Model Predictive Control
Glossary
ANCBI: Acronym for time-invariant dynamic linear system which are asymptotically null-control-label bounded (arbitrarily small) inputs.
ARE: Acronym for algebraic Riccati equation relevant to the linear- quadratic optimal control
CG: Acronym for a nonlinear device, called command governor or reference governor. Which is added to a pre- compensated control system so as to satisfy prescribed constraints on the system variables,
COLOC: Acronym for constrained open-loop optimal control problem underlying any model- based predictive control algorithm.
Feasible control sequence: A sequence of control satisfying control . state and terminal- state constraints in a COLOC problem .
H control :A min- max control problem whereby the control law minimizes an operator gain from the disturbance L2 �norm.
LQ: Acronym for an optimal regulation of the linear- quadratic type where the plant to be regulated is a dynamic linear system and the performance index to be minimized quadratic.
LQG: Acronym for an optimal regulation of the linear-quadratic type where in addition to the features of LQ regulation the plant to be regulated is a dynamic stochastic linear system whit state and disturbances jointly Gaussian distributed.
MBPC: Acronym for model- based predictive control
PCG: Acronym for predictive command governor, viz a CG designed on the grounds of conceptual tools of MBPC.
Receding horizon control: A control strategy according to which the effective control action applied to the plant at the current times is the first control of a string of possible controls solving a COLOC problem over a time-horizon beginning from the current time onwards.
Summary
MBPC is a feedback-control methodology dutiable to enforce efficiently hard constraints on the variables of the controlled system. It is shown that the method hinges upon a constrained open-loop option of the so-called receding �horizon control strategy. In the important case of time-invariant linear saturated ANCBI system.MBPC algorithms can be devised with the property of ensuring global feasibility/stability. Consideration on how to deal with disturbance and model uncertainties are also given. A presentation of a simplified form of MBPC, VIZ the PCG. Is finally discussed.
1. Introduction
Model-Based predictive control (MBPC) is conceptually a natural method for generating feedback control actions for linear and nonlinear plants subject to pointwise-in-time input and/or state-related constraints. A human being, for instanc, while driving a vehicle, generates steering-wheel commands by forecasting or predicting over a finite time-horizon the (possible)vehicle state-evolutions on the basic of vehicle current state and a virtual or potential steering-wheel command sequence. Then one among such sequences, then one among such sequences, is sorted out, which fulfills safety constraints and meets performance requirements. Only a short initial portion of such a sequence is applied by the driver to the steering-wheel, while its remaining part is discarded. After such initial portion is applied, the driver repeats the whole operation by restarting predictions over a moved-ahead or receded time-horizon from the updated vehicle state by as determined by the applied command. MBPC complies with the same logical scheme: the control sequence is computed. By solving on-line , over a finite control horizon an open-loop optimal control problem, given the plant dynamical model and current state. Though this computation hinges upon an open-loop problem, MBPC yields a feedback-control action. Indeed, similarly to the driver behavior in a discrete-time setting only the first control of the open-loop control sequence is applied to the plant, and according to the receding-horizon control philosophy, the whole optimization cycle is repeated at the subsequent time-instant based on the new plant-state. Because it involves a control horizon made up by only a finite number of time-steps,MBPC can be often calculated on-line by existing optimization routines so as to minimize a performance index in the presence of hard constraints on the time evolutions of input and/or state. MBPC ability of handling constraints is of paramount importance whenever constraints are part of the control design specifications,. In fact, constraints are typically present in applications ,as they stem from actuators saturations and/physically , safety or economical requirements. Despite the importance of constraints, there is a shortage of control methods for handling them effectively. The main reason for the interest of control engineering in MBPC is there its ability to systematically and effectively handle hard constraints. An important observation in this connection is that, in contrast to MBPC, in feedback-control system of more traditional type , e.g., LQG or H control. Constraints are indirectly enforced, by imposing, whenever possible, a conservative behavior at a performance-degradation expense. Other instances ehere MBPC can be advantageously used comprise unconstrained plants for which off-line computation of a control law is a difficult task as compared with on-line computations via receding-horizon control.
The presentation of MBPC given hereafter aims at enlightening the main features of the approach , related well- established risibility/ stability constructive arguments ,and current open problems. Con.
Sideration will be also give to the command governor, a specific control architecture of practical interest, which though introduced independently of MBPC in its recent developments has taken advantage of using conceptual tools of predictive control. For more specialized topics, the reader is referred to the three article level contribution dealing with MBPC viz:6.43.16.1 � MBPC for linear system �. 6.43.16.2 �MBPC for nonlinear system�, and 6.43.15.4 �adaptive predictive control�
The presentation is organized as follows : sect. 2 sets up the general ingredients of the constrained open-loop optimal control problem underlying any MBPC scheme. Sect. 3 describes the earliest and simplest form of a stabilizing MBPC algorithm. Sect. 4 introduces a convenient form of a membership(ellipsoidal) terminal state- constraint devised so as to improve in terms of feasibility the algorithm, of sect . 3 sect.5 extends the scheme of sect.4 by considering a state-dependent ellipsoidal constraint which allows one to get global feasibility/stability whenever such a property is achievable in principle . sect. 6 describes how to deal with constant disturbances and nonzero set-points, as well as model uncertainties of polytypic type. Sect . 7 describes predictive reference governors. In sect. 8 a brief assessments of the current status of MBPC concludes the contribution.
2. The constrained open-loop optimal control (COLOC) problem
In MBPC, the system to be controlled (plant) is usually represented by an ordinary differential equation. However as the control is normally piecewise constant the plant is most of the times described in terms of a difference equation
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