Controllability of Discrete-Time Linear Switched Systems with Constrains on Switching Signal

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Book Title Intelligent Information and Database SystemsSeries Title

Chapter Title Controllability of Discrete-Time Linear Switched Systems with Constrains on Switching Signal

Copyright Year 2015

Copyright HolderName Springer International Publishing Switzerland

Author Family Name BabiarzParticle

Given Name ArturPrefix

Suffix

Division Faculty of Automatic Control, Electronics and Computer Science, Instituteof Automatic Control

Organization Silesian University of Technology

Address Akademicka 16 Street, 44-101, Gliwice, Poland

Email [email protected]

Author Family Name CzornikParticle

Given Name AdamPrefix

Suffix





Author Family Name KlamkaParticle

Given Name JerzyPrefix

Suffix





Corresponding Author Family Name NiezabitowskiParticle

Given Name MichałPrefix

Suffix





Abstract In this paper we consider the controllability problem for discrete-time linear switched systems. Theproblem consists of finding a control signal that steers any initial condition to a given final state regardlessof the switching signal. In the paper a necessary and sufficient conditions for this type of controllability arepresented. Moreover, we consider problems of controllability from zero initial condition and to zero finalstate.

Keywords (separated by '-') Controllability - Switched systems - Discrete-time - Hybrid systems

Controllability of Discrete-Time Linear SwitchedSystems with Constrains on Switching Signal

Artur Babiarz, Adam Czornik,Jerzy Klamka, and Micha�l Niezabitowski(B)

Faculty of Automatic Control, Electronics and Computer Science,Institute of Automatic Control, Silesian University of Technology,

Akademicka 16 Street, 44-101 Gliwice, Poland{artur.babiarz,adam.czornik,jerzy.klamka,michal.niezabitowski}@polsl.pl

http://www.polsl.pl

Abstract. In this paper we consider the controllability problem fordiscrete-time linear switched systems. The problem consists of findinga control signal that steers any initial condition to a given final stateregardless of the switching signal. In the paper a necessary and sufficientconditions for this type of controllability are presented. Moreover, weconsider problems of controllability from zero initial condition and tozero final state.

Keywords: Controllability · Switched systems · Discrete-time · Hybridsystems

1 Introduction

Since the early work on state-space approaches to control systems analysis, it wasrecognized that certain nondegeneracy assumptions were useful, in particular inthe context of optimal control. However, it was until Kalman’s work [1] thatthe property of controllability was isolated as of interest in and of itself, asit characterizes the degrees of freedom available when attempting to control asystem.

The study of controllability for linear systems has spanned a great number ofresearch directions, and topics such as testing degrees of controllability, and theirnumerical analysis aspects, are still a subject of intensive research. This paperis devoted to controllability of discrete-time linear switched systems (see [2] fordefinition and motivation). Such a system can be seen as a collection of discretestationary linear systems between which is followed by the switching signal. The

The research presented here were done by the authors as parts of the projectsfunded by the National Science Centre granted according to decisions DEC-2014/13/B/ST7/00755, DEC-2012/05/B/ST7/00065, DEC-2012/07/B/ST7/01404and DEC-2012/07/N/ST7/03236, respectively. The calculations were performedwith the use of IT infrastructure of GeCONiI Upper Silesian Centre for Compu-tational Science and Engineering (NCBiR grant no POIG.02.03.01-24-099/13).

c© Springer International Publishing Switzerland 2015N.T. Nguyen et al. (Eds.): ACIIDS 2015, Part I, LNAI 9011, pp. 304–312, 2015.DOI: 10.1007/978-3-319-15702-3 30

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Controllability of Discrete-Time Linear Switched Systems 305

switching signal may model the phenomenon over which we have control (changeof regulator parameters, gear ratio) or uncontrolled events (failures, changing ofthe operating point). Most of the works (see [3] - [19] and the references therein)on the controllability of hybrid systems is related to the first case and then,the controllability problem is formulated as a problem of finding control andswitching signal which steers an initial condition to a given final state. In thiscase the switching signal plays a role of additional control. The work is connectedto the second case and then, we are looking for a control, that regardless of theswitching signal, steers an initial condition to a given final state. This situationis similar to a problem of controllability of jump linear systems ([20], [21]) butin our framework we do not have a probabilistic model of the switching signal.In addition, a new contribution of the paper is that we take into account thesituation in which certain switching sequences are not possible. This situationoften occurs in engineering practice.

2 Notation and Definitions

We consider a class of switched systems given by

x (k + 1) = A (r(k))x (k) + B (r(k))u (k) , k ≥ 0 (1)

where x(k) ∈ Rn denotes the state vector, r(k) ∈ {1, ..., N} =: S is the switch-

ing signal, u(k) ∈ Rm is the control input, k = 0, 1, ... . Furthermore, for

r(k) = i, Ai := A(i) and Bi := B(i) are constant matrices of appropriatesizes. Denote by x (k, x0, i0, u) the solution of (1) under the control u withinitial condition x0 at time k = 0 and switching signal satisfying r(0) = i0.The control u = (u(0), u(1), ...) is assumed to be such that u(k) is of the formfk (r (0) , r (1) , ..., r (k)) . Each control of this form we will call as an admissiblecontrol. Let us introduce the following notation:

F (k, k) = In×n

F (k, l, ik−1, ..., il) = A (ik−1)A (ik−2) ...A (il) , k > l ≥ 0, ik−1, ..., il ∈ S

Fr(k, l) = A (r (k − 1))A (r (k − 2)) ...A (r (l)) , k > l ≥ 0,

Using this notation we can write the solution of (1) in the following form

x (k, x0, i0, u) = Fr(k, 0)x0 +k−1∑

t=0

Fr (k, t + 1)B (r (t))u (t) , k ≥ 1 (2)

orx (k, x0, i0, u) = F (k, 0, r(k − 1), ..., r(0))x0+ (3)

k−1∑

t=0

F (k, t + 1, r(k − 1), ..., r(t + 1))B (r (t))u (t) , k ≥ 1.

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306 A. Babiarz et al.

We will also use the following notation

S(N)i0

= {(i0, ..., iN−1) : i0, ..., iN−1 ∈ S} . (4)

It will be convenient to have the elements of S(N)i0

ordered in a sequence. Inthat purpose let order the elements of S(N)

i0in lexicographical order i.e. they are

ordered as follows

(i0, 1, 1, ..., 1, 1) , (i0, 1, 1, ..., 1, 2) , ..., (i0, 1, 1, ..., 1, s) ,

(i0, 1, 1, ..., 2, 1) , (i0, 1, 1, ..., 2, 2) , ..., (i0, 1, 1, ..., 2, s) , ...

(i0, s, s, ..., s, 1) , (i0, s, s, ..., s, 2) , ..., (i0, s, s, ..., s, s) .

In many practical situations certain switches are impossible. It means that wehave certain set A of pairs (i, j) ∈ S ×S such that it is impossible that r(k) = i,

r(k+1) = j for a k = 0, 1, ... . Withdraw from S(N)i0

all the elements (i0, ..., iN−1)such that

(il, il+1) ∈ A for certain l = 0, 1, ..., N − 1.

and denote by S(N)

i0 the set obtained in this way. In this notation S(N)

i0 is a

sequence of all possible switching paths of the length N . By s(N)i0 we will denote

the number of elements of S(N)

i0 .Fix a number N > 0 and a sequence (i0, i1, ..., iN−1) of elements of S. Con-

sider a matrix column blocks which are numbered successively by sequences:i0, S

(2)

i0 , ..., S(N)

i0 and the block (i0, i1, ..., ik) , k = 0, 1, .., N − 1 is given by

F (N, k + 1, iN−1, ..., ik+1)Bik

and the others are equal to 0. Denote the matrix obtained in this way byC (i0, i1, ..., iN−1) and by G(i0) - the matrix consisting of all C(i0, i1, ..., iN−1)(as row blocks numbered by S

(N)

i0 ) for (i0, i1, ..., iN−1) ∈ S(N)

i0 . Moreover, by

H(i0) ∈ Rns(N)i0

×m let denote a matrix row blocks of which are numbered by thesequence S

(N)

i0 , the block (i0, i1, ..., iN−1) is given by F (N, 0, iN−1, ..., i0). Forexample in the case when S = {1, 2}, N = 3, A = {(2, 1)} and i0 = 1, we have

G(1) =

⎡

⎣C(1, 1, 1)C(1, 1, 2)C(1, 2, 2)

⎤

⎦ =

(1) (1, 1) (1, 2) (1, 1, 1) (1, 1, 2) (1, 2, 2)[A2

1B1 A1B1 0 B1 0 0][A2A1B1 A2B1 0 0 B2 0][A2

2B1 0 A2B2 0 0 B2]

and

H(1) =

⎡

⎣A3

1

A2A21

A22A1

⎤

⎦ .

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Moreover, let us denote by f(k)1 , ..., f

(k)n ∈ R

nk the vectors defined by

f(k)l =

⎡

⎢⎢⎢⎣

elel...el

⎤

⎥⎥⎥⎦

⎫⎪⎪⎪⎬

⎪⎪⎪⎭k times el, l = 1, ..., n

where e1, ..., en is the standard basis of Rn.We have the following definition:

Definition 1. We say that system (1) is i0− controllable at time N if, for allx0, x1 ∈ R

n there exists an admissible control u such that

x (N,x0, i0, u) = x1. (5)

Analogically, we say that system (1) is i0− controllable at time N to zero (fromzero) if, for all x0 ∈ R

n (x1 ∈ Rn) there exists a control u such that

x (N,x0, i0, u) = 0 (x (N, 0, i0, u) = x1). (6)

If the system (1) is i0− controllable at time N (i0− controllable at time N tozero, i0− controllable at time N from zero) for all i0 ∈ S then we say that (1)is controllable at time N (controllable at time N to zero, controllable at time Nfrom zero).

Observe that the controllability of each time-varying system correspondingto switching paths of the length N is only the necessary, but not the sufficientcondition for controllability at time N of the system (1).

3 Main Results

The next theorem contains necessary and sufficient conditions for i0− control-lability at time N as well as i0− controllability at time N from zero and tozero.

Theorem 1. System (1) is i0− controllable at time N from zero if and only if

rankG(i0) = rank

[G(i0) f

(s(N)i0

)

l

], for all l = 1, ..., n. (7)

System (1) is i0− controllable at time N to zero if and only if

ImH (i0) ⊂ ImG (i0) , (8)

and it is i0− controllable at time N if and only if

rankG(i0) = rank

[G(i0) f

(s(N)i0

)

l

], for all l = 1, ..., n, (9)

andImH (i0) ⊂ ImG (i0) . (10)

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Before entering the formal proof let us briefly discuss the main idea. Sincethe set S

(N)

i0 of all possible switching paths is finite, therefore the question abouti0− controllability can be reformulated, similarly as for classical time-varyingsystems, as a question about existence of a solution of a finite set of linear equa-tions. Nevertheless, now we must take into account the constrain that controlu(k) at time k may depend only on the variables r(0), ..., r(k) and must be inde-pendent of r(k+1), ..., r(N). In the proof we obtain this by the proper definitionof matrices G(i0) and H (i0) .

Proof. Suppose that system (1) is i0− controllable at time N from zero. Thenfor each y ∈ R

n there exists a control sequence u(0), ..., u(N − 1) such that

u (k) = gk (i0, r (1) , ..., r (k)) , k = 0, ..., N − 1,

andx (N, 0, i0, u) = y, (11)

where gk is a function from S(k)

i0 to Rm for k = 0, ..., N − 1. It means that for

any (i0, ..., iN−1) ∈ S(N)

i0 the following

N−1∑

t=0

F (N, t + 1, iN−1, ..., it+1)B (it) gt (i0, ..., it) = y

holds. This clearly forces that the system of equations

G(i0)v = z,

where

z =

⎡

⎢⎣t...t

⎤

⎥⎦

⎫⎪⎬

⎪⎭s(N)i0 times

has a solution for each t ∈ Rn. Since vectors f

(s(N)i0

)

l , l = 1, ..., n form a basis ofthe space ⎧

⎪⎨

⎪⎩

⎡

⎢⎣t...t

⎤

⎥⎦ ∈ Rn s(N)i0 : t ∈ Rn

⎫⎪⎬

⎪⎭,

Kronecker-Capelli Theorem (see e.g. [22]) implies that (7) holds.Suppose now that (7) holds. Again by the Kronecker-Capelli Theorem the

set of equationsG(i0)v = z

where

z =

⎡

⎢⎣y...y

⎤

⎥⎦

⎫⎪⎬

⎪⎭x times

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has a solution for each y ∈ Rn. The above-mentioned fact implies that for each

y ∈ Rn and each (i0, ..., iN−1) ∈ S

(N)

i0 there exists a sequence gk (i0, ..., ik) ,k = 0, ..., N − 1 such that

N−1∑

t=0

F (N, t + 1, iN−1, ..., it+1)B (it) gt (i0, ..., it) = y.

If we define the control

u(k) = gk (i0, r(1)..., r(k))

thenx (N, 0, i0, u) = y

and consequently system (1) is i0− controllable at time N from zero.Suppose that the system (1) is i0− controllable at time N to zero. Then for

each y ∈ Rn there exists a control sequence u(0), ..., u(N − 1) such that

u (k) = gk (i0, r (1) , ..., r (k)) , k = 0, ..., N − 1,

andx (N, y, i0, u) = 0, (12)

where gk is a function from S(k)

i0 to Rm, k = 0, ..., N − 1. From the last equation

we get

N−1∑

t=0

F (N, t + 1, iN−1, ..., it+1)B (it) gt (i0, ..., it) = −F (N, 0, iN−1, ..., i0) y.

and therefore−H (i0) y ∈ ImG (i0)

what implies that (8) holds.Assume now that the condition (8) holds. It means that for each x0 ∈ R

n

there exists v such that

−HX (i0)x0 = GX (i0) v.

This in turn implies that for each (i0, ..., iN−1) ∈ S(N)

i0 there exists a sequencegk (i0, ..., ik) , k = 0, ..., N − 1 such that

N−1∑

t=0

F (N, t + 1, iN−1, ..., it+1)B (it) gt (i0, ..., it) =

−F (N, 0, iN−1, ..., i0)x0.

If we define the control

u(k) = gk (i0, r(1)..., r(k))

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thenx (N,x0, i0, u) = 0

and system (1) is i0− controllable at time N to zero. In the same way we canshow the part about i0− controllability at time N .

When we consider the time-varying system it is well known that the con-trollability from zero implies the controllability to zero [23] and that inverseimplication is not true. The next example shows that for the switched systemthe controllability from zero does not imply the controllability to zero.

Example 1. Consider the system (1) with S = {1, 2}, N = 2, A = ∅.

A1 =[

1 23 1

], A2 =

[−1 21 −1

], B1 =

[02

], B2 =

[01

]

According to the notation we have

G(1) =[C(1, 1)C(1, 2)

]=

[A1B1 B1 0A2B1 0 B2

]=

⎡

⎢⎢⎣

4 0 02 2 04 0 0

−1 0 1

⎤

⎥⎥⎦

and

G(2) =[C(2, 1)C(2, 2)

]=

[A1B2 B1 0A2B2 0 B2

]=

⎡

⎢⎢⎣

2 0 01 2 02 0 0

−1 0 1

⎤

⎥⎥⎦ .

Now it is clear that that the condition (8) is satisfied. The control which steers

the zero initial condition to

[x(0)1

x(0)2

]at time N = 2 is given by

u (0) =

{x(0)14 if r(0) = 1

x(0)12 if r(0) = 2

u (1) =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

x(0)22 − x

(0)14 if r(0) = 1, r(1) = 1

x(0)2 + x

(0)14 if r(0) = 1, r(1) = 2

x(0)2 + x

(0)12 if r(0) = 2, r(1) = 2

x(0)22 − x

(0)14 if r(0) = 2, r(1) = 1

.

From the other hand the system is not controllable to zero at time 2. In fact wehave

H(1) =

⎡

⎢⎢⎣

7 46 75 0

−2 1

⎤

⎥⎥⎦ , H(2) =

⎡

⎢⎢⎣

3 −4−2 3

1 0−2 5

⎤

⎥⎥⎦

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and ⎡

⎢⎢⎣

111351

⎤

⎥⎥⎦ ∈ H(1),

⎡

⎢⎢⎣

−1113

⎤

⎥⎥⎦ ∈ H(2),

but ⎡

⎢⎢⎣

111351

⎤

⎥⎥⎦ /∈ G(1),

⎡

⎢⎢⎣

−1113

⎤

⎥⎥⎦ /∈ G(2).

4 Conclusions

In the paper we presented the necessary and sufficient conditions for controllabil-ity (controllability to zero and controllability from zero) for linear discrete-timeswitched linear systems. These conditions are given in terms of relations con-sisting of ranks and images of matrices constructed on the base of the systemcoefficients. The proposed controllability concept is appropriate to the situationwhen the switching signal models unpredictable events, for example systems fail-ures. Additionally, a new contribution of the paper is that we took into accountthe situation in which certain switching sequences are not possible. This situationoften occurs in engineering practice.

Acknowledgments. The research presented here were done by the authors as parts ofthe projects funded by the National Science Centre granted according to decisions DEC-2014/13/B/ST7/00755, DEC-2012/05/B/ST7/00065, DEC-2012/07/B/ST7/01404and DEC-2012/07/N/ST7/03236, respectively. The calculations were performed withthe use of IT infrastructure of GeCONiI Upper Silesian Centre for ComputationalScience and Engineering (NCBiR grant no POIG.02.03.01-24-099/13).

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Controllability of Discrete-Time Linear Switched Systems with Constrains on Switching Signal