滚动时域控制原理.pdf

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The Receding Horizon Control Principle

Receding Horizon Control Mar´ıa M. Seron September 2004 Centre for Complex Dynamic Systems and Control

The Receding Horizon Control Principle Fixedhorizonoptimisationleads to a control sequence fui; : : : ; ui+N1g, which begins at the current time i and ends at some future time i + N 1. This ﬁxed horizon solution suffers from two potential drawbacks: (i) Something unexpected may happen to the system at some time over the future interval [i; i + N 1] that was not predicted by (or included in) the model. This would render the ﬁxed control choices fui; : : : ; ui+N1g obsolete. (ii) As one approaches the ﬁnal time i + N 1, the control law typically “gives up trying” since there is too little time to go to achieve anything useful in terms of objective function reduction. Centre for Complex Dynamic Systems and Control

The Receding Horizon Control Principle The above two problems are addressed by the idea of receding horizonoptimisation. This idea can be summarised as follows: (i) At time i and for the current state xi, solve an optimal control problem over a ﬁxed future interval, say [i; i + N 1], taking into account the currentand futureconstraints. (ii) Apply only the ﬁrst step in the resulting optimal control sequence. (iii) Measure the state reached at time i + 1. (iv) Repeat the ﬁxed horizon optimisation at time i + 1 over the future interval [i + 1; i + N], starting from the (now) current state xi+1. Centre for Complex Dynamic Systems and Control

The Receding Horizon Control Principle In the absence of disturbances, the state measured at step (iii) will be the same as that predicted by the model. Nonetheless, it seems prudent to use the measuredstate rather than the predicted state just to be sure. The above description assumes that the state is measured at time i + 1. In practice, one would use some form of observer to estimate xi+1 based on the available data. More will be said about the use of observers in the next lecture. For the moment, we will assume that the full state vector is measured and we will ignore the impact of disturbances. Centre for Complex Dynamic Systems and Control

The Receding Horizon Control Principle If the model and objective function are time invariant, then the same input ui will result whenever the state takes the same value. That is, the receding horizon optimisation strategy is really a particular time-invariantstatefeedbackcontrollaw: uk xk+1 = f(xk; uk) xk PSfrag replacements RHC In particular, we can set i = 0 in the formulation of the open loop control problem. Centre for Complex Dynamic Systems and Control

The Receding Horizon Control Principle More precisely, at the current time, and for the current state x, we solve: PN(x) : for k = 0; : : : ; N 1; Vopt N (x) , min VN(fxkg; fukg); subject to: xk+1 = f(xk; uk) x0 = x; uk 2 U for k = 0; : : : ; N 1; xk 2 X for k = 0; : : : ; N; xN 2 Xf X; where VN(fxkg; fukg) , F(xN) + N1 X k=0 L(xk; uk): (1) (2) (3) (4) (5) (6) (7) Centre for Complex Dynamic Systems and Control

The Receding Horizon Control Principle The sets U Rm, X Rn, and Xf Rn are the input, state and terminal constraint set, respectively. All sequences fukg = fu0; : : : ; uN1g and fxkg = fx0; : : : ; xNg satisfying the constraints (2)–(6) are called feasible sequences. A pair of feasible sequences fu0; : : : ; uN1g and fx0; : : : ; xNg constitute a feasiblesolution. The functions F and L in the objective function (7) are the terminal stateweightingand the per-stageweighting, respectively. Centre for Complex Dynamic Systems and Control

The Receding Horizon Control Principle In the sequel we make the following assumptions: f, F and L are continuous functions of their arguments; U Rm is a compact set, X Rn and Xf Rn are closed sets; there exists a feasible solution to problem (1)–(7). Because N is ﬁnite, these assumptions are sufﬁcient to ensure the existence of a minimum by Weierstrass’ theorem. Centre for Complex Dynamic Systems and Control

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