ASSIGNMENT-1

ASSIGNMENT-1Advance Topic In ControlSubmitted to:Dr.Rana IftkharSubmitted By:Jahanzaib zafarReg#63972

How Passivity does play its role in studying the stability of some nonlinear system.

Passivity pay very important role in stability of linear as well nonlinear systems. To understand how and why, we first define what it is. Passivity:The passivity is the Inflow of electrical power from source to the System which must always non-negative or its inability to increase its own energy by input u(t). Mathematically it is defined as :

Graphically:

Passivity of system is correlated to transfer function of system and State space Model of Dynamical system by PR, KYP Lemma If Transfer function is Positive Real (PR) then System is Passive. If Transfer function is Strictly Positive Real (SPR) then System is Strictly Passive.

How Passivity play a Vital Role of in study of Stability of Nonlinear Systems:Passivity is very important role in stability of single open loop as well as closed loop nonlinear systems. The compositional property (for instance, negative feedback connection of two passive systems remains passive) makes passivity a powerful tool to analyze complicated, coupled systems, such as Cyber-physical systems.now point (Vicinity) if its linearization is asymptotically stable. Passivity is closely related to stability of non-linear system.Excess and Shortage of Passivity To extend the passivity-based stability conditions to more general cases for both passive and no passive systems, we need to define the passivity indices that quantify the degree of passivity. The passivity indices can be defined in terms of an excess or shortage of passivity.

By using concept of passivity we can define the stability of following types: Stability Asymptotically Stability Globally Asymptotically Stability Stability of Feed Back Systems To study the role of Passivity let take a system :Equation 1

1) Stability of a System:now the origin is stable if system is Passive and positive .

now we state another which lemma which state :2)Asymptotically Stability:if the system which is defined in equation 1 is Strictly Passive then the origin system is Asymptotically Stability.

3) Globally Asymptotically Stability: In another lemma it is stated that If the above mentioned system is Strictly Passive and the radially unbounded then the system is Globally Asymptotically Stability.4) Stability of Feedback Systems:Lets consider the feedback connection of two dynamical systems:

The feedback connection of two dynamical systems is Globally Asymptotically Stability if the systems have following property:Strictly Passive or output strictly passive and zero state observable.