DSCC2012-MOVIC

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1 ASME 22 5th Annual Dynamic Systems and Control Conference joint with the JSME 22 th Motion and Vibration Conference DSCC22-MOVIC22 October 7-9, 22, Fort Lauderdale, Florida, USA DSCC22-MOVIC ADAPTIVE DECENTRALIZED NOISE AND VIBRATION CONTROL WITH CONFLICTING PERFORMANCE OBJECTIVES E. Dogan Sumer Department of Aerospace Engineering University of Michigan Ann Arbor, Michigan Dennis S. Bernstein Department of Aerospace Engineering University of Michigan Ann Arbor, Michigan ABSTRACT Decentralized control is a longstanding challenge in systems theory. A decentralized controller may consist of multiple local controllers, connected to disjoint or overlapping sets of sensors and actuators, and each local controller has limited ability to communicate directly with the remaining local controllers and, in addition, may lack global knowledge of the plant and operation of the remaining local controllers. In the present paper we apply adaptive control to investigate the ability of the local controllers to cooperate globally despite uncertainty, communication constraints, and possibly conflicting performance objectives. The approach we apply in this paper is based on retrospective cost adaptive control (RCAC). The development of RCAC assumes a centralized controller structure; the goal of the present paper is to investigate the stability and performance of RCAC in a decentralized setting. INTRODUCTION In most applications of control, a single processing unit is used to update the control input; this is the case of centralized control. In many applications, however, the plant is spatially distributed, possibly in the form of a network, and communication constraints make it impractical to update all control inputs using a centralized controller. For example, the electrical power grid involves multiple sectors, which are controlled independently; the challenge is to ensure that overall decentralized control of this system is reliable. Decentralized control is a longstanding challenge in systems theory [ 4]. A decentralized controller may consist of multiple local controllers, connected to disjoint or overlapping sets of sensors and actuators, and each local controller has limited ability to communicate directly with the remaining local controllers and, in addition, may lack global knowledge of the plant and operation of the remaining local controllers. The local performance objectives of the local controllers are typically consistent in terms of a single global performance objective, although it may be the case that the local performance objectives may conflict, and the challenge is to ensure that the local controllers cooperate so that a measure of global performance is not unnecessarily degraded. The Witsenhausen counterexample [5, 6] illustrates the difficulties that arise in ensuring global performance objectives based on local information. In particular, communication constraints on the controller structure induce a nonconvex optimization problem that may be computationally intractable. In the present paper we apply adaptive control to investigate the ability of the local controllers to cooperate globally despite uncertainty, communication constraints, and possibly conflicting performance objectives. This investigation consists of simulations under various decentralized controller architectures to experimentally assess the interplay between local decisions and global performance. The application for this study is active noise control, it is desirable to implement an active noise control system with independent local controllers that lack the ability to communicate with the other local controllers. The approach we apply in this paper is based on retrospective cost adaptive control (RCAC). RCAC has the ability to control MIMO systems that are possibly unstable and nonminimum phase under limited modeling information [7 ]. The development of RCAC in [7, 8, 3] assumes a centralized controller structure; the goal of this paper is to investigate the stability and performance of RCAC in a decentralized setting. Adaptive techniques for decentralized control are considered in [4 6]. Copyright 22 by ASME

2 2 Disturbance Rejection Problem Consider the MIMO discrete-time system x(k+)= Ax+Bu+D w, () y= Cx, (2) z=e x, (3) k, x R n is the state variable, z R l z is the measured performance variable to be minimized, y R l y contains additional measurements that are available for control, u R l u is the input signal, and w R l w is the external disturbance signal to be rejected. The system () (3) can represent a sampled-data application arising from a continuous-time system with sample and hold operations with the sampling period h, y represents y(kh), z represents z(kh), and so on. The plant () (3) is represented by the transfer matrices G yu (z) = C(zI A) B, G yw (z) = C(zI A) D, G zu (z) = E (zi A) B and G zw (z) = E (zi A) D. Furthermore, for a positive integer i, H i = E A i B is the i th Markov parameter of G zu. Now, consider the n th c -order strictly proper output feedback controller x c (k+)= A c x c +B c y, (4) u= C c x c, (5) The closed-loop system (6) (8) is described by [ z y ] T [ T G = G(z)w= zw w, G yw (z)] G yw (z)= C(zI Ã) D, () G zw (z)=ẽ (zi Ã) D. (2) The goal of the adaptive disturbance rejection problem is to develop an adaptive output feedback controller of the form (4), (5) that minimizes the performance measure z T z in the presence of the exogenous signal w with limited modeling information about the dynamics and the exogenous signal. For the adaptive system, the matrices in (4), (5) may be time varying, and thus the transfer function models (), (2) may not be valid during controller adaptation. However, (), (2) illustrates the structure of the closed-loop system after controller convergence. Decentralized adaptive control problem is an extension of the adaptive disturbance rejection problem, the goal is to have multiple adaptive controllers G c,i which generate a control signal u i using a measurement y i. The objective of each controller is to minimize a performance measure z T i z i in the presence of the exogenous signal w as well as the control signals generated by the remaining decentralized controllers which may have conflicting performance objectives. Decentralized adaptive architecture is shown in Figure for two controllers. x c R n c. The feedback control (4) (5) is described by u=g c (q)y, G c (q)= C c (qi A c ) B c, and q is the forward shift operator. The closed-loop system with output feedback (4), (5) is thus given by x = [ x T x T ] T, c x(k+)= Ã x+ D w, (6) y= C x, (7) z=ẽ x, (8) [ ] [ ] A BCc D Ã=, D B c C A =, (9) c nc l w C= [ ] C ly n c, Ẽ = [ ] E lz n c. () Figure. DECENTRALIZED ADAPTIVE CONTROL ARCHITECTURE WITH TWO CONTROLLERS. 3 RETROSPECTIVE COST ADAPTIVE CONTROL 3. Control Law We represent Eqs. (4), (5) by u=θ T φ(k ), (3) 2 Copyright 22 by ASME

3 discussed in Section 4. θ=[ N T NT nc MT MT nc ]T, (4) φ(k )=[ y T (k ) y T (k n c ) u T (k ) u T (k n c )] T, (5) and, for all i n c, N i R l y l u, M i R l u l u. The control law (3) can be reformulated as u=φ(k )Θ, (6) Φ(k ) = I lu φ T (k ) R l u l u n c (l u +l y ), (7) Θ = vec(θ) R l un c (l u +l y ), (8) 3.3 Cumulative Cost and Update Law For k>, we define the cumulative cost function J( ˆΘ,k) = + k i= k i= λ k i ẑ T ( ˆΘ,i)ẑ( ˆΘ,i) λ k i η(i) ˆΘ T Φ T f (i )Φ f(i ) ˆΘ + λ k ( ˆΘ Θ ) T P ( ˆΘ Θ ), (24) λ (,], P R l un c (l u +l y ) l u n c (l u +l y ) is positive definite, η, and Θ R l un c (l u +l y ). In this paper, we choose denotes the Kronecker product, and vec is the columnstacking operator. 3.2 Retrospective Performance For a positive integer n f, we define η = p c η z T (k j)z(k j). (25) j= η, and p c. The following result provides the global minimizer of the cost function (24). G f (q) = D f (q)n f (q) R l z l u [q], (9) Proposition 3.. Let P()=P and Θ()=Θ. Then, for all k, the cumulative cost function (24) has a unique global minimizer Θ. Furthermore, Θ is given by N f (q) = K q n f + K 2 q n f + + K nf, D f (q) = I lz q n f + A q n f + A 2 q n f + + A nf, (2) Θ=Θ(k ) +η P(k )ΦT f (k )Λ ε, (26) K i R l z l u for i r, A j R l z l z for j r, n f is the order of G f, and each polynomial entry of D f (q) is asymptotically stable. Next, for k, we define the retrospective performance variable with ẑ( ˆΘ,k) = z+φ f (k ) ˆΘ u f, (2) Φ f (k ) = G f (q)φ(k ) R l z l u n c (l u +l y ), (22) u f = G f (q)u R l z, (23) ˆΘ is determined by optimization below. In this paper, we choose G f as a finite-impulse-response filter so that A i = for all i {,...,n f }. The choice of G f is further Λ = λ +η I l z + Φ f (k )P(k )Φ T f (k ), ε = z u f +(+η)û f, û f = Φ f (k )Θ(k ), and P satisfies P= λ [P(k ) P(k )Φ T f (k )Λ Φ f (k )P(k )]. (27) Proof 3.. The result follows from RLS theory [7, 8]. 3 Copyright 22 by ASME

4 4 CONSTRUCTION OF G f BASED ON PHASE MATCH- ING The phase-matching-based construction of G f is applicable to plants that are either minimum-phase or Lyapunov stable, and does not require knowledge of the NMP zeros of the plant. For NMP systems, this construction requires that η be positive. For unstable, NMP plants, knowledge of the NMP zero locations may be necessary [8, 2]. Let G zi u j denote the transfer function from the j th input u j R to the i th performance output z i R. Furthermore, let G f,i j denote the i j th entry of G f. Then, for Ω [,π] rad/sample, the phase mismatch i j (Ω) between G f,i j and G zi u j is defined as [ ] i j (Ω) = Re G cos zi u j (e jω )G f,i j (e jω ) Gzi u j (e jω ) Gf,i j (e jω ) [,8]. (28) For the phase-matching-based construction, we construct G f such that, for each input-output pair i j, i l z, j l u, the channel-wise phase mismatch is smaller than 9 deg, that is, i j (Ω) 9 deg,for all Ω [, π] rad/sample, (29) A weaker condition is sufficient when G zu is asymptotically stable, and the exogenous signal w is harmonic. In this case, the phase-mismatch-based construction requires (Ω i ) 9 deg,ω spec(w), (3) spec(w) is the frequency spectrum of w. Two methods for minimizing phase mismatch are presented in [9]. These methods fit the IIR plant G zi u i with an FIR transfer function G f,i j to bound or minimize the phase mismatch. In particular, one method solves a constrained linear least squares problem to bound i j (Ω), while the other method solves a nonlinear least squares problem to minimize i j (Ω). The former requires an estimate of the frequency response G zi u i (e jω ) for Ω [, π] (or for Ω spec(w)), while the latter requires an estimate of the phase G zi u i (e jω ) for Ω [, π] (or for Ω spec(w).) Although these methods parameterize the fit with an FIR plant, greater flexibility may be obtained by allowing G f to be a stable IIR filter. When the nonlinear fitting method is used, only the phase difference is minimized, therefore, the magnitude of the FIR fit may be arbitrary. In this case, considering the robustness margins presented in [], we typically normalize the leading coefficient of G f to H. 5 SIMULATION RESULTS 5. Decentralized Disturbance Rejection on an Acoustic Duct Consider the acoustic duct shown in Fig. 2 with five equally spaced speakers and microphones. The equations of motion for the duct are given by [2] c 2 p tt(ξ,t)= p ξξ (ξ,t)+ ζ i (t)= p ξi,t, 5 i= ρ v νi (t)δ(ξ ξ i ), p(ξ,t) is the acoustic pressure, c is the phase speed of the acoustic wave (343 m/sec in air at room conditions), v νi are the speaker cone velocities (m/sec), and ρ is the equilibrium density of air (.2 kg/m 3 at room conditions). By using separation of variables, retaining r modal frequencies, p(ξ, t) is given by p(ξ,t)= r i= q i (t)v i (ξ). The state space realization of the system is given by and χ(t)=āχ(t)+ Bν(t) ζ(t)=ē χ(t) χ(t) = [ t q (τ)dτ q (t) t q r(τ)dτ q r (t) ] T, ν(t) = [ ν (t) ν 5 (t) ] T [ = As vν (t) v ν5 (t) ] T, ζ(t) = [ ζ (t) ζ 5 (t) ] ω 2 n µ ω n Ā=..... ω 2 nr µ rω nr B = [ ] [ b b 5, b i = ρ A s V (ξ i ) ρ A s V r (ξ i ) ] T ē Ē = [., ē i = V (ξ i ) V r (ξ i ) ] ē 5 iπ ω ni = L c, V i(ξ) = 2 c L sin(iπ ξ), i=, 2,, r L 4 Copyright 22 by ASME

5 A s is the cross-sectional area of the speaker, µ i is the damping ratio of the i th acoustic mode, and L is the length of the duct. In the examples of this section, we let A s =.25 m 2, L=3 m, µ j =., j =,,4, and ξ i =.3+(i ).6 m, i=,,5. In each example, we sample the continuous-time system with the sampling period h =.5 sec/sample using zero-order hold. The sampling rate is chosen to be large enough to capture the fast modes of the system. In all examples, we assume that the performance measurement z i is the input to the i th decentralized controller G c,i, that is, y i = z i for all controllers. Finally, in each case, we assume that the phase plot and the first Markov parameter of G zi u i is available for each decentralized controller. constructed with nonlinear fitting method. However, for reasons explained below, we choose κ=2h Pole Zero Map.6.8 Figure POLES AND ZEROS OF G z u (z) AND G z2 u 2 (z). Figure 2. ACOUSTIC DUCT. Example 5.. [Decentralized Disturbance Rejection with Shared Performance Measure.] Consider two RCAC controllers G c,, G c,2 connected to the acoustic duct as in Figure. The controllers are placed symmetrically so that G c, actuates on speaker S2, that is, u =ν 2 (kh), G c,2 actuates on speaker S4, that is, u 2 = ν 4 (kh), and the performance objective for both controllers is to minimize the acoustic pressure ζ 3 (kh), therefore, z = z 2 = ζ 3 (kh), in the presence of the disturbance w=ν 3 (kh) generated by speaker S3. Because of the symmetry between the controllers, the sampled-data transfer functions G z u (z) and G z2 u 2 (z) are equal, whose pole-zero configurations are shown in Figure 3. Furthermore, H = for both transfer functions. Using the phase information, we apply the nonlinear fitting method [9] to minimize the phase mismatch and obtain G f, (q)=g f,2 (q)=k f [ q q 5] T, (3) K f = κ[ ] (32) and κ >. Note that the choice of κ does not affect the phase of G f and thus the phase mismatch, and, as stated in Section 4, κ is typically normalized to H when one centralized controller is We consider the sinusoidal disturbance ν 3 (t) = 5 (sinω t + (t )sinω 2 t) m 3 /sec, ω =.257 khz and ω 2 =.838 khz. Furthermore, we assume that at t =.5 sec, speaker S2 experiences damage and loses the ability to actuate. Both controllers are chosen to be of order n c = 5, and the RLS covariance matrix for each controller is initialized from P = I. The plant is allowed to operate in open-loop for time steps, and at k= (that is, t =.), both controllers are turned on. The acoustic pressure ζ 3 (kh), which is the performance measurement, is driven to zero, G c,2 readapts to counter the speaker damage at k =, and it further adapts when the second sinusoidal component of w appears at k = 2, as shown in Figure 4. In (32), contrary to the description in Section 4, we chose the leading coefficient κ to be equal to 2H, instead of H. Poor results are obtained with κ=h, and, if κ is further reduced below H, instability is observed. This is because the total control input u=u +u 2 generated by the two symmetrically placed, uncoordinated controllers is twice as large as the required control level. Therefore, to reduce the control effort generated by each decentralized controller, we multiply G f, and G f,2 by 2H, that is, twice the constant that is used with a centralized controller. Example 5.2. [Decentralized Disturbance Rejection with Separate Performance Measures.] Consider two RCAC controllers G c,, G c,2 connected to the acoustic duct as in Figure. The controllers are placed on both edges of the duct so that G c, actuates speaker S, that is, u = ν (kh), G c,2 actuates speaker S5, that is, u 2 = ν 5 (kh). The controllers now have separate performance objectives; z = ζ (kh), as z 2 = ζ 5 (kh), and the disturbance signal w is generated by speaker S2. Because of the symmetry between the controllers and the performance measurements, the sampled-data transfer functions G z u (z) and G z2 u 2 (z) are equal, whose pole-zero configurations are shown in Figure 5. Furthermore, H = 3383 for 5 Copyright 22 by ASME

6 Figure 4. acoustic pressure (Pa) θ control input (m 3 /sec) 2 ζ3(kh) x 5 ν2(kh) θ 2 control input (m 3 /sec) 2 x 4 ν4(kh) 2 3 time ste SIMULATION RESULTS FOR EX. 5.: TWO DECENTRAL- IZED RCAC CONTROLLERS ARE TURNED ON AT k = TO MIN- IMIZE THE SHARED PERFORMANCE MEASURE z = ζ 3 (kh). AT k =, SPEAKER S2 EXPERIENCES DAMAGE AND G c, LOSES THE ABILITY TO ACTUATE. G c,2 READAPTS TO COUNTER THE SPEAKER FAILURE AND THE ADDITIONAL COMPONENT OF THE DISTURBANCE THAT APPEARS AT k=2. both transfer functions. Using the phase information, we apply the nonlinear fitting method [9] to minimize the phase mismatch and obtain G f, (q)=g f,2 (q) = κ q q 3.27q 2.627q.4933 q 5, κ is chosen to be H, since the performance measures are separate Pole Zero Map of G zu (z).2566 rad/sample. The controllers interact with the duct as follows. First, for 5 s, the duct is allowed to run open-loop. Next, at k = 5, G c, is turned on for minimizing the acoustic pressure ζ (kh). At k = 5, the parameters θ of G c, are frozen, and G c,2 is turned on for minimizing the acoustic pressure ζ 5 (kh). Finally, at k = 25, G c, is allowed to adapt again, and both controllers adapt simultaneously. The tuning parameters are chosen to be n c, = 4, n c,2 = 2, P, =.I, P,2 =.5I, η, =.5, p c, =, η,2 =., and p c,2 =. Even though G zi u i are minimum-phase for i =,2, we choose nonzero η,i to bound the control amplitude during the transient period for better transient performance. The closedloop response is illustrated in Figure 6. The closed-loop suppression levels at k = 5,25 and 3 are compared with the open-loop level in Figure 7. Figure 6. θ acoustic pressure (Pa) control input (m 3 /sec) 5 5 ζ(kh) x 4.5 ν(kh) 2 3 acoustic pressure (Pa) control input (m 3 /sec) θ ζ5(kh) x 4 ν5(kh) EX. 5.2: TWO DECENTRALIZED RCAC CONTROLLERS AC- TUATING SPEAKERS S AND S5 WITH SEPARATE PERFORMANCE OBJECTIVES. AT k = 5, G c, IS ACTIVATED AND DRIVES THE ACOUSTIC PRESSURE ζ (kh) AT MICROPHONE M TO ZERO. NEXT, AT k = 5, THE PARAMETERS OF G c, ARE FROZEN, G c,2 IS ACTIVATED AND DRIVES ζ 5 (kh) TO ZERO, BUT DISRUPTS ζ (kh). FINALLY, AT k = 25, G c, IS ALLOWED TO ADAPT AGAIN TO DRIVE ζ (kh) BACK TO ZERO Figure POLES AND ZEROS OF G z u (z) AND G z2 u 2 (z). We consider the sinusoidal disturbance ν 2 (t) = 3 sin ωt m 3 /sec, ω=2.53 khz, hence, w= 3 sinωk, Ω= 5.2 Decentralized Disturbance Rejection for a Lumped-Parameter Structure Consider the multiple degrees-of-freedom (MDOF) r-mass lumped parameter structure with shown in Fig. 8. The equations of motion for this system are given by M q+c d q+kq=b f + D w w, (33) 6 Copyright 22 by ASME

7 Gzw Gzw k= k= Gzw Gzw k= k= Gzw Gzw k= k= Figure 7. OPEN-LOOP AND CLOSED-LOOP BODE MAGNITUDE PLOTS IN EX. 5.2 AT TIME STEPS k = 5, 25 AND 3. BLUE REPRESENTS OPEN-LOOP, RED REPRESENTS CLOSED- LOOP. THE FIRST ROW SHOWS G z w, WHEREAS THE SECOND ROW SHOWS G z2 w. COMPARED TO THE OPEN-LOOP LEVELS, ABOUT 45 db MORE SUPPRESSION IS OBTAINED AT THE COM- MAND FREQUENCY.2566 RAD/SAMPLE IN BOTH CHANNELS AT k=3. (33) (35) in state-space form as [ ] Ā= I r M K M C d C = [ C C ], Ē = [ Ep E v ] χ(t)=āχ(t)+ Bū(t)+ D w(t), (36) y(t)= Cχ(t), (37) z(t)=ē χ(t), (38) ] [ ] B =[ r lu r lw M, D B = M, D w χ(t) = [ q(t) q(t) ], ū(t) = f(t). q = [ q q r ] R r is the displacement vector, M, C d and K are the mass, damping and stiffness matrices respectively, and are given by M = diag(m,...,m r ), Figure 8. AN r-mass LUMPED PARAMETER STRUCTURE. c + c 2 c 2 c 2 c 2 + c 3 c 3 C d = c r c r + c r c r c r c r + c r+ k + k 2 k 2 k 2 k 2 + k 3 k 3 K = k r k r + k r k r k r k r + k r+ The control input to the system is the force f R l u, and the disturbance signal is given by w R l w. The outputs are y(t)= [ ] [ ] q C C, (34) q z(t)= [ ] [ ] q E p E v, (35) q z(t) R l z is the performance measurement, and y(t) R l y contains measurements available for feedback. We can write In each example, we sample the continuous-time system (36) (38) with the sampling period h =.2 sec/sample using zero-order hold to obtain the sampled-data plant () (3). Each example is constructed so that the sampling rate is sufficiently large to capture the fastest mode of the plant, and the disturbance frequency is chosen smaller than the Nyquist rate to prevent aliasing. Finally, in all examples, we assume that the performance measurement z i is the input to the i th decentralized controller G c,i, that is, y i = z i for all controllers. Example 5.3. [Decentralized Disturbance Rejection with Separate Performance Measures.] Consider a 3DOF lumped parameter structure with the masses m =. kg, m 2 =.2kg, m 3 = kg; spring constants k =.7 kg/sec 2, k 2 = 3 kg/sec 2, k 3 = 2.3 kg/sec 2, k 4 = kg/sec 2 ; and the damping coefficients c =.5 kg/sec, c 2 =.5 kg/sec, c 3 =.8 kg/sec, and c 4 = kg/sec. With these parameters, every eigenvalue of Ā lies in the open left-half plane, and thus the structure is asymptotically stable. Consider two RCAC controllers G c,, G c,2 connected to the structure as in Figure. The controller G c, applies a force f (kh) on m, and the controller G c,2 applies a force f 3 (kh) on m 3, that is, u = f (kh), u 2 = f 3 (kh) (see Figure 8). The controllers have separate performance objectives; z = q (kh), as z 2 = q 3 (kh), and there is a disturbance force w 2 (t) 7 Copyright 22 by ASME

8 exerted on the mass m 2. The goal of each decentralized controller is to reject the disturbance force from the corresponding performance variable. Pole-zero configurations of the sampleddata plants G z u (z) and G z2 u 2 (z) are shown in Figure 9. Both channels have a minimum-phase sampling zero near.7. Furthermore, the first Markov parameter corresponding to G z u is H, =.32, and the first Markov parameter corresponding to G z2 u 2 is H,2 =.44. Using the phase information of the sampled-data channels, we apply the nonlinear fitting method [9] to minimize the phase mismatch in both channels, and obtain Figure 9. G c, (q)=h, (q +.938q ), G c,2 (q)=h,2 (q q ). Pole Zero Map Gzu Pole Zero Map Gz2u2.5.5 POLES AND ZEROS OF G z u (z) AND G z2 u 2 (z). We consider the sinusoidal disturbance force w 2 (t) = sinωt N, ω = Hz, hence, w = sinωk, Ω =.472 rad/sample. The controllers interact with the structure as follows. First, for 5 s, the duct is allowed to run open-loop. Next, at k= 5, G c, is turned on for minimizing the displacement q (kh). At k=5, the parameters θ of G c, are frozen, and G c,2 is turned on for minimizing the displacement q 3 (kh). Finally, at k = 25, G c, is allowed to adapt again, and both controllers adapt simultaneously. The tuning parameters are chosen to be n c = 4, P = I, η = for both controllers. The closed-loop response is illustrated in Figure. Unlike Ex. 5.2, the decentralized controllers do not significantly disrupt the performance measure of each other. This is because the bode gains of G z u 2 and G z2 u at the disturbance frequency are significantly lower than the gains of G z u and G z2 u 2. The closed-loop suppression levels at k = 5,25 and 3 are compared with the open-loop level in Figure. Example 5.4. [Decentralized Disturbance Rejection in the Presence of Unknown NMP Zeros.] Consider a 4DOF lumped parameter structure with the masses m =.5 kg, m 2 =.2kg, m 3 =.4kg, m 4 =.4 kg; spring constants k =.5 kg/sec 2, k 2 = 3.5 kg/sec 2, k 3 = 2 kg/sec 2, k 4 = 3 kg/sec 2, k 5 = kg/sec 2 ; and the damping coefficients c =.7 kg/sec, c 2 = 2.6 kg/sec, c 3 = 2.3 kg/sec, c 4 = 2.5 kg/sec, and c 5 =.8 kg/sec. With Figure. force (N) θ displacement (m) 2 q(kh) f(kh) force (N) θ 2 displacement (m) 2 q3(kh) f3(kh) EX. 5.3: TWO DECENTRALIZED RCAC CONTROLLERS EXERTING FORCES f AND f 3 WITH SEPARATE PERFORMANCE OBJECTIVES. AT k = 5, G c, IS ACTIVATED AND DRIVES THE PERFORMANCE DISPLACEMENT q (kh) TO ZERO. NEXT, AT k = 5, THE PARAMETERS OF G c, ARE FROZEN, G c,2 IS ACTIVATED AND DRIVES q 3 (kh) TO ZERO. THE ACTIVATION OF G c,2 DOES NOT SIGNIFICANTLY AFFECT THE DISPLACEMENT q. 5 k=5 Gzw Gzw k= k=25 Gzw Gzw k= k=3 Gzw Gzw k= Figure. OPEN-LOOP AND CLOSED-LOOP BODE MAGNITUDE PLOTS IN EX. 5.3 AT TIME STEPS k = 5, 25 AND 3. BLUE REPRESENTS OPEN-LOOP, RED REPRESENTS CLOSED- LOOP. THE FIRST ROW SHOWS G z w, WHEREAS THE SECOND ROW SHOWS G z2 w. COMPARED TO THE OPEN-LOOP LEVELS, ABOUT 6 db MORE SUPPRESSION IS OBTAINED AT THE COM- MAND FREQUENCY.472 RAD/SAMPLE IN BOTH CHANNELS AT k=3. these parameters, every eigenvalue of Ā lies in the open left-half plane, and thus the structure is asymptotically stable. Consider two RCAC controllers G c,, G c,2 connected to the structure as in Figure. The controller G c, applies a force f (kh) on m, and the controller G c,2 applies a force f 4 (kh) on m 4, and thus, u = f (kh), u 2 = f 4 (kh). The controllers have separate performance objectives; z = q 2 (kh), as z 2 = q 3 (kh), and the disturbance forces w 2 (t) and w 3 (t) perturb the structure. The goal of each decentralized controller 8 Copyright 22 by ASME

9 is to reject the disturbance forces from the corresponding performance variable. As shown in Figure 2, both channels are nonminimum-phase; G z u (z) has a NMP sampling zero near.3, as G z2 u 2 (z) has a NMP sampling zero near.2. Furthermore, the first Markov parameter corresponding to G z u is H, =.35, and the first Markov parameter corresponding to G z2 u 2 is H,2 =.34. Using the frequency response information of the sampled-data channels, we apply the linear fitting method [9] to bound the phase mismatch for each channel from above by 6 deg. This gives G c, (q)=.5886q q, (39) G c,2 (q)=.5438q q. (4) Note that since the linear method is used to obtain (39), (4), there is no need to normalize the leading coefficients G zu Pole Zero Map 3 Figure G z2u2 Pole Zero Map 3 POLES AND ZEROS OF G z u (z) AND G z2 u 2 (z). We consider the sinusoidal disturbance forces w 2 (t) = sinω t N and w 3 (t) = 8sinω 2 t, ω = Hz and ω 2 = Hz. Hence, we have w [ w w 2 ], w = sinω k, w 2 = 8sinΩ 2 k, Ω =.8976 rad/sample, and Ω 2 =.572 rad/sample. The controllers interact with the structure as follows. First, for 5 s, the duct is allowed to run open-loop. Next, at k=5, G c, is turned on for minimizing the displacement q 2 (kh). At k = 3, the parameters θ of G c, are frozen, and G c,2 is turned on for minimizing the displacement q 3 (kh). Finally, at k = 6, G c, is allowed to adapt again, so that both controllers adapt simultaneously. The time intervals for individual controller adaptation are chosen to be larger than the previous Examples 5.2 and 5.3, because, typically, it takes longer for RCAC to converge when the plant is nonminimum-phase. The tuning parameters are chosen to be n c = 5 and P =.I for both controllers. Furthermore, since both channels have unknown NMP zeros, we let η =., p c = for G c, ; η =.2, p c = for G c,2. The closed-loop response is illustrated in Figure 3. The closed-loop suppression levels at k = 3, k = 6 and k = are compared with the open-loop level in Figure 4. Figure 3. displacement (m) force (N) θ 2 q2(kh) f(kh) displacement (m) force (N) θ 2 2 q3(kh) f4(kh) EX. 5.4: TWO DECENTRALIZED RCAC CONTROLLERS TO MINIMIZE q 2 (kh), q 3 (kh) IN THE PRESENCE OF NMP SAMPLING ZEROS. AT k = 5, G c, IS ACTIVATED AND DRIVES THE PER- FORMANCE DISPLACEMENT q 2 (kh) TO ZERO. NEXT, AT k = 3, THE PARAMETERS OF G c, ARE FROZEN, G c,2 IS ACTIVATED AND DRIVES q 3 (kh) TO ZERO. THE PERFORMANCE q 2 (kh) DEGRADES DUE TO THE ACTIVATION OF G c,2. HOWEVER, AT k=6, G c, IS ALLOWED TO ADAPT AGAIN AND DRIVES q 2 (kh) BACK TO ZERO. 5 k= k= k=3 k= k= k= k= k= k= k= k= k= Figure 4. OPEN-LOOP AND CLOSED-LOOP BODE MAGNITUDE PLOTS IN EX. 5.4 AT TIME STEPS k = 3, 6 AND. BLUE REPRESENTS OPEN-LOOP, RED REPRESENTS CLOSED- LOOP. FIRST ROW SHOWS G z w, SECOND ROW SHOWS G z w 2, THIRD ROW SHOWS G z2 w, AND FOURTH ROW SHOWS G z2 w 2. DIS- TURBANCE FREQUENCY IS ATTENUATED AT EACH CHANNEL. 6 CONCLUSION In this paper, we investigated the stability and performance of retrospective cost adaptive control in a decentralized setting. 9 Copyright 22 by ASME

10 We applied adaptive control to investigate the ability of the local controllers to cooperate globally despite uncertainty, lack of communication, and possibly conflicting performance objectives. The application for this study was noise rejection, it is desirable to implement an active noise control system with independent local controllers that lack the ability to communicate with the other local controllers. When the decentralized controllers share the same performance measure, stability was maintained by scaling the leading coefficient of the filter that is used in retrospective cost optimization. When the decentralized controllers have conflicting performance objectives, noise attenuation was obtained by properly scheduling the adaptation periods of the decentralized controllers. REFERENCES [] Siljak, D. D., 99. Decentralized Control of Complex Systems. Academic Press. [2] Wang, S. H., and Davison, E., 973. On the Stabilization of Decentralized Control Systems. IEEE Trans. Autom. Contr., 8(5), pp [3] Jr., N. S., Varaiya, P., Athans, M., and Safonov, M., 978. Survey of Decentralized Control Methods for Large Scale Systems. IEEE Trans. Autom. Contr., 23(2), pp [4] Corfmat, J. P., and Morse, A. S., 976. Decentralized Control of Linear Multivariable Systems. Automatica, 2(5), pp [5] Witsenhausen, S., 968. A Counterexample in Stochastic Optimum Control. SIAM J. Control, 6(), pp [6] Sahai, A., and Grover, P., 2. Demystifying the Witsenhausen Counterexample. IEEE Contr. Sys., 3(6), pp [7] Venugopal, R., and Bernstein, D., 2. Adaptive Disturbance Rejection Using ARMARKOV System Representation. IEEE Trans. Contr. Sys. Tech., 8, pp [8] Santillo, M. A., and Bernstein, D. S., 2. Adaptive Control Based on Retrospective Cost Optimization. J. Guid. Contr. Dyn., 33, pp [9] Hoagg, J. B., and Bernstein, D. S., 2. Cumulative Retrospective Cost Adaptive Control with RLS-Based Optimization. In Proc. Amer. Contr. Conf., pp [] D Amato, A. M., Sumer, E. D., Mitchell, K. S., Morozov, A. V., Hoagg, J. B., and Bernstein, D. S., 2. Adaptive Output Feedback Control of the NASA GTM Model with Unknown Nonminimum-Phase Zeros. In AIAA Guid. Nav Contr. Conf. [] Hoagg, J. B., Santillo, M. A., and Bernstein, D. S., 28. Discrete-Time Adaptive Command Following and Disturbance Rejection with Unknown Exogenous Dynamics. IEEE Trans. Autom. Contr., 53, pp [2] Hoagg, J. B., and Bernstein, D. S., 2. Retrospective Cost Adaptive Control for Nonminimum-Phase Discrete- Time Systems Part : The Ideal Controller and Error System; Part 2: The Adaptive Controller and Stability Analaysis. In Proc. Conf. Dec. Contr., pp [3] D Amato, A. M., Sumer, E. D., and Bernstein, D. S., 2. Frequency-Domain Stability Analysis of Retrospective- Cost Adaptive Control for Systems with Unknown Nonminimum-Phase Zeros. In Proc. Conf. Dec. Contr., pp [4] Ioannou, P., 986. Decentralized Adaptive Control of Interconnected Systems. IEEE Trans. Autom. Contr., 3(4), pp [5] Jain, S., and Khorrami, F., 997. Decentralized Adaptive Control of a Class of Large-Scale Interconnected Nonlinear Systems. IEEE Trans. Autom. Contr., 42(2), pp [6] Narendra, K. S., and Oleng, N. O., 22. Exact Output Tracking in Decentralized Adaptive Control Systems. IEEE Trans. Autom. Contr., 47(2), pp [7] Astrom, K. J., and Wittenmark, B., 995. Adaptive Control. Addison-Wesley. [8] Goodwin, G. C., and Sin, K. S., 984. Adaptive Filtering, Prediction and Control. Prentice Hall. [9] Sumer, E. D., Holzel, M. H., D Amato, A. M., and Bernstein, D. S., 22. FIR-Based Phase Matching for Robust Retrospective Cost Adaptive Control. In Amer. Contr. Conf. [2] Hong, J., Akers, J. C., Venugopal, R., Lee, M. N., Sparks, A. G., Washabaugh, P. D., and Bernstein, D. S., 996. Modeling, Identification, and Feedback Control of Noise in an Acoustic Duct. IEEE Trans. Contr. Sys. Tech., 4, pp Copyright 22 by ASME