Subject Name Control Systems Subject Code 10 ES

Subject Name: Control Systems Subject Code: 10 ES 43 Prepared. UNIT By: 4 M. Brinda, Sreepriya Kurup, Robina Gujral STABILITY ANALYSIS Department: ECE Date: 30/3/2015

UNIT 4 STABILITY ANALYSIS

TOPICS Concepts of stability Necessary conditions for stability Routh Stability Criterion Relative stability analysis More on Routh Stability Criterion

The Concept of Stability The concept of stability can be illustrated by a cone placed on a plane horizontal surface. A necessary and sufficient condition for a feedback system to be stable is that all the poles of the system transfer function have negative real parts. A system is considered marginally stable if only certain bounded inputs will result in a bounded output.

STABLE SYSTEM Response or output is predictable. A system is said to be stable if for a bounded disturbing input signal, the output vanishes ultimately as t infinity. A system is unstable if for a bounded disturbing input signal the output is of infinite amplitude or oscillatory. i) For a bounded i/p, it produces unbounded o/p. ii) In the absence of i/p, o/p may not return to zero. It shows certain o/p without i/p.

STABLE SYSTEM

UNSTABLE SYSTEM

UNCONTROLLABLE RESPONSE

DEFINITIONS OF STABILITY Bounded Input, Bounded Output (BIBO) Stability: A system is said to be BIBO Stable if i) For Bounded i/p, we have Bounded o/p; o/p – Controllable. ii) In the absence of i/p, o/p must tend to zero irrespective of initial conditions. Relaxed system: A System in which initial conditions are zero.

DEFINITIONS OF STABILITY Critically or Marginally Stable system: for a bounded i/p, o/p oscillates with constant frequency and amplitude. Such oscillations are called Damped or sustained oscillations. Conditionally Stable system: o/p is bounded only for certain condition. If this condition is violated, o/p is unbounded. Stability depends on condition of parameter of the system.

CRITICALLY OR MARGINALLY STABLE SYSTEM

DEFINITIONS OF STABILITY Zero input stability: If the zero input response of the system subjected to finite initial conditions, reaches to zero as t infinity, then the system is zero input stable. Asymptotic Stability: As magnitude of zero input response reaches zero as t approaches infinity, then zero input stability is also called asymptotic stability. If in the absence of i/p, the o/p tends to zero or to the equilibrium state irrespective of initial conditions.

DEFINITIONS OF STABILITY Absolutely Stable system: If the system o/p is stable for all variations of its parameters then the system is called absolutely stable system.

TRANSFER FUNCTION When order of the denominator polynomial is greater than the numerator polynomial the transfer function is said to be ‘proper’. Otherwise ‘improper’ 14

STABILITY OF CONTROL SYSTEM Roots of denominator polynomial of a transfer function are called ‘poles’. And the roots of numerator polynomials of a transfer function are called ‘zeros’. 15

STABILITY OF CONTROL SYSTEM Poles of the system are represented by ‘x’ and zeros of the system are represented by ‘o’. System order is always equal to number of poles of the transfer function. Following transfer function represents nth order plant. 16

STABILITY OF CONTROL SYSTEM Poles is also defined as “it is the frequency at which system becomes infinite”. Hence the name pole where field is infinite. And zero is the frequency at which system becomes 0. 17

STABILITY OF CONTROL SYSTEM Poles is also defined as “it is the frequency at which system becomes infinite”. Like a magnetic pole or black hole. 18

STABILITY OF CONTROL SYSTEMS The poles and zeros of the system are plotted in s-plane to check the stability of the system. LHP RHP s-plane 19

STABILITY OF CONTROL SYSTEMS If all the poles of the system lie in left half plane the system is said to be Stable. If any of the poles lie in right half plane the system is said to be unstable. If pole(s) lie on imaginary axis the system is said to be marginally stable. LHP RHP If all the poles s-plane 20

STABILITY OF CONTROL SYSTEMS For example Then the only pole of the system lie at LHP RHP X -3 s-plane 21

LOCATION OF ROOTS ON S PLANE

CONCLUSIONS BASED ON THE LOCATION OF ROOTS OF CHARACTERISTIC EQUATION Roots-LHS – negative real parts – Response – Bounded- BIBO Stable. Roots-RHS – Positive real parts –Response – Unbounded- Unstable. Repeated roots on Imaginary axis –Response – Unbounded- unstable. Single root at origin – Bounded –Unstable. Repeated roots at origin –Unbounded, unstable. Non repeated roots on imaginary axis or single pole at origin- Limitedly or marginally stable system.

OBSERVATIONS All the co efficients –Positive => roots –LHS If any co efficient is zero=> roots. Imaginary axis or RHS If any co efficient is negative => atleast one root -RHS

NECESSARY CONDITIONS FOR STABILITY All the co efficients of a characteristic polynomial be positive. If any co efficient is zero or negative, we can immediately say that the system is unstable. But not sufficient condition s 3+ s 2+2 s+8 = (s+2) (s- 0. 5 – 1. 93 j) (s 0. 5+1. 93 j) Co efficients –positive but roots –RHS So s/m – Unstable.

ROUTH HURWITZ CRITERION Sufficient conditions for stability. Hurwitz – investigated stability interms of determinants. Routh – in terms of array formulation. Routh Stability criterion: The necessary and sufficient condition for stability is that all the elements in the first column of the routh array must be positive. If this condition is not met, the system is unstable and the no of sign changes in the elements of the first column of the routh array corresponds to the no of roots of characteristic equation in RHS of s plane.