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If the @@Polyhedron@@ object exists, but it is not known which representation it has, it can be figured out using the following properties:
to:
If the @@Polyhedron@@ object exists, but it is not known which representation it has, it can be figured out using the following methods:
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to:
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to:
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to:
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which requires two arguments: the symbolic variables and the constraint set.
to:
which requires two arguments: the symbolic variables and the constraint set.
To construct a polyhedral set in YALMIP can be achieved by providing the corresponding data. For instance, to create a H-representation can be achieved as
(:source lang=MATLAB -getcode:) [@
x = sdpvar(2,1);
A = [ -0.37, 0.81; 0.79, 0.12; 0.57, 0.41; -0.98, 0.75];
b = [1; 2.3; 1.4; 2.8]
constraints = [A*x <= b ];
Y = YSet(x, constraints)
@]
Note that the variables must be provided in as a vector. For instance, to model a set given by linear matrix inequality [@ A*X + X*A' <= -I @] can be done as follows
(:source lang=MATLAB -getcode:) [@
X = sdpvar(3);
A = randn(3);
constraints = [A*X + X*A' <= eye(3) ];
Y = YSet(X(:), constraints)
@]
The input data for the @@YSet@@ object can be retrieved by referring the appropriate properties
(:source lang=MATLAB -getcode:) [@
Y.vars
Y.constraints
@]
Similarly as with the @@Polyhedron@@ object, two properties are inherited from the @@ConvexSet@@ class. In particular, the dimension of the set can be invoked using
(:source lang=MATLAB -getcode:) [@
Y.Dim
@]
and the user data stored with the set can be found under
(:source lang=MATLAB -getcode:) [@
Y.Data
@]
property. The @@Data@@ property can be modified after the object has been created.
YALMIP allows creation of various sets, including cones that can be imported to MPT
(:source lang=MATLAB -getcode:) [@
x = sdpvar(2,1);
F = [cone(x(1),x(2)), -1<= x <= 5];
Y = YSet(x, F)
@]
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The @@ConvexSet@@ object cannot be constructed directly, it is higher level object for sharing common properties in convex sets. The above properties are accessible in the objects derived from this class, such as @@Polyhedron@@ and @@YSet@@.
to:
The @@ConvexSet@@ object cannot be constructed directly, it is higher level object for sharing common properties in convex sets. The properties are accessible in the objects derived from this class, such as @@Polyhedron@@ and @@YSet@@.
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The @@YSet@@ object is an MPT interface to import convex sets described with the help of %newwin% [[ http://users.isy.liu.se/johanl/yalmip/ | YALMIP ]] %%.
to:
The @@YSet@@ object is an MPT interface to import convex sets described with the help of %newwin% [[ http://users.isy.liu.se/johanl/yalmip/ | YALMIP ]]%%. To construct the @@YSet@@ object one needs to define the symbolic variables for representing the sets and consequently to create appropriate constraint sets using YALMIP. Consider an interval of the form [@ -1 <= x <= 1 @]. In YALMIP this interval can be modeled as follows:
(:source lang=MATLAB -getcode:) [@
x = sdpvar(1);
interval = [ -1 <= x <= 1 ];
@]
which is better described in "Constraints" part of %newwin% [[http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Tutorials.Basics | Yalmip Wiki ]]%%. After creating the constraint set, the @@YSet@@ object can be constructed as follows
(:source lang=MATLAB -getcode:) [@
Y = YSet(x, interval);
@]
which requires two arguments: the symbolic variables and the constraint set.
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* [[Geometry.Sets#Polyhedron | The '''Polyhedron''' object]]
* [[Geometry.Sets#YSet | The '''YSet''' object]]
to:
* [[Geometry.Sets#Polyhedron | The '''Polyhedron''' object - representation of polyhedra ]]
* [[Geometry.Sets#YSet | The '''YSet''' object - representation of general convex sets]]
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!!! The '''YSet''' object - representation of convex sets
to:
!!! The '''YSet''' object - representation of general convex sets
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The polyhedron is in minimal (irredundant) H-representation, if the are no redundant inequalities/equalities describing the set. As an example, consider the polyhedron build by two intervals [@ -1 <= x <= 1, -2 <= x <= 2 @]. Obviously, the first interval is completely contained in the second interval, so it is considered as redundant. To query if the minimal representation of the polyhedron has been computed, one can use the following properties
to:
The polyhedron is in minimal (irredundant) H-representation, if the are no redundant inequalities/equalities describing the set. To query if the minimal representation of the polyhedron has been computed, one can use the following properties
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that apply either for H- or V-representation.
to:
that apply either for H- or V-representation. To compute the minimal H- or V-representation, one can employ the methods
(:source lang=MATLAB -getcode:) [@
P.minHRep
P.minVRep
@]
As an example, consider the polyhedron build by two intervals [@ -1 <= x <= 1, -2 <= x <= 2 @]. The first interval is completely contained in the second interval and to describe the set only the first interval is sufficient. We can check this by constructing the polyhedron from the inequality description
(:source lang=MATLAB -getcode:) [@
A = [-1; 1; -1; 1]; b = [1; 1; 2; 2];
P = Polyhedron('A', A, 'b', b);
P.irredundantHRep
@]
One can see in the output that the minimal representation of the polyhedron has not been computed yet. Computing the minimal representation for the above example gives only the first interval which can be checked by calling
(:source lang=MATLAB -getcode:) [@
P.minHRep;
P.irredundantHRep
P.H
@]
To quickly construct @@Polyhedron@@ objects, one can resort to fast syntax that comprises only of inequalies or vertices. The fast syntax for inequality description is given as
(:source lang=MATLAB -getcode:) [@
A = [-1, 2, 0; -0.1, -3.1, 1.8]; b = [5.5; 3.8];
P = Polyhedron(A, b);
@]
and for vertex description
(:source lang=MATLAB -getcode:) [@
V = [0, 1, -2; -1, 0.5 -4; 1, -1, 0.8; -4, -5, -0.9];
P = Polyhedron(V);
@]
Deleted line 4:
Added lines 74-87:
If the @@Polyhedron@@ object exists, but it is not known which representation it has, it can be figured out using the following properties:
(:source lang=MATLAB -getcode:) [@
P.hasHRep
P.hasVRep
@]
The output is a logical variable that indicates in which form is the polyhedron stored.
The polyhedron is in minimal (irredundant) H-representation, if the are no redundant inequalities/equalities describing the set. As an example, consider the polyhedron build by two intervals [@ -1 <= x <= 1, -2 <= x <= 2 @]. Obviously, the first interval is completely contained in the second interval, so it is considered as redundant. To query if the minimal representation of the polyhedron has been computed, one can use the following properties
(:source lang=MATLAB -getcode:) [@
P.irredundantHRep
P.irredundantVRep
@]
that apply either for H- or V-representation.
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!!! High-level @@ConvexSet@@ object
to:
!!! High-level '''ConvexSet''' object
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which returns the following result
[@
Properties for class ConvexSet:
Dim
Data
@]
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!!! The object for representation of polyhedra - @@Polyhedron@@
to:
!!! The '''Polyhedron''' object - representation of polyhedra
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!! Fast construction
!!! The object for representation of convex sets - @@YSet@@
to:
!!! The '''YSet''' object - representation of convex sets
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The V-representation of the polyhedron can be constructed by providing a set of vertices and rays. Vertices are accepted in a matrix form stored row-wise (each row of a matrix corresponds to a vertex):
to:
The data can be extracted in a more compact form - using the properties @@H@@ and @@He@@. The @@H@@ property collects the matrices for inequality description [@ H = [A, b] @] and the @@He@@ property collects the matrices of equality description [@ He = [Ae, be] @]:
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V = [-1.0 -0.8; -2.9 1.4; 0.3 -0.7];
R = [-0.5, 1.9];
P = Polyhedron('V', V, 'R', R);
to:
Changed line 49 from:
The data can be retrieved from the corresponding fields @@V@@ and @@R@@:
to:
The V-representation of the polyhedron can be constructed by providing a set of vertices and rays. Vertices are accepted in a matrix form stored row-wise (each row of a matrix corresponds to a vertex):
Changed lines 51-52 from:
to:
V = [-1.0 -0.8; -2.9 1.4; 0.3 -0.7];
R = [-0.5, 1.9];
P = Polyhedron('V', V, 'R', R);
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The dimension of the polyhedral set is available in @@Dim@@ property that is inherited from the higher level @@ConvexSet@@ object
to:
The data can be retrieved from the corresponding fields @@V@@ and @@R@@:
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to:
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The user can store arbitrary data with the @@Polyhedron@@ object with the help of @@Data@@ property that is inherited from from @@ConvexSet@@ object. As an example, consider that you want to store additional data that generated the V-representation. One possibility is to create a structure with the user data, e.g.
to:
The dimension of the polyhedral set is available in @@Dim@@ property that is inherited from the higher level @@ConvexSet@@ object
Added lines 64-68:
P.Dim
@]
The user can store arbitrary data with the @@Polyhedron@@ object with the help of @@Data@@ property that is inherited from from @@ConvexSet@@ object. As an example, consider that you want to store additional data that generated the V-representation. One possibility is to create a structure with the user data, e.g.
(:source lang=MATLAB -getcode:) [@
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Added lines 54-72:
@]
The dimension of the polyhedral set is available in @@Dim@@ property that is inherited from the higher level @@ConvexSet@@ object
(:source lang=MATLAB -getcode:) [@
P.Dim
@]
The user can store arbitrary data with the @@Polyhedron@@ object with the help of @@Data@@ property that is inherited from from @@ConvexSet@@ object. As an example, consider that you want to store additional data that generated the V-representation. One possibility is to create a structure with the user data, e.g.
(:source lang=MATLAB -getcode:) [@
data.name = 'filename01.dat';
data.size = 5;
V = [1, -2; -1, 2; 3, 3];
P = Polyhedron('V', V, 'Data', data);
@]
The user provided data are accessible in @@Data@@ property and can be modified after construction of the object:
(:source lang=MATLAB -getcode:) [@
P.Data
P.Data.size = 6;
Changed line 20 from:
The @@Polyhedron@@ object represents a polyhedron given as the intersection of inequalities and equalities (referred to as H-representation)
to:
The @@Polyhedron@@ object represents a polyhedron given as the intersection of inequalities and equalities (referred to as '''H-representation''')
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or as the convex combination of vertices and rays (referred to as V-representation)
to:
or as the convex combination of vertices and rays (referred to as '''V-representation''')
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The data are stored as they are provided, no automatic scaling or conversion is performed. The stored data can be retrieved in the appropriate fields @@A@@, @@b@@, @@Ae@@, @@be@@
to:
The data are stored as they are provided, no automatic scaling or conversion is performed unless a given operation is performed on the polyhedron. The stored data can be retrieved in the appropriate fields @@A@@, @@b@@, @@Ae@@, @@be@@
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@]
The V-representation of the polyhedron can be constructed by providing a set of vertices and rays. Vertices are accepted in a matrix form stored row-wise (each row of a matrix corresponds to a vertex):
(:source lang=MATLAB -getcode:) [@
V = [-1.0 -0.8; -2.9 1.4; 0.3 -0.7];
R = [-0.5, 1.9];
P = Polyhedron('V', V, 'R', R);
@]
The data can be retrieved from the corresponding fields @@V@@ and @@R@@:
(:source lang=MATLAB -getcode:) [@
P.V
P.R
Changed line 20 from:
The @@Polyhedron@@ object represents a polyhedron given as the intersection of inequalities and equalities
to:
The @@Polyhedron@@ object represents a polyhedron given as the intersection of inequalities and equalities (referred to as H-representation)
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or as the convex combination of vertices and rays
to:
or as the convex combination of vertices and rays (referred to as V-representation)
Added lines 25-43:
Both representations of polyhedra can be easily constructed in MPT3 providing the corresponding data. For instance, to construct the H-representation it suffices to provide the inequality and equality description in matrix form:
(:source lang=MATLAB -getcode:) [@
A = [2.7694 -1.3499; 3.0349 0.7254; -0.0631 0.7147];
b = [0.5675; 1.6870; 3.6629];
Ae = [0.1246, -1.012];
be = 2.45;
P = Polyhedron('A', A, 'b', b, 'Ae', Ae, 'be', be);
@]
The data are stored as they are provided, no automatic scaling or conversion is performed. The stored data can be retrieved in the appropriate fields @@A@@, @@b@@, @@Ae@@, @@be@@
(:source lang=MATLAB -getcode:) [@
P.A
P.b
P.Ae
P.be
@]
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The @@ConvexSet@@ object cannot be constructed directly, it is higher level object for sharing common properties in convex sets. The above properties are accessible in the objects derived from this class, such as @@YSet@@ and @@Polyhedron@@.
!!! The object for representation of convex sets - @@YSet@@
to:
The @@ConvexSet@@ object cannot be constructed directly, it is higher level object for sharing common properties in convex sets. The above properties are accessible in the objects derived from this class, such as @@Polyhedron@@ and @@YSet@@.
Changed lines 20-25 from:
to:
The @@Polyhedron@@ object represents a polyhedron given as the intersection of inequalities and equalities
or as the convex combination of vertices and rays
!!! The object for representation of convex sets - @@YSet@@
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In MPT3 there are new classes of objects to represent convex sets that are derived from a common @@ConvexSet@@ object. The @@ConvexSet@@ object contains the information on the dimension of the set and it can store arbitrary user data. The dimension is available under @@Dim@@ property and the user date can be stored in @@Data@@ property:
to:
!!! High-level @@ConvexSet@@ object
In MPT3 there are new classes of objects to represent convex sets that are derived from a common @@ConvexSet@@ object. The @@ConvexSet@@ object contains the information on the dimension of the set and it can store arbitrary user data. The dimension is available under @@Dim@@ property and the user data can be stored in @@Data@@ property. These properties are visible when typing
(:source lang=MATLAB -getcode:) [@
properties('ConvexSet')
@]
which returns the following result
[@
Properties for class ConvexSet:
Dim
Data
@]
The @@ConvexSet@@ object cannot be constructed directly, it is higher level object for sharing common properties in convex sets. The above properties are accessible in the objects derived from this class, such as @@YSet@@ and @@Polyhedron@@.
!!! The object for representation of convex sets - @@YSet@@
!!! The object for representation of polyhedra - @@Polyhedron@@