# Active Vibration Control of PID Based on Receptance Method

Authors: Lijuan Peng; Jian Wang; Guicheng Yu; Zuoxue Wang; Aijun Yin; Hongji Ren
Journal: Journal of Sensors
 (2020)

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## Abstract

Active vibration control approaches have been widely applied on improving reliability of robotic systems. For linear vibratory systems, the vibration features can be altered by modifying poles and zeros. To realize the arbitrary assignment of the closed-loop system poles and zeros of a linear vibratory system, in this paper, an active PID input feedback vibration control method is proposed based on the receptance method. The establishment and verification of the proposed method are demonstrated. The assignable poles during feedback control are calculated and attached with importance to expand the application of the integral control. Numerical simulations are conducted to verify the validity of the proposed method in terms of the assignment of closed-loop poles, zeros, and both. The results indicate that the proposed method can be used to realize the active vibration control of closed-loop system and obtain the desired damping ratio, modal frequency, and dynamic response.


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## Body

## 1. Introduction

Nowadays, the demand for lower vibration and higher reliability in robotic systems is increasing [1]. Extensive studies have been conducted to optimize the controlling of various nonlinear systems [2–6]. A common vibration control approach is to modify system dynamics using controllers [7]. PID-based approaches have been widely applied on motion control of robotic systems [8, 9]. Generally, the vibration characteristics of a linear vibratory system are determined by its zeros and poles, and the assignment of eigenvalues in a system is of great importance in improving system reliability.Traditionally, the solutions of pole assignment are solved by using the information of mass, damping, and stiffness matrices acquired from the finite element (FE) models [10]. However, FE models are generally limited by the incapability in obtaining precise damping models of real structures. Ram and Mottershead [11] developed an active vibration control method for linear system based on receptance method to accurately assign the system poles and zeros to specified values. One considerable advantage of this method is that it is entirely based on data from modal testing rather than the exact mass, damping, and stiffness matrices. Mottershead et al. assigned part of the system poles to the predetermined values while keeping nonimportant poles unchanged by making the nonimportant poles uncontrollable or unobservable [12]. Zhang et al. presented the theory of applying acceleration feedback and position feedback to the nondamping system to assign part of poles [13]. Ouyang proposed a method of using acceleration feedback and velocity feedback in active vibration control [14]. Recently, neural network- (NN-) based pole-zero assignment methods have gained a lot of attention [15–18]. However, NN-based approaches may fail to yield desired results on practice due to the black-box nature of NN and the requirement of NN parameter optimization skills.Previous studies indicate that various combinations of acceleration, velocity, and position make it flexible and applicable to select the type of feedback when implementing active vibration control. However, a vibration system will inevitably become a singular system when the acceleration feedback is applied, which leads to the unavailability of the receptance method. In this paper, integral control is introduced to overcome this limitation. The contributions of this paper are summarized as follows: (1) A PID input feedback active vibration control method is proposed based on the receptance method. (2) The numerical solutions for assignment of closed-loop poles and zeros by applying the proposed method are demonstrated.The rest of the paper is organized as follows: The assignments of poles and zeros in PID control systems are demonstrated in Section2 and Section 3, respectively. Then, numerical simulations are conducted to verify the validity of the proposed method in Section 4. Finally, Section 5 concludes the paper.


## 2. Pole Assignment of PID Control Systems

### 2.1. Formulation of Pole Assignment

The equation of an degree freedom linear system can be described in second-order form as follows according to [19]:
(1)Mx¨t+Cẋt+Kxt=0,where M,C,K∈Rn×n are mass, damping, and stiffness matrices, respectively; xt∈Rn×1 is the displacement vector. M=MT, C=CT, K=KT, and for any nonzero vector ν∈Rn×1, there is νTMν≥0, νTCν≥0, and νTKν≥0.Integrate PID feedback control to the system and the closed-loop system can be written as follows:(2)Mx¨t+Cẋt+Kxt=but,where b∈Rn×1 is the force distribution vector and ut is the input control vector:
(3)ut=g1Txt+g2T∫0txτdτ+g3Tẋt,where g1,g2,g3∈Rn×1 are input control gain vectors for displacement, differential, and velocity, respectively.Express Equation (2) in the Laplace space:
(4)Ms2+Cs+Kxs=bg1T+g2Ts+sg3Txs.Then,(5)Ms2+Cs+K−bg1T+g2Ts+sg3Txs=0.Equation (5) shows that the highest order of the system stiffness matrix is a positive definite matrix, so the closed-loop system must be a nonsingular system. Therefore, the system receptance matrix exists.The receptance matrix can be obtained by applying the Sherman–Morrison formula for Equation (5):
(6)H^s=Hs+Hsbg1+g2/s+sg3THs1−g1+g2/s+sg3THsb,where Hs=Ms2+Cs+K−1 is the receptance matrix of the open-loop system, which can be obtained in practice by measuring the receptance Hiω [20]. The characteristic polynomial ps can be derived from Equation (6) at the same time:
(7)ps=1−g1+g2s+sg3THsb.Thus, the roots of Equation (8) are the pole of the closed-loop system:
(8)ps=0.GivenHs, b, and Equation (9):
(9)pμi=0.The problem of assigning the poles of a closed-loop system to setpointsμ1,μ2,⋯,μN can be transformed to the solution of input control gain vectors g1,g2, and g3. Where μi∈μii=1N, μi≠0, μii=1N is closed under conjugation, N is the number of closed-loop system poles that can be assigned.


### 2.2. Minimum Number of Assignable PolesN

Given a positive definite system with closed-loop conjugate polesμii=1N, suppose that the adjoint matrix of Hs is As and the determinant is Ds. ps=0 can be expressed as follows:
(10)1−g1+g2s+sg3TAsDsb=0.Since for anyμi∈μii=1N, Dμi≠0, Equation (10) can be written as follows:
(11)sDs−sg1+g2+s2g3TAsb=0.Set(12)qs=sg1+g2+s2g3TAsb.The highest degree ofs in qs is 2n−2+2=2n and the highest degree of s in Ds is 2n. Since the number of closed-loop poles N is equal to the degree of Equation (10) and the highest degree of s in Equation (10) is 2n+1, the number of characteristic polynomial root is N=2n+1. It should be noted that due to the introduction of integral control, the closed-loop system will produce an extra nonzero pole.In fact, the introduction of integral control into closed-loop system will lead to a nonregular and generalized system. The closed-loop system is stable if (1) it is regular and (2) all of its assigned poles have negative real components [21]. The regularity of the closed-loop system in this method is guaranteed by the positive definiteness of the mass matrix. The values of the assigned poles can be arranged to stabilize the closed-loop system given that their real components are negative and μii=1N is self-conjugate.


### 2.3. Calculate Vectorsg1,g2,g3 and Minimum Sensor Layout Method

The solution ofg1,g2,g3 can be realized by the following equation:
(13)Gg=γ,where
(14)G=Ψ1TΨ1Tμ1μ1Ψ1TΨ2TΨ2Tμ2μ2Ψ2T⋮ΨNTΨNTμNμNΨNT,Ψi=Hμib,gT=g1Tg2Tg3TγT=11⋯1and G∈RN×3n,g∈R3n×1, and γ∈RN×1.The pole assignment problem raised by Equation (2) can be solved by applying Equation (13) and Equation (14). One solution for Equation (13) is as follows:
(15)g=G+γ,where G+ is the Moore–Penrose generalized inverse of G. When the assigned closed-loop poles are self-conjugate, g1,g2, and g3 can be calculated according to Equations (13)–(15).Considering the limitation of the additional mass in the actual control system, the number of sensors used to measure the motion parameters of the system should be as small as possible. Therefore, the value of some items ing1,g2,g3 can be assigned to zero by setting the values of the corresponding columns of the G to be zero. In practice, the solution means that fewer sensors are involved. Due to the integration of PID input feedback control and N=2n+1, the number that can be allocated to zero in G has at most 3n−N=n−1 columns. Therefore, the number of sensors can be reduced accordingly in practice to minimize the effect of the additional mass on the system.


### 2.4. Validity Analysis

When the value of the preset polesμii=1N and the optional k0≤k≤n−1 column vector to be replaced by the 0 in the G are determined, a specific PID input feedback solution is derived, identified as Wk. From Equations (13)–(15), g1,g2,g3 can be calculated in this specific PID input feedback solution. Below we will show how to verify whether Wk is valid or not.The closed-loop receptance matrixH^s can be derived from Equation (5) as follows:
(16)H^s=Ωs−1,where Ωs is as follows:
(17)Ωs=Ms2+Cs+K−bg1T+g2Ts+sg3T.The problem of verifying whether a particularWk is valid is equivalent to verifying whether Ωs is a nonsingular matrix for any s∉μii=1N under the determined case. The verification process of Wk is as follows: Firstly, the real vectors g1,g2,g3 are calculated according to Equations (13)–(15). Then, substitute g1,g2,g3 into Equation (17) and obtain the determinant ws of Ωs. For any s∉μii=1N, if ws is zero, the closed-loop system poles cannot be assigned to the designated values, meaning that Wk is invalid. Otherwise, the closed-loop system poles can be assigned to the specified desired values, indicating that Wk is valid.


## 2.1. Formulation of Pole Assignment

The equation of an degree freedom linear system can be described in second-order form as follows according to [19]:
(1)Mx¨t+Cẋt+Kxt=0,where M,C,K∈Rn×n are mass, damping, and stiffness matrices, respectively; xt∈Rn×1 is the displacement vector. M=MT, C=CT, K=KT, and for any nonzero vector ν∈Rn×1, there is νTMν≥0, νTCν≥0, and νTKν≥0.Integrate PID feedback control to the system and the closed-loop system can be written as follows:(2)Mx¨t+Cẋt+Kxt=but,where b∈Rn×1 is the force distribution vector and ut is the input control vector:
(3)ut=g1Txt+g2T∫0txτdτ+g3Tẋt,where g1,g2,g3∈Rn×1 are input control gain vectors for displacement, differential, and velocity, respectively.Express Equation (2) in the Laplace space:
(4)Ms2+Cs+Kxs=bg1T+g2Ts+sg3Txs.Then,(5)Ms2+Cs+K−bg1T+g2Ts+sg3Txs=0.Equation (5) shows that the highest order of the system stiffness matrix is a positive definite matrix, so the closed-loop system must be a nonsingular system. Therefore, the system receptance matrix exists.The receptance matrix can be obtained by applying the Sherman–Morrison formula for Equation (5):
(6)H^s=Hs+Hsbg1+g2/s+sg3THs1−g1+g2/s+sg3THsb,where Hs=Ms2+Cs+K−1 is the receptance matrix of the open-loop system, which can be obtained in practice by measuring the receptance Hiω [20]. The characteristic polynomial ps can be derived from Equation (6) at the same time:
(7)ps=1−g1+g2s+sg3THsb.Thus, the roots of Equation (8) are the pole of the closed-loop system:
(8)ps=0.GivenHs, b, and Equation (9):
(9)pμi=0.The problem of assigning the poles of a closed-loop system to setpointsμ1,μ2,⋯,μN can be transformed to the solution of input control gain vectors g1,g2, and g3. Where μi∈μii=1N, μi≠0, μii=1N is closed under conjugation, N is the number of closed-loop system poles that can be assigned.


## 2.2. Minimum Number of Assignable PolesN

Given a positive definite system with closed-loop conjugate polesμii=1N, suppose that the adjoint matrix of Hs is As and the determinant is Ds. ps=0 can be expressed as follows:
(10)1−g1+g2s+sg3TAsDsb=0.Since for anyμi∈μii=1N, Dμi≠0, Equation (10) can be written as follows:
(11)sDs−sg1+g2+s2g3TAsb=0.Set(12)qs=sg1+g2+s2g3TAsb.The highest degree ofs in qs is 2n−2+2=2n and the highest degree of s in Ds is 2n. Since the number of closed-loop poles N is equal to the degree of Equation (10) and the highest degree of s in Equation (10) is 2n+1, the number of characteristic polynomial root is N=2n+1. It should be noted that due to the introduction of integral control, the closed-loop system will produce an extra nonzero pole.In fact, the introduction of integral control into closed-loop system will lead to a nonregular and generalized system. The closed-loop system is stable if (1) it is regular and (2) all of its assigned poles have negative real components [21]. The regularity of the closed-loop system in this method is guaranteed by the positive definiteness of the mass matrix. The values of the assigned poles can be arranged to stabilize the closed-loop system given that their real components are negative and μii=1N is self-conjugate.


## 2.3. Calculate Vectorsg1,g2,g3 and Minimum Sensor Layout Method

The solution ofg1,g2,g3 can be realized by the following equation:
(13)Gg=γ,where
(14)G=Ψ1TΨ1Tμ1μ1Ψ1TΨ2TΨ2Tμ2μ2Ψ2T⋮ΨNTΨNTμNμNΨNT,Ψi=Hμib,gT=g1Tg2Tg3TγT=11⋯1and G∈RN×3n,g∈R3n×1, and γ∈RN×1.The pole assignment problem raised by Equation (2) can be solved by applying Equation (13) and Equation (14). One solution for Equation (13) is as follows:
(15)g=G+γ,where G+ is the Moore–Penrose generalized inverse of G. When the assigned closed-loop poles are self-conjugate, g1,g2, and g3 can be calculated according to Equations (13)–(15).Considering the limitation of the additional mass in the actual control system, the number of sensors used to measure the motion parameters of the system should be as small as possible. Therefore, the value of some items ing1,g2,g3 can be assigned to zero by setting the values of the corresponding columns of the G to be zero. In practice, the solution means that fewer sensors are involved. Due to the integration of PID input feedback control and N=2n+1, the number that can be allocated to zero in G has at most 3n−N=n−1 columns. Therefore, the number of sensors can be reduced accordingly in practice to minimize the effect of the additional mass on the system.


## 2.4. Validity Analysis

When the value of the preset polesμii=1N and the optional k0≤k≤n−1 column vector to be replaced by the 0 in the G are determined, a specific PID input feedback solution is derived, identified as Wk. From Equations (13)–(15), g1,g2,g3 can be calculated in this specific PID input feedback solution. Below we will show how to verify whether Wk is valid or not.The closed-loop receptance matrixH^s can be derived from Equation (5) as follows:
(16)H^s=Ωs−1,where Ωs is as follows:
(17)Ωs=Ms2+Cs+K−bg1T+g2Ts+sg3T.The problem of verifying whether a particularWk is valid is equivalent to verifying whether Ωs is a nonsingular matrix for any s∉μii=1N under the determined case. The verification process of Wk is as follows: Firstly, the real vectors g1,g2,g3 are calculated according to Equations (13)–(15). Then, substitute g1,g2,g3 into Equation (17) and obtain the determinant ws of Ωs. For any s∉μii=1N, if ws is zero, the closed-loop system poles cannot be assigned to the designated values, meaning that Wk is invalid. Otherwise, the closed-loop system poles can be assigned to the specified desired values, indicating that Wk is valid.


## 3. Zero Assignment of PID Control Systems

### 3.1. Formulation of Zero Assignment

The characteristic polynomial for the zeros of receptanceH^ij can be realized when the denominator matrix (i, j) of Equation (6) is zero:
(18)H^ijs=eiTHs+Hsbg1+g2/s+sg3THs1−g1+g2/s+sg3THsbej,where ei,ej are the unit vectors obtained from the corresponding columns of the identity matrix.Therefore, the problem of assigning zeros of the closed-loop system receptanceH^ij to the setpoints ξ1,ξ1,⋯,ξr can be realized as follows:GivenHs, b, i, j, and
(19)eiT1−g1+g2ξk+ξkg3THξkb+Hξkbg1+g2ξk+ξkg3THξkej=0Calculate the input control gain vectorsg1,g2,g3. Where ξk∉ξii=1r, r≤2n−1, r is the number of poles in a closed-loop system that can be assigned.Equation (19) can be written in the following form:
(20)Hijξkg1+g2ξk+ξkg3THξkb−eiTHξkbg1+g2ξk+ξkg3THξkej=Hijξk,where Hijs is the open-loop receptance value at the open-loop system coordinate of (i, j). Transform Equation (20) to g1+g2/ξk+ξkg3THijξkHξkb−eiTHξkbHξkej=Hijξk. Since eiTHξkb is a scalar, set
(21)tk=HijξkHξkb−eiTHξkbHξkej.We haveg1+g2/ξk+ξkg3Ttk=Hijξk. Since Hijξk is scalar, after transposing we get tkTg1+tkTg2/ξk+ξktkTg3=Hijξk, i.e., tkTtkT/ξkξktkTg1g2g3=Hijξk. Therefore, for preset closed-loop system zeros ξ1,ξ1,⋯,ξr, we have
(22)Tg1g2g3=Hijξ1Hijξ2⋮Hijξr,where
(23)T=t1Tt1Tμ1t1Tξ1t2Tt2Tμ2t2Tξ2⋮trTtrTμrtrTξr.It can be seen from Equation (22) that the inverse of the matrix T can be obtained when the preset zeros ξ1,ξ1,⋯,ξr are given, and then the real vectors g1,g2,g3 can also be obtained. In particular, when i and j are overlapped, the distribution of zeros will affect the closed-loop pole assignment of the system, which is significant [22].


### 3.2. Pole and Zero Assignment

We can assign the zeros and poles in the closed-loop system at the same time according to Equation (13) and Equation (22). In this case, the total number of zeros and poles follows r+N≤2n+1. Similarly, by assigning k0≤k≤n−1 columns in Equation (13) and Equation (22) to zero, we can get the optimized allocation scheme.


## 3.1. Formulation of Zero Assignment

The characteristic polynomial for the zeros of receptanceH^ij can be realized when the denominator matrix (i, j) of Equation (6) is zero:
(18)H^ijs=eiTHs+Hsbg1+g2/s+sg3THs1−g1+g2/s+sg3THsbej,where ei,ej are the unit vectors obtained from the corresponding columns of the identity matrix.Therefore, the problem of assigning zeros of the closed-loop system receptanceH^ij to the setpoints ξ1,ξ1,⋯,ξr can be realized as follows:GivenHs, b, i, j, and
(19)eiT1−g1+g2ξk+ξkg3THξkb+Hξkbg1+g2ξk+ξkg3THξkej=0Calculate the input control gain vectorsg1,g2,g3. Where ξk∉ξii=1r, r≤2n−1, r is the number of poles in a closed-loop system that can be assigned.Equation (19) can be written in the following form:
(20)Hijξkg1+g2ξk+ξkg3THξkb−eiTHξkbg1+g2ξk+ξkg3THξkej=Hijξk,where Hijs is the open-loop receptance value at the open-loop system coordinate of (i, j). Transform Equation (20) to g1+g2/ξk+ξkg3THijξkHξkb−eiTHξkbHξkej=Hijξk. Since eiTHξkb is a scalar, set
(21)tk=HijξkHξkb−eiTHξkbHξkej.We haveg1+g2/ξk+ξkg3Ttk=Hijξk. Since Hijξk is scalar, after transposing we get tkTg1+tkTg2/ξk+ξktkTg3=Hijξk, i.e., tkTtkT/ξkξktkTg1g2g3=Hijξk. Therefore, for preset closed-loop system zeros ξ1,ξ1,⋯,ξr, we have
(22)Tg1g2g3=Hijξ1Hijξ2⋮Hijξr,where
(23)T=t1Tt1Tμ1t1Tξ1t2Tt2Tμ2t2Tξ2⋮trTtrTμrtrTξr.It can be seen from Equation (22) that the inverse of the matrix T can be obtained when the preset zeros ξ1,ξ1,⋯,ξr are given, and then the real vectors g1,g2,g3 can also be obtained. In particular, when i and j are overlapped, the distribution of zeros will affect the closed-loop pole assignment of the system, which is significant [22].


## 3.2. Pole and Zero Assignment

We can assign the zeros and poles in the closed-loop system at the same time according to Equation (13) and Equation (22). In this case, the total number of zeros and poles follows r+N≤2n+1. Similarly, by assigning k0≤k≤n−1 columns in Equation (13) and Equation (22) to zero, we can get the optimized allocation scheme.


## 4. Numerical Simulations

### 4.1. Pole Assignment Simulation

Consider a three-degree-of-freedom system with matrices:(24)M=223,C=2.5−20−23−10−11,K=10−3−4−330−404.The open-loop poles are−0.0305±0.5894i, −0.8503±1.0119i, and −0.6609±2.1200i.Now use pole assignment method proposed in this paper to modify the closed-loop response of the system. Setb=111T. For n=3, N=7 according to Section 2.2. Since the original system has a lower characteristic frequency and a smaller damping ratio, the closed-loop system characteristic values ate assigned as μ1;2=−1±0.5i, μ3;4=−1±i, μ5;6=−2±i, and μ7=−3 to increase the damping of the system and enhance the stability.Obtaing1,g2,g3 according to Equations (13)–(15):
(25)g1=−38.1817−6.781011.3765T,g2=3.7396−14.44872.4203T,g3=−4.1357−4.5904−10.6608T.The above PID input feedback solution is valid according to Section2.4. In order to visualize the effect of this method, the frequency response curve, zero-pole distribution, and Nyquist plot of the H3,3 of original system and the modified system are given, shown in Figures 1–5. Figures 2 and 3 show that the poles of the modified system are assigned to the specified positions, and an additional pole is added to the modified system due to the integration of integral feedback. The introduction of the extra pole improves the flexibility to handle the closed-loop systems in engineering applications. We can see from Figure 1, Figure 4, and Figure 5 that after pole reassignment, system features such characteristic frequency, damping, and stability have been changed, and the overall performances have been improved.Figure 1
The frequency response for the original systemH3,3 (solid line) and the modified system H3,3 (dotted line).Figure 2
The zero-pole map for the original systemH3,3.Figure 3
The zero-pole map for the modified systemH3,3.Figure 4
The Nyquist curve for the original systemH3,3.Figure 5
The Nyquist curve for the modified systemH3,3.


### 4.2. Minimum Sensor Configuration for Pole Assignment

Based on the pole assignment in Section4.1 and the minimum sensor layout in Section 2.3, a minimum sensor configuration scheme for pole assignment simulation is developed and verified according to Section 2.4. Table 1 shows all solutions that can assign the poles of the closed-loop system to preset values μ1;2=−1±0.5i, μ3;4=−1±i, and μ5;6=−2±i, while keeping the closed-loop system stable at the same time for the case in Section 4.1. The PID feedback gains for the 2 solutions are given in Table 2 (only the control gains of the first method in each Wk are given). The system performance may destabilize when the number of assigned poles is less than N (i.e., when the lacked poles are assigned, the number of assigned poles in Table 3 is 2n). This highlights the importance of determining the number of assignments. The specific results are demonstrated in Tables 1–3.Table 1
Pole assignment solutions.
PlanColumn replaced by a zero vector inGW1{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}W2{1,2}, {1,3}, {2,3}, {1,4}, {2,4}, {3,4}, {1,5}, {2,5}, {3,5}, {4,5}, {1,6}, {2,6} {3,6}, {4,6}, {5,6}, {1,7}, {2,7}, {3,7}, {4,7}, {5,7}, {6,7}, {1,8}, {2,8}, {3,8}, {4,8}, {5,8}, {6,8}, {7,8}, {1,9}, {2,9}, {3,9}, {4,9}, {5,9}, {6,9}, {7,9}, {8,9}Table 2
The control gains that correspond to Table1.
PlanControl gainW1g1=0−64.649933.3755Tg2=55.1590−68.253518.7660Tg3=5.1115−31.213115.4023TW2g1=00−23.1931Tg2=301.6367−56.1317−191.4356Tg3=37.43659.1931−93.6944TTable 3
Results under lacked poles.
Preset polesControl gainActual polesμii=12n=−1±0.5i−1±i−2±ig1=−8.84550.9987−11.5646Tg2=4.90751.1948−1.5619Tg3=−13.09631.72485.5819T−1±0.5i−1±i−2±i1.0915


### 4.3. Zero Assignment Simulation

Considering the three-degree-of-freedom system in Section4.1, we use zero assignment method in Section 3 to modify the closed-loop system zeros and the closed-loop response of the system.Setb=111T and the closed-loop system H3,3s zeros as ξ1;2=−1±0.5i, ξ3;4=−2±0.5i. Obtain g1,g2,g3 according to Equation (22):
(26)g1=1.9623−3.04480T,g2=−1.3215−0.49120T,g3=−6.3791−1.46820T.The closed-loop poles calculated are−0.1439±0.5047i, −0.6195±1.6824i, −1.0199±0.9522i, and −3.4404. The poles are distributed in the left half plane of the imaginary axis and the closed-loop system is stable. The frequency response curve, the zero-pole distribution diagram, and the Nyquist diagram of the modified system H3,3 are illustrated in Figure 6, Figure 7, and Figure 8, respectively. Comparing Figure 2 with Figure 7, we can see that the zeros of the system are accurately assigned to the desired values. Figure 4, Figure 6, and Figure 8 indicate that, after the assignment of zeros, the system stability increases.Figure 6
The frequency response for the original systemH3,3 (solid line) and the modified system H3,3 (dotted line).Figure 7
The pole-zero map for the modified systemH3,3.Figure 8
The Nyquist curve for the modified systemH3,3.


### 4.4. Pole and Zero Assignment Simulation

Based on the three-degree-of-freedom system in Section4.1, set b=111T, the closed-loop system H3,3s zeros as ξ1;2=−1±0.5i, and the closed-loop system poles as μ1;2=−1±0.5i. Obtain g1,g2,g3 according to Equations (13)–(15) and Equation (22):
(27)g1=1.9623−3.04480T,g2=−1.3215−0.49120T,g3=−6.3791−1.46820T.The closed-loop poles calculated at this time are−0.3563±1.6213i, −0.9778±0.9769i, −1±0.5i, and −3. The poles are distributed in the left half plane of the imaginary axis and the closed-loop system is stable. The frequency response curve, the zero-pole distribution diagram, and the Nyquist diagram of the modified system H3,3 are illustrated in Figure 9, Figure 10, and Figure 11, respectively. Figure 2 and Figure 10 indicate that the system zeros and poles are accurately assigned to the desired values. After the assignment, an increased system stability is observed according to Figure 4, Figure 9, and Figure 11. The validity of the proposed method is verified by the simulation results.Figure 9
The frequency response for the original systemH3,3 (solid line) and modified system H3,3 (dotted line).Figure 10
The pole-zero map for the modified systemH3,3.Figure 11
The Nyquist curve for the modified systemH3,3.


## 4.1. Pole Assignment Simulation

Consider a three-degree-of-freedom system with matrices:(24)M=223,C=2.5−20−23−10−11,K=10−3−4−330−404.The open-loop poles are−0.0305±0.5894i, −0.8503±1.0119i, and −0.6609±2.1200i.Now use pole assignment method proposed in this paper to modify the closed-loop response of the system. Setb=111T. For n=3, N=7 according to Section 2.2. Since the original system has a lower characteristic frequency and a smaller damping ratio, the closed-loop system characteristic values ate assigned as μ1;2=−1±0.5i, μ3;4=−1±i, μ5;6=−2±i, and μ7=−3 to increase the damping of the system and enhance the stability.Obtaing1,g2,g3 according to Equations (13)–(15):
(25)g1=−38.1817−6.781011.3765T,g2=3.7396−14.44872.4203T,g3=−4.1357−4.5904−10.6608T.The above PID input feedback solution is valid according to Section2.4. In order to visualize the effect of this method, the frequency response curve, zero-pole distribution, and Nyquist plot of the H3,3 of original system and the modified system are given, shown in Figures 1–5. Figures 2 and 3 show that the poles of the modified system are assigned to the specified positions, and an additional pole is added to the modified system due to the integration of integral feedback. The introduction of the extra pole improves the flexibility to handle the closed-loop systems in engineering applications. We can see from Figure 1, Figure 4, and Figure 5 that after pole reassignment, system features such characteristic frequency, damping, and stability have been changed, and the overall performances have been improved.Figure 1
The frequency response for the original systemH3,3 (solid line) and the modified system H3,3 (dotted line).Figure 2
The zero-pole map for the original systemH3,3.Figure 3
The zero-pole map for the modified systemH3,3.Figure 4
The Nyquist curve for the original systemH3,3.Figure 5
The Nyquist curve for the modified systemH3,3.


## 4.2. Minimum Sensor Configuration for Pole Assignment

Based on the pole assignment in Section4.1 and the minimum sensor layout in Section 2.3, a minimum sensor configuration scheme for pole assignment simulation is developed and verified according to Section 2.4. Table 1 shows all solutions that can assign the poles of the closed-loop system to preset values μ1;2=−1±0.5i, μ3;4=−1±i, and μ5;6=−2±i, while keeping the closed-loop system stable at the same time for the case in Section 4.1. The PID feedback gains for the 2 solutions are given in Table 2 (only the control gains of the first method in each Wk are given). The system performance may destabilize when the number of assigned poles is less than N (i.e., when the lacked poles are assigned, the number of assigned poles in Table 3 is 2n). This highlights the importance of determining the number of assignments. The specific results are demonstrated in Tables 1–3.Table 1
Pole assignment solutions.
PlanColumn replaced by a zero vector inGW1{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}W2{1,2}, {1,3}, {2,3}, {1,4}, {2,4}, {3,4}, {1,5}, {2,5}, {3,5}, {4,5}, {1,6}, {2,6} {3,6}, {4,6}, {5,6}, {1,7}, {2,7}, {3,7}, {4,7}, {5,7}, {6,7}, {1,8}, {2,8}, {3,8}, {4,8}, {5,8}, {6,8}, {7,8}, {1,9}, {2,9}, {3,9}, {4,9}, {5,9}, {6,9}, {7,9}, {8,9}Table 2
The control gains that correspond to Table1.
PlanControl gainW1g1=0−64.649933.3755Tg2=55.1590−68.253518.7660Tg3=5.1115−31.213115.4023TW2g1=00−23.1931Tg2=301.6367−56.1317−191.4356Tg3=37.43659.1931−93.6944TTable 3
Results under lacked poles.
Preset polesControl gainActual polesμii=12n=−1±0.5i−1±i−2±ig1=−8.84550.9987−11.5646Tg2=4.90751.1948−1.5619Tg3=−13.09631.72485.5819T−1±0.5i−1±i−2±i1.0915


## 4.3. Zero Assignment Simulation

Considering the three-degree-of-freedom system in Section4.1, we use zero assignment method in Section 3 to modify the closed-loop system zeros and the closed-loop response of the system.Setb=111T and the closed-loop system H3,3s zeros as ξ1;2=−1±0.5i, ξ3;4=−2±0.5i. Obtain g1,g2,g3 according to Equation (22):
(26)g1=1.9623−3.04480T,g2=−1.3215−0.49120T,g3=−6.3791−1.46820T.The closed-loop poles calculated are−0.1439±0.5047i, −0.6195±1.6824i, −1.0199±0.9522i, and −3.4404. The poles are distributed in the left half plane of the imaginary axis and the closed-loop system is stable. The frequency response curve, the zero-pole distribution diagram, and the Nyquist diagram of the modified system H3,3 are illustrated in Figure 6, Figure 7, and Figure 8, respectively. Comparing Figure 2 with Figure 7, we can see that the zeros of the system are accurately assigned to the desired values. Figure 4, Figure 6, and Figure 8 indicate that, after the assignment of zeros, the system stability increases.Figure 6
The frequency response for the original systemH3,3 (solid line) and the modified system H3,3 (dotted line).Figure 7
The pole-zero map for the modified systemH3,3.Figure 8
The Nyquist curve for the modified systemH3,3.


## 4.4. Pole and Zero Assignment Simulation

Based on the three-degree-of-freedom system in Section4.1, set b=111T, the closed-loop system H3,3s zeros as ξ1;2=−1±0.5i, and the closed-loop system poles as μ1;2=−1±0.5i. Obtain g1,g2,g3 according to Equations (13)–(15) and Equation (22):
(27)g1=1.9623−3.04480T,g2=−1.3215−0.49120T,g3=−6.3791−1.46820T.The closed-loop poles calculated at this time are−0.3563±1.6213i, −0.9778±0.9769i, −1±0.5i, and −3. The poles are distributed in the left half plane of the imaginary axis and the closed-loop system is stable. The frequency response curve, the zero-pole distribution diagram, and the Nyquist diagram of the modified system H3,3 are illustrated in Figure 9, Figure 10, and Figure 11, respectively. Figure 2 and Figure 10 indicate that the system zeros and poles are accurately assigned to the desired values. After the assignment, an increased system stability is observed according to Figure 4, Figure 9, and Figure 11. The validity of the proposed method is verified by the simulation results.Figure 9
The frequency response for the original systemH3,3 (solid line) and modified system H3,3 (dotted line).Figure 10
The pole-zero map for the modified systemH3,3.Figure 11
The Nyquist curve for the modified systemH3,3.


## 5. Conclusions

In this paper, a PID-active vibration control method is proposed based on the receptance method. The proposed method can be utilized to assign the poles and zeros of closed-loop systems in order to improve the performances. Due to the integration of integral control, the proposed method guarantees the positive definiteness of the system and avoids the appearance of a nonsingular system. The proposed method also extends the application of the dynamic flexibility method. The procedures of assigning poles and zeros to a system are demonstrated. Numerical simulations for pole and zero assignments are conducted. Simulation results indicate that the proposed method is generally effective. The proposed method in this paper can be utilized to improve performance and reliability on robotic systems such as flexible robots, robot arms, and intelligent vehicles. In future work, the proposed method will be expanded to handle vibration control tasks for nonlinear systems with multi inputs and outputs.



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