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The circle criterion and Tsypkin's criterion for systems with several nonlinearities without the use of the V. A. Kamenetskiy V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia
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     § 1. Introduction1.1.The problem of absolute stability was originally formulated in [1]; it is one of the main problems in automated control theory. No complete solution of this problem has yet been obtained despite numerous publications (see the bibliography in [2]). The theory of stability of switched systems (see [3]) and absolute stability theory (see [4]) are the main instruments in the study of the stability of control systems with uncertainty (see [5]). Important results in this area of research are the circle criterion in the continuous-time case and Tsypkin’s criterion in the discrete-time case1 (see [4] and [6]–[9]). These criteria provide sufficient conditions for the existence of a quadratic Lyapunov function in the case of Lurie systems with several nonlinearities (see [4]). 1.2.Sufficiency is achieved by use of special technique, called the $S$-procedure (see [10]). On the other hand, the existence of a quadratic Lyapunov function in the case of several nonlinearities depends on the feasibility of the system of linear matrix inequalities, which was established in [11] in the continuous-time case and in [12] in the discrete-time case. To work with systems of matrix inequalities Pyatnitskii proposed a theorem, which originally appeared (with attribution) in [13], and then in [11]. Pyatnitskii’s theorem describes a method to obtain a single matrix inequality equivalent to a system of two matrix inequalities. The operation of transition from a system of two matrix inequalities to an equivalent single inequality specified in this theorem was called a convolution in [11]. Using the convolution operation, necessary and sufficient conditions for the existence of a quadratic Lyapunov function were obtained in the cases of two (see [13]) and several (see [11]) nonlinearities in the continuous-time case and several nonlinearities in the discrete-time case (see [12]). In § 2 we present two formulations of Pyatnitskii’s theorem, which were given in [13] and [11], and distinguish an important particular case of these theorems which was considered in [11]. In § 3 we show how the circle criterion in the case of several nonlinearities can be obtained on the basis of the convolution operation without the use of the $S$-procedure. A similar result for Tsypkin’s criterion is obtained in § 4. In § 5 we use the convolution operation to reduce substantially the number of matrix inequalities in connected (see [11] and [14]) systems of linear matrix inequalities, which determine the existence of a quadratic Lyapunov function for linear switched systems with arbitrarily many subsystems. Thus, the purpose of this work is both to demonstrate the capabilities of Pyatnitskii’s theorem by presenting new proofs of two classical results and to use this theorem to derive more efficient conditions for the existence of quadratic Lyapunov functions for a wide class of Lurie systems and switched systems. § 2. Pyatnitskii’s convolution theoremIn what follows the symbol $\{\,\cdot\, \}^{\top}$ is used to denote the transpose of a matrix. The inequality $W < 0$ ($>0$) for a matrix $W=W^{\top}$ means that the quadratic form $x^{\top}Wx$ is negative (positive) definite, that is, $x^{\top}Wx<0$ ($>0$) whenever $x\neq 0$. In a similar way, $W \leqslant 0$ ($\geqslant0$) means that the quadratic form $x^{\top}Wx$ is negative (positive) semidefinite, that is, $x^{\top}Wx\leqslant0$ ($\geqslant0$). The inequality $W_{1} \leqslant W_{2}$ for symmetric matrices $W_{1}$ and $W_{2}$ is equivalent to the matrix inequality $W_{2}-W_{1} \geqslant 0$. Theorem 1 (see [13]). For the consistency of the system of two matrix inequalities it is necessary and sufficient that there exist a presentation $I_{2}-I_{1}=Q=Q^{+}-Q^{-}$, $Q^{\pm} \geqslant 0$, such that the following single inequality holds: In [13] a general parametric presentation for $Q^{\pm}=Q^{\pm}(\nu)$ was obtained. Theorem 2 (see [11]). For the consistency of the system of two matrix inequalities it is necessary and sufficient that there exist a tuple of parameters $\widetilde{\nu}$ satisfying the following single inequality: Here we use the particular case of Theorems 1 and 2 which can be formulated as follows (see [11], [14]). Theorem 3. For the consistency of the system of two matrix inequalities it is necessary and sufficient that there exist a positive number $\widetilde{\varepsilon}$ such that the following single matrix inequality holds: It is obvious that the matrix inequality (2.2) for some $\varepsilon >0$ implies the consistency of system (2.1). Formally, the convolution operation based on Theorem 3 was called in [11] the convolution of rank 2 or $r_{2}$-convolution. Below we refer to Theorem 3 as the convolution theorem and employ the convolution operation based on Theorem 3. Theorem 3 was presented in [11] in a slightly different statement, with a reference to [13]. Note that the general parametric representation obtained in [13] for the matrices $Q^{\pm}=Q^{\pm}(\nu)$ is rather cumbersome and can hardly be used to derive the general representation (2.2) in the particular case (2.1). In [14] a direct proof of Theorem 3 was presented, which is based on the losslessness of the $S$-procedure in the case of one constraint. Here we give a simple direct proof of Theorem 3 without using the properties of the $S$-procedure. Proof of Theorem 3. It has already been noted above that sufficiency is obvious. Let us prove necessity. Suppose that (2.1) is satisfied. Then there exists a nonsingular linear transform with matrix $T_{1}$ ($\det T_{1}\neq 0$) such that
$$
\begin{equation*}
I_{1} \to T_{1}^{\top}I_{1}T_{1}=- E_{n}\quad\text{and} \quad I_{2}\to T_{1}^{\top}I_{2}T_{1}=\widetilde{I}_{2},
\end{equation*}
\notag
$$
where $E_{n}$ is the $n\times n$ identity matrix. Further, there exists a nonsingular orthogonal transform with matrix $T_{2}$ ($T_{2}^{\top}=T_{2}^{-1}$) such that where the diagonal matrix $\Lambda$ has the form
If $p \neq \lambda q$ ($\lambda \in \mathbb R$), then the matrix $pq^{\top}+qp^{\top}$ has rank $2$ and signature $0$. The corresponding quadratic form is a product of linear forms: $x^{\top}(pq^{\top}+qp^{\top})x= 2\langle p,x \rangle \langle q,x \rangle$ ($\langle \,\cdot\,{,} \,\cdot\, \rangle$ denotes the inner product in $\mathbb{R}^{n}$), and, consequently, it can be reduced to a difference of squares. This explains the form of the matrix $\Lambda$. It follows from the consistency of (2.1) that $\Lambda< 0$, and therefore $-1+\lambda_{2}<0$. Thus, it follows from the consistency of system (2.1) that for appropriate $a,b \in \mathbb{R}^{n}$. The same arguments apply to prove Theorem 1 for an arbitrary matrix $Q$. As the matrices $T_{1}$ and $T_{2}$ are unknown, we need to obtain a general expression for $a$ and $b$ from (2.3) in terms of $p$ and $q$. Let us compare the expressions for $Q$ in (2.1) and (2.3):
$$
\begin{equation*}
Q=I_{2}-I_{1}=aa^{\top}-bb^{\top}=\frac{1}{2}(a+b)(a-b)^{\top}+ \frac{1}{2}(a-b)(a+b)^{\top}= pq^{\top}+qp^{\top}.
\end{equation*}
\notag
$$
Two quadratic forms that are products of linear forms coincide if and only if the corresponding vectors are proportional. Hence we have $(a+b)/\sqrt{2}=\varepsilon p$ and ${(a-b)}/\sqrt{2}=q/\varepsilon$, where $\varepsilon \neq 0$ is the proportionality coefficient. Thereby we obtain a general representation for the vectors $a$ and $b$:
$$
\begin{equation*}
a=\frac{\varepsilon}{\sqrt{2}}\, u^{+}(\varepsilon)\quad\text{and} \quad b=\frac{\varepsilon}{\sqrt{2}}\, u^{-}(\varepsilon),
\end{equation*}
\notag
$$
where the vectors $u^{\pm}(\varepsilon)$ are defined in (2.2).
The case $p=\lambda q$ must be considered separately. If $\lambda \!>\! 0$, then ${I_{1} \!\leqslant\! I_{2}\!=\!I_{1}\!+\!2\lambda q q^{\top}}$. Consequently, (2.1) is equivalent to the matrix inequality $I_{1}+2\lambda q q^{\top} < 0$. This matrix inequality coincides with the one in (2.2), provided that we set $\varepsilon^{2} =1/\lambda$ in (2.2). In the case $\lambda < 0$ we must set $\varepsilon^{2}=-1/\lambda$. If $\lambda=0$, then $I_{1}=I_{2}$. In this case it follows from (2.2) that $Q^{+}=(1/(2\varepsilon^{2}))q q^{\top}$. Since the matrix inequality $I_{1} < 0$ is strict, there exists a sufficiently large $\varepsilon$ such that $I_{1}+(1/(2\varepsilon^{2}))q q^{\top} < 0$. The proof of Theorem 3 is complete. § 3. Circle criterion for systems with several nonlinearities3.1. The problem of absolute stabilityA Lurie system with several nonlinearities has the form
$$
\begin{equation}
\dot{x}=Ax+\sum^{m}_{j=1}b_{j}\varphi_{j}(t,\sigma_{j}), \qquad \sigma_{j}=\langle c_{j}, x \rangle, \quad A\in \mathbb R^{n\times n}, \quad b_{j},c_{j} \in \mathbb R^{n},
\end{equation}
\tag{3.1}
$$
where the nonlinearities $\varphi_{j}(t,\sigma_{j})$ satisfy the conditions of existence of an absolutely continuous solution $x(t)$, $j =1,\dots,m$. The absolute stability of system (3.1) in the class $N_{\varphi}$ of nonlinearities $\varphi= \|\varphi_{j}\|^{m}_{j=1}$ satisfying the sector constraints
$$
\begin{equation}
0 \leqslant \varphi_{j}\sigma_{j} \leqslant \sigma_{j}^{2}, \qquad j=1,\dots,m,
\end{equation}
\tag{3.2}
$$
means that this system is globally asymptotically stable for any such nonlinearities. 3.2. The circle criterion with the use of the $S$-procedureWe recall briefly, in a convenient form, the arguments based on the $S$-procedure and establish the circle criterion in the case of system (3.1) for an arbitrary finite $m$ (see [6] and [7]). The $S$-procedure is a special technique that allows passing from an inequality which is not imposed on the quadratic form in the whole space, but only in a bounded domain determined by quadratic constraints, to an inequality for the quadratic form in the whole space, that is, to a matrix inequality. The $S$-procedure can be formulated in different ways. In the formulation used in this work the $S$-procedure establishes a relation between the following conditions:
$$
\begin{equation}
x^{\top}G_{0}x < 0 \quad\text{for } x^{\top}G_{1}x \geqslant 0, \dots, x^{\top}G_{m}x \geqslant 0, \qquad x \neq 0,
\end{equation}
\tag{3.3}
$$
and
$$
\begin{equation}
\exists\,\tau_{j} > 0 \quad (j=1,\dots,m)\colon \quad x^{\top}G_{0}x+\sum_{j=1}^{m}\tau_{j}x^{\top}G_{j}x <0, \quad x\in \mathbb R^{n}, \quad x \neq 0,
\end{equation}
\tag{3.4}
$$
where $G_{j}\in \mathbb R^{n\times n}$ and $G_{j}^{\top}=G_{j}$, $j=0,\dots,m$. It is obvious that (3.4) implies (3.3). In the case $m=1$ it is known (see [15], Ch. I, § 1.7.2) that condition (3.4) is not only sufficient, but also necessary for the validity of condition (3.3) (the losslessness of the $S$-procedure). For various formulations of the $S$-procedure, the history of this technique, the introduction of the term itself and some other information see [4], [5], [10] and [15]. The sector constraints (3.2) are equivalent to the quadratic constraints
$$
\begin{equation}
F_{j}(x,\varphi_{j})=\varphi_{j}(\langle c_{j}, x \rangle- \varphi_{j}) \geqslant 0, \qquad j= 1,\dots,m.
\end{equation}
\tag{3.5}
$$
The derivative $\dot{v}(x)$ of the Lyapunov function $v(x)=x^{\top}L x$ ($L \in \mathbb{R}^{n\times n}$, $L^{\top}=L$) by virtue of system (3.1) is subject to the inequality
$$
\begin{equation}
x^{\top}(A^{\top}L+LA)x+2\sum_{j=1}^{m}\varphi_{j}\langle Lb_{j},x \rangle < 0, \qquad (x,\varphi) \neq 0,
\end{equation}
\tag{3.6}
$$
which must hold for all $(x,\varphi)$ satisfying (3.5). In compliance with the trick of the $S$-procedure we consider the quadratic form
$$
\begin{equation}
x^{\top}(A^{\top}L+LA)x+2\sum_{j=1}^{m}\varphi_{j}\langle Lb_{j},x \rangle+\sum_{j=1}^{m}\tau_{j}\varphi_{j} (\langle c_{j},x \rangle-\varphi_{j}),
\end{equation}
\tag{3.7}
$$
where the $\tau_{j} > 0$, $j=1,\dots,m$, are unknown parameters. Inequality (3.6) in the matrix form looks like
$$
\begin{equation*}
(Ax+B\varphi)^{\top}Lx+x^{\top}L(Ax+B\varphi) < 0, \qquad (x,\varphi) \neq 0,
\end{equation*}
\notag
$$
where $B=(b_{1}\,b_{2}\,\dots\,b_{m})$, and the constraint function $F(x,\varphi,\mathcal T)$ can be represented in the form
$$
\begin{equation}
F(x,\varphi,\mathcal T)=\sum_{j=1}^{m}\tau_{j}\varphi_{j} ( \langle c_{j},x \rangle-\varphi_{j})= \begin{pmatrix} x \\ \varphi \end{pmatrix}^{\top} \begin{pmatrix} 0 & \dfrac12 C\mathcal T \\ \dfrac12\mathcal T C^{\top} & -\Gamma \end{pmatrix} \begin{pmatrix} x \\ \varphi \end{pmatrix},
\end{equation}
\tag{3.8}
$$
where
$$
\begin{equation}
C= \begin{pmatrix} c_{1} & c_{2}& \dots & c_{m} \end{pmatrix}\quad\text{and} \quad \Gamma=\mathcal T =\operatorname{diag}\{ \tau_{1}, \dots , \tau_{m}\}.
\end{equation}
\tag{3.9}
$$
In the newly adopted notation the negative definiteness of the form (3.7) is equivalent to the matrix inequality
$$
\begin{equation}
\begin{pmatrix} A^{\top}L+LA & LB+\dfrac12C\mathcal T \\ B^{\top}L+\dfrac12\mathcal TC^{\top} & -\Gamma \end{pmatrix} <0.
\end{equation}
\tag{3.10}
$$
The feasibility of the matrix inequality (3.10) follows from the frequency theorem (the KYP lemma) (see [10] and [15]). The formulation of the frequency theorem which is most convenient in our case is presented in [15], Ch. I, § 1.2.2, Corollary 1. It states that if $A$ is a Hurwitz matrix and $\Gamma > 0$, then the feasibility of inequality (3.10) is equivalent to the frequency inequality
$$
\begin{equation}
\Gamma+\operatorname{Re}\mathcal T W(i\omega) > 0 , \quad\text{where } W(i\omega)= C^{\top}( A-i\omega E_{n})^{-1}B, \quad \omega \in [-\infty,\infty ],
\end{equation}
\tag{3.11}
$$
$\operatorname{Re}W=(W+W^{*})/2$ and $W^{*}=\overline{W}^{\top}$ is the Hermitian transpose of $W$. Thus, the circle criterion for system (3.1) with several nonlinearities consists in verifying the frequency condition (3.11), which is sufficient (for $m>1$) for the existence of a quadratic Lyapunov function for such systems. 3.3. The circle criterion without the use of the $S$-procedureIn [14], Theorem 3 was used to derive sufficient conditions for the existence of a quadratic Lyapunov function for system (3.1), which coincide with the circle criterion for the absolute stability of control systems with two nonlinearities. In this work we obtain a similar result for control systems with an arbitrary (finite) number of nonlinearities. The existence of a quadratic Lyapunov function $v(x)=x^{\top}L x$ for the Lurie system (3.1) depends (see [11]) on the feasibility of the system of linear matrix inequalities
$$
\begin{equation}
I_{s}=A^{\top}_{s}L+LA_{s}< 0, \qquad s=1,\dots,N, \quad N=2^{m},
\end{equation}
\tag{3.12}
$$
with matrices $A_{s}$ of the form (see [11])
$$
\begin{equation}
A_{s}=A+\sum^{m}_{j=1}h_{sj}b_{j}c_{j}^{\top}, \quad h_{s}=\|h_{sj}\|^{m}_{j=1}, \qquad s=1,\dots,N,
\end{equation}
\tag{3.13}
$$
where the $h_{sj}$ take one of the two values $0$ and $1$ independently. We assume that $h_{1}=(0, \dots, 0)$, which means that $A_{1}=A$. It follows from representation (3.13) that
$$
\begin{equation*}
I_{s}=A^{\top}L+LA+\sum^{m}_{j=1}h_{sj}(Lb_{j}c_{j}^{\top}+ c_{j}b_{j}^{\top}L)=A^{\top}L+LA+\sum^{m}_{j=1}h_{sj}Q_{j},
\end{equation*}
\notag
$$
where
$$
\begin{equation}
Q_{j}=p_{j}q_{j}^{\top}+q_{j}p_{j}^{\top}, \quad p_{j} \triangleq Lb_{j}, \quad q_{j} \triangleq c_{j}, \qquad j=1,\dots,m.
\end{equation}
\tag{3.14}
$$
We represent the matrices $Q_{j}$ in the form of the difference $Q_{j}=Q_{j}^{+}-Q_{j}^{-}$ as in Theorem 3. Then the matrix inequality
$$
\begin{equation}
Q_{j} \leqslant Q_{j}^{+}(\varepsilon_{j})= \frac{\varepsilon_{j}^{2}}{2}u_{j}^{+}(\varepsilon_{j})u_{j}^{+}(\varepsilon_{j})^{\top}, \qquad u_{j}^{+}(\varepsilon_{j}) \triangleq p_{j}+ \frac{1}{\varepsilon_{j}^{2}}q_{j}, \qquad j=1,\dots,m,
\end{equation}
\tag{3.15}
$$
holds for all $\varepsilon_{j} > 0$. Consider the following matrix inequality:
$$
\begin{equation}
I_{\mathrm {cir}} \triangleq A^{\top}L+LA+\sum^{m}_{j=1}Q^{+}_{j}(\varepsilon_{j}) < 0.
\end{equation}
\tag{3.16}
$$
Since $h_{sj}=0$ or $h_{sj}=1$, we have
$$
\begin{equation}
I_{s} \leqslant I_{\mathrm{cir}}, \qquad s=1,\dots,N.
\end{equation}
\tag{3.17}
$$
It follows from (3.17) that the matrix inequality (3.16) guarantees the validity of system (3.12). We substitute the expressions for $p_{j}$ and $q_{j}$ from (3.14) into the expression for $Q_{j}^{+}(\varepsilon_{j})$ and introduce the new parameters $ \tau_{j} \triangleq 2/ \varepsilon_{j}^{2}$ in (3.15). Then by Schur’s lemma the matrix inequality (3.16) is equivalent to (3.10). Thus, we have derived the circle criterion without the use of the $S$-procedure. Actually, the ‘necessary’ part of Theorem 3 is only required for $m=1$, to show that in this case system (3.12) is equivalent to the matrix inequality $ I_{\mathrm{cir}} < 0$. Below the matrix inequality (3.10), as well as the matrix inequality (3.16), is called the matrix inequality of the circle criterion. § 4. Tsypkin’s criterion for systems with several nonlinearities4.1. The problem of absolute stabilityA discrete-time Lurie system with several nonlinearities has the form
$$
\begin{equation}
x(t+1)=Ax(t)+\sum^{m}_{j=1}b_{j}\varphi_{j}(t,\sigma_{j}), \qquad \sigma_{j}=\langle c_{j}, x \rangle,
\end{equation}
\tag{4.1}
$$
where the $\varphi_{j}(t,\sigma_{j})$ and $t>0$ satisfy the sector constraints (3.2) for all $\sigma_{j}=\langle c_{j},x \rangle $. The absolute stability of system (4.1) means that it is globally asymptotically stable for any such nonlinearities. By Tsypkin’s criterion of the absolute stability of system (4.1) we mean a frequency condition for the existence of a quadratic Lyapunov function obtained with the use of the $S$-procedure and the generalized Kalman–Szegő–Popov lemma (see [8] and [9]). 4.2. Tsypkin’s criterion with the use of the $S$-procedureWe briefly recall how Tsypkin’s criterion can be derived using the $S$-procedure in the case of system (4.1) for an arbitrary finite value of $m$. In the matrix form system (4.1) can be written as where $B=(b_{1}\,b_{2}\,\dots\,b_{m})$ and $\varphi^{\top}=(\varphi_{1}\,\varphi_{2}\,\dots\, \varphi_{m})$. The inequality for the first difference $\triangle v(x(t))=x^{\top}(t+1)Lx(t+1)-x^{\top}(t)Lx(t)$ of the Lyapunov function $v(x)=x^{\top}L x$ along a solution of system (4.2) has the form
$$
\begin{equation*}
\triangle v(x,\varphi)=(Ax+B\varphi)^{\top}L(Ax+B\varphi)-x^{\top}Lx < 0;
\end{equation*}
\notag
$$
it must hold for all $(x,\varphi)\neq 0 $ satisfying constraints (3.2) (which are equivalent to (3.5)). In accordance with the $S$-procedure we consider the quadratic form
$$
\begin{equation}
(Ax+B\varphi)^{\top}L(Ax+B\varphi)-x^{\top}Lx +\sum_{j=1}^{m}\tau_{j}\varphi_{j} ( \langle c_{j},x \rangle -\varphi_{j}),
\end{equation}
\tag{4.3}
$$
where the $\tau_{j} $, $s=1,\dots,m$, are unknown positive parameters. In the discrete-time case the same expressions (3.8) and (3.9) as in the continuous-time case hold for the constraint function. Thus, the negative definiteness of the form (4.3) is equivalent to the matrix inequality
$$
\begin{equation}
{I}_{Ts}^{m}= \begin{pmatrix} A^{\top}LA-L & A^{\top}LB+\dfrac12C\mathcal T \\ B^{\top}LA+\dfrac12\mathcal T C^{\top} & B^{\top}LB -\Gamma \end{pmatrix} <0,
\end{equation}
\tag{4.4}
$$
where $C=( c_{1}\, c_{2}\,\dots\,c_{m})$ and $\Gamma=\mathcal T=\operatorname{diag}\{ \tau_{1}, \dots ,\tau_{m}\}$. Below we refer to inequality (4.4) as Tsypkin’s matrix inequality. The feasibility of the matrix inequality (4.4) follows from the generalized Kalman–Szegő–Popov lemma (see [8] and [9]) (in our case the most convenient formulation of this lemma is the one in [16], Pt. VI, Appendix H) in the form of the frequency inequality which must hold for all $\lambda \in \mathbb C$ such that $|\lambda|=1$, where the transfer matrix $W(\,\cdot\,)$ was defined in (3.11). Tsypkin’s criterion for system (4.1) with several nonlinearities consists in checking the frequency condition (4.5), which is sufficient (for $m>1$) for the existence of a quadratic Lyapunov function for such systems.4.3. Tsypkin’s criterion without use of the $S$-procedureThe existence of a quadratic Lyapunov function $v(x)=x^{\top}L x$ for the Lurie system (4.1) under constraints (3.2) depends (see [12]) on the feasibility of the following system of linear matrix inequalities:
$$
\begin{equation}
I_{s}=A^{\top}_{s}LA_{s}-L< 0, \qquad s=1, \dots ,N, \quad N=2^{m},
\end{equation}
\tag{4.6}
$$
where the matrices $A_{s}$ are determined by the relation (3.13). Note that the discrete-time case is much more complicated than the continuous-time one, because the matrix inequalities in (4.6) are quadratic in $A_{s}$, whereas in the continuous-time case the dependence is linear. Theorem 4. The feasibility of Tsypkin’s matrix inequality (4.4) with some additional parameters $\mathcal T=\operatorname{diag}\{\tau_{1},\dots ,\tau_{m}\}$, $\tau_{j} > 0$, implies the feasibility of system (4.6). Proof. The idea of the proof presented below consists in deducing Theorem 4 from Theorem 3 without the use of the $S$-procedure and the results of the previous subsection. For $m=1$ this result was obtained in [17] and for $m=2$, in [18].
We can assume by the inductive hypothesis that a sufficient condition for the feasibility of (4.6) is the feasibility of Tsypkin’s matrix inequality (4.4). When the number of nonlinearities in the system increases by one, the number of inequalities in (4.6) doubles, which means that for $N=2^{m+1}$ system (4.6) can be represented in the form
$$
\begin{equation}
A^{\top}_{s}LA_{s}-L< 0, \qquad s=2^{m}+1, \dots ,2^{m+1}.
\end{equation}
\tag{4.8}
$$
Assume that system (4.7) coincides exactly with (4.6) and that the matrices $A_{s}$ in (4.8) are numbered so that $A_{s+2^{m}}=A_{s}+ b_{m+1}c_{m+1}^{\top}$, $s=1, \dots ,2^{m}$. Thus, system (4.8) coincides with (4.7) if in the expression for $A_{s}$ in (4.7) we replace the matrix $A$ by $A_{1+2^{m}}=A+b_{m+1}c_{m+1}^{\top}$. For the sake of convenience set $1+2^{m} \triangleq \xi$, which means that $A_{\xi}= A_{1+2^{m}}$. Then by the inductive hypothesis a sufficient condition for the feasibility of (4.8) is the feasibility of the matrix inequality
$$
\begin{equation}
\widehat{I}_{Ts}^{m}= \begin{pmatrix} A_{\xi}^{\top}LA_{\xi}-L & A_{\xi}^{\top}LB+\dfrac12C\mathcal T \\ B^{\top}LA_{\xi}+\dfrac12\mathcal T C^{\top} & B^{\top}LB -\Gamma \end{pmatrix} <0,
\end{equation}
\tag{4.9}
$$
where $B$ and $C$ are the same as in (4.4). Consider the difference $\widehat{I}_{Ts}^{m}-I_{Ts}^{m}$ under the assumption that the additional parameters $\mathcal T$ have the same value in both matrices:
$$
\begin{equation*}
\widehat{I}_{Ts}^{m} -{I}_{Ts}^{m}= \begin{pmatrix} A_{\xi}^{\top}LA_{\xi}-A^{\top}LA & c_{m+1}b_{m+1}^{\top}LB \\ B^{\top}Lb_{m+1}c_{m+1}^{\top} & 0_{m\times m} \ \end{pmatrix},
\end{equation*}
\notag
$$
where $0_{n\times m}$ is an $n\times m$ matrix with all entries equal to $0$. Since $A_{\xi}^{\top}LA_{\xi}-A^{\top}LA= p_{m+1}c_{m+1}^{\top}+c_{m+1}p_{m+1}^{\top}$, where $p_{m+1}= A^{\top}Lb_{m+1}+(b_{m+1}^{\top}Lb_{m+1}/2)c_{m+1}$, it can be verified directly that
$$
\begin{equation*}
\widehat{I}_{Ts}^{m} -{I}_{Ts}^{m} =\widetilde{p}\,\widetilde{q}^{\top}+\widetilde{q}\,\widetilde{p}^{\top}, \qquad \widetilde{p}= \begin{pmatrix} p_{m+1} \\ B^{\top}Lb_{m+1} \end{pmatrix}, \qquad \widetilde{q}=\begin{pmatrix} c_{m+1} \\ 0_{m \times 1} \end{pmatrix}.
\end{equation*}
\notag
$$
This means that we can apply Theorem 3 to the system of two matrix inequalities (4.4) and (4.9). Hence the feasibility of this system is equivalent to the existence of $\varepsilon_{m+1} >0$ such that the single matrix inequality
$$
\begin{equation}
{I}_{Ts}^{m+1} ={I}_{Ts}^{m}+\frac{\varepsilon_{m+1}^{2}}{2} \biggl(\widetilde{p}+\frac{1}{\varepsilon_{m+1}^{2}}\, \widetilde{q} \biggr) \biggl(\widetilde{p}+\frac{1}{\varepsilon_{m+1}^{2}}\, \widetilde{q}\biggr)^{\top} <0
\end{equation}
\tag{4.10}
$$
is solvable. Set $\tau_{m+1} \triangleq b_{m+1}^{\top}Lb_{m+1}+ (2/\varepsilon_{m+1}^{2})$. Then
$$
\begin{equation*}
\widetilde{p}+\biggl(\frac{1}{\varepsilon_{m+1}^{2}}\biggr)\widetilde{q} = \begin{pmatrix} p_{m+1}+\dfrac{1}{\varepsilon_{m+1}^{2}}\, c_{m+1} \\ B^{\top}Lb_{m+1} \end{pmatrix}= \begin{pmatrix} A^{\top}Lb_{m+1}+\dfrac{\tau_{m+1}}{2}c_{m+1} \\ B^{\top}Lb_{m+1} \end{pmatrix}.
\end{equation*}
\notag
$$
By Schur’s lemma the matrix inequality (4.10) is equivalent to the following inequality in the extended space:
$$
\begin{equation*}
\begin{aligned} \, {I}_{Ts}^{m+1} &\cong \widetilde{I}_{Ts}^{m+1} =\begin{pmatrix} {I}_{Ts}^{m} & \widetilde{p}+\dfrac{1}{\varepsilon_{m+1}^{2}}\, \widetilde{q} \\ (\bullet)^{\top} & -\dfrac{2}{\varepsilon_{m+1}^{2}} \end{pmatrix} \\ &=\begin{pmatrix} A^{\top}LA-L & A^{\top}LB+\dfrac12C\mathcal T & A^{\top}Lb_{m+1}+ \dfrac{\tau_{m+1}}{2}c_{m+1} \\ (\bullet)^{\top} & B^{\top}LB -\Gamma & B^{\top}Lb_{m+1} \\ (\bullet)^{\top} & (\bullet)^{\top} & b_{m+1}^{\top}Lb_{m+1} -\tau_{m+1} \end{pmatrix} \\ &=\begin{pmatrix} A^{\top}LA-L & A^{\top}LB_{m+1}+\dfrac12C_{m+1}\tau_{m+1} \\ (\bullet)^{\top} & B_{m+1}^{\top}LB_{m+1} -\Gamma_{m+1} \end{pmatrix} < 0, \end{aligned}
\end{equation*}
\notag
$$
where $B_{m+1}=( b_{1} \, b_{2} \, \dots \, b_{m} \, b_{m+1})$, $C_{m+1}=( c_{1} \, c_{2} \, \dots \, c_{m} \,c_{m+1})$ and $\Gamma_{m+1}\!=\!\mathcal T_{m+1}\!=\operatorname{diag}\{ \tau_{1}, \dots, \tau_{m}, \tau_{m+1} \}$. Here and below the symbols “$\bullet$” denote the entries under the main diagonal of a symmetric matrix, which coincide with the corresponding entries above the main diagonal.
Thus, the matrix inequality $\widetilde{I}_{Ts}^{m+1} < 0$ is reduced to the form of Tsypkin’s matrix inequality for a system with $m+1$ nonlinearities. The proof of Theorem 4 is complete. § 5. Reducing the dimension of a system of linear matrix inequalities5.1. Systems of two and three linear matrix inequalitiesEven when the matrix inequalities in system (2.1) are linear, the matrix inequality (2.2) equivalent to this system is not. The following assertion, which combines Theorem 3 and Schur’s lemma, allows us to reduce a system of two linear matrix inequalities to an linear equivalent matrix inequality. Theorem 5. Suppose that the inequalities in system (2.1) are linear in the variable $\nu$, that is, $I_{1}=I_{1}(\nu)$ and $I_{2}= I_{2}(\nu)$, and $Q=Q(\nu)=I_{2}(\nu)-I_{1}(\nu)= p(\nu)q^{\top}+qp^{\top}(\nu)$, where $p=p(\nu)$ depends linearly on $\nu$ and $q$ does not depend on $\nu$. Then (2.1) is equivalent to the single matrix inequality which is a linear matrix inequality in $(\nu,\tau)$. A similar result holds for the connected (see [14]) system of three linear matrix inequalities
$$
\begin{equation}
I_{2}=I_{1}+Q_{1} < 0, \qquad I_{1} < 0, \qquad I_{3}=I_{1}+Q_{2} < 0.
\end{equation}
\tag{5.1}
$$
Theorem 6. Suppose that the inequalities in system (5.1) are linear in the unknown variable $\nu$, that is, $I_{s}=I_{s}(\nu)$, $s=1,2,3$, and $Q_{j}(\nu)= p_{j}(\nu)q_{j}^{\top}+q_{j}p_{j}^{\top}(\nu)$, where $p_{j}=p_{j}(\nu)$ depends linearly on $\nu$ and $q_{j}$ does not depend on $\nu$, $j=1,2$. Then (5.1) is equivalent to the single matrix inequality Proof. Applying Theorem 5 to the first two and then to the last two inequalities in (5.1), we reduce (5.1) to an equivalent system of two linear matrix inequalities:
$$
\begin{equation}
\widetilde{I}_{1}= \begin{pmatrix} I_{1}(\nu) & p_{1}(\nu)+\dfrac{\tau_{1}}{2}q_{1} \\ (\bullet)^{\top} & -\tau_{1} \end{pmatrix} <0, \qquad \widetilde{I}_{2} = \begin{pmatrix} I_{1}(\nu) & p_{2}(\nu)+\dfrac{\tau_{2}}{2}q_{2} \\ (\bullet)^{\top} & -\tau_{2} \end{pmatrix} < 0.
\end{equation}
\tag{5.3}
$$
The difference matrix has the form
$$
\begin{equation}
\widetilde{I}_{2}-\widetilde{I}_{1}= \begin{pmatrix} 0_{n\times n} & \widehat{p} \\ (\bullet)^{\top} & 2\gamma \end{pmatrix},
\end{equation}
\tag{5.4}
$$
where $\widehat{p} \triangleq p_{2}(\nu)-p_{1}(\nu)+({\tau_{2}}/{2})q_{2} - ({\tau_{1}}/{2})q_{1}$ and $\gamma \triangleq (\tau_{1}-\tau_{2})/2$. It is easily seen that
$$
\begin{equation*}
\widetilde{I}_{2}-\widetilde{I}_{1}=\widetilde{p}\,\widetilde{q}^{\top} + \widetilde{q}\,\widetilde{p}^{\top}, \qquad \widetilde{p}=\begin{pmatrix} \widehat{p} \\ \gamma\end{pmatrix}, \qquad \widetilde{q}=\begin{pmatrix} 0_{n\times 1} \\ 1 \end{pmatrix}.
\end{equation*}
\notag
$$
Thus, we can apply Theorem 5 to system (5.3). As a result, system (5.3), and therefore system (5.1), proves to be equivalent to the matrix inequality
$$
\begin{equation}
\widehat{\widetilde{I}}= \begin{pmatrix} \widetilde{I}_{1} & \widetilde{p}+\dfrac{\tau_{3}}{2}\widetilde{q} \\ (\bullet)^{\top} & -\tau_{3} \end{pmatrix} =\begin{pmatrix} I_{1} & p_{1}(\nu)+\dfrac{\tau_{1}}{2}q_{1} & \widehat{p} \\ (\bullet)^{\top} & -\tau_{1} & \gamma+\dfrac{\tau_{3}}{2} \\ (\bullet)^{\top} & \bullet&-\tau_{3} \end{pmatrix} < 0.
\end{equation}
\tag{5.5}
$$
Substituting the expressions for $\widehat{p}$ and $\gamma$ from (5.4) into (5.5), we obtain just the matrix inequality (5.2). The proof of the theorem is complete. 5.2.In this subsection we show how Theorem 5 can be used to reduce the dimension of system (3.12) for deciding about the existence of a quadratic Lyapunov function for the Lurie system (3.1) for an arbitrary finite $m$ in the continuous-time case. System (3.12) in $n(n+1)/2$ unknowns, which has general dimension $2^{m}n$ in this case, can be reduced to an equivalent system of dimension $2^{m-1}(n+1)$ in $n(n+1)/2+2^{m-1}$ unknowns. Theorem 7. For $N=2^m$ the system of matrix inequalities (3.12) is equivalent to the system of linear matrix inequalities with $2^{m-1}$ additional parameters $\tau_{s} > 0$. Proof. Suppose that for $N=2^m$ the matrix inequalities in system (3.12) are numbered in such a way that the first $2^{m-1}$ matrix inequalities $I_{s} < 0$, $s=1,\dots,2^{m-1}$, coincide with the inequalities in system (3.12) for $N=2^{m-1}$, while the remaining $2^{m-1}$ matrix inequalities $I_{s} < 0$, $s=2^{m-1}+1,\dots,2^{m}$, are assigned indices in the following way:
$$
\begin{equation*}
I_{s+2^{m-1}}= I_{s}+(Lb_{m}c_{m}^{\top}+ c_{m}b_{m}^{\top}L) < 0, \qquad s=1,\dots,2^{m-1}.
\end{equation*}
\notag
$$
Then Theorem 5 can be applied to the pairs of matrix inequalities
$$
\begin{equation}
I_{s} < 0, \quad I_{s+2^{m-1}} < 0, \qquad s=1,\dots,2^{m-1}.
\end{equation}
\tag{5.7}
$$
Applying Theorem 5 to each system (5.7) of two matrix inequalities, we reduce the system of matrix inequalities (3.12) to the equivalent system of matrix inequalities (5.6) with $2^{m-1}$ additional positive parameters $\tau_{s} > 0$. The proof of Theorem 7 is complete. 5.3.In this subsection we show how Theorem 5 can be applied to reduce the dimension of system (4.6) for deciding the existence of a quadratic Lyapunov function for the Lurie system (4.1) in the discrete-time case. Theorem 8. For $N=2^m$ the system of matrix inequalities (4.6) is equivalent to the system of linear matrix inequalities which involves $2^{m-1}$ additional positive parameters $\tau_{s} $. Proof. Suppose that for $N=2^m$ the matrices $A_{s}$ in (4.6) are numbered in such a way that the first $2^{m-1}$ matrices $A_{s}$, $s=1,\dots,2^{m-1}$, coincide with the matrices $A_{s}$ of system (4.6) for $N=2^{m-1}$, while the remaining $2^{m-1}$ matrices $A_{s}$, $s=2^{m-1}+1,\dots,2^{m}$, are assigned indices as follows:
Theorem 5 can be applied to the pairs of matrix inequalities
$$
\begin{equation}
I_{s} < 0, \quad I_{s+2^{m-1}} < 0, \qquad s=1,\dots,2^{m-1},
\end{equation}
\tag{5.9}
$$
from system (4.6), since $I_{s+2^{m-1}}-I_{s}=A_{s+2^{m-1}}^{\top}LA_{s+2^{m-1}}-A_{s}^{\top}LA_{s}= p_{s}c_{m}^{\top}+c_{m}p_{s}^{\top}$, where $p_{s}= A_{s}^{\top}Lb_{m}+(b_{m}^{\top}Lb_{m}/2)c_{m}$. Applying Theorem 5 to each system (5.9) of two matrix inequalities we see that the system of matrix inequalities (4.6) is equivalent to system (5.8) with the $2^{m-1}$ additional positive parameters $\tau_{s} $. The proof of Theorem 8 is complete. 5.4. Practical suggestionsThe circle criterion and Tsypkin’s criterion are obtained as feasibility conditions for the circle-criterion matrix inequality (3.10) and Tsypkin’s matrix inequality (4.4), which are linear matrix inequalities with respect to the unknowns $L$ and $\tau_{j}$, $j=1,\dots,m$, and can be solved numerically using standard software. Thus, instead of systems (3.12) or (4.6), which have dimension $2^{m}n$, one can consider the single matrix inequality (3.10) or the matrix inequality (4.4) of dimension $n+m$ with $m$ additional parameters. At the same time there can be losses in the domain of existence of a quadratic Lyapunov function due to the lossness of the $S$-procedure. Nevertheless, in the case of high-dimensional problems, it is reasonable to solve these matrix inequalities, since this can yield practically acceptable results. The fact that the matrix inequality (3.4) obtained as a result of the $S$-procedure is linear in the parameters $\zeta, \tau_{1},\dots,\tau_{m}$, whenever $G_{0}=G_{0}(\zeta)$ linearly depends on $\zeta$ and $G_{1},\dots,G_{m}$ do not depend on $\zeta$ was observed previously (see, for example, [19; § 2.6.3]). On the one hand each application of Theorem 5 reduces the number of matrix inequalities in the system by one, but on the other hand it increases the number of unknowns by one. Theorem 5 allows one to proceed without losses in the domain of existence of a quadratic Lyapunov function from systems (3.12) and (4.6) in $n({n+1})/2$ unknowns to systems (5.6) and (5.8) of dimension $2^{m-1}(n+1)$ in $n({n+1})/2+2^{m-1}$ unknowns. In the context of the problem of the stability of linear switching systems under arbitrary switchings the problem of the existence of a quadratic Lyapunov function is equivalent (see [3] and [19]) to the problem of the feasibility of systems (3.12) and (4.6) in the case of continuous and discrete time. Here the matrices $A_{s}$ can be arbitrary (and not necessarily of the form (3.13)). If these matrices determine connected switching systems (see [14] and [17]), then the corresponding systems (3.12) and (4.6) obey analogues of Theorems 7 and 8, which allow one to pass to equivalent systems of linear matrix inequalities of lower dimension. The form of these systems of linear matrix inequalities, their dimension and the number of additional parameters are determined by particular characteristics of the matrices $A_{s}$. For example, in the case of even $N$ Theorem 5 provides an equivalent system of dimension $nN/2$ with $N/2$ additional parameters. If Theorem 5 is used in combination with Theorem 6, then various cases are possible. Theorems 5 and 6 can be used to reduce the dimension of arbitrary systems of linear matrix inequalities containing appropriate pairs or triples of linear matrix inequalities. The efficiency of the use of the approaches proposed above, which are based on the reduction of the dimension of systems of linear matrix inequalities, is determined by the performance of the available software for solving systems of linear matrix inequalities in the problems of various complexity. 
Citation in format AMSBIB
\Bibitem{Kam24} |
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