Matrix Inequalities in the Stability Theory: New Results Based on the Convolution Theorem

Abstract

Using Pyatnitskiy’s convolution theorem, the circle criterion of absolute stability for Lurie systems with several nonlinearities is obtained without use of the S-lemma. For connected systems with switching between three linear subsystems, a new criterion for the existence of a quadratic Lyapunov function is proposed. On the basis of the convolution theorem, two theorems are proved which lead to a substantial reduction in the dimensionality of connected systems of linear matrix inequalities. Issues of improving the circle criterion for Lurie systems with two nonlinearities are also discussed.

APPENDIX

Proof of Theorem 5. Let N = 2^m and let the MIs in (2.13) be numerated in such a way that the first 2^m–1 inequalities I_s < 0, s = $\overline {{{{1,2}}^{{m - 1}}}}$, coincide with the inequalities in (2.13) for N = 2^m–1, and the rest ^m – 1 inequalities I_s < 0, s = $\overline {{{2}^{{m - 1}}} + {{{1,2}}^{m}}}$, are numerated as follows:

$${{I}_{{s + {{2}^{{m - \,1}}}}}} = {{I}_{s}} + \left( {L{{b}_{m}}c_{m}^{{\text{T}}} + {{c}_{m}}b_{m}^{{\text{T}}}L} \right) < 0,\quad s = \overline {{{{1,2}}^{{m - 1}}}} .$$

Then, Pyatnitskiy’s theorem is applicable to the pairs of inequalities

$${{I}_{s}} < 0,\quad {{I}_{{s + {{2}^{{m - \,1}}}}}} < 0,\quad s = \overline {{{{1,2}}^{{m - 1}}}} .$$

(A.1)

As a result, the system of inequalities (2.13) is equivalent to the set of MIs

$${{I}_{s}} + \frac{{\varepsilon _{s}^{2}}}{2}\left( {L{{b}_{m}} + \frac{1}{{\varepsilon _{s}^{2}}}{{c}_{m}}} \right){{\left( {L{{b}_{m}} + \frac{1}{{\varepsilon _{s}^{2}}}{{c}_{m}}} \right)}^{{\text{T}}}} < 0,\quad s = \overline {{{{1,2}}^{{m - 1}}}} ,$$

(A.2)

with 2^m–1 additional parameters ε_s > 0. Letting ε_s = ε_m > 0, s = $\overline {{{{1,2}}^{{m - 1}}}}$, in (A.2), we arrive at yet another proof, by induction, of the transition from (2.13) to the MICC (2.16). Application of Theorem 4 to every pair of inequalities (A.1) provides the equivalence of the set of MIs (2.13) to the set of MIs (4.2) with 2^m–1 extra parameters τ_s $\mathop = \limits^\Delta$ 2/$\varepsilon _{s}^{2}$. Theorem 5 is proved.

Proof of Theorem 7. We show that the fulfilment of the MICC (5.9) implies the existence of τ₄ such that the MI (5.10) holds. Similarly to [12], to determine the conditions of negative definiteness of a parameter-dependent matrix I_b(ν), given the negative definiteness of the matrix I_a(ν), we use the following obvious sufficient condition: if I_a(ν) < 0 and I_b(ν) ≤ I_a(ν), then I_b(ν) < 0.

To simplify derivations, we return to the notation used in the proof of Theorem 2 and transform the MI (5.10) (see Lemma A4 [13, p. 253]) as follows:

$$\begin{gathered} {{{\tilde {\bar {\bar {I}}}}}_{1}} = \left( {\begin{array}{*{20}{c}} {{{I}_{1}}}&{u_{1}^{ + }}&{u_{3}^{ + } - u_{1}^{ + }} \\ {{{{( \bullet )}}^{{\text{T}}}}}&{ - {{\tau }_{1}}}&{{{\delta }_{1}}} \\ {{{{( \bullet )}}^{{\text{T}}}}}& \bullet &{ - {{\tau }_{4}}} \end{array}} \right) < 0 \cong {{{\hat {I}}}_{1}} = \left( {\begin{array}{*{20}{c}} {{{I}_{1}}}&{u_{3}^{ + } - u_{1}^{ + }} \\ {{{{( \bullet )}}^{{\text{T}}}}}&{ - {{\tau }_{4}}} \end{array}} \right) + \frac{1}{{{{\tau }_{1}}}}\left( \begin{gathered} u_{1}^{ + } \\ {{\delta }_{1}} \\ \end{gathered} \right){{\left( \begin{gathered} u_{1}^{ + } \\ {{\delta }_{1}} \\ \end{gathered} \right)}^{{\text{T}}}} \\ = \left( {\begin{array}{*{20}{c}} {{{I}_{1}}}&{u_{3}^{ + } - u_{1}^{ + }} \\ {{{{( \bullet )}}^{{\text{T}}}}}&{ - {{\tau }_{4}}} \end{array}} \right) + \frac{1}{{{{\tau }_{1}}}}\left( {\begin{array}{*{20}{c}} {u_{1}^{ + }{{{(u_{1}^{ + })}}^{{\text{T}}}}}&{{{\delta }_{1}}u_{1}^{ + }} \\ {{{{( \bullet )}}^{{\text{T}}}}}&{\delta _{1}^{2}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{{I}_{1}} + \frac{1}{{{{\tau }_{1}}}}u_{1}^{ + }{{{(u_{1}^{ + })}}^{ \top }}}&{\frac{{{{\delta }_{1}} - {{\tau }_{1}}}}{{{{\tau }_{1}}}}u_{1}^{ + } + u_{3}^{ + }} \\ {{{{( \bullet )}}^{{\text{T}}}}}&{\frac{{\delta _{1}^{2}}}{{{{\tau }_{1}}}} - {{\tau }_{4}}} \end{array}} \right) < 0, \\ \end{gathered} $$

where, for brevity, the notation δ₁ $\mathop = \limits^\Delta$ (τ₁ – τ₃ + τ₄)/2 is introduced and the arguments of the vectors $u_{j}^{ \pm }$ = $u_{j}^{ \pm }$(τ_j) are omitted. Adopting yet another simplifying notations α₁ $\mathop = \limits^\Delta$ (δ₁ – τ₁)/τ₁ and β₁ $\mathop = \limits^\Delta$ τ₁/(τ₁τ₄ – $\delta _{1}^{2}$), we arrive at

$${{\hat {I}}_{1}} < 0 \cong {{\hat {\hat {I}}}_{1}} = {{I}_{1}} + \frac{1}{{{{\tau }_{1}}}}u_{1}^{ + }{{(u_{1}^{ + })}^{{^{{\text{T}}}}}} + {{\beta }_{1}}\left( {{{\alpha }_{1}}u_{1}^{ + } + u_{3}^{ + }} \right){{\left( {{{\alpha }_{1}}u_{1}^{ + } + u_{3}^{ + }} \right)}^{{^{{\text{T}}}}}} < 0$$

(A.3)

via use of the Schur complement.

For τ₁ = 2/$\varepsilon _{{1{\text{cir}}}}^{2}$ and τ₃ = 2/$\varepsilon _{{2{\text{cir}}}}^{2}$, the difference between the quadratic forms associated with the matrices I_cir in (5.9) and ${{\hat {\hat {I}}}_{1}}$ in (A.3) is the difference of squares

$${{\hat {\hat {I}}}_{1}} - {{I}_{{{\text{cir}}}}}\mathop {\,\, = }\limits^\Delta \,\,{{\Delta }_{1}} = {{\beta }_{1}}\left( {{{\alpha }_{1}}u_{1}^{ + } + u_{3}^{ + }} \right){{\left( {{{\alpha }_{1}}u_{1}^{ + } + u_{3}^{ + }} \right)}^{{\text{T}}}} - \frac{1}{{{{\tau }_{3}}}}u_{3}^{ + }{{(u_{3}^{ + })}^{{\text{T}}}}.$$

Inequality Δ₁ ≤ 0 for the difference of squares is valid if the corresponding (squared) linear forms are proportional; i.e., α₁$u_{1}^{ + }$ + $u_{3}^{ + }$ = λ₁$u_{3}^{ + }$, which is only possible if α₁ = 0 or τ₄ = τ₁ + τ₃. In that case, Δ₁ = 0.

We next determine the values of τ₅ which guarantee the feasibility of (5.11) provided that the MICC (5.9) is feasible. To this end, we perform the manipulations with the MI (5.11) similar to those performed above with the MI (5.10). As a result, we obtain

$${{\tilde {\bar {\bar {I}}}}_{2}} < 0 \cong {{\hat {\hat {I}}}_{2}} = {{I}_{3}} + \frac{1}{{{{\tau }_{2}}}}u_{2}^{ + }{{(u_{2}^{ + })}^{{\text{T}}}} + {{\beta }_{2}}\left( {{{\alpha }_{2}}u_{2}^{ + } + u_{3}^{ - }} \right){{\left( {{{\alpha }_{2}}u_{2}^{ + } + u_{3}^{ - }} \right)}^{{\text{T}}}} < 0,$$

(A.4)

where δ₂ $\mathop = \limits^\Delta$ (τ₂ – τ₃ + τ₅)/2, α₂ $\mathop = \limits^\Delta$ (δ₂ – τ₂)/τ₂, and β₂ $\mathop = \limits^\Delta$ τ₂/(τ₂τ₅ – $\delta _{2}^{2}$). With account for $u_{2}^{ + }$(τ) = $u_{1}^{ + }$(τ) and using the relation I₁ + $\frac{1}{{{{\tau }_{3}}}}u_{3}^{ + }{{(u_{3}^{ + })}^{{\text{T}}}}$(τ₃) = I₃ + $\frac{1}{{{{\tau }_{3}}}}u_{3}^{ - }{{(u_{3}^{ - })}^{{\text{T}}}}$(τ₃), we arrive at the conclusion that, for τ₂ = 2/$\varepsilon _{{1{\text{cir}}}}^{2}$ and τ₃ = 2/$\varepsilon _{{2{\text{cir}}}}^{2}$, the difference between the quadratic forms associated with the matrices I_cir in (5.9) and ${{\hat {\hat {I}}}_{2}}$ in (A.4) is nothing but the difference of squares

$${{\hat {\hat {I}}}_{2}} - {{I}_{{{\text{cir}}}}}\mathop = \limits^\Delta {{\Delta }_{2}} = {{\beta }_{2}}\left( {{{\alpha }_{2}}u_{2}^{ + } + u_{3}^{ - }} \right){{\left( {{{\alpha }_{2}}u_{2}^{ + } + u_{3}^{ - }} \right)}^{{\text{T}}}} - \frac{1}{{{{\tau }_{3}}}}u_{3}^{ - }{{(u_{3}^{ - })}^{{\text{T}}}}.$$

The inequality Δ₂ ≤ 0 for the difference of squares is valid if the corresponding linear forms are proportional; i.e.,

$${{\alpha }_{2}}u_{2}^{ + } + u_{3}^{ - } = {{\lambda }_{2}}u_{3}^{ - };$$

this takes place only if α₂ = 0 or τ₅ = τ₂ + τ₃. In that case Δ₂ = 0. Theorem 7 is proved.