Systems

In the course, we will develop adaptive control methods for three types of systems: ARX, ARMAX, and state space models. ARX and ARMAX models provide a direct relation between the system inputs and outputs whereas in state space models, they are related through a state variable.

ARX models

We will consider auto-regressive models with exogenous (i.e., manipulated) inputs (ARX). They are in the form

\[ y_t + a_1 y_{t-1} + \cdots + a_{n_a} y_{t-n_a} = b_0 u_t + b_1 u_{t-1} + \cdots + b_{n_b} u_{t - n_b} + e_t, \]

where \(y_t\) and \(u_t\) are the output and input at time \(t\) and the noise term \(e_t\) is a normally distributed random variable. By introducing the backshift operator, \(q^{-1}\), we can write the ARX model as

\[ y_t + a_1 q^{-1} y_t + \cdots + a_{n_a} q^{-n_a} y_t = b_0 u_t + b_1 q^{-1} u_{t-1} + \cdots + b_{n_b} q^{-n_b} u_t + e_t. \]

Furthermore, we can introduce the polynomials \(A(q^{-1})\) and \(B(q^{-1})\) in the backshift operator and write the ARX model in the form

\[ A(q^{-1}) y_t = B(q^{-1}) u_t + e_t. \]

Finally, it is convenient to write the ARX model in the form

\[ y_t = \phi_t^T \theta + e_t, \]

where \(\phi_t\) is a vector of regressors and \(\theta\) is a vector of parameters (i.e., the polynomial coefficients):

\[ \phi_t = [-y_{t-1}   \cdots   -y_{t - n_a}   u_t   \cdots   u_{t - n_b}]^T, \quad \theta = [a_1   \cdots   a_{n_a}   b_0   \cdots   b_{n_b}]^T. \]

ARMAX models

We will also consider auto-regressive moving average models with exogenous inputs (ARMAX):

\[ y_t + a_1 y_{t-1} + \cdots + a_{n_a} y_{t-n_a} = b_0 u_t + b_1 u_{t-1} + \cdots + b_{n_b} u_{t - n_b} + c_0 e_t + c_1 e_{t-1} + \cdots + c_{n_c} e_{t - n_c}. \]

It can also be written in the form

\[ A(q^{-1}) y_t = B(q^{-1}) u_t + C(q^{-1}) e_t. \]

State space models

Finally, we will consider state space models in the form

\[ x_{t+1} = A x_t + B u_t + v_t, \]

\[ y_t = C x_t + D u_t + e_t, \]

where \(x_t\) is the system state at time \(t\), \(u_t\) is the manipulated input, and \(y_t\) is the output. Furthermore, \(A\), \(B\), \(C\), and \(D\) are the system matrices, and the process noise \(v_t\) and the measurement noise \(e_t\) are normally distributed. Finally, the initial state \(x_0\) is also normally distributed.