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.