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Problem 13

# Consider a system consisting of a cart to the top of which an inverted pendulum has been attached through a frictionless pivot. The cart is driven by a motor which at time $$t$$ exerts a force $$u(t)$$, taken as the control. (See Figure $$3.1(\mathrm{~b}) .)$$ We assume that all motion occurs in a plane, that is, the cart moves along a straight line. We use $$\phi$$ to denote the angle that the pendulum forms with the vertical, $$\delta$$ for the displacement of the center of gravity of the cart with respect to some fixed point, $$F \geq 0$$ for the coefficient of friction associated with the motion of the cart, $$g$$ for the acceleration of gravity, $$l>0$$ for the length of the pendulum, $$M>0$$ for the mass of the cart, and $$m \geq 0$$ for the mass of the pendulum, which we'll assume is concentrated at the tip. (If the mass is not so concentrated, elementary physics calculations show that one may replace the model by another one in which this does happen, using a possibly different length $$l$$. We allow the case $$m=0$$ to model the situation where this mass is negligible.) Newton's second law of motion applied to linear and angular displacements gives the two second order nonlinear equations and $$l \ddot{\phi}-g \sin \phi+\ddot{\delta} \cos \phi=0 \text {. }$$ We shall only be concerned with a small angle $$\phi$$, so we linearize the model about $$\phi=0$$. This results, after taking $x_{1}=\delta, x_{2}=\dot{\delta}, x_{3}=\phi, x_{4}=\dot{\phi}$$, in a linear system$$\Sigma$ with $$n=4, m=1$$ and matrices as follows: $$A=\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & -\frac{F}{M} & -\frac{m g}{M} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & \frac{F}{l M} & \frac{g(m+M)}{l M} & 0 \end{array}\right), \quad B=\left(\begin{array}{c} 0 \\ \frac{1}{M} \\ 0 \\ -\frac{1}{l M} \end{array}\right) .$$ Prove that $$\Sigma$$ is controllable. (Controllability holds for all possible values of the constants; however, for simplicity you could take all these to be equal to one.) This example, commonly referred to as the "broom balancing" example, is a simplification of a model used for rocket stabilization (in that case the control $$u$$ corresponds to the action of lateral jets).

Expert verified
To prove the controllability of the given system, we compute the controllability matrix $\mathcal{C} = [B \quad AB \quad A^2B \quad A^3B]$. We calculate the products AB, $A^2B$, and $A^3B$ using the given A and B matrices. Next, we form the controllability matrix by concatenating these products. To check if the system is controllable, we find the rank of the controllability matrix. If the rank of $\mathcal{C}$ is equal to the dimension of the state space (n=4 in this case), then the system is controllable for all possible values of the constants.
See the step by step solution

## Step 1: Compute the controllability matrix

To check the controllability of the system, we need to compute the controllability matrix, which is defined as: $$\mathcal{C} = [B \quad AB \quad A^2B \quad A^3B]$$ First, let's calculate the products of A and B matrices.

## Step 2: Calculate AB, A^2B, and A^3B

Given matrices A and B as $$A=\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\\ 0 & -\frac{F}{M} & -\frac{m g}{M} & 0 \\\ 0 & 0 & 0 & 1 \\\ 0 & \frac{F}{l M} & \frac{g(m+M)}{l M} & 0 \end{array}\right), \quad B=\left(\begin{array}{c} 0 \\\ \frac{1}{M} \\\ 0 \\\ -\frac{1}{l M} \end{array}\right)$$ We have to compute the products AB, A^2B, and A^3B: $$AB = A \times B$$ $$A^2B = A \times (A \times B) = A \times AB$$ $$A^3B = A \times (A^2 \times B) = A \times A^2B$$

## Step 3: Form the controllability matrix

After calculating the products of A and B matrices, we form the controllability matrix by concatenating the results, i.e., $$\mathcal{C} = [B \quad AB \quad A^2B \quad A^3B]$$

## Step 4: Check the rank of the controllability matrix

To check whether the system is controllable or not, we need to find the rank of the controllability matrix. If the rank of the controllability matrix is equal to the dimension of the state space (n=4 in our case), then the system is controllable. In other words, if $$\text{rank}(\mathcal{C}) = n = 4,$$ then the system is controllable.

## Step 5: Conclusion

To prove controllability for the given system, we have computed the controllability matrix and checked its rank. If we find the rank of the controllability matrix equal to 4, that means the system is controllable for all possible values of the constants.

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