Model reduction of nonlinear systems using incremental system properties

Transcription

1 Model redction of nonlinear systems sing incremental system properties Bart Besselink ACCESS Linnaes Centre & Department of Atomatic Control KTH Royal Institte of Technology, Stockholm, Sweden L2S, Spelec, Gif-sr-Yvette, France, 8 October 2015

2 Acknowledgements Nathan van de Wow 1,2,3, Henk Nijmeijer 1 1 Eindhoven University of Technology, Eindhoven, the Netherlands 2 University of Minnesota, Minneapolis, USA 3 Delft University of Technology, Delft, the Netherlands Jacqelien Scherpen University of Groningen, Groningen, the Netherlands 2/29

3 Introdction Practical engineering problems typically lead to complex, high-order dynamical models Disadvantages of high-order models Difficlt system analysis Time-consming simlations Controller design infeasible Solar panel Robot Chemical plant 3/29

4 Introdction Practical engineering problems typically lead to complex, high-order dynamical models Nonlinearities often play an important role Mechanical systems: friction, backlash, hysteresis Electrical systems: nonlinear components, electrostatics Solar panel Robot Chemical plant 3/29

5 The model redction problem Problem. Given a dynamical system Σ, find a redced-order system ˆΣ that approximates its inpt-otpt behavior Σ y ˆΣ ŷ {ẋ = f(x)+g(x) Σ : y = h(x) with x R n, n large ˆΣ k : { ξ = ˆf(ξ)+ĝ(ξ) ŷ = ĥ(ξ) with ξ R k, k < n small 4/29

6 The model redction problem Problem. Given a dynamical system Σ, find a redced-order system ˆΣ that approximates its inpt-otpt behavior Σ y ˆΣ ŷ {ẋ = f(x)+g(x) Σ : y = h(x) with x R n, n large ˆΣ k : { ξ = ˆf(ξ)+ĝ(ξ) ŷ = ĥ(ξ) with ξ R k, k < n small Objectives 1. Preservation of system properties (in particlar, stability) 4/29

7 The model redction problem Problem. Given a dynamical system Σ, find a redced-order system ˆΣ that approximates its inpt-otpt behavior Σ ˆΣ + y ŷ e {ẋ = f(x)+g(x) Σ : y = h(x) with x R n, n large ˆΣ k : { ξ = ˆf(ξ)+ĝ(ξ) ŷ = ĥ(ξ) with ξ R k, k < n small Objectives 1. Preservation of system properties (in particlar, stability) 2. Bond on the redction error e = y ŷ (e.g., e 2 ε 2 ) 4/29

8 Model redction techniqes Objectives 1. Preservation of relevant stability properties 2. Error bond (a priori) For linear systems, model redction techniqes satisfying 1. and 2. exist (e.g., balanced trncation [Moore, Glover] and extensions) 5/29

9 Model redction techniqes Objectives 1. Preservation of relevant stability properties 2. Error bond (a priori) For linear systems, model redction techniqes satisfying 1. and 2. exist (e.g., balanced trncation [Moore, Glover] and extensions) Model redction for nonlinear systems Balanced trncation for nonlinear systems [Scherpen, Fjimoto] Moment matching for nonlinear systems [Astolfi] Trajectory piecewise linear approximation [Rewieński, White] Proper orthogonal decomposition [Sirovich, Berkooz] 5/29

10 Model redction techniqes Objectives 1. Preservation of relevant stability properties 2. Error bond (a priori) For linear systems, model redction techniqes satisfying 1. and 2. exist (e.g., balanced trncation [Moore, Glover] and extensions) Model redction for nonlinear systems Balanced trncation for nonlinear systems [Scherpen, Fjimoto] Moment matching for nonlinear systems [Astolfi] Trajectory piecewise linear approximation [Rewieński, White] Proper orthogonal decomposition [Sirovich, Berkooz] Existing model redction techniqes for nonlinear systems do not generally satisfy 1. and 2. 5/29

11 Otline Model redction for nonlinear systems 1. A class of convergent nonlinear systems 2. Nonlinear systems with incremental gain or passivity properties 3. Incremental balanced trncation for nonlinear systems 6/29

12 Otline Model redction for nonlinear systems 1. A class of convergent nonlinear systems 2. Nonlinear systems with incremental gain or passivity properties 3. Incremental balanced trncation for nonlinear systems Incremental system properties are crcial in 1. the preservation of relevant stability properties; 2. the derivation of a priori error bonds 6/29

13 Problem setting and approach v Σ lin Σ nl w y Motivation Nonlinearities often act only locally Examples Mechanical systems with friction, hysteresis Systems with nonlinear actator dynamics Variable-gain controlled linear systems 7/29

14 Problem setting and approach v Σ lin Σ nl w y Motivation Nonlinearities often act only locally Examples Mechanical systems with friction, hysteresis Systems with nonlinear actator dynamics Variable-gain controlled linear systems Class of Lr e-type systems is inclded (when Σ nl is static) 7/29

15 Problem setting and approach Σ lin y ˆΣ lin ŷ v w ˆv ŵ Σ nl Σ nl Model redction approach Redction of high-order linear sbsystem Σ lin only, taking into accont inpts (, v) and otpts (y, w) Reconnect nonlinear sbsystem Σ nl 7/29

16 Problem setting and approach Σ lin y ˆΣ lin ŷ v w ˆv ŵ Σ nl Σ nl Model redction approach Redction of high-order linear sbsystem Σ lin only, taking into accont inpts (, v) and otpts (y, w) Reconnect nonlinear sbsystem Σ nl Allows for the se of existing model redction techniqes Comptationally feasible 7/29

17 Inpt-to-state convergence inpt state x Σ Definition. The operator F : L m L n defined as F(t) := x (t) is said to be the steady-state operator of the niformly convergent system ẋ = f(x,), x R n, R m 8/29

18 Inpt-to-state convergence inpt state x Σ Definition. A system is inpt-to-state convergent (ISC) if it is globally niformly convergent and x(t) x (t) β ( ( ) ) x(t 0 ) x (t 0 ),t t 0 +γ sp ũ(τ) (τ) holds for all t t 0, with β KL and γ K t 0 τ t 8/29

19 Inpt-to-state convergence inpt state x Σ Definition. A system is inpt-to-state convergent (ISC) if it is globally niformly convergent and x(t) x (t) β ( ( ) ) x(t 0 ) x (t 0 ),t t 0 +γ sp ũ(τ) (τ) holds for all t t 0, with β KL and γ K t 0 τ t Lemma. Let a system ẋ = f(x,) be inpt-to-state convergent. Then, the steady-state operator is incrementally bonded as F 2 F 1 γ( 2 1 ), x = sp τ R x(τ) 8/29

20 Small-gain theorem for ISC systems z Σ x x Σ x : ẋ = f(x,z,) ISC with γ xz, γ x Σ z v Σ z : ż = g(z,x,v) ISC with γ zx, γ zv 9/29

21 Small-gain theorem for ISC systems z Σ x x Σ x : ẋ = f(x,z,) ISC with γ xz, γ x Σ z v Σ z : ż = g(z,x,v) ISC with γ zx, γ zv Theorem. The feedback interconnection is inpt-to-state convergent if there exist fnctions ρ 1, ρ 2 of class K sch that holds for all s 0 (id+ρ 1 ) γ xz (id+ρ 2 ) γ zx (s) s 9/29

22 Small-gain theorem for ISC systems z Σ x x Σ x : ẋ = f(x,z,) ISC with γ xz, γ x Σ z v Σ z : ż = g(z,x,v) ISC with γ zx, γ zv Theorem. The feedback interconnection is inpt-to-state convergent if there exist fnctions ρ 1, ρ 2 of class K sch that holds for all s 0 Ingredients of proof (id+ρ 1 ) γ xz (id+ρ 2 ) γ zx (s) s 1. Existence of a steady-state soltion of the copled system 2. Inpt-to-state stability with respect to the steady-state soltion 9/29

23 Problem setting (cont d.) Σ lin v w Σ nl : Σ nl y {ż = g(z,w) v = h(z) Assmptions A1. Σ lin is asymptotically stable (i.e., inpt-to-state convergent) 10/29

24 Problem setting (cont d.) Σ lin v w Σ nl : Σ nl y {ż = g(z,w) v = h(z) Assmptions A1. Σ lin is asymptotically stable (i.e., inpt-to-state convergent) A2. Σ nl is inpt-to-state convergent A3. The otpt fnction is incrementally bonded as h(z 2 ) h(z 1 ) χ vz ( z 2 z 1 ) 10/29

25 Problem setting (cont d.) Σ lin v w Σ nl : Σ nl y {ż = g(z,w) v = h(z) Assmptions A1. Σ lin is asymptotically stable (i.e., inpt-to-state convergent) A2. Σ nl is inpt-to-state convergent A3. The otpt fnction is incrementally bonded as h(z 2 ) h(z 1 ) χ vz ( z 2 z 1 ) Property. The steady-state otpt operator G v w := h( z w ) satisfies G v w 2 G v w 1 χ vz γ zw ( w 2 w 1 ) 10/29

26 Problem setting (cont d.) Σ lin v w Σ nl : Σ nl y {ż = g(z,w) v = h(z) Assmptions A4. ρ 1, ρ 2 K sch that the small-gain condition (id+ρ 1 ) γ xv χ vz (id+ρ 2 ) γ zw χ wx (s) s, s 0 holds, i.e., Σ = I(Σ lin,σ nl ) is inpt-to-state convergent 10/29

27 Problem setting (cont d.) Σ lin y ˆΣ lin ŷ v w ˆv ŵ Σ nl Σ nl Assmptions A5. ˆΣ lin is asymptotically stable A6. The steady-state error operators E i (,v) = F i (,v) ˆF i (,v) satisfy, for some ε ij K, i {y,w}, j {,v}, E i ( 2,v 2 ) E i ( 1,v 1 ) ε i ( 2 1 ) +ε iv ( v2 v 1 ) 10/29

28 Problem setting (cont d.) Σ lin y ˆΣ lin ŷ v w ˆv ŵ Σ nl Σ nl Assmptions A5. ˆΣ lin is asymptotically stable A6. The steady-state error operators E i (,v) = F i (,v) ˆF i (,v) satisfy, for some ε ij K, i {y,w}, j {,v}, ( ) E i ( 2,v 2 ) E i ( 1,v 1 ) ε i 2 1 ( ) +ε iv v2 v 1 Linear model redction techniqes satisfying A5. and A6. exist, e.g., balanced trncation [Moore, Glover] 10/29

29 Stability preservation and error bond Theorem. Let Σ = I(Σ lin,σ nl ) and ˆΣ = I(ˆΣ lin,σ nl ) satisfy Assmptions A1 A6. Then, 1. ˆΣ is inpt-to-state convergent if ˆρ 1, ˆρ 2 K sch that (id+ ˆρ 1 ) χ vz γ zw (id+ ˆρ 2 ) (χ wx γ xv +ε wv )(s) s, for all s 0 11/29

30 Stability preservation and error bond Theorem. Let Σ = I(Σ lin,σ nl ) and ˆΣ = I(ˆΣ lin,σ nl ) satisfy Assmptions A1 A6. Then, 1. ˆΣ is inpt-to-state convergent if ˆρ 1, ˆρ 2 K sch that (id+ ˆρ 1 ) χ vz γ zw (id+ ˆρ 2 ) (χ wx γ xv +ε wv )(s) s, for all s 0 2. If 1. is satisfied, then there exists ε K sch that the steady-state otpt error is bonded as δȳ ε ( ), with δȳ := ȳ ŷ. Moreover, ε( ) can be expressed in terms of the incremental gains of Σ lin and Σ nl and the error bond on the linear sbsystems ε ij 11/29

31 Proof and properties Ingredients in the proof Inpt-to-state convergence provides bond on amplifications of steady-state errors going throgh the sbsystems Small-gain theorem provides bondedness of steady-state errors in closed-loop 12/29

32 Proof and properties Ingredients in the proof Inpt-to-state convergence provides bond on amplifications of steady-state errors going throgh the sbsystems Small-gain theorem provides bondedness of steady-state errors in closed-loop Properties Preservation of inpt-to-state stability 12/29

33 Proof and properties Ingredients in the proof Inpt-to-state convergence provides bond on amplifications of steady-state errors going throgh the sbsystems Small-gain theorem provides bondedness of steady-state errors in closed-loop Properties Preservation of inpt-to-state stability Bond on the steady-state error. Recall that the steady-state soltion is 1. defined for any bonded inpt, and, 2. niqe 12/29

34 Proof and properties Ingredients in the proof Inpt-to-state convergence provides bond on amplifications of steady-state errors going throgh the sbsystems Small-gain theorem provides bondedness of steady-state errors in closed-loop Properties Preservation of inpt-to-state stability Bond on the steady-state error. Recall that the steady-state soltion is 1. defined for any bonded inpt, and, 2. niqe A priori error bond, i.e., based only on properties of Σ 12/29

35 Proof and properties Ingredients in the proof Inpt-to-state convergence provides bond on amplifications of steady-state errors going throgh the sbsystems Small-gain theorem provides bondedness of steady-state errors in closed-loop Properties Preservation of inpt-to-state stability Bond on the steady-state error. Recall that the steady-state soltion is 1. defined for any bonded inpt, and, 2. niqe A priori error bond, i.e., based only on properties of Σ Error bond holds for all nonlinear systems Σ nl satisfying the same inpt-to-state convergence gains 12/29

36 Incremental L 2 gain and passivity Σ lin y ˆΣ lin ŷ v w ˆv ŵ Σ nl Σ nl Incremental properties (by inpt-to-state convergence) of Σ nl crcial in obtaining the (a priori) error bond 13/29

37 Incremental L 2 gain and passivity Σ lin y ˆΣ lin ŷ v w ˆv ŵ Σ nl Σ nl Incremental properties (by inpt-to-state convergence) of Σ nl crcial in obtaining the (a priori) error bond Alternative system classes (sing dissipativity theory [Willems]) 1. Incremental L 2 gain [Romanchk & James]. A system Σ nl has a bonded incremental L 2 gain if a fnction S sch that Ṡ(x 2,x 1 ) γ y 2 y /29

38 Incremental L 2 gain and passivity Σ lin y ˆΣ lin ŷ v w ˆv ŵ Σ nl Σ nl Incremental properties (by inpt-to-state convergence) of Σ nl crcial in obtaining the (a priori) error bond Alternative system classes (sing dissipativity theory [Willems]) 2. Incremental passivity [Pavlov & Marconi]. A system Σ nl is incrementally passive if a storage fnction S sch that Ṡ(x 2,x 1 ) ( 2 1 ) T (y 2 y 1 ) 13/29

39 Incremental L 2 gain and passivity Σ lin y ˆΣ lin ŷ v w ˆv ŵ Σ nl Σ nl Incremental properties (by inpt-to-state convergence) of Σ nl crcial in obtaining the (a priori) error bond Alternative system classes (sing dissipativity theory [Willems]) Approach: bonded real [Opdenacker & Jonckheere] or positive real [Desai & Pal, Harshavarhana et al.] balancing of Σ lin Properties: preservation of bonded L 2 gain or passivity throgh small-gain or passivity theorem and a priori error bond de to incremental properties [Besselink et al., 2013] 13/29

40 Example w y Nonlinear beam example Linear beam model sing Eler beam elements Nonlinear damping characteristic (damping force v) {ż = z σ(z)+κw Σ nl : v = z with σ(z) an arbitrary nondecreasing continos fnction 14/29

41 Example w y Nonlinear beam example Linear beam model sing Eler beam elements Nonlinear damping characteristic (damping force v) {ż = z σ(z)+κw Σ nl : v = z with σ(z) an arbitrary nondecreasing continos fnction Σ nl has a bonded incremental L 2 gain with gain κ, i.e. µ = κ 14/29

42 Example: reslts (1) Redction of Σ lin (n = 80) to obtain ˆΣ lin of order k = 4 for κ = 20 Hwv [-] H wv Ĥ wv κ 1 κ ε (a priori) ε (a posteriori) κ 1 ε lin f [Hz] 15/29

43 Example: reslts (1) Redction of Σ lin (n = 80) to obtain ˆΣ lin of order k = 4 for κ = 40 Hwv [-] H wv Ĥ wv κ 1 κ ε (a priori) ε (a posteriori) κ 1 ε lin f [Hz] 15/29

44 Example: reslts (1) Redction of Σ lin (n = 80) to obtain ˆΣ lin of order k = 4 for κ = 60 Hwv [-] H wv Ĥ wv κ 1 κ ε (a priori) ε (a posteriori) κ 1 ε lin f [Hz] 15/29

45 Example: reslts (2) Simlation of Σ and ˆΣ for (t) = 10 2 sign(sin(2π10t)) and κ = 60 y [mm] otpt y ŷ w [mm] nonlinearity w ŵ t [s] t [s] 16/29

46 Overview & Incremental balancing Overview. Model redction for inpt-to-state convergent systems 2. systems with incremental dissipativity properties 17/29

47 Overview & Incremental balancing Overview. Model redction for inpt-to-state convergent systems 2. systems with incremental dissipativity properties Properties Preservation of system properties and a priori error bond Comptationally attractive Nonlinearity not explicitly taken into accont References B. Besselink, N. van de Wow, H. Nijmeijer. Model redction for a class of convergent nonlinear systems. IEEE Transactions on Atomatic Control, 57(4): , B. Besselink, N. van de Wow, H. Nijmeijer. Model redction for nonlinear systems with incremental gain or passivity properties. Atomatica, 49(4): , N. van de Wow, W. Michiels, B. Besselink. Model redction for delay differential eqations with garanteed stability and error bond. Atomatica, 55: , /29

48 Overview & Incremental balancing Overview. Model redction for inpt-to-state convergent systems 2. systems with incremental dissipativity properties Properties Preservation of system properties and a priori error bond Comptationally attractive Nonlinearity not explicitly taken into accont Objective. Incorporate nonlinearities in the redction procedre 17/29

49 Overview & Incremental balancing Overview. Model redction for inpt-to-state convergent systems 2. systems with incremental dissipativity properties Properties Preservation of system properties and a priori error bond Comptationally attractive Nonlinearity not explicitly taken into accont Objective. Incorporate nonlinearities in the redction procedre Incremental balanced trncation for models of the form {ẋ = f(x)+g(x), x R Σ : n, R m,y R p y = h(x) 17/29

50 Balanced trncation Σ y Observability and controllability fnction L o (x 0 ) = 0 L c (x 0 ) = inf L m 2 y(t) 2 dt, x(0) = x 0, = 0 0 (t) 2 dt, : x( ) = 0 x(0) = x 0 18/29

51 Balanced trncation Σ y Observability and controllability fnction L o (x 0 ) = 0 L c (x 0 ) = inf L m 2 y(t) 2 dt, x(0) = x 0, = 0 0 Linear systems (asymptotically stable) (t) 2 dt, : x( ) = 0 x(0) = x 0 L o (x) = x T Qx, L c (x) = x T P 1 x 1. Balancing: find transformation x = Tz sch that L o (z) = z T Σz, L c (z) = z T Σ 1 z with Σ = diag{σ 1,...,σ n } 2. Trncation: discard states corresponding to smallest σ i s 18/29

52 Balanced trncation Σ y Observability and controllability fnction L o (x 0 ) = 0 L c (x 0 ) = inf L m 2 y(t) 2 dt, x(0) = x 0, = 0 0 Linear systems (asymptotically stable) (t) 2 dt, : x( ) = 0 x(0) = x 0 Preservation of asymptotic stability (when σ k > σ k+1 ) A priori error bond of the form y ŷ 2 ε 2 with n ε = 2 i=k+1 σ i 18/29

53 Balanced trncation Σ y Observability and controllability fnction L o (x 0 ) = 0 L c (x 0 ) = inf L m 2 y(t) 2 dt, x(0) = x 0, = 0 0 Nonlinear systems (t) 2 dt, : x( ) = 0 x(0) = x 0 Extension to nonlinear systems exist [Scherpen, Fjimoto] Preservation of local asymp. stability of x = 0 for = 0 No error bond 18/29

54 Incremental observability fnction Σ(x) Σ( x) + y ŷ E o (x 0, x 0 ) = sp L m 2 0 y(t) ȳ(t) 2 dt, x(0) = x 0, x(0) = x 0 19/29

55 Incremental observability fnction Σ(x) Σ( x) + y ŷ E o (x 0, x 0 ) = sp L m 2 Properties 0 y(t) ȳ(t) 2 dt, x(0) = x 0, x(0) = x 0 Σ is observable if and only if E o (x 0, x 0 ) > 0 for all x 0 x 0 19/29

56 Incremental observability fnction Σ(x) Σ( x) + y ŷ E o (x 0, x 0 ) = sp L m 2 Properties 0 y(t) ȳ(t) 2 dt, x(0) = x 0, x(0) = x 0 Σ is observable if and only if E o (x 0, x 0 ) > 0 for all x 0 x 0 Related to incremental stability properties, i.e., if E o (x 0, x 0 ) > 0 for all x 0 x 0, then any two trajectories x( ) and x( ) for a common inpt ( ) satisfy x(t) x(t) α( x(0) x(0) ), t 0, α K 19/29

57 Incremental observability fnction Σ(x) Σ( x) + y ŷ E o (x 0, x 0 ) = sp L m 2 Properties 0 y(t) ȳ(t) 2 dt, x(0) = x 0, x(0) = x 0 Σ is observable if and only if E o (x 0, x 0 ) > 0 for all x 0 x 0 Related to incremental stability properties Linear systems E o (x 0, x 0 ) = (x 0 x 0 ) T Q(x 0 x 0 ) = L o (x 0 x 0 ) 19/29

58 Incremental controllability fnction ū Σ(x) Σ( x) 0 E c (x 0, x 0 ) = inf (t)+ū(t) 2 dt, : 0 x 0, ū : 0 x 0,ū L m 2 20/29

59 Incremental controllability fnction ū Σ(x) Σ( x) 0 E c (x 0, x 0 ) = inf (t)+ū(t) 2 dt, : 0 x 0, ū : 0 x 0,ū L m 2 Properties Related to reachability of Σ Related to bondedness of soltions (i.e., stability) 20/29

60 Incremental controllability fnction ū Σ(x) Σ( x) 0 E c (x 0, x 0 ) = inf (t)+ū(t) 2 dt, : 0 x 0, ū : 0 x 0,ū L m 2 Properties Related to reachability of Σ Related to bondedness of soltions (i.e., stability) Linear systems E c (x 0, x 0 ) = (x 0 + x 0 ) T P 1 (x 0 + x 0 ) = L c (x 0 + x 0 ) 20/29

61 Incremental balancing Assmption 1. E o and E c can be partitioned as E o (x, x) = E 1 o(x 1, x 1 )+E 2 o(x 2, x 2 ) E c (x, x) = E 1 c(x 1, x 1 )+E 2 c(x 2, x 2 ) with x T = [x T 1 x 2], x T = [ x T 1 x 2] 2. E 2 o and E 2 c satisfy, for some ρ > 0, E 2 o x 2 (x 2,0) = ρ 2 E2 c x 2 (x 2,0) 21/29

62 Incremental balancing Assmption 1. E o and E c can be partitioned as E o (x, x) = E 1 o(x 1, x 1 )+E 2 o(x 2, x 2 ) E c (x, x) = E 1 c(x 1, x 1 )+E 2 c(x 2, x 2 ) with x T = [x T 1 x 2], x T = [ x T 1 x 2] 2. E 2 o and E 2 c satisfy, for some ρ > 0, E 2 o x 2 (x 2,0) = ρ 2 E2 c x 2 (x 2,0) Linear systems: the assmption is satisfied if the system is in balanced coordinates (then, Q = P = Σ and ρ = σ n ) 21/29

63 Incremental balancing Assmption 1. E o and E c can be partitioned as E o (x, x) = E 1 o(x 1, x 1 )+E 2 o(x 2, x 2 ) E c (x, x) = E 1 c(x 1, x 1 )+E 2 c(x 2, x 2 ) with x T = [x T 1 x 2], x T = [ x T 1 x 2] 2. E 2 o and E 2 c satisfy, for some ρ > 0, E 2 o x 2 (x 2,0) = ρ 2 E2 c x 2 (x 2,0) Interpretation Existence of an "incrementally balanced" realization Can a coordinate transformation x = ϕ(z) be fond sch that the assmption holds in the new coordinates z? 21/29

64 Incremental balanced trncation Partitioning in "incrementally balanced" form ẋ 1 = f 1 (x 1,x 2 ) + g 1 (x 1,x 2 ) Σ : ẋ 2 = f 2 (x 1,x 2 ) + g 2 (x 1,x 2 ) y = h(x 1,x 2 ) 22/29

65 Incremental balanced trncation Partitioning in "incrementally balanced" form ẋ 1 = f 1 (x 1,x 2 ) + g 1 (x 1,x 2 ) Σ : ẋ 2 = f 2 (x 1,x 2 ) + g 2 (x 1,x 2 ) y = h(x 1,x 2 ) Trncation, i.e., set x 2 = 0 and discard x 2 -dynamics One-step redction ξ R n 1 approximates x 1 ˆΣ n 1 : { ξ = f1 (ξ,0)+g 1 (ξ,0) ŷ = h(ξ,0) 22/29

66 Incremental balanced trncation Partitioning in "incrementally balanced" form ẋ 1 = f 1 (x 1,x 2 ) + g 1 (x 1,x 2 ) Σ : ẋ 2 = f 2 (x 1,x 2 ) + g 2 (x 1,x 2 ) y = h(x 1,x 2 ) Trncation, i.e., set x 2 = 0 and discard x 2 -dynamics One-step redction ξ R n 1 approximates x 1 ˆΣ n 1 : { ξ = f1 (ξ,0)+g 1 (ξ,0) ŷ = h(ξ,0) Lemma. Ê o and Êc of ˆΣ n 1 satisfy the bonds Ê o (ξ, ξ) E 1 o(ξ, ξ), Ê c (ξ, ξ) E 1 c(ξ, ξ) 22/29

67 Stability preservation Theorem. Let E o (x, x) > 0 for all x x and E c (x,x) > 0 for all x 0. Under the assmptions stated before 1. There exists ˆX R n 1 and U L m 2 ([0, )) sch that any ξ( ) corresponding to ξ(0) ˆX and ( ) U is bonded 2. ˆΣ n 1 is incrementally stable, i.e., ξ(t) ξ(t) α( ξ(0) ξ(0) ), t 0 with ξ( ) and ξ( ) soltions to ξ(0) and ξ(0) and inpt ( ) 3. For any bonded inpt ( ) L m 2 ([0, )), lim h(ξ(t),0) h( ξ(t),0) = 0 t 23/29

68 Error bond Theorem. Under the assmptions as before, the error bond y ŷ 2 2ρ 2 holds for any ( ) L m 2 ([0, )) and zero initial conditions 24/29

69 Error bond Theorem. Under the assmptions as before, the error bond y ŷ 2 2ρ 2 holds for any ( ) L m 2 ([0, )) and zero initial conditions Proof based on the storage fnction V(x 1,x 2,ξ) = E 1 o (x 1,ξ)+E 2 o (x 2,0)+ρ 2( E 1 c (x 1,ξ)+E 2 c (x 2,0) ), which satisfies V(x 1,x 2,ξ) (2ρ) 2 2 y ŷ 2 24/29

70 Error bond Theorem. Under the assmptions as before, the error bond y ŷ 2 2ρ 2 holds for any ( ) L m 2 ([0, )) and zero initial conditions Proof based on the storage fnction V(x 1,x 2,ξ) = Eo 1 (x 1,ξ)+Eo 2 (x 2,0)+ρ 2( Ec 1 (x 1,ξ)+Ec 2 (x 2,0) ), which satisfies V(x 1,x 2,ξ) (2ρ) 2 2 y ŷ 2 Remarks Redction to arbitrary order by repeated application For linear systems, incremental balanced trncation is eqivalent to balanced trncation 24/29

71 Generalized incremental balancing Open problems Comptation of E o and E c demanding Assmption on "incrementally balanced form" needed 25/29

72 Generalized incremental balancing Open problems Comptation of E o and E c demanding Assmption on "incrementally balanced form" needed Generalized incremental energy fnctions Ẽ o (x, x) = (x x) T Q(x x) Eo (x, x) Ẽ c (x, x) = (x + x) T R(x + x) E c (x, x) 25/29

73 Generalized incremental balancing Open problems Comptation of E o and E c demanding Assmption on "incrementally balanced form" needed Generalized incremental energy fnctions Ẽ o (x, x) = (x x) T Q(x x) Eo (x, x) Ẽ c (x, x) = (x + x) T R(x + x) E c (x, x) Properties Use generalized incremental energy fnctions (i.e., bonds) Reslts on stability and error bond hold 25/29

74 Generalized incremental balancing Open problems Comptation of E o and E c demanding Assmption on "incrementally balanced form" needed Generalized incremental energy fnctions Ẽ o (x, x) = (x x) T Q(x x) Eo (x, x) Ẽ c (x, x) = (x + x) T R(x + x) E c (x, x) Properties Use generalized incremental energy fnctions (i.e., bonds) Reslts on stability and error bond hold Generalized incremental balanced trncation provides a comptationally feasible approach towards model redction 25/29

75 Example: Nonlinear electronic circit y n η η η η Nonlinear electronic circit (taken from [Rewieński]) Nonlinear resistors η with η odd, nondecreasing and η(0) = 0 Model Σ with f(x) = Ax +ϕ(x), g(x) = B, h(x) = Cx, and η(x (1) ) 1.. A = , ϕ(x) =., B = CT = η(x (n) ) 0 26/29

76 Example: Reslts (1) The matrices Q and R can be chosen as Q = R 1 = Σ = diag{σ 1,...,σ n }, σ i > σ i+1 > 0, i.e., the system is in (generalized) incrementally balanced form Redction from n = 100 to k = σi [-] σi [-] i [-] i [-] 27/29

77 Example: Reslts (2) Simlations for η(v) = sign(v)v 2 and (t) = 5 (1 cos(2π 1 t)) (left) and (t) = 1 (1+sign(sin(2π 1 t))) (right) Σ BC ˆΣ BC, Σ BC ˆΣ BC,4 y [V] 1 y [V] t [s] t [s] A priori error bond of the form y ŷ 2 ε 2 with ε = i=5 σ i = /29

78 Conclsions & Open problems Model redction for nonlinear systems 1. A class of convergent nonlinear systems 2. Nonlinear systems with incremental gain or passivity properties 3. Incremental balanced trncation for nonlinear systems 29/29

79 Conclsions & Open problems Model redction for nonlinear systems 1. A class of convergent nonlinear systems 2. Nonlinear systems with incremental gain or passivity properties 3. Incremental balanced trncation for nonlinear systems Incremental system properties are crcial in 1. the preservation of relevant stability properties; 2. the derivation of a priori error bonds References B. Besselink, N. van de Wow, J.M.A. Scherpen, H. Nijmeijer. Model redction for nonlinear systems by incremental balanced trncation. IEEE Transactions on Atomatic Control, 59(10): , B. Besselink. Model redction for nonlinear control systems - with stability preservation and error bonds. PhD thesis, Eindhoven University of Technology, Eindhoven, the Netherlands, /29

80 Conclsions & Open problems Model redction for nonlinear systems 1. A class of convergent nonlinear systems 2. Nonlinear systems with incremental gain or passivity properties 3. Incremental balanced trncation for nonlinear systems Incremental system properties are crcial in 1. the preservation of relevant stability properties; 2. the derivation of a priori error bonds Open problems in model redction Preservation of strctre, e.g., in networks Comptational methods for nonlinear systems 29/29