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This paper deals with the solution of dynamical systems in state space. Complicated differential equations are converted into a simpler form by using state variables in vector matrix. It is used for multi-input and multi-output systems, and the solution is performed using matrix notation. It describes systems with complex internal structure. It allows state models to be manipulated using matrix calculus. Systems described by a state model are characterized by the fact that it is easier to design state control for them.
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State-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a vector within that space.
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Mathematical models of control systems are mathematical expressions which describe the relationships among system inputs, outputs and other inner variables. Establishing the mathematical model describing the control system is the foundation for analysis and design of control systems. Systems can be described by differential equations including mechanical systems, electrical systems, thermodynamic systems, hydraulic systems or chemical systems etc. The response to the input (the output of the system) can be obtained by solving the differential equations, and then the characteristic of the system can be analyzed. The mathematical model should reflect the dynamics of a control system and be suitable for analysis of the system. Thus, when we construct the model, we should simplify the problem to obtain the approximate model which satisfies the requirements of accuracy. Mathematical models of control systems can be established by theoretical analysis or practical experiments. The theoretical analysis method is to analyze the system according to physics or chemistry rules (such as Kirchhoff's voltage laws for electrical systems, Newton's laws for mechanical systems and Law of Thermodynamics). The experimental method is to approximate the system by the mathematical model according to the outputs of certain test input signals, which is also called system identification. System identification has been developed into an independent subject. In this chapter, the theoretical analysis method is mainly used to establish the mathematical models of control system. There are a number of forms for mathematical models, for example, the differential equations, difference equations and state equations in time domain, the transfer functions and block diagram models in the complex domain, and the frequency characteristics in the frequency domain. In this chapter, we shall study the differential equation, transfer function and block diagram formulations.
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