Linear System Theory

Instructor: Zhenhua Li Associate Professor zhenhua.li@sdu.edu.cn

Mobile ： 18660166181

School of Control Science and Engineering, Shandong University

Course descriptionThis course is about control

An example: Control an electric motor

We want to spin a motor at a given angular velocity. We can apply a fixed voltage to it, and never check to see if it is rotating properly.

Called open loop.

Electric Motor

Power Amplifier

Electric Motor

Computer

Vol

tage

Angular Velocity

What if there is a changing load on the motor?

– Our output velocity will change!

torque

speed no torque at max speed

stall torque

Course description

Closing the loop

Let’s measure the actual angular velocities.

Now we can compensate for changes in load by

feeding back some information.Power

AmplifierElectric Motor

Computer

Vol

tag

e

AngularVelocity

Tachometer

Course description

Classic Feedback Diagram

PowerAmplification

Actuatoror

PlantController

Sensor

Command inputx(t)

Errore(t) System

ouputy(t)

ExternalDisturbance

d(t)Actuatorcommand

u(t)

SensorReading

b(t)

Course description

Laplace Transform of Classic Feedback System

ActuatorA(s)

ControllerC(s)

SensorS(s)

Command inputX(s)

ErrorE(s) System

ouputY(s)

ExternalDisturbance

D(s)Actuator

commandU(s)

SensorReading

B(s)

Course description

This course is the Transition from classical control to modern control

by introducing the notion of state spaceLinear control systems

Linear system theory

Nonlinear control

Robust control Adaptive control

Optimal control

Course description

Course organizationTime: Wednesday 2:00 pm−3:50 pmInstructor: Dr. Zhenhua Li

– (Office)

– (Email) zhenhua.li@sdu.edu.cn

– (Phone) 18660166181

– (Office hours) Tuesday 4 pm−7 pm;

visits at other times are also welcome

Course organization Time: Wednesday 2:00 pm−3:50 pm Instructor: Dr. Zhenhua Li Text book

– Chi-Tsong Chen, Linear System Theory and Design, 3rd Edition, Oxford University Press, Oxford, UK, 1999.

Lecture notes

– http://www.??? Email list

– Important or emergent notice will be sent to you via emails

– Please provide me an email address that is most convenient with you

Time: Wednesday 5:20 pm-7:50 pm Instructor: Dr. Zhi-Hong Mao Text book Lecture notes Email list

Course evaluation

– Homework and class participation: 30% (late homework will not be accepted)

– Final exam: 70%

Course organization

Contents

Chapter 1 Mathematical Descriptions of Systems

Chapter 2 Controllability and Observability of Linear Dynamical Equations Chapter 3 Canonical Form and Irreducible Realization of Linear Time-invariant Systems

Chapter 4 Pole Placement and Decoupling by State FeedbackChapter 5 Static Output Feedback and

EstimatorsChapter 6 Stability Analysis of Linear Systems

I. Control System Design Steps

The design of the controller that can alter or modify the behavior and response of a plant to meet certain performance required can be a tedious and challenging problem. “Plant” here means any process characterized by a certain number of inputs u and outputs yc, as shown below.

u cy

Introduction

For example, we consider the following physical system, which is usually complex, i.e., it may consist of various of mechanical, electronic, hydraulic parts, etc. The design of u is in general not a straightforward work, because the plant process is usually complex.

If we know nothing about the system, what we can do is to take a series of typical input signals and observe its corresponding outputs. For instance,

u cyt t

u cyt t

Thought the physical system may be very complex, from the above responses, the system can be described approximately by the following first-order system:

1( ) ( )c

c cdy

T y u Y s U sdt Ts

aa

If the system does not meet our requirements, the traditional way is to design a compensator, or a feedback or simply adjust the parameter of the system. This design method has been successfully applied to controller design of many systems.

However, if a high performance is required for a system, the above-mentioned traditional design method may not give satisfactory results. Consequently, the internal states of the system should be analyzed, that is, the state-space description should be considered.

The following control steps are often followed by most control engineers in designing the control law u.

Step 1. Modeling

The task of control engineer in this step is to understand the processing mechanism of the plant. By taking a given input signal u(t) and measuring the output response y(t), he or she can describe the plant in the form of some mathematical equations. These equations constitute the mathematical model of the plant.

Step 2. System Analysis based on the Model

The analysis is twofold: qualitative analysis and quantitative analysis. Qualitative analysis includes stability, controllability and observability, etc., while

Step 3. Controller Design based on the model

If the system cannot achieve the asked performance, we have to design a controller or change the control low. Generally speaking, system controller design is a more complex issue.

the quantitative analysis needs to compute the response with the help of a computer.

Step 4. Implementation

In this step, a controller designed in Step 3, which is shown to meet the performance requirements for the plant model and is robust with respect to possible plant model uncertainties, is ready to be applied to the unknown plant.

Another important aspect of implementation is the final adjustment, or as often called the tuning, of the controller to improve performance by compensating for the plant model uncertainties that are not accounted for during the design process.

Modeling

System analysis

(controllability, stability, etc.)

System design

(state feedback, observer, etc.)

Implementation

II. Linear systems

Many physical systems can be treated as linear systems with finite dimension at their operating points due to the following reasons:

1. Linear systems can be handled by using some powerful mathematical tools;

2. Linear systems in most cases can faithfully describe the behavior of the controlled plants.

As a matter of fact, linear system theory is the cornerstone of modern control theory.