ANALYSIS OF ABSOLUTE STABILITY OF PULSE-WIDTH MODULATION SYSTEMS

I. A. GONCHAR, O. S. VADUTOV

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

Recently, a need has arisen for new solutions in the field of power supplies. The use, in voltage stabilizers, of regulating elements operating in switching mode appears promising. This operating mode of the regulating element makes it possible to obtain a comparatively high efficiency of the source, since the losses in the regulating element in switching mode are insignificant.

The block diagram of a pulse voltage stabilizer is shown in Fig. 1.

Fig. 1.

The regulating element, periodically closing with a period equal to $T$, passes current during the time $t_i$, and then opens.

The rectangular pulses after the RE are smoothed by an ordinary filter consisting of capacitance $C$ and inductance $L$. The diode $D$ serves to create a circuit through which the inductor current closes after the regulating element opens.

Rectangular control pulses are supplied to the RE from the pulse-width modulator PWM; the pulse duration depends on the difference between the reference and output voltages. Regulation of the output voltage is achieved by changing the ratio $t_i/T$.

Thus, a pulse voltage stabilizer is essentially an automatic control system with pulse-width modulation. Such a system includes a pulse

the pulse element IE, the continuous part CP, and a mismatch element. The block diagram of a pulse-width system (PWS) is shown in Fig. 2.

Fig. 2.

The equation of a pulse element with pulse-width modulation has the form

\[ y[\bar t]= \begin{cases} k_i, & n<\bar t<n+\gamma[n],\\ 0, & n+\gamma[n]<\bar t<n+1. \end{cases} \tag{1} \]

Here \(k_i\) is the pulse height, and \(\gamma[n]\) is the relative duration of the pulses. The dependence of \(\gamma[n]\) on the input quantity \(x[n]\) is shown in Fig. 3; analytically this dependence is written as

\[ \gamma[n]= \begin{cases} \chi x[n], & \text{for } 0<x[n]<\dfrac{1}{\chi},\\ 1, & \text{for } x[n]>\dfrac{1}{\chi},\\ 0, & \text{for } x[n]<0. \end{cases} \tag{2} \]

Pulse systems with PWM are nonlinear systems, since the superposition principle is not valid for them and, moreover, the input characteristic of the pulse element has saturation. Therefore, the investigation of this class of systems in the general case is associated with certain difficulties.

The dynamics of pulse-width systems (PWS) has been considered by a number of authors in a linear approximation, i.e., for small modulation. In [1], graphical and tabular methods are presented for constructing transient processes at any modulation depth; however, these methods do not make it possible to judge the stability of these systems, nor to investigate self-oscillatory regimes [2].

Fig. 3.

V. M. Popov proposed new methods for investigating the absolute stability of continuous nonlinear automatic systems [3], which differ from Lyapunov’s second method.

Analogous methods for nonlinear pulse automatic systems were developed by Ya. Z. Tsypkin [4, 5], and are used in the present work in analyzing the stability of second-order PWS.

Equation of the Closed-Loop Pulse-Width System

A sequence of pulses acts on the continuous part of the PWS; their amplitude remains constant and equal to \(k_{\mathrm{и}}\), while their duration varies according to law (2).

The response of the continuous part to a pulse of duration \(\gamma[m]\), applied at the instant \(\bar t=m\), is equal to

\[ \left\{ \begin{array}{ll} k_{\mathrm{и}} h(\bar t-m), & m \leqslant \bar t \leqslant m+\gamma[m],\\[3pt] k_{\mathrm{и}}\bigl[h(\bar t-m)-h(\bar t-m-\gamma[m])\bigr], & m+\gamma[m] \leqslant \bar t, \end{array} \right. \tag{3} \]

where \(h(\bar t)\) is the response of the continuous part to a unit step, which is determined by the known expansion formula [6].

The output quantity \(z(\bar t)\) of the pulse-width system is equal to the sum of the responses of the continuous part to each pulse:

\[ z(\bar t)=k_{\mathrm{и}}\sum_{m=0}^{n-1} h(\bar t-m)-h(\bar t-m-\gamma[m]) +k_{\mathrm{и}}h(\bar t-n), \]

\[ m \leqslant \bar t \leqslant m+\gamma[m], \tag{4} \]

\[ z(\bar t)=k_{\mathrm{и}}\sum_{m=0}^{n} h(\bar t-m)-h(\bar t-m-\gamma[m]), \]

\[ m+\gamma[m]\leqslant \bar t \leqslant m+1. \]

Replacing, in the last expression, \(\bar t=n+\varepsilon\), we pass to lattice functions:

\[ z[n,\varepsilon]= \sum_{m=0}^{n-1} k_{\mathrm{и}}\bigl(h[n-m,\varepsilon]-h[n-m,\varepsilon-\gamma[m]]\bigr) +k_{\mathrm{и}}h[0,\varepsilon], \tag{5} \]

\[ 0\leqslant \varepsilon \leqslant \gamma[m], \]

\[ z[n,\varepsilon]= \sum_{m=0}^{n} k_{\mathrm{и}}\bigl(h[n-m,\varepsilon]-h[n-m,\varepsilon-\gamma[m]]\bigr), \]

\[ \gamma[m]\leqslant \varepsilon \leqslant 1. \]

Taking into account that

\[ k_{\mathrm{и}}h(\bar t)=c_{00}+\sum_{\nu=1}^{l} c_{\nu 0} e^{q_\nu \bar t}, \tag{6} \]

where

\[ c_{00}=\frac{k_{\mathrm{и}}P_{\mathrm{н}}(0)}{Q_{\mathrm{н}}(0)}; \qquad c_{\nu 0}=\frac{k_{\mathrm{и}}P_{\mathrm{н}}(q_\nu)} {Q'_{\mathrm{н}}(q_\nu)q_\nu}, \]

we obtain the equation of the PWS under consideration with respect to the originals:

\[ z[n,\varepsilon]= \sum_{m=0}^{n-1}\sum_{\nu=1}^{l} c_{\nu 0}\bigl(1-e^{-q_\nu\gamma[m]}\bigr) e^{q_\nu(n-m)}e^{q_\nu\varepsilon} +c_{00}+\sum_{\nu=1}^{l}c_{\nu 0}e^{q_\nu\varepsilon}, \]

\[ 0\leqslant \varepsilon \leqslant \gamma[m], \tag{7} \]

\[ z[n,\varepsilon]= \sum_{m=0}^{n}\sum_{\nu=1}^{l} c_{\nu 0}\bigl(1-e^{-q_\nu\gamma[m]}\bigr) e^{q_\nu(n-m)}, \]

\[ \gamma[m]\leqslant \varepsilon \leqslant 1. \]

Passing to transforms with use of the convolution theorem, we find for the open-loop PWMS

\[ Z^{*}(q,\varepsilon)=\sum_{\nu=1}^{l} c_{\nu 0}\frac{e^{q_\nu}}{e^{q}-e^{q_\nu}} D\{1-e^{-q_\nu\gamma[n]}\}e^{q_\nu\varepsilon} + D\{k_i h[0,\varepsilon]\}, \]

\[ 0\leq \varepsilon \leq \gamma[n], \tag{8} \]

\[ Z^{*}(q,\varepsilon)=\sum_{\nu=1}^{l} c_{\nu 0}\frac{e^{q_\nu}}{e^{q}-e^{q_\nu}} D\{1-e^{-q_\nu\gamma[n]}\}e^{q_\nu\varepsilon}, \]

\[ \gamma[n]\leq \varepsilon \leq 1. \]

For \(\varepsilon=0\), \(h[0,\varepsilon]=0\); then

\[ Z^{*}(q,0)=Z^{*}(q)=\sum_{\nu=1}^{l} c_{\nu 0}\frac{e^{q_\nu}}{e^{q}-e^{q_\nu}} D\{1-e^{-q_\nu\gamma[n]}\}. \tag{9} \]

The transforms are determined from their originals by means of the discrete Laplace transform, or \(D\)-transform,

\[ F^{*}(q)=\sum_{n=0}^{\infty} e^{-qn}f[n]=D\{f[n]\}, \tag{10} \]

where \(q\) is the transform parameter.

Equation (9) 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 series-connected linear pulse part (LPP) and a nonlinear element (NE). In this case the transfer function of the LPP will be

\[ W_{\nu}^{*}(q)=c_{\nu 0}\frac{e^{q_\nu}}{e^{q}-e^{q_\nu}}, \quad (\nu=1,2,\ldots,l), \tag{11} \]

and the characteristic of the nonlinear element is

\[ \Phi_{\nu}(x[n])=1-e^{-q_\nu\gamma[n]}. \tag{12} \]

Fig. 4. Block diagram with input \(f[n]\), summing junction producing \(x[n]\), parallel branches \(\Phi_1(x[n]) \to W_1^{*}(q)\), \(\Phi_2(x[n]) \to W_2^{*}(q)\), \(\ldots\), \(\Phi_l(x[n]) \to W_l^{*}(q)\), output \(z[n]\), and feedback to the summing junction.

The equation of the closed-loop PWMS (Fig. 4) with respect to transforms is written in the form

\[ X^{*}(q)=F^{*}(q)-\sum_{\nu=1}^{l} W_{\nu}^{*}(q)D\{\Phi_{\nu}(x[n])\}, \tag{13} \]

, respectively, with respect to the originals

\[ x[n]=f[n]-\sum_{m=0}^{n}\sum_{\nu=1}^{l} w_{\nu}[m]\,\Phi_{\nu}(x[n-m]), \tag{14} \]

where

\[ W_{\nu}^{*}(q)=D\{w_{\nu}[n]\}. \]

Absolute stability of a second-order PWM system

Consider a pulse-width system whose continuous part is described by a second-order differential equation (Fig. 1).

The stability problems of a nonlinear impulsive automatic system have been developed by Ya. Z. Tsypkin [4, 5] for the case when the NIAS consists of one nonlinear element and one linear impulsive part. We shall show that the criteria for absolute stability of the equilibrium position obtained in these works also extend to a second-order PWM system. For this case, equations (13) and (14) take the form

\[ X^{*}(q)=F^{*}(q)-W_{1}^{*}(q)\cdot D\{\Phi_{1}(x[n])\} -W_{2}^{*}(q)\cdot D\{\Phi_{2}(x[n])\}, \tag{15} \]

\[ x[n]=f[n]-\sum_{m=0}^{n} w_{1}[m]\Phi_{1}(x[n-m]) \]

\[ -\sum_{m=0}^{n} w_{2}[m]\Phi_{2}(x[n-m]). \tag{16} \]

We shall carry out the reasoning analogously to that presented in [4]. Introduce the auxiliary functions

\[ \varphi_{1N}[n]= \begin{cases} \Phi_{1}(x[n]), & 0\leq n\leq N,\\ 0, & n<0;\ n>N; \end{cases} \tag{17} \]

\[ \varphi_{2N}[n]= \begin{cases} \Phi_{2}(x[n]), & 0\leq n\leq N,\\ 0, & n<0,\ n>N \end{cases} \]

and

\[ \psi_{1N}[n]=x_{N}[n]-\frac{1}{k_{1}}\varphi_{1N}[n], \]

\[ \psi_{2N}[n]=x_{N}[n]-\frac{1}{k_{2}}\varphi_{2N}[n]. \tag{18} \]

The function \(x_{N}[n]\) is determined by equation (16) when \(\Phi_{1}(x[n])\) and \(\Phi_{2}(x[n])\) are replaced in it by \(\varphi_{1N}[n]\) and \(\varphi_{2N}[n]\), i.e.,

\[ x[n]=f[n]-\sum_{m=0}^{n} w_{1}[m]\varphi_{1N}[n-m]- \]

\[ -\sum_{m=0}^{n} w_{2}[m]\varphi_{2N}[n-m]. \tag{19} \]

Construct the expression

\[ \rho_N=\sum_{n=0}^{\infty}\varphi_{1N}[n]\psi_{1N}[n] +\sum_{n=0}^{\infty}\varphi_{2N}[n]\psi_{2N}[n]. \tag{20} \]

Substituting into it the values of \(\varphi_{1N}, \varphi_{2N}, \psi_{1N}, \psi_{2N}\), we obtain

\[ \rho_N=\sum_{n=0}^{N}\Phi_1(x[n]) \left[x[n]-\frac{\Phi_1(x[n])}{k_1}\right]+ \]

\[ +\sum_{n=0}^{N}\Phi_2(x[n]) \left[x[n]-\frac{\Phi_2(x[n])}{k_2}\right]. \tag{21} \]

If the characteristics of the nonlinear elements \(\Phi_1(x[n])\) and \(\Phi_2(x[n])\) belong respectively to the sectors \((0,k_1)\) and \((0,k_2)\), i.e.,

\[ 0<\frac{\Phi_1(x[n])}{x[n]}<k_1, \]

\[ 0<\frac{\Phi_2(x[n])}{x[n]}<k_2, \tag{22} \]

then the equilibrium position of the second-order PWM system will be absolutely stable under the condition

\[ \rho_N<c,\qquad (c>0). \tag{23} \]

Let us find the conditions for satisfaction of inequality (23), for which we use the Lyapunov–Parseval formula [3]

\[ \rho_N=\frac{1}{2\pi}\int_{-\pi}^{\pi} \Phi^*_{1N}(-j\bar{\omega})\Psi^*_{1N}(j\bar{\omega})\,d\bar{\omega} + \]

\[ +\frac{1}{2\pi}\int_{-\pi}^{\pi} \Phi^*_{2N}(-j\bar{\omega})\Psi^*_{2N}(j\bar{\omega})\,d\bar{\omega}, \tag{24} \]

where

\[ \Phi^*_{1N}(j\bar{\omega}),\quad \Phi^*_{2N}(j\bar{\omega}),\quad \Psi^*_{1N}(j\bar{\omega})\ \text{and}\ \Psi^*_{2N}(j\bar{\omega}) \]

are spectral functions.

\[ \Phi^*_{1N}(j\bar{\omega}) =D\{\varphi_{1N}[n]\}_{q=j\bar{\omega}} =\sum_{n=0}^{N}e^{-j\bar{\omega}n}\Phi_1(x[n]), \]

\[ \Phi^*_{2N}(j\bar{\omega}) =D\{\varphi_{2N}[n]\}_{q=j\bar{\omega}} =\sum_{n=0}^{N}e^{-j\bar{\omega}n}\Phi_2(x[n]), \tag{25} \]

\[ \Psi^*_{1N}(j\bar{\omega}) =D\{x_N[n]\}_{q=j\bar{\omega}} -\frac{1}{k_1}D\{\varphi_{1N}[n]\}_{q=j\bar{\omega}}, \]

\[ \Psi^*_{2N}(j\bar{\omega}) =D\{x_N[n]\}_{q=j\bar{\omega}} -\frac{1}{k_2}D\{\varphi_{2N}[n]\}_{q=j\bar{\omega}}. \tag{26} \]

Denote

\[ D\{x_N[n]\}_{q=j\bar{\omega}}=X_N^*(j\bar{\omega}), \tag{27} \]

Then equations (26), under condition (27), take the form

\[ \Psi_{1N}^*(j\bar{\omega})= X_N^*(j\bar{\omega})-\frac{1}{k_1}\Phi_{1N}^*(j\bar{\omega}), \]

\[ \Psi_{2N}^*(j\bar{\omega})= X_N^*(j\bar{\omega})-\frac{1}{k_2}\Phi_{2N}^*(j\bar{\omega}). \tag{28} \]

We determine \(X_N^*(j\bar{\omega})\) by subjecting (18) to the \(D\)-transform for \(q=j\bar{\omega}\),

\[ X_N^*(j\bar{\omega})= F^*(j\bar{\omega}) - W_1^*(j\bar{\omega})\Phi_{1N}^*(j\bar{\omega}) - W_2^*(j\bar{\omega})\Phi_{2N}^*(j\bar{\omega}). \tag{29} \]

From equation (24), under conditions (28) and (29), we obtain

\[ \begin{aligned} \rho_N &= \frac{1}{2\pi}\int_{-\pi}^{\pi} \left\{ \Phi_{1N}^*(j\bar{\omega})F^*(j\bar{\omega}) - \Pi_1^*(j\bar{\omega})\left|\Phi_{1N}^*(j\bar{\omega})\right|^2 \right.\\ &\qquad\qquad\left. - W_2^*(j\bar{\omega})\Phi_{1N}^*(-j\bar{\omega})\Phi_{2N}^*(j\bar{\omega}) \right\}\,d\bar{\omega} \\ &\quad+ \frac{1}{2\pi}\int_{-\pi}^{\pi} \left\{ \Phi_{2N}^*(j\bar{\omega})F^*(j\bar{\omega}) - \Pi_2^*(j\bar{\omega})\left|\Phi_{2N}^*(j\bar{\omega})\right|^2 \right.\\ &\qquad\qquad\left. - W_1^*(j\bar{\omega})\Phi_{1N}^*(j\bar{\omega})\Phi_{2N}^*(-j\bar{\omega}) \right\}\,d\bar{\omega}. \end{aligned} \tag{30} \]

Here it is denoted

\[ \Pi_1^*(j\bar{\omega})=W_1^*(j\bar{\omega})+\frac{1}{k_1}, \]

\[ \Pi_2^*(j\bar{\omega})=W_2^*(j\bar{\omega})+\frac{1}{k_2}. \tag{31} \]

Since the integral of the imaginary part of expression (30) is equal to zero, it is meaningful to retain in the integrand only the real parts.

In [5] it is proved that the first two terms in both integrals are bounded under the condition

\[ \operatorname{Re}\Pi_1^*(j\bar{\omega})>0, \]

\[ \operatorname{Re}\Pi_2^*(j\bar{\omega})>0. \tag{32} \]

Therefore we shall consider only the last terms, which we denote

\[ S= -\frac{1}{2\pi}\int_{-\pi}^{\pi} \operatorname{Re}\Phi_{1N}^*(j\bar{\omega})\Phi_{2N}^*(j\bar{\omega})W_2^*(j\bar{\omega})\,d\bar{\omega} - \]

\[ -\frac{1}{2\pi}\int_{-\pi}^{\pi} \operatorname{Re}\Phi_{1N}^*(j\bar{\omega})\Phi_{2N}^*(j\bar{\omega})W_1^*(j\bar{\omega})\,d\bar{\omega}. \tag{33} \]

For the special case in which the continuous part is an oscillatory circuit, the roots of whose characteristic equation...

whose roots are complex conjugates, the functions \(\Phi_{1N}^*(j\bar{\omega})\) and \(\Phi_{2N}^*(j\bar{\omega})\) will also be complex conjugates. Then one may write

\[ S=-\frac{1}{2\pi}\int_{-\pi}^{\pi} \left|\Phi_{1N}^*(j\bar{\omega})\right|^2 \operatorname{Re} W_2^*(j\bar{\omega})\,d\bar{\omega} - \frac{1}{2\pi}\int_{-\pi}^{\pi} \left|\Phi_{1N}^*(j\bar{\omega})\right|^2 \operatorname{Re} W_1^*(j\bar{\omega})\,d\bar{\omega}. \tag{34} \]

Under condition (32), the quantities \(\operatorname{Re} W_1^*(j\bar{\omega})\) and \(\operatorname{Re} W_2^*(j\bar{\omega})\) are bounded; therefore

\[ \rho_N<c \qquad (c>0). \tag{35} \]

Thus, a second-order pulse-width system will be absolutely stable if the characteristics of the nonlinear elements \(\Phi_1(x[n])\) and \(\Phi_2(x[n])\) belong, respectively, to the sectors \((0,k_1)\) and \((0,k_2)\), and

\[ \operatorname{Re}\Pi_1^*(j\bar{\omega}) = \operatorname{Re} W_1^*(j\bar{\omega})+\frac{1}{k_1}>0, \]

\[ \operatorname{Re}\Pi_2^*(j\bar{\omega}) = \operatorname{Re} W_2^*(j\bar{\omega})+\frac{1}{k_2}>0. \tag{36} \]

This stability criterion is only sufficient; moreover, it is valid for pulse-width systems whose continuous part is described by a second-order equation and has complex-conjugate roots.

REFERENCES

1. Ya. Z. Tsypkin. Calculation of processes in nonlinear sampled-data control systems. Avtomatika i telemekhanika, vol. XVII, no. 6, 1956.

2. Ya. V. Pypkin. Self-oscillations in systems with a pulse-width modulator. In: Theory and Application of Discrete Automatic Systems. Publishing House of the USSR Academy of Sciences, Moscow, 1960.

3. V. M. Popov. On the absolute stability of nonlinear automatic-control systems. Avtomatika i telemekhanika, vol. XXII, no. 8, 1961.

4. Ya. Z. Tsypkin. On global stability of nonlinear sampled-data automatic systems. Doklady of the USSR Academy of Sciences, vol. 145, no. 1, 1962.

5. Ya. Z. Tsypkin. Absolute stability of the equilibrium position and of processes in nonlinear sampled-data automatic systems. Avtomatika i telemekhanika, vol. XXIV, no. 12, 1963.

6. Ya. Z. Tsypkin. Theory of Linear Sampled-Data Systems. Fizmatgiz, Moscow, 1963.