1. Introduction
The stability analysis of dynamical systems and its applications to control theory have attracted many researchers [
1,
2,
3,
4,
5,
6,
7,
8]. The contribution to some classical problems in Lyapunov stability, LQ control and controllability has given in eguation [
2] by simply exploring theorems of alternative for linear time invariant systems and the determination of qualitative properties for the possible solutions corresponding to linear time invariant systems.
For linear time invariant systems, the computation of the spectrum of the system matrix gives sufficient information for stability conditions. However, the computation of spectrum of system matrix for linear time varying systems may not determine it’s stability and instability. The exponential stability of linear time varying systems with
assumed to vary slowly [
9,
10]. The necessary and sufficient conditions for exponential stability of such systems are studied in [
1,
7,
11,
12,
13].
In 1990, Lur’e, Postinkov and others in the Soviet Union applied lyapunov theory and had established methods based on Lyapunov theory to solve practical problems arising in control engineering, particularly the problems of stability of control system having non-linearity [
14]. The control system was studied with the help of matrix inequality and the stability of control system was studied with linear matrix inequality techniques. These linear matrix inequalities were reduced into a class of polynomial inequalities and then solved by hand for a small control system. For a complete detail we refer on linear matrix inequalities [
15] and the references therein.
In the early 1960’s the next major breakthrough came when Yakuborich, Popov, Kalman and other researchers reduced the linear matrix inequalities in Lure’s theory for simple graphical criterion by using positive real (PR) lemma [
14]. The graphical criterion resulted in Popov criterion, Tsypkin criterion and circle criterion are useful to apply on higher order systems in control.
By 1970’s several researchers knew methods to solve linear matrix inequalities and all of these methods are analytic or closed form solutions to solve linear matrix inequalities.
In 1976’s paper, both Horisberger and Belarber [
16] had showed the existence of the quadratic Lyapunov functions which uses linear matrix inequalities to check control systems stability analysis. The idea of making use of computer search for Lyapunov function appears in the paper of Schultz et al. [
17].
The linear matrix inequalities acts an important tool to find the quadratic Lyapunov function. However, the standard interior point methods may become not much effective with the increase of modes. An interactive gradient descent algorithm is proposed in [
18] which converges to quadratic Lyapunov function in finite steps. However, the convergence rate of gradient descent algorithm proposed in [
18] could be imposed by introducing some randomness. The existence of Lyapunov function is the sufficient condition for the stability of linear time invariant system [
19].
A various number of problems appearing in systems and control are reduceable to standard convex and quasi convex problems involving linear matrix inequalities [
20]. These linear matrix inequalities problems possesses analytical solution up to a few special cases but fortunately such problems are solveable with existing numerical techniques. These inequalities appears in the form of Lyapunov or algebraic Riccati inequalities which signify the computational cost of control theory based on the top of solutions of algebraic Riccati equations to a theory based on the solution of Lyapunov inequalities.
Overview of the Article
Section 2 provides the preliminaries of our article. In particular, we present the definitions of positive definite and positive semi definite matrices, negative definite and negative semi definite matrices, matrix inequality and linear matrix inequality.
In
Section 3 we provide an equivalent linear time invariant system with the bounded perturbation. Furthermore, in this section we discuss the quadratic stability of linear time invariant system. In
Section 4 of this article, we present a gradient system of ordinary differential equations to relocate the smallest eigenvalue
from the spectrum of the perturbed matrix
with
.
Section 5 of our article is devoted on the localization of eigenvalues
from the spectrum of perturbed matrix. Furthermore an optimization problem is formulated and solved with the help of a gradient system of ordinary differential equations to discuss the quadratic stability.
In
Section 6 we give the conclusion of our article.
2. Preliminaries
Definition 1. A symmetric matrix is called a positive definite matrix if
Definition 2. A symmetric matrix is called positive semi definite matrix if
Definition 3. A symmetric matrix is called negative definite matrix if
Definition 4. A symmetric matrix is called negative semi-definite matrix if
Definition 5. A matrix inequality in is defined as , with
Definition 6. A linear matrix inequality in the variable is defined as , with
Definition 7. The solution of dynamical system is stable if for such that all solutions satisfies
Definition 8. The solution of dynamical system is asymptotically stable (weak) if it is stable and as and is small enough.
Definition 9. The dynamical system is asymptotically stable (strong) if as and is small enough.
Definition 10. A square matrix is stable if for all i and denotes the spectrum of the matrix
Definition 11. A linear time invariant system with and is called asymptotically stale if the matrix M is a stability matrix.
Definition 12. A Lyapunov function or generalized energy to a system is a scalar valued function satisfying:
- (i)
a scalar function
- (ii)
positive definite
- (iii)
dissipativity.
3. Linear Time Invariant System with Bounded Perturbation
Consider a linear time invariant system with a non-linear bounded perturbation
as,
In Equation (
1), the perturbation
is a time varying matrix so that the vector 2-norm of output vector
p is bounded above by vector 2-norm of input vector
q. If the 2-norm of perturbation matrix is unify, then both
p and
q are equal. In such case, we have following equivalent system to linear time invariant system described in Equation (
1).
3.1. Equivalent System
The system equivalent to linear time invariant system in Equation (
1) is of the form
Next, we discuss the quadratic stability of linear time invariant system described in Equation (
2).
3.2. Quadratic Stability
The linear time invariant system in Equation (
2) is quadratically stable if there exists a positive definite matrix
P such that
and satisfying the matrix inequality
In Equation (
3), the expression
is negative if
Unfortunately it’s highly possible that
In turn, this implies that the perturbed matrix
Furthermore, all the eigenvalues of the matrix
are negative, that is,
Next, we perturbed the spectrum of by using an ordinary differential equation so that
5. A System of ODE’s to Shift ,
In this section, we aim to shift simultaneously eigen values and from the spectrum of the perturbed matrix such that , Here, and The matrix D is a diagonal matrix such that possesses unit entries along it’s main diagonal. The matrix is such that it has all zero entries along it’s main diagonal.
5.1. Optimization Problem
The following optimization problem allow us to compute the direction
such that the solution of the system of ODE’s obtained after solving optimization problem gives maximum growth to increase
and
The optimization problem to increase both eigen values
and
is
Next, we give the solution to optimization problem in Equation (
11).
5.2. System of ODE’s
The solution to optimization in Equation (
11) is given by the system of ODE’s
The system of ODE’s in Equation (
12) can be written as a function
, where
For
sufficiently large enough, we have
with
For
, we have that
We fix
and for
, we have
The perturbed matrix
has the form
Since,
Thus the matrix
has the structure
The matrix
can be decomposed into the upper triangular matrix
and the lower triangle matrix
as,
Similarly, the matrix
has the structure
and
Remark 1. The Frobenius norm of and are exactly same, that is, To minimize the eigen values
and
we compute matrix
while taking projection of
Z onto
. This gives following result for
The matrices
and
are obtained as
The computation of
gives the increase of
in the following optimization problem,
The solution of optimization problem in Equation (
15) is given by ODE
as,
with
and is given by
The above matrix can be decomposed into the upper triangular matrix
and the lower triangular matrix
as
Remark 2. The matrix Finally, the solution of the optimization problem in Equation (11) takes the form The solution
in Equation (
16) is given by Euler’s method
Thus, finally we compute all positive eigen values from the eigen value problem
5.3. Stability of Equivalent System
In this section, the aim is to show that the equivalent system in Equation (
2) is stable and furthermore, the linear time invariant system in Equation (
1) is also stable.
Since,
This implies that the equivalent system in Equation (
2) is stable if ∃ a positive definite matrix
satisfying the linear matrix inequality