Stability and controller design


The Nyquist stability criterion is a widely used technique for determining the stability of a dynamical system with feedback in the complex s-plane. This page presents the practical method introduced by Garcia-Sanz to compute the Nyquist stability criterion directly in the Nichols (magnitude/phase) chart (see paper [1] and chapter 3 in book [3]) .

The method has been implemented in the QFT Control Toolbox (QFTCT), also available in this website (see [2]). Additionally, a Matlab file with the method can be downloaded at the end of this page. Garcia-Sanz's method calculates the Nyquist stability criterion according to the parameters "p" and "N = Na + Nb + Nc + Nd", as it is shown in the next figures.
    

[1]. Garcia-Sanz, Mario, (2016). The Nyquist Stability Criterion in the Nichols Chart.  International Journal of Robust and Nonlinear Control, Wiley, Vol. 26, No. 12, pp. 2643-2651.

[2]. Garcia-Sanz, Mario, (2008-present). The QFT Control Toolbox (QFTCT) for Matlab Click: "Controller design window"; "File"; "Check stability".

[3]. Garcia-Sanz, Mario, (2017). Robust Control Engineering: practical QFT solutions", CRC Press, Taylor & Francis, USA, 2017. ISBN: 978-1-138-03207-1.

The Nyquist/Garcia-Sanz's method can be applied to linear time invariant (LTI) closed-loop systems with minimum and non-minimum phase zeros, stable and unstable poles, poles at the origin with diverse multiplicity and systems defined by non-rational functions, such as plants with time delay. The method also gives guidelines to design controllers to stabilize unstable plants when dealing with frequency domain techniques like the QFT robust control (see the next figures).
    

Matlab file: Nyquist stability criterion with Garcia-Sanz's method    

Syntax in Matlab.
Type: [zc,N,p,Na,Nb,Nc,Nd,zpCancel,k,sigma,alpha,gamma] = nyquistGS(L)

Inputs:
L = open loop transfer function: C*P*H

Outputs:
zc (= 0, Stable), (= 1, Unstable)
zc = N + p
N = Na + Nb + Nc + Nd
zpCancel = RHP zero-pole cancelations. "0" = no cancelations, "1" = there are cancelations
Nd = 2*(k+1) + sigma + alpha*gamma
 

 


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