How to Derive a State Space Model for a Pantograph System?

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I have pantograph simulation on simulink. The dynamic model of the system is given as a C code. So the input is torque and output is motor angles and velocities.

I need to build a QLR controller and to do so I need to come up with state space model of the system, right ?

I have the following equation of motion;

M(q)q'' = H(q,u) + Bf(q)Fext + Torque

where M and Bf are 2x2 matrices and H is 2x1

My states would be q1,q2,q1',q2' where q1' and q2' are the joint velocities.

But I don't know how can I obtain the state space model when my equation of motion consists of matrices

 

[Achy47]

and vectorsHello,

Thank you for sharing your question on the forum. It seems like you are trying to build a QLR controller for a pantograph simulation in Simulink. To answer your question, yes, you will need to come up with a state space model of the system in order to build the controller.

The state space model is a mathematical representation of a system in terms of its state variables, input variables, and output variables. In your case, the state variables would be q1, q2, q1', and q2', as you have correctly identified. The input variable would be the torque, and the output variables would be the motor angles and velocities.

To obtain the state space model, you will need to manipulate your equation of motion into the standard state space form:

x' = Ax + Bu
y = Cx + Du

Where x is the state vector, u is the input vector, and y is the output vector. A, B, C, and D are matrices that represent the dynamics of the system.

In your equation of motion, you have M(q)q'' on the left side, which is equivalent to the state vector x'. On the right side, you have H(q,u), which can be expressed as Ax + Bu. The matrices A and B are dependent on the system dynamics and can be obtained from the equation of motion.

Similarly, the output equation y = Cx + Du can be obtained by expressing the output variables (motor angles and velocities) as a function of the state variables (q1, q2, q1', q2').

I would recommend looking into some resources on state space modeling to get a better understanding of the process. Simulink also has built-in tools to help you convert your equations into a state space model.

I hope this helps. Good luck with your project!