Stability of Switched Systems in Sense of Lyapunov’s Theory M.Khudaydus M.Khudaydus Supervisor: Dr.Wajdi Kallel Department of Mathematics, a fulfillment requirement for the degree of MSc in Control Theory April 8, 2018 Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 1 /Contr 38 What is Stability ? M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 2 /Contr 38 What is Stability ? Motions of a ball in a bowl Figure: The trajectory of the ball is stable when it goes to steady state (origin) M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 2 /Contr 38 What is Stability ? Motions of a ball in a bowl Figure: The trajectory of the ball is stable when it goes to steady state (origin) We say a system is stable when it comes back to its original state after it was affected with an external disturbance. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 2 /Contr 38 Equilibria M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 3 /Contr 38 Equilibria Equilibrium Points Figure: The pendulum motion has only two equilibrium positions M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 3 /Contr 38 Equilibria Equilibrium Points Equation of motion Pendulum 2nd order ODE Equation θ̈ = − gl sin θ − k m θ̇ Figure: The pendulum motion has only two equilibrium positions M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 3 /Contr 38 Stability Theory of a Dynamical System Whose the foundation of the stability theorems in dynamical systems? M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 4 /Contr 38 Stability Theory of a Dynamical System Whose the foundation of the stability theorems in dynamical systems? Figure: a Russian mathematician and engineer who laid the foundation of the stability theory M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 4 /Contr 38 Stability Theory of a Dynamical System Whose the foundation of the stability theorems in dynamical systems? Stability of equilibrium points are characterized in the sense of Lyapunov Figure: a Russian mathematician and engineer who laid the foundation of the stability theory M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 4 /Contr 38 Stability Theory of a Dynamical System Whose the foundation of the stability theorems in dynamical systems? Stability of equilibrium points are characterized in the sense of Lyapunov Stability of equilibrium Pt. Theorem 1 An equilibrium point is stable if all solutions of the dynamical system starting at nearby points stay nearby; otherwise, it is unstable Figure: a Russian mathematician and engineer who laid the foundation of the stability theory M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 4 /Contr 38 State-Space Models Pendulum Equation 2nd Order ODE Equation M.Khudaydus g k θ̈ = − sin θ − θ̇ l m Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 5 /Contr 38 State-Space Models Pendulum Equation 2nd Order ODE Equation g k θ̈ = − sin θ − θ̇ l m 2nd ODE Dynamical system can be transformed into a finite number of 1st ODE equations by carefully choosing the state variables: State-Space Model ( ẋ1 = x2 ẋ2 = − gl sin x1 − k m x2 where we set the state variables as x1 = θ and x2 = θ̇. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 5 /Contr 38 Autonomous and Non-autonomous Dynamical systems M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 6 /Contr 38 Autonomous and Non-autonomous Dynamical systems Dynamical system is represented in a Vector-notation as ẋ = f (t, x) M.Khudaydus (1) Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 6 /Contr 38 Autonomous and Non-autonomous Dynamical systems Dynamical system is represented in a Vector-notation as ẋ = f (t, x) (1) Autonomous system (LTI dynamical systems): Definition 0.1 The system (1) is said to be autonomous when the function f does not depend on t; that is, ẋ = f (x) M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 6 /Contr 38 Autonomous and Non-autonomous Dynamical systems Dynamical system is represented in a Vector-notation as ẋ = f (t, x) (1) Autonomous system (LTI dynamical systems): Definition 0.1 The system (1) is said to be autonomous when the function f does not depend on t; that is, ẋ = f (x) Non-autonomous system: Definition 0.2 The system (1) is said to be non-autonomous when the function f depend on both t and x ; that is, ẋ = f (t, x) M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 6 /Contr 38 Stability and Asymptotic Stability of Autonomous Sys. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 7 /Contr 38 Stability and Asymptotic Stability of Autonomous Sys. Theorem 2 Let x = 0 be an equilibrium point of the nonlinear system ẋ = f (x) where f : D 7→ R n is continuously differentiable and D is a neighborhood of the origin.Let ∂f (x) A= ∂x x=0 Then M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 7 /Contr 38 Stability and Asymptotic Stability of Autonomous Sys. Theorem 2 Let x = 0 be an equilibrium point of the nonlinear system ẋ = f (x) where f : D 7→ R n is continuously differentiable and D is a neighborhood of the origin.Let ∂f (x) A= ∂x x=0 Then The origin is asymptotically stable if Re(λi ) < 0 for all eigenvalues of A. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 7 /Contr 38 Stability and Asymptotic Stability of Autonomous Sys. Theorem 2 Let x = 0 be an equilibrium point of the nonlinear system ẋ = f (x) where f : D 7→ R n is continuously differentiable and D is a neighborhood of the origin.Let ∂f (x) A= ∂x x=0 Then The origin is asymptotically stable if Re(λi ) < 0 for all eigenvalues of A. The origin is unstable if ∃ λi such that Re(λi ) > 0. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 7 /Contr 38 Example: Consider the pendulum equation M.Khudaydus ẋ1 = x2 ẋ2 = − g k sin x1 − x2 l m Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 8 /Contr 38 Example: Consider the pendulum equation ẋ1 = x2 g k sin x1 − x2 l m has two equilibrium points at (x1 , x2 ) = (0, 0) and (x1 , x2 ) = (π, 0). Using linearization,the Jacobian matrix is given by: " # ∂f1 ∂f1 ∂f 0 1 ∂x1 ∂x2 = ∂f2 ∂f2 = k − gl cos x1 − m ∂x ∂x1 ∂x2 ẋ2 M.Khudaydus = − Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 8 /Contr 38 Example: Consider the pendulum equation ẋ1 = x2 g k sin x1 − x2 l m has two equilibrium points at (x1 , x2 ) = (0, 0) and (x1 , x2 ) = (π, 0). Using linearization,the Jacobian matrix is given by: " # ∂f1 ∂f1 ∂f 0 1 ∂x1 ∂x2 = ∂f2 ∂f2 = k − gl cos x1 − m ∂x ∂x1 ∂x2 ẋ2 = − Evaluating the Jacobian at x = 0 ∂f A= ∂x M.Khudaydus x=0 0 = − gl 1 k −m Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 8 /Contr 38 Example: Consider the pendulum equation ẋ1 = x2 g k sin x1 − x2 l m has two equilibrium points at (x1 , x2 ) = (0, 0) and (x1 , x2 ) = (π, 0). Using linearization,the Jacobian matrix is given by: " # ∂f1 ∂f1 ∂f 0 1 ∂x1 ∂x2 = ∂f2 ∂f2 = k − gl cos x1 − m ∂x ∂x1 ∂x2 ẋ2 = − Evaluating the Jacobian at x = 0 ∂f A= ∂x x=0 0 = − gl 1 k −m The eigenvalues of A are : s λ1,2 k 1 =− ± 2m 2 k 2 4g − m l If the eigenvalues satisfy Re(λi ) < 0, the equilibrium point at the origin is asymptotically stable. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 8 /Contr 38 Stability based on the Position of Eigenvalues on complex-plane Consider the LTI autonomous system ẋ = A x where the matrix A is n × n and assume that the eigenvalues of A take the form M.Khudaydus λ = a ± jb Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 9 /Contr 38 Stability based on the Position of Eigenvalues on complex-plane Consider the LTI autonomous system ẋ = A x where the matrix A is n × n and assume that the eigenvalues of A take the form λ = a ± jb (1) if eigenvalues are real and Re(λ) < 0 regardless of the imaginary part then, the equilibrium point has a stable-node. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 9 /Contr 38 Stability based on the Position of Eigenvalues on complex-plane Consider the LTI autonomous system ẋ = A x where the matrix A is n × n and assume that the eigenvalues of A take the form λ = a ± jb (1) if eigenvalues are real and Re(λ) < 0 regardless of the imaginary part then, the equilibrium point has a stable-node. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory for April the degree 8, 2018ofMScin 9 /Contr 38 Stability based on the Position of Eigenvalues on complex-plane M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 10 /Contr 38 Stability based on the Position of Eigenvalues on complex-plane (2) if eigenvalues are real and Re(λ) ≥ 0 then, the equilibrium point has a unstable-node. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 10 /Contr 38 Stability based on the Position of Eigenvalues on complex-plane (2) if eigenvalues are real and Re(λ) ≥ 0 then, the equilibrium point has a unstable-node. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 10 /Contr 38 Stability based on the Position of Eigenvalues on complex-plane (3) if eigenvalues are complex then, (2) if eigenvalues are real and Re(λ) ≥ 0 then, the equilibrium point has a unstable-node. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 10 /Contr 38 Stability based on the Position of Eigenvalues on complex-plane (3) if eigenvalues are complex then, (2) if eigenvalues are real and Re(λ) ≥ 0 then, the equilibrium point has a unstable-node. M.Khudaydus (I) when Re(λ) < 0, the equilibrium point has a stable-focus and the trajectory takes a spiral shape converges to the origin Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 10 /Contr 38 Stability based on the Position of Eigenvalues on complex-plane (3) if eigenvalues are complex then, (2) if eigenvalues are real and Re(λ) ≥ 0 then, the equilibrium point has a unstable-node. M.Khudaydus (I) when Re(λ) < 0, the equilibrium point has a stable-focus and the trajectory takes a spiral shape converges to the origin Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 10 /Contr 38 Stability based on the Position of Eigenvalues on a complex-plane M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 11 /Contr 38 Stability based on the Position of Eigenvalues on a complex-plane (3) if eigenvalues are complex then, M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 11 /Contr 38 Stability based on the Position of Eigenvalues on a complex-plane (3) if eigenvalues are complex then, (II) when Re(λ) > 0, the equilibrium point has a unstable-focus and the trajectory takes a spiral shape diverges away from the origin M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 11 /Contr 38 Stability based on the Position of Eigenvalues on a complex-plane (3) if eigenvalues are complex then, (II) when Re(λ) > 0, the equilibrium point has a unstable-focus and the trajectory takes a spiral shape diverges away from the origin M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 11 /Contr 38 Stability based on the Position of Eigenvalues on a complex-plane (3) if eigenvalues are complex then, (III) when Re(λ) = 0, the (II) when Re(λ) > 0, the equilibrium point has a center equilibrium point has a unstable-focus and the and the trajectory takes a circle trajectory takes a spiral shape shape (closed orbits) diverges away from the origin M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 11 /Contr 38 Stability based on the Position of Eigenvalues on a complex-plane (3) if eigenvalues are complex then, (III) when Re(λ) = 0, the (II) when Re(λ) > 0, the equilibrium point has a center equilibrium point has a unstable-focus and the and the trajectory takes a circle trajectory takes a spiral shape shape (closed orbits) diverges away from the origin M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 11 /Contr 38 Lyapunov Exponential Stability M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 12 /Contr 38 Lyapunov Exponential Stability Theorem 3 The equilibrium point x̄ = 0 of the autonomous system ẋ = f (x(t)) (2) where f : D 7→ R n is a locally Lipschitz and x(t) is the system state victor, is exponentially stable, if there exist positive constants α, β and δ such that if ||x(0)|| < δ, then ||x(t)|| ≤ αe −βt ||x(0)|| ∀t ≥ 0 M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 12 /Contr 38 Lyapunov Exponential Stability Theorem 3 The equilibrium point x̄ = 0 of the autonomous system ẋ = f (x(t)) (2) where f : D 7→ R n is a locally Lipschitz and x(t) is the system state victor, is exponentially stable, if there exist positive constants α, β and δ such that if ||x(0)|| < δ, then ||x(t)|| ≤ αe −βt ||x(0)|| ∀t ≥ 0 M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 12 /Contr 38 Lyapunov Exponential Stability Theorem 3 The equilibrium point x̄ = 0 of the autonomous system ẋ = f (x(t)) (2) where f : D 7→ R n is a locally Lipschitz and x(t) is the system state victor, is exponentially stable, if there exist positive constants α, β and δ such that if ||x(0)|| < δ, then ||x(t)|| ≤ αe −βt ||x(0)|| ∀t ≥ 0 M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 12 /Contr 38 Pros and Cons of the Exponential Stability theorem Pros. (1) The exponential function converges to zero very fast. (2) Provide a robust measurement tool box to prove many modern theorems in stability. M.Khudaydus Cons. (I) Its need to give a solution of the ODEs of the state models (II) Satisfying the inequality is too tedious Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 13 /Contr 38 Lyapunov Functions In 1892, Lyapunov showed that other functions could be used to determine stability of an equilibrium points.He said, that if Φ(t, x) is a solution of the autonomous ẋ = f (x(t)) (3) then, the derivative of the Lyapunov function V (x) along the trajectories of (3) which is denoted by V̇ (x) = ∂V f (x) ∂ xi should be dependent on the system’s equation .In addition, if the solution Φ(t, x) starts at initial state x at time t = 0, then it should be V̇ (x) = d V (Φ(t, x)) dt <0 t=0 Therefore, V (x) will decrease along the solution of (3). M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 14 /Contr 38 Lyapunov Functions Theorem 4 Let x = 0 be an equilibrium point for (3) and D ⊂ R n be a domain containing x = 0.Let V : D → R n be a continuously differentiable function, such that V (0) = 0 ; V (x) > 0 ∀x ∈ D, x 6= 0 (4) V̇ (x) ≤ 0 ∀ x ∈ D (5) Then,x(t) = 0 is stable.Moreover, if V̇ (x) < 0 x ∈ in D , x 6= 0 (6) Then x = 0 is asymptotically stable.For proof, see Khalil book,p100. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 15 /Contr 38 Example on Lyapunov Function Candidate Consider the pendulum equation without friction ẋ1 = x2 ẋ2 = − M.Khudaydus g l sin x1 Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 16 /Contr 38 Example on Lyapunov Function Candidate Consider the pendulum equation without friction ẋ1 = x2 ẋ2 = − g l sin x1 A Lyapunov function candidate is: g 1 (1 − cos x1 ) + x22 V (x) = l 2 M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 16 /Contr 38 Example on Lyapunov Function Candidate Consider the pendulum equation without friction ẋ1 = x2 ẋ2 = − g l sin x1 A Lyapunov function candidate is: g 1 (1 − cos x1 ) + x22 V (x) = l 2 So,V (0) = 0 and V (x) is positive definite over the domain −2π < x1 < 2π. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 16 /Contr 38 Example on Lyapunov Function Candidate Consider the pendulum equation without friction ẋ1 = x2 ẋ2 = − g l sin x1 A Lyapunov function candidate is: g 1 (1 − cos x1 ) + x22 V (x) = l 2 So,V (0) = 0 and V (x) is positive definite over the domain −2π < x1 < 2π. The derivative of V (x) along the trajectories of the system is given by g g g x1 sin x1 + x2 x2 = x2 sin x1 − x2 sin x1 = 0 V̇ (x) = l l l M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 16 /Contr 38 Example on Lyapunov Function Candidate Consider the pendulum equation without friction ẋ1 = x2 ẋ2 = − g l sin x1 A Lyapunov function candidate is: g 1 (1 − cos x1 ) + x22 V (x) = l 2 So,V (0) = 0 and V (x) is positive definite over the domain −2π < x1 < 2π. The derivative of V (x) along the trajectories of the system is given by g g g x1 sin x1 + x2 x2 = x2 sin x1 − x2 sin x1 = 0 V̇ (x) = l l l Thus, conditions (4) and (5) are satisfied and we conclude that the origin is stable. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 16 /Contr 38 Example on Lyapunov Function Candidate Consider the pendulum equation without friction ẋ1 = x2 ẋ2 = − g l sin x1 A Lyapunov function candidate is: g 1 (1 − cos x1 ) + x22 V (x) = l 2 So,V (0) = 0 and V (x) is positive definite over the domain −2π < x1 < 2π. The derivative of V (x) along the trajectories of the system is given by g g g x1 sin x1 + x2 x2 = x2 sin x1 − x2 sin x1 = 0 V̇ (x) = l l l Thus, conditions (4) and (5) are satisfied and we conclude that the origin is stable. Since V̇ (x) = 0 , we can conclude that the origin is not asymptotically stable. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 16 /Contr 38 Lyapunov’s Equation based on Lyapunov Quadratic Form Consider the autonomous system M.Khudaydus ẋ = A x Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 17 /Contr 38 Lyapunov’s Equation based on Lyapunov Quadratic Form Consider the autonomous system ẋ = A x Assume that we have a quadratic Lyapunov function candidate M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 17 /Contr 38 Lyapunov’s Equation based on Lyapunov Quadratic Form Consider the autonomous system ẋ = A x Assume that we have a quadratic Lyapunov function candidate 1 V (x) = x T Px M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 17 /Contr 38 Lyapunov’s Equation based on Lyapunov Quadratic Form Consider the autonomous system ẋ = A x Assume that we have a quadratic Lyapunov function candidate 1 V (x) = x T Px 2 P is a real symmetric positive definite matrix M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 17 /Contr 38 Lyapunov’s Equation based on Lyapunov Quadratic Form Consider the autonomous system ẋ = A x Assume that we have a quadratic Lyapunov function candidate 1 V (x) = x T Px 2 P is a real symmetric positive definite matrix 3 V̇ (x) = x T P ẋ + ẋ T Px = x T (PA + AT P)x = −x T Qx M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 17 /Contr 38 Lyapunov’s Equation based on Lyapunov Quadratic Form Consider the autonomous system ẋ = A x Assume that we have a quadratic Lyapunov function candidate 1 V (x) = x T Px 2 P is a real symmetric positive definite matrix 3 V̇ (x) = x T P ẋ + ẋ T Px = x T (PA + AT P)x = −x T Qx 4 V 0 (x)f (x) = −x T Qx < 0 M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 17 /Contr 38 Lyapunov’s Equation based on Lyapunov Quadratic Form Consider the autonomous system ẋ = A x Assume that we have a quadratic Lyapunov function candidate 1 V (x) = x T Px 2 P is a real symmetric positive definite matrix 3 V̇ (x) = x T P ẋ + ẋ T Px = x T (PA + AT P)x = −x T Qx 4 V 0 (x)f (x) = −x T Qx < 0 5 which is so-called Lyapunov-Equation M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 17 /Contr 38 Lyapunov’s Equation based on Lyapunov Quadratic Form Consider the autonomous system ẋ = A x Assume that we have a quadratic Lyapunov function candidate 1 V (x) = x T Px 2 P is a real symmetric positive definite matrix 3 V̇ (x) = x T P ẋ + ẋ T Px = x T (PA + AT P)x = −x T Qx 4 V 0 (x)f (x) = −x T Qx < 0 5 which is so-called Lyapunov-Equation 6 Solving PA + AT P < −Q for P M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 17 /Contr 38 Lyapunov’s Equation based on Lyapunov Quadratic Form Consider the autonomous system ẋ = A x Assume that we have a quadratic Lyapunov function candidate 1 V (x) = x T Px 2 P is a real symmetric positive definite matrix 3 V̇ (x) = x T P ẋ + ẋ T Px = x T (PA + AT P)x = −x T Qx 4 V 0 (x)f (x) = −x T Qx < 0 5 which is so-called Lyapunov-Equation 6 Solving PA + AT P < −Q for P 7 we conclude that the origin is asymptotically stable; which is similar to say that, Reλi < 0 for all eigenvalues of A. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 17 /Contr 38 Example on Lyapunov Candidate Function Consider the autonomous system : ẋ = A x where −1 0 A= 0 −1 M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 18 /Contr 38 Example on Lyapunov Candidate Function Consider the autonomous system : ẋ = A x where −1 0 A= 0 −1 Solving the LMI AT P + PA < −Q where Q = [I ] M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 18 /Contr 38 Example on Lyapunov Candidate Function Consider the autonomous system : ẋ = A x where −1 0 A= 0 −1 Solving the LMI AT P + PA < −Q where Q = [I ] Based on MATLAB Calculations! M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 18 /Contr 38 Example on Lyapunov Candidate Function M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 19 /Contr 38 Example on Lyapunov Candidate Function M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 19 /Contr 38 Example on Lyapunov Candidate Function M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 19 /Contr 38 Example on Lyapunov Candidate Function M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 19 /Contr 38 Example on Lyapunov Candidate Function M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 19 /Contr 38 Example on Lyapunov Candidate Function M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 19 /Contr 38 Example on Lyapunov Candidate Function M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 19 /Contr 38 Example on Lyapunov Candidate Function M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 19 /Contr 38 Example on Lyapunov Candidate Function M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 19 /Contr 38 Example on Lyapunov Candidate Function M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 19 /Contr 38 Stability under Switching Signals M.Khudaydus Switched Systems Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 20 /Contr 38 Hybrid Systems Switched System Switched systems are combination of dynamical systems whose governed by a constructed Switching-Signals (switching events). M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 21 /Contr 38 Hybrid Systems Switched System Switched systems are combination of dynamical systems whose governed by a constructed Switching-Signals (switching events). dynamical system may be represented by a linear system ẋ = Ax + Bu M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 21 /Contr 38 Hybrid Systems Switched System Switched systems are combination of dynamical systems whose governed by a constructed Switching-Signals (switching events). dynamical system may be represented by a linear system ẋ = Ax + Bu x ∈ R n is state vector and u ∈ R n is control input A x denotes a Continuous-Dynamic System. B u denotes a Discrete-dynamic System. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 21 /Contr 38 Hybrid Systems Switched System Switched systems are combination of dynamical systems whose governed by a constructed Switching-Signals (switching events). dynamical system may be represented by a linear system ẋ = Ax + Bu x ∈ R n is state vector and u ∈ R n is control input A x denotes a Continuous-Dynamic System. B u denotes a Discrete-dynamic System. Hybrid Systems Hybrid systems are an interaction between the continuous and discrete dynamic systems that are form a complete dynamical systems. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 21 /Contr 38 Hybrid Systems in our Life M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 22 /Contr 38 Hybrid Systems in our Life Continuous systems M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 22 /Contr 38 Hybrid Systems in our Life Continuous systems M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 22 /Contr 38 Hybrid Systems in our Life Continuous systems 1 M.Khudaydus The engine that provide the change in position x and velocity are the continuous state variables Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 22 /Contr 38 Hybrid Systems in our Life Continuous systems 1 The engine that provide the change in position x and velocity are the continuous state variables 2 The continuous dynamical systems described with ODE Motion Equations M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 22 /Contr 38 Hybrid Systems in our Life Continuous systems Discrete Systems 1 The engine that provide the change in position x and velocity are the continuous state variables 2 The continuous dynamical systems described with ODE Motion Equations M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 22 /Contr 38 Hybrid Systems in our Life Continuous systems Discrete Systems 1 The engine that provide the change in position x and velocity are the continuous state variables 2 The continuous dynamical systems described with ODE Motion Equations M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 22 /Contr 38 Hybrid Systems in our Life Continuous systems Discrete Systems 1 2 M.Khudaydus The engine that provide the change in position x and velocity are the continuous state variables 1 The Transmission Box (GearBox) is the discrete state The continuous dynamical systems described with ODE Motion Equations Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 22 /Contr 38 Hybrid Systems in our Life Continuous systems Discrete Systems 1 2 M.Khudaydus The engine that provide the change in position x and velocity are the continuous state variables The continuous dynamical systems described with ODE Motion Equations 1 The Transmission Box (GearBox) is the discrete state 2 The discrete state affects the continuous trajectory of the continuous dynamical system Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 22 /Contr 38 Recasting Hybrid Systems into Switched Systems Equations The motion of an automobile might takes the form M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 23 /Contr 38 Recasting Hybrid Systems into Switched Systems Equations The motion of(an automobile might takes the form ẋ1 = x2 ẋ2 = f (a, i) M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 23 /Contr 38 Recasting Hybrid Systems into Switched Systems Equations The motion of(an automobile might takes the form ẋ1 = x2 ẋ2 = f (a, i) x1 is the position , x2 is the velocity and a is the acceleration input wheres i = {1, 2, 3, 4, 5, R ≡ −1} is the gear shift position. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 23 /Contr 38 Recasting Hybrid Systems into Switched Systems Equations The motion of(an automobile might takes the form ẋ1 = x2 ẋ2 = f (a, i) x1 is the position , x2 is the velocity and a is the acceleration input wheres i = {1, 2, 3, 4, 5, R ≡ −1} is the gear shift position. x1 , x2 are the continuous states M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 23 /Contr 38 Recasting Hybrid Systems into Switched Systems Equations The motion of(an automobile might takes the form ẋ1 = x2 ẋ2 = f (a, i) x1 is the position , x2 is the velocity and a is the acceleration input wheres i = {1, 2, 3, 4, 5, R ≡ −1} is the gear shift position. x1 , x2 are the continuous states i is the discrete state M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 23 /Contr 38 Recasting Hybrid Systems into Switched Systems Equations The motion of(an automobile might takes the form ẋ1 = x2 ẋ2 = f (a, i) x1 is the position , x2 is the velocity and a is the acceleration input wheres i = {1, 2, 3, 4, 5, R ≡ −1} is the gear shift position. x1 , x2 are the continuous states i is the discrete state The switching events of the discrete states are called switching signal M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 23 /Contr 38 Recasting Hybrid Systems into Switched Systems Equations The motion of(an automobile might takes the form ẋ1 = x2 ẋ2 = f (a, i) x1 is the position , x2 is the velocity and a is the acceleration input wheres i = {1, 2, 3, 4, 5, R ≡ −1} is the gear shift position. x1 , x2 are the continuous states i is the discrete state The switching events of the discrete states are called switching signal The Complete Form of Hybrid Switched system M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 23 /Contr 38 Recasting Hybrid Systems into Switched Systems Equations The motion of(an automobile might takes the form ẋ1 = x2 ẋ2 = f (a, i) x1 is the position , x2 is the velocity and a is the acceleration input wheres i = {1, 2, 3, 4, 5, R ≡ −1} is the gear shift position. x1 , x2 are the continuous states i is the discrete state The switching events of the discrete states are called switching signal The Complete Form of Hybrid Switched system The switched systems can be described by ẋ(t) = fσ(t) (x(t)) , σ(t) ∈ P = {1, 2, · · · } M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 23 /Contr 38 The Switching Signal The switching events can be classified into State Dependent switching Time Dependent switching M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 24 /Contr 38 The Switching Signal The switching events can be classified into State Dependent switching Time Dependent switching M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 24 /Contr 38 The Switching Signal The switching events can be classified into State Dependent switching Time Dependent switching The thick curves denotes the switching surfaces M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 24 /Contr 38 The Switching Signal The switching events can be classified into State Dependent switching Time Dependent switching The thick curves denotes the switching surfaces The thin curves denotes the continuous dynamical system trajectories M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 24 /Contr 38 The Switching Signal The switching events can be classified into State Dependent switching Time Dependent switching The thick curves denotes the switching surfaces The thin curves denotes the continuous dynamical system trajectories M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 24 /Contr 38 The Switching Signal The switching events can be classified into State Dependent switching Time Dependent switching The thick curves denotes the switching surfaces The thin curves denotes the continuous dynamical system trajectories M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 24 /Contr 38 The Switching Signal The switching events can be classified into State Dependent switching Time Dependent switching The thick curves denotes the switching surfaces The thin curves denotes the continuous dynamical system trajectories M.Khudaydus The switching signal σ(t) generates a finite number to specify which subsystem should be activate till time reach specified time Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 24 /Contr 38 The Switching Signal The switching events can be classified into State Dependent switching Time Dependent switching The thick curves denotes the switching surfaces The thin curves denotes the continuous dynamical system trajectories M.Khudaydus The switching signal σ(t) generates a finite number to specify which subsystem should be activate till time reach specified time Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 24 /Contr 38 Stable to Unstable Switched Systems and Vice-Versa Consider the following systems ẋ = fσ(t) x(t) , σ(t) = q, ∀ q ∈ P = {1, 2} M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 25 /Contr 38 Stable to Unstable Switched Systems and Vice-Versa Consider the following systems ẋ = fσ(t) x(t) , σ(t) = q, ∀ q ∈ P = {1, 2} Assume that the two systems are asymptotically stable M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 25 /Contr 38 Stable to Unstable Switched Systems and Vice-Versa Consider the following systems ẋ = fσ(t) x(t) , σ(t) = q, ∀ q ∈ P = {1, 2} Assume that the two systems are asymptotically stable M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 25 /Contr 38 Stable to Unstable Switched Systems and Vice-Versa Consider the following systems ẋ = fσ(t) x(t) , σ(t) = q, ∀ q ∈ P = {1, 2} Assume that the two systems are asymptotically stable Applying switching signal to activate each swiched systems might give us stable or unstable switched systems M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 25 /Contr 38 Stable to Unstable Switched Systems and Vice-Versa Consider the following systems ẋ = fσ(t) x(t) , σ(t) = q, ∀ q ∈ P = {1, 2} Assume that the two systems are asymptotically stable Applying switching signal to activate each swiched systems might give us stable or unstable switched systems M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 25 /Contr 38 Switching Between two Unstable Switched Systems M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 26 /Contr 38 Switching Between two Unstable Switched Systems M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 26 /Contr 38 Switching Between two Unstable Switched Systems Both switched systems are unstable M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 26 /Contr 38 Switching Between two Unstable Switched Systems Both switched systems are unstable Applying the strategy of switching between them, give a complete stable switched systems M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 26 /Contr 38 A Way to Guarantee Own a Stable Switched Systems Theorem 5 Consider the autonomous switched systems ẋ = Ai x with Ai = {1, 2}, assume that, M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 27 /Contr 38 A Way to Guarantee Own a Stable Switched Systems Theorem 5 Consider the autonomous switched systems ẋ = Ai x with Ai = {1, 2}, assume that, 1 A1 , A2 are asymptotically stable matrices. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 27 /Contr 38 A Way to Guarantee Own a Stable Switched Systems Theorem 5 Consider the autonomous switched systems ẋ = Ai x with Ai = {1, 2}, assume that, 1 A1 , A2 are asymptotically stable matrices. 2 A1 , A2 are commute. Then, M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 27 /Contr 38 A Way to Guarantee Own a Stable Switched Systems Theorem 5 Consider the autonomous switched systems ẋ = Ai x with Ai = {1, 2}, assume that, 1 A1 , A2 are asymptotically stable matrices. 2 A1 , A2 are commute. Then, The complete switched systems are stable under any arbitrary switching. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 27 /Contr 38 A Way to Guarantee Own a Stable Switched Systems Theorem 5 Consider the autonomous switched systems ẋ = Ai x with Ai = {1, 2}, assume that, 1 A1 , A2 are asymptotically stable matrices. 2 A1 , A2 are commute. Then, The complete switched systems are stable under any arbitrary switching. There is a positive symmetric matrix P solution to the Lyapunov Equation: Ai T P + PA < −Q M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 27 /Contr 38 A Way to Guarantee Own a Stable Switched Systems Theorem 5 Consider the autonomous switched systems ẋ = Ai x with Ai = {1, 2}, assume that, 1 A1 , A2 are asymptotically stable matrices. 2 A1 , A2 are commute. Then, The complete switched systems are stable under any arbitrary switching. There is a positive symmetric matrix P solution to the Lyapunov Equation: Ai T P + PA < −Q There is a quadratic Lyapunov function V (x) = x T Px working as a common function for both individual switched systems. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 27 /Contr 38 Example M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 28 /Contr 38 Stability Under State-Dependent Switching M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 29 /Contr 38 Stability Under State-Dependent Switching 1 M.Khudaydus Standing requires balancing between the lift switched system and the right switched system. Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 29 /Contr 38 Stability Under State-Dependent Switching 1 Standing requires balancing between the lift switched system and the right switched system. 2 Both switched system are unstable(individual). M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 29 /Contr 38 Stability Under State-Dependent Switching 1 Standing requires balancing between the lift switched system and the right switched system. 2 Both switched system are unstable(individual). 3 The ACCELEROMETER representing the switching surfaces. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 29 /Contr 38 State-Dependent Example Consider the autonomous switched system ẋ = Aσ(t) x(t) M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 30 /Contr 38 State-Dependent Example Consider the autonomous switched system ẋ = Aσ(t) x(t) 0 10 1.5 2 A1 = A2 = 0 0 −2 −0.5 M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 30 /Contr 38 State-Dependent Example Consider the autonomous switched system ẋ = Aσ(t) x(t) 0 10 1.5 2 A1 = A2 = 0 0 −2 −0.5 √ the eigenvalues are, λ(A1 ) = 0 and λ1,2 (A2 ) = 0.5 ± i 3 Thus, both systems are unstable. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 30 /Contr 38 State-Dependent Example Consider the autonomous switched system ẋ = Aσ(t) x(t) 0 10 1.5 2 A1 = A2 = 0 0 −2 −0.5 √ the eigenvalues are, λ(A1 ) = 0 and λ1,2 (A2 ) = 0.5 ± i 3 Thus, both systems are unstable. The phase portrait for two unstable systems as follow: M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 30 /Contr 38 State-Dependent Example Consider the autonomous switched system ẋ = Aσ(t) x(t) 0 10 1.5 2 A1 = A2 = 0 0 −2 −0.5 √ the eigenvalues are, λ(A1 ) = 0 and λ1,2 (A2 ) = 0.5 ± i 3 Thus, both systems are unstable. The phase portrait for two unstable systems as follow: M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 30 /Contr 38 Cont.. State-Dependent Example Applying the state-dependent switching M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 31 /Contr 38 Cont.. State-Dependent Example Applying the state-dependent switching and x2 (t) = −0.25x1 (t) 1, if σ(t − ) = 2 σ(t) = 2, if σ(t − ) = 1 and x2 (t) = +0.50x1 (t) M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 31 /Contr 38 Cont.. State-Dependent Example Applying the state-dependent switching and x2 (t) = −0.25x1 (t) 1, if σ(t − ) = 2 σ(t) = 2, if σ(t − ) = 1 and x2 (t) = +0.50x1 (t) M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 31 /Contr 38 Stability Under Time-Dependent Switching The concept of Dwell Time τ [DT]. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 32 /Contr 38 Stability Under Time-Dependent Switching The concept of Dwell Time τ [DT]. Theorem 6 Consider the switched system ẋ = Ai x ∀i ∈ P (7) If we assume that all the individual subsystems are asymptotically stable (i.e when all subsystem matrices Ai are Hurwitz stable) then the switched system is exponentially stable if and only if the dwell − time is sufficiently large to allow each subsystem reaching the steady-state. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 32 /Contr 38 Stability Under Time-Dependent Switching The concept of Dwell Time τ [DT]. Theorem 6 Consider the switched system ẋ = Ai x ∀i ∈ P (7) If we assume that all the individual subsystems are asymptotically stable (i.e when all subsystem matrices Ai are Hurwitz stable) then the switched system is exponentially stable if and only if the dwell − time is sufficiently large to allow each subsystem reaching the steady-state. The ADT (average dwell time) [Enhancement DT]. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 32 /Contr 38 Stability Under Time-Dependent Switching The concept of Dwell Time τ [DT]. Theorem 6 Consider the switched system ẋ = Ai x ∀i ∈ P (7) If we assume that all the individual subsystems are asymptotically stable (i.e when all subsystem matrices Ai are Hurwitz stable) then the switched system is exponentially stable if and only if the dwell − time is sufficiently large to allow each subsystem reaching the steady-state. The ADT (average dwell time) [Enhancement DT]. ADT might be applied to stabilize between stable matrices and unstable matrices. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 32 /Contr 38 Determining the Average Dwell Time A1 , A2 are unstable and stable Hurwitz matrices,respectively. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 33 /Contr 38 Determining the Average Dwell Time A1 , A2 are unstable and stable Hurwitz matrices,respectively. Determine λ1 , λ2 , .. such that the eigenvalues of (A1 − λ1 I ) , (A2 + λ2 I ) are Hurwitz stable matrices. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 33 /Contr 38 Determining the Average Dwell Time A1 , A2 are unstable and stable Hurwitz matrices,respectively. Determine λ1 , λ2 , .. such that the eigenvalues of (A1 − λ1 I ) , (A2 + λ2 I ) are Hurwitz stable matrices. Solve for Pi such that (A1 − λ1 I )T P1 + P1 (A1 − λ1 I ) < 0 (A2 + λ2 I )T P2 + P2 (A2 + λ2 I ) < 0 M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 33 /Contr 38 Determining the Average Dwell Time A1 , A2 are unstable and stable Hurwitz matrices,respectively. Determine λ1 , λ2 , .. such that the eigenvalues of (A1 − λ1 I ) , (A2 + λ2 I ) are Hurwitz stable matrices. Solve for Pi such that (A1 − λ1 I )T P1 + P1 (A1 − λ1 I ) < 0 (A2 + λ2 I )T P2 + P2 (A2 + λ2 I ) < 0 Calculate Dwell-Time: M.Khudaydus τADT = ln µ 2(λ∗ −λ) , µ= λM (P1,P2) λm (P1,P2) Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 33 /Contr 38 Determining the Average Dwell Time A1 , A2 are unstable and stable Hurwitz matrices,respectively. Determine λ1 , λ2 , .. such that the eigenvalues of (A1 − λ1 I ) , (A2 + λ2 I ) are Hurwitz stable matrices. Solve for Pi such that (A1 − λ1 I )T P1 + P1 (A1 − λ1 I ) < 0 (A2 + λ2 I )T P2 + P2 (A2 + λ2 I ) < 0 Calculate Dwell-Time: The Switching Law: τADT = Ts Tu ≥ ln µ 2(λ∗ −λ) λu −λ∗ λs −λ∗ , µ= , λ∗ ∈ (λ, λs ) λM (P1,P2) λm (P1,P2) , λ ∈ (0, λs ) where T s , T u denote the total activate time of Hurwitz stable subsystems to hurwitz unstable subsystems. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 33 /Contr 38 Example A1 = M.Khudaydus −9 10 −20 10 and A2 = −10 11 10 −20 Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 34 /Contr 38 Example A1 = −9 10 −20 10 and A2 = −10 11 10 −20 The λ(A1 ) = 1, 1 (Unstable) system and λ(A2 ) = −10, −30 (Stable) systems M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 34 /Contr 38 Example A1 = −9 10 −20 10 and A2 = −10 11 10 −20 The λ(A1 ) = 1, 1 (Unstable) system and λ(A2 ) = −10, −30 (Stable) systems The trajectories of both systems M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 34 /Contr 38 Cont.. No ADT Switching Strategy 1 The result of V (x), V̇ (x) according to time t: M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 35 /Contr 38 Cont.. No ADT Switching Strategy 1 The result of V (x), V̇ (x) according to time t: M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 35 /Contr 38 Cont.. No ADT Switching Strategy 1 The result of V (x), V̇ (x) according to time t: 2 V̇ (x) < 0 not satisfied M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 35 /Contr 38 Cont... with ADT Switching Strategy M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 36 /Contr 38 Cont... with ADT Switching Strategy 1 Evaluate µ = M.Khudaydus 0.6 0.4 = 1.5 Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 36 /Contr 38 Cont... with ADT Switching Strategy 1 2 Evaluate µ = 0.6 0.4 = 1.5 Evaluate = 10, λ− = 9 such that (A1 − λI ), (A2 + λI ) are Hurwitz, respectively M.Khudaydus λ+ Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 36 /Contr 38 Cont... with ADT Switching Strategy 1 2 3 Evaluate µ = 0.6 0.4 = 1.5 Evaluate = 10, λ− = 9 such that (A1 − λI ), (A2 + λI ) are Hurwitz, respectively λ+ let λ ∈ (0, λ− , we choose λ = 1 ⇒ λ∗ ∈ (λ, λ− ) ⇒ λ∗ = 6. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 36 /Contr 38 Cont... with ADT Switching Strategy 1 2 Evaluate µ = 0.6 0.4 = 1.5 Evaluate = 10, λ− = 9 such that (A1 − λI ), (A2 + λI ) are Hurwitz, respectively λ+ 3 let λ ∈ (0, λ− , we choose λ = 1 ⇒ λ∗ ∈ (λ, λ− ) ⇒ λ∗ = 6. 4 τd = M.Khudaydus ln 1.5 2(6−1) = 0.04s. Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 36 /Contr 38 Cont... with ADT Switching Strategy 1 2 Evaluate µ = 0.6 0.4 = 1.5 Evaluate = 10, λ− = 9 such that (A1 − λI ), (A2 + λI ) are Hurwitz, respectively λ+ 3 let λ ∈ (0, λ− , we choose λ = 1 ⇒ λ∗ ∈ (λ, λ− ) ⇒ λ∗ = 6. 4 τd = 5 So, we need to activate A2 for more than 0.04.Let say 0.08. M.Khudaydus ln 1.5 2(6−1) = 0.04s. Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 36 /Contr 38 Cont... with ADT Switching Strategy 1 2 Evaluate µ = 0.6 0.4 = 1.5 Evaluate = 10, λ− = 9 such that (A1 − λI ), (A2 + λI ) are Hurwitz, respectively λ+ 3 let λ ∈ (0, λ− , we choose λ = 1 ⇒ λ∗ ∈ (λ, λ− ) ⇒ λ∗ = 6. 4 τd = 5 So, we need to activate A2 for more than 0.04.Let say 0.08. 6 According to M.Khudaydus ln 1.5 2(6−1) = 0.04s. T− T+ ≥ λ+ −λ∗ λ− −λ∗ = 16 3 = 5.3 . Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 36 /Contr 38 Cont... with ADT Switching Strategy 1 2 Evaluate µ = 0.6 0.4 = 1.5 Evaluate = 10, λ− = 9 such that (A1 − λI ), (A2 + λI ) are Hurwitz, respectively λ+ 3 let λ ∈ (0, λ− , we choose λ = 1 ⇒ λ∗ ∈ (λ, λ− ) ⇒ λ∗ = 6. 4 τd = 5 So, we need to activate A2 for more than 0.04.Let say 0.08. 6 According to 7 ln 1.5 2(6−1) T+ ≤ 0.015 M.Khudaydus T− 5.3 , = 0.04s. T− T+ ≥ λ+ −λ∗ λ− −λ∗ = 16 3 = 5.3 . need to activate the unstable system for equal or less than Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 36 /Contr 38 Successful Results based on applying ADT The result of V (x), V̇ (x) according to time t : M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 37 /Contr 38 Successful Results based on applying ADT The result of V (x), V̇ (x) according to time t : M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 37 /Contr 38 Successful Results based on applying ADT The result of V (x), V̇ (x) according to time t : V̇ (x) < 0 Satisfied. M.Khudaydus Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 37 /Contr 38 M.Khudaydus Thank You! ( ˆ . ˆ ) Supervisor: Dr.Wajdi Kallel Stability Department of SwitchedofSystems Mathematics, in Sense a fulfillment of Lyapunov’s requirement Theory forApril the 8, degree 2018 ofMScin 38 /Contr 38