The latter notation denoting the state at time k resulting from state x at time t less than k and a control sequence u. the terminal time t+N increases with time t and, consequently , is referred to as a receding horizon. Various choices for the instantaneous loss j and the terminal loss L are in principle possible. However, according to the usual MBPC choice, hereafter the loss functions will be taken to be quadratic
It is to underlined that MBPC computes numerically on-line at the even (X t) the optimal control action (16) rather than computing off-line the optimal control law cN(.). it would be more convenient to explicitly compute off-line, once for all,cN(-) via dynamic programming. As this is usually very hard of even impossible, MBPC computes at the event(x.t) the optimal control action cN(x) rather than pre-computing the optimal control law cN(.).
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