On the Stability of Pulse-Width Automatic Control Systems
Abstract Generated abstract
This paper examines the absolute stability of pulse-width automatic control systems with first-order single-cycle modulation, treated as nonlinear sampled-data systems. It reformulates the pulse-width system as an equivalent structure with a single nonlinear element whose characteristic lies in a sector and an equivalent linear sampled-data part, allowing application of Tsypkin’s frequency stability criterion and a related criterion using a derivative constraint. Expressions are derived for the critical sector bound and gain coefficient, including an analytic evaluation for a second-order system. Numerical comparisons with machine modeling indicate that the criteria give reasonably accurate stability estimates, with the derivative-constrained criterion less conservative than the basic frequency criterion.
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ON THE STABILITY OF PULSE-WIDTH AUTOMATIC CONTROL SYSTEMS
I. A. GONCHAR
Presented by the scientific seminar of the Department of Automation and Telemechanics of the Tomsk Polytechnic Institute
Pulse-width automatic control systems (PWCS) belong to the class of nonlinear sampled-data automatic systems (NSAS). In the work of Ya. Z. Tsypkin [1], a frequency criterion for the absolute stability of NSAS was formulated, the characteristics of whose nonlinear elements belong to the sector \((r, k)\). It is assumed here that the nonlinear sampled-data system consists of a nonlinear element (NE) and a linear sampled-data part (LSP) (Fig. 1a).
Fig. 1.
Let us consider a PWCS with first-order single-cycle modulation. The dependence of the image of the system output quantity \(Z^{*}(q)\) on the input quantity of the modulator \(X[n]\) can be represented in the form:
\[ Z^{*}(q)=\sum_{\nu=1}^{l} c_{\nu 0}\frac{e^{q_{\nu}}-1}{e^{q}-e^{q_{\nu}}}\,D\left\{\frac{1-e^{-q_{\nu}\gamma[n]}}{1-e^{-q_{\nu}}}\right\}, \tag{1} \]
where the relative pulse width \(\gamma[n]\) at the output of the pulse-width modulator is determined by the relation:
\[ \gamma[n]= \begin{cases} \varkappa x[n] & \text{for } 0 \le x[n]\le \dfrac{1}{\varkappa};\\ 1 & \text{for } x[n]\ge \dfrac{1}{\varkappa};\\ 0 & \text{for } x[n]\le 0. \end{cases} \tag{2} \]
\(l\) is the number of roots of the continuous part of the PWCS.
The roots of the characteristic equation of the continuous part, \(q_\nu\), and the coefficients \(c_{\nu 0}\) \((\nu=1,2,\ldots,l)\) are determined by formulas known from the theory of pulse systems [2].
Equation (1) makes it possible to represent the open-loop pulse-width system in the form of \(l\) parallel-connected branches, each of which in turn consists of a linear pulse part and a nonlinear element connected in series (Fig. 1b).
The transfer function of an elementary linear pulse part is
\[ W_\nu^*(q)=c_{\nu 0}\frac{e^{q_\nu}-1}{e^q-e^{q_\nu}};\qquad (\nu=1,2,\ldots,l); \tag{3} \]
and the characteristic of each nonlinear element is
\[ \Phi_\nu(x)=\frac{1-e^{-q_\nu \gamma[n]}}{1-e^{-q_\nu}};\qquad (\nu=1,2,\ldots,l). \tag{4} \]
Consequently, the block diagram of pulse-width systems differs from the diagram shown in Fig. 1a. This complicates the application of Ya. Z. Tsypkin’s frequency stability criterion in analyzing the stability of PWCS of second, third, and higher order.
The family of nonlinear characteristics constructed according to formula (4) for \(\nu=1,2,\ldots,l\) is shown in Fig. 2a. If an action \(f[n]=\mathrm{const}\) has been applied to the closed-loop PWCS, then a stable system will arrive at some equilibrium position \(x_0\). Taking this position as the origin of coordinates (Fig. 2b), we obtain, in the new coordinate system, a family of nonlinear characteristics \(\Phi'_\nu(x)\), determined by the formula
\[ \Phi'_\nu(x)= \frac{e^{-q_\nu \chi x_0}\left(1-e^{-q_\nu \chi x}\right)} {1-e^{-q_\nu}}; \tag{5} \]
for
\[ -x_0 \leq x \leq \frac{1}{\chi}-x_0. \]
Fig. 2.
In addition, it is always possible to find a sector \((r; k)\) that would include all nonlinear characteristics \(\Phi'_\nu(x)\), i.e., such that the conditions
\[ \Phi'_\nu(0)=0;\qquad r<\frac{\Phi'_\nu(x)}{x}<k. \tag{6} \]
are satisfied.
When studying the stability of a NIAS by the frequency criterion of Ya. Z. Tsypkin, the specific form of the characteristic of the nonlinear element is of no importance; it is sufficient only that conditions (6) be satisfied. Therefore, all elementary nonlinear elements of the pulse-width system with characteristic \(\Phi'_\nu(x)\) may be replaced by a single equivalent nonlinear element with characteristic \(\Phi_{\mathrm{e}}(x)\), belonging to the same sector \((r;k)\) to which the characteristics \(\Phi'_\nu(x)\) \((\nu=1,2,\ldots,l)\) belong.
The upper bound \(k\) of this sector can be determined from the nonlinear characteristic \(\Phi'_l(x)\) corresponding to the smallest root, i.e.,
\[ k=\frac{e^{-q_l z x_{\nu0}}-e^{-q_l}} {(1-e^{-q_l})(1-zx_{\nu0})}\,z; \tag{7} \]
for \(q_l<q_\nu;\quad (\nu=1,2,\ldots,l-1)\).
When determining the critical gain coefficient of the pulse-width system, the lower bound \(r\) of the sector may be taken equal to zero.
The equation of the open-loop pulse-width system can now be written as
\[ Z^*(q)=W^*_{\mathrm{e}}(q)D\{\Phi_{\mathrm{e}}(x)\}, \tag{8} \]
where the equivalent transfer function of the linear part of the pulse-width system, \(W^*_{\mathrm{e}}(q)\), is determined by the relation
\[ W^*_{\mathrm{e}}(q)=\sum_{\nu=1}^{l} c_{\nu0}\, \frac{e^{q_\nu}-1}{e^q-e^{q_\nu}}; \qquad (\nu=1,2,\ldots,l). \tag{9} \]
The structural diagram of the equivalent pulse-width system obtained now corresponds to the diagram shown in Fig. 1a. It should be emphasized that the adopted transformation of the pulse-width system is admissible only when investigating the stability of the pulse-width system by the frequency criterion developed by Ya. Z. Tsypkin for NIAS.
According to this criterion, the equilibrium position of a pulse-width system with the characteristic of the nonlinear element belonging to the sector \((0,K)\) will be absolutely stable if the frequency characteristic of the equivalent linear part satisfies the condition
\[ \operatorname{Re} W^*_{\mathrm{e}}(j\omega)>-\frac{1}{k}. \tag{10} \]
From this one can find the maximum possible, or critical, value of the upper bound of the sector:
\[ k_{\mathrm{cr}}=\frac{-1}{\min \operatorname{Re} W^*_{\mathrm{e}}(j\omega)}, \tag{11} \]
and the critical value of the gain coefficient of the pulse-width system
\[ k_{\mathrm{i}} x_{\mathrm{cr}}= \frac{(1-e^{-q_l})(zx_0-1)k_{\mathrm{i}}} {(e^{-q_l z x_0}-e^{-q_l})\min \operatorname{Re} W^*_{\mathrm{e}}(j\omega)}. \tag{12} \]
where \(k_{\mathrm{I}}\) is the height of the pulses at the output of the width modulator.
The frequency criterion (10) gives a comparatively small value for the upper bound of the sector \(k\). To increase the value of \(k\), in [3, 4] restrictions are imposed on the derivative characteristic of the nonlinear element, i.e., the conditions must be satisfied
\[ \Phi_{\mathrm{e}}(0)=0;\quad 0<\frac{\Phi_{\mathrm{e}}(x)}{x}<k; \tag{13} \]
\[ \sup \Phi'_{\mathrm{e}}(x)=k'. \]
In this case the upper bound of the sector can be increased if
\[ \left|1-k'\left(e^{j\bar{\omega}}-1\right)W^*_{\mathrm{e}}(j\bar{\omega})\right|<1. \tag{14} \]
Having found from inequality (14) the maximum value of \(k'\), one can also determine the gain coefficient of the pulse-width system
\[ k_{\mathrm{I}}x_{\mathrm{cr}}=k'\frac{e^{q_l}-1}{q_l}. \tag{15} \]
As an example, let us consider a second-order pulse-width system whose transfer function of the continuous part is equal to:
\[ k_{\mathrm{H}}(q)=\frac{\beta^2}{q^2+2\xi\beta q+\beta^2};\quad \beta=\frac{T}{T_0}. \tag{16} \]
The frequency characteristic of the equivalent LIP of this system can be represented as:
\[ W^*_{\mathrm{e}}(j\bar{\omega})=\frac{c_1}{e^{j\bar{\omega}}-e^{q_1}}+\frac{c_2}{e^{j\bar{\omega}}-e^{q_2}}, \tag{17} \]
where \(c_1=c_{10}(e^{q_1}-1)\); \(c_2=c_{20}(e^{q_2}-1)\).
The minimum value of the real part of the frequency characteristic can be determined analytically:
\[ \min Re W^*_l(j\bar{\omega})= \frac{\left[\sqrt{c_2(1-e^{2q_1})}+\sqrt{c_1(e^{2q_2}-1)}\right]^2} {2(e^{q_1}-e^{q_2})(e^{q_1+q_2}-1)} + \]
\[ +\frac{c_1-c_2}{e^{q_2}-e^{q_1}}. \tag{18} \]
Substituting the values \(\min Re W^*_{\mathrm{e}}(j\bar{\omega})\) into formula (10), one can obtain the dependence of the critical gain coefficient on the system parameters. This gain coefficient depends substantially on the equilibrium position \(x_0\).
Table 1 gives the values of \(k_{\mathrm{I}}x_{\mathrm{cr}}\) of the system, computed by the frequency criterion of Ya. Z. Tsypkin (formulas 12, 18), by the criterion with restriction on the derivative of the NE characteristic (formulas 14, 15) for \(\beta=0.3\) and \(x_0\approx 1\). Also given here are the values of \(k_{\mathrm{I}}x_{\mathrm{cr}}\) obtained in modeling the system on the MNБ-1 machine.
Table 1
| Damping coefficient $\xi$ | Damping coefficient $\xi$ | Damping coefficient $\xi$ | |
|---|---|---|---|
| 2 | 1.33 | 1.11 | |
| $k_{\mathrm{I}}\tau_{\mathrm{cr}}$ by the frequency criterion of Ya. Z. Tsypkin | 8.2 | 6.25 | 5.1 |
| $k_{\mathrm{I}}\tau_{\mathrm{cr}}$ by the criterion with a derivative constraint | 14.5 | 9.0 | 8 |
| $k_{\mathrm{I}}\tau_{\mathrm{cr}}$ by the model | 17 | 11 | 9 |
In comparing the results obtained, one may conclude that the considered criteria of absolute stability of NIAS make it possible to assess the stability of pulse-width systems with a sufficient degree of accuracy.
REFERENCES
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Ya. Z. Tsypkin. On stability in the large of nonlinear pulse automatic systems. Dokl. AN SSSR, vol. 145, no. 1, 1962.
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Ya. Z. Tsypkin. Theory of Linear Pulse Systems. Fizmatgiz, 1963.
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Ya. Z. Tsypkin. On absolute stability of one class of nonlinear pulse automatic systems. Avtomatika i Telemekhanika, vol. 25, no. 7, 1964.
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E. I. Jury, B. W. Lee. On the stability of a certain class of nonlinear sampled-data systems. “IEEE Trans. Automat. Control,” 1964, 9, no. 1, 51–61.