Nonlinear Control Lecture # 14 Input-Output Stability Nonlinear - PowerPoint PPT Presentation

Nonlinear Control Lecture # 14 Input-Output Stability Nonlinear Control Lecture # 14 Input-Output Stability

L Stability Input-Output Models: y = Hu u ( t ) is a piecewise continuous function of t and belongs to a linear space of signals The space of bounded functions: sup t ≥ 0 � u ( t ) � < ∞ The space of square-integrable functions: � ∞ 0 u T ( t ) u ( t ) dt < ∞ Norm of a signal � u � : � u � ≥ 0 and � u � = 0 ⇔ u = 0 � au � = a � u � for any a > 0 Triangle Inequality: � u 1 + u 2 � ≤ � u 1 � + � u 2 � Nonlinear Control Lecture # 14 Input-Output Stability

L p spaces: L ∞ : � u � L ∞ = sup � u ( t ) � < ∞ t ≥ 0 �� ∞ L 2 : � u � L 2 = u T ( t ) u ( t ) dt < ∞ 0 �� ∞ � 1 /p � u ( t ) � p dt L p : � u � L p = < ∞ , 1 ≤ p < ∞ 0 Notation L m p : p is the type of p -norm used to define the space and m is the dimension of u Nonlinear Control Lecture # 14 Input-Output Stability

Extended Space: L e = { u | u τ ∈ L , ∀ τ ∈ [0 , ∞ ) } � u ( t ) , 0 ≤ t ≤ τ u τ is a truncation of u : u τ ( t ) = 0 , t > τ L e is a linear space and L ⊂ L e Example � t, 0 ≤ t ≤ τ u ( t ) = t, u τ ( t ) = 0 , t > τ u / ∈ L ∞ but u τ ∈ L ∞ e Nonlinear Control Lecture # 14 Input-Output Stability

Causality: A mapping H : L m e → L q e is causal if the value of the output ( Hu )( t ) at any time t depends only on the values of the input up to time t ( Hu ) τ = ( Hu τ ) τ Definition 6.1 A scalar continuous function g ( r ) , defined for r ∈ [0 , a ) , is a gain function if it is nondecreasing and g (0) = 0 A class K function is a gain function but not the other way around. By not requiring the gain function to be strictly increasing we can have g = 0 or g ( r ) = sat( r ) Nonlinear Control Lecture # 14 Input-Output Stability

Definition 6.2 A mapping H : L m e → L q e is L stable if there exist a gain function g , defined on [0 , ∞ ) , and a nonnegative constant β such that ∀ u ∈ L m � ( Hu ) τ � L ≤ g ( � u τ � L ) + β, e and τ ∈ [0 , ∞ ) It is finite-gain L stable if there exist nonnegative constants γ and β such that ∀ u ∈ L m � ( Hu ) τ � L ≤ γ � u τ � L + β, e and τ ∈ [0 , ∞ ) In this case, we say that the system has L gain ≤ γ . The bias term β is included in the definition to allow for systems where Hu does not vanish at u = 0 . Nonlinear Control Lecture # 14 Input-Output Stability

Example 6.1: Memoryless function y = h ( u ) Suppose | h ( u ) | ≤ a + b | u | , ∀ u ∈ R Finite-gain L ∞ stable with β = a and γ = b If a = 0 , then for each p ∈ [1 , ∞ ) � ∞ � ∞ | h ( u ( t )) | p dt ≤ ( b ) p | u ( t ) | p dt 0 0 Finite-gain L p stable with β = 0 and γ = b For h ( u ) = u 2 , H is L ∞ stable with zero bias and g ( r ) = r 2 . It is not finite-gain L ∞ stable because | h ( u ) | = u 2 cannot be bounded γ | u | + β for all u ∈ R Nonlinear Control Lecture # 14 Input-Output Stability

Example 6.2: SISO causal convolution operator � t y ( t ) = h ( t − σ ) u ( σ ) dσ, h ( t ) = 0 for t < 0 0 � ∞ Suppose h ∈ L 1 ⇔ � h � L 1 = | h ( σ ) | dσ < ∞ 0 � t | y ( t ) | ≤ 0 | h ( t − σ ) | | u ( σ ) | dσ � t ≤ 0 | h ( t − σ ) | dσ sup 0 ≤ σ ≤ τ | u ( σ ) | � t = 0 | h ( s ) | ds sup 0 ≤ σ ≤ τ | u ( σ ) | � y τ � L ∞ ≤ � h � L 1 � u τ � L ∞ , ∀ τ ∈ [0 , ∞ ) Finite-gain L ∞ stable Also, finite-gain L p stable for p ∈ [1 , ∞ ) (see textbook) Nonlinear Control Lecture # 14 Input-Output Stability

Small-signal L Stability Example 6.3 y = tan u The output y ( t ) is defined only when the input signal is restricted to | u ( t ) | < π/ 2 for all t ≥ 0 � tan r � u ( t ) ∈ {| u | ≤ r < π/ 2 } ⇒ | y | ≤ | u | r � tan r � � y � L p ≤ � u � L p , p ∈ [1 , ∞ ] r Nonlinear Control Lecture # 14 Input-Output Stability

Definition 6.3 A mapping H : L m e → L q e is small-signal L stable (respectively, small-signal finite-gain L stable) if there is a positive constant r such that the condition for L stability ( respectively, finite-gain L stability ) is satisfied for all u ∈ L m e with sup 0 ≤ t ≤ τ � u ( t ) � ≤ r Nonlinear Control Lecture # 14 Input-Output Stability

L Stability of State Models x = f ( x, u ) , ˙ y = h ( x, u ) , 0 = f (0 , 0) , 0 = h (0 , 0) Case 1: The origin of ˙ x = f ( x, 0) is exponentially stable Theorem 6.1 Suppose, ∀ � x � ≤ r , ∀ � u � ≤ r u , c 1 � x � 2 ≤ V ( x ) ≤ c 2 � x � 2 � � ∂V ∂V ∂x f ( x, 0) ≤ − c 3 � x � 2 , � � � ≤ c 4 � x � � � ∂x � � f ( x, u ) − f ( x, 0) � ≤ L � u � , � h ( x, u ) � ≤ η 1 � x � + η 2 � u � � Then, for each x 0 with � x 0 � ≤ r c 1 /c 2 , the system is small-signal finite-gain L p stable for each p ∈ [1 , ∞ ] . It is finite-gain L p stable ∀ x 0 ∈ R n if the assumptions hold globally [see the textbook for β and γ ] Nonlinear Control Lecture # 14 Input-Output Stability

Proof V = ∂V ∂x f ( x, 0) + ∂V ˙ ∂x [ f ( x, u ) − f ( x, 0)] √ V ≤ − c 3 � x � 2 + c 4 L � x � � u � ≤ − c 3 V + c 4 L ˙ � u � V √ c 1 c 2 � W ( x ) = V ( x ) a = c 3 , b = c 4 L ˙ W ≤ − aW + b � u ( t ) � , 2 √ c 1 2 c 2 U = e at ˙ ˙ U ( t ) = e at W ( x ( t )) ⇒ W + ae at W ≤ be at � u � � t be aτ � u ( τ ) � dτ U ( t ) ≤ U (0) + 0 Nonlinear Control Lecture # 14 Input-Output Stability

� t W ( x ( t )) ≤ e − at W ( x (0)) + e − a ( t − τ ) b � u ( τ ) � dτ 0 √ c 1 � x � ≤ W ( x ) ≤ √ c 2 � x � � t � c 2 � x (0) � e − at + c 4 L e − at � u ( τ ) � dτ � x ( t ) � ≤ c 1 2 c 1 0 � y ( t ) � ≤ η 1 � x ( t ) � + η 2 � u ( t ) � � t � y ( t ) � ≤ k 0 � x (0) � e − at + k 2 e − a ( t − τ ) � u ( τ ) � dτ + k 3 � u ( t ) � 0 Nonlinear Control Lecture # 14 Input-Output Stability

Example 6.4 x = − x − x 3 + u, ˙ y = tanh x + u ˙ V = 1 2 x 2 V = x ( − x − x 3 ) ≤ − x 2 ⇒ c 1 = c 2 = 1 2 , c 3 = c 4 = 1 , L = η 1 = η 2 = 1 Finite-gain L p stable for each x (0) ∈ R and each p ∈ [1 , ∞ ] Example 6.5 x 1 = x 2 , ˙ x 2 = − x 1 − x 2 − a tanh x 1 + u, ˙ y = x 1 , a ≥ 0 V ( x ) = x T Px = p 11 x 2 1 + 2 p 12 x 1 x 2 + p 22 x 2 2 Nonlinear Control Lecture # 14 Input-Output Stability

˙ − 2 p 12 ( x 2 V = 1 + ax 1 tanh x 1 ) + 2( p 11 − p 12 − p 22 ) x 1 x 2 − 2 ap 22 x 2 tanh x 1 − 2( p 22 − p 12 ) x 2 2 p 11 = p 12 + p 22 ⇒ the term x 1 x 2 is canceled p 22 = 2 p 12 = 1 ⇒ P is positive definite ˙ V = − x 2 1 − x 2 2 − ax 1 tanh x 1 − 2 ax 2 tanh x 1 V ≤ −� x � 2 + 2 a | x 1 | | x 2 | ≤ − (1 − a ) � x � 2 ˙ a < 1 ⇒ c 1 = λ min ( P ) , c 2 = λ max ( P ) , c 3 = 1 − a, c 4 = 2 c 2 L = η 1 = 1 , η 2 = 0 For each x (0) ∈ R 2 , p ∈ [1 , ∞ ] , the system is finite-gain L p stable γ = 2[ λ max ( P )] 2 / [(1 − a ) λ min ( P )] Nonlinear Control Lecture # 14 Input-Output Stability

Case 2: The origin of ˙ x = f ( x, 0) is asymptotically stable Theorem 6.2 Suppose that, for all ( x, u ) , f is locally Lipschitz and h is continuous and satisfies � h ( x, u ) � ≤ g 1 ( � x � ) + g 2 ( � u � ) + η, η ≥ 0 for some gain functions g 1 , g 2 . If ˙ x = f ( x, u ) is ISS, then, for each x (0) ∈ R n , the system x = f ( x, u ) , ˙ y = h ( x, u ) is L ∞ stable Nonlinear Control Lecture # 14 Input-Output Stability

Proof � � �� � x ( t ) � ≤ max β ( � x 0 � , t ) , γ sup � u ( t ) � 0 ≤ t ≤ τ � � � ��� � y ( t ) � ≤ g 1 max β ( � x 0 � , t ) , γ sup 0 ≤ t ≤ τ � u ( t ) � + g 2 ( � u ( t ) � ) + η g 1 (max { a, b } ) ≤ g 1 ( a ) + g 1 ( b ) � y τ � L ∞ ≤ g ( � u τ � L ∞ ) + β 0 g = g 1 ◦ γ + g 2 and β 0 = g 1 ( β ( � x 0 � , 0)) + η Nonlinear Control Lecture # 14 Input-Output Stability

Theorem 6.3 Suppose f is locally Lipschitz and h is continuous in some neighborhood of ( x = 0 , u = 0) . If the origin of ˙ x = f ( x, 0) is asymptotically stable, then there is a constant k 1 > 0 such that for each x (0) with � x (0) � < k 1 , the system x = f ( x, u ) , ˙ y = h ( x, u ) is small-signal L ∞ stable Proof Use Lemma 4.7 (asymptotic stability is equivalent to local ISS) Nonlinear Control Lecture # 14 Input-Output Stability

Example 6.6 x = − x − 2 x 3 + (1 + x 2 ) u 2 , y = x 2 + u ˙ ISS from Example 4.13 g 1 ( r ) = r 2 , g 2 ( r ) = r, η = 0 L ∞ stable Nonlinear Control Lecture # 14 Input-Output Stability

Example 6.7 x 1 = − x 3 x 2 = − x 1 − x 3 ˙ 1 + x 2 , ˙ 2 + u, y = x 1 + x 2 ˙ V = ( x 2 1 + x 2 V = − 2 x 4 1 − 2 x 4 2 ) ⇒ 2 + 2 x 2 u x 4 1 + x 4 2 � x � 4 2 ≥ 1 −� x � 4 + 2 � x �| u | ˙ V ≤ − (1 − θ ) � x � 4 − θ � x � 4 + 2 � x �| u | , = 0 < θ < 1 � 1 / 3 � 2 | u | − (1 − θ ) � x � 4 , ≤ ∀ � x � ≥ ⇒ ISS θ √ g 1 ( r ) = 2 r, g 2 = 0 , η = 0 L ∞ stable Nonlinear Control Lecture # 14 Input-Output Stability