*Article* **Absolute Stability Condition Derivation for Position Closed-Loop System in Hydraulic Automatic Gauge Control**

**Yong Zhu 1, Shengnan Tang 1,\*, Chuan Wang 1,2,\*, Wanlu Jiang 3, Jianhua Zhao 3,4 and Guangpeng Li <sup>1</sup>**


Received: 18 September 2019; Accepted: 12 October 2019; Published: 18 October 2019

**Abstract:** In the metallurgical industry, hydraulic automatic gauge control (HAGC) is a core mechanism for thickness control of plates used in the rolling process. The stability of the HAGC system's kernel position closed-loop is key to ensuring a process with high precision, speed and reliability. However, the closed-loop position control system is typically nonlinear, and its stability is affected by several factors, making it difficult to analyze instability in the system. This paper describes in detail the functioning of the position closed-loop system. A mathematical model of each component was established using theoretical analysis. An incremental transfer model of the position closed-loop system was also derived by studying the connections between each component. In addition, based on the derived information transfer relationship, a transfer block diagram of disturbance quantity of the system was established. Furthermore, the Popov frequency criterion method was introduced to ascertain its absolute stability. The results indicate that the absolute stability conditions of the position closed-loop system are derived in two situations: when spool displacement is positive or negative. This study lays a theoretical foundation for research on the instability mechanism of an HAGC system.

**Keywords:** rolling mill; hydraulic automatic gauge control system; position closed-loop system; absolute stability condition; Popov frequency criterion; flow control

#### **1. Introduction**

The development of "intelligent" and "green" manufacturing equipment has propelled the metallurgical industry to pursue intelligence in their rolling equipment, and to ensure high quality of plates and strips used in the industry [1]. However, it has been demonstrated that mass production often results in instability in the rolling process, hindering high-precision and intelligent development. The hydraulic automatic gauge control (HAGC) system is a core mechanism for thickness control of plates used in the rolling process. Its function is to automatically adjust the roll gap of a rolling mill when external disturbance factors change, so as to ensure that the target thickness of the strip is within the index range. The stability of the system's closed-loop kernel position is key to ensuring a process with high precision, speed and reliability.

The HAGC system is complex with multiple links—it is nonlinear and has several parameters that influence its functioning. Because the system's working mechanism is multifaceted, it is difficult to analyze its instability, a problem that researchers in the engineering field have been trying to solve [2–4]. Scholars are currently studying the dynamic characteristics of the HAGC system. Roman et al. [5] researched the thickness control of cold-rolled strips and proposed a system that compensates for errors caused by the hydraulic servo-system used for positioning of the rolls. Hu et al. [6] analyzed the rolling characteristics of the tandem cold-rolling process and proposed an innovative multivariable optimization strategy for thickness and tension based on inverse linear quadratic optimal control. Sun et al. [7] proposed a dynamic model of a cold rolling mill based on strip flatness and thickness integrated control. They conducted dynamic simulation of the rolling process, obtaining information on thickness and flatness. Prinz et al. [8] compared two different AGC setups and developed a feed forward approach for lateral asymmetry of entry thickness. They also developed a new feed forward control approach for the thickness profile of strips in a tandem hot rolling mill [9]. Kovari [10] studied the effect of internal leakage in a hydraulic actuator on dynamic behaviors of the hydraulic positioning system. Li et al. [11] presented a robust output-feedback control algorithm with an unknown input observer for the hydraulic servo-position system in a cold-strip rolling mill with uncertain parameters, immeasurable states and unknown external load forces. Sun et al. [12] introduced key technical features and new technology of the improved cold strip mill process control system: system architecture, hardware configuration and new control algorithms. Yi et al. [13] analyzed HAGC's step response test process: they simulated and established a transfer function model of the test using matrix and laboratory (MATLAB). Liu et al. [14] built a vibration system dynamic model with hydraulic-machinery coupling for four-stand tandem cold rolling mills. The model integrated MATLAB software with automatic dynamic analysis of mechanical systems (ADAMS). Wang et al. [15] established a mathematical model for position–pressure master–slave control of a hydraulic servo system, then simulated the system with AMESim and MATLAB. Hua et al. [16] provided rigorous proof of the exponential stability of the HAGC system by implementing the Lyapunov stability theory. Zhang et al. [17] studied the control strategy of a hydraulic shaking table based on its structural flexibility. Qian and Wang et al. [18,19] researched the effects of important elements, such as valves [20–22], pumps [23–26] and rotors [27], on stability. The influence of excitation forces on the vibration of a pump and measures of noise reduction were studied by Ye et al. [28,29]. Bai et al. [30–32] studied the vibration in a pump under varied conditions. These researchers have had great results with their experiments, providing the basis for further study. However, theoretical derivation and research on the instability mechanism of an HAGC system is still relatively rare.

Scholars have previously applied the Lyapunov method to study the absolute stability of a nonlinear closed-loop control system [33,34]. However, this method has certain reservations, and application of the required Lyapunov function is difficult [35,36]. In 1960, V. M. Popov created a frequency criterion method to determine absolute stability of a nonlinear closed-loop control system. It relied on a classical transfer function and eliminated the dilemma of reconstructing a decision function. This method is of great applicatory value and has been widely recognized by scholars worldwide [37,38]. However, there are still rare results via applying the Popov frequency criterion method to the stability of the HAGC system. The HAGC system is a typical nonlinear closed-loop control system with many influence parameters, and its dynamic characteristics are complex and changeable. When the system is in certain working states, the nonlinear vibration may be induced [39,40]. If the instability mechanism cannot be effectively mastered and controlled in time, a major vibration accident may occur in the system [41–43]. Therefore, it is very important and urgent to explore the instability mechanism of the HAGC system and solve the problem of dynamic instability and inhibition from the source. Conducting the theoretical derivation and in-depth study of the instability mechanism of the HAGC system by using Popov frequency criterion method, is a new technique which needs to be further explored.

In this paper, the Popov frequency criterion method is introduced to theoretically deduce the absolute stability condition for key position closed-loop system in HAGC. The purpose is to lay a theoretical foundation for the study on the instability mechanism of the HAGC system. In Section 2, the mathematical model of the position closed-loop system is established. In Section 3, the incremental transfer model of the position closed-loop system is deduced. In Section 4, the absolute stability condition for the position closed-loop system is deduced. In Section 5, some conclusions are provided.

#### **2. Mathematical Model of Position Closed-Loop System**

The HAGC system is mainly controlled by electro-hydraulic servo valve and oil cylinder to realize the setting and adjustment of roll gap or rolling pressure. In terms of control function, a complete HAGC system is composed of several automatic control systems. The most important three control loops are as follows: cylinder position closed loop, rolling pressure closed loop and thickness gauge monitoring closed loop, as shown in Figure 1.

**Figure 1.** Function diagram of the hydraulic automatic gauge control (HAGC) system.

As the basis of the whole thickness control, cylinder position closed loop is used to control the displacement in a timely and accurate manner with the change of rolling conditions, so as to achieve the setting and controlling of the roll gap. In the position closed-loop system, the measured displacement value is negatively fed back to the signal input end and compared with the given displacement value. If there is a deviation, it will be adjusted by the displacement adjuster and converted into current signal by the power amplifier and further sent to the electro-hydraulic servo valve. After the servo valve obtains the current signal, it will control the flow into the working chamber of the cylinder through the movement of valve spool and then adjust the piston displacement of the cylinder until the feedback value is equal to the set value.

#### *2.1. Mathematical Model of Controller*

The controller generally adopts proportion-integration-differentiation (PID) adjuster and its dynamic transfer function can be expressed as:

$$\mathbf{G}\_{\mathbf{c}}(\mathbf{s}) = K\_p (1 + \frac{1}{T\_{\mathbf{i}}\mathbf{s}} + T\_{\mathbf{d}}\mathbf{s}),\tag{1}$$

where *Kp* is proportionality coefficient, *Ti* is integral time constant, *Td* is differential time constant and *s* is the Laplace operator.

## *2.2. Mathematical Model of Servo Amplifier*

The function of the servo amplifier is to convert voltage signal into current signal and then control the servo valve to realize flow regulation. Since the response time of the servo amplifier is extremely short, it can be treated as a proportional component and its dynamic transfer function is:

$$K\_d = \frac{I}{\mathcal{U}}\tag{2}$$

where *I* is the output current (A), *U* is the input voltage (V) and *Ka* is amplification coefficient (A/V).

### *2.3. Mathematical Model of Hydraulic Power Mechanism*

The hydraulic power mechanism of HAGC system is mainly realized by controlling the motion of the hydraulic cylinder with the electro-hydraulic servo valve. Its structural principle is displayed in Figure 2. In order to improve the response performance of the system, the servo valve is generally used to control the rodless chamber of the hydraulic cylinder, and the rod chamber of the hydraulic cylinder is supplied with oil at a constant pressure.

**Figure 2.** Schematic diagram of the servo valve control hydraulic cylinder.

When the servo valve works in the right position, the high pressure oil directly enters into the rodless chamber of the hydraulic cylinder. At this time, the piston rod of the cylinder drives the load to realize the pressing down action. When the servo valve operates in the left position, the fast lifting action of the roll can be achieved. During the rolling process, oil at a constant pressure of 1 MPa is always passed through the rod chamber to increase the damping of the system.
