Calculation of controllability and observability matrices for special case of continuous-time multi-order fractional systems

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Calculation of controllability and observability matrices for special case of continuous-time multi-order fractional systems Iman Hassanzadeh, Mohammad Tabatabaei n Department of Electrical Engineering, Khomeinishahr Branch, Islamic Azad University, Isfahan, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 22 September 2016 Received in revised form 7 March 2017 Accepted 14 March 2017

In this paper, controllability and observability matrices for pseudo upper or lower triangular multi-order fractional systems are derived. It is demonstrated that these systems are controllable and observable if and only if their controllability and observability matrices are full rank. In other words, the rank of these matrices should be equal to the inner dimension of their corresponding state space realizations. To reduce the computational complexities, these matrices are converted to simplified matrices with smaller dimensions. Numerical examples are provided to show the usefulness of the mentioned matrices for controllability and observability analysis of this case of multi-order fractional systems. These examples clarify that the duality concept is not necessarily true for these special systems. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Controllability Observability Multi-order fractional systems Fractional order systems

1. Introduction Utilizing non-integer derivatives and integrals for describing real phenomena has increased the modelling precision and design flexibility [1]. This is the reason why the fractional calculus has been considered by the control engineers for modelling and control of industrial plants [2]. Thus, analysis of fractional order systems and designing appropriate controllers for these systems has been considered in the literature [3]. To apply modern control strategies to fractional order systems, these systems should be described with state space realizations. A variety of state space realizations including commensurate or incommensurate, minimal or non-minimal ones have been proposed for fractional order systems [4–9]. Implementation of fractional order controllers could be achieved by fractional order capacitors or Fractances. Implementation of Fractances using integrated technologies or Field Programmable Gate Arrays (FPGA) is complicated [10,11]. Thus, proposing state space realizations with minimum state variables or minimum inner dimension is important. In [5–7], minimal state space realizations for a class of fractional order systems have been extracted. Moreover, some classes of commensurate fractional order systems have been introduced which could be implemented with the same number of Fractances [8]. Employing state space equations with different fractional orders or multi-order fractional systems for obtaining minimal state n Correspondence to: Khomeinishahr Branch, Islamic Azad University, Manzarieh, Daneshjoo Boulvard, Khomeinishahr, Isfahan, Iran. E-mail addresses: [email protected] (I. Hassanzadeh), [email protected] (M. Tabatabaei).

space realizations is inevitable. Thus, analysis of multi-order fractional systems has been considered in the literature. In [12], the solution of a set of continuous-time fractional order systems with two different fractional orders has been presented. This solution has been extended for multi-order fractional systems in [13]. In [14], the solution of the discrete-time fractional order systems with two fractional orders has been presented. Stability analysis of continuous time multi-order fractional systems and nonlinear discrete time fractional order systems with multiple fractional orders has been presented in [15] and [16], respectively. To control the dynamical systems described with state space realizations, controllability and observability of these realizations should be investigated. Controllability and observability of linear commensurate fractional order systems have been verified [17,18]. In this case, the obtained controllability and observability matrices are independent of the commensurate order. In [19], constrained controllability of semilinear dynamical systems has been studied. In [20], sufficient conditions for the controllability of nonlinear fractional order systems have been provided. In [21], controllability of fractional order neutral systems has been studied. In [22,23], necessary and sufficient conditions for controllability and minimum energy control of fractional discrete-time systems have been proposed. In [24], local controllability of fractional discretetime semilinear systems with constant coefficients has been investigated. In [25], controllability of discrete-time switched fractional order systems has been verified. In [26], sufficient conditions for the approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay in Hilbert space have been derived. In [27], approximate controllability of fractional stochastic differential equations driven by

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Please cite this article as: Hassanzadeh I, Tabatabaei M. Calculation of controllability and observability matrices for special case of continuous-time multi-order fractional systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.006i

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2

mixed fractional Brownian motion in Hilbert space using resolvent operators has been investigated. Limited research has been performed about controllability and observability of multi-order fractional systems. In [28], controllability of a discrete-time fractional order system with two fractional orders has been studied. The controllability and observability problem for h-difference linear control systems with n fractional orders has been studied [29]. The controllability and observability gramians for continuous-time fractional order systems with two fractional orders have been extracted in [30] and [31]. In [32], these gramians for a general multi-order fractional system have been obtained. Moreover, the controllability and observability matrices for special case of continuous time multi-order fractional systems have been calculated [32]. In all previously published works, there are no explicit relations for controllability and observability matrices of continuous-time multi-order fractional systems. The previous works have been devoted to discrete time multi-order fractional systems. Only in [32], two complex algorithms for finding the controllability and observability matrices for special case of continuous-time multiorder fractional systems have been proposed. This paper continuous the work published in [32]. The main drawback of the controllability and observability matrices proposed in [32] is that their non-singularity is only the sufficient condition for controllability and observability. In other words, a multi-order realization could be controllable or observable but its corresponding controllability or observability matrix could be singular. In this paper, the multi-order fractional systems that their state space matrices are pseudo upper or lower triangular are introduced. Then, the controllability and observability matrices of these systems are calculated. It is shown that these pseudo upper or lower triangular realizations are controllable or observable if and only if their corresponding controllability or observability matrices are full rank. Unlike the matrices proposed in [32], the mentioned matrices are independent of the fractional orders. Another weakness of the proposed matrices in [32] is that their computation is not straightforward and requires solving some algebraic equations. While in this paper, explicit relations for controllability and observability matrices for this special case of multi-order fractional systems are presented. Moreover, an appropriate counterexample is presented to show that the duality concept is not necessarily true for this kind of systems. The remainder of this paper is organized as follows. In Section 2, a brief review on multi-order fractional systems is given. In Section 3, the pseudo upper and lower triangular multi-order fractional systems and their specifications are introduced. The controllability and observability matrices for these systems are calculated in Section 4. To show the effectiveness of the proposed controllability and observability matrices, some numerical examples are presented in Section 5. Finally, concluding remarks and future research directions are given in Section 6.

There are different definitions for fractional order derivative in the literature. In this paper, all the analysis is based on the Caputo definition. According to this definition, the fractional order derivative of an arbitrary function f (t ) with order α (0 Dtαf ( t )) is defined as [1]

( t) =

1 Γ (n − α )

∫0

t

f n (τ )

dτ , (t − τ )1 + α − n

(n − 1 ≤ α < n)

where Γ( . ) is the Gamma function defined as

∫0

∞

e−z z x − 1dz.

(2)

Fractional order systems could be described with various state space realizations. The commensurate state space realization for a single-input single-output fractional order system is represented with [1]

⎧ Dv x̲ (t ) = Ax̲ (t ) + Bu(t ) ⎨ ⎩ y(t ) = Cx̲ (t )

(3)

where x̲ (t ) ∈ R N , u(t ) and y(t ) are the state vector, the system input and the output, respectively. The commensurate order and the inner dimension are denoted by v and N , respectively. Generally, attaining a state space realization with minimum inner dimension is not possible with commensurate realizations like (3). Thus, utilizing multi-order state space realizations is inevitable. A multi-order state space realization is described as [13]

⎧ ⎡ Dα1 x̲ ⎤ ⎡ A A 1 ⎪⎢ ⎥ ⎢ 11 12 α2 ⎪ ⎢ D x̲ 2 ⎥ ⎢ A21 A22 ⎪⎢ ⎥ = ⎢⋮ ⋮ ⎨ ⋮ ⎢ α ⎪ ⎣ D n x̲ ⎥⎦ ⎢⎣ A A n2 n n 1 ⎪ ⎪ y = ⎡⎣ C C … C ⎤⎦ x̲ 1 2 n ⎩

⎡B ⎤ … A1n ⎤ ⎥ ⎢ 1⎥ … A2n ⎥ B x̲ + ⎢⎢ 2 ⎥⎥u ⋮ ⋮ ⎥ ⋮ ⎥ ⎢ ⎥ … A nn ⎦ ⎣ Bn ⎦ (4)

where αi ∈ (0, 1) , i = 1, ..., n are the fractional orders (αi ≠ αj, T i ≠ j, i, j = 1, …, n) and x̲ (t ) = ⎡⎣ x̲ 1(t ) … x̲ n(t )⎤⎦ . The state vector corresponding with αi is denoted by x̲ i ∈ Rni . The state space

matrices are described by Aij ∈ Rni × nj , i, j = 1, … , n, Bi ∈ Rni × 1 , and

Ci ∈ R1 × ni . The summation of all the state vector dimensions is the n

inner dimension. Or N = ∑i = 1 ni . The transfer function of the multi-order fractional system (4) ( H (s )) is given by [32]

⎤−1⎡ B ⎤ ⎥ ⎢ 1⎥ … −A2n ⎥ ⎢ B2 ⎥ ⎥ ⎢ ⋮⎥ ⋮ ⋮ ⎥ ⎢ ⎥ … s αnI − A nn ⎦ ⎣ Bn ⎦

⎡ s α1I − A −A12 11 ⎢ α2 − A s I − A22 ⎢ 21 H (s ) = C ⎢ ⋮ ⋮ ⎢ −A n2 ⎣ −A n1

−A1n

…

(5)

where C = ⎡⎣ C1 C2 … Cn ⎤⎦.The state space solution of (4) could be obtained as [32]

x̲ (t ) = φ0(t ) x̲ 0 +

∫0

t

n

F (t − τ )u(τ )dτ ,

F (t ) =

∑ φp(t )Bp0 p=1

(6)

where x̲ 0 , φ0(t ), φp(t ) and Bp0 ( p = 1, … , n ) are given by [13] T T x̲ 0 = ⎡⎣ x̲ 1(0) … x̲ n(0)⎤⎦ , B10 = ⎡⎣ B1 0 … 0⎤⎦ , …,

2. Review on multi-order fractional systems

α 0 Dt f

Γ( x) =

T Bn0 = ⎡⎣ 0 0 … Bn ⎤⎦ , ∞

φ0(t ) =

∑ k1= 0

∑ k1= 0

∑ kn = 0

∞

φp(t ) =

n

∞

…

Tk1… k n

t ∑ j = 1 k jα j

(

∑ kn = 0

)

,

n

t ( ∑ j = 1, k jαj)+ αp − 1

∞

…

n

Γ ∑ j = 1 kjαj + 1

Tk1… k n

Γ

(( ∑

n kα j=1 j j

) + αp)

(7)

(1) where Tk1, … , kn is defined as [13]

Please cite this article as: Hassanzadeh I, Tabatabaei M. Calculation of controllability and observability matrices for special case of continuous-time multi-order fractional systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.006i

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Tk1, … , kn

⎧I ⎪ N ⎪⎡ A … A ⎤ 1n ⎥ ⎪ ⎢ 11 ⎥ ⎪ ⎢ 0⎛ n ⎞ ⎪ ⎢ ⎜ ∑ n ⎟×N ⎥ i ⎟ ⎜ ⎢ ⎥ ⎪ ⎝ ⎠ ⎦ ⎪ ⎣ i =2 ⎪⎡0 ⎤ ⎪ ⎢ ⎛⎜ n − 1 ⎞⎟ ⎥ = ⎨ ⎢ ⎜⎜ ∑ n i⎟⎟ × N ⎥ ⎪ ⎢ ⎝ i =1 ⎠ ⎥ ⎪⎢ ⎥ ⎪ ⎣ A n1 … A nn ⎦ ⎪ ⎪0 ⎪ ⎪T ⎪ 10 … 0Tk1− 1, k2, … , kn + … ⎪ ⎩ + T0 … 01Tk1, … , kn − 1, kn − 1,

3. Pseudo upper or lower triangular multi-order fractional systems

for k1 = … = k n = 0

for k1 = 1, k 2 = … = k n = 0

In the remainder of this paper, the special case of multi-order fractional system (4) with the following property is considered

(Aij = 0, for: i > j )

for k1 = … = k n − 1 = 0, k n = 1

n

∑ k i > 0. i =1

(8)

Lemma 1. The following relations are true for any integer number i . i i i Ti0 … 00 = T10 … 00, T0i … 00 = T01 … 00, … , T00 … 0i = T00 … 01.

or

(Aij = 0, for: i < j ).

(13)

We call the fractional order systems (4) with property (13) as pseudo upper (or lower) triangular multi-order fractional systems. The properties of this kind of multi-order fractional systems are illustrated in this section.

for at least one k i < 0, i = 1, …, n . for

3

(9)

Proof. It could be easily proved using the mathematical induction principle and (8).

Remark 1. All the Theorems and Lemmas are stated for both pseudo upper and lower triangular multi-order systems while their proofs are presented only for the upper triangular case. All the Theorems and Lemmas could be proved for the lower-triangular case, similarly. Lemma 2. The relations (14) and (15) are valid for pseudo upper and lower triangular multi-order fractional systems, respectively

T010 … 00T100 … 00 = T001 … 00T100 … 00 = … = T000 … 01T100 … 00 = 0.

(14)

T10 … 00T00 … 01 = T01 … 00T00 … 01 = … = T00 … 10T00 … 01 = 0.

(15)

Definition 1. The system (4) is called controllable if there exists an input (u(t ) , 0 < t ≤ t f ) that transfers any arbitrary initial state x̲ 0 to any arbitrary final state x̲ f in the limited time interval (0, t f ].

Proof. Proof is trivial according to (8) and condition (13).

Theorem 1. The necessary and sufficient condition for controllability of system (4) is that its controllability gramian (WC (t )) is full rank for all positive values of t [32]

Lemma 3. For pseudo upper and lower triangular multi-order fractional systems the matrix Tk1… kn could be obtained according to relations (16) and (17), respectively

WC (t ) =

∫0

t

F (t − τ )FT (t − τ )dτ =

∫0

t

F (τ )FT (τ )dτ .

(10)

Definition 2. The system (4) is said to be observable if the initial state x̲ 0 could be uniquely obtained in terms of output ( y(t ) , 0 < t ≤ t f ) and input ( u(t ) , 0 < t ≤ t f ) data in a finite time interval (0, t f ]. Theorem 2. The system (4) is observable if and only if the following observability gramian (WO(t )) is non-singular for all positive values of t [32]

WO(t ) =

∫0

t

GT (τ )G(τ )dτ

(11)

where

G(t ) = Cφ0(t ).

(12)

Assumption 1. In [32], a special case of multi-order fractional system (4) with αi = mi v, i = 1, … , n , mi ∈ N , m1 < m2 < … < mn is considered ( v is the commensurate order). For this special case of multi-order fractional systems, the controllability and observability matrices were extracted [32].

k1 k2 kn Tk1k2 … k n = T10 … 00T010 … 00 … T00 … 01.

(16)

kn kn −1 k1 Tk1k2 … k n = T00 … 01T00 … 10 … T10 … 00.

(17)

Proof. It is enough to verify that relation (16) satisfies the recursive relation (8). Substituting (16) in (8) gives k1 k2 kn k1− 1 k2 kn T10 … 00T010 … 00 … T00 … 01 = T10 … 0T10 … 00T010 … 00 … T00 … 01 + … k1 k2 kn − 1 + T0 … 01T10 … 00T010 … 00 … T00 … 01.

(18)

According to (14), all the terms in the right side of (18) are zero except the first one. Thus, relation (18) is an evident relation. This completes the proof. Lemma 4. Relations (19) and (20) are true for pseudo upper and lower triangular multi-order fractional systems, respectively.

⎤ ⎡ 0⎛ p − 1 ⎞ ⎥ ⎢ ⎜ ∑ ni⎟⎟ × N ⎜ ⎥ ⎢ ⎝ i =1 ⎠ ⎥ ⎢ kp kp − 1 kp − 1 ⎢ 0 p − 1 A pp A pp A p(p + 1) … A pp A pn ⎥ kp T00 … 1 … 00 = ⎢ np × ∑ ni ⎥. ↓ ⎥ ⎢ i =1 p ⎥ ⎢ 0 ⎛ ⎞ n ⎥ ⎢ ⎜ ⎟ × n N ∑ i ⎜⎜ ⎟⎟ ⎥ ⎢ ⎝ i =p +1 ⎠ ⎦ ⎣

⎤ ⎡ 0⎛ n − p ⎞ ⎜ ∑ n ⎟ ×N ⎥ ⎢ i⎟ ⎜ ⎥ ⎢ ⎝ i =1 ⎠ ⎥ ⎢ k −1 kp − 1 kp p ⎥ ⎢ A (n − p + 1)(n − p + 1) A (n − p + 1)1 A (n − p + 1)(n − p + 1) A (n − p + 1)2 … A (n − p + 1)(n − p + 1) 0 n kp T00 … 1 … 00 = ⎢ nn − p + 1× ∑ ni ⎥ ↓ i =n −p +2 ⎥ ⎢ n −p +1 ⎥ ⎢ 0 ⎛ ⎞ n ⎥ ⎢ ⎜ ⎟ n i⎟ × N ∑ ⎜ ⎥ ⎢ ⎜ ⎟ ⎝ i =n −p +2 ⎠ ⎦ ⎣

(19)

(20)

Please cite this article as: Hassanzadeh I, Tabatabaei M. Calculation of controllability and observability matrices for special case of continuous-time multi-order fractional systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.006i

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where the index 1 is placed in the p-th location and n − p + 1-th location for upper and lower triangular forms, respectively. Proof. The mathematical induction principle is employed for the Proof It is obvious that relation (19) is true for k p = 1. Now, consider that this relation is true for k p . This relation should be investigated for k p + 1. Thus, we have k +1

k

T 00p… 1 … 00 = T00 … 1 … 00T 00p… 1 … 00 ↓ p

↓ p

⎤ ⎡ 0⎛ p − 1 ⎞ ⎥ ⎢ ⎜ ⎟ ⎜⎜ ∑ n i⎟⎟ × N ⎥ ⎢ ⎝ i =1 ⎠ ⎥ ⎢ ⎢0 A pp A p( p + 1) … A pn ⎥ p −1 ⎥ = ⎢ np × ∑ n i ⎥ ⎢ i =1 ⎥ ⎢ 0⎛ n ⎞ ⎥ ⎢ ⎟ ⎜ ⎥ ⎢ ⎜ ∑ n i⎟ × N ⎟ ⎜ ⎥⎦ ⎢⎣ ⎝ i =p +1 ⎠

⎡ ⎢ ⎢ ⎢ ⎢ 0 p −1 =⎢⎢ np × ∑ n i i =1 ⎢ ⎢ ⎢ ⎢ ⎣

Example 1. Consider a fractional order system with the following transfer function

H (s ) =

↓ p

⎡ ⎢ ⎢ ⎢ ⎢ 0 p −1 × ⎢⎢ np × ∑ n i i =1 ⎢ ⎢ ⎢ ⎢ ⎣

described with pseudo upper or lower triangular multi-order fractional systems. Obtaining a pseudo upper or lower multi-order realization for transfer functions in form (22) is a challenge and is not addressed in this paper. See the following example.

⎝ i =1

⎠

k −1 A ppp A p(p + 1)

…

k −1 A ppp A pn

0⎛

⎞ n ⎜ ⎟ ⎜ ∑ n i⎟ × N ⎜ ⎟ ⎝ i =p +1 ⎠

0⎛ p − 1 ⎞ ⎜ ⎟ ⎜⎜ ∑ n i⎟⎟ × N ⎝ i =1

k +1

⎠

k

A ppp

A ppp A p(p + 1) 0⎛

(24)

where u(t ) and y(t ) are the input and the output of the system, respectively. It is obvious that a commensurate order realization in form (3) with commensurate order 0.1 and inner dimension 19 could be written for this system. However, we will show that this system could be described with a pseudo upper triangular multiorder realization with inner dimension 5. Consider the following state variables

0⎛ p − 1 ⎞ ⎜ ⎟ ⎜⎜ ∑ n i⎟⎟ × N k A ppp

Y (s ) 1 = 1.9 U (s ) s − s1.6 − s1.3 + s + s 0.9 − s 0.6 − s 0.3 + 1

…

k

A ppp A pn

⎞ ⎜ ⎟ ⎜ ∑ n i⎟ × N ⎜ ⎟ ⎝ i =p +1 ⎠ n

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

z1 = y , z2 = D0.3y , z3 = D0.6y , z4 = D1.4 y − D1.1y − D0.8y + D0.5y , z5 = D0.3y + D0.6y − D0.9y − y , x1 = [z1 z2 z3]T , T x2 = [z4 z5]T , x̲ = ⎡⎣ x1 x2 ⎤⎦ .

(25)

According to (24) and (25), the following pseudo upper triangular multi-order realization is obtained

⎧ ⎡0 ⎪ ⎢ ⎤ ⎡ ⎪ D0.3 x̲ ⎢0 1⎥ ⎢ ⎪ = ⎢ −1 0.5 ⎥ ⎢ ⎨ ⎣ D x̲ 2 ⎦ ⎢ 0 ⎪ ⎢ ⎣0 ⎪ ⎪ ⎪ ⎩ y = ⎣⎡ 1 0 0 0

1 0 1 0 0

0 1 1 0 0

0 0 0 0 −1

⎡ 0⎤ 0 ⎤ ⎥ ⎢ ⎥ 0 ⎥ ⎢ 0⎥ −1⎥ x̲ + ⎢ 0⎥u ⎢ 1⎥ 1 ⎥ ⎥ ⎢ ⎥ ⎣ 0⎦ ⎦ 0

0⎤⎦ x̲ .

(26)

(21)

This means that relation (19) is true for k p + 1, too. Thus, proof is completed.

4. Controllability and observability analysis of pseudo upper or lower triangular multi-order fractional systems

Theorem 3. The transfer function of a pseudo upper or lower triangular multi-order fractional system could be described with.

In this section, the controllability and observability matrices for pseudo upper or lower triangular multi-order fractional systems have been extracted. It is shown that the non-singularity of these matrices is the necessary and sufficient condition for controllability and observability of the mentioned systems. Some Lemmas and Theorems are given to prove this claim.

n

H (s ) = where

n

s j1 α1+… jn αn 1 1 j2 … jn , nj n ∏ j = 1 ∑i = 0 q( j, i)s iαj

∑ j n = 0 … ∑ j 1= 0 p j n

pj

1j2 … jn

n

j1 + … + jn <

∑ nk k=1

(22)

, ( ji = 1, … , ni ) , (i = 1, … , n) and q( j, i ) , j = 1, … ,

n, i = 0, … , nj are real numbers. Proof. According to (5), the transfer function of a pseudo upper triangular multi-order system could be written as.

H (s ) =

⎡ s α1I − A −A12 11 ⎢ 0 s α2I − A22 ⎢ Cadj⎢ ⋮ ⋮ ⎢ 0 0 ⎣ n

⎤⎡ B ⎤ ⎥⎢ 1⎥ … −A2n ⎥⎢ B2 ⎥ ⎥⎢ ⋮ ⎥ ⋮ ⋮ ⎥⎢ ⎥ αn … s I − A nn ⎦⎣ Bn ⎦ …

∏i = 1 det(s αiI − Aii )

−A1n

4.1. Calculation of the observability matrix Remark 2. In [32], it is shown that observability of system (4) is independent of the input signal u(t ). Thus, in the remainder of this subsection, we assume u(t ) = 0. Definition 3. The observability matrix for a pseudo upper (or lower) triangular multi-order fractional system ( MO ) is a N n × N matrix defined as.

(23)

where adj(M ) is the adjugate (adjoint) matrix of M . It is obvious that the denominator of (23) could be rewritten as the denominator of (22). Moreover, according to the adjoint matrix definition, the numerator of (23) is a pseudo polynomial in which its degree n is smaller than ∑k = 1 nkαk . Thus, it could be written as the numerator of (22). It means that the transfer function (23) could be written as (22). This completes the proof. Theorem 3. gives a special form for transfer functions that could be

T MO = ⎡⎣ m0 … 0 … mk1… k n … m(N − 1)…(N − 1) ⎤⎦

(27)

where mk1… kn could be obtained for pseudo upper or lower triangular forms according to relations (28) and (29), respectively k1 k2 kn mk1… k n = CT10 … 00T01 … 00 … T00 … 01,

0 ≤ k1, …, k n ≤ N − 1

(28)

kn kn −1 k1 mk1… k n = CT00 … 01T00 … 10 … T10 … 00,

0 ≤ k1, …, k n ≤ N − 1

(29)

For example, the observability matrix for a two-order pseudo

Please cite this article as: Hassanzadeh I, Tabatabaei M. Calculation of controllability and observability matrices for special case of continuous-time multi-order fractional systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.006i

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upper triangular fractional system with inner dimension 3 is a 9 × 3 matrix that could be written as T 2 2 2 2 2 2 ⎤ . MO = ⎡⎣ C CT10 CT10 CT01 CT01 CT10T01 CT10T01 CT10 T01 CT10 T01⎦

(30)

Lemma 5. Consider constant vector z ∈ Rm and constant scalar value r and constant matrix Am × k ( m > k ). If the following equation has only one solution named x̲ ∈ Rk then matrix A is full rank. T

(31)

z̲ Ax̲ = r.

5

λN − 1, … , N − 1(t )⎤⎦ . On the other hand, the system is observable. Thus, according to Definition 2, there exists t1 > 0 such that the initial state x̲ 0 is uniquely determined from relation

(∫

t1 T

0

)

t

1 h̲ (t )dt MOx̲ 0 = ∫ y(t )dt . 0

Now, according to Lemma 5, the observability matrix MO is full rank and the proof is completed. Remark 3. For a general multi-order fractional system, matrix Tk1, … , kn, (k1, … , k n ≥ N ) could not necessarily be interpreted in terms of Ti1, … , in, (0 ≤ i1, … , in ≤ N − 1). This is the reason why the

Proof. The proof is based on reduction to the absurd. Assume that matrix A is singular. Thus, there exists a non-zero vector w ≠ 0 ( w ∈ Rn) fulfilling the following relation.

(33)

non-singularity of the controllability and observability matrices is only the sufficient condition for controllability and observability of this system. However, for pseudo upper or lower triangular multiorder fractional systems, relations (16) and (17) could be employed to obtain the necessary conditions for the observability and controllability of these systems in terms of their observability and controllability matrices.

This means that w̲ + x̲ is also a solution of (31).This is in contradiction with the solution uniqueness of (31). This completes the proof.

Lemma 6. The successive fractional order derivative of G(t ) for multi-order fractional system (4) under Assumption 1 in the time t = 0+ could be obtained as [32]

Theorem 4. The necessary condition for observability of a pseudo upper or lower triangular multi-order fractional system is that its observability matrix is full rank.

Dtv … Dtv Dtα1G(0+) = C ∑ To1(i, r )… on(i, r )   

Proof. According to Remark 2, u(t ) = 0 is considered. Incorporating relations (16), (6) and (4) gives the following relation for the upper triangular form.

where oj (i, r ) , j = 1, …, n, r = 1, …, β (i ) are all the possible integer solutions of the following equations ( β(i ) is the number of solutions for derivative number i ).

⎛ ∞ ∞ ∞ ⎜ t k nα n k1 k2 kn y (t ) = C ⎜ ∑ T10 t k1α1 ∑ T 01 t k 2α 2 … ∑ T 00 … 01 … 00 … 00 ⎜ k Γ ∑nj = 1 k jα j + 1 k 2= 0 kn = 0 ⎝ 1= 0

Λ(i):

(32)

Aw̲ = 0. According to (31) and (32), we have T

z̲ A(w̲ + x̲ ) = z̲ T Ax̲ = r.̲

β (i ) r=1

i

(40)

n

(

)

⎞ ⎟ ⎟ x̲ 0 . ⎟ ⎠

⎛ ⎜ …⎜ ⎜ ⎝

N −1

∑ k1= 0

∞ k1 T10 t k1α1 + …0

N −1

∑ kn = 0

n T 0k… 01

∑ k1= N

(

N −1

∑ k 2= 0

∞ k2 T 01 t k2α 2+ …0

∞

)

Γ ∑nj = 1 k jα j + 1

+

∑ kn = N

n T 0k… 01

∑ k 2= N

⎞ k2 k 2α 2⎟ T 01 t ⎟ …0 ⎟ ⎠

⎞ ⎟ ⎟ x̲ 0 . Γ ∑nj = 1 k jα j + 1 ⎟⎠ t knα n

(

)

(35)

On the other hand, the Cayley-Hamilton Theorem gives N

k

T00p… 1 … 00 = ↓ p

∑ θ kp, p, jT00j −…1 1 … 00,

kp ≥ N,

p = 1, …, n

↓ p

j=1

(36)

where θ kp, p, j, p = 1, … , n, j = 1, … , N , k p ≥ N are real numbers. The following form could be obtained by substituting (36) in (35) ⎛ y(t ) = C ⎜⎜ ⎝

⎞⎛ k1 ⎟⎜ ∑ β1, k1(t )T10 … 0⎟⎜ k1= 0 ⎠⎝ N −1

⎞ ⎛ k2 ⎟ ⎜ ∑ β 2, k2(t )T01 … 0⎟…⎜ k 2= 0 ⎠ ⎝ N −1

⎞ ⎟ ̲ n ∑ βn, kn(t )T0k… 01⎟ x 0 kn = 0 ⎠

y(t ) =

∑ k1= 0

∑

Proof. The necessary condition is already proved in Theorem 4. Thus, it is enough to prove the sufficiency. Assume that the observability matrix is full rank. Now, we should demonstrate that the system is observable. The proof is based on the reduction to the absurd. Assume that the system is not observable. This means that the observability gramian (WO(t1)) should be singular for at least a positive time t1. Thus, there exists a non-zero vector V ≠ 0 such that

V T WO(t1)V = 0.

(42)

Eqs. (11) and (42) give

V T GT (t )G(t )Vdt =

∫0

t1

‖G(t )V ‖2 dt =0.

(43)

According to (43), we have

0 < t ≤ t1.

(44)

(38)

T

h̲ (t )MO x̲ 0 = y(t ) MO is the observability matrix and

∫0

t1

G(t )V = 0,

k1 k2 kn ̲ λ k1, … , k n(t )CT10 … 0T01 … 0 … T0 … 01x0

kn = 0

where λk1, … , kn(t ) = β1, k (t )β2, k (t )… βn, k (t ). Now, relation (38) could 1 2 n be rewritten as

where

Theorem 5. A pseudo upper or lower triangular multi-order fractional system under Assumption 1 is observable if and only if its observability matrix given in (27) is non-singular.

(37)

N−1

…

i

the Dtα1+ ivG(0+).

N −1

where the parameters βj, i(t ) , j = 1, … , n, i = 0, … , N − 1 are time-varying functions. Eq. (37) could be rewritten as N−1

(41)

(34) Remark 4. The relation Dtv … Dtv Dtα1G(0+) isn’t necessarily equal to   

⎞⎛ k1 k1α1⎟⎜ T10 t ⎟⎜ …0 ⎟⎜ ⎠⎝

t knα n

r = 1, …, β(i).

j=1

Eq. (34) could be rewritten as ⎛ ⎜ y (t ) = C ⎜ ⎜ ⎝

∑ oj(i, r )mj = i + m1,

(39) h̲ (t ) = ⎡⎣ λ 0, … ,0(t )… T

Relation. (44) is true for lim+G(t ) = C )

t = 0+. Thus, we have (note that

t→0

CV = 0.

(45)

Successive fractional order differentiation of (44) yields

Please cite this article as: Hassanzadeh I, Tabatabaei M. Calculation of controllability and observability matrices for special case of continuous-time multi-order fractional systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.006i

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6

Dtv … Dtv Dtα1G(0+)V = 0,   

0 < t ≤ t1 (46)

i

where i is any non-negative integer number. Now, relations (40), (16) and (46) yield

⎛ ⎜ ⎜ ⎝

β (i )

⎞

r=1

⎠

∑ CT10o1(⋯i, r0) … T0o…n(i01,r )⎟⎟V = 0.

(47)

Consider that in (47) and (41), at least one of the oj (i, r ) , j = 1, … , n should be non-zero. The first row of the observability matrix is appeared in relation (45). Now, we can select the derivative number i so that the other rows of the observability matrix MO will be appeared as the summation terms in (47). It is n obvious that if we select i = ∑ j = 1 mj kj − m1 where k1, … , k n are the indexes of mk1… kn in (27), then the corresponding row appears in (47). Note that all of k i, i = 1, … , n could not be zero, simultan neously. Thus, the relation ∑ j = 1 mj kj − m1 is non-negative. This means that all of the rows of the observability matrix will be appeared as the summation terms in (47) with at most N n − 1 appropriate differentiations. On the other hand, each of the observability matrix rows could be appeared in (47) only in one differentiation stage. This could be proved easily. Because if we consider that the corresponding row with indices k1, … , k n is appeared in both derivative numbers i and i′, then the following relations should be satisfied n

n

∑ mjkj = m1 + i,

∑ mjkj = m1 + i′.

j=1

j=1

(48)

Thus, i = i′. For example, CT10 … 0 is appeared in (47) only for i = 0. Consider that some of the observability rows could be appeared simultaneously in (47) in one differentiation stage. For example, consider m1 = 2, m2 = 4, N = 3, i = 2. Then, Eq. (47) be2 + CT01)V = 0. This means that two rows of the obcomes as (CT10 2 servability matrix ( CT10 , CT01) are appeared in (47) for i = 2, simultaneously. Except the observability matrix rows, some other terms with powers greater than N − 1 could be appeared in (47). For example, for the case m1 = 2, m2 = 4, N = 3, i = 4 , Eq. (47) becomes as 3 3 (CT10 + CT10T01)V = 0. The power in term CT10 is greater than 2. However, according to the Cayley-Hamilton Theorem (see (36)), these terms could be represented in terms of the observability 3 CT10 could be written as matrix rows. For example 3 2 CT10 = θ3,1,1C + θ3,1,2CT10 + θ3,1,3CT10 . This means that, this term is a 2 linear combination of C , CT10, CT10 . Finally, the following results are valid.

a) All the observability matrix rows are appeared by at most N n − 1 appropriate differentiation stages in relation (47). b) Each of the observability matrix rows only appears once in (47). c) According to the Cayley-Hamilton Theorem, all the terms in (47) could be described in terms of the observability matrix rows. Now, by appropriate choosing of the derivative number i , in N n − 1 differentiation stages, the following relation could be obtained

M′O V = 0

(49)

where matrix M′O has the same rank as the observability matrix MO (the first row of the M′O is C ). In other words, the matrix MO could be converted to M′O by elementary row operations. According to (49), the matrix M′O is singular. This means that the observability matrix MO should be singular which contradicts the

assumption. Thus, system (4) under Assumption 1 should be observable. This completes the proof. Corollary 1. Unlike the observability matrix presented in [32], the observability matrix proposed in this paper is independent of the fractional orders. This independence could be considered as a great advantage for further analysis. The same property will be extracted for the controllability matrix. 4.2. Calculation of the controllability matrix

Definition 4. The controllability matrix for a pseudo upper (or lower) triangular multi-order fractional system ( MC ) is a N × nN n matrix given by

MC = ⎡⎣ c1(0, …, 0)… c1(N − 1, …, N − 1)… cn(0, …, 0)… cn(N − 1, …, N − 1)⎤⎦ (50) where cp(k1, … , k n) , 0 ≤ k1, … , k n ≤ N − 1, p = 1, … , n for pseudo upper or lower triangular forms are computed in accordance with relations (51) and (52), respectively k1 k2 kn cp(k1, …, k n) = T10 … 00T01 … 00 … T00 … 01Bp0.

(51)

kn kn −1 k1 cp(k1, …, k n) = T00 … 01T00 … 10 … T10 … 00Bp0.

(52)

Theorem 6. If a pseudo upper or lower triangular multi-order fractional system is controllable, then its controllability matrix is full rank. Proof. The proof is presented for the pseudo upper triangular case. According to the controllability definition, the state vector should be transformed from any initial value to any final value by the control signal in a finite time. Thus, the initial state x̲ (0) = 0 should be transformed to x(t f ) = x̲ f in a finite time t f for any arbitrary value for x̲ f . According to (6), (7) and (16), we have ⎛

x̲ f =

⎞ ⎛ ∞ ∞ ⎟ ⎜ kn ⎜ ∑ T k1 ⎟ ⎜ … T B ∑ p 0 10 … 00 00 … 01 ⎜ ⎟ ⎜ p = 1 ⎜ k1= 0 kn = 0 ⎠ ⎝ ⎝ n ⎜

∑

t

∫0 f

( ∑nj =1, k jαj)+αp−1 ⎞⎟ u(t )dt ⎟. ⎟ Γ (( ∑nj = 1 k jα j ) + α p ) ⎟ ⎠

(t f − t )

(53)

Considering relation (36) and the proof presented in Theorem 4, we have

MC × f̲ u (t f ) = x̲ f

(54)

where f̲ u (t f ) is a nN n × 1 time-varying vector depending on the control signal u and final time t f . This vector is given by

f̲ u (t f ) = ⎡⎣ γ1(0, …, 0)… γ1(N − 1, …, N − 1)… γn(0, …, 0)… T γn(N − 1, …, N − 1)⎤⎦

(55)

where γp(k1, … , k n) , 0 ≤ k1, … , k n ≤ N − 1, p = 1, … , n are unknown real numbers depending on the control signal u and final time t f . Eq. (54) should be solvable for any arbitrary final state x̲ f . Therefore, the column space of the controllability matrix MC should span R N . This means that the rank of MC should be equal to N or this matrix should be full rank. This completes the proof. Lemma 7. The sequential fractional order derivative of R(t ) = tF (t ) for multi-order fractional system (4) under Assumption 1 in the time t = 0+ is given by [32]

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Dtv … Dtv Dtα1R(0+) = v(i + m1)    i

λ (i , p )

∑ ∑

Tl1(i, p, r )… ln(i, p, r )Bp0

p=1 r=1

(56)

where l j(i, p, r ) , j, p = 1, … , n, r = 1, … , λ(i, p) are all the possible integer solutions of the following equations and λ(i, p) is the number of solutions for the derivative number i and the state vector number p. n

Ω( i , p ) :

∑ l j(i, p, r )mj = i − mp + m1,

p = 1, …, n, r = 1, …, λ(i, p)

j =1

(57)

Theorem 7. The necessary and sufficient condition for controllability of a pseudo upper or lower triangular multi-order fractional system under Assumption 1 is the non-singularity of its controllability matrix. Proof. The necessary condition already was proved in Theorem 6. Consider that the controllability matrix for a pseudo upper-triangular multi-order system is full rank. Now, the controllability of this system should be proved. The reduction to the absurd is employed for the proof. Assume that the system is not controllable. This means that there exists t1 > 0 such that the controllability gramian in this time (WC (t1)) is singular. This means that there exists a non-zero vector V ≠ 0 that

V T WC (t1)V = 0.

t1

V T F (τ )FT (τ )Vdτ =

Remark 5. Another common definition for the fractional order derivative is the Riemann–Liouville definition [1]. The fractional order derivative of a constant is zero according to the Caputo definition but it is not zero according to the Riemann–Liouville definition [1]. Thus, if the Riemann–Liouville definition is employed instead of the Caputo definition, Theorems 4 and 6 are still valid. However, Theorems 5 and 7 are not true for this case. 4.3. Simplified forms of the controllability and observability matrices Although for controllability or observability analysis of a pseudo upper or lower triangular multi-order fractional system, the linear independence of only N vectors should be investigated, the dimension of these vectors may be very large ( N n for the observability matrix or nN n for the controllability matrix). To reduce this complexity, it will be proved that the observability and controllability matrices could be converted to simplified matrices with reduced number of rows and columns, respectively. To attain this goal, some Lemmas and Theorems are presented in this subsection.

(59)

Lemma 8. The relations (28) and (29) given for the observability matrix rows for pseudo upper or lower multi-order fractional systems could be simplified as relations (65) and (66), respectively (consider that at least one of the indices k i, i = 1, … , n is non-zero or the row C is excluded)

∫0

t1

‖V T F (τ )‖2 dτ =0.

(60)

⎡ ⎤ k k d −1 0 0 Cd1ξr A d ddr C ξ A k dr − 1A … ⎥ r m k1… kn = ⎢  r r d1 r drdr drdr + 1 … Cd1ξr A drdt A drn . ⎢⎣ dr − 1 ⎥⎦

(65)

⎤ ⎡ k d −1 kd k d −1 1 1 0…0 1  ⎥ mk1… kn = ⎢ C drΨrA d1d1 A d1(1) … C drΨrA d1d1 A d1(d1− 1) C drΨrA d1d1  ⎥ ⎢⎣ n − d1 ⎦

(66)

Eq. (59) gives

V T F (t ) = 0,

these matrices have similar ranks. On the other hand, according to (64), M′C should be singular. Thus, the controllability matrix MC should be singular which is in contradiction with the assumption. Hence, the multi-order system (4) under Assumption 1 is controllable and the proof is completed.

(58)

Incorporating relations (10) and (58) gives

∫0

7

0 < t ≤ t1.

Multiplying both sides of (60) by t > 0 gives T

V R(t ) = 0,

0 < t ≤ t1.

(61)

The following relation could be obtained by consecutive fractional order differentiation of both sides of (61)

V T Dtv … Dtv Dtα1R(t ) = 0,   

0 < t ≤ t1, i = 1, …, N .

i

(62)

Eqs. (56), (16) and (62) give n

VT

where r is the number of non-zero indices, dl , l = 1, … , r , 0 < d1 1 dldl l (l + 1 ) ξr = ⎨ , ⎪ l=1 ⎪ 1, r=1 ⎩

⎧ r −1 −1 k ⎪ ⎪ ∏ A dr − l + 1 A , r>1 dr − l + 1dr − l + 1 dr − l + 1dr − l Ψr = ⎨ . ⎪ l=1 ⎪ r=1 ⎩ 1,

(67)

λ (i , p )

∑ ∑ p=1 r=1

l1(i, p, r ) l2(i, p, r ) l n (i , p , r ) T10 … 00T010 … 00 … T00 … 01 Bp0 = 0.

(63)

If nN n appropriate derivative numbers i are substituted in (63) such that all the controllability matrix columns could be appeared, then, (63) could be converted to the following relation (the proof details are similar to those stated for Theorem 5)

V T M′C = 0

(64)

where the columns of the matrix M′C have the following properties (The details of the proof are similar to the proof given for Theorem 5) a) All columns of the controllability matrix ( MC ) are appeared in (63) by at most nN n appropriate differentiation stages. b) Each of the controllability matrix columns only appears once in (63). c) According to the Cayley-Hamilton Theorem, all the terms in (63) could be written in terms of the controllability matrix columns. This means that matrix M′C is the result of the elementary column operations performed on controllability matrix MC . Therefore,

Proof. It is obvious that relation (28) could be rewritten as k

k

k

⏟

⏟

⏟

d1 d2 dr mk1… k n = CT 0 … 010 … 0T 0 … 010 … 0 … T 0 … 010 … 0. d1− 1

d2− 1

(68)

dr − 1

Thus, it is enough to prove that relation (68) is equal to relation (65). We prove it by the mathematical induction principle. For the case r = 1, according to (19), we have mk1… kn = ⎡⎣ C1 … C d1 … C n ⎤⎦ ⎡ ⎢ ⎢ ⎢ ⎢ d −1 ⎢ 0 × ⎢ n × 1∑ n i d1 ⎢ i =1 ⎢ ⎢ ⎢ ⎢⎣

0⎛ d1− 1 ⎞ ⎜ ⎟ ⎜⎜ ∑ n i⎟⎟ × N ⎝ i =1

k A d dd1 11

⎡ k 0 0 C d1A d dd1 … =⎢  11 ⎢ d −1 ⎣ 1

⎠

k −1 A d dd1 A d1(d1+ 1) 11

…

0⎛

⎞ n ⎜ ⎟ ⎜ ∑ n i⎟ × N ⎜ ⎟ ⎝ i = d1+ 1 ⎠

k

−1

C d1A d dd1 A d1(d1+ 1) 11

…

k

−1

A d dd1 A d1n 11

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

⎤ k −1 C d1A d dd1 A d1n ⎥ . ⎥ 11 ⎦ (69)

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It is obvious that (69) is equal to (65) for r = 1 (consider that in this case ξ1 = 1). Now, consider that (65) is true for r non-zero indices. We should prove it for r + 1 non-zero indices. Or k

k

k

  

  

  

kd r +1 … 0 0 10 … 0 

d1− 1

d2 − 1

dr − 1

dr + 1− 1

CT 0 d…1 0 10 … 0T 0 d…20 10 … 0 … T 0 d…r 0 10 … 0T

Corollary 2. A pseudo upper or lower triangular multi-order fractional system is observable if and only if its simplified observability matrix ( MOS ) is full rank.

⎡ ⎤ k kd −1 0 0 Cd1ξrA d ddr C A k dr − 1A … ⎥ r =⎢  r r d1ξr drdr drdr + 1 … C d1ξrA drdr A drn ⎢ d −1 ⎥ ⎣ r ⎦

Lemma 9. The relations (51) and (52) presented for the controllability matrix columns for pseudo upper or lower triangular multiorder fractional systems could be converted to relations (76) and (77), respectively ⎧ 0, p < dr ⎪ ⎤ ⎪⎡ 0(d1− 1)× 1 ⎥ ⎢ ⎪ ⎪⎢ k −1 dr ≤ p ≤ n cp(k1, …, k n) = ⎨ ξrA d ddr A d p Bp ⎥, r ⎥ r r ⎪⎢ ⎥ ⎢ ⎪⎣ 0(n − d1)× 1 ⎦ ⎪ ⎪ BP 0, dr = 0. ⎩ (76)

⎡ ⎤ 0⎛ dr ⎞ ⎢ ⎥ ⎜ ⎟ n i⎟ × N ∑ ⎜ ⎢ ⎥ ⎜ ⎟ ⎝ i =1 ⎠ ⎢ ⎥ ⎢ ⎥ kd kd kd −1 −1 r +1 A r +1 A r +1 0 A A A … d r dr + 1dr + 1 dr + 1(dr + 1+ 1) dr + 1dr + 1 dr + 1n ⎥ dr + 1dr + 1 ×⎢⎢ n × n ∑ ⎥ dr + 1 i i =1 ⎢ ⎥ ⎢ ⎥ 0⎛ n ⎞ ⎢ ⎥ ⎟ ⎜ ⎜ ∑ n i⎟ × N ⎢ ⎥ ⎟ ⎜ i d = + 2 ⎠ ⎝ ⎣ ⎦ r ⎡ ⎤ k −1 −1 kd kd 0 0 Cd1ξr + 1A d dr +d1 … ⎢ r +1 ⎥ C A … ξ r + 1 r + 1 C d ξr + 1A d r +d1 d + r d d 1 1 1 r +1 r +1 r + 1 r + 1⎥ . =⎢ d − 1 ⎢ r +1 ⎥ A dr + 1dr + 2 A dr + 1n ⎣ ⎦

(70)

According to (70), relation (65) is true for r + 1 and the proof is completed. Definition 5. The simplified observability matrix for a multi-order n fractional system (MOS ) is a ∏i = 1 (ni + 1) × N matrix which its rows for pseudo upper and lower triangular forms are defined as relations (71) and (72), respectively k1 k2 kn mk1… k n = CT10 … 00T01 … 00 … T00 … 01,

0 ≤ k i ≤ ni .

(71)

kn kn −1 k1 mk1… k n = CT00 … 01T00 … 10 … T10 … 00,

0 ≤ k i ≤ ni .

(72)

Theorem 8. The simplified observability matrix ( MOS ) and the observability matrix (MO ) given in (27) have the same rank. Proof. According to the Cayley-Hamilton Theorem, we have k −1

A d ddl

l l

l

∑

(

)

ρ i, dl , k dl A di ldl ,

k dl ≥ 1

i=0

(73)

where ρ(i, dl , k dl ) are the coefficients obtained from the CayleyHamilton Theorem. Consider that relation (73) is valid not only for k dl ≥ ndl + 1 but also for all k dl ≥ 1. For example for a 3 × 3 matrix

D , we have D = 0 × I + 1 × D + 0 × D2. Inserting (73) in (65) gives ⎡ ⎞ ⎛ nd −1 ⎞ r − 1 ⎛ n dl − 1 ⎢ ⎟ ⎜ r ⎟ ⎜ m k1… k = ⎢  0 … 0 Cd1 ∏ ⎜ ∑ ρ(i , dl , k d )Adi d ⎟Ad d + 1⎜ ∑ ρ(i , dr , k d )Adi d ⎟Ad d n l r l l⎟ l l ⎜ r r⎟ r r ⎜ ⎢ l=1 ⎝ i =0 ⎠ ⎠ ⎝ i =0 ⎣ dr − 1 ⎞ ⎛ nd −1 ⎞ r − 1 ⎛ n dl − 1 ⎟ ⎜ r ⎟ ⎜ Cd1 ∏ ⎜ ∑ ρ(i , dl , k d )Adi d ⎟Ad d + 1⎜ ∑ ρ(i , dr , k d )Adi d ⎟Ad d + 1 l r l l⎟ l l ⎜ r r⎟ r r ⎜ l=1 ⎝ i =0 ⎠ ⎠ ⎝ i =0 ⎤ ⎞ ⎛ nd −1 ⎞ r − 1 ⎛ n dl − 1 ⎟ ⎥ ⎜ r ⎟ ⎜ … Cd1 ∏ ⎜ ∑ ρ(i , dl , k d )Adi d ⎟Ad d + 1⎜ ∑ ρ(i , dr , k d )Adi d ⎟Ad n ⎥ . l r l l⎟ l l ⎜ r r⎟ r ⎥ ⎜ l=1 ⎝ i =0 ⎠ ⎦ ⎠ ⎝ i =0

(74)

If at least one of the k i, i = 1, … , n is greater than ni , relation (74) could be written as the weighted summation of the rows given in (65) in which all k dl ≤ ndl , l = 1, … , r , Or n1

mk1… k n =

n2

∑∑ i1= 0 i2 = 0

nn

…

∑ in = 0

σ (i1, …, in)mi1… in

⎧ 0, p > d1 ⎪ ⎪ ⎡ 0(d − 1)× 1 ⎤ r ⎥ ⎢ ⎪ ⎪ k −1 1 ≤ p ≤ d1 cp(k1, …, k n) = ⎨ ⎢ ΨrA d dd1 A d p Bp ⎥, 1 ⎥ 1 1 ⎪⎢ ⎥ ⎢ ⎪ ⎣ 0(n − d )× 1 ⎦ r ⎪ ⎪ BP 0, dr = 0 ⎩

(75)

(77)

where r is the number of non-zero indices, dl , l = 1, …, r , 0 < d1< d2 < … < dr ≤ n are indices that k dl ≠ 0 and ξr , Ψr are defined in (67). Proof. We should prove that (51) is equal to (76). Relation (51) could be rewritten as k

k

k

⏟

⏟

⏟

d1 d2 dr cp(k1, …, k n) = T 0 … 010 … 0T 0 … 010 … 0 … T 0 … 010 … 0Bp0. d1− 1

nd − 1

=

where σ(i1, … , in) are real coefficients. This means that the rank of matrix MOS is equal to the rank of MO and the proof is completed for pseudo upper triangular case.

d2− 1

(78)

dr − 1

It is obvious that if dr = 0, then relation (78) is converted to Bp0 . Now, consider that p < dr . In this case, in accordance with (19), we have ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ 0 dr − 1 k T 0 d…r 0 10 … 0B p0 = ⎢ n × ∑ n    dr i ⎢ i =1 ⎢ dr − 1 ⎢ ⎢ ⎢ ⎢⎣ ⎡ 0⎤ ⎢ ⋮⎥ ⎢ ⎥ ⎢ Bp ⎥ ⎢ ⎥ ⋮ ×⎢ ⎥=0. ⎢ 0⎥ ⎢ ↓ ⎥ ⎢ dr ⎥ ⎢ ⋮⎥ ⎢ ⎥ ⎣ 0⎦

0⎛ dr − 1 ⎞

⎜ ⎟ ⎜⎜ ∑ n i⎟⎟ × N ⎝ i =1 ⎠

k

A d ddr

r r

k

A d ddr

−1

r r

0⎛

A dr (dr + 1)

⎞ n ⎜ ⎟ ⎜ ∑ n i⎟ × N ⎜ ⎟ ⎝ i = dr + 1 ⎠

…

⎤ ⎥ ⎥ ⎥ ⎥ k d −1 A d dr A drn ⎥ r r ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

(79)

This means that relation (78) is zero for p < dr . Now, it is enough to prove that (78) is equal to (76) for dr ≤ p ≤ n. This proof is based on the mathematical induction principle. For the case r = 1, according to (78), we have

Please cite this article as: Hassanzadeh I, Tabatabaei M. Calculation of controllability and observability matrices for special case of continuous-time multi-order fractional systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.006i

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k

controllability matrix (MC ) defined in (50) have similar ranks.

⏟

Proof. According to (76) and (77), the columns of the controllability matrix defined in (51) and (52) for pseudo upper and lower triangular multi-order fractional systems are zero for p < dr and p > d1, respectively. On the other hand, in accordance with the Cayley-Hamilton Theorem, there is no need to employ k i > ni in these relations (The details of the proof are similar to those stated in Theorem 8 and are omitted for brevity). This completes the proof.

d1 cp(k1, …, k n) = T 0 … 010 … 0Bp0 d1− 1

⎤ ⎡ 0⎛ d1− 1 ⎞ ⎜ ⎥ ⎢ ∑ ni⎟⎟ × N ⎜ ⎥ ⎢ ⎝ i =1 ⎠ ⎥ ⎢ k k k − − 1 1 d d d ⎢0 A d d1 A d d1 A d1(d1+ 1) … A d d1 A d1n ⎥ d1 − 1 1 1 1 1 1 1 ⎥ =⎢ nd × ∑ ni 1 ⎥ ⎢ i =1 ⎥ ⎢ 0⎛ n ⎞ ⎥ ⎢ ⎜ ⎟ ⎥ ⎢ ⎜⎜ ∑ ni⎟⎟ × N ⎝ i = d1+ 1 ⎠ ⎦ ⎣

Corollary 3. The necessary and sufficient condition for controllability of a pseudo upper or lower triangular multi-order fractional system is that its simplified controllability matrix ( MCS ) is nonsingular.

⎡0 ⎤ ⎢⋮ ⎥ ⎢ ⎥ ⎤ ⎢ 0 ⎥ ⎡ 0(d1− 1)× 1 ⎥ ⎢↓ ⎥ ⎢ d1 − k 1 × ⎢ ⎥ = ⎢ A d dd1 A d p Bp ⎥. 1 ⎥ ⎢⋮ ⎥ ⎢ 1 1 ⎥ ⎢ B ⎥ ⎢⎣ 0(n − d )× 1 ⎦ p 1 ⎢ ⎥ ⎢⋮ ⎥ ⎢⎣ 0 ⎥⎦

5. Illustrative examples

(80)

Thus, (76) is true for r = 1( ξ1 = 1). Now, it should be proved that if (76) is true for r non-zero indices, then it is also true for r + 1 non-zero indices (dr + 1 ≤ p ≤ n). Or k

k

k

  

  

  

kd r +1 … 0 0 10 … 0 

d1− 1

d2 − 1

dr − 1

dr + 1− 1

T 0 d…1 0 10 … 0T 0 d…20 10 … 0 … T 0 d…r 0 10 … 0T

k

B p0 = T 0 d…1 0 10 … 0   

d1− 1

and v2. The inductors current are denoted by i1, i2 and i3. The current of the resistor R4 is denoted by i′. Consider R1 = R2 = R3 = R4 = R5 = R6 = 1Ω , C1 = C2 = 1F , L1 = L2 = L3 = 1H and

⎡ ⎤ 0(d2− 1)× 1 ⎢ ⎥ ⎢ r ⎥ − 1 k kd −1 dr + 1 l ⎢ × ∏ A d d A dld(l + 1) A d d A dr + 1p B p ⎥ r +1 r +1 ⎢ l=2 l l ⎥ ⎢ ⎥ 0(n − d2)× 1 ⎢⎣ ⎥⎦

∏ l=2

select

⎤ ⎥ ⎥

0(d2− 1)× 1 r

⎡ ⎤ 0(d2− 1)× 1 ⎢ ⎥ ⎢ ⎥ −1 kd + 1 r = ⎢ ξr + 1A A B . dr + 1dr + 1 dr + 1p p ⎥ ⎥ ⎥ ⎢⎢ 0(n − d2)× 1 ⎥⎦ ⎥⎦ ⎣ (81)

−1 k k −1 A d ddl A dld(l + 1) A d dr +d1 A dr + 1p B p ⎥ l l r +1 r +1 ⎥

0(n − d2)× 1

Thus, relation (76) is true for r + 1 and the proof is completed. Definition 6. The simplified controllability matrix for a multi-order fractional system (MCS ) could be defined for pseudo upper and lower triangular forms according to relations (82) and (83), respectively ⎡ k1 ⎤ = ⎣⎢ T10 B ... T k1 ...T kn − 1 B (n − 1)0 T k1 ...T kn B n0 ⎦⎥, 0 ≤ k i ≤ n i . … 0 10 10 … 0 0 … 010 10 … 0 0 … 01

MCS

⎡ n ⎤ k1 kn k2 kn MCS = ⎢⎣ T 0k… ⎥, 01...T10 … 0B10 T 0 … 01...T 010 … 0B 20 ... T 0 … 01B n0 ⎦

It could be verified that

N×

n j ∑ j = 1 ∏i = 1 (nn − i + 1

MCS is

In this section, some examples are given to show the effectiveness of the simplified controllability and observability matrices in controllability and observability analysis of multi-order fractional systems. Fractional order capacitors and fractional order inductors could better describe the behavior of real capacitors or inductors [33,34]. Thus, two electrical circuits with fractional order elements as real plants are considered in the following examples. Example 2. Consider the electrical circuit shown in Fig. 1. In this circuit, the inductors L2 and L3 are considered as the fractional order ones with fractional order α1, the capacitors are considered as the fractional capacitors with order α2 and the inductor L1 is a fractional order inductor with fractional order α3. Consider that the fractional orders are selected such that Assumption 1 is fulfilled. In this circuit, the independent voltage source u(t ) and the voltage of the inductor L2 ( y(t )) are considered as the input and output, respectively. The voltage of the capacitors are represented with v1

⎡ ⎤ 0(d2− 1)× 1 ⎢ ⎥ ⎢ r ⎥ −1 kd k −1 × ⎢ ∏ A d ddl A dld(l + 1) A d r +d1 A dr + 1p B p ⎥ l l r +1 r +1 ⎢ = ⎥ ⎢l 2 ⎥ 0(n − d2)× 1 ⎢⎣ ⎥⎦

⎡ ⎤ 0⎛ d1− 1 ⎞ ⎢ ⎥ ⎜ ⎟ ⎜⎜ ∑ n i⎟⎟ × N ⎢ ⎥ ⎝ i =1 ⎠ ⎢ ⎥ ⎢ ⎥ k k −1 k −1 ⎢0 A d dd1 A d dd1 A d1(d1+ 1) … A d dd1 A d1n ⎥ d1 − 1 11 11 11 ⎥ =⎢ n × ∑ n i ⎢ d1 ⎥ i =1 ⎢ ⎥ ⎢ ⎥ 0⎛ n ⎞ ⎢ ⎥ ⎜ ⎟ ⎜ ∑ n i⎟ × N ⎢ ⎥ ⎜ ⎟ ⎢⎣ ⎥⎦ ⎝ i = d1+ 1 ⎠

⎡ ⎢ ⎢ k −1 =⎢ A d dd1 A d1d2 ⎢ 11 ⎢ ⎢⎣

9

N×

0 ≤ k i ≤ ni .

n j ∑ j = 1 ∏i = 1 (ni

(82)

+ 1) for pseudo upper or lower triangular

cases, respectively. Theorem 9. The simplified controllability matrix ( MCS ) and the

state

variables

as

T T x1 = ⎡⎣ i2 i3⎤⎦ , x2 = ⎡⎣ v1 v2 ⎤⎦ ,

T x 3 = i1, x̲ = ⎡⎣ x1 x 2 x 3⎤⎦ . Now, according to the Kirchhoff's circuit laws, the following pseudo upper triangular multi-order state space realization will be obtained

⎤ ⎡ 0.5 ⎤ ⎥ ⎥ ⎢ −1 0 ⎥ 1 0 ⎢0 ⎥ ⎥ 0 −1.5 −0.5 0 x̲ + ⎢ 0.5 ⎥u( t ) ⎥ ⎢ 0.5 ⎥ 0 −0.5 −0.5 0 ⎥ ⎥ ⎢ ⎥ ⎣1 ⎦ 0 0 0 − 1⎦ y = ⎡⎣ 0 −1 0 1 0 ⎤⎦ x̲ .

⎡ −1 ⎡ Dα1 x ⎤ ⎢ 0 ⎢ α 1⎥ ⎢ ⎢ D 2 x2 ⎥ = ⎢ 0 ⎢ α3 ⎥ ⎢ ⎣ D x3 ⎦ ⎢ 0 ⎢ ⎣ 0

0

0.5

0.5 0

(84)

According to (82), the simplified controllability matrix for system (84) is a 5 × 30 matrix. However, some of its columns are zero. By deleting the zero columns, the following reduced controllability matrix is obtained

(83)

+ 1) or

the

MCS

⎡ .5 −.5 ⎢ ⎢0 0 =⎢0 0 ⎢ ⎢0 0 ⎣0 0

.5 0 0 0 .5 −0.75 1.25 −.5 0.75 − 1.25 0 0 0 0 .5 −0.5 0.75 −.5 0.5 −0.75 0 0.5 −1 1.75 0 0 0 0 0 0 0 0.5 −0.5 0.75 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1

0⎤ ⎥ 0⎥ 0 ⎥. ⎥ 0⎥ −1⎦

(85)

It could be verified that the controllability matrix (85) is full rank ( rank(MCS ) = 5). Thus, system (84) is controllable.

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The simplified observability matrix is a 18 × 5 matrix that some of its rows are zero. By deleting these zero rows, the following simplified observability matrix is obtained

MOS

⎡ 0 0 0 0 ⎢ 0 1 ⎢ −1 0 = ⎢ 0 −0.5 1 0 ⎢ 1 −0.5 0.5 −1 ⎢ ⎣ 0 0 0 0

0 0 0 0 0.5 −1 0.5 −0.5 0 0

T 0 0 0 ⎤ ⎥ 0 ⎥ −1 0 0 −0.5 1 ⎥ . 1 −0.5 0.5⎥ ⎥ 0 0 0 ⎦

Definition 7. The dual system of a multi-order fractional system (4) is defined as

(86)

The observability matrix (86) is singular ( rank(MOS ) = 3). This means that system (84) is not observable. Example 3. In this example, the electrical circuit shown in Fig. 2 is considered. In this circuit, all the fractional order inductors have the fractional order α1 and the Fractances have the fractional order α2. Assume that the fractional orders are appropriately considered such that Assumption 1 is satisfied. The current of inductors are denoted by i1, i2 and the voltage of capacitors are denoted by v1, v2. The current of resistor R3 is i′. The voltage source u(t ) is the input. v2 is selected as the output ( y = v2). Considering R1 = R2 = R3 = R4 = R5 = 1Ω , C1 = C2 = 1F , L1 = L2 = 1H T T and assuming the state variables as x1 = ⎡⎣ i1 i2 ⎤⎦ , x 2 = ⎡⎣ v1 v2 ⎤⎦ , T

x = ⎡⎣ x1 x 2 ⎤⎦ yields the following pseudo lower triangular multiorder state space realization

⎡ −1.5 ⎡ Dα1 x1 ⎤ ⎢ 0 ⎢ α ⎥=⎢ ⎣ D 2 x2 ⎦ ⎢ −0.5 ⎢⎣ −0.5

0 −1 0 −1

0 0 −2 −1

⎡ 0.5⎤ 0 ⎤ ⎥ ⎢ ⎥ 0 ⎥ 0 ⎥ x+⎢ u( t ) ⎢ 0.5⎥ −1⎥ ⎢⎣ 0.5⎥⎦ −1⎥⎦ (87)

The simplified controllability matrix (a 4 × 12 matrix) is calculated as ⎡ .5 0 0 − 0.75 0 0 1.12 0 0 0 0 0 ⎤ ⎢ ⎥ 0 0 0 0 0 0 0 0 0 0 0 0 ⎥ . MCS = ⎢ ⎢ 0 − 0.25 0.75 0 0.37 − 1.12 0 − 0.56 1.68 0.5 − 1.5 4 ⎥ ⎢⎣ 0 − 0.25 0.5 0 0.37 − 0.75 0 − 0.56 1.12 0.5 − 1 2.5⎥⎦

(88)

The system (87) is not controllable because MCS is not full rank (rank(MCS ) = 3). The simplified observability matrix is a 9 × 4 matrix calculated as

MOS

1 1 3 2

0 0.75 −1.5 0 1 −1 0 0 0 0 0 0

0 0 0 0

T −1.12 2.25⎤ ⎥ 1 ⎥ −1 . 0 0 ⎥ ⎥ 0 0 ⎦

⎧ ⎡ α1 ⎤ ⎡ T ⎪ D x̲ 1 ⎢ A11 ⎪ ⎢ Dα2 x̲ ⎥ ⎢ T A12 ⎥ ⎢ 2 ⎪ ⎪⎢ ⎥=⎢ ⎨⎢ ⋮ ⎥ ⎢ ⋮ ⎪ ⎣ Dαn x̲ n ⎦ ⎢ T ⎣ A1n ⎪ ⎪ ⎡ T T ⎪ ⎩ y = ⎣ B1 B2 …

⎡ CT ⎤ T … A nT1⎤⎥ A21 ⎢ 1⎥ T T ⎥ ⎢ CT ⎥ A22 … A n2 ̲ ⎥ x + ⎢ 2 ⎥u ⋮ ⋮ ⋮ ⎥ ⎢ ⋮⎥ ⎢CT⎥ T ⎥ A2Tn … A nn ⎣ n⎦ ⎦ BnT ⎤⎦ x.̲

(90)

It could be easily investigated that systems (4) and (90) have the same transfer functions. For ordinary systems, the duality Theorem states that a system is controllable if and only if its dual is observable and vice versa. We show by a counterexample that this fact is not necessarily true for multi-order fractional systems. See the next example. Example 4. Consider the following multi-order fractional system ( n1 = 2, n2 = 3)

y = ⎡⎣ 0 0 0 1⎤⎦ x .

⎡ 0 −0.5 ⎢ 0 −1 =⎢ ⎢ 0 −1 ⎢⎣ 1 −1

observability analysis of state space equations is the duality concept. Dual systems have different state space realizations and the similar transfer functions. In the following definition, the dual system of a multi-order fractional system is introduced.

⎧ ⎡1 ⎪ ⎢ ⎪ ⎡ Dα1 x ⎤ ⎢ 0 ⎪⎢ α 1⎥ = ⎢0 ⎨ ⎣ D 2 x2 ⎦ ⎢ 0 ⎪ ⎢ ⎪ ⎣0 ⎪ ⎩ y = [ 1 −1 1

It could be verified that rank(MOS ) = 4 and the system is observable. One of the main concepts in the controllability and

1 0 −2 −3 0

0 1 0 1 1

⎡1 ⎤ −1⎤ ⎥ ⎢ ⎥ 0 ⎥ ⎢ −1⎥ + x ⎥ 0 ⎢1 ⎥ ⎢0 ⎥ 0 ⎥ ⎥ ⎢⎣ ⎦⎥ 1 0 ⎦

0 1] x

(91)

T where x = ⎡⎣ x1 x 2 ⎤⎦ . The simplified controllability matrix for (91) is obtained as

MCS

(89)

2 0 0 0 0

⎡ 1 ⎢ ⎢ −1 =⎢ 0 ⎢ 0 ⎢ ⎣ 0

−1 0 0 0 0

−1 0 0 0 0

0 0 0 0 0 − 2 7 − 11 0 − 8 0 0 0 0 0 −3 3 −9 0 0 1 −2 4 −8 0 0 0 0 0 0 0 −3 3 −9 0 0 0 0 0 0 1 1 −3 3 0 0 0 0 0 0

13 − 29 ⎤ ⎥ 0 0 ⎥ 0 0 ⎥ 0 0 ⎥ ⎥ 0 0 ⎦

(92)

The controllability matrix (92) is full rank (rank(MCS ) = 5). Thus, system (91) is controllable. The dual state space realization for (91) is computed as

Fig. 1. The electrical circuit of Example 2.

Please cite this article as: Hassanzadeh I, Tabatabaei M. Calculation of controllability and observability matrices for special case of continuous-time multi-order fractional systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.006i

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Fig. 2. The electrical circuit of Example 3.

⎧ ⎡1 0 0 ⎪ ⎢ α ⎪ ⎡ D 1x ⎤ ⎢ 2 0 0 1 ⎪ ⎢ α ⎥ = ⎢ 1 0 −2 ⎨ ⎣ D 2 x2 ⎦ ⎢ 0 1 0 ⎪ ⎢ ⎪ ⎣ −1 0 0 ⎪ ⎩ y = [ 1 −1 1 0 1] x .

0 0 −3 1 0

⎡1 ⎤ 0⎤ ⎥ ⎢ ⎥ 0⎥ ⎢ −1⎥ 0⎥ x + ⎢ 1 ⎥ ⎢0 ⎥ 1⎥ ⎥ ⎢⎣ ⎥⎦ ⎦ 1 0 (93)

The observability matrix for the dual system (93) is given by

MOS

⎡ 1 ⎢ ⎢ −1 =⎢ 1 ⎢ 0 ⎢ ⎣ 1

0 −2 7 −1 0 −3 3 0 −2 4 −8 0 −3 3 −9 0 0 −3 3 0

0 −8 0 0 0 0 0 0 0 0

13 0 0 0 0

−1 0 0 0 0

0 −8 0 0 0 0 0 0 0 0

T 13⎤ ⎥ 0⎥ 0⎥ . 0⎥ ⎥ 0⎦

(94)

The observability matrix (94) is not full rank ( rank(MOS ) = 4 ). This means that the dual system (94) is not observable. Thus, the common duality Theorem is not necessarily true for multi-order fractional systems.

6. Conclusions and future works The controllability and observability matrices for pseudo upper and lower triangular multi-order fractional systems are calculated in this paper. The non-singularity of these matrices is the necessary and sufficient condition for controllability and observability of these systems. Although, the general form of transfer functions that could be described by this special case of multi-order systems are derived, a systematic approach for extracting a pseudo upper or lower triangular state space realization for these transfer functions could be considered as a future work. Obtaining pseudo controller or observer forms for these systems and designing state feedback and state observer for them are other future research fields. Moreover, considering constraints on control signal or constrained controllability analysis of multi-order fractional systems is another research filed.

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Please cite this article as: Hassanzadeh I, Tabatabaei M. Calculation of controllability and observability matrices for special case of continuous-time multi-order fractional systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.006i