8.4 Implementing state observer
8.4.3 Pole placement
By placing the poles in desired positions, we can obtain the desired response. The poles placed further to the left plane are more stable thus leading to better control dynamics.
The eigen values of the system was used as the starting point for the pole placement and was moved to the left to obtain faster response and stability.
The poles chosen are:
p1 = -15;
p2 = -116.88+ 008.08i;
p3 = -116.88 - 008.08i;
The plot of the poles placed are be seen in the figure below:
Figure 55:State space control pole placement
The feedback gain matrix K is calculated for the poles chosen. The matrix is defined as:
K =[ 8.6172 0.1468 0.0097]
The observer gains were set almost 5 times the dominant pole of the system op1 = -300;
op2 = -301;
op3 = -302;
The observer matrix L is given by:
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L= [886.832588895279 243738.404448511 11157510.8337544
]
The output of the system with state space control is described in graphs. The outputs from the simulation of the state model in Simulink and the output obtained when the same observer is used on the test rig are compared.
Figure 56: Position sensor output
The graph indicates the response good as the P controller. Placing the poles bit further to left would yield faster response. The observer acts as virtual sensor through which the system variables can be estimated if one or more variable is available for measurement. Let’s take a look at the position, velocity and acceleration estimates given by the observer.
VŠB – Technical University of Ostrava 60 Figure 57: (a) Position estimate, (b) Position output from state space model
The position estimate is similar to the output from the position sensor as seen from comparing both graphs.
Figure 58: (a) Estimated Velocity output from the drive, (b) Velocity estimate from the state space model
The observer estimation of velocity variable of the system is illustrated the Figure above.
The velocity signal is noise riddled but the dynamics looks good. The second part of the figure is the acceleration estimate from the state space model.
Figure 59: (a) Estimated Acceleration output, (b) Acceleration estimate from state space model
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The graph above indicates the dynamics of acceleration of the system (piston).
Figure 60: (a) Error dynamics of the drive and observer, (b) Error dynamics of the state space model and observer
The error is calculated to indicate how the observer reacts to the system changes i.e., how fast observer estimates converge to the actual state variable and track the state variable well in steady-state. Since it converges to 0 fast, the observer is fast enough to track the variables, which in our case is the position of the piston. The beginning overshoot is the observer trying to converge to real values of the system.
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Conclusion
Hydraulic systems are an integral part of various industries today. Due to their various advantages discussed so far, regarding power output and high speeds, it is not surprising to find hydraulic systems employed everywhere from robotics to agriculture. The popularity of hydraulics has drawn attention of control engineers to come up with innovative and robust control strategies which range from simple system identification to complex model-based control.
We reasoned, to understand and implement control strategies there is a need to analyze the components of the system to be used. Creation of mathematical model to simulate the working of the components of the systems is employed to test the control strategies that was chosen.
The eigen frequency identification method was employed to identify the critical parameters of the test rig and the simulation model due to its simplicity and effectiveness. The eigen frequency identification is effective as the dynamics of the hydraulic drive can be obtained as a result, which in turn helps in tuning of controllers. This method is simplistic in nature as it employs a single component with relay characteristics to obtain oscillations as the output, which can be used to calculate the critical parameters of the drive.
This identification method is first tested on the simulation model that is similar to the drive installed in the laboratory. Once this was a success, identification method was used on the test rig with the help of a multifunction I/O card for interfacing the PC and the drive.
The process of identifying the oscillating amplitude and respective time period is then automated to directly calculate the critical gain, eigen frequency and the damping ratio of the drive. Since the signal obtained from the drive was very noisy, a digital filter was employed and other signal processing tools from the MATLAB signal processing toolbox was employed too to precisely calculated the required parameters. The dependence of the frequency and damping ratio to the change in piston position was noted during the course of the project.
The program responsible for all the aforementioned functions and tasks is compiled and an Application is created in MATLAB App designer to make it all hassle free.
The obtained parameters are then used to tune the controller that was decided to be used.
The PID controller tuning was done by calculating the gains referring to the table. The P and PD controller gave the best control without any overshoot, which was not the case with PI and PID controller due to the integrating part inherent to the drive.
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The state space controller tuning was done by assuming the valve dynamics to be a proportional gain and the using the obtained eigen frequency to describe the dynamics of the cylinder.
Since not all the state variables are available to measure and control, an observer was designed to act as a virtual sensor and control the position of the piston.
The result of comparing the use of both the controllers, the state space control similar to the P control. The P controller, PD controller and the state space control with the observer provided suitable control which was fast and stable.
The use of observer allowed the estimate of all other state variables that were not available to measure.
Implementation of position control of the hydraulic test rig was accomplished by using digital controllers. There are provisions available to use the analog controller that come with this test kit.
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Acknowledgment
I would like to express my immense gratitude to my supervisor Prof. Ing. Petr Noskievič, CSc for his unceasing assistance of my thesis project, for his patience, motivation, and immense knowledge. His guidance was of great help from the beginning till the end of this thesis.
Besides my supervisor I would like to thank all the professors of the department, my family and friends for all the support and understanding.
This thesis has been supported by the project SP2019/51 Applied Research in the Area of Machine and Process Control supported by the Ministry of Education, Youth and Sports.