control of polynomial nonlinear systems using higher degree lyapunov functions

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Control of Polynomial Nonlinear Systems usingHigher Degree Lyapunov Functions

Shuowei Yang and Fen Wu∗

Department of Mechanical and Aerospace EngineeringNorth Carolina State University

Raleigh, NC 27695E-mail: [email protected], [email protected]

ABSTRACT

In this paper, we propose a new control design approach for polynomial nonlinear systems based on higherdegree Lyapunov functions. To derive higher degree Lyapunov functions and polynomial nonlinear controllerseffectively, the original nonlinear systems are augmentedunder the rule of power transformation. The augmentedsystems have more state variables and the additional variables represent higher order combinations of the originalones. As a result, the stabilization andL2 gain control problems with higher degree Lyapunov functions can berecast to the search of quadratic Lyapunov functions for augmented nonlinear systems. The Sum-of-Squares (SOS)programming is then used to solve the quadratic Lyapunov function of augmented state variables (higher degreein terms of original states) and its associated nonlinear controllers through convex optimization problems. Theproposed control approach has also been extended to polynomial nonlinear systems subject to actuator saturationsfor better performance including domain of attraction (DOA) expansion and regionalL2 gain minimization. Severalexamples are used to illustrate the advantages and benefits of the proposed approach for unsaturated and saturatedpolynomial nonlinear systems.

1 IntroductionThe analysis and control of nonlinear systems have been continually listed among the most challenging problems in

the area of systems and control. Lyapunov method is one of themost important methods to study nonlinear problems. It iswell known that quadratic Lyapunov functions are appropriate for Lyapunov-based analysis and control of linear systems.However, for more complicated systems such as polynomial nonlinear systems and those with saturated actuators, it isbelieved that quadratic Lyapunov function form is not adequate and would lead to conservative results. Therefore, the studyof these nonlinear systems often requires higher degree Lyapunov functions as well as polynomial nonlinear controllers toimprove the capability of nonlinear control designs.

Our study will focus on a class of nonlinear systems with polynomial vector field. In the past decade, Sum-of-Squares(SOS) decomposition [24] has become a powerful technique for the Lyapunov stability and reachability analysis of poly-nomial nonlinear systems [11, 17, 22, 23, 26, 30, 31,33, 40] and provided an effective algorithm for linear parameter-varyingsystems [37]. With the help of higher degree Lyapunov functions, convex analysis conditions have been derived to enlargestability domains of nonlinear systems [24]. On the other hand, most synthesis problems of polynomial control systemspursuing higher degree Lyapunov functions can be formulated as

∂V∂x

[A(x)+B(x)F(x)]x< 0,

which is a bilinear matrix inequality of the unknownsV(x) andF(x) and not reducible to convex problems. Therefore,simultaneous searching of the polynomial Lyapunov function and nonlinear controller gain is computationally cumbersome.

∗Corresponding author. Phone: (919) 515-5268, Fax: (919) 515-7968.

Journal of Dynamic Systems, Measurement and Control. Received March 08, 2013;Accepted manuscript posted December 06, 2013. doi:10.1115/1.4026172Copyright (c) 2013 by ASME

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 02/04/2014 Terms of Use: http://asme.org/terms

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There are currently several synthesis approaches of solving nonlinear control problems for polynomial nonlinear sys-tems. By representing the open-loop nonlinear systems in a pre-selected state dependent linear-like form, sufficient condi-tions for nonlinear control problems were formulated as state-dependent linear matrix inequalities (LMIs) [28]. Specifically,the nonlinear dynamics was written in the form of ˙x = A(x)Z(x)+B(x)u using a restricted polynomial vectorZ(x) of thestate variables. Nonlinear control problems were then solved using a higher degree Lyapunov function in the form ofV(x) = ZT(x)PZ(x). An alternative approach in [9,10] employed a transformed vectorZ(x) =W(x)x of the same dimensionas state variables andW(x) as a square, lower triangular polynomial matrix with all diagonal elements equal to 1. Furtherworks along this line have been done in [15, 16] by removing the restriction ofZ(x) and focusing on optimal control anddisturbance rejection problems. A non-iterative two-stepdesign procedure was also proposed with the first step aimingtofind a feasible solution, while the second step utilizing polynomial annihilators to reduce the conservatism in the firststep.Although the methods of [9,10,28] could work for some examples, one of the drawbacks is that they may lead to infeasibilityfor some pre-selected polynomial vectorZ(x). Besides, the resulting performance of the control design would depend onthe selection ofZ(x). This pre-selection process often requires insight of the nonlinear dynamics and there is no systematicapproach currently available to guide the selection of better vectors.

On the other hand, a convex parametrization of nonlinear output-feedbackL2 gain control condition was derived in [21]based on a pair of positive definite matrix functionsP(x),Q(x). Unfortunately, it is difficult, if not impossible, to specifythe functional form ofP(x) such that∂V(x)/∂x= 2xTP(x) except for the trivial case whenP(x) is a constant matrix. There-fore, the proposed synthesis condition does not naturally lead to computationally tractable solution algorithms for nonlinearinducedL2-gain control. As an improvement to [21], an iterative algorithm was proposed in [39] to solve polynomial Lya-punov functions and their associated nonlinear controllers starting from a quadratic Lyapunov function. To this end, thenon-convex synthesis problem was overcome by a computational scheme similar to theD−K iteration. The iterative algo-rithm, however, may not converge to its global minimum and the result could be conservative. Since the set of polynomialLyapunov functionsV(x) and stabilizing control lawsK(x) are not jointly convex, a dual approach to the state feedbackstabilization problem was proposed in [29] using density functions. Nevertheless, it is unclear how to choose the densityfunction with optimized denominator. In addition, a performance criterion cannot be incorporated into the density-basedapproach to optimize nonlinear control designs.

As one of the most common nonlinearities in engineering, saturation has received increasing attention from control re-searchers. Comparing with large number of literatures dealing with linear saturated systems [3,18,32], there are onlylimitedworks addressing analysis and synthesis problems of nonlinear systems subject to saturation nonlinearity. The tools devel-oped for linear systems can not be applied directly to the nonlinear systems, thus hinders the research on saturation control ofnonlinear systems. Reference [20] obtained a ‘universal’ construction of Artstein’s theorem and derived a nonlinear controlformula for smooth nonlinear systems with bounded control input. Nevertheless, the construction of such a stabilizingcon-trol law relies on a given control Lyapunov function, which may not available at hand. In [12], an optimality-based nonlinearcontrol approach was proposed for nonlinear systems with sector-bounded input nonlinearities by solving the HJB equationof unsaturated nonlinear systems. Then, a family of globally stabilizing controllers parameterized by the cost functionalminimization is utilized. In [5], a special class of nonlinear systems involving linear saturated systems affected by sector-bounded nonlinearities was studied. The saturated nonlinear systems were stabilized regionally using nonlinearity-dependentsaturation control laws. Recently, [35] proposed the design of nonlinear state-feedback control laws for polynomial systems(unsaturated or saturated) by reformulatingquadratic stabilizationconditions as polynomial inequalities. They are then re-laxed to SOS programs in solving quadratic Lyapunov functions and controller gains simultaneously. It is worth to mentionthat most of existing works have focused on the constructionof quadratic Lyapunov functions for nonlinear saturation controlproblems.

The concept of power transformation was first introduced forlinear systems in [4]. Since then, power transformation hasbeen used to analyze stability of nonlinear systems [2]. Utilizing this transformation in representing higher order polynomialsand higher degree Lyapunov functions, reference [38] improved the estimation of robust stability region for uncertainlinearsystems. Inspired by the analysis results using power transformation, we will employ this transformation for nonlinearcontrol synthesis problems. Our proposed approach will lead to a systematic procedure of augmenting polynomial nonlinearsystems to their higher-order equivalent forms, which havemore variables and rich representations than the original nonlinearmodels. With the help of power transformation, it can be shown that if a polynomial nonlinear system is quadraticallystabilizable, there always exists stabilizing higher degree Lyapunov functions for the system. Consequently, the designof higher degree Lyapunov functions for original nonlinearsystems can be recast to the search of quadratic Lyapunovfunctions for augmented systems. By combining power transformation and SOS programming techniques, it is possible tosynthesize such Lyapunov functions (higher order in terms of original states) and their associated nonlinear control lawsthrough computationally efficient SOS programming. Moreover, by incorporating aL2 performance index, the controlledperformance can be optimized using higher degree Lyapunov functions. The derived solvability conditions of nonlinearstabilization andL2-gain control problems are both in convex optimizations which are no more difficult to solve than aquadratic stabilization problem. We will also extend this approach to synthesize higher degree Lyapunov functions andnonlinear control laws for saturated polynomial nonlinearsystems with enlarged domain of attraction (DOA) and optimized



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regionalL2 gain.The method proposed in this paper could provide a new perspective to the works in [9, 16, 28]. From a power transfor-

mation viewpoint, we represent open-loop nonlinear systems using a new state vector as higher degree transformation oftheoriginal state vectorx. This constitutes a novel approach to augment nonlinear dynamics in its most extensive form. Then weconstruct general 2r-th degree Lyapunov functions which include the Lyapunov functions used in [9,10,15,16,28] as specialcases. Consequently the synthesis conditions of polynomial nonlinear controllers can be formulated as convex optimizationsand solved efficiently by SOS programming. It is clear that the proposed approach would always render less conservativeresults than those in [9,16,28]. Moreover, an optional iteration procedure similar to the one in [16] can be used to determinethe most suitable representation of augmented nonlinear systems for achieving better performance.

The notations used in this paper are quite standard.R stands for the set of real numbers andZ+ denotes the set ofpositive integers.Rm×n is the set of realm×n matrices. We useSn×n to denote real, symmetricn×n matrices, andSn×n

+ forpositive-definite matrices. Then×n identity matrix is denoted byIn. For a real matrixM, we define a Hermitian operatorHe{·} asHe{M}= M+MT . Given two matricesA∈ Rm×n andB∈ Rp×q, their Kronecker product is of dimensionpm×qnand defined as

A⊗B=

a11B · · · a1nB...

. . ....

am1B · · · amnB

.

In large symmetric matrix expressions, terms denoted as⋆ will be induced by symmetry. For two integersk1,k2, k1 < k2,we denoteI[k1,k2] = {k1,k1+1, · · · ,k2}. The space of square integrable functions is denoted byL2, that is, for any signal

x∈L2, ‖x‖2 :=(∫ ∞

0 xT(t)x(t)dt) 1

2 <∞. A multivariate polynomialp(x) is a Sum-of-Squares (SOS) if there exist polynomialsp1(x), . . . , pℓ(x) such thatp(x) = ∑ℓ

i=1 p2i (x). We will denote the set of SOS polynomials ofx asΣ[x].

This paper is organized as follows: In Section 2, we provide some background knowledge on power transformation andSOS programming. The nonlinear stabilization andL2-gain control problems for polynomial nonlinear systems are solvedin Section 3. In Section 4, we will extend the results in the previous section to saturated nonlinear systems. Then threeexamples are provided in Section 5 to illustrate the proposed nonlinear control design approach. Finally, we will concludethe paper in Section 6.

2 Preliminaries2.1 Power Transformation

For a vectorx= [x1 · · · xn]T ∈ Rn, its rth degree power transformation is a nonlinear change of coordinates that forms

a homogeneous vectorx[r] consisting of all integer powered monomials of degreer. The total number of independentmonomials inx[r] of degreer is determined by

Nrn =

(n+ r −1

r

)=

(n+ r −1)!r!(n−1)!

.

The ℓ-th monomial ofx[r] can be represented asx[r]ℓ := xkℓ11 xkℓ2

2 . . .xkℓnn , ℓ ∈ I[1,Nr

n], wherekℓ j ∈ {0,1, . . . , r} such that

∑nj=1kℓ j = r,∀ℓ, and the ordering ofx[r]ℓ terms is lexicographical in thekℓ js. For example, whenn = 2, r = 3,N3

2 = 4,the ordered list of degree 3 monomials isx3

1,x21x2,x1x2

2,x32.

Power transformation is useful to transform states of a dynamic system to any arbitrary high degree [4]. Moreover, itpreserves certain linear relation which makes the transformation attractive for studying linear systems. Applying the powertransformation a linear ordinary differential equation (ODE) x(t) = A(t)x(t), we obtain

ddt

x[r](t) = A[r](t)x[r](t), r ∈ Z+,

whereA[r] to denote the state matrix of the augmented differential equation. Therefore, the set of allrth degree monomials inx1,x2, . . . ,xn also satisfies a linear differential equation with a coefficient matrix derived fromA. It is not difficult to see thatA[r] depends linearly on the elements ofA. From the result of [4], ifA is Hurwitz, thenA[r] is Hurwitz as well. Therefore, thestability of a linear ODE is preserved under the power transformation. In sequel, we will extend the power transformationidea to polynomial nonlinear systems for nonlinear controldesigns.



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2.2 SOS DecompositionGiven a positive integerr, x(r) will be used to denote an augmented vector containing all possible monomials of degree

k ≤ r of the state vectorx. Thenx(r) has the dimension ofNrn = ∑r

k=1Nkn. As an example, the monomials of a two-variable

vector with degree no more than 2 are

x(2) =[x1 x2 x2

1 x1x2 x22

]T:=

[x1 x2 x3 x4 x5

]T∈ R5.

The augmented variables are not independent from each other. Clearly, we have the following coupling constraints amongthe elements ofx(2) vector

φ1(x) = x3− x21 = 0, φ2(x) = x4− x1x2 = 0, φ3(x) = x5− x2

2 = 0,

φ4(x) = x2x4− x1x5 = 0, φ5(x) = x2x3− x1x4 = 0, φ6(x) = x24− x3x5 = 0.

These constraints reflect the equivalence relationship among the multiple combinations of augmented state variables [17].Also, the number of coupling constraints can be found from [6].

If a polynomial f (x) is SOS decomposable, then it impliesf (x)≥ 0 for anyx∈ Rn. Given the polynomial of degree 2r,the SOS decomposition condition is equivalent to the existence of a positive semi-definite matrixG such that

p(x) = MT(x)GM(x),

whereM(x) is a monomial vector ofx with all components of degree no more thanr andG is called the Gram matrix [7].Therefore, testing whether a given polynomialp(x) is an SOS is related to the non-negativity of Gram matrix associatedto the polynomial and is solvable as a semi-definite programming problem. Specifically, one can solve the semi-definiteproblem

p(x)+∑i

λi(x(r))φi(x

(r)) ∈ Σ[x],

in which λi(x(r)) are multipliers to enforce the coupling constraintsφi(x(r)) = 0 among augmented state variables.The main advantages of SOS programming are the resulting computational tractability and the algorithmic character of

its solution procedure [24]. As a convenient tool to determine polynomial positivity, SOS programming has been used forpolynomial nonlinear system analysis problems. It could also help provide coherent methodology of synthesizing versatileLyapunov functions and control laws for nonlinear systems.

3 State-Feedback Control of Polynomial Nonlinear Systems3.1 Dynamic System Augmentation

Consider an affine-input,nth-order polynomial nonlinear system of the following form

{x = A(x)x+B1(x)d+B2(x)ue= C(x)x+D1(x)d+D2(x)u

(1)

with the statex ∈ Rn, the disturbanced ∈ Rnd , and the control inputu∈ Rnu. All state variables are assumed measurableand the state-space entries are polynomial functions of thestatex. It is assumed that the polynomial nonlinear systemis quadratically stabilizable. Our design objective is to find a higher degree Lyapunov function as well as its associatedpolynomial state-feedback controlleru= F(x)x to stabilize the closed-loop nonlinear system and optimizeits performance.

In order to synthesize high degree Lyapunov function and itsassociated nonlinear controller for the nonlinear system(1), we will augment the state vector tox(r) (as defined in subsection 2.2). Then, the augmented nonlinear system becomes

{˙x = A(r)(x)x+B1(r)(x)d+B2(r)(x)ue= C(r)(x)x+D1(r)(x)d+D2(r)(x)u

(2)

wherex= x(r), the augmented state matrixA(r) ∈ RNrn×Nr

n and the input matrixB2(r)(x) ∈ RNrn×nu. B1(r),C(r) andD2(r),D1(r)

are other relevant matrices after the state augmentation. Consequently, the goal of the nonlinear controller design becomes



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computing the controller gain matrixF(x(r)) such thatu= F(x(r))x(r) would stabilize the closed-loop system with optimizedperformance.

The augmented state-space matrices can be derived using thefollowing lemma, which is a special case of Lemma 1in [6].

Lemma 1. Given a polynomial nonlinear systemx= A(x)x+B(x)u, we define A[r](x) ∈ RNrn×Nr

n and B[r](x) ∈ RNrn×nu for

any r∈ Z+ such that

ddt

x[r] =∂x[r]

∂x[A(x)x+B(x)u] := A[r](x)x

[r]+B[r](x)u. (3)

Moreover, let x〈r〉 = x⊗ x⊗ . . .⊗ x︸︷︷︸r

and Tr ∈ Rnr×Nrn be the matrix satisfying x〈r〉 = Trx[r]. Then A[r],B[r] can be determined by

A[r](x) = (TTr Tr)

−1TTr

r−1

∑i=0

(Inr−1−i ⊗A(x)⊗ Ini)Tr

B[r](x) = (TTr Tr)

−1TTr

r−1

∑i=0

(x〈r−1−i〉⊗ In⊗ x〈i〉)B(x).

Proof. First, we introduceA〈r〉(x) ∈ Rnr×nrandB〈r〉(x) ∈ Rnr×nu as

∂x〈r〉

∂x[A(x)x+B(x)u] := A〈r〉(x)x

〈r〉+B〈r〉(x)u. (4)

Using the relation∂x〈r〉

∂x = ∑r−1i=0 x〈r−1−i〉⊗ In⊗x〈i〉 and the mixed product property of Kronecker product, it can be shown that

r−1

∑i=0

(x〈r−1−i〉⊗ In⊗ x〈i〉)A(x)x=r−1

∑i=0

(x〈r−1−i〉⊗ In⊗ x〈i〉)(In0 ⊗A(x)x⊗ In0)

=r−1

∑i=0

[(x〈r−1−i〉⊗ In) · (In0 ⊗A(x)x)]⊗ [x〈i〉In0]

=r−1

∑i=0

[(x〈r−1−i〉In0]⊗ [InA(x)x)]⊗ [x〈i〉In0]

=r−1

∑i=0

(Inr−1−i x〈r−1−i〉)⊗A(x)x⊗ (Inix〈i〉)

=r−1

∑i=0

[(Inr−1−i ⊗A(x)) · (x〈r−1−i〉⊗ x)]⊗ (Inix〈i〉)

=r−1

∑i=0

(Inr−1−i ⊗A(x)⊗ Ini) · (x〈r−1−i〉⊗ x⊗ x〈i〉)

=r−1

∑i=0

(Inr−1−i ⊗A(x)⊗ Ini)x〈r〉 = A〈r〉(x)x〈r〉. (5)

Moreover, we have

r−1

∑i=0

(x〈r−1−i〉⊗ In⊗ x〈i〉)B(x)u= B〈r〉(x)u. (6)

From eqn. (4), we have

Tr∂x[r]

∂x[A(x)x+B(x)u] = A〈r〉(x)Trx

[r]+B〈r〉(x)u. (7)



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Note thatTr has full column rank and containsNrn linearly independent columns. Multiplying from the left-hand of (7) by

(TTr Tr)

−1TTr and following eqns. (3) and (5)-(6), we obtain the expressions ofA[r],B[r] as desired.

The format of augmented system matrices provided in this lemma can help efficiently construct one representation ofthe augmented system during the control synthesis process.As an example, consider a 2nd-order nonlinear system with

A(x) =

[a11(x) a12(x)a21(x) a22(x)

], B1(x) =

[b11(x)b12(x)

], B2(x) =

[b21(x)b22(x)

],

C(x) =[c1(x) c2(x)

], D1(x) = d1(x), D2(x) = d2(x).

Through the power transformation, the original system state x can be augmented to a vector ˜x = x(2) of its power no more

than 2 withT2 =

1 0 00 1 00 1 00 0 1

. Applying Lemma 1 with simple calculations, we obtain one representation of the augmented

system

˙x1˙x2˙x3˙x4˙x5

=

a11(x) a12(x) 0 0 0a21(x) a22(x) 0 0 0

0 0 2a11(x) 2a12(x) 00 0 a21(x) a11(x)+a22(x) a12(x)0 0 0 2a21(x) 2a22(x)

x1

x2

x3

x4

x5

+

b11(x)b12(x)

2b11(x)x1

b11(x)x2+b12(x)x1

2b12(x)x2

d+

b21(x)b22(x)

2b21(x)x1

b21(x)x2+b22(x)x1

2b22(x)x2

u (8)

:= A(2)(x)x+B1(2)(x)d+B2(2)(x)u

e=[c1(x) c2(x) 0 0 0

]

x1

x2

x3

x4

x5

+D1(x)d+D2(x)u (9)

:=C(2)(x)x+D1(2)(x)d+D2(2)(x)u.

Following the same procedure, the original nonlinear system can be extended to its higher-degree (r > 2) equivalence aswell.

3.2 Existence of Higher Degree Lyapunov FunctionsUsing Picard iteration, it was shown in [25] that exponential stability of a polynomial vector field on a bounded set

implies the existence of a Lyapunov function which is a sum-of-squares polynomials. [1] has provided a proof of existenceof higher degree Lyapunov functions for quadratic stable systems by consideringV(x) = (xTPx)2r andV = 2r(xTPx)V. Inthis subsection, by considering the system augmentation, we will prove the existence of higher degree Lyapunov functions forquadratically stabilizable polynomial nonlinear systems. For space reason, the dependency of nonlinear state-spacematricesonx will often be omitted.

Theorem 1. For the polynomial nonlinear systemx= A(x)x+B(x)u, if there exists a quadratic Lyapunov function V(x) =xTPx> 0 and a control u= F(x)x such that xT(PA(x)+AT(x)P+PB(x)F(x)+FT(x)BT(x)P)x < 0, then for any r∈ Z+,the closed-loop system also admits2r-th degree Lyapunov function V(x[r]) = x[r]TP[r]x

[r] > 0 and a matrix functionF[r](x)satisfying

x[r]T[P[r]A[r](x)+AT

[r](x)P[r]+P[r]B[r](x)F[r](x)+ FT[r](x)B

T[r](x)P[r]

]x[r] < 0. (10)



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Proof. Substituting the control law into the nonlinear equation, we getx= A(x)x+B(x)F(x). Similar to the proof of lemma1, it can be shown that this equation can be augmented to its higher degree as

ddt

x[r] = A[r](x)x[r]+ B[r](x)F[r](x)x

[r] (11)

with

B[r](x) = (TTr Tr)

−1TTr

[(Inr−1 ⊗B(x)⊗ In0) · · · (In0 ⊗B(x)⊗ Inr−1)

](12)

F[r](x) =

(Inr−1 ⊗F(x)⊗ In0)

...(In0 ⊗F(x)⊗ Inr−1)

Tr (13)

for anyr ∈ Z+.SincexTPx> 0 andxT(PA+ATP+PBF+FTBTP)x< 0, based on the mixed-product property of a Kronecker product,

we have

[xTPx]⊗ [xT(PA+ATP+PBF+FTBTP)x]< 0

⇔ [xTP]⊗ [xT(PA+ATP+PBF+FTBTP)] · [x⊗ x]< 0

⇔ [xT ⊗ xT ] · [P⊗ (PA+ATP+PBF+FTBTP)] · [x⊗ x]< 0

⇔ x〈2〉T{(P⊗P) [(In⊗A)+ (In⊗BF)]+

[(In⊗A)T +(In⊗BF)T](P⊗P)

}x〈2〉 < 0. (14)

Similarly, from [xT(PA+ATP+PBF+FTBTP)x] · [xTPx]< 0 one can derive

x〈2〉T{(P⊗P) [(A⊗ In)+ (BF⊗ In)]+

[(A⊗ In)

T +(BF⊗ In)T](P⊗P)

}x〈2〉 < 0. (15)

Adding eqns. (14) and (15) together, we obtain

x〈2〉T[(P⊗P)(A⊗ In+ In⊗A)+ (A⊗ In+ In⊗A)T(P⊗P)

]x〈2〉

+ x〈2〉T{(P⊗P) [(BF⊗ In)+ (In⊗BF)]+ [(BF⊗ In)+ (In⊗BF)]T (P⊗P)

}x〈2〉 < 0,

which is equivalent to

x〈2〉T[(P⊗P)

1

∑i=0

(In1−i ⊗A⊗ Ini)+1

∑i=0

(In1−i ⊗A⊗ Ini)T(P⊗P)

]x〈2〉

+ x〈2〉T(P⊗P)1

∑i=0

(In1−i ⊗B⊗ Ini) · (In1−i ⊗F ⊗ Ini )x〈2〉

+ x〈2〉T1

∑i=0

(In1−i ⊗F ⊗ Ini )T · (In1−i ⊗B⊗ Ini)T(P⊗P)x〈2〉 < 0. (16)

Substitutingx〈r〉 = Trx[r] into eqn. (16) and using eqns. (12)-(13), one obtains

x[2]T(

P[2]A[2]+AT[2]P[2]+P[2]B[2]F[2]+ FT

[2]BT[2]P[2]

)x[2] < 0

with P[2] = TT2 (P⊗P)(T2TT

2 )−1T2(TT2 T2) = TT

2 (P⊗P)T2 > 0. For anyr > 2, the similar procedure can be applied to theaugmented system (11) by multiplying additional(xTPx)i ,(xTPx) j , i + j = r −1 on the left and right sides of (14) and (15)respectively.



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The implication of Theorem 1 to the nonlinear system (2) is that it is always feasible to construct a Lyapunov functionV(x) = xTP(r)x > 0 with P(r) = diag

{P,P[2], · · ·P[r]

}and its associated stabilizing control gains. These Lyapunov functions

will be beneficial for the stabilization andL2-gain control of polynomial nonlinear systems. Since this type of Lyapunovfunction includes the quadratic form as a special case, it isguaranteed to be no worse than a quadratic Lyapunov functioninachieving optimal performance.

3.3 State-Feedback Stabilization and L2-Gain ControlUsing higher degree Lyapunov functions, we will solve the state-feedback stabilization andL2-gain control problems

for polynomial nonlinear systems. The derived conditions in the following theorem share a similar form to their quadraticcounterpart and pertain in convex optimization problems asthe original ones.

Theorem 2. For the polynomial nonlinear system (1) and its r-th degree augmented system,

1. the system (1) is stabilizable by a polynomial state-feedback control if there exists a positive definite matrix Q∈ SNrn×Nr

n+

and a rectangular matrix function M(x) : RNrn → Rnu×Nr

n such that

−zTR(x)z+∑i

λi(x, z)φi(x) ∈ Σ[x, z], (17)

where R(x) = He{

A(r)(x)Q+B2(r)(x)M(x)}

, and z is an auxiliary vector of the same dimension as matrix R;λi aremultipliers to enforce the coupling constraintsφi(x) = 0 among augmented state variables.

2. the system (1) is stabilizable with itsL2-gain performance less than a given levelγ > 0 if there exists a positive definitematrix Q, a matrix function M(x) and multipliersλi(x, z) such that

−zTS(x)z+∑i

λi(x, z)φi(x) ∈ Σ[x, z] (18)

with

S(x) =

He{A(r)(x)Q+B2(r)(x)M(x)} B1(r)(x) QCT(r)(x)+MT(x)DT

2(r)(x)BT

1(r)(x) −γI DT1(r)(x)

C(r)(x)Q+D2(r)(x)M(x) D1(r)(x) −γI

.

Moreover, an admissible polynomial state-feedback control law is given by

u= F(x(r))x(r) = M(x(r))Q−1x(r).

Proof. 1. For a Lyapunov functionV(x) = xTQ−1x> 0, its derivative along the augmented system (2) is given by

V(x) = ˙xTQ−1x+ xTQ−1 ˙x

= xT{[

A(r)(x)+B2(r)(x)F(x)]T

Q−1+Q−1[A(r)(x)+B2(r)(x)F(x)

]}x

= zTR(x)z,

wherez= Q−1x andM(x) = F(x)Q. If φi(x) = 0 holds for all indicesi, the augmented system becomes an equivalentdescription of the original nonlinear dynamics. Thereforecondition (17) implies ˜zTR(x)z< 0, indicating the originalnonlinear system is stabilized.

2. By definingM(x) = F(x)Q and multiplying diag{Q−1, I , I} from left-hand side and its transpose from right-hand sideof S(x), we obtain

Sm(x) := diag{Q−1, I , I} S(x) diag{Q−1, I , I}

=

He{Q−1[(A(r)+B2(r)F ]} ⋆ ⋆

BT1(r)Q

−1 −γI ⋆

[C(r)+D2(r)F ] D1(r) −γI

.



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It is clear from eqn. (18) thatSm(x) < 0 if the coupling constraintφi(x) = 0 holds for alli. Then for the LyapunovfunctionV(x) = xTQ−1x, we have

V(x)+1γ

eTe− γdTd

= xT{[

A(r)(x)+B2(r)(x)F(x)]T

Q−1+Q−1[A(r)(x)+B2(r)(x)F(x)]}

x

+dTB1(r)(x)TQ−1x+ xTQ−1B1(r)(x)d− γdTd

+1γ{[

C(r)(x)+D2(r)(x)F(x)]x+D1(r)(x)d

}T {[C(r)(x)+D2(r)(x)F(x)]x+D1(r)(x)d

}

=[xT dT

]

{He{Q−1[(A(r)+B2(r)F]}

+ 1γ [C(r)+D2(r)F]

T [C(r)+D2(r)F ]

}⋆

BT1(r)Q

−1+ 1γ DT

1(r)[C(r)+D2(r)F ] −γI + 1γ DT

1(r)D1(r)

[

xd

]< 0. (19)

The last inequality can be verified by taking Schur complement from Sm(x)< 0. Integrating on both sides of the (19), itconfirms thatL2 gain of the original nonlinear system is bounded byγ.

Theorem 2 provides solvability conditions of robust state-feedback control problems by constructing higher degreeLyapunov functions as well as polynomial controller gains.They can be solved effectively using SOS programming tools[27]. Note that the conditions (17) and (18) synthesize quadratic Lyapunov functions for augmented systems which are higherdegree Lyapunov functions for the original nonlinear systems. By solving a series of convex optimization problems withincreased degree of augmentation, we will circumvent the difficulty in finding higher degree Lyapunov function directlyfrom∂V∂x [A(x)+B(x)F(x)]x< 0. For systems which are not quadratically stabilizable, however, since we cannot introduceM =F(x)Q and make the resulting condition convex, extending the results of the above theorems to non-quadratic stabilizationcases is very difficult.

The proposed approach can be thought as a generalization of existing methods in [9, 16, 28] when choosingZ(x) =x(r) and could provide less conservative results. Note that the synthesis condition is also different from previous resultsby including coupling constraint terms. The coupling constraints play two roles here: One is to ensure the equivalenceof augmented systems to original nonlinear dynamics. On theother hand, it helps to find a more appropriate nonlinearrepresentation in solving the synthesis condition. For some nonlinear systems, it is not necessary to include all of themonomials to guarantee the solvability of synthesis conditions (17)-(18).

3.4 Non-Uniqueness of Matrix RepresentationsThe linear like state-space representation (1) of a polynomial nonlinear dynamic system is not unique. Since the derived

form of A(r)(x) is just one among many possible representations, it is likely that there exists anNA(x) with NA(x)x= 0 suchthat,A(r)(x)+NA(x) is a more appropriate one to achieve a better optimization result when certain performance requirementsinvolved. The termNA(x) is called a polynomial annihilator of ˜x. Note that polynomial annihilators were also used in [16]to reduce the conservatism of nonlinear control designs. Byincorporating entries ofNA(x) as new variables, however, it willlead to bilinear matrix inequalities with two undeterminedmatrices multiplication, like[A(r)(x)+NA(x)]Q+Q[A(r)(x)+NA(x)]T . The iterative process described below can be utilized to resolve this issue.

In rewritingA(r)(x) asA(r)(x)+NA(x), with NA(x) satisfyingNA(x)x= 0, a better optimization result might be obtainedby solving for a suitableNA(x) iteratively. In each iteration, the suboptimal result is obtained using fixedNA(x) from theprevious cycle. Then, the results are substituted into the conditions to solve the same optimization problem for a newNA(x).TakingL2-gain optimization problems for example, during each iteration, we will try to minimize aγ value, with eitherNA(x)or Q fixed, and the other one withM(x) as unknowns, alternatively. The iterative process will lead to monotonic decreasing ofγ values, and at the end determine a suitable representation of augmented nonlinear system model with (sub)optimalL2-gainperformance.

4 Control of Polynomial Nonlinear Systems with Actuator SaturationsWe have proposed a method in Section 3 to solve nonlinear control problems with higher degree Lyapunov functions.

Nevertheless, the application of this method is not limitedto smooth polynomial nonlinear systems and many other perfor-mance related nonlinear control problems may benefit from the proposed approach. In the following, we will extend thisapproach to saturated nonlinear systems to enlarge the stability domain and optimize theL2 performance in a local region.



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To this end, we consider the following saturated polynomialnonlinear system

{x = A(x)x+B1(x)d+B2(x)sat(u)e= C(x)x+D1(x)d+D2(x)sat(u).

(20)

where sat(·) is a vectorized saturation function with the saturation levels given by a vector ¯u ∈ Rnu, ui > 0, i ∈ I[1,nu]. Inparticular,

sat(u) =

sat(u1)...

sat(unu)

, sat(ui) = sign(ui)min{ui, |ui |}.

The dead-zone nonlinearity is closely related to saturation function and is given by dz(u) := u− sat(u) for all u∈ Rnu. Thefollowing lemma provides a connection between the nonlinear saturation and dead-zone functions. Its proof can be foundin [14] and is omitted here.

Lemma 2. [13,14] Let the matrix function h(x) be a nonlinear continuous map. Suppose thatℓTj h(x) ∈ [−u j , u j ], whereℓ j

denotes the jth column of the unity matrix. For any uj , we havesat(u j) = Co{

u j , ℓTj h(x)

}whereCodenotes the convex hull

anddz(u j) = λ j(u j − ℓTj h(x)) for someλ j ∈ [0,1].

By incorporating the dead-zone function, the augmented polynomial nonlinear system with input saturation can bewritten as

˙x = A(r)(x)x−B2(r)(x)p+B1(r)(x)d+B2(r)(x)uu = F(x)xe= C(r)(x)x−D2(r)(x)p+D1(r)(x)d+D2(r)(x)up = dz(u)

(21)

It can be shown that quadratic stabilization of saturated nonlinear systems also implies the existence of stabilizing higherdegree Lyapunov functions. For disturbance attenuation ofsaturated nonlinear systems, we are mainly concerned with aclass of energy-bounded disturbances

Ws =

{d : R+ → Rnd ,

∫ ∞

0dT(τ)d(τ)dτ ≤ s2, d ∈ L2

}.

The level of disturbance attenuation will be measured by theregionalL2 gain supd∈Ws, x(0)=0

‖e‖2

‖d‖2.

For a matrix functionH(x) : RNrn → Rnu×Nr

n , we will define a local region as

L(H(x)) :={

x∈ RNrn : |ℓT

j H(x)x| ≤ u j , j ∈ I[1,nu]},

whereℓ j denotes thejth column of the unitary matrix. Different from theL(H) used in linear saturation control studies [14],L(H(x)) specifies a nonlinear map of original state variables which could help to improve the controlled performance.Following Lemma 2, the dead-zone functionp= dz(u) would satisfy

pTU−1(x)(u−H(x)x− p)< 0

with a diagonal positive-definite matrix functionU(x) : RNrn → Snu×nu

+ . Moreover, we will define an ellipsoidE(Q−1) = {x∈

RNrn : xTQ−1x≤ 1} for a givenQ∈ SNr

n×Nrn

+ .



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Using the Lyapunov functionV(x) = xTQ−1x> 0 andU(x), the regional performance condition for bounded disturbanced ∈ Ws will be

V +1γ2eTe−dTd+2pTU−1(x)(u−H(x)x− p)< 0 (22)

sE(Q−1)⊆ L(H(x)). (23)

The regional stabilization problem can also be addressed byremoving the term1γ2 eTe−dTd from condition (22). Conse-

quently, we obtain regional stabilization and performanceoptimization conditions for the saturated nonlinear system (20) asfollows.

Theorem 3. For the polynomial nonlinear system (20) with actuator saturation,φi(x) = 0 denotes the coupling constraintsof augmented state variables.

1. Given s> 0, if there exist a positive definite matrix Q, matrix functions Y(x),K(x) ∈ Rnu×Nrn, a diagonal matrix function

U(x) ∈ Snu×nu+ , and multipliersλ1i(x, z1), λ2i j (x, z2) satisfying

− zT1

[He

{A(r)(x)Q+B2(r)(x)K(x)

}⋆

K(x)−Y(x)−U(x)BT2(r)(x) −2U(x)

]z1+∑

iλ1i(x, z1)φi(x) ∈ Σ[x, z1] (24)

zT2

[u2

j

s2 ℓTj Y(x)

YT(x)ℓ j Q

]z2+∑

iλ2i j (x, z2)φi(x) ∈ Σ[x, z2], ∀ j ∈ I[1,nu] (25)

then the controller gainF(x) = K(x)Q−1 ensures that sE(Q−1) is a contractively invariant ellipsoid in the augmentedstate space.

2. Givenγ,s> 0, if there exist a positive definite matrix Q, matrix functions Y(x),K(x), a diagonal matrix function U(x),and multipliersλ2i j (x, z2), λ3i(x, z3) satisfying condition (25) and

− zT3

He{

A(r)(x)Q+B2(r)(x)K(x)}

⋆ ⋆ ⋆BT

1(r)(x) −I ⋆ ⋆

C(r)(x)Q+D2(r)(x)K(x) D1(r)(x) −γ2I ⋆K(x)−Y(x)−U(x)BT

2(r)(x) 0 −U(x)DT2(r)(x) −2U(x)

z3

+∑i

λ3i(x, z3)φi(x) ∈ Σ[x, z3] (26)

then the controller gainF(x) = K(x)Q−1 will render the regionalL2 gain less thanγ.

Proof. If d = 0 andx(0) ∈ sE(Q−1), we substitute the augmented dynamics (21) into the condition

V + pTU−1(x)(u−H(x)x− p)< 0,

this can be shown equivalent to condition (24) whenφi(x) = 0 for all i. Moreover, the set inclusion condition will hold if

[u2

j

s2 ℓTj H(x)

HT(x)ℓ j Q−1

]≥ 0.

LettingY(x) = H(x)Q, it can be shown that the above eqn. is equivalent to condition (25) when the coupling conditions aresatisfied. The regional performance condition can be shown similarly.

We reiterate that conditions in both Theorem 2 and Theorem 3 are in the forms of convex optimizations. The quadraticversions of these conditions (using original system matrices and removingφ terms) were often used to solve nonlinear controlproblems in the past. From the computational complexity perspective, solving these higher order version conditions are nomore difficult than solving a nonlinear quadratic stabilization problem.



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Remark 1. Suppose that the state-dependent inequalityzTH(x)z+∑i λi(x, z)φi(x)> 0ensures a specific stabilization/performancecontrol condition for the nonlinear system (1) or (20). Thena corresponding condition to guarantee its stabilization/performancewithin a restricted state-space region can be written as

zTH(x)z+∑i

λi(x, z)φi(x)+∑j

θ j(x, z)ψ j(x) ∈ Σ[x, z], (27)

whereψ j(x)< 0 are regional constraints.θ j(x, z)> 0 are SOS multipliers to enforce the regional constraints.

Although the regional constraints are optional, the richness of these terms will be helpful to improve solvability of SOSconditions. Specifically, to render the polynomial synthesis conditions SOS decomposable, it is often desired to amendtheconditions with regional constraints. The number and the form of the regional constraints are problem dependent. If theaugmented system hasmstates, the most commonly used regional constraints will bein the form ofψ j(x) = xq

j −c j < 0, j ∈I[1,m], in whichc j andq are pre-defined positive constants. Consequently, the local stabilization region is specified by theintersection of constraintsψ j(x)< 0, j ∈ I[1,m].

5 ExamplesIn the first example, we will compute and compare minimalL2 gains of the following nonlinear system using both

quadratic and quartic Lyapunov functions.

x=

[x1 1+ 1

5x2

x21−1 −x2− x1x2

]x+

[0.2 0.250.25 0.2

]d+

[11

]u (28)

e=

0.5 00 20 0

x+

00

0.2

u. (29)

Its 2nd-degree augmented system can be derived using the formula (8)-(9) and letting ˜x= x(2). Applying Theorem 2, robustcontrollers can be designed by minimizingL2 performance indexγ with quatic Lyapunov functions. Moreover, as describedin subsection 3.4,A(2)(x) can also be rewritten asA(2)(x)+NA(x) with

NA(x) =

−5

∑i=1

αi x2i 0 α1x1+α3x1x3 α4x1x4 α2x1+α5x1x5

0 −5

∑i=1

αi+5x2i α6x2+α8x2x3 α9x2x4 α7x2+α10x2x5

−α11x1−α12x1x3 0

(α11+α12x2

1+α13x2

2

)0 −α13x2

1

−α16x2x5 −α14x1 −α15x1x2

(α14+α15x2

1+α16x2

2

)0

−α18x1x5 −α17x2−α19x2x5 0 0

(α17+α18x2

1+α19x2

2

)

and NA(x)x = 0. By solving condition (18) iteratively using the LMI toolbox in Matlab, we obtain the results of eachiteration as listed in Table 1. At each odd step,NA(x) is fixed. Alternatively,Q is fixed in even steps. It is clear from thetable that theγ value in each iteration is gradually improved. After eight iterations, theL2-gain performance converges toa (sub)optimal valueγ4 = 1.7173 with a 4th-degree Lyapunov function. As a comparison,γ2 = 2.1362 is obtained using2nd-degree Lyapunov functions. For both cases, regional contraintsφ1(x) = x2

1 − 0.64< 0 andφ2(x) = x22− 0.64< 0 are

included. It is worth to mention that theγ4 value derived in the first iteration is even larger thanγ2. However, the iterativescheme will reduceγ dramatically. This example clearly indicates that a suitable representation of the nonlinear system maylead to a better synthesis result.

The nonlinear control laws corresponding to quartic and quadratic cases are given by

u4(x) =−107.3641x1+52.5632x2−65.0999x21−61.4079x1x2+32.3301x2

2−33.4939x31

−8.6229x1x22−12.9297x2

1x2−23.8567x32,

u2(x) =−42.4962x1+21.2374x2.



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Iteration # 1 2 3 4 5 6 7 8

γ value 2.4918 2.0566 1.8686 1.8261 1.7741 1.7607 1.7252 1.7173

Table 1. Iterative optimization results.

Note that the nonlinear terms inu2 are omitted because their coefficients are very small (< 10−2). Using the disturbance inFig. 1(a), the simulation results of closed-loop systems based on both control laws are provided in Fig. 1(b)-(d). The smallertransient response fromu4 clearly indicates better disturbance attenuation property.

Fig. 2 displays theL2 gains derived by using quadratic and quartic Lyapunov functions respectively under differentregional constraint sizes. From the figure we can see that, the advantage of using quartic Lyapunov functions are increasingwhen the area of investigated region becomes larger.

0 1 2 3 4 5 6 70

2

4

6

8

10

12

t(sec)

Dis

turb

ance

d1d2

(a). disturbance

0 1 2 3 4 5 6 7

−60

−50

−40

−30

−20

−10

0

t(sec)

Inpu

t

u2 using 4th degree Lyapunov function

u4 using 2nd degree Lyapunov function

u2(0)

u4(0)

(b). input

0 2 4 6−0.05

0

0.05

0.1

0.15

0.2

0.25

t(sec)

e(1)

using 4th degree Lyapunov function

using 2nd degree Lyapunov function

(c). outpute1

0 1 2 3 4 5 6 7−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t(sec)

e(2)

using 4th degree Lyapunov function

using 2nd degree Lyapunov function

(d). outpute2

Fig. 1. Simulation and comparison results.

On the other hand, using the method in [28], we failed to find feasible solutions based onZ(x) = [x1 x2 x21]

T ,[x1 x2 x1x2]

T , [x1 x2 x22]

T or [x1 x2 x21 x1x2 x2

1]T . By multiplying the derivative matrices of theseZ(x)’s with any

form of A(x) andB(x), it can be shown that some of ˜z2i , i > 2 are missing from condition (18). Without these critical even

power terms, the SOS condition will become infeasible. These issues also arise when the approach of [15, 16] was con-cerned. Moreover, references [9, 10] introducedZ(x) of the same dimension asx such thatZ(x) = W(x)x, in whichW(x)



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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.5

1

1.5

2

edge length of square regional constraints φ1 and φ

2

γ

2nd

4th

Fig. 2. L2 gain comparison under different regional constraint sizes.

is a pre-specified square polynomial matrix asW(x) =

1 0 . . . 0w11(x) 1 . . . 0

......

.. ....

wn1(x) wn2(x) . . . 1

. It is not difficult to verify that the lower

triangularW(x) is invertible and det(W(x)) = 1. An obvious drawback of this method is that it cannot provide all monomialsof augmented state variables. Specifically, theW(x) simply could not generate thex2

2 term for this particular system, whichis crucial to render the synthesis condition feasible.

The second example will utilize higher degree Lyapunov functions to improve the command tracking performance ofan axi-symmetric spacecraft with two control torques [34, 39]. As displayed in Fig. 3,b3 specifies the symmetry axis ofspacecraft. The control torquesT1 andT2 actuate along theb1 andb2 axis, and span a two-dimensional plane orthogonal tothe symmetric axis.

u1

2

2u

b

1b

^

^

1n

3n

2n

3b

Fig. 3. Axi-symmetric rigid body with two controls.

In the spacecraft model,ω1,ω2,ω3 andI1, I2, I3 denote the angular velocity vector and moment of inertia with respect tothree body axes, andI1 = I2. Moreover, a sterographic projection [8] with two kinematic parametersw1 andw2 is introducedto describe relative orientation of the body frame to the inertial frame through two perpendicular rotations. By defining the



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state variables as[x1 x2 x3 x4

]T=[ω1 ω2 w1 w2

]T, a state-space model of the axi-symmetric spacecraft is derived as

x1

x2

x3

x4

=

0 am 0 0−am 0 0 0

1+x23−x2

42 x3x4 0 m

x3x41+x2

4−x23

2 −m 0

x1

x2

x3

x4

+

1 00 10 00 0

[u1

u2

], (30)

wherea= I2−I3I1

,m= ω30,u1 =T1I1,u2 =

T2I2

, and−1< a < 1 is assumed from physical consideration. We assumea= 0.5andm=−0.5rad/sec.

To eliminate the tracking error asymptotically, we will introduce an additional state variablext =τs(x3−x3d), in whichxt

reflects the accumulated effect of tracking error andτ is a pre-specified integration constant. By incorporating the dynamics ofxt into the original spacecraft model (30) and redefining output variables, the modified nonlinear system including disturbanced and controlled output variablee becomes

x1

x2

x3

x4

xt

=

0 am 0 0 0−am 0 0 0 0

1+x23−x2

42 x3x4 0 m 0

x3x41+x2

4−x23

2 −m 0 00 0 τ 0 0

x1

x2

x3

x4

xt

+

0.5 0 00 1 00 0 00 0 00 0 −τ

d1

d2

d3

+

1 00 10 00 00 0

[u1

u2

](31)

e=

0 0 0 0 10 0 0 0 00 0 0 0 0

x1

x2

x3

x4

xt

+

0 00.1 00 0.1

[u1

u2

]. (32)

The disturbance vectord includes disturbancesd1,d2 along b1 and b2 axes and the desiredx3 trajectoryx3d = d3. Thenonlinear control design objective is to minimize the effect of these disturbances on the tracking error(x3 − x3d) withreasonable control forceu.

Following the same design procedure as in the first example, quadratic and quartic Lyapunov functions can be synthe-sized with their associated controller gains respectively. The only difference lies in the selection of augmented terms. Sincethe system (31)-(32) has five states, a full augmentation maylead to significant computational burden in nonlinear controlsynthesis. To resolve the problem, only nonlinear terms relating to the statesx3 andx4 are augmented. Then we create anew state vector ˜x= [x1 x2 x3 x4 x5 x6 x7 xt ]

T by adding three augmented states, namely, ˜x5 = x23, x6 = x3x4 andx7 = x2

4.The feasible state space region is also specified in terms of spacecraft angular velocity and attitude as the intersection ofψ1(x) = x2

3+ x24−0.5< 0 andψ2(x) = x2

1+ x22−0.5< 0.

With τ = 1 and using 2nd-degree Lyapunov function, we obtainγ = 1.1798 with the control law

u2 =

[−19.8702x1+2.2617x2−123.6381x3+34.1515x4−119.3556xt

2.2617x1−15.5288x2+38.4788x3−30.7421x4+31.0434xt

].

While using 4th-degree Lyapunov function, theL2 gain reduces toγ = 0.9156 and the controller input is given by

u4 =

[−22.1944x1+3.8570x2−122.5282x3+24.1453x4−151.4306xt

3.8570x1−8.5651x2+28.5428x3−25.9567x4+35.2549xt

].

Figure 4 provides the simulation results usingu2 andu4 respectively without extraneous disturbances (d1 = d2 = 0). Thecomparison clearly shows the advantage of the controller computed using 4th-degree Lyapunov function:u4 delivers moreaccurate tracking performance and uses much less control effort.

For the third example, we would achieve regional stabilization and optimize regional performance for the saturated



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0 5 10 15 20−1.5

−1

−0.5

0

0.5

u 1

0 5 10 15 20−1

−0.5

0

0.5

time (sec)

u 2

(a). control inputs using 2nd degree Lyapunov function

0 5 10 15 20−0.4

−0.2

0

0.2

u 1

0 5 10 15 20

−0.2

−0.1

0

0.1

0.2

time (sec)

u 2

(b). control inputs using 4th degree Lyapunov function

0 5 10 15 20−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

time (sec)

x 3 & x

3d

tracking command

2nd degree

4th degree

(c). tracking performance comparison

Fig. 4. Comparison of trajectories using 2nd and 4th degree Lyapunov functions.

nonlinear system

x=

[1.5x1−0.5x2

1 x2

x2 −1

]x+

[0.30.3

]d−

[1

0.5

]sat(u) (33)

e=

1 01 10 0

x+

002

sat(u) (34)

with the saturation level ¯u= 0.25.

Using both quadratic and quartic Lyapunov functions, we will synthesize nonlinear control laws to fullfil two differentdesign objectives. 1. Without the disturbance, we want to expand the DOA as large as possible. 2. We will optimize thedisturbance attenuation for bounded disturbances of energy level range 0.25≤ s≤ 0.5. The state-space region is in the rangeof ψ1(x) = x2

1−0.25≤ 0 andψ2(x) = x22−0.49≤ 0. The state and input matrices can be augmented similar to eqns. (8)-(9)

with the sameNA(x) as described in Example 1.

To enlarge DOA in desired directions, the method in [13] willbe adopted to reshape the Lyapunov functions. Since thelevel set of higher degree Lyapunov functions may have different shape from ellipsoid, this could help to expand level setsunder the pre-selected directions. By incorporating directional constraints into Theorem 3, we can formulate an optimization



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problem

infQ>0

γ (35)

s.t.

[γ vT

ivi Q

]≥ 0, i = 1,2, . . .

and(24)− (25) are satisfied,

wherevi ∈ RNrn are vectors specifying the desired directions. Solving theoptimization problem (35) for the original system

(33) and its 2nd-degree augmented system respectively, the nonlinear state-feedback controllers can be obtained as

u2(x) = 100.40x1+14.32x2−3.16x21−4.83x1x2−1.44x2

2

u4(x) = 2.17x1−0.95x2+2.22x21−1.48x1x2+1.39x2

2.

The optimized level sets by using 2nd and 4th-degree Lyapunov functions are shown in Fig. 5. The arrows represent pre-definedvi directional vectors. It can be seen clearly that the level set of higher degree Lyapunov function has more flexibleshape and covers the ellipsoid of quadratic function entirely. As a result, the DOA estimation of the saturated system hasbeen expanded.

−2 −1 0 1 2

−2

−1

0

1

2

x1

x 2

Fig. 5. Estimated DOA as level sets of V ≤ 1: 2nd-degree Lyapunov function (dash), 4th-degree Lyapunov function (solid).

Next, we will address disturbance attenuation problem for the same nonlinear system. Using Theorem 3, both 2nd and4th-degree Lyapunov functions are sought to minimizeγ under different disturbance levels. Figure 6 illustrates the curvesof theL2 gain evolution with the increasing disturbance energy level s. It is observed the controller derived from a higherdegree Lyapunov function always performs better and can improve the disturbance attenuation performance more for largerdisturbance.

The highest tolerable disturbance level under ¯u = 0.25 is s= 0.5. In this case, the regionalL2 gains obtained from2nd and 4th-degree Lyapunov functions areγ2 = 2.1051 andγ4 = 1.6507, respectively. Also, their corresponding nonlinearcontrol laws are

u2(x) = 64.57x1−42.01x2

u4(x) = 39.77x1−15.48x2−4.38x21+2.13x1x2−3.59x2

2+2.71x31+0.79x2

1x2+2x1x22+0.18x3

2.



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0.25 0.3 0.35 0.4 0.45 0.50.8

1

1.2

1.4

1.6

1.8

2

2.2

s

γ

2nd

4th

Fig. 6. L2 gain comparison under different disturbance energy levels.

Again, the nonlinear terms inu2 (with coefficients smaller than 10−3) are omitted. The nonlinear simulation was conductedusing a disturbance input

d(t) =

{2(t −1.5), 1.5sec≤ t < 2sec2(2.5− t), 2sec≤ t ≤ 2.5sec.

Fig. 7 illustrates the simulation results using the both controllers. With a smaller performance value, these results clearlyshow that a higher degree Lyapunov function could improve the disturbance attenuation capability without significant in-creasing the control input magnitude.

6 Concluding RemarksIn this paper, we developed a novel nonlinear control designtechnique by combining power transformation and SOS

programming to synthesize higher degree Lyapunov functions for polynomial nonlinear systems with or without saturation.The existence of higher degree Lyapunov functions has also been proved for quadratically stabilizable polynomial nonlinearsystems. It has been shown that power transformation provides a general procedure to augment polynomial nonlinear systemsto their higher order equivalence and facilitate the optimization of controlled performance. Consequently, the nonlinearcontrol synthesis conditions are formulated as polynomialinequalities in convex forms and can be solved effectively bySOS programming. Furthermore, taking the advantage of flexible form of the augmented system matrixA(r)(x)+NA(x), therobust performance or DOA estimation can be improved iteratively.

Comparing with existing nonlinear control approaches thatdeal with higher degree Lyapunov functions, the proposedmethod has three advantages: First, the higher order polynomial optimization problems are convex and lead to computa-tionally effective synthesis conditions for polynomial nonlinear systems. By augmenting nonlinear dynamics, the design ofhigher degree Lyapunov functions for original nonlinear systems can be recast to the search of quadratic Lyapunov functionsfor augmented systems. Second, different from existing approaches [9,16,28], the proposed method provides the most exten-sive transformed state vectors. This new state vector includes all possible combinations of the original states withina givendegree and could lead to better control synthesis results. In fact, the performance obtained using the proposed method isguaranteed to be no worse than the ones derived from existingmethods. Third, it is conceivable that different representationscould lead to different results and the given system matrices may not be the most appropriate ones. Using a polynomial an-nihilator, the proposed method is capable of selecting a suitable system representation for further improvement of controlledperformance.

Nevertheless, the proposed approach also has the limitation as other state-dependent approaches. This is due to the factthat zTH(x)z> 0 impliesxTH(x)x > 0 but the converse is not true. Another potential drawback ofthe proposed methodis the large number of state variables and multipliers resulted from the augmentation of nonlinear systems. Hence thecomputational costs could be high. Some simple solutions toaddress this issue include using less number of multipliersand restricting the form of multipliers. By doing so, the computing time can be reduced significantly with an acceptablesacrifice of optimalγ value. With increased computational power and developmentof more efficient algorithms tailoredto specific class of nonlinear systems, it is expected that these nonlinear control design methods would be very useful topractical problems in the future.



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0 1 2 3 4 5 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

t(sec)

e(1)

using 4 th degree Lyapunov function

using 2 nd degree Lyapunov function

(a). outpute1

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

t(sec)

e(2)



(b). outpute2

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

t(sec)

Inpu

t



(c). inputu

Fig. 7. Comparison of output and control input trajectories.

AcknowledgementsThe authors would like to acknowledge the financial support by the NSF Grant CMMI-0800044.

References[1] Ahmadi, A.A., and Parrilo, P.A., 2001. “Converse Results on Existence of Sum Of Squares Lyapunov Functions”. In

50th IEEE CDC Conf. Dec. Control, pp. 6516-6521.[2] Barkin, A., and Zelentsovsky, A., 1983. “Method of PowerTransformations for Analysis and Stability of Nonlinear

Control Systems”. Syst. Contr. Lett.,3, pp. 303-310.[3] Bernstein, D.S., and Michel, A.N., 1995. “A Chronological Bibliography on Saturating Actuators”. Int. J. Robust

Nonlinear Control,5, pp. 375-380.[4] Brockett, R.W., 1973. “Lie algebras and Lie groups in control theory”.Geometric Methods in System Theory(Mayne,

D.Q. and Brockett, R.W. Eds.), Dordrecht, Reidel, pp. 213-225.[5] Castelan, E.B., Tarbouriech, S., and Queinnec, I., 2008. “Control Design for A Class of Nonlinear Continuous-Time

Systems”. Automatica,44(8), pp. 2034-2039.[6] Chesi, G., Garulli, A., Tesi, A., and Vicino, A., 2003. “Homogeneous Lyapunov Functions for Systems with Structured

Uncertainties”. Automatica,39(6), pp. 1027-1035.[7] Choi, M.D., Lam, T.Y., and Reznick, B., 1995. “Sums Of Squares of Real Polynomials”. Symp. in Pure Math.,58, pp.

103- 126.[8] Conway, J.B., 1978. “Functions of One Complex Variable”. Springer, New York, NY.[9] Ebenbauer, C., and Allgower, F., 2006. “Analysis and Design of Polynomial Control Systems using Dissipation In-

equalities and Sum Of Squares”. Comput. Chem. Engineering,30, pp. 1590-1602.



Acce

pted

Man

uscr

ipt N

ot C

opye

dite

d

[10] Ebenbauer, C., Renz, J., and Allgower, F., 2005. “Polynomial Feedback and Observer Design using NonquadraticLyapunov Functions”. in Proc. Joint IEEE Conf. Dec. Contr. &Eur. Contr. Conf., pp. 7587-7592.

[11] Franze, G., Famularo, D., and Casavola, A., 2012. “Constrained Nonlinear Polynomial Time-Delay Systems: A Sum-Of-Squares Approach to Estimate the Domain of Attraction”.IEEE Trans. Autom. Control,57(10), pp. 2673-2679.

[12] Haddad, W.M., Fausz, J.L., and Chellaboina, V.-S., 1999. “Nonlinear Controllers for Nonlinear Systems with InputNonlinearities”. J. Franklin Institute,336, pp. 649-664.

[13] Hu, T., Lin, Z., and Chen, B., 2002. “An Analysis and Design Method for Linear Systems Subject to Actuator Saturationand Disturbance”. Automatica,38(2), pp. 351-359.

[14] Hu, T., Teel, A., and Zaccarian, L., 2006. “Stability and Performance for Saturated Systems via Quadratic and Non-quadratic Lyapunov Functions”. IEEE Trans. Autom. Control, 51(11), pp. 1133-1168.

[15] Ichihara, H., 2008. “State Feedback Synthesis for Polynomial Systems with Bounded Disturbances”. in Proc. IEEEConf. Dec. Contr., pp. 2520-2525.

[16] Ichihara, H., 2009. “Optimal Control for Polynomial Systems using Matrix Sum Of Squares Relaxations”. IEEE Trans.Autom. Control,54(5), pp. 1048-1053.

[17] Jarvis-Wloszek, Z., and Packard, A., 2002. “An LMI Method to Demonstrate Simultaneous Stability using Non-Quadratic Polynomial Lyapunov Functions”. in Proc. IEEE Conf. Dec. Contr., pp. 287-292.

[18] Kapila, V., and Grigoriadis, K.M. (eds.), 2002.Actuator Saturation Control. Marcel Dekkar, New York, NY.[19] Krstic, M., Kanellakopoulos, I., and Kokopotic, P.V.,1995.Nonlinear and Adaptive Control Design. John Wiley and

Sons, New York, NY.[20] Lin, Y., and Sontag, E.D., 1991. “A Universal Formula for Stabilization with Bounded Controls”. Syst. Contr. Lett.,

16(6), pp. 393-397.[21] Lu, W.-M., and Doyle, J.C., 1995. “H∞ Control of Nonlinear Systems: A Convex Characterization”.IEEE Trans.

Autom. Control,40(9), pp. 1668-1675.[22] Narimani M., and Lam, H., 2010. “SOS-Based Stability Analysis of Polynomial Fuzzy-Model-Based Control Systems

via Polynomial Membership Functions”. IEEE Trans. Fuzzy Systems,18(5), pp. 862-871.[23] Papachristodoulou, A., Peet, M.M., and Lall. S.K., 2009. “Analysis of Polynomial Systems with Time Delays via the

Sum of Squares Decomposition”. IEEE Trans. Autom. Control,54(5), pp. 1058-1064.[24] Parrilo, P., 2000.Structured Semidefinite Programs and Semi-Algebraic Geometry Methods in Robustness and Opti-

mization. Ph.D. dissertation, California Institute of Technology,Pasadena, CA.[25] Peet, M.M., and Papachristodoulou, A., 2012. “A Converse Sum Of Squares Lyapunov Result with A Degree Bound”.

IEEE Trans. Autom. Control,57(9), pp. 2281-2293.[26] Prajna, S., 2005.Optimization-Based Methods for Nonlinear and Hybrid Systems Verification. Ph.D. Dissertation,

California Institute of Technology, Pasadena, CA.[27] Prajna, S., Papachristodoulou, A., Seiler, P., and Parrilo, P.A., 2004.SOSTOOLS: Sum Of Squares Optimization Toolbox

for MATLAB, ver. 2. URLhttp://www.cds.caltech.edu/sostools.[28] Prajna, S., Papachristodoulou, A., and Wu, F., 2004. “Nonlinear Control Synthesis by Sum Of Squares Optimization:

A Lyapunov-Based Approach”. in Proc. Asian Contr. Conf., pp. 157-165.[29] Prajna, S., Parrilo, P.A., and Rantzer, A., 2004. “Nonlinear Control Synthesis by Convex Optimization”. IEEE Trans.

Autom. Control,49(2), pp. 310-314.[30] Summers, E., Chakraborty, A., Tan, W., Topcu, U., Seiler, P., Balas, G., and Packard, A., 2013. “Quantitative Local

L2-Gain and Reachability Analysis for Nonlinear Systems”. Int. J. Robust Nonlinear Control,23(10), pp. 1115-1135.[31] Tan, W., and Packard, A., 2008. “Stability Region Analysis using Polynomial and Composite Polynomial Lyapunov

Functions and Sum-Of-Squares Programming”. IEEE Trans. Autom. Control,53(2), pp. 565-571.[32] Tarbouriech, S., and Garcia, G.(eds), 1997.Control of Uncertain Systems with Bounded Inputs. Springer, London, UK.[33] Topcu, U., and Packard, A., 2009. “Linearized Analysisversus Optimization-Based Nonlinear Analysis for Nonlinear

Systems”. in Proc. of Amer. Contr. Conf., pp. 790-795.[34] Tsiotras, P., and Longuski, J.M., 1994. “Spin-axis Stabilization of Symmetric Spacecraft with Two Control Torques”.

Syst. Contr. Lett.,23, pp. 395-402.[35] Valmorbida, G., Tarbouriech, S., and Garcia, G., 2013.“Design of Polynomial Control Laws for Polynomial Systems

Subject to Actuator Saturation”. IEEE Trans. Autom. Control, 58(7), pp. 1758-1770.[36] Vandenberghe, L., and Boyd, S.P., 1996. “Semidefinite Programming”. SIAM Review,38, pp. 49-95.[37] Wu, F., and Prajna, S., 2005. “SOS-based Solution Approach for Polynomial LPV System Analysis and Synthesis

Problems”. Int. J. Control,78(8), pp. 600-611.[38] Zelentsovsky, A.L., 1994. “Nonquadratic Lyapunov Functions for Robust Stability Analysis of Linear Uncertain Sys-

tems”. IEEE Trans. Autom. Control,39(1), pp. 135-138.[39] Zheng, Q., and Wu, F., 2009. “NonlinearH∞ Control Designs with Axi-Symmetric Rigid Body Spacecraft”. AIAA J.

Guidance Nav. Control,32(3), pp. 850-859.[40] Zheng, Q., and Wu, F., 2009. “Stabilization of Polynomial Nonlinear Systems using Rational Lyapunov functions”. Int.



Acce

pted

Man

uscr

ipt N

ot C

opye

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d

J. Control,82(9), pp. 1605-1615.



control of polynomial nonlinear systems using higher degree lyapunov functions

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