Suggested languages for you:

Americas

Europe

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.

Expert verified
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.

We value your feedback to improve our textbook solutions.

Access millions of textbook solutions in one place

• Access over 3 million high quality textbook solutions
• Access our popular flashcard, quiz, mock-exam and notes features

Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

• Flashcards & Quizzes
• AI Study Assistant
• Smart Note-Taking
• Mock-Exams
• Study Planner