Stabilizing Control for Nonlinear Switched Systems in Two Dimensions with a Geometric Approach

The purpose of this paper is to give a sufficient condition for the existence and stability of a hybrid limit cycle for the stabilizing control of a class of switched dynamical systems in stable limit cycle is also important [7, 8]. In this paper we consider the following time invariant switched dynamic system (SDS) in IR: ẏ = v1Y (y) + v2Y (y) + v3Y (y) (1) where Yi is in vector fields of class C , the control vi ∈ {0, 1} for i ∈ {1, 2, 3} verifying the condition 3 ∑ i=1 vi = 1 and the state y = (y1, y2) is in IR . The main objective is to propose a new constructive method for synthesizing a hybrid limit cycle for the SDC (1). In this paper we use essentially the following result: we consider the nonlinear SDS in the plan [7] ẋ = u1X (x) + u2X (x) with ui ∈ {0, 1}, i ∈ {1, 2} (2) The solution of the differential equation ẋ = Xi(x), i ∈ {1, 2} after elapsed time t with initial condition x(0) = x0 is denoted X i t(x0). Definition 1 [7] Let us consider xc1 and xc2 two points in IR , with xc1 = xc2. CC(xc1 , xc2) is the hybrid limit cycle of the SDS (2) ẋ = X (x), i ∈ {1, 2}, between the switching points xc1 and xc2, if and only if (δc1 , δc2) ∈ R+ exists such that: xc1 = X 1 δc1 (xc2) and xc2 = X 2 δc2 (xc1). Then CC(xc1 , xc2) = {X δ (xc2)/0 ≤ δ ≤ δc1} ∪ {X δ (xc1)/0 ≤ δ ≤ δc2}. We recall a sufficient condition for existence and stability of a hybrid limit cycle: Let consider two maps enough smooth γj : I ⊂ IR −→ IR, j ∈ {1, 2}. Suppose that γ1(t0) = γ2(t0) (the two trajectories intersect at t0) with γ ′ 1(t0) = 0 and γ′ 2(t0) = 0 Definition 2 [7] We call that curves γ1 and γ2 are transverse if and o ly if γ1 cross the curve γ2 in x0 = γ1(t0). We explain this property in the following figure: case when curves γ1 and γ2 are transverse Example of γ1 and γ2 not transverse x0 x0 2 2. This is then illustrated an Inductio Heating Appliance. Case when curves γ 1 and γ 2 are transverse Example of γ 1 and γ 2 not transverse x 0 x 0 Figure 1: If γ1 cross the curve γ2 in x0=γ1(t0). Citation: Omri F (2018) Stabilizing Control for Nonlinear Switched Systems in Two Dimensions with a Geometric Approach. J Appl Computat Math 7: 396. doi: 10.4172/2168-9679.1000396


Introduction
Stability theory plays a central role in systems theory and engineering. There are different kinds of stability problems that arise in the study of dynamical systems. In recent years, the problem of stability and stabilization of switched systems has attracted a considerable attention from control community [1][2][3][4][5][6][7][8][9].
Switched dynamical systems (SDS) are an important class of hybrid systems, which consist of a family of continuous-time or discrete-time subsystems and a switching law that specifies the switching between them [10]. The SDS [2] are found in many fields of application: transport, embedded systems, electronics power, aeronautics, chemical engineering, pharmaceutical, etc. It can be seen in these applications that interactions between discrete events and continuous phenomena give rise to complex system behavior that can only be properly controlled if the hybrid phenomena (continuous and discrete features, and interactions between them) are fully taken into consideration [3,5].
Limit cycles are one of the most important phenomena in nonlinear dynamical systems, and applied in many engineering fields [6]. While stability analysis of limit cycle is a fundamental problem and many theories such as Lyapunov function methods have been proposed, the problem of synthesizing a nonlinear system which has a stable limit cycle is also important [7,8].
In this paper we consider the following time invariant switched dynamic system (SDS) in cle is also important [7,8].
er we consider the following time invariant switched dynamic system vector fields of class C 1 , the control v i ∈ {0, 1} for i ∈ {1, 2, 3} veri- v i = 1 and the state y = (y 1 , y 2 ) is in IR 2 .
ctive is to propose a new constructive method for synthesizing a hyfor the SDC (1).
w constructive method for synthesizing a hye following result: we consider the nonlinear 2}, between and only if (δ c1 , δ c2 ) ∈ R 2 + exists such that: hen stence and stability of a hybrid limit cycle: Suppose es intersect at t 0 ) with γ 1 (t 0 ) = 0 and γ 2 (t 0 ) = γ 1 and γ 2 are transverse if and only if γ 1 cross in this property in the following figure: erse Example of γ 1 and γ 2 not transverse The main objective is to propose a new constructive method for synthesizing a hybrid limit cycle for the SDC (1).
In this paper we use essentially the following result: we consider the nonlinear SDS in the plan [7] ( ) ( ) The solution of the differential equation after elapsed time t with initial condition x(0)=x 0 is denoted

Definition
Let us consider x c1 and x c2 two points in ℝ 2 , with x c1 ≠x c2 . CC(x c1 ,x c2 ) is the hybrid limit cycle of the SDS (2) , c c δ δ ∈ � stable limit cycle is also important [7,8].
In this paper we consider the following time invariant switched dynam (SDS) in where Y i is in vector fields of class C 1 , the control v i ∈ {0, 1} for i ∈ {1, fying the condition 3 i=1 v i = 1 and the state y = (y 1 , y 2 ) is in IR 2 .
The main objective is to propose a new constructive method for synthesiz brid limit cycle for the SDC (1).
In this paper we use essentially the following result: we consider the SDS in the plan [7] The solution of the differential equationẋ = X i (x), i ∈ {1, 2} after elaps with initial condition x(0) = x 0 is denoted X i t (x 0 ). Definition 1 [7] Let us consider x c1 and x c2 two points in IR 2 , with x CC(x c1 , x c2 ) is the hybrid limit cycle of the SDS (2) We recall a sufficient condition for existence and stability of a hybrid limi Let consider two maps enough smooth γ j : I ⊂ IR −→ IR 2 , j ∈ {1, 2}. that γ 1 (t 0 ) = γ 2 (t 0 ) (the two trajectories intersect at t 0 ) with γ 1 (t 0 ) = 0 and 0 Definition 2 [7] We call that curves γ 1 and γ 2 are transverse if and only the curve γ 2 in x 0 = γ 1 (t 0 ). We explain this property in the following figu case when curves γ 1 and γ 2 are transverse Example of γ 1 and γ 2 n We recall a sufficient condition for existence and stability of a hybrid limit cycle: Let consider two maps enough smooth γ j : I ⊂ stable limit cycle is also important [7,8].
In this paper we consider the following time invariant switched dynamic syste The main objective is to propose a new constructive method for synthesizing a h brid limit cycle for the SDC (1).
In this paper we use essentially the following result: we consider the nonline SDS in the plan [7] We recall a sufficient condition for existence and stability of a hybrid limit cycle: Let consider two maps enough smooth γ j : Definition 2 [7] We call that curves γ 1 and γ 2 are transverse if and only if γ 1 cro the curve γ 2 in x 0 = γ 1 (t 0 ). We explain this property in the following figure: case when curves γ 1 and γ 2 are transverse Example of γ 1 and γ 2 not trans x 0 x 0 2 → stable limit cycle is also important [7,8].
In this paper we consider the following time invariant switched dynami (SDS) in The main objective is to propose a new constructive method for synthesizi brid limit cycle for the SDC (1).
In this paper we use essentially the following result: we consider the n SDS in the plan [7] The solution of the differential equationẋ = We recall a sufficient condition for existence and stability of a hybrid limit Let consider two maps enough smooth γ j : Definition 2 [7] We call that curves γ 1 and γ 2 are transverse if and only if the curve γ 2 in x 0 = γ 1 (t 0 ). We explain this property in the following figure case when curves γ 1 and γ 2 are transverse Example of γ 1 and γ 2 no

Definition
We call that curves γ 1 and γ 2 are transverse if and only if γ 1 cross the curve γ 2 in x 0= γ 1 (t 0 ). We explain this property [7] following Figure 1.

Notation
Let us denote as with p ij ≥ 1 and i≠j . We also denote p ij (x) is the smallest positive integer such that (with γ 1 (t)=X t

(x)
and γ 2 (t)=X t 2 (x)): In this paper we consider the following time invariant switched dynamic system (SDS) in where Y i is in vector fields of class C 1 , the control v i ∈ {0, 1} for i ∈ {1, 2, 3} verifying the condition 3 i=1 v i = 1 and the state y = (y 1 , y 2 ) is in IR 2 .
The main objective is to propose a new constructive method for synthesizing a hybrid limit cycle for the SDC (1).
In this paper we use essentially the following result: we consider the nonlinear SDS in the plan [7] The solution of the differential equationẋ = X i (x), i ∈ {1, 2} after elapsed time t with initial condition x(0) = x 0 is denoted X i t (x 0 ). Definition 1 [7] Let us consider x c1 and x c2 two points in IR 2 , with x c1 = x c2 . CC(x c1 , x c2 ) is the hybrid limit cycle of the SDS (2)ẋ = X i (x), i ∈ {1, 2}, between the switching points x c1 and x c2 , if and only if (δ c1 , δ c2 ) ∈ R 2 + exists such that: We recall a sufficient condition for existence and stability of a hybrid limit cycle: Let consider two maps enough smooth γ j :
objective is to propose a new constructive method for synthesizing a hycycle for the SDC (1). paper we use essentially the following result: we consider the nonlinear plan [7] on of the differential equationẋ = X i (x), i ∈ {1, 2} after elapsed time t l condition x(0) = x 0 is denoted X i t (x 0 ). n 1 [7] Let us consider x c1 and x c2 two points in IR 2 , with x c1 = x c2 .
2 ) is the hybrid limit cycle of the SDS (2)ẋ = X i (x), i ∈ {1, 2}, between ing points x c1 and x c2 , if and only if (δ c1 , δ c2 ) ∈ R 2 + exists such that: }. a sufficient condition for existence and stability of a hybrid limit cycle: sider two maps enough smooth γ j : I ⊂ IR −→ IR 2 , j ∈ {1, 2}. Suppose = γ 2 (t 0 ) (the two trajectories intersect at t 0 ) with γ 1 (t 0 ) = 0 and γ 2 (t 0 ) = n 2 [7] We call that curves γ 1 and γ 2 are transverse if and only if γ 1 cross γ 2 in x 0 = γ 1 (t 0 ). We explain this property in the following figure: curves γ 1 and γ 2 are transverse Example of γ 1 and γ 2 not transverse x 0 x 0 2 2 /det(X i (x),X j (x))=0, ⟨X i (x),X j (x)⟩< 0 and p ij (z) is even} the set of points with collinear and opposite vector fields X i (z) and X j (z) and these orbits of z under X i and X j are not transverse [7]. Let x d the desired operating point of SDS. It is shown in ref. [7], that for x d ∈ E 12 there exists an infinity of hybrids limits cycles around the desired point x d . This result can be summarized in the following theorem:

Sufficient Condition of Existence and Stability of a Hybrid Limit Cycle
Let us consider the time invariant switched dynamic system (SDS) (1) y vY y vY y vY y = + +  .
the following time invariant switched dynamic system ntially the following result: we consider the nonlinear re transverse Example of γ 1 and γ 2 not transverse the set of all points y such that the vector Y 1 (y) is represented as a linear combination of (Y 2 (y),Y 3 (y)) with positive coefficient.

Theorem
For all y ∈ E and the desired point x d of SDC (1) is in E 23 , there exists a hybrid limit cycle CC (x c ,y c ) such that
In this paper we consider the following time invariant switched dynamic system S) in e main objective is to propose a new constructive method for synthesizing a hylimit cycle for the SDC (1).
In this paper we use essentially the following result: we consider the nonlinear S in the plan [7] x e solution of the differential equationẋ = X i (x), i ∈ {1, 2} after elapsed time t h initial condition x(0) = x 0 is denoted X i t (x 0 ). finition 1 [7] Let us consider x c1 and x c2 two points in IR 2 , with x c1 = x c2 . (x c1 , x c2 ) is the hybrid limit cycle of the SDS (2)ẋ = X i (x), i ∈ {1, 2}, between switching points x c1 and x c2 , if and only if (δ c1 , δ c2 ) ∈ R 2 + exists such that: recall a sufficient condition for existence and stability of a hybrid limit cycle: Let consider two maps enough smooth γ j : I ⊂ IR −→ IR 2 , j ∈ {1, 2}. Suppose t γ 1 (t 0 ) = γ 2 (t 0 ) (the two trajectories intersect at t 0 ) with γ 1 (t 0 ) = 0 and γ 2 (t 0 ) = finition 2 [7] We call that curves γ 1 and γ 2 are transverse if and only if γ 1 cross curve γ 2 in x 0 = γ 1 (t 0 ). We explain this property in the following figure: e when curves γ 1 and γ 2 are transverse Example of γ 1 and γ 2 not transverse x 0 x 0 2 2 Thus, the system (1) becomes By using the fact that x d ∈ E 23 and by using the theorem (2) there exists a hybrid limit cycle around of x d .

Remark
In the case where the control v 1 =0, the SDS (1) rest around of point x d . Otherwise, we can reach the point x d by the vector field Y 1 (y).

Three Control Strategies of an Induction Heating Appliance
An induction heating appliance (induction hob) is made of adaptable-diameter inductors. Currently, one resonant inverter is dedicated to supplying each winding [6,10]. The current variations in the inductor produce heat energy in the metal vessel placed on the winding. The inductor and the load (vessel) are each equivalent to a resistor and an inductor in series. In the system studied here, the global inductor (R, L) is in series with a capacitor C to compose the resonant inverter (voltage inverter). The voltage source, E, provides adjustable DC voltage through a thyristors or diodes rectifier. An adequat DC voltage is applied to the series RLC circuit, with the control of the opening and closure of the four switches 1  The objective is to control the u C voltage on a hybrid limit whatever the load.
The standard state-space representation of the induction heating appliance, with power state variables, is the following equation (3): with ρ the control signal of the switches. If The set of possible points of equilibrium for this system is {y=(y 1 ,y 2 ) ∈ stable limit cycle is also important [7,8].
In this paper we consider the following time invariant switched dynamic system (SDS) in where Y i is in vector fields of class C 1 , the control v i ∈ {0, 1} for i ∈ {1, 2, 3} verifying the condition The main objective is to propose a new constructive method for synthesizing a hybrid limit cycle for the SDC (1).
We have.  are each equivalent to a resistor and an inductor in series. With power state variables, ρ the control signal of the switches. If 1 2 ρ = , the configuration 1 is active, if ρ=1, the configuration 2 is active and, if ρ=0, the configuration 3 is active. It can be seen in these applications that interactions between discrete events and continuous phenomena give rise to complex system behavior that can only be properly controlled if the hybrid phenomena (continuous and discrete features, and interactions between them) are fully taken into consideration. E 23 ={z ∈ ℝ 2 /A −1 B< z< −A −1 B and p 23= 2} is the straight line that connects the two stable point x e1 et x e2 of vector fields Y 2 (y) and Y 3 (y). It follow that for each point in E 23 there exists a hybrid limit cycle. If we Figure 3 gives the dynamics of the two modes of the SDS, associated with vector fields Y 2 (y) (red curves) and Y 3 (y) (blue curves) as well as the set E (straight line in black).

Conclusion
The current variations in the inductor produce heat energy in the metal vessel placed on the winding. The inductor and the load (vessel)