UDC 517.941+517.948

V. A. Yakubovich

On Pulse Control Systems with Width Modulation

(Presented by Academician V. I. Smirnov on July 8, 1967)

Control systems with pulse-width modulation (p.-w.m.), despite their great applied importance, have been little studied theoretically. Stability conditions for systems with p.-w.m. were obtained only quite recently in the works of A. Kh. Gelig \((^1)\), V. M. Kuntsevich and Yu. M. Chekhovoi \((^2)\). One special case was considered by E. Jury and B. Lee in \((^3)\). Below it will be shown that the general quadratic stability criterion established in \((^{4,5})\) can, after small refinements (see Theorems 1 and 2 below), be applied to systems with p.-w.m. This makes it possible, in contrast to \((^{1,2})\), to obtain frequency-domain stability conditions at once for systems containing, in addition to pulse modulators, also nonlinear or linear nonstationary blocks of the usual types (see Section \(4^\circ\)). A stability criterion for a system with one pulse modulator is established by Theorem 3.

\(1^\circ\). A large class of control systems (in particular, systems with p.-w.m.) is described by the equations

\[ \sigma_t=\alpha_t+\int_0^t \Omega(t-\tau)\varphi_\tau d\tau-R\varphi_t,\qquad \varphi_t=\varphi[t,\sigma_\tau|_{\tau=0}^{t},\psi_0]. \tag{1} \]

The first equation (1) describes the linear part of the system, and the second the nonlinear part. In (1) \(\alpha_t,\sigma_t,\varphi_t\) are vector functions of orders \(m,m\), and \(n\), respectively; \(\Omega(t)\) is a rectangular matrix function of order \(n\times m\) of the impulse characteristics of the linear part of the system; \(R\) is a constant \(n\times m\) matrix of the so-called tachometric feedback coefficients. In many cases \(R=0\). All quantities in (1) are real. The second equation (1) means that the outputs of the nonlinear blocks (the components of the vector \(\varphi_t\)) may depend on the values of the inputs (the components of the vector \(\sigma_\tau\)) at all preceding instants of time \(0\le \tau\le t\), and also, possibly, on some vector parameter \(\psi_0\).

The dependence of \(\varphi_t\) on \(\sigma_\tau|_{\tau=0}^{t}\) and \(\psi_0\) occurs for nonlinearities of hysteresis type. For systems with p.-w.m. some of the components of the vector function \(\varphi[t,\sigma_\tau|_{\tau=0}^{t},\psi_0]\) depend only on \(\sigma_\tau|_{\tau=0}^{t}\), and this dependence has the special form indicated below (see \(3^\circ\)).

Below it is assumed that a solution of the system (1) exists on \((0,\infty)\).* We shall consider the condition

\[ \int_0^{t_k} F(\varphi_t,\sigma_t)\,dt \ge -\gamma,\qquad t_k\to\infty. \tag{2} \]

Here \(\varphi_t,\sigma_t\) is a solution of the system (1); \(F(\varphi_t,\sigma_t)\) is some real quadratic form of its arguments, and \(t_k\) is some unboundedly increasing sequence of times, \(t_0=0\). We extend \(F(\varphi_t,\sigma_t)\), preserving Hermiticity, to complex values of \(\varphi_t,\sigma_t\) (this extension is unique).

Theorem 1. Suppose that: \((a_1)\) the linear part of the system (1) is stable in the following sense: \(|\alpha_t|\in L_2(0,\infty)\), \(|\Omega(t)|\le Ce^{-\varepsilon t}\), \(\varepsilon>0\);

* If the second equation (1) describes only pulse-width modulators (their equations have the form (6)—see below), then the solution obviously exists on \((0,\infty)\). We note that in \((^4)\) only local existence of a solution is assumed.

\((b_1)\) either \(|\varphi_t| \leqslant \mathrm{const}\), or \(F(0,\sigma_t) \geqslant 0\) for all \(\sigma_t\). Define the matrix of frequency characteristics of the linear part of the system by the formula

\[ \chi(i\omega)=R-\int_0^\infty \Omega(t)e^{-i\omega t}\,dt \tag{3} \]

and put \(\widetilde F(i\omega,\widetilde\varphi)=F(\widetilde\varphi,\widetilde\sigma)\), where \(\widetilde\sigma=-\chi(i\omega)\widetilde\varphi\), and \(\widetilde\varphi\) is a complex vector of order \(n\). Suppose that: \((c_1)\) the form \(\widetilde F(i\omega,\widetilde\varphi)\) is a negative definite form in \(\widetilde\varphi\) for all \(-\infty \leqslant \omega \leqslant +\infty\). Then: \((A_1)\) \(|\varphi_t|\in L_2(0,\infty)\), \(|\sigma_t|\in L_2(0,\infty)\); \((B_1)\) for some constants \(\delta>0\), \(\varkappa>0\), independent of \(a_t\), the estimate* holds

\[ \delta\{\|\varphi_t\|^2+\|\sigma_t\|^2\}\leqslant \gamma+\varkappa\|a_t\|^2 . \]

Remark. The condition \(|\Omega(t)|\leqslant Ce^{-\varepsilon t}\), \(\varepsilon>0\), can be weakened. In the case when \(F(0,\sigma)\geqslant 0\), it is sufficient that \(|\Omega(t)|\in L(0,\infty)\); in the case when \(|\varphi_t|\leqslant \mathrm{const}\), it is sufficient that \(|\Omega(t)|\in L(0,\infty)\),

\[ \int_t^\infty |\Omega(\tau)|\,d\tau \in L_2(0,\infty). \]

The proof of Theorem 1 for the case when \(F(0,\sigma)\geqslant 0\) repeats word for word the proof of Theorem 1 in \((^4)\). In doing so one should take \(T=t_k\). Let \(|\varphi_t|\leqslant \mathrm{const}\). Assertion \((A_1)\) is proved word for word in the same way as Theorem 2 in \((^4)\)**. To prove assertion \((B_1)\), one should in formula (4.8) of \((^4)\) pass to the limit \(T=t_k\to\infty\) and use the fact that \(\gamma_0=\gamma\) for \(t_0=0\) and \(J\leqslant 0\).

\(2^\circ\). Let \(R=0\) in equations (1). Suppose that condition (2) is replaced by the more general condition

\[ \int_0^{t_k} F(\varphi_t,\sigma_t,\dot\sigma_t)\,dt\geqslant -\gamma,\qquad t_k\to\infty, \tag{4} \]

where \(t_k\) is, as before, an unbounded increasing sequence, \(t_0=0\), and \(F(\varphi_t,\sigma_t,\dot\sigma_t)\) is a Hermitian form of its arguments. Put

\[ \widetilde F(i\omega,\widetilde\varphi)=F(\widetilde\varphi,\widetilde\sigma,i\omega\widetilde\sigma),\quad \text{where } \widetilde\sigma=-\chi(i\omega)\widetilde\varphi \tag{5} \]

and \(\chi(i\omega)\) is determined from (3) for \(R=0\).

Theorem 2. Let the conditions \((a_1)\), \((c_1)\) of Theorem 1 be satisfied (where \(\widetilde F(i\omega,\widetilde\varphi)\) is determined from (5)), and also: \((a_2)\) \(|\dot a_t|\in L(0,\infty)\), \(|\dot\Omega(t)|\leqslant Ce^{-\varepsilon t}\), \(\varepsilon>0\); \((b_2)\) either \(F(0,\sigma_t,\dot\sigma_t)\geqslant 0\) for all possible \(\sigma_t,\dot\sigma_t\), or \(|\varphi_t|\leqslant \mathrm{const}\). Then: \((A_2)\) \(|\varphi_t|\in L_2(0,\infty)\), \(|\sigma_t|\in L_2(0,\infty)\), \(|\dot\sigma_t|\in L_2(0,\infty)\), and, consequently, \(|\sigma_t|\to 0\) as \(t\to\infty\); \((B_2)\) the estimate

\[ \delta\{\|\varphi_t\|^2+\|\sigma_t\|^2+\|\dot\sigma_t\|^2\}\leqslant \gamma+\varkappa\{\|a_t\|^2+\|\dot a_t\|^2\} \]

holds, where the constants \(\varkappa>0\), \(\delta>0\) do not depend on \(a_t\); in particular, \(\max_{t\geqslant 0}|\sigma_t|\to 0\) if \(\gamma\to 0\), \(\|a_t\|\to 0\), \(\|\dot a_t\|\to 0\).

The proof of Theorem 2 is carried out by means of the device set forth in § 5 of \((^4)\). System (1) is reduced to system (5.1), (5.2) of \((^4)\), to which Theorem 1 is applied.

\(3^\circ\). The use of Theorems 1, 2 for a system with P.W.M. is based on the lemma given below, which establishes, for the pulse-width modulator, a relation of the form (2). If the system, besides the pulse-width modulator, contains a number of nonlinear blocks of ordinary types, then, as shown in \((^{4,5})\), quadratic relations of the form

\[ F_j=(\varphi_t,\sigma_t,\dot\sigma_t)=0,\quad j=1,\ldots,p, \]

\[ \int_0^{t_k} F_j(\varphi_t,\sigma_t,\dot\sigma_t)\,dt\geqslant -\gamma_j,\quad j=p+1,\ldots,p+q. \]

* By \(|\varphi_t|\) is denoted the square root of the sum of the squares of the components of the vector \(\varphi_t\), and

\[ \|\varphi_t\|^2=\int_0^\beta |\varphi_t|^2\,dt. \]

** The assumption \(|a_t|\leqslant \mathrm{const}\) in Theorem 2 of \((^4)\) is superfluous.

Here \(t_k\) is any sequence, \(t_k \to \infty\), and \(\gamma_j\) are certain numbers. Among the last couplings one may also include a coupling of the form (2) for a pulse-width modulator. Setting

\[ F(\varphi_t,\sigma_t,\dot{\sigma}_t)=\sum_{j=1}^{p+q}\tau_j F_j, \]

where \(\tau_j\) are arbitrary for \(j=1,\ldots,p\) and \(\tau_j\ge 0\) for \(j=p+1,\ldots,p+q\), we find that (4) is satisfied. Theorem 2 or 1 will give frequency-domain stability conditions. In doing so one should, if possible, find the “envelope” of these conditions over all possible \(\tau_j\) of the indicated form (see the examples in (5)). One can proceed in a completely analogous way when there are several modulators in the system. Let us derive relations (2) for a pulse-width modulator. Let \(\sigma_t=\sigma(t)\), and let \(\varphi_t\) be the scalar input and output of the modulator; \(\Delta\) is the dead-zone magnitude, and \(t_k\) is the instant at which the \(k\)-th pulse is sent. The operation of the modulator is described by the equations

\[ \sigma^{(k)}=\sigma(t_k),\qquad t_{k+1}=\Psi(t_k,\sigma^{(k)}), \]

\[ \varphi_t=0\quad \text{for } t_k\le t<t_{k+1},\ \text{if } |\sigma^{(k)}|<\Delta, \]

\[ \varphi_t=s_k(t)\quad \text{for } t_k\le t<t_{k+1},\ \text{if } |\sigma^{(k)}|\ge \Delta. \tag{6} \]

The function \(s_k(t)\) determines the shape of the \(k\)-th pulse and depends in some way on \(\sigma^{(k)}\). In the case of a rectangular pulse we have

\[ s_k(t)=\operatorname{sign}\sigma^{(k)} \quad \text{for } t_k\le t<t_k+T(|\sigma^{(k)}|), \]

\[ s_k(t)=0 \quad \text{for } t_k+T(|\sigma^{(k)}|)\le t<t_{k+1}, \tag{7} \]

where \(T(\sigma)\) is some monotone and bounded function.

Lemma. Let the input \(\sigma_t\) and the output \(\varphi_t\) of the modulator with equations (6) be related by the relation

\[ \sigma_t=\xi(t)+\eta(t)+a\varphi_t, \]

where the functions \(\xi(t)\) and \(\eta(t)\) satisfy the conditions

\[ |\xi(t)|\le b,\qquad |\eta(t)|\in L(0,\infty). \]

Denote

\[ S_k(t)=\int_t^{t_{k+1}} s_k(t)\,dt,\qquad S_k=S(t_k),\qquad Q_k=\int_{t_k}^{t_{k+1}} |s_k(t)|\,dt,\qquad M_k=\int_{t_k}^{t_{k+1}} [s_k(t)]^2\,dt \]

and suppose that \(|S_k(t)|\le c\) for all \(k\), \(\sigma^{(k)}\), and \(t_k\le t\le t_{k+1}\), and

\[ M_k^{-1}\bigl[aS_{k/2}^2+\sigma^{(k)}S_k-bQ_k\bigr]\ge \nu \quad \text{when } |\sigma^{(k)}|\ge \Delta, \tag{8} \]

where \(\nu\) is a constant independent of \(k\). Then, for any \(t_k\),

\[ \gamma=C\int_0^\infty |\eta(t)|\,dt \]

satisfies (2) with the form \(F=(\sigma_t-\nu\varphi_t)\varphi_t\).

From (8) we obtain that, in the case of rectangular pulses (7), the value of \(\nu\) is found from the formula

\[ \nu=\inf_{\sigma\ge \Delta}\bigl[\sigma+\tfrac12(a-b)T(\sigma)\bigr]. \]

Proof. Denote

\[ I_k=\int_{t_k}^{t_{k+1}}(\sigma_t-\nu\varphi_t)\varphi_t\,dt. \]

When \(|\sigma^{(k)}|<\Delta\), we have \(I_k=0\). Let \(|\sigma^{(k)}|\ge \Delta\). Integrating by parts, we find

\[ I_k=aS_{k/2}^2+\sigma^{(k)}S_k-\nu M_k+ \int_{t_k}^{t_{k+1}}(\xi+\eta)S_k(t)\,dt. \]

If (8) is fulfilled, we have

\[ I_k\ge -c\int_{t_k}^{t_{k+1}}|\eta|\,dt. \]

Summing these inequalities, we obtain that (2) is fulfilled with the indicated values of \(F\) and \(\gamma\).

Theorem 3. Consider system (1), where all quantities are scalar \((m=n=1)\), \(R=0\), and the second equation (1) has the form (6). Let the pulses of the pulse-width modulator be bounded:

\[ |s_k(t)|\le c_0,\qquad |S_k(t)|\le c. \]

Let the conditions \((a_1)\), \((a_2)\) of Theorems 1, 2 be satisfied, and also \(|a_t| \in L(0,\infty)\).

Define \(\nu\) from relation (8), where

\[ a=\Omega(0), \qquad b=c_0\int_0^\infty |d\Omega(t)|. \]

Suppose that \(\nu+\operatorname{Re}\chi(i\omega)>0\) for \(-\infty\leq \omega\leq +\infty\), where \(\chi(i\omega)\) is defined from (3). Then assertion \((A_2)\) of Theorem 2 is valid, as is the estimate

\[ \|\varphi_t\|^2+\|\sigma_t\|^2+\|\dot{\sigma}_t\|^2 \leq \chi_1\int_0^\infty |a_t|\,dt +\chi_2\bigl(\|\alpha_t\|^2+\|\dot{\alpha}_t\|^2\bigr), \]

where \(\chi_j\) do not depend on \(a_t\).

Proof. From (1) it follows that

\[ \dot{\sigma}_t=\xi(t)+\eta(t)+a\varphi_t, \]

\(\eta(t)=\alpha_t,\ |\xi(t)|\leq b\) for the values of \(a\) and \(b\) indicated in Theorem 3. Applying the lemma and Theorem 2, we obtain the assertion of Theorem 3.

4°. In a completely analogous way one can obtain frequency conditions for the stability of a system with several modulators and, possibly, with several nonlinear blocks of the usual type. Consider, for example, a single-loop system with one modulator and one nonlinear block, which is described by the scalar equations

\[ \sigma_{1t}=a_{1t}+\int_0^t \Omega_1(t-\tau)\varphi_{2\tau}\,d\tau,\qquad \sigma_{2t}=a_{2t}+\int_0^t \Omega_2(t-\tau)\varphi_{1\tau}\,d\tau-\rho\varphi_{1t}, \tag{9} \]

where \(\alpha_{jt}, \Omega_{jt}\) satisfy condition \((a_1)\) of Theorem 1 and \(|a_{1t}|\in L(0,\infty)\). Let \(\varphi_{1t}=\varphi_1(\sigma_{1\tau}\mid_{\tau=0}^{t})\) be a pulse modulator with equations (6) and bounded pulses (see Theorem 3), and let \(\varphi_{2t}=\Phi(\sigma_{2t},t)\), where \(\Phi(\sigma_2,t)\) is a function satisfying the conditions

\[ |\Phi(\sigma_2,t)|\leq \Phi_0,\qquad 0\leq \frac{\Phi(\sigma_2,t)}{\sigma_2}\leq \mu_0,\quad \sigma_2\neq 0. \]

From the first equation (9) we find that the hypothesis of the lemma is fulfilled for

\[ \xi(t)=\Omega_1(0)\varphi_{2t}+\int_0^t \dot{\Omega}(t-\tau)\varphi_{2\tau}\,d\tau, \]

\[ \eta(t)=\dot{a}_{1t},\qquad a=0,\qquad b=\Phi_0\left\{|\Omega_1(0)|+\int_0^\infty |d\Omega_1(t)|\right\}. \]

Using the lemma, we obtain that (2) is fulfilled, where

\[ F(\varphi_t,\sigma_t)= \tau_1(\sigma_{1t}-\nu\varphi_{1t})\varphi_{1t} +\tau_2(\sigma_{2t}-\mu_0^{-1}\varphi_{2t})\varphi_{2t}, \]

and the number \(\nu\) is determined from (8) for the indicated values of \(a\) and \(b\). Theorem 1 gives the following result. If for some numbers \(\tau_1>0,\ \tau_2>0\) and all \(-\infty\leq\omega\leq+\infty\) the condition

\[ \nu\mu_0^{-1}\tau_1\tau_2> \left|\tau_1\chi_1(i\omega)+\tau_2\chi_2(i\omega)\right|^2 \]

is fulfilled, then system (9) is stable in the following sense:

\[ \varphi_{jt}\in L_2(0,\infty),\qquad \sigma_{jt}\in L_2(0,\infty) \]

and

\[ \|\varphi_{jt}\|\to 0,\qquad \|\sigma_{jt}\|\to 0,\quad \text{if }\|\alpha_{jt}\|\to 0,\quad j=1,2,\qquad \int_0^\infty |a_{1t}|\,dt\to 0. \]

Here

\[ \chi_1(i\omega)=-\int_0^\infty \Omega_1(t)e^{-i\omega t}\,dt,\qquad \chi_2(i\omega)=\rho-\int_0^\infty \Omega(t)e^{-i\omega t}\,dt. \]

It is easy to obtain (see (6), p. 53, § 8) the necessary and sufficient conditions for the existence of the indicated numbers \(\tau_1,\tau_2\). If \(\rho=0\), the conditions \((a_1)\), \((a_2)\) of Theorems 1, 2 are fulfilled, as well as the frequency condition indicated above, then according to Theorem 2 system (9) is stable in a stronger sense: assertions \((A_2)\), \((B_2)\) of Theorem 2 are valid with

\[ \gamma=c\int_0^\infty |a_{1t}|\,dt. \]

Leningrad State University
named after A. A. Zhdanov

Received
1 VII 1967

CITED LITERATURE

1. A. Kh. Geligt, DAN, 178, No. 4 (1968).
2. V. M. Kuntsevich, Yu. M. Chekhovoi, Avtomatika i telemekh., 28, No. 2 (1967).
3. É. Dzhuri, B. P., Avtomatika i telemekh., 26, No. 6 (1965).
4. V. A. Yakubovich, Vestn. LGU, No. 7, issue 2 (1967).
5. V. A. Yakubovich, Avtomatika i telemekh., 28, No. 6 (1967).
6. F. R. Gantmakher, V. A. Yakubovich, Tr. II Vsesoyuzn. s”ezda po teoret. i prikl. mekh., vol. 1, “Nauka,” 1965.