From acceleration-based semi-active vibration reduction control to functional observer design

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1 MITSUBISHI ELECTRIC RESEARCH LABORATORIES From acceleration-based semi-active vibration reduction control to unctional observer design Wang, Y; Utsunomiya, K TR18-35 March 18 Abstract This work investigates a unctional estimation problem or single input single output linear and nonlinear systems It inds application in enabling accelerationbased semi-active control Solvability o a linear unctional estimation problem is studied rom a geometric approach, where the unctional dynamics are derived, decomposed, and transormed to expose structural properties This approach is extended to solve a challenging nonlinear unctional observer problem, combining with the exact error linearization Existence conditions o nonlinear unctional observers are established Simulation veriies existence conditions and demonstrates the eectiveness o the proposed unctional observer designs Automatisierungstechnik This work may not be copied or reproduced in whole or in part or any commercial purpose Permission to copy in whole or in part without payment o ee is granted or nonproit educational and research purposes provided that all such whole or partial copies include the ollowing: a notice that such copying is by permission o Mitsubishi Electric Research Laboratories, Inc; an acknowledgment o the authors and individual contributions to the work; and all applicable portions o the copyright notice Copying, reproduction, or republishing or any other purpose shall require a license with payment o ee to Mitsubishi Electric Research Laboratories, Inc All rights reserved Copyright c Mitsubishi Electric Research Laboratories, Inc, 18 1 Broadway, Cambridge, Massachusetts 139

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3 Automatisierungstechnik 18; aop Article Yebin Wang* and Kenji Utsunomiya From acceleration-based semi-active vibration reduction control to unctional observer design DOI Received June 15, 17; accepted Abstract: This work investigates a unctional estimation problem or single input single output linear and nonlinear systems It inds application in enabling accelerationbased semi-active control Solvability o a linear unctional estimation problem is studied rom a geometric approach, where the unctional dynamics are derived, decomposed, and transormed to expose structural properties This approach is extended to solve a challenging nonlinear unctional observer problem, combining with the exact error linearization Existence conditions o nonlinear unctional observers are established Simulation veriies existence conditions and demonstrates the eectiveness o the proposed unctional observer designs Keywords: Semi-active, vibration reduction, estimation, nonlinear system, unctional observer Communicated by: 1 Introduction Transportation systems are subject to external disturbances which compromise passenger s ride comort A common way to mitigate such a situation is to have a vibration reduction subsystem in place The use o semi-active actuators, such as Magneto-Rheological (MR) dampers, was originally proposed in [1] to balance the perormance o vibration reduction and the system cost Thanks to the dissipative nature o semi-active actuators, the resultant system enjoys guaranteed stability, and receives longstanding interests rom both academia and industry Nonlinear actuators and inadequate measurements however pose a challenging control problem *Corresponding author: Yebin Wang, Mitsubishi Electric Research Laboratories, 1 Broadway, Cambridge, MA 139, USA yebinwang@ieeeorg Kenji Utsunomiya, Advanced Technology R&D Center, Mitsubishi Electric Corporation, 8-1-1, Tsukaguchihonmachi, Amagasaki City, , Japan UtsunomiyaKenji@dbMitsubishiElectriccojp With accelerations and external disturbances, the system is not state observable This violates the undamental assumption made in a majority o existing semi-active control designs, eg, clipping control [1 4], optimal control [5, 6], Lyapunov-based control [7], nonlinear H control [8], Acceleration Driven Damping (ADD) [9], etc Switching on or o a semi-active actuator according to the sign o its relative velocity, acceleration-based control [1] avoids the pitall o state unobservability Its eectiveness is crucially contingent on an accurate estimation o the relative velocity, which is solved as a unctional estimation problem This work extends discussions on acceleration-based control to a general unctional observer design problem Relevant work includes, but not limited to, unctional observer design or linear time-invariant (LTI) systems [11], minimum order unctional observer or LTI systems [1], etc Work [11] establishes necessary and suicient existence conditions o an rth order unctional observer, where r is the dimension o unctionals By leveraging additional virtual unctionals, work [1] lits the restriction on the order o the unctional observer Main contributions o this work on the unctional estimation are three-old First, resorting to a geometric perspective, this work oers a crystal interpretation regarding the solvability o the linear unctional estimation problem Apparent advantages o this approach is that it permits a natural generalization to nonlinear unctional observer problem Dierent rom the linear case, nonlinear unctional estimation is more challenging, and its existence conditions are typically suicient Second this work employs the exact error linearization [13, 14], and establishes existence conditions o nonlinear unctional observers Third, simulation is conducted to veriy the existence conditions and unctional observers This paper is organized as ollows Section introduces the semi-active vibration reduction system, and illustrates how the unctional estimation enables the acceleration-based control Section 3 investigates the unctional estimation or single input single output (SISO) linear/nonlinear systems Section 4 presents simulation results Section 5 concludes this paper

4 Y Wang, Acceleration-based semi-active control and unctional observer design Preliminary This section introduces the semi-active vibration reduction system, and ormulates the control problem Acceleration-based control and the related unction observer design are presented 1 System Architecture and Modeling For simplicity, consider a quarter car model which is simpliied as a two degree o reedom (DOF) system as shown in Figure 1 The DOF system consists o a irst mass (m 1 ), a second mass (m ), a road proile, a controller (C), a sensor (S), two passive dampers with viscous damping coeicients (b 1, b ), one semi-active damper with adjustable viscous damping coeicient (u), and springs (k 1, k ) The system dynamics are given by w = and u = is not state observable Given that system state can not be reconstructed rom measurements, neither the disturbance w can be estimated Readers are reerred to [15, 16] or uniorm state observability and input observability The Control Problem Vibration reduction is typically ormulated as a disturbance attenuation problem as illustrated by Figure, where G is the plant, and z is the controlled variable, respectively This work considers the vibration reduction w u G C z y k k 1 Fig : Disturbance attenuation problem setup [17] Road proile m m 1 S problem roughly stated as ollows b + u b w x 1 C Fig 1: A DOF quarter car model x 1 Problem 1 Given the system (1) and a bounded control set U : [b min, b max ] with b min and b max representing the lower and upper bounds o the control, ind control u U which minimizes the cost unction J ẋ = Ax + B 1 w + B (x, ẇ)u, y = ẍ = h(x, u, w, ẇ), (1) where x = [x 1, x, ẋ 1, ẋ ] = [x 1, x, x 3, x 4 ], w is the displacement disturbance rom the road proile, and A = B 1 = 1 1 k1 k 1 m 1 m 1 k 1 m k m k1+k b 1 m m, B (x, ẇ) = b1 b 1 m 1 m 1 b1 m x4 ẇ m, The system setup is non-unique For instance, a semiactive actuator can be placed between m 1 and m The aorementioned setup is chosen because or certain systems, its dominant resonant mode is result rom m and k, and an actuator between m and the road proile is more eective in suppressing the vibration System (1) is not uniormly state observable This act can be established by veriying that system (1) with where t is constant J(u) = t ẍ 1 (t) dt, () The cost unction () does not penalize control eorts This is not unusual, or example, the time optimal control problem uses the inal time as the cost unction without penalizing control As shown later, Problem 1 admits a bang-bang control as the time optimal case 3 Acceleration-Based Control A natural treatment o Problem 1 is optimal control theory Optimal open-loop or state eedback control requires to solve an optimization problem recursively [18] or the Hamilton-Jacobi-Bellman equations [19, ], respectively, neither o which is trivial and realistic An accelerationbased control has been proposed in [1], and demonstrates good perormance, mimicking the structure o the optimal control Its implementation entails the reconstruction o a relative velocity a unctional estimation problem The

5 Y Wang, Acceleration-based semi-active control and unctional observer design 3 acceleration-based control and the related unctional estimation are recalled or completeness and motivation 31 Control Structure Provided that w(t) is known, Problem 1 can be ormulated as a constrained optimal control problem Deining the Hamiltonian H(x, λ, u, w, ẇ) = ẍ 1 (t) + λ (Ax + B 1 w + B (x, ẇ)u), we have the structure o the optimal control which minimizes H(x, λ, u, t) { u b max, i λ B (x, ẇ), = b min, i λ B (x, ẇ) Given the costate λ = [λ 1,, λ 4 ], the optimal control can be written as { u b max, i λ 4 (x 4 ẇ), = (3) b min, i λ 4 (x 4 ẇ) Because system (1) is not state observable, the optimal control (3) is not realizable Instead, [1] approximates (3) as the ollowing acceleration-based control { b max, i (c 1ˆẍ 1 + c ẋ 4 + ˆx 4 )(ˆx 4 ˆẇ), u = (4) b min, otherwise, where c 1, c are constants to be tuned, ˆẍ 1 is the estimated acceleration o the irst mass, ˆx 4 is the estimated velocity o the second mass, and ˆx 4 ˆẇ is the estimated relative velocity o the semi-active actuator Readers are reerred to [1] or the essence o (4) 3 Estimation o the Relative Velocity We rearrange the dynamics o x 4 m ẋ 4 = m 1 ẍ 1 k (x w) u(t)(ẋ ẇ), and derive the dynamics o the relative position η = x w as ollows η = 1 u(t) (k η + m 1 ẍ 1 + m y) (5) Given the transer unction rom the second mass displacement to the irst mass displacement G = X 1(s) X (s) = b 1 s + k 1 m 1 s + b 1 s + k 1 we design a linear time-varying ilter ˆx 1 = ˆx 3, ˆx 3 = k 1 m 1 ˆx 1 b 1 m 1 ˆx m 1 y, ˆη = 1 u(t) (k ˆη + m 1 (k 1ˆx 1 + b 1ˆx 3 ) + m y), (6) where the output ˆη is globally exponentially convergent to η Since ˆx 1, ˆx 3, ˆη exponentially converge to x 1, x 3, η, respectively, η(t) ˆ η(t) as t The ilter is open-loop, and the convergent rate o the estimation error cannot be designed 3 Functional Estimation The aorementioned estimation method can be generalized to solve a class o problems: unctional estimation or unobservable systems This section proposes a unctional estimation method, based on deriving and transorming the dynamics o the unctional Without loss o generality, this work considers the unctional estimation o SISO systems The idea can be extended to systems having multi-input and multi-output, or unknown inputs 31 Linear System Case Consider an SISO linear time-invariant (LTI) system ẋ = Ax + Bu, y = Cx, z = Lx, (7) where x R n is the state, u R is the input, y R is the output, z R is the unctional to be estimated, and A, B, C, L are matrices with appropriate dimensions Deinition 31 Given an SISO LTI system (7), the z- dynamics takes the ollowing state space orm ξ = Qξ + Mu, z = ξ 1, (8) where ξ = [ξ 1,, ξ r ] and Q R r r, M R r with its ith element denoted by M i System (7) is a trivial representation o the z-dynamics The z-dynamics are generally non-unique The ollowing result states how to derive the z-dynamics with the minimal order r, abbreviated to the minimal z-dynamics

6 4 Y Wang, Acceleration-based semi-active control and unctional observer design Proposition 3 Given system (7), its minimal z- dynamics are represented by the rth order ξ-system (8) i and only i (i) dim(span {L, LA,, LA r 1 }) = r; (ii) LA r span {L, LA,, LA r 1 } Proo Suiciency: With Condition (i), one can deine state variables ξ as ollows ξ = [Lx, LAx,, LA r 1 x] Taking the time derivative o ξ gives ξ 1 = LAx + LBu = ξ + M 1 u, ξ r = LA r x + LA r 1 Bu = LA r x + M r u With Condition (ii), we have LA r = r 1 k= a kla k, and r 1 ξ r = a k ξ k+1 + M r u k= Condition (i) implies that or any 1 k r 1, LA k / span {L,, LA k 1 } Hence, or any subsystem ξ [1,k] = [ξ 1,, ξ k ] with 1 k r 1, its state space representation is given by ξ [1,k] = A [1,k] ξ [1,k] + B [1,k] ξ k+1 + M [1,k] u, where M [1,k] = [M 1,, M k ], and [ ] Ik 1 A [1,k] = ξ [1,k], B [1,k] = k 1 [ k 1 Clearly, ξ [1,k] -system is not in the orm o (8), and we conclude r is the minimal order Necessity: Similar to the proo o Suiciency, one can veriy that: i Condition (i) does not hold, then ξ-system (8) is not the minimal z-dynamics; and i Condition (ii) does not hold, ξ-system is not in the orm (8), and does not represent z-dynamics Remark 33 Proo o Proposition 3 indicates that the minimal z-dynamics exactly represent the observable subsystem o the ollowing system ẋ = Ax + Bu, The observable space is denoted by y = Lx O z = span {L,, LA r 1 } Remark 34 The z-dynamics can be alternatively deined as ollows ξ = Qξ + Mu + Hy, z = ξ 1, 1 ] (9) where H R r The existence conditions can be similarly derived by ollowing the proo o Proposition 3 I Q is Hurwitz, there exists an open-loop estimator ˆξ = Qˆξ + Mu, ẑ = ˆξ 1 (1) The zero solution o the resultant estimation error dynamics converges exponentially The convergence rate is dominated by the maximum eigenvalue o Q, eg λ max (Q), which is a key limitation Next, we treat the case when Q is not Hurwitz, and establish existence conditions o a stable unctional estimator Following notations are introduced to make subsequent discussions succinct For system (7) with output y, its observable subspace is O = span {C,, CA ro 1 }, and its r -dimension observable subspace is O r = span {C,, CA r 1 } A unctional observer orm is deined as ollows ξ 1 = Q 11 ξ1 + Q 1 ξ + M 1 u, ξ = Q ξ + M u, z = C z ξ1, y = C y ξ, (11) where ξ1 R r1, ξ R r, Q 11 is Hurwitz, M 1 = [M 1 1,, M 1 r 1 ], M = [M 1,, M r ], C z = [1, r1 1], C y = [1, r 1], and (C y, Q ) is observable Given (11), a unctional observer is designed as ollows ˆ ξ 1 = Q 11 ˆ ξ1 + Q 1 ˆ ξ + M 1 u, ˆ ξ = Q ˆ ξ + M u + G(y ŷ), ẑ = C z ˆ ξ1, ŷ = C y ˆ ξ, (1) where G makes Q c = Q GC y Hurwitz The resultant estimation error dynamics are ξ 1 = Q 11 ξ1 + Q 1 ξ, ξ = (Q GC y ) ξ, (13) where ξ1 = ξ 1 ˆ ξ1, ξ = ξ ˆ ξ The zero solution o (13) is globally exponentially stable (GES), which implies ẑ(t) z(t) as t We have the ollowing result regarding whether the unctional can be estimated

7 Y Wang, Acceleration-based semi-active control and unctional observer design 5 Proposition 35 Given system (7) and the minimal z- dynamics (8), the unctional z can be estimated i and only i (8) is a subsystem o the system (11) with r = r o, and the state is given by ξ 1 = [ ξ 1 1,, ξ 1 r 1 ] = [Lx,, LA r1 1 x], ξ = [ ξ 1,, ξ r o ] = [Cx,, CA ro 1 x] (14) Remark 36 I O z O =, z-dynamics are completely unobservable The unctional z can be reconstructed, at best, by the open-loop estimator (1) Proposition 35 establishes necessary and suicient conditions or unctional estimation The order o the resultant estimator is likely higher than the z-dynamics Proo Suiciency: Assume that z-dynamics are part o system (11), ie, ξ = T ξ with T being a linear map The unctional observer (1) yields convergent estimates ˆ ξ o ξ, which implies ˆξ(t) ξ(t) as t Necessity: assume the existence o a unctional observer to estimate z Given system (7), one can always deine a set o coordinates o its observable subspace as ξ = [ ξ 1,, ξ r o ] = [Cx,, CA ro 1 x] On the other hand, Proposition 3 shows that coordinates o the z-dynamics (8) are given by ξ = [Lx,, LA r 1 x] It is always possible to pick r 1 r states out o ξ to orm ξ 1, which are linearly independent o ξ For system (7), one can deine a change o coordinates as ollow ξ = [( ξ 1 ), ( ξ ), ( ξ 3 ) ] = T n x R n, where ξ 3 (O z O) That is: ξ 3 inluences neither z- dynamics nor the observable subsystem ξ We ought to derive the expression o system (7) in the new coordinates ξ It is clear that the ξ -dynamic should have the structure as speciied in (11) The state ξ 1 evolves in the subspace O z O, which is not impacted by ˆξ 3 Hence, ξ 1 -dynamics takes the orm as (11) State ξ 3 evolves in the subspace (O z O), and thus its dynamics has to be written as ξ3 = Q 33 ξ 3 + M 3 u Meanwhile, system (11) includes z-dynamics as a subsystem Next we show that [Lx,, LA r1 1 x] has to be one representation o ξ 1 The state space corresponding to the z-dynamics, or the observable space O z, can be decomposed into two subspaces: S o = O z O, and S u = O z \S o It is clear that dim S u = r 1 We only need to show that the coordinates o S u are given in (14) One can readily veriy that i LA k span {L,, LA k 1 } O, so does LA k+1 or k r That is, i dim(o z O) = r1 + r o, then span {L,, LA r 1 } O = span {L,, LA r1 1 } O Hence, [Lx,, LA r1 1 x] has to be one representation o ξ 1 This completes the proo Example 37 Consider the ollowing LTI system 1 1 ẋ = x + 1 u, y = [ 1 ] x, z = [ 1 1 ] x Veriication o Proposition 3 conirms r = 4 and L = [ 1 1 ], LA = [ 1 1 ], LA = [ 1 ], LA 3 = [ 3 ], LA 4 = [ 3 4 ], (15) where LA 4 = LA LA 3 span {L, LA, LA, LA 3 } We thereore have the minimal z-dynamics 1 ξ = 1 1 ξ + Mu, } 1 {{ } Q z = ξ 1, where Q is not Hurwitz The unctional z can not be reconstructed by the open-loop estimator (1) Proposition 35 holds with state variables ξ 1 = [Lx, LAx], ξ = [Cx, CAx, CA x], and the dynamics are given by [ ] [ ] 1 ξ 1 = ξ ξ = 1 ξ + u 1 [ ] ξ + u,

8 6 Y Wang, Acceleration-based semi-active control and unctional observer design The unctional can be reconstructed by the estimator [ ] [ ] [ ] 1 ˆ ξ 1 = ˆ ξ 1 + ˆ ξ + u, ˆ ξ = 1 ˆ ξ + u + G(y ŷ), 1 ŷ = ˆ ξ 1, ẑ = ˆ ξ1 1, where G stabilizes the zero solution o ξ -dynamics 3 Nonlinear System Case We irst introduce a ew notations Given a C vector ield : R n R n, and a C unction α : R n R, the unction L α = α x is the Lie derivative o α along Repeated Lie derivatives are deined as L k α = L (L k 1 α), k 1 with L α = α Consider the ollowing nonlinear system ẋ = (x) + g(x)u, y = c(x), z = l(x), (16) where x R n, u R, y R, (x), g(x) : R n R n are smooth vector ields, c(x), l(x) : R n R are smooth scalar unctions, and z R is the unctional We deine z-dynamics or system (16) Deinition 38 Given system (16), the z-dynamics take the ollowing state space orm ξ = Q(ξ) + M(ξ)u, z = ξ 1, (17) where ξ = [ξ 1,, ξ r ] and Q, M are smooth vector ields in the orm o ξ M 1 (ξ) Q(ξ) =, M(ξ) = ϕ(ξ) M r (ξ) Let O z = span {dl(x),, dl r 1 l(x)} be the observable space o system (16) with output z = l(x) The ollowing result establishes existence conditions o z-dynamics (17) Proposition 39 Given system (16), its z-dynamics are represented by an r-th order system (17) i and only i (i) dim(o z ) = r; (ii) dl g L k l(x) O z or k r 1; (iii) dl r l(x) O z Proo Suiciency: Condition (i) ensures that ξ(x) = [l(x),, L r 1 l(x)] deines coordinates o the space O z We have ξ 1 = ξ + L g l(x)u, ξ r = L r l(x) + L gl r 1 l(x)u Conditions (ii) and (iii) assure that L g L k l(x) and Lr l(x) is a unction o ξ, respectively Suiciency is shown Necessity: Assume that the z-dynamics are given by (17) One can veriy Conditions (i)-(iii) Remark 31 The z-dynamics (17) can be generalized to the ollowing orm ξ = Q(ξ, y) + M(ξ, y)u, z = ξ 1, (18) where Q(ξ, y) = [ξ,, ϕ(ξ, y)], M(ξ, y) = [M 1 (ξ, y),, M r (ξ, y)] Similar to Proposition 39, existence conditions o the z-dynamics in the orm o (18) can be derived Remark 311 Functional estimation or nonlinear systems is signiicantly challenging, compared with the linear case To be consistent with the linear case where the Q matrix in (8) is Hurwitz, we assume that the zero solution o the uncontrolled z-dynamics ξ = Q(ξ) is GES Given an estimator ˆξ = Q(ˆξ) + M(ˆξ)u, ẑ = ˆξ 1, the stability analysis o the estimation error ξ ˆξ is hardly conclusive The z-dynamics (17) do not necessarily permit the unctional estimation It is however always possible i the z- dynamics are dieomorphic to an observer orm ξ = Qξ + Mu, z = ξ 1, (19) where Q and M are constant matrices with Q Hurwitz Recall the well-established exact error linearization (EEL) to perorm observer design or nonlinear systems, eg [13] Applying the EEL to the z-dynamics with z being treated

9 Y Wang, Acceleration-based semi-active control and unctional observer design 7 as the virtual output, existence conditions o the observer orm (19) can be readily established The observer orm (19) guarantees the existence o a stable open-loop estimator or estimation o z Proposition 31 System (16) induces the z-dynamics in the observer orm (19), i and only i (i) dim(o z ) = r; (ii) L r l(x) = r 1 i= α il i l(x) with the characteristics equation r 1 i= α is i = being Hurwitz; (iii) or k r 1, dl g L k l(x) = where Q 11 is Hurwitz, M 1 is constant, and ξ 1 = [l(x),, L r1 1 l(x)], ξ = [c(x),, L r 1 c(x)], Q 1 (ξ ) = [,, Q 1 r 1 (ξ )], ξ Q (ξ ) = with r 1 + r = r ϕ(ξ ), M (ξ ) = M1 (ξ 1 ), M r (ξ ) Proo Suiciency: Condition (i) suggests that ξ = [l(x),, L r 1 l(x)] are coordinates o the observable subsystem associated with the virtual output z Conditions (ii) ascertains that Q is Hurwitz in the ξ-coordinates; and Condition (iii) guarantees that M is constant Necessity: Given system (16) and its subsystem in the observer orm (19), one can always choose (n r) independent state variables ξ(x) rom the original state x, such that [ξ, ξ (x)] deine new coordinates o system (16) Under the new coordinates, system (16) is written as ollows ξ = Qξ + Mu, ξ = (ξ, ξ) + ḡ(ξ, ξ)u, y = c(ξ, ξ), z = ξ 1 Vector ields, g in the new coordinates are given by (ξ, ξ) = [(Qξ), ((ξ, ξ)) ], g(ξ, ξ) = [M, (g(ξ, ξ)) ] With (ξ, ξ), g(ξ, ξ), Conditions (i)- (iii) are veriied Necessity is shown Proposition 31 is restrictive, not only because it requires the z-dynamics being exactly linearized, but also because it needs the linearized z-dynamics are stable to allow unctional estimation Like Proposition 35, the restriction can be lited by decomposing the z-dynamics into two parts: observable and unobservable through y Generalize the observer orm (19) to the partial observer orm: ξ 1 = Q 11 ξ 1 + Q 1 (ξ ) + M 1 u, ξ = Q (ξ ) + M (ξ )u, y = ξ 1, z = ξ 1 1, () Proposition 313 System (16) induces the z-dynamics in the orm () or a pair (r 1, r ) i and only i, (i) dim(o z,r1 ) + dim(o r ) = r where O z,r1 = span {dl(x),, dl r1 1 l(x)}, O r = span {dc(x),, dl r 1 c(x)}; (ii) There exists α i or i r 1 such that the characteristics equation r 1 i= α is i = is Hurwitz; and r 1 1 i= α i dl i l(x) = dlr1 l(x) mod {dc(x),, dl r 1 c(x)}; (iii) or k r 1 1, dl g L k l(x) = ; (iv) or k r 1 1, dl g L k c(x) span {dc(x),, dlk c(x)}; (v) dl r c(x) O r Proo Suiciency: From Conditions (i), one known ξ 1 and ξ are independent variables For ξ 1 -subsystem, using Conditions (ii)-(iii), we have ξ 1 1 = L l(x) + L g l(x)u = ξ + M 1 1 u, ξ r 1 1 = L r1 l(x) + L gl r1 1 l(x)u = r 1 1 i= α i dl i l(x) + Q1 r 1 (ξ ) + Mr 1 1 u For ξ -subsystem, using Conditions (iv)-(v), we have ξ 1 = L c(x) + L g c(x)u = ξ + M 1 (ξ 1 ), ξ r = L r c(x) + L gl r 1 c(x)u = ϕ(ξ ) + Mr (ξ )u The system with state variables ξ 1 and ξ admits the partial observer orm ()

10 8 Y Wang, Acceleration-based semi-active control and unctional observer design Necessity: Given (), ξ 1, ξ are independent variables One can construct new coordinates or system (16) ξ = [(ξ 1 ), (ξ ), (ξ 3 ) ] R n, where ξ 3 are selected rom x to ensure the non-singularity o ξ/ x Vector ields, g, in the new coordinates, are represented by Q 11 ξ 1 + Q 1 (ξ ) M 1 (ξ) = Q (ξ ) 3 (ξ), g(ξ) = M (ξ ) g 3 (ξ) where 3 and g 3 are vectors o certain unctions With (ξ) and g(ξ), one can compute L k l(ξ) or k r 1 L l(ξ) = ξ 1, L r1 1 l(ξ) = ξ 1 r 1, r 1 1 L r1 1 l(ξ) = α i ξi Q 1 r 1 (ξ ) i= and L k c(ξ) or k r L c(ξ) = ξ, L r 1 c(ξ) = ξ r, L r 1 c(ξ) = ϕ(ξ ) Conditions (i)-(ii) and (v) are veriied Calculating L g L k l(ξ) = M 1 k+1, k r 1 1,, L g L k c(ξ) = M k+1 (ξ 1,, ξ k+1 ), k r 1, one veriies Conditions (iii)-(iv) Necessity is proven Based on (), we propose the ollowing estimator ˆξ 1 = Q 11 ˆξ1 + Q 1 (ˆξ ) + M 1 u, ˆξ = Q (ˆξ ) + M (ˆξ )u + S 1 Cy (y ŷ), ŷ = C y ˆξ, ẑ = ˆξ 1 1, where C y = [1,,, ] R r and S is the solution o θs + A y S + SA y C y C y =, with a suiciently large θ and [ ] Ir 1 A y = (1) Under certain assumptions, (1) yields convergent estimation error dynamics We have the ollowing result Proposition 314 For system, assume that () is globally deined and ξ is uniormly observable through y or all u, and Q 1 r 1 (ξ ), ϕ(ξ ), and M (ξ ) are globally Lipschitz with respect to their corresponding arguments, ie, Q 1 (ξ ) Q 1 (ˆξ ) L 1 ξ, ϕ(ξ ) ϕ(ˆξ ) L ξ, M (ξ ) M (ˆξ ) L 3 ξ, where L 1, L, L 3 are positive constants Given the unctional estimator (1), the zero solution o the resultant estimation error dynamics is GES Proo Write the estimation error dynamics as ollows ξ 1 = Q 11 ξ1 + Q 1 (ξ ) Q 1 (ˆξ ), ξ = Q (ξ ) Q (ˆξ ) + (M (ξ ) M (ˆξ ))u ŷ = C y ˆξ + S 1 C y (y ŷ), () The ξ -dynamics are shown GES in [15] Since Q 11 is Hurwitz, ξ1 = Q 11 ξ 1 is GES Rearrange ξ 1 as ξ 1 = Q 11 ξ1 + d(t), where d(t) L ξ exponentially decays We conclude that ξ 1 (t) converges to zero exponentially or any ξ 1 () R r1 Estimation error dynamics () are GES 4 Simulation This section presents two examples The semi-active example demonstrates the eectiveness o the accelerationbased control, where the unctional observer estimates the relate velocity A numerical example veriies existence conditions o partial observer orm and the proposed unctional observer design 41 Semi-Active Example Extensive simulation and experiments were carried out to validate the proposed design No experimental details are oered here or the protection o proprietary inormation The simulated system (1) has parameter values: m 1 = 8116kg, m = 1754kg, k 1 = 754e4N/m, k = 3e4N/m, b 1 = 58Ns/m, b = 9Ns/m, b min = 175Ns/m, b max = 9e3Ns/m The system is subject to a disturbance shown in Figure 3 We compare the perormance o three cases: Case 1: the passive, ie, system (1) with u = ; Case : system (1)

11 Y Wang, Acceleration-based semi-active control and unctional observer design 9 Disturbance (m) Time(sec) Fig 3: the disturbance data with the control (4) and ull state; Case 3: the control (4) and the estimator (6) Simulation results, provided in Table 1 and Figure 4, show that Case 3 achieves almost the same perormance as Case, and leads to 46% vibration reduction over the passive system In experiment, control (4) and the estimator (6) are implemented in embedded platorms and compared with the passive case Both experiment and simulation results coincide Table 1: Simulation results Acceleration (m/s ) Control Cost ẍ 1 Passive 6496m/s Semi-active with (4) and ull state 145m/s Semi-active with (4) and (6) 1497m/s Passive (Case 1) Semi active (Case ) Semi active (Case 3) Time(sec) Fig 4: the perormance o the passive and semi-active vibration reduction systems 4 Numerical Example Consider a nonlinear system ẋ = (x) + g(x)u, y = x 1 1, z = x 1, (3) where x = [(x 1 ), (x ) ] with x 1 R 3, x R, and x 1 + x1 1 x cos(x ) (x) = x sin(x ) + cos(x1 3 ), g(x) = 1 x 1 arctan(x 1 ) 1x 1 x 1 To apply Proposition 39, one computes ξ(x) = [x 1,, L4 x 1 ] and the jacobian matrix O z = ξ(x)/ x Since det(o z ) = 1/(1 + (x 1 ) ) 3 >, ξ(x) is a global dieomorphism The dimension o z-dynamics is ive Conditions (ii)-(iii) in Proposition 39 can be readily validated The z-dynamics are equivalent to system (3), and the homogenous part is unstable Due to the presence o terms sin(x ), cos(x1 x1 3 ), and arctan(x1 ), the z-dynamics are not transormable to the observer orm (19) Next, we try to apply Proposition 313 With ξ 1 = [x 1, L x 1 ], ξ = [x 1 1, L x 1 1, L x1 1], the system representation in the new coordinates is [ ] [ ] 1 ξ 1 = ξ arctan(ξ ξ 1 ) + M 1 u, x ξ = ξ3 + M u, ϕ(ξ) y = ξ 1, z = ξ 1 1, where M 1 and M are constant, and ϕ = 1 sin(ξ 1 )(ξ1 1 arctan(ξ ξ 1 )) + ξ 1 + ξ 3 + sin(ξ 1 )ξ1 + sin(ξ1 ) + cos(ξ + cos(ξ1 ) ξ 3 ) It is clear that both ξ -dynamics and ξ 1 -dynamics are globally Lipschitz A unctional observer can be designed as given by (1), and the resultant error dynamics are GES, as shown in Proposition 314 Simulation is perormed, in ξ-coordinates, to validate the design Initial conditions o the original system and the observer are ξ = [1, 1, 1, 1, 1], ˆξ = ξ

12 1 Y Wang, Acceleration-based semi-active control and unctional observer design The observer gain is SCy 1 = [3θ, 3θ, θ 3 ] Results are shown in Figs 5-8, where Figs 5-6 correspond to the case θ = 4, and Figs 7-8 or θ = The observer results in convergent estimation error dynamics or both cases, albeit convergent rates are dierent The homogenous part o ξ 1 -dynamics has poles at 1 With θ = 4, ξ converges slowly, and dominates the ξ 1 -dynamics On the contrary, with θ =, ξ converges aster than ξ 1, which is dominated by its homogenous part Fig 5: y(t),ŷ(t),ỹ(t) or θ = Fig 7: y(t),ŷ(t),ỹ(t) or θ = Fig 6: z(t),ẑ(t), z(t) or θ = Fig 8: z(t),ẑ(t), z(t) or θ = 5 Conclusion This work began with the acceleration-based control o a semi-active vibration reduction system, where the reconstruction o the relative velocity is tackled by unction observer design The idea was urther generalized to investigate unction estimation problem or a class o single input single output linear and nonlinear systems Speciically, necessary and suicient existence conditions o linear unctional observer or LTI systems were established For nonlinear system, existence conditions o nonlinear unctional observers were established The proposed designs were validated by simulation

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