Electromechanical Systems Dynamics

DingTalk’s stupid bug still exists, damn.

Lec 1. Introduction to system dynamics

Brief introduction, seems nothing special. Some phenomenon and corresponding results。
Role of simulation in development work.
Some modeling examples and life cycle stages.

Lec 2. Mathematic modelling of dynamic systems

Static

Output determined only by current input, reacts instantaneously
Relationship does not change
Relationship is represented by an algebraic equation

Dynamic

Output takes time to react;
Relationship changes with time, depends on past inputs and initial conditions;
Relationship is represented by a differential equation

Ways of modelling of dynamic systems

Linear and non linear, which can be transformed by linearization in some quilibirium points.

Recall of Laplace transformation

Transfer functions

Linear dynamic system has at least one input and at least one output which connected by linear differential equations.
It is also called linear time invariant (LTI) system

Lec 3. Typical dynamic blocks

State-space modelling

• A state-space model represents a system by a series of first-order differential state equation and algebraic output equations
• State-space models are numerically efficient to solve, can handle complex systems, allow for a more geometric understanding of dynamic systems and form the basis for much of modern control theory.

(Followed by a lot of examples in slides. Remember to check them)

Lec 4. Mathematic modelling of electrical systems dynamic

It is more like a generalization and summary to the previous lecture.

Algo for getting the state-space model:

1. Write component equations
2. Write topological equations
3. State-space model.
   1. Select the required number of equations for the model
   2. Express the remaining parameters in terms of the state vector parameters and substitute it to the equation.
   3. Express the output vector in terms of the state vector and substitute input vector
   4. Write the state-space model
4. Transfer fucntion
   1. Perform Laplace transformation for the state-space model (zero initial conditions)
   2. Express the output in terms of input
   3. Write the transfer function

Lec 5. Mathematic modelling of mechanical systems dynamic

Introduction

❖ Mechanical systems are modelled, basically, as systems with lumped-parameter elements;

❖ From the energy point of view mechanical systems can be described with dissipative elements, potential energy storage elements, kinetic energy storage elements;

❖ Forces and moments which drive mechanical system are typically applied by actuators but might represent other loads applied by the environment.

Basic elements

❖ Inertia elements

1. Mass moment of inertia
2. Each inertia element with independent motion needs its own differential equation
3. Stores kinetic energy

❖ Spring elements

1. Force (toeque) is generated to resist deflection
2. Translational and rotational springs
3. Store potential energy

❖ Damper elements

1. Force (torque) is generated to resist motion
2. Dashpots, friction, wind drag
3. Dissipate energy

Basic principles of mechanical model designing

1. Choose coordinates and orientation
2. Draw free-body diagrams for each inertia
3. Generate equations of motion using Newton’s 2nd law and Euler’s 2nd law
4. Double check.

Some examples

check them in slides.

Lec 6. Dynamical models of mechanical systems

Two types of dynamical model of mechanical system

Model for control system analysis

A model must correspond to the most of dynamical properties of real system

Model for control system synthesis

A model must correspond to dynamical properties of real system only in bandwidth of the control system

Some examples

Mainly about one-mass, two-mass and three-mass systems.

Resonant frequency.

$$\omega_{R1}=\sqrt{\frac{K_{12}(J_1+J_2)}{J_1J_2}}=\sqrt{\frac{K_{12}\gamma}{J_2}}$$

where $\gamma=\frac{J_1+J_2}{J_1}$ - mass ratio

Note that only in laplace domain is the notation’s unit in $rad\cdot s^{-1}$, while in real domain, it’s $\frac{1}{s}$

The denominator is the resonant part of the system, while the numerator is the anti-resonant part.

Lec 7. Mechanical systems dynamic with nonlinearities

Notations

• Dead band
• Saturation zone
• Dead band and Saturation zone
• Ideal relay
  • Ideal replay with deadband zone

• Relay with delay

  • Discontinuous switching delay in switching

• Relay with delay and deadband zone

Friction Nonlinearity

1. Static Friction
2. Dynamic Friction
3. Limiting Friction

The coulomb friction force model

Benson exponential friction model

Smooth coulomb friction model

Velocity-based friction model

Karnopp friction model

Dahl friction model

LuGre friction model

Elasto-plastic friction model

Stick-slip friction model

Gonthier friction model

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Lec 8. Analysis and control of linear systems

Introduction

Automatic control systems:

•give new properties to technological equipment;

•allow you to increase the speed of operating modes ;

•saving consumables and electricity;

•improving the quality of products due to the precision measuring subsystem;

•reducing the number of service personnel and reducing the requirements for their qualifications ;

•reducing the influence of subjective factors , in particular, the psychological characteristics of the operator;

•payroll savings .

Basic concepts

The dimension of the vector x is the order of the object.

Types of control

• Open loop control
• Disturbance control
• Feedback control

Criteria for the quality of control systems

The control system must be stable and also must have a certain quality of the transition process .

Errors in steady state conditions should not exceed admissible values.

Numerical characteristics are needed to evaluate the effectiveness of control systems.

Quality criteria make it possible to quantify various automatic control systems.

Classification of quality indicators

•Direct indicators. Determined directly from the transient response of the process.

•Root indicators. They are determined by the roots of the characteristic polynomial.

•Frequency indicators. Determined by frequency characteristics.

•Integral indicators. Obtained by integrating functions.

Direct indicator

• absolute error:

$\varepsilon(t)=g(t)-y(t)$

• relative error:

$\delta\varepsilon(t)=\frac{g(t)-y(t)}{g(t)}$

Also, there is maximum version of this two value.

• Regulation error

• Transient time

$$t_{tr}=t_1-t_0\\ t_0=begining\\ t_1=max(t),\varepsilon(t_1)\ge D\\ D=0.05\cdot\abs{y_0-y_{ss}}$$

• Overshoot

  $$\delta y_{max}=\abs{\frac{\triangle y_{max}}{y_{ss}-y_0}}$$

• Oscillation (fluctuation)

$$M=\frac{10^{(L(\omega_c)-L(0))/20}}{10^{L(0)/20}}$$

• Steady-state error

Limit value theorem：

$$\lim_{s\rightarrow\infty}(s\cdot F(s))=\lim_{s\rightarrow0}(f(t))=f(0)\\ \lim_{s\rightarrow0}(s\cdot F(s))=\lim_{s\rightarrow\infty}(f(t))=f(\infty)\\\\ E(s)=\frac{1}{1+W_1(s)W_2(s)}G(s)-\frac{W_2(s)}{1+W_1(s)W_2(s)}F(s)\\\\ E_G(s)=W_G(s)G(s), \ F(s)=0\\\\ \varepsilon_g(t)|_{t\rightarrow\infty}=\lim_{s\rightarrow0}(s\cdot E_G(s))=c_{g0}\cdot g_0\\\\ c_{g0}=W_G(s)|_{s=0}$$

The formulas are only valid if the stability condition is satisfied for F(s)

PID regulators

• P

  $$u(t)=K_p\cdot e(t)\\W_{reg}(s)=\frac{U(s)}{E(s)}=K_p$$

• I

  $$u(t)=K_i\int e(t)dt\\W_{reg}(s)=\frac{U(s)}{E(s)}=K_i\frac{1}{s}$$

• D

  $$u(t)=K_d\frac{de(t)}{dt}\\W_{reg}(s)=\frac{U(s)}{E(s)}=K_ds\\REAL\ ONE:\\W_{reg}=K_d\frac{s}{T_{\mu r}s+1}$$

Ziegler-Nichols method

In most cases, a properly tuned PID controller satisfies all system requirements. There are two methods.

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Fin.