Robust H 2 / H ∞ Control for the Electrohydraulic Steering System of a Four-Wheel Vehicle

To shorten the steer diameter and to improve the maneuverability flexibility of a construction vehicle, four wheels’ steering system is presented. This steering system consists of mechanical-electrical-hydraulic assemblies. Its diagram and principle are depicted in detail. Then the mathematical models are derived step by step, including the whole vehicle model and the hydraulic route model. Considering the nonlinear and time-varying uncertainty of the steering system, robust H 2 /H ∞ controller is put forward to guarantee both the system performance and the robust stability. The H ∞ norm of the sensitive function from the parameter perturbation of the hydraulic system to the yaw velocity of the vehicle is taken as the evaluating index of the robustness and the


Introduction
Traditionally a wheel vehicle is steered by two front wheels or by two rear wheels, which is controlled by a driver's steering wheel.The shortage of the two wheels' steering system is that the steering diameter is large, and it is not easy to satisfy steering requirement under narrow space, which constrict its application, especially for a large and heavy-duty construction vehicle.To improve the direction maneuverability and stability during the driving, furthermore to improve the safety and comfort, four wheels' steering (4WS) system is studied recently.4WS systems for automobiles have been actively studied to improve the maneuverability of vehicles at low speeds and enhance their stability at high speeds.Many automobile companies had developed concept vehicles with 4WS system, for example, Honda, Nissan, and Mazda [1,2].Borrowing the idea from the automobile industry, 4WS has been applied in the construction vehicle, for example the concrete spreading machine "SF-3004" from CMI Terex Company and ditch cutter "560" from Case Company [3].The uncertainty of a construction vehicle is more serious than of a car, which has a large power hydraulic system and works in bad and dirty environment.
The control of the 4WS system is complicated and sometimes may not be effective due to the nonlinear characteristics and unknown environmental parameters.During the last 20 years, many different control methods have been applied on 4WS system [4][5][6].Early for a 4WS vehicle, a simple speeddependent ratio between rear and front wheels has been used in an open-loop controller to achieve a zero constant sideslip angle during directional maneuvers [7].Ackermann and Sienel [8] used a proportional controller in their nonlinear 3 DOF (degrees of freedom) model, while Ji et al. [9] used proportional and compensator controllers in their control strategy.Lv et al. [10] and You and Chai [11] used fuzzy logic method to investigate the performance on controlling the wheel angle, and they did not give out the experimental validation.With the development of sliding mode control [12,13], it has been applied to control four-wheel vehicle [14].However it should be noted that the aforementioned controllers are based on the fairly concise model, where the uncertainty is not considered.The parameters of a construction vehicle are subject to a vast range of uncertainties such as external disturbances, unmodeled dynamics, road roughness, wind gusts, load fluctuations, and braking/accelerating forces.Thus, a serious robust stability problem for the 4WS vehicle control has been raised; namely, the vehicle controller has to cope with these uncertainties to keep maneuvering stability and ensure that the system performance does not deteriorate too much.With the uncertainty the linear design model cannot express the exact behavior, which is usually required for the controller design.So that the classical control method is ineffective to guarantee the control performance.Robust  2 / ∞ control has been proven to be effective for controls related to nonlinear dynamic systems, which is robust to uncertainty.In [15], a LMI approach to the robust state feedback  ∞ control for linear discrete singular systems with norm-bounded uncertainty was developed.The robust  ∞ controller design algorithm was presented and an explicit expression for the desired state feedback control law was also given [16].So it can be concluded that the benefits of a 4WS system are often described but not quantified, and most of the studies carry out only the simulations not the experiments.This paper reviews the purposes, methods, and advantages of 4WS; firstly, the uncertainty model is put forward and secondly the robust  2 / ∞ controller is designed to attenuate the parameter perturbation and the outside disturbance of the steering system.Furthermore experiments are carried out.Finally the conclusions are summarized.

The Diagram of the 4WS Vehicle
To realize the function of 4WS for a construction vehicle, the drive-by-wire hydraulic system is proposed, as shown in Figure 1, which includes a hydraulic pump, an electrohydraulic servo valve, a hydraulic cylinder, and a controller.The hydraulic cylinder is connected to the wheel by the traditional double suspension fork guidance mechanism, as shown in the dotted line of Figure 1, which includes the planar mechanism of 4WS from front view and top view.The cylinder rod propels the suspension fork and furthermore steers the wheel to the desired angle.The hydraulic cylinder of each wheel is connected in parallel to hydraulic route to propel the four wheels, preventing them from interfering with each other.The pressure of the pump is set up by the spillover valve and keeps constant during running.The controller acquaints the command signal from the driver's steering wheel and output PWM control signals for each electrohydraulic servo valve.The PWM signal is proportional to the flow rate running into the cylinder.Provided that the hydraulic flow cannot be compressed, the displacement of the cylinder rod is proportional to the PWM signals.The steer degree of the wheel is controlled by the controller indirectly.
For a wheel vehicle, the requirements for the steering system are high following accuracy, quick response velocity, and  good stability.For a construction vehicle, which is running at serious condition, good disturbance attenuation ability is extra required.To shorten the power consumption, tire wear, and ground friction and to improve maneuverability flexibility of a wheel vehicle during steering and running, it is better that all the wheels roll only on the ground without producing any sliding (including side sliding, longitudinal sliding, and slippage).There are three kinds of steer modes for a construction vehicle: two front wheels' steering, four wheels' steering, and sideways steering modes.The Ackerman steer triangle is shown in Figure 2(a).At front wheel steering mode, the steering centerline lies in the rear wheel axis and the steering angles of the rear wheels are 0 degrees.At four wheels' steering mode, the steering centerline is in the middle of front and rear axles.The steering directions of the front wheels are contrary to those of rear wheels.At sideways steering mode, the steering directions of four wheels are the same, and there is no steering centerline.The relationship between the rod of the hydraulic cylinder and the steering wheel is shown in Figure 2(b).Without any input signal from the steering wheel, the cylinder rod locates at initial position .When the wheel steering angle is , the rod locates at   .The distance between   is expressed as   .Because Δ  ∽ Δ  , so   /  = /.Assume that    = ,  = ; we can get that where  = arc sin(/) is the angle between  and ,  is the steering angle of the wheel.According to the cosine theory, the length between two joints can be expressed as We can get that where  is the initial length of the cylinder when the steering angle is 0 degrees.y is the stretching length of the cylinder rod, and The relationships between two front wheels are as follows at front wheel steering mode: The relationships between the four front wheels are shown as follows at four wheels' steering mode: The relationships between the four wheels are shown as follows at sideways steering mode: where, in the above equations,  FR and  FL are steer angles of FL (front left) or FR (front right) wheel. RL and  RR are steer angles of RL (rear left) or RR (rear right) wheel;  1 ,  2 are distances between two front or rear wheels;  1 ,  2 are the distances from front axle or rear axle to steer centerline;  is the distance from front axle to rear axle.The structure parameters of the steering system are shown in Table 1.
Table 1: Structure parameters of the 4ws system (unit: m).

Mathematical Models of the 4WS Vehicle
There are two parts for the models.One is the whole vehicle model and the other is the hydraulic route model.
The former aims at the yaw velocity dynamic response of the whole vehicle.The latter consists of the controller, the electrohydraulic servo valve, the hydraulic cylinder, and the wheel.

The Dynamic Vehicle Model.
The dynamic vehicle model is shown in Figure 3, where  is the slip angle of the vehicle,  is the yaw velocity, V is the lateral velocity, and  is the forward velocity.  ( =  or ;  =  or ) is the lateral force of different wheel.  is the steer angle of different wheel,   is the equivalent steer angle   = (  +   )/2, and   is the slip angle of different wheel. is the angle between the longitudinal center line of the vehicle and the coordinate .  and   are the distance from the mass center of the vehicle to the front and rear axles [17].
The motion differential equation of the vehicle can be expressed as where   is cornering stiffness coefficient of different wheel,  =  or  represents the front or rear axle, and  =  or  represents the left or right wheel. is the vehicle mass,   is the yaw inertia moment, and, for other parameters, one can refer to Figure 3.If the steer angle   and the slip angle of the mass center  are small enough, we can assume that  = V/, so (1) can be simplified as At the same time we can get that   = (  +   )/2.If we assume that   = (  +   )/2, according to (7) and ( 8), it can be concluded that In ( 9),  = V/, where if  is a constant value, it can be derivated that β = V /.The state equation of the vehicle can be gotten as where the state vector is  = [ ]  and the input vector is According to state equation (10) and derivate equation (10), we can get the second-order differential equation as where The equation can be simplified further as So by the natural frequency  2 0 =  and the subsidence 2 0  = , we can get the solution of (12) as Supposed that the input angle  1 is a step input, which can be shown in ( 14) where  0 is a constant value.The initial condition can be expressed as that () = 0,  1 =  0 , ω =  1  0 .
We can get If the travelling velocity of the vehicle is 80 km/h, after the unit conversion to international standard unit, the region of  is (0 m/s, 25 m/s).The damping coefficient response and the natural frequency of the vehicle are shown in Figures 4(a) and 4(b).In Figure 4, the damping coefficient increases with the velocity and arrives at the maximum 1.67 m/s when the vehicle is 25 m/s.However the natural frequency decreases quickly with the velocity increase.When the vehicle velocity is more than 5 m/s, the natural frequency response is less than 10 Hz.The minimum value is nearly 2 Hz when the velocity is 25 m/s.
Combining the condition of velocity region and time region, we can get the yaw velocity gain as in Figure 5.The yaw velocity gain increases with the velocity firstly and then decreases.The peak value is nearly 1 when the velocity is 10 m/s.

The Electrohydraulic System
Model.The electrohydraulic system mainly consists of the servo valve and the cylinder, as shown in Figure 6.The PWM signals from the controller are modified to change the current through the magnetic coil.Further, the displacement of the rod of valve core is controller.As a result, the hydraulic flow from and to the cylinder is consist with the reference value.The following is a summary of the assumption that has been made in developing the model of a hydraulic cylinder.(1) The proportional valve is a symmetrical 3-way and 4port valve.The dead band of valve is also symmetrical, and the flow in it is turbulent.
(2) Possible dynamic behavior of the pressure in the transmission lines between valve and cylinder is assumed to be negligible.
(3) Pressure is equal everywhere in one volume of hydraulic cylinder, and the temperature and the bulk modulus are constants.
(4) The leakage of flows is laminar.
According to the continuous equation of compressible oil, where  is the initial volume of liquid subjected to compression. and  are the changes in pressure and volume, respectively.∑  in is the input flows of liquid, and ∑  out is the output flows of liquid. is the bulk modulus.
Considering the internal and external leakage of a cylinder, the equations of the left and right chambers of a cylinder are defined as where  1 is the input flow to cylinder, and  2 is the output flows from cylinder, as shown in Figure 6. ic is the internal leakage coefficient. ec is the external leakage coefficient.  is the valid bulk modulus (including liquid and the air in the oil) and  1 ,  2 are the volume of fluid flow from and to the hydraulic cylinder. 1 ,  2 can be got as follows: where  01 is the initial volume of cylinder side into where the fluid flows. 02 is the initial volume of cylinder side from which the fluid flows out. is the displacement of piston.So the derivation of ( 18) can be given as Because of the development of sealing technology, the influence of external leakage can be neglected, which means the leakage between the piston rod and external seals. can be expressed as a function of the wheel steering angle .Then (17) can be rebuilt as where  ic is assumed to be a constant to simplify the system model.
The flow equation of electrohydraulic servo valve is that where   is the pressure of return oil.Δ is the pressure of spring.  is the flow coefficient. is the area grads of orifice. V is the displacement of spool. is the density of oil.  is the current coefficient of the servo valve.() is the control current on the servo valve, which is proportional to the duty cycle of PWM signals.
There is some dead band in any servo valve, and   is proved to be nonlinear by experiment [18].The valid displacement of spool can be given as follows: where   is the dead band of valve.
In Figure 7, the flow plus coefficient of valve can be approximated by two lines to simplify the model.To be controlled conveniently,  (the flow of valve) can be expressed as where   is the simplified projection function of flow.Q is the model error of the flow projection.Generally speaking,  1 and  2 are constant for a stable working state.According to the model of 4WS, the main difficulties to control are that (1) the system dynamics is strongly nonlinear and that (2) the system parameters are uncertain, for example, the uncertainty of   (inertia load),  (valid bulk modulus), and   (the total disturb of system), which is brought by external disturb and unmodeled friction.
In this paper, the nonlinearity and uncertainty of parameters are treated by a robust controller.In Figure 7, the polyfit line is used to compensate for the dead band of the valve, which is the symmetrical line of plus curve based on the axis that is through the coordinate origin and makes an angle of 45 degrees with the  axis.The flow projection error is compensated by robust feedback.Then ( 16) and ( 20) can be transform to where and  min ,  max ,   , and  1 ,  2 are given value.It is assumed that q and q are the estimated value and estimated error of . V is assumed to be proportional to (); that is  V = (); then

Robust 𝐻 2 /𝐻 ∞ Control
First, because the 4Ws system is composed of mechanical and electrical units, the coupling of two subsystems is a serious disturbance for the electrical control system.Secondly, the driver decisions are stochastic, resulting in the variation of certain parameters.Thirdly the stochastic variation of the pump pressure is uncertainty element for the model, so is the rolling resistance.On the other hand, the electrohydraulic system is a strongly nonlinear system for the inherent PWM modulation method and the dead band "born in nature, " and the variation of parameters will deteriorate the system performance also.Considering the nonlinear and time-varying uncertainty of the steering system, robust  2 / controller is put forward to guarantee both the system performance and the robust stability.
A model in terms of state space equation is denoted as where  is the state vector,  is the control input vector,  is the outer disturbance vector,  1 and  2 are the control output vector, and ,  1 ,  2 ,  1 ,  2 ,  11 ,  12 ,  21 , and  22 are known certain matrices defined by the nominal model.The robust  2 / ∞ controller to be designed should guarantee that (1) the closed-loop system is stable; (2) the performance index for robustness satisfies that ‖( 1 )‖ ∞ <  1 with the parameter perturbation and outer disturbance, defined by the  ∞ norm of the transfer function ( 1 ) from  to  1 ; (3) the performance index for linear quadratic Gaussian (LQG) is as small as possible satisfying that ‖( 2 )‖ 2 <  2 , defined by the  2 norm of the transfer function ( 2 ) from  to  2 .The design of the controller is meant to minimize  2 with the situation of (  ) ⊂  − , ‖  1  ‖ ∞ <  1 , ‖  2  ‖ 2 <  2 , where  is the controller to be designed [19,20].
To design the state feedback controller, The robust hybrid controller design for (28) is difficult because of the solution of the nonlinear matrix inequality.According to the variable substitution method [21], the state feedback controller is solvable with the existence of the optimal solution ,  of (27).The robust hybrid controller can be expressed as (29), Mathematical Problems in Engineering The controller can guarantee that the  2 performance index is ‖( 2 )‖ 2 < 12 and the  ∞ index ‖( 1 )‖ ∞ < 10.The nominal closed-loop model is robust with its gain margin 38 dB and phase margin 52 deg with the designed controller.The continuous transfer function of the controller is shown in A linear continuous model control system is stable if all poles of the closed-loop transfer function () lie in the left half of the -plane.The -plane is related to the -plane by the transformation  =   =  (+) .In the left-hand -plane,  < 0, and therefore the related magnitude of  varies between 0 and 1. Therefore the imaginary axis of the -plane corresponds to the unit circle in the -plane, and the inside of the unit circle corresponds to the left half of the -plane.Therefore we can state that a sampled system is stable if all the poles of the closed-loop transfer function () lie within the unit circle of the -plane, as shown in Figure 8.
The discretization/interpolation methods include zeroorder hold, first-order hold, impulse invariant mapping, bilinear approximation, and matched poles and zeros [24,25].The zero-order hold and first-order hold methods are generally accurate for systems driven by smooth inputs.For the PWM modulation system, the impulse invariant mapping and the bilinear transform methods are used.The impulse invariant mapping matches the discretized impulse response with that of the continuous time system.Note that the impulse responses match and the frequency responses do not match, however, because of the scaling factor   , the sample time.Although the impulse invariant transform is ideal when you are interested in matching the impulse response, it may not be a good choice if you are interested in matching the frequency response of the continuous system, because it is susceptible to aliasing.As the sampling time increases, you can see the effects of aliasing.In general, if you are interested in matching the frequency response of the continuous system, a bilinear transform (such as Tustin approximation) is a better choice.So the bilinear transform method is selected here.Choosing the right sample time involves many factors, including the performance, the fastest time constant, and the running speed.With the sample time   = 5 ms, the controller by bilinear transform is shown in (32).The discrete system is stable:

The Experiment
First the control hardware is introduced.Then the experiments are carried out, including two aspects, the time domain response and the frequency domain experiments.At last the analysis is given out.5.1.The Hardware Controler for 4WS.The configuration of the hardware controller for 4WS is shown in Figure 9.In the experimental implementation, the required time for the controller to execute all the tasks is approximately 60 ms during one period.Designer always moved to a DSP when the application requires a great deal of mathematical computations since the DSPs perform calculations faster than the single chip controller.The core processor of the experimental system is a DSP TMS320LF2407 with clock frequency of 40 MHz, produced by TI Company.The servo valve is driven by a voltage-fed PWM converter with a switching frequency of 200 Hz.The angles of each wheel are detected by the Hall-Effect device and the detected analogue signals are converted to digital values by using the A/D converter with a 12bit resolution.The discrete controller is programmed by C language and loaded into the DSP.

The Time Domain Experiments.
The standard velocity is 80 km/h, the acceleration velocity of the steering wheel is as quick as possible, and it is better more than 200 degree/sec.The driver turned the steering wheel to the left 80 degrees and then to the right 80 degrees.Then return to 0 degrees.Repeat

Figure 3 :
Figure 3: The diagram of the 4WS vehicle.

Figure 4 :
Figure 4: The vehicle response under different velocity.

Figure 5 :
Figure 5: The yaw velocity gain of the vehicle.

Figure 6 :
Figure 6: The diagram of hydraulic system after retrofitting.

Figure 7 :
Figure 7: Nonlinear characteristic flow gain of the proportional valve.

Figure 9 :
Figure 9: The control diagram based on DSP.
1  + ( +  1 )   2 ( 1  +  11 ) The Proposed  2 / ∞ Controller for 4WS.For the 4WS vehicle,  1 is defined as the yaw velocity of the vehicle . 2 is defined as the tire steering angle  FL , which is ratio to the sensor voltage.The sensor voltage is decided by the angular displacement of the steering wheel.Now define the performance index for robustness as the  ∞ norm of the sensitivity function from the parameter perturbation w to the control output  1 .Define the performance index for LQG as the  2 norm of the transfer function from the control input  to the control output  2 .Then the state vectors [ 1 ,  2 ] = [, ,  1 ,  2 ] are defined as the angles of the whole vehicle and the pressures of the hydraulic route, respectively.All the four state variables can be measured directly by physical sensor.