The structure of servo valve usually includes electromagnetic torque motor, nozzle baffle mechanism, slide valve and feedback mechanism. In the process of modeling, it is necessary to comprehensively consider the coupling relationship between electromagnetic, mechanical and hydraulic fields. The basic idea of state equation is to divide the variables in the system into state variables, input variables and output variables, and describe the dynamic behavior of the system through a set of first-order differential equations.
First, the choice of state variables
When establishing the state equation of servo valve, the state variables should be selected reasonably first. Common state variables include:
-armature displacement $ x $;
-armature speed $ dot {x} $;
-coil current $ i $;
-hydraulic chamber pressures of $p_1$ and $p_2$ (if any);
-spool displacement $y$.
These variables can completely describe the dynamic process inside the servo valve, and they are independent of each other.
Second, system dynamics modeling
Taking a typical force feedback servo valve as an example, its working principle is: the input current generates electromagnetic force through the coil, which drives the armature to move, then controls the gap between the nozzle baffle, changes the pressure difference of the hydraulic chamber, pushes the spool to move, and finally adjusts the flow of the hydraulic actuator.
According to Newton’s second law and the basic law of circuit, the following dynamic equations can
be established:
1. Electromagnetic part:
[ Lfrac{di}{dt} + Ri =
u – K_f x ]
Where $L$ is the coil inductance, $R$ is the resistance, $u$ is the input voltage, and $K_f$ is the electromagnetic stiffness coefficient.
2. Mechanical part:
[ mddot{x} + cdot{x} + kx = K_f i – A(p_1 – p_2) ]
Where $m$ is the mass, $c$ is the damping coefficient, $k$ is the spring stiffness coefficient and $A$ is the effective area.
3. Hydraulic part:
[ frac{d(p_1 – p_2)}{dt} = frac{Q}{C_h} ]
Where $Q$ is the flow through the valve port and $C_h$ is the hydraulic volume.
Third, the expression of the equation of state
Arrange the above differential equation into the standard form of the state equation:
[
dot{mathbf{x}} = mathbf{A}mathbf{x} + mathbf{B}u
]
[
mathbf{y} = mathbf{C}mathbf{x} + mathbf{D}u
]
Among them:
-$ mathbf {x} = [x, dot {x}, I, (p _ 1-p _ 2)] t $ is the state vector;
-$u$ is the input vector (voltage);
-$mathbf{y}$ is the output vector (such as spool displacement or flow);
-$ mathbf {a}, mathbf {b}, mathbf {c} and mathbf {d} $ are state matrix, input matrix, output matrix and direct transfer matrix respectively.
The specific matrix form can be obtained by sorting out the above differential equations, and it usually needs linearization (such as approximation in a small signal range) to facilitate the design of the control system.
Fourth, the application and significance
The state equation model of servo valve is of great significance in modern control theory. Through the equation of state, it is convenient to:
-System stability analysis;
-State observer design;
-Realization of optimal control strategy;
-Numerical simulation and parameter optimization.
In addition, the state equation can also be used in the design of digital controller to improve the responsiveness and anti-interference ability of servo system.
tag
To sum up, the state equation of the servo valve is a mathematical abstraction of the dynamic behavior of the system and an important bridge connecting the physical model with the control design. With the development of hydraulic servo technology, the application of state equation will be more extensive, which plays an irreplaceable role in improving the performance of the system.