1. Linear System Theory Instructor: Zhenhua Li Associate Professor zhenhua.li@sdu.edu.cn Mobile ： 18660166181 School of Control Science and Engineering, Shandong University

2. Course description This 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.

3. 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

4. Closing the loop Let’s measure the actual angular velocities. Now we can compensate for changes in load by feeding back some information. Course description

5. Classic Feedback Diagram Course description

6. Laplace Transform of Classic Feedback System Course description

7. This course is the Transition from classical control to modern control by introducing the notion of state space Linear control systems Linear system theory Nonlinear control Robust controlAdaptive control Optimal control Course description

8. Course organization Time: Wednesday 2:00 pm−3:50 pm Instructor: 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

9. 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.???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

10. 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

11. 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 Feedback Chapter 5 Static Output Feedback and Estimators Chapter 6 Stability Analysis of Linear Systems

12. 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 y c, as shown below. Introduction

13. 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.

14. 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, tttt

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

16. 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.

17. 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

18. 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.

19. 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.

20. Modeling System analysis (controllability, stability, etc.) System design (state feedback, observer, etc.) Implementation

21. 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.