of piecewise constant functions. In addition, the so-called sampling times , i.e. the jump points of these functions, are specified in advance. This is motivated by the digital implementation using computers that have fixed clock rates and that cannot be undercut. ${\displaystyle {\mathcal {U}}}$

The process must continue with an initial value

${\displaystyle x(0)=x_{0}}$

initialized in order to guarantee the existence of solution trajectories according to Caratheodory .

literature

• M. Alamir: Stabilization of Nonlinear Systems Using Receding-horizon Control Schemes. Springer Berlin 2006, ISBN 1-84628-470-8 .
• R. Dittmar, B.-M. Pfeiffer: Model-Based Predictive Control: An Introduction for Engineers. Oldenbourg, 2004, ISBN 3-486-27523-2 .
• DQ Mayne, JB Rawlings, CV Rao, PO Scokaert: Constrained model predictive control: Stability and optimality. In: Automatica. vol. 36, 2000, pp. 789-814.