User:Santiago andresp/sandbox

= Anti-windup synthesis problem =

The static linear anti-windup synthesis is the problem of optimizing different frequency domain or time domain properties of a feedback-controlled dynamical system subjet to input saturation by setermining thegains of a static linear anti-windup strategy. Overshoot reduction, stabilization and convergence rate can be addressed within this problem.

Problem statement
Given a feedbak control system subject to imput saturation


 * $$ \left | \begin{matrix}\begin{array}{ccl} \dot{x}_p & = & A_p x_p + B_p \textrm{sat} (u)\\

y & = & C_p x_p + D_p \textrm{sat} (u) \\ \dot{x}_c & = & A_c x_c + B_c y + E_c \textrm{dz} (u) \\ u & = & C_c x_c + D_c y + F_c \textrm{dz} (u) \end{array}\end{matrix} \right .$$,

where $$\textrm{sat}(u)$$ is the actuator saturation function and the deadzone of the control signal $$u$$ is $$ \textrm{dz}(u) = u - \textrm{sat}(u)$$. Assuming that the controlled system is exponentially stable when there is no anti-windup action let $$ E_c $$ and $$ F_c $$ be the anti-windup gains to be designed. After some extensive algebraic derivarions, it can be found that


 * $$ \left | \begin{matrix} \begin{array}{ccl}

\dot{x} & = & A_{cl}x + \left ( B_0 + B_{aw} \begin{bmatrix} E_c \\ F_c \end{bmatrix} \right ) \textrm{dz}(u) \\ u & = & K_1 x + \left ( D_0 + D_{aw} \begin{bmatrix} E_c \\ F_c \end{bmatrix} \right ) \textrm{dz}(u) \end{array} \end{matrix} \right. $$, with $$ x = \begin{bmatrix} x_p^T & x_c^T \end{bmatrix}^T $$.

Solution based on Lyapunov stability certificates
Consider the common quadratic form $$ V = x^T P x $$, with $$ P = P^T $$ positive definite. Hence, if there exists matrix $$ P $$ such that $$ V > 0 $$ and


 * $$ \dot{V}(x) := \left \langle \nabla W(x), A_{cl} x + \left ( B_0 + B_{aw} \begin{bmatrix} E_c \\ F_c \end{bmatrix} \right ) \textrm{dz}(u) \right \rangle < 0 $$,

then, the closed loop system with input saturation and anti-windup action is exponentially stable.

Globally exponentially stable system
Consider the case in which the controlled system is globally exponentially stable, or similarly, when matrix $$A_p$$ is Hurwitz. In this case, static linear anti-windup is expected to better condition the characteristical overshoot and convergence rate of the closed loop dynamics.

Locally exponentially stable system
Now, consider the case in which the closed loop system is stable when there is no anti-windup action nor input saturation, but not globally stable. Here, the anti-windup action is in charge of expanding the volume of the basin of attraction of the origin of the dynamic system, while reducing overshoot and time response.