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50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Necessary and Sufficient Conditions for Input-Output Finite-Time Stability of Linear Time-Varying Systems Francesco Amato 1 Giuseppe Carannante 2 Gianmaria De Tommasi 2 Alfredo Pironti 2 1 Universit` a degli Studi Magna Græcia di Catanzaro, Catanzaro, Italy, 2 Universit` a degli Studi di Napoli Federico II, Napoli, Italy Joint 50 th IEEE Conference on Decision and Control & European Control Conference December 12–15, 2011, Orlando, Florida

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Outline Outline 1 Motivations 2 Preliminaries Notation Problem Statement Preliminary result 3 Main Theorem 4 Numerical example

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Motivations Input-output finite-time stability vs classic IO stability IO stability A system is said to be IO L p -stable if for any input of class L p , the system exhibits a corresponding output which belongs to the same class IO-FTS A system is defined to be IO-FTS if, given a class of norm bounded input signals over a specified time interval T , the outputs of the system do not exceed an assigned threshold during T

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Motivations Main features of IO-FTS IO-FTS: involves signals defined over a finite time interval does not necessarily require the inputs and outputs to belong to the same class specifies a quantitative bounds on both inputs and outputs IO stability and IO-FTS are independent concepts

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Motivations Contribution of the paper In this paper we show that, in the case of L 2 inputs, the sufficient condition given in F. Amato, R. Ambrosino, G. De Tommasi, C. Cosentino Input-output finite-time stabilization of linear systems Automatica , 2010 is also necessary . To prove this result, a machinery involving the teachability gramian is used.

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Preliminaries Notation Notation L p denotes the space of vector-valued signals whose p -th power is absolutely integrable over [0 , + ∞ ). The restriction of L p to Ω := [ t 0 , t 0 + T ] is denoted by L p (Ω). Given the time interval Ω, a symmetric positive definite matrix-valued function R ( · ), bounded on Ω, and a vector-valued signal s ( · ) ∈ L p (Ω), the weighted signal norm � 1 �� � p p 2 d τ s T ( τ ) R ( τ ) s ( τ ) � , Ω will be denoted by � s ( · ) � p , R . If p = ∞ � 1 2 . s T ( t ) R ( t ) s ( t ) � � s ( · ) � ∞ , R = ess sup t ∈ Ω

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Preliminaries Notation LTV systems as Linear Operator Let us consider a LTV system in the form � ˙ x ( t ) = A ( t ) x ( t ) + G ( t ) w ( t ) , x ( t 0 ) = 0 Γ : (1) y ( t ) = C ( t ) x ( t ) Γ can be viewed as a linear operator mapping input signals ( w ( · )’s) into output signals ( y ( · )’s). Φ( t , τ ) denotes the state transition matrix of system (1).

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Preliminaries Notation Reachability Gramian The reachability Gramian of system (1) is defined as � t Φ( t , τ ) G ( τ ) G T ( τ )Φ T ( t , τ ) d τ . W r ( t , t 0 ) � t 0 W r ( t , t 0 ) is symmetric and positive semidefinite for all t ≥ t 0 . Given system (1), W r ( t , t 0 ) is the unique solution of the matrix differential equation ˙ W r ( t , t 0 ) = A ( t ) W r ( t , t 0 ) + W r ( t , t 0 ) A T ( t ) + G ( t ) G T ( t ) , (2a) W r ( t 0 , t 0 ) = 0 (2b)

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Preliminaries Problem Statement IO-FTS of LTV systems Given a positive scalar T , a class of input signals W defined over Ω = [ t 0 , t 0 + T ], a positive definite matrix-valued function Q ( · ) defined in Ω, system (1) is said to be IO-FTS with � � respect to W , Q ( · ) , Ω if w ( · ) ∈ W ⇒ y T ( t ) Q ( t ) y ( t ) < 1 , t ∈ Ω . In this work we consider the class of norm bounded square integrable signals over Ω � � � � W 2 Ω , R ( · ) := w ( · ) ∈ L 2 (Ω) : � w � 2 , R ≤ 1 , where R ( · ) denotes a continuous positive definite matrix-valued function.

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Preliminaries Problem Statement Linear operator The LTV system (1) is regarded as a linear operator that maps signals from the space L 2 (Ω) to the space L ∞ (Ω) Γ : w ( · ) ∈ L 2 (Ω) �→ y ( · ) ∈ L ∞ (Ω) . (3) If we equip the L 2 (Ω) and L ∞ (Ω) spaces with the weighted norms � · � 2 , R and � · � ∞ , Q , respectively, the induced norm of the linear operator (3) is given by � � � Γ � = sup � y ( · ) � ∞ , Q , � w ( · ) � 2 , R =1 Theorem 1 Given a time interval Ω, the class of input signals W 2 , and a continuous positive definite matrix-valued function Q ( · ), � � system (1) is IO-FTS with respect to W 2 , Q ( · ) , Ω if and only if � Γ � < 1.

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Preliminaries Problem Statement Dual operator Given the linear operator (3), its dual operator is ¯ Γ : z ( · ) ∈ L 1 (Ω) �→ v ( · ) ∈ L 2 (Ω) , with � � � Γ � = sup � v ( · ) � 2 , R . � z ( · ) � 1 , Q =1 By definition it holds � Γ � = � ¯ Γ � , (4) and � z , Γ w � = � ¯ Γ z , w � , (5) where z ( · ) ∈ L 1 (Ω) and w ( · ) ∈ L 2 (Ω).

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Preliminaries Preliminary result Theorem 2 Given the LTV system (1), the norm of the corresponding linear operator (3) is given by 1 � � 1 1 2 ( t ) C ( t ) W ( t , t 0 ) C T ( t ) Q 2 ( t ) � Γ � = ess sup λ 2 Q , (6) max t ∈ Ω for all t ∈ Ω, where λ max ( · ) denotes the maximum eigenvalue, and W ( t , t 0 ) is the positive semidefinite matrix-valued solution of ˙ W ( t , t 0 ) = A ( t ) W ( t , t 0 ) + W ( t , t 0 ) A T ( t ) + G ( t ) R ( t ) − 1 G T ( t ) (7a) W ( t 0 , t 0 ) = 0 (7b)

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Preliminaries Preliminary result Sketch of proof - 1 For the sake of simplicity, the weighting matrices R ( t ) and Q ( t ) are set equal to the identity; it follows that the solution of (7) is given by the reachability gramian W r ( t , t 0 ); Considering the dual operator ¯ Γ, proving (6) is equivalent to show 1 � C ( t ) W r ( t , t 0 ) C T ( t ) � � ¯ Γ � = ess sup λ 2 . max t ∈ Ω We denote with H ( t , τ ) = G T ( t )Φ T ( τ , t ) C T ( τ ) δ − 1 ( τ − t ) ¯ the impulsive response of the dual system � ˙ x ( t ) = − A T ( t )˜ x ( t ) − C T ( t ) z ( t ) ˜ ¯ Γ : . v ( t ) = G T ( t )˜ x ( t )

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Preliminaries Preliminary result Sketch of proof - 2 Using ¯ H ( t , τ ) it is possible to show that � � � 1 C ( t ) W r ( t , t 0 ) C T ( t ) � � ¯ � � 2 � v ( · ) � 2 = H ( · , τ ) z ( τ ) d τ ≤ ess sup λ · � z ( · ) � 1 � � max � � t ∈ Ω Ω 2 Hence 1 � C ( t ) W r ( t , t 0 ) C T ( t ) � � ¯ 2 Γ � ≤ ess sup λ max t ∈ Ω Exploiting similar arguments as in D. A. Wilson Convolution and hankel operator norms for linear IEEE Trans. on Auto. Contr. , 1989 it is possible to show that 1 � C ( t ) W r ( t , t 0 ) C T ( t ) � � ¯ Γ � = ess sup 2 λ , max t ∈ Ω which proofs the theorem.

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Preliminaries Preliminary result Remark If the system matrices in (1) and the weighting matrices R ( · ) and Q ( · ) are assumed to be continuous, in the closed time interval Ω the condition (6) is equivalent to 1 � 1 1 � 2 ( t ) C ( t ) W ( t , t 0 ) C T ( t ) Q 2 ( t ) � Γ � = max t ∈ Ω λ 2 Q . max

50 th IEEE Conf. on Decision and Control – European Control Conf. – Orlando, Florida 12–15 Dec. 2011 Main Theorem Theorem 3 The following statements are equivalent: i) System (1) is IO-FTS with respect to � W 2 , Q ( · ) , Ω � . ii) The inequality 1 1 2 ( t ) C ( t ) W ( t , t 0 ) C T ( t ) Q � � 2 ( t ) λ max Q < 1 (8) holds for all t ∈ Ω, where W ( · , · ) is the positive semidefinite solution of the Differential Lyapunov Equality (DLE) (7). iii) The coupled DLMI/LMI � ˙ P ( t ) + A T ( t ) P ( t ) + P ( t ) A ( t ) � P ( t ) G ( t ) < 0 (9a) G T ( t ) P ( t ) − R ( t ) P ( t ) > C T ( t ) Q ( t ) C ( t ) , (9b) admits a positive definite solution P ( · ) over Ω.