I agree with Kranthi Kumar Deveerasetty's comment on Kharitonov theorem "The Kharitonov theorem fails to give stability for an asymptotically stable system (or plant)."

(1) Kharitonov’s theorem is used to assess the stability of a dynamical system when the physical parameters of the system are uncertain (see the following book published by J. Ackermann in 1993 [Ackermann1993]).

When the coefficients of the characteristic polynomial are known, the Routh-Hurwitz stability criterion can be used to check if the system is stable (i.e. if all roots have negative real parts). Kharitonov's theorem can be used in the case where the coefficients are only known to be within specified ranges. It reduces checking stability of an infinite number of polynomials to checking stability of only four so-called Kharitonov polynomials.

http://en.wikipedia.org/wiki/Kharitonov's_theorem

J. Ackermann et al regarded Kharitonov’s theorem as the Parameter Space Approach [ABBGKKMO02].

(2) However, when the dependency among polynomial coefficients is nonlinear (e.g., a asymptotically stable system), J. Ackermann demonstrated in the following IEEE TAC 1992 paper that checking a subset of a polynomial family generally can NOT guarantee the stability of the entire family.

J. Ackermann, "Does it suffice to check a subset of multilinear parameters in robustness analysis?" IEEE Transactions on Automatic Control, 37(4), April 1992, pp.487-488.

"For polynomials with general multilinear dependence on uncertain real parameters it does not suffice to check a subset for Q for stability."

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6117295

(3) J. Ackermann published the following book to present methods for robustness analysis that guarantee the desired properties for all possible values of the plant uncertainty.

[Ackermann93] J. Ackermann, Robust Control: Systems with Uncertain Physical Parameters, Springer-Verlag, New York, 1993.

http://www.springer.com/engineering/robotics/book/978-1-4471-3365-0

Short review on J. Ackermann's 1993 book by IEEE TCST 2001 paper:

W.K. Ho, T.H. Lee, H.P. Han, and Y. Hong, "Self-Tuning IMC-PID Control with Interval Gain and Phase Margin Assignment," IEEE Transactions on Control Systems Technology, 9(3), May 2001, pp. 535-541. Available from the following RG Link.

"A controller may be found such that all the closed-loop poles are located in a specified region—a pole-region assignment problem—and remain in this region if the plant parameters remain within the upper and lower bounds [Ackermann93]."

H. Nyquist (Sweden) --> K.J. Astrom (Sweden) --> W.K. Ho (Sweden)

     | Nyquist plot (published in 1932)

     |  Bell Labs

    V  Bode plot (published in 1940)

H.W. Bode (Harvard) --> K.S. Narendra (Harvard, Yale) --> T.H. Lee (Yale)

https://www.researchgate.net/publication/3332273_Self-tuning_IMC-PID_control_with_interval_gain_and_phase_marginsassignment

J. Ackermann and his peer researchers co-published the 2nd edition of the previously published book [Ackermann93] in 1993.

[ABBGKKMO02] J. Ackermann et al., Robust Control: The Parameter Space Approach, Springer-Verlag, New York, 2002.

http://www.amazon.com/Robust-Control-Parameter-Communications-Engineering/dp/1852335149

(4) Quote Kranthi Kumar Deveerasetty's comment "Is there any other method to overcome from this problem?"

The introduction section of the following Automatica 2009 paper provides an extensive reviews on the follow-up work which aims at overcome the limitation of Kharitonov’s theorem.

J. Fisher and R. Bhattacharya, "Linear quadratic regulation of systems with stochastic parameter uncertainties," Automatica, 45(12), December 2009, pp. 2831-2841,

"Kharinotov’s work renewed interested in this area and led to various results that extended well known classical control techniques in the frequency domain (Ackermann, 2002, Bhattacharyya et al., 1995 and Mansour et al., 1992). ....... "

http://www.sciencedirect.com/science/article/pii/S0005109809004622

(5) Iterative feedback tuning (IFT) was proposed to tune controller parameters for system with unknown parameters. A linear control model can be used to approximate non-linear system with unknown parameters roughly, and such control plant approximation may exist large deviation from the real non-linear system, which would cause classical tuning method of PID controller ineffective. IFT was proposed to solve controller tuning issues caused by plant uncertainty. Iterative feedback tuning has been integrated with PID controller to solve controller tuning issues caused by plant uncertainty.

W.K. Ho, Y. Hong, A. Hansson, H. Hjalmarsson, and J.W. Deng, "Relay auto-tuning of PID controllers using iterative feedback tuning," Automatica 39 (1), January 2003, pp. 149-157. Available from the following RG Link.

https://www.researchgate.net/publication/223504459_Relay_auto-tuning_of_PID_controllers_using_iterative_feedback_tuning

(6) Recent discussions on control system design

"What are trends in control theory and its applications in physical systems (from a research point of view)?"

https://www.researchgate.net/post/What_are_trends_in_control_theory_and_its_applications_in_physical_systems_from_a_research_point_of_view2

Article Self-tuning IMC-PID control with interval gain and phase mar...

Article Relay auto-tuning of PID controllers using iterative feedback tuning

I agree with Kranthi Kumar Deveerasetty's comment on Kharitonov theorem "The Kharitonov theorem fails to give stability for an asymptotically stable system (or plant)."

(1) Kharitonov’s theorem is used to assess the stability of a dynamical system when the physical parameters of the system are uncertain (see the following book published by J. Ackermann in 1993 [Ackermann1993]).

When the coefficients of the characteristic polynomial are known, the Routh-Hurwitz stability criterion can be used to check if the system is stable (i.e. if all roots have negative real parts). Kharitonov's theorem can be used in the case where the coefficients are only known to be within specified ranges. It reduces checking stability of an infinite number of polynomials to checking stability of only four so-called Kharitonov polynomials.

http://en.wikipedia.org/wiki/Kharitonov's_theorem

J. Ackermann et al regarded Kharitonov’s theorem as the Parameter Space Approach [ABBGKKMO02].

(2) However, when the dependency among polynomial coefficients is nonlinear (e.g., a asymptotically stable system), J. Ackermann demonstrated in the following IEEE TAC 1992 paper that checking a subset of a polynomial family generally can NOT guarantee the stability of the entire family.

J. Ackermann, "Does it suffice to check a subset of multilinear parameters in robustness analysis?" IEEE Transactions on Automatic Control, 37(4), April 1992, pp.487-488.

"For polynomials with general multilinear dependence on uncertain real parameters it does not suffice to check a subset for Q for stability."

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6117295

(3) J. Ackermann published the following book to present methods for robustness analysis that guarantee the desired properties for all possible values of the plant uncertainty.

[Ackermann93] J. Ackermann, Robust Control: Systems with Uncertain Physical Parameters, Springer-Verlag, New York, 1993.

http://www.springer.com/engineering/robotics/book/978-1-4471-3365-0

Short review on J. Ackermann's 1993 book by IEEE TCST 2001 paper:

W.K. Ho, T.H. Lee, H.P. Han, and Y. Hong, "Self-Tuning IMC-PID Control with Interval Gain and Phase Margin Assignment," IEEE Transactions on Control Systems Technology, 9(3), May 2001, pp. 535-541. Available from the following RG Link.

"A controller may be found such that all the closed-loop poles are located in a specified region—a pole-region assignment problem—and remain in this region if the plant parameters remain within the upper and lower bounds [Ackermann93]."

H. Nyquist (Sweden) --> K.J. Astrom (Sweden) --> W.K. Ho (Sweden)

     | Nyquist plot (published in 1932)

     |  Bell Labs

    V  Bode plot (published in 1940)

H.W. Bode (Harvard) --> K.S. Narendra (Harvard, Yale) --> T.H. Lee (Yale)

https://www.researchgate.net/publication/3332273_Self-tuning_IMC-PID_control_with_interval_gain_and_phase_marginsassignment

J. Ackermann and his peer researchers co-published the 2nd edition of the previously published book [Ackermann93] in 1993.

[ABBGKKMO02] J. Ackermann et al., Robust Control: The Parameter Space Approach, Springer-Verlag, New York, 2002.

http://www.amazon.com/Robust-Control-Parameter-Communications-Engineering/dp/1852335149

(4) Quote Kranthi Kumar Deveerasetty's comment "Is there any other method to overcome from this problem?"

The introduction section of the following Automatica 2009 paper provides an extensive reviews on the follow-up work which aims at overcome the limitation of Kharitonov’s theorem.

J. Fisher and R. Bhattacharya, "Linear quadratic regulation of systems with stochastic parameter uncertainties," Automatica, 45(12), December 2009, pp. 2831-2841,

"Kharinotov’s work renewed interested in this area and led to various results that extended well known classical control techniques in the frequency domain (Ackermann, 2002, Bhattacharyya et al., 1995 and Mansour et al., 1992). ....... "

http://www.sciencedirect.com/science/article/pii/S0005109809004622

(5) Iterative feedback tuning (IFT) was proposed to tune controller parameters for system with unknown parameters. A linear control model can be used to approximate non-linear system with unknown parameters roughly, and such control plant approximation may exist large deviation from the real non-linear system, which would cause classical tuning method of PID controller ineffective. IFT was proposed to solve controller tuning issues caused by plant uncertainty. Iterative feedback tuning has been integrated with PID controller to solve controller tuning issues caused by plant uncertainty.

W.K. Ho, Y. Hong, A. Hansson, H. Hjalmarsson, and J.W. Deng, "Relay auto-tuning of PID controllers using iterative feedback tuning," Automatica 39 (1), January 2003, pp. 149-157. Available from the following RG Link.

https://www.researchgate.net/publication/223504459_Relay_auto-tuning_of_PID_controllers_using_iterative_feedback_tuning

(6) Recent discussions on control system design

"What are trends in control theory and its applications in physical systems (from a research point of view)?"

https://www.researchgate.net/post/What_are_trends_in_control_theory_and_its_applications_in_physical_systems_from_a_research_point_of_view2

Article Self-tuning IMC-PID control with interval gain and phase mar...

Article Relay auto-tuning of PID controllers using iterative feedback tuning

Prof. Yang Hong,

Thank you very much for providing complete details, it will helpful for my research.

Kharitonov's Theorem holds for Hurwitz stability, but in general does not apply to arbitrary regions.

Ref: S.P. Bhattacharyya, H. Chapellat and L.H. Keel,"Robust Control The Parametric Approach", Chapter 5, Prentice Hall PTR, 1995.