LQR Feedback Controller For An Active Suspension System Of A Rail Vehicle Save
Modelling and control of a railway vehicle active suspension system to improve the reduction in vibrations in the vehicle.
An LQR feedback controller for an active suspension system of a rail vehicle
Modelling and control of a railway vehicle active suspension system to improve the reduction in vibrations in the vehicle.
What is this?
For high-speed railway vehicles, passenger safety is the top priority when boarding a railway vehicle but also ride comfort and the quality of a journey. However, when moving from one location to the next, railway vehicles usually encounter various vibrations and shocks from the tracks, and prolonged exposure to low-frequency vibrations can be harmful and cause health effects for passengers such as discomfort, fatigue and irritation. For actual railway vehicle, the different disturbances and vibrations result in decreased durability of the railway car, which increases maintenance costs and reduces its performance. For the vehicle drivers, low-frequency vibrations are also an issue, but railway handling is also affected by vibrations caused by the railway disturbance, and good handling always demands stiff suspensions in the system to control the vehicle. There is always a compromise between the railway handling of the vehicle and the passengers ride quality when it comes to the conventional suspension system, which is the passive suspension system. To alleviate these problems, this project introduces and an active suspension system with an electrohydraulic actuator to apply a controlled force to control the motion of the railway vehicle body stabilising the system and reducing the vibrations thus, providing a better ride quality for passengers. The control strategy used to control the actuators is a Linear Quadratic Regulator (LQR) feedback controller.
Aims and Objectives
The main aim of this dissertation is to develop a reliable active suspension system for railway vehicle applications that can alienate the harmful low-frequency vibrations caused by railway disturbances and transmitted through the suspension systems and railway vehicle body to the passengers and driver, as well as provide good handling of the vehicle. To achieve this aim, the following objectives must be met:
1. The development of a reliable control strategy for the active suspension system using the LQR method will be laid out, which will also be cost-effective regarding the actuator force.
2. Develop a state-space model of the suspension system using a quarter model to simulate both the passive and active suspension systems and find appropriate railway vehicle model parameters.
3. Investigate the parameters of the active suspension system that affect the ride comfort and rail handling, which will be it suspension displacement or travel, the railway vehicles body acceleration was sprung mass acceleration and the vehicle’s wheel deflection. The body displacements of the vehicle will also be analysed.
4. Evaluate the performance of both the passive and active suspension system through simulation tests and compare results and analyse the performance change of including an LQR feedback controller.
5. Carry out a robustness assessment to see if the control strategy is robust in different operating conditions such as a different rail disturbance and vehicle parameters such as an increase in the railcar body’s mass.
Method
The LQR controller designed will enable the active suspension system to dynamically change in different operating conditions compared to the passive suspensions without compromising on the rail handling ability and ride comfort. The electrohydraulic actuator is used for the active suspension system, and an LQR control strategy has been chosen. MATLAB will be used to develop and analyse the state-space model for both the passive and active systems, and a Simulink will be used to create block diagrams and simulate the systems.
MATLAB/Simulink
MATLAB software will be used to develop the state-space model of the railway suspension system and the LQR controller for the active suspension system; it will also house the model parameters which will be used in conjunction with Simulink. For theoretical calculations, MATLAB is a popular software when it comes to the analysis of experimental data. The MATLAB environment will be of benefit when it comes to analysing the suspension parameters inside the simulation model. The results obtained using MATLAB will be used for comparison and discussion. Simulink is a software package for simulating, analysing and modelling dynamic systems. The software can be utilised to analyse the behaviour of real-world dynamic systems and will be used in this dissertation to analyse the quarter-car model suspension system for the railway vehicle.
MATHEMATICAL MODEL
When designing a control system, the establishment of the mathematical model of the system is needed. Like most engineering control systems, a mathematical model will typically rely on known laws which will then be derived, Newton second law of motion will be the law focus for the quarter car suspension model. The mathematical model's primary purpose is to provide an equation or equations describing how the system behaves.
QUARTER CAR MODEL
In order to analyse the parameters related to the suspension system, a simplified quarter car model shown in Figure 1 (a) was utilised for analysis. The quarter car model was chosen due to it being the most common and straightforward of the vehicle dynamic vibration models, for the analysis of the vertical vibration caused by the railway disturbance. The railway vehicles mass is divided into two, the railway vehicle body which is referred to as the sprung mass and the railway vehicles bogie which is referred to as the unsprung mass. Connected in between the sprung and unsprung and the track are suspension springs and dampers. Both the transversal and longitudinal deflections considered insignificant in comparison to the vertical deflections of the suspension system. For the passive suspension system, the actuator force will not be considered due to not having a control element shown in Figure 1 (b).
Figure 1: A quarter model for the active suspension system (a) and the passive suspension system (b)
Where:
QUARTER RAIL VEHICLE MODEL DESIGN PARAMETERS
The input parameter values of both the passive and active suspension systems for the railcar model which will be used for analysis and produce results are given in Table 1.
| Parameters | Values |
|---|---|
| 5333 kg | |
| 906.5 kg | |
| 430,000 N/m | |
| 2,440000 N/m | |
| 20,000 sec/m | |
| 40,000 sec/m |
Table 1: Quarter rail model design parameters
EQUATION OF MOTION
The equations of motion considered for both the passive and the active suspension systems are derived using Newton second law of motion.
Equation 1
Transposed for the acceleration, the equation now becomes:
Equation 2
So, from the figure above, Newton's law of motion, the dynamic equation of the active suspension system and with the forces acting on the sprung mass, the following equation is given:
Equation 3
Transposed for the sprung mass acceleration, the equation now becomes:
Equation 4
The forces acting on the unsprung mass are the following:
Equation 5
Transposed for the unsprung mass acceleration, the equation now becomes:
Equation 6
STATE SPACE REPRESENTATION
The general state-space representation is given by the following:
Equation 7
Where:
A state-space model describing the active suspension system will be created using the two equations of motion derived earlier.
Let the state variables representing the system be:
Where:
The input u into the system will be the railway disturbance velocity Z ̇_r and the actuator input F_c.
The desired outputs y from the system are the suspension travel Z_s-Z_us and the railway vehicles body acceleration Z ̈_s.
Using the equations of motion found in equations 3, 4, 5 and 6, the state-space model of the active suspension system can easily be obtained and written in the form given in equation 7, which will then be written matrix form shown below:
Equation 8
STATE-SPACE CONTROLLABILITY
There are many problems when it comes to control systems such as stabilising unstable systems using feedback control, and controllability is an important factor when attempting to deal with these problems. If the system consists of a single control input, u, that can move a system state around in its entire configuration to another state; then the system can be controlled. For a linear time-invariant state system to be controllable, the controllability matrix P must be full rank and equalled the n number of states in the system.
Equation 9
Where:
FULL STATE VARIABLE FEEDBACK CONTROL
A full state feedback controller, which is also referred to as a pole placement controller, is shown in Figure 2 and is an optimal solution for achieving the desired pole locations of a closed-loop system, as it allows all state variables to be known to the controller all times and available for feedback.
The state space matrix is the plant with each state variable being fed back to the control input ,u, through the gain K represented by feedback vector, which can be adjusted to reach the desired closed-loop pole values. So, the system input is given by:
Equation 10
Substituting this into equation 7 will give the state space equation for the closed-loop system:
Equation 11
Figure 2: A full-state feedback block diagram
LINEAR QUADRATIC REGULATOR (LQR)
A popular type of state feedback control that provides a systematic way of determining the control gain, K, is the LQR controller. The LQR approach will be used for the controller design for the active suspension system, as it is one of more classic control options for linear MIMO time-invariant systems and is simple enough to design. The advantages of using an LQR controller is that the factors affecting the performance index can be weighted depending on the desired result of the individual. Improving the rail handling ability in the ride comfort of the quarter rail model will be the focus of the LQR approach.
The primary function of an LQR controller is to minimise the cost function, J, the performance index seen in equation 12, and calculate the optimal gain K:
Equation 12
Where:
Equation 13
The performance index or quadratic cost function J must be minimised by adjusting both the weighting Q and R matrices, where Q is a diagonal positive definite and R is a positive constant. The desired closed-loop performance is then obtained by tuning the weighting matrices, by penalising bad performance by adjusting the Q matrix or penalising actuator effort by adjusting the R matrix until suitable results regarding the cost function are reached for the plant.
Based on equation 10, the feedback regulator and solution and to the performance index in is given by:
Equation 14
Meaning that both the A and B matrices must correspond to the actuator control force in the feedback regulator, giving the matrices shown in equation 15:
Equation 15
LQR CONTROLLER MATLAB IMPLEMENTATION
Since it has been determined that the A and B matrices are both controllable, an LQR controller can now be designed and be implemented into MATLAB. the system matrix A, the input matrix B, the output matrix C and the feedforward matrix D must first be represented in MATLAB shown below:
A = [ 0 1 0 -1 ;
-ks/ms -bs/ms 0 bs/ms;
0 0 0 1;
ks/mus bs/mus -kus/mus -(bs+bus)/mus];
B = [0 0 ;
0 1/ms ;
-1 0 ;
bus/mus -1/mus ];
C = [ 1 0 0 0 ;
-ks/ms -bs/ms 0 bs/ms ];
D = [0 0;
0 0;
0 0;
0 0;
0 0;
0 1/ms];
The controllability of the open-loop system has already been determined so the weighting matrices Q and R of the quadratic performance index can now be tuned for and obtained in MATLAB. The performance measures that were of relative importance and needed more of a focus, were of the states related to the suspension travel and the railway vehicle body acceleration. After trial and error of changing the nonzero elements in the Q matrix and the input weighting of the R matrix, the final weighting matrices Q and R are shown below:
Equation 16
By entering the Q and R weighting matrices, the feedback gain vector K was obtained using the following MATLAB command:
lqr( A, B(:,2), Q, R )
Q = diag([1760*10^6, 11.6*10^6, 1, 1]);
R = 0.01;
K = lqr( A, B(:,2), Q, R )
K =
1.0e+05 *
1.7075 0.3637 0.7759 0.0052
So, the feedback gain vector obtained is:
Equation 17
SIMULINK MODEL
Both the active and passive suspension systems can now be modelled in Simulink using the state-space model and the LQR controller. The block diagrams below describe the modelling process of the active and passive suspension models and the function of the various subsystems.
The first step when designing the system was to create the rail disturbance that will excite both the passive and active suspensions shown in Figure 3. The first rail disturbance (RD1) is in the form of a simple step input with a step disturbance height of 0.06m. The step block gives a step input between two definable values at a specific time, for the simulated model a step time of 0 seconds was chosen with an initial value of 0 also selected as shown in Figure 13.
The second rail disturbance (RD2), also shown in Figure 3, is presented in the form of a pulse width modulation, attributing square waved impulses at six-second intervals. The amplitude disturbance height chosen was 0.1m with a pulse width of 50% of the period, which is three seconds and a phase delay of 0.1 seconds. The second disturbance was chosen to be the worst-case scenario for the active suspension system and is used in the robustness test, which will be discussed later in this dissertation in the results and discussion section.
The next step was to create a first-order filter for the rail disturbances and derivative block to get their velocities. For railway vehicle, the railway disturbance would not necessarily be a sharp step disturbance but rather a smoother edge disturbance with some angle present, which is why a filter will be used to smooth out the disturbance. A first-order filter in the form of a transfer function was used to smooth out the results generated from both the step input in RD1 and the pulse width generator for RD2, as their derivatives will be taken later and to ensure the simulation is producing accurate results.
Figure 3: Rail disturbance Simulink model
To model the open-loop plant, a state-space block will be used, as shown in Figure 4. The state-space block is used to implement the suspension systems behaviour, which was defined in equation 7.
Figure 4: Simulink model of the passive suspension system
Both the state-space models for the active and passive suspension systems are all made into two subsystems, as shown below in Figure 5.
Figure 5: Simulink model of both the active suspension system and the passive suspension system
The active suspension system, on the other hand, has a control element, and all the state variables outputted from the suspension system is feedback to the control input through the gain K, generating the actuator force to stabilise the system shown in Figure 6.
Figure 6:A Simulink model inputting the actuator control force into the active suspension subsystem
The results for the suspension travel in the sprung mass acceleration are inputted into a workspace through workspace blocks that write signal data into the workspace module in MATLAB.
In order to get the displacements for the sprung mass and the unsprung mass of the railway vehicle for the active suspension system, calculations were carried out. A summation of the suspension travel, the wheel deflection and the rail disturbance displacement were calculated.
Results and Analysis
Analysis of the results of the passive suspension system of the quarter car model and then analyse the response of the active suspension system with the state space controller. The impact of the LQR controller will also be assessed to see the significance of its impact on the system. When discussing the performance of the two suspension systems the, ride quality and the railcar handling will be monitored, with the parameters of focus being the railcar body acceleration, the travel of the suspension system and the wheel deflection as well as the body displacements of the quarter car model.
When comparing the two systems, the criterions that will be compared are the following;
• The Settling Time: it is a measured period the system oscillates until it reaches a specific limit where it begins to match the desired value.
• The Rise Time: how fast system will respond and the time for the system response to reach a specific percentage within the specified value.
• Overshoot: a measure of how the system starts to move away from the desired response at the beginning.
• Steady-State Error: the measure of the final error from the desired response.
NOTE: The mathematical quarter railcar model for the suspension system derived in the methodology is used, and the parameters also mentioned in methodology will be utilised for the generation of the results. The quarter railcar model is assumed to be at a constant speed along the railway within 10 seconds with a disturbance with a rail disturbance height of 0.06m.
Figure 7 shows the response of the force generated by the actuator in the active suspension controller once the step input rail disturbance is met. What is noticed is that the force being generated from the actuator is being applied in the opposite direction of the sprung mass or railcar body and overshoots, this is due to the tendency of the sprung mass to move in the upward direction. The figure shows that the response of the actuator is stable and gives an acceptable response to the rail disturbance in the form of a step input with a value of 0.06m.
Figure 7: Time response of the force generated from the actuator
Figure 8 shows the effect of adding a feedback LQR controller has on the system in regards to suspension travel when compared to the passive suspension in Figure 35, a similar overshoot of 0.05 m is present in contrast to the passive suspensions. Compared to the passive system, however, the active suspensions see a significant reduction in the settling time at 1.86 seconds which is a 50% decrease of the response rate for the passive suspensions, therefore giving a faster response and improving the ride comfort and lowering the vibrations to the passengers.
Figure 8: Time response of the suspension travel for the active suspension system
Figure 9 shows the effects of the acceleration in the railcar body after an LQR controller is implemented. As was shown in Figure 36 for the suspension travel of the active suspension system, a similar overshoot of 7.5 m/s2 is present when compared to the response of the passive suspension in Figure 37. However, the system demonstrates a 47% improvement with a settling time of 1.75 seconds, thus improving the ride comfort and road handling.
Figure 9: Time response of the sprung mass acceleration for the active suspension system
The results for when the LQR controller is applied for the active suspension system is shown below in Figure 10. Compared to the passive suspension, the active suspension experiences a similar overshoot in wheel deflection of 0.0255m. However, like the previous results for the active suspension system, the settling time has improved noticeably at 1.74 seconds, a 42% improvement for the response rate. While the deflection value at the starting point was slightly higher than the passive suspensions when the rail disturbance is applied, the vibration of the wheel deflection was significantly reduced over the next 1.74 seconds with system stability being reached faster. Both the amplitude and the settling time were both optimised with the LQR controller implemented, thus minimising vibrations and preventing irregular movements of the railcar on the railway when in contrast to the passive suspension which had a longer settling time resulting in more considerable deflections and less quality rail handling.
Figure 10: Time response of the wheel deflection for the active suspension system
Figure 11 shown below is for the active suspensions and the comparison between the body displacements of the sprung mass and unsprung mass, with the displacement of the rail disturbance. It is shown that with the LQR control method applied, the output response of both the railcar body and the bogie were able to reach a steady-state close to the same timeframe. The overshoot for the sprung mass is slightly lower at 0.0143, and the unsprung mass does not change significally when compared to the passive suspension (Figure 12), but the amplitude and settling time has reduced considerably to 1.85 seconds and 1.42 seconds for the unsprung mass, which is an improvement of 51% and 43% respectively . For the active suspension system with LQR control implemented, significant improvement is demonstrated the system able to reduce the railway vehicles body displacement in comparison to the passive suspension, which in turn provides good ride comfort for the passengers.
Figure 11: Time response for the body displacements for the active suspension system
Figure 12: Time response of the body displacements for the passive suspension system
Robustness Assessment
After the design and testing of the LQR controller for the active suspension system, a robustness test is carried out. The primary reason for this test is that when using an LQR control for a system, there is no guarantee that the stability margins will stay in place if the parameters of the system are modified. In order to ensure that a change in the parameters does not result in a considerable decline in the performance of the active suspension system, performing a robustness assessment is essential. The test also demonstrated that the system could have perhaps some flexibility in the changed parameters.
The parameters of the modified values are shown in Table 2 below. The variation of parameters is possibly on the more severe side, and perhaps not conceivable in practice, but this test is meant to test the robustness of the system and for the worst-case scenario.
A square signal amplitude representing a railway disturbance height of 0.1m with a period of six seconds was used to simulate the rail disturbance for this robustness test, as a worst-case scenario.
| Modified Quarter rail vehicle design parameters | |
|---|---|
| Parameters | Values |
| 8532.8 kg | |
| 906.5 kg | |
| 361,200 N/m | |
| 2,440000 N/m | |
| 14,600 sec/m | |
| 40,000 sec/m |
Table 2: Modified Quarter rail vehicle design parameters
The feedback gain vector, K, obtained for this robustness assessment is:
Equation 18
The results of the robustness test are shown below:
Summary
The test demonstrated very little change in the performance in terms of ride comfort, with the only significant change being in the force results for the actuator showing that 70% increases in force is needed in order to control the system and produce similar results as the LQR controller with the initial parameters. Moreover, it shows that with the modified parameters selected, the active suspension system, with LQR control strategy in place, demonstrates its robustness and can adapt relatively well with variations of parameters chosen for robustness check.
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