Coordinated Decentralized Adaptive Output ... - Semantic Scholar
FrA11.4
Proceeding of the 2004 American Control Conference Boston, Massachusetts June 30 - July 2, 2004
Coordinated Decentralized Adaptive Output Feedback Control of Interconnected Systems Naira Hovakimyan, Eugene Lavretsky, Bong-Jun Yang and Anthony Calise
A BSTRACT A decentralized adaptive output feedback control design is proposed for large-scale interconnected systems. It is assumed that all the controllers share prior information about the system reference models. A linearly parameterized neural network is introduced for each subsystem to partially cancel the effect of the interconnections on tracking performance. Boundedness of error signals is shown through Lyapunov’s direct method. I. I NTRODUCTION With the advent of complex engineering systems, interest in design of decentralized controllers has especially increased. The problem can be briefly formulated as a control design for a system composed of several dynamically interconnected subsystems, such that the output of each subsystem has to track a prescpecified reference trajectory, while no communication is allowed between the controllers. The problem was first introduced in [1] for weakly interconnected subsystems having regulated outputs with relative degree 1 or 2. In [2] a framework for model reference adaptive control has been developed under restrictive assumptions, like positive definiteness of an Mmatrix involving unknown constants, relative degrees of outputs being one or two, and matched uncertainties. These conditions were further relaxed in [3]–[10]. A detailed review of the cited literature one can find in [11]. Here we formulate and solve the problem of decentralized adaptive output feedback control for a class of nonlinear subsystems with known relative degrees, subject to unknown interconnections with known upper bound. We depart from attempting to obtain global results, and restrict the synthesis approach to a domain over which the interconnections and nonlinearities can be approximated by a linearly parameterized neural network. Similar attempts Research of the first author is supported in part by AFOSR under Grant F49620-03-1-0443, and MURI subcontract F49620-03-1-0401. Research of third and fourth authors is sponsored in part by AFOSR under Grant F4960-01-1-0024. Naira Hovakimyan is an Associate Professor of Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061-0203, e-mail: [email protected], Senior Member IEEE, Corresponding author Eugene Lavretsky is an Associate Technical Fellow of The Boeing Company - Phantom Works, Huntington Beach, CA 92647-2099, Associate Fellow, AIAA Bong-Jun Yang is a postdoctoral fellow of the School of Aerospace Engineering, Georgia Tech, Atlanta, GA 30332-0150 Anthony Calise is a Professor of the School of Aerospace Engineering, Georgia Tech, Atlanta, GA 30332-0150
0-7803-8335-4/04/$17.00 ©2004 AACC
of incorporating neural networks into decentralized adaptive control have have been reported in [12], [13] for design of state feedback controllers. Following [14]–[16], we assume that the desired trajectories are known to all the controllers, i.e. the controllers share prior information about their goals, and we develop an adaptive output feedback synthesis approach that achieves ultimate boundedness of tracking errors. As in [14], [15], we will say that the controllers are engaged in implicit cooperation. While most of the existing results in decentralized control literature rely on the definition of a robust controller for dominating the interconnections, we show through Lyapunov’s direct method that a linearly parameterized neural network, operating over reference model states, can partially cancel the interconnection effects. Ultimate boundedness of error signals is shown using Lyapunov’s direct method. This paper should be viewed as the extension of adaptive output feedback control approach developed in [17] for centralized control to a decentralized setup, using the viewpoint of [14], [16] for definition of implicit cooperation. The paper is organized as follows. In Section II we state the problem formulation and assumptions about the subsystem dynamics. In Section III we present the approach and define the error dynamics. In Section IV, we define the adaptive controller for each subsystem and derive associated bounds. Section V has a proof on ultimate boundedness of error signals of the large-scale system. In Section VI, we illustrate the theoretical results on non-minimum phase system like three inverted pendulums. Throughout the manuscript bold symbols are used for column vectors, small letters for scalars, capital letters for matrices, || · || denotes 2-norm unless otherwise noted. II. S YSTEM D ESCRIPTION AND P ROBLEM F ORMULATION Let the large-scale system be composed of m stabilizable nonlinear single-input single-output (SISO) subsystems, represented in the following normal form: x˙ i z˙ i
= Ai xi + Bi z i + bi [ui + fi (x1 , z 1 , . . . , xm , z m )] = Ci xi + Di z i + g i (x1 , z 1 , . . . , xm , z m ) (1)
yi
= cTi xi , i = 1, · · · , m,
where [xTi z Ti ]T ∈ Rri +(ni −ri ) is the state vector of the realization of the ith subsystem in normal form, ri representing the relative degree, ui ∈ R and yi ∈ R are the control and measurement of the ith subsystem, fi : Rn1 +···+nm → R , g i : Rn1 +···+nm → Rni −ri are sufficiently smooth
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unknown functions, representing the modeling errors and interconnection effects, while Ai , Bi , bi , ci are matrices and vectors corresponding to the normal realization:
0 .. = . ai1 0 .. = .
Ai
Bi
ci1
1 .. . ai2 ··· .. . ···
··· .. . ···
0 .. .
0 0
airi
cini −ri
,
and bi = [ 0 · · · bi ]T , ci = [ 1 · · · 0 ]T . Assumption 2.1: The functions g i (x1 , z 1 , . . . , xm , z m ) are bounded for all i = 1, · · · , m: kg i (·)k ≤
m X j=1
T
αi xTj z Tj
,
αi > 0
(2)
The objective is to synthesize decentralized adaptive output feedback control laws ui , such that yi (t) tracks a smooth bounded reference trajectory yli (t) with bounded errors for all i = 1, · · · , m, under the assumption that the ith controller knows the desired states of all the subsystems j = 1, · · · , m, while having access only to its own measurement yi (t). As in [14]–[16], we introduce the following assumption. Assumption 2.2: The signals yli (t) are assumed to be generated by the following stable linear closed-loop reference models ξ˙ li = A¯i ξ li + bri yci ,
¯Ti ξ li yli = c
(3)
consisting of an open loop system x˙ li z˙ li
= Ai xli + Bi z li + bi uli = Ci xli + Di z li
y li
= cTi xli ,
i = 1, · · · , m
= Aci xci + bci (yci − yli )
u li
= cTci xci + dci (yci − yli ) ,
(4)
i = 1, · · · , m , (5)
where Ai − bi dci cTi ¯ Ci Ai = −bci cTi
Bi Di 0
bi cTci 0 Aci
(7) k[ xTli z Tli ]T k ≤ βi , i = 1, · · · , m Remark 2.1: In [14], this problem formulation has been addressed for linear subsystems, and, by a proper choice of robustifying signal, it has been shown that global asymptotic tracking can be achieved if the robustifying gain satisfies a lower bound, depending upon the number of subsystems and the apriori known bound on the interconnection effects. In [15], these results have been extended to nonlinear interconnections, modeled by known nonlinear functions. Moreover, output feedback has been formulated and solved for the case of subsystems having regulated outputs with relative degree 1. Our approach is different in two perspectives: i) we formulate the problem in output feedback for arbitrary relative degree by extending the results of [17] for centralized control, ii) we use an adaptive signal for overcoming the effect of interconnections on the tracking performance. On the other hand it should be understood that, due to results in [18], one cannot expect global results while using dynamic output feedback compensators with the class of nonlinear systems presented here. III. C ONTROLLER D ESIGN , E RROR DYNAMICS The control design for each of the subsystems will be based on the logic of combining a linear controller, that stabilizes the nominal linear model in the absence of interconnections, with a neural network (NN) that approximately cancels the interconnection effects in the controllable range. Towards this end, introduce the following control signal ui = uci − uadi , where uci is the output of the following dynamic compensator η˙ ci
= Aci η ci + bci (yci − yi )
u ci
= cTci η ci + dci (yci − yi ) ,
i = 1, · · · , m , (8)
nci
and a stabilizing dynamic compensator: x˙ ci
open-loop subsystem. The following bounds are assumed to be known
(6)
= bri = [ xTli z Tli xTci ]T , ξ li T T T ¯i = [ cTi 0 0 ]T , and yci [ bi dci 0 bci ] , c is a bounded input of interest to track. The matrices Ai , Bi , bi , ci are assumed to correspond to the normal realization, as defined in (2), so that dim xli = dim xi , and dim z li = dim z i . Notice that this choice of the open loop system in (4) implies that the relative degree of the ith open-loop reference model equals that of the ith
where η ci ∈ R , Aci , bci , cci , dci are introduced in (5), and the adaptive signal uadi will be defined later. This results in the following closed-loop subsystem dynamics: ξ˙ i yi
¯i (uad − fi ) + g ¯i = A¯i ξ i + bri yci − b i T ¯i ξ i , = c i = 1, · · · , m ,
(9) (10)
¯i = [bT 0 0]T , g ¯i = where ξ i = [xTi z Ti η Tci ]T , b i T T [0 g i 0] . Following [15], define the error vector E i = ξ li − ξ i , and write the tracking error dynamics for the ith subsystem: ¯i (uad − fi ) − g ˙ i = A¯i E i + b ¯i, y ¯ i = C¯i E i , (11) E i T T ¯i I where C¯i = c separates the signals available for feedback. IV. N EURAL N ETWORK A PPROXIMATION OF N ONLINEARITIES AND A DAPTIVE C ONTROL Following [19], given arbitrary ∗ > 0 and a continuous function f (x) : Rn → Rm , defined on a compact set x ∈ D ⊂ Rn , there exists a set of bounded constant
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weights W , and a set of basis functions φ(x), such that the following representation holds ∀x ∈ D: f (x) = W T φ(x) + (x), k(x)k < ∗ . Thus, one can model the interconnections fi (x1 , z 1 , · · · , xm , z m ) = W Ti φi (Y ) + i (Y )
(12)
using the following input vector Y = [xT1 z T1 · · · xTm z Tm ]T ∈ D ⊂ Rn1 +···+nm and a vector of the radial basis functions φi (Y ) = [φi1 (Y ) · · · φiNi (Y )]T , where Ni is the number of basis functions to be used by the ith subsystem, 2 φik (Y ) = e−kY −Y ick k /2σik , Y ick is the vector of centers of the basis functions used by the ith subsystem, having the same dimension as Y , σik specifies the width of the k th basis function in the ith subsystem, and |i | < ∗i , kW i k ≤ Wi∗ . Since our interest is in decentralized design, the states of other subsystems are not available to individual controllers, therefore the input vector Y cannot be used in designing adaptive elements. Based on the assumption that the controllers share prior information about their reference models, the adaptive control signal for the ith subsystem can be designed following the same logic as in [15], ˆ T φ (Y l ) uadi = W i i
(13)
where the vector Y l is defined as Y l = [xTl1 z Tl1 · · · xTlm z Tlm ]T having the states of all the subsystems replaced by their corresponding reference states when compared to Y . Notice that due to boundedness of reference model states there exists a set Dl in the extended ˆ i are space such that Y l ∈ Dl . The adaptive laws for W similar to those in [20]: ¯i + ki W ˆ i] ˆ T Pi b ˆ˙ i = −Fi [2φi (Y l )E W i
(14)
in which Pi is the solution of the Lyapunov equation A¯Ti Pi + Pi A¯i = −Qi for some Qi > 0, and Fi , ki > 0 ˆ i propagates according to the are adaptation gains, while E following dynamics: ˆ i, ˆ i + Ki (¯ ˆ˙ i = A¯i E ˆi) , y ˆ i = C¯i E yi − y E
(15)
where Ki is a gain matrix, and should be chosen such that A¯i − Ki C¯i is asymptotically stable, while i = 1, · · · , m. ˜ i = Ei − E ˆ i , i = 1, · · · , m. Let A˜i = A¯i − Ki C¯i , E Then ¯i (uad − fi ) − g ˜i +b ˜˙ i = A˜i E ¯i, E i
i = 1, · · · , m
(16)
Through several algebraic manipulations and, using the mean value theorem, one can obtain T
˜ φi (Y l ) + W T φ0 (Y ∗ )Y˜ − i uadi − fi = W i i i
˜ T1 · · · x ˜ Tm z ˜ Tm ]T is comprised of the where Y˜ = [˜ xT1 z ˜i = tracking errors of all the subsystems (˜ xi = xli − xi , z z li − z i ), φ0i (Y ∗ ) is the bounded derivative of the basis function in an intermediate point Y ∗ = Y l + (1 − λ)Y , ˜i ,W ˆ i − W i is the parameter error 0 < λ < 1, and W vector. From the definition of E j and Y˜ it follows that kY˜ k ≤
j=m X
kE j k
(19)
j=1
V. S TABILITY A NALYSIS In this section we show through Lyapunov’s ˜ i, W ˜ i, direct method that the error signals E i , E i = 1, · · · , m, are ultimately bounded. To this end, introduce the composite error ivector h T ∆ T ˜T ··· E ˜T W ˜ T ··· W ˜T ζ = E · · · ET E ∈ 1
m
1
m
1
¯i y
= C¯i E i .
(17)
m
R2(n1 +···+nm ) × RN1 +···+Nm , and consider the following ∆ positive definite function V (ζ) = ζ T T ζ, where −1 ]. T = blockdiag[P1 · · · Pm P˜1 · · · P˜m 12 F1−1 · · · 12 Fm Further, notice that the RBF network approximation can be defined over arbitrarily large compact set D. Based on the definition of the compact set Dl , and the boundedness of xci and η ci , in the subspace of the error variables consider the following compact set ΩE of n T T possible initial errors: ΩE = ∈ E 1 · · · E Tm o n1 +···+nm : Y ∈ D, Y l ∈ Dl , xc ∈ Dxc , η c ∈ Dηc . R In the expanded space of the error variable ζ ∈ R2(n1 +···+nm ) × RN1 +···+Nm , consider the largest level set of V (ζ) = ζ T T ζ
(20)
corresponding to [ E T1 · · · E Tm ]T ∈ ΩE and introduce the largest ball that lies inside this level set: BR = {ζ | kζk ≤ R}
(21) T
Let α be the minimum value of V (ζ) = ζ T ζ on the boundary of BR : ∆
α = min V (ζ) = R2 λmin (T ) kζk=R
(22)
where λmin (T ) is introduced for the minimum eigenvalue of T . Introduce the set ∆
We immediately note that for arbitrary positive definite ˜ i > 0 there exists a unique solution P˜i = P˜ T > 0 such Q i ˜ i . The error dynamics in (11) can that A˜Ti P˜i + P˜i A˜i = −Q be expressed as: h T i ¯i W ˆ φi (Y l ) − W T φi (Y ) − i − g ˙ i = A¯i E i + b ¯i E i i
(18)
Ωα = {ζ ∈ BR | V (ζ) ≤ α}
(23)
Assumption 5.1: Let p R > γ λmax (T )/λmin (T )
(24)
where λmax (T ) is introduced for the maximum eigenvalue q q q ω ω ω of T , while γ = max ˜ , λmin (D) , λmin (Λ) , λ (D) min
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∆ in which D = diag θ1 · · · θm , θi = λmin (Qi ) − ¯i k − Pj=m (2kPj b ¯j k + 2mαi λmax (Pi ) − (mφ∗i + 1)kPi b j=1 ∆ ˜ = > 0, D Θj )φ∗j − αi λmax (Pi ) + λmax (P˜i ) ˜ i )−2mαi λmax (P˜i )− diag θ˜1 · · · θ˜m , θ˜i = λmin (Q ∆ ¯i k + µi − µi kφ (Y l )k > 0, Λ = (mφ∗i + 1) kPi b i diag λ1 · · · λm , λi = k2i − µi kφi (Y l )k > 0, ∆ Pm ¯i k+µi )(∗ )2 +αi (λmax (Pi )+ ω = i=1 ( k2i (Wi∗ )2 +(2kPi b i P j=m ¯i + λmax (P˜i )) j=1 βj2 ), φ∗i , Wi∗ kφ0i (Y ∗ )k, µi , kP˜i b ¯ Pi bi k. Theorem 5.1: Let Assumptions 2.1, 2.2 and 5.1 hold. If the initial errors lie in Ωα , defined in (23), then all ˜ i, W ˜ i , i = 1, · · · , m, in the closed loop the signals E i , E system are ultimately bounded. Proof. Consider the following Lyapunov function candidate for each of the subsystems: ˜i ˜ T F −1 W ˜ i, W ˜ i ) = E T Pi E i + E ˜ i + 1W ˜ T P˜i E Vi (E i , E i i 2 i i Substituting the adaptive laws from (14) implies: V˙ i
=
˜i ˜TQ ˜iE −E Ti Qi E i − E i h T ¯i W T φ0 (Y ∗ )Y˜ ˆ Pi b +2E i i i
i
Notice that using the bound in (7), the upperbound in (2) can be represented: j=m X
αi [kE j k + βj ]
V˙
(25)
j=1
=
m X
V˙ i
(26)
i=1
Then, using the notations from Assumption 5.1, the upper bound reduces to: Pm V˙ = i=1 V˙ i ≤ h Pm i=1 − λmin (Qi ) − 2mαi λmax (Pi ) ¯i k kE i k2 − θ˜i kE ˜ i k2 −(mφ∗i + 1)kPi b h ¯i k + µi )φ∗ ˜ i k2 + Pj=m (2kPi b −λi kW i j=1 i i +αi λmax (Pi ) + λmax (P˜i ) kE j k2 + ω Regrouping, this can be written: V˙
=
m X
V˙ i ≤
i=1 m h X
¯i 2E Ti Pi g
− i + h ¯ i + Pi b ¯i ) W ˜ T (P˜i b ˜ T φ (Y l ) +2E i i i i h i ˆi ˜ iW ˜ T P˜i g ¯ i − ki W +W Ti φ0i (Y ∗ )Y˜ − i + 2E i
kg i k ≤
Now introduce the following Lyapunov function for the whole system:
i=1
(27)
i ˜ i k2 − λi kW ˜ i k2 + ω − θi kE i k2 − θ˜i kE
Following an argument similar to that in [12], T ∆ kE 1 k · · · kE m k define the vectors E = , T ∆ ∆ ˜ ˜ ˜ ˜ E = , W = kE 1 k · · · kE m k ˜ 1 k · · · kW ˜ m k T . Then the expression in kW (27) can be put into the following form: V˙
≤
˜TD ˜ −W ˜ T ΛW ˜ +ω ˜E −E T DE − E
The following upper bound
Then using (19), the following upper bound can be derived: ˜ i k2 ˜ i )kE V˙ ≤ −λmin (Qi )kE i k2 − λmin (Q i h P i j=m ∗ ¯ i k φ∗ ˜ i k + kE i k kPi b +2 kE i j=1 kE j k + i Pj=m +2αi λmax (Pi )kE i k j=1 [kE j k + βj ] h i ˜ i kkφi (Y l )k + φ∗ Pj=m kE j k + ∗ ˜ i kµi kW +2kE i i j=1 h i P j=m ˜ ik ˜ iW ˆi +2αi λmax (P˜i )kE [kE j k + βj ] − ki W j=1
Completing the squares twice and regroupong, the following upper bound can be derived: V˙ i ≤ − λmin (Qi ) − 2mαi λmax (Pi ) − (mφ∗i ¯i k kE i k2 − λmin (Q ˜ i ) − 2mαi λmax (P˜i ) +1)kPi b ¯i k + µi − µi kφ (Y l )k kE ˜ i k2 −(mφ∗i + 1) kPi b i h ¯i k ˜ i k2 + Pj=m (2kPi b − k2i − µi kφi (Y l )k kW j=1 i +µi )φ∗i + αi λmax (Pi ) + λmax (P˜i ) kE j k2 ¯i k + µi (∗ )2 + k2i (Wi∗ )2 + 2kPi b P i j=m 2 +αi λmax (Pi ) + λmax (P˜i ) j=1 βj
˜ 2 ˜ Ek V˙ ≤ −λmin (D)kEk2 − λmin (D)k 2 ˜ k +ω −λmin (Λ)kW implies that either of the following conditions r ω kEk > λmin (D)
r ω
˜
E > ˜ λmin (D) r ω ˜ k> kW λmin (Λ) will render V˙ < 0 outside the compact set Bγ = {ζ | kζk ≤ γ}
(28)
Let Γ be the maximum value of the function V (ζ) on the boundary of Bγ : ∆
Γ = max V = γ 2 λmax (T ) kζk=γ
(29)
Assumption 5.1 ensures Ωγ , {ζ| V = Γ} ⊂ Ωα
(30)
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Thus, if the initial error ζ 0 = ζ(0) belongs to Ωα , then there exists a time instant tζ (ζ 0 ), such that ζ(t) will enter the set Ωγ at tζ and remain inside it for all t > tζ . This implies ultimate boundedness of ζ, completing the proof. Remark 5.1: The results obtained above can be extended to the case where the modeling errors also depend upon the control signal, i.e. in (1) one can have fi (ui , x1 , z 1 , . . . , xm , z m ), subject to ∂fi /∂ui 6= 0. Notice then that the adaptive signal will be introduced to cancel a function fi (ui (uadi (·), ·) of itself. To avoid this algebraic loop, one way of implementing this is to use a one step delayed value of the control signal ui (t − d), where d > 0 is sufficiently small. Remark 5.2: Assumption 5.1 may be interpreted as placing both upper and lower bounds on the adaptation gains. ∆ ∆ ∆ ¯ = Let γ¯ = max(λmax (Fi )), γ = min(λmin (Fi )), λ ∆ max(λmax (Pi ), λmax (P˜i )), λ = min(λmin (Pi ), λmin (P˜i )), i = 1, · · · , m. Then an upper bound for the adaptation gains ¯ > 1 and 2λ¯ γ > 1, for which the relation results when 2λγ ¯ A lower bound for in (24) reduces to γ¯ < R2 /(2γ 2 λ). ¯ < 1 and 2λ¯ γ < 1, the adaptation gains results when 2λγ for which the relation in (24) reduces to γ > γ 2 /(2R2 λ). Notice that the upper bound for the adaptation gain has R in the numerator, while the lower bound has R in the denominator. Therefore, R can be selected sufficiently large to ensure that γ < γ¯ . VI. S IMULATIONS We consider three inverted pendulums mounted on carts, as depicted in Figure 1. The carts are connected by springs
m, I
q1
q3
q2
u1
k
M
u2
k
M
x1
u3
M
x2
x3
Three inverted pendulums on three carts
and dampers. In each subsystem, we assume that the position of the cart(xi ) and the angle of the pendulum(θi ) are measured and the cart is regulated by input forces(ui ). The equations of motion for the system are described as follows: (M + m)¨ x1 + mlp θ¨1 cos θ1 − mlp θ˙12 sin θ1
= u1 + s1
mlp cos θ1 x ¨1 + (I + mlp2 )θ¨1 − mglp sin θ1
=
0
(M + m)¨ x2 + mlp θ¨2 cos θ2 − mlp θ˙22 sin θ2 = u2 − s1 + s2 mlp cos θ2 x ¨2 + (I + mlp2 )θ¨2 − mglp sin θ2
=
0
= u3 − s2
mlp cos θ3 x ¨3 + (I + mlp2 )θ¨3 − mglp sin θ3
=
0
where u1 , u2 , u3 are input forces to the carts(N), M is the mass of the cart (kg), m is the mass of the rod(kg), lp is the distance from the pivot on the cart to the center of gravity of the rod(half of full length)(m), I(= 31 mlp2 ) is the moment of inertia of the rod with respect to its center of mass (kg·m2 ), g is the gravitational acceleration (kg·m/sec2 ), k is the spring constant (N/m), c is the damping constant (N·sec/m), s1 = k(x2 − x1 ) + c(x˙ 2 − x˙ 1 ), s2 = k(x3 − x2 ) + c(x˙ 3 − x˙ 2 ) are interconnection forces due to springs and dampers. The parameter values are: M = 0.9, m = 0.18, lp = 0.305, g = 9.8, k = 1, c = 2 × 10−5 . Our control objective is to regulate the displacements of the carts xi while balancing the inverted rods on the carts without velocity measurements. The open loop subsystem in (4) is derived after the dynamics are first linearized with respect to equilibrium position xi = θi = 0, and then put into a normal form by the transformation: xli1 = xi , xli2 = x˙ i , zli1 = θi , zli2 = xl˙pi + θ˙i . The linear subsystems, for i = 1, 2, 3, is described by the following system matrices: 0 0 0 0 1 , bi = Ai = , Bi = 1 m ˆ 0 −M 0 0 ˆ ˆ M 0 1 0 − l1p 1 Ci = , Di = g 0 , ci = 0 0 0 lp (31) ˆ = 0.815, m The constants M ˆ = 0.21 represent parameter estimates for M, m respectively. Further, in this linear model, the inverted rod is treated as a lumped mass located on its center of mass, i.e., I = 0. Putting each subsystem into normal form leads to the following modelling errors and interconnection effects defined in (1): fi = ˆ M (ui + mlp θ˙i2 sin θi − 43 mg sin θi cos θi + M +m(1−3/4 cos θi2 ) " # − coslpθi xi2 − 13 θ˙i + l1p xi2 τi ) + mgθ ˆ i − ui , g i = g sin θi − 1 xi θ˙i sin θi − g θi lp
c
c
Fig. 1.
m, I
m, I
(M + m)¨ x3 + mlp θ¨3 cos θ3 − mlp θ˙32 sin θ3
lp
2
lp
where τ1 = s1 , τ2 = −s1 + s2 , τ3 = −s2 . The τi terms imply that the spring and the damper are not considered in the open loop model. The term ui means that the modelling error also depends on the control signal as in Remark 5.1. Note that the interconnections between the two carts and the modelling errors contain velocity terms which are not measured. This implies that the existing adaptive output feedback approaches in the decentralized control literature, such as the ones developed in [8], [21] and many others, although establishing global results, cannot be applied. Moreover, regulation of xi using ui to the carts renders the control problem nonminimum phase -linearization of each subsystem about vertical-up position leads to unstable pg . These issues make the control problem even zero L more challenging. The dynamic compensator in (8) for each subsystem is designed as a LQG controller based on the open loop model in (31), in which two measured outputs
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output plot:x
output plot:x
1
output plot:x
2
0.4
3
3
0.4
0.2
0.2 2.5
0
0 2
reference without adaptation with adaptation
−0.2
−0.4
x2(m)
x1(m)
reference without adaptation with adaptation
−0.6
−0.2
x3(m)
1.5
−0.4
reference without adaptation with adaptation
−0.6
1
−0.8
−0.8 0.5
−1
−1 0
−1.2
−1.4
−1.2
0
10
20
30
40 time(sec)
50
60
70
80
−0.5
0
10
20
(a) x1
30
40 time(sec)
50
60
70
80
−1.4
0
10
20
(b) x2
30
40 time(sec)
50
60
70
80
(c) x3
Fig. 2. Comparison of the cart displacements without and with adaptive signal uadi .
output plot:θ
output plot:θ
1
3
40
reference without adaptation with adaptation
30
40
reference without adaptation with adaptation
30
20
20
3
0
0
2
1
10
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θ (degree)
10
0
reference without adaptation with adaptation
30
20
10 θ (degree)
output plot:θ
2
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−10
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−20
−20
−20
−30
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−30
0
10
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30
40 time(sec)
(a) θ1 Fig. 3. uadi .
50
60
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−30
0
10
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30
40 time(sec)
(b) θ2
50
60
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−40
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40 time(sec)
50
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(c) θ3
Comparison of the rod angles without and with adaptive signal
xi , θi are available for control design. The error observer in (15) is designed to have the smallest eigenvalue of A˜i equal approximately five times the smallest eigenvalue of A¯i . The basis functions have the following structure for three 2 subsystems: φik (Y l ) = e−kY l −Y ick k /2σik , σik = 1, i = 1, 2, 3, k = 1, . . . , Ni , where N1 = N3 = 7, N2 = 9. The centers Y ick are randomly selected over a grid of possible values for the vector Y l . All of the NN inputs are normalized using an estimate for their maximum values. Since the dynamics of the first and third carts are coupled only through the dynamics of the middle cart, and the modelling error contains a control signal, we choose the NN input vectors as: Y Tl1 = [u1 xl11 xl12 zl11 zl12 xl21 xl22 0 0 0 0 0 0]T , Y Tl2 = [u2 xl11 xl12 0 0 xl21 xl22 zl21 zl22 xl31 xl32 0 0]T , Y Tl3 = [u3 0 0 0 0 xl21 xl22 0 0 xl31 xl32 zl31 zl32 ]T , where Y li represents the NN input vector for the ith subsystem. Adaptation gains are chosen as: Fi = 0.5I, ki = 0.05. Figure 2 compares output tracking performances when the reference command yci , i = 1, 2, 3 is a square wave signal of magnitude 0.15m and 0.05 Hz. The pendulum angles are shown in Figure 3. The initial conditions are: x1 (0) = x˙ 1 (0) = 0, θ1 (0) = −30o , θ˙1 (0) = −10o /sec, x2 (0) = x˙ 2 (0) = 0, θ2 (0) = 30o , θ˙2 (0) = 10o /sec, x3 (0) = x˙ 3 (0) = 0, θ3 (0) = 20o , θ˙2 (0) = −10o /sec. Without adaptive control compensation, the system goes unstable. When each control is augmented with an adaptive term, the three carts are in synchronous motion with the pendulums balanced, implying implicit cooperation for output tracking.
VII. C ONCLUSIONS A methodology is presented for adaptive output feedback decentralized control design under the assumption that the reference trajectories are known to all subsystems. A linearly parameterized neural network is used to model the interconnection effects on-line. Boundedness of error signals is shown using Lyapunov’s direct method. The methodology is applicable to non-minimum phase subsystems. R EFERENCES [1] P. Ioannou. Decentralized adaptive control of interconnected systems. IEEE Trans. Autom. Contr., 31(4):291–298, 1986. [2] P. Ioannou and P. Kokotovic. Decentralized adaptive control of interconnected systems with reduced order models. Automatica, 21(4):401–412, 1985. [3] D. Gavel and D. Siljak. Decentralized adaptive control: structural conditions for stability. IEEE Trans. Autom. Contr., 34(4):413–426, 1989. [4] A. Datta and P. Ioannou. Decentralized adaptive control. Adavances in Control and Dynamic Systems, Academic Press, 1992. [5] R. Ortega. An energy amplification condition for decentralized adaptive stabilization. IEEE Trans. Autom. Contr., 41:285–288, 1996. [6] M. Krstic, I. Kanellakopoulos, and P. Kokotovic. Nonlinear design of adaptive controllers for linear systems. IEEE Trans. Autom. Contr., 39(4):738–752, 1994. [7] L. Shi and S.K. Singh. Decentralized adaptive controller design for large scale systems with higher order interconnections. IEEE Trans. Autom. Contr., 37(8):1106–1118, 1992. [8] S. Jain and F. Khorrami. Decentralized adaptive output feedback design for large scale nonlinear systems. IEEE Trans. Autom. Contr., 42(4):729–735, 1997. [9] J.C. Geromel, J. Bernussou, and M.C. Oliveira. Norm optimization with constrained dynamic output feedback controllers: Decentralized and reliable control. IEEE Trans. Autom. Contr., 44(7):1449–1454, 1999. [10] G.H. Yang, J. Wang, C.B. Soh, and J. Lam. Decentralized controller design for nonlinear systems. IEEE Trans. Autom. Contr., 44(3):578– 583, 1999. [11] Z.P. Jiang. Decentralized disturbance attenuating output feedback trackers for large-scale nonlinear systems. Automatica, 38(8):1407– 1415, 2002. [12] J.T. Spooner and K.M. Passino. Decentralized adaptive control of nonlinear systems using radial basis neural networks. IEEE Trans. Autom. Contr., 44(11):2050–2057, 1999. [13] J. Spooner, M. Maggiore, R. Ordonez, and K. Passino. Stable Adaptive Control and Estimation for Nonlinear Systems: Neural and Fuzzy Approximator Techniques. John Wiley & Sons, New York, 2002. [14] K. Narendra and N. Oleng. Exact output tracking in decentralized adaptive control systems. IEEE Trans. Autom. Contr., 47(2):390–395, 2002. [15] K. Narendra and N. Oleng. Decentralized adaptive control. Proceedings of American Control Conference, 2002. [16] B.. Mirkin. Comments on “Exact Output Tracking in Decentralized Adaptive Control Systems”. IEEE Trans. Autom. Contr., 48(2):348– 350, 2003. [17] N. Hovakimyan, B.-J. Yang, and A.J. Calise. An adaptive output feedback control methodology for non-minimum phase systems. Conference on Decision and Control 2002. [18] F. Mazenc, L. Praly, and Dayawansa W. Global stabilization by output feedback: examples and counterexamples. Syst. Contr. Lett., 23:119–125, 1994. [19] R. Sanner and J.J. Slotine. Gaussian networks for direct adaptive control. IEEE Transactions on Neural Networks, 3(6):837–864, 1992. [20] N. Hovakimyan, F. Nardi, A. Calise, and N. Kim. Adaptive Output Feedback Control of Uncertain Systems using Single Hidden Layer Neural Networks. IEEE Transactions on Neural Networks, 13(6):1420–1431, 2002. [21] Z.P. Jiang. Decentralized and adaptive nonlinear tracking of largescale systems via output feedback. IEEE Trans. Autom. Contr., 45(11):2122 –2128, 2000.
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Proceeding of the 2004 American Control Conference Boston, Massachusetts June 30 - July 2, 2004
Coordinated Decentralized Adaptive Output Feedback Control of Interconnected Systems Naira Hovakimyan, Eugene Lavretsky, Bong-Jun Yang and Anthony Calise
A BSTRACT A decentralized adaptive output feedback control design is proposed for large-scale interconnected systems. It is assumed that all the controllers share prior information about the system reference models. A linearly parameterized neural network is introduced for each subsystem to partially cancel the effect of the interconnections on tracking performance. Boundedness of error signals is shown through Lyapunov’s direct method. I. I NTRODUCTION With the advent of complex engineering systems, interest in design of decentralized controllers has especially increased. The problem can be briefly formulated as a control design for a system composed of several dynamically interconnected subsystems, such that the output of each subsystem has to track a prescpecified reference trajectory, while no communication is allowed between the controllers. The problem was first introduced in [1] for weakly interconnected subsystems having regulated outputs with relative degree 1 or 2. In [2] a framework for model reference adaptive control has been developed under restrictive assumptions, like positive definiteness of an Mmatrix involving unknown constants, relative degrees of outputs being one or two, and matched uncertainties. These conditions were further relaxed in [3]–[10]. A detailed review of the cited literature one can find in [11]. Here we formulate and solve the problem of decentralized adaptive output feedback control for a class of nonlinear subsystems with known relative degrees, subject to unknown interconnections with known upper bound. We depart from attempting to obtain global results, and restrict the synthesis approach to a domain over which the interconnections and nonlinearities can be approximated by a linearly parameterized neural network. Similar attempts Research of the first author is supported in part by AFOSR under Grant F49620-03-1-0443, and MURI subcontract F49620-03-1-0401. Research of third and fourth authors is sponsored in part by AFOSR under Grant F4960-01-1-0024. Naira Hovakimyan is an Associate Professor of Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061-0203, e-mail: [email protected], Senior Member IEEE, Corresponding author Eugene Lavretsky is an Associate Technical Fellow of The Boeing Company - Phantom Works, Huntington Beach, CA 92647-2099, Associate Fellow, AIAA Bong-Jun Yang is a postdoctoral fellow of the School of Aerospace Engineering, Georgia Tech, Atlanta, GA 30332-0150 Anthony Calise is a Professor of the School of Aerospace Engineering, Georgia Tech, Atlanta, GA 30332-0150
0-7803-8335-4/04/$17.00 ©2004 AACC
of incorporating neural networks into decentralized adaptive control have have been reported in [12], [13] for design of state feedback controllers. Following [14]–[16], we assume that the desired trajectories are known to all the controllers, i.e. the controllers share prior information about their goals, and we develop an adaptive output feedback synthesis approach that achieves ultimate boundedness of tracking errors. As in [14], [15], we will say that the controllers are engaged in implicit cooperation. While most of the existing results in decentralized control literature rely on the definition of a robust controller for dominating the interconnections, we show through Lyapunov’s direct method that a linearly parameterized neural network, operating over reference model states, can partially cancel the interconnection effects. Ultimate boundedness of error signals is shown using Lyapunov’s direct method. This paper should be viewed as the extension of adaptive output feedback control approach developed in [17] for centralized control to a decentralized setup, using the viewpoint of [14], [16] for definition of implicit cooperation. The paper is organized as follows. In Section II we state the problem formulation and assumptions about the subsystem dynamics. In Section III we present the approach and define the error dynamics. In Section IV, we define the adaptive controller for each subsystem and derive associated bounds. Section V has a proof on ultimate boundedness of error signals of the large-scale system. In Section VI, we illustrate the theoretical results on non-minimum phase system like three inverted pendulums. Throughout the manuscript bold symbols are used for column vectors, small letters for scalars, capital letters for matrices, || · || denotes 2-norm unless otherwise noted. II. S YSTEM D ESCRIPTION AND P ROBLEM F ORMULATION Let the large-scale system be composed of m stabilizable nonlinear single-input single-output (SISO) subsystems, represented in the following normal form: x˙ i z˙ i
= Ai xi + Bi z i + bi [ui + fi (x1 , z 1 , . . . , xm , z m )] = Ci xi + Di z i + g i (x1 , z 1 , . . . , xm , z m ) (1)
yi
= cTi xi , i = 1, · · · , m,
where [xTi z Ti ]T ∈ Rri +(ni −ri ) is the state vector of the realization of the ith subsystem in normal form, ri representing the relative degree, ui ∈ R and yi ∈ R are the control and measurement of the ith subsystem, fi : Rn1 +···+nm → R , g i : Rn1 +···+nm → Rni −ri are sufficiently smooth
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unknown functions, representing the modeling errors and interconnection effects, while Ai , Bi , bi , ci are matrices and vectors corresponding to the normal realization:
0 .. = . ai1 0 .. = .
Ai
Bi
ci1
1 .. . ai2 ··· .. . ···
··· .. . ···
0 .. .
0 0
airi
cini −ri
,
and bi = [ 0 · · · bi ]T , ci = [ 1 · · · 0 ]T . Assumption 2.1: The functions g i (x1 , z 1 , . . . , xm , z m ) are bounded for all i = 1, · · · , m: kg i (·)k ≤
m X j=1
T
αi xTj z Tj
,
αi > 0
(2)
The objective is to synthesize decentralized adaptive output feedback control laws ui , such that yi (t) tracks a smooth bounded reference trajectory yli (t) with bounded errors for all i = 1, · · · , m, under the assumption that the ith controller knows the desired states of all the subsystems j = 1, · · · , m, while having access only to its own measurement yi (t). As in [14]–[16], we introduce the following assumption. Assumption 2.2: The signals yli (t) are assumed to be generated by the following stable linear closed-loop reference models ξ˙ li = A¯i ξ li + bri yci ,
¯Ti ξ li yli = c
(3)
consisting of an open loop system x˙ li z˙ li
= Ai xli + Bi z li + bi uli = Ci xli + Di z li
y li
= cTi xli ,
i = 1, · · · , m
= Aci xci + bci (yci − yli )
u li
= cTci xci + dci (yci − yli ) ,
(4)
i = 1, · · · , m , (5)
where Ai − bi dci cTi ¯ Ci Ai = −bci cTi
Bi Di 0
bi cTci 0 Aci
(7) k[ xTli z Tli ]T k ≤ βi , i = 1, · · · , m Remark 2.1: In [14], this problem formulation has been addressed for linear subsystems, and, by a proper choice of robustifying signal, it has been shown that global asymptotic tracking can be achieved if the robustifying gain satisfies a lower bound, depending upon the number of subsystems and the apriori known bound on the interconnection effects. In [15], these results have been extended to nonlinear interconnections, modeled by known nonlinear functions. Moreover, output feedback has been formulated and solved for the case of subsystems having regulated outputs with relative degree 1. Our approach is different in two perspectives: i) we formulate the problem in output feedback for arbitrary relative degree by extending the results of [17] for centralized control, ii) we use an adaptive signal for overcoming the effect of interconnections on the tracking performance. On the other hand it should be understood that, due to results in [18], one cannot expect global results while using dynamic output feedback compensators with the class of nonlinear systems presented here. III. C ONTROLLER D ESIGN , E RROR DYNAMICS The control design for each of the subsystems will be based on the logic of combining a linear controller, that stabilizes the nominal linear model in the absence of interconnections, with a neural network (NN) that approximately cancels the interconnection effects in the controllable range. Towards this end, introduce the following control signal ui = uci − uadi , where uci is the output of the following dynamic compensator η˙ ci
= Aci η ci + bci (yci − yi )
u ci
= cTci η ci + dci (yci − yi ) ,
i = 1, · · · , m , (8)
nci
and a stabilizing dynamic compensator: x˙ ci
open-loop subsystem. The following bounds are assumed to be known
(6)
= bri = [ xTli z Tli xTci ]T , ξ li T T T ¯i = [ cTi 0 0 ]T , and yci [ bi dci 0 bci ] , c is a bounded input of interest to track. The matrices Ai , Bi , bi , ci are assumed to correspond to the normal realization, as defined in (2), so that dim xli = dim xi , and dim z li = dim z i . Notice that this choice of the open loop system in (4) implies that the relative degree of the ith open-loop reference model equals that of the ith
where η ci ∈ R , Aci , bci , cci , dci are introduced in (5), and the adaptive signal uadi will be defined later. This results in the following closed-loop subsystem dynamics: ξ˙ i yi
¯i (uad − fi ) + g ¯i = A¯i ξ i + bri yci − b i T ¯i ξ i , = c i = 1, · · · , m ,
(9) (10)
¯i = [bT 0 0]T , g ¯i = where ξ i = [xTi z Ti η Tci ]T , b i T T [0 g i 0] . Following [15], define the error vector E i = ξ li − ξ i , and write the tracking error dynamics for the ith subsystem: ¯i (uad − fi ) − g ˙ i = A¯i E i + b ¯i, y ¯ i = C¯i E i , (11) E i T T ¯i I where C¯i = c separates the signals available for feedback. IV. N EURAL N ETWORK A PPROXIMATION OF N ONLINEARITIES AND A DAPTIVE C ONTROL Following [19], given arbitrary ∗ > 0 and a continuous function f (x) : Rn → Rm , defined on a compact set x ∈ D ⊂ Rn , there exists a set of bounded constant
4280
weights W , and a set of basis functions φ(x), such that the following representation holds ∀x ∈ D: f (x) = W T φ(x) + (x), k(x)k < ∗ . Thus, one can model the interconnections fi (x1 , z 1 , · · · , xm , z m ) = W Ti φi (Y ) + i (Y )
(12)
using the following input vector Y = [xT1 z T1 · · · xTm z Tm ]T ∈ D ⊂ Rn1 +···+nm and a vector of the radial basis functions φi (Y ) = [φi1 (Y ) · · · φiNi (Y )]T , where Ni is the number of basis functions to be used by the ith subsystem, 2 φik (Y ) = e−kY −Y ick k /2σik , Y ick is the vector of centers of the basis functions used by the ith subsystem, having the same dimension as Y , σik specifies the width of the k th basis function in the ith subsystem, and |i | < ∗i , kW i k ≤ Wi∗ . Since our interest is in decentralized design, the states of other subsystems are not available to individual controllers, therefore the input vector Y cannot be used in designing adaptive elements. Based on the assumption that the controllers share prior information about their reference models, the adaptive control signal for the ith subsystem can be designed following the same logic as in [15], ˆ T φ (Y l ) uadi = W i i
(13)
where the vector Y l is defined as Y l = [xTl1 z Tl1 · · · xTlm z Tlm ]T having the states of all the subsystems replaced by their corresponding reference states when compared to Y . Notice that due to boundedness of reference model states there exists a set Dl in the extended ˆ i are space such that Y l ∈ Dl . The adaptive laws for W similar to those in [20]: ¯i + ki W ˆ i] ˆ T Pi b ˆ˙ i = −Fi [2φi (Y l )E W i
(14)
in which Pi is the solution of the Lyapunov equation A¯Ti Pi + Pi A¯i = −Qi for some Qi > 0, and Fi , ki > 0 ˆ i propagates according to the are adaptation gains, while E following dynamics: ˆ i, ˆ i + Ki (¯ ˆ˙ i = A¯i E ˆi) , y ˆ i = C¯i E yi − y E
(15)
where Ki is a gain matrix, and should be chosen such that A¯i − Ki C¯i is asymptotically stable, while i = 1, · · · , m. ˜ i = Ei − E ˆ i , i = 1, · · · , m. Let A˜i = A¯i − Ki C¯i , E Then ¯i (uad − fi ) − g ˜i +b ˜˙ i = A˜i E ¯i, E i
i = 1, · · · , m
(16)
Through several algebraic manipulations and, using the mean value theorem, one can obtain T
˜ φi (Y l ) + W T φ0 (Y ∗ )Y˜ − i uadi − fi = W i i i
˜ T1 · · · x ˜ Tm z ˜ Tm ]T is comprised of the where Y˜ = [˜ xT1 z ˜i = tracking errors of all the subsystems (˜ xi = xli − xi , z z li − z i ), φ0i (Y ∗ ) is the bounded derivative of the basis function in an intermediate point Y ∗ = Y l + (1 − λ)Y , ˜i ,W ˆ i − W i is the parameter error 0 < λ < 1, and W vector. From the definition of E j and Y˜ it follows that kY˜ k ≤
j=m X
kE j k
(19)
j=1
V. S TABILITY A NALYSIS In this section we show through Lyapunov’s ˜ i, W ˜ i, direct method that the error signals E i , E i = 1, · · · , m, are ultimately bounded. To this end, introduce the composite error ivector h T ∆ T ˜T ··· E ˜T W ˜ T ··· W ˜T ζ = E · · · ET E ∈ 1
m
1
m
1
¯i y
= C¯i E i .
(17)
m
R2(n1 +···+nm ) × RN1 +···+Nm , and consider the following ∆ positive definite function V (ζ) = ζ T T ζ, where −1 ]. T = blockdiag[P1 · · · Pm P˜1 · · · P˜m 12 F1−1 · · · 12 Fm Further, notice that the RBF network approximation can be defined over arbitrarily large compact set D. Based on the definition of the compact set Dl , and the boundedness of xci and η ci , in the subspace of the error variables consider the following compact set ΩE of n T T possible initial errors: ΩE = ∈ E 1 · · · E Tm o n1 +···+nm : Y ∈ D, Y l ∈ Dl , xc ∈ Dxc , η c ∈ Dηc . R In the expanded space of the error variable ζ ∈ R2(n1 +···+nm ) × RN1 +···+Nm , consider the largest level set of V (ζ) = ζ T T ζ
(20)
corresponding to [ E T1 · · · E Tm ]T ∈ ΩE and introduce the largest ball that lies inside this level set: BR = {ζ | kζk ≤ R}
(21) T
Let α be the minimum value of V (ζ) = ζ T ζ on the boundary of BR : ∆
α = min V (ζ) = R2 λmin (T ) kζk=R
(22)
where λmin (T ) is introduced for the minimum eigenvalue of T . Introduce the set ∆
We immediately note that for arbitrary positive definite ˜ i > 0 there exists a unique solution P˜i = P˜ T > 0 such Q i ˜ i . The error dynamics in (11) can that A˜Ti P˜i + P˜i A˜i = −Q be expressed as: h T i ¯i W ˆ φi (Y l ) − W T φi (Y ) − i − g ˙ i = A¯i E i + b ¯i E i i
(18)
Ωα = {ζ ∈ BR | V (ζ) ≤ α}
(23)
Assumption 5.1: Let p R > γ λmax (T )/λmin (T )
(24)
where λmax (T ) is introduced for the maximum eigenvalue q q q ω ω ω of T , while γ = max ˜ , λmin (D) , λmin (Λ) , λ (D) min
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∆ in which D = diag θ1 · · · θm , θi = λmin (Qi ) − ¯i k − Pj=m (2kPj b ¯j k + 2mαi λmax (Pi ) − (mφ∗i + 1)kPi b j=1 ∆ ˜ = > 0, D Θj )φ∗j − αi λmax (Pi ) + λmax (P˜i ) ˜ i )−2mαi λmax (P˜i )− diag θ˜1 · · · θ˜m , θ˜i = λmin (Q ∆ ¯i k + µi − µi kφ (Y l )k > 0, Λ = (mφ∗i + 1) kPi b i diag λ1 · · · λm , λi = k2i − µi kφi (Y l )k > 0, ∆ Pm ¯i k+µi )(∗ )2 +αi (λmax (Pi )+ ω = i=1 ( k2i (Wi∗ )2 +(2kPi b i P j=m ¯i + λmax (P˜i )) j=1 βj2 ), φ∗i , Wi∗ kφ0i (Y ∗ )k, µi , kP˜i b ¯ Pi bi k. Theorem 5.1: Let Assumptions 2.1, 2.2 and 5.1 hold. If the initial errors lie in Ωα , defined in (23), then all ˜ i, W ˜ i , i = 1, · · · , m, in the closed loop the signals E i , E system are ultimately bounded. Proof. Consider the following Lyapunov function candidate for each of the subsystems: ˜i ˜ T F −1 W ˜ i, W ˜ i ) = E T Pi E i + E ˜ i + 1W ˜ T P˜i E Vi (E i , E i i 2 i i Substituting the adaptive laws from (14) implies: V˙ i
=
˜i ˜TQ ˜iE −E Ti Qi E i − E i h T ¯i W T φ0 (Y ∗ )Y˜ ˆ Pi b +2E i i i
i
Notice that using the bound in (7), the upperbound in (2) can be represented: j=m X
αi [kE j k + βj ]
V˙
(25)
j=1
=
m X
V˙ i
(26)
i=1
Then, using the notations from Assumption 5.1, the upper bound reduces to: Pm V˙ = i=1 V˙ i ≤ h Pm i=1 − λmin (Qi ) − 2mαi λmax (Pi ) ¯i k kE i k2 − θ˜i kE ˜ i k2 −(mφ∗i + 1)kPi b h ¯i k + µi )φ∗ ˜ i k2 + Pj=m (2kPi b −λi kW i j=1 i i +αi λmax (Pi ) + λmax (P˜i ) kE j k2 + ω Regrouping, this can be written: V˙
=
m X
V˙ i ≤
i=1 m h X
¯i 2E Ti Pi g
− i + h ¯ i + Pi b ¯i ) W ˜ T (P˜i b ˜ T φ (Y l ) +2E i i i i h i ˆi ˜ iW ˜ T P˜i g ¯ i − ki W +W Ti φ0i (Y ∗ )Y˜ − i + 2E i
kg i k ≤
Now introduce the following Lyapunov function for the whole system:
i=1
(27)
i ˜ i k2 − λi kW ˜ i k2 + ω − θi kE i k2 − θ˜i kE
Following an argument similar to that in [12], T ∆ kE 1 k · · · kE m k define the vectors E = , T ∆ ∆ ˜ ˜ ˜ ˜ E = , W = kE 1 k · · · kE m k ˜ 1 k · · · kW ˜ m k T . Then the expression in kW (27) can be put into the following form: V˙
≤
˜TD ˜ −W ˜ T ΛW ˜ +ω ˜E −E T DE − E
The following upper bound
Then using (19), the following upper bound can be derived: ˜ i k2 ˜ i )kE V˙ ≤ −λmin (Qi )kE i k2 − λmin (Q i h P i j=m ∗ ¯ i k φ∗ ˜ i k + kE i k kPi b +2 kE i j=1 kE j k + i Pj=m +2αi λmax (Pi )kE i k j=1 [kE j k + βj ] h i ˜ i kkφi (Y l )k + φ∗ Pj=m kE j k + ∗ ˜ i kµi kW +2kE i i j=1 h i P j=m ˜ ik ˜ iW ˆi +2αi λmax (P˜i )kE [kE j k + βj ] − ki W j=1
Completing the squares twice and regroupong, the following upper bound can be derived: V˙ i ≤ − λmin (Qi ) − 2mαi λmax (Pi ) − (mφ∗i ¯i k kE i k2 − λmin (Q ˜ i ) − 2mαi λmax (P˜i ) +1)kPi b ¯i k + µi − µi kφ (Y l )k kE ˜ i k2 −(mφ∗i + 1) kPi b i h ¯i k ˜ i k2 + Pj=m (2kPi b − k2i − µi kφi (Y l )k kW j=1 i +µi )φ∗i + αi λmax (Pi ) + λmax (P˜i ) kE j k2 ¯i k + µi (∗ )2 + k2i (Wi∗ )2 + 2kPi b P i j=m 2 +αi λmax (Pi ) + λmax (P˜i ) j=1 βj
˜ 2 ˜ Ek V˙ ≤ −λmin (D)kEk2 − λmin (D)k 2 ˜ k +ω −λmin (Λ)kW implies that either of the following conditions r ω kEk > λmin (D)
r ω
˜
E > ˜ λmin (D) r ω ˜ k> kW λmin (Λ) will render V˙ < 0 outside the compact set Bγ = {ζ | kζk ≤ γ}
(28)
Let Γ be the maximum value of the function V (ζ) on the boundary of Bγ : ∆
Γ = max V = γ 2 λmax (T ) kζk=γ
(29)
Assumption 5.1 ensures Ωγ , {ζ| V = Γ} ⊂ Ωα
(30)
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Thus, if the initial error ζ 0 = ζ(0) belongs to Ωα , then there exists a time instant tζ (ζ 0 ), such that ζ(t) will enter the set Ωγ at tζ and remain inside it for all t > tζ . This implies ultimate boundedness of ζ, completing the proof. Remark 5.1: The results obtained above can be extended to the case where the modeling errors also depend upon the control signal, i.e. in (1) one can have fi (ui , x1 , z 1 , . . . , xm , z m ), subject to ∂fi /∂ui 6= 0. Notice then that the adaptive signal will be introduced to cancel a function fi (ui (uadi (·), ·) of itself. To avoid this algebraic loop, one way of implementing this is to use a one step delayed value of the control signal ui (t − d), where d > 0 is sufficiently small. Remark 5.2: Assumption 5.1 may be interpreted as placing both upper and lower bounds on the adaptation gains. ∆ ∆ ∆ ¯ = Let γ¯ = max(λmax (Fi )), γ = min(λmin (Fi )), λ ∆ max(λmax (Pi ), λmax (P˜i )), λ = min(λmin (Pi ), λmin (P˜i )), i = 1, · · · , m. Then an upper bound for the adaptation gains ¯ > 1 and 2λ¯ γ > 1, for which the relation results when 2λγ ¯ A lower bound for in (24) reduces to γ¯ < R2 /(2γ 2 λ). ¯ < 1 and 2λ¯ γ < 1, the adaptation gains results when 2λγ for which the relation in (24) reduces to γ > γ 2 /(2R2 λ). Notice that the upper bound for the adaptation gain has R in the numerator, while the lower bound has R in the denominator. Therefore, R can be selected sufficiently large to ensure that γ < γ¯ . VI. S IMULATIONS We consider three inverted pendulums mounted on carts, as depicted in Figure 1. The carts are connected by springs
m, I
q1
q3
q2
u1
k
M
u2
k
M
x1
u3
M
x2
x3
Three inverted pendulums on three carts
and dampers. In each subsystem, we assume that the position of the cart(xi ) and the angle of the pendulum(θi ) are measured and the cart is regulated by input forces(ui ). The equations of motion for the system are described as follows: (M + m)¨ x1 + mlp θ¨1 cos θ1 − mlp θ˙12 sin θ1
= u1 + s1
mlp cos θ1 x ¨1 + (I + mlp2 )θ¨1 − mglp sin θ1
=
0
(M + m)¨ x2 + mlp θ¨2 cos θ2 − mlp θ˙22 sin θ2 = u2 − s1 + s2 mlp cos θ2 x ¨2 + (I + mlp2 )θ¨2 − mglp sin θ2
=
0
= u3 − s2
mlp cos θ3 x ¨3 + (I + mlp2 )θ¨3 − mglp sin θ3
=
0
where u1 , u2 , u3 are input forces to the carts(N), M is the mass of the cart (kg), m is the mass of the rod(kg), lp is the distance from the pivot on the cart to the center of gravity of the rod(half of full length)(m), I(= 31 mlp2 ) is the moment of inertia of the rod with respect to its center of mass (kg·m2 ), g is the gravitational acceleration (kg·m/sec2 ), k is the spring constant (N/m), c is the damping constant (N·sec/m), s1 = k(x2 − x1 ) + c(x˙ 2 − x˙ 1 ), s2 = k(x3 − x2 ) + c(x˙ 3 − x˙ 2 ) are interconnection forces due to springs and dampers. The parameter values are: M = 0.9, m = 0.18, lp = 0.305, g = 9.8, k = 1, c = 2 × 10−5 . Our control objective is to regulate the displacements of the carts xi while balancing the inverted rods on the carts without velocity measurements. The open loop subsystem in (4) is derived after the dynamics are first linearized with respect to equilibrium position xi = θi = 0, and then put into a normal form by the transformation: xli1 = xi , xli2 = x˙ i , zli1 = θi , zli2 = xl˙pi + θ˙i . The linear subsystems, for i = 1, 2, 3, is described by the following system matrices: 0 0 0 0 1 , bi = Ai = , Bi = 1 m ˆ 0 −M 0 0 ˆ ˆ M 0 1 0 − l1p 1 Ci = , Di = g 0 , ci = 0 0 0 lp (31) ˆ = 0.815, m The constants M ˆ = 0.21 represent parameter estimates for M, m respectively. Further, in this linear model, the inverted rod is treated as a lumped mass located on its center of mass, i.e., I = 0. Putting each subsystem into normal form leads to the following modelling errors and interconnection effects defined in (1): fi = ˆ M (ui + mlp θ˙i2 sin θi − 43 mg sin θi cos θi + M +m(1−3/4 cos θi2 ) " # − coslpθi xi2 − 13 θ˙i + l1p xi2 τi ) + mgθ ˆ i − ui , g i = g sin θi − 1 xi θ˙i sin θi − g θi lp
c
c
Fig. 1.
m, I
m, I
(M + m)¨ x3 + mlp θ¨3 cos θ3 − mlp θ˙32 sin θ3
lp
2
lp
where τ1 = s1 , τ2 = −s1 + s2 , τ3 = −s2 . The τi terms imply that the spring and the damper are not considered in the open loop model. The term ui means that the modelling error also depends on the control signal as in Remark 5.1. Note that the interconnections between the two carts and the modelling errors contain velocity terms which are not measured. This implies that the existing adaptive output feedback approaches in the decentralized control literature, such as the ones developed in [8], [21] and many others, although establishing global results, cannot be applied. Moreover, regulation of xi using ui to the carts renders the control problem nonminimum phase -linearization of each subsystem about vertical-up position leads to unstable pg . These issues make the control problem even zero L more challenging. The dynamic compensator in (8) for each subsystem is designed as a LQG controller based on the open loop model in (31), in which two measured outputs
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output plot:x
output plot:x
1
output plot:x
2
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3
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0.2
0.2 2.5
0
0 2
reference without adaptation with adaptation
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x1(m)
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1.5
−0.4
reference without adaptation with adaptation
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−1
−1 0
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30
40 time(sec)
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80
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0
10
20
(b) x2
30
40 time(sec)
50
60
70
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(c) x3
Fig. 2. Comparison of the cart displacements without and with adaptive signal uadi .
output plot:θ
output plot:θ
1
3
40
reference without adaptation with adaptation
30
40
reference without adaptation with adaptation
30
20
20
3
0
0
2
1
10
θ (m)
θ (degree)
10
0
reference without adaptation with adaptation
30
20
10 θ (degree)
output plot:θ
2
40
−10
−10
−10
−20
−20
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−30
−40
−30
0
10
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30
40 time(sec)
(a) θ1 Fig. 3. uadi .
50
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−30
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40 time(sec)
(b) θ2
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(c) θ3
Comparison of the rod angles without and with adaptive signal
xi , θi are available for control design. The error observer in (15) is designed to have the smallest eigenvalue of A˜i equal approximately five times the smallest eigenvalue of A¯i . The basis functions have the following structure for three 2 subsystems: φik (Y l ) = e−kY l −Y ick k /2σik , σik = 1, i = 1, 2, 3, k = 1, . . . , Ni , where N1 = N3 = 7, N2 = 9. The centers Y ick are randomly selected over a grid of possible values for the vector Y l . All of the NN inputs are normalized using an estimate for their maximum values. Since the dynamics of the first and third carts are coupled only through the dynamics of the middle cart, and the modelling error contains a control signal, we choose the NN input vectors as: Y Tl1 = [u1 xl11 xl12 zl11 zl12 xl21 xl22 0 0 0 0 0 0]T , Y Tl2 = [u2 xl11 xl12 0 0 xl21 xl22 zl21 zl22 xl31 xl32 0 0]T , Y Tl3 = [u3 0 0 0 0 xl21 xl22 0 0 xl31 xl32 zl31 zl32 ]T , where Y li represents the NN input vector for the ith subsystem. Adaptation gains are chosen as: Fi = 0.5I, ki = 0.05. Figure 2 compares output tracking performances when the reference command yci , i = 1, 2, 3 is a square wave signal of magnitude 0.15m and 0.05 Hz. The pendulum angles are shown in Figure 3. The initial conditions are: x1 (0) = x˙ 1 (0) = 0, θ1 (0) = −30o , θ˙1 (0) = −10o /sec, x2 (0) = x˙ 2 (0) = 0, θ2 (0) = 30o , θ˙2 (0) = 10o /sec, x3 (0) = x˙ 3 (0) = 0, θ3 (0) = 20o , θ˙2 (0) = −10o /sec. Without adaptive control compensation, the system goes unstable. When each control is augmented with an adaptive term, the three carts are in synchronous motion with the pendulums balanced, implying implicit cooperation for output tracking.
VII. C ONCLUSIONS A methodology is presented for adaptive output feedback decentralized control design under the assumption that the reference trajectories are known to all subsystems. A linearly parameterized neural network is used to model the interconnection effects on-line. Boundedness of error signals is shown using Lyapunov’s direct method. The methodology is applicable to non-minimum phase subsystems. R EFERENCES [1] P. Ioannou. Decentralized adaptive control of interconnected systems. IEEE Trans. Autom. Contr., 31(4):291–298, 1986. [2] P. Ioannou and P. Kokotovic. Decentralized adaptive control of interconnected systems with reduced order models. Automatica, 21(4):401–412, 1985. [3] D. Gavel and D. Siljak. Decentralized adaptive control: structural conditions for stability. IEEE Trans. Autom. Contr., 34(4):413–426, 1989. [4] A. Datta and P. Ioannou. Decentralized adaptive control. Adavances in Control and Dynamic Systems, Academic Press, 1992. [5] R. Ortega. An energy amplification condition for decentralized adaptive stabilization. IEEE Trans. Autom. Contr., 41:285–288, 1996. [6] M. Krstic, I. Kanellakopoulos, and P. Kokotovic. Nonlinear design of adaptive controllers for linear systems. IEEE Trans. Autom. Contr., 39(4):738–752, 1994. [7] L. Shi and S.K. Singh. Decentralized adaptive controller design for large scale systems with higher order interconnections. IEEE Trans. Autom. Contr., 37(8):1106–1118, 1992. [8] S. Jain and F. Khorrami. Decentralized adaptive output feedback design for large scale nonlinear systems. IEEE Trans. Autom. Contr., 42(4):729–735, 1997. [9] J.C. Geromel, J. Bernussou, and M.C. Oliveira. Norm optimization with constrained dynamic output feedback controllers: Decentralized and reliable control. IEEE Trans. Autom. Contr., 44(7):1449–1454, 1999. [10] G.H. Yang, J. Wang, C.B. Soh, and J. Lam. Decentralized controller design for nonlinear systems. IEEE Trans. Autom. Contr., 44(3):578– 583, 1999. [11] Z.P. Jiang. Decentralized disturbance attenuating output feedback trackers for large-scale nonlinear systems. Automatica, 38(8):1407– 1415, 2002. [12] J.T. Spooner and K.M. Passino. Decentralized adaptive control of nonlinear systems using radial basis neural networks. IEEE Trans. Autom. Contr., 44(11):2050–2057, 1999. [13] J. Spooner, M. Maggiore, R. Ordonez, and K. Passino. Stable Adaptive Control and Estimation for Nonlinear Systems: Neural and Fuzzy Approximator Techniques. John Wiley & Sons, New York, 2002. [14] K. Narendra and N. Oleng. Exact output tracking in decentralized adaptive control systems. IEEE Trans. Autom. Contr., 47(2):390–395, 2002. [15] K. Narendra and N. Oleng. Decentralized adaptive control. Proceedings of American Control Conference, 2002. [16] B.. Mirkin. Comments on “Exact Output Tracking in Decentralized Adaptive Control Systems”. IEEE Trans. Autom. Contr., 48(2):348– 350, 2003. [17] N. Hovakimyan, B.-J. Yang, and A.J. Calise. An adaptive output feedback control methodology for non-minimum phase systems. Conference on Decision and Control 2002. [18] F. Mazenc, L. Praly, and Dayawansa W. Global stabilization by output feedback: examples and counterexamples. Syst. Contr. Lett., 23:119–125, 1994. [19] R. Sanner and J.J. Slotine. Gaussian networks for direct adaptive control. IEEE Transactions on Neural Networks, 3(6):837–864, 1992. [20] N. Hovakimyan, F. Nardi, A. Calise, and N. Kim. Adaptive Output Feedback Control of Uncertain Systems using Single Hidden Layer Neural Networks. IEEE Transactions on Neural Networks, 13(6):1420–1431, 2002. [21] Z.P. Jiang. Decentralized and adaptive nonlinear tracking of largescale systems via output feedback. IEEE Trans. Autom. Contr., 45(11):2122 –2128, 2000.
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