Dynamic System Modeling - State Space Equations
The state-space equation is the basis of modern control theory, which expresses the relationship between system state variables, input and output in the form of matrix. It can describe and process multiple input multiple output (Multiple Input Multiple Output, MIMO) system.
state space equation
Single Input Single Output (SISO) system
After Laplace transform
The state-space equation is a set, which includes the input, output and state variables of the system, and expresses them with a series of first-order differential equations .
For the second-order system in this example, in order to write it as a state-space equation, it is necessary to select appropriate state variables (State Variables) to transform the second-order system into a series of first-order systems.
Find the reciprocal of the state variable.
specific derivation process
This shows that when using state space equations to describe the system, there are n state variables, m outputs and p inputs. It can express multi-state, multi-output, multi-input systems. Among them, matrix A is an n × n matrix, which represents the relationship between system state variables, and is called state matrix or system matrix. Matrix B is an n × p matrix, which represents the influence of input on state variables, and is called input matrix or control matrix. Matrix C is an m × n matrix, which represents the relationship between the output of the system and the system state variables, called the output matrix. Matrix D is an m × p matrix, which means that the input of the system directly acts on the output of the system, which is called the direct transfer matrix.
Multiple Inputs Multiple Outputs (MIMO) system
To establish the state-space equations of the above system, one must first master its dynamic differential equations. The system can be considered as two closed loops, using Kirchhoff's voltage law in each closed loop.
So that
whoever's second-order derivative (or the relational expression of the first-order derivative) needs to be eliminated is made to be the state variable.
Find the reciprocal of the state variable.
Written in matrix form
The state-space equation of the system
The specific derivation process.
Relationship between Equation of State and Transfer Function
