J
Hi there!
I am developing a controller for my final career project. My problem is the swing-up and balance of an inverted pendulum mounted on a cart.
I am at the stage of design and simulate the different control strategies which I am going to implement in a real time target using Lab-view.
I have some questions which I´d like to share with you in order to get some kind of orientation, regarding the control strategies of the balance problem. We are going to use the State Space formulation and theory and the state selected is going to be (x,x_dot,theta,theta_dot), this is, the position and velocity of both the cart along the rail and the pendulum (angular), from the vertical.
First of all,I have to stabilize the pendulum in his vertical position using the movement of the cart, which I can control by a small voltage controlled DC motor. I have available for measuring (with relative precision) only 2 variables of the state for measuring, (x and theta).
If I choose Theta as my output, I have a SISO system and an obvious regulator problem.
My doubt here is when doing this, a mode cancellation occurs and the representation loses his observability (although is still controllable), so what is the point to develop a state feedback controller by pole placement or LQR technique if the system is not observable? I am asking that because my tutor told me to do it, but I am reluctant because I am not able to have all of the state available to measure. Despite of the fact that I think that the condition that leads to the existence of a solution if that you have controllability + observability.
Another thing I would want to ask is that in the literature I have consulted (K.Ogata books, mostly) it states that in the Inverted Pendulum the plant has no an integrator, but in my system appears a pole at the origin. (It cancels with a zero when you choose Theta as the output, but not when you choose the X of the cart).
And finally, and I am sorry for the size of this post!, choosing X as the output leads to a SISO system which is controllable and Observable so a solution can be obtained, and therefore you can implement a state feedback plus observer (full or reduced order) controller.
Here the objective is that the position of the cart must follow a reference while maintaining the pendulum vertical. My problem here is with the pre-compensation constant used to adequate the reference prior to the sum block with the feedback branch. It seems that it tends to infinity so no useful control action is obtained.
If the formula is Kr=-1/(C*(A-B*K)^-1*B), that matrix in the denominator tends to a non singular matrix so the inverse dose not exists.
I am trying to do a fine tuning of the poles in closed loop in order to avoid this problem but I am not be able to obtain a valid solution.
I am also trying to do that by choosing a properly stated optimum root loci that leads to a LQR design.
So any help would be very appreciated. Best regards!
I am developing a controller for my final career project. My problem is the swing-up and balance of an inverted pendulum mounted on a cart.
I am at the stage of design and simulate the different control strategies which I am going to implement in a real time target using Lab-view.
I have some questions which I´d like to share with you in order to get some kind of orientation, regarding the control strategies of the balance problem. We are going to use the State Space formulation and theory and the state selected is going to be (x,x_dot,theta,theta_dot), this is, the position and velocity of both the cart along the rail and the pendulum (angular), from the vertical.
First of all,I have to stabilize the pendulum in his vertical position using the movement of the cart, which I can control by a small voltage controlled DC motor. I have available for measuring (with relative precision) only 2 variables of the state for measuring, (x and theta).
If I choose Theta as my output, I have a SISO system and an obvious regulator problem.
My doubt here is when doing this, a mode cancellation occurs and the representation loses his observability (although is still controllable), so what is the point to develop a state feedback controller by pole placement or LQR technique if the system is not observable? I am asking that because my tutor told me to do it, but I am reluctant because I am not able to have all of the state available to measure. Despite of the fact that I think that the condition that leads to the existence of a solution if that you have controllability + observability.
Another thing I would want to ask is that in the literature I have consulted (K.Ogata books, mostly) it states that in the Inverted Pendulum the plant has no an integrator, but in my system appears a pole at the origin. (It cancels with a zero when you choose Theta as the output, but not when you choose the X of the cart).
And finally, and I am sorry for the size of this post!, choosing X as the output leads to a SISO system which is controllable and Observable so a solution can be obtained, and therefore you can implement a state feedback plus observer (full or reduced order) controller.
Here the objective is that the position of the cart must follow a reference while maintaining the pendulum vertical. My problem here is with the pre-compensation constant used to adequate the reference prior to the sum block with the feedback branch. It seems that it tends to infinity so no useful control action is obtained.
If the formula is Kr=-1/(C*(A-B*K)^-1*B), that matrix in the denominator tends to a non singular matrix so the inverse dose not exists.
I am trying to do a fine tuning of the poles in closed loop in order to avoid this problem but I am not be able to obtain a valid solution.
I am also trying to do that by choosing a properly stated optimum root loci that leads to a LQR design.
So any help would be very appreciated. Best regards!