Switched Linear Systems

faculteit Wiskunde en Natuurwetenschappen

Controllability of

Switched Linear Systems

Bacheloronderzoek Wiskunde

Augustus 2010 Student: H. Jongsma

Begeleider: Dr. M.K. Camlibel

Abstract

Switched linear systems consist of a collection of linear systems and a switching law that governs the switching between them. In this paper it is proven that the reachable set of a switched linear system is a subspace of the entire state space. A geometric characterization of the reachable subspace is given.

Contents

1 Introduction 2

1.1 Notation . . . 3 1.2 Switched linear systems . . . 3

2 Controllability 5

2.1 Controllability and reachability . . . 5 2.2 Reachable set of switched systems . . . 5

3 Examples 11

4 Conclusions 16

A Computer program 18

Chapter 1

Introduction

In system theory the properties of mathematical models that describe physical systems are studied. An important class of models that are analyzed are linear systems. These systems exhibit simpler, more conve- nient properties and behaviour than non-linear systems. Some well-studied notions in the analysis of linear systems are stability, controllability and reachability. A great number of fundamental concepts and tools have been developed to make a systematic analysis of these properties possible. For example, it is well known that the controllable and reachable set of linear systems coincide and are a subspace of the entire state space (e.g. [3, p. 39]). A through understanding of the properties and the behaviour of systems is necessary when, for instance, designing (optimal) controllers for physical systems.

In the past decades the research on linear systems has been extended to switched linear systems. These systems consist of a collection of ordinary linear systems and a switching law that determines the switching between them. Switched systems and controllers have many applications in engineering. For instance, switching occurs in power systems and switched controllers have been designed to control aircrafts and satellites [2]. It was discovered that some powerful methods used to examine linear systems can also be applied to switched linear systems. This has produced important results about stability, controllability and reachability of switched systems.

Although switched systems provide numerous new possibilities to tackle increasingly complicated prob- lems, they are extensively more complicated to analyze. It was shown that both the controllable and the reachable set of switched linear systems are unions of uncountably many subspaces of the entire state space instead of simple subspaces. In 2002 it was proven by Sun et al. that the controllable and reachable set coincide and that they are a subspace of the entire state space. They proved that the reachable space is the smallest subspace that is invariant under all state matrices and that contains the image of the input matrix of every subsystem. In this paper we present a concise and simpler proof for this result, that is, the reachable set of switched linear systems is a subspace of the entire state space. We provide a full geometric characterization of the reachable subspace. For simplicity, the proof will be restricted to switched linear systems consisting of two subsystems. Since it is known that the reachable and controllable set of switched linear systems coincide, we will only consider the reachable set in our proof.

The paper has the following structure. First, we will give a formal definition of switched linear systems.

In Chapter 2, we provide definitions for controllability and reachability of switched linear systems. We will show that the reachable set of a switched system contains a subspace of a certain maximal dimension and that it is equal to this subspace. Chapter 3 presents some illustrative examples of different switched systems and the evolution of their reachable set with each switching. In Chapter 4 we provide some concluding remarks and give a sketch of interesting problems that could be addressed in the future.

1.1 Notation

We will denote the real numbers by R and the natural numbers by N. We define R+:= {x : x ∈ R, x ≥ 0}

which is the set of all positive real numbers. The n-ary Cartesian product of a set X will be denoted by Xⁿ, Xⁿ:= X × X × · · · × X

| {z }

n times

.

Let V and W be two linear subspaces such that V ⊆ W. Also let A : W → W be a linear transformation.

We say V is A-invariant if Av ∈ V for all v ∈ V. The smallest A-invariant subspace that contains the subspace B is denoted by hA|Bi. The smallest subspace that contains the subspace B and is both A1- invariant and A2-invariant is denoted by h{A1, A2}|Bi. If A is a linear transformation of set X to Y , the set {y : y = Ax; x ∈ X} is called the image of A and is denoted by im A.

1.2 Switched linear systems

A switched linear system is of the form

˙

x(t) = Aσ(t)x(t) + Bσ(t)u(t) (1.1)

where the switching signal σ : R+→ {1, 2} is piecewise constant and has only finitely many discontinuities on every finite interval, x ∈ Rⁿ, u : R+→ R^m, A_i∈ R^n×n and B_i∈ R^n×m. The state trajectory with initial state x₀, input u and switching signal σ is denoted by x(t; 0, x₀, u, σ).

We define

L = {ρ : ρ = i1i2· · · ik; ij∈ {1, 2}; ij6= ij+1 ∀j = 1, 2, . . . , k − 1; k ∈ N}. (1.2) The length of ρ ∈ L is defined as the number of elements in ρ and is denoted by |ρ|. There is an obvious correspondence between switching signals σ and pairs (ρ, τ ) where ρ ∈ L and τ ∈ R^|ρ|+.

Example 1.1 (Boost converter).

In electronics engineering, a boost converter is used to increase the voltage from a DC source to a higher voltage [1, pp. 9–10]. The circuit of a boost converter is shown in figure 1.1

V

V

V

S D C R

L

I

I

I

Figure 1.1: Boost converter

In this figure L is the inductance, C is the capacitance, R is the load resistance and Vi(t) is the source voltage at time t. The voltage over the the load R at time t is denoted by Vo(t) and the current through the circuit

is denoted by IL(t). The switch in this example has two modes, off and on, denoted by 0 and 1. The mode at time t is denoted by s : R⁺→ {0, 1}. The differential equations that describe the system then become

V˙_o(t) = − 1

RCV_o(t) + (1 − s(t))1 CI_L(t) I˙_L(t) = −(1 − s(t))1

LV_o(t) + s(t)1 LV_i(t).

Let x₁= V_o, x₂= I_L, u = V_i, σ = s + 1 and A1=

 −_RC¹ _C¹

−_L¹ 0



and B1=

 0 0



A₂=

 −_RC¹ 0

0 0



and B₂=

 0

1 L

 .

(1.3)

Now we can describe the system by

˙

x = A_σx + B_σu, σ : R+→ {1, 2}

which is exactly the definition of a switched system with two subsystems.

Chapter 2

Controllability

2.1 Controllability and reachability

The controllability and reachability of linear systems are well-known and well-studied notions. Similar to the linear case, we define controllability and reachability for switched systems [2] in what follows.

Definition 2.1 (Controllability of switched systems).

A state x ∈ Rⁿ is said to be controllable at time t0 if there exist a time instant tf ≥ t0, a switching signal σ : [t0, tf] → {1, 2} and input u : [t0, tf] → R^msuch that x(tf; t0, x, u, σ) = 0.

Definition 2.2 (Reachability of switched systems).

A state x ∈ Rⁿ is said to be reachable at time t0 if there exist a time instant tf ≥ t0, a switching signal σ : [t0, tf] → {1, 2} and input u : [t0, tf] → R^msuch that x(tf; t0, 0, u, σ) = x.

2.2 Reachable set of switched systems

The state equation at time t of switched system (1.1) for a fixed switching signal σ, input u and initial state x(t0) = x0is given by

x(t; t0, x0, u, σ) = e^Aik(t−tk)· · · e^Ai0(t1−t0)x0

+ e^Aik(t−tk)· · · e^Ai1(t2−t1)× Z t1

t₀

e^{Ai0(t1−τ )}B_i0u(τ )dτ + · · ·

+ Z t

t_k

e^{Aik(t−τ )}Bi_ku(τ )dτ

(2.1)

where t₀, t₁, . . . , t_k is the switching time sequence of σ and i_j = σ(t_j). The complete set of reachable states of the system (1.1) is given by

R = {x : x = x(t; t₀, 0, u, σ), t ≥ t₀, u : [t₀, t] → R^m, σ : [t₀, t] → {1, 2}}. (2.2) The subsystems of the switched system are linear systems and can be described by using the matrix pair (Ai, Bi). The set of all state vectors that can be reached within finite time for subsystem i is a subspace that will be denoted by Ri. It is well-known [3] that Ri is the smallest Ai-invariant subspace that contains im Bi, i.e.

R_i: = im[B_iA_iB_i A²_iB_i · · · Aⁿ⁻¹_i B_i]

= hAi| im Bii. (2.3)

We define

R := R1+ R2. (2.4)

We can write the reachable set of switched system (1.1) in the following way

R = [

l∈N i∈{1,2}^l

h∈R^l+

e^Ailhl· · · e^Ai1h1R_i0+ · · · + e^AilhlR_il−1+ R_il
. (2.5)

We also define

P(ρ, τ ) =

|ρ|

X

k=1 k−1

Y

l=1

e^Aρlτl

!

R_ρk (2.6)

where ρ ∈ L and τ ∈ R^|ρ|. Here, we use the convention

0

Y

l=1

X_l:= I.

Example 2.1.

Let ρ = 121, τ = τ1τ2τ3. Then, the set P(ρ, τ ) is of the form

P(ρ, τ ) = e^A1τ1e^A2τ2R₁+ e^A1τ1R₂+ R₁. Now, if we take ρ = 212121, τ = τ₁τ₂· · · τ₆. We obtain

P(ρ, τ ) = e^A2τ1e^A1τ2· · · e^A2τ5R1+ · · · + e^A2τ1e^A1τ2R2+ e^A2τ1R1+ R2. We can now write the reachable set of the switched system (1.1) as

R = [

ρ∈L τ ∈R^|ρ|₊

P(ρ, τ ). (2.7)

We now define

Q(ρ, τ ) =

|ρ|−1

X

k=1 k

Y

l=1

e^Aρlτl

!

R. (2.8)

Example 2.2.

Let ρ = 121, τ = τ₁τ₂τ₃. Now, the set Q(ρ, τ ) is of the form

Q(ρ, τ ) = e^A1τ1e^A2τ2R + e^A1τ1R.

It can be verified that P(ρ, τ ) and Q(ρ, τ ) are the same. Indeed P(121, τ1τ2τ3) = e^A1τ1e^A2τ2R1+ e^A1τ1R2+ R1

= e^A1τ1e^A2τ2R1+ e^A1τ1e^A2τ2R2+ e^A1τ1R2+ e^A1τ1R1 [since Ri is Ai invariant]

= e^A1τ1e^A2τ2R + e^A1τ1R

= Q(121, τ1τ2τ3).

Now, we will show that this holds not only for this specific example, but also in general.

Theorem 2.1 (Equivalence of P- and Q-sets).

For any ρ ∈ L and τ ∈ R^|ρ|

P(ρ, τ ) = Q(ρ, τ ). (2.9)

Proof. We will prove the statement by induction on the length of the switching sequence ρ. Let |ρ| = 2.

Because Riis Ai-invariant we can write

P(ρ, τ ) = e^Aρ1τ1Rρ2+ Rρ1

= e^Aρ1τ1R_ρ2+ e^Aρ1τ1R_ρ1

= e^Aρ1τ1R

= Q(ρ, τ ).

(2.10)

Now suppose the statement holds for all ρ with |ρ| = r. We will show that it also holds for any ρ with

|ρ| = r + 1. Let ρ be such that |ρ| = r + 1. Define ¯ρ = ρ₂· · · ρ_r+1and ¯τ = τ₂· · · τ_r+1. Note that P(ρ, τ ) =

r+1

X

k=1 k−1

Y

l=1

e^Aρlτl

!

Rρ_k [since |ρ| = r + 1]

= Rρ₁+ e^Aρ1τ1Rρ₂+

r+1

X

k=2 k−1

Y

l=1

e^Aρlτl

! Rρ_k

= Rρ₁+ e^Aρ1τ1Rρ₂+ e^Aρ1τ1

r+1

X

k=2 k−1

Y

l=2

e^Aρlτl

! Rρ_k

!

= Rρ₁+ e^Aρ1τ1Rρ₂+ e^Aρ1τ1

r

X

p=1 p−1

Y

q=1

e^Aρq+1τq+1

! Rρ_p+1

!

[p := k − 1, q := l − 1]

= Rρ₁+ e^Aρ1τ1Rρ₂+ e^Aρ1τ1P( ¯ρ, ¯τ )

= e^Aρ1τ1Rρ₁+ e^Aρ1τ1Rρ₂+ e^Aρ1τ1Q( ¯ρ, ¯τ ) [by induction hypothesis]

= e^Aρ1τ1R + e^Aρ1τ1Q( ¯ρ, ¯τ )

= e^Aρ1τ1R + e^Aρ1τ1

r−1

X

k=1 k

Y

l=1

e^Aρl¯τ¯l

! R

!

= e^Aρ1τ1R +

r−1

X

k=1 k+1

Y

l=1

e^Aρlτl

! R

!

=

r−1

X

k=0 k+1

Y

l=1

e^Aρlτl

! R

=

r

X

p=1 p

Y

l=1

e^Aρlτl

!

R [p := k + 1]

= Q(ρ, τ ).

Next, we present an auxiliary result that relates the Q sets of two switching signals and their concatena- tion.

Lemma 2.2.

For any ρ, ˆρ ∈ L, τ ∈ R^|ρ|+ and ˆτ ∈ R^{| ˆ}+^ρ|

Q(ρ ˆρ, τ ˆτ ) ⊇ e^Aρ1τ1· · · e^{Aρ|ρ|τ|ρ|}Q( ˆρ, ˆτ ) + Q(ρ, τ ).

Proof. Let ρ, ˆρ ∈ L, τ ∈ R^|ρ|+ and ˆτ ∈ R^{| ˆ}+^ρ|. Now Q(ρ ˆρ, τ ˆτ ) is of the form

Q( ¯ρ, ¯τ ) =

| ¯ρ|−1

X

k=1 k

Y

l=1

e^Aρl¯τ¯l

! R

where ¯ρ = ρ ˆρ and ¯τ = τ ˆτ . Define l = |ρ|. Note that

Q( ¯ρ, ¯τ ) =

| ¯ρ|−1

X

p=1 p

Y

q=1

e^A¯ρq¯τq

! R

=

| ¯ρ|−1

X

p=l+1 p

Y

q=1

e^{Aρq¯ τ¯q}

! R +

l

Y

q=1

e^{Aρq¯ τ¯q}

! R +

l−1

X

p=1 p

Y

q=1

e^A¯ρq¯τq

! R

⊇ e^{Aρ1¯ τ¯1}· · · e^Aρl¯τ¯l





| ¯ρ|−1

X

p=l+1





p

Y

q=l+1

e^{Aρq¯ ¯τq}



R



+

|ρ|−1

X

p=1 p

Y

q=1

e^Aρqτq

! R

= e^Aρ1τ1· · · e^Aρlτl





| ¯ρ|−l−1

X

k=1 k

Y

r=1

e^{Aρr+l¯ τ¯r+l}

! R



+ Q(ρ, τ ) [k := p − l, r := q − l]

= e^Aρ1τ1· · · e^Aρlτl





| ˆρ|−1

X

k=1 k

Y

r=1

e^{Aρrˆ τˆr}

! R



+ Q(ρ, τ )

= e^Aρ1τ1· · · e^AρlτlQ( ˆρ, ˆτ ) + Q(ρ, τ ).

The following theorem plays a key role for what follows.

Theorem 2.3.

There exist a ρ ∈ L and τ ∈ R^|ρ| such that

Q(ρ, τ ) = h{A1, A2}|Ri.

Proof. We define

r := max

ρ∈L τ ∈R^|ρ|

dim Q(ρ, τ ). (2.11)

This number is well-defined as dim Q(ρ, τ ) ≤ n for all ρ and τ . From Lemma 2.2 it follows that

dim Q(ρ, τ ) ≤ dim Q(ρ⁰, τ⁰) (2.12)

for each ρ⁰ and τ⁰ of the form ρ⁰= ¯ρρ and τ⁰ = ¯τ τ . Now let (ρ, τ ) and ( ˆρ, ˆτ ) be such that

dim Q(ρ, τ ) = dim Q( ˆρ, ˆτ ) = r.

It follows from (2.12) that we can assume ρ ˆρ ∈ L without loss of generality. Define l = |ρ|, [|ρ| ≥ 2]

Q = Q(ρ, τ ) and Q = Q( ˆˆ ρ, ˆτ ).

Take T1, T2∈ R+. It follows from Lemma 2.2 that

Q(ρ ˆρ1ρˆ2ρ, τ Tˆ 1T2τ ) ⊇ eˆ ^Aρ1τ1· · · e^Aρlτle^{Aρ1ˆ T1}e^{Aρ2ˆ T2}Q + Q.ˆ (2.13) Define

S = e^Aρ1τ1· · · e^Aρlτl.

Since dim(Se^{Aρ1ˆ T1}e^{Aρ2ˆ T2}Q) = dim ˆˆ Q = r, from (2.13) we obtain dim Q(ρ ˆρ₁ρˆ₂ρ, τ Tˆ ₁T₂τ ) ≥ r.ˆ It follows from (2.11) that

dim Q(ρ ˆρ₁ρˆ₂ρ, τ Tˆ ₁T₂τ ) = r.ˆ (2.14) Since dim(Se^{Aρ1ˆ T1}e^{Aρ2ˆ T2}Q) = dim Q = r, we get from (2.13)ˆ

Se^{Aρ1ˆ T1}e^{Aρ2ˆ T2}Q = Qˆ for any T1, T2∈ R+. (2.15) Now suppose

e^A1t1e^A2t2V = W (2.16)

for any t1, t2> 0 and for two subspaces V and W. If we fix t1we get e^A2t2V = W⁰ ∀t2> 0

where W⁰ = e^−A1t1W. This implies V is A2-invariant, c.q. e^A2t2V = V. From (2.16) it follows that e^A1t1V = W ∀t1> 0

which implies V is also A₁-invariant. We can now conclude that ˆQ is both A₁- and A₂-invariant, and

Q = Q.ˆ (2.17)

By definition

Q = e^Aρ1τ1· · · e^AρlτlR + · · · + e^A1τ1R.

Since Q is Aρ₁ invariant

Q = e^−Aρ1τ1Q = e^Aρ2τ2· · · e^AρlτlR + · · · + R which shows R ⊆ Q. Because Q is {A1, A2}-invariant and contains R

h{A1, A2}|Ri ⊆ Q. (2.18)

From the definition of Q-sets it follows that, for any ρ ∈ L and τ ∈ R^|ρ|, Q(ρ, τ ) ⊆ h{A₁, A₂}|Ri and so

Q = h{A₁, A₂}|Ri. (2.19)

The following lemma gives an alternative characterization of the reachable set of a switched linear system.

Lemma 2.4.

h{A1, A2}|Ri = h{A1, A2}| im B1+ im B2i

Proof. By definition, Riis the smallest Ai-invariant subspace that contains im Bi. So im Bi⊆ Rifor i = 1, 2.

This implies

h{A1, A2}| im B1+ im B2i ⊆ h{A1, A2}|R1+ R2i. (2.20) For the reverse inclusion, note that

R_i= hA_i| im Bii ⊆ h{A1, A₂}| im B1+ im B₂i.

Hence

R₁+ R₂⊆ h{A₁, A₂}| im B₁+ im B₂i.

Since the right hand side is {A1, A2}-invariant, we obtain

h{A1, A₂}|R1+ R₂i ⊆ h{A1, A₂}| im B1+ im B₂i.

From (2.20) it follows that

h{A₁, A₂}| im B₁+ im B₂i = h{A₁, A₂}|R₁+ R₂i

= h{A1, A2}|Ri.

With these preparations, we are in a position to present the main result of this project.

Theorem 2.5 (Reachable set of switched system).

The reachable set for switched system (1.1) is the smallest {A₁, A₂}-invariant subspace that contains im B₁ and im B2.

R = h{A1, A2}| im B1+ im B2i Proof. From (2.7) and Theorem 2.1 it follows that

R = [

ρ∈L τ ∈R^|ρ|₊

Q(ρ, τ ).

From Theorem 2.3 we know there exist a ˆρ ∈ L and a ˆτ ∈ R^{| ˆρ|} with Q( ˆρ, ˆτ ) = h{A1, A2}|Ri.

It follows from the definition of Q that

h{A1, A2}|Ri = Q(ˆρ, ˆτ ) ⊆ R = [

ρ∈L τ ∈R^|ρ|+

Q(ρ, τ ) ⊆ h{A1, A2}|Ri.

From Lemma 2.4 we obtain

R = h{A1, A₂}| im B1+ im B₂i.

Chapter 3

Examples

In this chapter some examples are provided in order to illustrate the evolution of the reachable set of a switched system after an increasing number of switches. For each system the reachable set converges to a subspace of the entire state space as proven in the previous chapter. The images were constructed with a MATLAB script of which the code is provided in the Appendices.

Example 3.1.

We take switched system (1.3) from Example 1.1 and set L = C = R = 1. The system matrices are of the form

A1=

 −1 1

−1 0



, B1=

 0 0

 , A2=

 −1 0

0 0



and B2=

 0 1

 .

(3.1)

The reachable subspaces of subsystems (A₁, B₁) and (A₂, B₂) are respectively R₁= im

 0 0



= {0}, R₂= im

 0 1

 .

The set of all states that are reachable from the origin within k switchings is denoted by R^k. If no switching occurs, then the reachable set is equal to the reachable subspace of (A2, B2),

R⁰=

 0 a

 : a ∈ R

 . The reachable set of the system after one switch is given by

R¹=



 [

t∈R+

e^A1tR⁰+ R₁



∪



 [

t∈R+

e^A2tR⁰+ R₂





=

 a b



: a, b ∈ R

 .

This is a subspace since this spans the entire state space. Figure 3.1 depicts the sets R⁰ and R¹.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

(a) Zero switchings (R⁰)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

(b) One switching (R¹)

Figure 3.1: Evolution of the reachable set of switched system (3.1) restricted to the unit circle.

Example 3.2.

Consider a switched system in R³with

A₁=





0 1 0

−1 0 0

0 0 0



, B₁=



 1 0 0



,

A2=





0 0 0 0 0 0 0 1 0



 and B2=



 0 0 0



.

(3.2)

The reachable subspaces of subsystems (A1, B1) and (A2, B2) are

R1= im



 1 0 0 1 0 0



, R2= im



 0 0 0



= {0}.

Without switching, the reachable set is equal to the reachable subspace of (A₁, B₁),

R⁰=









 a b 0



: a, b ∈ R





 .

The reachable set of the system after one switch is given by

R¹=



 [

t∈R+

e^A2tR⁰+ R₂



∪



 [

t∈R+

e^A1tR⁰+ R₁





=









 a b bt



: a, b ∈ R, t ∈ R⁺





 .

Repeating the above process, we obtain

R²=



 [

t∈R+

e^A1tR¹+ R1



∪



 [

t∈R+

e^A2tR¹+ R2





=











a cos t₃+ b sin t₃+ c

−a sin t₃+ b cos t₃+ d bt2



: a, b, c, d ∈ R, t2, t₃∈ R+







which spans the whole R³ and is indeed a subspace. These R^k sets are depicted in figure 3.2.

−1

−0.5 0

0.5

1 −1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

(a) Zero switchings (R⁰)

−1−0.5 0 0.5

1 −0.5 0 0.5

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8

(b) One switching (R¹)

Figure 3.2: Evolution of the reachable set of switched system (3.2) restricted to the unit sphere.

Example 3.3.

We define a switched system in R³ with two subsystems. Let

A1=





0 0 0 0 0 0 0 1 0



, B1=



 1 0 0



,

A₂=





0 0 0 1 0 0 0 0 0



 and B₂=



 0 0 0



.

(3.3)

Computing the reachable subspaces of both (A₁, B₁) and (A₂, B₂), we obtain

R1= im



 1 0 0



, R2= im



 0 0 0



= {0}.

Without switching, the reachable set is equal to R1:

R⁰=









 a 0 0



: a ∈ R





 .

After one switch the reachable set is given by

R¹=



 [

t∈R+

e^A2tR⁰+ R2



∪



 [

t∈R+

e^A1tR⁰+ R1





=









 a at

0



: a ∈ R, t ∈ R+





 .

Note that this is not a subspace, but a union of uncountably many subspaces. Repeating the above process, we get

R²=



 [

t∈R+

e^A1tR¹+ R₁



∪



 [

t∈R+

e^A2tR¹+ R₂





=









 a b bt



: a, b ∈ R, t ∈ R⁺





 ,

R³=



 [

t∈R+

e^A2tR²+ R2



∪



 [

t∈R+

e^A1tR²+ R1





=









 a at₃+ b

bt₂



: a, b ∈ R, t², t3∈ R+





 .

Switching once more yields

R⁴=



 [

t∈R+

e^A1tR³+ R1



∪



 [

t∈R+

e^A2tR³+ R2





=









 a b + ct3

bt2



: a, b, c ∈ R, t2, t₃∈ R+







which spans the whole R³and is indeed a subspace. Restrictions of each R^k to the unit sphere can be seen in figure 3.3.

−1

−0.5 0

0.5 1

−1

−0.5 0 0.5

−11

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

(a) Zero switchings (R⁰)

−1

−0.5 0

0.5 1

−1

−0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

(b) One switching (R¹)

−1 0

1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

(c) Two switchings (R²)

−1 −0.5

0 0.5

1 −1

−0.5 0

0.5 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

(d) Three switchings (R³)

−1 −0.5 0 0.5 1

−0.5 −1 0.5 0

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8

(e) Three switchings (R³) from alternative angle

Figure 3.3: Evolution of the reachable set of switched system (3.3) restricted to the unit sphere.

Chapter 4

Conclusions

It has been proven that both the controllable and reachable set of switched linear systems are a subspace of the entire state space and that they always coincide with each other [2]. The details of this proof were highly technical. In this paper we provided a simpler, compact proof for the same result. For simplicity we restricted the proof to switched linear systems containing two subsystems. We showed that the reachable set of a switched system is the smallest subspace that is invariant under all state matrices and that contains the image of the input matrix for both subsystems. Since the controllable and reachable set of switched linear systems coincide, the controllable set is also a subspace.

The proof in this paper could be extended to include switched linear systems with an arbitrary number of subsystems. Future research could determine whether there exists a supremum for the number of switching required to obtain the entire reachable subspace. A closely related problem is designing a controller that is able to reach a controllable state in the least number of switchings.

Bibliography

[1] Z. Sun and S.S. Ge. Switched Linear Systems: Control and Design. Springer, London, 2005.

[2] Z. Sun, S.S. Ge, and T.H. Lee. Controllability and reachability criteria for switched linear systems.

Automatica, 38(5):775–786, 2002.

[3] H.L. Trentelman, A.A. Stoorvogel, and M.L.J. Hautus. Control Theory for Linear Systems. Springer, London, 2001.

Appendix A

Computer program

The following program was written in MATLAB. It was used for the creation of the examples in chapter 3.

SwitchedSystem.m

1 c l a s s d e f S w i t c h e d S y s t e m

%SWITCHEDSYSTEM S w i t c h e d L i n e a r System

% T h i s c l a s s makes i t p o s s i b l e t o e x p l o r e t h e o v e r a l l s h a p e o f

% t h e r e a c h a b l e s e t o f a s w i t c h e d s y s t e m and t h e e v o l u t i o n

5 % t h e r e o f w i t h r e s p e c t t o t h e number o f s w i t c h i n g s . properties

A 1 % A m a t r i x f o r f i r s t s u b s y s t e m

9 B 1 % B m a t r i x f o r f i r s t s u b s y s t e m A 2 % A m a t r i x f o r s e c o n d s u b s y s t e m B 2 % B m a t r i x f o r s e c o n d s u b s y s t e m P % P r o j e c t i o n o f S t a t e s p a c e on Rˆ3

13 t s t e p % Nr o f s t e p s i n t i m e t e n d % Endtime

r s t e p s % T o t a l r a d i a n s t e p s end

17

properties ( S e t A c c e s s=p r i v a t e )

R 1 % R e a c h a b l e s p a c e o f f i r s t s y s t e m R 2 % R e a c h a b l e s p a c e o f s e c o n d s y s t e m

21 R % R e a c h a b l e s e t

t s t e p s % a r r a y o f p o i n t s i n t i m e s i n t % For d i s p l a y i n g p u r p o s e c o s t % Idem

25 end

methods

function s y s = S w i t c h e d S y s t e m ( A 1 , B 1 , A 2 , B 2 , t s t e p , tend , r s t e p s )

29

% V a l i d a t i n g i n p u t

i f s i z e ( A 1 , 1 ) ˜= s i z e ( A 1 , 2 )

33 throw ( MException ( ’ S w i t c h e d S y s t e m : nonSquareA ’ , ’ A 1 s h o u l d be s q u a r e ’ ) ) ; end

i f s i z e ( A 2 , 1 ) ˜= s i z e ( A 2 , 2 )

37 throw ( MException ( ’ S w i t c h e d S y s t e m : nonSquareA ’ , ’ A 2 s h o u l d be s q u a r e ’ ) ) ; end

s i z e s = [ s i z e ( A 1 , 1 ) s i z e ( B 1 , 1 ) s i z e ( A 2 , 1 ) s i z e ( B 2 , 1 ) ] ;

41

i f max( s i z e s ) ˜= min( s i z e s )

throw ( MException ( ’ S w i t c h e d S y s t e m : i n v a l i d D i m e n t i o n s ’ , ’ The s u p p l i e d m a t r i c e s do n o t have v a l i d d i m e n t i o n s ’ ) ) ;

end

45

% A s s i g n i n g v a l u e s t o p r o p e r t i e s s y s . A 1 = A 1 ;

s y s . B 1 = B 1 ;

49 s y s . A 2 = A 2 ;

s y s . B 2 = B 2 ;

s y s . P = [ eye ( s i z e ( A 1 , 1 ) , 3 ) zeros ( s i z e ( A 1 , 1 ) , s i z e ( A 1 , 2 ) − 3 ) ] ; s y s . t s t e p = t s t e p ;

53 s y s . t e n d = t e n d ;

s y s . t s t e p s = [ 0 exp ( log ( . 0 1 ) : ( log ( t e n d )−log ( . 0 1 ) ) / t s t e p : log ( t e n d ) ) ] ; s y s . R 1 = orth ( c t r b ( A 1 , B 1 ) ) ;

s y s . R 2 = orth ( c t r b ( A 2 , B 2 ) ) ;

57 s y s . r s t e p s = r s t e p s ;

s y s . s i n t = s i n ( 0 : 2* pi / r s t e p s : 2 * pi ) ; s y s . c o s t = cos ( 0 : 2* pi / r s t e p s : 2 * pi ) ;

61 end

function [ Sp ac e ] = S p a c e A f t e r S w i t c h i n g s ( s y s , s w i t c h i n g s )

% P l o t s t h e r e a c h a b l e s e t o f t h e s w i t c h e d s y s t e m a f t e r t h e

65 % s p e c i f i e d number o f s w i t c h i n g s . R e t u r n s a c e l l s t r u c t u r e

% c o n t a i n i n g a l l s u b s p a c e c o n t a i n e d i n t h e s e t .

69 % C r e a t e a new f i g u r e i n w h i c h we p l o t t h e r e a c h a b l e s e t f i g u r e ( ) ;

hold on ;

73 S pa ce = { s y s . R 1 } ;

i f s w i t c h i n g s == 0

s y s . DrawSpace ( Sp ac e { 1 } ) ;

77 end

f o r i = 1 : s w i t c h i n g s

81 f p r i n t f ( ’ S t e p %d\n ’ , i ) ; i f mod( i , 2 )

85 S pa ce = s y s . S p a c e A f t e r S w i t c h ( s y s . A 2 , Space , s y s . R 2 , ˜mod ( i , s w i t c h i n g s ) ) ;

e l s e

89 S pa ce = s y s . S p a c e A f t e r S w i t c h ( s y s . A 1 , Space , s y s . R 1 , ˜mod ( i , s w i t c h i n g s ) ) ;

end

93 end

end

97 function [ Sp ac e ] = S p a c e A f t e r S w i t c h ( s y s , A, V 1 , V 2 , draw )

% A p p l i e s m a t r i x A t o s e t V 1 f o r t h e t i m e s s p e c i f i e d by

% t s t e p s . R e t u r n s a c e l l o f a l l s u b s p a c e s spanned by V 1

% and V 2

101

S pa ce = { } ;

f o r s = s y s . t s t e p s

105

% Apply s h i f t on e a c h c e l l o f s e t V 1

W = c e l l f u n (@(M) expm(A * s ) * M, V 1 , ’ UniformOutput ’ , 0) ;

109 f o r i = 1 : s i z e (W, 2 )

V = orth ( [W{ i } V 2 ] ) ;

i f draw

113 s y s . DrawSpace (V) ;

end

S pa ce = [ Sp ac e V ] ;

117 end

end

121 end

function DrawSpace ( s y s , W)

% P l o t s t h e c e l l o f s u b s p a c e s W

125

W = orth ( s y s . P * W) ;

i f s i z e (W, 2 ) == 3

129

f p r i n t f ( ’ F u l l s p h e r e \n ’ ) ;

e l s e i f s i z e (W, 2 ) == 2

133

X = W( 1 , 1 ) * sys . c o s t + W( 1 , 2 ) * sys . s i n t ; Y = W( 2 , 1 ) * sys . c o s t + W( 2 , 2 ) * sys . s i n t ; Z = W( 3 , 1 ) * sys . c o s t + W( 3 , 2 ) * sys . s i n t ;

137

plot3 (X, Y, Z , ’ k ’ ) ;

e l s e i f s i z e (W, 2 ) == 1

141

l i n e ( [W( 1 ) −W( 1 ) ] ’ , [W( 2 ) −W( 2 ) ] ’ , [W( 3 ) −W( 3 ) ] ’ , ’ C o l o r ’ , ’ k ’ ) ;

end

145

end end

149 end