Observer-Based Robust Control for Hydraulic Velocity Control System

Abstract

This paper investigates the problems of robust stabilization and robust control for the secondary component speed control system with parameters uncertainty and load disturbance. The aim is to enhance the control performance of hydraulic system based on Common Pressure Rail (CPR). Firstly, a mathematical model is presented to describe the hydraulic control system. Then a novel observer is proposed, and an observed-based control strategy is designed such that the closed-loop system is asymptotically stable and satisfies the disturbance attenuation level. The condition for the existence of the developed controller can by efficiently solved by using the MATLAB software. Finally, simulation results are provided to demonstrate the effectiveness of the proposed method.

1. Introduction

It is well known that construction machinery has been widely employed in many occasions, but the efficiency of hydraulic system which is the main transmission model is low. This is primarily why the construction machinery has the drawbacks of high oil consumption and bad emission [1]. In recent years, energy-saving research of the hydraulic system becomes hotspot due to growing energy crisis. If the element which converts the mechanical energy into hydraulic energy could be called primary element, the component which can interchange between mechanical energy and hydraulic energy (hydraulic motor/pump) would be named as secondary component [2, 3]. The hydraulic secondary regulation system is that it can regulate the hydraulic secondary components which interconvert between mechanical energy and hydraulic energy to realize the energy transformation and transmission. Moreover, this kind of system includes energy-storage components. Hence, it can recover braking energy and gravity potential energy. The secondary regulation system can be divided into Common Pressure Rail (CPR) system and flow coupled system [4, 5]. CPR system is suit for construction machinery since it could not only recuperate energy but also keep the engine work in the high efficiency area. Therefore, this technique can reduce the oil consumption. Moreover, it can drive multiple uncorrelated loads simultaneously because the pressure is almost constant in theory. However, in practice hydraulic system always includes parameter uncertainty, nonlinearity, and load disturbance [6, 7]. It is difficult to obtain ideal performance since the precise model is difficult to investigate and then the classical control method is not used well. In the construction machinery area especially, the good performance is difficult to obtain because the bad working condition also results in that the load fluctuates frequently and greatly. For example, the load of swing system in an excavator as the disturbance changes along with different mass of soil, and when mechanical arms work simultaneously, it will also cause the load change. The hydraulic speed control system should meet all of these requirements and it is hard for the traditional PID controller to get a good performance [8]. Energy saving and good performance hydraulic system is the development trend, especially for construction machinery. In recent years, the sensors are used widely in the construction machinery to pursue the good performance; they also help improve the control performance by using advanced control algorithms.

In order to suppress interference, we can adopt novel control methods such as adaptive control [9, 10], fuzzy control [11], and robust control [12–26]. Among the existing approaches, adaptive control method requires large computation and the real-time capability is poor. The precision of fuzzy control is sensitive and is dependent on high experiences. Robust control approach can convert the error between mathematical model and realistic plant into the uncertainty of the mathematical model. Robust controller can guarantee the performances of the resulting closed-loop systems and satisfy all admissible uncertainties. In [14], the author deals with the problem of robust control for delayed singular systems with parametric uncertainties. 𝐻_∞ control of uncertain systems with time delay in state or input is investigated in [18, 19], and the result shows much less conservativeness. Reference [25] is concerned with the problem of robust output feedback control for a class of uncertain discrete time delayed nonlinear stochastic systems. However, there are few results on robust control for velocity control system to improve the systems performance. Furthermore, more sensors are required by collecting signals to obtain the higher performance, but this is improper for construction machinery because of the low reliability due to the serious working condition which is harmful for the sensors.

Motivated by the previous discussion, this paper investigates the problem of robust control for the hydraulic velocity control system with parameters uncertainty and load disturbance. First, we select the swing system of one excavator as the research object and build the mathematical model. Then, by constructing a novel proportional and derivative observer, an observer-based controller is designed such that the closed-loop system is asymptotically stable and satisfies the disturbance attenuation level. Moreover, MATLAB software is used to solve the controller according to the new method. Finally, simulation results are provided to illustrate the effectiveness of the proposed method.

2. System Modeling

In CPR, the constant pressure variable pump and hydraulic accumulator constitute the high pressure oil sources, and multiple different loads connect in parallel between the high press and low press oil passage. When the load changes, the pressure of system has small fluctuation while the flow varies with the load, and the control target such as position, velocity, or power can be reached by regulating the displacement of hydraulic pump/motor. The velocity control system block diagram based on CPR is depicted in Figure 1. The current signal output to the electric-hydraulic servo valve is processed by controller, and the inclination angle of the swash plate can be changed as desired by adjusting the position of variable displacement cylinder. The aim of driving or braking the load is then achieved by the adjustment of rotational velocity. We choose the swing system of 20𝑡 excavator as the model to design the controller; the detailed period is written in the following.

Figure 1 

Velocity control system block diagram.

2.1. Servo Valve Flow Rate Equation

The linear servo valve flow equation is 𝑞_𝐿=𝐾_𝑞𝐾_𝑖𝐾_𝑠𝑢_𝑖−𝐾_𝑐𝑝, (1) where 𝑞_𝐿 and 𝑢_𝑖 are, respectively, cylinder flow and input control voltage, 𝐾_𝑖, 𝐾_𝑠, 𝐾_𝑞, and 𝐾_𝑐 are different kinds of gain: they are the gain of the amplifier, servo valve coefficient, servo valve flow gain, and servo valve flow-pressure coefficient, respectively, and  𝑝  represents cylinder pressure difference.

2.2. Continuity Equation in the Hydraulic Cylinder

Applying the continuity equation to each of the cylinder chambers and the flow rate of cylinder yields 𝑞_𝐿=𝐴_𝑔𝑑𝑦𝑑𝑡+𝐶_𝑖𝑐𝑝₁−𝑝₂+𝐶_𝑒𝑐2𝑝₁−𝑝₂+12𝛽_𝑒𝑉_𝑡2𝑑𝑝₁−𝑉𝑑𝑡2^𝑡𝑑𝑝₂+𝐴𝑑𝑡_𝑔𝑦2𝛽_𝑒𝑑𝑝₁+𝑑𝑡𝑑𝑝₂,𝑑𝑡 (2) where 𝐴_𝑔 represents area of variables cylinder,  𝑦  is the displacement of variables cylinder, 𝛽_𝑒, 𝐶_𝑖𝑐, 𝐶_𝑒𝑐, and 𝑉_𝑡 are coefficients; they are the effective bulk modulus of system, internal or cross-port leakage coefficient of cylinder, external leakage coefficient of cylinder, and total volume of fluid under compression in both chambers, respectively. The volume 𝐴_𝑔𝑦 is usually much smaller than 𝑉_𝑡, and 𝑑𝑝₁/𝑑𝑡+𝑑𝑝₂/𝑑𝑡≈0 because 𝑝=𝑝₁−𝑝₂; thus, 𝑞_𝐿=𝐴_𝑔𝑑𝑦𝑑𝑡+𝐶_𝑡𝑐𝑉𝑝+^𝑡4𝛽_𝑒𝑑𝑝,𝑑𝑡 (3) where 𝐶_𝑡𝑐 is total leakage coefficient of variables cylinder.

The equilibrium of forces equation by applying Newton’s second law to the forces on the cylinder yields 𝐴_𝑔𝑑𝑃=𝑚²𝑦𝑑𝑡²+𝐵_𝑐𝑑𝑦𝑑𝑡+𝐾𝑦+𝐹, (4) where 𝑚 is total mass of cylinder and load referred to cylinder, 𝐵_𝑐 represents viscous damping coefficient of cylinder (N·s/m), and 𝐾 and 𝐹 are load spring gradient and load disturbance, respectively.

2.3. Torque Motion Equation at the Load

The relationship between the instantaneous volume of the secondary component and displacement of the cylinder can be deduced as follows 𝑦𝐷=𝑦_max𝐷_max, (5) where 𝐷_max is maximum volume of the secondary component and 𝑦_max is maximum displacement of cylinder.

Another basic relation is the torque balance equation for the secondary component which is expressed as 𝑝_𝑠𝑦𝑦_max𝐷_max=𝐽_𝑝/𝑚𝑑²𝜃𝑑𝑡²+𝐵𝑑𝜃+𝑀𝑑𝑡𝑟^𝑓, (6) where 𝜃 denotes angular position of the secondary component, 𝑝_𝑠 is pressure of the high pressure pipeline in CPR, 𝐽_𝑝/𝑚 represents total inertia of secondary component and transformational load, 𝑟 is reduction ratio, and 𝑀_𝑓 is the total including friction torque of swing system, wind resistance torque, and slope resistance torque.

2.4. System State-Space Model

State variables are selected as 𝑋=[𝑥₁𝑥₂𝑥₃𝑥₄]^𝑇, where 𝑥₁ is output angular velocity of secondary component, 𝑥₂ is displacement of cylinder, 𝑥₃ is velocity of cylinder, and 𝑥₄ is pressure difference of cylinder. Thus, the state equation can be given by 𝑥̇(𝑡)=−𝐵𝐽_𝑝/𝑚𝑥₁+𝑃_𝑠𝐷_max𝐽_𝑝/𝑚⋅𝑦_max𝑥₂𝑥₃−𝐾𝑚𝑥₂−𝐵_𝑐𝑚𝑥₃+𝐴_𝑔𝑚𝑥₄−4𝛽_𝑒𝐴_𝑔𝑉_𝑡𝑥₃−4𝛽_𝑒𝐶_𝑡𝑐+𝐾_𝑐𝑉_𝑡𝑥₄+0004𝛽_𝑒𝑉_𝑡𝐾_𝑖𝐾_𝑠𝐾_𝑞𝑢(𝑡)+−𝑀_𝑓𝑟⋅𝐽_𝑝/𝑚0−𝐹𝑚0,𝑦(𝑡)=[1000]^𝑇𝑥(𝑡). (7)

The previous constructed model is based on ideal condition. However, the real parameters are variable due to the hard working period of excavator. We consider the parameters variation in 𝐴. Moreover, we add 𝐶_𝑑𝑑(𝑡) to the output because the accuracy of sensor is also influenced with temperature variation and electrical equipments. Finally, since 𝐵_𝑑 is changing with the different load, the time varying function 𝑓(𝑥) is presented to simulate the real status. Hence, the state equations are proposed as follows: 𝑥̇(𝑡)=((𝑡𝑥(𝑡)(𝑡)(),𝑦(𝑡)(𝑡)𝐴+Δ𝐴))+𝐵𝑢+𝑓𝑡,𝑥=𝐶𝑥+𝐶_𝑑𝑑(𝑡), (8) where 𝐴∈𝑅^𝑛×𝑛,  𝐵∈𝑅^𝑛×𝑚,   𝐶∈𝑅^𝑃×𝑛, and 𝐶_𝑑∈𝑅^𝑃×𝑑.

Throughout this paper, we first provide the following assumptions for the following proof(i)(𝐴,𝐵) is a stabilizable pair, (𝐴,𝐶) is an observable pair, and 𝐶_𝑑 is a full column rank;(ii)𝑓(𝑡,𝑥) satisfies the following Lipschitz condition: 𝑓(𝑡,𝑥)−𝑓(𝑡,𝑥)≤𝜃‖𝑥−𝑥‖,for∀𝑥(𝑡),𝑥(𝑡)∈𝑅^𝑛; (9)(iii)𝑑(𝑡) satisfies the norm-bounded condition ‖𝑑(𝑡)‖≤𝛿, where  𝛿>0 is known constant;(iv)Δ𝐴(𝑡) is unknown time-varying uncertainties satisfying Δ𝐴(𝑡)=𝐷₁⋅𝐹(𝑡)⋅𝐷₂, with 𝐷₁∈𝑅^𝑛×𝑛, 𝐷₂∈𝑅^𝑛×𝑛, and 𝐹(𝑡)∈𝑅^𝑛×𝑛 is the unknown time-varying and satisfies 𝐹^𝑇(𝑡)⋅𝐹(𝑡)≤𝐼_𝑛.

3. Main Results

Consider the following traditional Luenberger observer: 𝑥̇(𝑡)=(𝐴−𝐿)⋅𝑥(𝑡)+𝐵𝑢(𝑡)+𝐿𝑦(𝑡)+𝑓(𝑡,𝑥), (10) where 𝑥(𝑡) is the estimation of 𝑥(𝑡) and  𝐿  is the observer gain to be designed. If one employs the observer (10) to plant (8) and defines 𝑒(𝑡)=𝑥(𝑡)−𝑥(𝑡), then one can obtain the following error dynamics: 𝑒̇(𝑡)=[𝐴−𝐿+Δ𝐴(𝑡)]⋅𝑒(𝑡)−Δ𝐴(𝑡)⋅𝑥(𝑡)+𝐿𝐶_𝑑𝑑(𝑡)+𝑓(𝑡,𝑒(𝑡)), (11) where 𝑓(𝑡,𝑒(𝑡))=𝑓(𝑡,𝑥(𝑡))−𝑓(𝑡,𝑥(𝑡)). It is observed that the disturbance 𝐶_𝑑𝑑(𝑡) in the error system (11) is multiplied by the gain matrix 𝐿. Hence, if 𝐿 is a high gain, the disturbance effect will be amplified unavoidably. As a result, ideal estimation performance of system state 𝑥(𝑡) cannot be obtained by using the conventional observer (10).

To begin the formulation of our approach, the following augmented variables and matrices are defined: 𝑥(𝑡)=𝑥(𝑡)𝐶_𝑑𝑑(𝑡),𝐴=𝐴00−𝐼_𝑃,𝐵0𝐵=_𝑝×𝑚,𝐶=𝐶𝐼_𝑃×𝑃,𝐼𝐸=^𝑛000_𝑃×𝑃,𝐹(𝑡)0𝐹=^𝑛×𝑃0_𝑃×𝑛0_𝑃×𝑃,𝐷₁=𝐷₁000_𝑃×𝑃,𝐷₂=𝐷₂000_𝑃×𝑃,Δ𝐴(𝑡)=Δ𝐴(𝑡)000_𝑃×𝑃,0𝐼𝑁=_𝑃,𝑓()=𝑓()0𝑡,𝑥𝑡,𝑥_𝑃×1, (12) and the following augmented plant is constructed: 𝐸𝑥̇(𝑡)=𝐴+Δ𝐴(𝑡)𝑥(𝑡)+𝐵𝑢(𝑡)+𝑓(𝑡,𝑒)+(𝑡),𝑦(𝑡)=𝑁𝑑𝐶𝑥(𝑡). (13)

System (13) is a singular system, where the state vector 𝑥(𝑡) and disturbance vector 𝑑(𝑡) are both the components of the augmented state vector.

It is noted that 𝐸 and 𝐶 defined previously have the following property: rank𝐸𝐶𝐼=rank^𝑛000_𝑃×𝑃𝐶𝐼_𝑃=𝑛+𝑃. (14)

Therefore, an appropriate matrix 𝐿_𝐷∈𝑅^{(𝑛+𝑝)×𝑝}  can always be selected such that the matrix 𝑆≜(𝐸+𝐿_𝐷𝐶) is nonsingular. Motivated by this observation, the following proportional and derivative observer is proposed for system (13): 𝑆𝑧̇(𝑡)=𝐴−𝐿_𝑃𝐶𝑧(𝑡)+𝐴𝑆⁻¹𝐿_𝐷𝑦(𝑡)+𝐵𝑢(𝑡)+𝑓(𝑡,𝑥),𝑥(𝑡)=𝑧(𝑡)+𝑆⁻¹𝐿_𝐷𝑦(𝑡), (15) where 𝑧(𝑡)=[𝑧^𝑇_𝑥(𝑡)𝑧^𝑇_𝑑(𝑡)]^𝑇 is the middle variable, 𝑥(𝑡)≜[𝑥^𝑇(𝑡)𝑑^𝑇(𝑡)]^𝑇 is the estimation of 𝑥(𝑡), and 𝐿_𝐷∈𝑅^{(𝑛+𝑝)×𝑝} and 𝐿_𝑝∈𝑅^{(𝑛+𝑝)×𝑝} are the observer gains to be designed. 𝑆=𝐸+𝐿_𝐷⋅𝐶 is designed to be invertible by selecting appropriate 𝐿_𝐷. It is now in the position to derive the error system. In fact, from (14)-(15), it is derived that 𝑆𝑥̇=𝐴−𝐿_𝑃𝐶𝑥(𝑡)+𝐿_𝑃𝑦(𝑡)+𝐿_𝐷𝑦̇(𝑡)+𝐵𝑢(𝑡)+𝑓(𝑡,𝑥). (16)

On the other hand, adding 𝐿_𝐷𝑦(̇𝑡) to both sides of the plant (13), the following equation is derived: 𝑆𝑥̇(𝑡)=𝐴−𝐿_𝑃𝐶+Δ𝐴(𝑡)𝑥(𝑡)+𝐿_𝑃𝑦̇(𝑡)+𝐿_𝐷𝑦̇‌(𝑡)+𝐵𝑢(𝑡)+𝑓(𝑡,𝑥)+𝑁𝑑(𝑡). (17)

Define the following error variable: 𝑒(𝑡)=𝑥(𝑡)−𝑥(𝑡),𝑓(𝑡,𝑒)=𝑓(𝑡,𝑥)−𝑓(𝑡,𝑥). (18)

Subtracting (17) from (16), the error system is derived as 𝑆𝑒̇(𝑡)=𝐴−𝐿_𝑃𝐶𝑒(𝑡)−Δ𝐴(𝑡)𝑥(𝑡)−𝑁𝑑(𝑡)+𝑓(𝑡,𝑒). (19)

On the other hand, we propose the following observer-based control strategy for plant (8): 𝑢(𝑡)=𝐾𝑥(𝑡)𝐾∈𝑅^{𝑚×(𝑛+𝑝)}, (20) where 𝐾=[𝐾,0_𝑚×𝑝]. Applying (20) to plant (8) yields 𝑥̇(𝑡)=(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+𝐵𝐾𝑥(𝑡)(𝑡)=((𝑡𝑥(𝑡)+𝑓𝐴+Δ𝐴))+𝐵𝐾(𝑒(𝑡)+𝑥(𝑡(𝑡)=((𝑡𝑥(𝑡)))+𝑓𝐴+Δ𝐴))+𝐵𝐾𝑒(𝑡)+𝐵𝐾𝑥(𝑡)+𝑓(𝑡). (21) Equation (21) together with (19) yields the following overall closed-loop system: 𝑥̇(𝑡)=(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+𝐵𝐾𝑥(𝑡)+𝑓(𝑡)+𝐵𝐾𝑒(𝑡),𝑒̇(𝑡)=𝑆⁻¹𝐴−𝐿_𝑃𝐶𝑒(𝑡)−Δ𝐴(𝑡)𝑥(𝑡)+𝑁𝑑(𝑡)+𝑓(𝑡,𝑒). (22)

In system (22), recall that 𝑆=(𝐸+𝐿_𝐷𝐶). We decompose 𝐿_𝐷 as 𝐿_𝐷=^𝐿𝐷1_𝐿𝐷2 with 𝐿_𝐷1∈𝑅^𝑛×𝑝, 𝐿_𝐷2∈𝑅^𝑃×𝑃. In this paper, let 𝐿_𝐷1=0, 𝐿_𝐷2=diag{𝛽₁,…,𝛽_𝑃}, where 𝛽₁⋯𝛽_𝑃 are reasonable large numbers. In this setting, the derivative gain 𝐿_𝐷=⁰_𝐿𝐷2 can thus guarantee that 𝑆=𝐸+𝐿_𝐷𝐶 is nonsingular and reduce the effect of disturbance 𝑓(𝑡) at the same time. In addition, design the state feedback gain 𝐾∈𝑅^𝑚×𝑛, such that 𝐴+𝐵𝐾 is Hurwitz. The following theorem provides a sufficient condition for the stability analysis of the overall closed-loop system (22).

Theorem 1. Consider the overall closed-loop system (22), where the observer gain 𝐿_𝐷 and feedback gain 𝐾 have been designed as previously discussed. If there exist positive and definite matrices 𝑃∈𝑅^𝑛×𝑛, 𝑅∈𝑅^{(𝑛+𝑝)×(𝑛+𝑝)}, and matrices 𝐿_𝑝∈𝑅^{(𝑛+𝑝)×𝑝}, such that the following matrix condition: Φ₁₁𝑃𝐵𝐾0𝑃𝐷₁𝜃𝑃0∗Φ₂₂𝑅𝑆⁻¹𝐷₁00𝑅𝑆⁻¹∗∗−𝜀₁𝐼_𝑛000∗∗∗−𝜀₂𝐼_𝑛+𝑝00∗∗∗∗−𝐼_𝑛0∗∗∗∗∗−𝐼_𝑛+𝑝<0, (23) with Φ₁₁=(𝐴+𝐵𝐾)^T𝑃+𝑃(𝐴+𝐵𝐾)+𝐼_𝑛+𝜃𝐼_𝑛+𝜀₂𝐷^T₂𝐷₂, Φ₂₂=𝑅𝑆⁻¹(𝐴−𝐿_𝑃𝐶)+(𝐴−𝐿_𝑃𝐶)𝑆^−T𝑅.

Proof. We define the following Lyapunov function: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡) with 𝑉₁(𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡),  𝑉₂(𝑡)=𝑒^𝑇(𝑡)𝑅𝑒(𝑡) for system (22). Along the trajectory of system (22), it follows that 𝑉̇₁(𝑡)=2𝑥^𝑇(𝑡)𝑃(𝑡𝑥(𝑡)(𝑡)(𝑡𝑉̇(𝐴+Δ𝐴))+𝐵𝐾𝑥+𝑓)₂(𝑡)=2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝐴−𝐿_𝑃𝐶−Δ𝐴(𝑡)𝑥(𝑡)+𝑁𝑑(𝑡)+𝑓(𝑡,𝑒)=2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝐴−𝐿_𝑃𝐶𝑒(𝑡)−Δ𝐴(𝑡)𝑥(𝑡)−2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝑁𝑑(𝑡)+2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝑓(𝑡,𝑒). (24)
Since 𝑆=^{𝐼𝑛+𝐿𝐷1𝐶𝐿𝐷1}_{𝐿𝐷2𝐶𝐿𝐷2}, one can calculate that 𝑆⁻¹=^𝐼𝑛0_{−𝐶(𝐿𝐷2)−1}.
Hence, it is calculated that 𝑆⁻¹𝑁𝑑(𝑡)=^𝐼𝑛0_{−𝐶(𝐿𝐷2)−1}⁰_𝐼𝑃𝑑(𝑡)=⁰_(𝐿𝐷2)⁻¹𝑑(𝑡). Since (𝐿_𝐷2)⁻¹=diag{𝛽₁⁻¹,…,𝛽_𝑝⁻¹}, the term 𝑆⁻¹𝑁𝑑(𝑡) is prevailed successfully and can be ignored in 𝑉̇₂(𝑡). Therefore, 𝑉̇₂(𝑡) becomes 𝑉̇₂(𝑡)=2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝐴−𝐿_𝑃𝐶𝑒^𝑇(𝑡)−Δ𝐴(𝑡)𝑥(𝑡)+2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝑓(𝑡,𝑒). (25)
Notice that 2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝑓(𝑡,𝑒)≤2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝑆^−𝑇𝑅𝑒(𝑡)+𝑓^𝑇(𝑡,𝑒)𝑓(𝑡,𝑒)≤2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝑆^−𝑇𝑅𝑒^𝑇(𝑡)+𝑓^𝑇(𝑡,𝑒)𝑓(𝑡,𝑒)≤2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝑆^−𝑇𝑅𝑒(𝑡)+𝜃²⋅𝑥^𝑇(𝑡)𝑥(𝑡). (26)
It is thus derived from (25) and (26) that 𝑉̇₂(𝑡)≤2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝐴−𝐿_𝑃𝐶𝑒(𝑡)−𝑆⁻¹Δ𝐴(𝑡)𝑥(𝑡)+2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝑆^−𝑇𝑅𝑒(𝑡)+𝜃²⋅𝑥^𝑇(𝑡)𝑥(𝑡). (27)
Notice that Δ𝐴(𝑡)𝑥(𝑡)=(𝑡)0𝑥(𝑡)𝐶Δ𝐴00_𝑠𝑑(𝑡)=𝐷₁𝐹(𝑡)𝐷₂0_𝑛×𝑝0_𝑝×𝑛0_𝑝×𝑝𝑥(𝑡)𝐶_𝑠𝑑(𝑡), (28) and also recall that 𝐷₁=^{𝐷10𝑛×𝑝}_{0𝑝×𝑛0𝑝×𝑝}, 𝐷₂=^{𝐷20𝑛×𝑝}_{0𝑝×𝑛0𝑝×𝑝}, then it is shown that Δ𝐴(𝑡)𝑥(𝑡)=𝐷₁0_𝑛×𝑝0_𝑝×𝑛0_𝑝×𝑝𝐹(𝑡)0_𝑛×𝑝0_𝑝×𝑛0_𝑝×𝑝×𝐷₂0_𝑛×𝑝0_𝑝×𝑛0_𝑝×𝑝=𝐷₁𝐹(𝑡)𝐷₂. (29)
And there exists 𝜀₁>0, such that −2𝑒^𝑇(𝑡)𝑅𝑆⁻¹Δ𝐴(𝑡)𝑥(𝑡)=−2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝐷₁𝐹(𝑡)𝐷₂𝑥(𝑡)≤𝜀₁𝑥^𝑇(𝑡)𝐷^𝑇₂𝐷₂𝑥(𝑡)+𝜀₁⁻¹𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝐷₁𝐷^𝑇₁𝑆^−𝑇𝑅𝑒(𝑡)≤𝜀₁𝑥^𝑇(𝑡)𝐷^𝑇₂𝐷₂𝑥(𝑡)+𝜀₁⁻¹𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝐷₁𝐷^𝑇₁𝑆^−𝑇𝑅𝑒(𝑡). (30)
As a result, it is derived from (27) and (30) that 𝑉̇₂(𝑡)≤2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝐴−𝐿_𝑃𝐶+𝜀₁𝑥^𝑇(𝑡)𝐷^𝑇₂𝐷₂𝑥(𝑡)+𝜀₁⁻¹𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝐷₁𝐷^𝑇₁𝑆^−𝑇𝑅𝑒(𝑡)+2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝑆^−𝑇𝑅𝑒(𝑡)+𝜃²⋅𝑥^𝑇(𝑡)𝑥(𝑡). (31)
On the other hand, for 𝑉̇₁(𝑡), we have 2𝑥^𝑇(𝑡)𝑃Δ𝐴(𝑡)𝑥(𝑡)=2𝑥^𝑇(𝑡)𝑃𝐷₁𝐹(𝑡)𝐷₂𝑥(𝑡)≤𝜀₂𝑥^𝑇(𝑡)𝐷^𝑇₂𝐷₂𝑥(𝑡)+𝜀₂⁻¹𝑥^𝑇(𝑡)𝑃𝐷₁𝐷^𝑇₁𝑃𝑥(𝑡), (32) and 2𝑥^𝑇(𝑡)𝑃𝑓(𝑡)≤2‖𝑥(𝑡)‖²‖𝑃‖⋅𝜃≤𝜃²⋅𝑥^𝑇(𝑡)𝑃𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑥(𝑡).
Then 𝑉̇₁(𝑡)≤2𝑥^𝑇(𝑡)𝑃(𝐴+𝐵𝐾)𝑥(𝑡)+𝜀₂𝑥^𝑇(𝑡)𝐷^𝑇₂𝐷₂𝑥(𝑡)+𝜀₂⁻¹𝑥(𝑡)𝑃𝐷₁𝐷^𝑇₁𝑃𝑥₁(𝑡)+𝜃²⋅𝑥^𝑇(𝑡)𝑃𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐵𝐾𝑥(𝑡). (33)
As a result, it is derived that 𝑉̇(𝑡)=𝑉̇₁(𝑡)+𝑉̇₂(𝑡)≤2𝑥^𝑇(𝑡)𝑃(𝐴+𝐵𝐾)𝑥(𝑡)+𝜀₂𝑥^𝑇(𝑡)𝐷^𝑇₂𝐷₂𝑥(𝑡)+𝜀₂⁻¹𝑥^𝑇(𝑡)𝑃𝐷₁𝐷^𝑇₁𝑃𝑥₁(𝑡)+𝜃²⋅𝑥^𝑇(𝑡)𝑃𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐵𝐾𝑥(𝑡)+2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝐴−𝐿_𝑃𝐶+𝜀₁𝑥^𝑇(𝑡)×𝐷^𝑇₂𝐷₂𝑥(𝑡)+𝜀₁⁻¹𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝐷₁𝐷^𝑇₁𝑆^−𝑇𝑅𝑒(𝑡)+2𝑒^𝑇(𝑡)𝑅𝑆⁻¹𝑆^−𝑇𝑅𝑒(𝑡)+𝜃²⋅𝑥^𝑇(𝑡)𝑥(𝑡)≤𝑥^𝑇(𝑡)𝑒^𝑇(𝑡)ΙΙ₁₁ΙΙ₁₂∗ΙΙ₂₂𝑥(𝑡)𝑒(𝑡), (34) with ΙΙ₁₁=(𝐴+𝐵𝐾)^𝑇𝑃+𝑃(𝐴+𝐵𝐾)+𝜀₂𝐷^𝑇₂𝐷₂+𝜀₂⁻¹𝑃𝐷₁𝐷^𝑇₁𝑃+𝜃²𝑃𝑃+𝐼_𝑛+𝜃²𝐼_𝑛,ΙΙ₂₂=𝑅𝑆⁻¹𝐴−𝐿_𝑃𝐶+𝐴−𝐿_𝑃𝐶^𝑇𝑆^−𝑇𝑅+𝜀₁⁻¹𝑅𝑆⁻¹𝐷₁𝐷^𝑇₁𝑆^−𝑇𝑅+𝑅𝑆⁻¹𝑆^−𝑇𝑅. (35)
By Schur complement lemma, ΙΙ<0 is equivalent to the matrix condition (23) in Theorem 1. This completes the proof.

It is noted that the matrix condition (23) in Theorem 1 is a nonlinear one since the observer gain 𝐿_𝑃 has not been designed. To this end, we provide the following theorem to design 𝐿_𝑃 and transform the condition of Theorem 1 into a linear version.

Theorem 2. For the closed-loop system (22), let the gain matrices 𝐿_𝐷 and 𝐾 be designed as previously discussed, if there exist positive and definite matrix 𝑃∈𝑅^𝑛×𝑛, 𝑅∈𝑅^{(𝑛+𝑝)×(𝑛+𝑝)}, and matrices 𝑌∈𝑅^{(𝑛+𝑝)×𝑝}such that the following LMI holds: Ψ₁₁𝑃𝐵𝐾0𝑃𝐷₁𝜃𝑃0∗Ψ₂₂𝑅𝑆⁻¹𝐷₁00𝑅𝑆⁻¹∗∗−𝜀₁𝐼_𝑛000∗∗∗−𝜀₂𝐼_𝑛+𝑝00∗∗∗∗−𝐼_𝑛0∗∗∗∗∗−𝐼_𝑛+𝑝<0, (36) where Ψ₁₁=(𝐴+𝐵𝐾)^𝑇𝑃+𝑃(𝐴+𝐵𝐾)+𝐼_𝑛+𝜃𝐼_𝑛+𝜀₂𝐷^𝑇₂𝐷₂,Ψ₂₂=𝑅𝑆𝐴−𝑌𝐶+𝐴^𝑇𝑆^𝑇𝑅−𝐶^𝑇𝑌^𝑇. (37)
Furthermore, the observer gain 𝐿_𝑃 is calculated as 𝐿_𝑃=𝑆𝑅⁻¹𝑌.

Proof. In the matrix condition of Theorem 1, let 𝑌=𝑅𝑆⁻¹𝐿_𝑝, and substituting it into the matrix condition of Theorem 1, one can easily obtain the LMI of Theorem 2. This completes the proof.

4. Simulation Results

Consider the plant and augmented form (8), the system data are chosen as shown in Table 1 as follows: 𝐴=−0.1734.59×10⁴0000100−4.43×10⁴−133.34.8×10⁻⁴00−7.14×10¹⁰−1.28×10³,000𝐵=3.3×10⁸1000,𝐶=^𝑇,𝐷₁=,𝐷0.10.20.050.100.20.10.100.10.130.10.130.120.140.15₂=,𝐹(𝑡)=(𝑡)(𝑡)(𝑡)0(𝑡),0.20.10.070.0200.10.20.30.10.10.130.10.140.120.140.150.1sin00000.15sin00000.14sin0000.1sin (38) and the Lipschitz condition is selected as 𝑓(𝑡,𝑥)=0.131sin−0.2𝑥₁(𝑡)+𝑥₂(𝑡)+𝑥₃(𝑡)sin𝑥₁(𝑡)−2𝑥₂(𝑡)+𝑥₃(𝑡)00. (39)

Table 1 

Miscellaneous coefficient value.

It is easy to check that (𝐴,𝐶) is observable pair, and the Lipschitz condition is given as 𝜃=0.420. It is assumed that 𝑑(𝑡) has the following form.

Case 1. Consider 𝑑(𝑡)=()()()0.1cos8𝑡,𝑡<2,0.1sin10𝑡+0.2cos10𝑡,2≤𝑡<5. (40)

Case 2. Consider 𝑑(𝑡)=()0.5,𝑡<3,0.3sin4𝑡+0.1,3≤𝑡<5. (41)
We select 𝛿=0.4, which is the norm upper bound of  𝑑(𝑡)  as defined in assumption (iii). In order to design the estimator, we choose the derivative gain 𝐿_𝐷 as 𝐿_𝐷=0001,suchthatthesystemmatrix𝑆=1000001000001000001010001isnonsingular. (42)
Design the state feedback gain as 𝐾=[−1.413,−0.286,−2.854,−0.423] such that  𝐴+𝐵𝐾 is Hurwitz. Solving the LMI condition of Theorem 2, one can obtain the following solution: ,.𝑃=369.0123−254.7902−107.2517−184.3887254.7902352.6723−104.852375.6133−107.2517−104.8523502.8899134.2559−184.388775.6133134.2559212.2635𝑅=3.6029−3.77815.8391−1.36491.2867−3.77819.2899−13.85480.7897−3.77745.8391−13.854829.0279−0.26896.1213−1.36490.7897−0.26891.08890.32381.2867−3.77746.12130.32382.2892 (43) According to Theorem 2, the proportional gain is selected as 𝐿_𝑝=10²×0.00000.00000.00006.36960.0000. (44)

The initial condition is chosen as 𝑒(1)=[00000],  𝑥(1)=[1−1−1.51]. The simulation results are provided in Figures 1–3. The state trajectory of 𝑥(𝑡) of the closed-loop system is shown in Figure 2. The trajectories of disturbances 𝑑(𝑡) and their estimation in Case 1 are shown in Figure 3. The trajectories of disturbances 𝑑(𝑡) and their estimation in Case 2 are shown in Figure 4. It can be seen that the asymptotic stability of the closed-loop system is guaranteed, and the tracking performance of system states and disturbances has achieved an ideal performance.

Figure 2 

Trajectory of the closed-loop system.

Figure 3 

𝑑(𝑡) and the estimation: Case 1.

Figure 4 

𝑑(𝑡) and the estimation: Case 2.

5. Conclusion

In this paper, the problem of robust control has been studied for the velocity control system based on CPR with parameters uncertainty and load disturbance. The hydraulic velocity control system model has been built firstly for the control design objective. By constructing a Luenberger observer and using Lyapunov stability theory, the observer-based controller has been designed such that the closed-loop system is asymptotically stable and satisfies the disturbance attenuation level. Finally, simulation results have shown the effectiveness of the proposed method. The future work will focus on constructing one test rig and trying to prove the effective performance for the proposed control method. And the long research object of the project is to apply this method to hydraulic velocity control system design for getting a good performance of construction machinery.

Acknowledgments

The authors acknowledge the contribution of the National Natural Science Foundation of China (50875054) and Foreign Visiting Project of Harbin Institute of Technology for Ph.D. Students.