US 2024/0402667 A1 · 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.

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:
Φ
=
{
v
⁢
❘
"\[LeftBracketingBar]"
v
=
B
·
u
,
u
=
(
u
1
,
…
,
u
m
)
T
,
u
∈
Ω
}
⁢
s
.
t
⁢
u
i
⁢
min
≤
u
i
≤
u
i
⁢
max
,
i
=
1
,
L
,
m
⁢
(
u
k
⁢
max
-
u
k
⁢
min
)
⁢
u
k
+
1
+
(
u
k
+
1
⁢
max
-
u
k
+
1
⁢
min
)
⁢
u
k
≤
u
k
+
1
⁢
max
⁢
u
k
⁢
max
-
u
k
+
1
⁢
min
⁢
u
k
⁢
min
,
k
=
1
,
3
,
…
,
2
[
m
2
]
-
1
(
1
)
wherein [m/2] represents an integer part obtained by dividing m by 2, u represents a control vector, the u is denoted as uELR, uELR=(u1 . . . , um)T, uimin≤ui≤uimax, i=1,L,m, (ukmax−ukmin)uk+1+(uk+1max−uk+1min)uk≤uk+1maxukmax−uk+1minukmin, and k=1,3, . . . , 2[m/2]−1;
wherein an ith control component ui is a control action of a corresponding ith actuator, wherein 1≤≤m, and m represents a quantity of actuators;
uimin represents a minimum constraint value of the control action of the ith actuator, and uimax represents a maximum constraint value of the control action of the ith actuator;
Ω represents a control set, and Ω={u};
v represents a control attainable vector of the redundant drive system, and v=(v1, v2, v3)T, which represents a control output of the redundant drive system;
Φ 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):
Φ
ELR
=
{
v
⁢
❘
"\[LeftBracketingBar]"
v
=
B
·
u
ELR
,
u
ELR
=
(
u
1
,
…
,
u
m
)
T
,
u
ELR
∈
Ω
ELR
}
⁢
s
.
t
.
u
i
⁢
min
≤
u
i
≤
u
i
⁢
max
,
i
=
1
,
L
,
m
;
⁢
(
u
k
⁢
max
-
u
k
⁢
min
)
⁢
u
k
+
1
+
(
u
k
+
1
⁢
max
-
u
k
+
1
⁢
min
)
⁢
u
k
≤
u
k
+
1
⁢
max
⁢
u
k
⁢
max
-
u
k
+
1
⁢
min
⁢
u
k
⁢
min
,
k
=
1
,
3
,
…
,
2
[
m
2
]
-
1
(
2
)
wherein uk and uk+1 represent a pair of control components with a linear constraint, ΦELR represents a control attainable set of a redundant drive system with each pair of control components being linear constraint control components, ΩELR represents a control set, uELR represents a control vector, and ΩELR={uELR};
2) classifying all boundary surfaces of the control set ΩELR obtained in the step 1) into four types, which comprises:
using, as a vertex of the control set ΩELR, a point obtained by taking a corresponding maximum or minimum constraint value of each component ui of the control vector uELR=(u1, . . . , um)T, wherein the m-dimensional control set ΩELR represented in the expression (2) has
3
[
m
2
]
·
2
(
m
2
)
vertices, and (m/2) represents a remainder of dividing m by 2;
and
setting ∂(ΩELR) to represent a boundary of the ΩELR, and classifying boundary surfaces of the ∂(ΩELR) 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 ΩELR into Cm2 groups, wherein
if values of two components in the uELR are between the corresponding minimum and maximum values, and values of other m−2 components are the corresponding minimum or maximum value, the m components form 2m−2 boundary surfaces of the control set;
and
if any two components in the uELR are denoted as a pth component and a qth component, values of the pth component and the qth component are between the corresponding minimum and maximum values, 1≤p≤m, 1≤q≤m, p<q, values of other m−2 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 Cm2 groups;
4) determining a key boundary surface for each of the Cm2 groups in the step 3), wherein
a boundary surface of an image mapped onto the ΦELR in the ΩELR of a boundary ∂(ΦELR) is denoted as the key boundary surface, the Ω is denoted as a set of key boundary surfaces, the Ω 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 pth control component and the qth 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 pth and qth 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 qth control component has no paired linear constraint component, performing a step 4-5);
4-3) when the pth control component and the qth 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 pth control component and the qth 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:
{
u
p
⁢
min
≤
u
p
≤
u
p
⁢
max
u
q
⁢
min
≤
u
q
≤
u
q
⁢
max
⁢
u
p
⁢
max
-
u
q
⁢
min
⁢
u
p
⁢
min
-
(
u
q
⁢
max
-
u
q
⁢
min
)
⁢
u
p
(
u
p
⁢
max
-
u
p
⁢
min
)
u
i
∈
{
u
i
⁢
min
,
u
i
⁢
max
}
,
i
∈
Z
+
,
1
≤
i
≤
m
,
i
≠
p
,
q
(
3
)
wherein a vertex of each boundary surface in the group is represented by an m-row and 3-column matrix Δ, and 3 columns are vectors corresponding to 3 vertices;
values of the pth and qth components of three vertices on each boundary surface are shown in pth and qth rows of the matrix Δ, while values of other components are all the uimax or the uimin;
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 pth row of the matrix Δ is (upmax, upmin, upmin), and the qth row is (uqmin, uqmin, uqmax);
Δ
=
[
…
…
…
u
p
⁢
max
u
p
⁢
mi𝔫
u
p
⁢
mi𝔫
u
q
⁢
mi𝔫
u
q
⁢
mi𝔫
u
q
⁢
max
…
…
…
]
;
4-3-2) constructing a coordinate rotation transformation 1T, such that a transformed coordinate axis v1 is perpendicular to an image of the triangular boundary surface of the group, which specifically comprises:
setting
1
T
=
[
1
t
1
⁢
1
1
t
1
⁢
2
1
t
1
⁢
3
1
t
2
⁢
1
1
t
2
⁢
2
1
t
2
⁢
3
1
t
3
⁢
1
1
t
3
⁢
2
1
t
3
⁢
3
]
,
B
=
[
b
1
⁢
1
⁢
…
⁢
b
1
⁢
p
⁢
…
⁢
b
1
⁢
q
⁢
…
⁢
b
1
⁢
m
b
2
⁢
1
⁢
…
⁢
b
2
⁢
p
⁢
…
⁢
b
2
⁢
q
⁢
…
⁢
b
2
⁢
m
b
3
⁢
1
⁢
…
⁢
b
3
⁢
p
⁢
…
⁢
b
3
⁢
q
⁢
…
⁢
b
3
⁢
m
]
,
and
⁢
1
C
=
[
1
c
1
⁢
1
⁢
…
⁢
1
c
1
⁢
p
⁢
…
⁢
1
c
1
⁢
q
⁢
…
⁢
1
c
1
⁢
m
1
c
2
⁢
1
⁢
…
⁢
1
c
2
⁢
p
⁢
…
⁢
1
c
2
⁢
q
⁢
…
⁢
1
c
2
⁢
m
1
c
3
⁢
1
⁢
…
⁢
1
c
3
⁢
p
⁢
…
⁢
1
c
3
⁢
q
⁢
…
⁢
1
c
3
⁢
m
]
,
wherein 1cij represents an element of an ith row and a jth column of the matrix 1C, 1tij represents an element of an ith row and a jth column of the matrix 1T, and bij represents an element of an ith row and a jth column of the matrix B;
setting 1C=1T·B, such that 1c1p=0 and 1c1q=0;
substituting the 1T and the B into the 1C=1T·B to obtain a following equation set:
{
∑
s
=
1
3
1
t
1
⁢
s
·
b
sp
=
0
∑
s
=
1
3
1
t
1
⁢
s
·
b
sq
=
0
(
4
)
solving the linear equation set (4) to obtain a first row (1t11, 1t12, 1t13) of the 1T;
and
calculating a first row (1c11, . . . , 1c1m) of the matrix 1C according to the 1C=1T·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 Ω, and 1≤i≤m, i is an integer, i≠p, q;
and
the rule is as follows:
when 1c1i·1c1i+1<0,
if 1c1i>0 and 1c1i+1<0, setting an ith row of the matrix Δ to the uimax and an i+1th row to ui+1 min;
or
if 1c1i<0 and 1c1i+1<0, setting an ith row of the matrix Δ to the uimin and an i+1th row to ui+1max;
when 1c1i<0and 1c1i+1<0,
setting an ith row of the matrix Δ to the uimin and an i+1th row to ui+1min;
or
when 1c1i>0 and 1c1i+1>0,
calculating 1z01=1c1i·uimax+1c1i+1·ui+1min and 1z02=1c1i·uimin+1c1i+1·ui+1max, and then performing determining as follows:
if 1z01>1z02, setting an ith row of the matrix Δ to the uimax and an i+1th row to ui+1min;
if 1z01<1z02 setting an ith row of the matrix Δ to the uimin and an i+1th row to ui+1max;
or if 1z01=1z02, setting an ith row and an i+1th row of the matrix Δ as follows:
setting the ith row of the matrix Δ to the uimax and the i+1th row to ui+1min;
or setting the ith row of the matrix Δ to the uimin and the i+1th row to ui+1max;
4-3-4) processing other rows than the rows p, q of the matrix Δ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω;
and then performing a step 4-6), wherein 1≤i≤m,i is an integer, i≠p, q;
and
the rule is as follows:
when 1c1i·1c1i+1<0,
if 1c1i>0 and 1c1i+1<0, setting the ith row of the matrix Δ to the uimin and the i+1th row to the ui+1max;
or
if 1c1i<0 and 1c1i+1>0, setting the ith row of the matrix Δ to the uimax and the i+1th row to ui+1 min;
when 1c1i>0 and 1c1i+1>0,
setting the ith row of the matrix Δ to the uimin and the i+1th row to the ui+1min;
or
when 1c1i<0 and 1c1i+1<0,
calculating 2z01=1c1i·uimax+1c1i+1·ui+1min and 2z02=1c1i·uimin+1c1i+1·ui+1max and performing determining as follows:
if 2z01>2z02, setting the ith row of the matrix Δ to the uimin and the i+1th row to the uimax;
if 2z01<2z02, setting the ith row of the matrix Δ to the uimax and the i+1th row to the ui+1min;
or if 1z01=1z02, setting the ith row and the i+1th row of the matrix Δ as follows:
setting the ith row of the matrix Δ to the uimax and the i+1th row to the ui+1min;
or setting the ith row of the matrix Δ to the uimin and the i+1th row to the ui+1max;
4-4) when the pth control component and the qth 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 pth component as a p component, and the component paired with the qth component as a q′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:
{
u
pmin
≤
u
p
≤
u
p
⁢
max
u
qmin
≤
u
q
≤
u
q
⁢
max
u
p
′
∈
{
u
p
′
⁢
min
,
u
p
′
⁢
max
⁢
u
p
⁢
max
-
u
p
′
⁢
min
⁢
u
p
⁢
min
-
(
u
p
′
⁢
max
-
u
p
′
⁢
min
)
⁢
u
p
(
u
p
⁢
max
-
u
p
⁢
min
)
}
u
q
∈
{
u
q
′
⁢
min
,
u
q
′
⁢
max
⁢
u
q
⁢
max
-
u
q
′
⁢
min
⁢
u
q
⁢
min
-
(
u
q
′
⁢
max
-
u
q
′
⁢
min
)
⁢
u
q
(
u
q
⁢
max
-
u
q
⁢
min
)
}
u
i
∈
{
u
i
⁢
min
,
u
i
⁢
max
}
,
1
≤
i
≤
m
,
i
≠
p
,
q
,
p
′
,
q
′
u
k
+
u
k
+
1
<
u
k
⁢
max
+
u
k
+
1
⁢
max
,
k
=
1
,
3
,
…
,
2
[
m
2
]
-
1
,
k
≠
p
,
q
,
p
′
,
q
′
(
5
)
when up′=up′min and uq′=uq′min, determining that a boundary surface formed by a point satisfying the formula (5) is the type-I rectangular boundary surface;
when up′=up′min and
u
q
′
=
u
q
′
⁢
mαx
⁢
u
q
⁢
max
-
u
q
′
⁢
min
⁢
u
q
⁢
min
-
(
u
q
′
⁢
max
-
u
q
′
⁢
min
)
⁢
u
q
(
u
q
⁢
max
-
u
q
⁢
min
)
,
determining that a boundary surface formed by a point satisfying the formula (5) is the type-II rectangular boundary surface;
when
u
p
′
=
u
p
′
⁢
mαx
⁢
u
p
⁢
max
-
u
p
′
⁢
min
⁢
u
p
⁢
min
-
(
u
p
′
⁢
max
-
u
p
′
⁢
min
)
⁢
u
p
(
u
p
⁢
max
-
u
p
⁢
min
)
and uq′=uq′min, determining that a boundary surface formed by a point satisfying the formula (5) also is the type-II rectangular boundary surface;
when
u
p
′
=
u
p
′
⁢
mαx
⁢
u
p
⁢
max
-
u
p
′
⁢
min
⁢
u
p
⁢
min
-
(
u
p
′
⁢
max
-
u
p
′
⁢
min
)
⁢
u
p
(
u
p
⁢
max
-
u
p
⁢
min
)
⁢
and
⁢
u
q
′
=
u
q
′
⁢
mαx
⁢
u
q
⁢
max
-
u
q
′
⁢
min
⁢
u
q
⁢
min
-
(
u
q
′
⁢
max
-
u
q
′
⁢
min
)
⁢
u
q
(
u
q
⁢
max
-
u
q
⁢
min
)
,
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 Λ represents a vertex of one type-I rectangular boundary surface, a matrix 1Π or 2Π 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 pth, p′th, qth, and q′th components of four vertices of each boundary surface are shown in a pth row, a p′th row, a qth row, and a q′th row of the matrix, and other components are all the uimax or the uimin;
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 Λ, the pth row is (upmax, upmin, upmin, upmax), the p′th row is (up′min, up′min, up′min, up′min), the qth row is (uqmin, uqmin, uqmax, uqmax), and the q′th row is (uq′min, uq′min, uq′min, uq′min);
for the 1Π, the pth row is (upmax, upmin, upmin, upmax), the p′th row is (up′min, up′max, up′max, up′min), the qth row is (uqmin, uqmin, uqmax, uqmax), and the q′th row is (uq′min, uq′min, uq′min, uq′min);
for the 2Π, the pth row is (upmax, upmin, upmin, upmax), the p′th row is (up′min, up′min, up′min, up′min, the qth row is (uqmin, uqmin, uqmax, uqmax), and the q′th row is (uq′max, uq′max, uq′min, uq′min);
and
for the T, the pth row is (upmax, upmin, upmin, upmax), the p′th row is (up′min, up′max, up′max, up′min), the qth row is (uqmin, uqmin, uqmax, uqmax), and the q′th row is (uq′max, uq′max, uq′min, uq′min);
Λ
=
[
…
…
…
…
u
p
⁢
max
u
p
⁢
min
u
p
⁢
min
u
p
⁢
max
u
p
′
⁢
min
u
p
′
⁢
min
u
p
′
⁢
min
u
p
′
⁢
min
…
…
…
…
u
q
⁢
min
u
q
⁢
min
u
q
⁢
max
u
q
⁢
max
u
q
′
⁢
min
u
q
′
⁢
min
u
q
′
⁢
min
u
q
′
⁢
min
…
…
…
…
]
,
1
Π
=
[
…
…
…
…
u
p
⁢
max
u
p
⁢
min
u
p
⁢
min
u
p
⁢
max
u
p
′
⁢
min
u
p
′
⁢
max
u
p
′
⁢
max
u
p
′
⁢
min
…
…
…
…
u
q
⁢
min
u
q
⁢
min
u
q
⁢
max
u
q
⁢
max
u
q
′
⁢
min
u
q
′
⁢
min
u
q
′
⁢
min
u
q
′
⁢
min
…
…
…
…
]
,
2
Π
=
[
…
…
…
…
u
p
⁢
max
u
p
⁢
min
u
p
⁢
min
u
p
⁢
max
u
p
′
⁢
min
u
p
′
⁢
min
u
p
′
⁢
min
u
p
′
⁢
min
…
…
…
…
u
q
⁢
min
u
q
⁢
min
u
q
⁢
max
u
q
⁢
max
u
q
′
⁢
max
u
q
′
⁢
max
u
q
′
⁢
min
u
q
′
⁢
min
…
…
…
…
]
,
T
=
[
…
…
…
…
u
p
⁢
max
u
p
⁢
min
u
p
⁢
min
u
p
⁢
max
u
p
′
⁢
min
u
p
′
⁢
max
u
p
′
⁢
max
u
p
′
⁢
min
…
…
…
…
u
q
⁢
min
u
q
⁢
min
u
q
⁢
max
u
q
⁢
max
u
q
′
⁢
max
u
q
′
⁢
max
u
q
′
⁢
min
u
q
′
⁢
min
…
…
…
…
]
(
6
)
wherein if P is an odd number, the pth row comes before the p′th row;
if p is an even number, the pth row comes after the p′th row;
if q is an odd number, the qth row comes before the q′th row;
and if q is an even number, the qth row comes after the q′th row;
4-4-2) constructing a coordinate rotation transformation G, such that a transformed coordinate axis v1 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 v1 axis, which specifically comprises:
setting
G
=
[
g
11
g
1
⁢
2
g
1
⁢
3
g
2
⁢
1
g
2
⁢
2
g
2
⁢
3
g
3
⁢
1
g
3
⁢
2
g
3
⁢
3
]
,
B
=
[
b
1
⁢
1
…
b
1
⁢
p
b
1
⁢
p
′
…
b
1
⁢
q
b
1
⁢
q
′
…
b
1
⁢
m
b
2
⁢
1
…
b
2
⁢
p
b
2
⁢
p
′
…
b
2
⁢
q
b
2
⁢
q
′
…
b
2
⁢
m
b
3
⁢
1
…
b
3
⁢
p
b
3
⁢
p
′
…
b
3
⁢
q
b
3
⁢
q
′
…
b
3
⁢
m
]
,
and
⁢
E
=
[
e
1
⁢
1
…
e
1
⁢
p
e
1
⁢
p
′
…
e
1
⁢
q
e
1
⁢
q
′
…
e
1
⁢
m
e
2
⁢
1
…
e
2
⁢
p
e
2
⁢
p
′
…
e
2
⁢
q
e
2
⁢
q
′
…
e
2
⁢
m
e
3
⁢
1
…
e
3
⁢
p
e
3
⁢
p
′
…
e
3
⁢
q
e
3
⁢
q
′
…
e
3
⁢
m
]
,
wherein gij represents an element of an ith row and a jth column of the matrix G, and eij represents an element of an ith row and a jth column of the matrix E;
setting E=G·B, such that
e
1
⁢
p
′
=
u
p
⁢
max
-
u
p
⁢
min
u
p
′
⁢
max
-
u
p
′
⁢
min
⁢
e
1
⁢
p
,
e
1
⁢
q
′
=
u
q
⁢
max
-
u
q
⁢
min
u
q
′
⁢
max
-
u
q
′
⁢
min
⁢
e
1
⁢
q
;
and substituting the G and the B into the E=G·B to obtain a following equation set:
{
∑
s
=
1
3
g
1
⁢
s
·
b
sp
=
e
1
⁢
p
∑
s
=
1
3
g
1
⁢
s
·
b
sp
′
=
u
p
⁢
max
-
u
p
⁢
min
u
p
′
⁢
max
-
u
p
′
⁢
min
⁢
e
1
⁢
p
∑
s
=
1
3
g
1
⁢
s
·
b
sq
=
e
1
⁢
q
∑
s
=
1
3
g
1
⁢
s
·
b
sq
′
=
u
q
⁢
max
-
u
q
⁢
min
u
q
′
⁢
max
-
u
q
′
⁢
min
⁢
e
1
⁢
q
(
7
)
solving the linear equation set (7) to obtain a first row of the G, and calculating a first row (e11, . . . , e1m) of the matrix E according to the E=G·B;
and if e1p′>0 and e1q′>0, performing a step 4-4-3);
if e1p′<0 and e1q′<0, performing a step 4-4-4);
or if e1p′·e1q′<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 Ω;
and then performing the step 4-4-5), wherein i=1, . . . , m, i≠p, p′, q, q′;
and
the rule is as follows:
when e1i·e1i+1<0,
if e1i>0 and e1i−1<0, setting an ith row of the matrix T to the uimax and an i+1th row to ui+1min;
or
if e1i<0 and e1i+1>0, setting an ith row of the matrix T to the uimin and an i+1th row to ui+1max;
when e1i<0 and e1i+1<0 (i=1, . . . , m,i≠p, p′,q,q′),
setting an ith row of the matrix T to the uimin and an i+1th row to ui+1min;
or
when e1i>0 and e1i+1>0,
calculating 1d01=e1i·uimax+e1i+1min and 1d02=e1i·uimin+e1i+1max, and performing determining as follows:
if 1d01>1d02, setting an ith row of the matrix T to the uimax and an i+1th row to ui+1min;
if 1d01<1d02, setting an ith row of the matrix T to the uimin and an i+1th row to ui+1max;
or if 1d01=1d02, setting an ith row and an i+1th row of the matrix T as follows:
setting the ith row of the matrix T to the uimax and the i+1th row to the ui+1min;
or setting the ith row of the matrix T to the uimin and the i+1th row to the ui+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 Ω;
and then performing the step 4-4-5), where i=1, . . . , m, i≠p, p′, q, q′;
and
the rule is as follows:
when e1i·e1i+1<0,
if e1i>0 and e1i+1<0, setting an ith row of the matrix T to the uimin and an i+1th row to ui+1max;
or
if e1i<0 and e1i+1>0, setting an ith row of the matrix T to the uimax and an i+1th row to ui+1min;
when e1i<0 and e1i+1<0,
calculating 2d01=2h1i·uimax+2h1i+1·+ui+1min and 2d02=2h1i·uimin+2h1i+1·ui+1max and if 2d01>2d02, setting an ith row of the matrix T to the uimin and an i+1th row to ui+1max;
if 2d01<2d02, setting an ith row of the matrix T to the uimax and an i+1th row to ui+1min;
or if 2d01=2d02, setting an ith row and an i+1th row of the matrix T as follows:
setting the ith row of the matrix T to the uimax and the i+1th row to ui+1min;
or setting the ith row of the matrix T to the uimin and the i+1th row to ui+1max;
or
when e1i>0 and e1i+1>0,
setting an ith row of the matrix T to the uimin and an i+1th row to ui+1min;
4-4-5) constructing a coordinate rotation transformation 2T, such that a transformed coordinate axis v1 is perpendicular to an image of the type-I rectangular boundary surface of the group, which specifically includes:
setting
2
T
=
[
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
]
,
B
=
[
b
1
⁢
1
⁢
…
⁢
b
1
⁢
p
⁢
…
⁢
b
1
⁢
q
⁢
…
⁢
b
1
⁢
m
b
21
⁢
…
⁢
b
2
⁢
p
⁢
…
⁢
b
2
⁢
q
⁢
…
⁢
b
2
⁢
m
b
31
⁢
…
⁢
b
3
⁢
p
⁢
…
⁢
b
3
⁢
q
⁢
…
⁢
b
3
⁢
m
]
,
and
2
C
=
[
2
c
11
⁢
…
⁢
2
c
1
⁢
p
⁢
…
⁢
2
c
1
⁢
q
⁢
…
⁢
2
c
1
⁢
m
2
c
21
⁢
…
⁢
2
c
2
⁢
p
⁢
…
⁢
2
c
2
⁢
q
⁢
…
⁢
2
c
2
⁢
m
2
c
31
⁢
…
⁢
2
c
3
⁢
p
⁢
…
⁢
2
c
3
⁢
q
⁢
…
⁢
2
c
3
⁢
m
]
,
wherein 2Cij represents an element of an ith row and a jth column of the matrix 2C, and 2tij represents an element of an ith row and a jth column of the matrix 2T;
setting 2C=2T·B, such that 2c1p=0 and 2C1q=0;
and substituting the 2T and the B into the 2C=2T·B to obtain a following equation set:
{
∑
s
=
1
3
2
t
1
2
·
b
sp
=
0
∑
s
=
1
3
2
t
1
2
·
b
sq
=
0
(
8
)
solving the linear equation set (8) to obtain a first row (2t11, 2t12, 2t13) of the 2T, and calculating a first row (2C11, . . . , 2C1m) of the matrix 2C according to the 2C=2T·B;
and
if 2c1p′<0 and 2C1q′<0 performing a step 4-4-6);
if 2c1p′>0 and 2C1q′>0, performing a step 4-4-7);
or if 2c1p′·2C1q′<0, performing a step 4-4-8);
4-4-6) processing other rows than the rows p, q, p′, q′ of the matrix Λ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, 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 2c1i·2c1i+1<0,
if 2c1i>0 and 2c1i+1<0 setting an ith row of the matrix Λ to the uimax and an i+1th row to ui+1min;
or
if 2c1i<0 and 2c1i+1>0, setting an ith row of the matrix Λ to the uimin and an i+1th row to ui+1max;
when 2c1i<0 and 2c1i+1<0,
setting an ith row of the matrix Λ to the uimin and an i+1th row to ui+1min;
or
when 2c1i>0 and 2c1i+1>0,
calculating 3d01=2c1i·uimax+2c1i+1·ui+1min, and 3d02=2c1i·uimin+2c1i+1·ui+1max;
and if 3d01>3d02, setting an ith row of the matrix Λ to the uimax and an i+1th row to ui+1min;
if 3d01<3d02, setting an ith row of the matrix Λ to the uimin and an i+1th row to ui+1max;
or if 3d01=3d02, setting an ith row and an i+1th row of the matrix Λ as follows:
setting the ith row of the matrix Λ to the uimax and the i+1th row to ui+1min;
or setting the ith row of the matrix Λ to the uimin and the i+1th row to ui+1max;
4-4-7) processing other rows than the rows p, q, p′, q′ of the matrix according Λ to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, 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 2c1i·2c1i+1<0,
if 2c1i>0 and 2c1i+1<0, setting an ith row of the matrix Λ to the uimin and an i+1th row to ui+1max;
or
if 2c1i<0 and 2c1i+1>0 setting an ith row of the matrix Λ to the uimax and an i+1th row to ui+1min;
when 2c1i<0 and 2c1i+1<0,
calculating 4d01=2c1i·uimax+2c1i+1·ui+1min and 4d02=2c1i·uimin+2c1i+1·ui+1max;
and if 4d01>4d02, setting an ith row of the matrix Λ to the uimin and an i+1th row to ui+1max;
if 4d01<4d02, setting an ith row of the matrix Λ to the uimax and an i+1th row to ui+1min;
or if 4d01=4d02, setting an ith row and an i+1th row of the matrix Λ as follows:
setting the ith row of the matrix Λ to the uimax and the i+1th row to ui+1min;
or setting the ith row of the matrix Λ to the uimin and the i+1th row to ui+1max;
or
when 2c1i>0 and 2c1i+1>0,
setting an ith row of the matrix Λ to the uimin and an i+1th row to ui+1 min;
4-4-8) constructing a coordinate rotation transformation 1F, such that a transformed coordinate axis v1 is Perpendicular to an Image of a Boundary Surface Whose Vertex Matrix is the 1Π in the group, and images of all points on the boundary surface have equal coordinate values on the v1 axis, which specifically includes:
setting
1
F
=
[
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
]
,
B
=
[
b
11
…
b
1
⁢
p
b
1
⁢
p
′
…
b
1
⁢
q
…
b
1
⁢
m
b
21
…
b
2
⁢
p
b
2
⁢
p
′
…
b
2
⁢
q
…
b
2
⁢
m
b
31
…
b
3
⁢
p
b
3
⁢
p
′
…
b
3
⁢
q
…
b
3
⁢
m
]
,
and
1
H
=
[
1
h
11
…
1
h
1
⁢
p
1
h
1
⁢
p
′
…
1
h
1
⁢
q
…
1
h
1
⁢
m
1
h
21
…
1
h
2
⁢
p
1
h
2
⁢
p
′
…
1
h
2
⁢
q
…
1
h
2
⁢
m
1
h
31
…
1
h
3
⁢
p
1
h
3
⁢
p
′
…
1
h
3
⁢
q
…
1
h
3
⁢
m
]
,
wherein 1fij represents an element of an ith row and a jth column of the matrix 1F, and 1hij represents an element of an ith row and a jth column of the matrix 1H;
setting 1H=1F·B, such that
1
h
1
⁢
p
′
=
u
p
⁢
max
-
u
p
⁢
min
u
p
′
⁢
max
-
u
p
′
⁢
min
⁢
1
h
1
⁢
p
and 11q=0;
and substituting the 1F and the B into the 1H= 1F·B to obtain a following equation set:
{
∑
s
=
1
3
1
f
1
⁢
s
·
b
sp
=
1
h
1
⁢
p
∑
s
=
1
3
1
f
1
⁢
s
·
b
sp
′
=
u
p
⁢
max
-
u
p
⁢
min
u
p
′
⁢
max
-
u
p
′
⁢
min
⁢
1
h
1
⁢
p
∑
s
=
1
3
1
f
1
⁢
s
·
b
sq
=
0
(
9
)
solving the linear equation set (9) to obtain a first row of the 1F, and calculating a first row (1h11, . . . , 1h1m) of the matrix 1H according to the 1H=1F·B;
and
if 1h1p′>0 1h1q′<0, performing a step 4-4-9);
if 1h1p′<0 and 1h1q′>0, performing a step 4-4-10);
or if 1h1p′·1h1q′>0, performing a step 4-4-11);
4-4-9) processing other rows than the rows p, q, P′, q′ of the matrix 1Π according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, 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 1h1i·1h1i+1<0,
if 1h1i>0 and 1h1i+1<0, setting an ith row of the matrix 1Π to the uimax and an i+1th row to ui+1min;
or
if 1h1i<0 and 1h1i+1>0, setting an ith row of the matrix 1Π to the uimin and an i+1th row to ui+1max;
when 1h1i<0 and 1h1i+1<0,
setting an ith row of the matrix 1Π to the uimin and an i+1th row to ui+1min;
or
when 1h1i>0 and 1h1i+1>0,
calculating 5d01=1h1i·uimax+1h1i+1·ui+1min and 5d02=1h1i·uimin+1h1i+1·ui+1max;
and if 5d01>5d02, setting an ith row of the matrix 1Π to the uimax and an i+1th row to ui+1min;
if 5d01<5d02, setting an ith row of the matrix 1Π to the uimin and an i+1th row to ui+1max;
or if 5d01=5d02, setting an ith row and an i+1th row of the matrix 1Π as follows:
setting the ith row of the matrix 1Π to the uimax and the i+1th row to ui+1min;
or setting the ith row of the matrix 1Π to the uimin and the i+1th row to ui+1max;
4-4-10) processing other rows than the rows p, q, p′, q′ of the matrix 1Π according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, 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 1h1i·1h1i+1<0,
if 1h1i>0 and 1h1i+1<0, setting an ith row of the matrix 1Π to the uimin and an i+1th row to ui+1max;
or
if 1h1i<0 and 1h1i+1>0, setting an ith row of the matrix 1Π to the uimax and an i+1th row to ui+1min;
when 1h1i>0 and 1h1i+1>0,
setting an ith row of the matrix 1Π to the uimin and an i+1th row to ui+1min;
or
when 1h1i<0 and 1h1i+1<0,
calculating 6d01=1h1i·uimax+1h1i+1·ui+1min and 6d02=1h1i·uimin+1h1i+1·ui+1max;
and if 6d01>6d02, setting an ith row of the matrix 1Π to the uimin and an i+1th row to ui+1max;
if 6d01<6d02, setting an ith row of the matrix 1Π to the uimax and an i+1th row to ui+1min;
or if 6d01=6d02, setting an ith row and an i+1th row of the matrix 1Π as follows:
setting the ith row of the matrix 1Π to the uimax and the i+1th row to ui+1min;
or setting the ith row of the matrix 1Π to the uimin and the i+1th row to ui+1max;
4-4-11) constructing a coordinate rotation transformation 2F, such that a transformed coordinate axis v1 is perpendicular to an image of a boundary surface whose vertex matrix is the 2Π in the group, and images of all points on the boundary surface have equal coordinate values on the v1 axis, which specifically includes:
setting
2
F
=
[
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
]
,
B
=
[
b
11
…
b
1
⁢
p
b
1
⁢
p
′
…
b
1
⁢
q
…
b
1
⁢
m
b
21
…
b
2
⁢
p
b
2
⁢
p
′
…
b
2
⁢
q
…
b
2
⁢
m
b
31
…
b
3
⁢
p
b
3
⁢
p
′
…
b
3
⁢
q
…
b
3
⁢
m
]
,
and
2
H
=
[
2
h
11
…
2
h
1
⁢
p
2
h
1
⁢
q
…
2
h
1
⁢
q
′
…
2
h
1
⁢
m
2
h
21
…
2
h
2
⁢
p
2
h
2
⁢
q
…
2
h
2
⁢
q
′
…
2
h
2
⁢
m
2
h
31
…
2
h
3
⁢
p
2
h
3
⁢
q
…
2
h
3
⁢
q
′
…
2
h
3
⁢
m
]
,
where 2fij represents an element of an ith row and a jth column of the matrix 2F, and 2hij represents an element of an ith row and a jth column of the matrix 2H;
setting 2H=2F·B, such that
2
h
1
⁢
q
′
=
u
q
⁢
max
-
u
q
⁢
min
u
q
′
⁢
max
-
u
q
′
⁢
min
⁢
2
h
1
⁢
q
and 2h1p=0 and substituting the 2F and the B into the 2H=2F·B to obtain a following equation set:
{
∑
s
=
1
3
2
f
1
⁢
s
·
b
sp
=
0
∑
s
=
1
3
2
f
1
⁢
s
·
b
sq
=
2
h
1
⁢
q
∑
s
=
1
3
2
f
1
⁢
s
·
b
sq
′
=
u
q
⁢
max
-
u
q
⁢
min
u
q
′
⁢
max
-
u
q
′
⁢
min
⁢
2
h
1
⁢
q
(
10
)
solving the linear equation set (10) to obtain a first row of the 2F, and calculating a first row (2h11, . . . , 2h1m) of the matrix 2H according to the 2H=2F·B;
and if 2h1p′<0 and 2h1q′>0, performing a step 4-4-12);
if 2h1p′>0 and 2h1q′<0 and, performing a step 4-4-13);
or if 2 h1p′·2h1q′>0, performing a step 4-6);
4-4-12) processing other rows than the rows p, q, p′, q′ of the matrix 2Π according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′;
and then performing the step 4-6), where
the rule is as follows:
when 2h1i·2h1i+1<0,
if 2h1i>0 and 2h1i+1<0, setting an ith row of the matrix 2Π to the uimax and an i+1th row to ui+1min;
or
if 2h1i<0 and 2h1i+1>0, setting an ith row of the matrix 2Π to the uimin and an i+1th row to ui+1max;
when 2h1i<0 and 2h1i+1<0,
setting an ith row of the matrix 2Π to the uimin and an i+1th row to ui+1min;
or
when 2h1i>0 and 2h1i+1>0,
calculating 7d01=2h1i·uimax+2h1i+1·ui+1min and 7d02=2h1i·uimin+2h1i+1·ui+1max;
and if 7d01>7d02, setting an ith row of the matrix 2Π to the uimax and an i+1th row to ui+1min;
if 7d01<7d02, setting an ith row of the matrix 2Π to the uimin and an i+1th row to ui+1max;
or if 7d01=7d02, setting an ith row and an i+1th row of the matrix 2Π as follows:
setting the ith row of the matrix 2Π to the uimax and the i+1th row to ui+1min;
or setting the ith row of the matrix 2Π to the uimin and the i+1th row to ui+1max;
4-4-13) processing other rows than the rows p, q, p′, q′ of the matrix 2Π according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′;
and then performing the step 4-6), where
the rule is as follows:
when 2h1i·2h1i+1<0,
if 2h1i>0 and 2h1i+1<0, setting an ith row of the matrix 2Π to the uimin and an i+1th row to ui+1max;
or
if 2h1i<0 and 2h1i+1>0, setting an ith row of the matrix 2Π to the uimax and an i+1th row to ui+1min;
when 2h1i<0 and 2h1i+1<0,
calculating 8d01=2h1i·uimax+2h1i+1·ui+1min and 8d02=2h1i·uimin+2h1i+1·ui+1max;
and if 8d01>8d02, setting an ith row of the matrix 2Π to the uimin and an i+1th row to ui+1max;
if 8d01<8d02, setting an ith row of the matrix 2Π to the uimax and an i+1th row to ui+1min;
or if 8d01=8d02, setting an ith row and an i+1th row of the matrix 2Π as follows:
setting the ith row of the matrix 2Π to the uimax and the i+1th row to ui+1min;
or setting the ith row of the matrix 2Π to the uimin and the i+1th row to ui+1max;
or
when 2h1i>0 and 2h1i+1>0,
setting an ith row of the matrix 2Π to the uimin and an i+1th row to ui+1min;
4-5) when the pth control component and the qth control component are not a pair of linear constraint components, and the qth 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 pth component as a p′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):
{
u
p
⁢
min
≤
u
p
≤
u
p
⁢
max
u
q
⁢
min
≤
u
q
≤
u
q
⁢
max
u
p
′
∈
{
u
p
′
⁢
min
,
u
p
′
⁢
max
⁢
u
p
⁢
max
-
u
p
′
⁢
min
⁢
u
p
⁢
min
-
(
u
p
′
⁢
max
-
u
p
′
⁢
min
)
⁢
u
p
(
u
p
⁢
max
-
u
p
⁢
min
)
}
u
i
∈
{
u
i
⁢
min
,
u
i
⁢
max
}
,
1
≤
i
<
m
,
i
≠
p
,
q
,
p
′
u
k
+
u
k
+
1
<
u
k
⁢
max
+
u
k
+
1
⁢
max
,
k
=
1
,
3
,
…
,
2
[
m
2
]
-
1
,
k
≠
p
,
1
,
p
′
(
11
)
when up′=up′min determining that a boundary surface formed by a point satisfying the formula (11) is the type-I rectangular boundary surface;
or when
u
p
′
=
u
p
′
⁢
max
⁢
u
p
⁢
max
-
u
p
′
⁢
min
⁢
u
p
⁢
min
-
(
u
p
′
⁢
max
-
u
p
′
⁢
min
)
⁢
u
p
(
u
p
⁢
max
-
u
p
⁢
min
)
,
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 Λ′ is used to represent a vertex of one type-I rectangular boundary surface, and a matrix Π′ represents a vertex of one type-II rectangular boundary surface;
values of the pth, p′th, and qth components of four vertices of each boundary surface are shown in a pth row, a p′th row, and a qth row of the matrix, and other components are all uimax or uimin;
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 Λ′, the pth row is (upmax, upmin, upmin, upmax), the p′th row is (up′min, up′min, up′min, up′min) the qth row is (uqmin, uqmin, uqmax, uqmax), and the q′th row is (uq′min, uq′min, uq′min, uq′min);
Λ
′
=
[
…
…
…
…
u
p
⁢
max
u
p
⁢
min
u
p
⁢
min
u
p
⁢
max
u
p
′
⁢
min
u
p
′
⁢
min
u
p
′
⁢
min
u
p
′
⁢
min
…
…
…
…
u
q
⁢
min
u
q
⁢
min
u
q
⁢
max
u
q
⁢
max
]
,
Π
′
=
[
…
…
…
…
u
p
⁢
max
u
p
⁢
min
u
p
⁢
min
u
p
⁢
max
u
p
′
⁢
min
u
p
′
⁢
max
u
p
′
⁢
max
u
p
′
⁢
min
…
…
…
…
u
q
⁢
min
u
q
⁢
min
u
q
⁢
max
u
q
⁢
max
]
in the matrices shown in the above formula, if p is an odd number, the pth row comes before the p′th row;
or if p is an even number, the pth row comes after the p′th row;
4-5-2) constructing a coordinate rotation transformation T, such that a transformed coordinate axis v1 is perpendicular to an image of the type-I rectangular boundary surface of the group, which specifically includes:
setting
3
T
=
[
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
]
,
B
=
[
b
1
⁢
1
⁢
…
⁢
b
1
⁢
p
⁢
…
⁢
b
1
⁢
q
b
21
⁢
…
⁢
b
2
⁢
p
⁢
…
⁢
b
2
⁢
q
b
31
⁢
…
⁢
b
3
⁢
p
⁢
…
⁢
b
3
⁢
q
]
,
and
3
C
=
[
3
c
11
⁢
…
⁢
3
c
1
⁢
p
⁢
…
⁢
3
c
1
⁢
q
3
c
21
⁢
…
⁢
3
c
2
⁢
p
⁢
…
⁢
3
c
2
⁢
q
3
c
31
⁢
…
⁢
3
c
3
⁢
p
⁢
…
⁢
3
c
3
⁢
q
]
,
wherein 3tij represents an element of an ith row and a jth column of the matrix 3T, and 3cij represents an element of an ith row and a jth column of the matrix 3C;
setting 3C=3T·B, such that 3c1p=0 and 3c1q=0;
and substituting the 3T and the B into the 3C=3T·B to obtain a following equation set:
{
∑
s
=
1
3
3
t
1
⁢
s
·
b
sp
=
0
∑
s
=
1
3
3
t
1
⁢
s
·
b
sq
=
0
(
12
)
solving the linear equation set (12) to obtain a first row (3t11, 3t12, 3t13) of the 3T, and calculating a first row (3c11, . . . , 3c1m) of the matrix 3C according to the 3C=3T·B;
and if 3 c1p′<0, performing a step 4-5-3);
or if 3c1p′>0, performing a step 4-5-4);
4-5-3) processing other rows than the rows p, q, p′ of the matrix Λ′ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω;
and then performing a step 4-5-5), where i=1, . . . , m, i≠p, p′, q;
and
the rule is as follows:
when 3c1i·3h1i+1<0,
if 3c1i>0 and 3c1i+1<0, setting an ith row of the matrix Π′ to the uimax and an i+1th row to ui+1min;
or
if 3c1i<0 and 3h1i+1>0, setting an ith row of the matrix Π′ to the uimin and an i+1th row to ui+1max;
when 3c1i<0 and 3c1i+1<0,
setting an ith row of the matrix Π′ to the uimin and an i+1th row to ui+1min;
or
when 3c1i>0 and 3h1i+1>0,
calculating 9d01=3c1i·uimax+3c1i+1·ui+1min, and 9d02=3c1i·uimin+3c1i+1·ui+1max;
and performing determining as follows:
if 9d01>9d02, setting an ith row of the matrix Λ′ to the uimax and an i+1th row to ui+1min;
if 9d01<9d02, setting an ith row of the matrix Λ′ to the uimin and an i+1th row to ui+1max;
or if 9d01=9d02, setting an ith row and an i+1th row of the matrix Λ′ as follows:
setting the ith row of the matrix Λ′ to the uimax and the i+1th row to ui+1min;
or setting the ith row of the matrix Λ′ to the uimin and the i+1th row to ui+1max;
4-5-4) processing other rows than the rows p, q, p′ of the matrix Λ′ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω;
and then performing a step 4-5-5), where i=1, . . . , m, i≠p, p′, q;
and
the rule is as follows:
when 3c1i·3h1i+1<0,
if 3c1i>0 and 3c1i+1<0, setting an ith row of the matrix Λ′ to the uimin and an i+1th row to ui+1max;
or
if 3c1i<0 and 3h1i+1>0, setting an ith row of the matrix Λ′ to the uimax and an i+1th row to ui+1min;
when 3c1i>0 and 3c1i+1>0,
setting an ith row of the matrix Λ′ to the uimin and an i+1th row to ui+1min;
or
when 3c1i<0 and 3h1i+1<0,
calculating 10d01=3c1i·uimax+3c1i+1·ui+1min, and 10d02=3c1i·uimin+3c1i+1·ui+1max;
and if 10d01>10d02, setting an ith row of the matrix Λ′ to the uimin and an i+1th row to ui+1max;
if 10d01<10d02, setting an ith row of the matrix Λ′ to the uimax and an i+1th row to ui+1min;
or if 10d01=10d02, setting an ith row and an i+1th row of the matrix Λ′ as follows:
setting the ith row of the matrix Λ′ to the uimax and the i+1th row to ui+1min;
or setting the ith row of the matrix Λ′ to the uimax and the i+1th row to ui+1min;
or setting the ith row of the matrix Λ′ to the uimin and the i+1th row to ui+1max;
4-5-5) constructing a coordinate rotation transformation 3F, such that a transformed coordinate axis v1 is perpendicular to an image of a boundary surface whose vertex matrix is the Π′ in the group, and images of all points on the boundary surface have equal coordinate values on the v1 axis, which specifically includes:
setting
3
F
=
[
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
]
,
B
=
[
b
11
…
b
1
⁢
p
b
1
⁢
p
′
…
b
1
⁢
q
b
21
…
b
2
⁢
p
b
2
⁢
p
′
…
b
2
⁢
q
b
31
…
b
3
⁢
p
b
3
⁢
p
′
…
b
3
⁢
q
]
,
and
3
H
=
[
3
h
11
…
3
h
1
⁢
p
3
h
1
⁢
p
′
…
3
h
1
⁢
q
3
h
21
…
3
h
2
⁢
p
3
h
2
⁢
p
′
…
3
h
2
⁢
q
3
h
31
…
3
h
3
⁢
p
3
h
3
⁢
p
′
…
3
h
3
⁢
q
]
,
wherein 3hij represents an element of an ith row and a jth column of the matrix 3H, and 3fij represents an element of an ith row and a jth column of the matrix 3F;
setting 3H=3F·B, such that 3h1p′=upmax−upmin/up′max−up′min 3h1p, and 3h1q=0;
and substituting the 3F and the B into the 3H=3F·B to obtain a following equation set:
{
∑
s
=
1
3
3
f
1
⁢
s
·
b
sp
=
3
h
1
⁢
p
∑
s
=
1
3
3
f
1
⁢
s
·
b
sp
′
=
u
p
⁢
max
-
u
p
⁢
min
u
p
′
⁢
max
-
u
p
′
⁢
min
⁢
3
h
1
⁢
p
∑
s
=
1
3
3
f
1
⁢
s
·
b
sq
=
0
(
13
)
solving the linear equation set (13) to obtain a first row of the 3F, and calculating a first row (3h11, . . . , 3h1m) of the matrix 3H according to the 3H=3F·B;
and if 3h1p′>0, performing a step 4-5-6);
or if 3h1p′<0, performing a step 4-5-7);
4-5-6) processing other rows than the rows p, q, p′ of the matrix Π′ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω;
and then performing a step 4-6), where i=1, . . . , m, i≠p, p′, q;
and
the rule is as follows:
when 3h1i·3h1i+1<0,
if 3h1i>0 and 3h1i+1<0, setting an ith row of the matrix Λ′ to the uimax and an i+1th row to ui+1min;
or
if 3h1i<0 and 3h1i+1>0, setting an ith row of the matrix Λ′ to the uimin and an i+1th row to ui+1max;
when 3h1i<0 and 3h1i+1<0,
setting an ith row of the matrix Λ′ to the uimin and an i+1th row to ui+1min;
or
when 3h1i;
>0 and 3h1i+1>0,
calculating 11d01=3c1i·uimax+3c1i+1·ui+1min, and 11d02=3c1i·uimin+3c1i+1·ui+1max;
and if 11d01>11d02, setting an ith row of the matrix Π′ to the uimax and an i+1th row to ui+1min;
if 11d01<11d02, setting an ith row of the matrix Π′ to the uimin and an i+1th row to ui+1max;
or if 11d01=11d02, setting an ith row and an i+1th row of the matrix Π′ as follows:
setting the ith row of the matrix Π′ to the uimax and the i+1th row to ui+1min;
or setting the ith row of the matrix Π′ to the uimin and the i+1th row to ui+1max;
4-5-7) processing other rows than the rows p, q, p′ of the matrix Π′ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω;
and then performing a step 4-6), where i=1, . . . , m, i≠p, p′, q;
and
the rule is as follows:
when 3h1i·3h1i+1<0,
if 3h1i>0 and 3h1i+1<0, setting an ith row of the matrix Π′ to the uimin and an i+1th row to ui+1max;
or
if 3h1i<0 and 3h1i+1>0, setting an ith row of the matrix Π′ to the uimax and an i+1th row to ui+1min;
when 3h1i<0 and 3h1i+1<0,
calculating 12d01=3c1i·uimax+3c1i+1·ui+1min, and 12d02=3c1i·uimin+3c1i+1·ui+1max;
and if 12d01>12d02, setting an ith row of the matrix Π′ to the uimin and an i+1th row to ui+1max;
if 12d01<12d02, setting an ith row of the matrix Π′ to the uimax and an i+1th row to ui+1min;
or if 12d01=12d02, setting an ith row and an i+1th row of the matrix Π′ as follows:
setting the ith row of the matrix Π′ to the uimax and the i+1th row to ui+1min;
or setting the ith row of the matrix Π′ to the uimin and the i+1th row to ui+1max;
or
when 3h1i>0 and 3h1i+1>0,
setting an ith row of the matrix Π′ to the uimin and an i+1th row to ui+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 Ω′;
and
then performing a step 6);
and
6) mapping all vertices of each key boundary surface in the Ω′ according to v=B·uELR 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 Ω′ 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 Ω′ constitute the boundary ∂(ΦELR) of the control attainable set.

3,913 more elements

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

Life of this patent

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

Life of this patent

Abstract

Claims1