Embedded Digital Control with Microcontrollers

9State-space Based Control System Analysis

State-space provides another option to analyze and design digital controllers besides transfer function based methods. Therefore, we will explore the state-space approach in detail here. To do so, we will start with the definition of state and emphasize its difference from the transfer function approach. Then, we will show how state-space representation of an LTI system can be constructed from its differential or difference equation in continuous- and discrete-time, respectively. We know that differential equations lead to transfer function representation in continuous-time from Chapter 6. Likewise, difference equations lead to transfer function representation in discrete-time as mentioned in Chapter 4. Hence, we will cover methods on converting state-space and transfer function representations. Finally, we will look at properties of a system from its state-space representation. While performing all these, we will benefit from MicroPython and Python control systems libraries. As the end of chapter application, we will simulate state-space representation of the DC motor.

9.1 State-space Approach

In order to grasp the concepts to be covered in this chapter, we should first explain what we understand from the state. Then, we should emphasize the difference of state-space approach from transfer function based methods. We will perform these in this section.

9.1.1 Definition of the State

States of the system can be represented as the ...