Linear Algebra & Linear Dynamical Systems
AA 203 Recitation #1
April 10th, 2020
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 1 / 37
Overview
1 Linear Algebra2 Linear Dynamical Systems (LDS)
a. Discrete time LDS.b. Continuous time LDS.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 2 / 37
Overview (cont.)
1 Linear Algebra
a. Vector Spaces.b. Matrices and linear functions.d. Matrix Multiplication and Matrix Inverse.e. Singular Value Decomposition.f. Eigenvalue Decomposition.
2 Discrete time Linear Dynamical Systems
a. Stabilization and tracking.b. Constraint structure.c. Operator norm & stability via linear feedback.
3 Continuous time Linear Dynamical Systems
a. Stabilization and tracking.b. State evolution.c. Diagonalization & stability via linear feedback.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 3 / 37
Overview (cont.)
1 Linear Algebra
a. Vector Spaces.b. Matrices and linear functions.d. Matrix Multiplication and Matrix Inverse.e. Singular Value Decomposition.f. Eigenvalue Decomposition.
2 Discrete time Linear Dynamical Systems
a. Stabilization and tracking.b. Constraint structure.c. Operator norm & stability via linear feedback.
3 Continuous time Linear Dynamical Systems
a. Stabilization and tracking.b. State evolution.c. Diagonalization & stability via linear feedback.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 3 / 37
Overview (cont.)
1 Linear Algebra
a. Vector Spaces.b. Matrices and linear functions.d. Matrix Multiplication and Matrix Inverse.e. Singular Value Decomposition.f. Eigenvalue Decomposition.
2 Discrete time Linear Dynamical Systems
a. Stabilization and tracking.b. Constraint structure.c. Operator norm & stability via linear feedback.
3 Continuous time Linear Dynamical Systems
a. Stabilization and tracking.b. State evolution.c. Diagonalization & stability via linear feedback.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 3 / 37
1. Linear Algebra
1. Linear Algebra
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 4 / 37
1. Linear Algebra - Vector Spaces
We will use Euclidean vector spaces in this course.
Rn is the set of n-vectors.
R is the set of scalars.
Vector addition is entrywise: x + y = z means xi + yi = zi for all1 ≤ i ≤ n. Example: [
25
]+
[92
]=
[117
]
Scalar multiplication is entrywise: cx = y means cxi = yi for all1 ≤ i ≤ n. Example:
2
[43
]=
[86
]
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 5 / 37
1. Linear Algebra - Vector Spaces
We will use Euclidean vector spaces in this course.
Rn is the set of n-vectors.
R is the set of scalars.
Vector addition is entrywise: x + y = z means xi + yi = zi for all1 ≤ i ≤ n. Example: [
25
]+
[92
]=
[117
]
Scalar multiplication is entrywise: cx = y means cxi = yi for all1 ≤ i ≤ n. Example:
2
[43
]=
[86
]
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 5 / 37
1. Linear Algebra - Vector Spaces
We will use Euclidean vector spaces in this course.
Rn is the set of n-vectors.
R is the set of scalars.
Vector addition is entrywise: x + y = z means xi + yi = zi for all1 ≤ i ≤ n. Example: [
25
]+
[92
]=
[117
]
Scalar multiplication is entrywise: cx = y means cxi = yi for all1 ≤ i ≤ n. Example:
2
[43
]=
[86
]
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 5 / 37
1. Linear Algebra - Vector Spaces
We will use Euclidean vector spaces in this course.
Rn is the set of n-vectors.
R is the set of scalars.
Vector addition is entrywise: x + y = z means xi + yi = zi for all1 ≤ i ≤ n. Example: [
25
]+
[92
]=
[117
]
Scalar multiplication is entrywise: cx = y means cxi = yi for all1 ≤ i ≤ n. Example:
2
[43
]=
[86
]
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 5 / 37
1. Linear Algebra - Linear Combinations
Definition (Linear Combination)
A linear combination of the vectors x1, x2, ..., xm is any vector of the form
m∑i=1
cixi where c1, ..., cm ∈ R.
Definition (Span)
The span of a collection of vectors x1, ..., xm, denoted by span(x1, ..., xm) isthe collection of all possible linear combinations.
span(x1, ..., xm) :=
{y : y =
m∑i=1
cixi for some c1, ..., cm ∈ R
}
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 6 / 37
1. Linear Algebra - Linear Combinations
Definition (Linear Combination)
A linear combination of the vectors x1, x2, ..., xm is any vector of the form
m∑i=1
cixi where c1, ..., cm ∈ R.
Definition (Span)
The span of a collection of vectors x1, ..., xm, denoted by span(x1, ..., xm) isthe collection of all possible linear combinations.
span(x1, ..., xm) :=
{y : y =
m∑i=1
cixi for some c1, ..., cm ∈ R
}
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 6 / 37
1. Linear Algebra - Linear Independence
Definition (Linear Independence)
A collection of vectors x1, ..., xm is linearly independent if no member canbe written as a linear combination of the other members.
Equivalently, the vectors x1, ..., xm are independent if
m∑i=1
cixi = 0 =⇒ ci = 0 for all 1 ≤ i ≤ m.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 7 / 37
1. Linear Algebra - Linear Subspaces
Definition (Linear Subspaces)
A set V ⊂ Rn is a linear subspace if it is
1 Closed under scalar multiplication: v ∈ V =⇒ cv ∈ V for all c ∈ R.
2 Closed under vector addition: u, v ∈ V =⇒ u + v ∈ V .
Example 1: Rn itself is a linear subspace!
Example 2: For any set of vectors V0, span(V0) is a linear subspace.
Example 3: {x ∈ Rn : x1 = 0} is a linear subspace.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 8 / 37
1. Linear Algebra - Linear Subspaces
Definition (Linear Subspaces)
A set V ⊂ Rn is a linear subspace if it is
1 Closed under scalar multiplication: v ∈ V =⇒ cv ∈ V for all c ∈ R.
2 Closed under vector addition: u, v ∈ V =⇒ u + v ∈ V .
Example 1: Rn itself is a linear subspace!
Example 2: For any set of vectors V0, span(V0) is a linear subspace.
Example 3: {x ∈ Rn : x1 = 0} is a linear subspace.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 8 / 37
1. Linear Algebra - Linear Subspaces
Definition (Linear Subspaces)
A set V ⊂ Rn is a linear subspace if it is
1 Closed under scalar multiplication: v ∈ V =⇒ cv ∈ V for all c ∈ R.
2 Closed under vector addition: u, v ∈ V =⇒ u + v ∈ V .
Example 1: Rn itself is a linear subspace!
Example 2: For any set of vectors V0, span(V0) is a linear subspace.
Example 3: {x ∈ Rn : x1 = 0} is a linear subspace.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 8 / 37
1. Linear Algebra - Linear Subspaces
Definition (Linear Subspaces)
A set V ⊂ Rn is a linear subspace if it is
1 Closed under scalar multiplication: v ∈ V =⇒ cv ∈ V for all c ∈ R.
2 Closed under vector addition: u, v ∈ V =⇒ u + v ∈ V .
Example 1: Rn itself is a linear subspace!
Example 2: For any set of vectors V0, span(V0) is a linear subspace.
Example 3: {x ∈ Rn : x1 = 0} is a linear subspace.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 8 / 37
1. Linear Algebra - Spanning sets, Basis, Dimension
Definition (Spanning Set)
Given a linear subspace V , a collection of vectors x1, ..., xm is a spanningset if span(x1, ..., xm) = V .
Definition (Basis)
A collection of vectors x1, ..., xm is a basis for a subspace V if a) it is aspanning set of V and b) it is linearly independent.
Definition (Dimension)
Every basis of a subspace V contains the same number of vectors. Thisnumber is the dimension of V .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 9 / 37
1. Linear Algebra - Spanning sets, Basis, Dimension
Definition (Spanning Set)
Given a linear subspace V , a collection of vectors x1, ..., xm is a spanningset if span(x1, ..., xm) = V .
Definition (Basis)
A collection of vectors x1, ..., xm is a basis for a subspace V if a) it is aspanning set of V and b) it is linearly independent.
Definition (Dimension)
Every basis of a subspace V contains the same number of vectors. Thisnumber is the dimension of V .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 9 / 37
1. Linear Algebra - Spanning sets, Basis, Dimension
Definition (Spanning Set)
Given a linear subspace V , a collection of vectors x1, ..., xm is a spanningset if span(x1, ..., xm) = V .
Definition (Basis)
A collection of vectors x1, ..., xm is a basis for a subspace V if a) it is aspanning set of V and b) it is linearly independent.
Definition (Dimension)
Every basis of a subspace V contains the same number of vectors. Thisnumber is the dimension of V .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 9 / 37
1. Linear Algebra - Vector Spaces
Definition
Given x , y ∈ Rn, their dot product 〈x , y〉 is defined as:
〈x , y〉 :=n∑
i=1
xiyi .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 10 / 37
1. Linear Algebra - Matrices
A matrix A ∈ Rm×n is an array of numbers with m rows and n columns.Aij is the number in the ith row and jth column.
Example: Below is a 4× 3 matrix, A. Note that A1,3 = 5.
A =
4 1 52 2 13 0 90 0 7
Vectors are also matrices! A vector x ∈ Rn is a n × 1 matrix.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 11 / 37
1. Linear Algebra - Matrices
A matrix A ∈ Rm×n is an array of numbers with m rows and n columns.Aij is the number in the ith row and jth column.
Example: Below is a 4× 3 matrix, A. Note that A1,3 = 5.
A =
4 1 52 2 13 0 90 0 7
Vectors are also matrices! A vector x ∈ Rn is a n × 1 matrix.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 11 / 37
1. Linear Algebra - Matrices
A matrix A ∈ Rm×n is an array of numbers with m rows and n columns.Aij is the number in the ith row and jth column.
Example: Below is a 4× 3 matrix, A. Note that A1,3 = 5.
A =
4 1 52 2 13 0 90 0 7
Vectors are also matrices! A vector x ∈ Rn is a n × 1 matrix.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 11 / 37
1. Linear Algebra - Matrix Transpose
If A ∈ Rm×n, then its transpose is A′ ∈ Rn×m and
Aij = A′ji .
Example:
A =
1 60 25 3
,A′ =
[1 0 56 2 3
]
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 12 / 37
1. Linear Algebra - Matrix Transpose
If A ∈ Rm×n, then its transpose is A′ ∈ Rn×m and
Aij = A′ji .
Example:
A =
1 60 25 3
,A′ =
[1 0 56 2 3
]
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 12 / 37
1. Linear Algebra - Matrix Multiplication
Matrix Multiplication: For A ∈ Rm×n,B ∈ Rp×q, AB exists if and only ifn = p. In this case, AB is a m × q matrix.
Furthermore, (AB)ij = 〈Ai ,Bj〉.Ai = ith row of ABj is the jth column of B.
Dot product as matrix multiplication: For x , y ∈ Rn,〈x , y〉 = x ′y = y ′x .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 13 / 37
1. Linear Algebra - Matrix Multiplication
Matrix Multiplication: For A ∈ Rm×n,B ∈ Rp×q, AB exists if and only ifn = p. In this case, AB is a m × q matrix.
Furthermore, (AB)ij = 〈Ai ,Bj〉.Ai = ith row of ABj is the jth column of B.
Dot product as matrix multiplication: For x , y ∈ Rn,〈x , y〉 = x ′y = y ′x .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 13 / 37
1. Linear Algebra - Matrix Multiplication
Matrix Multiplication: For A ∈ Rm×n,B ∈ Rp×q, AB exists if and only ifn = p. In this case, AB is a m × q matrix.
Furthermore, (AB)ij = 〈Ai ,Bj〉.Ai = ith row of ABj is the jth column of B.
Dot product as matrix multiplication: For x , y ∈ Rn,〈x , y〉 = x ′y = y ′x .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 13 / 37
1. Linear Algebra - Linear functions
Definition (Linear functions)
A function f : Rn → Rm is linear if for any x , y ∈ Rn and any c1, c2 ∈ R,
f (c1x + c2y) = c1f (x) + c2f (y).
Example: Any matrix A ∈ Rm×n defines a linear function f (x) := Ax .
f (c1x + c2y) = A(c1x + c2y) = A(c1x) + A(c2y) = c1Ax + c2Ay
= c1f (x) + c2f (y).
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 14 / 37
1. Linear Algebra - Linear functions
Definition (Linear functions)
A function f : Rn → Rm is linear if for any x , y ∈ Rn and any c1, c2 ∈ R,
f (c1x + c2y) = c1f (x) + c2f (y).
Example: Any matrix A ∈ Rm×n defines a linear function f (x) := Ax .
f (c1x + c2y) = A(c1x + c2y) = A(c1x) + A(c2y) = c1Ax + c2Ay
= c1f (x) + c2f (y).
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 14 / 37
1. Linear Algebra - Linear Representation
All linear functions are matrices!
Theorem
For every linear f : Rn → Rm, there exists a matrix A so that f (x) = Axfor every x ∈ Rn.
Idea: Let e1, e2, ..., en be a basis of Rn. By linearity,
f (x) = f
(n∑
i=1
xiei
)=
n∑i=1
xi f (ei ).
=⇒ f is entirely determined by its behavior on a basis.
A matrix A specifies the image of the standard basis vectors.
ei is mapped to the ith column of A.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 15 / 37
1. Linear Algebra - Linear Representation
All linear functions are matrices!
Theorem
For every linear f : Rn → Rm, there exists a matrix A so that f (x) = Axfor every x ∈ Rn.
Idea: Let e1, e2, ..., en be a basis of Rn. By linearity,
f (x) = f
(n∑
i=1
xiei
)=
n∑i=1
xi f (ei ).
=⇒ f is entirely determined by its behavior on a basis.
A matrix A specifies the image of the standard basis vectors.
ei is mapped to the ith column of A.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 15 / 37
1. Linear Algebra - Linear Representation
All linear functions are matrices!
Theorem
For every linear f : Rn → Rm, there exists a matrix A so that f (x) = Axfor every x ∈ Rn.
Idea: Let e1, e2, ..., en be a basis of Rn. By linearity,
f (x) = f
(n∑
i=1
xiei
)=
n∑i=1
xi f (ei ).
=⇒ f is entirely determined by its behavior on a basis.
A matrix A specifies the image of the standard basis vectors.
ei is mapped to the ith column of A.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 15 / 37
1. Linear Algebra - Linear Representation
All linear functions are matrices!
Theorem
For every linear f : Rn → Rm, there exists a matrix A so that f (x) = Axfor every x ∈ Rn.
Idea: Let e1, e2, ..., en be a basis of Rn. By linearity,
f (x) = f
(n∑
i=1
xiei
)=
n∑i=1
xi f (ei ).
=⇒ f is entirely determined by its behavior on a basis.
A matrix A specifies the image of the standard basis vectors.
ei is mapped to the ith column of A.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 15 / 37
1. Linear Algebra - Linear Representation
All linear functions are matrices!
Theorem
For every linear f : Rn → Rm, there exists a matrix A so that f (x) = Axfor every x ∈ Rn.
Idea: Let e1, e2, ..., en be a basis of Rn. By linearity,
f (x) = f
(n∑
i=1
xiei
)=
n∑i=1
xi f (ei ).
=⇒ f is entirely determined by its behavior on a basis.
A matrix A specifies the image of the standard basis vectors.
ei is mapped to the ith column of A.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 15 / 37
1. Linear Algebra - Linear Representation
All linear functions are matrices!
Theorem
For every linear f : Rn → Rm, there exists a matrix A so that f (x) = Axfor every x ∈ Rn.
Idea: Let e1, e2, ..., en be a basis of Rn. By linearity,
f (x) = f
(n∑
i=1
xiei
)=
n∑i=1
xi f (ei ).
=⇒ f is entirely determined by its behavior on a basis.
A matrix A specifies the image of the standard basis vectors.
ei is mapped to the ith column of A.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 15 / 37
1. Linear Algebra - Operator Subspaces
For A ∈ Rm×n, let
1 R ⊂ Rn be the set of rows of A.
2 C ⊂ Rm be the set of columns of A.
Definition (Row space)
The row space of A is denoted row(A) := span(R).
Definition (Column space)
The column space of A is denoted col(A) := span(C ).
Definition
The kernel of A is denoted ker(A) := {x ∈ Rn : Ax = 0}.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 16 / 37
1. Linear Algebra - Operator Subspaces
For A ∈ Rm×n, let
1 R ⊂ Rn be the set of rows of A.
2 C ⊂ Rm be the set of columns of A.
Definition (Row space)
The row space of A is denoted row(A) := span(R).
Definition (Column space)
The column space of A is denoted col(A) := span(C ).
Definition
The kernel of A is denoted ker(A) := {x ∈ Rn : Ax = 0}.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 16 / 37
1. Linear Algebra - Operator Subspaces
For A ∈ Rm×n, let
1 R ⊂ Rn be the set of rows of A.
2 C ⊂ Rm be the set of columns of A.
Definition (Row space)
The row space of A is denoted row(A) := span(R).
Definition (Column space)
The column space of A is denoted col(A) := span(C ).
Definition
The kernel of A is denoted ker(A) := {x ∈ Rn : Ax = 0}.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 16 / 37
1. Linear Algebra - Operator Subspaces
For A ∈ Rm×n, let
1 R ⊂ Rn be the set of rows of A.
2 C ⊂ Rm be the set of columns of A.
Definition (Row space)
The row space of A is denoted row(A) := span(R).
Definition (Column space)
The column space of A is denoted col(A) := span(C ).
Definition
The kernel of A is denoted ker(A) := {x ∈ Rn : Ax = 0}.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 16 / 37
1. Linear Algebra - Operator Subspaces
row(A), col(A), ker(A) are all subspaces.
row(A), col(A) have the same dimension. This dimension is also the rankof A.
A is full rank if its rank is min(n,m).
Rank-Nullity: rank(A) + dim(ker(A)) = n.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 17 / 37
1. Linear Algebra - Operator Subspaces
row(A), col(A), ker(A) are all subspaces.
row(A), col(A) have the same dimension. This dimension is also the rankof A.
A is full rank if its rank is min(n,m).
Rank-Nullity: rank(A) + dim(ker(A)) = n.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 17 / 37
1. Linear Algebra - Operator Subspaces
row(A), col(A), ker(A) are all subspaces.
row(A), col(A) have the same dimension. This dimension is also the rankof A.
A is full rank if its rank is min(n,m).
Rank-Nullity: rank(A) + dim(ker(A)) = n.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 17 / 37
1. Linear Algebra - Operator Subspaces
row(A), col(A), ker(A) are all subspaces.
row(A), col(A) have the same dimension. This dimension is also the rankof A.
A is full rank if its rank is min(n,m).
Rank-Nullity: rank(A) + dim(ker(A)) = n.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 17 / 37
1. Linear Algebra - Identity Matrix
In ∈ Rn×n is the identity matrix:
(In)ij =
{1 if i = j0 otherwise.
In is a multiplicative identity: AIn = A and InB = B whenever A,B havecompatible dimensions to multiply with In.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 18 / 37
1. Linear Algebra - Matrix Inverse & Orthogonal Matrices
Definition (Matrix Inverse)
The inverse of a matrix A ∈ Rn×n is a matrix A−1 such thatAA−1 = A−1A = In.
Remark: Not all matrices have inverses. A matrix has an inverse if andonly if it is full rank.
Definition (Orthogonal Matrix)
A matrix A ∈ Rn×n is orthogonal if and only if A′ = A−1.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 19 / 37
1. Linear Algebra - Matrix Inverse & Orthogonal Matrices
Definition (Matrix Inverse)
The inverse of a matrix A ∈ Rn×n is a matrix A−1 such thatAA−1 = A−1A = In.
Remark: Not all matrices have inverses. A matrix has an inverse if andonly if it is full rank.
Definition (Orthogonal Matrix)
A matrix A ∈ Rn×n is orthogonal if and only if A′ = A−1.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 19 / 37
Orthogonal Matrices are Isometries
Theorem (Orthogonal Matrices preserve length)
If U ∈ Rn×n is orthogonal, then for every x ∈ Rn,
||Ux ||2 = ||x ||2 .
Proof:
||x ||22 = x ′x = x ′Inx = x ′U ′Ux = (Ux)′(Ux) = ||Ux ||22 .
Take square roots of both sides.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 20 / 37
Orthogonal Matrices are Isometries
Theorem (Orthogonal Matrices preserve length)
If U ∈ Rn×n is orthogonal, then for every x ∈ Rn,
||Ux ||2 = ||x ||2 .
Proof:
||x ||22 = x ′x = x ′Inx = x ′U ′Ux = (Ux)′(Ux) = ||Ux ||22 .
Take square roots of both sides.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 20 / 37
1. Linear Algebra - Singular Value Decomposition
For any matrix A ∈ Rm×n, we can factorize
A = UΣV ′
where U ∈ Rm×m,V ∈ Rn×n are orthogonal matrices,
Σ ∈ Rm×n is diagonal:
Σij = 0 if i 6= j .
Σii ≥ 0.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 21 / 37
1. Linear Algebra - Eigenvalue Decomposition
For any symmetric matrix A ∈ Rn×n, we can factorize
A = UΛU ′
where U ∈ Rn×n is orthogonal and Λ ∈ Rn×n is a diagonal matrix.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 22 / 37
2. Discrete Linear Dynamical Systems
2. Discrete Linear Dynamical Systems
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 23 / 37
2. Discrete Linear Dynamical Systems
Dynamics:
xk+1 = Axk + Buk
Time is discrete k ∈ {1, 2, 3, ...}.
xk ∈ Rn describes the system state at timestep k .
uk ∈ Rm is the control/action that is applied at timestep k .
A ∈ Rn×n,B ∈ Rn×m.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 24 / 37
2. Discrete Linear Dynamical Systems
Dynamics:
xk+1 = Axk + Buk
Time is discrete k ∈ {1, 2, 3, ...}.xk ∈ Rn describes the system state at timestep k .
uk ∈ Rm is the control/action that is applied at timestep k .
A ∈ Rn×n,B ∈ Rn×m.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 24 / 37
2. Discrete Linear Dynamical Systems
Dynamics:
xk+1 = Axk + Buk
Time is discrete k ∈ {1, 2, 3, ...}.xk ∈ Rn describes the system state at timestep k .
uk ∈ Rm is the control/action that is applied at timestep k .
A ∈ Rn×n,B ∈ Rn×m.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 24 / 37
2. Discrete Linear Dynamical Systems
Dynamics:
xk+1 = Axk + Buk
Time is discrete k ∈ {1, 2, 3, ...}.xk ∈ Rn describes the system state at timestep k .
uk ∈ Rm is the control/action that is applied at timestep k .
A ∈ Rn×n,B ∈ Rn×m.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 24 / 37
2. Discrete Linear Dynamical Systems - Stabilizing
Stabilization Objective: Try to get xk to converge to the origin as fast aspossible.
minimize{uk}T−1
k=1
T∑k=1
||xk ||22
s.t. xk+1 = Axk + Buk for 1 ≤ k ≤ T − 1
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 25 / 37
2. Discrete Linear Dynamical Systems - Stabilizing
Stabilization Objective:
We can penalize deviations in different coordinates by differentamounts.
We can add penalty for using large control uk .
minimize{uk}T−1
k=1
T∑k=1
x ′kRxk + u′kQuk
s.t. xk+1 = Axk + Buk for 1 ≤ k ≤ T − 1
where R,Q � 0.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 26 / 37
2. Discrete Linear Dynamical Systems - Stabilizing
Stabilization Objective:
We can penalize deviations in different coordinates by differentamounts.
We can add penalty for using large control uk .
minimize{uk}T−1
k=1
T∑k=1
x ′kRxk + u′kQuk
s.t. xk+1 = Axk + Buk for 1 ≤ k ≤ T − 1
where R,Q � 0.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 26 / 37
2. Discrete Linear Dynamical Systems - Stabilizing
Stabilization Objective:
We can penalize deviations in different coordinates by differentamounts.
We can add penalty for using large control uk .
minimize{uk}T−1
k=1
T∑k=1
x ′kRxk + u′kQuk
s.t. xk+1 = Axk + Buk for 1 ≤ k ≤ T − 1
where R,Q � 0.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 26 / 37
2. Discrete Linear Dynamical Systems - Optimization
The relation between xk+1, xk , uk is linear. We can incorporate thedynamics constrains as one large matrix equation: Az = 0.
A B −In 0 0 ... 0 0 00 0 A B −In ... 0 0 0...
......
...... ...
......
...0 0 0 0 0 ... A B −In
︸ ︷︷ ︸
A
x1
u1
x2
u2
x3...
xT−1
uT−1
xT
︸ ︷︷ ︸
z
=
00...0
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 27 / 37
2. Discrete Linear Dynamical Systems - Optimization
The relation between xk+1, xk , uk is linear. We can incorporate thedynamics constrains as one large matrix equation: Az = 0.
A B −In 0 0 ... 0 0 00 0 A B −In ... 0 0 0...
......
...... ...
......
...0 0 0 0 0 ... A B −In
︸ ︷︷ ︸
A
x1
u1
x2
u2
x3...
xT−1
uT−1
xT
︸ ︷︷ ︸
z
=
00...0
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 27 / 37
2. Discrete Linear Dynamical Systems - Optimization
Thus the stabilization and tracking problems can be solved by optimizingan objective function subject to linear equality constraints:
minimizez
J(z)
s.t. Az = 0.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 28 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
How do we choose u so that limk→∞ xk = 0?
Idea: Let’s try linear state feedback, where uk = −Kxk for someK ∈ Rm×n. Then the dynamics become
xk+1 = Axk + Buk
= Axk − BKxk = (A− BK )xk
If (A− BK ) is symmetric, then we can factor it (A− BK ) = UΛU ′ Thus
xk+1 = (A− BK )xk
⇐⇒ xk+1 = UΛU ′xk
=⇒ U ′xk+1 = U ′U︸︷︷︸In
ΛU ′xk
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 29 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
How do we choose u so that limk→∞ xk = 0?
Idea: Let’s try linear state feedback, where uk = −Kxk for someK ∈ Rm×n.
Then the dynamics become
xk+1 = Axk + Buk
= Axk − BKxk = (A− BK )xk
If (A− BK ) is symmetric, then we can factor it (A− BK ) = UΛU ′ Thus
xk+1 = (A− BK )xk
⇐⇒ xk+1 = UΛU ′xk
=⇒ U ′xk+1 = U ′U︸︷︷︸In
ΛU ′xk
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 29 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
How do we choose u so that limk→∞ xk = 0?
Idea: Let’s try linear state feedback, where uk = −Kxk for someK ∈ Rm×n. Then the dynamics become
xk+1 = Axk + Buk
= Axk − BKxk = (A− BK )xk
If (A− BK ) is symmetric, then we can factor it (A− BK ) = UΛU ′ Thus
xk+1 = (A− BK )xk
⇐⇒ xk+1 = UΛU ′xk
=⇒ U ′xk+1 = U ′U︸︷︷︸In
ΛU ′xk
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 29 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
How do we choose u so that limk→∞ xk = 0?
Idea: Let’s try linear state feedback, where uk = −Kxk for someK ∈ Rm×n. Then the dynamics become
xk+1 = Axk + Buk
= Axk − BKxk = (A− BK )xk
If (A− BK ) is symmetric, then we can factor it (A− BK ) = UΛU ′
Thus
xk+1 = (A− BK )xk
⇐⇒ xk+1 = UΛU ′xk
=⇒ U ′xk+1 = U ′U︸︷︷︸In
ΛU ′xk
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 29 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
How do we choose u so that limk→∞ xk = 0?
Idea: Let’s try linear state feedback, where uk = −Kxk for someK ∈ Rm×n. Then the dynamics become
xk+1 = Axk + Buk
= Axk − BKxk = (A− BK )xk
If (A− BK ) is symmetric, then we can factor it (A− BK ) = UΛU ′ Thus
xk+1 = (A− BK )xk
⇐⇒ xk+1 = UΛU ′xk
=⇒ U ′xk+1 = U ′U︸︷︷︸In
ΛU ′xk
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 29 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
How do we choose u so that limk→∞ xk = 0?
Idea: Let’s try linear state feedback, where uk = −Kxk for someK ∈ Rm×n. Then the dynamics become
xk+1 = Axk + Buk
= Axk − BKxk = (A− BK )xk
If (A− BK ) is symmetric, then we can factor it (A− BK ) = UΛU ′ Thus
xk+1 = (A− BK )xk
⇐⇒ xk+1 = UΛU ′xk
=⇒ U ′xk+1 = U ′U︸︷︷︸In
ΛU ′xk
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 29 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
How do we choose u so that limk→∞ xk = 0?
Idea: Let’s try linear state feedback, where uk = −Kxk for someK ∈ Rm×n. Then the dynamics become
xk+1 = Axk + Buk
= Axk − BKxk = (A− BK )xk
If (A− BK ) is symmetric, then we can factor it (A− BK ) = UΛU ′ Thus
xk+1 = (A− BK )xk
⇐⇒ xk+1 = UΛU ′xk
=⇒ U ′xk+1 = U ′U︸︷︷︸In
ΛU ′xk
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 29 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
Defining yk := U ′xk gives:
U ′xk+1 = ΛU ′xk ⇐⇒ yk+1 = Λyk
So yk+1,i = Λiiyk,i for each coordinate.
Iterate: yk,i = (Λii )ky0,i .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 30 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
Defining yk := U ′xk gives:
U ′xk+1 = ΛU ′xk ⇐⇒ yk+1 = Λyk
So yk+1,i = Λiiyk,i for each coordinate.
Iterate: yk,i = (Λii )ky0,i .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 30 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
Defining yk := U ′xk gives:
U ′xk+1 = ΛU ′xk ⇐⇒ yk+1 = Λyk
So yk+1,i = Λiiyk,i for each coordinate.
Iterate: yk,i = (Λii )ky0,i .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 30 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
Defining yk := U ′xk gives:
U ′xk+1 = ΛU ′xk ⇐⇒ yk+1 = Λyk
So yk+1,i = Λiiyk,i for each coordinate.
Iterate: yk,i = (Λii )ky0,i .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 30 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
U orthogonal =⇒ ||xk ||2 = ||U ′xk ||2 = ||yk ||2.
So xk → 0 ⇐⇒ yk → 0.
Next, yk → 0 ⇐⇒ yk,i → 0 for all i .
yk,i = (Λii )ky0,i → 0 ⇐⇒ |Λii | < 1.
So xk → 0 if all eigenvalues of A− BK have magnitude less than 1.
If we can choose K so that all eigenvalues of A− BK have magnitude lessthan 1, then u(t) = −Kx(t) will be a stabilizing controller!
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 31 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
U orthogonal =⇒ ||xk ||2 = ||U ′xk ||2 = ||yk ||2.
So xk → 0 ⇐⇒ yk → 0.
Next, yk → 0 ⇐⇒ yk,i → 0 for all i .
yk,i = (Λii )ky0,i → 0 ⇐⇒ |Λii | < 1.
So xk → 0 if all eigenvalues of A− BK have magnitude less than 1.
If we can choose K so that all eigenvalues of A− BK have magnitude lessthan 1, then u(t) = −Kx(t) will be a stabilizing controller!
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 31 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
U orthogonal =⇒ ||xk ||2 = ||U ′xk ||2 = ||yk ||2.
So xk → 0 ⇐⇒ yk → 0.
Next, yk → 0 ⇐⇒ yk,i → 0 for all i .
yk,i = (Λii )ky0,i → 0 ⇐⇒ |Λii | < 1.
So xk → 0 if all eigenvalues of A− BK have magnitude less than 1.
If we can choose K so that all eigenvalues of A− BK have magnitude lessthan 1, then u(t) = −Kx(t) will be a stabilizing controller!
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 31 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
U orthogonal =⇒ ||xk ||2 = ||U ′xk ||2 = ||yk ||2.
So xk → 0 ⇐⇒ yk → 0.
Next, yk → 0 ⇐⇒ yk,i → 0 for all i .
yk,i = (Λii )ky0,i → 0 ⇐⇒ |Λii | < 1.
So xk → 0 if all eigenvalues of A− BK have magnitude less than 1.
If we can choose K so that all eigenvalues of A− BK have magnitude lessthan 1, then u(t) = −Kx(t) will be a stabilizing controller!
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 31 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
U orthogonal =⇒ ||xk ||2 = ||U ′xk ||2 = ||yk ||2.
So xk → 0 ⇐⇒ yk → 0.
Next, yk → 0 ⇐⇒ yk,i → 0 for all i .
yk,i = (Λii )ky0,i → 0 ⇐⇒ |Λii | < 1.
So xk → 0 if all eigenvalues of A− BK have magnitude less than 1.
If we can choose K so that all eigenvalues of A− BK have magnitude lessthan 1, then u(t) = −Kx(t) will be a stabilizing controller!
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 31 / 37
2. Discrete Linear Dynamical Systems - Control Synthesis
U orthogonal =⇒ ||xk ||2 = ||U ′xk ||2 = ||yk ||2.
So xk → 0 ⇐⇒ yk → 0.
Next, yk → 0 ⇐⇒ yk,i → 0 for all i .
yk,i = (Λii )ky0,i → 0 ⇐⇒ |Λii | < 1.
So xk → 0 if all eigenvalues of A− BK have magnitude less than 1.
If we can choose K so that all eigenvalues of A− BK have magnitude lessthan 1, then u(t) = −Kx(t) will be a stabilizing controller!
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 31 / 37
3. Continuous Linear Dynamical Systems
3. Continuous Linear Dynamical Systems
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 32 / 37
3. Continuous Linear Dynamical Systems
Dynamics:
d
dtx(t) = Ax(t) + Bu(t)
(shorthand ver.) x = Ax + Bu
Time is continuous t ∈ R.
x(t) ∈ Rn describes the system state at time t.
u(t) ∈ Rm is the control/action that is applied at time t.
A ∈ Rn×n,B ∈ Rn×m.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 33 / 37
3. Continuous Linear Dynamical Systems
Dynamics:
d
dtx(t) = Ax(t) + Bu(t)
(shorthand ver.) x = Ax + Bu
Time is continuous t ∈ R.
x(t) ∈ Rn describes the system state at time t.
u(t) ∈ Rm is the control/action that is applied at time t.
A ∈ Rn×n,B ∈ Rn×m.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 33 / 37
3. Continuous Linear Dynamical Systems
Dynamics:
d
dtx(t) = Ax(t) + Bu(t)
(shorthand ver.) x = Ax + Bu
Time is continuous t ∈ R.
x(t) ∈ Rn describes the system state at time t.
u(t) ∈ Rm is the control/action that is applied at time t.
A ∈ Rn×n,B ∈ Rn×m.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 33 / 37
3. Continuous Linear Dynamical Systems
Dynamics:
d
dtx(t) = Ax(t) + Bu(t)
(shorthand ver.) x = Ax + Bu
Time is continuous t ∈ R.
x(t) ∈ Rn describes the system state at time t.
u(t) ∈ Rm is the control/action that is applied at time t.
A ∈ Rn×n,B ∈ Rn×m.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 33 / 37
3. Continuous Linear Dynamical Systems - Stabilizing
Stabilizing Objective: Guide x(t) to zero.
minimizeu:[0,T ]→Rm
∫ T
0x ′(t)Rx(t) + u′(t)Qu(t)dt
s.t. x(t) = Ax(t) + Bu(t).
where R,Q � 0.
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 34 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
Objective: How do we choose u so that limt→∞ x(t)→ 0?
Idea: Let’s try linear state feedback, where u(t) = −Kx(t) for someK ∈ Rm×n. The dynamics become
x(t) = Ax(t) + Bu(t)
= Ax(t)− BKx(t) = (A− BK )x(t).
If A− BK is symmetric, then A− BK = UΛU ′ for some diagonal Λ andorthogonal U. Thus
x(t) = UΛU ′x(t)
=⇒ U ′x(t) = ΛU ′x(t).
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 35 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
Objective: How do we choose u so that limt→∞ x(t)→ 0?
Idea: Let’s try linear state feedback, where u(t) = −Kx(t) for someK ∈ Rm×n.
The dynamics become
x(t) = Ax(t) + Bu(t)
= Ax(t)− BKx(t) = (A− BK )x(t).
If A− BK is symmetric, then A− BK = UΛU ′ for some diagonal Λ andorthogonal U. Thus
x(t) = UΛU ′x(t)
=⇒ U ′x(t) = ΛU ′x(t).
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 35 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
Objective: How do we choose u so that limt→∞ x(t)→ 0?
Idea: Let’s try linear state feedback, where u(t) = −Kx(t) for someK ∈ Rm×n. The dynamics become
x(t) = Ax(t) + Bu(t)
= Ax(t)− BKx(t) = (A− BK )x(t).
If A− BK is symmetric, then A− BK = UΛU ′ for some diagonal Λ andorthogonal U. Thus
x(t) = UΛU ′x(t)
=⇒ U ′x(t) = ΛU ′x(t).
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 35 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
Objective: How do we choose u so that limt→∞ x(t)→ 0?
Idea: Let’s try linear state feedback, where u(t) = −Kx(t) for someK ∈ Rm×n. The dynamics become
x(t) = Ax(t) + Bu(t)
= Ax(t)− BKx(t) = (A− BK )x(t).
If A− BK is symmetric, then A− BK = UΛU ′ for some diagonal Λ andorthogonal U. Thus
x(t) = UΛU ′x(t)
=⇒ U ′x(t) = ΛU ′x(t).
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 35 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
Define y(t) := U ′x(t).
Then
U ′x(t) = ΛU ′x(t) ⇐⇒ y(t) = Λy(t).
Since Λ is diagonal, yi (t) = Λiiy(t) for all i . Solve this ODE!
yi (t)
y(t)= Λii
=⇒ ln (yi (t)) = Λii t + C
=⇒ yi (t) = eCeΛii t = yi (0)eΛii t .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 36 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
Define y(t) := U ′x(t). Then
U ′x(t) = ΛU ′x(t) ⇐⇒ y(t) = Λy(t).
Since Λ is diagonal, yi (t) = Λiiy(t) for all i . Solve this ODE!
yi (t)
y(t)= Λii
=⇒ ln (yi (t)) = Λii t + C
=⇒ yi (t) = eCeΛii t = yi (0)eΛii t .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 36 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
Define y(t) := U ′x(t). Then
U ′x(t) = ΛU ′x(t) ⇐⇒ y(t) = Λy(t).
Since Λ is diagonal, yi (t) = Λiiy(t) for all i .
Solve this ODE!
yi (t)
y(t)= Λii
=⇒ ln (yi (t)) = Λii t + C
=⇒ yi (t) = eCeΛii t = yi (0)eΛii t .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 36 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
Define y(t) := U ′x(t). Then
U ′x(t) = ΛU ′x(t) ⇐⇒ y(t) = Λy(t).
Since Λ is diagonal, yi (t) = Λiiy(t) for all i . Solve this ODE!
yi (t)
y(t)= Λii
=⇒ ln (yi (t)) = Λii t + C
=⇒ yi (t) = eCeΛii t = yi (0)eΛii t .
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 36 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
U orthogonal =⇒ ||x(t)||2 = ||U ′x(t)||2 = ||y(t)||2.
So x(t)→ 0 ⇐⇒ y(t)→ 0.
Next, y(t)→ 0 ⇐⇒ yi (t)→ 0 for all i .
yi (t) = yi (0)eΛii t → 0 ⇐⇒ Λii < 0.
So x(t)→ 0 if all eigenvalues of A− BK are negative.
If we can choose K so that A− BK ≺ 0, then u(t) = −Kx(t) will be astabilizing controller!
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 37 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
U orthogonal =⇒ ||x(t)||2 = ||U ′x(t)||2 = ||y(t)||2.
So x(t)→ 0 ⇐⇒ y(t)→ 0.
Next, y(t)→ 0 ⇐⇒ yi (t)→ 0 for all i .
yi (t) = yi (0)eΛii t → 0 ⇐⇒ Λii < 0.
So x(t)→ 0 if all eigenvalues of A− BK are negative.
If we can choose K so that A− BK ≺ 0, then u(t) = −Kx(t) will be astabilizing controller!
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 37 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
U orthogonal =⇒ ||x(t)||2 = ||U ′x(t)||2 = ||y(t)||2.
So x(t)→ 0 ⇐⇒ y(t)→ 0.
Next, y(t)→ 0 ⇐⇒ yi (t)→ 0 for all i .
yi (t) = yi (0)eΛii t → 0 ⇐⇒ Λii < 0.
So x(t)→ 0 if all eigenvalues of A− BK are negative.
If we can choose K so that A− BK ≺ 0, then u(t) = −Kx(t) will be astabilizing controller!
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 37 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
U orthogonal =⇒ ||x(t)||2 = ||U ′x(t)||2 = ||y(t)||2.
So x(t)→ 0 ⇐⇒ y(t)→ 0.
Next, y(t)→ 0 ⇐⇒ yi (t)→ 0 for all i .
yi (t) = yi (0)eΛii t → 0 ⇐⇒ Λii < 0.
So x(t)→ 0 if all eigenvalues of A− BK are negative.
If we can choose K so that A− BK ≺ 0, then u(t) = −Kx(t) will be astabilizing controller!
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 37 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
U orthogonal =⇒ ||x(t)||2 = ||U ′x(t)||2 = ||y(t)||2.
So x(t)→ 0 ⇐⇒ y(t)→ 0.
Next, y(t)→ 0 ⇐⇒ yi (t)→ 0 for all i .
yi (t) = yi (0)eΛii t → 0 ⇐⇒ Λii < 0.
So x(t)→ 0 if all eigenvalues of A− BK are negative.
If we can choose K so that A− BK ≺ 0, then u(t) = −Kx(t) will be astabilizing controller!
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 37 / 37
3. Continuous Linear Dynamical Systems - ControlSynthesis
U orthogonal =⇒ ||x(t)||2 = ||U ′x(t)||2 = ||y(t)||2.
So x(t)→ 0 ⇐⇒ y(t)→ 0.
Next, y(t)→ 0 ⇐⇒ yi (t)→ 0 for all i .
yi (t) = yi (0)eΛii t → 0 ⇐⇒ Λii < 0.
So x(t)→ 0 if all eigenvalues of A− BK are negative.
If we can choose K so that A− BK ≺ 0, then u(t) = −Kx(t) will be astabilizing controller!
AA 203 Recitation #1 Linear Algebra & Linear Dynamical Systems April 10th, 2020 37 / 37