Common diagonal solutions to the Lyapunov inequalities for interval systems
- Bengi Yıldız^{1},
- Taner Büyükköroğlu^{2}Email author and
- Vakif Dzhafarov^{2}
https://doi.org/10.1186/s13660-016-1194-x
© Yıldız et al. 2016
Received: 9 June 2016
Accepted: 3 October 2016
Published: 19 October 2016
Abstract
In this paper for interval systems we consider the problem of existence and evaluation of common diagonal solutions to the Lyapunov inequalities. For second order systems, we give necessary and sufficient conditions and exact solutions, that is, complete theoretical solutions. For third order systems, an algorithm for the evaluation of common solutions in the case of existence is given. In the general case a sufficient condition is obtained for a common diagonal solution in terms of the center and upper bound matrices of an interval family.
Keywords
1 Introduction
Diagonal stability problems have many applications (see [4–10]). The existence of diagonal type solutions is considered in [3, 5, 8, 11–14] and the references therein.
Common diagonal stability problems arise, for example, in the study of large-scale dynamic systems (see [10]).
- (1)
Full theoretical solutions of the common diagonal matrix problems for the case \(n=2\) (Section 2). Note that for second order systems the existence and evaluation of common nondiagonal matrix solutions to the Lyapunov inequalities have been considered in [15, 16].
- (2)
A numerical algorithm for the case \(n=3\) where the proposed algorithm gives almost all common diagonal solutions (Section 3).
- (3)
Sufficient condition for a common diagonal solution in the general case (Section 4).
2 Full solutions for second order systems
2.1 Hurwitz case
Theorem 1
The matrix (4) is Hurwitz diagonally stable if and only if \(a_{1}<0\), \(a_{4}<0\), and \(a_{1}a_{4}-a_{2}a_{3}>0\).
As noted above a necessary condition for the existence of a common diagonal solution is the robust diagonal stability of (5). From Theorem 1, we have the following.
Proposition 1
Assume that the family (5) is robust diagonally stable, that is, (6) is satisfied and we are looking for conditions of the existence of a common diagonal solution.
Analogously the exists an open interval \((\beta_{1},\beta_{2})\) which is the solution set of the second condition in (8) (the discriminant is positive by (6)).
Now we give the main result of this section.
Theorem 2
Example 1
Example 2
2.2 Schur case
To have a common diagonal solution a family must be robust diagonally stable.
Proposition 2
Theorem 3
- (i)
(10) is satisfied,
- (ii)
\(\gamma_{1} < \gamma_{2}\),
- (iii)
\((\alpha,\infty) \cap(\gamma_{1},\gamma_{2}) \neq \emptyset\).
Example 3
3 Solution algorithm for third order systems
In this section for \(3 \times3\) interval family, we give necessary and sufficient condition for the existence of a common diagonal solution and the corresponding solution algorithm in the Hurwitz case.
- (i)
\(a_{1}<0\),
- (ii)
\((a_{2}t+a_{4})^{2}-4a_{1}a_{5}t<0\),
- (iii)
\(d_{0}(t,a_{1},\ldots,a_{9})+d_{1}(t,a_{1},\ldots ,a_{9})s+ d_{2}(t,a_{1},\ldots,a_{9})s^{2}<0\).
We suggest the following approach to check (14) numerically. This approach is based on the openness property of the solution set of (13) [1]. In other words, the following proposition is true.
Proposition 3
If there exists a common \(D=\operatorname{diag}(t_{*},1,s_{*})\) then there exist intervals \([t_{1},t_{2}]\) and \([s_{1},s_{2}]\) which contain \(t_{*}\) and \(s_{*}\), respectively, such that the matrix \(D=\operatorname{diag}(t,1,s)\) is a common solution for all \(t \in[t_{1},t_{2}]\), \(s \in[s_{1},s_{2}]\).
Due to this proposition, we suggest the following algorithm for a common diagonal solution.
Algorithm 1
- (s1)
Using the results on \(2 \times2\) interval systems from Section 2 calculate the interval \((\alpha,\beta)\) for t.
- (s2)
Determine an upper bound s̅ for the variable s from the positive definiteness condition of a suitable submatrix of \(-(A^{T}D+DA)\).
- (s3)
Divide the interval \([\alpha, \beta]\) into k equal parts \([\alpha_{i}, \beta_{i}]\) and the interval \([0, \overline{s}]\) into m equal parts \([s_{j}^{-},s_{j}^{+}]\).
- (s4)On each boxconsider the maximization of the polynomial function \(f(t,s,a_{1},\ldots ,a_{9})\). If there exist indices \(i_{*}\) and \(j_{*}\) such that the maximum is negative then stop. The whole interval \([\alpha_{i_{*}},\beta_{i_{*}}] \times[s_{j_{*}}^{-},s_{j_{*}}^{+}]\) defines family of common diagonal solutions.$$[\alpha_{i},\beta_{i}] \times\bigl[s_{j}^{-},s_{j}^{+} \bigr] \times\bigl[a_{1}^{-},a_{1}^{+}\bigr] \times \cdots\times \bigl[a_{9}^{-},a_{9}^{+}\bigr] $$
As can be seen, the above game problem (14) is reduced to a finite number of maximization problems in which low order multivariable polynomials are maximized over boxes. Instead of a single problem (14) we consider a sequence of solvable problems from step s4) where low order multivariable polynomial functions are maximized over 11-dimensional boxes \([\alpha_{i},\beta_{i}] \times[s_{j}^{-},s_{j}^{+}] \times[a_{1}^{-},a_{1}^{+}] \times\cdots\times[a_{9}^{-},a_{9}^{+}]\). These optimizations can be carried out by MAPLE program or by the Bernstein expansion. The Bernstein expansion is an effective method for testing positivity or negativity of a multivariable polynomial over a box. The following example shows that Algorithm 1 is sufficiently effective.
Example 4
- (a)
Fix \((t,s)\).
- (b)
Solve parametric maximization of f with respect to \((q_{1},q_{2},q_{3})\). Denote \({\phi(t,s)=\max_{q_{1},q_{2},q_{3}} f}\).
- (c)
Solve minimization of \(\phi(t,s)\) with respect to \((t,s)\).
It should be noted that the sufficient condition from [3], Theorem 1, is not satisfied for this example, since for the matrix U from [3], Theorem 1, the maximum real eigenvalue is positive.
4 Sufficient condition for the general case
In this section, we give sufficient condition for a common diagonal solution in the general case. This condition is expressed in terms of the center and upper bound matrices of interval family.
Theorem 4
Proof
The left-hand side of (16) is a convex function in the entries of diagonal matrix D. For \(D=\operatorname{diag}(x_{1},x_{2},\ldots ,x_{n})\) denote the left-hand side by \(f(x_{1},x_{2},\ldots,x_{n})\). Then from Theorem 4, it follows that, by minimization of the function \(f(x_{1},x_{2},\ldots,x_{n})\) over the unit rectangle, we can arrive at a common solution.
Example 5
The minimization procedure of the convex function \(f(x_{1},\ldots,x_{5})\) evaluated by the Kelley cutting-plane method starting with the values \(x_{1}=\cdots=x_{5}=1\) after 5 steps gives negative value −0.088 for the function f. This value is attained for \(x_{1}=1\), \(x_{2}=0.247\), \(x_{3}=0.161\), \(x_{4}=0.219\), \(x_{5}=0.093\). Therefore by Theorem 4, the matrix \(D=\operatorname{diag}(1,0.247,0.161,0.219,0.093)\) is a common diagonal solution. Note again that the LMIs method cannot give a solution: the number of vertices equals 2^{25}.
5 Conclusion
In this paper, the problem of diagonal stability of interval matrices is considered. This problem is investigated in the framework of the existence of common diagonal Lyapunov functions. For second and third order systems necessary and sufficient conditions for the existence of common diagonal solutions are given. In the general case, a sufficient condition for the existence of a common diagonal solution is given.
Declarations
Acknowledgements
The authors thank the reviewer for his valuable comments. This work is supported by the Anadolu University Research Fund under Contract 1605F464.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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