Question

Simulation of linear and nonlinear dynamic systems (a) For a linear dynamic system shown in state representation x˙=Ax+Bu,y = Cx+Du, we could use the MATLAB command lsim() for simulations. Suppose we have a linear system with 3 states (x1​,x2​ and x3​), with the dynamics

x₁​=x2​,
x₂​= x3​,
x₃​= −2x₁​−3x₂​−6x₃+u​
Generate the response of the system under the input signal u=10.0∗sin(2π it), for t∈.

Answer 1

The question is about simulating a linear dynamic system using MATLAB's lsim() function. The simulation will reveal the system's dynamic response to a given nonlinear input signal over time.

Step-by-step explanation:

The student asks about creating a simulation for a linear dynamic system using MATLAB, specifically using the lsim() function. Given the state-representation equations of motion for a system with three states, the dynamics are defined as x₁ = x₂, x₂ = x₃, and x₃ = −2x₁ −3x₂ −6x₃ + u. The question involves generating the system's response to the input signal u = 10.0*sin(2π*t), a nonlinear function of time.

To simulate this system in MATLAB using lsim(), one needs to define matrices A, B, C, and D according to the system equations provided. The matrix A represents the system dynamics, B represents the input effect on the system, C links the state variables to the output, and D represents the direct input-to-output mapping. For this particular set of equations, these matrices would be configured as follows:

A = [0 1 0; 0 0 1; -2 -3 -6]

B = [0; 0; 1]C = [1 0 0]D = [0]

Once the matrices are defined, the student can use the lsim() function along with a time vector and the sinusoidal input vector to produce the output response of the system. This process will yield a graphical representation of the system's output over time, providing insight into the dynamic behavior of the system.