State variable analysis of state variables

Methods to describe a system are usually input-output methods and state variable analysis, also known as state-space methods. Generally, the analysis of the time domain or frequency domain of the system is based on the input-output method, that is, the main concern is the relationship between the input and output of the system, and the relevant problems within the system are not considered. For the simple general single-input single-output system, it is very convenient to use the input-output method, but for the multiple-input multiple-output system, especially for the research of more and more nonlinear systems or time-varying systems encountered in modern engineering, it is almost impossible to adopt the input-output description law.

With the rapid development of system theory and computer technology, the state variable method, which is the basis of modern control theory, has been widely used in system analysis since the 60s of the 20th century. The main feature of this method is to replace the system functions that only describe the external characteristics of the system by using state variables that describe the internal characteristics of the system, and to apply this description to multiple-input-multiple-output systems very conveniently. In addition, state-space methods are also successfully used to describe nonlinear or time-varying systems, and are easy to solve with the help of computers.

State: The state of a dynamic system is a minimum set of physical quantities that represent the system by which the behavior of the system can be completely determined.

State variables: Those variables that represent the state of a system are called state variables.

State vectors: state variables that can fully describe the behavior of a system, and can be regarded as the components of the vector.

Equation of state: A set of first-order differential equations that describe the law of change in state variables. On the left side of each program is the first derivative of the state variable, and on the right is the general function expression containing the system parameters, state variables, and excitations, and the differential and integral operations without the variables.

Output Equations: A system of equations that describes the relationship between the output of a system and state variables. On the left side of each program are the output variables, on the right are the general function expressions including system parameters, state variables, and excitations, and the differentiation and integration operations without variables.

For discrete-time systems, the state variables are described similarly to the equation of state, except that the state variables are discrete quantities, so the equation of state is a set of first-order difference equations, and the output equation is a linear algebraic equation of a set of discrete variables. Step 1: Select the state variables, write the equation of state and the output equation in columns;

Step 2: Solve the solution of the equation of state and the output equation using the initial state and input excitation of the system.

The state variable method solves a time-domain circuit, see the following example.

The amount of change in the state function is related to the initial and final states of the system.

A state function is a property determined by the state of the system. When the state is constant, the value of the state function is also constant, and if the state changes, the change value of the corresponding state function is only related to the initial and final states of the system, regardless of the specific process experienced between the initial and final states. Temperature, pressure, volume, internal energy, etc., are all state functions.

State functions

The state function, that is, the macroscopic properties that characterize the characteristics of the system, mostly refers to the energy dimension.

thermodynamic functions such as internal energy, enthalpy, Gibbs free energy.

Helmholtz Free Energy). The state function only has a definite value for the system in equilibrium state, and its change value depends only on the beginning and end states of the system. In addition, state functions are interrelated and mutually restrictive.

The state function characterizes and determines the macroscopic properties of the state of the system. The state function has a definite value only for systems in equilibrium and no definite value for systems in non-equilibrium states. When solving various thermodynamic functions, it is usually necessary to do path integrals.

If the integration result is not path-independent, the function is called a state function, and if it is not a fight, it is called a non-state function.

The above content reference:Encyclopedia – State Functions

The output is the effect of the system on the environment, and the state is a set of information that completely describes the motion behavior of the system. In addition, the output is always measurable, whereas state variable information is not always measurable.

A state variable is used to describe a given variable in a range of states, which can be regarded as exogenous variables, and the value of the state variable is certain when the state is unchanged.

Control variables. It's a model analysis.

when mentioning the concept. When there are multiple variables in a model, the impact of each variable on the final outcome is usually assumed that the other variables remain unchanged and only one variable changes the impact on the final result, which is the control variable.

Paraphrase. The morphomorphic variables are a set of variables that describe the motion of a system in full. It should be able to determine the future evolutionary behavior of the system. For example, ideal gases.

The state variables are temperature t, pressure p, and volume v, and the state variables of one-dimensional particle motion are its position and velocity. The system should evolve continuously or jumpingly, and its state variables can also be continuous or discrete.

The variable group with the fewest number of variables that can fully describe the time-domain behavior of a dynamic system is called the state variable of the system. The so-called full description of the time-domain behavior of the system means that if the state x(t0) and the input function u(t) on t0,t i are given the initial moment t0 i, then any instantaneous behavior of the system at t0,t)i is uniquely determined.

For the above content, please refer to: Encyclopedia - State Variables.

A state variable is a variable that describes the state of a system, and it has the following properties:

1.Uniqueness: After determining the initial state, the state variables of the system can uniquely describe the state of the system.

2.Independence: Each state variable is independent, i.e., it is not affected by other state variables.

3.Changeability: The state of the system changes with the time of reply, so the state variables will also change with time.

4.Observability: The state value of the system can be obtained through measurements.

5.Controllability: refers to the state value of the system that can be changed by controlling certain inputs in the control system.

6.Normalization: In order to facilitate analysis and calculation, each state variable is usually limited to a certain range and normalized.

In short, state variables are the key indicators to describe the operating state and evolution process of the system, and their characteristics determine their wide application in control engineering, signal processing, automation and other fields.

State Variables in a Circuit: A set of independent dynamic variables in a circuit, which is a set of responses at any given moment in a circuit that is linearly independent, necessary, sufficient, and minimal.

As shown in the figure above, the state variables of the circuit are:

For linear time-invariant dynamic networks, independent capacitance voltage and inductor current are a set of variables that can meet the above conditions, and can be used as a set of state variables of the network.