# LMIs in Control/pages/KYP Lemma without Feedthrough

**The Concept**Edit

It is assumed in the Lemma that the state-space representation **(A, B, C, D)** is minimal. Then Positive Realness (PR) of the transfer function *C(SI − A) ^{-1}B + D * is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where

*(A, B, C, D)=(−α, 0, 0, 1),*with

*α > 0*. The transfer function is

*H(s) = 0*that is PR

**The System**Edit

Consider a contiuous-time LTI system, , with minimal state-space relization **(A, B, C, 0)**, where and .

**The Data**Edit

The matrices The matrices and

**LMI : KYP Lemma without Feedthrough**Edit

The system is positive real (PR) under either of the following equivalet necessary and sufficient conditions.

- 1. There exists where such that

- 2. There exists where such that

This is a special case of the *KYP* Lemma for QSR dissipative systems with **Q = 0, Q = 0.5** and **R = 0**.

The system is strictly positive real (SPR) under either of the following equivalet necessary and sufficient conditions.

- 1. There exists where such that

- 2. There exists where such that

This is a special case of the *KYP* Lemma for QSR dissipative systems with **Q = ε1, Q = 0.5** and **R = 0**. where ε

**Conclusion:**Edit

If there exist a positive definite for the the selected **Q,S** and **R** matrices then the system is **Positive Real**.

**Implementation**Edit

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

**Related LMIs**Edit

KYP Lemma

State Space Stability

Discrete Time KYP Lemma with Feedthrough

**References**Edit

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational
Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.

2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac-
tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.

3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2

4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London