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...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.展开更多
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 mechanizati...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.展开更多
(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 re...(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.展开更多
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)...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.展开更多
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 close...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.展开更多
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 com...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.展开更多
基金supported by the Natural Science Foundation of Zhejiang Province(LR17F030002)the Science Fund for Creative Research Groups of the National Natural Science Foundation of China(61621002)
文摘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.
文摘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.
基金supported by“863”project and the National Natural Science Foundation of China.
文摘(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.
基金supported by EPSRC Research Grant"Turing Definability"No.GR/M 91419(UK)+1 种基金partially supported by NSF Grant No.69973048supported by NSF Major Grant No.19931020 of P.R.CHINA
文摘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.
文摘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.
文摘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.
