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Abstract

An adaptive backstepping fuzzy sliding mode control is proposed to approximate the unknown system dynamics for a cantilever beam in this paper. The adaptive backstepping fuzzy sliding mode control is developed by combining the backstepping method with adaptive fuzzy strategy, where backstepping design approach is used to drive the trajectory tracking errors to converge to zero rapidly with global asymptotic stability and fuzzy logic system is designed to approximate the unknown nonlinear function in the adaptive backstepping fuzzy sliding mode control. The proposed backstepping controllers can ensure proper tracking of the reference trajectory, and impose a desired dynamic behavior, giving robustness and insensitivity to parameter variations. Numerical simulation for cantilever beam is investigated to verify the effectiveness of the proposed adaptive backstepping fuzzy sliding mode control scheme and demonstrate the satisfactory vibration suppression performance.

Keywords

Cantilever beam backstepping control adaptive fuzzy control sliding mode control vibration suppression

Introduction

Recently, vibration control using piezoceramic material has been widely studied and applied in flexible structures. The advantages of piezoceramic material include low-power consumption, rapid response, compactness, and easy implementation. Usually flexible structures are lightly damped owing to small material damping and lack of air damping in space, making vibration inevitable. Therefore, active vibration control for space flexible structures is an important concern, a challenging problem for both academic and industrial researchers for many years. During the past few years, advanced control approaches have been proposed to control the vibration of flexible structures. Active vibration controllers of the smart beams with piezoelectric sensors and actuators have been proposed in Ge et al.¹ and Fallah and Ebrahimnejad.² Ge et al.¹ used an adaptive boundary control technique to suppress the vibration of an Euler–Bernoulli beam system. Fallah et al.² proposed a finite volume formulation for the free vibration analysis and active vibration control of the smart beams with piezoelectric sensors and actuators. Adaptive pole placement controller³ and adaptive controllers^4–6 have been developed for the vibration control of flexible structures using piezoceramic sensor and actuator. Zhang et al.³ developed a finite element model of a piezoelectric actuator and cantilever in a thermal environment to suppress vibration effectively.

As we know, sliding mode control (SMC) has many attractive features such as robustness to parameter variations and insensitivity to disturbance. The basic idea of the SMC is to drive and maintain the system trajectory on a sliding surface designed a priori in the state space.

It is reasonable to combine the adaptive control with sliding mode control in practical application. Some adaptive sliding mode controllers using smart materials^7–11 have been investigated for vibration control of flexible structures. Wang et al.⁷ presented an experimental study of an adaptive robust sliding-mode control scheme based on the Lyapunov’s direct method for active vibration control of a flexible beam using PZT sensor and actuator. Oveisi et al.¹⁰ proposed an adaptive sliding mode vibration controller for a nonlinear smart beam and implemented a comparison with self-tuning Ziegler–Nichols proportional–integral–derivative controller. Azadi et al.¹¹ presented a hybrid adaptive sliding mode controller for both the rotational maneuver and the vibration control of smart flexible appendages of a satellite moving in a circular orbit.

In the last few years, fuzzy control has been extensively applied in a wide variety of industrial systems and consumer products because of its model-free approach. Fuzzy control has achieved great practical successes in industrial processes and many other fields. Lee et al.¹² presented active vibration control of a cantilever beam structural system by combining the adaptive input estimation method with the fuzzy robust controller. Nasser et al.¹³ used a nonlinear fuzzy controller for active vibration damping of composite structures. Adaptive fuzzy sliding mode based active vibration controller for flexible structures were discussed in Li et al.,¹⁴ Nguyen et al.,¹⁵ and Qiu et al.^16,17 Li et al.¹⁴ presented an adaptive fuzzy sliding mode based active vibration control where fuzzy logic-based controller was applied to active vibration control of a smart flexible beam with mass uncertainty through experimental studies. Qiu et al.¹⁸ designed the design and implementation of a fuzzy sliding mode control algorithm and a composite controller to dampen the vibration of a flexible manipulator with a flexible link and a harmonic drive gear. Adaptive neuro fuzzy system and fuzzy least square support vector machine based on adaptive variable chaos immune algorithm have been discussed in Watany and Eltantawie¹⁹ and Qian and Zhu²⁰ for flexible structures. Adaptive sliding mode control with new double-loop recurrent neural network for dynamic systems was investigated in Fei and Lu.²¹

Backstepping control is an important tool for nonlinear system owing to the recursive and systematic controller design procedure for strict-feedback systems. However, robust adaptive fuzzy control using backstepping design for flexible structure has not been investigated, the ABFSMC approach has not been applied to cantilever beam yet. In this paper, an adaptive backstepping fuzzy sliding mode control (ABFSMC) approach is designed to realize the vibration tracking for cantilever beam. The dynamic equations of cantilever beam are transformed into analogically cascade system that can be implemented using backstepping control. The advantages of the proposed controller compared to the existing ones can be summarized as follows:

Backstepping control, adaptive control, fuzzy control, and sliding mode control are combined and applied to cantilever beam. The fuzzy control method combined with the adaptive control for cantilever beam not only avoids dependence on the system model, but also makes the algorithm obtain the self-learning ability and adjust the fuzzy parameters. Hence, the ABFSMC approach not only removes some of the fundamental limitations of the traditional approach but also provides improved tracking accuracy.

The proposed sliding mode control adds additional compensator for achieving and improving the system stability, hence obtaining desired system behavior and performance. Thus, the entire closed-loop system meets the expectations indicators of dynamic and static performance and achieves accurate vibration tracking performance.

Backstepping design for a class of systems satisfying the strict feedback form can relax the matching condition appeared in the design of sliding mode control. In addition, backstepping design can avoid the cancellation of useful nonlinearities. The ideal of backstepping is to design a controller recursively by considering some of the state variables as “virtual controls” and designing intermediate control laws to improve the robustness of the cantilever beam. Therefore the proposed ABFSMC approach attenuates the model uncertainties and external disturbances.

The paper is organized as follows. In the upcoming section, the dynamic equation of piezoceramic cantilever beam is established. Next, an adaptive backstepping fuzzy sliding mode controller method is derived to guarantee the stability of the closed-loop system and good tracking performance. Later, simulation examples are presented to illustrate the excellent performance of the proposed controller. Conclusions are given in the last section.

Model of piezoceramic cantilever beam

As shown in Figure 1, the length, width, and thickness of the cantilever beam is $L_{b}, b$ _, and $t_{b}$ _, respectively. The piezoceramic actuator and sensor are symmetrically pasted on the top and bottom of cantilever beam, the length, width and thickness of the piezoceramic element are respectively $L_{p}, b$ and $t_{p}$ . $X_{1}$ and $X_{2}$ are the distances between both ends of the piezoceramic and fixed ends of cantilever beam, respectively.

Figure 1.

Model of piezoceramic cantilever beam.

We assume that the piezoceramic patches are pasted well on the cantilever beam and the stickup layer of piezoceramic patches piezoceramic has no effect on the dynamic characteristics of the cantilever beam. According to the direct piezoceramic effect, the output charge of piezoceramic sensor, which is caused by the beam deflection, is

Q (t) = \frac{1}{2} d_{_{31}} t_{b} E_{p} \int_{x_{1}}^{x_{2}} \frac{\partial^{2} w (x, t)}{\partial x^{2}} d x

(1)where

d_{31}

is the piezoceramic constant,

E_{p}

is the spring modulus of piezoceramic patches, and

w

is the beam deflection.

Under the action of the input voltage, the piezoceramic actuator torque for the cantilever beam is

M (x, t) = K_{a} U_{a} (t) [h (x - x_{1}) - h (x - x_{2})]

(2)where

h (x)

is the Heaviside step function,

U_{α} (t)

is the input voltage,

K_{a} = \frac{1}{2} b d_{31} E_{_{p}} (t_{b} + t_{p})

is the coupling coefficient.

The dynamics of cantilever beam vibration is as follows

ρ_{1} \frac{\partial^{2} w (x, t)}{\partial t^{2}} + c \frac{\partial w (x, t)}{\partial t} + \frac{\partial^{2}}{\partial x^{2}} (E I \frac{\partial^{2} w (x, t)}{\partial t^{2}}) = \frac{\partial^{2}}{\partial x^{2}} M (x, t)

(3)where

ρ_{_{1}}, c, E I

are the linear density, damping coefficient, and flexural rigidity of the beam, respectively.

When the cantilever beam does flexural vibration in the xy plane, by means of the preceding N modals of the cantilever beam, the deflection of the beam $w$ can be expressed as

w (x, t) = \sum_{j = 1}^{n} φ_{i} (x) q_{i} (t) = φ q

(4)where

φ = [φ_{1} (x), φ_{2} (x), … φ_{n} (x)]

is the mass-normalized orthogonal modal matrix,

q = [q_{1} (x), q_{2} (x), q_{n} (x)]

is the modal coordinates vector, thus the vibration of the free end of the cantilever beam is

y = w (L_{b}, t) = \sum_{i = 1}^{n} φ_{i} (L_{b}) q_{i} (t) = φ_{i} q

(5)

The output voltage of piezoceramic sensors can be expressed as

U_{s} (t) = \frac{Q (t)}{C_{p}} = \frac{d_{31} t_{b} E_{p}}{2 C_{p}} \sum_{i = 1}^{n} [φ_{i}^{'} (x_{2}) - φ_{i}^{'} (x_{1})] q_{i} (t) = \sum_{i = 1}^{n} G_{i} q_{i} (t) = G q

(6)where

C_{p}

is the capacitance of piezoceramic patch,

G = [G_{1}, G_{2}, … G_{n}]

G_{i} = K_{s} [φ_{i}^{'} (x_{2}) - φ_{i}^{'} (x_{1})]

, and

K_{s} = \frac{d_{31} t_{b} E_{p}}{2 C_{p}}

Using equations (3) and (4), modal equation of motion of piezoceramic smart cantilever beam under the action of a piezoceramic actuator can be expressed as

{\ddot{q}}_{i} (t) + 2 ξ_{i} w_{i} {\dot{q}}_{i} (t) + w_{i}^{2} q_{i} (t) = B_{i} U_{a}

(7)where

ξ_{i}

is the ith order structure damping,

ω_{i}

is the ith natural frequency,

B_{i} = K_{a} [φ_{i}^{'} (x_{2}) - φ_{i}^{'} (x_{1})]

Design of adaptive backstepping fuzzy sliding mode controller

In this section, the ABFSMC approach is designed for the vibration control of cantilever beam in the presence of the unknown system parameters of cantilever beam. The control target is to achieve real-time compensation for the model uncertainties and external disturbance. The ABFSMC shown in Figure 2 is proposed.

Figure 2.

Block diagram of an adaptive backstepping fuzzy sliding mode vibration control.

The vector form of piezoceramic cantilever beam dynamics model can be written as

\ddot{q} + C \dot{q} + K q = u

(8)where

C, K \in R^{i \times i}

C

is the damping term,

K

is the frequency term,

u

is the input term, and

u = B_{i} U_{a}

Defining $x_{1} = q, x_{2} = \dot{q}$ , the cantilever beam model (8) can be rewritten as

{\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = - C x_{2} - K_{b} x_{1} + u \end{matrix}

(9)

Considering the model uncertainties and external disturbance, equation (9) can be expressed as

\begin{matrix} {\dot{x}}_{2} = [- C + Δ A_{1}] x_{2} + (- K_{b} + Δ A_{2}) x_{1} + (1 + Δ B) u + η = f (x) + u + H (t) \end{matrix}

(10)where

f (x) = - C x_{2} - K_{b} x_{1}

, ΔA₁, ΔA₂, ΔB are the model uncertainties of the dynamic model for cantilever beam,

η

is the external disturbances of cantilever beam.

H (t)

includes model uncertainties and external disturbances of cantilever beam, which can be expressed as

H (t) = Δ A_{1} x_{2} + Δ A_{2} x_{1} + Δ B u + η

Then, the reference model is defined as $r$ , and the tracking error is defined as

e_{1} = x_{1} - r

(11)

Defining virtual control force $α$ as

α = - c_{1} e_{1} + \dot{r}

(12)where

c_{1} > 0

Then defining virtual error as

e_{2} = x_{2} - α

(13)

The first Lyapunov function candidate is chosen as $V_{1} = \frac{1}{2} e_{1}^{T} e_{1}$

The time derivative of the $V_{1}$ is

{\dot{V}}_{1} = e_{1}^{T} {\dot{e}}_{1} = e_{1}^{T} (x_{2} - \dot{r}) = e_{1}^{T} (e_{2} - c_{1} e_{1}) = - c_{1} e_{1}^{T} e_{1} + e_{1}^{T} e_{2}

(14)

When $e_{2} = 0$ , it is easy to know that the equation ${\dot{V}}_{1} = - c_{1} e_{1}^{T} e_{1} \leq 0$ . So the system is global asymptotic stable and the error $e_{1} = x_{1} - r$ asymptotically converges to zero.

Define the second Lyapunov function as

V_{2} = V_{1} + \frac{1}{2} s^{T} s

(15)where

s

is the function of sliding surface defined as

s = λ e_{1} + e_{2}

(16)where

λ

is a positive constant.

The time derivative of $s$ is

\begin{array}{l} \dot{s} = λ {\dot{e}}_{1} + {\dot{e}}_{2} \\ = λ (e_{2} - c_{1} e_{1}) + ({\dot{x}}_{2} - \dot{α}) \\ = λ (e_{2} - c_{1} e_{1}) - C x_{2} - K_{b} x_{1} + u + H (t) + c_{1} (e_{2} - c_{1} e_{1}) - \ddot{r} \\ = (λ + c_{1}) (e_{2} - c_{1} e_{1}) - C x_{2} - K_{b} x_{1} + u + H (t) - \ddot{r} \end{array}

(17)

The time derivative of $V_{2}$ is

\begin{array}{l} {\dot{V}}_{2} = - c_{1} e_{1}^{T} e_{1} + e_{_{1}}^{T} e_{2} + s^{T} \dot{s} \\ = - c_{1} e_{1}^{T} e_{1} + e_{1}^{T} e_{2} + s^{T} ((λ + c_{1}) (e_{2} - c_{1} e_{1}) - C x_{2} - K_{b} x_{1} + u + H (t) - \ddot{r}) \\ = - c_{1} e_{1}^{T} e_{1} + s^{T} (\frac{s}{{‖ s ‖}^{2}} e_{1}^{T} e_{2} + (λ + c_{1}) (e_{2} - c_{1} e_{1}) + f (x) + u + H (t) - \ddot{r}) \end{array}

(18)where

f (x) = - C x_{2} - K_{b} x_{1}

In the design of the sliding mode controller, we use exponential reaching law as

\dot{s} = - ρ s - k sgn (s)

(19)where

ρ

and

k

are positive constants.

By equations (18) and (19), we design a sliding mode control law as

u = - (\frac{s}{{‖ s ‖}^{2}} (e_{1}^{T} e_{2} + φ) + (λ + c_{1}) (e_{2} - c_{1} e_{1}) - \ddot{r}) - H_{\max} - ϕ - ρ s - k sgn (s)

(20)where

φ > 0

H_{\max}

is the upper limit of

H (t)

, and

ϕ

is the fuzzy function .

Remark. In order to eliminate the chattering, the discontinuous control component in equation (20) can be replaced by a smooth sliding mode component $\tanh (s)$ , which can create a small boundary layer about the switching surface in which the system trajectory will remain.

Substituting equations (20) into (¹⁸) yields

\begin{array}{l} {\dot{V}}_{2} = = - c_{1} e_{1}^{T} e_{1} - φ + s^{T} [H (t) - H_{\max} + f (x) - ϕ - ρ s - k sgn (s)] \\ = - c_{1} e_{1}^{T} e_{1} - φ + s^{T} [H (t) - H_{\max}] + s^{T} (f (x) - ϕ) - ρ s^{T} s - k s^{T} sgn (s) \end{array}

(21)

The ABFSMC system still requires detailed information of cantilever beam explicitly in terms of nonlinear compensation, which makes it difficult to be implemented in practical situations. Hence, in allusion to the vibration control of cantilever beam with unknown model uncertainties and external disturbances, an adaptive fuzzy control algorithm is synthesized in this section, which utilizes the advantages of fuzzy control to approximate and compensate for the unknown uncertainties.

Observing the expression $f (x)$ , it is a nonlinear function containing the dynamic model information of cantilever beam. In order to control the system without model information, fuzzy system is used to approach $f (x)$ . Designing $ϕ$ which is a fuzzy system used to approach $f (x)$ through the means of a single value fuzzy, multiplied reasoning machine and center of gravity average defuzzification.

The fuzzy system is constituted by $N$ fuzzy rules, where the $i$ fuzzy rule as

$R^{i}$ : IF $x_{1}$ is $μ_{1}^{i}$ and …. $x_{n}$ is $μ_{n}^{i}$ , then $y$ is $B^{i}$ $(i = 1, 2, ……., N)$ where $μ_{j}^{i}$ is the membership function of $x_{j} (j = 1, 2, ……., n)$ .

So the output of the fuzzy system is

y = \frac{\sum_{i = 1}^{N} θ_{i} \prod_{j = 1}^{n} μ_{j}^{i} (x_{j})}{\sum_{i = 1}^{N} \prod_{j = 1}^{n} μ_{j}^{i} (x_{j})} = ξ^{T} θ

(22)where

ξ = {[\begin{matrix} ξ_{1} (x) & ξ_{2} (x) & … & ξ_{N} (x) \end{matrix}]}^{T}

ξ_{i} (x) = \frac{\prod_{j = 1}^{n} μ_{j}^{i} (x_{j})}{\sum_{i = 1}^{N} \prod_{j = 1}^{n} μ_{j}^{i} (x_{j})}

θ = {[\begin{matrix} θ_{1} & θ_{2} & … & θ_{N} \end{matrix}]}^{T}

Defining the fuzzy function as

ϕ = ξ^{T} (x) θ

(23)where

ξ^{T} (x)

is the membership function and

θ

is the adaptive fuzzy parameter.

We define the optimal approximation constant $θ^{*}$ . For a given small arbitrarily constant $ε (ε > 0)$ , inequality expression $‖ f - ξ^{T} θ^{*} ‖ \leq ε$ holds.

Theorem 1. If the updated control law (20), with the adaptation law of the fuzzy system designed as equation (24), is applied to the nonlinear uncertain system such as flexible structure defined by equation (8), then the system’s tracking error can converge to zero and the vibration can be suppressed . And the unknown model uncertainties and external disturbances can be approximated by fuzzy system so as to improve the robustness of the dynamic system.

\dot{θ} = τ ξ (x) s

(24)where

τ

is the scalar design parameter and

τ > 0

Proof. Let us consider the following positive definite function as a Lyapunov function candidate

\begin{array}{l} V = \frac{1}{2} e_{1}^{T} e_{1} + \frac{1}{2} s^{T} s + \frac{1}{2 τ} {\tilde{θ}}^{T} \tilde{θ} \\ = V_{2} + \frac{1}{2 τ} {\tilde{θ}}^{T} \tilde{θ} \end{array}

(25)where

\tilde{θ} = θ^{*} - θ

Differentiating equation (25) yields

\begin{array}{l} \dot{V} = {\dot{V}}_{2} - \frac{1}{τ} {\tilde{θ}}^{T} \dot{θ} \\ = - c_{1} e_{1}^{T} e_{1} - φ + s^{T} [H (t) - H_{\max}] + s^{T} (f (x) - ϕ) \\ - ρ s^{T} s - k s^{T} sgn (s) - \frac{1}{τ} {\tilde{θ}}^{T} \dot{θ} \\ = - c_{1} e_{1}^{T} e_{1} - φ - ρ s^{T} s - k s^{T} sgn (s) + s^{T} [H (t) - H_{\max}] \\ + s^{T} (f (x) - ξ^{T} (x) θ^{*}) + s^{T} (ξ^{T} (x) θ^{*} - ξ^{T} (x) θ) - \frac{1}{τ} {\tilde{θ}}^{T} \dot{θ} \\ = - c_{1} e_{1}^{T} e_{1} - φ - ρ s^{T} s - k s^{T} sgn (s) + s^{T} [H (t) - H_{\max}] \\ + s^{T} (f (x) - ξ^{T} (x) θ^{*}) + s^{T} ξ^{T} (x) \tilde{θ} - \frac{1}{τ} {\tilde{θ}}^{T} \dot{θ} \\ = - c_{1} e_{1}^{T} e_{1} - φ - ρ s^{T} s - k s^{T} sgn (s) + s^{T} [H (t) - H_{\max}] \\ + s^{T} (f (x) - ξ^{T} (x) θ^{*}) + {\tilde{θ}}^{T} (ξ (x) s - \frac{1}{τ} \dot{θ}) \end{array}

(26)

Since

| (\vec{x}, \vec{y}) | \leq | \vec{x} | | \vec{y} |

(27)

Then equation (26) becomes

\begin{matrix} \dot{V} \leq - c_{1} e_{1}^{T} e_{1} - φ - ρ s^{T} s - k s^{T} sgn (s) + | s^{T} | \cdot | H (t) - H_{\max} | + | s^{T} | \cdot | f (x) - ξ^{T} (x) θ^{*} | + {\tilde{θ}}^{T} (ξ (x) s - \frac{1}{τ} \dot{θ}) \end{matrix}

(28)

Since

| \vec{x} | \cdot | \vec{y} | \leq \frac{1}{2} ({| \vec{x} |}^{2} + {| \vec{y} |}^{2})

(29)

Then equation (28) becomes

\begin{matrix} \dot{V} \leq - c_{1} e_{1}^{T} e_{1} - φ - ρ s^{T} s - k s^{T} sgn (s) + \frac{1}{2} {| s^{T} |}^{2} + \frac{1}{2} {| H (t) - H_{\max} |}^{2} + \frac{1}{2} {| s^{T} |}^{2} + \frac{1}{2} {| f (x) - ξ^{T} (x) θ^{*} |}^{2} + {\tilde{θ}}^{T} (ξ (x) s - \frac{1}{τ} \dot{θ}) \\ \leq - c_{1} e_{1}^{T} e_{1} - φ - ρ s^{T} s - k s^{T} sgn (s) + s^{T} s + \frac{1}{2} {| H (t) - H_{\max} |}^{2} + \frac{1}{2} ε^{2} + {\tilde{θ}}^{T} (ξ (x) s - \frac{1}{τ} \dot{θ}) \\ \leq - c_{1} e_{1}^{T} e_{1} - (ρ - 1) s^{T} s - k s^{T} sgn (s) - (φ - \frac{1}{2} {| H (t) - H_{\max} |}^{2} - \frac{1}{2} ε^{2}) + {\tilde{θ}}^{T} (ξ (x) s - \frac{1}{τ} \dot{θ}) \end{matrix}

(30)

Substituting the parameter adaptation law (24) into equation (30), we obtain

\dot{V} \leq - c_{1} e_{1}^{T} e_{1} - (ρ - 1) s^{T} s - k s^{T} sgn (s) - (φ - \frac{1}{2} {| H (t) - H_{\max} |}^{2} - \frac{1}{2} ε^{2})

(31)which is guaranteed positive as long as

ρ \geq 1

φ \geq \frac{1}{2} {| H (t) - H_{\max} |}^{2} + \frac{1}{2} ε^{2}

Consequently, $\dot{V}$ is the negative semi-definite ensuring that $V$ , $e_{1}$ , $s$ , $θ$ are all bounded. Therefore, asymptotical stability of the designed system can be guaranteed. Thus, the ABFSMC approach can adaptively control the cantilever beam and make the cantilever beam system track the desired reference trajectory quickly.

Simulation study

According to the dynamic characteristics, the main vibration is determined by the structure’s first several modes. To illustrate the simulation of this flexible structure vibration suppression, here we only choose the first mode system. In this section, we will evaluate the proposed ABFSMC scheme, which will require no knowledge of the dynamics, not even their structure on the cantilever beam model. The control objective is to make the cantilever beam system track the desired reference trajectory and estimate the unknown function $f (x)$ . The parameters of the cantilever beam are chosen as: $C = 0.18, K = 56.4$ . The desired trajectories are described by $q = 0, \dot{q} = 0, \ddot{q} = 0$ . Initial conditions are $q (0) = {[\begin{matrix} 0.1 & 0.1 \end{matrix}]}^{T}$ , other parameters are selected as $c_{1} = 20000$ , $λ = 20000$ , $τ = 2$ , $k = 1000, φ = 250, ρ = 10, H_{\max} = 10$ . The membership functions are selected as $μ_{F_{i}}^{1} = \exp [- 0.5 {((x_{i} + A_{i} / 2) / (A_{i} / 4))}^{2}]$ , $μ_{F_{i}}^{2} = \exp [- 0.5 {(x_{i} / (A_{i} / 4))}^{2}]$ , $μ_{F_{i}}^{3} = \exp [- 0.5 {((x_{i} - A_{i} / 2) / (A_{i} / 4))}^{2}]$ , $(i = 1, 2)$ .

where $A_{i}$ is the amplitude of the reference trajectory, and $A_{1} = 1, A_{2} = 1.2$ .

Figure 3 shows the membership function of the control system.

Figure 3.

The membership function of fuzzy system.

Simulation study with steady and transient disturbances

White noise chosen as $H (t) = r a n d n (1, 1)$ is added all the time as the model uncertainties and external disturbances of the cantilever beam, and the control action is implemented after 10 s. Figure 3 shows the membership function of the control system. Figures 4 and 5 depict the vibration trajectories of cantilever beam using the ABFSMC law under steady disturbance. Figure 6 shows the adaptive fuzzy parameter under steady disturbance, which can reach a stable value in finite time.

Figure 4.

Vibration tracking under steady disturbance.

Figure 5.

A comparison of the vibration control performance between ABFSMC and without control under steady disturbance.

Figure 6.

The adaptive fuzzy parameter under steady disturbance.

Then white noise chosen as $H (t) = r a n d n (1, 1)$ is added during 0–10 s, and the control action is implemented after 10 s. Figures 7 and 8 show the vibration trajectories of cantilever beam using the ABFSMC law under transient disturbance. It can be observed intuitively from Figures 4, 5, 7, 8 that the actual motion trajectory of the cantilever beam is consistent with the desired reference trajectory in a shorter finite time, demonstrating that the adaptive backstepping tracking performance with fuzzy compensator is satisfactory as expected. Figure 9 shows the adaptive fuzzy parameter under transient disturbance which can reach a stable value in finite time. Figure 10 depicts vibration tracking comparisons of ABFSMC under steady disturbance with ABFSMC under transient disturbance. It can be observed that the trajectory under transient disturbance is much smoother.

Figure 7.

Vibration tracking under transient disturbance.

Figure 8.

A comparison of the vibration control performance between the ABFSMC and without control under transient disturbance.

Figure 9.

The adaptive fuzzy parameter under transient disturbance.

Figure 10.

A comparison of the vibration control performance between ABFSMC under steady disturbance and ABFSMC under transient disturbance.

In order to demonstrate the advantages of the proposed controller, a comparable investigation is accomplished between the proposed ABFSMC and conventional proportional–derivative (PD) control applied to the uncertain cantilever beam, where the vibration tracking corresponding to ABFSMC decrease obviously in contrast with the conventional PD control. Besides, from Figures 11 and 12, it can be also seen that the vibration tracking of ABFSMC converge to zero in about 1 s and the trajectory is much smoother, improving the dynamic behavior of the cantilever beam and verifying that the designed control law can ensure the stability of the vibration control system.

Figure 11.

Vibration tracking comparisons of PD control with ABFSMC under steady disturbance.

Figure 12.

Vibration tracking comparisons of PD control with ABFSMC under transient disturbance.

Simulation study with large steady and transient disturbances

In order to testify the vibration control performance and robustness of the vibration control system in the presence of large disturbance white noise disturbance $H (t) = 1000 r a n d n (1, 1)$ is added all the time, and the control action is implemented after 10 s. Figures 13 and 14 depict the cantilever beam tracking trajectories using the ABFSMC law under large steady disturbance. Figure 15 shows the adaptive fuzzy parameter under large steady disturbance, which can reach a stable value in finite time.

Figure 13.

Vibration tracking under steady disturbance.

Figure 14.

A comparison of the vibration control performance between ABFSMC and without control under steady disturbance.

Figure 15.

The adaptive fuzzy parameter under steady disturbance.

Similarly as in the previous section, white noise disturbance $H (t) = 1000 r a n d n (1, 1)$ is added during 0–10 s, and the control action is implemented after 10 s. Figures 16 and 17 draw the trajectories of cantilever beam using the ABFSMC law under large transient disturbance. It can be observed intuitively from Figures 13, 14, 16, and 17 that the actual motion trajectory of the cantilever beam is consistent with the desired reference trajectory in a shorter finite time, demonstrating that the adaptive backstepping tracking performance with fuzzy compensator is satisfactory as expected. Figure 18 shows the adaptive fuzzy parameter under large transient disturbance, which can reach a stable value in finite time. Figure 19 depicts vibration tracking comparisons of ABFSMC under steady disturbance with ABFSMC under large transient disturbance. It can be observed that the trajectory under large transient disturbance is much smoother.

Figure 16.

Vibration tracking under transient disturbance.

Figure 17.

A comparison of the vibration control performance between ABFSMC and without control under transient disturbance.

Figure 18.

The adaptive fuzzy parameter under transient disturbance.

Figure 19.

A comparison of the vibration control performance between ABFSMC under steady disturbance and ABFSMC under transient disturbance.

In comparison to PD control, from Figures 20 and 21, it can be also seen that the vibration tracking of ABFSMC converge to zero in about 1 s and the trajectory is much smoother, which improves the dynamic behavior of the cantilever beam and verifies that the designed control law can ensure the stability of the system.

Figure 20.

Vibration tracking comparisons of PD control with ABFSMC under steady disturbance.

Figure 21.

Vibration tracking comparisons of PD control with ABFSMC under transient disturbance.

Conclusion

An adaptive backstepping sliding mode controller using a fuzzy system to approximate the unknown system dynamics for a cantilever beam is presented. In the presence of unknown model uncertainties and external disturbances, sliding mode controller is employed to compensate such system nonlinearities and improve the tracking performance. The fuzzy algorithm is designed to guarantee bounded tracking errors. A backstepping controller is designed recursively by considering some of the state variables as “virtual controls” and designing intermediate control law to improve the robustness of the cantilever beam. Numerical simulation for cantilever beam is investigated to verify the effectiveness of the proposed ABFSMC scheme and demonstrate the satisfactory tracking performance and robustness. It is clearly shown that the function of the cantilever beam can be effectively approximated using the ABFSMC and the tracking error of the proof mass can be greatly improved.

Generally, controller spillover phenomenon occurs because the unmodeled dynamics, a robust control strategy based on H-infinity mixed sensitivity can suppress spillover instability. In the case of the unmeasured state, a state observer can be designed to get the feedback signals. In the next step, vibration control results will be presented in the frequency domain to check the stiffness and damping effect of the proposed controller. Other future works will be the experimental investigation and performance evaluation of the proposed controller in real-time system.

Footnotes

Acknowledgement

The authors thank the anonymous reviewers for their informative comments that have improved the quality of this paper.

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) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work was partially supported by National Science Foundation of China [Grant No. 61374100]; Natural Science Foundation of Jiangsu Province [Grant No. BK20171198].

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Adaptive backstepping fuzzy sliding mode vibration control of flexible structure

Abstract

Keywords

Introduction

Model of piezoceramic cantilever beam

Design of adaptive backstepping fuzzy sliding mode controller

Simulation study

Simulation study with steady and transient disturbances

Simulation study with large steady and transient disturbances

Conclusion

Footnotes

Acknowledgement

Declaration of conflicting interests

Funding

References