電氣工程及其自動(dòng)化專業(yè)畢設(shè)外文翻譯--采樣數(shù)據(jù)模型預(yù)測(cè)控制-電氣類(文件)
【正文】 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。 see [8] for a discussion. 6 Robust Stability In the last years the synthesis of robust MPC laws is considered in different works [14]. The framework described below is based on the one in [9], extended to timevarying systems. Our objective is to drive to a given target set the state of the nonlinear system subject to bounded disturbances Since is finite, we conclude that the function is bounded and then that is also bounded. Therefore is bounded and, since f is continuous and takes values on bounded sets of is also bounded. Using the fact that x? is absolutely continuous and coincides with ?x at all sampling instants, we may deduce that and are also bounded. We are in the conditions to apply the previously established Generalization of Barbalat’s Lemma 3, yielding the assertion of the theorem. 7 Finite Parameterizations of the Control Functions The results on stability and robust stability were proved using an optimal control problem where the controls are functions selected from a very general set (the set of measurable functions taking values on a set U, subset of Rm). This is adequate to prove theoretical stability results and it even permits to use the results on existence of a minimizing solution to optimal control problems (. [7,Proposition 2]). However, for implementation, using any optimization algorithm, the control functions need to be described by a finite number of parameters (the so called finite parameterizations of the control functions). The control can be parameterized as piecewise constant controls (. [13]), polynomials or splines described by a finite number of coeficients, bangbang controls (. [9, 10]), that we are not considering discretization of the model or the dynamic equation. The problems of discrete approximations are discussed in detail . in [16] and [12]. But, in the proof of stability, we just have to show at some point that the optimal cost (the value function) is lower than the cost of using another admissible control. So, as long as the set of admissible control values U is constant for all time, an easy, but nevertheless important, corollary of the previous stability results follows If we consider the set of admissible control functions (including the auxiliary control law) to be a finitely parameterizable set such that the set of admissible control values is constant for all time, then both the nominal stability and robust stability results here described remain valid. An example, is the use of discontinuous feedback control strategies of bangbang type, which can be described by a small number of parameters and so make the problem putationally tractable. In bangbang feedback strategies, the
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