On the Existence of a Periodic Mode in a Nonlinear System

We consider a nonlinear control system with a bang-bang hysteresis control, which is a simplified model of a thermal energy harvester. We obtain conditions on the controller and the system parameters guaranteeing the existence of a periodic mode in the system.


INTRODUCTION
The paper [1] dealt with the nonlinear control system on the half-line t ≥ 0 with the initial conditions y(0) = y 0 ,ẏ(0) = 0, T (0) = T 0 (2) and the bang-bang hysteresis feedback control ( Fig. 1) u if y(t) ∈ (y 1 , y 2 ) and there exists an s ∈ [0, t) such that y(s) = y 1 and y(τ ) ∈ (y 1 , y 2 ) for all τ ∈ (s, t] u if y(t) ∈ (y 1 , y 2 ) and there exists an s ∈ [0, t) such that y(s) = y 2 and y(τ ) ∈ (y 1 , y 2 ) for all τ ∈ (s, t]. ( System (1) is a simplified model of a thermal energy harvester [2], where the following notation is used: y is an output variable characterizing the strain of a shape memory material [3] with 0 < y 0 < ∆; T is the material temperature with T 0 ≥ 0; the positive numbers m, β, k, α, l, ∆, and γ are physical parameters of the thermal energy harvester; g is the acceleration due to gravity; u is an output feedback control (u = u(y)); and E is the material Young modulus, which is described by a nonlinear characteristic with hysteresis and saturation (see Fig. 2).  The mapping E(T ) can be viewed as a set-valued mapping E : R + → R + , Here A 1 , A 2 , M 1 , M 2 , E 1 , and E 2 are positive constants determined by the physical properties of the shape memory material. In what follows, we assume that these constants are related by the inequalities Note that, given a specific continuous function T (t), a single-valued branch of E(T ) is selected, which is a continuous function ranging in the interval [E 1 , E 2 ] (see [1]).
We assume that the threshold values y 1 and y 2 characterizing the control (3) satisfy the condition 0 < y 1 < y 2 < ∆.
Efficiently verifiable conditions on the coefficients and initial values of the state variables in system (1), (2) and on the parameters of the controller (3) ensuring the onset of oscillatory motions [4, pp. 10] in the closed-loop system were obtained in [1], where the following definition of oscillatory motion (oscillatory mode) was used. A solution of system (1), (2), that is, a pair of functions (y(t), T (t)) satisfying the system and the initial conditions with the control u given by (3) is called an oscillatory mode if there exist positive constants (mode parameters) t * 1 , t * 2 , y, and y (where 0 < t * 1 < t * 2 and 0 < y < y < ∆) such that the following conditions are satisfied: 1. There exists a t ≥ 0 such that y(t) = y.

2.
For each t such that y(t) = y, there exists a ξ ∈ [t + t * 1 , t + t * 2 ] such that y(ξ) = y and y(τ ) = y for all τ ∈ (t, t + t * 1 ). 3. For each t such that y(t) = y, there exists a ξ ∈ [t + t * 1 , t + t * 2 ] such that y(ξ) = y and y(τ ) = y for all τ ∈ (t, t + t * 1 ). An analysis of the results in [1] shows that the problem of finding conditions on the parameters of the closed-loop system (1)-(3) guaranteeing the existence of oscillatory modes turns out to be very difficult. However, from the viewpoint of applications, the problem of determining conditions for the existence of a periodic mode in the closed-loop system is more important.
In the present paper, based on the results in [1], we obtain sufficient conditions for the existence of a periodic mode in the closed-loop system (1)-(3).

STATEMENT OF THE PROBLEM
We write system (1), (2) in the normal Cauchy forṁ where The corresponding closed-loop system with the controller (3) for t ≥ 0 has the forṁ where 0 < y 0 < ∆ and T 0 ≥ 0; we assume that y 0 < y 1 and T 0 < M 1 . We write the closed-loop system (5) in the vector formẋ Now we say that the systemẋ is active if the bang-bang relay output is u and the systeṁ is active if the bang-bang relay output is u.

(8)
Here the solutions of systems (6) and (7) are matched by continuity at the points of discontinuity of the right-hand side of the system (on the hyperplanes x 1 = y 1 and Now let us state a control problem for system (4).
Problem. Find constraints on the number parameters m, k, α, l, β, ∆, and γ of system (4) as well as on the parameters y 1 , y 2 , u, and u of the controller (3) under which there exists a periodic solution (periodic mode) in the closed-loop system (5).

PERIODIC MODE IN SYSTEM WITH PROGRAMMED CONTROL
We divide the solution of the problem about the existence of a periodic solution in system (5) into several steps. First, we study the existence of a periodic solution of system (4) with the programmed control Here u, u, Θ 1 , and Θ are positive parameters of the programmed control. Now consider the problem of finding control parameter values ensuring the existence of a periodic solution of the closed-loop system (4), (9).
First, note that this control u(t) is a piecewise constant Θ-periodic function. Let us show that the equationẋ has a Θ-periodic solution for any u, u, and Θ > Θ 1 > 0. Indeed, the Cauchy formula holds for the solutions of Eq. (10) for any t ≥ s ≥ 0. Let Further, since u p (t) ≡ u on the interval [Θ 1 , Θ), we obtain In view of the Θ-periodicity of the function u p (t), we conclude that the solution x 3 (t) is Θ-periodic if and only if x 3 (Θ) = T 0 . Then we find the initial condition for the Θ-periodic solution x Θ 3 (t) from the representation (13), It follows from (13) and (14) that Now assume that the parameters u, u, Θ 1 , and Θ of the controller (9) have been chosen so that Assuming that the component x 3 of the solution of the closed-loop system (4), (9) satisfies the initial condition x 3 (0) = T , consider the subsysteṁ where ϕ(t) = E(x Θ 3 (t)). By virtue of conditions (15) and the fact that x Θ 3 (t) is a Θ-periodic solution of Eq. (10), the function ϕ(t) will be Θ-periodic as well, and we can write where and the inequality T 0 < M 1 has been taken into account. Figures 3 and 4 schematically depict the graphs of the functions x Θ 3 (t) and ϕ(t) for t ∈ [0, Θ].  It follows from the preceding that system (16) is actually a linear time-varying system of the formẋ where x = (x 1 , x 2 ) T and the matrix A(t) and the column vector f (t) being continuous Θ-periodic functions; i.e., in particular, It is well known [5, p. 215] that if the linear homogeneous Θ-periodic systeṁ does not have a nontrivial Θ-periodic solution, then the corresponding inhomogeneous system (18) has a unique Θ-periodic solution. Now note that if the homogeneous system (20) is asymptotically stable, then it cannot have a nontrivial Θ-periodic solution.
Let us obtain conditions under which the linear homogeneous system (20) is asymptotically stable. To this end, we use the method proposed in [5, p. 197]. Consider system (20) with coefficient matrix A(t) given by (19). This system is equivalent to the second-order differential equation where Let us make the standard change of dependent variable y = e −at/2 z in Eq. (21); theṅ Therefore, this change of variable reduces Eq. (21) to the form where According to the results in the monograph [5, p. 202], if the Θ-periodic function p(t) satisfies the inequalities then all solutions z(t) of Eq. (22) are bounded together with their first derivatives. However, the boundedness of z(t) andż(t) implies that the solutions y(t) of Eq. (21), together with their derivativesẏ(t), tend to zero, and consequently, system (20) is asymptotically stable. Since E 1 ≤ ϕ(t) ≤ E 2 , we obtain a sufficient condition for system (20) with coefficient matrix A(t) given by (19) to be asymptotically stable in the form Consequently, the linear inhomogeneous system (18) has a unique Θ-periodic solution under condition (23). We denote this solution by x Θ (t) = (x Θ 1 (t), x Θ 2 (t)) T . Thus, we have proved the following assertion. Theorem 1. Let the parameters u, u, Θ 1 , and Θ of the programmed control (9) for system (4) satisfy the following conditions: 1. One has the inequalities

PERIODIC MODE IN THE SYSTEM WITH A FEEDBACK
Let us return to the original problem (see Sec. 2) of constructing a feedback control (3) ensuring the existence of a periodic mode in the closed-loop system (5). The main idea for solving this problem is to choose the parameters of the feedback (3) based on the results in the paper [1] guaranteeing the existence of an oscillatory mode in the closed-loop system (5) and also based on the programmed control (9) calculated in accordance with Theorem 1 and the periodic solution x Θ (t) produced by this control.
Thus, the following assertion holds based on Theorem 1 and sufficient conditions obtained in [1] for the existence of oscillatory modes.
Theorem 2. Assume that 1. The parameters of system (4) satisfy the inequality 2. The spectrum of each of the matrices lies on the negative half-axis and is simple.
3. The parameters u, u, Θ 1 , and Θ satisfy the conditions in Theorem 1 and, in addition, 4. The numbers y 1 = x Θ 1 (0) and y 2 = x Θ 1 (Θ 1 ) satisfy the inequalities ) T is the periodic solution of system (4) supplemented by the programmed control (9) with parameters u, u, Θ 1 , and Θ. 5. The following inequalities hold for system (4) supplemented by the feedback (3) with parameters u, u, y 1 , and y 2 : , C 4 = mg k + ∆ + y 1 ; 6. The conditions 0 < x Θ 1 (t 0 ) < ζ + , x Θ 2 (t 0 ) = 0, t 0 < Θ A2 , are satisfied at time t 0 , where ζ + is the positive root of the quadratic trinomial Then the solutionx(t) of the closed-loop system (5) with the initial conditions is an oscillatory mode with the parameters t * 1 , t * 2 , y 1 , and y 2 , where t * 1 = (y 2 − y 1 ) m/H max . (An algorithm for calculating the constant t * 2 is rather awkward ; it is presented in full in [1].) Further, the solutionx(t) satisfies the identityx(t) ≡ x Θ (t + t 0 ) for t ∈ [0, t * 1 ]. Now let us show that, under certain additional conditions on the constant t * 1 , the oscillatory modex(t) of system (5) with the initial conditions (24) is actually a periodic mode. To this end, it suffices to establish that if the parameters u, u, y 1 , and y 2 of the bang-bang feedback control (3) satisfy the assumptions of Theorem 2, then, under the initial conditions (24), the control switches occur on the interval [0, Θ] at t 1 = Θ 1 − t 0 and t 2 = Θ − t 0 .
Thus, let the assumptions of Theorem 2 be satisfied, and let the inequality hold for t * 1 . Since the solutionx(t) of the closed-loop system (5) with the initial conditions (24) is oscillatory, we have the inequalityx It follows by Theorem 2 that the identityx(t) ≡ x Θ (t + t 0 ) holds for t ∈ [0, Θ A2 − t 0 ]. Further, consider the behavior of the functionsx 1 (t) andx 2 (t) on the interval [Θ A2 − t 0 , Θ 1 − t 0 ]. First, note that, by the definition of the feedback (3), the identitiesx , it follows that the functions x Θ 1 (t + t 0 ) and x Θ 2 (t + t 0 ) identically coincide on this interval with the respective componentsx 1 (t) andx 2 (t) of the solutionx(t) of the linear time-invariant systeṁ with the initial conditions By condition 2 in Theorem 2, the coefficient matrix of system (27) is stable and has simple spectrum. Let us show that the solutionx(t) of this system with the initial conditions (28) satisfies the relationx Indeed, the first component x 1 (t) of each solution x(t) of system (27) has the property where x * 1 = ∆ − (mg + k∆)/(k + αE 2 /l); the convergence in (29) was proved in [1], while the inequality in (29) is the second inequality in condition 4 in Theorem 2.
Consider the equationx Since the vector function (x 1 (t),x 2 (t)) T on the interval [Θ A2 − t 0 , Θ 1 − t 0 ] is a solution of system (27) whose coefficient matrix has negative distinct eigenvalues λ 1 and λ 2 , we have the representationx 1 (t) = C 1 e λ1t + C 2 e λ2t + x * 1 , where C 1 and C 2 are some constants, which are not zero simultaneously, becausê Therefore, Eq. (30) can be written in the form Note that the function on the left-hand side in this equation may have at most one point of extremum for t > 0, and hence Eq. (30) may have at most one root by virtue of property (29). Consequently, the equation x Θ 1 (t + t 0 ) = y 2 may have at most one root on the interval t ∈ [Θ A2 − t 0 , Θ 1 − t 0 ]. Since the function x Θ 1 (t + t 0 ) takes the value y 2 at the point t = Θ 1 − t 0 by a condition in Theorem 2, we conclude that x Θ 1 (t + t 0 ) = y 2 for t ∈ [Θ A2 − t 0 , Θ 1 − t 0 ). Thus, Now assume that the condition is additionally satisfied for t * 1 . Since the solutionx(t) of the closed-loop system (5) with the initial conditions (24) is an oscillatory mode, we have the inequalityx 1 It follows in view of relations (31) and (26) that Now consider the behavior of the functionsx 1 (t) andx 2 (t) on the interval [Θ M1 − t 0 , Θ − t 0 ]. By analogy with the preceding argument, note that the identitiesx 1 , it follows that the functions x Θ 1 (t + t 0 ) and x Θ 2 (t + t 0 ) identically coincide on this interval with the respective componentsx 1 (t) andx 2 (t) of the solutionx(t) of the linear time-invariant systeṁ with the initial conditions By condition 2 in Theorem 2, the coefficient matrix of system (34) is stable and has simple spectrum. Let us show that the solutionx(t) of this system with the initial conditions (35) satisfies the relationx Indeed, the first component x 1 (t) of each solution x(t) of system (34) has the property where x * * = ∆ − (mg + k∆)/(k + αE 1 /l); the convergence in (36) was proved in the paper [1], while the inequality is the first inequality in condition 4 in Theorem 2. Consider the equationx Since the vector function (x Θ 1 (t), x Θ 2 (t)) T on the interval [Θ M1 − t 0 , Θ − t 0 ] is a solution of system (34) whose coefficient matrix has distinct negative eigenvalues λ 1 and λ 2 , one has the representationx 1 (t) = C 1 e λ 1 t + C 2 e λ 2 t + x * * 1 , where C 1 and C 2 are some constants, which are not zero simultaneously, becausê Therefore, Eq. (37) can be written in the form C 1 e λ 1 t + C 2 e λ 2 t = y 1 − x * * 1 , where λ 1 < 0, λ 2 < 0, C 2 1 + C 2 2 > 0, t ≥ Θ M1 − t 0 . Note that the function on the left-hand side in this equation may have at most one point of extremum for t > 0, and then Eq. (37) may have at most one root by virtue of property (36). Consequently, the equation x Θ 1 (t + t 0 ) = y 1 may have at most one root on the interval t ∈ [Θ M1 − t 0 , Θ − t 0 ]. Since the function x Θ 1 (t + t 0 ) takes the value y 1 at the point t = Θ − t 0 by a condition in Theorem 2, we conclude that x Θ 1 (t + t 0 ) = y 1 for t ∈ [Θ M1 − t 0 , Θ − t 0 ). Thus,

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