#### 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.

#### 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 Section 2 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 a degree freedom linear system can be described in second-order form as follows according to [19]: where are mass, damping, and stiffness matrices, respectively; is the displacement vector. , , , and for any nonzero vector , there is , , and .

Integrate PID feedback control to the system and the closed-loop system can be written as follows: where is the force distribution vector and is the input control vector: where are input control gain vectors for displacement, differential, and velocity, respectively.

Express Equation (2) in the Laplace space:

Then,

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): where is the receptance matrix of the open-loop system, which can be obtained in practice by measuring the receptance [20]. The characteristic polynomial can be derived from Equation (6) at the same time:

Thus, the roots of Equation (8) are the pole of the closed-loop system:

Given , , and Equation (9):

The problem of assigning the poles of a closed-loop system to setpoints can be transformed to the solution of input control gain vectors , and . Where , , is closed under conjugation, is the number of closed-loop system poles that can be assigned.

##### 2.2. Minimum Number of Assignable Poles

Given a positive definite system with closed-loop conjugate poles , suppose that the adjoint matrix of is and the determinant is . can be expressed as follows:

Since for any , , Equation (10) can be written as follows:

Set

The highest degree of in is and the highest degree of in is . Since the number of closed-loop poles is equal to the degree of Equation (10) and the highest degree of in Equation (10) is , the number of characteristic polynomial root is . 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 is self-conjugate.

##### 2.3. Calculate Vectors and Minimum Sensor Layout Method

The solution of can be realized by the following equation: where and ,, and .

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: where is the Moore–Penrose generalized inverse of . When the assigned closed-loop poles are self-conjugate, , and 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 in can be assigned to zero by setting the values of the corresponding columns of the to be zero. In practice, the solution means that fewer sensors are involved. Due to the integration of PID input feedback control and , the number that can be allocated to zero in has at most 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 poles and the optional column vector to be replaced by the 0 in the are determined, a specific PID input feedback solution is derived, identified as . From Equations (13)–(15), can be calculated in this specific PID input feedback solution. Below we will show how to verify whether is valid or not.

The closed-loop receptance matrix can be derived from Equation (5) as follows: where is as follows:

The problem of verifying whether a particular is valid is equivalent to verifying whether is a nonsingular matrix for any under the determined case. The verification process of is as follows: Firstly, the real vectors are calculated according to Equations (13)–(15). Then, substitute into Equation (17) and obtain the determinant of . For any , if is zero, the closed-loop system poles cannot be assigned to the designated values, meaning that is invalid. Otherwise, the closed-loop system poles can be assigned to the specified desired values, indicating that is valid.

#### 3. Zero Assignment of PID Control Systems

##### 3.1. Formulation of Zero Assignment

The characteristic polynomial for the zeros of receptance can be realized when the denominator matrix (*i*, *j*) of Equation (6) is zero:
where are the unit vectors obtained from the corresponding columns of the identity matrix.

Therefore, the problem of assigning zeros of the closed-loop system receptance to the setpoints can be realized as follows:

Given , *b*, *i*, *j*, and

Calculate the input control gain vectors . Where , , is the number of poles in a closed-loop system that can be assigned.

Equation (19) can be written in the following form:
where is the open-loop receptance value at the open-loop system coordinate of (*i*, *j*). Transform Equation (20) to . Since is a scalar, set

We have . Since is scalar, after transposing we get , i.e., . Therefore, for preset closed-loop system zeros , we have where

It can be seen from Equation (22) that the inverse of the matrix can be obtained when the preset zeros are given, and then the real vectors can also be obtained. In particular, when and 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 . Similarly, by assigning 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:

The open-loop poles are , , and .

Now use pole assignment method proposed in this paper to modify the closed-loop response of the system. Set . For , 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 , , , and to increase the damping of the system and enhance the stability.

Obtain according to Equations (13)–(15):

The above PID input feedback solution is valid according to Section 2.4. In order to visualize the effect of this method, the frequency response curve, zero-pole distribution, and Nyquist plot of the 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.

##### 4.2. Minimum Sensor Configuration for Pole Assignment

Based on the pole assignment in Section 4.1 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 , , and , 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 are given). The system performance may destabilize when the number of assigned poles is less than (i.e., when the lacked poles are assigned, the number of assigned poles in Table 3 is ). This highlights the importance of determining the number of assignments. The specific results are demonstrated in Tables 1–3.

##### 4.3. Zero Assignment Simulation

Considering the three-degree-of-freedom system in Section 4.1, we use zero assignment method in Section 3 to modify the closed-loop system zeros and the closed-loop response of the system.

Set and the closed-loop system zeros as , . Obtain according to Equation (22):

The closed-loop poles calculated are , , , and . 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 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.

##### 4.4. Pole and Zero Assignment Simulation

Based on the three-degree-of-freedom system in Section 4.1, set , the closed-loop system zeros as , and the closed-loop system poles as . Obtain according to Equations (13)–(15) and Equation (22):

The closed-loop poles calculated at this time are , , , and . 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 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.

#### 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.

#### Data Availability

No data was used to support this study.

#### Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

#### Acknowledgments

This work is supported by the Key Science and Technology Research Project of Chongqing (cstc2018jszx-cyztzxX0032).