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

LQR CONTROL FORMULATION

LQR Linear Quadratic Regulator is a state feedback control technique applied for linear systems in state space form as given below:

(39)
where are the systems states, inputs. , are system, input matrices.

It computes the optimal state feedback control gains based on the cost function given by:

(40)
Where are positive definite symmetric matrices to be designed.
The feedback control law that minimizes the cost function is given by:
(41)
And is given by:
(42)
Where is a positive definite matrix found by solving the forced Lyapunov inequality which is also called algebraic Ricatti equation given by:
(43)

The matrices are design weighting matrices used to penalize the state variables and the control inputs. The size of the matrix depends on the number of states and the size of the matrix depends on the number of inputs.

The performance of the LQR can be changed dramatically by adjusting the weighting matrices. is chosen to be big if there is a limit in the control input signal because of actuator saturation and is chosen small if the physical system does not have limitations.
In our application because we have 4 states
(44)

The terms penalizes the states respectively.

is a scalar because we only have one input which is the force .

Figure 5 is illustration of the closed loop system

Fig. 5. Block Diagram of the LQR Controller.