Question

A linear time-invariant discrete-time system is given by the following state-space representation:

x(k+1)=Ax(k)+Bu(k) y(k)=Cx(k)+Du(k)
a) x(k+1)=Ax(k)
b) y(k)=Cx(k)
c) u(k)=Du(k)
d) x(k+1)=Bu(k)

Answer 1

The question deals with the state-space representation of a linear time-invariant discrete-time system used in engineering to describe how the system evolves. It involves equations that represent the dynamics and output of the system in terms of matrices that define the system's behavior concerning states and inputs. The correct answer is a) x(k+1)=Ax(k).

Step-by-step explanation:

The student's question pertains to a linear time-invariant discrete-time system, which is a concept used in the field of engineering, more specifically in control systems and signal processing.

A state-space representation describes how a system evolves over time, and comprises of equations that represent the system dynamics.

The given equations can be broken down as follows: x(k+1) = Ax(k) + Bu(k) describes how the state of the system x changes with each discrete time step k based on the current state and the input u(k). Equation y(k) = Cx(k) + Du(k) defines the output y of the system based on the current state and input at time k.

Each part of the state-space representation has its own significance. A is the state matrix that defines the system dynamics. Matrix B models the way inputs affect the state, C is the output matrix that maps the state to the output, and D represents the direct transmission of the input to the output.