Observability and Controllability of Fractional Linear Dynamical Systems

Observability and Controllability ofFractional Linear Dynamical Systems

K. Balachandran ∗ V. Govindaraj ∗ M. D. Ortigueira ∗∗

M. Rivero ∗∗∗ J. J. Trujillo ∗∗∗∗

∗Department of Mathematics, Bharathiar University, Coimbatore 641046, India; (e-mail: [email protected];

[email protected]).∗∗UNINOVA and Department of Electrical Engineering, Faculdade de

Ciencias e Tecnologia da Universidade Nova de Lisboa, Campus daFCT da UNL, Quinta da Torre, 2829, 516 Caparica, Portugal;

(e-mail: [email protected]).∗∗∗Departamento de Matematica Fundamental, Universidad de La

Laguna, 38271 La Laguna, Tenerife, Spain; (e-mail: [email protected]).∗∗∗∗Departamento de Analisis Matematico, Universidad de La Laguna,

38271 La Laguna, Tenerife, Spain; (e-mail: [email protected]).

Abstract: In this paper we study the observability and controllability of fractional lineardynamical systems in finite dimensional spaces. Examples are included to illustrate thetheoretical results proved in this manuscript.

Keywords: Controllability; Observability; Fractional Differential Equations; Mittag LefflerMatrix Function.

1. INTRODUCTION

Fractional differential equations (FDEs) are considered asa emergent branch of applied mathematics with many ap-plications in the field of physical and engineering to modelthe dynamics of different processes through anomalousmedia, but also introduce more efficient model in fieldsas signal processing or control theory (see, for example,the survey [29], the books [5, 10, 17, 21] and the follow-ing special issues [6, 20, 24, 25, 28]). FDEs capture nonlocal relations in space and time with power-law memorykernels, due to this fact, research in this topic has grownsignificantly all around the world. Fractional differentialsand integrals provide more accurate models of systemsunder consideration. Few examples of how many authorshave demonstrated the application of fractional calculusare the following: in electrochemistry [9], thermal systemsand heat conduction [3], viscoelastic materials [1], fractalelectrical networks [22] and many others areas. Differentialequations with fractional order have recently proved to bevaluable tools to the modelling of many physical phenom-ena [12, 23].

In particular, an increasing interest in issues related tofractional dynamical systems oriented towards the field ofcontrol theory can be observed in the literature. The studyof the observability and controllability of the fractionaldynamical systems are two important issues for manyapplied problems. It is well known that the problem ofcontrollability of dynamical systems are widely used inanalysis and the design of control system. Any system issaid to be controllable if every state corresponding to thisprocess can be affected or controlled in respective time by

some controller. Observability is a measure for how wellinternal states of a system can be inferred by knowledge ofits external outputs. The observability and controllabilityof a system are mathematical duals. The concept ofobservability and controllability were introduced by R.E.Kalman for linear dynamical systems. Several authors [2,8, 27, 30] studied the controllability results for linear andnonlinear dynamical systems in finite dimensional spaces.This is not the case of fractional linear systems. In factthere are few works reporting the study of observabilityand controllability of fractional linear systems (see, forexample, [4, 7, 15, 16, 26]).

In this paper we study the observability and controlla-bility of the linear fractional dynamical system in finitedimensional spaces. The observability and controllabilityGrammian matrices are obtained by using Mittag-Lefflermatrix function. Examples are constructed to verify theresults proved in this paper.

2. THE FRACTIONAL DERIVATIVE

To work in a general context we will use the generalformulation of the incremental ratio derivative valid forany order, real or complex. In the following we will considerthe real case.

Definition 1. Similarly to the classic case, we define frac-tional derivative by the limit of the fractional incrementalratio ([18])

Dαθ f(t) = e−jαθ lim

|h|→0

∑∞k=0(−1)k

(αk

)f(t− kh)

hα, (1)

where h = ejθ is a complex number, with θ ∈ (−π, π].

6th Workshop on Fractional Differentiation and Its ApplicationsPart of 2013 IFAC Joint Conference SSSC, FDA, TDSGrenoble, France, February 4-6, 2013

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The above defined derivative is a general incrementalratio based derivative that generalizes classical Grunwald-Letnikov fractional derivative. To understand and give aninterpretation to the above formula, assume that t ∈ R isa time and that h is real, θ = 0 or θ = π. If θ = 0 , onlythe present and past values are being used (2), while, ifθ = π, only the present and future values are used. Thismeans that if we look at (1) as a linear system, the firstcase is causal, while the second is anti-causal [21].

In general, if θ = 0, we call (1) the forward Grunwald-Letnikov 1 derivative, which is well known; however itsproperties are not so well studied:

Dαf f(t) = lim

|h|→0+

∑∞k=0(−1)k

(αk

)f(t− kh)

hα. (2)

In particular, if we assume that f(t) = 0 for t < a and let[·] be the ”integer part of the argument”, we obtain from(2).

Dαf f(t) = lim

|h|→0+

∑[ t−ah ]k=0 (−1)k

(αk

)f(t− kh)

hα, (3)

that is, the formulation we find frequently [23]. Withθ = π we would obtain the corresponding called backwardderivative Dα

b f(t).

In the following, we will use mainly the forward derivative.So we will remove, in such case, the subscript “f ”.

2.1 Simple examples

(1) The exponential

Applying the above definitions to the function f(t) =est, s ∈ C. we obtain see [18]

Dαf(t) = limh→0+

(1− e−sh

)αhα

est = | h |αejθαest (4)

iff θ ∈ (−π2 ,π2 ) which corresponds to be working with

the principal branch of (·)α and assuming a branchcut line in the left hand complex half plane.

This result can be used to generalize a well knownproperty of the Laplace transform. If we return backto equation (2) and apply the bilateral Laplace trans-form

F (s) =

∫ +∞

−∞f(t)e−stdt (5)

to both sides we conclude that:

LT [Dαf(t)] = sαF (s) (Re(s) > 0) (6)

where in sα we assume the principal branch and a cutline in the left half plane. This result is valid for allfunctions that have Laplace transform with region ofconvergence on the right half complex plane.

(2) The causal power function

We start by computing the fractional derivative of the

Heaviside unit step function, ε(t) =

{1 t ≥ 0

0 t < 0. From

(2) we have:

1 The terms forward and backward are used here in agreement tothe way the time flows, from past to future or the reverse.

Dαε(t) = limh→0+

∑[ th ]k=0(−1)k (αk )

hα.

where we assumed h > 0. On the other hand we canput with k = t/h, with t > 0

(−1)k (αk ) =(−α)kk!

=Γ(−α+ k)

Γ(−α)Γ(k + 1)

=Γ(−α+ t/h)

Γ(−α)Γ(t/h+ 1)

Using a well-known property of the gamma function,when h→ 0+ we can write

(−1)k (αk ) =(−α)kk!

≈ (t/h)−α−1

Γ(−α)

Inserting this expression above we obtain

Dαε(t) = limh→0+

t

h

(t/h)−α−1

Γ(−α)hα.

allowing us to write:

Dαε(t) =t−αε(t)

Γ(−α)(7)

In the following we will be interested in negativevalues of the order, meaning powers of the typetα

Γ(α)ε(t). Its Laplace transform is equal to 1sα+1 [21]

3. THE LINEAR SYSTEMS AND THEIR INITIALCONDITIONS

The linear systems we will consider assume the generalformat

Dαx(t) = Ax(t) + f(t), (8)

with the derivative defined in (2) and where 0 < α < 1,t ∈ R, x, f ∈ Rn, A is a n× n constant matrix and f is acontinuous function on J = [0, T ] ∈ R.The above equation could be consider on R. However,we shall be concerned with the system behaviour afterthe application of f(t). This means that our observationwindow is in R+. This leads us to the need for consideringa initial condition of the system. This can be done byintroducing a change in the above equation making theinitial condition appear explicitly - see [19]. As it wasshown there the definition of the system to account foran initial condition x0 = x(t0) is

Dαx(t) = Ax(t) + x(t0)δ(α−1)(t− t0) + f(t), (9)

where δ(t) is the Dirac delta function. The fractionalderivative of the delta is given by the first order derivativeof (7), accordingly to the theory presented in [18].Applying the Laplace transform to this equation, weobtain:

sαX(s) = AX(s) + sα−1x(t0) + F (s),

andX(s) = [sαI −A]

−1 [sα−1x(t0) + F (s)

](10)

To obtain a general expression to the output x(t) we must

compute the inverse of [sαI −A]−1

. To do it, proceedformally to get the series expansion

[sαI −A]−1

=

∞∑n=1

An−1s−nα

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that inverted gives

LT−1{

[sαI −A]−1]

=

∞∑n=1

An−1 tnα−1

Γ(nα)ε(t)

This function is called α−exponential [11]. However itis commonly expressed in terms of the Mittag-Lefflerfunction.

Definition 2. The Mittag-Leffler function Eα,β(z) is acomplex function which depends on two complex parame-ter. It is defined by (see, for example, [11, 14])

Eα,β(z) =

∞∑k=0

zk

Γ(αk + β), α, β ≥ 0. (11)

When β = 1, Eα,1(z) = Eα(z) converges for all valuesof the argument z. Thus the Mittag-Leffler function is anentire function. For a n×n matrix A, the matrix extensionof the above Mittag-Leffler function is

Eα,β(A) =

∞∑k=0

Ak

Γ(αk + β).

Finally, using the Mittag-Leffler function the solution ofthe system (9) is given by

x(t) =Eα(Atα)x0

+

∫ t

0

(t− s)α−1Eα,α(A(t− s)α)f(s)ds. (12)

with x0 = x(t0).

4. OBSERVABILITY RESULT

Consider the fractional order linear time invariant system

Dαx(t) = Ax(t), (13)

where again x ∈ Rn and A is a n × n constant matrix.Along with (13) we have a linear observation in the intervalt ∈ [t0, t1],

y(t) = Hx(t), (14)

where y ∈ Rm and H is an m× n constant matrix.

Definition 3. The system (13), (14) is observable on aninterval [t0, t1] if

y(t) = Hx(t) = 0, t ∈ [t0, t1],

implies

x(t) = 0, t ∈ [t0, t1].

Theorem 1. The observed linear system (13), (14) is ob-servable on [t0, t1] if and only if the observability Gram-mian matrix

W =

∫ t1

t0

Eα(AT (t− t0)α)HTHEα(A(t− t0)α)dt (15)

is positive definite, where the T denotes the transposematrix.

Proof. The solution x(t) of (13) corresponding to theinitial condition x(t0) = x0 is given by

x(t) = Eα(A(t− t0)α)x0

and we have, for y(t) = Hx(t) = HEα(A(t− t0)α)x0

‖y‖2 =

∫ t1

t0

yT (t)y(t)dt

= xT0

∫ t1

t0

Eα(AT (t− t0)α)HTHEα(A(t− t0)αdt x0

= xT0 W x0,

a quadratic form in x0. Clearly W is an n× n symmetricmatrix. If W is positive definite, then y = 0 impliesxT0 W x0 = 0. Therefore x0 = 0. Hence (13), (14) isobservable on [t0, t1]. If W is not positive definite, thenthere is some x0 6= 0 such that xT0 W x0 = 0. Thenx(t) = Eα(A(t− t0)α)x0 6= 0, for t ∈ [t0, t1] but ‖y‖2 = 0,so y = 0 and we conclude that (13), (14) is not observableon [t0, t1].

5. CONTROLLABILITY RESULT

Consider the fractional order linear time invariant system

Dαx(t) =Ax(t) +Bu(t) (16)

x(t0) = x0,

where x ∈ Rn, u ∈ Rm and A, B are n×n, n×m matricesrespectively. Let u(t) ∈ L2([t0, t1],Rm), the space of allsquare integrable Rm valued measurable functions definedon [t0, t1].

Definition 4. The system (16) is controllable on [t0, t1] iffor every pair of vectors x0, x1 ∈ Rn, there is a controlu(t) ∈ L2([t0, t1],Rm) such that the solution x(t) of (16)which satisfies

x(t0) = x0, (17)

also satisfies

x(t1) = x1. (18)

We say that u steers the system form x0 to x1 during theinterval [t0, t1].

Lemma 2. The system (16) is controllable on [t0, t1] ifand only if for each vector x1 ∈ Rn there is a controlu ∈ L2([t0, t1],Rm) which steers 0 to x1 during [t0, t1].

Proof. Suppose the system (16) is controllable on [t0, t1],then by taking x0 = 0 we see that u steers form 0 to x1

during [t0, t1]. To prove the sufficiency it is only necessaryto choose two vectors x0, x1 ∈ Rn and put

x1 = x1 − Eα(A(t1 − t0)α)x0.

If u steers 0 to x1 during [t0, t1], then

x1 =

∫ t1

t0

(t1 − t)α−1Eα,α(A(t1 − t)α)Bu(t)dt.

By the above we have,

2013 IFAC FDAGrenoble, France, February 4-6, 2013

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x1 − Eα(A(t1 − t0)α)x0

=

∫ t1

t0

(t1 − t)α−1Eα,α(A(t1 − t)α)Bu(t)dt

x1 =Eα(A(t1 − t0)α)x0

+

∫ t1

t0

(t1 − t)α−1Eα,α(A(t1 − t)α)Bu(t)dt

= x(t1),

this implies that the solution of (16) for this control usatisfies both (17) and (18), and steers the system formx0 to x1. Hence the given system (16) is controllable on[t0, t1].

Theorem 3. The linear control system (16) is controllableon [t0, t1] if and only if the controllability Grammianmatrix

M =

∫ t1

t0

(t1 − τ)α−1Eα,α(A(t1 − τ)α)BBT

×Eα,α(AT (t1 − τ)α)dτ, (19)

is positive definite, for some t1 > t0.

Proof. Since M is positive definite, that is, it is non-singular and therefore its inverse is well-defined. Definethe input as,

u(t) =BTEα,α(AT (t1 − t)α)

×M−1[x1 − Eα(A(t1 − t0)α)x0]. (20)

The solution of (16) with initial condition x(t0) = x0 is

x(t) =Eα(A(t− t0)α)x0

+

∫ t

t0

(t− τ)α−1Eα,α(A(t− τ)α)Bu(τ)dτ. (21)

From (20), we have

x(t1) =Eα(A(t1 − t0)α)x0

+

∫ t1

t0

(t1 − τ)α−1Eα,α(A(t1 − τ)α)BBT

×Eα,α(AT (t1 − τ)α)

×M−1[x1 − Eα(A(t1 − t0)α)x0]dτ

=Eα(A(t1 − t0)α)x0

+MM−1[x1 − Eα(A(t1 − t0)α)x0] = x1.

Thus (16) is controllable.

On the other hand, if it is not positive definite, there existsa nonzero y such that

yTMy = 0

that is,

yT∫ t1

t0

(t1 − τ)α−1Eα,α(A(t1 − τ)α)BBT

×Eα,α(AT (t1 − τ)α)y dτ = 0,

which implies that yTEα,α(A(t1 − τ)α)B = 0 on [t0, t1].

Let x0 = [Eα(A(t1 − t0)α)]−1y. By the assumption, thereexists a control u such that it steers x0 to the origin in theinterval [t0, t1]. It follows that

x(t1) = 0 = Eα(A(t1 − t0)α)x0

+

∫ t1

t0

(t1 − τ)α−1Eα,α(A(t1 − τ)α)Bu(τ)dτ.

Then,

0 = yT y

+

∫ t1

t0

(t1 − τ)α−1yTEα,α(A(t1 − τ)α)Bu(τ)dτ.

But the second term is zero leading to the conclusion thatyT y = 0. This is a contradiction to y 6= 0. Hence M ispositive definite.

Theorem 4. The system (16) is controllable on [t0, t1] ifand only if the adjoint linear observed system

Dαb y(t) =AT y (22)

w(t) =BT y (23)

is observable on [t0, t1].

Proof. Define a linear subspace R(t0, t1) ⊂ Rn by

R(t0, t1) = {x1 ∈ Rn ; x1 = I(t0, t1)} , (24)

with

I(t0, t1) =

∫ t1

t0

(t1 − t)α−1Eα,α(A(t1 − t)α)Bu(t)dt,

and the control u ∈ L2 ([t0, t1],Rm). Thus R(t0, t1) is thesubspace of states reachable from the origin using thecontrol u ∈ L2([t0, t1],Rm). Suppose y1 ∈ Rn has theproperty

yT1 x1 = 0, for x1 ∈ R(t0, t1). (25)

Therefore, using (24)

yT1

∫ t1

t0

(t1 − t)α−1Eα,α(A(t1 − t)α)Bu(t)dt = 0,

and since u(t) is an arbitrary element of L2([t0, t1],Rm).So, conclude that

yT1 Eα,α(A(t1 − t)α)B = 0,

or

w(t) = BTEα,α(AT (t1 − t)α)y1 = 0, t ∈ [t0, t1]. (26)

Now y(t) = Eα,α(AT (t1 − t)α)y1 is a solution of (22) on[t0, t1] and w(t) is the associated observation (23). If (22),(23) is observable, then (26) implies y(0) = 0, which givesy1 = 0. Then (25) shows that y1 = 0 and we concludethat R(t0, t1) = Rn. Hence the system (16) is controllableon [t0, t1]. If (22), (23) is not observable on [t0, t1], there issome y1 = y(0) 6= 0 such that (26) holds. Then we concludethat (25) holds for this nonzero y1 and R(t0, t1) 6= Rn. So(16) is not controllable on [t0, t1]. Hence the pair (22), (23)is observable, whenever (16) is controllable.

2013 IFAC FDAGrenoble, France, February 4-6, 2013

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6. EXAMPLES

The examples consider in this section involve the generalderivative defined above.

Example 1. Consider the sequential linear fractional dy-namical equation of order 2α and 0 < α < 1

D2αx(t) + x(t) = 0 (t ∈ [0, t1]) (27)

x(0) = x0,

Dαx(0) = limt→0

x1−αx′(t) = x′0, (28)

with the linear observation y = x′0.

Let us introduce the auxiliary variables x1(t) = x(t) andx2(t) = Dαx1(t). Then

Dαx1(t) = Dαx(t) = x2(t)

Dαx2(t) = D2αx(t) = −x1(t),

and therefore the problem (27) can be expressed as

Dαx(t) = Ax(t), where A =

[0 1−1 0

]and x(t) =

[x1(t)x2(t)

].

From the linear observation

y = x1 = [ 1 0 ] x(t), H = [ 1 0 ] .

The Mittag-Leffler matrix function of the given matrix Ais

Eα(Atα) =

(L1 L2

−L2 L1

)where

L1 =

∞∑n=0

(−1)nt2nα

Γ(2αn+ 1)

L2 =

∞∑n=0

(−1)nt(2n+1)α

Γ((2n+ 1)α+ 1).

The observability Grammian of this system is

W =

∫ T

0

Eα(AT tα)HTHEα(Atα)dt

=

∫ T

0

(L2

1 L1L2

L1L2 L22

)dt,

which is non-singular if t1 > 0. Hence the given system isobservable.

Example 2. Consider the sequential linear control frac-tional dynamical equation of order 2α and 0 < α < 1

D2αx(t)− x(t) = u(t) (29)

observed in the interval [0, t1]. Let us introduce the follow-ing auxiliary variables x1(t) = x(t) and x2(t) = Dαx1(t).Then

Dαx1(t) =Dαx(t) = x2(t)

Dαx2(t) =D2αx(t) = x1(t) + u(t).

Therefore the problem (29) can be expressed as Dαx(t) =Ax(t) +Bu(t),

where A =

[0 11 0

], B =

[01

]and x(t) =

[x1(t)x2(t)

].

We will assume the following boundary conditions[x1(0)x2(0)

]=

[11

]and

[x1(T )x2(T )

]=

[22

].

The Mittag-Leffler matrix function of the given matrix Ais

Eα,α(A(T − τ)α) =

(N1 N2

N2 N1

)where

N1 =

∞∑k=0

(T − τ)2kα

Γ(2kα+ α)

N2 =

∞∑k=0

(T − τ)(2k+1)α

Γ(2kα+ 2α).

The controllability Grammian of this system is

M =

∫ t1

0

(t1 − τ)α−1Eα,α(A(T − τ)α)BBT

×Eα,α(AT (t1 − τ)α)dτ

=

∫ T

0

(t1 − τ)α−1

(N2

2 N1N2

N1N2 N21

)dτ.

Therefore M is non-singular if T > 0, and the controldefined by

u(t) = BTEα,α(AT (t1 − t)α)M−1 × [x1 − Eα(A(T )α)x0]

steers the given system from

(11

)to

(22

).

This example is interesting, because the system as definedin (29) is unstable.

7. CONCLUSIONS

Observability is a measure for how well internal states ofa system can be inferred by knowledge of its externaloutputs, while the controlability informs us about theability to change the state of a system in order to assumea pre-specified value in a given time interval. The study ofboth the observability and controllability of the fractionaldynamical systems are important issues for many practicaldaily applications. In this paper we have proved twonew main results connected with the observability andcontrollability of fractional linear system by the use ofso called Grammian matrix. The results are formally verysimple and open a new way into interesting possibilities forapplications. Also, we included two application examplesillustrating the presented theory. In the second examplewe can see a formal structure of the controller.

ACKNOWLEDGEMENTS

This work was partially funded by National Funds throughthe FCT (Foundation for Science and Technology) of

2013 IFAC FDAGrenoble, France, February 4-6, 2013

897

Portugal, under the project PEst-OE/EEI/UI0066/2011and by project MTM2010-16499 from the Government ofSpain.

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