Robust stability and stabilization of uncertain fractional order systems subject to input saturation

Abstract

This paper studies the stability and the stabilization for a class of uncertain fractional order (FO) systems subject to input saturation. The Lipschitz condition and the Gronwall–Bellman lemma are adopted and sufficient conditions are derived to stabilize systems by designing a state feedback controller. Numerical examples demonstrate the effectiveness of the proposed method.

Keywords

Fractional order systems input saturation stability Gronwall–Bellman lemma Lipschitz condition state feedback

1. Introduction

Fractional calculus and its applications in different fields of science and engineering have received considerable attention during the last two decades (Faieghi et al., 2014; Feng et al., 2015; Ionescu et al., 2015; Jiang and Tian, 2015; Jakovljević et al., 2016; Lopes and Tenreiro Machado, 2016; Tenreiro Machado, 2016). The dynamic models that contain fractional derivatives and integrals are referred to as fractional order (FO) systems. The key attribute of FO differential equations in the study of dynamics is to embed long-range memory effects in the system modeling (Baleanu et al., 2011). Similarly to integer systems, stability analysis is a significant topic in the control of FO systems.

The stability problem of FO linear time-invariant (FO-LTI) systems with interval uncertainties has been extensively addressed in recent years (Petrás et al., 2004; Chen et al., 2006; Ahn et al., 2007; Deng et al., 2007; Qian et al., 2010; Rivero et al., 2013). In Petrás et al. (2004), the stability of uncertain FO systems with interval coefficients in the state-space representation was studied to find the range of the interval eigenvalues using matrix perturbation theory. In Chen et al. (2006), the Lyapunov inequality was adopted for interval uncertain FO-LTI systems with the FO, v, so that $1 < v < 2$ , to reduce the conservatism of the robust stability test. In Xing and Lu (2009), the robust stability and the stabilization of uncertain linear FO systems with order lying in the range $1 < v < 2$ was addressed via linear matrix inequality (LMI). In Sabatier et al. (2013), bounded input–bounded output stability analysis was derived for incommensurate linear FO systems. In Lu and Chen (2010), sufficient conditions for robust stability and stabilization of the linear FO interval systems with $0 < v < 1$ were investigated.

Control input saturation is a significant topic in physical control systems since real actuators cannot provide unlimited energy. Controllers that ignore the actuator saturation may cause undesirable responses and even closed-loop instability. Due to this reason, it is important to develop methods that take into account actuator saturation at the control design procedure, which makes it more difficult. In studies by Lim et al. (2013), Alaviyan Shahri et al. (2015), and Alaviyan Shahri and Balochian (2015a, 2016), the stability conditions of such systems were provided using the state feedback controller. However, the model uncertainty has not been taken into account in the controller design. The key motivations for the current work come from several sources, namely from uncertainty in the parameters and the control action under actuator saturation. Large values of the control inputs are not satisfactory, particularly in the presence of saturation. This imposes significant restrictions in real-world applications. Therefore, the robust controller design of uncertain FO-LTI systems subject to saturation remains to be investigated. The main contribution of this paper is to address the robust stability analysis and stabilization of uncertain FO-LTI systems subject to input saturation. The Lipschitz condition and the Gronwall–Bellman lemma are applied to obtain an explicit bound to an unknown function as a powerful tool in the study of quantitative properties and stability of solutions of FO equations (N’Doye et al., 2011; Lim and Ahn, 2012). Sufficient conditions are also derived to stabilize such systems by designing a state feedback controller.

This paper is organized as follows. The fundamental concepts are described in Section 2. The main results are developed and discussed in Section 3. Simulation results are represented in Section 4. Finally, the main conclusions are given in Section 5.

Notation: $E_{v} (\cdot)$ denotes the Mittag–Leffler function with one parameter, $Re (z)$ is real part of complex number of z, $spec (A)$ stands the spectrum of a matrix A, $‖ x ‖ = \sqrt{x^{T} x}$ and $‖ A ‖ = \sqrt{{eig}_{max} (A^{T} A)}$ are respectively the Euclidean vector and the spectral matrix norms, where ${eig}_{max} (A^{T} A)$ represents the maximal eigenvalue of the symmetric matrix $A^{T} A$ and ${eig}_{k} (A)$ is for the kth eigenvalue of A.

2. Fundamental concepts

There are several definitions for the fractional differential-integration operators (Sabatier et al., 2007; Monje et al., 2010; Das, 2011; Valério et al., 2013). The Caputo definition is considered in the current work.

Definition 1

(Sabatier et al., 2007) The Caputo derivative of order $v \in R$ of a continuous function $f : R^{+} \to R$ is defined as follows where $n \in N, v \in R^{+}, Γ (\cdot)$ is the Gamma function, $f^{(n)} (\cdot)$ is n^th the derivative of function $f (\cdot)$ , and $n - 1 < v < n$ .

Definition 2

(Jiang and Tian, 2015) The Mittag–Leffler function with two parameters is expressed as

E_{v, β} (z) = \sum_{k = 0}^{\infty} \frac{z^{k}}{Γ (kv + β)}

(2)

where

v > 0, β > 0, z \in C

If $β = 1$ , the Mittag–Leffler function with one parameter is obtained.

Remark 1

(Lim and Ahn, 2012): From Definition 2, the Mittag–Leffler function can be stated as follows

E_{v, β} ({At}^{v}) = \sum_{k = 0}^{\infty} \frac{A^{k} t^{vk}}{Γ (vk + β)} = \sum_{k = 0}^{\infty} \frac{A^{(\frac{(v - 1) k}{v})} t^{(v - 1) k} k!}{Γ (vk + β)} \cdot \frac{A^{\frac{1}{v} k} t^{k}}{k!}

(3)

The norm of the Mittag–Leffler function is bounded and given by

‖ E_{v, β} ({At}^{v}) ‖ \leq {Sup}_{τ \in (1, \infty) \cap R} ({Sup}_{k \in Z +} (\frac{‖ A^{(\frac{(v - 1) k}{v})} ‖ k!}{τ^{(1 - v) k} Γ (vk + β)})) ‖ e^{A^{\frac{1}{v} t}} ‖ = Q ‖ e^{A^{\frac{1}{v}} t} ‖, t \in (1, \infty) \cap R_{+}

(4)

where

Q > 0

Furthermore, since $max {Re (spec (A^{\frac{1}{v}}))} < 0$ , then it follows that $‖ E_{v} ({At}^{v}) ‖ \leq {Qe}^{- ct}$ with $c > 0$ .

Remark 2

(Alaviyan Shahri and Balochian, 2015b) The nonlinear function satisfies the Lipchitz condition if and only if

‖ f_{1} (x_{1}) - f_{1} (x_{2}) ‖ \leq λ_{1} ‖ x_{1} - x_{2} ‖

(5)

where

λ_{1} > 0

is the Lipschitz constant.

Remark 3

(Alaviyan Shahri and Balochian, 2016) The saturation function denoted by $Sat (u) = sign (u) min {u, 1}$ satisfies the Lipschitz condition.

Lemma 1

(N’Doye et al., 2011; the Gronwall–Bellman lemma) Assume that $u_{1} (t)$ and f(t) are real-valued piecewise-continuous functions defined on the real interval $[a, b]$ , H(t) is real-valued and $H (t) \in L (a, b), u_{1} (t)$ and H(t) are nonnegative on this interval. If for all $t \in [a, b]$

u_{1} (t) \leq f (t) + \int_{a}^{t} H (τ) u_{1} (τ) d τ

(6)

then for all

t \in [a, b]

, the following inequality holds

Corollary 1

(Qian et al., 2010; N’Doye et al., 2011) Let $H : R^{+} \to R$ , locally inferable on R⁺. If $u_{1} : R^{+} \to R$ satisfies

u_{1} (t) \leq C + \int_{0}^{t} H (τ) u_{1} (τ) d τ \forall t \geq 0

(8)

where

f (t) = C

is a constant function, then

u_{1} (t) \leq C exp (\int_{0}^{t} H (τ) d τ)

(9)

3. Main results for the uncertain FO-LTI system

Consider the uncertain FO-LTI system subject to input saturation

C D_{t}^{v} x (t) = (A + δ A) x (t) + (B + δ B) Sat (u)

(10)

where A and B are known constant real-value matrices and δA and δB denote the uncertainties satisfying

‖ δ A ‖ \leq 1

and

‖ δ B ‖ \leq 1

. Applying the state feedback controller defined as

u = - Kx, K \in R^{m \times n}

, the entire closed-loop system is given by

C D_{t}^{v} x (t) = (A + δ A) x (t) + (B + δ B) Sat (- Kx) x (0) = x_{0}

(11)

Theorem 1

Consider the closed-loop system given by equation (11). If exists K such that $| \arg (spec (A - BK)) | \geq \frac{v π}{2}$ , then the closed-loop system is robustly asymptotically stable.

Proof

Using the Laplace transfer function on equation (11), we obtain

X (s) = (I_{n} s^{v} - A_{c})^{- 1} (s^{v - 1} x_{0} + L ((δ A + BK) x (t) + (B + δ B) Sat (- Kx))

(12)

where

A_{c} = A - BK

By taking the Laplace inverse transform for equation (12), we have

x (t) = E_{v, 1} (A_{c} t^{v}) x_{0} + \int_{0}^{t} (t - τ)^{v - 1} E_{v, v} (A_{c} (t - τ)^{v}) ((δ A + BK) x (t) + (B + δ B) Sat (- Kx)) d τ

(13)

Using the norm operator on both sides of equation (13) yields

‖ x (t) ‖ \leq ‖ E_{v} (A_{c} t^{v}) x_{0} ‖ + \int_{0}^{t} (t - τ)^{v - 1} ‖ E_{v, v} (A_{c} (t - τ)^{v}) ‖ \times (‖ (δ A + BK) ‖ ‖ x (τ) ‖ + ‖ (B + δ B) ‖ ‖ Sat (- Kx) ‖) d τ

(14)

In the sequel of the proof, we need to establish a bounded value for $‖ x (t) ‖$ . Using the triangular inequality and the specification of δA and δB, i.e. $‖ δ A ‖ \leq 1$ and $‖ δ B ‖ \leq 1$ , we obtain

‖ x (t) ‖ \leq ‖ E_{v} (A_{c} t^{v}) x_{0} ‖ + \int_{0}^{t} (t - τ)^{v - 1} ‖ E_{v, v} (A_{c} (t - τ)^{v}) ‖ (1 + ‖ BK ‖ + (‖ B ‖ + 1) λ_{1}) ‖ x (τ) ‖ d τ

(15)

where

λ_{1} > 0

is the Lipschitz constant of the saturation function. For simplification, let us define

G = (1 + ‖ BK ‖ + (‖ B ‖ + 1) λ_{1})

, which is a constant matrix. Then, equation (15) can be written as

‖ x (t) ‖ \leq ‖ E_{v} (A_{c} t^{v}) x_{0} ‖ + G \int_{0}^{t} (t - τ)^{v - 1} ‖ E_{v, v} (A_{c} (t - τ)^{v}) ‖ ‖ x (τ) ‖ d τ

(16)

To guarantee the system stability, assume that A_c satisfies the stability condition, i.e. $| \arg (spec (A - BK)) | \geq \frac{v π}{2}$ . Now, we want to demonstrate that $x (t) \to 0$ as $t \to \infty$ . Assume that $t > 1$ . Then, from equation (3) and knowing that $k! = Γ (k + 1)$ , the above inequality can be written as

‖ x (t) ‖ \leq ‖ E_{v} (A_{c} t^{v}) x_{0} ‖ + G \int_{0}^{t} (e^{A^{\frac{1}{v}} (t - τ)} \sum_{k = 0}^{\infty} \frac{‖ A_{c}^{(\frac{(v - 1) k}{v})} ‖ Γ (k + 1)}{Γ (vk + v)} \times (t - τ) (v - 1) (k + 1) ‖ x (τ) ‖) d τ

(17)

Appling equation (4)

‖ x (t) ‖ \leq Q ‖ x_{0} ‖ e^{- ct} + G \int_{0}^{t} (e^{- c (t - τ)} \sum_{k = 0}^{\infty} \frac{‖ A_{c}^{(\frac{(v - 1) k}{v})} ‖ Γ (k + 1)}{Γ (vk + v)} \times (t - τ) (v - 1) (k + 1) ‖ x (τ) ‖) d τ

(18)

where $Q > 0$ and $c = k = 1 n | {eig}_{k} (A_{c}) | > 0$ . Multiplying both sides of equation (18) by e ^ct yields

‖ x (t) ‖ e^{ct} \leq Q ‖ x_{0} ‖ + G \int_{0}^{t} (\sum_{k = 0}^{\infty} \frac{‖ A_{c}^{(\frac{(v - 1) k}{v})} ‖ Γ (k + 1)}{Γ (vk + v)} \cdot (t - τ) (v - 1) (k + 1) ‖ x (τ) ‖ e^{c τ}) d τ

(19)

Using Corollary 1, i.e. $u_{1} (t) = ‖ x (t) ‖ e^{ct}, C = Q ‖ x_{0} ‖$ , and

H (τ) = G k = 0 \infty \frac{‖ A_{c}^{(\frac{(v - 1) k}{v})} ‖ Γ (k + 1)}{Γ (vk + v)} (t - τ) (v - 1) (k + 1)

we have

‖ x (t) ‖ e^{ct} \leq Q ‖ x_{0} ‖ \times \exp {G \sum_{k = 0}^{\infty} \frac{‖ A_{c}^{(\frac{(v - 1) k}{v})} ‖ Γ (k + 1)}{Γ (vk + v)} \int_{0}^{t} (t - τ)^{(v - 1) (k + 1)} d τ}

(20)

By solving the above integral, we obtain the following inequality

‖ x (t) ‖ e^{ct} \leq Q ‖ x_{0} ‖ \times \exp {G \sum_{k = 0}^{\infty} \frac{‖ A_{c}^{(\frac{(v - 1) k}{v})} ‖ Γ (k + 1)}{Γ (vk + v)} \cdot \frac{t^{(v - 1) (k + 1) + 1}}{(v - 1) (k + 1) + 1}}

(21)

Multiplying both sides of the above inequality by $e^{- ct}$ , the following result is obtained

‖ x (t) ‖ \leq Q ‖ x_{0} ‖ \times \exp {- ct + G \sum_{k = 0}^{\infty} \frac{‖ A_{c}^{(\frac{(v - 1) k}{v})} ‖ Γ (k + 1)}{Γ (vk + v)} \times \frac{t^{(v - 1) (k + 1) + 1}}{(v - 1) (k + 1) + 1}}

(22)

Based on the results given in Lim and Ahn (2012), if time tends to infinity, then $‖ x ‖$ converges to zero, i.e. $lim {no}_{t \to \infty} ‖ x (t) ‖ = 0$ , which implies the robust stability of the closed-loop system. This completes the proof.

Corollary 2

Assume that $‖ δ A ‖ \equiv 0$ in the closed-loop system given by equation (11). If K exists such that $| \arg (spec (A - BK)) | \geq \frac{v π}{2}$ , then the closed-loop system is robustly asymptotically stable.

Proof

The proof is similar to that of Theorem 1 by considering $‖ δ A ‖ \equiv 0$ .

Corollary 3

Suppose that $‖ δ B ‖ \equiv 0$ in the closed-loop system given by equation (11). If K exists such that $| \arg (spec (A - BK)) | \geq \frac{v π}{2}$ , then the closed-loop system is robustly asymptotically stable.

Proof

The proof is similar to that of Theorem 1 by considering $‖ δ B ‖ \equiv 0$ .

Remark 4

Many systems involve nonlinear relationships between the variables. In order to apply the linear control theory, it is required to linearize the equations of the system around an equilibrium point. Therefore, there is a mismatch between the real and nominal models. In the following, we develop the control method for a class of uncertain nonlinear FO systems. In this point of view, the system model can be described by the sum of the two terms, namely a linear term and a nonlinear term, as follows

C D_{t}^{v} x (t) = (A + δ A) x (t) + (B + δ B) Sat (u) + g (x, t)

(23)

where

| g (x_{1}, t) - g (x_{2}, t) | \leq λ_{2} | x_{1} - x_{2} |

(24)

In equation (24), $λ_{2} > 0$ is the Lipschitz constant of the nonlinear term. The following theorem can be applied to check the robust stability of the closed-loop systems given in equation (23).

Theorem 2

Consider the system described by equation (23) using the state feedback controller $u = - Kx$ . The overall closed-loop system is robustly asymptotically stable if $| \arg (spec (A - BK)) | \geq \frac{v π}{2}$ .

Proof

The entire closed-loop system with the feedback control $u = - Kx$ can be expressed as

C D_{t}^{v} x (t) = (A_{c} + δ A + BK) x (t) + (B + δ B) Sat (- Kx) + g (x, t)

(25)

Using the Laplace transform on equation (25), we have

X (s) = (I_{n} s^{v} - A_{c})^{- 1} \times (s^{v - 1} x_{0} + L ((δ A + BK) x (t) + (B + δ B) Sat (- Kx) + g (x, t)))

(26)

If the Laplace inverse is taken, then we obtain

x (t) = E_{v, 1} (A_{c} t^{v}) x_{0} + \int_{0}^{t} (t - τ)^{v - 1} E_{v, v} (A_{c} (t - τ)^{v}) ((δ A + BK) x (t) + (B + δ B) Sat (- Kx) + g (x, t)) d τ

(27)

Applying the norm operator on both sides of the above equation, it can be concluded that

‖ x (t) ‖ \leq ‖ E_{v} (A_{c} t^{v}) x_{0} ‖ + \int_{0}^{t} (t - τ)^{v - 1} ‖ E_{v, v} (A_{c} (t - τ)^{v}) ‖ \times (‖ (δ A + BK) ‖ ‖ x (τ) ‖ + ‖ (B + δ B) ‖ ‖ Sat (- Kx) ‖ + ‖ g (x, t) ‖) d τ

(28)

Using the triangular inequality and the Lipschitz condition of saturation function given in equations (24) and (5), respectively, we obtain

‖ x (t) ‖ \leq ‖ E_{v} (A_{c} t^{v}) x_{0} ‖ + \int_{0}^{t} (t - τ)^{v - 1} ‖ E_{v, v} (A_{c} (t - τ)^{v}) ‖ (‖ (δ A + BK) ‖ + ‖ (B + δ B) ‖ λ_{1} + λ_{2}) ‖ x (τ) ‖ d τ

(29)

In the sequel of the proof, we need to establish a bounded value for $‖ x (t) ‖$ , i.e.

‖ x (t) ‖ \leq ‖ E_{v} (A_{c} t^{v}) x_{0} ‖ + \int_{0}^{t} (t - τ)^{v - 1} ‖ E_{v, v} (A_{c} (t - τ)^{v}) ‖ (1 + ‖ BK ‖ + (‖ B ‖ + 1) λ_{1} + λ_{2}) ‖ x (τ) ‖ d τ

(30)

For simplification, let us define $G_{1} = (1 + ‖ BK ‖ + (‖ B ‖ + 1) λ_{1} + λ_{2})$ , which is a constant matrix. Then, equation (30) can be written as

‖ x (t) ‖ \leq ‖ E_{v} (A_{c} t^{v}) x_{0} ‖ + G_{1} \int_{0}^{t} (t - τ)^{v - 1} ‖ E_{v, v} (A_{c} (t - τ)^{v}) ‖ ‖ x (τ) ‖ d τ

(31)

The rest of the proof is the same as for Theorem 1. From the results reported in Lim and Ahn (2012), it can be deduced that if time t tends to infinity, then $‖ x (t) ‖$ converges to zero. In other words, $lim {no}_{t \to \infty} ‖ x (t) ‖ = 0$ , which implies the robust stability of the closed-loop system. This completes the proof.

4. Simulations

In this section, the performance of the proposed control method is investigated by means of two examples. It is worth mentioning that since this is the first research work regarding the design of a robust controller for uncertain FO systems under input saturation, there are no related works reported in the literature for comparison.

4.1. Example 1

Consider the mechanical system composed of an oscillator with both classical fluid damping of order one and fractional damping of order one-half as follows (Alaviyan Shahri and Balochian, 2014)

\overset{··}{z} + α \overset{\cdot}{z} + β (D_{t}^{v} z) + γ z = Sat (u)

(32)

In this example, the behavior of the system is studied in the presence of uncertainties based on Theorem 1, where the stability condition, i.e. $| \arg (spec (A - BK)) | \geq \frac{v π}{2}$ , must be held. With $v = 0.5$ and state vector $x = [z D_{t}^{0.5} z \overset{\cdot}{z} D_{t}^{1.5} z]$ , the dimension of the system is $n = 4$ . The resulting state-space representation is

A = [010000100001 - γ - β - α 0], B = [0001]

(33)

In order to consider the uncertainties, we assume that

δ A = 0.35 \times [1000010000100001] \sin (5 t) δ B = 0.20 \times [1111]^{T} \cos (t)

(34)

With $γ = 1, β = - 1.4$ , and $α = 0.8$ , the open-loop system is unstable. To stabilize the system, we consider the gain of the state feedback controller as $K = [510121]$ . The time responses of the open-loop and closed-loop system are shown in Figures 1 and 2, respectively.

Figure 1.

Trajectory of the open-loop system, $v = 0.5$ .

Figure 2.

Trajectory of the closed-loop system, $v = 0.5$ .

4.2. Example 2

Consider the parameters of the deterministic linear system adopted from Lim et al. (2013) as

A = [0.1 - 3 12], B = [5001]

(35)

The open-loop system is unstable. We consider the state feedback controller as $u = - Kx$ , while $K = [13 - 2 1]$ . It can be seen that the stability condition, i.e. $| \arg (spec (A - BK)) | \geq \frac{v π}{2}$ , is satisfied. We assume that $v = 0.45$ and the uncertainties into matrices A and B as

δ A = [1001] \sin (5 t), δ B = [1001] \cos (t)

(36)

and the Lipschitz nonlinear is

g (x, t) = 3 \sin (Cx)

where

C = [10]

. The behavior of the closed-loop system is depicted in Figure 3.

Figure 3.

Trajectory of the close-loop system, $v = 0.45$ .

5. Conclusion

In this paper we have investigated the robust stability and the stabilization for a class of uncertain FO systems with order lying in (0,1) under input saturation. Using the Lipschitz condition and the Gronwall–Bellman lemma, the sufficient condition for designing a state feedback controller to stabilize such a type of systems was derived. Numerical examples demonstrated the validity of the proposed approach.

Footnotes

Declaration of Conflicting Interests

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) received no financial support for the research, authorship, and/or publication of this article.

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