MATHEMATICS

Yu. A. DMITRIEV

ABSOLUTE STABILITY OF AUTOMATIC SYSTEMS WITH ONE PULSE CONTROLLER

(Presented by Academician V. I. Smirnov on 20 VII 1964)

\(1^\circ\). In the present paper the method of matrix inequalities of V. A. Yakubovich \((^{1-5})\) is extended to the case of difference equations of the form

\[ x_{t+1}=Px_t+q\varphi(\sigma_t),\qquad \sigma_t=r^*x_t,\qquad t=0,1,2,\ldots, \tag{1} \]

which describe pulse control systems. Here \(x,q,r\) are \(\nu\times 1\)-vectors; \(P\) is a nonsingular \(\nu\times\nu\)-matrix whose spectrum lies inside the unit circle\(^*\); \(\varphi(\sigma)\) is a differentiable function satisfying the conditions:

\[ \text{a) }\ \varphi(0)=0,\qquad 0\leq \sigma\varphi(\sigma)\leq \mu_0\sigma^2; \qquad \text{b) }\ \lim_{|\sigma|\to\infty}|\sigma|^{-2} \left|\int_0^\sigma \varphi\,d\sigma-\frac12\varphi\sigma\right|=0; \tag{2} \]

\[ a_1\leq \varphi'(\sigma)\leq a_2,\qquad a_1\leq 0,\qquad a_2\geq \mu_0>0. \tag{3} \]

For the first time for systems (1), a frequency condition for absolute stability in the class of nonlinearities satisfying relation (2a) was obtained by Ya. Z. Tsypkin \((^6)\). Under assumptions (2a), (3), R. Kalman and G. Szegő obtained \((^{7-9})\) new criteria. Below, under some additional inessential assumptions, criteria of absolute stability will be obtained that are broader than conditions \((^{6-9})\).

Introduce the notation

\[ \chi(\lambda)=r^*(P-\lambda I)^{-1}q,\qquad \chi_0=\chi(0),\qquad \beta_j=\alpha_j(\alpha_j\chi+2),\qquad j=1,2; \]

\[ \Phi(\lambda,\vartheta,\xi)=\mu_0^{-1}+\operatorname{Re}\chi(\lambda) -\vartheta\left[2\operatorname{Re}(1-\lambda)\chi(\lambda) +\alpha_0|(1-\lambda)\chi(\lambda)|^2\right] + \xi|1-\lambda|^2\left[\chi_0-2\operatorname{Re}\chi(\lambda)-\beta_0|\chi(\lambda)|^2\right], \tag{4} \]

where the quantity \(a_0\) may be assigned the values \(a_0=a_1\) or \(a_0=a_2\), while the choice of \(\beta_0\) is restricted by the conditions: \(\beta_0=\beta_1\), if \((\beta_2-\beta_1)(\chi_0+a_1^{-1})\leq 0\), \(\beta_0=\beta_2\), if \((\beta_2-\beta_1)(\chi_0+a_2^{-1})\geq 0\)\(^{**}\).

The main result of the paper can be formulated as follows:

Theorem. If for system (1) there exist real numbers \(\vartheta,\xi\) satisfying the inequalities: \(\vartheta\leq 0\) for \(a_0=a_1\), \(\vartheta\geq 0\) for \(a_0=a_2\), \(0\leq \xi(\chi_0+a_1^{-1})^{-1}\) for \(\beta_0=\beta_1\), \(0\leq \xi(\chi_0+a_2^{-1})^{-1}\) for \(\beta_0=\beta_2\), and such that \(\Phi(\lambda,\vartheta,\xi)>0\) for all \(\lambda,\ |\lambda|=1\), then the trivial solution of this system is stable in the large for any differentiable functions \(\varphi(\sigma)\) satisfying conditions (2), (3).

For a given frequency response \(\chi(e^{i\omega})\), \(0\leq \omega\leq 2\sigma\), the question of the existence of numbers \(\vartheta,\xi\) satisfying the condition of the theorem is easily solved geometrically, similarly to how this was done in paper \((^5)\).

\(^*\) All matrices and vectors here and below are real, unless complex numbers explicitly enter into their expressions. The asterisk denotes transposition (in the case of complex expressions also conjugation).

\(^ {**}\) It is easy to check that in the case when \(a_2^{-1}\geq -\chi_0\geq a_1^{-1}\), both values \(\beta_0=\beta_1,\beta_2\) are possible.

Let us note three special cases that simplify the conditions of the theorem:

1) If in relations (3) \(\alpha_2=-\alpha_1=\alpha_0>0\), then the expression \(\Phi(\lambda,\vartheta,\xi)\) has the form

\[ \Phi(\lambda,\vartheta,\xi)=\mu_0^{-1}+\xi\chi_0|1-\lambda|^2+\operatorname{Re}\{[1-2\vartheta(1-\lambda)- \]

\[ -2\xi|1-\lambda|^2]\chi(\lambda)\}-\alpha_0(|\vartheta|+2|\xi|+\xi\alpha_0\chi_0)|(1-\lambda)\chi(\lambda)|^2, \]

where \(\vartheta,\xi\) are some arbitrary fixed real numbers, the choice of the sign of \(\xi\) being restricted by the conditions: \(\xi\leq 0\) for \(\chi_0\leq-\alpha_0^{-1}\), \(\xi\geq 0\) for \(\chi_0\geq\alpha_0^{-1}\), and \(\xi\) arbitrary for \(-\alpha_0^{-1}<\chi_0<\alpha_0^{-1}\).

We note that in this case, and for \(\xi=0\), the conditions of the theorem coincide exactly with the Segé criterion \((^7)\), if in the latter the boundary of the stability region is excluded.

2) If in relations (3) \(\alpha_1=0\) and \(\chi_0\geq -2\alpha_2^{-1}\), then, restricting ourselves to the choice \(\alpha_0=\alpha_1=0\), \(\beta_0=\beta_1=0\), the conditions of the theorem may be written in the form

\[ \mu_0^{-1}-\frac{1}{2}\xi\chi_0|1-\lambda|^2+\operatorname{Re}\{[1+\vartheta_1(1-\lambda)+\xi_1|1-\lambda|^2]\chi(\lambda)\}>0 \]

for all \(\lambda\), \(|\lambda|=1\), and for some fixed positive numbers \(\xi_1,\vartheta_1\).

3) If in relations (3) \(\alpha_1=-\infty,\ \alpha_2=+\infty\), then it is necessary that \(\xi=\vartheta=0\), and the conditions of the theorem coincide with the criteria of Tsypkin \((^6)\) and Segé–Kalman \((^{8,9})\).

2°. Lemma 1. Suppose that for the system

\[ x_{t+1}=f(x_t),\qquad f(0)=0,\qquad t=0,1,2,\ldots, \tag{5} \]

where \(f(x)\) is a vector function continuous in the whole space \(\{x\}\), there exists a continuous scalar function \(V(x)\) having the following properties:

I. \(V(0)=0\). II. \(\lim_{|x|\to\infty}V(x)=\infty\). III. \(\Delta V(x)\equiv V[f(x)]-V(x)\leq0\) for all \(x\). IV. From \(\Delta V(x_t)\equiv0\) it follows that \(x_t\equiv0\), if \(x_t\) is a solution of equations (5). Then \(V(x)>0\) for all \(x\neq0\), and the trivial solution of system (5) is stable in the large in the sense of Lyapunov, i.e., \(x_t\to0\) as \(t\to\infty\), and for every \(\varepsilon>0\) there exists \(\delta>0\) such that \(|x_t|<\varepsilon\) for \(|x_0|<\delta\) and for all \(t\geq0\).

The proof of Lemma 1 is carried out according to scheme (5). By III and (5) we have \(V(x_t)\leq V(x_0)\); therefore, from II it follows that \(|x_t|\leq\mathrm{const}\). Let \(y\) be an arbitrary limit point of the sequence \(x_t\). There exists a subsequence \(t'\) such that \(\lim_{t'\to\infty}x_{t'}=y\), and, by virtue of the continuity of \(V(x)\),

\[ \lim_{t'\to\infty} V(x_{t'})=V(y). \]

Since the sequence \(V(x_t)\) is monotone, it follows that

\[ \lim_{t\to\infty} V(x_t)=V(y). \]

Obviously, \(x_t=x(t,x_0)\) is a continuous function of the initial values \(x_0\). Therefore

\[ \lim_{t'\to\infty}x_{t'+\tau}=\lim_{t'\to\infty}x(\tau,x_{t'})=x(\tau,y), \]

\[ V(y)=\lim_{t'\to\infty}V[x_{t'+\tau}]=V[x(\tau,y)], \]

whence we have \(\Delta V[x(\tau,y)]\equiv0\), \(\tau=0,1,\ldots\). Consequently, by property IV, \(x(\tau,y)\equiv0\). In particular, \(x(0,y)=y=0\), and, since \(y\) is an arbitrary limit point, \(x_t\to0\) as \(t\to\infty\).

Suppose that for some vector \(z\) one has \(V(z)\leq0\). Setting in (5) \(x_0=z\), we have, according to III,

\[ V[x(t+\tau,z)]\leq V[x(t,z)]\leq V(z)\leq0,\qquad t=0,1,\ldots,\quad \tau=0,1,\ldots \]

Passing to the limit as \(\tau\to\infty\), by what has been proved we obtain

\[ \lim_{\tau\to\infty}x(t+\tau,z)=0 \]

and, by virtue of I,

\[ \lim_{t\to\infty}V[x(t+\tau,z)]=0. \]

Therefore \(0\leq V[x(t,z)]\leq0,\ t=0,1,\ldots\), i.e. \(V[x(t,z)]\equiv0\). By property IV, we have \(x(t,z)\equiv0\); in particular, \(x(0,z)=z=0\). Thus, for all \(z\neq0\) one has \(V(z)>0\).

Let \(\varepsilon>0\) be given. By what has been proved,

\[ V_\varepsilon=\operatorname{Inf}_{|y|\geq\varepsilon} V(z)>0. \]

By virtue of properties I, III and the continuity of the function \(V(x)\), there exists \(\delta>0\) such that \(V(x_t)\leq V(x_0)<V_\varepsilon\) in the case \(|x|<\delta\). If for some \(t\) at

if for \(|x_0|<\delta\) the inequality \(|x_t|\geqslant\varepsilon\) holds, then \(V(x_t)\geqslant V_\varepsilon\). A contradiction shows that for \(|x_0|<\delta\), for all \(t\) one has \(|x_t|<\varepsilon\).

Lemma 2. Let \(H=H^*\) and let \(P\) be some \(v\times v\) matrix, with the spectrum of \(P\) lying inside the unit circle; \(q,r_1\) are \(v\times 1\) vectors; \(\gamma_0,\varphi\) are real numbers. Denote
\[ \Psi(x,\varphi)=x^*Gx+2g^*x\varphi+\gamma\varphi^2, \]
where \(G=H-P^*HP\), \(-g=\frac12 r_1+P^*Hq\), \(\gamma=\gamma_0-q^*Hq\). For the existence of a matrix \(H=H^*>0\) such that \(\Psi(x,\varphi)>0\) for any \(x,\varphi\), \(|x|+|\varphi|\ne0\), it is necessary and sufficient that
\[ \Phi_0(\lambda)\equiv \gamma_0+\operatorname{Re} r_1^*(P-\lambda I)^{-1}q>0 \]
for all \(\lambda,\ |\lambda|=1\).

Proof. Introduce the matrix \(A=(P-I)(P+I)^{-1}\), which, by the properties of the matrix \(P\), is Hurwitz, and perform the nonsingular transformation
\[ x=\sqrt2(P+I)^{-1}(x_1-q\varphi_1),\qquad \varphi=\sqrt2\varphi_1. \]
Then
\[ \Psi(x,\varphi)=\Psi_1(x_1,\varphi_1)=x_1^*G_1x_1+2g_1^*x_1\varphi_1+\gamma_1\varphi_1^2, \]
where
\[ -G_1=A^*H+HA,\qquad -g_1=Ha+b, \]
\[ \gamma_1=2[\gamma_0+r_1^*(P+I)^{-1}q],\qquad a=-2(P+I)^{-1}q,\qquad b=-(P^*+I)^{-1}r_1, \]
and the condition \(\Phi_0(\lambda)>0\) for all \(\lambda,\ |\lambda|=1\), is equivalent to the inequality
\[ \Phi_1(i\omega)=\gamma_1+2\operatorname{Re} b^*(A-i\omega I)^{-1}a>0, \]
satisfied for all \(\omega\geqslant0\), which, by theorem 1 \((^1)\), is necessary and sufficient for the existence of a matrix \(H>0\) satisfying the relation
\[ \gamma_1G_1-g_1g_1^*>0, \]
which, together with the inequality \(\gamma_1=2\Phi_0(-1)>0\), proves Lemma 2.

Lemma 3. If the spectrum of the matrix \(P\) lies inside the unit circle and
\[ G=H-P^*HP>0, \]
then also \(H>0\).

The proof follows from the formula
\[ H=\sum_{t=0}^{\infty} P^{*t}GP^t, \]
which gives the unique, as is readily checked, solution of the equation \(G=H-P^*HP\).

Proof of the theorem. Suppose that the conditions of the theorem are satisfied. Consider the function
\[ V(x)=x^*H_1x+2h^*x\varphi(\sigma)+\varkappa\varphi(\sigma)^2+2\vartheta\int_0^\sigma \varphi(\zeta)\,d\zeta,\qquad \sigma=r^*x, \tag{6} \]
where \(h=\xi(I-P^{-1})^*r\), \(\varkappa=\xi\chi_0\), \(H_1=H+H_0\); \(H_0\) is determined from the equation
\[ H_0-P^*H_0P=R\equiv(\vartheta\alpha_0+\xi\beta_0)(I-P^*)rr^*(I-P); \]
\(\chi_0,\alpha_0,\beta_0\) are the same as in (4), and the parameters \(\xi,\vartheta\) are to be chosen in accordance with the conditions of the theorem.

It is obvious that the function (6) has property I of Lemma 1. Denoting \(\sigma_{t+1}-\sigma_t=\Delta\sigma\), \(\varphi(\sigma_t)=\varphi_t\), \(\varphi_{t+1}-\varphi_t=\Delta\varphi\), we obtain from (6), (1), (4)
\[ -\Delta V(x_t)=\Psi(x_t,\varphi_t)+\Omega, \]
where
\[ \Omega=(\sigma_t\varphi_t-\mu_0^{-1}\varphi_t^2)+2\vartheta\left[\varphi_t\Delta\sigma+\frac12\alpha_0\Delta\sigma^2-\int_{\sigma_t}^{\sigma_{t+1}}\varphi\,d\sigma\right] -\xi\left[\chi_0\left(\frac{\Delta\varphi}{\Delta\sigma}\right)^2+2\frac{\Delta\varphi}{\Delta\sigma}-\beta_0\right]\Delta\sigma^2\geqslant0 \]
by virtue of the conditions on \(\varphi(\sigma)\), \(\alpha_0,\beta_0,\xi,\vartheta\), and \(\Psi(x,\varphi)\) has the same form as in Lemma 2, if \(r_1\) and \(\gamma_0\) are defined by the relations
\[ \begin{aligned} r_1={}&2P^*H_0q+\left[1-2\vartheta(1-\alpha_0r^*q)-2\xi(2+\beta_0r^*q)\right]r\\ &+2\left[\vartheta(1+\alpha_0r^*q)+\xi(1+\beta_0r^*q)\right]P^*r+2\xi P^{*-1}r, \end{aligned} \tag{7} \]
\[ \gamma_0=-q^*H_0q+\mu_0^{-1}-\vartheta r^*q(2+\alpha_0r^*q)+\xi\left[2\chi_0-2r^*q-\beta_0(r^*q)^2\right]. \]
For convenience in estimating, in the expression \(-\Delta V(x_t)\), obtained from (6), (1), the terms
\[ (\sigma_t\varphi_t-\mu_0^{-1}\varphi_t^2)+2\vartheta(\varphi_t\Delta\sigma+\alpha_0\Delta\sigma^2/2)+\xi\beta_0\Delta\sigma^2 \]
have been added and subtracted.

From the equation for \(H_0\) it follows that
\[ 2\operatorname{Re} q^*H_0P(P-\lambda I)^{-1}q-q^*H_0q =-q^*(P^*-\lambda^* I)^{-1}R(P-\lambda I)^{-1}q \]
for \(|\lambda|=1\). With the help of this identity and the relations (7), (4), it is easy to verify that the expressions for \(\Phi(\lambda,\vartheta,\xi)\) of the theorem and \(\Phi_0(\lambda)\) of Lemma 2 coincide. Therefore, in the conditions of the theo-

therefore, by Lemma 2, there exists a matrix \(H=H^*>0\) for which \(\Psi(x,\varphi)>0\) for all \(x,\varphi,\ |x|+|\varphi|\ne0\). Substituting this matrix into expression (6), we obtain that function (6) has properties III, IV of Lemma 1 and, moreover, \(-\Delta V(x)>0\) for all \(x\ne0\).

Let us now show that function (6) also satisfies condition II of Lemma 1. Denote

\[ P_\mu=P+\mu qr^*,\qquad H_1(\mu)=H_1+\mu(hr^*+rh^*)+\mu(\mu x+\vartheta)rr^*, \]
\[ 0\le \mu\le \mu_0. \tag{8} \]

For the special case \(\varphi(\sigma)=\mu\sigma\), from (1), (6), (8) we have \(x_{t+1}=P_\mu x_t\), \(V(x)=x^*H_1(\mu)x\), \(-\Delta V(x)=x^*[H_1(\mu)-P_\mu^*H_1(\mu)P_\mu]x\). According to what has been set forth, \(-\Delta V(x)>0\) for all \(x\ne0\), i.e. \(H_1(\mu)-P_\mu^*H_1(\mu)P_\mu>0\). It is easy to prove (by contradiction) that, under the conditions of the theorem, the hodograph \(\chi(\lambda)\) for \(|\lambda|=1\) does not intersect the real axis on the interval \((-\infty,-\mu_0^{-1}]\), which, in view of the properties of the matrix \(P\), is necessary and sufficient for the absolute stability of linear systems of the form (1) for \(\varphi(\sigma)=\mu\sigma,\ 0\le\mu\le\mu_0\). Thus, the spectra of all matrices \(P_\mu\) (see (8)) lie inside the unit circle. By Lemma 3, the last inequality implies \(H_1(\mu)>0\). Therefore there exists \(\varepsilon>0\) such that \(x^*H_1(\mu)x\ge\varepsilon |x|^2\) for \(0\le\mu\le\mu_0\). Then from (6), (8), for any \(\varphi(\sigma)\) satisfying conditions (2), we easily obtain

\[ V(x)=x^*H_1(\varphi/\sigma)x +2\vartheta\left[\int_0^\sigma \varphi\,d\sigma-\frac12\varphi\sigma\right] \ge \varepsilon |x|^2 -2|\vartheta|\cdot \left|\int_0^\sigma \varphi\,d\sigma-\frac12\varphi\sigma\right| \]

\[ V(x)\ge \frac{\varepsilon}{2}|x|^2-\mathrm{const},\qquad V(x)\to\infty\ \text{as } |x|\to\infty. \]

Thus, function (6) satisfies all the conditions of Lemma 1, from which the assertion of the theorem follows.

Siberian Scientific Research
Institute of Power Engineering

Received
17 VII 1964

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