On the well-posedness of cascades of analytic nonlinear input-output - PowerPoint PPT Presentation

On the well-posedness of cascades of analytic nonlinear input-output systems driven by noise ∗ Luis A. Duffaut Espinosa Department of Electrical and Computer Engineering The Johns Hopkins University, Baltimore, Maryland USA ∗ This research was supported in part by NSF grant DMS 0960589.

RPCCT 2011 Overview ∗ 1. Fliess Operators 1.1. Formal Power Series 1.2. Fliess Operators with Stochastic Inputs 2. Convergence of Fliess Operators with Stochastic Inputs 2.1. Global Convergence 2.2. Local Convergence 2.3. Solving a type of polynomial differential equations 3. System Interconnections with Stochastic Inputs 3.1. Formal Interconnections 3.2. Parallel and Product Interconnections 3.3. Cascade Interconnection ∗ See www.ece.odu.edu/ ∼ sgray/RPCCT2011/duffautespinosaslides.pdf 2

RPCCT 2011 1. Fliess Operators • Functional series expansions of nonlinear input-output operators have been utilized since the early 1900’s in engineering, mathematics and physics ( V. Volterra, N. Wiener, etc ). • A broad class of deterministic nonlinear systems can be described by Fliess operators, which are input-output maps constructed using the Chen-Fliess formalism ( Fliess (1981)). • Such operators are described by a summation of Lebesgue iterated integrals codified using the theory of noncommutative formal power series. 3

RPCCT 2011 1.1 Formal Power Series • Let X = { x 0 , x 1 , . . . , x m } be an alphabet and X ∗ the set of all words over X (including the empty word ∅ ). • A formal power series is any mapping c : X ∗ → R ℓ . Typically, c is written as a formal sum � c = ( c, η ) η. η ∈ X ∗ • The set of all such series is denoted by R ℓ �� X �� , and the subset denoted by R ℓ � X � is the set of polynomials. • A series c is rational if it belongs to the rational closure of R ℓ � X � . • A series c is rational if and only if ( c, η ) = λµ ( η ) γ, ∀ η ∈ X ∗ , where µ : X ∗ → R n × n is a monoid morphism, and γ , λ T ∈ R n × 1 . • c is called globally convergent when | ( c, η ) | ≤ KM | η | , ∀ η ∈ X ∗ . • c is called locally convergent when | ( c, η ) | ≤ KM | η | | η | !, ∀ η ∈ X ∗ . 4

RPCCT 2011 • For a measurable function u : [ a, b ] → R m with finite L 1 -norm, define E η : L m 1 [ t 0 , t 0 + T ] → C [ t 0 , t 0 + T ] by E ∅ [ u ] = 1, and � t E x i η ′ [ u ]( t, t 0 ) = u i ( τ ) E η ′ [ u ]( τ, t 0 ) dτ, t 0 (1) where x i ∈ X , η ′ ∈ X ∗ and u 0 = 1. • Note that to each letter x i is assigned a function u i . • Each c ∈ R ℓ �� X �� is associated with an m -input, ℓ -output system, � F c [ u ]( t ) = ( c, η ) E η [ u ]( t, t 0 ) , η ∈ X ∗ called a Fliess operator ( Fliess (1981)). 5

RPCCT 2011 Example 1 A linear input-output system F : u → y with u ( t ) ∈ R m and y ( t ) ∈ R ℓ can be described by a convolution integral involving its impulse response H ( t, τ ) = ( H 1 ( t, τ ) , . . . , H m ( t, τ )) ′ and the system input � t H ( t, τ ) u ( τ ) dτ, t ≥ t 0 . y ( t ) = F [ u ]( t ) = (2) t 0 If each H i is real analytic on D = { ( t, τ ) ∈ R 2 : t 0 ≤ τ ≤ t ≤ t 0 + T } , then its Taylor series at ( τ, t 0 ) is ∞ � c ( n 2 , i, n 1 )( t − τ ) n 2 ( τ − t 0 ) n 1 H i ( t, τ ) = , (3) n 2 ! n 1 ! n 1 ,n 2 =0 where c ( n 2 , i, n 1 ) ∈ R ℓ . 6

RPCCT 2011 Substituting (3) into (2) and using the uniform convergence of the series on D , it follows that � t ∞ ,m � ( t − τ ) n 2 u i ( τ )( τ − t 0 ) n 1 y ( t ) = c ( n 2 , i, n 1 ) dτ . (4) n 2 ! n 1 ! t 0 n 1 ,n 2 =0 ,i =1 � �� � E x n 2 0 [ u ]( t, t 0 ) x i x n 1 0 Thus, (4) can be written as ∞ ,m � y ( t ) = c ( n 2 , i, n 1 ) E x n 2 0 [ u ]( t, t 0 ) . x i x n 1 0 n 1 ,n 2 =0 ,i =1 Observe that the formal power series associated with system (2) is   : η = x n 2 0 x i x n 1 c ( n 2 , i, n 1 ) 0 , n 1 , n 2 ≥ 0 , i � = 0 ( c, η ) =  0 : otherwise . 7

RPCCT 2011 1.2 Fliess Operators with Stochastic Inputs • System inputs in applications usually have noise. • Several authors have formulated approaches where Wiener processes are admissible inputs to a Fliess operators ( G. B. Arous (1989), Fliess (1977 , 1981), Fliess and Lamnabhi (1981), Sussmann (1988)). • A suitable mathematical formulation will use Stratonovich integrals: i. They obey the rules of ordinary differential calculus. ii. When schemes for solving stochastic differential equations use smooth functions to approximate white Gaussian noise, the appropriate model will use Stratonovich integrals. 8

RPCCT 2011 Example 2 Let W be a Wiener process. Consider a system modeled by the stochastic differential equation (SDE) in Stratonovich form � t � t z t = z 0 + f ( z s ) ds + S g ( z s ) dW ( s ) , (5) 0 0 where f ( z ) and g ( z ) are suitably defined functions. For a C 2 function F , the Stratonovich differential chain rule gives � t � t f ( z s ) ∂ g ( z s ) ∂ F ( z t ) = F ( z t ) + ∂z F ( z s ) ds + S ∂z F ( z s ) dW ( s ) . (6) 0 0 Identifying operators L f = f ( z ) ∂ ∂z and L g = g ( z ) ∂ ∂z , (6) becomes � t � t F ( z t ) = F ( z 0 ) + L f F ( z s ) ds + S L g F ( z s ) dW ( s ) . 0 0 9

RPCCT 2011 Now let F ( z ) in (6) be replaced by either f or g from (5) and substitute f ( z t ) and g ( z t ) back into (5). This yields � t � t ds + g ( z 0 ) S z t = z 0 + f ( z 0 ) dW ( s ) 0 0 � t � s � t � s S + L f f ( z r ) drds + L g f ( z r ) dW ( r ) ds 0 0 0 0 � t � s � t � s + S L f g ( z r ) drdW ( s ) + S S L g g ( z r ) dW ( r ) dW ( s ) 0 0 0 0 � t � t z t = z 0 + f ( z 0 ) ds + g ( z 0 ) S dW ( s ) + R 1 ( z ( t )) . 0 0 � �� � � �� � E x 0 [0]( t, 0) E y 0 [0]( t, 0) Continuing this way produces the usual Peano-Baker formula. 10

RPCCT 2011 Let I be the identity map and define X = { x 0 } , Y = { y 0 } , XY = X ∪ Y , L g x 0 η = L g η L g x 0 and L g y 0 η = L g η L g y 0 , where g x 0 = f , g y 0 = g and η ∈ XY ∗ . Thus, the solution of the SDE (5) in series form is � y ( t ) � z ( t ) = η ∈ XY ∗ L g η I ( z (0)) E η [0]( t ) (7) Here, ( f, g, I, z (0)) realizes F c when ( c, η ) = L g η I ( z (0)), ∀ η ∈ XY ∗ . Remarks: • The output of this nonlinear input-output system is in general not a Wiener process. For example, equation (7) can be written as � t � y ( t ) = ( c, ∅ ) + L g x 0 η I ( z (0)) E x 0 η [0]( s, 0) ds 0 η ∈ XY ∗ � t � + S L g y 0 η I ( z (0)) E y 0 η [0]( s, 0) dW ( s ) . 0 η ∈ XY ∗ • Note that y ( t ) is not well-defined unless the integrands converge. 11

RPCCT 2011 • Consider a Wiener process, denoted by W ( t ), defined over (Ω , F , P ). • Let u : Ω × [ t 0 , t 0 + T ] → R m be a predictable function, and � u � p = max {� u i � L p : 1 ≤ i ≤ m } . Definition 1 ( Duffaut et al. 2009) Consider the set of all m -dimensional m [ t 0 , t 0 + T ], which stochastic processes over [ t 0 , t 0 + T ], denoted by � UV can be written as � � t t w ( t ) = u ( s ) ds + S v ( s ) dW ( s ) . t 0 t 0 m [ t 0 , t 0 + T ] will refer to processes The set UV m [ t 0 , t 0 + T ] ⊂ � UV satisfying: i. Each m -dimensional integrand has E [ u i ( t )] < ∞ , E [ v i ( t )] < ∞ , t ∈ [ t 0 , t 0 + T ] and are mutually independents. ii. Also, � u � L 2 , � v � L 2 , � v � L 4 ≤ R ∈ R + . 12

RPCCT 2011 � t Definition 2 ( Duffaut et al. 2010) Let X ( t ) = S v ( s ) dW ( s ), where v is 0 o process. The set UV m [0 , τ R ] is defined as the an m -dimensional L 2 -Itˆ set of processes w ∈ UV m [0 , T ] stopped at � � � � � t � � � � τ R � i ∈{ 0 , 1 , ··· ,m } inf min t ∈ T : � S v i ( s ) dW ( s ) � = R . 0 X t ( ) R ! R R t Figure 1: First time process X ( t ) hits the barrier R . Remark: τ R is a strictly positive stopping time for any real R > 0. 13

RPCCT 2011 • Let X = { x 0 , x 1 , . . . , x m } , Y = { y 0 , y 1 , . . . , y m } and XY = X ∪ Y . • An iterated integral over UV m [ t 0 , t 0 + T ] is defined recursively by � t − u i ( s ) E η ′ [ w ]( s ) ds, x i ∈ X, E x i η ′ [ w ]( t, t 0 ) = t 0 � t − E y i η ′ [ w ]( t, t 0 ) = S v i ( s ) E η ′ [ w ]( s ) dW ( s ) , y i ∈ Y, t 0 where η ′ ∈ XY ∗ , E ∅ = 1 and u 0 = v 0 = 1. Definition 3 ( Duffaut et al. 2009) An m -input, ℓ -output Fliess operator F c , c ∈ R ℓ �� XY �� , driven by w ∈ UV m [0 , T ] is formally defined as � F c [ w ]( t ) = ( c, η ) E η [ w ]( t, t 0 ) . η ∈ XY ∗ (8) 14

RPCCT 2011 Definition 4 For any T > 0, w ∈ UV m [0 , T ] and t ∈ [0 , T ], the Chen series associated with a formal power series in R ℓ �� XY �� is defined as � P [ w ]( t, t 0 ) = η E η [ w ]( t, t 0 ) . η ∈ XY ∗ • The Chen series satisfies the stochastic differential equation � m � � dP [ w ]( t, t 0 ) = x i u i ( t ) dt + y i v i ( t ) dW ( t ) P [ w ]( t, t 0 ) . i =0 ⊔ ν ) = ( P [ u ] , ξ ) ( P [ u ] , ν ) , ∀ ξ, ν ∈ XY ∗ . • For any t , ( P [ u ] , ξ ⊔ Therefore, from Ree’s theorem P [ u ], is an exponential Lie series. • The Fliess operator (8) can be written as F c [ w ]( t ) = ( c, P [ w ]( t, 0)) • P [ w ] satisfies P [ w ]( t, t 0 ) = P [ w ]( t, t ′ ) P [ w ]( t ′ , t 0 ) (Chen’s identity). 15