linear algebra & linear dynamical...

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.

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.

Overview (cont.)

1 Linear Algebra

Overview (cont.)

1 Linear Algebra

1. Linear Algebra

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: [

Scalar multiplication is entrywise: cx = y means cxi = yi for all1 ≤ i ≤ n. Example:

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

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

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.

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.

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 .

Definition (Basis)

Definition

Given x , y ∈ Rn, their dot product 〈x , y〉 is defined as:

〈x , y〉 :=n∑

xiyi .

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.

4 1 52 2 13 0 90 0 7

Vectors are also matrices! A vector x ∈ Rn is a n × 1 matrix.

4 1 52 2 13 0 90 0 7

1. Linear Algebra - Matrix Transpose

If A ∈ Rm×n, then its transpose is A′ ∈ Rn×m and

Aij = A′ji .

Example:

1 60 25 3

,A′ =

[1 0 56 2 3

1. Linear Algebra - Matrix Transpose

If A ∈ Rm×n, then its transpose is A′ ∈ Rn×m and

Aij = A′ji .

Example:

1 60 25 3

,A′ =

[1 0 56 2 3

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 .

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).

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).

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

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.

Theorem

f (x) = f

n∑i=1

xi f (ei ).

Theorem

f (x) = f

n∑i=1

xi f (ei ).

Theorem

f (x) = f

n∑i=1

xi f (ei ).

Theorem

f (x) = f

n∑i=1

xi f (ei ).

Theorem

f (x) = f

n∑i=1

xi f (ei ).

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}.

Definition

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.

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.

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.

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.

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.

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.

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.

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.

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.

Dynamics:

xk+1 = Axk + Buk

Time is discrete k ∈ {1, 2, 3, ...}.xk ∈ Rn describes the system state at timestep k .

Dynamics:

xk+1 = Axk + Buk

Dynamics:

xk+1 = Axk + Buk

2. Discrete Linear Dynamical Systems - Stabilizing

Stabilization Objective: Try to get xk to converge to the origin as fast aspossible.

minimize{uk}T−1

T∑k=1

||xk ||22

s.t. xk+1 = Axk + Buk for 1 ≤ k ≤ T − 1

Stabilization Objective:

We can penalize deviations in different coordinates by differentamounts.

We can add penalty for using large control uk .

minimize{uk}T−1

T∑k=1

x ′kRxk + u′kQuk

where R,Q � 0.

minimize{uk}T−1

T∑k=1

where R,Q � 0.

minimize{uk}T−1

T∑k=1

where R,Q � 0.

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

︸︷︷︸

xT−1

uT−1

︸︷︷︸

00...0

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

︸︷︷︸

xT−1

uT−1

︸︷︷︸

00...0

Thus the stabilization and tracking problems can be solved by optimizingan objective function subject to linear equality constraints:

minimizez

s.t. Az = 0.

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

Idea: Let’s try linear state feedback, where uk = −Kxk for someK ∈ Rm×n.

Then the dynamics become

xk+1 = Axk + Buk

xk+1 = (A− BK )xk

=⇒ U ′xk+1 = U ′U︸︷︷︸In

ΛU ′xk

xk+1 = Axk + Buk

xk+1 = (A− BK )xk

=⇒ U ′xk+1 = U ′U︸︷︷︸In

ΛU ′xk

xk+1 = Axk + Buk

If (A− BK ) is symmetric, then we can factor it (A− BK ) = UΛU ′

xk+1 = (A− BK )xk

=⇒ U ′xk+1 = U ′U︸︷︷︸In

ΛU ′xk

xk+1 = Axk + Buk

xk+1 = (A− BK )xk

=⇒ U ′xk+1 = U ′U︸︷︷︸In

ΛU ′xk

xk+1 = Axk + Buk

xk+1 = (A− BK )xk

=⇒ U ′xk+1 = U ′U︸︷︷︸In

ΛU ′xk

xk+1 = Axk + Buk

xk+1 = (A− BK )xk

=⇒ U ′xk+1 = U ′U︸︷︷︸In

ΛU ′xk

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 .

U orthogonal =⇒ ||xk ||2 = ||U ′xk ||2 = ||yk ||2.

So xk → 0 ⇐⇒ yk → 0.

Next, yk → 0 ⇐⇒ yk,i → 0 for all i .