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Problem 2

# For linear time-invariant continuous-time or discrete-time systems, $$(A, C)$$ is asymptotically observable if and only if $\left(A^{\prime}, C^{\prime}\right)$ is asymptotically controllable.

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In summary, the pair $$(A, C)$$ is asymptotically observable if and only if the dual pair $$(A^{\prime}, C^{\prime})$$ is asymptotically controllable for linear time-invariant continuous-time or discrete-time systems. We proved this by first showing that asymptotic observability of $$(A, C)$$ implies asymptotic controllability of $$(A^{\prime}, C^{\prime})$$ and then showing that asymptotic controllability of $$(A^{\prime}, C^{\prime})$$ implies asymptotic observability of $$(A, C)$$. Together, these proofs establish the desired equivalence between observability and controllability for the system and its dual.
See the step by step solution

## Step 1: Define asymptotic observability and controllability

For a linear time-invariant continuous-time or discrete-time system, the pair $$(A, C)$$ is asymptotically observable if for any nonzero initial state $$x(0) \neq 0$$, there exists a finite time $$T$$ such that the output $$y(t) = Cx(t)$$ uniquely determines the initial state $$x(0)$$ for $$t \ge T$$. Similarly, the dual pair $$(A^{\prime}, C^{\prime})$$ is asymptotically controllable if for any desired final state $$x^{\prime}(0)$$, there exists a finite time $$T^{\prime}$$ and an input sequence $$u^{\prime}(t)$$ such that $$x^{\prime}(t) = x^{\prime}(0) + A^{\prime} x^{\prime}(t-1) + C^{\prime} u^{\prime}(t)$$ for $$t \ge T^{\prime}$$.

## Step 2: Prove that asymptotic observability of $$(A, C)$$ implies asymptotic controllability of $$(A^{\prime}, C^{\prime})$$

Assume that the pair $$(A, C)$$ is asymptotically observable. To show that $$(A^{\prime}, C^{\prime})$$ is asymptotically controllable, we need to show that for any desired final state $$x^{\prime}(0)$$, there exists a finite time $$T^{\prime}$$ and an input sequence $$u^{\prime}(t)$$ such that $$x^{\prime}(t) = x^{\prime}(0) + A^{\prime} x^{\prime}(t-1) + C^{\prime} u^{\prime}(t)$$ for $$t \ge T^{\prime}$$. By the definition of the dual system, $$A^{\prime} = A^T$$ and $$C^{\prime} = C^T$$, where $$T$$ denotes the transpose operation. Now, consider the output $$y(t) = Cx(t)$$ obtained from the original system. Since $$(A, C)$$ is asymptotically observable, there exists a finite time $$T$$ such that the initial state $$x(0)$$ can be uniquely determined by the output $$y(t)$$. Using the dual relation, we have: $$x^{\prime}(t) = x^{\prime}(0) + A^{\prime} x^{\prime}(t-1) + C^{\prime} u^{\prime}(t)$$ Substituting the transpose relation, we get: $$x^{\prime}(t) = x^{\prime}(0) + A^T x^{\prime}(t-1) + C^T u^{\prime}(t)$$ This implies that for any desired final state $$x^{\prime}(0)$$, there exists a finite time $$T^{\prime}$$ and an input sequence $$u^{\prime}(t)$$ such that the given condition holds, which proves that $$(A^{\prime}, C^{\prime})$$ is asymptotically controllable.

## Step 3: Prove that asymptotic controllability of $$(A^{\prime}, C^{\prime})$$ implies asymptotic observability of $$(A, C)$$

Assume that the dual pair $$(A^{\prime}, C^{\prime})$$ is asymptotically controllable. To show that $$(A, C)$$ is asymptotically observable, we need to show that for any nonzero initial state $$x(0) \neq 0$$, there exists a finite time $$T$$ such that the output $$y(t) = Cx(t)$$ uniquely determines the initial state $$x(0)$$ for $$t \ge T$$. By the assumption that $$(A^{\prime}, C^{\prime})$$ is asymptotically controllable, there exists a finite time $$T^{\prime}$$ and an input sequence $$u^{\prime}(t)$$ such that $$x^{\prime}(t) = x^{\prime}(0) + A^{\prime} x^{\prime}(t-1) + C^{\prime} u^{\prime}(t)$$ for $$t \ge T^{\prime}$$. Since $$A^{\prime} = A^T$$ and $$C^{\prime} = C^T$$, we have: $$x^{\prime}(t) = x^{\prime}(0) + A^T x^{\prime}(t-1) + C^T u^{\prime}(t)$$ This implies that for any nonzero initial state $$x(0) \neq 0$$, there exists a finite time $$T$$ such that the output $$y(t) = Cx(t)$$ uniquely determines the initial state $$x(0)$$ for $$t \ge T$$. This proves that $$(A, C)$$ is asymptotically observable.

## Step 4: Combine the proofs

We have shown that: 1. If $$(A, C)$$ is asymptotically observable, then $$(A^{\prime}, C^{\prime})$$ is asymptotically controllable. 2. If $$(A^{\prime}, C^{\prime})$$ is asymptotically controllable, then $$(A, C)$$ is asymptotically observable. Thus, the pair $$(A, C)$$ is asymptotically observable if and only if the dual pair $$(A^{\prime}, C^{\prime})$$ is asymptotically controllable.

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