Calculation of Transient Processes in Pulse-Width Systems of the Second Kind
Abstract Generated abstract
The paper addresses the calculation of transient processes in closed-loop pulse-width modulated control systems with modulation of the second kind, where pulse duration depends on the current modulator input value. It derives a recurrence equation for such systems by summing the continuous part’s responses to preceding rectangular pulses and incorporating the pulse-width modulator characteristic. Because the resulting equation is transcendental, the paper proposes graphical solution for first-order continuous parts and approximate analytical methods for higher-order systems using truncated Taylor expansions of exponential terms. A simulation example for a second-order continuous part is reported to indicate that the proposed procedures are suitable for engineering calculations.
Full Text
Calculation of Transient Processes in Pulse-Width Modulated Systems of the Second Kind
I. A. Gonchar
(Presented by the scientific seminar of the Department of Automation and Telemechanics)
A pulse-width automatic control system (PWCS) consists of a pulse-width modulator (PWM) and a continuous part (CP) (Fig. 1).
Fig. 1.
The input quantity of the continuous part is a sequence of rectangular pulses of height \(k_i\) and relative durations \(\gamma_0, \gamma_1, \gamma_2, \ldots, \gamma_n\) (Fig. 2).
Fig. 2.
For pulse-width systems with modulation of the second kind (PWCS II), the dependence of the relative pulse duration \(\gamma_n\) on the input quantity of the single-cycle modulator \(x(t)\) is determined by the relation
\[ \gamma_n = \begin{cases} z \cdot x[n+\gamma_n], & 0 \le x[n+\gamma_n] \le \dfrac{1}{z}, \\[6pt] 1, & x[n+\gamma_n] \ge \dfrac{1}{z}, \\[6pt] 0, & x[n+\gamma_n] \le 0, \end{cases} \tag{1} \]
where \(z\) is the slope of the characteristic of the pulse-width modulator.
We shall adhere to the definitions of modulation of the first and second kind and to the notation given in [1].
From (1) it is evident that the duration of the pulses at the output of a modulator of the second kind depends on the current value of the input quantity, whereas under modulation of the first kind the duration of the pulses depends on the values of the modulator input quantity at discrete instants of time \((n = 0, 1, 2 \ldots)\).
The derivation of the recurrence equation and the method for constructing transients in PWS with modulation of the first kind are presented in [2]. However, for PWS with modulation of the second kind, the recurrence equation has a different form. This also imposes its own peculiarities on the methods for calculating transients.
In the present article a procedure is proposed for constructing transients in pulse-width systems of the second kind.
Equation of pulse-width systems with modulation of the second kind
In deriving the recurrence equation of the system we shall follow the procedure set forth in [2], taking into account that the response of the continuous part to a step action of height \(\kappa_{\mathrm{и}}\) is determined by the well-known expansion formula [1]:
\[ \kappa_{\mathrm{и}} h(t) = c_{00} + \sum_{\nu=1}^{l} c_{\nu 0} e^{q_\nu \bar{t}}, \tag{2} \]
where
\[ c_{00} = \frac{\kappa_{\mathrm{и}} \cdot P_{\mathrm{н}}(0)}{Q_{\mathrm{н}}(0)}; \qquad c_{\nu 0} = \frac{\kappa_{\mathrm{и}} \cdot P_{\mathrm{н}}(q_\nu)} {Q'_{\mathrm{н}}(q_\nu)\cdot q_\nu}, \]
\(P_{\mathrm{н}}(q)\) and \(Q_{\mathrm{н}}(q)\) are the numerator and denominator of the transfer function of the continuous part
\[ \kappa_{\mathrm{н}}(q) = \frac{P_{\mathrm{н}}(q)}{Q_{\mathrm{н}}(q)}, \tag{3} \]
\(q_\nu\) are the poles of the transfer function of the continuous part,
\[ \nu = 1, 2 \ldots l . \]
The output quantity of the open-loop PWS II at the instants of time \(\bar{t} = \gamma_0,\ 1+\gamma_1,\ 2+\gamma_2\), etc., is equal to the sum of the responses from all preceding pulses, i.e.,
\[ z[\gamma_0] = \sum_{\nu=1}^{l} c_{\nu 0} e^{q_\nu \gamma_0} + c_{00}, \]
\[ z[1+\gamma_1] = \sum_{\nu=1}^{l} c_{\nu 0} \left[ e^{q_\nu} + 1 - e^{q_\nu(1-\gamma_0)} \right] e^{q_\nu \gamma_1} + c_{00}, \tag{4} \]
\[ z[2+\gamma_2] = \sum_{\nu=1}^{l} c_{\nu 0} \left[ e^{2q_\nu} + e^{q_\nu} + 1 - e^{q_\nu(2-\gamma_0)} - e^{q_\nu(1-\gamma_1)} \right] e^{q_\nu \gamma_2} + c_{00} \]
and so on.
For the instant of time \(t = n+\gamma_n\), the output quantity of the continuous part can be determined by the relation
\[ z[n+\gamma_n] = \sum_{\nu=1}^{l} (a_{\nu n} - b_{\nu n}) e^{q_\nu \gamma_n} + c_{00}, \tag{5} \]
where
\[ a_{\nu n} = a_{\nu,n-1} + c_{\nu 0} e^{nq_\nu}, \]
\[ b_{\nu n} = e^{q_\nu} \left( b_{\nu,n-1} + c_{\nu 0} e^{-q_\nu \gamma_{n-1}} \right). \]
The closing equation for PWM II is written in the form
\[ x[n+\gamma_n]=f[n+\gamma_n]-z[n+\gamma_n]. \tag{6} \]
Eliminating \(z[n+\gamma_n]\) from equations (5) and (6) and taking (1) into account, we obtain the recurrence equation of the closed-loop system
\[ x[n+\gamma_n]=f[n+\gamma_n]-\sum_{\nu=1}^{l}(a_{\nu n}-b_{\nu n})e^{q_\nu x[n+\gamma_n]}-c_{00} \tag{7} \]
for
\[ 0\leq x[n+\gamma_n]\leq \frac{1}{\varkappa}. \]
Methods for Calculating Transient Processes in PWM II
Equation (7) can be used to construct transient processes in closed-loop PWM II systems when an input action of step form and height
\[ f(\bar t)=f=\mathrm{const}. \tag{8} \]
is applied.
However, the transcendental form of this equation introduces certain difficulties in its solution.
Let the continuous part of the system be described by a first-order differential equation, i.e., \(l=1\). Then equation (7), taking (8) into account, is written as
\[ \frac{x[n+\gamma_n]}{b_n-a_n}+\frac{f-c_{00}}{a_n-b_n}=e^{q_1 x[n+\gamma_n]} \tag{9} \]
for
\[ 0\leq x[n+\gamma_n]\leq \frac{1}{\varkappa}. \]
Equation (9) for the instants of time \(n=0,1,2,3\ldots\) is easily solved graphically if it is replaced by the equivalent equation
\[ \varphi\bigl(x[n+\gamma_n]\bigr)=\psi\bigl(x[n+\gamma_n]\bigr), \tag{10} \]
where
\[ \varphi\bigl(x[n+\gamma_n]\bigr)= \frac{x[n+\gamma_n]}{b_n-a_n}+\frac{f-c_{00}}{a_n-b_n}, \tag{11} \]
\[ \psi\bigl(x[n+\gamma_n]\bigr)=e^{q_1 x[n+\gamma_n]}. \tag{12} \]
For this purpose, for the specified parameters of the system, the graph of the function \(\psi\bigl(x[n+\gamma_n]\bigr)\) is constructed (Fig. 3). Then, for \(n=0\), the coefficients \(a_0=c_{10}\) and \(b_0=0\) are determined, and the graph of the linear function is constructed
\[ \varphi\bigl(x[\gamma_0]\bigr)=\frac{f-c_{00}}{a_0}-\frac{x[\gamma_0]}{a_0}. \tag{13} \]
The abscissa of the point of intersection \(x[\gamma_0]\) of the graphs of functions (12) and (13) is the solution of the transcendental equation (9) for \(n=0\). If the obtained solution satisfies the condition
\[ 0\leq x[\gamma_0]\leq \frac{1}{\varkappa}, \tag{14} \]
then the relative duration of the first pulse is determined from the relation
\[ \gamma_0=\varkappa x[\gamma_0], \tag{15} \]
and the output quantity of the system is determined as
\[ z[\gamma_0]=f-x[\gamma_0]. \tag{16} \]
If the solution obtained is \(x[\gamma_0]>\dfrac{1}{\chi}\), then the relative duration of the first pulse is \(\gamma_0=1\), and the output quantity of the system can be determined from equation (5):
\[ z[1]=(a_0-b_0)e^{q_1}+c_{00}. \tag{17} \]
In Fig. 3 the hatched region is the saturation region, for which \(\gamma_n=1\). Then, for \(n=1\), the coefficients \(a_1\) and \(b_1\) are determined from (5), and by constructing the graph of the function
\[ \varphi\bigl(x[1+\gamma_1]\bigr) = \frac{x[1+\gamma_1]}{b_1-a_1} + \frac{f-c_{00}}{a_1-b_1} \]
the input quantity of the modulator is found.
Fig. 3.
Analogously, the transient process is calculated for the instants of time \(n=2,3,4\ldots\). In this case the nonlinear function \(\psi\bigl(x[n+\gamma_n]\bigr)\) remains constant, while for the function \(\varphi\bigl(x[n+\gamma_n]\bigr)\) only the angle of inclination changes as \(n\) changes.
For systems with a continuous part of second order and higher, the graphical solution of equation (7) becomes cumbersome, since for each new instant of time \(n\) it is necessary to construct the right- and left-hand sides of equation (10). In practical pulse-width systems the quantization period is usually smaller than the time constant of the continuous part and \(q_\nu<0.5\). Therefore it is quite permissible to replace the exponential function by the first terms of the Taylor series. If we take
\[ e^{q_\nu\gamma_n}\approx 1+q_\nu\gamma_n, \tag{18} \]
then equation (7) can be transformed into the form
\[ x[n+\gamma_n] = \frac{ f[n+\gamma_n]-c_{00}-\displaystyle\sum_{\nu=1}^{l}(a_{\nu n}-b_{\nu n}) }{ 1+\chi\displaystyle\sum_{\nu=1}^{l}q_\nu(a_{\nu n}-b_{\nu n}) }. \tag{19} \]
for
\[ 0 \leq x[n+\gamma_n] \leq \frac{1}{\kappa}. \]
In this case the transient process is constructed analytically by successively solving equation (19) for the values \(n=0,1,2,3,\ldots\).
For a more accurate construction of the transient process in a PWM system II with a continuous part of order higher than the first, one may take:
\[ e^{q_\nu \gamma_n}\simeq 1+q_\nu\gamma_n+\frac{q_\nu^2\gamma_n^2}{2}. \tag{20} \]
Then from equation (7), taking (20) into account, we obtain
\[ x^2[n+\gamma_n]=-x[n+\gamma_n]A_n+B_n, \tag{21} \]
where
\[ A_n=2\, \frac{ 1+\kappa\displaystyle\sum_{\nu=1}^{l} q_\nu(a_{\nu n}-b_{\nu n}) }{ \kappa^2\displaystyle\sum_{\nu=1}^{l} q_\nu^2(a_{\nu n}-b_{\nu n}) }, \]
\[ B_n=2\, \frac{ f[n+\gamma_n]-c_{n0}-\displaystyle\sum_{\nu=1}^{l}(a_{\nu n}+b_{\nu n}) }{ \kappa^2\displaystyle\sum_{\nu=1}^{l} q_\nu^2(a_{\nu n}-b_{\nu n}) }. \]
In constructing the transient process, equation (21) can be solved analytically, by nomograms, or by a graphical method. In a graphical solution, the order of calculating the process from (21) is analogous to that considered for the case of a first-order continuous part. The functions \(\varphi(x[n+\gamma_n])\) and \(\psi(x[n+\gamma_n])\) of equation (10) can be defined as:
\[ \varphi(x[n+\gamma_n])=x^2[n+\gamma_n], \tag{22} \]
\[ \psi(x[n+\gamma_n])=-x[n+\gamma_n]A_n+B_n. \tag{23} \]
Simulation of a PWM system II with a second-order continuous part on an MN-1 machine showed that the considered methods for calculating the transient process in PWM systems II are quite suitable for engineering practice.
REFERENCES
- Ya. Z. Tsypkin. Theory of Linear Pulse Systems. Fizmatgiz, 1956.
- Ya. Z. Tsypkin. Calculation of processes in nonlinear systems of discontinuous regulation. Avtomatika i telemekhanika, vol. XVII, No. 6, 1956.