Fundamentals of Feedback Control using Microcontroller [Preparation]

Generally speaking, automatic control is broadly classified into Sequence control and Feedback control. Sequence control is a process that functions according to the conditions set in a combined or programmed process using relay circuits, etc. It is also called open-loop control or open-loop control.

A PLC (Programmable Logic Controller) that controls industrial machinery is a typical sequence control. Familiar examples include traffic signals, which function by means of programs that use timers and conditional branching in sequence control.

In contrast, Feedback control refers to automatic control in which the output of the controlled object is fed back (returned to the input side) to make corrective actions to bring the measured value by a sensor, etc. closer to a set target value. It is also called closed-loop control. A familiar example is the temperature setting of an air conditioner. Driving a car can also be considered feedback control, since the brain makes judgments based on visual and other information and controls the car.

Instead of a person controlling the autocruise function of an automobile by sensing the speed from the meter or engine noise, an on-board sensor detects the speed and provides feedback to the engine to adjust the power output to the engine.

Feedback control has advantages that sequence control does not have, but because it uses feedback information, it can become very unstable if there is a defect in the way it is corrected. In other words, if the signals and information to be corrected are always close to the target values, the system is stable, but if they move away from the target values due to various factors (disturbances), the system may become unstable in a so-called vicious circle due to the closed-loop nature of the system.

Control theory is academically classified into Classical control and Modern control.
Classical control is simply put, a control theory that evaluates and analyzes in the frequency domain (complex number space) based on a transfer function model. On the other hand, Modern control theory is a control theory that evaluates and analyzes in the time domain (real number space) based on a state space model.

These are not to be taken as old or new classifications as the names imply, but rather different approaches to control, both of which have their advantages and disadvantages. Other derivatives include Robust control and Adaptive control, which are control methods that have evolved from classical and modern approaches.

This site covers what is called Classical control, which is analyzed in the frequency domain (complex number space). In addition to the fact that I have a background in electrical engineering, the idea of using transfer functions as a base is to construct a kind of special filter between inputs and outputs, and the fact that it can be designed physically, rather than mathematically, suits my nature.

The appearance is intuitive, as it can be constructed as a block diagram by a transfer function transformed into complex number space with complex number s as a variable. The point is complex numbers, but you just need to get used to complex numbers as a convenient tool rather than treating them as a mathematical concept. The great advantage of the classical control method based on transfer functions is that the modeling parameters can be done with a certain degree of accuracy, but not so strictly because of the filter concept.

Modern control theory is based on analysis in the time domain, which is rather mathematical and involves modeling the control object in terms of differential equations called Equations of state. Although it is difficult to solve differential equations, nowadays, this is not a problem because the computer can analyze the equations instead of human hard works. Compared to classical control, the modeling parameters require a certain degree of precision. Since the state variables used in the state equation are handled, the system can be analyzed as a multi-input, multi-output system.

To summarize the characteristics of the two, classical control theory is based on transfer functions in complex space in the frequency domain for design evaluations such as system stabilization, while modern control theory is based on equations of state, which are differential equations, for mathematical analysis and design in actual time space.
Classical control theory is easier to understand if the emphasis is on physical evaluation, while modern control theory is better suited for advanced analysis of internal parameters with an emphasis on calculation.

For example, in the case of a cart-type inverted pendulum, the angle of the pendulum and the position of the cart are used as state variables to control the angle of the output pendulum by operating the actuator's torque, etc. The relationship between input and output is complicated, and the characteristics cannot be easily predicted with a single transfer function.

In this example, there are many variables from input to output, and modern control theory is well suited for capturing their states. With appropriate state feedback, the system can be stabilized very easily. Classical control theory using transfer functions can also be used, but it seems to be a little too difficult.

Modern control theory has some advantages that classical control theory does not have, but it is more of a mathematical analysis, so if you are a beginner in control and good at mathematical interpretation, you can learn from modern control theory, but if you want to focus on physical interpretation, it would be better to start from classical control theory. I think it is a good idea. The key is not so difficult.

In my personal view, the best way to utilize the advantages of both modern and classical control theory is not to divide them in two, but to merge them according to the application. For example, modern control theory is utilized to optimally manage and control a large, complex system as a whole, while classical control theory is utilized to control the individual parts used within that system to ensure performance.

Many control theory explanations often make extensive use of mathematical formulas, but this site focuses on the minimum necessary to realize motion control using DC motors, but it is still sufficient for practical use.

In mathematics, an imaginary number is simply a number j that, when squared, becomes j²=-1 (in electrical and control engineering, j is used to distinguish it from the current i), however, when this concept of imaginary numbers is applied to physics as a tool, it can be turned into something very useful.

A complex number Z is a combination of the real part a and the imaginary part jb, Z=a+jb. In real number space, for example, current i(t) is a scalar quantity (magnitude itself) that only increases or decreases with parameter t (time), whereas in complex number space, it can be expressed as a vector quantity (having magnitude and phase) with Z=a+jb.

In other words, Z=a+jb means that the imaginary part b is on the imaginary axis counterclockwise by 90° relative to a on the real axis. Multiplying a complex number by an imaginary number j has the property that the vector of that complex number will be in a position rotated 90° counterclockwise around the origin.

Taking advantage of this property, electric circuits and equations of motion can be converted to complex number space, which is much easier to analyze and handle than when expressed in real number space (the world of differential equations). Incidentally, the concept of complex numbers is not used in physics up to high school, but first appears in college.

Complex numbers used in engineering should be thought of as a useful tool that makes use of their properties rather than as a mathematical essential.

In electrical engineering, the complex number plane is used to analyze circuits. A DC circuit with only resistance R can be handled using only real numbers, which can be understood intuitively by anyone because Ohm's law holds true. On the other hand, in the case of an AC circuit combined with an inductor L and a capacitor C, the operation cannot be explained simply by Ohm's law of real numbers.

This is because the phase of the voltage across the coil L and capacitor C with respect to the AC current (sinusoidal) flowing through them is advanced in the coil and delayed in the capacitor C. Therefore, using the complex number plane, which can include not only the effective values of current and voltage but also phase information, makes it possible to handle the circuit as a vector and simplifies it.

The phase shift of the inductor L and capacitor C can be expressed by the following equation. This formula is valid for time-varying circuits. We can see that for a sinusoidal alternating current flowing through a coil and a capacitor, the phase of the voltage on each end advances by 90° (π/2) for the coil and delays by 90° for the capacitor, respectively. I will represent this in complex number space.

In so-called classical control, which treats the transfer function between input and output as a transfer function, the input is the target set value and the output is the physical quantity to be controlled, and the entire system including the transfer function is transformed into complex number space in the frequency domain for analysis.

To create a transfer function, the control object must first be modeled in mathematical form. This is true for both classical and modern control theory. Whether it is an electric circuit or the equations of motion of dynamics, modeling in the real time domain results in a differential equation with time as a parameter.

Differential equations must be solved to analyze the control target, but the more complex the control object, the more complicated it is to solve and the more difficult it is to grasp the actual situation because it is a numerical calculation. In addition, when complex numbers are included in the solution, it is difficult to understand how the equation behaves because it is an abstraction that does not actually exist, even if it is expressed as a function of time t.

Therefore, one means of solving differential equations is the Laplace transform, which utilizes the complex number s. This transform transforms difficult differential equations into algebraic equations, and by representing the parameters in complex number space, the behavior can be intuitively grasped. By actually modeling the control object and Laplace transforming it, I will try to create a transfer function in the frequency domain for a model of differential equations.

In these examples, a model in real time space with differential equations and a model in complex number space were obtained. In classical control theory, all of these models are analyzed in complex number space, so if the model is a differential equation in real time space, it is transformed using a tool called Laplace transform to represent it as a transfer function in complex number space.

The purpose of Laplace transform in control engineering is to transform differential equations in real time t-space into algebraic equations in complex s-space, which can be easily handled for computation, analysis, and evaluation. The variable s in the Laplace transform is the Laplace operator, which is equivalent to the differential operator, replacing jω with s. After Laplace transform, the mathematical model becomes a block of transfer functions composed of polynomial equations, which can be used to verify input-output and frequency characteristics, and to determine stability.

The Laplace transform itself can be used without understanding the mathematical content, and it is acceptable to separate it as a means of analysis only. If you want to deepen your understanding of the mathematical meaning of the Laplace transform after becoming familiar with it, you may follow the content of the Laplace transform without getting too caught up in the mathematical formulas at first.

After design evaluation of the transfer function transformed into complex space by Laplace transform, the final step is to convert it back to a time function by inverse Laplace transform to check the actual behavior. If the inverse Laplace transform is performed by manual calculation and returned to the time function, it can be evaluated by simulation using Microsoft Excel, etc. However, nowadays, simulation applications can be used to perform the final evaluation without doing any mathematical work.

From a practical standpoint, simulation using manual formulas is convenient and reliable for systems with at most a 2nd-order lag (second-order delay), and for more complex systems, simulation applications are more convenient and reliable. In this case, you don't have to be very conscious of the mathematical part to use it, and the user can concentrate on the essential aspects of the control. In other words, you can design the control with awareness only in the s-space of complex numbers.

In the Laplace transform from the time function f(t) to the complex function F(s), f(t) is a function of time t, whereas F(s) after the Laplace transform is a function of the complex number s. In electric circuit modeling(Model1), the differential equation (1) is replaced by a complex function with "s" as the derivative term, and in (2), which has already been created in complex number space, "jω" is replaced by "s", then the result is the same as the Laplace transform, and the both of them match.

There are several Laplace transform formulas, but they are not something you should memorize unless it is for an exam or research. For now, there are a limited number of Laplace transforms that you need to know in order to use them in motion control.

The Laplace transform of differential equations in electrical and mechanical systems can be modeled using 1st- and 2nd-order derivatives and integrals, and the properties of 1st- and 2nd-order lag transfer functions should be understood as the basic forms of transfer functions. 1st and 2nd order refers to the degree of the transfer function denominator polynomial.

In real-time space simulations, the following formulas are often evaluated using step or ramp signals as input signals, so it is good to know the following formulas.

The inverse Laplace transform is used to turn the transfer function G(s) into a real-time space time function g(t) for simulation.

For manual calculation, for example, if the input signal u(t) is a step signal, inverse Laplace transform is performed as G(s)U(s)=G(s)/s, or G(s)/s² for a ramp function. The step response time functions for 1st-order lag and 2nd-order lag systems are described for manual simulation.

The mathematical model created by modeling the control object is converted to a model in complex s-space by Laplace transform to represent the control object as a transfer function in a block diagram. In a block diagram in s-space, the signal transfer from input to output is classified by function like a flowchart and expressed in a visually easy-to-understand manner.