Stable polynomial

<h2 id="properties">Properties</h2>
<ul><li>The <a href="/facts/Routh%25E2%2580%2593Hurwitz_theorem/SGsE7USb">Routh–Hurwitz theorem</a> provides an algorithm for determining if a given polynomial is Hurwitz stable, which is implemented in the <a href="/facts/Routh%25E2%2580%2593Hurwitz_stability_criterion/YNYfQr3F">Routh–Hurwitz</a> and <a href="/facts/Li%25C3%25A9nard%25E2%2580%2593Chipart_criterion/S1L8fge0">Liénard–Chipart</a> tests.</li>
<li>To test if a given polynomial <i>P</i> (of <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> <i>d</i>) is Schur stable, it suffices to apply this theorem to the transformed polynomial</li></ul>

Q
        (
        z
        )
        =
        (
        z
        −
        1
        
          )
          
            d
          
        
        P
        
          (
          
            
              
                z
                +
                1
              
              
                z
                −
                1
              
            
          
          )
        
      
    
    {\displaystyle Q(z)=(z-1)^{d}P\left({{z+1} \over {z-1}}\right)}

obtained after the <a href="/facts/M%25C3%25B6bius_transformation/YEkTTLfK">Möbius transformation</a> 
  
    
      
        z
        ↦
        
          
            
              z
              +
              1
            
            
              z
              −
              1
            
          
        
      
    
    {\displaystyle z\mapsto {{z+1} \over {z-1}}}
  
 which maps the left half-plane to the open unit disc: <i>P</i> is Schur stable if and only if <i>Q</i> is Hurwitz stable and 
  
    
      
        P
        (
        1
        )
        ≠
        0
      
    
    {\displaystyle P(1)\neq 0}
  
. For higher degree polynomials the extra computation involved in this mapping can be avoided by testing the Schur stability by the Schur-Cohn test, the <a href="/facts/Jury_stability_criterion/8hh2uGYN">Jury test</a> or the <a href="/facts/Bistritz_stability_criterion/gYYuLf7t">Bistritz test</a>.
<ul><li>Necessary condition: a Hurwitz stable polynomial (with <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a>) has coefficients of the same sign (either all positive or all negative).</li>
<li>Sufficient condition: a polynomial 
  
    
      
        f
        (
        z
        )
        =
        
          a
          
            0
          
        
        +
        
          a
          
            1
          
        
        z
        +
        ⋯
        +
        
          a
          
            n
          
        
        
          z
          
            n
          
        
      
    
    {\displaystyle f(z)=a_{0}+a_{1}z+\cdots +a_{n}z^{n}}
  
 with (real) coefficients such that</li></ul>

a
          
            n
          
        
        >
        
          a
          
            n
            −
            1
          
        
        >
        ⋯
        >
        
          a
          
            0
          
        
        >
        0
        ,
      
    
    {\displaystyle a_{n}>a_{n-1}>\cdots >a_{0}>0,}

is Schur stable.
<ul><li>Product rule: Two polynomials <i>f</i> and <i>g</i> are stable (of the same type) <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the product <i>fg</i> is stable.</li>
<li>Hadamard product: The Hadamard (coefficient-wise) product of two Hurwitz stable polynomials is again Hurwitz stable.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li></ul>
<h2 id="examples">Examples</h2>
<ul><li>
  
    
      
        4
        
          z
          
            3
          
        
        +
        3
        
          z
          
            2
          
        
        +
        2
        z
        +
        1
      
    
    {\displaystyle 4z^{3}+3z^{2}+2z+1}
  
 is Schur stable because it satisfies the sufficient condition;</li>
<li>
  
    
      
        
          z
          
            10
          
        
      
    
    {\displaystyle z^{10}}
  
 is Schur stable (because all its roots equal 0) but it does not satisfy the sufficient condition;</li>
<li>
  
    
      
        
          z
          
            2
          
        
        −
        z
        −
        2
      
    
    {\displaystyle z^{2}-z-2}
  
 is not Hurwitz stable (its roots are −1 and 2) because it violates the necessary condition;</li>
<li>
  
    
      
        
          z
          
            2
          
        
        +
        3
        z
        +
        2
      
    
    {\displaystyle z^{2}+3z+2}
  
 is Hurwitz stable (its roots are −1 and −2).</li>
<li>The polynomial 
  
    
      
        
          z
          
            4
          
        
        +
        
          z
          
            3
          
        
        +
        
          z
          
            2
          
        
        +
        z
        +
        1
      
    
    {\displaystyle z^{4}+z^{3}+z^{2}+z+1}
  
 (with positive coefficients) is neither Hurwitz stable nor Schur stable. Its roots are the four primitive fifth <a href="/facts/Root_of_unity/EqH0ho1Y">roots of unity</a></li></ul>

z
          
            k
          
        
        =
        cos
        ⁡
        
          (
          
            
              
                2
                π
                k
              
              5
            
          
          )
        
        +
        i
        sin
        ⁡
        
          (
          
            
              
                2
                π
                k
              
              5
            
          
          )
        
        ,
        
        k
        =
        1
        ,
        …
        ,
        4
        
        .
      
    
    {\displaystyle z_{k}=\cos \left({{2\pi k} \over 5}\right)+i\sin \left({{2\pi k} \over 5}\right),\,k=1,\ldots ,4\,.}

Note here that

cos
        ⁡
        (
        
          
            2
            π
          
          
            /
          
          5
        
        )
        =
        
          
            
              
                
                  5
                
              
              −
              1
            
            4
          
        
        >
        0.
      
    
    {\displaystyle \cos({{2\pi }/5})={{{\sqrt {5}}-1} \over 4}>0.}

It is a "boundary case" for Schur stability because its roots lie on the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a>. The example also shows that the necessary (positivity) conditions stated above for Hurwitz stability are not sufficient.
<h2 id="stable-matrices">Stable matrices</h2>
<p>Just as stable polynomials are crucial for assessing the stability of systems described by polynomials, stability matrices play a vital role in evaluating the stability of <a href="/facts/State-space_representation/Jf4EanCP">systems represented by matrices</a>.
</p>
<h3>Hurwitz matrix</h3>
<p class="note">Main article: <a href="/facts/Hurwitz-stable_matrix/Jme6EsXz">Hurwitz-stable matrix</a></p>
<p>A <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> <i>A</i> is called a Hurwitz matrix if every <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> of <i>A</i> has strictly negative <a href="/facts/Real_part/2w7mqI7e">real part</a>.
</p>
<h3>Schur matrix</h3>
<p>Schur matrices is an analogue of the Hurwitz matrices for discrete-time systems. A matrix <i>A</i> is a Schur (stable) matrix if its eigenvalues are located in the <a href="/facts/Open_unit_disk/phrUu7yC">open unit disk</a> in the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Kharitonov_region/LfKtm24U">Kharitonov region</a></li>
<li><a href="/facts/Stability_criterion/AMSTVwWT">Stability criterion</a></li>
<li><a href="/facts/Stability_radius/hePJ6FUE">Stability radius</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://mathworld.wolfram.com/StablePolynomial.html">Mathworld page</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Garloff, Jürgen; Wagner, David G. (1996). "Hadamard Products of Stable Polynomials Are Stable". Journal of Mathematical Analysis and Applications. 202 (3): 797–809. doi:10.1006/jmaa.1996.0348. <a href="https://doi.org/10.1006%2Fjmaa.1996.0348" target="_blank">https://doi.org/10.1006%2Fjmaa.1996.0348</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Stable polynomial open-in-new

Stable polynomial