# 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 WuDepartment of Mechanical and Aerospace Engineering

North Carolina State UniversityRaleigh, NC 27695

E-mail: syang9@ncsu.edu, fwu@eos.ncsu.edu

ABSTRACTIn this paper, we propose a new control design approach for polynomial nonlinear systems based on higher

degree Lyapunov functions. To derive higher degree Lyapunov functions and polynomial nonlinear controllerseffectively, the original nonlinear systems are augmented under 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 and L2 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 regional L2 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 the most 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

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

which is a bilinear matrix inequality of the unknowns V (x) and F(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 vector Z(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 vector Z(x) =W (x)x of the same dimensionas state variables and W (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 of Z(x) and focusing on optimal control anddisturbance rejection problems. A non-iterative two-step design procedure was also proposed with the first step aiming tofind a feasible solution, while the second step utilizing polynomial annihilators to reduce the conservatism in the first step.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 vector Z(x). Besides, the resulting performance of the control design would depend onthe selection of Z(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-feedback L2 gain control condition was derived in [21]based on a pair of positive definite matrix functions P(x),Q(x). Unfortunately, it is difficult, if not impossible, to specifythe functional form of P(x) such that V (x)/x = 2xT P(x) except for the trivial case when P(x) is a constant matrix. There-fore, the proposed synthesis condition does not naturally lead to computationally tractable solution algorithms for nonlinearinduced L2-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 the DK iteration. The iterative algo-rithm, however, may not converge to its global minimum and the result could be conservative. Since the set of polynomialLyapunov functions V (x) and stabilizing control laws K(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 only limitedworks 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 Artsteins theorem and derived a nonlinear controlformula for smooth nonlinear systems with bounded control input. Nevertheless, the construction of such a stabilizing con-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 reformulating quadratic stabilization conditions 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 construction of quadratic Lyapunov functions for nonlinear saturation controlproblems.

The concept of power transformation was first introduced for linear 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 uncertain linearsystems. 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 have more 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 nonlinear systems 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 a L2 performance index, the controlledperformance can be optimized using higher degree Lyapunov functions. The derived solvability conditions of nonlinearstabilization and L2-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 nonlinear systems with enlarged domain of attraction (DOA) and optimized

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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regional L2 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 of theoriginal state vector x. 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 and Z+ denotes the set ofpositive integers. Rmn is the set of real mn matrices. We use Snn to denote real, symmetric nn matrices, and Snn+ forpositive-definite matrices. The n n identity matrix is denoted by In. For a real matrix M, we define a Hermitian operatorHe{} as He{M}= M+MT . Given two matrices A Rmn and B Rpq, their Kronecker product is of dimension pmqnand defined as

AB =

a11B a1nB...

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am1B amnB

.In large symmetric matrix expressions, terms denoted as will be induced by symmetry. For two integers k1,k2, k1 < k2,we denote I[k1,k2] = {k1,k1 + 1, ,k2}. The space of square integrable functions is denoted by L2, that is, for any signalxL2, x2 :=

(

0 xT (t)x(t)dt

) 12

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

k r of the state vector x. Then x(r) has the dimension of Nrn = rk=1 Nkn . As an example, the monomials of a two-variablevector with degree no more than 2 are

x(2) =[x1 x2 x

21 x1x2 x

22]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 of x(2) vector

1(x) = x3 x21 = 0, 2(x) = x4 x1x2 = 0, 3(x) = x5 x22 = 0,4(x) = x2x4 x1x5 = 0, 5(x) = x2x3 x1x4 = 0, 6(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 implies f (x) 0 for any x Rn. Given the polynomial of degree 2r,the SOS decomposition condition is equivalent to the existence of a positive semi-definite matrix G such that

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

where M(x) is a monomial vector of x with all components of degree no more than r and G is called the Gram matrix [7].Therefore, testing whether a given polynomial p(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

i(x(r))i(x(r)) [x],

in which i(x(r)) are multipliers to enforce the coupling constraints i(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 state x Rn, the disturbance d Rnd , and the control input u Rnu . All state variables are assumed measurableand the state-space entries are polynomial functions of the state x. It is assumed that the polynomial nonlinear syste...