Discrete-time Chen–Fliess Series for Learning and Adaptive Control ∗ W. Steven Gray Old Dominion University, Norfolk, Virginia USA ACPMS Seminar August 21, 2020 ∗ Joint work with Luis A. Duffaut Espinosa and G. S. Venkatesh . Supported by NSF grants CMMI-1839378 and CMMI-1839387.
ACPMS Seminar Overview 1. Motivation - The canonical control problem 2. Chen series and Chen–Fliess series 3. Discrete-time analogues 4. Implementation of a learning unit 5. Application to adaptive control 2
ACPMS Seminar 1. Motivation - The canonical control problem y u F � plant Fig. 1.1: Open-loop control • Suppose F is an operator mapping a set of input functions U to a set of output functions Y . • Select some desired y d ∈ Range( F ) ⊆ Y . • The canonical control problem is to determine a right inverse u = F − 1 [ y d ] ∈ U such that F [ u ] = ( F ◦ F − 1 )[ y d ] = y d . • Whenever F − 1 can be computed explicitly, this is called open-loop control. 3
ACPMS Seminar • In many applications F is known to be realized by some finite dimensional state space model m � z = g 0 ( z ) + ˙ g i ( z ) u i , z (0) = z 0 i =1 y j = h j ( z ) , j = 1 , . . . , ℓ, where the g i are vector fields in local coordinates, and h j maps the state to the j -th output. • If ( g, h, z 0 ) are known, then under certain conditions (well defined relative degree), F − 1 exists on a neighborhood of z 0 and is computable. • But normally a state space model is only known approximately, so this option is not always feasible. 4
ACPMS Seminar u e y d y H � F � + _ controller plant Fig. 1.2: Closed-loop control • A more practical approach is to design a mapping H : Y → U so that lim t → ∞ | y d ( t ) − y ( t ) | = 0. • This is closed-loop control via dynamic inversion. • In general closed-loop control is known to reduce the sensitivity of the output to the plant, ∂y/∂F . • The design of H still relies on some knowledge of F , usually a nominal state space model ( g, h, z 0 ). 5
ACPMS Seminar u e y d y H � F � + _ plant adaptive � controller learning unit Fig. 1.3: Adaptive closed-loop control • When F is unknown, the idea is to learn F as the system operates and then tune H in real-time. This is known as adaptive control. • All such control systems are implemented in discrete-time. • The goal is to describe an adaptive control system whose implementation utilizes a discrete-time Chen–Fliess series. 6
ACPMS Seminar 2. Chen series and Chen–Fliess series Definition 2.1: (Chen, 1952) Let X = { x 0 , x 1 , . . . , x m } . For any fixed u ∈ L m 1 [ t 0 , t 1 ] and t ∈ [ t 0 , t 1 ] one can associate the formal power series in R �� X �� � P [ u ]( t, t 0 ) = η E η [ u ]( t, t 0 ) , η ∈ X ∗ where the map E η : L m 1 [ t 0 , t 1 ] → C [ t 0 , t 1 ] is defined inductively by setting E ∅ [ u ] = 1 and letting � t E x i η [ u ]( t, t 0 ) = u i ( τ ) E η [ u ]( τ, t 0 ) dτ, t 0 with x i ∈ X , η ∈ X ∗ , and u 0 ( t ) := 1. Such a series is called a Chen series. Remark: For any fixed t ≥ 0, P [ u ]( t ) := P [ u ]( t, 0) is an exponential Lie series satisfying � � m d � dtP [ u ] = x 0 + x i u i P [ u ] , P [ u ](0) = 1 . i =1 7
� � ACPMS Seminar u u v# � u v v t t t c t b t d t c t b t d t a t a 0 0 Fig. 2.1 The catenation of two inputs u and v at t = τ The set of functions � L m L m 1 (0) := 1 [0 , T ] 0 ≤ T < ∞ is a monoid under this catenation operator. Theorem 2.1: (Chen’s identity) Given ( u, v ) ∈ L m 1 [ t a , t b ] × L m 1 [ t c , t d ], τ ∈ [ t a , t b ], and t ∈ [ τ, τ + ( t d − t c )] it follows that P [ v ](( t − τ ) + t c , t c ) P [ u ]( τ, t a ) = P [ v # τ u ]( t, t a ) . 8
ACPMS Seminar Remarks: • The set of Chen series G C ( X ) = { P [ u ]( t ) ∈ R �� X �� : u ∈ L m 1 [0 , T ] , 0 ≤ t ≤ T < ∞} defines a monoid under the Cauchy product. • P : L m 1 (0) → G C ( X ) acts as a monoid homomorphism. • G C ( X ) constitutes a group if the drift letter x 0 is omitted. Definition 2.2: (Fliess, 1981) For c ∈ R ℓ �� X �� , the corresponding Chen–Fliess series is � y ( t ) = F c [ u ]( t ) := ( c, η ) E η [ u ]( t ) η ∈ X ∗ � = ( c, η )( P [ u ]( t ) , η ) η ∈ X ∗ =: ( c, P [ u ]( t )) . 9
ACPMS Seminar Remarks: • If there exists real numbers K, M ≥ 0 such that | ( c, η ) | ≤ KM | η | | η | ! , ∀ η ∈ X ∗ then the series defining F c converges. • F c defined is said to be realizable when there exists a state space model m � z = g 0 ( z ) + ˙ g i ( z ) u i , z ( t 0 ) = z 0 i =1 y j = h j ( z ) , j = 1 , 2 , . . . , ℓ, such that y j = F c j [ u ] = h j ( z ), j = 1 , 2 , . . . , ℓ . • In this case, for any word η = x i k · · · x i 1 ∈ X ∗ ( c j , η ) = L g η h j ( z 0 ) := L g i 1 · · · L g ik h j ( z 0 ) , where L g i h j is the Lie derivative of h j with respect to g i . 10
ACPMS Seminar 3. Discrete-time analogues Here inputs are sequences from the normed linear space l m +1 ( N 0 ) := { ˆ u = (ˆ u ( N 0 ) , ˆ u ( N 0 + 1) , . . . ) : � ˆ u � ∞ < ∞} , ∞ u m ( N )] T , N ≥ N 0 . where ˆ u ( N ) := [ˆ u 0 ( N ) , ˆ u 1 ( N ) , . . . , ˆ u ∈ l m +1 Definition 3.1: Given any N ≥ N 0 and ˆ ( N 0 ), a discrete-time ∞ Chen series is defined as � S [ˆ u ]( N, N 0 ) = ηS η [ˆ u ]( N, N 0 ) , η ∈ X ∗ where N � S x i η [ˆ u ]( N, N 0 ) = u i ( k ) S η [ˆ ˆ u ]( k, N 0 ) k = N 0 with x i ∈ X , η ∈ X ∗ , and S ∅ [ˆ u ]( N, N 0 ) := 1. Remark: If N 0 = 0 then S [ˆ u ]( N, 0) is abbreviated as S [ˆ u ]( N ). 11
ACPMS Seminar The discrete-time Chen series S [ˆ u ] satisfies a difference equation. For η = x i k · · · x i 1 ∈ X ∗ , define ˆ u i k ( N ) · · · ˆ u η ( N ) = ˆ u i 1 ( N ) and � c u ( N ) = u η ( N ) η. ˆ η ∈ X ∗ Example 3.1: If X = { x 1 } , then ˆ u x 1 ( N ) = ˆ u 1 ( N ) and ∞ u 1 ( N ) x 1 ) k = (1 − ˆ � u 1 ( N ) x 1 ) − 1 . c u ( N ) = (ˆ k =0 u ∈ l m +1 Theorem 3.1: For any ˆ ( N 0 ) and N ≥ N 0 , ∞ S [ˆ u ]( N + 1 , N 0 ) = c u ( N + 1) S [ˆ u ]( N, N 0 ) with S [ˆ u ]( N 0 , N 0 ) = c u ( N 0 ). In addition, ← − − N � S [ˆ u ]( N, N 0 ) = c u ( i ) . i = N 0 12
ACPMS Seminar Remarks: • There is a discrete-time analogue of Chen’s identity. • Both l m +1 ∞ ,e (0) := l m +1 (0) ∪ { ˆ 0 } and the set of discrete-time Chen ∞ series, M C , form monoids. • S : l m +1 ∞ ,e (0) → M C is a monoid homomorphism. • Given c ∈ R ℓ �� X �� , the corresponding discrete-time Chen–Fliess series is defined as y ( N ) = ˆ � ˆ F c [ˆ u ]( N ) := ( c, η ) S η [ˆ u ]( N, N 0 ) η ∈ X ∗ = ( c, S [ˆ u ]( N, N 0 )) . • ˆ F c [ˆ u ] approximates its continuous-time counterpart, F c [ u ], with computable error bounds (Duffaut Espinosa, Ebrahimi-Fard, G., 2017). 13
ACPMS Seminar Theorem 3.2: The monoid M C ( X ) has a faithful infinite dimensional ← − − − � N real representation Π given by Π( S [ˆ u ]( N )) = i =0 S ( i ), where S ( i ) is any matrix representation of the R -linear map on R �� X �� given by the catenation map C : d �→ c u ( i ) d . Example 3.2: If X = { x 1 } then for all i ≥ 0 1 0 0 0 · · · u 1 ( i ) ˆ 1 0 0 · · · u 2 ˆ 1 ( i ) u 1 ( i ) ˆ 1 0 · · · S ( i ) = u 3 u 2 ˆ 1 ( i ) ˆ 1 ( i ) u 1 ( i ) ˆ 1 · · · . . . . ... . . . . . . . . u ]( N )) = S ( N ) · · · S (0). and Π( S [ˆ Remark: The goal is to find a convenient monoid representation of M C ( X ) when X has more than one letter. 14
� � ACPMS Seminar 4. Implementation of a learning unit y p discrete-time Chen-Fliess series � MSE parameter u estimator y Fig. 4.1 Learning unit based on a discrete-time Chen–Fliess series • The main idea is to approximate some unknown plant y = F c [ u ] by a truncated discrete-time Chen–Fliess series, y ( N ) = ˆ � F J ˆ c [ˆ u ]( N ) := ( c, η ) S η [ˆ u ]( N ) , η ∈ X ≤ J in order to predict future outputs. • All that is available to the learning unit is input-output data. 15
ACPMS Seminar • The first step is to express ˆ F J c [ˆ u ]( N ) in the form of a regression y ( N ) = φ T ( N ) θ 0 , N ≥ 1 , ˆ where u ]( N )] T φ ( N ) = [ S η 1 [ˆ u ]( N ) S η 2 [ˆ u ]( N ) · · · S η l [ˆ θ 0 = [( c, η 1 ) ( c, η 2 ) · · · ( c, η l )] T with l = card( X ≤ J ) and assuming some fixed order ( η 1 , η 2 , . . . , η l ). • If an estimate of θ 0 is available at time N − 1, say ˆ θ ( N − 1), a corresponding prediction of ˆ y ( N ) is y p ( N ) := φ T ( N )ˆ θ ( N − 1) . ˆ • ˆ θ ( N − 1) can be generated using any textbook recursive MSE estimation algorithm. The objective is to find an analogous implementation for the regressor, φ ( N ). 16
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