FREQUENCY CONDITIONS FOR ABSOLUTE STABILITY OF PULSE AUTOMATIC SYSTEMS WITH ONE NONLINEAR BLOCK

Yu. A. DMITRIEV

MATHEMATICS

(Presented by Academician L. S. Pontryagin on 15 III 1965)

Pulse control systems with a finite number of degrees of freedom and one nonlinear block can be described by difference equations of the form

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

where \(x, q, r\) are real \(v\times 1\) vectors (the asterisk denotes transposition); \(\varphi(\sigma)\) is a scalar, continuous, real function of \(\sigma\); \(P\) is a real \(v\times v\) matrix.

Assume that:

A. All eigenvalues \(\pi_j\) of the matrix \(P\) satisfy the conditions
\[ |\pi_j|\leq 1,\quad j=1,\ldots,v. \]

B. The vectors \(q, Pq,\ldots,P^{v-1}q\) are linearly independent.

C. \(\varphi(0)=0\), and for any \(\sigma_1\ne\sigma_2\) the inequalities
\[ \alpha_1\leq(\sigma_2-\sigma_1)^{-1}\,[\varphi(\sigma_2)-\varphi(\sigma_1)]\leq\alpha_2 \]
hold, where \(\alpha_1\leq 0,\ \alpha_2>0\).

D. If \(|\pi_j|<1,\ j=1,\ldots,v\) (the noncritical case), then
\[ 0\leq \sigma^{-1}\varphi(\sigma)\leq \mu_0 \]
for all \(\sigma\ne 0\), where \(0<\mu_0\leq\alpha_2\); if for at least one value of \(j\) one has \(|\pi_j|=1\) (the critical case), then
\[ 0<\varepsilon\leq \sigma^{-1}\varphi(\sigma)\leq \mu_0 \]
for all \(\sigma\ne 0\), where \(\varepsilon<\mu_0\leq\alpha_2\).

By the amplitude-phase characteristic \(\chi(\lambda)=r^*(P-\lambda I)^{-1}q\) (where \(|\lambda|=1\)) of the linear part of system (1), define two functions

\[ \begin{aligned} \Phi_k(\lambda,\vartheta,\xi) &=\mu_0^{-1}+\operatorname{Re}\chi(\lambda) -\vartheta(-1)^k\left[\operatorname{Re}(1-\lambda)\chi(\lambda) +\frac{1}{2}\alpha_k |(1-\lambda)\chi(\lambda)|^2\right] \\ &\quad+\xi |1-\lambda|^2\left[1+(\alpha_1+\alpha_2)\operatorname{Re}\chi(\lambda) +\alpha_1\alpha_2|\chi(\lambda)|^2\right], \qquad k=1,2; \end{aligned} \tag{2} \]

here \(\alpha_1,\alpha_2,\mu_0\) are the same as in conditions C, D.

Theorem 1. Let \(|\pi_j|<1,\ j=1,\ldots,v\). Suppose that, for some \(\vartheta\geq 0,\ \xi\geq 0\) and all \(\lambda,\ |\lambda|=1\), either
\[ \Phi_1(\lambda,\vartheta,\xi)>0, \]
or
\[ \Phi_2(\lambda,\vartheta,\xi)>0 \]
is satisfied.

Then the trivial solution \(x_t\equiv 0\) of system (1) is asymptotically stable in the large.

Theorem 2. Let \(|\pi_j|\leq 1,\ j=1,\ldots,v\), and suppose there exists \(\mu_*\), \(\varepsilon\leq \mu_*\leq \mu_0\), for which the spectrum of the matrix
\[ P_* = P+\mu_*qr^* \]
is entirely located inside the unit circle\(^*\).

Suppose that, for some \(\vartheta\geq 0,\ \xi\geq 0\) and all \(\lambda,\ |\lambda|=1\), either
\[ \Phi_1(\lambda,\vartheta,\xi)\geq 0, \]
or
\[ \Phi_2(\lambda,\vartheta,\xi)\geq 0 \]
is satisfied. Then the trivial solution \(x_t\equiv 0\) of system (1) is asymptotically stable in the large.

The question of the existence of the numbers \(\vartheta,\xi\) indicated in Theorems 1 and 2 can be resolved graphically, analogously to \((^1)\).

\[ \text{---} \]

\(^*\) The existence of the indicated number \(\mu_*\) is necessary for the assertion of Theorem 2 to be valid.

Consider special cases. 1) If, in conditions C, \(\alpha_1=-\infty\), \(\alpha_2=+\infty\), then necessarily \(\vartheta=\xi=0\), and the inequality \(\Phi_k(\lambda,0,0)\equiv\mu_0^{-1}+\operatorname{Re}\chi(\lambda)>0\) is the well-known criterion of Ya. Z. Tsypkin \((^2)\).* 2) If, in relations C, \(-\alpha_1=\alpha_2<+\infty\), then the frequency condition \(\Phi_k(\lambda,\vartheta,0)\geqslant0\) (\(k=1\) or \(k=2\)) coincides with the criterion of Segré \((^4)\), obtained for the noncritical case under the assumptions \(0\leqslant\sigma^{-1}\varphi(\sigma)\leqslant\mu_0,\;0\leqslant d\varphi/d\sigma<\alpha_2\). 3) For the case \(\mu_0=\alpha_2<+\infty\), the stability condition \(\Phi_2(\lambda,\vartheta,0)\geqslant0\) was obtained by Segré in \((^5)\). 4) If, in relations C, \(\alpha_2=+\infty,\;\alpha_1>-\infty\), then Segré’s criteria \((^4,^5)\) necessarily coincide with the Segré–Kalman condition \((^3)\) and are less broad than the frequency conditions of Theorems 1 and 2 of the present paper. 5) In the case \(\alpha_1=0\), Theorems 1 and 2 include, as is easily verified, the corresponding results \((^6,^7)\) (Theorem 5 \((^7)\)) of the author and, consequently (see \((^7)\)), the results \((^{3-5})\).

It can be shown that, in cases 2), 3), the conditions \(\Phi_k(\lambda,\vartheta,\xi)\geqslant0\) (\(\Phi_k>0\)), \(\vartheta>0\), \(\xi>0\), may be satisfied when the conditions \(\Phi_k(\lambda,\vartheta,0)\geqslant0\) are certainly not valid for any \(\vartheta\geqslant0\).

The proofs of Theorems 1 and 2 are carried out by the method of matrix inequalities of V. A. Yakubovich (see, for example, \((^1,^8,^9)\)), modified for the difference-equation case under consideration, and are based on the following algebraic proposition:

Theorem 3. Let \(e,s\), and \(Q,G_0=G_0^*\) be, respectively, some given real \(\nu\times1\) vectors and \(\nu\times\nu\) matrices, and let \(\xi\) be a real number. Suppose that: 1) one eigenvalue of the matrix \(Q\) is equal to unity, while the others are located inside the unit circle; 2) the vectors \(e,Qe,\ldots,Q^{\nu-1}e\) are linearly independent. Define, in terms of the unknown real \(\nu\times\nu\) matrix \(H=H^*\), the quadratic form \(\Psi(z,\zeta)\), where \(z\) and \(\zeta\) are a real \(\nu\times1\) vector and a number, by the relations

\[ G=H-Q^*HQ,\qquad -g=Q^*He+s,\qquad \gamma=\xi-e^*He, \]

\[ \Psi(z,\zeta)=z^*(G+G_0)z+2g^*z\zeta+\gamma\zeta^2 \tag{3} \]

and denote

\[ e_\lambda=(Q-\lambda I)^{-1}e,\qquad \Phi_0(\lambda)=\xi+2\operatorname{Re}s^*e_\lambda+e_\lambda G_0e_\lambda . \tag{4} \]

For the existence of a matrix \(H=H^*\) satisfying the inequality \(\Psi(z,\zeta)>0\) for all \(z,\zeta\), \(|z|+|\zeta|\ne0\), it is necessary and sufficient that the following hold: a) \(\Phi_0(\lambda)>0\) for all \(\lambda\ne1,\;|\lambda|=1\); b) \(\lim_{\lambda\to1}|1-\lambda|^2\Phi_0(\lambda)>0\).

The proof of Theorem 3 can be obtained with the aid of Lemma 1 \((^8)\), if one sets \(Q=(I+A)(I-A)^{-1}\), \(\lambda=(1+i\omega)(1-i\omega)^{-1}\), \(\omega\geqslant0\), and carries out the nonsingular transformation \(z=\sqrt{2}(Q+I)^{-1}(z_1-\zeta_1e)\), \(\zeta=i\sqrt{2}\zeta_1\). The problem under consideration is thereby reduced to the solution of a matrix inequality (Theorem 4, \((^1)\)).

Proof of Theorem 1. Denote

\[ \varphi_t=\varphi(\sigma_t),\quad \Delta\sigma_t=\sigma_{t+1}-\sigma_t,\quad \zeta_t=\zeta(z_t)=\Delta\varphi_t=\varphi_{t+1}-\varphi_t, \tag{5} \]

\[ z_t=\left\|\begin{array}{c}x_t\\ \varphi_t\end{array}\right\|,\quad Q=\left\|\begin{array}{cc}P&q\\ 0&1\end{array}\right\|,\quad e=\left\|\begin{array}{c}0\\ 1\end{array}\right\|,\quad r_1=\left\|\begin{array}{c}r\\ 0\end{array}\right\|,\quad r_2=\left\|\begin{array}{c}(P^*-I)r\\ r^*q\end{array}\right\|. \]

In view of (5), system (1) is equivalent to the equations

\[ z_{t+1}=Qz_t+e\zeta(z_t),\qquad t=0,1,\ldots \tag{6} \]

* For systems of the form (1), under the assumptions \(|\pi_j|<1,\;j=1,\ldots,\nu,\;0\leqslant\sigma^{-1}\varphi(\sigma)<\mu_0\), an analogous condition with the strict inequality replaced by \(\mu_0^{-1}+\operatorname{Re}\chi(\lambda)\geqslant0\) was obtained by Kalman and Segré \((^3)\). (Here and below, the notation introduced in the cited papers has been replaced by the notation of the present paper.)

Choose a real \((\nu+1)\times(\nu+1)\) matrix \(H=H^*\) and a number \(\vartheta \geqslant 0\) such that the function

\[ V_k(z)=z^*Hz+\vartheta(-1)^k\int_0^\sigma \varphi(\sigma')\,d\sigma' \quad(\sigma=r_1^*z), \tag{7} \]

where \(k=1\) or \(2\), satisfies the conditions of Lemma 1 \((^6)\). Condition (I) of Lemma 1 \((^6)\) is, obviously, satisfied.

From (7), by virtue of (5), (6), we have

\[ \Delta V_k(z_t)\equiv V(z_{t+1})-V(z_t)= \]

\[ =-\Psi(z_t,\xi_t)-\Omega_1(\sigma_t^{-1}\varphi_t)\sigma_t^2-\vartheta\Omega_1^{(k)}(\sigma_t,\Delta\sigma_t) -\xi\Omega_3(\Delta\sigma_t^{-1}\Delta\varphi_t)\Delta\sigma_t^2. \]

Here

\[ \Omega_1(\mu)=\mu(1-\mu_0^{-1}\mu);\quad \Omega_1^{(k)}(\sigma,\Delta\sigma)= \]

\[ =(-1)^k\left[\varphi(\sigma)\Delta\sigma+\frac{1}{2}\alpha_k\Delta\sigma^2 -\int_\sigma^{\sigma+\Delta\sigma}\varphi(\sigma')\,d\sigma'\right]; \quad \Omega_3(\alpha)=(\alpha-\alpha_1)(\alpha_2-\alpha); \]

the numbers \(\mu_0,\alpha_1,\alpha_2\) are the same as in conditions C, D; the real parameter \(\xi \geqslant 0\) is to be chosen, and the quadratic form \(\Psi(z,\zeta)\) is determined by relations (3), where

\[ s=\tfrac{1}{2}\xi(\alpha_1+\alpha_2)r_2,\quad G_0\equiv G_0^{(k)}=\mu_0^{-1}ee^*-\tfrac{1}{2}(er_1^*+r_1e^*)- \]

\[ -\tfrac{1}{2}\vartheta(-1)^k(er_2^*+r_2e^*)+ \left[\xi\alpha_1\alpha_2-\tfrac{1}{2}\vartheta(-1)^k\alpha_k\right]r_1r_2^*, \quad k=1,2. \tag{8} \]

For convenience of the estimates, in the expression for \(\Delta V_k(z_t)\), obtained from (6), (7), the quantity

\[ \left[\varphi_t(\sigma_t-\mu_0^{-1}\varphi_t) +\vartheta(-1)^k(\varphi_t\Delta\sigma_t+\tfrac{1}{2}\alpha_k\Delta\sigma_t^2) +\right. \]

\[ \left. +\xi(\Delta\varphi_t-\alpha_1\Delta\sigma_t)(\alpha_2\Delta\sigma_t-\Delta\varphi_t)\right], \]

transformed taking into account relations (5), has been added and subtracted.

Suppose that there exist numbers \(\vartheta\geqslant 0,\ \xi\geqslant 0\), satisfying the frequency condition \(\Phi_k>0\) of Theorem 1 (\(k=1\), or \(k=2\)). According to requirements C, D we have \(\Omega_j\geqslant 0,\ j=1,2,3\). From relations (2), (4), by virtue of (5), (8), we obtain \(|1-\lambda|^2\Phi_0(\lambda)\equiv \Phi_k(\lambda,\vartheta,\xi)\). It is easy to verify that the matrix \(Q\) and the vector \(e\) satisfy requirements 1), 2) of Theorem 3, and, by the assertion of this theorem, from the condition \(\Phi_k>0\) there follows the existence of a matrix \(H=H^*\) for which \(\Psi(z,\zeta)>0\) for any \(z,\zeta,\ |z|+|\zeta|\ne0\). Taking (5) and the estimates for \(\Omega_j\) into account, we obtain \(\Delta V_k(z)<0\) for all \(z\ne0,\ \Delta V_k(0)=0\). Thus, the function (7) satisfies conditions (III), (IV) of Lemma 1 \((^6)\).

Denote \(Q_\mu=Q+\mu er_2^*,\ H_k(\mu)=H+\tfrac{1}{2}\vartheta(-1)^k\mu r_1r_2^*,\ G_k(\mu)=H_k(\mu)-Q_\mu^*H_k(\mu)Q_\mu,\ k=1,2\), and consider the particular case \(\varphi(\sigma)=\mu\sigma,\ \mu\in[0,\mu_0]\)^*. From (5), (6), (7) we have \(z_{t+1}=Q_\mu z_t,\ V_k(z)=z^*H_k(\mu)z,\ \Delta V_k(z)=-z^*G_k(\mu)z\). By what has been proved, \(G_k(\mu)>0\) for arbitrary \(\mu,\ 0\leqslant \mu\leqslant\mu_0\), and, by continuity, \(G_k^{(\varepsilon)}(\mu)\equiv H_k(\mu)-Q_\mu(\varepsilon)^*H_k(\mu)Q_\mu(\varepsilon)>0\) for all \(\mu\in[0,\mu_0]\), where \(Q_\mu(\varepsilon)=Q_\mu-\varepsilon ee^*\), and \(\varepsilon\) is a sufficiently small positive number. From the frequency condition \(\Phi_k(\lambda,\vartheta,\xi)>0,\ \vartheta\geqslant0,\ \xi\geqslant0\) of Theorem 1 it follows that the hodograph \(\chi(\lambda)\) does not intersect the real axis on the interval \((-\infty,-\mu_0^{-1}]\) when \(|\lambda|=1\). Therefore the spectra of all matrices \(P_\mu=P+\mu qr^*,\ \mu\in[0,\mu_0]\), lie inside the unit circle (see also the conditions of Theorem 1 on \(\pi_j\)). With the aid of the rela—

\(^*\) Obviously, the function \(\varphi(\sigma)=\mu\sigma\) satisfies conditions C, D for any \(\mu,\ 0\leqslant\mu\leqslant\mu_0\). Therefore, for the case under consideration, all the estimates obtained above are valid.

relations (5) it is easy to show that the spectra of all matrices \(Q_\mu(\varepsilon)\), \(\mu \in [0,\mu_0]\), for sufficiently small \(\varepsilon > 0\) are also located inside the unit circle. From the inequality \(G_k^{(\varepsilon)}(\mu) > 0\), by Lemma 3 \(\left({}^{6}\right)\), it follows that \(H_k(\mu) > 0\) for all \(\mu,\ 0 \le \mu \le \mu_0\). Represent the function (7) in the form

\[ V_k(z)=z^*H_k(\mu_k)z+ \vartheta\int_0^{r_1^*z}\bigl[\mu_k+(-1)^k\sigma^{-1}\varphi(\sigma)\bigr]\sigma\,d\sigma, \]

where \(\mu_1=\mu_0,\ \mu_2=0\). By condition D the second term in the expression \(V_k(z)\) is nonnegative, and, since \(H_k(\mu)>0,\ \mu\in[0,\mu_0]\), we obtain \(V_k(z)>0\) for \(z\ne0\), \(\lim_{|z|\to\infty}V(z)=+\infty\), i.e., the function (7) satisfies condition II of Lemma 1 \(\left({}^{6}\right)\), from which the assertion of Theorem 1 follows.

The proof of Theorem 2 can be obtained by reducing the problem to the noncritical case (Theorem 1) by means of the substitution \(\varphi(\sigma)=\mu_0\sigma-\varphi_1(\sigma)\), as was done, for example, in \(\left({}^{10}\right)\).

Siberian Scientific Research
Institute of Power Engineering

Received
4 III 1965

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