# What happens when we give an input that is a sum of two other inputs?

simulate <- function( pulse1, pulse2 ) {
  
  
  # Define the time of simulation
  maxTime <- 6
  dt <- 0.1 # Precision of simulation
  
  # Prepare variable for results of simulation
  ts <- seq(0,maxTime,dt)
  N <- length(ts) # Number of samples
  x <- numeric( N ) # The results will go here
  
  # Run the simulation
  x[1] <- 0 # Initial condition
  for( i in 2:N ) {
    # The input for the current time:
    if( ts[i] > 1 & ts[i] <= 2 ) {
      u <- pulse1
    } else if( ts[i] > 3 & ts[i] <= 4 ) {
      u <- pulse2
    } else {
      u <- 0
    }
    
    # The dynamical system: DEFINED OUTSIDE THIS FUNCTION
    dx <- system( x[i-1], u )
    
    # Numerical integration:
    x[i] <- x[i-1] + dt*dx
  }
  return( list( ts, x ) )
}


# Define the dynamical system
system <- function( x, u ) {
  -0.5 * x + u
}


# Simulate the dynamical system
results <- simulate( 1, 0 )

# Store results for first pulse
x1 <- results[[2]]

# Repeat for second pulse
results <- simulate( 0, 1 )
x2 <- results[[2]]

# Repeat for both pulses
results <- simulate( 1, 1 )
xboth <- results[[2]]

# Compare
xcalc <- x1 + x2

# Plot the results
matplot( results[[1]], xboth, xlab = "Time", ylab = "x", type = "l",col="red",add = FALSE )
matplot( results[[1]], x1, xlab = "Time", ylab = "x", type = "l",col="blue",add = TRUE )
matplot( results[[1]], x2, xlab = "Time", ylab = "x", type = "l",col="green",add = TRUE )
matplot( results[[1]], xcalc, xlab = "Time", ylab = "x", type = "p",col="black",add = TRUE )