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The following 3 web pages describe the system model of the linear actuator in mathematical terms. I must emphasize now that this web site is a DIY version of my Master's thesis. This presentation is more tailored for hobbyists with a basic background in electronics, computer programming, and embedded systems. However, I needed to have an advanced understanding of control theory and automation before I was able to complete this project. I am going to simplify some of the more theoretical concepts and equations but it will help if you are well versed in the following math and engineering subjects:
- Calculus
- Linear Algebra
- Differential equations
- Probability and Statistics
- Control Systems
If you have not had exposure to these subjects, you might briefly skim this section and skip to the formulas at the end, which are implemented in the Arduino sketch. I will be putting plenty of links to various web sites that will provide further learning on these concepts.
Before we begin, I would like to define some basic control systems terms:
- The plant is the system that is to be automatically controlled. It could be mechanical, chemical, electronic, etc. In this case, it is the solenoids and payload.
- The controller or compensator is the system that will be connected to the plant to automatically control it and correct its shortcomings.
- The state is a collection of various mathematical quantities that can be used to describe the control system.
- An observer is one half of the compensator. It is connected to the output of the plant. More often then not, we will not be able to measure the entire state of the plant. In this control system, the payload position, velocity, and acceleration are the 3 state variables. However, the Metro M4 will only be able to measure the payload position using the linear potentiometer. An observer can use the available measurements in order to estimate the other state variables.
- A control law is the other half of the compensator. It is connected to the input of the plant. The control law will use the estimated state variables to calculate the correct input to exert on the plant to produce stable, robust performance. In this control system, the input is the net force the solenoids should exert on the payload.
- An exogenous input is an input that cannot be controlled by the compensator. It can either be (1) a disturbance caused by nature, (2) a shortcoming in the construction of the plant, or (3) some sort of reference input set by an outside user. In this control system, the exogenous input will be the friction from the linear potentiometer and the reaction force from the springs attached to the payload.
For a more thorough discussion of control systems and automation, please see references [5] and [6]. These are the college text books used in the control systems courses at NJIT. There will be far more citations in the subsequent pages.
Figure 1: Top-Level View of System Model
Figure 1: Top-Level View of System Model
Figure 1 shows a top-level view of the entire system. Note that this signal-flow diagram was created using the Simulink package found in Matlab. Let's look at each module individually.
A. Plant Dynamics - This section is a mathematical modeling of the motion plant. This is a variant of the standard kinematic modeling of the movement of an object along a single axis.
B. Linearization Module - The controller demands that the motion plant be a linear time-invariant system. However, the solenoids have a nonlinear relationship between the drive current and the magnetic force. This module inverts the solenoids' nonlinear dynamics.
C. Compensator Design - This section discusses the compensator that controls the motion plant.
i. FSFB Control - This web page discusses the basics of the full-state feedback controller. This includes a discussion of stability.
ii. Continuous-Time Analysis - This web page appropriates the mathematical model stated in Plant Dynamics so that a copy of this model may be stored in the linear controller.
iii. Discrete-Time Analysis - Because the controller samples and processes data in discrete-time, we will need to discretize the mathematical model of the motion plant. Note that a lot of calculus is involved in the discretization process.
iv. Kalman Filter intro - This web page discusses the basics of the Kalman filter, the optimum observer equation, and the observation error used to correct the estimated state.
v. Calculating the Kalman gains - This web page discusses the iterative algorithm used to calculate the gain coefficients used in the Kalman filter. This includes a discussion of white noise.
vi. Gain Selection - This web page details some addition constraints used in calculating the gains for the Kalman filter and FSFB control law.
vii. Simulation - This web page discusses some nonideal factors that could impact performance.
[5] B. Friedland, Control System Design: An Introduction to State-Space Methods, Mineola, New York, Dover Publications, Inc., 1986,
[6] B. Friedland, Advanced Control System Design, Englewood Cliffs, NJ: Prentice Hall, Inc., 1996.
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