7.4: Stability of Sampled-Data Systems
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Stability Region in the Complex Plane
In continuous-time control system design, \(s=j\omega\) defines the stability boundary.
The \(z\)-plane stability boundary is obtained from the transform: \(z=e^{j\omega T} =1\angle \omega T\); it maps the \(j\omega\)-axis to the unit circle and the open left-half plane to the inside of the unit circle: \(|z|<1\).
Thus, the sampled-data system is stable if and only if the pulse characteristic polynomial \(\Delta (z)\) has its roots inside the unit circle, i.e., \(|z_\rm i |<1.\)where \(z_\rm i\) is the root of \(\Delta (z)\).
The analytical conditions for a \(z\)-domain polynomial, \(A\left(z\right)\), to have its roots inside the unit circle are given by the Schur–Cohn stability test. When applied to real polynomials, the Schur–Cohn test results in a criterion similar to the Ruth’s test, and is known as Jury’s stability test.
Jury’s Stability Test
Assume that the \(n\)th order \(z\)-domain polynomial to be investigated is given as:
\[A(z)=a_{0} z^{n} +a_{1} z^{n-1} +\ldots +a_{n-1} z+a_{n} ;a_{0} >0. \nonumber \]
For stability determination, the Jury’s table is built as follows:
\[\left|\begin{array}{c} {\begin{array}{ccc} {a_{n} } & {a_{n-1} } & {\begin{array}{cc} {\ldots } & {a_{0} } \end{array}} \\ {a_{0} } & {a_{1} } & {\begin{array}{cc} {\ldots } & {a_{n} } \end{array}} \end{array}} \\ {\begin{array}{ccc} {b_{n-1} } & {b_{n-2} } & {\begin{array}{cc} {\ldots } & {b_{0} } \end{array}} \\ {b_{0} } & {b_{1} } & {\begin{array}{cc} {\ldots } & {b_{n-1} } \end{array}} \end{array}} \\ {\begin{array}{ccc} {c_{n-2} } & {c_{n-1} } & {\begin{array}{cc} {\ldots } & {c_{0} } \end{array}} \\ {c_{0} } & {c_{1} } & {\begin{array}{cc} {\ldots } & {c_{n-2} } \end{array}} \end{array}} \\ {\vdots } \end{array}\right. \nonumber \]
where the first two rows reflect the coefficients of the polynomial; the coefficients of the third and subsequent rows are computed as:
\[b_{k} =\left|\begin{array}{cc} {a_{n} } & {a_{n-1-k} } \\ {a_{0} } & {a_{k+1} } \end{array}\right|,\; \; k=0,\ldots ,n-1 \nonumber \]
\[c_{k} =\left|\begin{array}{cc} {b_{n-1} } & {b_{n-2-k} } \\ {b_{0} } & {b_{k+1} } \end{array}\right|,\; \; k=0,\ldots ,n-2 \nonumber \]
and so on. The necessary conditions for polynomial stability are:
\[A(1)>0,(-1)^{n} A(-1)>0. \nonumber \]
The sufficient conditions for stability, given by the Jury’s test, are:
\[a_{0} >|a_{n} |,\; |b_{n-1} |>|b_{0} |,\; |c_{n-2} |>|c_{0} |,\ldots \rm (n-1) constraints. \nonumber \]
Second-Order Polynomial
Let \(A(z)=z^{2} +a_{1} z+a_{2}\); then, the Jury’s table is given as:
\[\left|\begin{array}{c} {\begin{array}{ccc} {a_{2} } & {a_{1} } & {1} \\ {1} & {a_{1} } & {a_{2} } \end{array}} \\ {\begin{array}{ccc} {b_{1} } & {b_{0} } & {} \\ {b_{0} } & {b_{1} } & {} \end{array}} \end{array}\right. \nonumber \]
The resulting necessary conditions are: \(1+a_{1} +a_{2} >0,\; \; 1-a_{1} +a_{2} >0.\)
The sufficient conditions are: \(|a_{2} |<1,|1+a_{2} |>|a_{1} |\).
Stability Determination through Bilinear Transform
The bilinear transform (BLT) defines a linear map between \(s\)-domain and \(z\)-domain.
The BLT is based on the first-order Pade’ approximation of \(z=e^{sT}\), given as:
\[z=\frac{e^{sT/2} }{e^{-sT/2} } \cong \frac{1+sT/2}{1-sT/2} ,\; \; s=\frac{2}{T} \frac{z-1}{z+1} \nonumber \]
Since \(T\) has no impact on stability determination, we may use \(T=2\) for simplicity. Further, to differentiate from continuous-time systems, a new complex variable, \(w\), is introduced. Thus \(z=\frac{1+w}{1-w} ,\; \; w=\frac{z-1}{z+1}\)
Let \(\Delta (z)\) represent the polynomial to be investigated; application of BLT to \(\Delta (z)\) returns a polynomial, \(\Delta (w)\), whose stability is determined through the application of Hurwitz criterion (Sec. 2.5).
Second-Order Polynomial
Let \(\Delta (z)=z^{2} +a_{1} z+a_{2}\); then, application of BLT, ignoring the denominator term, results in:
\[\Delta (w)=\Delta (z)|_{z=\frac{1+w}{1-w} } =(1-a_{1} +a_{2} )w^{2} +2(1-a_{2} )w+1+a_{1} +a_{2} \nonumber \]
Application of the Hurwitz criterion results in the following stability conditions for \(\Delta (z)\):
\[a_{2} +a_{1} +1>0, a_{2} -a_{1} +1>0, 1-a_{2} >0 \nonumber \]
The above conditions are similar those obtained from the application of Jury’s stability test.
Stability of the Closed-Loop System
Let the pulse transfer function be given as: \(G\left(z\right)=\frac{n\left(z\right)}{d\left(z\right)}\); then, for a static controller, the closed-loop pulse characteristic polynomial is given as: \(\mathit{\Delta}\left(z\right)=d\left(z\right)+Kn(z)\).
The stability of the closed-loop characteristic polynomial can be determined by applying Jury’s stability test. Alternatively, BLT can be used to determine stability through the application of Routh’s test to the transformed polynomial, \(\mathit{\Delta}(w)\).
Let \(G(s)=\frac{1}{s+1}\), \(T=0.2\rm s\); then, the pulse transfer function is given as: \(G(z)=\frac{0.181}{z-0.819}\).
The closed-loop characteristic polynomial is: \(\mathit{\Delta}\left(z\right)=z-0.819+0.181K\).
The polynomial is stable for \(\left|0.819+0.181K\right|<1\), or \(-1<K<10\) for stability.
Let \(\Delta (z)=z^{2} +z+K\); then, \(\Delta (w)=\Delta (z)|_{z=\frac{1+w}{1-w} } =Kw^{2} +2(1-K)w+1+K\).
The application of the Hurwitz criteria to \(\Delta (w)\) reveals \(0<K<1\) for stability.
Let \(G(s)=\frac{1}{s(s+1)} ,\; \; T=0.2s\); then, the pulse transfer function is given as: \(G(z)=\frac{0.0187z+0.0175}{(z-1)(z-0.181)}\).
The characteristic polynomial is:
\[\Delta (z)=z^{2} +(0.0187K-1.819)z+0.0175K+0.819. \nonumber \]
The \(w\)-polynomial obtained through BLT is given as:
\[\Delta (w)=(3.637-0.0012K)w^{2} +(0.363-0.035K)w+0.036K \nonumber \]
The application of the Hurwitz criteria to \(\Delta (w)\) gives \(0<K<10.34\) for stability.