Polyhedral Feasible Set Computation of MPC-Based Optimal Control Problems

Feasible sets play an important role in model predictive control(MPC) optimal control problems(OCPs). This paper proposes a multi-parametric programming-based algorithm to compute the feasible set for OCP derived from MPC-based algorithms involving both spectrahedron(represented by linear matrix inequalities) and polyhedral(represented by a set of inequalities) constraints. According to the geometrical meaning of the inner product of vectors, the maximum length of the projection vector from the feasible set to a unit spherical coordinates vector is computed and the optimal solution has been proved to be one of the vertices of the feasible set. After computing the vertices,the convex hull of these vertices is determined which equals the feasible set. The simulation results show that the proposed method is especially efficient for low dimensional feasible set computation and avoids the non-unicity problem of optimizers as well as the memory consumption problem that encountered by projection algorithms.

A SYMBOLIC COMPUTATION METHOD TO DECIDE THE COMPLETENESS OF THE SOLUTIONS TO THE SYSTEM OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS

A symbolic computation method to decide whether the solutions to the system Of linear partial differential equation is complete via using differential algebra and characteristic set is presented. This is a mechanization method, and it can be carried out on the computer in the Maple environment.

Representations of Co-regular Subsets on Euclidean Spaces

For the computability of co-regular subsets in metric spaces, the properties of the co-regular subsets and several reasonable representations on co-regular sets have been suggested in this paper. As last, the ＇weaker or stronger＇ relations of these representations have been revealed.

Accurate level set method for simulations of liquid atomization

Computational fluid dynamics is an efficient numerical approach for spray atomization study, but it is challenging to accurately capture the gas-liquid interface. In this work, an accurate conservative level set method is intro- duced to accurately track the gas-liquid interfaces in liquid atomization. To validate the capability of this method, binary drop collision and drop impacting on liquid film are investigated. The results are in good agreement with experiment observations. In addition, primary atomization （swirling sheet atomization） is studied using this method. To the swirling sheet atomization, it is found that Rayleigh-Taylor instability in the azimuthal direction causes the primary breakup of liquid sheet and complex vortex structures are clustered around the rim of the liq- uid sheet. The effects of central gas velocity and liquid-gas density ratio on atomization are also investigated. This work lays a solid foundation for further studvin~ the mechanism of s^rav atomization.

An extension of Harrington's noncupping theorem

(i) Call a c.e. degree b anti-cupping relative to x, if there is a c.e. a < b such that for any c.e. w, w x implies a ∪ w b ∪ x.(ii) Call a c.e. degree b everywhere anti-cupping (e.a.c.), if it is anti-cupping relative to x for each c.e. degree x.By a tree method, we prove that every high c.e. degree has e.a.c. property by extending Harrington's anti-cupping theorem.

Non-Uniformity and Generalised Sacks Splitting

We show that there do not exist computable fimetions f_1(e,i).f_2(e,i).g_1(e,i),g_2(e,i)such that for all e,i ∈ω, (1)(W_(f_1)(e,i)-W_(f_2)(e,i))≤T(W_e-W_1): (2)(W_(g_1)(e,i)-W_(g_2)(e,i))≤T(W_e-W_i): (3)(W_w-W_i)≤T(W_(f_1)(e,i)-W_(f_2)(e,i))⊕(W_(g_1)(e,i)-W_(g_2)(e,i)): (4)(W_e-W_i)T(W_(f_1)(e,i)-W_(f_2)(e,i))uuless(W_e-W_i)≤T:and (5)(W_e-W_i)T(E_(g_1)(e,i)-W_(g_2)(e,i))unless(W_w-W_i)≤T. It follows that the splitting theorems of Sacks and Cooper cannot be combined uniformly.

Plus cupping degrees do not form an ideal

A computably enumerable (c.e.,for short)degree a is called plus cupping,if every c.e. degree x with 0<x≤a is cuppable.Let PC be the set of all plus cupping degrees.In the present paper,we show that PC is not closed under the join operation ∨ by constructing two plus cupping degrees which join to a high degree. So by the Harringtons noncupping theorem,PC is not an ideal of ε.

A Non-Recursive Algorithm Computing Set Expressions

This paper concerns set expressions satisfying some constraints. Although algorithms processing expressions (for every kind) have been proposed in compilers, almost all of them use stacks and recursions, and adopt complicated parsing techniques. However, the simplification of set expressions can be done easily by its special characteristics. In this paper, we propose linear, nonrecursive, and thus efficient algorithms simplifying set expressions. The method can be well used in proposition calculus.