a linear second order single input continuous time system is described by following set of differential equation:x1(t) = -2x1(t) + 4x2 (t)x2(t) = 2x1(t) - x2 (t) + u(t)where x1(t) and x2(t) are state variable and u(t) is the control variable..................the system is:(a)controllable and stable(b)controllable but unstable(c)uncontrollable and unstable(d)uncontrollable but stable
Question
a linear second order single input continuous time system is described by following set of differential equation:x1(t) = -2x1(t) + 4x2 (t)x2(t) = 2x1(t) - x2 (t) + u(t)where x1(t) and x2(t) are state variable and u(t) is the control variable..................the system is:(a)controllable and stable(b)controllable but unstable(c)uncontrollable and unstable(d)uncontrollable but stable
Solution
To answer this question, we need to determine two things: the controllability and the stability of the system.
- Controllability: A system is controllable if we can drive the system's state to any desired state in a finite time using an appropriate control input. For a linear time-invariant system described by the state-space representation, the system is controllable if and only if the controllability matrix is full rank.
The controllability matrix (C) for the given system is given by:
C = [B AB]
where A is the system matrix and B is the input matrix. For the given system, A = [[-2, 4], [2, -1]] and B = [[0], [1]].
Calculating AB gives us [[4], [-2]]. So, the controllability matrix becomes [[0, 4], [1, -2]]. The determinant of this matrix is -4, which is not equal to zero. Therefore, the system is controllable.
- Stability: A system is stable if the real parts of all eigenvalues of the system matrix A are negative.
The eigenvalues of A are the roots of the characteristic equation, which is given by det(A - λI) = 0, where I is the identity matrix and λ are the eigenvalues.
For the given system, the characteristic equation becomes (-2 - λ)((-1) - λ) - (4*2) = 0. Solving this equation gives us the roots λ = -1 ± √7. Since both roots have negative real parts, the system is stable.
Therefore, the system is controllable and stable, so the correct answer is (a) controllable and stable.
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