This time, let's talk about vibration control.
1. History of Control
The history of control systems is long, with servo control and process control being considered as early as the 1950s. Control systems can be broadly divided into classical control and modern control.
Classical control theory is a design method that primarily emphasizes transient characteristics, steady-state characteristics, and steady-state errors. Classical control aimed to stabilize the controlled system by shaping the open-loop frequency characteristics.
In contrast, there is a control method called modern control. With the increased speed of computer calculations, vibration control using modern control methods has become widely used, but vibration control using classical control methods is still widely used. One reason for this is that in modern control, the controlled object must be described using equations of motion and equations of state, and if the equations of state are unknown, it is difficult to design the control system.
In the 1960s, it became possible to construct multi-input, multi-output control systems. The aerospace industry was among the first to adopt this modern control theory. While classical control theory primarily deals with single-input, single-output systems, solving aircraft flight problems required a description of how to centrally control control systems that handle multiple inputs, such as rudder and flap thrust.
Modern control theory can handle multiple variables compared to classical control theory, making it an effective control theory for vibration control involving multi-degree-of-freedom systems.
In modern control theory, the aforementioned equations of motion can be handled uniformly by describing them in a state-space representation using matrices. However, control performance is affected by model errors in these state equations and noise such as disturbances, so care must be taken. Therefore, robust control methods that can respond to changes in the characteristics of the controlled system have been proposed. Subsequently, postmodern control theories such as H∞ control theory have emerged, and other rule-based control methods such as sliding mode control and fuzzy control for nonlinear systems have also been proposed.
Figure: Flow from classical control to modern control
Now, modern control theory includes pole placement theory, optimal regulator theory (LQR, LQG), and observer theory. A common characteristic of these is that their design domain is based on time-domain data.
Later, theories like H2 control theory emerged that performed design on the frequency axis.
The evaluation primarily involves using a quadratic evaluation function and determining the state feedback gain to minimize this function. Therefore, optimal design theory requires feedback across all state variables, and the accuracy of the model must be precisely understood. Furthermore, determining design parameters often involves trial and error, requiring repeated selection of the final control system. For example, in vibration control, targeting low-order vibrations results in a lower model order, but this can lead to instability due to the omission of higher-order modes. This is called spillover, and various methods have been reported to minimize its effects.
Subsequently, by incorporating design on the frequency axis, as in H2 control theory, it became possible to design robust control systems. Furthermore, in the design of the robust H∞ control system, design is performed on the frequency axis and using a singular value Bode diagram. This has made it possible to satisfy the desired performance while ensuring higher-order stability of the controlled system. We will now explain the overview in order.
2. Equations of Motion and Equations of State
In the previous lesson, we expressed the equations of motion using matrices. Now, let's introduce the following state variable vectors into this equation as follows:
X＝┏ x1’┓
┃ x2’┃
┃ x1 ┃ (7)
┗ x2 ┛
Let's describe it using a state equation.
Let's rearrange the equation of motion from last time, equation (1).
x1”−｛-k1／ m1・x1＋k1／m1・x2｝＝0
x2”−｛k1／m2・x1-（k1＋k2）／m2・x2｝＝0
┏ x1''┓ ┏0 0 -k1/m1 k1/m1 ┓ ┏ x1' ┓ ┏ 0 ┓
┃ x2‘’┃− ┃0 0 k1/m2 (k1+k2)/m2 ┃┃ x2 '┃＝┃ 0 ┃
┃ x1’ ┃ ┃1 0 0 0 ┃ ┃ X1 ┃ ┃ 0 ┃
┗ x2’ ┛ ┗0 1 0 0 ┛ ┗ X2 ┛ ┗ 0 ┛ (8)
this is
X‘― AX＝0 (9)
If a disturbance U is input here,
X’＝AX＋bU (10)
b＝ ┏ 1 ┓
┃ 0 ┃
┃ 0 ┃
┗ 0 ┛
This is how it is expressed.
Next, define the observation matrix C, and let the output be Y.
Y = CX + d (d: disturbance) (11)
Equations (10) and (11) can be represented as a pair in the equation of state.
■Supplement
○ Definitions of state variables, transfer functions, and frequency response functions
Assume that a certain shape receives an input that changes with time, frequency, etc., and outputs some kind of signal quantity in a corresponding shape. State variables are quantities that determine the state of the shape, such as displacement and velocity. In vibration engineering, excitation force is often used as the input, and displacement, velocity, and acceleration are often used as the output (response), and the ratio of the input to the output is called the transfer function. A transfer function defined as a function of angular frequency or frequency as the independent variable is specifically called the frequency response function (FRF). Supplementary source: "Introduction to Modal Analysis" by Akio Nagamatsu
(Excerpt from the email newsletter issued on March 18, 2004)
