direct adaptive control for nonlinear …haddad.gatech.edu/journal/adaptive_control_msos_tv.pdfasian...

Asian Journal of Control, Vol. 9, No. 1, pp. 11-19, March 2007 11

Manuscript received December 9, 2004; revised May 13, 2005; accepted December 8, 2005.

W.M. Haddad and M.C. Stasko are with the School of Aero-space Engineering Georgia Institute of Technology, Atlanta, GA 30332-0150 (e-mail: [email protected]; [email protected]).

T. Hayakawa is with the Dept. of Mechanical and Environ-mental Informatics, Tokyo Institute of Technology, Tokyo 152- 8552, Japan (e-mail: hayakawa@ mei.titech.ac.jp).

This research was supported in part by the Air Force Office of Scientific Research under Grant F49620-03-1-0178 and the National Aeronautics and Space Administration under Grant NGT4-52434.

DIRECT ADAPTIVE CONTROL FOR NONLINEAR MATRIX SECOND-ORDER SYSTEMS WITH TIME-VARYING AND

SIGN-INDEFINITE DAMPING AND STIFFNESS OPERATORS

Wassim M. Haddad, Tomohisa Hayakawa, and Michael C. Stasko

ABSTRACT

A direct adaptive control framework for a class of nonlinear ma-trix second-order systems with time-varying and sign-indefinite damping and stiffness operators is developed. The proposed frame-work guarantees global asymptotic stability of the closed-loop system states associated with the plant dynamics without requiring any knowledge of the system nonlinearities other than the assumption that they are continuous and bounded. The proposed adaptive control ap-proach is used to design adaptive controllers for suppressing ther-moacoustic oscillations in combustion chambers.

KeyWords: Adaptive nonlinear control, disturbance rejection, nonlinear matrix second-order systems, nonlinear state- and time-dependent uncertainty, parametrization free uncertainty models.

I. INTRODUCTION

In the recent paper [1], the authors present a novel adaptive control framework for scalar second-order linear time-varying systems. In this paper we generalize the result in [1] in several directions. In particular, for a class of nonlinear multivariable matrix second-order uncertain dy-namical systems with time-varying and sign-indefinite damping and stiffness operators, we develop a nonlinear adaptive control framework that guarantees global partial asymptotic stability of the closed-loop system, that is, global asymptotic stability with respect to part of the closed-loop system states associated with the plant. This is

achieved without requiring any knowledge of the system nonlinearities other than the assumption that they are con-tinuous and bounded. Furthermore, these bounds need not be known. Hence, unlike standard adaptive control meth-ods [2-4], the proposed adaptive control framework does not require any parametrization of the state-dependent sys-tem uncertainty.

The class of systems represented by our framework includes nonlinear vibrational systems, as well as multi-variable nonlinear matrix second-order dynamical systems with sign-varying, that is, nondissipative, generalized stiff-ness and damping time-varying operators. In the special case of scalar second-order systems with linear time- varying coefficients, our results specialize to the results in [1]. A similar adaptive control framework for nonlinear uncertain matrix second-order systems and nonlinear scalar second-order systems was considered in [5,6], respectively. The results presented in [5,6], however, only address time-invariant, sign-indefinite stiffness and damping op-erator uncertainty, with the damping operator uncertainty being a partial function of the system state. Finally, we note that related, but different, approaches to the present framework are given in [7-9]. Specifically, the adaptive control papers discussed in [7] consider matrix sec-ond-order rigid link manipulators with sign-definite damp-ing and stiffness operators, while the authors in [8,9] con-

12 Asian Journal of Control, Vol. 9, No. 1, March 2007

sider single input, strict feedback systems with unmeasured disturbances.

The notation used in the paper is fairly standard. Spe-cifically, (resp., n

× n) denotes the set of real numbers

(resp., n × n real matrices), (⋅)T denotes transpose, and In denotes the n × n identity matrix. Furthermore, we write tr(⋅) for the trace operator, λmin(M) (resp., λmax(M)) for the minimum (resp., maximum) eigenvalue of the Hermitian matrix M, and M ≥ 0 (resp., M > 0) to denote the fact that the Hermitian matrix M is nonnegative (resp., positive) definite. Finally, M ⊗ N denotes the Kronecker product of matrices M and N [10].

II. ADAPTIVE CONTROL OF NONLINEAR TIME-VARYING MATRIX SECOND-

ORDER DYNAMICAL SYSTEMS

In this section we consider the problem of adaptive stabilization of nonlinear time-varying matrix second-order dynamical systems with exogenous disturbances. Specifi-cally, consider the controlled nonlinear time-varying un-certain matrix second-order dynamical system G given by

( ) ( ( ) ( ) ) ( ) ( ( ) ) ( )Mq t C q t q t t q t K q t t q t+ , , + ,

0 00( ) ( ) (0) (0)u t Dw t q q q t tq= + , = , = , ≥ , (1)

where ( ) ( ) ( ) nq t q t q t, , ∈ , t ≥ t0, represent generalized position, velocity, and acceleration coordinates, respec-tively, u(t) ∈ n, t ≥ t0, is the control input, w(t) ∈ d, t ≥ t0, is a known bounded disturbance, M∈ n×n, C: n× n × → n×n, K: n× → n×n, and D∈ n×d. In general, for matrix second-order mechanical systems the inertia matrix M can depend on the generalized position q in n. However, for rigid body translational mechanical systems M is a constant, positive-definite matrix. Here, we assume that M > 0 and C(⋅, ⋅, ⋅) and K(⋅, ⋅) are piecewise continuous in t and locally Lipschitz continuous in q and q for all t ≥ t0 and all ( ) n nq q, ∈ × . Furthermore, we assume that C(⋅, ⋅, ⋅) and K(⋅, ⋅) are symmetric maps. Other-wise, we assume that M, C(⋅, ⋅, ⋅), K(⋅, ⋅), and D are un-known. It is important to note here that C(⋅, ⋅, ⋅) and K(⋅, ⋅) need not be sign-definite operators, and hence, (1) does not correspond to a Lagrangian system. Thus, standard adap-tive controllers cannot be derived by putting the physics in the model as in [7]. Furthermore, note that even though w(t), t ≥ t0, is assumed to be known, the disturbance signal Dw(t), t ≥ t0, is an unknown bounded disturbance. The con-trol input u(⋅) in (1) is restricted to the class of admissible controls consisting of measurable functions such that u(t) ∈ n, t ≥ t0. Finally, for the uncertain dynamical sys-tem G we assume that the required properties for the exis-tence and uniqueness of solutions are satisfied, that is,

C(⋅, ⋅, ⋅), K(⋅, ⋅), u(⋅), and w(⋅) satisfy sufficient regularity conditions such that (1) has unique solution forward in time.

Next, with x1 q, 2x q , and T T T1 2[ ]x x x , it fol-

lows that the state space representation of (1) is given by

1

2

( )( )txtx

⎡ ⎤⎢ ⎥⎣ ⎦

( )( )2

11 1 1 2 2

( )( ( ) ) ( ) ( ) ( ) ( ) ( ) ( )

x tM K x t t x t C x t x t t x t u t Dw t−

⎡ ⎤= ,⎢ ⎥− , + , , − −⎢ ⎥⎣ ⎦

010

2 0

(0)(0)

qxt t

qx⎡ ⎤⎡ ⎤

= , ≥ .⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

(2)

For the statement of our main result let P∈ 2×2 be positive

definite, where 1 12

12 2

p pP

p p

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= and P12 > 0, and define

B0 [0n In]T.

Theorem 2.1. Consider the nonlinear time-varying matrix second-order dynamical system G given by (1), or, equiva-lently, the nonlinear time-varying dynamical system given by (2). Assume there exist scalars α1, α2, β1, β2 ∈ such that

1 1 2 1 1 2 2( ) ( )n n n nI K x t I I C x x t Iα ≤ , ≤ α , β ≤ , , ≤ β ,

1 2 0( ) n nx x t t, ∈ × , ≥ . (3)

Let Q1, Q2 ∈ n×n, Y∈ 2n×2n, and Z∈ d×d be positive definite. Then the adaptive feedback control law

( ) ( ) ( ) ( ) ( )u t t x t t w t= Ψ + Φ , (4)

where Ψ(t) ∈ n×2n, t ≥ t0, and Φ(t)∈ n×d, t ≥ t0, with up-date laws

1 0( ) ( ) ( ) ( )T Tnt Q B P I x t x t YΨ = − ⊗ , (5)

2 0( ) ( ) ( ) ( )T Tnt Q B P I x t w t ZΦ = − ⊗ , (6)

guarantees that the solution (x(t), Ψ(t), Φ(t)) ≡ (0, Kg, −D), where Kg ∈ n×2n, of the closed-loop system given by (2), (4) to (6) is uniformly Lyapunov stable and x(t) → 0 as t → ∞ for all x0 ∈ 2n.

Proof. Define γ1 (p2α1+ p12β1)/λmax(M), γ2 (p2α2+ p12β2)/λmin(M), 2 2

1 2max{ }γ γ , γ , and define the set K ⊂ n×2n by

21 2{[ ] n nk M k M ×∈ :K

1 max 1 2 max 1 1 2 1 12 2( ) ( ) 0,k M k M p p k p kλ < α , λ < β , + + <

W.M. Haddad et al.: Direct Adaptive Control for Nonlinear Matrix Second-Order Systems 13

( ) ( )12 1 max 1 2 1 max 2 12 1( ) ( ) 2( )p M k p M k p⎡ ⎤α /λ − β /λ − − ≥ γ − γ γ ,⎣ ⎦

( ) ( )12 1 max 1 2 1 max 2 12( ) ( ) }p M k p M k p⎡ ⎤α /λ − β /λ − − > γ .⎣ ⎦

(7)

Note that since all of the inequalities in (7) may be rewrit-ten as upper bounds on k1 and k2, K is not empty. Next, let Kg [K1gM K2gM]∈ K and define g( ) ( )t t KΨ Ψ − ,

( ) ( )t t DΦ Φ + , 1 2 2g( ) ( )C x t C x x t k M, , , − , and 1 1 1g( ) ( )K x t K x t k M, , − . Furthermore, define 1p̂ −

1 2 1g 12 2g( )p p k p k+ + and 1 1ˆ( ) ( )H x t p Mp, − + + 12 2 1( ) ( )p C x t p K x t, + , , and note that 1ˆ 0p > . Now, it

follows from the definitions of γ1 and γ2 that

21 2 0( ) nM H x t M x t tγ ≤ , ≤ γ , ∈ , ≥ . (8)

Moreover, it follows from (8) and the definition of γ that

1 200 ( ) ( ) nH x t M H x t M x t t−≤ , , ≤ γ , ∈ , ≥ . (9)

Next, with u(t), t ≥ t0, given by (4), (2) becomes

1

2

( )( )txtx

⎡ ⎤=⎢ ⎥

⎣ ⎦

( )2

11 1 2

( )

( ( ) ) ( ) ( ( ) ) ( ) ( ) ( ) ( ) ( )

x t

M K x t t x t C x t t x t t x t t w t−

⎡ ⎤⎢ ⎥ ,− , + , − Ψ − Φ⎢ ⎥⎣ ⎦

010

2 0

(0)(0)

qxt t

qx⎡ ⎤⎡ ⎤

= , ≥ .⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

(10)

To show uniform Lyapunov stability of the closed- loop system (5), (6), and (10), consider the Lyapunov func-tion candidate

T T1 11

1 ˆ( ) [( )2

V x x P M x x Mxp, Ψ, Φ = ⊗ +

1 1 T 1 1 T1 2tr( ) tr( )] .Q Y Q Z− − − −+ Ψ Ψ + Φ Φ (11)

Note that V(0, Kg, −D) = 0 and, since P, M, Q1, Q2, Y, and Z are positive definite, V(x, Ψ, Φ) > 0 for all (x, Ψ, Φ) ≠ (0, Kg, −D). Furthermore, V(x, Ψ, Φ) is radially unbounded. Now, letting x(t), t ≥ t0, denote the solution to (10) and us-ing (3), (5), and (6) it follows that the Lyapunov derivative along the closed-loop system trajectories is given by

( ) T( ) ( ) ( ) ( )( ) ( )V x t t t t x t P M x t, Ψ ,Φ , = ⊗ T 1 1 T

11 11ˆ ( ) ( ) tr[ ( ) ( )]x t M t Q t Y tp x − −+ Ψ Ψ+1 1 T

2tr[ ( ) ( )]Q t Z t− −+ Φ Φ T T

1 1 2 12 2 2( ) ( ) ( ) ( )p x t Mx t p x t Mx t= +T T

12 1 1 1 12 1 2( ) ( ( ) ) ( ) ( ) ( ( ) ) ( )p x t K x t t x t p x t C x t t x t− , − ,

T T2 2 1 1 2 2 2( ) ( ( ) ) ( ) ( ) ( ( ) ) ( )p x t K x t t x t p x t C x t t x t− , − ,

T T T12 1 2 2 12 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )p x t t x t p x t t x t p x t t w t+ Ψ + Ψ + Φ

T T2 2 1 21ˆ( ) ( ) ( ) ( ) ( )p x t t w t x t Mx tp+ Φ +

1 1 T 1 1 T1 2tr[ ( ) ( )] tr[ ( ) ( )]Q t Y t Q t Z t− − − −+ Ψ Ψ + Φ Φ

T T12 2 2 12 1 1 1( ) ( ) ( ) ( ( ) ) ( )p x t Mx t p x t K x t t x t= − ,

T T2 2 2 1 2( ) ( ( ) ) ( ) ( ) ( ( ) ) ( )p x t C x t t x t x t H x t t x t− , − ,

T 1 T 10 1tr{ ( )[ ( ) ( )( ) ( ) ]}nt x t x t P I B Y t Q− −+ Ψ ⊗ + Ψ

T 1 T 10 2tr{ ( )[ ( ) ( )( ) ( ) ]}nt w t x t P I B Z t Q− −+ Φ ⊗ + Φ

T T12 1 1 1 1 2( ) ( ( ) ) ( ) ( ) ( ( ) ) ( )p x t K x t t x t x t H x t t x t= − , − ,

T2 2 12 2( )( ( ( ) ) ) ( )x t p C x t t p M x t− , −

T12 1 max 1g 1 1( ( ) ) ( ) ( )p M k x t Mx t≤ − α /λ −

( ) ((T1 2 2 1 max( ) ( ) ( ) ( )x t H x t t x t p M− , − β /λ

) ) T2g 12 2 2( ) ( )k p x t Mx t− −

( )T0( ) ( ) ( )x t x t t x t t t= − , , ≥ ,R (12)

where

( )( )( )

12 1 max 1g

2 1 max 2g 12

( ) ( ) 2( )

( ) 2 ( )

p M k M H x tx t

H x t p M k p M

⎡ ⎤α /λ − , /⎢ ⎥, .⎢ ⎥, / β /λ − −⎣ ⎦

R

Next, define

( )( )

12 1 max 1g

max1 2g2 12

( )1( )2

p M kR

p M k p

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

α /λ − γγ β /λ − −

and note that, using (7), R > 0. Hence, it follows from (7) to (9) that R(x, t) ≥ (R ⊗ M) > 0, x∈ 2n, t ≥ t0. Thus, it follows from (12) that

( ) T( ) ( ) ( ) ( ) ( ( ) ) ( )V x t t t t x t x t t x t, Ψ , Φ , ≤ − ,R T

0( )( ) ( ) 0x t R M x t t t≤ − ⊗ ≤ , ≥ , (13)

which proves that the solution (x(t), Ψ(t), Φ(t)) ≡ (0, Kg, −D) of the closed-loop system (5), (6), and (10) is uniformly Lyapunov stable. Furthermore, since (R ⊗ M) > 0, it follows from Theorem 8.4 of [11] that x(t) → 0 as t → ∞ for all x0∈ 2n. ■

Remark 2.1. Note that the conditions in Theorem 2.1 im-ply that x(t) → 0 as t → ∞, and hence, it follows from (5) and (6) that (x(t), Ψ(t), Φ(t)) → M {(x, Ψ, Φ)∈ n ×

n×2n × n×d : x = 0, Ψ = 0, Φ = 0} as t → ∞.


Remark 2.2. It is important to note that the bounds α1, α2, β1, and β2 for K(x1, t), (x1, t) ∈ n × , and C(x1, x2, t), (x1, x2, t) ∈ n × n × , do not need to be known in order to implement the adaptive controller (4). All that is required is that K(⋅, ⋅) and C(⋅, ⋅, ⋅) are continuous, symmetric, and bounded; otherwise they are unknown. Likewise, M∈ n×n needs to be positive definite but is otherwise unknown.

Remark 2.3. It is important to note that even though Theorem 2.1 seems similar to Theorem 2.1 of [5], these results are in fact very different. Specifically, the dynami-cal model in [5] requires that C(⋅, ⋅, ⋅) and K(⋅, ⋅) are time invariant, C(⋅, ⋅, ⋅) be only a partial function of the system state, and C(⋅, ⋅, ⋅) = C(⋅) ∈ S and K(⋅, ⋅) = K(⋅) ∈ S, where

( )T

1( ) ( ) ( )

n k jn n nk

k i

FS F F q F q q q

q,×

=

∂⎧: → : = ,⎨ ∂⎩

∑

( )

1( ) 1 ,

n k ik

k j

Fq q i j n

q,

=

⎫∂ ⎪= , , = , , ⎬∂ ⎪⎭∑ (14)

and where qi denotes the ith element of q and F(k, j) (⋅) de-notes the (k, j) th element of F(⋅). However, the operators C(⋅) and K(⋅) in [5] need only be lower bounded. Further-more, since the Lyapunov function construction in [5] in-volves path integrals, the results of [5] cannot be extended to the case of time-varying operators C(⋅, ⋅, ⋅) and K(⋅, ⋅). Hence, the adaptive controller framework in the present paper and in [5] result in disjoint controller architectures in the sense that the adaptive controller (4) to (6) does not specialize to the adaptive controller presented in [5] in the special case where C(⋅, ⋅, ⋅) and K(⋅, ⋅) are time-invariant, nor can the results of [5] be generalized to the time-varying case.

Remark 2.4. Although K(⋅, ⋅) and C(⋅, ⋅, ⋅) are assumed to be symmetric, Theorem 2.1 also holds for the more general case where K(⋅, ⋅) and C(⋅, ⋅, ⋅) are nonsymmetric operators. In this case, however, the inequalities in (3) involving K(⋅, ⋅) and C(⋅, ⋅, ⋅) should be replaced with the symmetric part of K(⋅, ⋅) and C(⋅, ⋅, ⋅), respectively. Furthermore, if M is known to be negative definite but otherwise unknown, then Theo-rem 2.1 holds with u(t) given by (4) replaced by u(t) = − Ψ(t) x(t) − Φ(t) w(t). A similar remark holds for Theorem 2.2 below.

Remark 2.5. It is important to note that our adaptive con-trol framework requires that the bounded disturbance w(t), t ≥ t0, can be accurately measured even though the distur-bance signal Dw(t), t ≥ t0, is an unknown bounded distur-bance since D is unknown. In the case where the distur-bance is not accurately known, persistent disturbances can drive up the gain to unsafe levels. In this case, one would have to use parameter projection or leakage methods to

assure that the nonlinear uncertainties lie within given bounds. For details see [9].

Theorem 2.1 is applicable to the case where ( )C q q t, , and K(q, t), nq q, ∈ , t ≥ t0, are bounded. In

practice, however, ( )C q q t, , and K(q, t), nq q, ∈ , t ≥ t0, are often unbounded. Next, we provide a corollary to Theorem 2.1 addresses the case where C(⋅, ⋅, ⋅) and K(⋅, ⋅) can be unbounded operators.

Corollary 2.1. Consider the nonlinear time-varying dy-namical system G given by (2), or, equivalently, the nonlinear matrix second-order dynamical system G given by (1). Assume there exist known, symmetric matrix func-tions Kb : n × → n×n and Cb : n × n × → n×n and scalars α1, α2, β1, β2 ∈ such that β1In ≤ C(x1, x2, t) − Cb(x1, x2, t) ≤ β2In and α1In ≤ K(x1, t) − Kb(x1, t) ≤ α2In, x1, x2 ∈ n, t ≥ t0. Let Q1, Q2 ∈ n×n, Y∈ 2n×2n, Z∈ d×d, and

P∈ 2×2 be positive definite, where 1 12

12 2

p pP

p p

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= and

P12 > 0. Then the adaptive feedback control law

b 1 1( ) ( ( ) ) ( )u t K x t t x t= ,

( )b 1 2 2( ) ( ) ( ) ( ) ( ) ( ) ( ) ,C x t x t t x t t x t t w t+ , , + Ψ + Φ (15)

where Ψ(t) ∈ n×2n, t ≥ t0, and Φ(t) ∈ n×d, t ≥ t0, with update laws (5) and (6) guarantees that the solution (x(t), Ψ(t), Φ(t)) ≡ (0, Kg, −D), where Kg ∈ n×2n, of the closed-loop system given by (2), (5), (6), and (15) is uni-formly Lyapunov stable and x(t) → 0 as t → ∞ for all x0 ∈ 2n.

Proof. Rewrite (1) as

( ) ( )ˆ ˆ( ) ( ) ( ) ( ) ( ) ( )Mq t C q t q t t q t K q t t q t+ , , + ,

0 00ˆ( ) ( ) (0) (0)u t Dw t q q q t tq= + , = , = , ≥ , (16)

where bˆ ( ) ( ) ( )C q q t C q q t C q q t, , , , − , , , ˆ ( )K q

b( ) ( )K q t K q t, − , , and b bˆ ( ) ( )u u C q q t q K q t q− , , − , . Now, the result is a direct consequence of Theorem 2.1.

Finally, we generalize Theorem 2.1 and Corollary 2.1 to the case where T

b( ) ( ) ( )c nC q q t I C q q t, , − θ ⊗ , , and T

b( ) ( ) ( )k nK q t I K q t, − θ ⊗ , are bounded, where θc∈ pc and θk∈ pk are unknown parameters and Cb : n × n × → pcn×n as well as Kb : n × → pkn×n are known functions. ■

Theorem 2.2. Consider the nonlinear time-varying dy-namical system G given by (2), or, equivalently, the nonlinear matrix second-order dynamical system G given by (1). Assume there exist scalars α1, α2, β1, β2 ∈ , vec-tors θc∈ pc and θk∈ pk, and symmetric matrix functions


Cb : n × n × → pcn×n and Kb : n × → pkn×n such that T

1 1 b 1 2( ) ( ) ( )n k n nI K x t I K x t Iα ≤ , − θ ⊗ , ≤ α and T

1 1 2 b 1 2 2( ) ( ) ( )n c n nI C x x t I C x x t Iβ ≤ , , − θ ⊗ , , ≤ β , x1, x2, ∈ n , t ≥ t0. Furthermore, let Cbi : n × n × →

n×n, i = 1, …, Pc, and Kbj : n × → n×n, j = 1, …, Pk, be symmetric maps such that Cb(x1, x2, t) = [Cb1(x1, x2, t), …, Cbpc (x1, x2, t)]T and Kb(x1, t) = [Kb1(x1, t), …, Kbpk (x1, t)]T. Let Q1, Q2 ∈ n×n, Q3 ∈

pk×pk, Q4 ∈pc×pc

Y∈ 2n×2n, Z∈ d×d, and P∈ 2×2 be positive definite,

where 1 12

12 2

p pP

p p

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= and P12 > 0. Then the adaptive

feedback control law

( ) ( )Tb 1 1( ) ( ) ( ) ( )k nu t t I K x t t x t= Θ ⊗ ,

( )Tb 1 2 2( ( ) ) ( ) ( ) ( )c nt I C x t x t t x t+ Θ ⊗ , ,

( ) ( ) ( ) ( )t x t t w t+Ψ +Φ (17)

where Θk(t)∈pk, t ≥ t0, Θc(t)∈

pc, t ≥ t0, Ψ(t)∈ n×2n, t ≥ t0, and Φ(t)∈ n×d, t ≥ t0, with update laws (5), (6), and

( ) ( )T T3 1 b 1 0( ) ( ) ( ) ) ( )

kp nk t Q I x t K x t t B P I x t= − ⊗ , ⊗ ,Θ

(18)

( ) ( )T T4 2 b 1 2 0( ) ( ) ( ) ( ) ) ( )

cp nc t Q I x t C x t x t t B P I x t= − ⊗ , , ⊗ ,Θ

(19)

guarantees that the solution (x(t), Ψ(t), Φ(t), Θc(t), Θk(t)) ≡ (0, Kg, −D, θc, θk), where Kg ∈ n×2n, of the closed-loop system given by (2), (5), (6), (17) to (19) is uniformly Lyapunov stable and x(t) → 0 as t → ∞ for all x0 ∈ 2n.

Proof. Let Kg [K1gM K2gM]∈ K. Furthermore, define T

1 2 1 2 b 1 2 2g( ) ( ) ( ) ( )c nC x x t C x x t I C x x t k M, , , , − θ ⊗ , , − , T

1 1 b 1 1g( ) ( ) ( ) ( ) ,k nK x t K x t I K x t k M, , − θ ⊗ , − ( )k tΘ ( )k ktΘ −θ , and ( ) ( )c cc t tΘ −θΘ , and consider the

Lyapunov function candidate

T T1 11

1 ˆ( ) [ ( )2k cV x x P M x x Mxp, Ψ, Φ, Θ , Θ = ⊗ +

1 1 T 1 1 T1 2tr( ) tr( )Q Y Q Z− − − −+ Ψ Ψ + Φ Φ

T 1 T 13 4 ]k ck cQ Q− −+ Θ +Θ .Θ Θ (20)

Now, the proof is identical to the proof of Theorem 2.1. ■

Remark 2.6. Theorem 2.2 generalizes Corollary 2.1 since 1 2 1 2( ) ( ) ( )T

c n bC x x t I C x x t, , − θ ⊗ , , and 1( )K x t, − 1( ) ( )T

k n bI K x tθ ⊗ , are bounded as opposed to 1 2 1 2( ) ( )bC x x t C x x t, , − , , and 1 1( ) ( )bK x t K x t, − ,

being bounded. This gives yet a larger class of nonlineari-

ties that can be considered in the operators C(⋅, ⋅, ⋅) and K(⋅, ⋅). See [5] for further details.

III. ILLUSTRATIVE NUMERICAL EXAMPLES

In this section we present two numerical examples to demonstrate the utility of the proposed direct adaptive con-trol framework for adaptive stabilization. For the first ex-ample, consider the nonlinear time-varying matrix second- order dynamical system with nonlinear damping and stiff-ness matrix functions given by (1), where n = 2, and M, C( , ,q q t ), and K( ,q t ) are unknown with M > 0, and C( , ,q q t ) and K( ,q t ) bounded for all ,q q ∈ 2 and t ≥ t0. Furthermore, assume w(t) ≡ 0. Now, with p2> 0 and p12> 0, it follows from Theorem 2.1 that the adaptive feedback controller (4) guarantees that x(t) → 0 as t → ∞.

For illustrative purposes, consider (1) with n = 2 and

21

22

1

2

5 0.5,

0.5 4

exp( )sin( ) sin( 4)( , , ) ,

sin( 4) exp( )sin( )

sin( ) sin( )( , ) .

sin( ) sin( )

M

q t tC q q t

t q t

q t tK q t

t q t

⎡ ⎤= ⎢ ⎥⎣ ⎦

⎡ ⎤− /= ⎢ ⎥

/ −⎢ ⎥⎣ ⎦

⎡ ⎤= ⎢ ⎥⎣ ⎦

Let p1 = 2, p2 = p12 = 1, Q1 = 2I2, Y = I4, and set the initial conditions q(0) = [1, −2]T, q (0) = [0, −1]T, and Ψ(0) = 02×4, Figure 1 shows the phase portraits of the controlled and uncontrolled systems. Note that the adaptive controller is switched on at t = 10 sec. Figure 2 shows the state tra-jectories and the control signals versus time. Finally, the adaptive gain history versus time is shown in Fig. 3.

Our second example involves the design of adaptive controllers for suppressing thermoacoustic oscillations in combustion chambers. High performance aeroengine af-terburners and ramjets often experience combustion insta-bilities at some operating condition. Combustion in these high energy density engines is highly susceptible to flow disturbances, resulting in fluctuations to the instantaneous rate of heat release in the combustor. This unsteady com-bustion provides an acoustic source resulting in self-excited oscillations [12,13]. In particular, unsteady combustion generates acoustic pressure and velocity oscillations which in turn perturb the combustion even further. These pressure oscillations, known as thermoacoustic instabilities, often lead to high vibration levels causing mechanical failures, high levels of acoustic noise, high burn rates, and even component melting. In the next example we apply the framework developed in Section 2 to suppress the effects of thermoacoustic instabilities in uncertain combustion processes.


Fig. 1. Phase portraits, controlled and uncontrolled systems.

Fig. 2. State trajectories and control signals versus time.

Fig. 3. Adaptive gain history versus time.

As shown in [14,15], a matrix second-order model with sign-indefinite damping and stiffness operators can be used to capture the coupling between unsteady combustion and acoustics in a combustion process. Specifically, using the mass, momentum, and energy conservation equations for a two phase mixture in a combustor, and using a Galerkin decomposition, the authors in [14,15] obtain

( )2

1( ) ( ) ( ) ( )i i ip ip pi p

pt t d t e t

∞

=+ω η + + ηη η∑

( )1 1

( ) ( ) ( ) ( ) ( ) ,ipq ipq p q ip qp q

a t t b t t u t∞ ∞

= =+ + η η =η η∑ ∑ (21)

where ηi denotes the ith modal combustion pressure, dip, eip, aipq, and bipq, i = 1, …, n, are constants depending on the unperturbed mode shapes and natural frequencies of the combustor [14], and ui(t), t ≥ 0, i = 1, …, n, is the control input to the ith mode and is given by

2

21

( ) ( ) ( )ˆm

ji i sjji

au t t xupE =

= ψ ,∑ (22)

where pa ργ is the local average sound velocity in-side the combustor, ρ is the average density in the two phase mixture, γ is the mixture ratio of specific heats, p is the average pressure inside the combustor, ψi(⋅), i = 1, … , n, are the normal modes of the system,

20 ( ) ( ) ,L

i i cE x x dx= ψ∫ A Ac(x) is the cross sectional area of the combustor, L is the combustor length, ( )ˆ j tu is a control excitation through an acoustic driver, and xsj corre-sponds to the location of the ith actuator.

To design a direct adaptive controller for combustion systems we use the nonlinear combustion model given by the matrix second-order system (21) with nonlinearities present due to the second-order gas dynamics. Furthermore, we assume that actuation is provided by loud speakers while we measure pressure fluctuations via pressure-type microphones. Assuming a two-mode, nonlinear combustion plant model, (21) and (22) yield

21 1 1 1 1 111 1 1 2( ) 2 ( ) ( 2 ) ( ) ( ) ( )t t t F t t= α − ω − θ ω η −η η η η

2

1 212 1 2 1 1 1 221

( ) ( ) ( ( ) ( ) ( ) ( ))ˆ ˆs sF t t x t x tu upEa− η η + ψ +ψ ,

1 10 1 10(0) (0) 0,tη = η , = , ≥η η (23)

22 2 2 2 22 2( ) 2 ( ) ( 2 ) ( )t t t= α − ω − θ ω ηη η

( )2

2 21 221 22 1 2 1 2 21 2

2

( ) ( ) ( ) ( ) ( ) ( )ˆ ˆs sF t F t x t x tu upEa− − η + ψ +ψ ,η

2 20 2 20(0) (0) ,η = η , =η η (24)


where ( )ˆ i tu , i = 1, 2, are control input signals,

12i iidα = − ∈ represents a growth/decay constant,

12

iii

i

eθ = − ∈ω

represents a frequency shift constant, ω1

and ω2 are the frequencies of the first and second modes,

113 2

2F − γ=

γ, 2

12 15( 1)

2F γ −= ω

γ, 21

32

F γ += −γ

, and

222 1

12

F γ −= ωγ

. In the case where we consider a cylindrical

combustor closed at both ends with pure longitudinal modes, it follows that the first two modes are given by

ψi(x) = cos(ki x), ik iLπ= , i = 1, 2. For the nondimension-

alized (using the time factor t L aτ = π / ) data parameters

[16] α1 = 0.0144, α2 = −0.0559, θ1 = 0.0062, θ2 = 0.0178, γ

= 1.2, ω1 = 1, ω2 = 2, and TT T T0 0[ ] [0 01 0 1 0 0]η = . , . , ,η , the

open- loop ( ( ) 0 1 2)ˆ i t iu ≡ , = , dynamics (23) and (24)

result in a limit cycle instability. Figure 4 shows the open-loop response versus time of the system.

Next, we assume that loud speakers are placed at

134sx L= and 2

12sx L= . It is important to note that our

proposed adaptive controller would stabilize any nonlinear time-varying, matrix second-order dynamical system with unknown nonlinear sign-indefinite damping and stiffness operators given by (1). Hence, we assume our combustion model is given by (23), (24) with n = 2, q = [η1, η2]T,

1 2[ ]ˆ ˆ Tu u u= , ,

21

2 22

2 00EpM

a E

⎡ ⎤= − ,⎢ ⎥

⎢ ⎥⎣ ⎦

21 11 2 1

2 2 22 21 1 2 2

2 ( 2 ) 0( )

2E F qpC q q t

E F q Ea

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

− α, , = − ,

− α

2 21 12 2 1 1 1

2 2 2 22 22 1 2 2 2 2

2 ( 2 ) 0( )

( 2 )E F qpK q t

E F q Ea

⎡ ⎤+ ω − θ ω, = − ,⎢ ⎥

ω − θ ω⎢ ⎥⎣ ⎦

where αi, θi, ωi, Fij, and 2

2 ( 0)ipE

a > , i, j = 1, 2, are un-

known. Next, let 2 2 T1 11 2 212

1[ 2 ]2c

p E F E Fa

θ = − , ,

2kp

aθ = − 2 2

1 12 2 221[ 2 ]2

TE F E F, , and

2b1

0( )

0 0q

C q q t⎡ ⎤

, , = ,⎢ ⎥⎣ ⎦

1b2

1

0( )

0q

C q q tq

⎡ ⎤, , = ,⎢ ⎥

⎣ ⎦

2b1

0( )

0 0q

K q t⎡ ⎤

, = ,⎢ ⎥⎣ ⎦

1b2

1

0( )

0q

K q tq

⎡ ⎤, = .⎢ ⎥

⎣ ⎦

Now, it follows from Theorem 2.2 that the adaptive feed-back controller (17) with update laws (5), (6), (18), and (19) guarantees that the closed-loop system is uniformly Lyapunov stable and q(t) → 0 as t → ∞.

To illustrate the dynamic behavior of the closed-loop system, let α1 = 0.0144, α2 = −0.0559, θ1 = 0.0062, θ2 =

0.0178, γ = 1.2, ω1 = 1, ω2 = 2, 2

2 0 4ipE

a = . , i = 1, 2, Q1 = Q3

= Q4 = 0.1I2, and Y = 0.5I2. The response of the controlled system (1) with the adaptive feedback control law (17) and initial conditions q0 = [0.01, 0.1]T, T

0 [0 0]q = , , Ψ(0) = 02×4, Θc(0) = 02×1, and Θk(0) = 02×1 is shown in Fig. 5 Uni-form Lyapunov stability of the closed-loop system (1), (5), (6), (18), (19), and (17) as well as attraction of q(t) is guaranteed by Theorem 2.2. Note that the adaptive con-troller is switched on at t = 300.

To illustrate the robustness of the proposed adaptive control law, we switch the growth constant of the first mode from α1 = 0.0144 to α1 = 0.0720 at t = 600. The closed-loop response is shown in Figs. 6 and 7 shows the same change in the growth constant of the first mode with the switch occurring at t = 350 while the control law is still in process of adapting. Finally, we change the transient parameters θ1 = 0.0062 and θ2 = 0.0178 to θ1 = 0.4998 and θ2 = 1.009 at t = 600. The closed-loop response is shown in Fig. 8. Note that this change corresponds to 8061% and 5669%, respectively, of the original values of the parame-ters.

IV. CONCLUSIONS

A direct adaptive control framework for a class of nonlinear matrix second-order systems with time varying and sign-indefinite damping and stiffness operators was developed. In particular, using a Lyapunov-based frame-work, global uniform asymptotic stability of the closed- loop system states associated with the plant dynamics was guaranteed without requiring any knowledge of the system nonlinearities other than the assumption that they are con-tinuous and bounded. The efficacy of the proposed ap-proach was demonstrated on two nonlinear systems with sign-varying stiffness and damping operators.


Fig. 4. Open-loop state response versus time.

Fig. 5. Closed-loop state response versus time.




REFERENCES

1. Roup, A.V. and D.S. Bernstein, “Adaptive Stabilization of Linear Second-Order Systems with Bounded Time- Varying Coefficients,” J. Vib. Contr., Vol. 10, No. 7, pp. 963-978 (2004).

2. Åström, K.J. and B. Wittenmark, Adaptive Control., Reading, Addison-Wesley, MA (1989).

3. Narendra, K.S. and A.M. Annaswamy, Stable Adaptive Systems, Prentice-Hall, Englewood Cliffs, NJ (1989).

4. Krstić, M., I. Kanellakopoulos, and P.V. Kokotović, Nonlinear and Adaptive Control Design, Wiley, New York (1995).

5. Chellaboina, V., W.M. Haddad, and T. Hayakawa, “Di-rect Adaptive Control for Nonlinear Matrix Second- Order Dynamical Systems with State-Dependent Uncer-tainty,” Sys. Contr. Lett., Vol. 48, pp. 53-67 (2003).


6. Roup, A.V. and D.S. Bernstein, “Adaptive Stabilization of a Class of Nonlinear Systems with Nonparametric Uncertainty,” IEEE Trans. Automat. Contr., Vol. 46, pp. 1821-1825 (2001).

7. Spong, M.W., F.L. Lewis, and C.T. Abdallah, Eds., Ro-bot Control: Dynamics, Motion Planning, and Analysis, IEEE Press, Piscataway, NJ (1993).

8. Marino, R. and P. Tomei, “Observer-Based Adaptive Stabilization for a Class of Nonlinear Systems,” Auto-matica, Vol. 28, pp. 787-793 (1992).

9. Freeman, R.A., M. Krstic, and P.V. Kokotovich, “Ro-bustness of Adaptive Nonlinear Control to Bounded Uncertainties,” Automatica, Vol. 34, pp. 1227-1230 (1998).

10. A. Graham, Kronecker Products and Matrix Calculus with Applications, Ellis Horwood, Chichester (1981).

11. H.K. Khalil, Nonlinear Systems, 3rd Ed., Prentice-Hall, Upper Saddle River, NJ (2002).

12. Culick, F.E.C., “Nonlinear Behavior of Acoustic Waves in Combustion Chambers I,” Acta Astronautica, Vol. 3, pp. 715-734 (1976).

13. Candel, S.M. “Combustion Instabilities Coupled by Pressure Waves and Their Active Control,” in Proc. 24th Int. Symp. Combust., Sydney, Australia, pp. 1277-1296 (1992).

14. Culick, F.E.C., “Combustion Instabilities in Liq-uid-Fueled Propulsion Systems ⎯ An Overview,” Proc. AGARD Conf., No. 450, Bath, UK (1988).

15. Haddad, W.M., A. Leonessa, J.R. Corrado, and V. Kapila, “State Space Modeling and Robust Reduced- Order Control of Combustion Instabilities,” J. Franklin Inst., Vol. 336, pp. 1283-1307 (1999).

16. Fung, Y.-T. and V. Yang, “Active Control of Nonlinear Pressure Oscillations in Combustion Chambers,” J. Propulsion Power, Vol. 8, pp. 1282-1289 (1992).

Wassim M. Haddad received the B.S., M.S., and Ph.D. degrees in mechanical engineering from Florida Institute of Technology, Melbourne, FL in 1983, 1984, and 1987, respectively, with spe-cialization in dynamical systems and control. From 1987 to 1994 he served as a consultant for the Structural Controls

Group of the Government Aerospace Systems Division, Harris Corporation, Melbourne, FL. In 1988 he joined the faculty of the Mechanical and Aerospace Engineering De-partment at Florida Institute of Technology where he founded and developed the Systems and Control Option within the graduate program. Since 1994 he has been a

member of the faculty in the School of Aerospace Engi-neering at Georgia Institute of Technology where he holds the rank of Professor. Dr. Haddad’s research contributions in linear and nonlinear dynamical systems and control are documented in over 450 archival journal and conference publications. His recent research is concentrated on nonlinear robust and adaptive control, nonlinear dynamical system theory, large-scale systems, hierarchical nonlinear switching control, analysis and control of nonlinear impul-sive and hybrid systems, system thermodynamics, thermo-dynamic modeling of mechanical and aerospace systems, network systems, nonlinear analysis and control for bio-logical and physiological systems, and active control for clinical pharmacology. Dr. Haddad is an NSF Presidential Faculty Fellow, a member of the Academy of Nonlinear Sciences, and a coauthor of the books Hierarchical Nonlinear Switching Control Design with Applications to Propulsion Systems (Springer-Verlag, 2000), Thermody-namics: A Dynamical Systems Approach (Princeton Uni-versity Press, 2005), and Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control (Princeton University Press, 2006). His secondary interests include the history of ancient Greek philosophy and its influence on modern science, mathematics, and engineering, and Hel-lenic music, particularly the traditional songs and dances of the Greek islands.

Tomoihsa Hayakawa received the B.Eng. degree in Aeronautical Engineer-ing from Kyoto University, Japan, in 1997, the M.S. degree in Aerospace En-gineering from the State University of New York (SUNY) at Buffalo, NY, in 1999, and the M.S. degree in Applied

Mathematics and the Ph.D. degree in Aerospace Engineer-ing both from Georgia Institute of Technology, GA, in 2001 and 2003, respectively. After serving as a research fellow at the Department of Aeronautics and Astronautics, Kyoto University, and at the Japan Science and Technology Agency (JST), Dr. Hayakawa joined the Department of Mechanical and Environmental Informatics, Tokyo Insti-tute of Technology, in 2006, where he holds the rank of associate professor. His research interests include stability of nonlinear systems, nonnegative and compartmental sys-tems, hybrid systems, nonlinear adaptive control, neural networks and intelligent control, stochastic dynamical sys-tems, and applications to aerospace vehicles, multiagent systems, robotic systems, financial dynamics, and biologi-cal/biomedical systems.

direct adaptive control for nonlinear …haddad.gatech.edu/journal/adaptive_control_msos_tv.pdfasian...

Documents