**EE114 System & Control Review Note**

**Author : Lance Cai （ **HANLIN.CAI.2021@MUMAIL.IE** ）**

**Update in 2022.04.07**

我是Lance, MIEC2020级本科生，RIDS专业

这份Review Guide是我在大二下学期复习EE114FZ[A]课程时整理的，主要内容来源于Lecture Slides（2021版），该学期的lecturer是Zhicong Chen先生

值得一提的是，本材料并不适用于系统的复习，仅仅只能提供一个复习的方向与思路，而这也是我的初衷：“授人以鱼不如授人以渔”——我希望引导大家去尽量完整地学会EE114的内容，同时掌握自主学习的方法。

**如果你发现了材料内容的重大缺陷与错误，非常期待你可以给我email（邮箱号已贴在最上方）。随信附上你的修改建议，我将不胜感激！另外，如果你需要其他的学习材料，可以到我的个人网站获取：**`www.mieclance.club`

最后的最后，祝你考试顺利、学业有成。我们高处见。

## EE114

- Introduction to Control Systems
- Introduction to Systems
- Modeling of Basic Systems
- Laplace Transforms & Transfer Functions
- Block Diagram Analysis
- System Analysis (Time Response) & Simulation
- Transient and Steady-state Characteristics
- Controller Design

## intro to control system

Open-loop control system

### Closed-loop feedback control system

- the system output can be made to follow a desired set point automatically.
- easier to achieve desired transient and steady-state response.
- less sensitive to external disturbances.
- less sensitive to variations in the parameters defining the process.

### Standard closed-loop control system

- The
*plant or process*is the system to be controlled - The
*sensor*is the device used to measure the output - The
*controller*is the device that outputs an appropriate control signal based on the difference between the measures output and the desired setpoint (or reference point). - The
*actuator*is the mechanism that changes the input to the plant. - The
*disturbance*represents any external factors that can affect the performance of the plant.

**Modeling, analysis, design and implementation**

## intro to Systems

*Categorizing systems*- Dynamical & Static
- Linear & Nonlinear
- Time-invariant & Time-varying
- Continuous & Discrete

*Mathematical modelling*

### Dynamical & Static

Dynamical —-> the current output depends on past inputs &

with memoryStatic —-> the current output depends only on the current input

no mry

### Linear & Nonlinear

- Superposition
- Homogeneity (or scaling)

### Time-invariant & Time-varying

The output of a time-invariant system does not explicitly depend on time whereas the output of a time-varying system does.

### Continuous v Discrete

Continuous-time systems involve signals that are continuous in time while discrete-time systems involve variables that only change at discrete instants of time.

### Mathematical modelling

differential equations for continuous-time systems Laplace transform

difference equations. for discrete-time systems Z-transform

## Modeling of Basic Systems

- Modeling of static systems
- Modeling of dynamical systems
- Physical Modeling
- Mathematical modelling
*Ex1. Modeling electrical systems**Ex2. Modeling mechanical systems**Ex3. Modelling a flow system*

### Static systems

- Easy Circuit analyse
- nodal analysis
- mesh analysis
- KCL & KVL & Ohm’s Law

### Dynamical systems

1.Physical modeling : This typically leads to a schematic representation showing the key system components and variables and how they are physically related.

2.Mathematical modeling

**Equilibrium relations** describe the balance of forces, of flow rates, of energy, of current, etc. which must exist for the system (conservation of energy).

**Compatibility relations** describe how motions of the system are interrelated because of the way they are connected.

- through variables (eg. current , flow)
- across variables (eg. voltage, pressure)

Equilibrium relations apply to through variables while

Compatibility relations apply to across variables.

*Ex1. Modeling electrical systems*

- Resistance(R)
`v = iR`

- Inductance(L)
`v = L(di/dt)`

- Capacitance(C)
`v=(1/C)*ƒ(i)dt`

1.Resistor/Capacitor filter circuit

2.RC filter

3.LRC filter

*Ex2. Modeling mechanical systems*

- Spring (K)
`Fs = Kx`

- Damper (or Dashpot) (B)
`Fd =Bx`

- Mass (M)
`Fm = Mx`

Fm = M*(d2x/dt2)

Fd = B*(dx/dt)

Fs = K*x

*Ex3. Modelling a flow system*

Change in Water Volume V = Flow-in – Flow-out

Fout = k*(h) (for laminar flow)

Fout = k*(h)^0.5 (for turbulent flow)

### Review of modeling physical (dynamical) systems

- A common generalized approach to the modeling of physical systems
- The first step is to select the fundamental variables
*mass**energy**momentum*- Most models are based on the conservation of these quantities.

- The fundamental variables are not conveniently measured. Hence, we use characterizing dependent variables instead.(such as temperature, pressure)
- The values of all the characterizing variables at any instance in time
*define the state of the system*

**Laplace Transforms & Transfer Functions**

- Laplace Transforms
- Inverse Laplace Transform
- Solve differential equations
- Transfer function representation

The transfer function is

a compact representation of the relationship between the input and an outputfor a linear system.

The transfer function model is the input-output relationship of a system in the Laplace Transform space.

It is defined as the ratio of the Laplace transforms of the output and input of a system for zero initial conditions.

因为拉普拉斯变化在EE112 & EE206课程中已经讲过许多，此处不在冗余介绍，如有需要，建议移步：

1⃣️https://tutorial.math.lamar.edu/

2⃣️https://space.bilibili.com/230105574/channel/seriesdetail?sid=1569595

两个链接都是我自己常用的学习资料，推荐给你

## Block Diagram Analysis

Blocks in seriesare simply combined by multiplying them together.

Blocks in parallelare simply combined by adding them together.

*Feedback connection* – when the output of a block is fed back to the input of an earlier block in the block diagram, a feedback mechanism is introduced.

- open-loop transfer function (OLTF)
- closed-loop transfer function (CLTF)

Lance附：这一章推荐学习链接🔗：`https://www.bilibili.com/video/BV1jt411M7QU?spm_id_from=333.999.0.0`

## System Analysis (Time Response) & Simulation

- Classical model
- RC circuit
- Mass-spring-damper
- Single tank

- Solving the differential equation model (first order)
- Solving the transfer function model
*System simulation*

Lance附：这一章推荐学习链接🔗：`https://www.bilibili.com/video/BV1is411c7u6?spm_id_from=333.999.0.0`

## Transient and Steady-state Characteristics

- Step response of a 1st order system (RC circuit & single tank)
- Step response of a 2nd order system (mass-spring-damper)

### 1st order system (RC circuit & single tank)

- KA
- (tao)
- Yes

The transient responserefers to theinitial part of the output response of the system, i.e.that part of the response before it settles to a steady conditionWhen the

response settles, it is said to be insteady-state

The response is characterized by two important parameters, namely the

steady-state gain Kand thetime constant t(tao)The steady-state output is the value at which the output settles and we will refer to this as

Yss

When t = (tao) —-> y=0.63(Yss)

In other words, the time constant (tao) is defined as the time at which the output has reached 63% of its final value.

The time constant effectively determines the responsiveness of the output.

when t = 2(tao) —-> y=0.88*KA

when t = 3(tao) —-> y=0.95*KA

when t = 4(tao) —-> y=0.98*KA

K = Yss/Uss

Yss = KA

Uss = A

## 2nd order system (mass-spring-damper)

- K is the steady-state gain
- ç is referred to as the damping ratio
- w is the undamped natural frequency

The damping factor or damping ratio ç indicates *the type of transient response of the system* (both in terms of speed and possible oscillations)

**系统瞬态响应的类型**

ç > 1 —-> poles of Y (s) from quadratic term are real and distinct

实数且可异ç < 1 —-> poles of Y (s) from quadratic term are complex

复数ç = 1 —-> poles of Y (s) from quadratic term are equal

相等ç = 0 —-> poles of Y (s) from quadratic term are imaginary

二次项是虚数

ç > 1 —-> critical damping, just no overshoot

ç < 1 —-> underdamped(and hence, some decaying oscillations)

ç = 1 —-> over-damped (no oscillations, similar to a first order response)

ç = 0 —-> undamped, oscillates (constant oscillations)

In general, the preferred damping range would be 0.5 < ç ≤ 1, as this gives a fast response consistent with only a small overshoot.

The undamped natural frequency w is defined as *the frequency at which the system oscillates in the absence of any damping*

**系统在无任何阻尼情况下振荡的频率**

Wd = Wm√(1-ç^2) damped natural frequency

Note, the terms *Wd* and *Wn* determine the frequency of oscillation of the output response（输出响应的振荡频率）

**significance of poles**

左上下 underdamped

左线分 over-damped

左线合 critically damped

中线分 undamped

右上下 unstable

### Measures of performance for a 2nd order system

- rise time
- settling time (PO)
- peak overshoot
- steady-state value

The rise time

Tris the time taken for the response to rise from 10% to 90% of its final value.响应从其最终值的10%上升到90%所花的时间

The settling time

Ts (PO)is the time for the response to settle within 2% of its final value and can be measured for all responses.响应在其最终值的2%以内的时间，可以对所有响应进行测量

The time to peak

Tpis the time it takes the step response to reach its peak-overshoot value.阶跃响应达到峰值超调值所花费的时间

The steady-state gain

Gssis a measure of the ratio of the steady-state output to the steady-state input for a given system.是给定系统稳态输出与稳态输入之比的量度

Analytically we can determine the time to peak using the following equation:

Tp = Pi/[(Wn)*√(1 -ç^2)]

Another Ex:

PO(%) = 100(0.18/0.75) = 24%

In the 1st and 2nd order models, the steady state gain is K.

For a general transfer function, G(s), the steady-state gain can be computed as:

Gss = lim G(s)

## Controller Design

- On/off Control
- Proportional Control
- PID Control
- Ziegler-Nichols Tuning Rules

### On/Off Control

bang bang control && hysteresis control

often use in heating systems

Ex for heating system

- relay
- Furnace valve
- Heating system(Furnace)
- Thermostat

This rapid on/off oscillation would accelerate wear of the actuator (in this case the flow valve) and hence shorten its life.

Hence, we introduce a gap between the upper and lower set-points (as shown previously) to prevent this rapid cycling between on and off.

This gap is known as the

differential gapor adeadbandand allows the temperature to change for a certain amount before the valve condition is altered.

This effect is also commonly referred to as *hysteresis effect（迟滞效应）*, where the output depends not only on the current input but also on knowledge of its past inputs.

### Proportional Control

This type of control is known as proportional control, where the control output is proportional to the error measurement.

Proportional control offers better flexibility and a smoother output then on/off control and works by changing the input to the process (or system) in proportion to the error between the desired output and the measured output.

For stability: `Re(s)<0`

### PID Control

The PID controller consists of the sum of three parts, namely proportional, integral and derivative

Thus, in summary, integral action allows us to eliminate the steady-state error in a given process or system, i.e. we can get the output of the system to match the desired set-point.（积分动作允许我们消除给定过程或系统中的稳态误差，也就是说，我们可以得到系统的输出，以匹配期望的设定值。）

Thus, in summary, derivative action helps in stabilizing a system.

（因此，总而言之，导数作用有助于稳定系统。）

#### Ziegler-Nichols Tuning Rules

##### Method 1

Ki = Kp/t¡

Kd = Kp/td

##### Method 2 (step response of the open-loop system)

Slope

R = A/(tao)

##### Lance说：关于PID控制器的推荐课程：

**Last update in 2022.04**

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