ABSTRACT
In this study, the behavior of a second-order dynamical system around its equilibrium point was analyzed based on the behavior of some appropriate equipotential curves which were considered around the same equilibrium point. In fact two sets of equipotential curves were considered so that a set of the equipotential curves had a role as the upper band of the system trajectory and another set played a role as the lower band. It was shown that stability of the system around its equilibrium point can be assessed using the behavior of these two set of equipotential curves. It was shown that asymptotically stability and instability analysis of the system only need the analysis of the upper band set of the equipotential curves but oscillation behavior analysis of the system need to analyze both the lower band set of the equipotential curves and the upper band set. Also as was shown the proposed method is a mathematical method that can even detect a stable limit cycle appearing in the oscillation systems and furthermore the method has many applications such as designing of oscillators that were presented.
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URL: https://scialert.net/abstract/?doi=tasr.2012.651.662
INTRODUCTION
The stability analysis of a nonlinear system is very important and difficult problem. In fact, there are not any assumptions to start the stability analysis of a nonlinear system. A very simple nonlinear system can be unstable while a very complex nonlinear system can be stable. When a system is designed, the first important problem is to guarantee the system stability. The nonlinearity of a system results many various behaviors that makes impossibility to classify the nonlinear systems in distinguished categories and this makes very difficult to analyze the stability of a nonlinear system. For this reason, researches on the stability analysis of a nonlinear system have been always interesting. To days, new methods for stability analysis of a nonlinear system are very appealing and necessary especially for system designers (Vidyasagar, 1993; Thomsen, 2003).
There are two classic methods which are essentially used to analyze the stability of a nonlinear system (Vidyasagar, 1993; Doyle et al., 1992). The first method is stability analysis using energy function and the second method is based on the linearization the system around its equilibrium point. Sometimes the first method presented by Lyapunov is called Direct Method and the second method is called Indirect Method. The complexity of the first method is to find the appropriate energy functions to assess the stability of a nonlinear system. The second method has two major weaknesses. Firstly, if the eigenvalues of the coefficient matrix in the linear system resulted of linearization have real parts that are equal zero, the original nonlinear system may be stable or instable around the equilibrium point. Secondly, the method can only assess stability as the form of local stability around the equilibrium point (Vidyasagar, 1993).
Also two basic methods are essentially used to analyze the behavior of limit cycles appearing in nonlinear system (Vidyasagar, 1993; Doyle et al., 1992). The first method is to draw the trajectories of the system using the softwares such as MATLAB in order to detect the limit cycles of the system. It is clear that the limit cycles detected by this approach can be recognized as stable, unstable or semi-stable (Vidyasagar, 1993; Doyle et al., 1992). The second method is based on the linearization of the system around its equilibrium point or points. The second method has two major weaknesses. Firstly, the equilibrium point of the system, which the system is linearized around it, must be on the limit cycle or in a small neighborhood of it otherwise the method can not detect the real behavior of the nonlinear system. Secondly, the method can only assess the behavior of the system around the limit cycle as the form of point to point if and only if these points all locate on the limit cycle (Vidyasagar, 1993). There are some new researches done in recent years that present some mathematical methods to analyze the stability and behavior of some special nonlinear systems (Nan and Weber, 2010; Mann and Shiller, 2006; Benkler et al., 2006; Marro and Zattoni, 2002; Ntogramatzidis et al., 2007; Martinez et al., 2003; Sheheitli and Rand, 2011; Thomsen, 2003). These methods can not be applied for analyzing of general second or higher order nonlinear systems.
The proposed method mentioned in this paper is a mathematical method which is suitable to analyze the stability of second-order nonlinear autonomous systems. These kinds of systems are very important because they model the behavior of some devices such as oscillators (Doyle et al., 1992). The proposed method does not have any limitations and it can even detect a stable limit cycle appearing in the oscillation systems and furthermore the method has many applications such as designing of oscillators.
EQUIPOTENTIAL CURVES
Consider the second-order autonomous system described by the following equation:
(1) |
Definition 1: A set is said compact if it is bounded and closed (Vidyasagar, 1993).
Definition 2: Consider the set called P so that P⊂R2, P is said invariant set if the trajectories of the system beginning in the P remain in it as t→∞ (Vidyasagar, 1993).
Definition 3: Suppose that the X = 0 is the equilibrium point of the second-order autonomous system described by the Eq. 1 and suppose that the compact set called M includes the equilibrium point (the origin). The closed curves belonging the M, which is described by u(x1, x2) = C so that CεR and enclosing the equilibrium point, are called equipotential curves because for each value of C there is a closed curve with the potential of C, so all points locating on the u(x1, x2) = C have the equal potential the numerical quantity of which is C.
STABILITY ANALYSIS
Theorem 1: The second-order autonomous system described by Eq. 1 is local asymptotic stable around the equilibrium point (X = 0) if there are equipotential curves u(x1, x2) = C with clockwise direction, enclosing the equilibrium point and further on the trajectories of the system 1:
(2) |
Proof: From u(x1, x2) = C, we have:
(3) |
and as a result, the dynamic of u(x1, x2) = C can be expressed as:
(4) |
where, and are the state variables of the dynamic of u(x1, x2) = C. The velocity vector on the u(x1, x2) = C symbolized by is define as , so from Eq. 4 we found that:
(5) |
where, and are respectively the unity vectors of the x1 axis and x2 axis. Also, the velocity vector of the system 1 is defined as , so it can be written as:
(6) |
The derivative:
on the trajectories of the system 1 can be expressed as:
(7) |
or:
(8) |
where:
Fig. 1: | Trajectories of an asymptotic stable system |
and α is the angle between and . So the inequality 2 can be written as:
(9) |
and this means that the direction of the trajectories of the system 1 are to inside of the equipotential curves u(x1, x2) = C as shown in Fig. 1, on the other hand CεR and C can be changed so that closed curves u(x1, x2) = C enclosing the equilibrium point could tend to be smaller and smaller and finally approach to the equilibrium point (the origin). This means that the direction of the trajectory of the system 1 will tend to the origin, so the system 1 is asymptotically stable.
Theorem 2: The second-order autonomous system described by Eq. 1 is unstable around the equilibrium point (X = 0) if there are equipotential curves u(x1, x2) = C with clockwise direction, enclosing the equilibrium point and further on the trajectories of the system 1:
(10) |
Proof: It follows from the proof of the theorem 1 that inequality 10 means that:
(11) |
and in the similar manner with the proof of the theorem 1, the direction of the trajectories of the system 1 are to outside of the equipotential curves u(x1, x2) = C as shown in Fig. 2, so the trajectories tend to infinity (far and farther of the equilibrium point) and this means that the system around the equilibrium point (X = 0) is unstable.
Definition 4: A limit cycle is said asymptotic stable if all trajectories in vicinity of the limit cycle converge to it as t→∞. Otherwise the limit is semi-stable or unstable (Doyle et al., 1992).
Theorem 3: Consider the second-order autonomous system 1, suppose that no equilibrium point belongs to the compact set M which encloses the origin (X = 0). There are equipotential curves u(x1, x2) = C1 and u(x1, x2) = C2 with clockwise directions that belong to M, does not intersect one with another, enclose the origin and satisfy the following inequalities on the trajectories of the system 1:
Fig. 2: | Trajectories of an unstable system |
(12) |
(13) |
if and only if there exists an asymptotic stable limit cycle L so that:
(14) |
where, Ω(C1, C2) is the region located between u1(x1, x2) = C1 and u2(x1, x2) = C2.
Proof: (⇒) Similar to the theorem 1, the inequality 12 can be rewritten as:
(15) |
This means that the direction of the trajectories of the system 1 is to the outside of the equipotential curves u1(x1, x2) = C1 as shown in Fig. 3. In the similar manner the inequality (13) can be expressed as:
(16) |
where, is the velocity vector on the u2(x1, x2) = C2 and this means that the direction of the trajectories of the system 1 is to the inside of the equipotential curves u1(x1, x2) = C2 as shown in Fig. 3. On the other hand there is no equilibrium points belonging to M and consequently to Ω(C1, C2), so there is an asymptotic stable limit cycle L so that L⊂intΩ.
(⇒) the necessary condition can similarly be proofed using above geometric concepts.
Remark 1: If on the trajectories of the system 1:
or:
the equipotential curve u1(x1, x2) = C1 or u2(x1, x2) = C2 itself is the limit cycle respectively.
Fig. 3: | A stable limit cycle appearing in the system |
Example 1: Consider the following system:
By choosing:
for C<1, it can be seen that not only the equipotential curves u(x1, x2) = C are closed but also on the trajectories of the system:
so the system is asymptotic stable.
Example 2: Consider the following system:
By choosing:
for all value of the C, it can be seen that not only the equipotential curves u1(x1, x2) = C are closed but also on the trajectories of the system:
so the system is unstable.
Example 3: Consider the following system
By choosing:
for 0<C1<2.5, it can be seen that not only the equipotential curves u1 (x1, x2) = C1 are closed but also we have:
Also by choosing:
for 2.5<C2, it can be seen that not only the equipotential curves u2 (x1, x2) = C2 are closed but also:
so there is an asymptotic stable limit cycle located between the u1 (x1, x2) = C1 and u2 (x1, x2) = C2. The area located between the u1 (x2, x2) = C1 and u2 (x1, x2) = C2 is an invariant set as the following set:
(17) |
where, 0<C1<2.5<C2. It is clear that the limit cycle can be estimated by varying the C1 and C2 in (17). In above set by increasing C1 and decreasing C2, the limit cycle can be found as:
FIRST APPLICATION-CONTROL OF LIMIT CYCLES
Consider the following nonlinear autonomous system:
(18) |
where, and are the control inputs as the form of state feedback presented by the following equations.
(19) |
Now, the question is that how h1(x1, x2) and h2(x1, x2) must be chosen so that an asymptotic stable limit cycle can be added to the system 18 The condition:
on the trajectories of the system 18 in the theorem 3 can be rewritten as following inequality:
(20) |
and in the similar manner the:
appeared in the theorem 3, can be expressed as:
(21) |
The inequalities 20 and 21 give the conditions which have to be satisfied by u*1, u*2, u1(x1, x2) and u2(x1, x2) in order to appear an asymptotic limit cycle in the system 18.
Example 4: Consider the following system:
(22) |
It is clear that the equilibrium point at the origin is asymptotic stable. Now, the state feedback lows (u1* and u2*) have to be determined so that an asymptotic stable limit cycle can be added to the resulted closed loop system. By choosing equipotential curves as:
(23) |
(24) |
and replacing Eq. 23-24 in inequalities 20-21, respectively the following inequalities are found:
(25) |
and
(26) |
It can be derived from inequalities 25-26 that:
(27) |
and
(28) |
By choosing the state feedback laws as the following forms:
(29) |
and by replacing Eq. 29 in inequalities 27-28, the following inequalities are found as the conditions to appear an asymptotic limit cycle in the system:
(30) |
and
(31) |
The inequalities 30-31 both are satisfied, when:
(32) |
It also follows from the theorem 3 that the asymptotic stable limit cycle L, which is added to the system 18 using state feedback, appears in the following region:
(33) |
SECOND APPLICATION-DESIGN OF OSCILLATION IN ELECTRONIC CIRCUITS
Consider the basic model of an oscillator shown in Fig. 4 including LC tank and dependent current source. The state equations of the circuit can be written as the following autonomous system:
Fig. 4: | Basic circuit of an oscillator including LC tank |
(34) |
where, x1 and x2 are defined as the voltage which appears across the capacitor and the current of the inductor respectively. As we see the dependent current source plays the role of the control input. By considering the Eq. 19 and defining the state feedback as the following equation:
(35) |
the Eq. 34 can be rewritten as:
(36) |
where, ||•|| ∞is infinite norm and f(·) is determined by the electronic elements such as BJT, MOSFET and used to design the oscillator. Now, the equipotential curves u1(x1, x2) = C1 and u2(x1, x2) = C2 are considered as the circles surrounding the origin and described by the following equations:
(37) |
(38) |
where, C1≤C2. By checking inequalities 12-13 of the theorem 3, we have:
(39) |
(40) |
It is clear that above inequalities can not both be satisfied unless:
(41) |
This is the necessary and sufficient condition to appear oscillation in the circuit.
Suppose that the oscillation appearing in the circuit is as the form of sinusoidal wave, in other word the voltage appearing across the capacitor of the tank is expressed by the following equation:
(42) |
It is clear that:
(43) |
and
(44) |
So the Eq. 41 can be rewritten as:
(45) |
CONCLUSION
In this study, the behavior of a second-order nonlinear dynamical system around its equilibrium point was analyzed based on the behavior of some appropriate equipotential curves which were considered around the same equilibrium point. In fact two sets of equipotential curves were considered so that a set of the equipotential curves had a role as the upper band of the system trajectory and another set played a role as the lower band. It was shown that stability of the system around its equilibrium point can be assessed using the behavior of these two set of equipotential curves. It was shown that the proposed mathematical method can detect a stable limit cycle appearing in the systems and also the method has some useful applications that described in section 4 and 5. For future works the method can be extended for analyzing of third order dynamical systems.
REFERENCES
- Benkler, S., N. Chavannes and N. Kuster, 2006. A new 3-D conformal PEC FDTD scheme with user-defined geometric precision and derived stability criterion. IEEE Trans. Antennas Propagation, 54: 1843-1849.
CrossRef - Martinez, S., J. Cortes and F. Bullo, 2003. Analysis and design of oscillatory control systems. IEEE Trans. Automatic Control, 48: 1164-1177.
CrossRef