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  • Model predictive control is an advanced method of process control that has been

  • in use in the process industries in chemical plants and oil refineries since

  • the 1980s. In recent years it has also been used in power system balancing

  • models. Model predictive controllers rely on dynamic models of the process,

  • most often linear empirical models obtained by system identification. The

  • main advantage of MPC is the fact that it allows the current timeslot to be

  • optimized, while keeping future timeslots in account. This is achieved

  • by optimizing a finite time-horizon, but only implementing the current timeslot.

  • MPC has the ability to anticipate future events and can take control actions

  • accordingly. PID and LQR controllers do not have this predictive ability. MPC is

  • nearly universally implemented as a digital control, although there is

  • research into achieving faster response times with specially designed analog

  • circuitry. Overview

  • The models used in MPC are generally intended to represent the behavior of

  • complex dynamical systems. The additional complexity of the MPC control

  • algorithm is not generally needed to provide adequate control of simple

  • systems, which are often controlled well by generic PID controllers. Common

  • dynamic characteristics that are difficult for PID controllers include

  • large time delays and high-order dynamics.

  • MPC models predict the change in the dependent variables of the modeled

  • system that will be caused by changes in the independent variables. In a chemical

  • process, independent variables that can be adjusted by the controller are often

  • either the setpoints of regulatory PID controllers or the final control

  • element. Independent variables that cannot be adjusted by the controller are

  • used as disturbances. Dependent variables in these processes are other

  • measurements that represent either control objectives or process

  • constraints. MPC uses the current plant measurements,

  • the current dynamic state of the process, the MPC models, and the process

  • variable targets and limits to calculate future changes in the dependent

  • variables. These changes are calculated to hold the dependent variables close to

  • target while honoring constraints on both independent and dependent

  • variables. The MPC typically sends out only the first change in each

  • independent variable to be implemented, and repeats the calculation when the

  • next change is required. While many real processes are not

  • linear, they can often be considered to be approximately linear over a small

  • operating range. Linear MPC approaches are used in the majority of applications

  • with the feedback mechanism of the MPC compensating for prediction errors due

  • to structural mismatch between the model and the process. In model predictive

  • controllers that consist only of linear models, the superposition principle of

  • linear algebra enables the effect of changes in multiple independent

  • variables to be added together to predict the response of the dependent

  • variables. This simplifies the control problem to a series of direct matrix

  • algebra calculations that are fast and robust.

  • When linear models are not sufficiently accurate to represent the real process

  • nonlinearities, several approaches can be used. In some cases, the process

  • variables can be transformed before and/or after the linear MPC model to

  • reduce the nonlinearity. The process can be controlled with nonlinear MPC that

  • uses a nonlinear model directly in the control application. The nonlinear model

  • may be in the form of an empirical data fit or a high-fidelity dynamic model

  • based on fundamental mass and energy balances. The nonlinear model may be

  • linearized to derive a Kalman filter or specify a model for linear MPC.

  • An algorithmic study by El-Gherwi, Budman, and El Kamel shows that

  • utilizing a dual-mode approach can provide significant reduction in online

  • computations while maintaining comparative performance to a non-altered

  • implementation. The proposed algorithm solves N convex optimization problems in

  • parallel based on exchange of information among controllers.

  • = Theory behind MPC = MPC is based on iterative,

  • finite-horizon optimization of a plant model. At time t the current plant state

  • is sampled and a cost minimizing control strategy is computed for a relatively

  • short time horizon in the future: . Specifically, an online or on-the-fly

  • calculation is used to explore state trajectories that emanate from the

  • current state and find a cost-minimizing control strategy until time . Only the

  • first step of the control strategy is implemented, then the plant state is

  • sampled again and the calculations are repeated starting from the new current

  • state, yielding a new control and new predicted state path. The prediction

  • horizon keeps being shifted forward and for this reason MPC is also called

  • receding horizon control. Although this approach is not optimal, in practice it

  • has given very good results. Much academic research has been done to find

  • fast methods of solution of EulerLagrange type equations, to

  • understand the global stability properties of MPC's local optimization,

  • and in general to improve the MPC method. To some extent the theoreticians

  • have been trying to catch up with the control engineers when it comes to MPC.

  • = Principles of MPC = Model Predictive Control is a

  • multivariable control algorithm that uses:

  • an internal dynamic model of the process a history of past control moves and

  • an optimization cost function J over the receding prediction horizon,

  • to calculate the optimum control moves. An example of a non-linear cost function

  • for optimization is given by: without violating constraints

  • With: = i -th controlled variable

  • = i -th reference variable = i -th manipulated variable

  • = weighting coefficient reflecting the relative importance of

  • = weighting coefficient penalizing relative big changes in

  • etc. Nonlinear MPC

  • Nonlinear Model Predictive Control, or NMPC, is a variant of model predictive

  • control that is characterized by the use of nonlinear system models in the

  • prediction. As in linear MPC, NMPC requires the iterative solution of

  • optimal control problems on a finite prediction horizon. While these problems

  • are convex in linear MPC, in nonlinear MPC they are not convex anymore. This

  • poses challenges for both NMPC stability theory and numerical solution.

  • The numerical solution of the NMPC optimal control problems is typically

  • based on direct optimal control methods using Newton-type optimization schemes,

  • in one of the variants: direct single shooting, direct multiple shooting

  • methods, or direct collocation. NMPC algorithms typically exploit the fact

  • that consecutive optimal control problems are similar to each other.

  • This allows to initialize the Newton-type solution procedure

  • efficiently by a suitably shifted guess from the previously computed optimal

  • solution, saving considerable amounts of computation time. The similarity of

  • subsequent problems is even further exploited by path following algorithms

  • that never attempt to iterate any optimization problem to convergence, but

  • instead only take one iteration towards the solution of the most current NMPC

  • problem, before proceeding to the next one, which is suitably initialized.

  • While NMPC applications have in the past been mostly used in the process and

  • chemical industries with comparatively slow sampling rates, NMPC is more and

  • more being applied to applications with high sampling rates, e.g., in the

  • automotive industry, or even when the states are distributed in space

  • Robust MPC Robust variants of Model Predictive

  • Control are able to account for set bounded disturbance while still ensuring

  • state constraints are met. There are three main approaches to robust MPC:

  • Min-max MPC. In this formulation, the optimization is performed with respect

  • to all possible evolutions of the disturbance. This is the optimal

  • solution to linear robust control problems, however it carries a high

  • computational cost. 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 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 stages 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.

  • 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 11 733–764.

  • See also System identification

  • Control theory Control engineering

  • Feed-forward References

  • Further reading Kwon, W. H.; Bruckstein, Kailath.

  • "Stabilizing state feedback design via the moving horizon method".

  • International Journal of Control 37: pp.631–643.

  • doi:10.1080/00207178308932998. CS1 maint: Extra text

  • Garcia, C; Prett, Morari. "Model predictive control: theory and

  • practice". Automatica 25: pp.335–348. doi:10.1016/0005-1098(89)90002-2. CS1

  • maint: Extra text Mayne, D.Q.; Michalska. "Receding

  • horizon control of nonlinear systems". IEEE Transactions on Automatic Control

  • 35: pp.814–824. doi:10.1109/9.57020. CS1 maint: Extra text

  • Mayne, D; Rawlings, Rao, Scokaert. "Constrained model predictive control:

  • stability and optimality". Automatica 36: pp.789–814.

  • doi:10.1016/S0005-1098(99)00214-9. CS1 maint: Extra text

  • Allgöwer; Zheng. Nonlinear model predictive control. Progress in Systems

  • Theory 26. Birkhauser. Camacho; Bordons. Model predictive

  • control. Springer Verlag. Findeisen; Allgöwer, Biegler. Assessment

  • and Future Directions of Nonlinear Model Predictive Control. Lecture Notes in

  • Control and Information Sciences 26. Springer.

  • Diehl, M; Bock, Schlöder, Findeisen, Nagy, Allgöwer. "Real-time optimization

  • and Nonlinear Model Predictive Control of Processes governed by

  • differential-algebraic equations". Journal of Process Control 12:

  • pp.577–585. doi:10.1016/S0959-1524(01)00023-3. CS1

  • maint: Extra text External links

  • Control Tuning and Best Practices P. Orukpe: Basics of Model Predictive

  • Control Case Study. Lancaster Waste Water

  • Treatment Works, optimisation by means of Model Predictive Control from

  • Perceptive Engineering ACADO Toolkit - Open Source Toolkit for

  • Automatic Control and Dynamic Optimization providing linear and

  • non-linear MPC tools. jMPC Toolbox - Open Source MATLAB

  • Toolbox for Linear MPC. Model Predictive Control Free book

  • edited by Tao Zheng, Publisher: Sciyo, 2010.

  • Study on application of NMPC to superfluid cryogenics.

  • Nonlinear Model Predictive Control Toolbox for MATLAB and Python

  • Model Predictive Control Toolbox from MathWorks for design and simulation of

  • model predictive controllers in MATLAB and Simulink

  • Pulse step model predictive controller - virtual simulator

  • Tutorial on MPC with Excel and MATLAB Examples

Model predictive control is an advanced method of process control that has been

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モデル予測制御 (Model predictive control)

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    Hsu Johnny に公開 2021 年 01 月 14 日
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