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= 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.