## Abstract

Feedforward control is widely used in control systems since it can achieve high tracking performance by effectively compensating for known disturbances before they affect the system. For traditional feedforward control methods, the performance improvement highly depends on the model quality of the system model and the accuracy of the model-inversion. However, on one hand, in practice, the modeling error is inevitable, especially for precision motion systems with complex dynamics, on the other hand, the nonminimum phase (NMP) systems often lead to a problem that plant inversion is unstable and non- causal. Therefore, this paper proposes an approach that combines the benefits of data-driven feedforward control and the gradient descent method. This integrated approach aims to address challenges related to laborious model identification and unstable plant inversion simultaneously. The main idea is to replace the model with dedicated experiments on the system and to avoid calculating plant inversion by applying the gradient descent method to the learning process. The simulation and experimental results show that the algorithm can achieve the optimal point-to-point tracking performance without relying on model-inversion.

## 1 Introduction

Feedforward control is widely used in the high-precision and high-speed motion control because it can effectively compensate for tracking errors induced by reference trajectories and disturbance [1–3]. There are two kinds of feedforward control commonly applied in the field of high-speed and high-precision motion control, one is iterative force injection-based feedforward control (IFIFC) and the other one is model-based feedforward control (MFC).

Iterative force injection-based feedforward control updates the feedforward force by learning from previous tasks and injects it into the control system to obtain optimal tracking performance. Article [4] analyzed three convergence rate design methods for IFIFC and conducted experiments on ABB industrial robots. It proves that the approximation degree between model inversions of the plant directly affects the convergence. Moreover, Articles [4–9] show that theoretically, IFIFC can achieve zero-error tracking performance for the entire trajectory in repetitive tasks. However, when the trajectory or the regular disturbances change, a reiterative calculation of the optimal feedforward force is necessary. Otherwise, the tracking performance will deteriorate.

The ideal controller of MFC is the inverse of the plant, aiming to compensate the tracking error caused by the trajectory. In contrast to IFIFC, MFC does not exhibit significant performance degradation due to changes in the reference trajectory. The parameterized feedforward control is one of the common MFC methods, including a filant, aiming to compensate the tracking error caused by the trajectory. In 10–14] introduced basis functions to parameterize the feedforward controller and converted the design of the controller to the identification of the controller parameters. In article [10], the optimal parameters are identified by the least squares method, and it was verified that the fixed structure parameterized feedforward control method could achieve high trajectory tracking performance. However, the premise of this method is that the system model has a unit molecule; otherwise, the parameter optimization process will lead to a nonanalytic problem. In view of this problem, a parameterization method, combined with input shaping and feedforward control, was proposed in Ref. [12]. The method solves the nonanalytic optimization problem caused by the system model with a nonunit molecule and ensures the stability of both the feedforward controller and input shaper. The optimal parameters were identified by the least squares method. However, in the process of iteration, the model inverse composed of basis functions needs to be calculated, too. The accuracy of the inverse will affect the performance [15].

No matter what design method, the extent of approximation between the nominal model and the true plant significantly affects the performance of the feedforward controller [16–23]. Actually, the acquisition of an accurate system mathematical model is often complex and expensive in practice. In addition, for NMP systems, the existence of NMP zeros often leads to a problem that the plant inversion is unstable and noncausal. Butterworth [24] et al. reviewed three preview-based stable inversion methods, including the nonminimum-phase zero ignore (NPZ-Ignore) method, the zero phase error tracking (ZPETC) method, and the zero amplitude error tracking (ZMETC) method. However, these methods use a stable approximate inverse ignoring or approximating the kinetic of the unstable part, which affects the actual feedforward compensation effect [25].

The gradient descent method with the first derivative could remove the need for calculating the model inverse in the iterative process. The data-driven methods update the feedforward force through the system input and output data collected in the previous task and avoid the process of tedious model identification. Therefore, an adjoint-based data-driven inverse-free iterative feedforward control method is proposed in this paper to solve the problem of complex model identification and unstable model inversion, simultaneously.

This paper will verify the tracking performance of the proposed method on a parameterized feedforward control framework. The main contributions can be summarized as follows:

A feedforward control algorithm based on the gradient descent method is introduced to address the parameter optimization challenges in fixed-structure feedforward control. The method avoids calculating the plant inverse throughout the entire learning process. Furthermore, the convergence of the algorithm is thoroughly analyzed.

A data-driven method based on an adjoint system is introduced in this paper to remove the need for model identification. This method eliminates the need for laborious model identification. Instead, before the commencement of each iteration, an additional experiment is conducted to gather data information from the system. Subsequently, this acquired data is incorporated into the learning process.

The simulation results and experimental findings on a brushless DC motor indicate that the proposed algorithm achieves optimal trajectory tracking without requiring model inversion information.

This article mainly analyzes the theoretical performance of the algorithm and conducts preliminary verification on brushless DC motors. This method will be applied in practical applications in servosystems on the x-y motion platform and industrial robotic arms in the future.

The article is organized as follows: In Sec. 1, the dependence of the existing inverse model feedforward control methods on the inverse quality of the plant is analyzed. Section 2 illustrates the process of parameter identification and proves the feasibility of the algorithm in theory. In Sec. 3, an adjoint-based data-driven method is proposed. The optimal parameters are identified and the convergence of the algorithm is analyzed. In Sec. 4, the simulation results show that the proposed algorithm has higher tracking performance when compared to the ZPETC method. Section 5 details experiments conducted with a brushless DC motor, demonstrating that the proposed algorithm achieves optimal point-to-point trajectory tracking without the necessity of calculating plant inversion. Section 6 provides the articles's conclusion.

*Notation*: Let $w\u2208RN\xd7N$ is defined a positive definite matrix. For a vector $x\u2208RN$, the $ith$ element of the vector *x* is denoted as $x(i)$, $||x||w=xTwx$.

## 2 Problem Formulation

The framework of parameterized input shaping and feedforward control is shown in Fig. 1. The control configuration consists of an input shaper $Cy$, a feedforward controller $Cff$, and a discrete-time stabilizing feedback controller $Cfb$ for ensuring the stability of a closed-loop system. $uff$ is the feedforward control signal and $ey$ is the error signal, $r$ is a known trajectory as shown in Fig. 2, in which is the execution section and is the settling section. Motion starts at $t1$ and ends at $t2$. $ry$ denotes the filtered reference signal after input shaping by $Cy$. The system executes point-to-point reference trajectory tasks, so the control goal is to obtain zero-settling behavior at the settling section.

*By introducing the basis functions, the input shaping filter*$Cy$

*and feedforward controller*$Cff$

*can be parameterized as*

In Fig. 1, the shaped reference trajectory $ry$ is equal to $Cyr$. Note that $ry$ is delayed with respect to $r$ in the settling section with the delay length of $na$, i.e., $ry=r$ when $t\u2208[t2+na,t3]$. Similarly, $e=r\u2212y=ry\u2212y=ey$ in this period of time. Moreover, the goal for point-to-point trajectory tasks is to minimize the positioning errors at settling section. Thus, *e* in the objective function can be replaced by *e _{y}*.

*r*to

*e*is given by

_{y}with $\Psi =r1+PCfb[\phi y,P\phi ff].$

where, $uff=r\phi ff\theta ff$.

*Remark 1*. According to Eq. (8), it becomes clear that $ey=0$ if $CyCff\u22121=P$. This implies that the numerator and denominator of $P$ are described by $Cy$ and $Cff$, respectively. Therefore, the process of determining the optimal parameter vector $\theta $ in $Cy$ and $Cff$ can be regarded as model identification for $P$.

*Remark 2*. The update law of $\theta $ is obtained through the data of the previous iteration, where *C* is polynomials composed of feedforward controllers and shapers. According to Remark 1, it can be seen that the exactness of *C*^{−1} affects the optimal value of $\theta $ and the accuracy of the approximate model of the plant. An article with a similar controller structure using Newton's method further verifies this view, see, e.g., [27].

It is well known that noncollocated sensors and actuators [28], and fast sampling to a continuous-time plant with a relative degree greater than or equal to 2 [29] can lead to NMP systems. For these conditions, the inverse dynamics of *C* are usually unstable. The existing algorithms cannot avoid the influence caused by approximate inverse on the results. Therefore, a feedforward control algorithm based on the gradient descent method is proposed in this paper to solve the problem. Actually, the proposed algorithm removes the need for computing inversion during the whole iterative learning process so as to eliminate the possible impact on the final result caused by approximate inverse. Meanwhile, a special experiment is carried out on the real system to replace the need for the mathematical model of the plant.

## 3 Parameter Identification

*v*for the sake of analysis convenience. $eyj$ and $yj$ in

*j*th task can be given by

*G*is the transfer from

*u*to

*y*

*M*and

_{c}*M*as follows:

_{r}*θ*can be expressed as the following equation:

where $\xi \u2208$$R(na+nb)\xd7(na+nb)$ is a gain diagonal matrix that determinates the iteration rate, and $\xi $ meets the condition that $\xi \u2208[0,\xi \xaf]$. The range of $\xi $ is discussed in Sec. 4.2.

According to Eq. (15), it is clear that *G* in Eq. (21) contains the unknown plant *P*. In the traditional inverse model feedforward control methods, *P* needs to be identified to obtain an exact mathematical model. The system performance is determined by the extent of approximation between the nominal model and the true plant. However, for complex systems, it is difficult to obtain exact models, and high uncertainty worsens the control performance. Therefore, in Sec. 4, a data-driven method based on an adjoint system is adopted to replace the plant with an experiment on the system.

## 4 Data-Driven Learning: Adjoint-Based Feedforward Control

Section 3 completed the introduction of the feedforward control algorithm based on gradient descent method. In this section, to eliminate the algorithm's dependence on the approximate model of the system, a data-driven method based on adjoint is first introduced. Then, the convergence of the algorithm is analyzed in detail, and the detailed process of algorithm implementation is summarized.

### 4.1 Adjoint System.

It can be seen that the data of $\theta j$ and $eyj$ in Eq. (21) can be obtained by the last experiment except for $GTweeyj.$ Therefore, a data-driven approach based on adjoint systems is discussed to get $GTweeyj$ in this section.

*G*, the adjoint

*G**satisfies the condition

*G**of

*G*satisfies $G*=GT$. Therefore, the gradient descent method shown in (18) can be elaborated by the adjoint method

It is expected that $GTweeyj$ can be measured without the parameters of the system mathematical model in this paper. As mentioned earlier, the method of adjoint operators can represent the transfer function related to the controlled object *G*. Using the above conditions, an extra experiment can be designed for the control system to obtain the data information about the system and complete the data required for parameter update formulas. The experiment steps are shown in Procedure 1.

(1) Get the time reverse of $eyj$ :$e\xafyj=\tau weyj$; |

(2) Take $e\xafyj$ is used as the input signal for the experiment. Without adding trajectory signals, it is injected into the closed-loop control system to obtain $Ge\xafyj$; |

(3) Calculate $GTweeyj=\tau Ge\xafyj$. |

(1) Get the time reverse of $eyj$ :$e\xafyj=\tau weyj$; |

(2) Take $e\xafyj$ is used as the input signal for the experiment. Without adding trajectory signals, it is injected into the closed-loop control system to obtain $Ge\xafyj$; |

(3) Calculate $GTweeyj=\tau Ge\xafyj$. |

### 4.2 Analysis of Algorithm Convergence.

With $\delta \u2208[0,1)$ See [35] for equivalent definitions.

*θ*can be given by

Set $\epsilon =I\u2212\xi wf$$\u2212\xi \Phi Twe\Phi $, and the following equation is obtained: $||\theta j+1\u2212\theta \u221e||\u2212||\theta j\u2212\theta \u221e||$$=||\epsilon \theta j\u2212\epsilon \theta \u221e||\u2212||\theta j\u2212\theta \u221e||$. According to the compatibility of matrix norm, $||\epsilon \theta j\u2212\epsilon \theta \u221e||\u2264$$||\epsilon ||||\theta j\u2212\theta \u221e||$. If $\delta =||\epsilon ||<1$, the convergence of the algorithm is guaranteed. According to the triangle inequality property of norm, it can be obtained that $||I\u2212\xi (wf+\Phi Twe\Phi )||\u22641\u2212||\xi (wf+\Phi Twe\Phi )||$, and the constraint on $\xi $ is given by $\xi \u2264\xi \xaf=$$2||wf+\Phi Twe\Phi ||\u22121$, the closer $\xi $ is to $\xi \xaf$, the faster the algorithm converges.

### 4.3 Algorithm Implementation Procedure.

Sections 3 and 4 implement a data-driven feedforward controller algorithm, which is divided into two parts. The first step is to obtain data information through a separate experiment on the system, as shown in Procedure 1. The second part is to feed the obtained data into the iterative learning process to find the optimal parameters of the feedforward controller, as shown in Procedure 2.

1) Initialization Procedure |

a) Select the initial value of the identification parameter vector $\theta $ |

b) Determine the appropriate w and _{e}w_{f} |

c) Calculate $\xi $ |

2) Measurement and calculation program |

a) Get $eyj$ in $jth$ task |

b) Get $GTweeyj$ according to Procedure 1 |

c) Update $\theta j+1$ in Eq. (18) |

3) Calculate $Cffj+1,Cyj+1$ |

$Cyj+1=1+\u2211k=1na\phi k(z\u22121)\theta kj+1$ |

$Cffj+1=1+\u2211k=na+1na+nb\phi k(z\u22121)\theta kj+1$ |

4) set $j\u2192j+1$, and repeat step 2) and 3) until the error is small enough. |

1) Initialization Procedure |

a) Select the initial value of the identification parameter vector $\theta $ |

b) Determine the appropriate w and _{e}w_{f} |

c) Calculate $\xi $ |

2) Measurement and calculation program |

a) Get $eyj$ in $jth$ task |

b) Get $GTweeyj$ according to Procedure 1 |

c) Update $\theta j+1$ in Eq. (18) |

3) Calculate $Cffj+1,Cyj+1$ |

$Cyj+1=1+\u2211k=1na\phi k(z\u22121)\theta kj+1$ |

$Cffj+1=1+\u2211k=na+1na+nb\phi k(z\u22121)\theta kj+1$ |

4) set $j\u2192j+1$, and repeat step 2) and 3) until the error is small enough. |

## 5 Simulation

In this section, Matlab/Simulink is applied to complete the simulation. ZPETC and the other two methods are commonly used feedforward model-inverse control techniques in discrete-time systems. Compared to the ZMETC method, ZPETC is more conducive to reducing the impact of high-frequency noise on the system. At the same time, compared to the NPZ ignore method, it has a higher order and retains more nonminimum phase zero dynamic characteristics. In addition, ZMETC may transform the overall function of the system into an infinite-impulse-response filter, which remains a finite-impulse-response filter when using ZPETC technology, making ZPETC advantageous for various applications [24]. Therefore, the ZPETC method is used for comparison to verify the theoretical control performance of the proposed feedforward control algorithm. The results show that the proposed method achieves higher tracking performance than ZPETC and avoids calculating approximate inverse (Fig. 3).

*P*(

*z*) is given by

*n*sampling points are the execution section, and the remaining

*N*−

*n*sampling points are the settling section trajectory, then the weighting matrix we can be described as

$\rho $ and $\lambda $ limit the influence weight of the errors in the execution section and the settling section, respectively. It can be seen from Fig. 4 that the smaller $\gamma $ is, the higher the tracking performance is.

In order to compare under the same conditions, the ZPETC method used in the comparison selects the same parameters as the proposed algorithm when setting the initial values. At the same time, to ensure the optimal performance that the two methods can achieve, the $wf$ and $wd$ in the objective function are set to 0, and only the convergence performance of the two algorithms is compared.

Both algorithms undergo 100 iterations. The simulation comparison results between the ZPETC method (black dashed line) and the proposed method (solid line) are shown in Fig. 5. Obviously, the proposed method has better convergence performance.

The curve of tracking error $ey$ in the whole section is shown in Fig. 6, and the curve of tracking error $e$ in the settling section is shown in Fig. 7. As can be seen from Fig. 7, the maximum tracking error in the first iteration is of the order of magnitude 10 deg, and is reduced to 10^{−2} at the 100th iteration. The variation trend of the error in the first, second and 100th iteration indicates that the tracking error has been significantly suppressed. Simulation results show that the proposed feedforward control algorithm can obtain the optimal trajectory tracking performance without calculating the inverse of the plant.

The change curve of $\theta $ is shown in Fig. 8. It can be seen from the figure that $\theta $ converges to the optimal value after 40 iterations. $\theta 1\u2212\theta 4$ are the identification parameters of the shaping filter, and $\theta 5\u2212\theta 7$ are the identification parameters of the feedforward controller. The optimal values are shown in Table 1. $\theta 5$ in feedforward controller corresponds to the parameter of acceleration. Since its order of magnitude is far larger than other parameters, it plays a key role in the whole simulation experiment. The simulation results prove the convergence performance of the algorithm.

Parameter | Optimal value |
---|---|

$\theta 1$ | 1.17 × 10^{−7} |

$\theta 2$ | 8.09 × 10^{−8} |

$\theta 3$ | 1.05 × 10^{−7} |

$\theta 4$ | 1.81 × 10^{−11} |

$\theta 5$ | 1.24 |

$\theta 6$ | 1.76 × 10^{−4} |

$\theta 7$ | −5.81 × 10^{−6} |

Parameter | Optimal value |
---|---|

$\theta 1$ | 1.17 × 10^{−7} |

$\theta 2$ | 8.09 × 10^{−8} |

$\theta 3$ | 1.05 × 10^{−7} |

$\theta 4$ | 1.81 × 10^{−11} |

$\theta 5$ | 1.24 |

$\theta 6$ | 1.76 × 10^{−4} |

$\theta 7$ | −5.81 × 10^{−6} |

## 6 Experiment

In the simulation, the following tasks have been accomplished: (1) selection of the reference trajectory and determination of basis functions; (2) investigation of the impact of different $\gamma $ values on the convergence of the norm of the final error, with the selection of appropriate parameters for experimentation; and (3) simulation of the proposed algorithm and comparison with other methods to validate the practical performance of the algorithm.

In this section, the chosen task trajectories and basis functions are consistent with those used in the simulation. The $\gamma $ value yielding the best results in the simulation is selected for experimentation.

The primary goal of this experiment is to further validate the tracking performance of the proposed algorithm in simulation task 3. The experimental platform is composed of a motion control card and a brushless DC motor, as shown in Fig. 9. The rated voltage of the motor is 24 V, the rated current is 1.17 A, and the rated speed is 6000 r/min. The maximum rotor torque is 104 mN·m, the torque constant is 27.8 mN·m/A, and the encoder is 2000 lines. The host is a PC, and the slave is a four-axis motion control board with a current-loop servodrive in it and designed with an advanced RISC machines chip.

The proportional-integral-derivative control controller serves as a feedback controller, and its parameters are shown in Table 2.

The initial value of *θ* is $\theta =[0,0,0,0,0,0,0]T$. The selected basis functions of $Cy$ and $Cff$ are consistent with the simulation. During the experiment, tracking error, output trajectory, control signal, and other data are collected through the experimental platform.

The two-norm of the error in the settling section is shown in Fig. 11. At the 20th iteration, the error converges to the optimal value. The convergence performance of the proposed algorithm is better than that of the approximate inverse method. Figures 12 and 13 show the variation curves of the entire trajectory tracking error $ey$ and the settling section trajectory positioning error $e$, respectively.

It becomes clear that the order of magnitude of the maximum error in the settling section in the first iteration is 10 deg, and drops to 10^{−2} at the 100th iteration. As can be seen from the variation trend of errors during the first, second, and last iterations in the figure, the error is significantly suppressed. Figure 14 shows the convergence learning process of the parameter vector $\theta $ and the optimal values are shown in Table 3. The experimental results are consistent with the simulation, which proves the convergence of the proposed algorithm.

Parameter | Optimal value |
---|---|

$\theta 1$ | 4.57 × 10^{−4} |

$\theta 2$ | 6.57 × 10^{−6} |

$\theta 3$ | 6.77 × 10^{−9} |

$\theta 4$ | 9.84 × 10^{−11} |

$\theta 5$ | 3.78 × 10^{−2} |

$\theta 6$ | 6.24 × 10^{−5} |

$\theta 7$ | −1.09 × 10^{−7} |

Parameter | Optimal value |
---|---|

$\theta 1$ | 4.57 × 10^{−4} |

$\theta 2$ | 6.57 × 10^{−6} |

$\theta 3$ | 6.77 × 10^{−9} |

$\theta 4$ | 9.84 × 10^{−11} |

$\theta 5$ | 3.78 × 10^{−2} |

$\theta 6$ | 6.24 × 10^{−5} |

$\theta 7$ | −1.09 × 10^{−7} |

## 7 Conclusion

In this article, a data-driven parameterized feedforward control algorithm based on the gradient descent method is proposed in this paper. It is verified that the proposed method can obtain higher tracking performance than the preview-based stable inversion methods through simulation and experiments on the servomotor platform.

The proposed feedforward control method combined the advantages of the data-driven control and the gradient descent method. On one hand, the complex model identification process is avoided, according to replacing the model with dedicated experiments; on the other hand, the proposed algorithm with gradient method removes the need for calculating plant inverse so that it can effectively eliminate the influence on the experimental results caused by approximate inverse when dealing with NMP systems.

In the subsequent work, we plan to continue the improvement of the algorithm from the following two points: first, the proposed algorithm is carried out under the premise of ignoring disturbances. In fact, most working environments cannot avoid the impact of colored noise on system control performance. Therefore, we will discuss how to improve the performance of the algorithm under the influence of external disturbances in the future. Second, the optimization of the parameters of the proposed algorithm feedforward controller is carried out independently, but there is coupling between the parameters. Subsequent work will discuss how to eliminate the coupling between the parameters.

## Funding Data

National Natural Science Foundation of China (Grant No. 5227068; Funder ID: 10.13039/501100001809).

Key Research and Development Program of Zhejiang Science and Technology Department (Grant Nos. 2023C01159, 2024C01230, and 2022C01242).

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.