Abstract
Full Text
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\tau_j F_j,}^{p+q
]
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
- A. Kh. Geligt, DAN, 178, No. 4 (1968).
- V. M. Kuntsevich, Yu. M. Chekhovoi, Avtomatika i telemekh., 28, No. 2 (1967).
- É. Dzhuri, B. P., Avtomatika i telemekh., 26, No. 6 (1965).
- V. A. Yakubovich, Vestn. LGU, No. 7, issue 2 (1967).
- V. A. Yakubovich, Avtomatika i telemekh., 28, No. 6 (1967).
- F. R. Gantmakher, V. A. Yakubovich, Tr. II Vsesoyuzn. s”ezda po teoret. i prikl. mekh., vol. 1, “Nauka,” 1965.