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Stability Analysis for Autonomous Dynamical Switched Systems through Nonconventional Lyapunov Functions
The stability of autonomous dynamical switched systems is analyzed by means of multiple Lyapunov functions. The stability theorems given in this paper have finite number of conditions to check. It is shown that linear functions can be used as Lyapunov functions. An example of an exponentially asymptotically stable switched system formed by four unstable systems is also given.
The switched system is a special class of hybrid or variable structure systems [1, 3, 6–10]. Similar to variable structure systems, the dynamics of switched systems is described by different differential equations in different space regions and the change of dynamics occurs when the trajectories pass through the boundaries between two regions. Variable structure systems may have special type of solutions, the so-called sliding mode solutions. The theory and different applications of sliding mode solutions in control are investigated in many books and papers (see, e.g., ). In contrast to sliding mode theory, the case when the variable structure system has no sliding mode solution is not sufficiently studied. To separate these two cases, we call variable structure systems without sliding mode solutions as switched systems. In the theory of hybrid systems the change of systems dynamics may occur by action of automata or by other reasons. In this paper, the stability of switched systems is studied. The problem of stability of switched systems is not simple. First of all, we give some examples illustrating different aspects of this problem and showing that stability of all subsystems is not sufficient to ensure the stability of the whole switched system. Namely, in one example the whole switched system formed by stable subsystems is unstable or asymptotically stable depending on the structure of the switched systems, that is, depending on the regions where these subsystems are acting. If some of subsystems are unstable and others are asymptotically stable, the switched system may be unstable or asymptotically stable or it may have periodic solutions. Also, in some cases the stability of the whole switched system depends neither on the structure of the switched system nor on the order of the switching.
The papers [11, 12] were dedicated essentially to the case of linear systems. In paper  the stability of switched and hybrid systems is investigated. However, for each trajectory, the theorem on stability from  imposes certain conditions on Lyapunov functions at the moment of passing through the switching lines. These conditions are possible to check only if the trajectory is known. Furthermore, typical trajectory of switched systems has an infinite number of switchings and it would be necessary to check an infinite number of conditions. Our previous works [14, 15] investigate some stability problems in switched systems and also investigate possibilities of appearance of chaotic solutions in such systems.
In present paper, some stability theorems using multiple Lyapunov functions are established. In contrast to  our conditions are imposed on values of Lyapunov functions in corresponding space regions and on switching lines, allowing in this way obtaining different stability conditions without knowing trajectories, while only a finite number of conditions depending only on Lyapunov functions are verified. The Lyapunov functions used in these theorems may be different from usual Lyapunov functions defined in the whole space, which allows extending the class of functions used as Lyapunov functions. For example, it is possible to use linear Lyapunov functions to investigate the stability of switched systems. The sum of quadratic and linear functions may be used also as Lyapunov functions. Some examples of such functions are also given. The using of unusual Lyapunov functions simplifies the search for convenient Lyapunov function. We present also one example (Example 15) where all subsystems are unstable by considering the whole space, but the switched system formed by these subsystems is exponentially asymptotically stable.
Our theorems which give conditions of asymptotic stability or instability of whole switched system in case when all subsystems are only stable are especially interesting. These results may be considered as a special type of parametric excitation or parametric stabilization of stable systems.
2. Description of Autonomous Two-Dimensional Switched Systems
Suppose the phase space , , with norm , is divided into a finite number of open 1-connected regions , , with smooth boundaries such that the origin of rectangular coordinate systems pertains to closure of any region . The boundary between two regions and is noted by and is called switching line. These switching lines are supposed to be smooth and let the normal to the switching lines existing in each point have direction from to . Suppose each region has points with norms such that , where is any number. The topological properties of plane conduce to the conclusion that each region has only two boundaries which go from origin to infinity without intersections.
In each region the dynamics of switched systems is described by a proper autonomous equation with Lipschitz continuous function :with initial condition
The trajectory of switched system (1)-(2) may pass from region to region only crossing the switching line . Suppose also the nonexistence of sliding modes in switched system (1)-(2). The sufficient condition for absence of sliding modes is the following transversality condition: the normal which goes from to and the trajectory velocities and from one to another side of switching line form acute angles; that is,for every point on switching line and all lines , (Figure 1). Transversality condition (3) guarantees the passage only from region to and not in opposite direction.
If at instant the trajectory of switched system arrives from region on the switching line , the trajectory passes across the switching line and at the following instants the system dynamic is described by equation acting in region with initial condition for the new equation coinciding with the final condition of the previous equation on the switching line , for ; that is, . Under this condition, all trajectories of switched system will be continuous for all . The regions together with equations completely define the switched system. The set of regions defines the geometrical structure while the set of equations defines the dynamical structure of switched system.
Any switched system for a chosen initial condition generates a sequence of continuous dynamical subsystems acting in regions , , and switching on lines ; that is,
The initial condition defines the first element in (4). Depending on , the first subsystem is acting from initial time to the first switching instant in region such that . The second subsystem is acting in from time to the second switching instant and so on. In other words, each subsystem is defined by three elements :(a)region where this subsystem is acting,(b)differential equation acting in ,(c)initial values and .
This definition is slightly different from definition given in [1, 3, 13]. The sequence (4) may contain finite or infinite number of dynamical subsystems although the total number of different regions is finite because trajectory can return to region after the whole rotation around origin, and therefore for all . If condition (3) holds for all switching lines, then the sequence (4) may contain an infinite number of subsystems . In case of infinite number of subsystems , in sequence (4) each region has two boundary switching lines: by one of them the trajectories enter from the precedent region , and by the other, after a finite time, the trajectories go out from region to region . After the whole rotation around origin the switched system returns to initial region and .
This property has, as a consequence, the following fundamental property of autonomous two-dimensional switched systems: the order of terms in sequence (4) does not depend on initial conditions.
In multidimensional phase space , , each region may have more than two switching surfaces. For that reason, there does not exist an analogy of announced fundamental property in multidimensional space. So, in the order of terms in (4) may depend on initial conditions. This complicates the stability investigation of switched systems in multidimensional space , .
The number of dynamical subsystems is finite, , if the switched system stays in the final region for all time after last switching , . The switched system may have more than one final region and the final region may depend on initial conditions . The sufficient condition for finiteness of sequence (4) is the existence of at least one line such that from one side of the line the normal and the trajectory velocities and form an obtuse and acute angle, respectively:for .
In this case, the subsystem will be the last subsystem in the sequence (4). If there exist more than one switching line satisfying condition (5) then the switched system may have more than one final region and the final region may depend on initial conditions . Let us consider two examples.
Example 1. Suppose the switched system is described by two pendulum equations and with different natural frequencies acting in regions and of phase plane :while equation isEvery trajectory of switched system has an infinite number of switchings and it is represented by an infinite number of continuous dynamical subsystems .
The subsystems are described by the same equation in the form of (6) acting in region and the subsystems are described by equation in the form of (7) acting in region . But the initial conditions for and are different: for , for , and so on. Therefore all subsystems are different (see Figure 2).
Example 2. Consider now the switched system with acting in the region , acting in the lower half-space , and acting in withThe equation is overdamped and its general solution is The trajectories of the whole switched system are presented in Figure 3 with . The region denoted as in Figure 3 is a final region of the switched system (8)–(10). Depending on initial conditions, the trajectory may has two, one, or zero commutations. If belongs to region then the trajectory has two commutations, if belongs to region then the trajectory has one commutation, and if belongs to region then the trajectory has no commutations.
3. Stability of Switched Systems
Let us introduce some definitions of Lyapunov stability.
Clearly, the stability (asymptotic stability) is uniform with respect to because the switched system is a stationary system with all elements independent of .
Obviously, in case of finite sequence (4) the stability or instability of the whole switched system origin depends only on stability of instability of origin for final subsystem .
In case of infinite sequence (4) the instability of one equation does not conduce automatically to the instability of the whole switched system. Also, the stability of all equations is not sufficient to conclude that the whole switched system is stable .
Let us give other corresponding examples.
Example 5. Consider the switched system with two equations and acting in two half-planes and , respectively, withSwitching lines and in this case are and . Equation is unstable and is asymptotically stable. The asymptotic stability of equation can be called stronger than the instability of . It is easy to verify that for any the subsequent points, , , of intersection with the axis are equal toTherefore, the trivial solution of switched system (13)-(14) is asymptotically stable.
Replace (14) in region byNow, the same calculations show that for all . Therefore all solutions of switched system formed by unstable equation (13) and by asymptotically stable equation (16) are periodic.
Example 6. Consider switched system of Example 1:Both equations and are stable. Also, it is easy to calculate for any the values of , and thatTherefore, if then the switched system origin is asymptotically stable, but if the switched system origin is unstable. Suppose now coincides with the first quadrant; that is, ; then analogous calculations show that the switched system origin is asymptotically stable if , but if the switched system origin is unstable.
Remark 7. These two examples show that stability of switched systems depends not only on the stability of equations but also on all other elements which define the switched system; that is, it depends also on the regions where different equations are acting and on the order of the switching.
Thus, the following definitions are justified.
Definition 8. The origin of switched system (4) formed by an infinite sequence of subsystems is stable (asymptotically stable) independently of geometrical structure of switched system if the origin is stable (asymptotically stable) for any choice of equations acting in arbitrary chosen regions .
Definition 9. The origin of switched system (4) formed by a sequence of subsystems is stable (asymptotically stable) for given geometrical and dynamical structure of switched system if the origin is stable (asymptotically stable) for a given choice of equations acting in corresponding regions .
4. General Theorems
The previous examples show that the stability properties of switched system origin need special investigation and they do not follow directly from simple stability conditions imposed on subsystems, but also they depend on how the switching occurs within the whole system.
The only interesting case when sequence (4) which determines the switched system is infinite is considered below.
Suppose equations are stable (asymptotically stable) and also suppose the existence of some smooth positive definite functions defined for . Each function is called Lyapunov function in region , if it is continuously differentiable in and fulfilling , , , and [16, 17].
Denote by scalar continuous nondecreasing functions (also called wedges) defined and positive for such that .
The symbols denote the derivatives of the functions along the trajectory of equation :
The existence of a common Lyapunov function for all equations simplifies the stability analysis of switched system origin. For this reason, Theorem 10 has been explicitly formulated but not proven, because this theorem is a direct consequence of the more general Theorem 12.
Theorem 10. Suppose there exists a common Lyapunov function defined in the whole space satisfying the following: (a), ,(b)for any equation the following conditions are fulfilled for , .Then, the trivial solution of switched system (1)-(2) is stable (asymptotically stable) independently of geometrical structure of switched system.
Example 11. Consider three equations:As a common Lyapunov function we can take the functionThe derivatives of these functions along the trajectories of equations , , and are not positive:Therefore, the trivial solution of the switched system formed by equations , , and is stable independently of geometrical structure of switched system determined by a choice of arbitrary regions , , and such that .
Theorem 12. Suppose conditions (3) hold for all switching lines and suppose that for each equation acting in there exists a Lyapunov function defined in such that (a), , ,(b), , , ,(c)on all switching lines where trajectories pass from region to , the following inequalities hold: , .Then the trivial solution of switched system (4) is stable (asymptotically stable) for given switched system.
Proof. Let us proof stability of trivial solution . Denote The wedge functions , , are scalar continuous nondecreasing functions positive for and satisfy (25). For a given we define a number such that . Evidently, such does exist. Using conditions (b) and (c) we obtain that the sequence of Lyapunov functions for arbitrary and is nonincreasing on the trajectories of switched system:Using now condition (a) of the theorem, we obtain for arbitrary and , the following inequalities:Therefore,Inequality (28) is equivalent to stability of trivial solution of switched system (1)-(2).
To proof asymptotic stability it is necessary to demonstrate that for any there exist numbers and such that for and . Take any and determine which corresponds to in demonstration of stability; that is, . Take also .
Let us demonstrate that on the interval there exists an instant such that .
If this is not so, that is, for all , then using continuity of and conditions (b) and (c) of Theorem 12 we haveInequality (29) contradicts the positiveness of Lyapunov function . Therefore, there exists an instant and , such that for . From condition and the stability of the trivial solution it follows that for all , . Therefore for all and for all . The asymptotic stability of the origin is proven.
Example 13. Consider once again first switched system of Example 6 where coincides with second quadrant, . Lyapunov functions for equations , are respectively.
On switching line where and , we have . On switching line where we have and . Therefore, condition (c) of Theorem 12 holds on both lines and if . Under this condition, the origin of the switched system described by (17)-(18) of Example 6 is stable.
Consider now the second switched system of Example 6, where coincides with the first quadrant . In this case, on line , where we have and on line where we have and . Therefore condition (c) of Theorem 12 holds if . In this case, the origin of the second switched system of Example 6 is stable under this situation. This conclusion coincides with results of Example 6 obtained by direct analytical computations. Analytical calculations show asymptotic stability of switched system origin (17)-(18), while by using Theorem 12 we can only establish stability but no asymptotic stability.
Replacing (18) in region by new equationand considering the same Lyapunov function , we can also establish stability of the trivial solution of this new switched system (17), (31), but analytical calculations cannot be used in this case.
Example 14. Consider the switched system with two equations and acting in two half-planes and :The switching lines are and . Normal going from to is , and condition (3) on line has the form On the other hand, normal going from to is , and condition (3) on line has the formAs Lyapunov functions for equations and in regions and takeThe strange function (35) satisfies condition (a) of Theorem 12 for unusual norm in of the form . The switching lines are and and on these switching lines . The derivatives of functions (35) and (36) in regions and are equal toAll conditions of Theorem 12 hold and therefore, the trivial solution of switched system (32) is stable.
Example 15. Consider a switched system with four equations acting in four quadrants of the plane as shown in Figure 4:where .
It is easy to observe that equations , , , and are unstable in the whole plane . To analyze the stability of the switched system origin consider the following four linear Lyapunov functions defined in corresponding regions:Clearly, functions (39) satisfy conditions (a) of Theorem 12 if the norm in is equal to . The derivatives of functions (39) areThe derivatives of Lyapunov functions , , , and are negative definite with respect to norm in the corresponding regions .
On switching lines , where or , the corresponding Lyapunov functions areFurthermore, all conditions of Theorem 12 are fulfilled; therefore, the trivial solution of the switched system is asymptotically stable (Figure 4). Moreover, as the trivial solution is exponentially asymptotically stable.
Example 16. Consider now the switched systems formed by the following nonlinear subsystems: with , , , and as in Example 15. As before, functions (39) fulfill conditions of Theorem 12 if the norm in is equal to . The derivatives of functions (39) areThe derivatives of Lyapunov functions , , , and are negative definite with respect to norm in the corresponding regions , and on switching lines , where or , the corresponding Lyapunov functions coincide with (41). Therefore, the trivial solution of the switched system (42) is exponentially asymptotically stable. The simulation results are shown in Figure 5.
Theorem 17. Suppose that sequence (4) is infinite and that for each equation , , there exists a Lyapunov function such that (a), , ,(b), , ,(c) on all switched lines , , where the trajectories pass from region to , the following inequalities hold:Furthermore, there exists at least one line , such thatThen, the trivial solution of the switched system (1)-(2) or (4) is asymptotically stable for a given switched system.
Proof. The trivial solution is stable because conditions of Theorem 12 are satisfied. It means that any Lyapunov function in region as function of does not increase. Moreover, the sequence of successive Lyapunov functions as function of does not increase either. More exactly, for any numbers , and any instants , such that , , , the following inequality holds: .
Suppose the trivial solution of switched system (4) is not asymptotically stable. It means that there exists the solution with sufficiently small , , such that it is bounded and does not tend to zero as . In this case there exists and a sequence such that , . The solution must satisfy conditionwhere is a number corresponding to in definition of stability for solution ; that is, . If (46) does not hold then there exists an instant such thatTaking now as a new initial moment and using stability of this solution must satisfy conditionCondition (48) contradicts condition (46) and this means that condition (46) holds. It follows from (46) that for an infinite number of moments , , when solution crosses line we have , , andInequality (49) contradicts condition and our supposition that does not tend to zero conducts to a contradiction. Furthermore, the asymptotic stability of trivial solution of switched system (4) is proven.
Theorem 18. Suppose that sequence (4) is infinite and that for all switching lines and that for each , , there exists a Lyapunov function such that (a), , ,(b), , ,(c) on all switched lines , , where the trajectories pass from region to , the following inequalities hold:furthermore, there exists at least one line , , such thatThen, the trivial solution of the switched system (4) is unstable for a given switched system.
Example 19. Consider the switched system formed by two nonlinear pendulums:Take as Lyapunov functions for (52) the following functions:The derivatives of functions (53) along the trajectories of (52) are equal to zero: . On switching line where , we have . On switching line where and we have and . Therefore, condition (c) of Theorem 18 holds on both lines and ifCondition (54) is a condition of instability of the trivial solution of switched system (52). Also, this condition may be considered as a condition of parametric discontinuous excitations for the nonlinear pendulum.
5. Autonomous Multidimensional Switched Systems
Consider multidimensional switched system acting in which is divided into a finite number of open 1-connected regions , , such that the origin of rectangular coordinates systems pertains to closure of any region . The boundaries of all regions are supposed to be smooth. Suppose, also, that each region has points fulfilling condition , where is any number. An equation is defined in each region satisfying conditions of existence and uniqueness of solutions. In multidimensional case, the switched system may have much more complicated behavior because region may have more than two boundaries separating it from other regions. To eliminate topological complications, consider only the case when all boundaries (switching surfaces) separating regions and are planes in . Suppose the trajectories of switched system pass through the boundary in all points only in direction from to and on all lines where touches other regions all trajectories pass from only to one determined region . Suppose also the existence of some Lyapunov functions for all equation , . In this case theorems similar to Theorems 10, 12, 17, and 18 may be established. The definitions of stability for switched systems in are the same as for the two-dimensional case. Now, consider the case of a switched system with infinite number of switchings. Let us formulate one of the theorems on stability.
Theorem 20. Suppose that for each equation acting in there exists a Lyapunov function defined in such that(a), , ,(b), , ,(c)on all switched surfaces and lines where the trajectories pass from region to , the following inequalities hold:Then, the trivial solution of the switched system acting in is stable (asymptotically stable) for a given switched structure.
Proof. The demonstration of this theorem is similar to proof of Theorem 12. Therefore it is omitted.
Example 21. In space consider two regions and . Suppose equations and are defined in and :Normals and to the surface separating and are and . Therefore, condition (3) holds on surface :