电气工程及其自动化专业毕设外文翻译

电气工程及其自动化专业毕设外文翻译内容摘要：

s, for example a carlike vehicle (see [8] for a discussion of this problem and this example). A similar approach can be used to deduce robust stability of MPC for systems allowing uncertainty. After establishing monotone decrease of the value function, we would want to guarantee that the state trajectory asymptotically approaches some set containing the origin. But, a difficulty encountered is thatthe predicted trajectory only coincides with the resulting trajectory at specificsampling instants. The robust stability properties can be obtained, as we show,using a generalized version of Barbalat’s lemma. These robust stability resultsare also valid for a very general class of nonlinear timevarying systems allowing discontinuous feedbacks. The optimal control problems to be solved within the MPC strategy are here formulated with very general admissible sets of controls (say, measurable control functions) making it easier to guarantee, in theoretical terms, the existence of solution. However, some form of finite parameterization of the control functionsis required/desirable to solve online the optimization problems. It can be shown that the stability or robustness results here described remain valid when the optimization is carried out over a finite parameterization of the controls, such as piecewise constant controls (as in [13]) or as bangbang discontinuous feedbacks (as in [9]). 2 A SampledData MPC Framework We shall consider a nonlinear plant with input and state constraints, where the evolution of the state after time t0 is predicted by the following model. The data of this model prise a set containing all possible initial states at the initial time t0, a vector xt0 that is the state of the plant measured at time t0, a given function of possible control values. We assume this system to be asymptotically controllable on X0 and that for all t ≥ 0 f(t, 0, 0) = 0. We further assume that the function f is continuous and locally Lipschitz with respect to the second argument. The construction of the feedback law is acplished by using a sampleddata MPC strategy. Consider a sequence of sampling instants π := {ti}i≥0 with a constant intersampling time δ 0 such that ti+1 = ti+δ for all i ≥ 0. Consider also the control horizon and predictive horizon, Tc and Tp, with Tp ≥ Tc δ, and an auxiliary control law kaux : IRIRn → IRm. The feedback control is obtained by repeatedly solving online openloop optimal control problems P(ti, xti, Tc, Tp) at each sampling instant ti ∈ π, every time using the current measure of the state of the plant xti . Note that in the interval [t + Tc, t + Tp] the control value is selected from a singleton and therefore the optimization decisions are all carried out in the interval [t, t + Tc] with the expected benefits in the putational time. The notation adopted here is as follows. The variable t represents real time while we reserve s to denote the time variable used in the prediction model. The vector xt denotes the actual state of the plant measured at time t. The process (x, u) is a pair trajectory/control obtained from the model of the system. The trajectory is sometimes denoted as s _→ x(s。 t, xt, u) when we want to make explicit the dependence on the initial time, initial state, and control function. The pair (ˉx, ˉu) denotes our optimal solution to an openloop optimal control problem. The process (x∗, u∗) is the closedloop trajectory and control resulting from the MPC strategy. We call design parameters the variables present in the openloop optimal control problem that are not from the system model (. variables we are able to choose)。 these prise the control horizon Tc, the prediction horizon Tp, the running cost and terminal costs functions L and W, the auxiliary control law kaux, and the terminal constraint set S ⊂ IRn. The resultant control law u∗ is a “samplingfeedback” control since during each sampling interval, the control u∗ is dependent on the state x∗(ti). More precisely the resulting trajectory is given by and the function t _→ _ t_π gi。