What Is an Equation of State?

State equations are expressions that characterize the relationship between system inputs and states. The vector ordinary differential equation satisfied by the state vector is called the state equation of the control system. The equation of state is an important part of the mathematical model of the control system. [1]

State equations are expressions that characterize the relationship between system inputs and states. The vector ordinary differential equation satisfied by the state vector is called the state equation of the control system. The equation of state is an important part of the mathematical model of the control system. [1]
The mathematical model of the classical control theory based on the transfer function is adapted to the limitations of manual calculations at that time. It focuses on the external connections of the system, and focuses on a single-input-single-output linear steady-state system. With the development of computers, the mathematical model of modern control theory based on state-space theory uses state-space equations, mainly based on time-domain analysis, focusing on the state of the system and its internal connections. The research on the mechatronic control system has expanded to multiple inputs- Multi-output time-varying system. The so-called state variables are a minimum number of variables sufficient to fully characterize the system's motion state, and the state equation is a first-order differential equation system composed of system state variables.
Chinese name
Equation of state
Foreign name
state equation
Application area
Control science

Interpretation of Equation of State

State equations describe the relationship between system inputs and states. The vector ordinary differential equation satisfied by the state vector is called the state equation of the control system. Such as a continuous linear time-varying control system:
(A) in the formula is called an equation of state. If the initial conditions of the state vector x (t 0 ) = x 0 and t t 0 are known, then all states x (t) at t t 0 can be completely determined from equation (a), so control The dynamic behavior of the system is completely determined. The algebraic relationship describing the relationship between the output and the state of the control system is called the output (or measurement) equation. Equation (b) is the output equation. The output equation provides people with information about system state changes through measurement data. State equations and output equations are important components of the mathematical model of the control system. [1]

General Equation of State Equation of State for Continuous Time Systems

The state equation of a continuous-time system is a system of first-order differential equations of state variables. Let the state variables of the nth-order system be x 1 (t), x 2 (t), ..., x n (t), and the excitation be e (t), then the general form of the state equation is as follows:
(2.1)
Each coefficient in the formula is determined by the system's element parameters. For linear non-time-varying systems, they are all constants; for linear time-varying systems, some of them can be time functions. Equation (2.1) is a single input case. If there are m inputs e 1 (t), e 2 (t), ..., en (t), the general form of the state equation is
It can be written as a matrix:
(2.2)
Define the state vector x (t) and the first derivative x '(t) of the state vector as
(2.3)
Redefine the input vector e (t) as
In addition, if a matrix of n rows and n columns composed of coefficients a ij is denoted as A, and a matrix of n rows and m columns composed of coefficients b ij is denoted as B, then
Substituting equations (2.3), (2.4), and (2.5) into equation (2.2), the equation of state can be abbreviated as
If the system has q outputs y 1 (t), y 2 (t), ..., y q (t), then the matrix form of the output equation is
Following the previous example, define the output vector y (t) as
And the matrix of q rows and n columns composed of coefficients c ij is denoted as C, and the matrix of q rows and m columns composed of coefficients d ij is denoted as D, that is,
So the output equation is abbreviated as
For a linear time-invariant system, all the coefficient matrices above are constant matrices. Equations (2.6) and (2.10) are matrix forms of state equations and output equations, respectively. By applying the concepts of state equations and output equations, many complex engineering problems can be studied.

The general form of the equation of state for discrete-time systems

The discrete time system state equation is [2] :
The output equation is:
If the system is a linear time-invariant system, the equations of state and output are linear combinations of state variables and input signals.
The equation of state is:
The output equation is:
It can be seen that the state equation and the output equation are determined by the input and output. The state variables and the A, B, C, and D matrices that connect them, that is, the state equation and output equation can be abbreviated as

Equation of State for Linear Equations of State

(1 ) Solution of Homogeneous Equation of State [3] :
Consider n-th order linear stationary homogeneous equation
Solution.
First analyze the solution of the scalar differential equation. Let the scalar differential equation be
The equation (2) is obtained by Laplace transform
;
Taking the inverse Laplace transform, we get
.
The scalar differential equation can be considered as the characteristic of the matrix differential equation when n = 1. Therefore, the solution of the matrix differential equation and the scalar differential equation should have the form invariance, so the following theorem is obtained:
[Theorem 1] The solution of the n-th order linear stationary homogeneous state equation (1) is:
In the formula:
.
[Corollary 1] n-th order linear stationary homogeneous state equation
The solution is
.
The physical meaning of the solution of the homogeneous state equation is that e A (tt 0 ) transfers the system from the initial state x 0 at the initial time t 0 to the state x (t) at the time t . Therefore, e A (t-t0) is also called the state transition matrix of the stationary system.
(There are four methods to find the state transition matrix: definition (matrix index definition) method, inverse Lagrangian method, eigenvector method, and Cayly-Hamilton method)
Get two equations from above
Among them, the first formula is a matrix index definition formula, and the second formula can be e At 's frequency domain method or inverse Laplace transform method.
(2 ) The solution of the non-homogeneous state equation [3] :
Let n order non-homogeneous equations
Multiplying the equation of state by e -At , there is
Shift the integral, and then multiply the term left by e At .
[Theorem 2] The solution of the n-th order linear stationary non-homogeneous equation (5) is
It can be seen from the expression of the solution of the non-homogeneous equation of state that the solution is a combination of the solution of the homogeneous equation and the action of controlling u (t).
(3 )
Calculation method [3]
(3.1 ) Definition method:
(3.2 ) Laplace transform method:
(3.3 ) Eigenvalue method:
This method is calculated in two cases.
First, when the eigenvalues of A are not heavy (mutually different), and if the eigenvalues of A are i (i = 1, 2, ... n), then A can be transformed into a diagonal normal form through non-singular transformation. which is:
Written according to the nature of e At 7
By definition
This gives:
(9) Formula is the formula for calculating e At when the eigenvalue of A is not heavy. Among them, the P matrix is an exchange matrix that transforms A into a diagonal normal form. How to find the eigenvectors of the P matrix:
If the matrix A has multiple roots, the above method can also be used to derive: the fraction of the matrix index e Ajt of the Jordan block Aj corresponding to the multiple roots is
Find the fraction of the matrix index e At :
Where P is a transformation matrix that transforms A j into an approximately normal form. When A has both j-roots and different roots:
How to find the eigenvectors of the P matrix:
Note: In the formula (13), the feature vectors p 1 , p 2 , ... p j corresponding to the double roots can be placed at the front of the P matrix or after it, without strict regulations.
(3.4 ) Cayley-Hamilton method:
Consider the characteristic polynomial of A
Obviously for n eigenvalues of A
,Have
.
According to Cayley-Hamilton theorem
It can be seen here that the matrix A and i have the same status.
Shift
The above formula shows that
Yes
Linear combination.
Therefore, you can set
Where
Is the coefficient to be determined,
.
There are two cases to determine the coefficient to be determined:
(1) A has n different eigenvalues
, The characteristic value of A
Has the same status as A, then
There are n equations here, and n unique coefficients can be uniquely determined.
.
(2) When the eigenvalues of A are heavy, let A have p mutually different eigenvalues, r different heavy eigenvalues, and each multiple is
,
. If
Yes
Eigenvalues, then
Satisfied equation
Correct
begging
Secondary leads, so there are
Independent equations. Generally, let A be the characteristic value
Is a single eigenvalue. among them,
Yes
Heavy eigenvalues,
for
Heavy eigenvalues.
Have
,then
Determined by the following n independent equations:

Equation of State for Linear Steady Discrete Systems

For n-th-order linear stationary discrete systems [2]
There are two ways to solve it:
(1 ) Recursive method [2] :
(2 ) Z transform method [2] :
Z is the frequency domain solution. Z-transform Eq. (17), we have
Shift term, get
Left multiplication
, Got
take
, Got
[Theorem 3] The solution of equation (17) of the n-th order linear stationary discrete system is

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