For fractional calculus, researchers have presented many fractional multiple definitions, which include the Riemann– Liouville, the Grunwald-Letnikov, and the Caputo definitions.

Definition 1 The Riemann– Liouville fractional derivative of function f(t) with respect to t is given by

where m is the first integer larger than α , i.e., m– 1 < α < m, and Γ is Euler’ s Gamma function, which is defined as

Definition 2 The Caputo fractional derivative of a continuous function f: R⁺ → R is defined as follows:

where m is the first integer larger than α

The Laplace transform depression of Eq. (1) is

When fⁱ(t)| _{t = 0} = 0, (i = 0, 1, … , n), we obtain

Additionally, the additivity is effective for fractional calculus, which can be described as

Westerlund^[34] presented the model of fractional capacitor, which is

where C is the capacitance, V(t) is the voltage, and m is the fractional order, which is a constant depending on the wastage of the capacitor.

Similarly, he also presented fractional order inductance as

where L is the inductance and m is related to the proximity effect.

A fractional order linear system usually can be described as

where D^α is the Caputo fractional derivative operator, 0 < α < 1; u(t) is the system input; the vector X(t) = (x₁ (t), … , x_n (t))^T is the state of each node; and matrices A and B are the system matrix and the input matrix, respectively.

The controllability matrix of the fractional order linear time-invariant system is

We can judge the controllability of the fractional order linear time-invariant system by the rank of the controllability matrix. If rank(Q_c) = n, the fractional order linear time-invariant system is controllable, otherwise uncontrollable. We give strictly mathematical proof in the following contents.

1) Sufficiency We assume that the final state is the original point of the state space, and t₀ = 0. Equation (5) can be rewritten by the Laplacian transformation as

Thus, we obtain

Using the inverse Laplace transformation, we have

Because the system is completely controllable, we have

or

From the Cayley– Hamilton theorem, any square matrix should meet its characteristic equation. Since A is an n× n square matrix, we havehave

Thus, we obtain

From the Cayley– Hamilton theorem, we can obtain

Equations (6) and (7) can be combined to yield the following equation:

Let , we have

where q is an n × 1 vector. So we have

If the system is fully controllable, for any initial state X(0), the sufficient and necessary condition for Eq. (9) to have a unique solution is that the matrix [BABA²B · · · A^{n– 1}B] has full rank, which means its rank is n.

2) Necessity Now that the known condition is that the system is fully controllable, we will prove rank(Q_c) = n. Here, we will use proof by contradiction.

First, we assume that rank(Q_c) ≤ n; that is, each row of Q_c is linearly correlated. So, there is ∃ β ∈ Rⁿ, β ≠ 0 meeting β ^TQ_c = β ^T [BAB · · · A^{n– 1}B] = 0, which means β ^TAⁱB = 0, i = 0, 1, … , n − 1. From the Cayley– Hamilton theorem, we have β ^TAⁱB = 0, i = 0, 1, 2, … Thus, we obtain

And we have β ^TΦ _{α , α} (t)B = 0. That is . Because β ≠ 0, W_c [0, t₁] is singular. From the Gram controllability theorem, the system is not controllable. Obviously, there is a contradiction between the conclusion and the known condition. Therefore, the proof is completed.

According to Refs. [35] and [36], we know that the Kalman rank condition is equivalent between the fractional order systems and the integer order systems. The controllable condition is applicative for arbitrary order. Therefore, the sufficient and necessary controllable condition of the fractional order system is the rank of Q_c = [B, AB, … , A^{n − 1}B] is n(n is the dimension of matrix A), that is rank(Q_c) = n.

Chua’ s circuit is a famous nonlinear circuit, which has lots of fluent nonlinear dynamic phenomena. From Fig. 1, we can obtain the system equation as

Equation (10) can be rewritten as

where there is a nonlinear resistance (NR), whose voltage current characteristic f(V₁ (t)) is

Fig. 1.

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	Fig. 1. (a) Diagram of fractional order Chua’ s circuit. (b) Voltage current characteristic of NR.

Let x = V₁/B_P, y = V₂/B_P, z = I_L/B_PG, α = C₂/C₁, β = C₂/(L₁G²), γ = C₂R/(LG), m₁ = G_b/G, m₀ = G_a/G, τ = tG/C₂, and u = I_u/B_PG. Equations (11) and (12) can be rewritten as

and

Equation (14) can be rewritten as

For simplification, we set x(t) = x₁ (t), y(t) = x₂(t), z(t) = x₃ (t). Equation (13) can then be written as

where A, B, and C are the system matrix, the input matrix, and the constant matrix, respectively. More specially,

Therefore, from the controllability theorem of a fractional linear system, we can obtain the following results. 1) When x(t) ≤ − 1 or x(t) ≥ 1, if and only if α (m₁ + 1) ≠ γ + 2, the fractional Chua’ s circuit is controllable, otherwise the fractional Chua’ s circuit is uncontrollable. 2) When − 1 < x(t) < 1, if and only if α (m₀ − 2m₁ − 1) ≠ 2 + γ , the fractional Chua’ s circuit is controllable, otherwise the fractional Chua’ s circuit is uncontrollable.