Input–output finite time stability of fractional order linear systems with 0 < α

Abstract

The input–output finite time stability (IO-FTS) for a class of fractional order linear time-invariant systems with a fractional commensurate order 0 < α < 1 is addressed in this paper. In order to give the stability property, we first provide a new property for Caputo fractional derivatives of the Lyapunov function, which plays an important role in the main results. Then, the concepts of the IO-FTS for fractional order normal systems and fractional order singular systems are introduced, and some sufficient conditions are established to guarantee the IO-FTS for fractional order normal systems and fractional order singular systems, respectively. Finally, the state feedback controllers are designed to maintain the IO-FTS of the resultant fractional order closed-loop systems. Two numerical examples are provided to illustrate the effectiveness of the proposed results.

Keywords

Fractional order systems linear time invariant systems finite time stability input–output

Introduction

Fractional calculus is understood to be the generalization of ordinary calculus to the non-integer order case and has been investigated with growing interest (Kilbas et al., 2006; Podlubny, 1999). This is mainly due to the fact that many systems are known to display fractional order dynamics. The semi-infinite lossy Resistance Capacitance (RC) transmission line is the first physical system to be widely recognized as one demonstrating fractional behavior, where the current into the line is equal to the half-derivative of the applied voltage. Other systems involving fractional order dynamics are viscoelasticity, coloured noise, electrode-electrolyte polarization, dielectric polarization, boundary layer effects in ducts and electromagnetic waves (Das, 2011). The stability and the control problems of fractional order systems have been studied in many contributions (Efe, 2011; Li et al., 2010; Matignon, 1996; Tan et al., 2010; Yeroglu and Tan, 2011). For instance, the stability region for fractional order linear time-invariant systems was discussed by Matignon (1996) and Li et al. (2010) studied the stability of fractional order nonlinear dynamic systems using the second method of Lyapunov.

In practical applications, one always needs to consider the behaviour of the system over a finite time, where large values of state are unacceptable. For instance, in the boost converter (Niu and Zhao, 2013; Wang et al., 2009; Zhang et al., 2014), if the current changes too quickly, the circuit will break down. Besides, a large amount of overshoot cannot be applied in many practical engineering situations (Sun and Wang, 2012; Wang et al., 2014; Xiang et al., 2012). To study the transient performance, finite time stability (FTS) was proposed by Kamenkov (1953). FTS concerns only the specified bounds on the state. However, sometimes it is just the output, rather than the state, that is required to be restrained within a bound. In this case, the input–output finite time stability (IO-FTS) of a system is very significant. The concept of IO-FTS of integer order systems was originally introduced by Amato et al. (2010). We say that a system is input–output (IO) finite time stable if, for a given class of input signals, the output of the system does not exceed an assigned threshold during a specified time interval. From then on, many efforts have been devoted to the research of IO-FTS (Amato et al., 2010, 2012, 2014; Yao et al., 2012). In Amato et al. (2014), two sufficient conditions which guarantee IO-FTS of a given linear system for two different input classes was introduced. The IO finite time stabilization problem for time-varying linear singular systems was studied in Yao et al. (2012). Nevertheless, it is worth noticing that these studies were mainly concerned with integer order systems. In fact, it is interesting to study the IO-FTS for a class of fractional order systems. As far as we know, there are some results concerning the FTS of fractional order systems (Lazarević and Spasić, 2009; Wu et al., 2013). However, until now, there are no results concerning the IO-FTS of fractional order systems. The problem of the IO-FTS for fractional order systems is challenging. On the one hand, this is because of the complexity of the fractional order calculus equation and the fact that integer order algorithms (Chen, 2014;Wang et al., 2013; Xu and Lam, 2006) cannot be directly applied to the fractional order system. On the other hand, if we want to provide the condition of the IO-FTS for fractional order systems by employing the Lyapunov function, we have to find a Lyapunov candidate function in the fractional order. Some authors have proposed Lyapunov functions for fractional order systems (Burton, 2011; Lakshmikantham et al., 2008), but they are intricate and limited.

The objective of the present paper is to study the IO-FTS of fractional order linear time-invariant systems. Such systems are researched extensively in the literature (Chen et al., 2006; Hartley and Lorenzo, 2002) and can be seen as the natural generalization of state-space models of conventional integer order systems. The novel contribution of this paper goes in several directions. First, we provide a lemma which admits a simple Lyapunov function in the fractional order case and we introduce a property of the fractional calculus. Then, the concept of the IO-FTS for fractional order systems is given, and some sufficient conditions are provided to guarantee the IO-FTS of fractional order linear normal and singular systems. Finally, state feedback controllers are designed such that the resultant fractional closed-loop systems are IO finite time stable.

The remainder of this paper is organized as follows. Some necessary definitions and lemmas are recalled in the next section. Sufficient conditions, which guarantee the IO-FTS of the given fractional order linear normal and singular systems, are introduced in the third section. In the fourth section, two examples are given to present the effectiveness of the results.

Notation. Throughout this paper, $R^{n}$ denotes the n-dimensional Euclidean space, and $R^{n \times m}$ represents the set of all $n \times m$ real matrices. For a matrix P, $P > 0$ means that P is symmetric positive definite, $P \geq 0$ means that P is symmetric positive semidefinite and $P^{T}$ and $P^{- 1}$ denote the transpose and the inverse of a matrix P. The symbol $L_{\infty}$ refers to the space of essentially bounded signals.

Fundamentals of fractional calculus

In order to utilize fractional calculus for the discussion in the current paper, the fundamental definition of the fractional calculus is recalled. The fractional integral with non-integer order $α > 0$ of function $x (t)$ is defined as follows

{{}_{t}I}_{a}^{α} x (t) = \frac{1}{Γ (α)} \int_{a}^{t} (t - τ)^{α - 1} x (τ) d τ

where $x (t)$ is an arbitrary integrable function, ${{}_{t}I}_{a}^{α}$ is the fractional integral of order $α$ on $[a, t]$ and $Γ (\cdot)$ is the Gamma function $Γ (s) = \int_{0}^{\infty} t^{s - 1} e^{- t} dt$ .

Based on the definition of the fractional order integral, the well known definition of the Caputo derivative of fractional order $α$ of function $x (t)$ with respect to t and the initial value a is given by

{{}_{a}^{C}D}_{t}^{α} x (t) = \frac{1}{Γ (k - α)} \int_{a}^{t} (t - τ)^{k - α - 1} x^{(k)} (τ) d τ

where k is the first integer larger than $α$ , i.e $k - 1 < α < k$ , $k \in Z^{+}$ .

Without loss of generality, it is assumed that the lower limit of the fractional integrals and derivatives is zero throughout this paper. For the sake of brevity, we shall use the notation $D^{α}$ for ${{}_{0}^{C}D}_{t}^{α}$ and $I^{α}$ for ${{}_{t}I}_{0}^{α}$ respectively.

Similar to the exponential function, the Mittag–Leffler function is the function frequently used in the solutions of fractional order systems, and defined as

E_{α, β} (z) = \sum_{i = 0}^{\infty} \frac{z^{i}}{Γ (i α + β)}

To proceed the discussion of the main results, the following lemmas are given.

Lemma 1. (Duarte-Mermoud et al., (2015) Assume $α \in (0, 1)$ and let $x (t) \in R^{n}$ be a vector of differentiable functions. Then, for any time instant $t \geq 0$ , the following relationship holds

\frac{1}{2} D^{α} [x^{T} (t) Px (t)] \leq x^{T} (t) P D^{α} x (t)

where $P \in R^{n \times n}$ , $P \geq 0$ .

Lemma 2. (Kilbas et al., 2006) From the definition of the fractional integral and the Caputo derivative, we have

I^{α} (D^{α} x (t)) = x (t) - \sum_{i = 0}^{k - 1} \frac{x^{i} (0)}{i!} t^{i}

In particular, when $0 < α < 1$

I^{α} (D^{α} x (t)) = x (t) - x (0)

Main results

In this section, sufficient conditions which guarantee the IO-FTS of the fractional order system are derived for the input signal belonging to $L_{\infty}$ . Then, the problem of IO finite time stabilization via state feedback is solved.

IO-FTS of a fractional order linear normal system

Consider the following fractional order linear normal system

{\begin{matrix} D^{α} x (t) = Ax (t) + Bw (t), & x (0) = 0 \\ y (t) = Cx (t) \end{matrix}

(1)

where $0 < α < 1$ is the fractional commensurate order of the system; $A \in R^{n \times n}$ , $B \in R^{n \times m}$ and $C \in R^{q \times n}$ are constant matrices; $x (t) \in R^{n}$ is the state; $y (t) \in R^{q}$ is the output; and $w \in R^{m}$ is the disturbance input. $W_{\infty} (T, R) : {w (\cdot) \in L_{\infty, [0, T]} : w^{T} (t) Rw (t) \leq 1, t \in [0, T]}$ . Throughout the paper, R denotes a positive definite symmetric matrix.

In the following, we introduce the concept of IO-FTS for the fractional order system.

Definition 1. Given a positive scalar T, a class of input signals $W_{\infty}$ defined over $[0, T]$ and a positive definite matrix Q, system (1) is said to be IO finite time stable with respect to $(W_{\infty}, Q, T)$ if $w (\cdot) \in W_{\infty} \Rightarrow y^{T} (t) Qy (t) < 1, t \in [0, T] .$

Remark 1. From the solution of the system (1)

y (t) = C \int_{0}^{t} (t - τ)^{α - 1} E_{α, α} [A (t - τ)^{α}] Bw (τ) d τ

it is clear that the output of the system (1) is well-defined for $t \in [0, T]$ once the boundary of the input is given. It seems that there is some parallelism between bounded input–bounded output stability and IO-FTS with respect to $(W_{\infty}, Q, T)$ , but they are two independent concepts since the latter involves signals defined over a finite time interval, and that quantitative bounds on both inputs and outputs must be specified, while bounded input–bounded output stability considers an infinite time interval and just requires the existence of the bounds.

The following theorem provides a sufficient condition on the IO-FTS of the fractional order linear normal system.

Theorem 1. The fractional order system (1) is IO finite time stable with respect to $(W_{\infty}, Q, T)$ for all $t \in [0, T]$ if there exists a matrix $P \in R^{n \times n}$ , $P > 0$ such that

(\begin{matrix} P^{T} A + A^{T} P & P^{T} B \\ B^{T} & P - R \end{matrix}) < 0

(2a)

\frac{T^{α}}{Γ (α + 1)} C^{T} QC < P

(2b)

Proof 1. Applying the Lyapunov function $V (x (t)) = x^{T} (t) Px (t)$ , and by Lemma 1, it follows that

\begin{matrix} D^{α} V (x (t)) = D^{α} (x^{T} (t) Px (t)) \\ \leq 2 x^{T} (t) P^{T} D^{α} x (t) \\ = 2 x^{T} (t) P^{T} (Ax (t) + Bw (t)) \\ = x^{T} (t) P^{T} (Ax (t) + Bw (t)) + {(Ax (t) + Bw (t))}^{T} Px (t) \\ = x^{T} (t) (P^{T} A + A^{T} P) x (t) + x^{T} (t) P^{T} Bw (t) \\ + w^{T} (t) B^{T} Px (t) \end{matrix}

(3)

Then, by the Schur complement, (2a) holds if and only if $P^{T} A + A^{T} P + P^{T} B R^{- 1} B^{T} P < 0 .$ This together with (3) implies

\begin{matrix} D^{α} V (x (t)) < x^{T} (t) P^{T} Bw (t) + w^{T} (t) B^{T} Px (t) \\ - x^{T} (t) P^{T} B R^{- 1} B^{T} Px (t) \\ = - {(R^{- \frac{1}{2}} B^{T} Px (t) - R^{\frac{1}{2}} w (t))}^{T} (R^{- \frac{1}{2}} B^{T} Px (t) \\ - R^{\frac{1}{2}} w (t)) + w^{T} (t) Rw (t) \\ \leq w^{T} (t) Rw (t) \end{matrix}

(4)

Integrating with order $α$ both sides of (4) from 0 to $t \leq T$ , it can be known from Lemma 2 that $I^{α} (D^{α} V (x (t))) = x^{T} (t) Px (t)$ . Note that $w (\cdot)$ belongs to $W_{\infty}$ . Then

\begin{matrix} x^{T} (t) Px (t) < I^{α} [w^{T} (t) Rw (t)] \\ = \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} w^{T} (τ) Rw (τ) d τ \\ \leq \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} d τ \\ \leq \frac{T^{α}}{Γ (α + 1)} \end{matrix}

(5)

From (2b) and (5), it can be derived that

y^{T} (t) Qy (t) = x^{T} (t) C^{T} QCx (t) < \frac{Γ (α + 1)}{T^{α}} x^{T} (t) Px (t) < 1

This ends the proof.□

Remark 2. When $α = 1$ , it is easy to obtain a sufficient condition of IO-FTS for the corresponding integer order normal system as follows

(\begin{matrix} P^{T} A + A^{T} P & P^{T} B \\ B^{T} P & - R \end{matrix}) < 0

(6a)

T C^{T} QC < P

(6b)

From the result, we know it is consistent with the condition in the integer order linear time-varying system degenerated to time invariance in Amato et al. (2010). Comparing with the condition of IO-FTS for the integer order linear normal system, the inequality (2b) is the specific condition of the fractional order system (1) and the IO-FTS for integer order linear normal systems is just the special case of IO-FTS for fractional order linear normal systems.

By Theorem 1, it is easy to obtain the following corollary.

Corollary 1. The fractional order system (1) is IO finite time stable with respect to $(W_{\infty}, Q, T)$ for all $t \in [0, T]$ if there exists a matrix $P \in R^{n \times n}$ , $P > 0$ such that

(\begin{matrix} AP + P^{T} A^{T} & B \\ B^{T} & - R \end{matrix}) < 0

(7a)

\frac{T^{α}}{Γ (α + 1)} P^{T} C^{T} QCP < P

(7b)

Based on the stability conditions presented in Corollary 1, the stabilizing controller design can be formulated as a convex optimization problem characterized by linear matrix inequalities. Here, we assume that all the state variables are available for state feedback. The purpose is to design a state feedback controller $u (t) = Kx (t)$ for the fractional order linear normal system

{\begin{matrix} D^{α} x (t) = Ax (t) + Fu (t) + Bw (t), & x (0) = 0 \\ y (t) = Cx (t) \end{matrix}

(8)

where $u (t) \in R^{l}$ is the control input and $F \in R^{n \times l}$ is a constant matrix, such that the closed-loop system

{\begin{matrix} D^{α} x (t) = A_{c} x (t) + Bw (t), & x (0) = 0 \\ y (t) = Cx (t) \end{matrix}

(9)

with $A_{c} = A + FK$ is IO finite time stable. Then, we have the following stabilization result.

Theorem 2. If there exist matrices $P \in R^{n \times n}$ , $P > 0$ and $Y \in R^{l \times n}$ such that

(\begin{matrix} AP + P^{T} A^{T} + FY + Y^{T} F^{T} & B \\ B^{T} & - R \end{matrix}) < 0

(10a)

(\begin{matrix} \frac{Γ (α + 1)}{T^{α}} P & P^{T} C^{T} \\ CP & Q^{- 1} \end{matrix}) > 0

(10b)

then the system (9) is IO finite time stable with respect to $(W_{\infty}, Q, T)$ under the feedback controller $u (t) = Kx (t) = Y P^{- 1} x (t)$ .

Proof 2. Applying the state feedback controller $K = Y P^{- 1}$ to (8), we get the closed-loop system (9) with $A_{c} = A + FY P^{- 1}$ . It is easy to see that (10a) is equivalent to

(\begin{matrix} A_{c} P + P^{T} A_{c}^{T} & B \\ B & - R \end{matrix}) < 0

By Corollary 1, we know that the closed-loop system (9) is IO finite time stable with respect to $(W_{\infty}, Q, T)$ . This concludes the proof.□

IO-FTS of a fractional order linear singular system

Consider the following fractional order linear singular system

{\begin{matrix} E D^{α} x (t) = Ax (t) + Bw (t), & x (0) = 0 \\ y (t) = Cx (t) \end{matrix}

(11)

where $E \in R^{n \times n}$ is a singular matrix such that rank(E)= $r < n$ .

By the assumption of rank(E) = r for the system $E D^{α} x (t) = Ax (t) + Bw (t)$ , we can choose two nonsingular matrices M and N such that

MEN = (\begin{matrix} I & 0 \\ 0 & 0 \end{matrix}), MAN = (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix})

(12)

Lemma 3. (Yao et al., 2013) The system $E D^{α} x (t) = Ax (t) + Bw (t)$ is impulse-free in the time interval $[0, T]$ if and only if $A_{4}$ is nonsingular for all $t \in [0, T] .$

The definition of the IO-FTS of the fractional order singular system is the same statement as the fractional order normal system.

Theorem 3. The fractional order system (11) is IO finite time stable with respect to $(W_{\infty}, Q, T)$ for all $t \in [0, T]$ if there exists a nonsingular matrix $P \in R^{n \times n}$ such that

(\begin{matrix} P^{T} A + A^{T} P & P^{T} B \\ B^{T} P & - R \end{matrix}) < 0

(13a)

\frac{T^{α}}{Γ (α + 1)} C^{T} QC \leq P^{T} E = E^{T} P

(13b)

Proof 3. Assume that there exists a matrix P such that (13a) and (13b) are satisfied. Under this condition, we first show that the system (11) is impulse-free.

Now, define

M^{- T} PN = (\begin{matrix} P_{1} & P_{2} \\ P_{3} & P_{4} \end{matrix})

(14)

where the partition is compatible with that of A in (12). Then, by (13b), $P^{T} E = E^{T} P$ , it can be shown that $P_{2} = 0$ and $P_{1} = P_{1}^{T} .$ It can be known from the 1–1 block in (13a) that

P^{T} A + A^{T} P < 0

(15)

Pre- and post-multiplying (15) by $N^{T}$ and N, respectively, then using the expressions in (12) and (14), we have

(\begin{matrix} U_{1} & U_{2} \\ U_{2}^{T} & U_{3} \end{matrix}) < 0

(16)

where

\begin{matrix} U_{1} = P_{1}^{T} A_{1} + A_{1}^{T} P_{1} + P_{3}^{T} A_{3} + A_{3}^{T} P_{3} \\ U_{2} = P_{1}^{T} A_{2} + P_{3}^{T} A_{4} + A_{3}^{T} P_{4} \\ U_{3} = P_{4}^{T} A_{4} + A_{4}^{T} P_{4} \end{matrix}

Then, the 2–2 block in (16) gives $P_{4}^{T} A_{4} + A_{4}^{T} P_{4} < 0,$ which implies $A_{4}$ is nonsingular. Thus, by Lemma 3, it can be seen that the system (11) is impulse-free.

Furthermore, by the Schur complement, it is easy to check that inequality (13a) holds if and only if the following inequality holds

P^{T} A + A^{T} P + P^{T} B R^{- 1} B^{T} P < 0

(17)

By inequality (13b), it follows that $P^{T} E \geq 0$ . According to the function $V (x (t)) = x^{T} (t) P^{T} Ex (t)$ and Lemma 1, we get

\begin{matrix} D^{α} V (x (t)) = D^{α} (x^{T} (t) P^{T} Ex (t)) \\ \leq 2 x^{T} (t) P^{T} E D^{α} x (t) \\ = 2 x^{T} (t) P^{T} (Ax (t) + Bw (t)) \\ = x^{T} (t) P^{T} (Ax (t) + Bw (t)) + {(Ax (t) + Bw (t))}^{T} Px (t) \\ = x^{T} (t) (P^{T} A + A^{T} P) x (t) + x^{T} (t) P^{T} Bw (t) \\ + w^{T} (t) B^{T} Px (t) \end{matrix}

(18)

It can be obtained from (17) that

\begin{matrix} D^{α} V (x (t)) < x^{T} (t) P^{T} Bw (t) + w^{T} (t) B^{T} Px (t) - x^{T} (t) P^{T} B R^{- 1} B^{T} Px (t) \\ = - {(R^{- \frac{1}{2}} B^{T} Px (t) - R^{\frac{1}{2}} w (t))}^{T} (R^{- \frac{1}{2}} B^{T} Px (t) \\ - R^{\frac{1}{2}} w (t)) + w^{T} (t) Rw (t) \\ \leq w^{T} (t) Rw (t) \end{matrix}

(19)

Integrating with order $α$ both sides of (19) from 0 to $t \leq T$ , by Lemma 2 and $x (0) = 0$ , one has $I^{α} (D^{α} V (x (t))) = x^{T} (t) P^{T} Ex (t)$ . Taking into account the fact that $w (\cdot)$ belongs to $W_{\infty}$ , then

\begin{matrix} x^{T} (t) P^{T} Ex (t) < I^{α} [w^{T} (t) Rw (t)] \\ = \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} w^{T} (τ) Rw (τ) d τ \\ \leq \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} d τ \\ \leq \frac{T^{α}}{Γ (α + 1)} \end{matrix}

(20)

Combining condition (13b) with the inequality (20) gives

\begin{matrix} y^{T} (t) Qy (t) = x^{T} (t) C^{T} QCx (t) \leq \frac{Γ (α + 1)}{T^{α}} x^{T} (t) P^{T} Ex (t) < 1 \end{matrix}

The proof is completed.□

Remark 3. If $α = 1$ , we can obtain the same conditions which relate to the integer order system (Yao et al., 2012) as follows

(\begin{matrix} P^{T} A + A^{T} P & P^{T} B \\ B^{T} P & - R \end{matrix}) < 0

T C^{T} QC \leq P^{T} E = E^{T} P

Therefore, IO-FTS of fractional order linear singular systems is the generalization of integer order linear singular systems.

The following corollary, which is analogous to Corollary 1, can be stated for a fractional order linear singular system.

Corollary 2. The fractional order system (11) is IO finite time stable with respect to $(W_{\infty}, Q, T)$ for all $t \in [0, T]$ if there exists a nonsingular matrix $P \in R^{n \times n}$ such that

(\begin{matrix} AP + P^{T} A^{T} & B \\ B^{T} & - R \end{matrix}) < 0

(22a)

\frac{T^{α}}{Γ (α + 1)} P^{T} C^{T} QCP \leq EP = P^{T} E^{T}

(22b)

Next, we define the stabilization problems we want to solve. Given the positive scalar T, the positive defined matrices R and Q, the fractional order singular system described by the state-space equation

{\begin{matrix} E D^{α} x (t) = Ax (t) + Fu (t) + Bw (t), & x (0) = 0 \\ y (t) = Cx (t) \end{matrix}

(23)

finds a sufficient condition which guarantees the interconnection of (23) with the controller $u (t) = Kx (t)$

{\begin{matrix} E D^{α} x (t) = A_{c} x (t) + Bw (t), & x (0) = 0 \\ y (t) = Cx (t) \end{matrix}

(24)

where $A_{c} = A + FK$ is IO finite time stable with respect to $(W_{\infty}, Q, T)$ . The solution of this problem is given by the following theorem.

Theorem 4. The fractional closed-loop system (24) is IO finite time stable with respect to $(W_{\infty}, Q, T)$ if there exist matrices $P \in R^{n \times n}$ and $Y \in R^{l \times n}$ such that

(\begin{matrix} AP + P^{T} A^{T} + FY + Y^{T} F^{T} & B \\ B^{T} & - R \end{matrix}) < 0

(25a)

\frac{T^{α}}{Γ (α + 1)} P^{T} C^{T} QCP \leq EP = P^{T} E^{T}

(25b)

Then, a stabilizing state feedback controller can be given as

u (t) = Kx (t) = Y P^{- 1} x (t)

(26)

Proof 4. Applying the state feedback controller (26) to (23) results in the closed-loop system

{\begin{matrix} E D^{α} x (t) = (A + FY P^{- 1}) x (t) + Bw (t), & x (0) = 0 \\ y (t) = Cx (t) \end{matrix}

(27)

It is easy to obtain that (25a) can be rewritten as

(\begin{matrix} P^{T} {(A + FY P^{- 1})}^{T} + (A + FY P^{- 1}) P & B \\ B^{T} & - R \end{matrix}) < 0

(28)

It follows from Corollary 2 that the closed-loop system (24) is IO finite time stable with respect to $(W_{\infty}, Q, T)$ . This completes the proof.□

Numerical example

In this section, two examples are presented to illustrate the effectiveness of the proposed method.

Example 1. Let us consider the fractional linear time-invariant normal system with disturbances and one output defined by

\begin{matrix} D_{t}^{0.9} x (t) = (\begin{matrix} 3 & 4 & - 2.8 \\ 5 & - 1 & 0.1 \\ 5 & 4 & - 0.8 \end{matrix}) x (t) + (\begin{matrix} 1 & - 1 & 7 \\ - 4.5 & - 0.5 & 3 \\ 1 & 3 & 9 \end{matrix}) u (t) \\ + (\begin{matrix} 1 \\ 0.6 \\ 0.2 \end{matrix}) w (t) \\ y (t) = (\begin{matrix} 1 & 1 & 2 \end{matrix}) x (t) \end{matrix}

(29)

with $x (0) = 0$ , and let $R = 1$ , $Q = 1$ , $T = 0.6$ . If we set $u (t) = 0$ and choose $w (t) = e^{- t} \in W_{\infty}$ , from Figure 1, it is straightforward to verify that the system (29) is not IO finite time stable with respect to $(W_{\infty}, Q, T)$ . Then we can exploit Theorem 2 to design a state feedback controller $u (t) = Kx (t)$ , such that the closed-loop system is IO finite time stable with respect to $(W_{\infty}, Q, T)$ . By using the Matlab LMI Toolbox to solve (10), we can obtain matrices P and Y

\begin{matrix} P = (\begin{matrix} 0.9537 & - 0.1533 & - 0.3065 \\ - 0.1533 & 0.9537 & - 0.3065 \\ - 0.3065 & - 0.3065 & 0.4938 \end{matrix}) \\ Y = (\begin{matrix} 218.6337 & 95.4245 & 124.8621 \\ - 23.2820 & - 188.3344 & - 41.1181 \\ - 35.1733 & 112.0394 & 0.1032 \end{matrix}) \end{matrix}

(30)

and the desired state feedback controller with

K = (\begin{matrix} 614.2807 & 502.9724 & 946.3318 \\ - 238.2581 & - 387.3679 & - 471.5963 \\ 25.1985 & 158.1917 & 114.0419 \end{matrix})

(31)

Figure 1.

Output of the open-loop system (29) versus time.

By applying the controller to system (29), it is easy to see the IO-FTS of the fractional closed-loop system from the output of the closed-loop system (Figure 2).

Figure 2.

Output of the closed-loop system (29) versus time.

Example 2. Consider the fractional linear time-invariant singular system (23) with parameters as follows

\begin{matrix} E = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}), A = (\begin{matrix} 4.5 & - 5 & 0 \\ 2 & 3 & 0 \\ 0 & 0 & 1 \end{matrix}), F = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) \\ B = (\begin{matrix} 6 & 0 \\ 8 & 0 \\ 0 & - 1 \end{matrix}), C = (\begin{matrix} 7 & - 1 & 0 \end{matrix}), α = 0.5 \end{matrix}

(32)

Let $R = I$ , $Q = 1$ and $T = 1$ . Set $u (t) = 0$ and take $w (t) = (0.5 \sin (t), 0.5 \sin (t))^{T} \in W_{\infty}$ , $t \in [0, 1]$ . From Figure 3, it is easy to see that the system (32) is not IO finite time stable with respect to $(W_{\infty}, Q, T)$ . We can design a state feedback controller such that the closed-loop system is IO finite time stable with respect to $(W_{\infty}, Q, T)$ . Solving the linear matrix inequalities in (25), we obtain the output of the closed-loop system performance (Figure 4). It is obvious that the closed-loop system is IO finite time stable.

Figure 3.

Output of the open-loop system (32) versus time.

Figure 4.

Output of the closed-loop system (32) versus time.

Conclusion

In this paper, we have considered the IO-FTS for the fractional order linear time-invariant normal systems and singular systems. Some sufficient conditions which guarantee the IO-FTS for a class of fractional order systems have been derived by applying the Lyapunov function. The main results are formulated in terms of linear matrix inequalities. For the sake of presentation simplicity, we have considered the case where the systems are time invariant and certain. The results can be extended to the uncertain framework.

Footnotes

Conflict of interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant numbers 11371233 and 61403241).

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Input–output finite time stability of fractional order linear systems with 0 < α < 1

Abstract

Keywords

Introduction

Fundamentals of fractional calculus

Main results

IO-FTS of a fractional order linear normal system

IO-FTS of a fractional order linear singular system

Numerical example

Conclusion

Footnotes

Conflict of interest

Funding

References