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

MATHEMATICAL MODELING IN STATE-SPACE

When analyzing using state-space, we are concerned with three types of variables, input variables, output variables, and state variables, what is meant by the state of the system is the smallest set of variables so that when we know these variables at time t=t₀ and know the value of the input at t ≥ t₀, we can fully know the dynamic response of the system at any time t ≥ t₀ .Thus, the dynamic system must contain elements that can act as a memory to store the input value of the time t≥ t₀. It has been found that the integrators in control systems work as a storage device, so the output of these integrators is considered to be the variables that define the internal state of the dynamic system, and thus the output of the integrators works as state variables. This indicates that the number of state variables required to fully describe the system’s dynamics is equal to the number of integrators the system contains. In the case of a multi-input multi-output system (M.I.M.O.) consisting of the number of integrals n and assuming that the number of inputs is r, they can be represented as a vector (r×1)

Where: is the axis of the matrix

Number of outputs (b×1)

Assume that state variables are equal to the number of integrators and are called state variables (n×1)

Thus, the system can be described by the following two equations

(25)

(26)

Where:

n- number of state variables

p- number of inputs

q- number of outputs