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I am given a linear system $\dot{x} = Ax + Bu, y = Cx$, which is given to be passive with storage function $S(t) = \frac{1}{2}x^T Q x$, where $Q=Q^T\geq 0$. I am now looking for a way that I can show that the system is also shifted passive with respect to any steady state $\bar{x}$ with storage function

$S_\bar{x}(x) = \frac{1}{2}(x-\bar{x})^T Q (x-\bar{x})$

such that

$\frac{d}{dt}S_\bar{x}(x) \leq (y-\bar{y})^T(u-\bar{u})$

For examining the passivity of the non-shifted LTI system, I recognize that we should consider

$\dot{S}(x(t)) = [Ax+Bu]^T \frac{\partial S}{\partial x}(x) \leq u^TCx$

which we can split into two cases ($u=0$ and $u\neq0$), finally resulting in

$A^TQ +QA \leq 0, \quad B^T Q = C^T$,

which satisfies the Kalman-Yakubich-Popov (KYP) conditions, implying that the transfer matrix is positive real and thus, the LTI system is passive.

I've got the feeling that for the shifted system, I should use a similar operation. However, I can't set $u$ to a certain value that eliminates it from the full equation above.

If I would be able to derive the KYP conditions for the shifted system, I recognize that I can again show passivity of the system with respect to the new storage function $S(x(t))$ and supply rate $s(t)$. Can anyone indicate how I should proceed solving this problem?

Thom
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1 Answers1

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I am assuming $\bar{x}$, $\bar{u}$ satisfies $$A\bar{x}+B\bar{u}=0$$, Let $X=x-\bar{x}$, $U=u-\bar{u}$. The following is true $$\dot{X}=AX+BU. $$

Consider the time derivative of the storage function

$S_\bar{x}(x) = \frac{1}{2}X^T Q X$

This gives $$ \dfrac{d}{dt}S_\bar{x}(x) = X^T Q \dot X= X^T Q \left(AX+bU\right)=X^T QAX+ X^TbU \leq Y^\top U $$ where $Y=y-\bar{y}$ and $Y=b^\top X$. Above we used $X^T QAX = \dfrac{1}{2}X^T QA+A^\top QX \leq 0$.

From here I think you can see what are the required conditions.

Conclusion For Linear time-invariant systems KYP implies Passivity, Shifted passivity, Differential Passivity and Incremental Passivity.

kosa
  • 185