Optimal design for fractional-order active isolation system

Abstract

The optimal control of fractional-order active isolation system is researched based on the optimal control theory, and the effect of fractional-order derivative on passive isolation system is also analyzed. The mechanical model is established where viscoelastic features of isolation materials are described by fractional-order derivative. The viscoelastic property of the fractional-order derivative in dynamical system is studied and the fractional-order derivative could be divided into linear stiffness and linear damping. It is found that both the fractional coefficient and the fractional order could affect not only the resonance amplitude through the equivalent linear damping coefficient but also the resonance frequency by the equivalent linear stiffness. Based on optimal control theory, the feedback gain of fractional-order active isolation system under harmonic excitation is obtained, which is changed with the excitation frequency. The statistical responses of the displacement and velocity for passive and active vibration isolation systems subjected to random excitation are also presented, which further verifies the excellent performance of fractional-order derivative in vibration control engineering.

Introduction

The concept of fractional-order derivative was first presented by Hospital and Leibniz in the late 1600s. Because there was no support of physics, mechanics, and other background disciplines, fractional-order derivative was merely studied as a purely theoretical question of mathematics. After 300 years of development, many scholars had tried their best to study the definition, approximation calculation, mathematical properties, and application of fractional-order derivative,^1–6 and fractional-order optimal control problems had been paid more and more attention.^7–11 In recent years, the field of the fractional-order derivative has attracted interest in several areas including physics, chemistry, engineering, and even finance and social sciences.^12–17 It has received an increasing interest due to the fact that fractional-order derivative could reflect a lot of natural phenomena more reasonably.

In the engineering problems, fractional-order derivative was generally introduced into the control system, so that it could improve the control performance and robustness of the system. The fractional-order derivative can also be used to describe viscoelastic features of advanced materials or dissipative forces in structural dynamics^18,19 and the processes with power-law memory. ²⁰ In 1991, VE Tarasov ²¹ proposed a dynamical system governed by fractional-order differential equations, which could be considered as an open (non-isolated) system with memory. It had been found that fractional-order derivative was more appropriate than traditional integer-order counterpart in describing the viscoelastic constitutive relation through numerous studies in recent years. The concept of fractional-order derivative was used in the development of a force–displacement relationship for viscous dampers by Oldham and Spanier ²² in 1974. And the fractional-order derivative within the context of viscoelasticity was used as early as 1936 by Gemant, ²³ who first proposed the fractional-order derivative models for viscoelastic materials. In 1991, Makris and Constantinou ²⁴ proposed a fractional-derivative Maxwell model for viscous dampers, and recently, Liu et al. ²⁵ proposed a fractional-derivative Bingham model for magnetorheological (MR) damper. All this suggested that fractional-order derivative could be used in modeling of the viscoelastic materials, and generally, the fractional-order derivative could demonstrate both stiffness and damping effects. Accordingly, the force–displacement relationship of viscoelastic materials in vertical motion could be expressed as $F_{\mathrm{vm}}=k_0{\mathrm{D}}_t^{p}x$ , in which $F_{\mathrm{vm}}$ is the force, x is the displacement, $k_0$ is the fractional coefficient, p is the fractional order, and ${\mathrm{D}}_t^{p}x$ is the p-order derivative of $x(t)$ to t.

Analytical research on fractional-order system is important. Recently, Shen et al.^14–16,26 obtained approximately analytical solutions of some fractional-order dynamical systems based on averaging method, and they verified that fractional-order derivative should be considered as a damping and stiffness factor simultaneously in dynamical system. Optimal control theory for the integer-order isolation system had acquired more mature development, and the implementation methods for optimal control had been provided in some literatures.^27–31 However, only limited work has been done in the area of fractional-order systems optimal control.^7–11 Because the calculation of fractional-order derivative was very complex, many scholars had adopted numerical method,^5,6,9 which was another main direction of study on fractional-order derivative.

In this article, the object is the optimal control of a fractional-order active isolation system. The dynamical model is established in section “Dynamic modeling of fractional-order active isolation system.” The damping and stiffness of fractional-order isolation system will change with the excitation frequency, which is different from the traditional integer-order isolation system, so that the feedback gain will also change with the excitation frequency. It is difficult to deal with fractional-order derivative when studying fractional-order system by optimal control theory. Based on the heuristic level in Shen et al.,^14–16,26 the fractional-order derivative is studied in section “Processing of fractional-order derivative.” The optimal control aim is to minimize the displacement and velocity amplitude with different excitation frequency. Fractional-order passive and active isolation systems are analyzed in sections “Analysis on fractional-order passive vibration isolation system” and “Analysis of fractional-order active isolation system,” respectively. Finally, in order to further illustrate the control performance, the response of active isolation system is compared with that of passive isolation system subjected to random excitation. The results show that active vibration isolation system can theoretically reach the designed control target and provide a simple idea for optimal control of the fractional-order dynamical system, which could be used to solve similar problems in engineering applications.

Dynamic modeling of fractional-order active isolation system

In this article, the study object is a single degree-of-freedom fractional-order active isolation system shown in Figure 1. According to Newtonian second law, one could obtain the motion differential equation

$$m\stackrel{··}{x}+c_1\stackrel{·}{x}+k_{1}x+k_2{\mathrm{D}}_t^{p}x+u=F$$

(1)

Figure 1.

Mechanical model of single degree-of-freedom isolation system.

where m, $k_1$ , and $c_1$ are the system mass, linear stiffness coefficient, and damping coefficient, respectively. Fractional-order derivative in the system may come from the viscoelasticity of isolation devices. The force–displacement relationship of viscoelastic materials in vertical motion could be expressed as $F_{\mathrm{vm}}=k_2{\mathrm{D}}_t^{p}x$ , in which $F_{\mathrm{vm}}$ is the force, x is the displacement of system, and ${\mathrm{D}}_t^{p}x$ is the p-order derivative of $x(t)$ to t with the fractional coefficient $k_2(k_2>0)$ and the fractional order $p(0\le p\le 1)$ . u is the active control force, and $F=F_0\sin (\omega t)$ is the harmonic excitation to the system. Based on the Introduction, the fractional-order derivative will exhibit both the stiffness and damping effect.

In order to use optimal control theory to find the explicit form of the optimal control force, equation (1) should be rewritten as the form of state equation. Considering state vector as

$$X={\left(\begin{array}{c}\hfill x\stackrel{·}{x}\hfill \end{array}\right)}^{\mathrm{T}}={\left(\begin{array}{c}\hfill x_1{x}_2\hfill \end{array}\right)}^{\mathrm{T}}$$

(2a)

one could obtain the system state equation

$$\stackrel{·}{X}=\mathrm{H}(X)$$

(2b)

where H is the function of state vector X.

The main objective of vibration isolation is to minimize the displacement and velocity, so that the controlled performance indicators can be selected as

$$J=\frac{1}{2}\underset{0}{\overset{\infty }{\int }}(q_1{x}_1^2+q_2{x}_2^2+r{u}^2)\mathrm{dt}$$

(3a)

where $q_1$ , $q_2$ , and r are the weighting coefficients. Defining the weighting coefficient matrix as

$$Q=\left(\begin{array}{c}\hfill q_{1}0\hfill \\ \multicolumn{1}{c}{0q_2}\end{array}\right)$$

one can establish another form of the objective function as

$$J=\frac{1}{2}\underset{0}{\overset{\infty }{\int }}(X^{\mathrm{T}}\mathrm{QX}+r{u}^2)\mathrm{d}t$$

(3b)

Different from traditional integer-order isolation system, this system contains fractional-order derivative. In order to get the explicitly optimal control force u by calculus of variations, first, the fractional-order derivative must be processed.

Processing of fractional-order derivative

Here, Caputo’s definition is adopted

$${\mathrm{D}}_t^{p}x(t)=\frac{1}{\Gamma (1-p)}\underset{0}{\overset{t}{\int }}\frac{\stackrel{·}{x}(\tau )}{{(t-\tau )}^p}d\tau$$

(4)

where $\Gamma (x)$ is the Gamma function satisfying $\Gamma (x+1)=x\Gamma (x)$ .

When it is in different form, the approximate expansion of the fractional-order derivative of $x(t)$ is researched in the following part.

$x(t)$ is a trigonometric function

When $x(t)=\sin (\omega t)$ , substituting it into equation (4), one can obtain

$${\mathrm{D}}_t^{p}x(t)=\frac{\omega }{\Gamma (1-p)}\underset{0}{\overset{t}{\int }}\frac{\cos (\omega \tau )}{{(t-\tau )}^p}d\tau$$

(5)

where $t-\tau =z$ . Expanding equation (5), one can obtain

$${\mathrm{D}}_t^{p}x(t)=\frac{\omega }{\Gamma (1-p)}\underset{0}{\overset{t}{\int }}\frac{\cos (\omega t)\cos (\omega z)+\sin (\omega t)\sin (\omega z)}{z^p}\mathrm{dz}$$

(6)

Assuming that ${\mathrm{D}}_t^{p}x(t)$ can be written as the following form

$${\mathrm{D}}_t^{p}x(t)=\mathrm{ax}(t)+b\stackrel{·}{x}(t)+\mathrm{cf}(t)$$

(7)

Differentiating equation (6), one can obtain

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\omega }{\Gamma (1-p)}}\\ \multicolumn{1}{c}{[-\omega \sin (\omega t)\underset{0}{\overset{t}{\int }}\frac{\cos (\omega z)}{z^p}\mathrm{d}z+\frac{1}{t^p}+\omega \cos (\omega t)\underset{0}{\overset{t}{\int }}\frac{\sin (\omega z)}{z^p}\mathrm{d}z]}\end{array}$$

(8)

Introducing the two formulae ¹¹

$$\begin{array}{c}\multicolumn{1}{c}{\underset{0}{\overset{t}{\int }}\frac{\cos (\omega z)}{z^p}\mathrm{d}z={\omega }^{p-1}}\\ \multicolumn{1}{c}{\{\Gamma (1-p)\sin \left(\frac{p\pi }{2}\right)+\frac{\sin (\omega t)}{{(\omega t)}^p}+\mathrm{O}\left[{(\omega t)}^{-p-1}\right]\}}\end{array}$$

(9a)

$$\begin{array}{c}\multicolumn{1}{c}{\underset{0}{\overset{t}{\int }}\frac{\sin (\omega z)}{z^p}\mathrm{d}z={\omega }^{p-1}}\\ \multicolumn{1}{c}{\{\Gamma (1-p)\cos \left(\frac{p\pi }{2}\right)-\frac{\cos (\omega t)}{{(\omega t)}^p}+\mathrm{O}\left[{(\omega t)}^{-p-1}\right]\}}\end{array}$$

(9b)

one can get

$$\begin{array}{c}\multicolumn{1}{c}{-{\omega }^{p+1}[\sin \left(\frac{p\pi }{2}\right)+\frac{\mathrm{O}\left({(\omega t)}^{-p-1}\right)}{\Gamma (1-p)}]\sin (\omega t)}\\ \multicolumn{1}{c}{+{\omega }^{p+1}[\cos \left(\frac{p\pi }{2}\right)+\frac{\mathrm{O}\left({(\omega t)}^{-p-1}\right)}{\Gamma (1-p)}]\cos (\omega t)}\end{array}$$

(10)

Differentiating equation (7), one can obtain

$$\frac{\mathrm{d}[{\mathrm{D}}_t^{p}x(t)]}{\mathrm{d}t}=a\omega \cos (\omega t)-b{\omega }^2\sin (\omega t)+c\stackrel{·}{f}(t)$$

(11)

Because equations (10) and (11) are equal, the following results could be established

$$a={\omega }^p\cos \left(\frac{p\pi }{2}\right)+\frac{\mathrm{O}(t^{-p-1})}{\omega \Gamma (1-p)}$$

(12a)

$$b={\omega }^{p-1}\sin \left(\frac{p\pi }{2}\right)+\frac{\mathrm{O}(t^{-p-1})}{{\omega }^2\Gamma (1-p)}$$

(12b)

$$c=0$$

(12c)

When $x(t)=\cos (\omega t)$ , one can obtain the same conclusion based on the similar procedure. And this conclusion could be extended to the more general case of trigonometric functions. For the general form of trigonometric function $x(t)=\sin (\omega t+\phi )$ , it can be written as $x(t)=\sin \omega t\cos \phi +\sin (\phi )\cos (\omega t)$ . According to the nature of fractional-order derivative, $x(t)=\sin (\omega t+\phi )$ is also satisfied in the above conclusion. Therefore, any trigonometric function meets the following conclusion

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{D}}_t^{p}x(t)=[{\omega }^p\cos \left(\frac{p\pi }{2}\right)+\frac{\mathrm{O}(t^{-p-1})}{\omega \Gamma (1-p)}]x(t)}\\ \multicolumn{1}{c}{+[{\omega }^{p-1}\sin \left(\frac{p\pi }{2}\right)+\frac{\mathrm{O}(t^{-p-1})}{{\omega }^2\Gamma (1-p)}]\stackrel{·}{x}(t)}\end{array}$$

(13)

$x(t)$ is a periodic function

For a periodic function satisfying Dirichlet condition, one can assume its period as T

$$x(t)=x(t\pm \mathrm{nT}),n=1,2,3,\dots$$

(14)

The Fourier series expansion of equation (14) is

$$x(t)=\frac{a_0}{2}+\sum _{n=1}^{\infty }{c}_n\sin (n{\omega }_{1}t+{\phi }_n)$$

(15)

where $a_0$ and ${\omega }_1=\frac{2\pi }{T}$ are the constants, and the other two parameters satisfy

$$c_n=\sqrt{a_n^2+b_n^2}$$

(16a)

$${\phi }_n=t{g}^{-1}\frac{a_n}{b_n}$$

(16b)

where

$$\{\begin{array}{c}\hfill a_n=\frac{2}{T}\underset{0}{\overset{T}{\int }}x(t)\cos (n{\omega }_{1}t)\mathrm{d}t\hfill \\ \multicolumn{1}{c}{b_n=\frac{2}{T}\underset{0}{\overset{T}{\int }}x(t)\sin (n{\omega }_{1}t)\mathrm{d}t}\end{array}$$

(17)

Therefore, $c_n$ and ${\phi }_n$ are also constants. Then, one can obtain

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{D}}_t^{p}x(t)={\mathrm{D}}_t^p[\frac{a_0}{2}+\sum _{n=1}^{\infty }{c}_n\sin (n{\omega }_{1}t+{\phi }_n)]}\\ \multicolumn{1}{c}{=\sum _{n=1}^{\infty }{c}_n{\mathrm{D}}_t^p\sin (n{\omega }_{1}t+{\phi }_n)}\end{array}$$

(18)

Moreover, using a finite number of terms in equation (18), one could get

$${\mathrm{D}}_t^{p}x(t)\approx \sum _{n=1}^k{c}_n{\mathrm{D}}_t^p{y}_n(t),k=1,2,3,\dots$$

(19)

where

$$y_n(t)=\sin (n{\omega }_{1}t+{\phi }_n)$$

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{D}}_t^p{y}_n(t)=[{(n{\omega }_1)}^p\cos \left(\frac{p\pi }{2}\right)+\frac{\mathrm{O}(t^{-p-1})}{n{\omega }_1\Gamma (1-p)}]y_n(t)}\\ \multicolumn{1}{c}{+[{(n{\omega }_1)}^{p-1}\sin \left(\frac{p\pi }{2}\right)+\frac{\mathrm{O}(t^{-p-1})}{{(n{\omega }_1)}^2\Gamma (1-p)}]{\stackrel{·}{y}}_n(t)}\end{array}$$

Obviously, the general periodic functions also satisfy the same property as trigonometric function.

Analysis on fractional-order passive vibration isolation system

Simplification of passive control system

For passive control system, equation (1) can be rewritten as

$$\begin{array}{c}\multicolumn{1}{c}{m\stackrel{··}{x}+[c_1+k_2{\omega }^{p-1}\sin \left(\frac{p\pi }{2}\right)]\stackrel{·}{x}}\\ \multicolumn{1}{c}{+[k_1+k_2{\omega }^p\cos \left(\frac{p\pi }{2}\right)]x=F}\end{array}$$

(20)

Defining two new parameters, that is, equivalent damping coefficient and equivalent stiffness coefficient ²⁶

$$C(p)=c_1+k_2{\omega }^{p-1}\sin \left(\frac{p\pi }{2}\right)$$

(21a)

$$K(p)=k_1+k_2{\omega }^p\cos \left(\frac{p\pi }{2}\right)$$

(21b)

one could study the dynamical property of fractional-order passive vibration isolation system. From equations (20) and (21), it can be seen that fractional-order derivative has viscoelastic characteristics, which means it has not only the damping effect but also the stiffness effect. The effects of fractional-order derivative on dynamical response are related with the fractional coefficient $k_2$ , the excitation frequency $\omega$ , and the fractional order p. From equation (21), it can be seen that $C(p)$ and $K(p)$ are monotonic increasing functions of the fractional coefficient $k_2$ . Accordingly, the increase in $k_2$ will lead to the increase in the equivalent damping coefficient and equivalent stiffness coefficient. However, the increase in the excitation frequency $\omega$ would make the linear damping smaller and the linear stiffness greater. Moreover, the fractional order p has a main impact on the distribution of viscoelasticity. When $p\to 0$ , the fractional-order derivative is almost the equivalent linear stiffness, and the fractional-order derivative will be almost the equivalent linear damping if $p\to 1$ .

Steady-state solution and stability analysis

Substituting equation (21) into equation (20) and rewriting it in the plural form, one can obtain

$$\stackrel{··}{x}+\frac{C(p)}{m}\stackrel{·}{x}+\frac{K(p)}{m}x=\frac{F_0}{m}{e}^{i\omega t}$$

(22a)

The characteristic equation of equation (22a) is

$${\gamma }^2+\frac{C(p)}{m}\gamma +\frac{K(p)}{m}=0$$

(22b)

The characteristic roots are

$${\gamma }_{1,2}=\frac{-C(p)}{2m}\pm \sqrt{\frac{C^2(p)}{4m^2}-\frac{K(p)}{m}}$$

(22c)

Due to

$$\sqrt{|\frac{C^2(p)}{4m^2}-\frac{K(p)}{m}|}<\frac{C(p)}{2m}$$

(22d)

the real parts of the characteristic roots are always less than 0. That means the solution of equation (22a) is always stable.

Assuming that the steady-state solution of equation (22a) can be written in the following form

$$x=B_1{e}^{i(\omega t-\phi )}$$

(23)

Substituting it into equation (22a), the steady-state amplitude can be obtained

$$B_1=\frac{F_0}{\sqrt{{\omega }^2{C}^2(p)+{[K(p)-m{\omega }^2]}^2}}$$

(24)

The phase–frequency curve equation is

$$\phi =\mathrm{t}{\mathrm{g}}^{-1}\left[\frac{\omega C(p)}{K(p)-m{\omega }^2}\right]$$

(25)

Equation (24) can be rewritten as

$$B_1=\frac{F_0}{k_1}\frac{k_1}{\sqrt{{\omega }^2{C}^2(p)+{[K(p)-m{\omega }^2]}^2}}$$

(26)

Defining $B_0=\frac{F_0}{k_1}$ as the static deformation of passive isolation system without fractional-order derivative, one can obtain the dimensionless amplitude amplification factor $\beta$ as

$$\beta =\frac{B_1}{B_0}=\frac{k_1}{\sqrt{{\omega }^2{C}^2(p)+{[K(p)-m{\omega }^2]}^2}}$$

(27)

Based on equation (27), one could easily find the relationship between the amplitude amplification factor $\beta$ and the excitation frequency $\omega$ . Then, it is easy to further investigate the effect of fractional-order derivative on the system response.

Numerical simulation

In order to study the effect of fractional-order derivative on the system, one could consider denominator of equation (27)

$$f(\omega ,k_2,p)={\omega }^2{C}^2(p)+{[K(p)-m{\omega }^2]}^2$$

(28)

Because the resonance has maximum impact on the system, we mainly research the influence of fractional-order derivative on resonance. The resonance point is the extreme point of equation (28), so that we discuss the extreme points in equation (28).

We take the first-order partial derivative of equation (28), and let it be equal to 0

$$\frac{\partial f}{\partial \omega }=\mathrm{g}(\omega ,k_2,p)=0$$

(29)

Then, one can take the second-order partial derivative of equation (28)

$$\frac{{\partial }^{2}f}{\partial {\omega }^2}=g_1(\omega ,k_2,p)$$

(30)

When any set of $(k_2,p)$ is substituted into equation (29), $\omega$ can be obtained. Because the highest order of variable $\omega$ in equation (29) is 3, equation (29) maybe has more than one root. If $g_1(\omega ,k_2,p)>0$ , the amplitude on $\omega$ is the maximum and we define $\omega$ as a positive resonance point. If $g_1(\omega ,k_2,p)<0$ , the amplitude on $\omega$ will be the minimum and we define this $\omega$ as a negative resonance point. When equation (29) has no real roots or extreme points, the amplitude–frequency characteristic curve of the system is a monotonic curve.

One could research equation (20) and select the basic parameters as $m=6$ , $k_1=12$ , $c_1=0.3$ , $k_2=2$ , and $p=0.5$ . The study focus is on the influence of the two fractional-order parameters on dynamical response. We change the value of fractional order p, which is selected as 0, 0.25, 0.5, 0.75, and 1.0. One could obtain the amplitude–frequency curves of the steady-state solution shown in Figure 2.

Figure 2.

Effect of fractional order p on the amplitude–frequency curves.

From the observation of Figure 2, it could be found that $\beta$ has only one extreme point which is the maximum point, no matter how the value of p is changed. With the gradual increase in p, the maximum amplitude will gradually decrease and the positive resonance point will move to the left, because the equivalent linear damping is also increased in this procedure. Different fractional coefficient $k_2$ is also studied, which is selected as 1, 1.5, 2, 2.5, and 3. Amplitude–frequency curves of the steady-state solution are shown in Figure 3 for different $k_2$ . From the observation of Figure 3, it could be concluded that $\beta$ has only one extreme point which is the maximum point, no matter how the value of $k_2$ is changed. When $k_2$ is gradually increasing, the maximum amplitude gradually decreases, and the positive resonance point will move rightwards. The reason lies in that the equivalent linear damping is also increased in this procedure. Accordingly, the effects of the fractional order p and fractional coefficient $k_2$ on the system response are reflected in two aspects, which are the resonance amplitude and resonance frequency.

Figure 3.

Effect of fractional coefficient $k_2$ on the amplitude–frequency curves.

Analysis of fractional-order active isolation system

Optimal control force of single degree-of-freedom active isolation system

Under harmonic excitation, fractional-order derivative can be written as equation (13). We substitute it into equation (1) and omit higher order infinitesimal, so that equation (2b) can be written as

$$\stackrel{·}{X}=\mathrm{AX}+\mathrm{Bu}+\mathrm{LF}$$

(31)

where

$$A=\left(\begin{array}{c}\hfill 01\hfill \\ \multicolumn{1}{c}{-\frac{c_1+k_2{\omega }^{p-1}\sin \left(\frac{p\pi }{2}\right)}{m}-\frac{k_1+k_2{\omega }^p\cos \left(\frac{p\pi }{2}\right)}{m}}\end{array}\right)$$

(32a)

$$B=\left(\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{\frac{-1}{m}}\end{array}\right)$$

(32b)

$$L=\left(\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{\frac{1}{m}}\end{array}\right)$$

(32c)

Based on optimal control theory, there is a real symmetric matrix E satisfying the Riccati equation ²⁷

$$\mathrm{EA}+A^{\mathrm{T}}E-\mathrm{EB}{r}^{-1}{B}^{\mathrm{T}}E+Q=0$$

(33)

The analytical expressions of E can be obtained using computation software, such as Mathematica. From the expression form, we can see that even if the other system parameters are fixed, the matrix E will change with the excitation frequency. However, with the one excitation frequency, there is a unique optimal control force as

$$u^{*}=-r^{-1}{B}^T{P}_{1}X=\mathrm{KX}=k_1^{*}{x}_1+k_2^{*}{x}_2$$

(34)

Numerical simulation of fractional-order active isolation system

We take a set of system parameters, including the system quality m = 30 kg, stiffness coefficient $k_1={10}^4\mathrm{N}/\mathrm{m}$ , damping coefficient $c_1=500\mathrm{N}\mathrm{s}/\mathrm{m}$ , fractional coefficient $k_2=5\times {10}^3$ , and fractional order $p=0.25$ . The main study objective is to reduce the displacement amplitude and velocity amplitude of the isolation system. Through a large number of trials, the weighting coefficients of the objective function J can be taken as $q_1={10}^8$ , $q_2={10}^7$ , and r = 1. The amplitude of the excitation force is $F_0=600$ . When the fractional order p is variable, we use Mathematica to calculate the system feedback gain

$$k_1^{*}=\frac{-3\times {10}^4-3\times {10}^5{\omega }^p\sin \left(\frac{p\pi }{2}\right)+\sqrt{3.6\times {10}^{11}{\omega }^2+{[3\times {10}^4\omega +3\times {10}^5{\omega }^p\sin \left(\frac{p\pi }{2}\right)]}^2}}{60\omega }$$

(35)

$$k_2^{*}=\frac{-6\times {10}^5\omega -3\times {10}^5{\omega }^{p+1}\cos \left(\frac{p\pi }{2}\right)+{\mathrm{kk}}_2^{*}}{60\omega }$$

(36)

where

$${\mathrm{kk}}_2^{*}=\sqrt{{[6\times {10}^5\omega +3\times {10}^5{\omega }^{p+1}\cos \left(\frac{p\pi }{2}\right)]}^2-{\mathrm{kk}}_3^{*}}$$

$$\begin{array}{c}k{k}_3^{\ast }=4\omega [-8.973\times {10}^9\omega +2.7\times {10}^8{\omega }^p\sin \left(\frac{p\pi }{2}\right)\\ -900\sqrt{3.6\times {10}^{11}{\omega }^2+{\left(3\times {10}^4\omega +3\times {10}^5{\omega }^p\sin \left(\frac{p\pi }{2}\right)\right)}^2}]\end{array}$$

From equations (35) and (36), it could be found that the system feedback gain is a function of the excitation frequency $\omega$ . In order to show that the relationship between feedback gain and excitation frequency $\omega$ more distinct, we present the curves between the gain and the excitation frequency $\omega$ . When p is taken as 0.25 and 0.5, respectively, the results are shown in Figures 4 and 5. From Figure 4, one can see that $k_1^{*}$ changes quickly in the low frequency. The change in $k_1^{*}$ gradually slows down and eventually tends to a constant value if the excitation frequency is large enough. When the fractional order p is smaller, $k_1^{*}$ is bigger. From Figure 5, it can be concluded that $k_2^{*}$ also changes quickly in the low frequency and has the similar characteristic as that of $k_1^{*}$ . The difference is that when the fractional order p is smaller $k_2^{*}$ is also smaller in the low frequency. But with the increase in the excitation frequency, the changing curves will intersect. After that $k_2^{*}$ will become bigger if the fractional order p is smaller. The reason for this phenomenon is that the change in the fractional order p gives rise to the change in the viscoelastic distribution.

Figure 4.

Relationship between $k_1^{*}$ and the excitation frequency.

Figure 5.

Relationship between $k_2^{*}$ and the excitation frequency.

After obtaining the feedback gain, one could research the performance of the fractional-order active isolation system. Theoretically speaking, the active isolation system is to add appropriately parallel spring and damper to the original passive isolation system. The change rule about the added stiffness and the damping coefficients are shown in Figures 4 and 5, respectively. We compare the amplitude–frequency curves of the passive isolation system and the active isolation system under different fractional order and could analyze the effect of active control force on control performance. Because the isolation system is subjected to harmonic excitation, the displacement amplitude is proportional to the velocity amplitude. One can find the optimization effect by comparing the amplitude–frequency curve of the displacement only. When the fractional order p is, respectively, taken as 0.25 and 0.5, the amplitude–frequency curves are shown in Figures 6 and 7. When $0\le \omega \le 40\mathrm{rad}/\mathrm{s}$ , it could be found that the performance of the active isolation system is superior to the passive isolation system. If $\omega >40\mathrm{rad}/\mathrm{s}$ , the performances of the two system are similar so that the active control is unnecessary.

Figure 6.

Amplitude–frequency curve for fractional order p = 0.5.

Figure 7.

Amplitude–frequency curve for fractional order p = 0.25.

In practical engineering, the isolation system may be subjected to random excitation more frequently. Therefore, one should also analyze the control performance of isolation system under random excitation. In this article, we research the control effect of the optimal control force when the isolation system is subjected to random excitation. From equation (29), we can obtain the resonance frequency $\omega =8.6\mathrm{rad}/\mathrm{s}$ or $\omega =20.6\mathrm{rad}/\mathrm{s}$ . From equation (30), we can find that $\omega =8.6\mathrm{rad}/\mathrm{s}$ is a negative resonance frequency and $\omega =20.6\mathrm{rad}/\mathrm{s}$ is a positive resonant frequency. From Figure 7, it could be found that the response amplitude is bigger when $\omega =20.6\mathrm{rad}/\mathrm{s}$ . Therefore, we create a random time series with frequency 20.6 rad/s, mean 0, variance 1, and acting time as 60 s. Because the amplitude is too small, we enlarge the random time series 200 times. The system responses of the passive isolation system and the active isolation system are shown in Figures 8 and 9, respectively. From Figures 8 and 9 and the mean square value in Table 1, one can see that both the displacement response and the velocity response of the active isolation system are all superior to those of passive isolation system.

Figure 8.

System response of passive isolation system.

Figure 9.

System response of active isolation system.

Table 1.

Root mean square (RMS) of the displacement and velocity.

	Velocity (cm)	Displacement (m/s)
Passive isolation system	0.2774	0.0708
Active isolation system	0.1952	0.0599

Conclusion

In this article, we reasonably deal with the fractional-order derivative and study the effect of fractional-order derivative on passive isolation system. It can be clearly found that the effect of the fractional order and fractional coefficient on the system response could be characterized by the equivalent damping coefficient and equivalent stiffness coefficient, and they could affect the resonance frequency and resonance amplitude. After processing the fractional-order derivative, the system optimal feedback gain is obtained based on the optimal control theory. It could be found that the optimal feedback gain will change with the excitation frequency. Finally, we study the control effect through numerical simulation, and the simulation results show that this method can improve the performance of isolation system, whenever the isolation system is subjected to harmonic excitation or random excitation. This article provides a guideline for dealing with fractional-order derivative in vibration isolation system and also provides a simple and easy method for solving the similar engineering problems.