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Problem 1

# $$\Sigma$$ is $$\delta$$-sampled controllable if and only if the pair $$\left(e^{\delta A}, B\right)$$ is controllable and $$A$$ has no eigenvalues of the form $$2 k \pi i / \delta$$ for any nonzero integer $$k$$.

Expert verified
To determine if the given system is $$\delta$$-sampled controllable, first compute the matrix exponential $$e^{\delta A}$$ using a suitable method. Next, find the eigenvalues of matrix A and check for any eigenvalues of the form $$2 k \pi i/ \delta$$ for any nonzero integer k. If such eigenvalues are present, the system is not $$\delta$$-sampled controllable. If the eigenvalue condition is satisfied, test the controllability of the pair $$\left(e^{\delta A}, B\right)$$ by forming the controllability matrix $$C$$ and determining its rank. If the rank of C is equal to the dimension of the system, the system is $$\delta$$-sampled controllable; otherwise, it is not.
See the step by step solution

## Step 1: Determine the Matrix Exponential

First, we need to find the matrix exponential $$e^{\delta A}$$. This can be computed using various methods such as matrix power series, Jordan decomposition, or numerical approaches. Choose a suitable method based on the given matrix A and calculate the matrix exponential.

## Step 2: Determine the Eigenvalues of A

Next, we need to check if A has any eigenvalues of the form $$2 k \pi i / \delta$$ for any nonzero integer k. To do this, first find the eigenvalues of matrix A using characteristic equation or any suitable numerical method.

## Step 3: Check the Eigenvalue Condition

Based on the computed eigenvalues, check for any eigenvalues of the form $$2 k \pi i/ \delta$$ for any nonzero integer k. If such eigenvalues are present, the system is not $$\delta$$-sampled controllable.

## Step 4: Test Controllability of the Pair

If the eigenvalue condition is satisfied, we need to test the controllability of the pair $$\left(e^{\delta A}, B\right)$$. To do this, form the controllability matrix $$C = \begin{bmatrix} B & e^{\delta A}B & e^{2\delta A}B & \dots & e^{(n-1)\delta A}B \end{bmatrix}$$, where $$n$$ is the dimension of matrix A.

## Step 5: Assess the Rank of the Controllability Matrix

Determine the rank of the controllability matrix C. If the rank of C is equal to $$n$$, which is the dimension of the system, then the pair $$\left(e^{\delta A}, B\right)$$ is controllable. In this case, the system is $$\delta$$-sampled controllable. If the rank of C is less than n, then the system is not $$\delta$$-sampled controllable.

## Step 6: Conclusion

Based on the results of the eigenvalue condition and controllability test, provide the conclusion on whether the given system is $$\delta$$-sampled controllable or not.

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