**Robust Control of Electro-Hydraulic Actuator Systems Using the Adaptive Back-Stepping Control Scheme**

Jong Shik Kim, Han Me Kim and Sung Hwan Park *School of Mechanical Engineering, Pusan National University Republic of Korea* 

## **1. Introduction**

18 Will-be-set-by-IN-TECH

188 Challenges and Paradigms in Applied Robust Control

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London, U.K.: Taylor & Francis.

Vol. (33): 995-1003.

168-173.

Conventional hydraulic actuator (CHA) systems have been widely used as power units because they can generate very large power compared to their size. In general, a CHA system consists of an electric motor, a pump, a reservoir, various valves, hoses, which are used to transfer the working fluid and an actuator. CHA systems, however, have some problems such as environmental pollution caused by the leakage of the working fluid, maintenance load, heavy weight and limited installation space. These shortcomings can be overcome by compactly integrating the components of CHA systems and by applying a suitable control scheme for the electric motor. To overcome these shortcomings of CHA systems, electro-hydraulic actuator (EHA) systems have been developed, having merits such as smaller size, higher energy efficiency and faster response than existing CHA systems (Kokotovic, 1999). However, for the robust position control of EHA systems, system uncertainties such as the friction between the piston and cylinder and the pump leakage coefficient have to be considered.

To solve these system uncertainty problems of EHA systems and to achieve the robustness of EHA systems with system disturbance and bounded parameter uncertainties, Wang et. al. presented a sliding mode control and a variable structure filter based on the variable structure system (Wang, 2005). Perron et. al proposed a sliding mode control scheme for the robust position control of EHA systems showing volumetric capacity perturbation of the pump (Perron, 2005). However, these control methods have some chattering problem due to the variable structure control scheme. The chattering vibrates the system and may reduce the life cycle of the system. Jun et. al. presented a fuzzy logic self-tuning PID controller for regulating the BLDC motor of EHA systems which has nonlinear characteristics such as the saturation of the motor power and dead-zone due to the static friction (Jun, 2004). Chinniah et. al. used a robust extended Kalman filter, which can estimate the viscous friction and effective bulk modulus, to detect faults in EHA systems (Chinniah, 2006). Kaddissi et. al. applied a robust indirect adaptive back-stepping control (ABSC) scheme to EHA systems having perturbations of the viscous friction coefficient and the effective bulk modulus due to temperature variations (Kaddissi, 2006). However, in spite of the variation of the effective bulk modulus due to the temperature and pressure variations, Chinniah et. al. considered

Robust Control of Electro-Hydraulic Actuator Systems

where *Q* is the flow rate in the actuator,

are very short and hard. Then, (2) can be expressed as

pump, *Cp* is the volumetric capacity of the pump,

chamber, respectively.

Using the Adaptive Back-Stepping Control Scheme 191

where *V* and *V*0 are the chamber volume and the initial chamber volume, respectively, *A* and *x* are the pressure area of a double rod hydraulic cylinder and displacement of the piston, respectively, and subscripts 'A' and 'B' denote the chamber notations of the actuator. Considering the fluid compressibility and continuity principle for the actuator, the flow rate

0

*A A A A e B B B B e*

*V Ax Q Ax P LP*

(2)

(3)

(4)

*<sup>p</sup>* is the rotational velocity of the electric

( ) *P P A Mx F F <sup>A</sup> <sup>B</sup> <sup>f</sup> ex* (5)

*p* .

*<sup>e</sup>* is the effective bulk modulus of the working

0

*V Ax Q Ax P LP* 

fluid, and *L* and *P* are the actuator external leakage coefficient and the pressure in the

It is assumed that there is no fluid leakage of conduits because the conduits of EHA systems

0

*V Ax Q Ax <sup>P</sup>*

 

*V Ax Q Ax P*

The electric motor, directly connected to the hydraulic pump, changes the flow direction and adjusts the flow rate through the ports. In addition, the pressure generated by the continuous supply of flow in the actuator can produce a minute fluid leakage of the pump.

*a pp f L*

*Q C LP*

where *Q* is the flow rate of the pump, whose subscripts a and b denote the ports of the

motor, *Lf* is the leakage factor of the pump and the load pressure *PPP L AB* . From (4), the inflow and outflow of the pump are expressed as functions of the rotational velocity

where *M* and *x* are the mass and displacement of the piston, respectively, *Ff* is the friction force between the cylinder and piston and *Fex* is the external disturbance force.

 () *P P A Mx F F <sup>A</sup> <sup>B</sup> <sup>f</sup> ex* (6) In addition, it is assumed that the conduits connected between the actuator ports and the pump ports are very short. Then, the flow rates in (3) and (4) can be represented as *Q Q A a*

*b a*

*Q Q* 

*A A A e B B B e*

0

equations of both ports of the actuator can be represented as (Merritt, 1967)

Hence, the equations for the fluid leakage of the pump are expressed as

 

In addition, the actuator dynamic equation of EHA systems is expressed as

In order to substitute (3) into (5), the derivative of (5) is expressed as

 

only the case of constant effective bulk modulus and Kaddissi et. al. used EHA systems that are not controlled by an electric motor but by a servo valve.

In this chapter, an ABSC scheme was proposed for EHA systems to obtain the desired tracking performance and the robustness to system uncertainties. Firstly, to realize a stable back-stepping control (BSC) system with a closed loop structure and to select new state variables, EHA system dynamics are represented with state equations and error equations. Defining the Lyapunov control functions, we can design a BSC system, which can guarantee exponential stability for the nominal system without system uncertainties. However, the BSC system cannot achieve robustness to system uncertainties. To overcome the drawback of the BSC system, an ABSC scheme for EHA position control systems with classical discrete disturbance observer was proposed. To evaluate the tracking performance and robustness of the proposed EHA position control system, both BSC and ABSC schemes were evaluated by computer simulation and experiment.

## **2. System modeling of EHA system**

Figure 1 shows the simplified schematic diagram of an EHA system that consists of an electric servo motor, bi-directional gear pump and actuator. The servo motor rotates the gear pump, which, in turn, generates the flow rate. The pressure generated by the flow rate changes the position of the piston rod. The movement direction of the piston is related to the rotational direction of the servo motor. The chamber volumes of the actuator depend on the cross sectional area and the displacement of the piston as follows

$$\begin{cases} V\_A(t) = V\_{0A} + A\mathbf{x}(t) \\ V\_B(t) = V\_{0B} - A\mathbf{x}(t) \end{cases} \tag{1}$$

Fig. 1. Simplified schematic diagram of an EHA system

190 Challenges and Paradigms in Applied Robust Control

only the case of constant effective bulk modulus and Kaddissi et. al. used EHA systems that

In this chapter, an ABSC scheme was proposed for EHA systems to obtain the desired tracking performance and the robustness to system uncertainties. Firstly, to realize a stable back-stepping control (BSC) system with a closed loop structure and to select new state variables, EHA system dynamics are represented with state equations and error equations. Defining the Lyapunov control functions, we can design a BSC system, which can guarantee exponential stability for the nominal system without system uncertainties. However, the BSC system cannot achieve robustness to system uncertainties. To overcome the drawback of the BSC system, an ABSC scheme for EHA position control systems with classical discrete disturbance observer was proposed. To evaluate the tracking performance and robustness of the proposed EHA position control system, both BSC and ABSC schemes were evaluated by

Figure 1 shows the simplified schematic diagram of an EHA system that consists of an electric servo motor, bi-directional gear pump and actuator. The servo motor rotates the gear pump, which, in turn, generates the flow rate. The pressure generated by the flow rate changes the position of the piston rod. The movement direction of the piston is related to the rotational direction of the servo motor. The chamber volumes of the actuator depend on the

> 0 0 () () () ()

(1)

*V t V Ax t V t V Ax t* 

*A A B B*

are not controlled by an electric motor but by a servo valve.

cross sectional area and the displacement of the piston as follows

Fig. 1. Simplified schematic diagram of an EHA system

computer simulation and experiment.

**2. System modeling of EHA system** 

where *V* and *V*0 are the chamber volume and the initial chamber volume, respectively, *A* and *x* are the pressure area of a double rod hydraulic cylinder and displacement of the piston, respectively, and subscripts 'A' and 'B' denote the chamber notations of the actuator. Considering the fluid compressibility and continuity principle for the actuator, the flow rate equations of both ports of the actuator can be represented as (Merritt, 1967)

$$\begin{cases} Q\_A = A\dot{\mathbf{x}} + \frac{V\_{0A} + A\mathbf{x}}{\mathcal{J}\_e} \dot{P}\_A + LP\_A\\ Q\_B = A\dot{\mathbf{x}} - \frac{V\_{0B} - A\mathbf{x}}{\mathcal{J}\_e} \dot{P}\_B - LP\_B \end{cases} \tag{2}$$

where *Q* is the flow rate in the actuator, *<sup>e</sup>* is the effective bulk modulus of the working fluid, and *L* and *P* are the actuator external leakage coefficient and the pressure in the chamber, respectively.

It is assumed that there is no fluid leakage of conduits because the conduits of EHA systems are very short and hard. Then, (2) can be expressed as

$$\begin{cases} Q\_A = A\dot{\mathbf{x}} + \frac{V\_{0A} + A\mathbf{x}}{\mathcal{J}\_e} \dot{P}\_A\\ Q\_B = A\dot{\mathbf{x}} - \frac{V\_{0B} - A\mathbf{x}}{\mathcal{J}\_e} \dot{P}\_B \end{cases} \tag{3}$$

The electric motor, directly connected to the hydraulic pump, changes the flow direction and adjusts the flow rate through the ports. In addition, the pressure generated by the continuous supply of flow in the actuator can produce a minute fluid leakage of the pump. Hence, the equations for the fluid leakage of the pump are expressed as

$$\begin{cases} \mathbb{Q}\_a = \mathbb{C}\_p \rho o\_p - \mathbb{L}\_f P\_L\\ \mathbb{Q}\_b = -\mathbb{Q}\_a \end{cases} \tag{4}$$

where *Q* is the flow rate of the pump, whose subscripts a and b denote the ports of the pump, *Cp* is the volumetric capacity of the pump, *<sup>p</sup>* is the rotational velocity of the electric motor, *Lf* is the leakage factor of the pump and the load pressure *PPP L AB* . From (4), the inflow and outflow of the pump are expressed as functions of the rotational velocity*p* . In addition, the actuator dynamic equation of EHA systems is expressed as

$$(P\_A - P\_B)A = M\ddot{\mathbf{x}} + F\_f + F\_{ex} \tag{5}$$

where *M* and *x* are the mass and displacement of the piston, respectively, *Ff* is the friction force between the cylinder and piston and *Fex* is the external disturbance force. In order to substitute (3) into (5), the derivative of (5) is expressed as

$$(\dot{P}\_A - \dot{P}\_B)A = M\ddot{x} + \dot{F}\_f + \dot{F}\_{ex} \tag{6}$$

In addition, it is assumed that the conduits connected between the actuator ports and the pump ports are very short. Then, the flow rates in (3) and (4) can be represented as *Q Q A a*

Robust Control of Electro-Hydraulic Actuator Systems

Now, let (11) represent state equations as follows

where *<sup>d</sup> x* is the desired position input, and

From (13), the state equation for 1*z* can be described as

systems.

where

**Step 1.**

1 1 

Then,

From (18), if 1 1 11 ( ) *<sup>d</sup>* 

is a design parameter.

form (Slotine, 1999) as follows

Using the Adaptive Back-Stepping Control Scheme 193

than in the inner loop to improve the performance and robustness of EHA position control

In this chapter, the BSC and ABSC schemes based on EHA system dynamics are considered as the position controller. Firstly, to design a BSC system, (7) is transformed to a general

> 1 11 <sup>2</sup> 1 1 , *<sup>e</sup> f ex e f <sup>L</sup> A B A B*

> > ( ) , *e AB <sup>p</sup> A B AV V C*

*MV V*

 

*f A x F F LA P M VV V V*

*u p*

2 2 11 *zx z* 

3 3 212 *z x zz* 

1 2 11 ( ) *<sup>d</sup> zz z x* 

( ) *z* is the virtual control which should be selected to guarantee the stability of the control

11 1 <sup>1</sup> ( ) <sup>2</sup>

1 1 11 1 1 1 1 2 () [() ] *V z zz z z x zz* 

2

*z kz x* , (16) can be exponentially stable when *t* . And 1 *k* ( 0)

1 and

*x x x x x f bu*

 

 

And, in order to design the BSC system, new state variables are defined as follows

variables, which can be obtained through the following BSC design procedure.

system through the Lyapunov control function(LCF) which is defined as

*b*

 

*x f bu* (11)

(12)

1 1 *<sup>d</sup> zxx* (13)

*Vz z* (17)

*<sup>d</sup>* (18)

( ) (14)

(,) (15)

2 are the functions for new state

(16)

and *Q Q B b* . Substituting (1) through (4) into (6), therefore, the dynamic equation of EHA systems can be represented as

$$\ddot{\mathbf{x}} = -\frac{1}{M} \left| \mathcal{J}\_{\varepsilon} A^2 \left( \frac{1}{V\_A} + \frac{1}{V\_B} \right) \dot{\mathbf{x}} + \dot{F}\_f + \dot{F}\_{\varepsilon\varepsilon} \right| + \frac{\mathcal{J}\_{\varepsilon} A}{M} \left( \frac{1}{V\_A} + \frac{1}{V\_B} \right) \left( L\_f P\_L - C\_p \alpha\_p \right) \tag{7}$$

To represent the characteristics of the friction *Ff* between the piston and cylinder, the LuGre friction model is considered. The LuGre friction model is based on bristles analysis, which is represented with the average deflection force of bristles stiffness. The deflection displacement equation of bristles *z* , which is actually unmeasurable by experiment, is expressed as (Choi, 2004)(Lee, 2004)

$$\frac{dz}{dt} = \dot{\mathbf{x}} - \frac{\sigma\_0 \left| \dot{\mathbf{x}} \right|}{g(\dot{\mathbf{x}})} \mathbf{z} \tag{8}$$

where 0 is the bristles stiffness coefficient, *z* is the unmeasurable internal state and the nonlinear function *g x*( ) depends on the material property, grade of lubrication and temperature; that is

$$\log(\dot{\mathbf{x}}) = F\_c + F\_s e^{-(\dot{\mathbf{x}}/v\_{sv})^2} \tag{9}$$

where *Fc* , *Fs* and *sv v* represent the Coulomb friction force, static friction force, and Stribeck velocity between the cylinder and piston, respectively.

If the relative velocity of the contact materials increases gradually, the friction force decreases instantaneously and then it increases gradually again; this effect is called the Stribeck effect and the relative velocity is the Stribeck velocity. This phenomenon depends on the material property, grade of lubrication and temperature. The friction force *Ff* can be represented with the average deflection *z* and the velocity of the piston *x* as follows

$$F\_f = \sigma\_0 z + \sigma\_1 \dot{z} + \mu \dot{x} \tag{10}$$

where 1 and represent the bristles damping and viscous friction coefficients, respectively.

## **3. Controller design for the EHA position control systems**

The EHA position control system consists of the inner loop for the angular velocity control of the servo motor/pump and the outer loop for the position control of the piston. For the velocity control of the motor in the inner loop, Kokotovic et al. applied an adaptive control scheme so that the EHA position control systems can have robustness (Kokotovic, 1999). Habibi et. al. presented that if the inner loop dynamics are stable, the control gains of the PID velocity controller in the inner loop can have relatively large values and then the disturbance effect can be sufficiently rejected (Habibi, 1999). The velocity controller in the inner loop is very important because it regulates the electric motor. However, the case of (Kokotovic, 1999) is very complicated and the case of (Habibi, 1999), although it is theoretically possible, has a physical limitation that increases the control gains of the inner loop controller. Therefore, it is desirable to handle the controller in the outer loop rather than in the inner loop to improve the performance and robustness of EHA position control systems.

In this chapter, the BSC and ABSC schemes based on EHA system dynamics are considered as the position controller. Firstly, to design a BSC system, (7) is transformed to a general form (Slotine, 1999) as follows

$$
\ddot{\mathbf{x}} = f + bu\tag{11}
$$

where

192 Challenges and Paradigms in Applied Robust Control

and *Q Q B b* . Substituting (1) through (4) into (6), therefore, the dynamic equation of EHA

To represent the characteristics of the friction *Ff* between the piston and cylinder, the LuGre friction model is considered. The LuGre friction model is based on bristles analysis, which is represented with the average deflection force of bristles stiffness. The deflection displacement equation of bristles *z* , which is actually unmeasurable by experiment, is

> *dz x x z dt g x*

<sup>2</sup> (/ ) ( ) *sv x v*

where *Fc* , *Fs* and *sv v* represent the Coulomb friction force, static friction force, and Stribeck

If the relative velocity of the contact materials increases gradually, the friction force decreases instantaneously and then it increases gradually again; this effect is called the Stribeck effect and the relative velocity is the Stribeck velocity. This phenomenon depends on the material property, grade of lubrication and temperature. The friction force *Ff* can be represented with the average deflection *z* and the velocity of the piston *x* as follows

> *F z zx <sup>f</sup>* 0 1

The EHA position control system consists of the inner loop for the angular velocity control of the servo motor/pump and the outer loop for the position control of the piston. For the velocity control of the motor in the inner loop, Kokotovic et al. applied an adaptive control scheme so that the EHA position control systems can have robustness (Kokotovic, 1999). Habibi et. al. presented that if the inner loop dynamics are stable, the control gains of the PID velocity controller in the inner loop can have relatively large values and then the disturbance effect can be sufficiently rejected (Habibi, 1999). The velocity controller in the inner loop is very important because it regulates the electric motor. However, the case of (Kokotovic, 1999) is very complicated and the case of (Habibi, 1999), although it is theoretically possible, has a physical limitation that increases the control gains of the inner loop controller. Therefore, it is desirable to handle the controller in the outer loop rather

 

represent the bristles damping and viscous friction coefficients,

 

 1 11 <sup>2</sup> 1 1 *<sup>e</sup> <sup>e</sup> <sup>f</sup> ex <sup>f</sup> <sup>L</sup> p p A B A B <sup>A</sup> x A xF F LP C M VV MV V*

> 0| | ( )

 0 is the bristles stiffness coefficient, *z* is the unmeasurable internal state and the nonlinear function *g x*( ) depends on the material property, grade of lubrication and

(7)

(8)

*c s g x F Fe* (9)

(10)

systems can be represented as

expressed as (Choi, 2004)(Lee, 2004)

where

where

1 and

respectively.

temperature; that is

velocity between the cylinder and piston, respectively.

**3. Controller design for the EHA position control systems** 

$$f = -\frac{1}{M} \left\{ \mathcal{J}\_c A^2 \left( \frac{1}{V\_A} + \frac{1}{V\_B} \right) \dot{\mathbf{x}} + \dot{F}\_f + \dot{F}\_{ex} - \mathcal{J}\_c L\_f A \left( \frac{1}{V\_A} + \frac{1}{V\_B} \right) P\_L \right\},$$

$$b = -\frac{\mathcal{J}\_c A (V\_A + V\_B) C\_p}{M V\_A V\_B},$$

$$u = a\_p$$

Now, let (11) represent state equations as follows

$$\begin{cases} \dot{\mathbf{x}}\_1 = \mathbf{x}\_2\\ \dot{\mathbf{x}}\_2 = \mathbf{x}\_3\\ \dot{\mathbf{x}}\_3 = f + bu \end{cases} \tag{12}$$

And, in order to design the BSC system, new state variables are defined as follows

$$\mathbf{x}\_1 = \mathbf{x}\_1 - \mathbf{x}\_d \tag{13}$$

$$z\_2 = x\_2 - a\_1(z\_1) \tag{14}$$

$$z\_3 = x\_3 - a\_2(z\_1, z\_2) \tag{15}$$

where *<sup>d</sup> x* is the desired position input, and 1 and 2 are the functions for new state variables, which can be obtained through the following BSC design procedure. **Step 1.**

From (13), the state equation for 1*z* can be described as

$$
\dot{z}\_1 = z\_2 + \alpha\_1(z\_1) - \dot{x}\_d \tag{16}
$$

1 1 ( ) *z* is the virtual control which should be selected to guarantee the stability of the control system through the Lyapunov control function(LCF) which is defined as

$$V\_1(z\_1) = \frac{1}{2}z\_1^2\tag{17}$$

Then,

$$
\dot{V}\_1(z\_1) = z\_1 \dot{z}\_1 = z\_1[a\_1(z\_1) - \dot{x}\_d] + z\_1 z\_2 \tag{18}
$$

From (18), if 1 1 11 ( ) *<sup>d</sup> z kz x* , (16) can be exponentially stable when *t* . And 1 *k* ( 0) is a design parameter.

Robust Control of Electro-Hydraulic Actuator Systems

then the BSC law can be selected as

guarantee exponential stability.

where ˆ

where ˆ *fff* .

can be obtained by substituting (31) into (29) as

Then,

the back-stepping control, the third LCF for (15) can be defined as

If the last term of (29) for satisfying the system stability is defined as

*b* 

of *f* , in which system uncertainties are included , should be estimated.

*b* 

> 1 2 2 3

*x x x x x f*

 

 

*f* is the estimator of the system uncertainties.

Substituting (33) into (12), (12) is modified as

Using the Adaptive Back-Stepping Control Scheme 195

Since (24) uses the information of 1*z* and 2 *z* due to the property of the design procedure of

3123 212 3 <sup>1</sup> (,,) (,) <sup>2</sup>

3 1 2 3 2 33 11 22 3 2 <sup>2</sup> *V z z z V z z k z k z z z f bu* (,,) (

33 2 <sup>2</sup> *k z z f bu*

2 33 2 <sup>1</sup> *u kz z f* ( )

From (31), if the information of *f* is assumed to be known, the negative semi-definite of *V*<sup>3</sup>

From (32), it is found that EHA position control systems using the BSC law of (31) can

If system uncertainties can be exactly known, the BSC law of (31) can achieve the desired tracking performance and the robustness to the system uncertainties of EHA systems. However, the BSC law of (31) will cause a tracking error and does not achieve the robustness to the system uncertainties because the value of *f* cannot be exactly known. To improve the tracking performance and the robustness to the system uncertainties, the value

Therefore, in this chapter, an ABSC scheme is proposed, which is the BSC scheme with the estimator of *f* . In order to design the ABSC system, the BSC law of (31) is modified as

> 2 33 2 <sup>1</sup> <sup>ˆ</sup> *u kz z f* ( )

> > 3 2 33 2

*kz z*

2 2

) (29)

222 3 1 2 3 2 33 11 22 33 *V z z z V zz kz kz kz* (,,) <sup>0</sup> (32)

2

*Vzzz Vzz z* (28)

(30)

(31)

(33)

(34)

#### **Step 2.**

From (14), the state equation for 2 *z* can be described as

$$
\dot{z}\_2 = z\_3 + a\_2(z\_1, z\_2) - \dot{a}\_1(z\_1) \tag{19}
$$

where

$$\dot{\alpha}\_1(z\_1) = \frac{\partial \alpha\_1}{\partial z\_1}\dot{z}\_1 + \frac{\partial \alpha\_1}{\partial \dot{x}\_d}\ddot{x}\_d = -k\_1\dot{z}\_1 + \ddot{x}\_d = -k\_1(z\_2 - k\_1z\_1) + \dddot{x}\_d$$

Since (19) includes the information of (16), the second LCF for obtaining the virtual control to guarantee the stability of the control system can be selected as

$$V\_2(z\_1, z\_2) = V\_1(z\_1) + \frac{1}{2}z\_2^2\tag{20}$$

Then,

$$\dot{V}\_2(z\_1, z\_2) = \dot{V}\_1(z\_1) + \dot{z}\_2 z\_2 = -k\_1 z\_1^2 + z\_2 z\_3 + z\_2 [z\_1 + a\_2(z\_1, z\_2) - \dot{a}\_1(z\_1)] \tag{21}$$

If the virtual control 2 in the last term of (21) is defined as

$$a\_2(z\_1, z\_2) = -k\_2 z\_2 - z\_1 + \dot{a}\_1(z\_1)$$

where 2 *k* ( 0) is a design parameter, then another expression of 2 can be rearranged as

$$
\alpha\_2(z\_1, z\_2) = -(k\_1 + k\_2)z\_2 - (1 - k\_1^2)z\_1 + \ddot{x}\_d \tag{22}
$$

Therefore

$$
\dot{V}\_2 = z\_2 z\_3 - k\_1 z\_1^2 - k\_2 z\_2^2 \tag{23}
$$

**Step 3.**

From (15), the state equation for 3 *z* is described as

$$
\dot{z}\_3 = \dot{x}\_3 - \dot{\alpha}\_2(z\_1, z\_2) = f + bu - \dot{\alpha}\_2(z\_1, z\_2) \tag{24}
$$

where

$$
\dot{\alpha}\_2(z\_1, z\_2) = \frac{\partial \alpha\_2}{\partial z\_1}\dot{z}\_1 + \frac{\partial \alpha\_2}{\partial z\_2}\dot{z}\_2 + \frac{\partial \alpha\_2}{\partial \dot{\alpha}\_1}\ddot{\alpha}\_1 = -\dot{z}\_1 - k\_2\dot{z}\_2 + \ddot{\alpha}\_1 \tag{25}
$$

and

$$
\ddot{\alpha}\_1 = \frac{\partial \dot{\alpha}\_1}{\partial z\_1} \dot{z}\_1 + \frac{\partial \dot{\alpha}\_1}{\partial z\_2} \dot{z}\_2 + \frac{\partial \dot{\alpha}\_1}{\partial \ddot{\alpha}\_d} \ddot{\mathbf{x}}\_d = k\_1^2 \dot{z}\_1 - k\_1 \dot{z}\_2 + \dddot{\mathbf{x}}\_d \tag{26}
$$

Substituting (16), (19), and (26) into (25), (25) can be rearranged as

$$
\dot{\alpha}\_2(z\_1, z\_2) = (k\_1^2 - 1)\dot{z}\_1 - (k\_1 + k\_2)\dot{z}\_2 + \dddot{x}\_d \tag{27}
$$

Since (24) uses the information of 1*z* and 2 *z* due to the property of the design procedure of the back-stepping control, the third LCF for (15) can be defined as

$$V\_3(z\_1, z\_2, z\_3) = V\_2(z\_1, z\_2) + \frac{1}{2}z\_3^2 \tag{28}$$

Then,

194 Challenges and Paradigms in Applied Robust Control

2 3 212 11 *z z zz z* 

11 1 1 1 1 2 11

Since (19) includes the information of (16), the second LCF for obtaining the virtual control

212 11 2 <sup>1</sup> (,) () <sup>2</sup>

2 1 2 22 1 1 1

212 1 22 11 ( , ) ( ) (1 ) *<sup>d</sup>*

3 3 212 <sup>212</sup> *z x z z f bu z z*

222 2 1 2 1 2 1 1 22 1 12 1 (,) *zz z z z kz*

111 2 112 11 12

(,) (,)

(25)

*d d*

*zz k z k kz x* (27)

(26)

*d z z x kz kz x*

212 1 1 1 22 ( , ) ( 1) ( ) *<sup>d</sup>*

*z z*

*zzx* 

2

1 2

Substituting (16), (19), and (26) into (25), (25) can be rearranged as

(,) *z z kz z z*

2 2 1 2 1 1 22 11 23 2 1 2 1 2 1 1 *V z z V z zz kz zz z z z z z* (,) () [ ( , ) ( )]

2 in the last term of (21) is defined as

( ) ( ) *dd d*

*z z x kz x k z kz x*

 

2

( )

2

*zz k kz kz x* (22)

*Vzz Vz z* (20)

2 2 *V zz kz kz* 2 23 11 22 (23)

 

2 can be rearranged as

(24)

  (21)

 (,) () 

(19)

From (14), the state equation for 2 *z* can be described as

1 1

*z x*

*d*

 

1

to guarantee the stability of the control system can be selected as

where 2 *k* ( 0) is a design parameter, then another expression of

From (15), the state equation for 3 *z* is described as

**Step 2.**

where

Then,

Therefore

**Step 3.**

where

and

If the virtual control

$$
\dot{V}\_3(z\_1, z\_2, z\_3) = \dot{V}\_2 + z\_3 \dot{z}\_3 = -k\_1 z\_1^2 - k\_2 z\_2^2 + z\_3(z\_2 + f + bu - \dot{a}\_2) \tag{29}
$$

If the last term of (29) for satisfying the system stability is defined as

$$-k\_3 z\_3 = z\_2 + f + b\mu - \dot{\alpha}\_2\tag{50}$$

then the BSC law can be selected as

$$
\mu = \frac{1}{b} (\dot{a}\_2 - k\_3 z\_3 - z\_2 - f) \tag{31}
$$

From (31), if the information of *f* is assumed to be known, the negative semi-definite of *V*<sup>3</sup> can be obtained by substituting (31) into (29) as

$$
\dot{V}\_3(z\_1, z\_2, z\_3) = \dot{V}\_2 + z\_3 \dot{z}\_3 = -k\_1 z\_1^2 - k\_2 z\_2^2 - k\_3 z\_3^2 \le 0 \tag{32}
$$

From (32), it is found that EHA position control systems using the BSC law of (31) can guarantee exponential stability.

If system uncertainties can be exactly known, the BSC law of (31) can achieve the desired tracking performance and the robustness to the system uncertainties of EHA systems. However, the BSC law of (31) will cause a tracking error and does not achieve the robustness to the system uncertainties because the value of *f* cannot be exactly known. To improve the tracking performance and the robustness to the system uncertainties, the value of *f* , in which system uncertainties are included , should be estimated.

Therefore, in this chapter, an ABSC scheme is proposed, which is the BSC scheme with the estimator of *f* . In order to design the ABSC system, the BSC law of (31) is modified as

$$
\mu = \frac{1}{b} (\dot{a}\_2 - k\_3 z\_3 - z\_2 - \hat{f}) \tag{33}
$$

where ˆ *f* is the estimator of the system uncertainties. Substituting (33) into (12), (12) is modified as

$$\begin{cases}
\dot{\mathbf{x}}\_1 = \mathbf{x}\_2 \\
\dot{\mathbf{x}}\_2 = \mathbf{x}\_3 \\
\dot{\mathbf{x}}\_3 = \tilde{f} + \dot{\alpha}\_2 - k\_3 \mathbf{z}\_3 - \mathbf{z}\_2
\end{cases} \tag{34}$$

where ˆ *fff* .

Robust Control of Electro-Hydraulic Actuator Systems

diagram of the EHA position control system.

Therefore,

**4. Computer simulation** 

*Cp*

Table 1. System parameters of the EHA

Using the Adaptive Back-Stepping Control Scheme 197

<sup>0</sup> 4 4 ( ) ( (0), (0)) ( ) *<sup>t</sup>* 

Applying Barbalat's Lemma(Krstic, 1995) to (42), we can obtain that () 0 *t* as *t* .

In order to evaluate the validity of the proposed control scheme for EHA position control

sin(0.25 ) sin(0.05 ) [cm] *<sup>d</sup> x tt* 

This sinusoidal reference input is suitable for evaluating the tracking performance and the robustness of EHA position control systems because it reflects the various changes in the magnitudes of the velocity and position of the piston. Table 1 shows the system parameters of the EHA system which are used to computer simulation. Figure 2 shows the block

*V*<sup>0</sup> Initial volume of the chamber 4 3 3.712 10 m

*A* Pressure area of the piston 3 2 5.58 10 m

*<sup>e</sup>* Effective bulk modulus <sup>3</sup> 1.7 10 MPa

*Lf* Leakage factor of the pump 16 3 3.16 10 m /Pa

<sup>0</sup> Bristles stiffness coefficient <sup>6</sup> 5.77 10 N/m

<sup>1</sup> Bristles damping coefficient <sup>4</sup> 2.28 10 N/m/s

*<sup>p</sup>*max Maximum speed of the motor 178 rad/s

<sup>0</sup> Coulomb friction coefficient 370 N

<sup>1</sup> Static friction 217 N

*sv v* Stribeck velocity 0.032 m/s

<sup>2</sup> Viscous friction coefficient 2318 N/m/s

Volumetric capacity of the pump

*M* Piston mass 5 kg

Notation Description Unit

<sup>0</sup> <sup>4</sup> lim ( ) ( (0), (0)) *<sup>t</sup>*

 

*d V <sup>f</sup> V t* **<sup>z</sup>** (42)

*dV f* **<sup>z</sup>** (43)

(44)

6 3 1.591 10 m /rad

 

*t*

systems, a sinusoidal reference input was considered as follows

From (13), (14), and (15), these equations are the error equations for the velocity, acceleration and jerk of the piston, which include 1 1 ( ) *z* and 212 (,) *z z* that guarantee the exponential stability of EHA position control systems. Substituting these equations into (34), the error dynamics can be represented as

$$\begin{cases} \dot{z}\_1 = z\_2 - k\_1 z\_1 \\ \dot{z}\_2 = z\_3 - k\_2 z\_2 - z\_1 \\ \dot{z}\_3 = \tilde{f} - k\_3 z\_3 - z\_2 \end{cases} \tag{35}$$

From (35), the LCF is defined as

$$V\_4 = \frac{1}{2}z\_1^2 + \frac{1}{2}z\_2^2 + \frac{1}{2}z\_3^2 + \frac{1}{2\gamma}\tilde{f}^2\tag{36}$$

where is a positive constant.

The derivative of (36) can be described as

$$\dot{V}\_4 = z\_1 \dot{z}\_1 + z\_2 \dot{z}\_2 + z\_3 \dot{z}\_3 + \frac{1}{\gamma} \dot{\tilde{f}} = -k\_1 z\_1^2 - k\_2 z\_2^2 - k\_3 z\_3^2 + \tilde{f} (z\_3 - \frac{1}{\gamma} \dot{\tilde{f}}) \le 0 \tag{37}$$

From (37), an estimation rule to guarantee the system stability can be obtained as

$$
\dot{\hat{f}} = \dot{f} - \chi z\_3 \tag{38}
$$

Equation (38) uses the information of 3 *z* , which depends on the information of 1*z* and 2 *z* . Therefore, (38) closely relates to 1 and <sup>2</sup> , which guarantee the stability of BSC systems. However, (38) cannot be used to the estimation rule because the value of *f* is unknown. On the other hand, if *f* is assumed as a lumped uncertainty, system uncertainty *f* can be estimated by 3 <sup>ˆ</sup> *f <sup>z</sup>* . However, since the value of *f* for the EHA system is changed according to the operating condition, it cannot be assumed as the lumped uncertainty. Therefore, to obtain the value of *f* , the classical discrete disturbance observer scheme was used. Assuming that the sampling rate of the control loop is very fast, the classical discrete disturbance observer expressed by the difference equation is induced from (31) as follows

$$f(k-1) = bu(k) - \dot{\alpha}\_2(k) + k\_3 z\_3(k) + z\_2(k)\tag{39}$$

To analyze the stability of the proposed control scheme, (38) is substituted into (37). Then,

$$\dot{V}\_4 = -k\_1 z\_1^2 - k\_2 z\_2^2 - k\_3 z\_3^2 = -\mathbf{z}^T \mathbf{K} \mathbf{z} < 0 \tag{40}$$

where **K** is the diagonal matrix of 1 *<sup>k</sup>* , 2 *<sup>k</sup>* and 3 *<sup>k</sup>* , <sup>T</sup> =[ ] <sup>123</sup> **<sup>z</sup>** *zzz* , and 4 *<sup>V</sup>* <sup>0</sup> if 0 **<sup>z</sup>** . If **z** is bounded, (40) can be defined as

$$\mathbf{d}\Phi(t) = \mathbf{z}^{\mathrm{T}} \mathbf{K} \mathbf{z} \ge 0 \tag{41}$$

Integrating (41) from 0 to *t* , the following result can be obtained

$$V\_0^\dagger \Phi(\tau)d\tau = V\_4(\mathbf{z}(0), \tilde{f}(0)) - V\_4(t) \tag{42}$$

Applying Barbalat's Lemma(Krstic, 1995) to (42), we can obtain that () 0 *t* as *t* . Therefore,

$$\lim\_{t \to \infty} \int\_0^t \Phi(\tau) d\tau \le V\_4(\mathbf{z}(0), \tilde{f}(0)) < \infty \tag{43}$$

## **4. Computer simulation**

196 Challenges and Paradigms in Applied Robust Control

From (13), (14), and (15), these equations are the error equations for the velocity, acceleration

stability of EHA position control systems. Substituting these equations into (34), the error

1 2 11 2 3 22 1 3 33 2

*z z kz z z kz z z f kz z*

 

4123 111 1 <sup>2222</sup> *Vzzz f*

4 11 22 33 11 22 33 3 1 1 *V z z z z z z ff k z k z k z f z f* ( )0

> <sup>3</sup> <sup>ˆ</sup> *f f*

Equation (38) uses the information of 3 *z* , which depends on the information of 1*z* and 2 *z* .

However, (38) cannot be used to the estimation rule because the value of *f* is unknown. On the other hand, if *f* is assumed as a lumped uncertainty, system uncertainty *f* can be

according to the operating condition, it cannot be assumed as the lumped uncertainty. Therefore, to obtain the value of *f* , the classical discrete disturbance observer scheme was used. Assuming that the sampling rate of the control loop is very fast, the classical discrete disturbance observer expressed by the difference equation is induced from (31) as follows

> 2 33 2 *f* ( 1) ( ) ( ) ( ) ( ) *k bu k k k z k z k*

> > 2 2 2T

To analyze the stability of the proposed control scheme, (38) is substituted into (37). Then,

where **K** is the diagonal matrix of 1 *<sup>k</sup>* , 2 *<sup>k</sup>* and 3 *<sup>k</sup>* , <sup>T</sup> =[ ] <sup>123</sup> **<sup>z</sup>** *zzz* , and 4 *<sup>V</sup>* <sup>0</sup> if 0 **<sup>z</sup>** .

From (37), an estimation rule to guarantee the system stability can be obtained as

1 and 

Integrating (41) from 0 to *t* , the following result can be obtained

 

 ( ) *z* and 212 

222 2

222

(37)

*<sup>z</sup>* . However, since the value of *f* for the EHA system is changed

(36)

 

*<sup>z</sup>* (38)

<sup>2</sup> , which guarantee the stability of BSC systems.

(39)

4 11 22 33 *V kz kz kz* **z Kz** <sup>0</sup> (40)

<sup>T</sup> () 0 *t* **z Kz** (41)

(,) *z z* that guarantee the exponential

(35)

and jerk of the piston, which include 1 1

dynamics can be represented as

From (35), the LCF is defined as

Therefore, (38) closely relates to

*f* 

If **z** is bounded, (40) can be defined as

estimated by 3 <sup>ˆ</sup>

 is a positive constant. The derivative of (36) can be described as

where  In order to evaluate the validity of the proposed control scheme for EHA position control systems, a sinusoidal reference input was considered as follows

$$\mathbf{x}\_d = \sin(0.25\pi t) + \sin(0.05\pi t) \text{ [cm]} \tag{44}$$

This sinusoidal reference input is suitable for evaluating the tracking performance and the robustness of EHA position control systems because it reflects the various changes in the magnitudes of the velocity and position of the piston. Table 1 shows the system parameters of the EHA system which are used to computer simulation. Figure 2 shows the block diagram of the EHA position control system.


Table 1. System parameters of the EHA

Robust Control of Electro-Hydraulic Actuator Systems

reference input

parameters

Using the Adaptive Back-Stepping Control Scheme 199

Fig. 4. Estimated value for the system uncertainties of the ABSC system for the sinusoidal

Fig. 5. Tracking errors of the BSC and ABSC systems with the perturbation of the system

Fig. 2. Block diagram of the EHA position control system

Figure 3 shows the tracking errors of the BSC and ABSC systems for the sinusoidal reference. This result shows that the ABSC system has better tracking performance than the BSC system and has error repeatability precision of higher reliability than the BSC system. In addition, in both position and control schemes relatively large tracking errors occur at the nearly zero velocity regions. This is due to the effect of dynamic friction characteristics, which produce an instantaneous large force at the nearly zero velocity regions. For the transient response region of the initial operation of EHA position control systems, the ABSC system with the estimator for system uncertainties yields approximately 40% improvement compared with the BSC system without the estimator because the *f* in (31) including system uncertainties is estimated by (43), as shown in Fig. 4. Figure 4 shows the estimated value ˆ *f* for the system uncertainties of EHA systems obtained by the proposed adaptive rule. The estimated value plat a role in the consideration of nonlinearity and uncertainties included in EHA systems.

Fig. 3. Tracking errors of the BSC and ABSC systems for the sinusoidal reference input

198 Challenges and Paradigms in Applied Robust Control

Figure 3 shows the tracking errors of the BSC and ABSC systems for the sinusoidal reference. This result shows that the ABSC system has better tracking performance than the BSC system and has error repeatability precision of higher reliability than the BSC system. In addition, in both position and control schemes relatively large tracking errors occur at the nearly zero velocity regions. This is due to the effect of dynamic friction characteristics, which produce an instantaneous large force at the nearly zero velocity regions. For the transient response region of the initial operation of EHA position control systems, the ABSC system with the estimator for system uncertainties yields approximately 40% improvement compared with the BSC system without the estimator because the *f* in (31) including system uncertainties is estimated by (43), as shown in Fig. 4. Figure 4 shows the estimated

*f* for the system uncertainties of EHA systems obtained by the proposed adaptive rule. The estimated value plat a role in the consideration of nonlinearity and uncertainties

Fig. 3. Tracking errors of the BSC and ABSC systems for the sinusoidal reference input

Fig. 2. Block diagram of the EHA position control system

value ˆ

included in EHA systems.

Fig. 4. Estimated value for the system uncertainties of the ABSC system for the sinusoidal reference input

Fig. 5. Tracking errors of the BSC and ABSC systems with the perturbation of the system parameters

Robust Control of Electro-Hydraulic Actuator Systems

nonlinear friction effects by using the estimated value ˆ

wave type reference input. The estimated value ˆ

shows the estimated value ˆ

reference input.

Using the Adaptive Back-Stepping Control Scheme 201

Figure 7 shows the tracking errors of the BSC and ABSC systems for the sinusoidal reference input, which was used in the computer simulation. The tracking error of the BSC system is relatively large when the direction of the piston is changed because the BSC system cannot compensate the friction of the EHA system. In addition, the tracking error of the BSC varies according to the direction of the piston because of the system uncertainties of the EHA system. However, the ABSC system has better tracking performance than the BSC system because the ABSC system can effectively compensate the system uncertainties as well as the

Figure 9 shows the speed of the motor as the control input for the sinusoidal reference input. Figure 10 shows the tracking errors of the BSC and ABSC systems for the square wave type reference input. The characteristics of the transient responses of the BSC and the ABSC systems are almost same. In the BSC system, however, steady-state error occurs relatively large in the backward direction. This shows that the BSC system cannot compensate the system uncertainties of the EHA system. But we can show that the ABSC system can effectively compensate the system uncertainties regardless of the piston direction. Figure 11

desired tracking performance and robustness to the EHA system with system uncertainties. Figure 12 shows the speed of the motor as the control input for the square wave type

Fig. 7. Tracking errors of the BSC and ABSC systems for the sinusoidal reference input

*f* , which is shown in Fig. 8.

*f* for the system uncertainties makes the

*f* for the system uncertainties of the ABSC system for the square

Figure 5 shows the tracking errors of the BSC and ABSC systems having perturbations of the system parameters such as the Coulomb friction, viscous friction and pump leakage coefficient in the EHA system for the sinusoidal reference input. It was assumed that the system parameters have a perturbation of 50%. From Fig. 5, it was found that the perturbations of the system parameters of the EHA system are closely related with the tracking performance of the EHA system. Table 2 shows the tracking RMS errors of the BSC and ABSC systems according to the perturbation of the system parameters. The variations of the tracking RMS errors due to the 50% perturbation of the system parameters for the BSC and ABSC systems are 17.6% and 3.02%, respectively. These results show that the proposed position control scheme has the desired robustness to system uncertainties such as the perturbation of the viscous friction, Coulomb friction and pump leakage coefficient.


Table 2. Tracking RMS errors of the BSC and ABSC systems according to the perturbations of the system parameters

## **5. Experimental results and discussion**

Figure 6 shows the experimental setup of the EHA system. To evaluate the effectiveness of the proposed control system, the PCM-3350(AMD Geode processor, 300MHz) was used. The control algorithms were programmed by Turbo-C++ language on MS-DOS, in order to directly handle the PCM-3718 as a data acquisition board. The PCM-3718 is a fully multifunctional card with PC/104 interface. In addition, to measure the position of the piston, an LVDT(linear variable differential transformer) sensor was used. The sampling rate was set to 1 kHz.

Fig. 6. Experimental setup of the EHA system

200 Challenges and Paradigms in Applied Robust Control

Figure 5 shows the tracking errors of the BSC and ABSC systems having perturbations of the system parameters such as the Coulomb friction, viscous friction and pump leakage coefficient in the EHA system for the sinusoidal reference input. It was assumed that the system parameters have a perturbation of 50%. From Fig. 5, it was found that the perturbations of the system parameters of the EHA system are closely related with the tracking performance of the EHA system. Table 2 shows the tracking RMS errors of the BSC and ABSC systems according to the perturbation of the system parameters. The variations of the tracking RMS errors due to the 50% perturbation of the system parameters for the BSC and ABSC systems are 17.6% and 3.02%, respectively. These results show that the proposed position control scheme has the desired robustness to system uncertainties such as the

perturbation of the viscous friction, Coulomb friction and pump leakage coefficient.

BSC

ABSC

**5. Experimental results and discussion** 

Fig. 6. Experimental setup of the EHA system

of the system parameters

rate was set to 1 kHz.

Control scheme Perturbation ratio RMS value

Table 2. Tracking RMS errors of the BSC and ABSC systems according to the perturbations

Figure 6 shows the experimental setup of the EHA system. To evaluate the effectiveness of the proposed control system, the PCM-3350(AMD Geode processor, 300MHz) was used. The control algorithms were programmed by Turbo-C++ language on MS-DOS, in order to directly handle the PCM-3718 as a data acquisition board. The PCM-3718 is a fully multifunctional card with PC/104 interface. In addition, to measure the position of the piston, an LVDT(linear variable differential transformer) sensor was used. The sampling

0% 1.878 mm 50% 2.209 mm

0% 0.265 mm 50% 0.273 mm Figure 7 shows the tracking errors of the BSC and ABSC systems for the sinusoidal reference input, which was used in the computer simulation. The tracking error of the BSC system is relatively large when the direction of the piston is changed because the BSC system cannot compensate the friction of the EHA system. In addition, the tracking error of the BSC varies according to the direction of the piston because of the system uncertainties of the EHA system. However, the ABSC system has better tracking performance than the BSC system because the ABSC system can effectively compensate the system uncertainties as well as the nonlinear friction effects by using the estimated value ˆ *f* , which is shown in Fig. 8. Figure 9 shows the speed of the motor as the control input for the sinusoidal reference input. Figure 10 shows the tracking errors of the BSC and ABSC systems for the square wave type reference input. The characteristics of the transient responses of the BSC and the ABSC systems are almost same. In the BSC system, however, steady-state error occurs relatively large in the backward direction. This shows that the BSC system cannot compensate the system uncertainties of the EHA system. But we can show that the ABSC system can effectively compensate the system uncertainties regardless of the piston direction. Figure 11 shows the estimated value ˆ *f* for the system uncertainties of the ABSC system for the square

wave type reference input. The estimated value ˆ *f* for the system uncertainties makes the desired tracking performance and robustness to the EHA system with system uncertainties. Figure 12 shows the speed of the motor as the control input for the square wave type reference input.

Fig. 7. Tracking errors of the BSC and ABSC systems for the sinusoidal reference input

Robust Control of Electro-Hydraulic Actuator Systems

input

wave type reference input

Using the Adaptive Back-Stepping Control Scheme 203

Fig. 10. Tracking errors of the BSC and ABSC systems for the square wave type reference

Fig. 11. Estimated value for the system uncertainties of the ABSC system for the square

Fig. 8. Estimated value for the system uncertainties of the ABSC system for the sinusoidal reference input

Fig. 9. Speed of the motor as the control input for the sinusoidal reference input

202 Challenges and Paradigms in Applied Robust Control

Fig. 8. Estimated value for the system uncertainties of the ABSC system for the sinusoidal

Fig. 9. Speed of the motor as the control input for the sinusoidal reference input

reference input

Fig. 10. Tracking errors of the BSC and ABSC systems for the square wave type reference input

Fig. 11. Estimated value for the system uncertainties of the ABSC system for the square wave type reference input

Robust Control of Electro-Hydraulic Actuator Systems

Control system Sinusoidal reference input

Table 3. Tracking RMS errors of the BSC and ABSC systems

**6. Conclusion** 

**7. References** 

*16(10)* , pp. 643-653.

Using the Adaptive Back-Stepping Control Scheme 205

Table 3 shows the tracking RMS errors of the BSC and ABSC systems for the sinusoidal reference input and the square wave type reference input at steady-state. From Table 3, it was found that using the ABSC system instead of the BSC system yields about 5 times

> BSC 1.762 mm 0.395 mm ABSC 0.309 mm 0.114 mm

A robust position control of EHA systems was proposed by using the ABSC scheme, which has robustness to system uncertainties such as the perturbation of viscous friction, Coulomb friction and pump leakage coefficient. Firstly, a stable BSC system based on the EHA system dynamics was derived. However, the BSC scheme had a drawback: it could not consider system uncertainties. To overcome the drawback of the BSC scheme, the ABSC scheme was proposed having error equations for the velocity, acceleration and jerk of the piston, respectively, which were induced by the BSC scheme. To evaluate the performance and robustness of the proposed EHA position control system, BSC and ABSC schemes were implemented in a computer simulation and experiment. It was found that the ABSC scheme can yield the desired tracking performance and the robustness to system uncertainties.

Y. Chinniah, R. Burton and S. Habibi (2006), Failure monitoring in a high performance

J. J. Choi, J. S. Kim and S. I. Han (2004), Pre-sliding friction control using the sliding mode

S. Habibi and A. Goldenberg (2000), Design of a new high-performance electro-hydraulic

L. Jun, F. Yongling, Z. Guiying, G. Bo and M. Jiming (2004), Research on fast response and

C. Kaddissi, J. P. Kenne and M. Saad (2006), Indirect adaptive control of an electro-hydraulic

V. V. Kokotovic, J. Grabowski, V. Amin and J. Lee (1999), Electro hydraulic power steering

M. Krstic, I. Kanellakopoulos and P. Kokotovic (1995), Nonlinear and Adaptive Control

system, *Int. Congress & Exposition*, Detroit, Michigan, USA, pp. 1-4.

actuator, IEEE Trans. *Mechatronics 5(2),* pp. 158-164.

*Robotics and Biomimetics*, Shenyang, China, pp. 807-810.

Montreal, Quebec, Canada, pp. 3147-3153.

Design, *Wiley Interscience*, New York, USA.

hydrostatic actuation system using the extended kalman filter, *Int. J. Mechatronics*

controller with hysteresis friction compensator, *KSME Int'l J. 18(10)*, pp. 1755-1762.

high accuracy control of an airborne brushless DC motor, *Proc. 2004 IEEE Int. Conf.* 

servo system based on nonlinear backstepping, *IEEE Int. Symposium Ind. Electron*,

Square wave type reference input at steady state

improvement in the tracking performance of the EHA position control system.

Fig. 12. Speed of the motor as the control input for the square wave type reference input

Table 3 shows the tracking RMS errors of the BSC and ABSC systems for the sinusoidal reference input and the square wave type reference input at steady-state. From Table 3, it was found that using the ABSC system instead of the BSC system yields about 5 times improvement in the tracking performance of the EHA position control system.


Table 3. Tracking RMS errors of the BSC and ABSC systems

## **6. Conclusion**

204 Challenges and Paradigms in Applied Robust Control

Fig. 12. Speed of the motor as the control input for the square wave type reference input

A robust position control of EHA systems was proposed by using the ABSC scheme, which has robustness to system uncertainties such as the perturbation of viscous friction, Coulomb friction and pump leakage coefficient. Firstly, a stable BSC system based on the EHA system dynamics was derived. However, the BSC scheme had a drawback: it could not consider system uncertainties. To overcome the drawback of the BSC scheme, the ABSC scheme was proposed having error equations for the velocity, acceleration and jerk of the piston, respectively, which were induced by the BSC scheme. To evaluate the performance and robustness of the proposed EHA position control system, BSC and ABSC schemes were implemented in a computer simulation and experiment. It was found that the ABSC scheme can yield the desired tracking performance and the robustness to system uncertainties.

## **7. References**


**10**

and Adam K. Piłat2

<sup>1</sup>*Finland* <sup>2</sup>*Poland*

**Discussion on Robust Control Applied to Active**

<sup>1</sup>*Dept. of Electrical Engineering, LUT Energy, Lappeenranta University of Technology*

Since the 1980s, a stream of papers has appeared on system uncertainties and robust control. The robust control relies on H<sup>∞</sup> control and *μ* synthesis rather than previously favored linear-quadratic Gaussian control. However, highly mathematical techniques have been difficult to apply without dedicated tools. The new methods have been consolidated in the practical applications with the appearance of software toolboxes, such as Robust Control Toolbox from Matlab. This chapter focuses on the application of this toolbox to the active

AMBs are employed in high-speed rotating machines such as turbo compressors, flywheels, machine tools, molecular pumps, and others (Schweitzer & Maslen, 2009). The support of rotors using an active magnetic field instead of mechanical forces of the fluid film, contact rolling element, or ball bearings enables high-speed operation and lower friction losses. Other major advantages of AMBs include no lubrication, long life, programmable stiffness and damping, built-in monitoring and diagnostics, and availability of automatic balancing. However, AMB rotor system forms an open-loop unstable, multiple-input multiple-output (MIMO) coupled plant with uncertain dynamics that can change over time and that can vary significantly at different rotational speeds. In practical systems, the sensors are not collocated with the actuators, and therefore, the plant cannot always be easily decoupled. Additionally,

The major drawback of an AMB technology is a difficulty in designing a high-performance reliable control and its implementation. For such systems, the *μ* and H<sup>∞</sup> control approaches

The high-performance and high-precision control for the nominal plant without uncertainties can be realized by using model-based, high-order controllers. In the case of control synthesis, which is based on the uncertain plant model, there is a tradeoff between the nominal performance (time- and frequency-domain specifications) and the robustness. The modeled uncertainties cannot be too conservative or otherwise obtaining practical controllers might be not feasible (Sawicki & Maslen, 2008). Moreover, too complex uncertainty models lead to increased numerical complexity in the control synthesis. The models applied for the control

offer useful tools for designing a robust control (Moser, 1993; Zhou et al., 1996).

magnetic bearing (AMB) suspension system for high-speed rotors.

the control systems face a plethora of external disturbances.

**1. Introduction**

**Magnetic Bearing Rotor System**

Rafal P. Jastrzebski1, Alexander Smirnov1, Olli Pyrhönen1

<sup>2</sup>*Dept. of Automatics, AGH University of Science and Technology, Krakow*

