We want to solve for the output y under the input signal u = sin(t) both analytically and numerically. The initial condition at time t0 = 0 is [1, 0, 1]T . In the questions below, we ask you to implement several numerical integration methods in MATLAB and observe how the choice of these methods can affect the solution.

Consider the following linear time-invariant system: xA=Ax+Bu,y=Cx+Du=000110011,B=001,C=[100],D=[0]. 2. The Euler method (or forward Euler method) is a first-order method based on the approximation x(t+t)=x(t)+tx(t)=x(t)+t(Ax+Bu) when t is small. Using MATLAB, find the numerical solutions of y with two different choices of t:t=0.05 and t=0.005. Plot both the numerical and analytical solutions. 3. Use the following "delayed" step rule x(t+t)=x(tt)+2tx(tt)=x(tt)+2t(Ax+Bu) to find the numerical solution of y with t=0.05 and t=0.005 in MATLAB and plot both the numerical and analytical solutions. For initial condition, assume that x at time (t) is the same as x at time 0. 4. The MATLAB function ode45 is based on an explicit Runge-Kutta (4,5) formula 1. Use ode45 to solve for y and plot both the numerical and analytical solutions. 5. Briefly describe how the choice of the step size t and the numerical integration method affects the accuracy of the solution. x(t)=2ettettetet+1ettettetetm(t) y(t)=1 f(t)=C(SIA)1x(0)+[CCSIA)1B+0]h =c(ettet)(1+s(x)t)+1 Consider the following linear time-invariant system: xA=Ax+Bu,y=Cx+Du=000110011,B=001,C=[100],D=[0]. 2. The Euler method (or forward Euler method) is a first-order method based on the approximation x(t+t)=x(t)+tx(t)=x(t)+t(Ax+Bu) when t is small. Using MATLAB, find the numerical solutions of y with two different choices of t:t=0.05 and t=0.005. Plot both the numerical and analytical solutions. 3. Use the following "delayed" step rule x(t+t)=x(tt)+2tx(tt)=x(tt)+2t(Ax+Bu) to find the numerical solution of y with t=0.05 and t=0.005 in MATLAB and plot both the numerical and analytical solutions. For initial condition, assume that x at time (t) is the same as x at time 0. 4. The MATLAB function ode45 is based on an explicit Runge-Kutta (4,5) formula 1. Use ode45 to solve for y and plot both the numerical and analytical solutions. 5. Briefly describe how the choice of the step size t and the numerical integration method affects the accuracy of the solution. x(t)=2ettettetet+1ettettetetm(t) y(t)=1 f(t)=C(SIA)1x(0)+[CCSIA)1B+0]h =c(ettet)(1+s(x)t)+1