LMIs in Control/Matrix and LMI Properties and Tools/Passivity and Positive Realness

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This section deals with passivity of a system.

The System

Given a state-space representation of a linear system

{\displaystyle {\begin{aligned}\ {\dot {x}}=Ax+Bu\\\ y=Cx+Du\\\end{aligned}}}

${\displaystyle x\in \mathbb {R} ^{n},y\in \mathbb {R} ^{m},u\in \mathbb {R} ^{r}}$ are the state, output and input vectors respectively.

The Data

${\displaystyle A,B,C,D}$ are system matrices.

Definition

The linear system with the same number of input and output variables is called passive if

{\displaystyle {\begin{aligned}\int \limits _{0}^{T}u^{T}y(t)dt\geq 0\\\end{aligned}}}

(1)

hold for any arbitrary ${\displaystyle T\geq 0}$, arbitrary input ${\displaystyle u(t)}$, and the corresponding solution ${\displaystyle y(t)}$ of the system with ${\displaystyle x(0)=0}$. In addition, the transfer function matrix

{\displaystyle {\begin{aligned}G(s)&=C(sI-A)^{-1}B+D\\\end{aligned}}}

(2)

of system is called is positive real if it is square and satisfies

{\displaystyle {\begin{aligned}\ G^{H}(s)+G(s)\geq 0\forall s\in \mathbb {C} ,Re(s)>0\\\end{aligned}}}

(3)

LMI Condition

Let the linear system be controllable. Then, the system is passive if an only if there exists ${\displaystyle P>0}$ such that

{\displaystyle {\begin{aligned}\ {\begin{bmatrix}A^{T}P+PA&PB-C^{T}\\B^{T}P-C&-D^{T}-D\end{bmatrix}}\leq 0\\\end{aligned}}}

(4)

Implementation

This implementation requires Yalmip and Mosek.

Conclusion

Thus, it is seen that passivity and positive-realness describe the same property of a linear system, one gives the time-domain feature and the other provides frequency-domain feature of this property.