D ynamical Modeling and Control of Motion System of the Gantry Crane to Minimize Swing Angle of the Payload

x- state vector,

y- output vector,

u- input or control vector,

[A]- state matrix,

[B]- input matrix,

[C]- output matrix,

[D]- direct transmission matrix,
Fig.4. shows the block diagram of a linear system represented by the state-space of the equations (25), (26)

Fig. 4. Block Diagram of a linear system represented by the state variables

Since the model of the crane subject of the study is of the Linearized Time-Invariant type, therefore it can be represented in the form of state variables as in the equations (25), (26). If the input of this system is the external forces F(t), the output is a cart displacement x(t), and the angle of the payload swing θ(t), which is a system of the (S.I.M.O.) type, and the crane system represented by the equations (21), (22), which are two differential equations of the second degree. This means that the system includes four integrators, two for each equation. Thus, the number of state variables is four, and they can be defined as x₁(t), x₂(t), x₃(t), x₄(t)

where

(27)

By removing the variable from equations 21 and 22, yields

(28)
By removing the variable from equations 21 and 22, yields

(29)