Method for calculating Control Attainable Set of redundant drive system under linear constraint

Abstract

Provided is a method for calculating a control attainable set of a redundant drive system under a linear constraint, which relates to the technical field of dynamics control allocation of the redundant drive system. The method first constructs a control attainable set problem for a redundant drive system under each pair of linear constraint control components, and then classifies boundary surfaces corresponding to a control set into three types of rectangular boundary surfaces and one type of triangular boundary surface. By grouping the boundary surfaces, the method determines a key boundary surface for each group to form a boundary surface set. After removing duplicate boundary surfaces from the boundary surface set, the method calculates a boundary surface of a control attainable set, and finally obtains the control attainable set.

Claims

1. A method for calculating a control attainable set of a redundant drive system under a linear constraint, comprising following steps: 1) establishing an expression of a control attainable set of a redundant drive system under each pair of linear constraint control components as follows: $$\begin{gathered} \begin{array}{cc}=\left\{v.Math.v=B.Math.u,u={\left(u_1,.Math.,u_m\right)}^T,u\right\} \\ s.t \\ {u}_{i\min }{u}_i{u}_{i\max },i=1,L,m \\ \left(u_{k\max }-u_{k\min }\right)u_{k+1}+\left(u_{k+1\max }-u_{k+1\min }\right)u_k{u}_{k+1\max }{u}_{k\max }-u_{k+1\min }{u}_{k\min }, \\ k=1,3,.Math.,2\left[\frac{m}{2}\right]-1& \left(1\right)\end{array} \end{gathered}$$ wherein [m/2] represents an integer part obtained by dividing m by 2, u represents a control vector, the u is denoted as u.sub.ELR, u.sub.ELR=(u.sub.1 . . . , u.sub.m).sup.T, u.sub.iminu.sub.iu.sub.imax, i=1,L,m, (u.sub.kmaxu.sub.kmin)u.sub.k+1+(u.sub.k+1maxu.sub.k+1min)u.sub.ku.sub.k+1maxu.sub.kmaxu.sub.k+1minu.sub.kmin, and k=1,3, . . . , 2[m/2]1; wherein an i.sup.th control component u.sub.i is a control action of a corresponding i.sup.th actuator, wherein 1m, and m represents a quantity of actuators; u.sub.imin represents a minimum constraint value of the control action of the i.sup.th actuator, and u.sub.imax represents a maximum constraint value of the control action of the i.sup.th actuator; represents a control set, and ={u}; v represents a control attainable vector of the redundant drive system, and v=(v.sub.1, v.sub.2, v.sub.3).sup.T, which represents a control output of the redundant drive system; represents the control attainable set; and B represents a 3-row and m-column control efficiency matrix; and the expression (1) is rewritten to a following expression (2): $$\begin{gathered} \begin{array}{cc}{}_{\mathrm{ELR}}=\left\{v.Math.v=B.Math.u_{\mathrm{ELR}},u_{\mathrm{ELR}}={\left(u_1,.Math.,u_m\right)}^T,u_{\mathrm{ELR}}{}_{\mathrm{ELR}}\right\} \\ s.t. \\ {u}_{i\min }{u}_i{u}_{i\max },i=1,L,m; \\ \left(u_{k\max }-u_{k\min }\right)u_{k+1}+\left(u_{k+1\max }-u_{k+1\min }\right)u_k{u}_{k+1\max }{u}_{k\max }-u_{k+1\min }{u}_{k\min }, \\ k=1,3,.Math.,2\left[\frac{m}{2}\right]-1& \left(2\right)\end{array} \end{gathered}$$ wherein u.sub.k and u.sub.k+1 represent a pair of control components with a linear constraint, .sub.ELR represents a control attainable set of a redundant drive system with each pair of control components being linear constraint control components, .sub.ELR represents a control set, u.sub.ELR represents a control vector, and .sub.ELR={u.sub.ELR}; 2) classifying all boundary surfaces of the control set .sub.ELR obtained in the step 1) into four types, which comprises: using, as a vertex of the control set .sub.ELR, a point obtained by taking a corresponding maximum or minimum constraint value of each component u.sub.i of the control vector u.sub.ELR=(u.sub.1, . . . , u.sub.m).sup.T, wherein the m-dimensional control set .sub.ELR represented in the expression (2) has $3^{\left[\frac{m}{2}\right]}.Math.2^{\left(\frac{m}{2}\right)}$ vertices, and (m/2) represents a remainder of dividing m by 2; and setting (.sub.ELR) to represent a boundary of the .sub.ELR, and classifying boundary surfaces of the (.sub.ELR) into a type-rectangular boundary surface, a type-rectangular boundary surface, a type=2\*ROMAN I rectangular boundary surface, and a triangular boundary surface based on a value of each component of each vertex, which specifically comprises: assuming that four vertices of the rectangular boundary surface are denoted as A, B, D, and C in clockwise order, and three vertices of the triangular boundary surface are denoted as A, C, and B in clockwise order, if any rectangular boundary surface satisfies following conditions: for the vertices A and B, one component has a different value, and other components have a same value; for the vertices A and C, one component of another pair of components has a different value, and other components have a same value; and for the vertices A and D, the above two components have different values, and other components have a same value, determining that the rectangular boundary surface is the type-rectangular boundary surface; if any rectangular boundary surface satisfies following conditions: for the vertex A and the vertex B, one pair of linear constraint components have different values, and other components have a same value; for the vertex C and the vertex D, the one pair of linear constraint components have different values, and other components have a same value; and for the vertex A and the vertex C, the one pair of linear constraint components have a same value, and other components have a same value except that one of the other components has a different value, determining that the rectangular boundary surface is the type-rectangular boundary surface; if any rectangular boundary surface satisfies following conditions: for the vertex A and the vertex B, one pair of linear constraint components have different values, and other components have a same value; for the vertex A and the vertex C, another pair of linear constraint components have different values, and other components have a same value; and for the vertex D and the vertex A, each of the above two pairs of linear constraint components has a different value, and other components have a same value, determining that the rectangular boundary surface is the type=2\*ROMAN I rectangular boundary surface; and if any right angle satisfies following conditions: for the vertex A and the vertex B, one pair of linear constraint components have different values, and other components have a same value; for the vertex A and the vertex C, one component of the one pair of linear constraint components has a different value, and other components have a same value; and for the vertex B and the vertex C, the other component of the one pair of linear constraint components has a different value, and other components have a same value, determining that the right angle is the triangular boundary surface; 3) dividing all the boundary surfaces of the control set .sub.ELR into C.sub.m.sup.2 groups, wherein if values of two components in the u.sub.ELR are between the corresponding minimum and maximum values, and values of other m2 components are the corresponding minimum or maximum value, the m components form 2.sup.m2 boundary surfaces of the control set; and if any two components in the u.sub.ELR are denoted as a p.sup.th component and a q.sup.th component, values of the p.sup.th component and the q.sup.th component are between the corresponding minimum and maximum values, 1pm, 1qm, p<q, values of other m2 components are the corresponding minimum or maximum value, and a formed boundary surface is denoted as a p-q group, all the boundary surfaces of the control set are divided into the C.sub.m.sup.2 groups; 4) determining a key boundary surface for each of the C.sub.m.sup.2 groups in the step 3), wherein a boundary surface of an image mapped onto the .sub.ELR in the .sub.ELR of a boundary (.sub.ELR) is denoted as the key boundary surface, the is denoted as a set of key boundary surfaces, the is initialized as an empty set, and the step comprises following substeps: 4-1) selecting any group for which no key boundary surface has been determined, denoting the group as a current p-q group, and performing a step 4-2); 4-2) determining whether the p.sup.th control component and the q.sup.th control component are a pair of linear constraint components; and if they are a pair of linear constraint components, performing a step 4-3); if they are not a pair of linear constraint components, and m is an even number, performing a step 4-4); if they are not a pair of linear constraint components, m is an odd number, and the p.sup.th and q.sup.th control components each have a paired linear constraint component, performing a step 4-4); or if they are not a pair of linear constraint components, m is an odd number, and the q.sup.th control component has no paired linear constraint component, performing a step 4-5); 4-3) when the p.sup.th control component and the q.sup.th control component are a pair of linear constraint components, determining a key boundary surface of the P-q group according to a following specific method: 4-3-1) when the p.sup.th control component and the q.sup.th control component are a pair of linear constraint components, determining that all boundary surfaces of the group are triangular boundary surfaces, wherein a value of each component of each point on the boundary surface satisfies a following formula: $\begin{array}{cc}\{\begin{array}{c}{u}_{p\min }{u}_p{u}_{p\max }\\ u_{q\min }{u}_q\frac{u_{q\max }{u}_{p\max }-u_{q\min }{u}_{p\min }-\left(u_{q\max }-u_{q\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)}\\ u_i\left\{u_{i\min },u_{i\max }\right\},i{Z}^{+},1im,ip,q\end{array}& \left(3\right)\end{array}$ wherein a vertex of each boundary surface in the group is represented by an m-row and 3-column matrix , and 3 columns are vectors corresponding to 3 vertices; values of the p.sup.th and q.sup.th components of three vertices on each boundary surface are shown in p.sup.th and q.sup.th rows of the matrix , while values of other components are all the u.sub.imax or the u.sub.imin; one pair of linear constraint components in the other components do not simultaneously take a maximum constraint value corresponding to the one pair of linear constraint components; and the p.sup.th row of the matrix is (u.sub.pmax, u.sub.pmin, u.sub.pmin), and the q.sup.th row is (u.sub.qmin, u.sub.qmin, u.sub.qmax); $=\left[\begin{array}{ccc}.Math.& .Math.& .Math.\\ u_{p\max }& u_{p\mathrm{mi}}& u_{p\mathrm{mi}}\\ u_{q\mathrm{mi}}& u_{q\mathrm{mi}}& u_{q\max }\\ .Math.& .Math.& .Math.\end{array}\right];$ 4-3-2) constructing a coordinate rotation transformation .sup.1T, such that a transformed coordinate axis v.sub.1 is perpendicular to an image of the triangular boundary surface of the group, which specifically comprises: setting ${}^{1}T=\left[\begin{array}{ccc}{}^1{t}_{11}& {}^1{t}_{12}& {}^1{t}_{13}\\ {}^1{t}_{21}& {}^1{t}_{22}& {}^1{t}_{23}\\ {}^1{t}_{31}& {}^1{t}_{32}& {}^1{t}_{33}\end{array}\right],B=\left[\begin{array}{c}{b}_{11}.Math.b_{1p}.Math.b_{1q}.Math.b_{1m}\\ b_{21}.Math.b_{2p}.Math.b_{2q}.Math.b_{2m}\\ b_{31}.Math.b_{3p}.Math.b_{3q}.Math.b_{3m}\end{array}\right],\mathrm{and}{}^{1}C=\left[\begin{array}{c}{}^1{c}_{11}.Math.{}^1{c}_{1p}.Math.{}^1{c}_{1q}.Math.{}^1{c}_{1m}\\ {}^1{c}_{21}.Math.{}^1{c}_{2p}.Math.{}^1{c}_{2q}.Math.{}^1{c}_{2m}\\ {}^1{c}_{31}.Math.{}^1{c}_{3p}.Math.{}^1{c}_{3q}.Math.{}^1{c}_{3m}\end{array}\right],$ wherein .sup.1c.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.1C, .sup.1t.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.1T, and b.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix B; setting .sup.1C=.sup.1T.Math.B, such that .sup.1c.sub.1p=0 and .sup.1c.sub.1q=0; substituting the .sup.1T and the B into the .sup.1C=.sup.1T.Math.B to obtain a following equation set: $\begin{array}{cc}\{\begin{array}{c}\underset{s=1}{\overset{3}{.Math.}}{}^1{t}_{1s}.Math.b_{\mathrm{sp}}=0\\ \underset{s=1}{\overset{3}{.Math.}}{}^1{t}_{1s}.Math.b_{\mathrm{sq}}=0\end{array}& \left(4\right)\end{array}$ solving the linear equation set (4) to obtain a first row (.sup.1t.sub.11, .sup.1t.sub.12, .sup.1t.sub.13) of the .sup.1T; and calculating a first row (.sup.1c.sub.11, . . . , .sup.1c.sub.1m) of the matrix .sup.1C according to the .sup.1C=.sup.1T.Math.B; 4-3-3) processing other rows than the rows p, q of the matrix according to a following rule, wherein a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set , and 1im, i is an integer, ip, q; and the rule is as follows: when .sup.1c.sub.1i.Math..sup.1c.sub.1i+1<0, if .sup.1c.sub.1i>0 and .sup.1c.sub.1i+1<0, setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1 min; or if .sup.1c.sub.1i<0 and .sup.1c.sub.1i+1<0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; when .sup.1c.sub.1i<0and .sup.1c.sub.1i+1<0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; or when .sup.1c.sub.1i>0 and .sup.1c.sub.1i+1>0, calculating .sup.1z.sub.01=.sup.1c.sub.1i.Math.u.sub.imax+.sup.1c.sub.1i+1.Math.u.sub.i+1min and .sup.1z.sub.02=.sup.1c.sub.1i.Math.u.sub.imin+.sup.1c.sub.1i+1.Math.u.sub.i+1max, and then performing determining as follows: if .sup.1z.sub.01>.sup.1z.sub.02, setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; if .sup.1z.sub.01<.sup.1z.sub.02 setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if .sup.1z.sub.01=.sup.1z.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix as follows: setting the i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max; 4-3-4) processing other rows than the rows p, q of the matrix according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set ; and then performing a step 4-6), wherein 1im,i is an integer, ip, q; and the rule is as follows: when .sup.1c.sub.1i.Math..sup.1c.sub.1i+1<0, if .sup.1c.sub.1i>0 and .sup.1c.sub.1i+1<0, setting the i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to the u.sub.i+1max; or if .sup.1c.sub.1i<0 and .sup.1c.sub.1i+1>0, setting the i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to u.sub.i+1 min; when .sup.1c.sub.1i>0 and .sup.1c.sub.1i+1>0, setting the i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to the u.sub.i+1min; or when .sup.1c.sub.1i<0 and .sup.1c.sub.1i+1<0, calculating .sup.2z.sub.01=.sup.1c.sub.1i.Math.u.sub.imax+.sup.1c.sub.1i+1.Math.u.sub.i+1min and .sup.2z.sub.02=.sup.1c.sub.1i.Math.u.sub.imin+.sup.1c.sub.1i+1.Math.u.sub.i+1max and performing determining as follows: if .sup.2z.sub.01>.sup.2z.sub.02, setting the i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to the u.sub.imax; if .sup.2z.sub.01<.sup.2z.sub.02, setting the i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to the u.sub.i+1min; or if .sup.1z.sub.01=.sup.1z.sub.02, setting the i.sup.th row and the i+1.sup.th row of the matrix as follows: setting the i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to the u.sub.i+1min; or setting the i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to the u.sub.i+1max; 4-4) when the p.sup.th control component and the q.sup.th control component are not a pair of linear constraint components, and each have the paired linear constraint component, determining a key boundary surface of the p-q group according to a following specific method: 4-4-1) denoting the component paired with the p.sup.th component as a p component, and the component paired with the q.sup.th component as a q.sup.th component, where the p-q group has three types of rectangular boundary surfaces, and each component of each point on the boundary surfaces of the group has a value that satisfies a following formula: $\begin{array}{cc}\{\begin{array}{c}{u}_{\mathrm{pmin}}{u}_p{u}_{p\max }\\ u_{\mathrm{qmin}}{u}_q{u}_{q\max }\\ u_{p^{}}\left\{u_{p^{}\min },\frac{u_{p^{}\max }{u}_{p\max }-u_{p^{}\min }{u}_{p\min }-\left(u_{p^{}\max }-u_{p^{}\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)}\right\}\\ u_q\left\{u_{q^{}\min },\frac{u_{q^{}\max }{u}_{q\max }-u_{q^{}\min }{u}_{q\min }-\left(u_{q^{}\max }-u_{q^{}\min }\right)u_q}{\left(u_{q\max }-u_{q\min }\right)}\right\}\\ u_i\left\{u_{i\min },u_{i\max }\right\},1im,ip,q,p^{},q^{}\\ u_k+u_{k+1}<u_{k\max }+u_{k+1\max },k=1,3,.Math.,2\left[\frac{m}{2}\right]-1,kp,q,p^{},q^{}\end{array}& \left(5\right)\end{array}$ when u.sub.p=u.sub.pmin and u.sub.q=u.sub.qmin, determining that a boundary surface formed by a point satisfying the formula (5) is the type-I rectangular boundary surface; when u.sub.p=u.sub.pmin and $u_{q^{}}=\frac{u_{q^{}\mathrm{mx}}{u}_{q\max }-u_{q^{}\min }{u}_{q\min }-\left(u_{q^{}\max }-u_{q^{}\min }\right)u_q}{\left(u_{q\max }-u_{q\min }\right)},$ determining that a boundary surface formed by a point satisfying the formula (5) is the type-II rectangular boundary surface; when $u_{p^{}}=\frac{u_{p^{}\mathrm{mx}}{u}_{p\max }-u_{p^{}\min }{u}_{p\min }-\left(u_{p^{}\max }-u_{p^{}\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)}$ and u.sub.q=u.sub.qmin, determining that a boundary surface formed by a point satisfying the formula (5) also is the type-II rectangular boundary surface; when $u_{p^{}}=\frac{u_{p^{}\mathrm{mx}}{u}_{p\max }-u_{p^{}\min }{u}_{p\min }-\left(u_{p^{}\max }-u_{p^{}\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)}\mathrm{and}{u}_{q^{}}=\frac{u_{q^{}\mathrm{mx}}{u}_{q\max }-u_{q^{}\min }{u}_{q\min }-\left(u_{q^{}\max }-u_{q^{}\min }\right)u_q}{\left(u_{q\max }-u_{q\min }\right)},$ determining that a boundary surface formed by a point satisfying the formula (5) is the type-III rectangular boundary surface; and representing a vertex of each rectangular boundary surface of the group by using an m-row and 4-column matrix, wherein 4 columns respectively represent vectors corresponding to 4 vertices; a matrix represents a vertex of one type-I rectangular boundary surface, a matrix .sup.1 or .sup.2 represents a vertex of one type-II rectangular boundary surface, and a matrix T represents a vertex of one type-III rectangular boundary surface; values of the p.sup.th, p.sup.th, q.sup.th, and q.sup.th components of four vertices of each boundary surface are shown in a p.sup.th row, a p.sup.th row, a q.sup.th row, and a q.sup.th row of the matrix, and other components are all the u.sub.imax or the u.sub.imin; one pair of linear inequality constraint components do not simultaneously take a maximum constraint value corresponding to the one pair of linear inequality constraint components; for the , the p.sup.th row is (u.sub.pmax, u.sub.pmin, u.sub.pmin, u.sub.pmax), the p.sup.th row is (u.sub.pmin, u.sub.pmin, u.sub.pmin, u.sub.pmin), the q.sup.th row is (u.sub.qmin, u.sub.qmin, u.sub.qmax, u.sub.qmax), and the q.sup.th row is (u.sub.qmin, u.sub.qmin, u.sub.qmin, u.sub.qmin); for the .sup.1, the p.sup.th row is (u.sub.pmax, u.sub.pmin, u.sub.pmin, u.sub.pmax), the p.sup.th row is (u.sub.pmin, u.sub.pmax, u.sub.pmax, u.sub.pmin), the q.sup.th row is (u.sub.qmin, u.sub.qmin, u.sub.qmax, u.sub.qmax), and the q.sup.th row is (u.sub.qmin, u.sub.qmin, u.sub.qmin, u.sub.qmin); for the .sup.2, the p.sup.th row is (u.sub.pmax, u.sub.pmin, u.sub.pmin, u.sub.pmax), the p.sup.th row is (u.sub.pmin, u.sub.pmin, u.sub.pmin, u.sub.pmin, the q.sup.th row is (u.sub.qmin, u.sub.qmin, u.sub.qmax, u.sub.qmax), and the q.sup.th row is (u.sub.qmax, u.sub.qmax, u.sub.qmin, u.sub.qmin); and for the T, the p.sup.th row is (u.sub.pmax, u.sub.pmin, u.sub.pmin, u.sub.pmax), the p.sup.th row is (u.sub.pmin, u.sub.pmax, u.sub.pmax, u.sub.pmin), the q.sup.th row is (u.sub.qmin, u.sub.qmin, u.sub.qmax, u.sub.qmax), and the q.sup.th row is (u.sub.qmax, u.sub.qmax, u.sub.qmin, u.sub.qmin); $\begin{array}{cc}=\left[\begin{array}{cccc}.Math.& .Math.& .Math.& .Math.\\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{}\min }& u_{p^{}\min }& u_{p^{}\min }& u_{p^{}\min }\\ .Math.& .Math.& .Math.& .Math.\\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\\ u_{q^{}\min }& u_{q^{}\min }& u_{q^{}\min }& u_{q^{}\min }\\ .Math.& .Math.& .Math.& .Math.\end{array}\right],{}^1=\left[\begin{array}{cccc}.Math.& .Math.& .Math.& .Math.\\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{}\min }& u_{p^{}\max }& u_{p^{}\max }& u_{p^{}\min }\\ .Math.& .Math.& .Math.& .Math.\\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\\ u_{q^{}\min }& u_{q^{}\min }& u_{q^{}\min }& u_{q^{}\min }\\ .Math.& .Math.& .Math.& .Math.\end{array}\right],{}^2=\left[\begin{array}{cccc}.Math.& .Math.& .Math.& .Math.\\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{}\min }& u_{p^{}\min }& u_{p^{}\min }& u_{p^{}\min }\\ .Math.& .Math.& .Math.& .Math.\\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\\ u_{q^{}\max }& u_{q^{}\max }& u_{q^{}\min }& u_{q^{}\min }\\ .Math.& .Math.& .Math.& .Math.\end{array}\right],T=\left[\begin{array}{cccc}.Math.& .Math.& .Math.& .Math.\\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{}\min }& u_{p^{}\max }& u_{p^{}\max }& u_{p^{}\min }\\ .Math.& .Math.& .Math.& .Math.\\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\\ u_{q^{}\max }& u_{q^{}\max }& u_{q^{}\min }& u_{q^{}\min }\\ .Math.& .Math.& .Math.& .Math.\end{array}\right]& \left(6\right)\end{array}$ wherein if P is an odd number, the p.sup.th row comes before the p.sup.th row; if p is an even number, the p.sup.th row comes after the p.sup.th row; if q is an odd number, the q.sup.th row comes before the q.sup.th row; and if q is an even number, the q.sup.th row comes after the q.sup.th row; 4-4-2) constructing a coordinate rotation transformation G, such that a transformed coordinate axis v.sub.1 is perpendicular to an image of the type-III rectangular boundary surface, and images of all points on the boundary surface have equal coordinate values on the v.sub.1 axis, which specifically comprises: setting $G=\left[\begin{array}{ccc}{g}_{11}& g_{12}& g_{13}\\ g_{21}& g_{22}& g_{23}\\ g_{31}& g_{32}& g_{33}\end{array}\right],B=\left[\begin{array}{ccccccccc}{b}_{11}& .Math.& b_{1p}& b_{1p^{}}& .Math.& b_{1q}& b_{1q^{}}& .Math.& b_{1m}\\ b_{21}& .Math.& b_{2p}& b_{2p^{}}& .Math.& b_{2q}& b_{2q^{}}& .Math.& b_{2m}\\ b_{31}& .Math.& b_{3p}& b_{3p^{}}& .Math.& b_{3q}& b_{3q^{}}& .Math.& b_{3m}\end{array}\right],\mathrm{and}E=\left[\begin{array}{ccccccccc}{e}_{11}& .Math.& e_{1p}& e_{1p^{}}& .Math.& e_{1q}& e_{1q^{}}& .Math.& e_{1m}\\ e_{21}& .Math.& e_{2p}& e_{2p^{}}& .Math.& e_{2q}& e_{2q^{}}& .Math.& e_{2m}\\ e_{31}& .Math.& e_{3p}& e_{3p^{}}& .Math.& e_{3q}& e_{3q^{}}& .Math.& e_{3m}\end{array}\right],$ wherein g.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix G, and e.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix E; setting E=G.Math.B, such that $e_{1p^{}}=\frac{u_{p\max }-u_{p\min }}{u_{p^{}\max }-u_{p^{}\min }}{e}_{1p},e_{1q^{}}=\frac{u_{q\max }-u_{q\min }}{u_{q^{}\max }-u_{q^{}\min }}{e}_{1q};$ and substituting the G and the B into the E=G.Math.B to obtain a following equation set: $\begin{array}{cc}\{\begin{array}{c}\underset{s=1}{\overset{3}{.Math.}}{g}_{1s}.Math.b_{\mathrm{sp}}=e_{1p}\\ \underset{s=1}{\overset{3}{.Math.}}{g}_{1s}.Math.b_{{\mathrm{sp}}^{}}=\frac{u_{p\max }-u_{p\min }}{u_{p^{}\max }-u_{p^{}\min }}{e}_{1p}\\ \underset{s=1}{\overset{3}{.Math.}}{g}_{1s}.Math.b_{\mathrm{sq}}=e_{1q}\\ \underset{s=1}{\overset{3}{.Math.}}{g}_{1s}.Math.b_{{\mathrm{sq}}^{}}=\frac{u_{q\max }-u_{q\min }}{u_{q^{}\max }-u_{q^{}\min }}{e}_{1q}\end{array}& \left(7\right)\end{array}$ solving the linear equation set (7) to obtain a first row of the G, and calculating a first row (e.sub.11, . . . , e.sub.1m) of the matrix E according to the E=G.Math.B; and if e.sub.1p>0 and e.sub.1q>0, performing a step 4-4-3); if e.sub.1p<0 and e.sub.1q<0, performing a step 4-4-4); or if e.sub.1p.Math.e.sub.1q<0, performing a step 4-4-5); 4-4-3) processing other rows than the rows p, q, p, q of the matrix T according to a following rule, wherein a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set ; and then performing the step 4-4-5), wherein i=1, . . . , m, ip, p, q, q; and the rule is as follows: when e.sub.1i.Math.e.sub.1i+1<0, if e.sub.1i>0 and e.sub.1i1<0, setting an i.sup.th row of the matrix T to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or if e.sub.1i<0 and e.sub.1i+1>0, setting an i.sup.th row of the matrix T to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; when e.sub.1i<0 and e.sub.1i+1<0 .sup.(i=1, . . . , m,ip, p,q,q), setting an i.sup.th row of the matrix T to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; or when e.sub.1i>0 and e.sub.1i+1>0, calculating .sup.1d.sub.01=e.sub.1i.Math.u.sub.imax+e.sub.1i+1min and .sup.1d.sub.02=e.sub.1i.Math.u.sub.imin+e.sub.1i+1max, and performing determining as follows: if .sup.1d.sub.01>.sup.1d.sub.02, setting an i.sup.th row of the matrix T to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; if .sup.1d.sub.01<.sup.1d.sub.02, setting an i.sup.th row of the matrix T to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if .sup.1d.sub.01=.sup.1d.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix T as follows: setting the i.sup.th row of the matrix T to the u.sub.imax and the i+1.sup.th row to the u.sub.i+1min; or setting the i.sup.th row of the matrix T to the u.sub.imin and the i+1.sup.th row to the u.sub.i+1max; 4-4-4) processing other rows than the rows p, q, p, q of the matrix T according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set ; and then performing the step 4-4-5), where i=1, . . . , m, ip, p, q, q; and the rule is as follows: when e.sub.1i.Math.e.sub.1i+1<0, if e.sub.1i>0 and e.sub.1i+1<0, setting an i.sup.th row of the matrix T to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if e.sub.1i<0 and e.sub.1i+1>0, setting an i.sup.th row of the matrix T to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; when e.sub.1i<0 and e.sub.1i+1<0, calculating .sup.2d.sub.01=.sup.2h.sub.1i.Math.u.sub.imax+.sup.2h.sub.1i+1.Math.+u.sub.i+1min and .sup.2d.sub.02=.sup.2h.sub.1i.Math.u.sub.imin+.sup.2h.sub.1i+1.Math.u.sub.i+1max and if .sup.2d.sub.01>.sup.2d.sub.02, setting an i.sup.th row of the matrix T to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; if .sup.2d.sub.01<.sup.2d.sub.02, setting an i.sup.th row of the matrix T to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or if .sup.2d.sub.01=.sup.2d.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix T as follows: setting the i.sup.th row of the matrix T to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix T to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max; or when e.sub.1i>0 and e.sub.1i+1>0, setting an i.sup.th row of the matrix T to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; 4-4-5) constructing a coordinate rotation transformation .sup.2T, such that a transformed coordinate axis v.sub.1 is perpendicular to an image of the type-I rectangular boundary surface of the group, which specifically includes: setting ${}^{2}T=\left[\begin{array}{ccc}{}_{}^2{t}_{11}& {}_{}^2{t}_{12}& {}_{}^2{t}_{13}\\ {}_{}^2{t}_{21}& {}_{}^2{t}_{22}& {}_{}^2{t}_{23}\\ {}_{}^2{t}_{31}& {}_{}^2{t}_{32}& {}_{}^2{t}_{33}\end{array}\right],B=\left[\begin{array}{c}{b}_{11}.Math.b_{1p}.Math.b_{1q}.Math.b_{1m}\\ b_{21}.Math.b_{2p}.Math.b_{2q}.Math.b_{2m}\\ b_{31}.Math.b_{3p}.Math.b_{3q}.Math.b_{3m}\end{array}\right],\mathrm{and}$ ${}^{2}C=\left[\begin{array}{c}{}_{}^2{c}_{11}.Math.{}_{}^2{c}_{1p}.Math.{}_{}^2{c}_{1q}.Math.{}_{}^2{c}_{1m}\\ {}_{}^2{c}_{21}.Math.{}_{}^2{c}_{2p}.Math.{}_{}^2{c}_{2q}.Math.{}_{}^2{c}_{2m}\\ {}_{}^2{c}_{31}.Math.{}_{}^2{c}_{3p}.Math.{}_{}^2{c}_{3q}.Math.{}_{}^2{c}_{3m}\end{array}\right],$ wherein .sup.2C.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.2C, and .sup.2t.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.2T; setting .sup.2C=.sup.2T.Math.B, such that .sup.2c.sub.1p=0 and .sup.2C.sub.1q=0; and substituting the .sup.2T and the B into the .sup.2C=.sup.2T.Math.B to obtain a following equation set: $\begin{array}{cc}\{\begin{array}{c}\underset{s=1}{\overset{3}{.Math.}}{}_{}^2{t}_{1_2}.Math.b_{\mathrm{sp}}=0\\ \underset{s=1}{\overset{3}{.Math.}}{}_{}^2{t}_{1_2}.Math.b_{\mathrm{sq}}=0\end{array}& \left(8\right)\end{array}$ solving the linear equation set (8) to obtain a first row (.sup.2t.sub.11, .sup.2t.sub.12, .sup.2t.sub.13) of the .sup.2T, and calculating a first row (.sup.2C.sub.11, . . . , .sup.2C.sub.1m) of the matrix .sup.2C according to the .sup.2C=.sup.2T.Math.B; and if .sup.2c.sub.1p<0 and .sup.2C.sub.1q<0 performing a step 4-4-6); if .sup.2c.sub.1p>0 and .sup.2C.sub.1q>0, performing a step 4-4-7); or if .sup.2c.sub.1p.Math..sup.2C.sub.1q<0, performing a step 4-4-8); 4-4-6) processing other rows than the rows p, q, p, q of the matrix according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set , and i=1, . . . , m, ip, p, q, q; and then performing determining as follows: if two key boundary surfaces have been found for the group, performing a step 4-6); otherwise, performing the step 4-4-8); where the rule is as follows: when .sup.2c.sub.1i.Math..sup.2c.sub.1i+1<0, if .sup.2c.sub.1i>0 and .sup.2c.sub.1i+1<0 setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or if .sup.2c.sub.1i<0 and .sup.2c.sub.1i+1>0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; when .sup.2c.sub.1i<0 and .sup.2c.sub.1i+1<0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; or when .sup.2c.sub.1i>0 and .sup.2c.sub.1i+1>0, calculating .sup.3d.sub.01=.sup.2c.sub.1i.Math.u.sub.imax+.sup.2c.sub.1i+1.Math.u.sub.i+1min, and .sup.3d.sub.02=.sup.2c.sub.1i.Math.u.sub.imin+.sup.2c.sub.1i+1.Math.u.sub.i+1max; and if .sup.3d.sub.01>.sup.3d.sub.02, setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; if .sup.3d.sub.01<.sup.3d.sub.02, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if .sup.3d.sub.01=.sup.3d.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix as follows: setting the i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max; 4-4-7) processing other rows than the rows p, q, p, q of the matrix according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set , and i=1, . . . , m, ip, p, q, q; and then performing determining as follows: if two key boundary surfaces have been found for the group, performing a step 4-6); otherwise, performing the step 4-4-8); where the rule is as follows: when .sup.2c.sub.1i.Math..sup.2c.sub.1i+1<0, if .sup.2c.sub.1i>0 and .sup.2c.sub.1i+1<0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if .sup.2c.sub.1i<0 and .sup.2c.sub.1i+1>0 setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; when .sup.2c.sub.1i<0 and .sup.2c.sub.1i+1<0, calculating .sup.4d.sub.01=.sup.2c.sub.1i.Math.u.sub.imax+.sup.2c.sub.1i+1.Math.u.sub.i+1min and .sup.4d.sub.02=.sup.2c.sub.1i.Math.u.sub.imin+.sup.2c.sub.1i+1.Math.u.sub.i+1max; and if .sup.4d.sub.01>.sup.4d.sub.02, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; if .sup.4d.sub.01<.sup.4d.sub.02, setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or if .sup.4d.sub.01=.sup.4d.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix as follows: setting the i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max; or when .sup.2c.sub.1i>0 and .sup.2c.sub.1i+1>0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1 min; 4-4-8) constructing a coordinate rotation transformation .sup.1F, such that a transformed coordinate axis v.sub.1 is Perpendicular to an Image of a Boundary Surface Whose Vertex Matrix is the .sup.1 in the group, and images of all points on the boundary surface have equal coordinate values on the v.sub.1 axis, which specifically includes: setting ${}^{1}F=\left[\begin{array}{ccc}{}_{}^1{f}_{11}& {}_{}^1{f}_{12}& {}_{}^1{f}_{13}\\ {}_{}^1{f}_{21}& {}_{}^1{f}_{22}& {}_{}^1{f}_{23}\\ {}_{}^1{f}_{31}& {}_{}^1{f}_{32}& {}_{}^1{f}_{33}\end{array}\right],B=\left[\begin{array}{cccccccc}{b}_{11}& .Math.& b_{1p}& b_{1p^{}}& .Math.& b_{1q}& .Math.& b_{1m}\\ b_{21}& .Math.& b_{2p}& b_{2p^{}}& .Math.& b_{2q}& .Math.& b_{2m}\\ b_{31}& .Math.& b_{3p}& b_{3p^{}}& .Math.& b_{3q}& .Math.& b_{3m}\end{array}\right],\mathrm{and}$ ${}^{1}H=\left[\begin{array}{cccccccc}{}^1{h}_{11}& .Math.& {}^1{h}_{1p}& {}^1{h}_{1p^{}}& .Math.& {}^1{h}_{1q}& .Math.& {}^1{h}_{1m}\\ {}^1{h}_{21}& .Math.& {}^1{h}_{2p}& {}^1{h}_{2p^{}}& .Math.& {}^1{h}_{2q}& .Math.& {}^1{h}_{2m}\\ {}^1{h}_{31}& .Math.& {}^1{h}_{3p}& {}^1{h}_{3p^{}}& .Math.& {}^1{h}_{3q}& .Math.& {}^1{h}_{3m}\end{array}\right],$ wherein .sup.1f.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.1F, and .sup.1h.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.1H; setting .sup.1H=.sup.1F.Math.B, such that ${}^1{h}_{1p^{}}=\frac{u_{p\max }-u_{p\min }}{u_{p^{}\max }-u_{p^{}\min }}{}^1{h}_{1p}$ and .sup.1.sub.1q=0; and substituting the .sup.1F and the B into the .sup.1H= .sup.1F.Math.B to obtain a following equation set: $\begin{array}{cc}\{\begin{array}{c}\underset{s=1}{\overset{3}{.Math.}}{}_{}^1{f}_{1s}.Math.b_{\mathrm{sp}}={}^1{h}_{1p}\\ \underset{s=1}{\overset{3}{.Math.}}{}_{}^1{f}_{1s}.Math.b_{{\mathrm{sp}}^{}}=\frac{u_{p\max }-u_{p\min }}{u_{p^{}\max }-u_{p^{}\min }}{}^1{h}_{1p}^{}\\ \underset{s=1}{\overset{3}{.Math.}}{}_{}^1{f}_{1s}.Math.b_{\mathrm{sq}}=0\end{array}& \left(9\right)\end{array}$ solving the linear equation set (9) to obtain a first row of the .sup.1F, and calculating a first row (.sup.1h.sub.11, . . . , .sup.1h.sub.1m) of the matrix .sup.1H according to the .sup.1H=.sup.1F.Math.B; and if .sup.1h.sub.1p>0 .sup.1h.sub.1q<0, performing a step 4-4-9); if .sup.1h.sub.1p<0 and .sup.1h.sub.1q>0, performing a step 4-4-10); or if .sup.1h.sub.1p.Math..sup.1h.sub.1q>0, performing a step 4-4-11); 4-4-9) processing other rows than the rows p, q, P, q of the matrix .sup.1 according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set , and i=1, . . . , m, ip, p, q, q; and then performing determining as follows: if two key boundary surfaces have been found for the group, performing a step 4-6); otherwise, performing the step 4-4-11); wherein the rule is as follows: when .sup.1h.sub.1i.Math..sup.1h.sub.1i+1<0, if .sup.1h.sub.1i>0 and .sup.1h.sub.1i+1<0, setting an i.sup.th row of the matrix .sup.1 to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or if .sup.1h.sub.1i<0 and .sup.1h.sub.1i+1>0, setting an i.sup.th row of the matrix .sup.1 to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; when .sup.1h.sub.1i<0 and .sup.1h.sub.1i+1<0, setting an i.sup.th row of the matrix .sup.1 to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; or when .sup.1h.sub.1i>0 and .sup.1h.sub.1i+1>0, calculating .sup.5d.sub.01=.sup.1h.sub.1i.Math.u.sub.imax+.sup.1h.sub.1i+1.Math.u.sub.i+1min and .sup.5d.sub.02=.sup.1h.sub.1i.Math.u.sub.imin+.sup.1h.sub.1i+1.Math.u.sub.i+1max; and if .sup.5d.sub.01>.sup.5d.sub.02, setting an i.sup.th row of the matrix .sup.1 to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; if .sup.5d.sub.01<.sup.5d.sub.02, setting an i.sup.th row of the matrix .sup.1 to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if .sup.5d.sub.01=.sup.5d.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix .sup.1 as follows: setting the i.sup.th row of the matrix .sup.1 to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix .sup.1 to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max; 4-4-10) processing other rows than the rows p, q, p, q of the matrix .sup.1 according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set , and i=1, . . . , m, ip, p, q, q; and then performing determining as follows: if two key boundary surfaces have been found for the group, performing a step 4-6); otherwise, performing the step 4-4-11); where the rule is as follows: when .sup.1h.sub.1i.Math..sup.1h.sub.1i+1<0, if .sup.1h.sub.1i>0 and .sup.1h.sub.1i+1<0, setting an i.sup.th row of the matrix .sup.1 to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if .sup.1h.sub.1i<0 and .sup.1h.sub.1i+1>0, setting an i.sup.th row of the matrix .sup.1 to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; when .sup.1h.sub.1i>0 and .sup.1h.sub.1i+1>0, setting an i.sup.th row of the matrix .sup.1 to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; or when .sup.1h.sub.1i<0 and .sup.1h.sub.1i+1<0, calculating .sup.6d.sub.01=.sup.1h.sub.1i.Math.u.sub.imax+.sup.1h.sub.1i+1.Math.u.sub.i+1min and .sup.6d.sub.02=.sup.1h.sub.1i.Math.u.sub.imin+.sup.1h.sub.1i+1.Math.u.sub.i+1max; and if .sup.6d.sub.01>.sup.6d.sub.02, setting an i.sup.th row of the matrix .sup.1 to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; if .sup.6d.sub.01<.sup.6d.sub.02, setting an i.sup.th row of the matrix .sup.1 to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or if .sup.6d.sub.01=.sup.6d.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix .sup.1 as follows: setting the i.sup.th row of the matrix .sup.1 to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix .sup.1 to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max; 4-4-11) constructing a coordinate rotation transformation .sup.2F, such that a transformed coordinate axis v.sub.1 is perpendicular to an image of a boundary surface whose vertex matrix is the .sup.2 in the group, and images of all points on the boundary surface have equal coordinate values on the v.sub.1 axis, which specifically includes: setting ${}^{2}F=\left[\begin{array}{ccc}{}_{}^2{f}_{11}& {}_{}^2{f}_{12}& {}_{}^2{f}_{13}\\ {}_{}^2{f}_{21}& {}_{}^2{f}_{22}& {}_{}^2{f}_{23}\\ {}_{}^2{f}_{31}& {}_{}^2{f}_{32}& {}_{}^2{f}_{33}\end{array}\right],B=\left[\begin{array}{cccccccc}{b}_{11}& .Math.& b_{1p}& b_{1p^{}}& .Math.& b_{1q}& .Math.& b_{1m}\\ b_{21}& .Math.& b_{2p}& b_{2p^{}}& .Math.& b_{2q}& .Math.& b_{2m}\\ b_{31}& .Math.& b_{3p}& b_{3p^{}}& .Math.& b_{3q}& .Math.& b_{3m}\end{array}\right],\mathrm{and}$ ${}^{2}H=\left[\begin{array}{cccccccc}{}^2{h}_{11}& .Math.& {}^2{h}_{1p}& {}^2{h}_{1q}& .Math.& {}^2{h}_{1q^{}}& .Math.& {}^2{h}_{1m}\\ {}^2{h}_{21}& .Math.& {}^2{h}_{2p}& {}^2{h}_{2q}& .Math.& {}^2{h}_{2q^{}}& .Math.& {}^2{h}_{2m}\\ {}^2{h}_{31}& .Math.& {}^2{h}_{3p}& {}^2{h}_{3q}& .Math.& {}^2{h}_{3q^{}}& .Math.& {}^2{h}_{3m}\end{array}\right],$ where .sup.2f.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.2F, and .sup.2h.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.2H; setting .sup.2H=.sup.2F.Math.B, such that ${}^2{h}_{1q^{}}^{}=\frac{u_{q\max }-u_{q\min }}{u_{q^{}\max }-u_{q^{}\min }}{}^2{h}_{1q}^{}$ and .sup.2h.sub.1p=0 and substituting the .sup.2F and the B into the .sup.2H=.sup.2F.Math.B to obtain a following equation set: $\begin{array}{cc}\{\begin{array}{c}\underset{s=1}{\overset{3}{.Math.}}{}_{}^2{f}_{1s}.Math.b_{\mathrm{sp}}=0\\ \underset{s=1}{\overset{3}{.Math.}}{}_{}^2{f}_{1s}.Math.b_{\mathrm{sq}}={}^2{h}_{1q}^{}\\ \underset{s=1}{\overset{3}{.Math.}}{}_{}^2{f}_{1s}.Math.b_{{\mathrm{sq}}^{}}=\frac{u_{q\max }-u_{q\min }}{u_{q^{}\max }-u_{q^{}\min }}{}^2{h}_{1q}^{}\end{array}& \left(10\right)\end{array}$ solving the linear equation set (10) to obtain a first row of the .sup.2F, and calculating a first row (.sup.2h.sub.11, . . . , .sup.2h.sub.1m) of the matrix .sup.2H according to the .sup.2H=.sup.2F.Math.B; and if .sup.2h.sub.1p<0 and .sup.2h.sub.1q>0, performing a step 4-4-12); if .sup.2h.sub.1p>0 and .sup.2h.sub.1q<0 and, performing a step 4-4-13); or if .sup.2 h.sub.1p.Math..sup.2h.sub.1q>0, performing a step 4-6); 4-4-12) processing other rows than the rows p, q, p, q of the matrix .sup.2 according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set , and i=1, . . . , m, ip, p, q, q; and then performing the step 4-6), where the rule is as follows: when .sup.2h.sub.1i.Math..sup.2h.sub.1i+1<0, if .sup.2h.sub.1i>0 and .sup.2h.sub.1i+1<0, setting an i.sup.th row of the matrix .sup.2 to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or if .sup.2h.sub.1i<0 and .sup.2h.sub.1i+1>0, setting an i.sup.th row of the matrix .sup.2 to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; when .sup.2h.sub.1i<0 and .sup.2h.sub.1i+1<0, setting an i.sup.th row of the matrix .sup.2 to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; or when .sup.2h.sub.1i>0 and .sup.2h.sub.1i+1>0, calculating .sup.7d.sub.01=.sup.2h.sub.1i.Math.u.sub.imax+.sup.2h.sub.1i+1.Math.u.sub.i+1min and .sup.7d.sub.02=.sup.2h.sub.1i.Math.u.sub.imin+.sup.2h.sub.1i+1.Math.u.sub.i+1max; and if .sup.7d.sub.01>.sup.7d.sub.02, setting an i.sup.th row of the matrix .sup.2 to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; if .sup.7d.sub.01<.sup.7d.sub.02, setting an i.sup.th row of the matrix .sup.2 to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if .sup.7d.sub.01=.sup.7d.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix .sup.2 as follows: setting the i.sup.th row of the matrix .sup.2 to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix .sup.2 to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max; 4-4-13) processing other rows than the rows p, q, p, q of the matrix .sup.2 according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set , and i=1, . . . , m, ip, p, q, q; and then performing the step 4-6), where the rule is as follows: when .sup.2h.sub.1i.Math..sup.2h.sub.1i+1<0, if .sup.2h.sub.1i>0 and .sup.2h.sub.1i+1<0, setting an i.sup.th row of the matrix .sup.2 to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if .sup.2h.sub.1i<0 and .sup.2h.sub.1i+1>0, setting an i.sup.th row of the matrix .sup.2 to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; when .sup.2h.sub.1i<0 and .sup.2h.sub.1i+1<0, calculating .sup.8d.sub.01=.sup.2h.sub.1i.Math.u.sub.imax+.sup.2h.sub.1i+1.Math.u.sub.i+1min and .sup.8d.sub.02=.sup.2h.sub.1i.Math.u.sub.imin+.sup.2h.sub.1i+1.Math.u.sub.i+1max; and if .sup.8d.sub.01>.sup.8d.sub.02, setting an i.sup.th row of the matrix .sup.2 to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; if .sup.8d.sub.01<.sup.8d.sub.02, setting an i.sup.th row of the matrix .sup.2 to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or if .sup.8d.sub.01=.sup.8d.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix .sup.2 as follows: setting the i.sup.th row of the matrix .sup.2 to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix .sup.2 to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max; or when .sup.2h.sub.1i>0 and .sup.2h.sub.1i+1>0, setting an i.sup.th row of the matrix .sup.2 to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; 4-5) when the p.sup.th control component and the q.sup.th control component are not a pair of linear constraint components, and the q.sup.th control component has no paired linear constraint component, determining a key boundary surface of the p-q group according to a following specific method: 4-5-1) denoting a component paired with the p.sup.th component as a p.sup.th component, wherein the p-q group contains both the type-I and the type-II rectangular boundary surfaces, and each component of each point on the boundary surfaces of the group has a value that satisfies a following formula (11): $\begin{array}{cc}\{\begin{array}{c}{u}_{p\min }{u}_p{u}_{p\max }\\ u_{q\min }{u}_q{u}_{q\max }\\ u_{p^{}}\left\{u_{p^{}\min },\frac{u_{p^{}\max }{u}_{p\max }-u_{p^{}\min }{u}_{p\min }-\left(u_{p^{}\max }-u_{p^{}\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)}\right\}\\ u_i\left\{u_{i\min },u_{i\max }\right\},1i<m,ip,q,p^{}\\ u_k+u_{k+1}<u_{k\max }+u_{k+1\max },k=1,3,.Math.,2\left[\frac{m}{2}\right]-1,kp,1,p^{}\end{array}& \left(11\right)\end{array}$ when u.sub.p=u.sub.pmin determining that a boundary surface formed by a point satisfying the formula (11) is the type-I rectangular boundary surface; or when $u_{p^{}}=\frac{u_{p^{}\max }{u}_{p\max }-u_{p^{}\min }{u}_{p\min }-\left(u_{p^{}\max }-u_{p^{}\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)},$ determining that a boundary surface formed by a point satisfying the formula (11) is the type-II rectangular boundary surface; representing a vertex of each rectangular boundary surface of the group by using an m-row and 4-column matrix, where 4 columns respectively represent vectors corresponding to 4 vertices; a matrix is used to represent a vertex of one type-I rectangular boundary surface, and a matrix represents a vertex of one type-II rectangular boundary surface; values of the p.sup.th, p.sup.th, and q.sup.th components of four vertices of each boundary surface are shown in a p.sup.th row, a p.sup.th row, and a q.sup.th row of the matrix, and other components are all u.sub.imax or u.sub.imin; one pair of linear inequality constraint components do not simultaneously take a maximum constraint value corresponding to the one pair of linear inequality constraint components; and for the , the p.sup.th row is (u.sub.pmax, u.sub.pmin, u.sub.pmin, u.sub.pmax), the p.sup.th row is (u.sub.pmin, u.sub.pmin, u.sub.pmin, u.sub.pmin) the q.sup.th row is (u.sub.qmin, u.sub.qmin, u.sub.qmax, u.sub.qmax), and the q.sup.th row is (u.sub.qmin, u.sub.qmin, u.sub.qmin, u.sub.qmin); ${}^{}=\left[\begin{array}{cccc}.Math.& .Math.& .Math.& .Math.\\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{}\min }& u_{p^{}\min }& u_{p^{}\min }& u_{p^{}\min }\\ .Math.& .Math.& .Math.& .Math.\\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\end{array}\right],$ ${}^{}=\left[\begin{array}{cccc}.Math.& .Math.& .Math.& .Math.\\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{}\min }& u_{p^{}\max }& u_{p^{}\max }& u_{p^{}\min }\\ .Math.& .Math.& .Math.& .Math.\\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\end{array}\right]$ in the matrices shown in the above formula, if p is an odd number, the p.sup.th row comes before the p.sup.th row; or if p is an even number, the p.sup.th row comes after the p.sup.th row; 4-5-2) constructing a coordinate rotation transformation T, such that a transformed coordinate axis v.sub.1 is perpendicular to an image of the type-I rectangular boundary surface of the group, which specifically includes: setting ${}^{3}T=\left[\begin{array}{ccc}{}_{}^3{t}_{11}& {}_{}^3{t}_{12}& {}_{}^3{t}_{13}\\ {}_{}^3{t}_{21}& {}_{}^3{t}_{22}& {}_{}^3{t}_{23}\\ {}_{}^3{t}_{31}& {}_{}^3{t}_{32}& {}_{}^3{t}_{33}\end{array}\right],B=\left[\begin{array}{c}{b}_{11}.Math.b_{1p}.Math.b_{1q}\\ b_{21}.Math.b_{2p}.Math.b_{2q}\\ b_{31}.Math.b_{3p}.Math.b_{3q}\end{array}\right],\mathrm{and}$ ${}^{3}C=\left[\begin{array}{c}{}_{}^3{c}_{11}.Math.{}_{}^3{c}_{1p}.Math.{}_{}^3{c}_{1q}\\ {}_{}^3{c}_{21}.Math.{}_{}^3{c}_{2p}.Math.{}_{}^3{c}_{2q}\\ {}_{}^3{c}_{31}.Math.{}_{}^3{c}_{3p}.Math.{}_{}^3{c}_{3q}\end{array}\right],$ wherein .sup.3t.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.3T, and .sup.3c.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.3C; setting .sup.3C=.sup.3T.Math.B, such that .sup.3c.sub.1p=0 and .sup.3c.sub.1q=0; and substituting the .sup.3T and the B into the .sup.3C=.sup.3T.Math.B to obtain a following equation set: $\begin{array}{cc}\{\begin{array}{c}\underset{s=1}{\overset{3}{.Math.}}{}_{}^3{t}_{1s}.Math.b_{\mathrm{sp}}=0\\ \underset{s=1}{\overset{3}{.Math.}}{}_{}^3{t}_{1s}.Math.b_{\mathrm{sq}}=0\end{array}& \left(12\right)\end{array}$ solving the linear equation set (12) to obtain a first row (.sup.3t.sub.11, .sup.3t.sub.12, .sup.3t.sub.13) of the .sup.3T, and calculating a first row (.sup.3c.sub.11, . . . , .sup.3c.sub.1m) of the matrix .sup.3C according to the .sup.3C=3T.Math.B; and if .sup.3 c.sub.1p<0, performing a step 4-5-3); or if .sup.3c.sub.1p>0, performing a step 4-5-4); 4-5-3) processing other rows than the rows p, q, p of the matrix according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set ; and then performing a step 4-5-5), where i=1, . . . , m, ip, p, q; and the rule is as follows: when .sup.3c.sub.1i.Math..sup.3h.sub.1i+1<0, if .sup.3c.sub.1i>0 and .sup.3c.sub.1i+1<0, setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or if .sup.3c.sub.1i<0 and .sup.3h.sub.1i+1>0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; when .sup.3c.sub.1i<0 and .sup.3c.sub.1i+1<0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; or when .sup.3c.sub.1i>0 and .sup.3h.sub.1i+1>0, calculating .sup.9d.sub.01=.sup.3c.sub.1i.Math.u.sub.imax+.sup.3c.sub.1i+1.Math.u.sub.i+1min, and .sup.9d.sub.02=.sup.3c.sub.1i.Math.u.sub.imin+.sup.3c.sub.1i+1.Math.u.sub.i+1max; and performing determining as follows: if .sup.9d.sub.01>.sup.9d.sub.02, setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; if .sup.9d.sub.01<.sup.9d.sub.02, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if .sup.9d.sub.01=.sup.9d.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix as follows: setting the i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max; 4-5-4) processing other rows than the rows p, q, p of the matrix according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set ; and then performing a step 4-5-5), where i=1, . . . , m, ip, p, q; and the rule is as follows: when .sup.3c.sub.1i.Math..sup.3h.sub.1i+1<0, if .sup.3c.sub.1i>0 and .sup.3c.sub.1i+1<0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if .sup.3c.sub.1i<0 and .sup.3h.sub.1i+1>0, setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; when .sup.3c.sub.1i>0 and .sup.3c.sub.1i+1>0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; or when .sup.3c.sub.1i<0 and .sup.3h.sub.1i+1<0, calculating .sup.10d.sub.01=.sup.3c.sub.1i.Math.u.sub.imax+.sup.3c.sub.1i+1.Math.u.sub.i+1min, and .sup.10d.sub.02=.sup.3c.sub.1i.Math.u.sub.imin+.sup.3c.sub.1i+1.Math.u.sub.i+1max; and if .sup.10d.sub.01>.sup.10d.sub.02, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; if .sup.10d.sub.01<.sup.10d.sub.02, setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or if .sup.10d.sub.01=.sup.10d.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix as follows: setting the i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max; 4-5-5) constructing a coordinate rotation transformation .sup.3F, such that a transformed coordinate axis v.sub.1 is perpendicular to an image of a boundary surface whose vertex matrix is the in the group, and images of all points on the boundary surface have equal coordinate values on the v.sub.1 axis, which specifically includes: setting ${}^{3}F=\left[\begin{array}{ccc}{}_{}^3{f}_{11}& {}_{}^3{f}_{12}& {}_{}^3{f}_{13}\\ {}_{}^3{f}_{21}& {}_{}^3{f}_{22}& {}_{}^3{f}_{23}\\ {}_{}^3{f}_{31}& {}_{}^3{f}_{32}& {}_{}^3{f}_{33}\end{array}\right],B=\left[\begin{array}{cccccc}{b}_{11}& .Math.& b_{1p}& b_{1p^{}}& .Math.& b_{1q}\\ b_{21}& .Math.& b_{2p}& b_{2p^{}}& .Math.& b_{2q}\\ b_{31}& .Math.& b_{3p}& b_{3p^{}}& .Math.& b_{3q}\end{array}\right],\mathrm{and}$ ${}^{3}H=\left[\begin{array}{cccccc}{}^3{h}_{11}& .Math.& 3^{}{h}_{1p}& {}^3{h}_{1p^{}}& .Math.& {}^3{h}_{1q}\\ {}^3{h}_{21}& .Math.& {}^3{h}_{2p}& {}^3{h}_{2p^{}}& .Math.& {}^3{h}_{2q}\\ {}^3{h}_{31}& .Math.& {}^3{h}_{3p}& {}^3{h}_{3p^{}}& .Math.& {}^3{h}_{3q}\end{array}\right],$ wherein .sup.3h.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.3H, and .sup.3f.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.3F; setting .sup.3H=.sup.3F.Math.B, such that .sup.3h.sub.1p=u.sub.pmaxu.sub.pmin/u.sub.pmaxu.sub.pmin .sup.3h.sub.1p, and .sup.3h.sub.1q=0; and substituting the .sup.3F and the B into the .sup.3H=.sup.3F.Math.B to obtain a following equation set: $\begin{array}{cc}\{\begin{array}{c}\underset{s=1}{\overset{3}{.Math.}}{}_{}^3{f}_{1s}.Math.b_{\mathrm{sp}}={}^3{h}_{1p}\\ \underset{s=1}{\overset{3}{.Math.}}{}_{}^3{f}_{1s}.Math.b_{{\mathrm{sp}}^{}}=\frac{u_{p\max }-u_{p\min }}{u_{p^{}\max }-u_{p^{}\min }}{}^3{h}_{1p}^{}\\ \underset{s=1}{\overset{3}{.Math.}}{}_{}^3{f}_{1s}.Math.b_{\mathrm{sq}}=0\end{array}& \left(13\right)\end{array}$ solving the linear equation set (13) to obtain a first row of the .sup.3F, and calculating a first row (.sup.3h.sub.11, . . . , .sup.3h.sub.1m) of the matrix .sup.3H according to the .sup.3H=.sup.3F.Math.B; and if .sup.3h.sub.1p>0, performing a step 4-5-6); or if .sup.3h.sub.1p<0, performing a step 4-5-7); 4-5-6) processing other rows than the rows p, q, p of the matrix according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set ; and then performing a step 4-6), where i=1, . . . , m, ip, p, q; and the rule is as follows: when .sup.3h.sub.1i.Math..sup.3h.sub.1i+1<0, if .sup.3h.sub.1i>0 and .sup.3h.sub.1i+1<0, setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or if .sup.3h.sub.1i<0 and .sup.3h.sub.1i+1>0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; when .sup.3h.sub.1i<0 and .sup.3h.sub.1i+1<0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; or when .sup.3h.sub.1i; >0 and .sup.3h.sub.1i+1>0, calculating .sup.11d.sub.01=.sup.3c.sub.1i.Math.u.sub.imax+.sup.3c.sub.1i+1.Math.u.sub.i+1min, and .sup.11d.sub.02=.sup.3c.sub.1i.Math.u.sub.imin+.sup.3c.sub.1i+1.Math.u.sub.i+1max; and if .sup.11d.sub.01>.sup.11d.sub.02, setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; if .sup.11d.sub.01<.sup.11d.sub.02, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if .sup.11d.sub.01=.sup.11d.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix as follows: setting the i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max; 4-5-7) processing other rows than the rows p, q, p of the matrix according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set ; and then performing a step 4-6), where i=1, . . . , m, ip, p, q; and the rule is as follows: when .sup.3h.sub.1i.Math..sup.3h.sub.1i+1<0, if .sup.3h.sub.1i>0 and .sup.3h.sub.1i+1<0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if .sup.3h.sub.1i<0 and .sup.3h.sub.1i+1>0, setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; when .sup.3h.sub.1i<0 and .sup.3h.sub.1i+1<0, calculating .sup.12d.sub.01=.sup.3c.sub.1i.Math.u.sub.imax+.sup.3c.sub.1i+1.Math.u.sub.i+1min, and .sup.12d.sub.02=.sup.3c.sub.1i.Math.u.sub.imin+.sup.3c.sub.1i+1.Math.u.sub.i+1max; and if .sup.12d.sub.01>.sup.12d.sub.02, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; if .sup.12d.sub.01<.sup.12d.sub.02, setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or if .sup.12d.sub.01=.sup.12d.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix as follows: setting the i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max; or when .sup.3h.sub.1i>0 and .sup.3h.sub.1i+1>0, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; and 4-6) returning to the step 4-1), selecting a next group for which no key boundary surface has been determined, until the key boundary surface has been determined for all groups, and then performing a step 5); 5) checking all key boundary surfaces in the 52, and performing determining as follows: if there are identical key boundary surfaces, retaining only one of the identical key boundary surfaces, and removing other key boundary surfaces that are the same as the retained one key boundary surface; after processing all the key boundary surfaces, forming a new set of key boundary surfaces, which is denoted as ; and then performing a step 6); and 6) mapping all vertices of each key boundary surface in the according to v=B.Math.u.sub.ELR to obtain all vertices of a boundary surface corresponding to the control attainable set, so as to determine the boundary surface of the control attainable set, where all key boundary surfaces in the are all the key boundary surfaces of the control set; the boundary surface of the control attainable set is a quadrilateral or triangle; and boundary surfaces that are of the control attainable set and determined based on all the key boundary surfaces in the constitute the boundary (.sub.ELR) of the control attainable set.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0229] FIG. 1 is an overall flowchart of a method for calculating a control attainable set of a redundant drive system under a linear constraint according to the present disclosure.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0230] The present disclosure provides a method for calculating a control attainable set of a redundant drive system under a linear constraint. The following provides further detailed description based on the accompanying drawings and specific embodiments.

[0231] The present disclosure proposes a method for calculating a control attainable set of a redundant drive system under a linear constraint. An overall process is shown in FIG. 1, and the method includes following steps.

[0232] 1) Establish an expression of a control attainable set of a redundant drive system under each pair of linear constraint control components as follows:

[00035]$\begin{array}{cc}=\left\{v.Math.v=B.Math.u,u={\left(u_1,.Math.,u_m\right)}^T,u\right\}s.t.& \left(1\right)\end{array}$$u_{\mathrm{imin}}{u}_i{u}_{\mathrm{imax}},i=1,L,m$$\left(u_{\mathrm{kmax}}-u_{\mathrm{kmin}}\right)u_{k+1}+\left(u_{k+1\max }-u_{k+1\min }\right)u_k{u}_{k+1\max }{u}_{\mathrm{kmax}}-u_{k+1\min }{u}_{\mathrm{kmin}},$$k=1,3,.Math.,2\left[\frac{m}{2}\right]-1$

[0233] In the above expression, [m/2] represents an integer part obtained by dividing m by 2, u represents a control vector, the u is denoted as u.sub.ELR, u.sub.ELR=(u.sub.1, . . . , u.sub.m).sup.T, u.sub.iminu.sub.iu.sub.imax, i=1,L,m, (u.sub.kmaxu.sub.kmin)u.sub.k+1+(u.sub.k+1maxu.sub.k+1min)u.sub.ku.sub.k+1maxu.sub.kmaxu.sub.k+1minu.sub.kmin, and k=1,3, . . . , 2[m/2]1.

[0234] An i.sup.th control component u.sub.i is a control action of a corresponding i.sup.th actuator, where 1im, and m represents a quantity of actuators; and u.sub.imin represents a minimum constraint value of the control action of the i.sup.th actuator, and u.sub.imax represents a maximum constraint value of the control action of the i.sup.th actuator. represents a control set, and ={u}; v represents a control attainable vector of the redundant drive system, and v=(v.sub.1, v.sub.2, v.sub.3).sup.T, which represents a control output of the redundant drive system; represents the control attainable set; and B represents a 3-row and m-column control efficiency matrix.

[0235] For convenience of subsequent description, the expression (1) is rewritten to an expression (2):

[00036]$\begin{array}{cc}{}_{\mathrm{ELR}}=\left\{v.Math.v=B.Math.u_{\mathrm{ELR}},u_{\mathrm{ELR}}={\left(u_1,.Math.,u_m\right)}^T,u_{\mathrm{ELR}}{}_{\mathrm{ELR}}\right\}s.t.& \left(2\right)\end{array}$$u_{\mathrm{imin}}{u}_i{u}_{\mathrm{imax}},i=1,L,m;$$\left(u_{\mathrm{kmax}}-u_{\mathrm{kmin}}\right)u_{k+1}+\left(u_{k+1\max }-u_{k+1\min }\right)u_k{u}_{k+1\max }{u}_{\mathrm{kmax}}-u_{k+1\min }{u}_{\mathrm{kmin}},$$k=1,3,.Math.,2\left[\frac{m}{2}\right]-1$

[0236] In the above expression, u.sub.k and u.sub.k+1 represent a pair of control components with a linear constraint, which is referred to as a pair of linear constraint control components, .sub.ELR represents a control attainable set of a redundant drive system with each pair of control components being linear constraint control components, .sub.ELR represents a control set, u.sub.ELR represents a control vector, and .sub.ELR={u.sub.ELR}. A boundary of the .sub.ELR is represented by (.sub.ELR), and a boundary of the .sub.ELR is represented by (.sub.ELR). In this way, determining of the control attainable set of the redundant drive system with each pair of control components being linear constraint control components is to determine (.sub.ELR) based on the given .sub.ELR and B. [0237] 2) Classify all boundary surfaces of the control set .sub.ELR obtained in the step 1) into four types.

[0238] In this embodiment, for the control set .sub.ELR={u.sub.ELR}, a point obtained by taking a corresponding maximum or minimum constraint value of each component u.sub.i(i=1, . . . , m) of the control vector u.sub.ELR=(u.sub.1, . . . , u.sub.m).sup.T is used as a vertex of the control set .sub.ELR. m represents the quantity of actuators, and m>3. One m-dimensional control set has at most 2.sup.m vertices. Because paired linear constraint control components cannot take the maximum value simultaneously, the m-dimensional control set .sub.ELR represented by the expression (2) has

[00037]$3^{\left[\frac{m}{2}\right]}.Math.2^{\left(\frac{m}{2}\right)}$

vertices. [m/2] represents the integer part obtained by dividing m by 2, and (m/2) represents a remainder of dividing m by 2.

[0239] .sub.ELR represents the control set of the redundant drive system with each pair of control components being linear constraint control components, and the boundary of the .sub.ELR is the (.sub.ELR) Based on a value of each component of each vertex, there are four types of boundary surfaces constituting the (.sub.ELR): a type-rectangular boundary surface, a type-rectangular boundary surface, a type-=2\*ROMAN I rectangular boundary surface, and a triangular boundary surface. The four types of boundary surfaces are defined as follows:

[0240] Four vertices of the rectangular boundary surface are denoted as A, B, D, and C in clockwise order, and three vertices of the triangular boundary surface are denoted as A, C, and B in clockwise order.

[0241] A rectangle that satisfies following three conditions is referred to as the type-I rectangular boundary surface: (i) for the vertices A and B, one component has a different value, and other components have a same value; (ii) for the vertices A and C, one component of another pair of components has a different value, and other components have a same value; and (iii) for the vertices A and D, the above two components have different values, and other components have a same value.

[0242] A rectangle that satisfies following three conditions is referred to as the type-II rectangular boundary surface: (i) for the vertex A and the vertex B, one pair of linear constraint components have different values, and other components have a same value; (ii) for the vertex C and the vertex D, the one pair of linear constraint components have different values, and other components have a same value; and (iii) for the vertex A and the vertex C, the one pair of linear constraint components have a same value, and other components have a same value except that one of the other components has a different value.

[0243] A rectangle that satisfies following three conditions is referred to as the type=2\*ROMAN I rectangular boundary surface: (i) for the vertex A and the vertex B, one pair of linear constraint components have different values, and other components have a same value; (ii) for the vertex A and the vertex C, another pair of linear constraint components have different values, and other components have a same value; and (iii) for the vertex D and the vertex A, each of the above two pairs of linear constraint components has a different value, and other components have a same value.

[0244] A right triangle that following three conditions is referred to as the triangular boundary surface: (i) for the vertex A and the vertex B, one pair of linear constraint components have different values, and other components have a same value; (ii) for the vertex A and the vertex C, one component of the one pair of linear constraint components have a different value, and other components have a same value; and (iii) for the vertex B and the vertex C, the other component of the one pair of linear constraint components has a different value, and other components have a same value.

[0245] 3) Divide all the boundary surfaces of the control set .sub.ELR into C.sub.m.sup.2 groups.

[0246] If values of two components of the u.sub.ELR are between the corresponding minimum and maximum values, and values of other m2 components are the corresponding minimum or maximum value, the m components form 2.sup.m2 boundary surfaces of the control set.

[0247] If any two components in the u.sub.ELR are denoted as a p.sup.th component and a q.sup.th component, values of the p.sup.th component and the q.sup.th component are between the corresponding minimum and maximum values, 1pm, 1qm, p<q, values of other m2 components are the corresponding minimum or maximum value, and a boundary surface formed in such a manner is added to one group that is referred to as a p-q group, all the boundary surfaces in the control set are divided into the C.sub.m.sup.2 groups.

[0248] 4) Determine a key boundary surface for each of the C.sub.m.sup.2 groups in the step 3).

[0249] A boundary surface of an image mapped onto the .sub.ELR in the .sub.ELR of the boundary (.sub.ELR) is denoted as the key boundary surface, the is denoted as a set of key boundary surfaces, and the is initialized as an empty set.

[0250] 4-1) Select any group for which no key boundary surface has been determined, denote the group as a current p-q group, and perform a step 4-2).

[0251] 4-2) Determine whether the p.sup.th control component and the q.sup.th control component are a pair of linear constraint components; and [0252] if they are a pair of linear constraint components, perform a step 4-3); [0253] if they are not a pair of linear constraint components, and m is an even number, perform a step 4-4);

[0254] if they are not a pair of linear constraint components, m is an odd number, and the p.sup.th and q.sup.th control components each have a paired linear constraint component, perform a step 4-4); or

[0255] if they are not a pair of linear constraint components, m is an odd number, and the q.sup.th control component has no paired linear constraint component, perform a step 4-5).

[0256] It should be noted that an m.sup.th component does not have a paired linear constraint component only when m is the odd number. Since it has been limited earlier that p<q the p.sup.th control component must have a paired linear constraint component.

[0257] 4-3) When the p.sup.th control component and the q.sup.th control component are a pair of linear constraint components, determine a key boundary surface of the p-q group according to a following specific method: [0258] 4-3-1) When the p.sup.th control component and the q.sup.th control component are a pair of linear constraint components, determine that all boundary surfaces of the group are triangular boundary surfaces, where a value of each component of each point on the boundary surface satisfies a following formula:

[00038]$\begin{array}{cc}\{\begin{array}{c}{u}_{\mathrm{pmin}}{u}_p{u}_{\mathrm{pmax}}\\ u_{\mathrm{qmin}}{u}_q\frac{u_{\mathrm{qmax}}{u}_{\mathrm{pmax}}-u_{\mathrm{qmin}}{u}_{\mathrm{pmin}}-\left(u_{\mathrm{qmax}}-u_{\mathrm{qmin}}\right)u_p}{\left(u_{\mathrm{pmax}}-u_{\mathrm{pmin}}\right)}\\ u_i\left\{u_{\mathrm{imin}},u_{\mathrm{imax}}\right\},i{Z}^{{}_{}+},1im,ip,q\end{array}& \left(3\right)\end{array}$

[0259] A vertex of each boundary surface in the group is represented by an m-row and 3-column matrix 4, and .sup.3 columns are vectors corresponding to 3 vertices. Values of the p.sup.th and q.sup.th components of three vertices on each boundary surface are shown in p.sup.th and q.sup.th rows of the matrix . A value of an i(i=1, . . . , m,ip,q).sup.th component is the u.sub.imax or the u.sub.imin, and an i(i=1, . . . , m, ip, q).sup.th row of the corresponding matrix is the u.sub.imax Or the u.sub.imin. However, one pair of linear constraint components in other components cannot simultaneously take a maximum constraint value corresponding to the one pair of linear constraint components, and values of the other components are uniformly represented by ellipses in the . The p.sup.th row of the matrix is (u.sub.pmax, u.sub.pmin, u.sub.pmin), and the q.sup.th row is (u.sub.qmin, u.sub.qmin, u.sub.qmax).

[00039]$=\left[\begin{array}{ccc}.Math.& .Math.& .Math.\\ u_{\mathrm{pmax}}& u_{\mathrm{pmin}}& u_{\mathrm{pmin}}\\ u_{\mathrm{qmin}}& u_{\mathrm{qmin}}& u_{\mathrm{qmax}}\\ .Math.& .Math.& .Math.\end{array}\right].$

[0260] 4-3-2) Construct a coordinate rotation transformation .sup.1T, such that a transformed coordinate axis v.sub.1 is perpendicular to an image of the triangular boundary surface of the group, where only a first row of the .sup.1T is required because only the V.sub.1 axis is considered.

[00040]${}^{1}T=\left[\begin{array}{ccc}{}^1{t}_{11}& {}^1{t}_{12}& {}^1{t}_{13}\\ {}^1{t}_{21}& {}^1{t}_{22}& {}^1{t}_{23}\\ {}^1{t}_{31}& {}^1{t}_{32}& {}^1{t}_{33}\end{array}\right],B=\left[\begin{array}{ccccccc}{b}_{11}& .Math.& b_{1p}& .Math.& b_{1q}& .Math.& b_{1m}\\ b_{21}& .Math.& b_{2p}& .Math.& b_{2q}& .Math.& b_{2m}\\ b_{31}& .Math.& b_{3p}& .Math.& b_{3q}& .Math.& b_{3m}\end{array}\right],\mathrm{and}$${}^{1}C=\left[\begin{array}{ccccccc}{}^1{c}_{11}& .Math.& {}^1{c}_{1p}& .Math.& {}^1{c}_{1q}& .Math.& {}^1{c}_{1m}\\ {}^1{c}_{21}& .Math.& {}^1{c}_{2p}& .Math.& {}^1{c}_{2q}& .Math.& {}^1{c}_{2m}\\ {}^1{c}_{31}& .Math.& {}^1{c}_{3p}& .Math.& {}^1{c}_{3q}& .Math.& {}^1{c}_{3m}\end{array}\right]$

are set, where .sup.1c.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.1C, .sup.1t.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix .sup.1T, and b.sub.ij represents an element of an i.sup.th row and a j.sup.th column of the matrix B.

[0261] .sup.1C=.sup.1T.Math.B is set, such that .sup.1c.sub.1p=0, and .sup.1c.sub.1q=0. The .sup.1T and the B are substituted into the .sup.1C=.sup.1T.Math.B to obtain a following equation set:

[00041]$\begin{array}{cc}\{\begin{array}{c}\underset{s=1}{\overset{3}{.Math.}}{}^1{t}_{1s}.Math.b_{sp}=0\\ \underset{s=1}{\overset{3}{.Math.}}{}^1{t}_{1s}.Math.b_{sq}=0\end{array}& \left(4\right)\end{array}$

[0262] The linear equation set (4) is solved to obtain the first row (.sup.1t.sub.11, .sup.1t.sub.12, .sup.1t.sub.13) of the .sup.1T. Then, a first row (.sup.1C.sub.11, . . . , .sup.1C.sub.1m) of the matrix .sup.1C is calculated according to the .sup.1C=.sup.1T.Math.B.

[0263] 4-3-3) Process other rows than the rows P, q of the matrix according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set , and 1im,i is an integer, ip,q.

[0264] The rule is as follows: [0265] when .sup.1c.sub.1i.Math..sup.1c.sub.1i+1<0, [0266] if .sup.1c.sub.1i>0 and .sup.1c.sub.1i+1<0, setting the i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or [0267] if .sup.1c.sub.1i<0 and .sup.1c.sub.1i+1>0, setting the i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or

[0268] when .sup.1c.sub.1i<0 and .sup.1c.sub.1i+1<0, [0269] setting the i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; or [0270] when .sup.1c.sub.1i>0 and .sup.1c.sub.1i+1>0, [0271] calculating .sup.1z.sub.01=.sup.1c.sub.1i.Math.u.sub.imax+.sup.1c.sub.1i+1.Math.u.sub.i+1min and .sup.1z.sub.02=.sup.1c.sub.1i.Math.u.sub.imin+.sup.1c.sub.1i+1.Math.u.sub.i+1max and then performing determining as follows: [0272] if .sup.1z.sub.01>.sup.1z.sub.02, setting an i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; if .sup.1z.sub.01<.sup.1z.sub.02, setting an i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or if .sup.1z.sub.01=.sup.1z.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix as follows: setting the i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max. [0273] 4-3-4) Process other rows than the rows p, q of the matrix according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set ; and then perform a step 4-6), where 1im, i is an integer, ip, q.

[0274] The rule is as follows: [0275] when .sup.1c.sub.1i.Math..sup.1c.sub.1i+1<0, [0276] if .sup.1c.sub.1i>0 and .sup.1c.sub.1i+1<0, setting the i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1max; or [0277] if .sup.1c.sub.1i<0 and .sup.1c.sub.1i+1>0, setting the i.sup.th row of the matrix to the u.sub.imax and an i+1.sup.th row to u.sub.i+1min; or [0278] when .sup.1c.sub.1i>0 and .sup.1c.sub.1i+1>0, [0279] setting the i.sup.th row of the matrix to the u.sub.imin and an i+1.sup.th row to u.sub.i+1min; or [0280] when .sup.1c.sub.1i<0 and .sup.1c.sub.1i+1<0, [0281] calculating .sup.2z.sub.01=.sup.1c.sub.1i.Math.u.sub.imax+.sup.1c.sub.1i+1.Math.u.sub.i+1min and .sup.2z.sub.02=.sup.1c.sub.1i.Math.u.sub.imin+.sup.1c.sub.1i+1.Math.u.sub.i+1max and performing determining as follows: if .sup.2z.sub.01>.sup.2z.sub.02, setting an i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to the u.sub.i+1max; if .sup.2z.sub.01<.sup.2z.sub.02, setting an i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or if .sup.1z.sub.01=.sup.1z.sub.02, setting an i.sup.th row and an i+1.sup.th row of the matrix as follows: setting the i.sup.th row of the matrix to the u.sub.imax and the i+1.sup.th row to u.sub.i+1min; or setting the i.sup.th row of the matrix to the u.sub.imin and the i+1.sup.th row to u.sub.i+1max. [0282] 4-4) When the p.sup.th control component and the q.sup.th control component are not a pair of linear constraint components, and each have the paired linear constraint component, determine a key boundary surface of the p-q group according to a following specific method: [0283] 4-4-1) The p-q group has three types of rectangular boundary surfaces. Each component of each point on the boundary surfaces of the group has a value that satisfies a following formula. The component paired with the p.sup.th component is denoted as a p.sup.th component, and the component paired with the q.sup.th component is denoted as a q.sup.th component.

[00042]$\begin{array}{cc}\{\begin{array}{c}{u}_{p\min }{u}_p{u}_{p\max }\\ u_{q\min }{u}_q{u}_{q\max }\\ u_{p^{}}\left\{u_{p^{}\min },\frac{u_{p^{}\max }{u}_{p\max }-u_{p^{}\min }{u}_{p\min }-\left(u_{p^{}\max }-u_{p^{}\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)}\right\}\\ u_{q^{}}\left\{u_{q^{}\min },\frac{u_{q^{}\max }{u}_{q\max }-u_{q^{}\min }{u}_{q\min }-\left(u_{q^{}\max }-u_{q^{}\min }\right)u_q}{\left(u_{q\max }-u_{q\min }\right)}\right\}\\ u_i\left\{u_{i\min },u_{i\max }\right\},1im,ip,q,p^{},q^{}\\ u_k+u_{k+1}<u_{k\max }+u_{k+1\max },k=1,3,.Math.,2\left[\frac{m}{2}\right]-1,kp,q,p^{},q^{}\end{array}& \left(5\right)\end{array}$