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Linear System Theory
Instructor: Zhenhua Li Associate Professor [email protected]
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) [email protected]
– (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.