Generalized Minimum Phase Property for Finite-Dimensional Continuous-Time SISO LTI Systems with Additive Disturbances

Zigang Pan, Tamer Basar

Research output: Chapter in Book/Report/Conference proceedingConference contribution


For finite-dimensional continuous-time single-input and single-output linear time-invariant systems, we introduce the concept of extended zero dynamics and generalize the concept of minimum phase, which accommodates the presence of disturbance inputs. By representing a SISO LTI system with a finite relative degree in its extended zero dynamics canonical form, we obtain its extended zero dynamics, which is simply its zero dynamics (according to [1]) driven by the noiseless output of the system and the disturbance input. We then say a system is minimum phase if its extended zero dynamics is absent or satisfies that, for any bounded admissible initial condition, any bounded noiseless output, and any bounded admissible disturbance input waveform, the zero dynamics state trajectory is bounded. The system is minimum phase (according to this extended notion) if its zero dynamics is asymptotically stable. It is proved that the converse holds under the additional condition that the system be stabilizable from the control input. For a system to be minimum phase, it is necessary that the transfer function from the control input to the output has all zeros with negative real parts. The converse holds when the system is both controllable (from the control input) and observable. It is further shown that the generalized minimum phase property is necessary for model reference control that achieves 1) perfect tracking of any bounded reference trajectories with bounded derivatives up to certain order without any disturbances and 2) the existence of bounded state trajectory for any admissible bounded initial condition, any admissible bounded disturbance input waveform, and any bounded reference trajectory with bounded derivatives up to a certain order.

Original languageEnglish (US)
Title of host publication2018 IEEE Conference on Decision and Control, CDC 2018
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages7
ISBN (Electronic)9781538613955
StatePublished - Jul 2 2018
Externally publishedYes
Event57th IEEE Conference on Decision and Control, CDC 2018 - Miami, United States
Duration: Dec 17 2018Dec 19 2018

Publication series

NameProceedings of the IEEE Conference on Decision and Control
ISSN (Print)0743-1546
ISSN (Electronic)2576-2370


Conference57th IEEE Conference on Decision and Control, CDC 2018
Country/TerritoryUnited States


  • continuous-time systems
  • extended zero dynamics
  • extended zero dynamics canonical form
  • minimum phase

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization


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