Nonlinear Control Lecture # 7 Stability of Feedback Systems Nonlinear Control Lecture # 7 Stability of Feedback Systems
u 1 e 1 y 1 ✲ ❧ ✲ ✲ H 1 + ✻ − + ❄ y 2 e 2 u 2 + ✛ ✛ ✛ ❧ H 2 x i = f i ( x i , e i ) , ˙ y i = h i ( x i , e i ) y i = h i ( t, e i ) Nonlinear Control Lecture # 7 Stability of Feedback Systems
Passivity Theorems Theorem 7.1 The feedback connection of two passive systems is passive Proof Let V 1 ( x 1 ) and V 2 ( x 2 ) be the storage functions for H 1 and H 2 ( V i = 0 if H i is memoryless ) i y i ≥ ˙ e T V i , V ( x ) = V 1 ( x 1 ) + V 2 ( x 2 ) e T 1 y 1 + e T 2 y 2 = ( u 1 − y 2 ) T y 1 + ( u 2 + y 1 ) T y 2 = u T 1 y 1 + u T 2 y 2 � u 1 � � y 1 � u = , y = u 2 y 2 2 y 2 ≥ ˙ V 1 + ˙ V 2 = ˙ u T y = u T 1 y 1 + u T V Nonlinear Control Lecture # 7 Stability of Feedback Systems
Asymptotic Stability Theorem 7.2 Consider the feedback connection of two dynamical systems. When u = 0 , the origin of the closed-loop system is asymptotically stable if one of the following conditions is satisfied: both feedback components are strictly passive; both feedback components are output strictly passive and zero-state observable; one component is strictly passive and the other one is output strictly passive and zero-state observable. If the storage function for each component is radially unbounded, the origin is globally asymptotically stable Nonlinear Control Lecture # 7 Stability of Feedback Systems
Proof H 1 is SP; H 2 is OSP & ZSO 1 y 1 ≥ ˙ e T V 1 + ψ 1 ( x 1 ) , ψ 1 ( x 1 ) > 0 , ∀ x 1 � = 0 2 y 2 ≥ ˙ e T V 2 + y T y T 2 ρ 2 ( y 2 ) , 2 ρ ( y 2 ) > 0 , ∀ y 2 � = 0 e T 1 y 1 + e T 2 y 2 = ( u 1 − y 2 ) T y 1 + ( u 2 + y 1 ) T y 2 = u T 1 y 1 + u T 2 y 2 V ( x ) = V 1 ( x 1 ) + V 2 ( x 2 ) ˙ V ≤ u T y − ψ 1 ( x 1 ) − y T 2 ρ 2 ( y 2 ) ˙ V ≤ − ψ 1 ( x 1 ) − y T u = 0 ⇒ 2 ρ 2 ( y 2 ) Nonlinear Control Lecture # 7 Stability of Feedback Systems
˙ V ≤ − ψ 1 ( x 1 ) − y T 2 ρ 2 ( y 2 ) ˙ V = 0 ⇒ x 1 = 0 and y 2 = 0 y 2 ( t ) ≡ 0 ⇒ e 1 ( t ) ≡ 0 ( & x 1 ( t ) ≡ 0 ) ⇒ y 1 ( t ) ≡ 0 y 1 ( t ) ≡ 0 ⇒ e 2 ( t ) ≡ 0 By zero-state observability of H 2 : y 2 ( t ) ≡ 0 ⇒ x 2 ( t ) ≡ 0 Apply the invariance principle Nonlinear Control Lecture # 7 Stability of Feedback Systems
Example 7.1 � � � � x 1 ˙ = x 2 x 3 ˙ = x 4 � � x 2 ˙ = − ax 3 1 − kx 2 + e 1 x 4 ˙ = − bx 3 − x 3 4 + e 2 � � y 1 = x 2 y 2 = x 4 � � � �� � � �� � � H 1 H 2 a, b, k > 0 V 1 = 1 4 ax 4 1 + 1 2 x 2 2 ˙ V 1 = ax 3 1 x 2 − ax 3 1 x 2 − kx 2 2 + x 2 e 1 = − ky 2 1 + y 1 e 1 With e 1 = 0 , y 1 ( t ) ≡ 0 ⇔ x 2 ( t ) ≡ 0 ⇒ x 1 ( t ) ≡ 0 H 1 is output strictly passive and zero-state observable Nonlinear Control Lecture # 7 Stability of Feedback Systems
V 2 = 1 2 bx 2 3 + 1 2 x 2 4 ˙ V 2 = bx 3 x 4 − bx 3 x 4 − x 4 4 + x 4 e 2 = − y 4 2 + y 2 e 2 With e 2 = 0 , y 2 ( t ) ≡ 0 ⇔ x 4 ( t ) ≡ 0 ⇒ x 3 ( t ) ≡ 0 H 2 is output strictly passive and zero-state observable V 1 and V 2 are radially unbounded The origin is globally asymptotically stable Nonlinear Control Lecture # 7 Stability of Feedback Systems
Theorem 7.3 Consider the feedback connection of a strictly passive dynamical system with a passive time-varying memoryless function. When u = 0 , the origin of the closed-loop system is uniformly asymptotically stable. if the storage function for the dynamical system is radially unbounded, the origin will be globally uniformly asymptotically stable Proof Let V 1 ( x 1 ) be (positive definite) storage function of H 1 . V 1 = ∂V 1 ˙ f 1 ( x 1 , e 1 ) ≤ e T 1 y 1 − ψ 1 ( x 1 ) = − e T 2 y 2 − ψ 1 ( x 1 ) ∂x 1 ˙ e T 2 y 2 ≥ 0 ⇒ V 1 ≤ − ψ 1 ( x 1 ) Nonlinear Control Lecture # 7 Stability of Feedback Systems
Example 7.4 Consider the feedback connection of a strictly positive real transfer function and a passive time-varying memoryless function From Lemma 5.4, we know that the dynamical system is strictly passive with a positive definite storage function V ( x ) = 1 2 x T Px From Theorem 7.3, the origin of the closed-loop system is globally uniformly asymptotically stable Nonlinear Control Lecture # 7 Stability of Feedback Systems
Theorem 7.4 Consider the feedback connection of a time-invariant dynamical system H 1 with a time-invariant memoryless function H 2 . Suppose H 1 is zero-state observable, V 1 ( x 1 ) is positive definite 1 y 1 ≥ ˙ e T V 1 + y T e T 2 y 2 ≥ e T 1 ρ 1 ( y 1 ) , 2 ϕ 2 ( e 2 ) Then, the origin of the closed-loop system (when u = 0 ) is asymptotically stable if v T [ ρ 1 ( v ) + ϕ 2 ( v )] > 0 , ∀ v � = 0 Furthermore, if V 1 is radially unbounded, the origin will be globally asymptotically stable Nonlinear Control Lecture # 7 Stability of Feedback Systems
Example 7.5 � � � � x ˙ = f ( x ) + G ( x ) e 1 � y 2 = σ ( e 2 ) � y 1 = h ( x ) � � �� � � � �� � H 2 � H 1 e T σ (0) = 0 , 2 σ ( e 2 ) > 0 , ∀ e 2 � = 0 Suppose H 1 is zero-state observable and there is a radially unbounded positive definite function V 1 ( x ) such that ∂V 1 ∂V 1 ∂x G ( x ) = h T ( x ) , ∀ x ∈ R n ∂x f ( x ) ≤ 0 , V 1 = ∂V 1 ∂x f ( x ) + ∂V 1 ˙ ∂x G ( x ) e 1 ≤ y T 1 e 1 Nonlinear Control Lecture # 7 Stability of Feedback Systems
Apply Theorem 7.4: ˙ V 1 ≤ e T 1 y 1 1 y 1 ≥ ˙ e T V 1 + y T 1 ρ 1 ( y 1 ) is satisfied with ρ 1 = 0 e T 2 y 2 = e T 2 σ ( e 2 ) e T 2 y 2 ≥ e T 2 ϕ 2 ( e 2 ) is satisfied with ϕ 2 = σ v T [ ρ 1 ( v ) + ϕ 2 ( v )] = v T σ ( v ) > 0 , ∀ v � = 0 The origin is globally asymptotically stable Nonlinear Control Lecture # 7 Stability of Feedback Systems
The Small-Gain Theorem u 1 e 1 y 1 ✲ ♥ ✲ ✲ H 1 + ✻ − + ❄ y 2 e 2 u 2 + ✛ ✛ ✛ ♥ H 2 ∀ e 1 ∈ L m � y 1 τ � L ≤ γ 1 � e 1 τ � L + β 1 , e , ∀ τ ∈ [0 , ∞ ) ∀ e 2 ∈ L q � y 2 τ � L ≤ γ 2 � e 2 τ � L + β 2 , e , ∀ τ ∈ [0 , ∞ ) Nonlinear Control Lecture # 7 Stability of Feedback Systems
� � � � � � u 1 y 1 e 1 u = , y = , e = u 2 y 2 e 2 Theorem 7.7 The feedback connection is finite-gain L stable if γ 1 γ 2 < 1 Nonlinear Control Lecture # 7 Stability of Feedback Systems
Absolute Stability + r = 0 u y ✎☞ ✲ ✲ ✲ G ( s ) ✍✌ ✻ − ✛ ψ ( · ) Definition 7.1 The system is absolutely stable if the origin is globally uniformly asymptotically stable for any nonlinearity in a given sector. It is absolutely stable with finite domain if the origin is uniformly asymptotically stable Nonlinear Control Lecture # 7 Stability of Feedback Systems
Circle Criterion Suppose G ( s ) = C ( sI − A ) − 1 B + D is SPR, ψ ∈ [0 , ∞ ] x ˙ = Ax + Bu y = Cx + Du u = − ψ ( t, y ) By the KYP Lemma, ∃ P = P T > 0 , L, W, ε > 0 − L T L − εP PA + A T P = C T − L T W PB = W T W D + D T = 2 x T Px V ( x ) = 1 Nonlinear Control Lecture # 7 Stability of Feedback Systems
x T Px ˙ 1 2 x T P ˙ x + 1 V = 2 ˙ 1 2 x T ( PA + A T P ) x + x T PBu = 2 x T L T Lx − 2 εx T Px + x T ( C T − L T W ) u − 1 1 = 2 x T L T Lx − − 1 1 2 εx T Px + ( Cx + Du ) T u = − u T Du − x T L T Wu u T Du = 1 2 u T ( D + D T ) u = 1 2 u T W T Wu ˙ 2 εx T Px − 2 ( Lx + Wu ) T ( Lx + Wu ) − y T ψ ( t, y ) V = − 1 1 ˙ y T ψ ( t, y ) ≥ 0 V ≤ − 1 2 εx T Px ⇒ The origin is globally exponentially stable Nonlinear Control Lecture # 7 Stability of Feedback Systems
What if ψ ∈ [ K 1 , ∞ ] ? ✲ ❢ ✲ G ( s ) ✲ ✲ ❢ ✲ ❢ ✲ G ( s ) ✲ + + + − − − ✻ ✻ ✻ ✛ K 1 ✛ ✛ ✛ ❢ + ψ ( · ) ψ ( · ) − ✻ ✛ ˜ K 1 ψ ( · ) ˜ ψ ∈ [0 , ∞ ] ; hence the origin is globally exponentially stable if G ( s )[ I + K 1 G ( s )] − 1 is SPR Nonlinear Control Lecture # 7 Stability of Feedback Systems
What if ψ ∈ [ K 1 , K 2 ] ? ❄ ✲ ❢ ✲ G ( s ) ✲ ✲ ❢ ✲ ❢ ✲ G ( s ) ✲ K ✲ ❢ ✲ + + + + − − − + ✻ ✻ ✻ ✛ K 1 ✛ ✛ ✛ ✛ ✛ ❢ + ❢ + ψ ( · ) ψ ( · ) K − 1 − + ✻ ✻ ✛ ˜ K 1 ψ ( · ) ˜ ψ ∈ [0 , ∞ ] ; hence the origin is globally exponentially stable if I + KG ( s )[ I + K 1 G ( s )] − 1 is SPR Nonlinear Control Lecture # 7 Stability of Feedback Systems
I + KG ( s )[ I + K 1 G ( s )] − 1 = [ I + K 2 G ( s )][ I + K 1 G ( s )] − 1 Theorem 7.8 (Circle Criterion) The system is absolutely stable if ψ ∈ [ K 1 , ∞ ] and G ( s )[ I + K 1 G ( s )] − 1 is SPR, or ψ ∈ [ K 1 , K 2 ] and [ I + K 2 G ( s )][ I + K 1 G ( s )] − 1 is SPR If the sector condition is satisfied only on a set Y ⊂ R m , then the foregoing conditions ensure absolute stability with finite domain Nonlinear Control Lecture # 7 Stability of Feedback Systems
Scalar Case: ψ ∈ [ α, β ] , β > α The system is absolutely stable if 1 + βG ( s ) is Hurwitz and 1 + αG ( s ) � 1 + βG ( jω ) � Re > 0 , ∀ ω ∈ [0 , ∞ ] 1 + αG ( jω ) Nonlinear Control Lecture # 7 Stability of Feedback Systems
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