Beijing University of Aeronautics & Astronautics Linear System Theory School of Automation Beijing University of Aeronautics & Astronautics

Contents Chapter 1 Concept of Linear Systems 6 periods Chapter 2 Controllability and Observability of Linear Systems 8 periods Chapter 3 Canonical Form and Irreducible Realization of LTI Systems 10 periods Chapter 4 State Feedback and Observer Design 10 periods Chapter 5 Stability Analysis of Linear Systems 6 periods

I. Control System Design Steps Introduction 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.

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, t t t 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: 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

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

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

Linear systems Many physical systems can be treated as linear systems with finite dimension at their operating points due to the following reasons: Linear systems can be handled by using some powerful mathematical tools; 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.