diff --git a/README.md b/README.md
index 9cd4261..8b6e91d 100644
--- a/README.md
+++ b/README.md
@@ -30,7 +30,14 @@ See also the [examples](examples) folder.
 
 Tested on Ubuntu 20.04 LTS.
 
-Git clone the repository and run the file `setup.py` to install the required dependencies.
+Firstly, `git clone` this repository. To install the dependencies in a venv virtual environment run:
+
+```
+python -m venv rlgainenv
+source ./rlgainenv/bin/activate
+```
+
+then, run the command `python3 setup.py install`or `pip install -r requirements.txt` to install the required dependencies.
 
 #### Maintainer
 
diff --git a/atoms/kalmanFilter.py b/atoms/kalmanFilter.py
index a208464..68909fd 100644
--- a/atoms/kalmanFilter.py
+++ b/atoms/kalmanFilter.py
@@ -1,5 +1,4 @@
-import numpy
-from numpy import dot, sum, tile
+import numpy as np
 from numpy.linalg import inv, det
 from atoms.atoms_helpers import Helpers
 
@@ -18,6 +17,7 @@ class KalmanFilter:
 
     def __init__(self, debug=False):
         self.variables = {}
+        self.debug = debug
 
         if debug:
             self.logger = Helpers.init_logger()
@@ -38,10 +38,10 @@ def setup(self, variables):
             if var in var_keys:
                 self.variables.update({var: variables[var]})
             else:
-                raise ValueError('Required variable', var, 'not found in the input dictionary.')
+                raise ValueError(f'Required variable {var} not found in the input dictionary.')
 
-        # add matrix P to the variables dictionary
-        self.variables.update({'P': numpy.zeros(variables['Q'].shape)})
+        # Initialize matrix P to the identity matrix scaled by a large number
+        self.variables.update({'P': np.eye(variables['Q'].shape[0]) * 1000})
 
     def predict(self):
         """
@@ -54,11 +54,14 @@ def predict(self):
         - Q = process noise covariance matrix
         """
         # calculate X(k) from (k-1) quantities
-        self.variables['X'] = dot(self.variables['A'], self.variables['X']) + dot(self.variables['B'],
-                                                                                  self.variables['U'])
+        self.variables['X'] = self.variables['A'] @ self.variables['X'] + self.variables['B'] @ self.variables['U']
+
         # calculate the state covariance P(k) from (k-1) quantities
-        self.variables['P'] = dot(self.variables['A'], dot(self.variables['P'],
-                                                           self.variables['A'].T)) + self.variables['Q']
+        self.variables['P'] = self.variables['A'] @ self.variables['P'] @ self.variables['A'].T + self.variables['Q']
+
+        if self.debug:
+            self.logger.info(f"Predicted state X: {self.variables['X']}")
+            self.logger.info(f"Predicted state covariance P: {self.variables['P']}")
 
     def update(self, y_measured):
         """
@@ -70,40 +73,30 @@ def update(self, y_measured):
         - R = measurements noise covariance matrix
         Returns the predictive probability (likelihood) of the measurements.
         """
-        y_mean_predicted = dot(self.variables['C'], self.variables['X'])
-        y_covariance = self.variables['R'] + dot(self.variables['C'], dot(self.variables['P'], self.variables['C'].T))
-        k_gain = dot(self.variables['P'], dot(self.variables['C'].T, inv(y_covariance)))
+        y_mean_predicted = self.variables['C'] @ self.variables['X']
+        y_covariance = self.variables['R'] + self.variables['C'] @ self.variables['P'] @ self.variables['C'].T
+        k_gain = self.variables['P'] @ self.variables['C'].T @ inv(y_covariance)
 
         # correct the predicted state and covariance matrix
-        self.variables['X'] = self.variables['X'] + dot(k_gain, (y_measured - y_mean_predicted))
-        self.variables['P'] = self.variables['P'] - dot(k_gain, dot(y_covariance, k_gain.T))
+        self.variables['X'] = self.variables['X'] + k_gain @ (y_measured - y_mean_predicted)
+        self.variables['P'] = self.variables['P'] - k_gain @ self.variables['C'] @ self.variables['P']
         x_estimated = self.variables['X']
 
         # calculate the predictive probability of the measurements
         y_predictive_prob = self.__gauss_pdf(y_measured, y_mean_predicted, y_covariance)
 
+        if self.debug:
+            self.logger.info(f"Updated state X: {self.variables['X']}")
+            self.logger.info(f"Updated state covariance P: {self.variables['P']}")
+            self.logger.info(f"Kalman Gain K: {k_gain}")
+            self.logger.info(f"Predictive probability: {y_predictive_prob}")
+
         return x_estimated, y_predictive_prob
 
     @staticmethod
     def __gauss_pdf(y_measured, y_mean_predicted, y_covariance):
-
-        # see also https://arxiv.org/pdf/1204.0375.pdf
-        if y_mean_predicted.shape[1] == 1:
-            delta_y = y_measured - tile(y_mean_predicted, y_measured.shape[1])
-            E = 0.5 * sum(delta_y * (dot(inv(y_covariance), delta_y)), axis=0)
-            E = E + 0.5 * y_mean_predicted.shape[0] * numpy.log(2 * numpy.pi) + 0.5 * numpy.log(det(y_covariance))
-            y_predictive_prob = numpy.exp(-E)
-
-        elif y_measured.shape[1] == 1:
-            delta_y = tile(y_measured, y_mean_predicted.shape[1]) - y_mean_predicted
-            E = 0.5 * sum(delta_y * (dot(inv(y_covariance), delta_y)), axis=0)
-            E = E + 0.5 * y_mean_predicted.shape[0] * numpy.log(2 * numpy.pi) + 0.5 * numpy.log(det(y_covariance))
-            y_predictive_prob = numpy.exp(-E)
-
-        else:
-            delta_y = y_measured - y_mean_predicted
-            E = 0.5 * dot(delta_y.T, dot(inv(y_covariance), delta_y))
-            E = E + 0.5 * y_mean_predicted.shape[0] * numpy.log(2 * numpy.pi) + 0.5 * numpy.log(det(y_covariance))
-            y_predictive_prob = numpy.exp(-E)
-
+        delta_y = y_measured - y_mean_predicted
+        exponent = -0.5 * delta_y.T @ inv(y_covariance) @ delta_y
+        normalization = 0.5 * len(y_mean_predicted) * np.log(2 * np.pi) + 0.5 * np.log(det(y_covariance))
+        y_predictive_prob = np.exp(exponent - normalization)
         return y_predictive_prob
diff --git a/examples/example_kalmanFilter.py b/examples/example_kalmanFilter.py
index f7a3de5..5153e15 100644
--- a/examples/example_kalmanFilter.py
+++ b/examples/example_kalmanFilter.py
@@ -4,81 +4,91 @@
 from atoms.kalmanFilter import KalmanFilter
 
 """
-In the example code we use KF to track a mobile user connected to a wireless network. The discrete-time dynamical 
-system is given by:
+Example of using Kalman Filter to track a mobile user connected to a wireless network. 
+The discrete-time dynamical system is given by:
 
  X(k+1) = A*X(k) + B*U(k)
- 
-The matrix of measurement Y describes the estimated position of the mobile:
- 
+
+The measurement matrix Y describes the estimated position of the mobile:
+
  Y(k) = C*X(k)
- 
-We plot the estimated, the real trajectory of the mobile user, and the measurements.
+
+We plot the estimated trajectory, the real trajectory, and the noisy measurements.
 """
+
+# Initialize logger
 logger = Helpers.init_logger()
 logger.info('Example of Kalman Filter estimation.')
 
-# time step of mobile user movement
-dt = 0.1
-
-# initialization of state matrices
-A = np.array([[1, 0, dt, 0], [0, 1, 0, dt], [0, 0, 1, 0], [0, 0, 0, 1]])
-X = np.array([[0.0], [0.0], [0.1], [0.1]])
+# Time step of mobile user movement
+dt = 0.01
+
+# Initialization of state matrices
+A = np.array([
+    [1, 0, dt, 0],
+    [0, 1, 0, dt],
+    [0, 0, 1, 0],
+    [0, 0, 0, 1]
+])
+X = np.array([[0.0], [0.0], [0.1], [0.1]])  # Initial state
 B = np.eye(X.shape[0])
-U = np.zeros((X.shape[0], 1))
+U = np.zeros((X.shape[0], 1))  # Control input
 
-# process noise covariance matrix and initial state covariance
+# Process noise covariance matrix
 Q = np.eye(X.shape[0])
-P = np.diag((0.01, 0.01, 0.01, 0.01))
 
-# measurement matrices (state X plus a random gaussian noise)
-Y = np.array([[X[0, 0] + abs(np.random.randn(1)[0])], [X[1, 0] + abs(np.random.randn(1)[0])]])
+# Measurement matrix (state X plus a random Gaussian noise)
 C = np.array([[1, 0, 0, 0], [0, 1, 0, 0]])
 
-# measurement noise covariance
-R = np.eye(Y.shape[0])
+# Measurement noise covariance
+R = np.eye(C.shape[0])
 
-# Set up the KF class and simulate the estimation process
+# Set up the Kalman Filter class
 logger.info('Setting up the KF class...')
+variables = {
+    'X': X, 'A': A, 'B': B, 'U': U, 'Q': Q, 'C': C, 'R': R
+}
+kf = KalmanFilter(debug=True)
+kf.setup(variables)
 
-var = {}
-var.update({'X': X})
-var.update({'A': A})
-var.update({'B': B})
-var.update({'U': U})
-var.update({'Q': Q})
-var.update({'C': C})
-var.update({'R': R})
-
-kf = KalmanFilter()
-kf.setup(var)
-
-# number of iterations in Kalman Filter
+# Number of iterations in Kalman Filter
 n_iter = 50
-y_predictive_prob = np.zeros((n_iter, 1))
+y_measured = np.zeros((C.shape[0], n_iter))
 x_estimated = np.zeros((X.shape[0], n_iter))
 
 # Applying the Kalman Filter
-for i in np.arange(0, n_iter):
+for i in range(n_iter):
     kf.predict()
-    x_est, y_predict = kf.update(Y)
-    x_estimated[0:4, i] = x_est.reshape(1, -1)
-    y_predictive_prob[i, 0] = y_predict.reshape(1, -1)
-    Y = np.array([[X[0, 0] + abs(0.1 * np.random.randn(1)[0])], [X[1, 0] + abs(0.1 * np.random.randn(1)[0])]])
+    # Simulate measurement with some noise
+    Y = C @ X + np.random.normal(0, 0.1, (C.shape[0], 1))
+    y_measured[:, i] = Y.flatten()
+
+    x_est, _ = kf.update(Y)
+    x_estimated[:, i] = x_est.flatten()
+
+    # Simulate real system's next state (for demonstration purposes)
+    X = A @ X + B @ U
 
 logger.info('Estimation complete.')
 
 # Plot the results
 logger.info('Plotting results ...')
-
-time = np.arange(0, 50*dt, dt)
-x_estimated = x_estimated.transpose()
-plt.figure()
-plt.plot(time, x_estimated)
-plt.xlabel('Time')
-plt.ylabel('State (X)')
-plt.title('KF results')
-plt.legend(['x1', 'x2', 'x3', 'x4'], loc='upper right')
+time = np.arange(0, n_iter * dt, dt)
+
+fig, axs = plt.subplots(2, 2, figsize=(10, 8))
+
+# Plot each state variable
+for i in range(X.shape[0]):
+    row, col = divmod(i, 2)
+    axs[row, col].plot(time, x_estimated[i, :], label='Estimated')
+    axs[row, col].scatter(time, y_measured[i % 2, :], s=10, color='r', label='Measured', alpha=0.6)
+    axs[row, col].set_xlabel('Time [s]')
+    axs[row, col].set_ylabel(f'State X{i + 1}')
+    axs[row, col].legend(loc='upper right')
+    axs[row, col].grid(True)
+
+fig.suptitle('Kalman Filter Results')
+plt.tight_layout(rect=[0, 0, 1, 0.96])
 plt.show()
 
 logger.info('Done!')
diff --git a/examples/example_linearMPC.py b/examples/example_linearMPC.py
index d0db76e..9ff8214 100644
--- a/examples/example_linearMPC.py
+++ b/examples/example_linearMPC.py
@@ -14,80 +14,77 @@
 Rewrite the problem in its state-space form:
 
   y(k)   = [x_1(k); x_2(k)]
-  y(k+1) = A*y(k) + B*u(k) = [1 1; 0 1]*y(k) + [0; 1]*u(k)
+  y(k+1) = A*y(k) + B*u(k) = [1 dt; 0 1]*y(k) + [0; dt]*u(k)
 
 Setup a constrained linear-quadratic MPC problem to stabilize the system using multiple shooting.
 """
 logger = Helpers.init_logger()
 logger.info('Example of linear MPC problem.')
 
-# set the state and input dimensions, initial time, and define matrices A and B
+# Define the time step
+dt = 0.025
+
+# Define state-space matrices for a double integrator
 n_x = 2
-n_u = 2
-time_init = 0
-A = np.block([[np.eye(n_x), np.eye(n_x)],
-              [np.zeros((n_x, n_x)), np.eye(n_x)]])
-B = np.block([[np.zeros((n_x, n_u))], [np.eye(n_u)]])
+n_u = 1
+A = np.array([[1, dt], [0, 1]])
+B = np.array([[0], [dt]])
 
-# define all variables needed for setting up the MPC problem
+# Define all variables needed for setting up the MPC problem
 var = {}
-var.update({'N': 6})
+var.update({'N': 20})
 var.update({'A': A})
 var.update({'B': B})
-var.update({'Q_N': np.diag([20, 20, 10, 10], k=0)})
-var.update({'Q': np.diag([2, 2, 1, 1], k=0)})
-var.update({'R': 15*np.eye(n_u)})
-var.update({'x_r': np.array([np.cos(2*np.pi*time_init), -np.cos(2*np.pi*time_init), 0, 0])})
-var.update({'x_0': np.array([0.8, -0.8, 0, 0])})
-var.update({'x_min': np.array([-2*np.pi, -2*np.pi, -100*np.pi/180, -100*np.pi/180])})
-var.update({'x_max': np.array([2*np.pi,  2*np.pi,  100*np.pi/180,  100*np.pi/180])})
-var.update({'u_min': np.array([-5, -5])})
-var.update({'u_max': np.array([5, 5])})
-
-# create the linear MPC object and setup
+var.update({'Q_N': np.diag([200, 200])})
+var.update({'Q': np.diag([2, 2])})
+var.update({'R': 0.1 * np.eye(n_u)})
+var.update({'x_r': np.array([1.0, 0.0])})
+var.update({'x_0': np.array([0.0, 0.5])})
+var.update({'x_min': np.array([-2, -10])})
+var.update({'x_max': np.array([2, 10])})
+var.update({'u_min': np.array([-5])})
+var.update({'u_max': np.array([5])})
+
+# Create the linear MPC object and setup
 opti = LinearMPC(debug=True)
 opti.setup(var)
 
-# simulate the problem in closed loop
-time = time_init
+# Simulate the problem in closed loop
+time = 0
 n_sim = 250
-y = np.zeros((n_sim, 2*n_x))
-y_r = np.zeros((n_sim, 2*n_x))
+y = np.zeros((n_sim, n_x))
+y_r = np.zeros((n_sim, n_x))
 logger.info('Running simulation ...')
 
 for i in range(n_sim):
-
-    # solve the problem
+    # Solve the problem
     u_star = opti.solve()
 
-    # apply first control input and update initial conditions
+    # Apply first control input and update initial conditions
     y[i, :] = var['x_0']
-    u = u_star[(var['N']+1)*2*n_x:(var['N']+1)*2*n_x+n_u]
+    u = u_star[n_x * (var['N']+1):n_x * (var['N']+1) + n_u]
     x_0 = A.dot(var['x_0']) + B.dot(u)
     var.update({'x_0': x_0})
 
-    # update the reference trajectory
-    time = time + 0.025
-    x_r = np.array([np.cos(2*np.pi*time), -np.cos(2*np.pi*time), 0, 0])
+    # Update the reference trajectory
+    time = time + dt
+    x_r = np.array([1.0, 0.0])
     y_r[i, :] = x_r
     var.update({'x_r': x_r})
     opti.update(x_r=x_r, x_0=x_0)
 
-# plot the results
+# Plot the results
 logger.info('Plotting results ...')
-fig, axs = plt.subplots(2, 2)
+fig, axs = plt.subplots(2, 1)
 cont = 0
 
 for i in range(n_x):
-    for j in range(n_x):
-        axs[i, j].plot(range(n_sim), y[:, cont], range(n_sim), y_r[:, cont])
-        axs[i, j].set_xlabel('iters')
-        axs[i, j].set_ylabel('data')
-        # set legend
-        axs[i, j].legend(['measured', 'reference'], loc='upper left')
-        axs[i, j].grid(True)
-        fig.tight_layout()
-        cont = cont + 1
+    axs[i].plot(range(n_sim), y[:, i], range(n_sim), y_r[:, i])
+    axs[i].set_xlabel('Iterations')
+    axs[i].set_ylabel(f'x_{i+1}')
+    axs[i].legend(['Measured', 'Reference'], loc='upper left')
+    axs[i].grid(True)
 
+fig.tight_layout()
 plt.show()
 logger.info('Done!')
diff --git a/setup.py b/setup.py
index 83308db..3bbf2d4 100644
--- a/setup.py
+++ b/setup.py
@@ -2,7 +2,7 @@
 
 setup(
     name='atoms',
-    version='1.0',
+    version='1.1',
     packages=find_packages(),
     install_requires=[
         'osqp',