Lab 8A: Dynamical Systems¶

0 Environment Setup¶

For this lab, we will need some external code. The next cell will clone a repository with the code we need.

Make sure you have git installed in your system, or use Colab.

1 Dynamical Systems¶

A dynamical system is a system that evolves in time. We will consider here a state vector $x(t)$ and a control action $u(t)$. In a dynamical system the state at time $t+1$ is given by

\begin{equation}\tag{1} x(t+1) = f ( x(t), u(t) ). \end{equation}

The interpretation of the previous equation is that the state of the system $x(t)$ and the control action $u(t)$ determine the state of the system at time $t+1$. Our goal is to choose control actions $u(t)$ to manipulate the state trajectory $x(t)$ to satisfy some stated goal.

We point out that the system is time invariant, deterministic, and memoryless. Dynamical systems may also have memory, be stochastic, or time varying.

1.1 Circuit Navigation¶

A simple characterization of a car is to model it as an ideal point mass whose acceleration we control. Thus, the state of the system at time $t$ includes the position $ p(t)$ and the velocity v(t) of the car. Positions and velocities are both vectors with horizontal and vertical coordinates. We therefore write the state of the car as,

\begin{equation}\tag{2} x(t) ~=~ [ \, p(t) ; \, v(t) \, ] ~=~ [ \, p_{H(t)} ; \, p_{V(t)} ; \, v_{H(t)} ; \, v_{V(t)} \, ]. \end{equation}

The control action of the car is the acceleration $ a(t)$. As is the case of positions and velocities, the acceleration contains horizontal and vertical components. We therefore write the control action as

\begin{equation}\tag{3} u(t) ~=~ a(t) ~=~ [\, a_{H(t)} ; \, a_{V(t)} \, ]. \end{equation}

As we have assumed that the car is an ideal point mass, we can readily write the dynamical system using the laws of motion. To do so we just need to specify the sampling time $T_s$ that elapses between subsequent time instances and write,

\begin{align} & p(t+1) ~=~ p(t) + T_s v(t) + \frac{T_s^2}{2} a(t), \\ & v(t+1) ~=~ v(t) + T_s a(t) \tag{4}. \end{align}

This is true if the acceleration $ a(t)$ is maintained between times $t T_s $ and $(t+1)T_s$. It just follows from the definitions of velocity and acceleration.

In reality, the dynamics of a car are more complex than the dynamics of an ideal point mass. The laws of motion in previous equation are therefore a model of the real dynamical system.

Task 1¶

Load the data. This data contains $Q=10$ trajectories, which correspond to different laps of the circuit shown in Figure below. For each of these trajectories we have positions $ p_q(t)$, velocities $ v_q(t)$, and accelerations $ a_q(t)$ for times $t$ ranging from $t=0$ to $t=T=10$. Positions are measured in meters ($m$), velocities in meters per second ($m/s$), and accelerations in meters per second squared ($m/s^2$). The sampling time is $T_s = 0.1s$.

Plot the positions $ p_q(t)$ for some chosen trajectories.

1.2 Model Predicted States¶

We expect the previous model to be a reasonable approximation of the actual car dynamics. Let us test this expectation.

As a first figure of merit we consider the state predictions that the model makes. For a given trajectory $q$ and a given point in time $t$ we consider the observed state of the car $ x_q(t)$ and the control action $ u_q(t)$. We then use the ideal point mass model to predict the state at time $t+1$,

\begin{align}\label{eqn_circuit_navigation_laws_of_motion_as_predictions_one_step} &\hat{p}_q(t+1) ~=~ p_q(t) + T_s v_q(t) + \frac{T_s^2}{2} a_q(t), \\ &\hat{v}_q(t+1) ~=~ v_q(t) + T_s a_q(t) \tag{5}. \end{align}

We compare predicted positions $\hat{p}_q(t+1)$ and predicted velocities $\hat{v}_q(t+1)$ with observed positions $ p_q(t+1)$ and observed velocities $ v_q(t+1)$ to test the accuracy of the model.

Task 2¶

For a trajectory $ x_q(t)$ define the state estimation errors as

\begin{align} e_q(t) = \| p_q(t) – \hat{p}_q(t)\| + \| v_q(t) – \hat{v}_q(t)\| \tag{6}. \end{align}

Plot trajectory errors $e_q(t)$ for some trajectories. Compute and plot the average error $ \bar{e}_q(t)=(1/Q)\sum_{q=1}^Q e_q(t)$. Compute the average error over time $\bar{\bar{e}}_q(t)=(1/T)\sum_{t=1}^T \bar{e}_q(t)$.

Report Deliverables:

• Plot of trajectory errors $e_q(t)$
• Average error

1.3 Model Predicted Trajectories¶

The errors in Task 2 are small. This is an indication that the ideal point mass model is accurate. To further test the accuracy of this model we compare observed trajectories with model predicted trajectories. This entails selecting the actions $ a_q(t)$ of a particular trajectory to generate the model predicted trajectory as

\begin{align} &\hat{p}_q(t+1) ~=~ \hat{p}_q(t) + T_s \hat{v}_q(t) + \frac{T_s^2}{2} a_q(t), \nonumber \\ &\hat{v}_q(t+1) ~=~ \hat{v}_q(t) + T_s a_q(t) \tag{7}. \end{align}

Observe how in this expression we use the observed accelerations $ a_q(t)$ as control actions to generate the trajectory $\hat{x}(t) = [ \hat{p}(t) ; \hat{v}(t) ]$ predicted by the model in (4). This differs from the previous method in (5) in which state predictions at time $t+1$ depend on the state that is observed at time $t$. In this model (7) we compound predictions.

Comparing $\hat{x}(t)$ with $ x(t)$ gives an alternative measurement of the accuracy of the ideal point mass model in (4).

Task 3¶

To compare trajectories $x_q(t)$ with model predicted trajectories $\hat{x}(t)$ we define the position errors

\begin{align} e_q(t) = \| p_q(t) – \hat{p}_q(t+1)\| \tag{8}. \end{align}

Plot trajectory errors $e_q(t)$ for some trajectories. Compute and plot the average error $\bar{e}_q(t)=(1/Q)\sum_{q=1}^Q e_q(t)$.

Plot the positions $p_q(t)$ along with the predicted positions $\hat{p}(t)$ for some chosen trajectories.

These plots are likely unexpected. Comment.

Report Deliverables:

• Plot of trajectory errors $e_q(t)$
• Plot of an observed trajectory and the corresponding model predicted trajectory
• Paragraph explaining differences between observed and model predicted trajectories.

2 Model Learning¶

To mitigate the limitations of the ideal point mass model we can use observed trajectories to learn a more accurate model. To that end we consider a linear parameterization to predict the state at time $t+1$ given observations of the state and control input at time $t$,

\begin{equation}\tag{9} \hat{x\;} (t+1)= A x(t) + B u(t) \end{equation}

In this equation the matrices $A$ and $B$ are parameters that we will learn. These parameters map the observed state $x(t)$ and the observed control input $u(t)$ into a prediction $\hat{x\;}(t+1)$ of the next state.

The prediction made by {9} $x(t+1)$ is, in general, different from the actual state $x(t+1)$ observed at time $t+1$. We therefore seek parameters $A$ and $B$ that make predictions and reality as close as possible. For a loss function $\ell(x(t+1),\hat{x\;}(t+1))$ this gives the empirical risk minimization (ERM) problem

\begin{align} (A^*, B^*) &~=~ argmin_{A,B}~ \frac{1}{Q}\sum_{q=1}^Q ~ \frac{1}{T}\sum_{t=1}^T ~ \ell\, \Big( \, x(t+1) , \, \hat{x\;} (t+1) \, \Big) \nonumber \\ &~=~ argmin_{A, B}~ \frac{1}{Q}\sum_{q=1}^Q ~ \frac{1}{T}\sum_{t=1}^T ~ \ell\, \Big( \, x(t+1) , \, A x(t) + B u(t) \, \Big) \tag{10} . \end{align}

This problem differs from standard ERM in that we sum over trajectories and points in time. This is a minimal difference and the same techniques we use for solving standard ERM can be used to solve (10).

Task 4¶

Create an object to represent the linear dynamical system model in (9). Instantiate this object to represent a car navigating a circuit. This requires a state x(t) with the car’s position and velocity (2) and a control action u(t) representing the car’s acceleration (3).

Use the trajectories loaded in Task 1 to train a linear model for a car navigating a circuit. Use a MSE loss and remember to save some trajectories for validation. We recommend that you save a single trajectory for this purpose.

Report Deliverable: Report the training loss and the test loss.

The test loss that we observe in Task 4 is smaller than the average error that we observed in Task 2. This indicates that the learned model is better than the ideal point mass model.

Observe that the loss computed in Task 4 compares the model’s state predictions in (9) with observed states. As we did in Section 1.3 we may also consider predicted trajectories. This are defined as

\begin{equation}\tag{11} x(t+1) = A\hat x(t) + Bu(t). \end{equation}

In both, (9) and (11), we apply observed actions u(t). The difference is that in (9) we start from the observed state x(t) whereas in (11) we start from the predicted state xˆ(t). Concatenating state predictions results in the accumulation of errors. We evaluate this next.

Task 5¶

As in Task 3 we compare trajectories $x_q(t)$ with model predicted trajectories $\hat{x\;}_q(t)$. We do that by evaluating the position error $e_q(t)$. Plot trajectory errors $e_q(t)$ for test trajectories. If you have more than one test trajectory, compute and plot the average error.

Plot the positions $p_q(t)$ along with the predicted positions $\hat{p\;}(t)$ for a test trajectory.

Compared the learned model with the ideal point mass model.

Report Deliverables:

• Plot of test trajectory errors $e_q(t)$
• Plot of an observed trajectory and the corresponding learned model predicted trajectory
• Paragraph comparing ideal point mass model with learned model.