Reports of the Academy of Sciences of the USSR

1. Volume 160, No. 2

MATHEMATICS

V. A. YAKUBOVICH

FREQUENCY CONDITIONS FOR ABSOLUTE STABILITY AND DISSIPATIVITY OF CONTROLLED SYSTEMS WITH ONE DIFFERENTIABLE NONLINEARITY

(Presented by Academician L. S. Pontryagin on 2 VII 1964)

1°. Consider the system*

\[ dx/dt=Px+q\varphi(\sigma), \qquad \sigma=r^*x, \tag{1} \]

where \(P\) is a Hurwitz matrix, \(\varphi(\sigma)\) is a differentiable function satisfying the conditions

\[ \text{a) } \ 0\leq \sigma\varphi(\sigma)\leq \mu_0\sigma^2; \qquad \text{b) } \ -\alpha_1\leq \varphi'(\sigma)\leq \alpha_2. \tag{2} \]

Here \(\mu_0,\alpha_1,\alpha_2\) are finite numbers and, without loss of generality, \(\alpha_1\geq 0\), \(\alpha_2\geq \mu_0\).** We shall also assume that the vectors \(q,Pq,\ldots,P^{\nu-1}q\) are linearly independent. Introduce the transfer function of the linear part of the system \(\chi(\lambda)=r^*(P-\lambda I)^{-1}q\). Let \(\tau_1,\tau_2,\vartheta\) be certain parameters. Put

\[ \pi(\omega)\tau_1[\mu_0^{-1}+\operatorname{Re}\chi(i\omega)] +\vartheta \operatorname{Re}[i\omega\chi(i\omega)] + \]

\[ +\tau_2\omega^2[1+(\alpha_2-\alpha_1)\operatorname{Re}\chi(i\omega) -\alpha_1\alpha_2|\chi(i\omega)|^2]. \tag{3} \]

Theorem 1. If, for some \(\tau_1>0,\tau_2\geq 0,\vartheta\) satisfying the condition

\[ +\infty \geq \varkappa_{\pm}= \tau_2 \lim_{\sigma\to \pm\infty} \frac{\vartheta}{\sigma^2} \left[ \int_0^\sigma \varphi(s)\,ds-\frac{\sigma\varphi(\sigma)}{2} \right]\geq 0 \tag{4} \]

and for all \(\omega\geq 0\) one has \(\pi(\omega)>0\), then the solution \(x\equiv 0\) of system (1) is asymptotically stable in the large.

Theorem 2. Let relation (2b) be satisfied in the strict sense, i.e. \(-\alpha_1<\varphi'(\sigma)<\alpha_2\), and let \(\mu_0^{-1}+\chi(i\omega)\ne 0\). If, for some \(\tau_1\geq 0\), \(\tau_2>0\), \(\vartheta\), satisfying condition (4), and all \(\omega\geq 0\), one has \(\pi(\omega)\geq 0\), then the solution \(x\equiv 0\) of system (1) is asymptotically stable in the large*.

The condition \(\pi(\omega)>0\) for \(\tau_2=0\) is the frequency condition of V. M. Popov \((^2)\). (Then the requirement that \(\varphi(\sigma)\) be differentiable and condition (2b) are absent.) Examples can be given in which the frequency condition of V. M. Popov—

* Here and below, capital Latin letters denote \(\nu\times \nu\) matrices, lowercase Latin letters denote \(\nu\times 1\) column vectors, and Greek letters denote numbers. The exceptions are \(t\), time, and \(V\), a Lyapunov function. Indices are denoted by any letters. Unless otherwise stated, matrices, vectors, and numbers are real. The asterisk denotes Hermitian conjugation. The notation \(H>0\) (\(H\geq 0\)) means that \(x^*Hx>0\) for \(x\ne 0\) (\(x^*Hx\geq 0\)); \(I\) is the identity matrix, \(0\) is the zero vector.

** The case in which one of these numbers is equal to \(\pm\infty\) is treated in another way in \((^1)\). The case in which, instead of \(0\leq \sigma\varphi\leq \mu_0\sigma^2\), one has \(\mu_1\sigma^2\leq \sigma\varphi\leq \mu_2\sigma^2\) and the matrix \(P+\mu_1qr^*\) is Hurwitz, is reduced to the case considered by the substitution \(\varphi=\varphi_1+\mu_1\sigma\).

*** Theorems 1 and 2 assert the absolute stability of the linear part of system (1) in the class of functions satisfying relations (2), and, for example, the requirement of the existence of the limits \(\lim_{\sigma\to\pm\infty}\varphi(\sigma)/\sigma\leq +\infty\) (then (4) is satisfied for all \(\vartheta\)).

... is not satisfied, but asymptotic stability in the large holds according to Theorems 1 and 2.

The condition \(\pi(\omega)\geqslant 0\) can be given the following geometric interpretation. Denote
\(\zeta(\omega)=\omega^2[1+(\alpha_2-\alpha_1)\operatorname{Re}\chi(i\omega)-\alpha_1\alpha_2|\chi(i\omega)|^2]\),
\(\xi(\omega)=[\zeta(\omega)]^{-1}[\mu_0^{-1}+\operatorname{Re}\chi(i\omega)]\),
\(\eta(\omega)=[\zeta(\omega)]^{-1}\operatorname{Re}\{i\omega\chi(i\omega)\}\), and construct in the plane \(\{\xi,\eta\}\) the curves
\(\Gamma_+\{\xi=\xi(\omega),\eta=\eta(\omega),\zeta(\omega)>0\}\),
\(\Gamma_-\{\xi=\xi(\omega),\eta=\eta(\omega),\zeta(\omega)<0\}\).
Any straight line in the plane \(\{\xi,\eta\}\) intersecting the half-axis \(\eta=0,\ \xi\leqslant 0\) will be called an admissible straight line. The closed half-plane bounded by an admissible straight line and containing the half-axis \(\eta=0,\ \xi\geqslant 0\) will be called positive; the closed half-plane bounded by an admissible straight line and not containing the half-axis \(\eta=0,\ \xi\leqslant 0\) will be called negative. The existence of parameters for which \(\pi(\omega)\geqslant 0\) when \(\zeta(\omega)\ne 0\) is equivalent to the existence of an admissible straight line such that the curve \(\Gamma_+\) lies in the positive, and the curve \(\Gamma_-\) in the negative, half-planes. (In the case \(\zeta(\omega)\equiv 0\), asymptotic stability in the large holds by V. M. Popov’s condition.)

Consider the system

\[ dx/dt=Px+q\varphi(\sigma)+f(t,x),\qquad \sigma=r^*x, \tag{5} \]

under the previous assumptions, supposing that instead of (2a) the weaker condition
\[ \lim \frac{\varphi(\sigma)}{\sigma}\left[\mu_0-\frac{\varphi(\sigma)}{\sigma}\right]\geqslant 0 \quad\text{as }|\sigma|\to\infty \]
is fulfilled. The function \(f(t,x)\) is a continuous function of \(t,x\), and
\[ \lim_{|x|\to\infty}|f(t,x)|/|x|=0 \]
uniformly in \(t\).

Theorem 3. If, for some \(\tau_1>0,\ \tau_2\geqslant 0,\ \vartheta\), one has \(\pi(\omega)>0\), then system (5) is dissipative, i.e., in the space \(\{x\}\) there exists a bounded closed set \(\mathfrak F\) such that

I. From \(x(t_0)\in\mathfrak F\) it follows that \(x(t)\in\mathfrak F\) for \(t\geqslant t_0\).

II. For every solution there is a time \(t_0\) such that \(x(t_0)\in\mathfrak F\).

III. There exists a solution \(x_0(t)\in\mathfrak F\) for \(-\infty<t<+\infty\).

If \(f(t,x)\) is periodic in \(t\) with period \(\chi\), then there exists a periodic solution \(x_0(t)\in\mathfrak F\) with period \(\chi\) (when \(\tau_2=0\)) or with a multiple of \(\chi\) (when \(\tau_2>0\)), \(-\infty<t<+\infty\).

The proof of Theorems 1–3 essentially uses the following algebraic proposition.

Theorem 4. Given \(\rho>0,\ a,\ b,\ K=K^*,\ A\), the equation \(\det(A-\lambda I)=0\) has a simple zero root and the remaining roots lie in the left half-plane, \(An=\bar 0,\ A^*m=\bar 0\) \((n\ne \bar 0,\ m\ne \bar 0)\), and the vectors \(a,Aa,\ldots,A^{\nu-1}a\) are linearly independent. With respect to the unknown matrix \(H=H^*\) define

\[ -G=A^*H+HA,\qquad -g=Ha+b,\qquad F=\rho(G+K)-gg^*, \tag{6} \]

and set \(A_\omega=A-i\omega I,\ a_\omega=A_\omega^{-1}a,\ \pi_0(\omega)=\rho+2\operatorname{Re} b^*a_\omega+a_\omega^*Ka_\omega\).
For the existence of a matrix \(H=H^*\) satisfying the quadratic inequality \(F>0\) \((F\geqslant 0)\), it is necessary and sufficient that \(\pi_0(\omega)>0\) for \(\omega>0\), \(n^*Kn=\lim_{\omega\to 0}\omega^2\pi_0(\omega)>0\) \((\pi_0(\omega)\geqslant 0)\). If \(\pi_0(\omega)\geqslant 0\), then there exists a matrix \(H=H^*\) satisfying the equation \(F=0\).

Below we give concise proofs of Theorems 1–4.

\(2^\circ.\) Proof of Theorem 4*. Necessity. We have
\[ a_\omega^*Ga_\omega=-a_\omega^*(A_\omega^*H+HA_\omega)a_\omega =2\operatorname{Re} a_\omega^*(g+b), \]
therefore from the last relation (6) it follows that

\[ a_\omega^*Fa_\omega=\rho\pi_0(\omega)-|g^*a_\omega-\rho|^2, \tag{7} \]

* We note that an analogous theorem for the case when \(A\) is a Hurwitz matrix is simply derived from \({}^{(3)}\) for \(F>0\) and from \({}^{(4)}\), \(F\geqslant 0\). Theorem 4 can be obtained from \({}^{(3)}\) by the corresponding limiting passage. The direct proof given below is simpler; the proof of necessity repeats \({}^{(3)}\), while in the proof of sufficiency Kalman’s idea \({}^{(4)}\) is used.

i.e., \(\pi_0(\omega)>0\) (\(\geqslant 0\)) for \(F>0\) (\(\geqslant 0\)). Since \(n^*Gn=0\), from (6) we have \(\rho n^*Kn=n^*Fn+(n^*g)^2>0\) (\(\geqslant 0\)) for \(F>0\) (\(\geqslant 0\)).

Sufficiency. The case \(F>0\). As \(\omega\to 0\) we have
\[ \pi_0(\omega)=-\omega^{-2}(m^*a)^2(m^*n)^{-2}n^*Kn+O(1), \]
\[ a_\omega^*Fa_\omega=\omega^{-2}(m^*a)^2(m^*n)^{-2}n^*Fn+O(1). \]
Therefore there exists \(F>0\) such that \(a_\omega^*Fa_\omega<\pi_0(\omega)\) for \(0<\omega<\infty\). As in (4) (see also \((^2)\), p. 131), it is easy to show that there exists a vector \(g\) satisfying identity (7). From the asymptotics as \(\omega\to 0\) it follows that \(n^*Gn=0\), where \(G\) is determined by the last equality (6). Therefore there exists a solution \(H_0=H_0^*\) of the first equation (6), and the general solution of (6) is
\[ H=H_0+\xi mm^* \]
with arbitrary \(\xi\). Let \(h=Ha+b+g\). From (7) it follows that \(\operatorname{Re} a_\omega^*h=0\). Let
\[ h^*(A-\lambda I)^{-1}a=\psi(\lambda)+\gamma/\lambda, \]
where \(\psi(\lambda)\) is holomorphic in the right closed half-plane. From the condition \(\operatorname{Re}\psi(i\omega)\equiv 0\) it follows, by the principle of analytic continuation, that \(\psi(\lambda)\equiv\mathrm{const}\), i.e. \(\psi(\lambda)\equiv 0\). Since
\[ \gamma=h^*n\cdot m^*a\cdot (m^*n)^{-1}, \]
one can choose \(\xi\) so that \(\gamma=0\). Then \(h^*(A-\lambda I)^{-1}a\equiv 0\), \(h=0\), i.e. the second relation (6) is fulfilled.

The case \(F\geqslant 0\) is considered similarly, with some simplifications.

\(3^\circ\). Proof of Theorems 1 and 2. It is enough to consider the case \(\tau_2>0\). Denote
\[ y=\binom{x}{\varphi},\qquad a=\binom{\bar 0}{1},\qquad c=\binom{P^*r}{q^*r},\qquad r_1=\binom{r}{0},\qquad A=\begin{pmatrix}P&q\\ \bar 0^*&0\end{pmatrix}. \]

Taking as Lyapunov function \(^*\)
\[ V(x)=y^*Hy+\vartheta\int_0^\sigma \varphi\,d\sigma, \]
we obtain \(^{**}\)
\[ -\dot V=\bigl[y^*(G+K)y+2y^*g\dot\varphi+\tau_2\dot\varphi^{\,2}\bigr]+ \]
\[ +\tau_1(\mu_0\sigma-\varphi)\varphi+\tau_2(\varphi'+\alpha_1\sigma)(\alpha_2-\varphi')\dot\sigma^{\,2}, \tag{8} \]
where
\[ y^*Ky=\vartheta y^*c\cdot a^*y+\tau_1a^*y\cdot(a^*y-\mu_0 r_1^*y)-\tau_2\alpha_1\alpha_2(c^*y)^2; \]
\(K=K^*\); \(G,g\) have the form (6);
\[ b=-\frac{\tau_2}{2}(\alpha_2-\alpha_1)c. \]
Using the expressions for \(a,b,K\) and taking \(\rho=\tau_2\), after simple transformations we obtain that the function \(\pi_0(\omega)\) in Theorem 4 has the form \(\pi_0(\omega)=\omega^{-2}\pi(\omega)\) (see (3)). Therefore, by Theorem 4, there exists a matrix \(H=H^*\) such that the expression in square brackets in (8) is a positive definite (Theorem 1) or nonnegative (Theorem 2) form in \(y\) and \(\dot\varphi\), i.e. \(\dot V<0\) for \(x\ne 0\) (Theorem 1) or \(\dot V\leqslant 0\) (Theorem 2). To prove Theorem 1 it remains, according to Theorem 4 \((^6)\), to show the absence of solutions for which \(|x(t)|\to\infty\). Arguing as in (7) (Appendix, § 2), we obtain that \(^{***}\)
\[ \Omega_0(x)\equiv y^*Hy+\vartheta\sigma\varphi/2\geqslant \varepsilon_0|x|^2 \]
for some \(\varepsilon_0>0\). If \(|x(t)|\to\infty\), on the one hand,
\[ \dot V\to-\infty,\qquad \dot V<0,\qquad V\to-\infty, \]
and, on the other hand, from (4) it follows that
\[ V=\Omega_0(x)+\vartheta\left[\int_0^\sigma \varphi\,d\sigma-\frac{\sigma\varphi}{2}\right]\to+\infty. \]
The contradiction proves Theorem 1.

To prove Theorem 2 we apply Lemma 5 \((^1)\) (Appendix, § 3). From \(\dot V\equiv 0\) it follows, according to (8), that \(\dot\sigma\equiv 0\), \(\sigma\equiv\sigma_0=\mathrm{const}\). From the ident—

\(^*\) From the results of \((^5)\) it follows that, irrespective of the use of additional information about the nonlinearity, Popov’s frequency condition cannot be improved in the class of Lyapunov functions
\[ V=x^*H_1x+\vartheta\int\varphi\,d\sigma \]
(\(H_1\) and \(\vartheta\) are parameters).

\(^{**}\) To the expression obtained after differentiation there has been added and subtracted the expression, nonnegative by virtue of (2),
\[ \tau_1(\mu_0\sigma-\varphi)\varphi+\tau_2(\varphi'+\alpha_1\sigma)(\alpha_2-\varphi')\dot\sigma^{\,2}. \]

\(^{***}\) The function \(\Omega_0(x)\) is obtained from \(V\) by replacing \(\varphi=\mu\sigma\), \(\mu=\mathrm{const}\), \(0\leqslant\mu\leqslant\mu_0\), and, after integration, replacing \(\mu=\varphi/\sigma\).

from the equality
\(\sigma_0 = r^*[\exp(Pt)x_0 - Pq^{-1}\varphi(\sigma_0)]\) we obtain that
\(f^*\exp(Pt)x_0 = 0\), \(\sigma_0 + \chi(0)\varphi(\sigma_0)=0\), which, for \(\varphi(\sigma_0)\ne 0\), contradicts assumptions (2a), \(\pi(0)\ge 0\), \(\chi(0)\ne \mu_0^{-1}\). Consequently, \(\varphi(\sigma_0)=0\), \(x(t)\to 0\). Thus, conditions (II), (III) of Lemma 5 \((^1)\) are satisfied. From the inequalities \(\pi(\omega)\ge 0\), \(\chi(i\omega)\ne \mu_0^{-1}\), it is easy to derive that the curve \(\chi(i\omega)\) does not intersect the segment \([-\mu_0^{-1},-\infty)\), i.e., the matrix \(P+\mu qr^*\) is Hurwitz for \(0\le \mu\le \mu_0\), whence follows the fulfillment of condition (I) of Lemma 5 \((^1)\) and the inequality \(\Omega_0(x)\ge \varepsilon_0|x|^2\). Hence, as above, we obtain the fulfillment of the last condition \((\mathrm{IV}')\) of Lemma 5 \((^1)\).

4°. The proof of Theorem 3 is carried out according to the same scheme as the proof of Theorem 4 \((^7)\) (which is obtained from Theorem 3 for \(\tau_2=0\) and in the absence of condition (26)); moreover, as the Lyapunov function one should take the function constructed above,

\[ V = y^*Hy + \vartheta \int_0^\sigma \varphi\,d\sigma . \]

Leningrad State University
named after A. A. Zhdanov

Received
26 VI 1964

CITED LITERATURE

\(^1\) V. A. Yakubovich, Avtomatika i telemekh., 25, No. 10 (1964).
\(^2\) M. A. Aizerman, F. R. Gantmakher, Absolute Stability of Nonlinear Controlled Systems, Publishing House of the Academy of Sciences of the USSR, 1963.
\(^3\) V. A. Yakubovich, DAN, 143, No. 6 (1962).
\(^4\) R. E. Kalman, Proc. Nat. Acad. Sci. U.S.A., 49, No. 2 (1963).
\(^5\) V. A. Yakubovich, DAN, 156, No. 2 (1964).
\(^6\) V. A. Yakubovich, Avtomatika i telemekh., 25, No. 5 (1964).
\(^7\) V. A. Yakubovich, Avtomatika i telemekh., 25, No. 7 (1964).