Stability in Control Systems: Routh-Hurwitz Method, Cheat Sheet of Automatic Controls

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Chapter ( 5 )

STABILITY

Learning Outcomes

After completing this chapter the student will be

able to:

1) Known the definition of stability. 2) Determine the stability conditions by evaluating poles of transfer function. 3) Make and interpret a basic Routh table to determine the stability of a system. 4) Make and interpret a basic Routh table where either the first element of a row is zero or an entire row is zero. 2

1. Introduction The stability of a cone.

1. Introduction

• Requirements for design of a control system
  • Transient Response
  • Stability
  • Steady State Errors
• Stability – most important parameter for design
• Total response of a system is = forced response + natural response, or

forced natural

c t = c t + c t

1. Introduction Stability in the s-plane.

How to define stability H(s) G(s) R(s) C(s) +

- Stability with respect to G(s)? All poles in the left half plane Stability with respect to 𝑮(𝒔) 𝟏+𝑮 𝒔 𝑯(𝒔)

Poles of 1 +G(s)H(s) in the left half.

❖ System Stability Definition – Stable System As time approaches infinity , the natural response approaches zero Bounded input yields bounded output Stable system have poles only in the left hand plane

❖ System Stability Definition – Unstable System Time approaches infinity the natural response approaches infinity Bounded input yields an unbounded output Unstable system have at least one pole in the right hand plane And/or poles of multiplicity greater than one on imaginary axis

❖ How do we determine if a system is stable? It is not always a simple matter to determine if a feedback control system is stable. Consider the system below We need a method to test for stability without having to solve for the roots of the denominator!! 13

❖ Routh-Hurwitz Method ➢ A method that yields stability information without the need to solve for the closed-loop system poles. ➢ Using this method, we can tell how many closed-loop system poles are in the left half-plane, in the right half-plane, and on the jw - axis (imaginary axis). ➢ Notice that we say how many, not where. We can find the number of poles in each section of the s-plane, but we cannot find their coordinates. ➢ The method requires two steps: ( 1 ) Generate a data table called a Routh table ( 2 ) Interpret the Routh table to tell how many closed- loop system poles are in the left half-plane, the right half-plane, and on the jw - axis.

Generating of Routh table

Generating of Routh table Example:

❖ Routh-Hurwitz Method Interpretation of Routh table Routh-Hurwitz criterion declares that the number of roots of the polynomial that are in the right half-plane is equal to the number of sign changes in the first column.

❖ Routh-Hurwitz Method Interpretation of Routh table Two sign changes in the first column! Two sign changes = two right half plane poles, therefore unstable system^20