IAA Flashcards

Key concepts and equations for Intelligence & Autonomy, covering aspects of control theory, machine learning, and more broadly mechatronics.

What is observability?

Observability refers to the ability of a control system to observe/measure the system state (conduct accurate state estimation).

In other words, the system’s complete internal state can be uniquely inferred from the system’s output measurements over a finite interval.

What happens if a system does not meet observibility conditions?

If a system is not observable, it is unable to conduct accurate state estimation (aka the robot doesn’t know where it is) which may lead to unstable or non-optimal control commands.

Additionally, control laws like LQR require state knowledge (x) to calculate motor commands (u); these controllers cannot perform when the system is not observable.

What is the mathematical condition for observability for an LTI system?

The discrete-time LTI system can be described by the state equation x_k+1 = Ax_k + Bu_k with output measurements y_k = Cx_{k .}

For any initial state x₀, outputs y_k (k = 0, 1, …) are sufficient to determine x₀.

The system is observable if the observability matrix H = [C, CA, CA², … , CA^n-1] is non-singular (det(H) not equal to zero).

If H is non-square, it must have full rank n where n is the number of states.

What is controllability?

Controllability refers to the ability fo a system to achieve a desired output by applying a specifc controlled input.

In other words, it’s the ability to control the state of the system.

What happens if a system does not meet controllability conditions?

If a system is not controllable, it cannot reach certain states or potentially even its goal, regardless of control commands inputted.

This means the system is reduced to a lower-dimensional subspace, and control laws like LQR cannot be applied.

What is the mathematical condition for controllability for an LTI system?

For a system to be controllable, it must be able to transfer the system from any initial state x(0) = x₀ to any desired final state x(T) = x_T in a finite time interval by applying a sequence of controls u₀, u₁, …, u_T-1.

For a discrete, LTI system described by the dynamics x_k+1 = Āx_k + Bu_k, it is controllable if the controllability matrix G = [B AB A²B … A^n-1B] is non-singular (det(G) is not zero).

If G is non-square, it must have full rank n where n is the number of states.

What is PCA?

Principal Component Analysis (PCA) is a statistical technique used for dimensionality reduction. It projects a dataset onto a lower-dimensional (principal) subspace such that the variance of the projected data is maximised.

The M-dimensional projection space is defined by the M eigenvectors corresponding to the M-largest eigenvalues of the covariance matrix ∑ of the data.

The eigenvector with the largest eigenvalue is the first principal component (1st axis), showing the direction of the most variance in the data. The second-largest eigenvalue-eigenvector pair shows the direction of the 2nd-greatest variance in the data, or the 2nd axis, which is orthogonal to the 1st. And so on for all eigenvectors of ∑.

What is the equation for the covariance matrix ∑?

$$∑=\frac{1}{1-n}\sum_{i=1}^{n}(x_{i}-\mu)(x_{i}-\mu)^{T}$$

What is the significance of the eigenvalues and eigenvectors of the covariance matrix in PCA?

The covariance matrtix in PCA can be used to inform dimensionality reduction. As ∑ is symmetric and positive semi-definite, its eigenvalues are real and non-negative, and its eigenvectors are orthogonal to each other.

The eigenvectors represent the directions of the principal components, and their eigenvalues represent the amount of variance explain by each component.

By discarding the components with the smallest eigenvalues, the dimensionality of the dataset can be reduced whilst variance is maximised.

What is LQR?

The Linear Quadratic Regulator (LQR) is an optimal control method that calculates the best feedback gain K (for optimal control law u = -Kx) using the Riccati Equation to stabilise/regulate a linear system (described by dynamics ẋ = Ax + Bu) by minimising a quadratic cost function.

What is the cost function minimised by LQR?

In the finite-horizon case, there is both cost rate $j(x,u) = \frac{1}{2}u^TRu + \frac{1}{2}x^TQx$ and final cost $h(x)=\frac{1}{2}x^{T}Q_{F}x$ .

In the infinite-horizon case, the quadratic cost function is $J = \int_{0}^{\infty} (x(t)^TQx(t) + u(t)^TRu(t))dt$.

$R$ is symmetric and positive-definite, $Q$and $Q_F$ are symmetric only. The R matrix weights how much you care about control effort (could correlate with energy usage) and the Q matrix weights how much you care about state errors / accuracy.

What does it mean for a matrix to be positive-definite (4 things)?

It is often symmetric.
Its eigenvalues are all strictly positive.
Its quadratic form must be greater than zero: $x^TAx\geq 0$ .
It is invertible (and non-singular).

What is the solution to the Riccati EQ?

Continuous-time: $u = -R^{-1}B^TV(t)x$ where $R^{-1}B^TV(x)$ is the feedback gain K.

Discrete-time: $u=-\left(R+B^{T}V_{k+1}B\right)^{-1}B^{T}V_{k+1}Ax$ where $\left(R+B^{T}V_{k+1}B\right)^{-1}B^{T}V_{k+1}A$ is the feedback gain K.

$u = -Kx$ is the optimal control law.