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Symmetry 2019, 11(4), 569; https://doi.org/10.3390/sym11040569
Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems
Institute of Applied Informatics, Automation and Mechatronics, Slovak University of Technology in Bratislava, Bottova 25, 917 01 Trnava, Slovakia
Received: 5 March 2019 / Accepted: 13 April 2019 / Published: 19 April 2019
In this paper we have established the sufficient conditions for asymptotic convergence of all solutions of nonlinear dynamical system (with potentially unknown and unbounded external disturbances) to zero with time We showed here that the symmetric part of linear part of nonlinear nominal system, or, to be more precise, its time-dependent eigenvalues, play important role in assessment of the robustness of systems.
Keywords:nonlinear system; perturbation; robustness; variation of constants formula; symmetric part of linear operator
1. Notations, Motivation and Introduction
Our purpose here is to prove a new result regarding the convergence of all solutions to the origin as for a nonlinear dynamical system subject to external disturbances,given that 0 is a solution for the nominal system and that f and satisfy certain conditions.
Let denote n-dimensional vector space endowed by the Euclidean norm and be an induced norm for matrices, We always assume that the function f is continuously differentiable, for all that perturbation is at least continuous both from to and it is not assumed that the zero function is a solution of (1). The nonlinear term aggregates all external disturbances which affect the state variable of the nominal system Let us denote by the linear part of nominal vector field at that is, where denotes the Jacobian matrix of f with respect to variable x and let us assume that for all and all where is the Taylor remainder. We also assume that the solutions of (1) are uniquely determined by for all Throughout the whole paper, the superscript “T” indicates the transpose operator.
1.2. Motivation and Introduction
For a motivation, let us consider the case when linear part of the nominal system is asymptotically stable. What can we say about the asymptotic behavior (as ) of solutions of perturbed system (1)? This question represents one of the fundamental problems in the area of robust stability and robustness of the systems in general and so the effect of (known or unknown) perturbations on the solutions of nominal system as a potential source of instability attracts the attention and interest of scientific community for a long time in the various contexts, recently for example [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. A comprehensive overview of the most significant results on robust control theory as a stand-alone subfield of control theory and its history is presented in [17,18].
On the other side, the analysis of the robustness of uncontrolled systems often merges with the mathematical theory of dynamical systems. As is traditional in perturbation theory of linear and nonlinear dynamical systems, the behavior of solutions of a perturbed system is determined in terms of the behavior of solutions of an unperturbed system [19,20,21,22,23]. In principle, to answer the question regarding behavior of the solutions of perturbed systems as , it usually makes a difference whether the origin remains an equilibrium for the perturbed system or not. If , then the origin is an equilibrium point of (1). In this case, we can analyze the stability property of the origin as an equilibrium point of the perturbed system. It is worth mentioning in this context the well-known Demidovich condition  on asymptotic stability of all solutions of a nonlinear system stating that if for some positive definite matrix the matrixis negative definite uniformly in then for any two solutions and is for all and some independent on and constants but this condition is not-very-well suited for reasoning about the convergence of all solutions to zero as if and so we can not set In this case, that is, if , the origin is no longer an equilibrium point of (1), and we usually analyze the ultimate boundedness of solutions of perturbed system. As have been shown in ( Chapter 9), if the point is an exponentially stable equilibrium point of the nominal system and perturbation term satisfieswhere are continuous, and is bounded, then for the origin is an exponentially stable equilibrium point of the perturbed system and the solutions of the perturbed system are ultimately bounded in the opposite case (if is not identically zero). These analyses are close and compatible with the concept input-to-state stability which has been introduced by E. Sontag [26,27]. In contrast to the case of exponential stability, the unperturbed system with uniformly asymptotically stable (but not exponentially stable) zero solution is not robust even for continuous perturbations with arbitrarily small linear bounds and see  for more details. The definitions of various types of stability for non-autonomous systems mentioned here can be found, for example, in ( Chapter 4). There are two useful and principally different methods for studying the qualitative behavior of the solutions of the perturbed nonlinear system: the second method of Lyapunov [25,28,29,30], and the use of a variation of constants formula [19,20,21,31]. The present paper is based on the second approach in combination with the eigenvalue techniques to prove, in Theorem 1 with the help of Lemma 1, the new sufficient conditions for global robustness of nonlinear (uncontrolled) systems, whose linear part is asymptotically stable and in general time-varying; the notion “global robustness” is meant in the sense of convergence of all solutions of system (1) to the origin as provided the perturbing term satisfies some growth constraints. At this point, we would also like to emphasize that we achieve our results without a priori assumption on the boundedness of perturbation More specifically, in (3) may not be bounded for and so our ambition here is to partially complement the mosaic of asymptotic behavior of the solutions of dynamical systems. The theory is illustrated by two nontrivial examples, Example 1 and Example 2.
In this section, we formulate the main result on the asymptotic behavior of solutions for (1) as
2.1. Auxiliary Lemma
The purpose of this subsection is to present the lemma upon which subsequent result will be based. The core part of the proof of the later result is contained in the proof of this lemma.
Let be a fundamental matrix of Denote the largest and smallest pointwise eigenvalues of by and Then
First note that the integrals in (4) are well defined since the eigenvalues of a matrix are continuous functions of the entries of the matrix, and because the entries of are continuous functions of time, the frozen-time eigenvalues and are also continuous functions of Suppose is a solution of corresponding to a given and nonzero Usingthe Rayleigh–Ritz inequality  gives
Dividing through by which is positive at each and integrating from to any yields
Exponentiation followed by taking the nonnegative square root gives
Now, given any and let be such that
Note that such exists from the definition of induced norm for matrices. Then the initial state yields a solution of that at time satisfies
In the last inequality, we used the right inequality from (5) for and the fact that Since and the last inequality gives that
Because such can be selected for any and the proof of the right inequality in (4) is complete. The other half of the theorem follows by analogous arguments. □
Taking into consideration the fact that linear system is uniformly asymptotically stable if and only if for some positive constants K and , we have the following
If for all then the linear system is uniformly asymptotically stable (⇔ uniformly exponentially stable).
2.2. Main Result
Let us consider the system (1), Assume that
- in some left neighborhood of where is the largest pointwise eigenvalue of
- for all is where function is continuous on and satisfies
Then all solutions of (1) converge to 0 as
The effect of the perturbation on the solutions of a system of nonlinear differential equations can be studied using the variation of constants formula identifying the Taylor remainder of f together with as an inhomogeneous term Thus we can rewrite the system (1) aswhere and Then for allwhere is a fundamental matrix of representing the linear part of nominal system Thus
Since by Assumption A2 is for all Lemma 1 yields the inequality
Due to Assumption A1, the first term on the right-hand side, namely the linear homogeneous response to the initial states, decays exponentially fast to zero as It remains to analyze the second term on the right-hand side of the last inequality, the estimate of response to nonlinear term We getand the L’Hospital rule yields
This result together with Assumption A3 gives the claim of Theorem 1 regarding the robustness of the system under consideration. □
Recall that for unperturbed linear time-varying system Assumption A1 of Theorem 1 extended on implies Demidovich condition (2) with P equal to the unit matrix, and taking into account that the eigenvalues of the symmetric matrix are uniformly negative; however, for the perturbed linear time-varying systems, the uniform negative definiteness of the matrix is difficult to verify, unlike Assumption A3, taking into account that Assumption A2 is reduced to
Let a constant matrix.
- When A is negative definite, then Assumption A1 of Theorem 1 is automatically satisfied because is also negative definite ( Corollary 14.2.7), and
- in connection with Assumption A3, it is worth noting that Assumption A3 reduces to as ensuring the vanishing at infinity of all solutions of perturbed system cf. Example 2 below, where non-constant allows convergence to zero of all solutions of perturbed system for a wider class of perturbations, where even unbounded perturbations are admissible.
3. Simulation Experiments in MATLAB®
To illustrate Theorem 1 with an example, let us consider the systemwith yet unspecified δ and parameter Obviously, in this specific example, and Assumptions A1–A3 are satisfied if and if that is, as
The condition in Example 1 is only sufficient condition for convergence to zero of all solutions. Thus, the solutions can theoretically converge to zero also for the perturbations that do not satisfy that constraint. In fact, the fundamental matrix of satisfiesand the solutions for some specific perturbations δ depending only on t can be calculated explicitly by using (6). For example, if is constant, the solution of (7) with and the initial stateas On the other side, for another bounded perturbationthe explicit solution forbecause
This example showed that the coupled conditions on the system and perturbation in Theorem 1 cannot be weakened too much. Note, that the sine and cosine functions contained in the fundamental matrix are responsible for the "wavy" shape of the solutions of (7).
The following example illustrates the possibility that the system with time-varying linear part of nominal system can be robust also in the case of unbounded external disturbances affecting the system.
Let us consider the linear time-varying systemwhere b is a real parameter. Stability analysis for time-varying linear systems and their robustness is of growing interest in the control community. One of the reasons for both researchers and practitioners is the growing importance of adaptive controllers for which underlying feedback closed-loop adaptive control system is time-varying and linear [35,36].
The eigenvalues of areand so Asumption A1 of Theorem 1 is satisfied for any fixed value of .
In this paper we have established in terms of eigenvalues of symmetric part of linear part of the nominal vector field the sufficient conditions to maintain the origin “attractive” for a wide class of perturbed nonlinear systems where the term aggregates all external disturbances affecting the system. As a result, we proved a new criterion for assessment of global robustness of nonlinear systems in the sense that all solutions of the perturbed system converge to zero as as long as the perturbing term satisfies certain constraints given in Theorem 1.
This publication was created thanks to the support of the Ministry of Education, Science, Research and Sport of the Slovak Republic in the framework of the call for subsidy for the development project No. 002STU-2-1/2018 with the title “STU as the Leader of the Digital Coalition”.
The author thanks the editors and the anonymous reviewers for their constructive comments and suggestions which improved the quality of this paper.
Conflicts of Interest
The author declares no conflict of interest.
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Figure 1. Solution of (7) for and initial state Obviously, as
Figure 2. Solution of (8) for and initial state Obviously, as
Figure 3. The solution component of (8) for initial state and with (a) satisfying Assumption 3 of Theorem 1, (b) that does not satisfy Assumption 3 of Theorem 1 and (c) the borderline case, .
Listing 1. MATLAB ® code for Figure 1 (The MathWorks, Inc., 3 Apple Hill Drive, Natick, Massachusetts 01760 USA).
f = @(t,x) [(-2)*x(1)+(exp(t))*x(2)+(atan(x(1)+x(2))/(t+1));
[t,xa] = ode45(f,[0 4],[50 -20]);
pbaspect([2 1 1])
plot(t,xa(:,1), ’k’, ’LineWidth’,1.5) % 1 or 2
ylabel(’x_1’) % 1 or 2
print(’example_first_x_1’,’-deps’) % 1 or 2
Listing 2. MATLAB® code for Figure 2.
f = @(t,x) [(-t^2+sin(t))*x(1)+(b)*x(2)+t^(1.5);
[t,xa] = ode45(f,[0 4],[10 -5]);
pbaspect([2 1 1])
plot(t,xa(:,1), ’k’, ’LineWidth’,1.5) % 1 or 2
ylabel(’x_1’) % 1 or 2
print(’example_second_x_1’,’-deps’) % 1 or 2
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