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1. Sistem berbilang input berbilang output (MIMO) digambarkan dalam Rajah 1. A...

Nov 19, 2024

Solution by Steps

step 1

To identify the state-space model of the MIMO system, we can represent the system in the form:

\dot{x}(t) = Ax(t) + Bu(t)

where

A

is the system matrix,

B

is the input matrix, and

u(t)

is the input vector. The outputs are given as

y_1 = x_1

and

y_2 = x_2

step 2

To check the stability of the system, we need to find the eigenvalues of the matrix

A

. The system is stable if all eigenvalues have negative real parts. This can be done by calculating the characteristic polynomial and finding its roots

step 3

The state-transition matrix

\Phi(t)

for a continuous-time system can be computed using the formula:

\Phi(t) = e^{At}

This matrix describes the evolution of the state over time

step 4

To find the state

x(t)

given a unit step input and initial conditions

x(0) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}

, we can use the formula:

x(t) = \Phi(t)x(0) + \int_0^t \Phi(t - \tau)Bu(\tau)d\tau

where

u(t)

is the unit step function

Answer

The state-space model, stability check, state-transition matrix, and state

x(t)

can be derived as described in the steps above.

Key Concept

The state-space representation is a mathematical model of a physical system that describes its dynamics using state variables.

Explanation

The solution involves identifying the state-space model, checking stability through eigenvalues, calculating the state-transition matrix, and determining the state response to a unit step input.