characteristic equation

Consider the state-space model $$ \begin{aligned} \dot{x}(t) & =A x(t)+B u(t) \\ y(t) & =C x(t) \end{aligned} $$ where $x(t), r(t), y(t)$ are the state, input and output, respectiv...

The open loop transfer function of a unity gain negative feedback system is given by $$G(s) = {k \over {{s^2} + 4s - 5}}$$. The range of k for which the system is stable, is

Consider a matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 4 & { - 2} \cr 0 & 1 & 1 \cr } } \right]$$. The matrix A satisfies the equation 6A $$-$$1 = A 2 + cA + dI, where c and d...

Let $p$ and $q$ be real numbers such that $p^2+q^2=1$. The eigen values of the matrix $\left[\begin{array}{cc}p & q \\ q & -p\end{array}\right]$ are

The characteristic equation of a linear time-invariant (LTI) system is given by $\Delta(s) = s^4 + 3s^3 + 3s^2 + s + k = 0$. The system is BIBO stable if

A closed loop system has the characteristic equation given by s³ + Ks² + (K+2)s + 3 = 0. For this system to be stable, which one of the following conditions should be satisfied?

The root locus of the feedback control system having the characteristic equation s² +6Ks +2s+5=0 where K > 0, enters into the real axis at

A closed loop system has the characteristic equation given by $${s^3} + K{s^2} + \left( {K + 2} \right)s + 3 = 0.$$ For this system to be stable, which one of the following conditi...

The eigen values of the matrix given below are $$\left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & { - 3} & { - 4} \cr } } \right]$$

The range of K for which all the roots of the equation $${s^3} + 3{s^2} + 2s + K = 0$$ are in the left half of the complex $$s$$-plane is

The root locus of the feedback control system having the characteristic equation $${s^2} + 6Ks + 2s + 5 = 0$$ where $$K>0,$$ enters into the real axis at

The solution of the differential equation, for $$t > 0,\,\,y''\left( t \right) + 2y'\left( t \right) + y\left( t \right) = 0$$ with initial conditions $$y\left( 0 \right) = 0$$ and...

The maximum value of $$'a'$$ such that the matrix $$\left[ {\matrix{ { - 3} & 0 & { - 2} \cr 1 & { - 1} & 0 \cr 0 & a & { - 2} \cr } } \right]$$ has three linearly independent real...

An open loop transfer function $$G(s)$$ of system is $$G\left( s \right) = {k \over {s\left( {s + 1} \right)\left( {s + 2} \right)}}$$ For a unity feedback system, the breakaway po...

A system with the open loop transfer function $$G\left( s \right) = {K \over {s\left( {s + 2} \right)\left( {{s^2} + 2s + 2} \right)}}$$ is connected in a negative feedback configu...

A system matrix is given as follows $$$A = \left[ {\matrix{ 0 & 1 & { - 1} \cr { - 6} & { - 11} & 6 \cr { - 6} & { - 11} & 5 \cr } } \right].$$$ The absolute value of the ratio of...

Given that $$A = \left[ {\matrix{ { - 5} & { - 3} \cr 2 & 0 \cr } } \right]$$ and $${\rm I} = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right],$$ the value of $${A^3}$$ is

The characteristic equation of a closed-loop system is $$s\left( {s + 1} \right)\left( {s + 3} \right) + \,\,k\left( {s + 2} \right) = 0,\,\,k > 0.$$ Which of the following stateme...

The characteristic equation of a $$3\,\, \times \,\,3$$ matrix $$P$$ is defined as $$\alpha \left( \lambda \right) = \left| {\lambda {\rm I} - P} \right| = {\lambda ^3} + 2\lambda...

If $$A = \left[ {\matrix{ { - 3} & 2 \cr { - 1} & 0 \cr } } \right]$$ then $$A$$ satisfies the relation

If the loop gain $$K$$ of a negative feedback system having a loop transfer function $$K\left( {s + 3} \right)/{\left( {s + 8} \right)^2}$$ is to be adjusted to induce a sustained...

A closed loop system has the characteristic function $$\left( {{s^2} - 4} \right)\left( {s + 1} \right) + K\left( {s - 1} \right) = 0.$$ Its root locus plot against $$K$$ is

The algebraic equation $$F\left( s \right) = {s^5} - 3{s^4} + 5{s^3} - 7{s^2} + 4s + 20$$ $$F\left( s \right) = 0$$ has

A unity feedback system, having an open loop gain becomes stable when $$G\left( s \right)H\left( s \right) = {{K\left( {1 - s} \right)} \over {\left( {1 + s} \right)}}$$

The state variable description of a linear autonomous system is, $$\mathop X\limits^ \bullet = AX,\,\,$$ where $$X$$ is the two dimensional state vector and $$A$$ is the system mat...

The loop gain $$GH$$ of a closed loop system is given by the following expression $${K \over {s\left( {s + 2} \right)\left( {s + 4} \right)}}.$$ The value of $$K$$ for which the sy...

A control system with certain excitation is governed by the following mathematical equation $$${{{d^2}x} \over {d{t^2}}} + {1 \over 2}{{dx} \over {dt}} + {1 \over {18}}x = 10 + 5{e...

A unity feedback system has open loop transfer function $$G\left( s \right) = {{K\left( {s + 5} \right)} \over {s\left( {s + 2} \right)}};K \ge 0$$ (a) Draw a rough sketch of the r...

The number of roots on the equation $$2{s^4} + {s^3} + 3{s^2} + 5s + 7 = 0$$ that lie in the right half of $$S$$ plane is:

Closed loop stability implies that $$\left[ {1 + G\left( s \right)H\left( s \right)} \right]$$ has only ____________ in the left half of the $$s$$-plane.

Given the matrix $$A = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr { - 6} & { - 11} & { - 6} \cr } } \right].\,\,$$ Its eigen values are

The eigen values of the matrix $$\left[ {\matrix{ a & 1 \cr a & 1 \cr } } \right]$$ are