# A Sum of Squares-Based Approach to Robust Nonlinear MixedH2/H∞ Output-Feedback Control for a Class of Uncertain Polynomial Systems

Authors: Huihui Bai; Lijun Wang; Qingchun Jiao
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper considers the robust nonlinear mixedH2/H∞ output-feedback control problem for a class of uncertain polynomial systems. Generally, the solvable conditions of such nonlinear output-feedback control problems are nonconvex, whose computations are challenging. By using the semidefinite programming relaxation technique based on sum of squares (SOS) decomposition, the solvable conditions of above control problem are formulated in terms of state-dependent linear matrix equality (LMI), which can be effectively solved. This conversion effectively overcomes the computational difficulties caused by the non-convexity of output-feedback H∞ control design for nonlinear systems. Furthermore, the state-observer and the controller can be designed simultaneously through a single-step SOS condition and constructed in a simple analytical form, thus reducing the computational complexity in a certain degree. In the simulation, the nonlinear mass-spring-damper system is considered to illustrate the effectiveness and feasibility of the proposed approach. The results show that the proposed method not only guarantees the stability of the system, but also has good transient performance and robust performance.


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## Body

## 1. Introduction

The mixedH2/H∞ control design has attracted considerable attention in recent years, because it combines the merits of both the H2 guaranteed cost control and the H∞ robust control. Mixed H2/H∞ control problems for linear systems have been extensively studied by many researchers [1–7]. The results of their works are formulated in terms of LMIs, which can be solved easily by efficient numerical algorithm. However, in practical physical systems, the plants are always nonlinear and the state variables might not be frequently measurable, and hence it is more meaningful to carry out research on the nonlinear output-feedback control design. However, it is well known that the nonlinear output-feedback control problems are generally related to some cross-coupled Hamilton-Jacobi-Isaacs equations or inequalities, whose solutions are very difficult to solve, because of the non-convex nature.A recent computational relaxation technique based on SOS provides a promising new way for the analysis and synthesis of nonlinear control problems. Under the framework of SOS, we can try to recast the nonlinear control problems as SOS convex programming problems, which may effectively overcome the computational challenges existing in traditional nonlinear control problems.As a powerful and promising technique, SOS decomposition has also been used in the analysis and synthesis of nonlinear control problems. For example, in [8–10], SOS technique is applied to investigate the state-feedback H2 and H∞ control synthesis problems for a class of polynomial nonlinear systems. In [11], the simultaneous stabilization and robust performance control problems are considered for a class of polynomial nonlinear systems by using SOS technique. In [12], an SOS-based approach is presented for the guaranteed cost control of polynomial discrete fuzzy systems. In [13], an SOS-based robust H∞ fuzzy dynamic output feedback control problem is discussed for the discrete-time nonlinear networked control systems. And in [14–16], based on an iterative SOS approach, a nonlinear dynamic output-feedback control scheme and a nonlinear robust static output feedback control scheme are proposed for a class of affine nonlinear polynomial systems.The resulting conditions corresponding to nonlinear output-feedback control problems are difficult to solve, since these conditions are generally non-convex [13–16]. Therefore, it is meaningful to formulate some convex and computationally tractable conditions for nonlinear output-feedback control problems, and thus can effectively overcome the numerical difficulty in solving the non-convex HJIs.In practice, we often encounter some control systems, such as the attitude dynamic system of flexible spacecraft [17, 18], the nonlinear mass-spring-damper system [14], and the Rossler chaotic systems [19]. All those nonlinear systems have some features in common, that is, their mathematical models can be formulated as the form of linear-like state-space equations and the resulting coefficient matrices are only related to partial states which are available. Furthermore, not only the nonlinearities but also the uncertainties are always encountered in practice. In recent years, to address the disturbances, uncertainties, saturation, and the other input nonlinearity, the intelligent-based control methods (such as adaptive-based control method [20, 21], fuzzy-based control method [22, 23], and neural networks-based control method [24]) and the nonlinear robust control methods (such as nonlinear H2/H∞ control method [25–27]) are usually adopted. Motived by the facts above, for those kinds of nonlinear systems, this paper considers the robust nonlinear mixed H2/H∞ output-feedback control design problem using SOS technique. This control design approach presents that the nonlinear mixed H2/H∞ output-feedback control problem is formulated as an SOS convex programming problem, and an output-feedback controller is derived in terms of a convex and tractable SOS conditions which can be directly solved by SOS decomposition. As a result, this conversion effectively overcomes the difficulties in constructing Lyapunov functions and implementing numerical computation caused by the non-convexity of output-feedback control design. Furthermore, the state-observer and the controller can be designed simultaneously through a single-step SOS condition and constructed in a simple analytical form, thus reducing the computational complexity in a certain degree.The rest of this paper is organized as follows. In Section2, the problem description is introduced. In Section 3, the sufficient conditions to robust nonlinear mixed H2/H∞ output-feedback control problem are derived. In Section 4, a numerical example is given to illustrate the effectiveness and advantages of the proposed approach. Finally, Section 5 draws the conclusions.The notations used are standard in this paper and are shown in Table1.Table 1
Notations used in this paper.
NotationDefinitionRnThe set ofn-dimensional real vectorsRm×n (R+m×n)The set of allm×n real (positive-define) matricesRxThe set of all real polynomial function of vectorx∈RnRxp×q (Rx+p×q)The set of all realp×q polynomial (positive-define) matrices of vector x∈Rn∑sosThe set of all SOS polynomials.I and 0The identity matrix and zero matrix with proper dimensions∗The symmetric entries in a symmetric matrixSymAAT+A


## 2. Problem Description

Consider the following uncertain polynomial system:(1)x˙=Ax1+ΔAx1x+B1x1+ΔB1x1u+B2x1w,y=x1=C0x,C0=I0,where x=x1Tx2TT∈Rn is the state vector with the measurable substate x1∈Rn0 and the immeasurable substate x2∈Rn−n0; u∈Rp is the control input vector; w∈Rq is the disturbance input vector; y∈Rn0 is the measured output vector; Ax1, B1x1, and B2x1 are the known polynomial matrices; ΔAx1 and ΔB1x1 denote the time-varying parameter uncertainties and satisfy the following assumption.Assumption 1.
(2)ΔAx1ΔB1x1=Ex1ΔtFx1F1x1,where Ex1, Fx1, and F1x1 are known polynomial matrices with appropriate dimensions that characterise the structures of the uncertainties; Δt∈Rp0×q0 is an unknown time-varying matrix satisfying ΔTtΔt≤I.
Define the controlled output equations as(3)z1=C1x1x+D1x1u,(4)z2=C2x1x+D2x1u,where z1∈Rn1, z2∈Rn2, Cix1, and Dix1 (i=1,2) are known polynomial matrices with appropriate dimensions.
Respectively introduce theH∞ and H2 performance measures as(5)J∞=∫0∞z1z1T−γ2wwTdt,and(6)J2=∫0∞z2z2Tdt,where γ is a positive scalar representing a desired H∞ disturbance attenuation level.
The robust nonlinear mixedH2/H∞ output-feedback control problem to be addressed in this paper is specifically described as follows:Problem 1.
(Robust nonlinear mixedH2/H∞ output-feedback control problem) For the system in (1), (3) and (4) our objective is to design a robust nonlinear output-feedback controller such that the following conditions are satisfied for all admissible uncertainties under Assumption 1:(i)
The resulting closed-loop system is asymptotically stable at zero equilibrium withw=0(ii)
TheH2 performance guarantees J2≤J2∗ under w=0, where J2∗ is a positive constant representing the guaranteed cost(iii)
TheL2-gain from the disturbance input w∈L20,∞ to the controlled output z is less than a given positive scalar γ, i.e., J∞<0 under zero initial condition.Remark 1.
In fact, the termsH2 and H∞ mentioned in Problem 1 are extensively defined for linear systems. In nonlinear case, these two notations are strictly not precise, as they should be called as “guaranteed cost” and L2-gain, respectively. In this paper, we refer to the proposed problem as H2/H∞ control problem, as it is the common practice in the literature nowadays [13–18, 25–27]. Therefore, for the sake of rigor, a note is made herein.
Before further analysis, we make some preliminaries as follows.Definition 1.
(see [28]). A multivariate polynomial px=px1,…,xn is an SOS polynomial, if there exist polynomials f1x,…,fmx such that(7)px=∑i=1mfi2x.
Therefore, if a polynomialpx can be decomposed into the form in (7), i.e, px∈∑sos, we have px≥0 for all x∈Rn. However, the converse is generally not true, that is to say a polynomial being an SOS is a sufficient condition for its non-negativity. Nevertheless, some researchers have proved that the gap between the non-negative polynomial and the SOS polynomial is not significant and they are even equivalent for some special cases [19].Lemma 1.
(see [29]). Let E and F be real matrices of appropriate dimensions. For any matrix Δt satisfying ΔTtΔt≤I and a positive scalar ε, we have(8)EΔtF+FTΔTtET≤εEET+ε−1FTF.Lemma 2.
(see [30]). For any x,y∈Rn and any positive definite matrix T∈Rn×n, the following inequality holds(9)2xTy≤xTTx+yTT−1y.Lemma 3.
(see [8]). Let Fx∈Rxn×n be a symmetric polynomial matrix of degree 2d in x∈Rn and Zx be a column vector whose entries are all monomials in x∈Rn with degree no greater than d, and consider the following conditions:(i)
Fx≥0, x∈Rn(ii)
vTFxv∈Φsos, where v∈Rn(iii)
There exists a positive semidefinite matrixQ satisfying vTFxv=v⊗ZxTQv⊗Zx, where ⊗ denotes the Kronecker product.
Then,iii⇔ii⇒i.Lemma 4.
(Generalized S-procedure) (see [17]). Given gixi=1m∈Rx, if there exist sixi=1m∈∑sos such that(10)g0x−∑i=1msixgix∈∑sos,then(11)x∈Rn|g1x≥0,…,gmx≥0⊆x∈Rn|g0x≥0.


## 3. Robust Nonlinear MixedH2/H∞ Output-Feedback Control Design

In this paper, for the system in (1), (3) and (4) an observer-based output feedback controller of the following form is considered:(12)x∧˙=Ax1x∧+B1x1u+Lx1,x∧y−C0x∧,(13)u=Kx1,x∧x∧,where x∧∈Rn is the estimate of x, Lx1,x∧, and Kx1,x∧ are, respectively, the observer gain matrix and controller gain matrix to be designed.Combining the system in (1), (3) and (4) with the controller in (12) and (13) produces the resulting closed-loop system:(14)ζ˙=Aζx1,x∧+ΔAζx1,x∧ζ+Bζx1w,z1=C1ζx1,x∧ζ,z2=C2ζx1,x∧ζ,where(15)ζ=xx−x^=xeC1ζx1,x^=C1x1+D1x1Kx1,x^−D1x1Kx1,x^,C2ζx1,x^=C2x1+D2x1Kx1,x^−D2x1Kx1,x^,Bζx1=B2Tx1B2Tx1T,Aζx1,x^=Ax1+B1x1Kx1,x^−B1x1Kx1,x^0Ax1−Lx1,x^C0,ΔAζx1,x^=E0x1ΔF0x1,x^,E0x1=Ex1Ex1,F0x1,x^=Fx1+F1x1Kx1,x^−F1x1Kx1,x^.Obviously, the task of the control problem is converted to design an output feedback controller of the form in (12) and (13) such that the closed-loop system in (14) satisfies the conditions i-iii in Problem 1 for all admissible uncertainties under Assumption 1.On the basis of Lyapunov stability theory, a solvable criterion for Problem1 is first established as follows.Lemma 5.
Consider the uncertain system in (1), (3) and (4). For the given scalars γ>0 and ε0>0, if there exist polynomial matrices PxJ∈RxJ+2n×2n, Lx1,x∧∈Rx1,x∧n×n0 ,and Kx1,x∧∈Rx1,x∧p×n, such that the closed-loop system in (14) satisfies the following inequality:(16)SymPxJAζx1,x∧+∑i∈J∂PxJ∂x1iAix1x1∗∗∗∗∗C1ζx1,x∧−I∗∗∗∗C2ζx1,x∧0−I∗∗∗ε0E0Tx1PxJ00−ε0I∗∗F0x1,x∧000−ε0I∗BζTx1PxJ0000−γ2I<0,where xJ=x1i1x1i2…x1ir, x1ii=i1,i2,…,ir denotes the i-th entry of x1, J=i1,i2,…,ir represents the row indices of ΔAx1B1x1+ΔB1x1B2x1 whose corresponding row is equal to zero; Aix1 is the i-th row of Ax1.Then, Problem 1 is solvable with the controller of the form in (12) and (13), and the guaranteed cost is given by(17)J2∗=ζT0PxJ0ζ0.Proof.
Choose Lyapunov function for the closed-loop system in (14) as(18)Vζ=ζTPxJζ.
DifferentiatingVζ along the trajectories of (14) and using Lemma 1 yield(19)V˙ζ≤ζTwTΦ0x1,x∧∗BζTx1PxJ−γ2Iζw+γ2wTw−ζTC1ζTx1,x∧C1ζx1,x∧+C2ζTx1,x∧C2ζx1,x∧ζ,Φ0x1,x∧=SymPxJAζx1,x∧+∑i∈J∂PxJ∂x1iAix1x1+C1ζTx1,x∧C1ζx1,x∧+C2ζTx1,x∧C2ζx1,x∧+ε0PxJE0x1E0Tx1PxJ+ε0−1F0Tx1,x∧F0x1,x∧.
From condition (16) and Schur complement Lemma, we have(20)V˙ζ≤−ζTC1ζTx1,x∧C1ζx1,x∧+C2ζTx1,x∧C2ζx1,x∧ζ+γ2wTw.
Whenw=0, from (20), we obtain(21)V˙ζ≤−ζTC2ζTx1,x∧C2ζx1,x∧ζ<0,which implies the asymptotic stability of the closed-loop system in (14).
Furthermore, by integrating both sides of the inequality (21) from t=0 to t=∞, we obtain(22)J2=∫0∞z2z2Tdt≤V0=ζ0T0Px0Jζ0=J2∗.
Whenζ0=0, w∈L20,∞, we have(23)J∞=∫0∞zTz−γ2wTwdt=∫0∞zTz−γ2wTw+V˙ζdt−Vζ∞≤∫0T0zTz−γ2wTw+V˙ζdt.
From inequality (20), we obtain(24)J∞≤−ζTC2ζTx1,x∧C2ζx1,x∧ζ<0,which implies that the L2-gain from the disturbance input w to the controlled output z of the closed-loop system in (14) is less than γ.
Obviously, the condition in (16) is difficult to solve, because the cross couplings among matrix variables PxJ, Lx1,x∧ and Kx1,x∧ make it non-convex. Therefore, in order to analytically determine the solution for (16), the main purpose in the following procedures is to convert the non-convex condition to a convex one. In addition, to simplify the design problem, in the following discussion, we select PxJ as a positive-definite diagonal matrix.Theorem 1.
Consider the uncertain system in (1) (3) and (4). For the given scalars γ>0 and ε0>0, there exists an observer-based output feedback controller of the form in (12) and (13) such that the closed-loop system in (14) satisfies the conditions i-iii in Problem 1, if there exist polynomial matrices P1xJ∈RxJ+n×n, P2xJ∈RxJ+n×n, X1x1,x∧∈Rx1,x∧p×n, and X2x1,x∧∈Rx1,x∧n×n0, such that the following inequality holds(25)Ωx1,x∧=Ω1x1,x∧∗∗Ω2x1,x∧−P1xJ0Ω30−P1xJ<0,Ω1x1,x∧=Ω11x1,x∧∗∗∗∗∗∗0Ω22x1,x∧∗∗∗∗∗C1x1P1xJ+D1x1X1x1,x∧0−I∗∗∗∗C2x1P1xJ+D2x1X1x1,x∧00−I∗∗∗ε0ETx1ε0ETx1P2xJ00−ε0I∗∗Fx1P1xJ+F1x1X1x1,x∧0000−ε0I∗B2Tx1B2Tx1P2xJ0000−γ2I,Ω11x1,x∧=SymAx1P1xJ+B1x1X1x1,x∧−∑i∈J∂P1xJ∂x1iAix1x1,Ω22x1,x∧=SymP2xJAx1−X2x1,x∧C0+∑i∈J∂P2xJ∂x1iAix1x1,Ω2x1,x∧=X1Tx1,x∧−B1Tx10−D1Tx1−D2Tx100−F1Tx1,Ω3=0I00000.
In this case, the corresponding observer and controller gain matrices are respectively given by(26)Kx1,x∧=X1x1,x∧P1−1xJ,and(27)Lx1,x∧=P2−1xJX2x1,x∧.
Moreover, the guaranteed cost is given by(28)J2∗=xT0P1−1xJ0x0+eT0P2xJ0e0.Proof.
ChoosingPxJ=diagP1−1xJ,P2xJ, from (16), we have(29)Ω¯11x1,x∧∗∗∗∗∗∗−KTx1,x∧B1Tx1P1−1xJΩ¯22x1,x∧∗∗∗∗∗C1x1+D1x1Kx1,x∧−D1x1Kx1,x∧−I∗∗∗∗C2x1+D2x1Kx1,x∧−D2x1Kx1,x∧0−I∗∗∗ε0ETx1P1−1xJε0ETx1P2xJ00−ε0I∗∗Fx1+F1x1Kx1,x∧−F1x1Kx1,x∧000−ε0I∗B2Tx1P1−1xJB2Tx1P2xJ0000−γ2I<0,Ω¯11x1,x∧=SymP1−1xJAx1+B1x1Kx1,x∧+∑i∈J∂P1−1xJ∂x1iAix1x1,Ω¯22x1,x∧=SymP2xJAx1−Lx1,x∧C0+∑i∈J∂P2xJ∂x1iAix1x1.
Multiplying both sides of (29) by diagP1xJ,I,I,I,I,I,I and denoting Kx1,x∧P1xJ=X1x1,x∧, P2xJLx1,x∧=X2x1,x∧ yield(30)Ω1x1,x∧+Ω2Tx1,x∧P1−1xJΩ3+Ω3TP1−1xJΩ2x1,x∧<0.
Using Lemma2 and choosing T=P1xJ, we have(31)Ω2Tx1,x∧P1−1xJΩ3+Ω3TP1−1xJΩ2x1,x∧≤Ω2Tx1,x∧P1−1xJΩ2x1,x∧+Ω3TP1−1xJΩ3.
Then, by Schur complement Lemma, (30) and (31) lead to the condition (25).
In Theorem1, the condition in (25) is a state-dependent LMI. By Lemma 3, it can be formulated as the corresponding SOS constraint that can be solved by SOS decomposition directly. Generally speaking, if the corresponding SOS constraint is solvable, a global nonlinear controller will be obtained. However, in many cases, it is too restrictive to synthesize a globally stable controller. Moreover, in a restricted region, a local controller will often perform better than a global controller. In such situations, in order to search for a local controller, the state-region constraints should be added into the solvable conditions in Theorem 1. For the convenience of design, we describe the state-region as follows:(32)BR=BR1∪BR2,BR1=x1∈Rm|gi≥0,i=1,2,…,m1,BR2=x∧=x∧1x∧2∈Rn|gm1+j≥0,j=1,2,…,m2,where gi and gm1+j are the polynomial functions of x1 and x∧, respectively.
Then, on the basis of SOS technique and the generalized S-procedure, the state-dependent LMI in (25) with the state-region constraint in (32) is formulated as the following SOS constraints which can be directly solved by SOS tools.Theorem 2.
Supposex1∈BR1 and x∧∈BR2. For the given scalars γ>0 and ε0>0, the polynomial functions ε1xJ>0, ε2xJ>0 and ε3x1,x∧>0, there exists an observer-based output feedback controller of the form in (12) and (13) such that the closed-loop system in (14) satisfies the conditions i-iii in Problem 1, if there exist polynomial matrices P1xJ∈RxJ+n×n, P2xJ∈RxJ+n×n, X1x1,x∧∈Rx1,x∧p×n, X2x1,x∧∈Rx1,x∧n×n0 and SOS polynomial functions μ1ixJ, μ2ixJ, κjx1,x∧i=1,2,…,m1,j=1,2,…,m1+m2 such that(33)υ1TP1xJ−ε1xJIυ1−ϑ1TΛ1ϑ1∈∑sos,υ2TP2xJ−ε2xJIυ2−ϑ2TΛ2ϑ2∈∑sos,−υ3TΩx1,x∧+ε3x1,x∧Iυ3−ϑ3TΛ3ϑ3∈∑sos,Λ1=diagμ11x1g1,μ12x1g2,…,μ1m1x1gm1,Λ2=diagμ21x1g1,μ22x1g2,…,μ2m1x1gm1,Λ3=diagκ1x1,x∧g1,κ2x1,x∧g2,…,κm1+m2x1,x∧gm1+m2,where υ1∈Rn, υ2∈Rn, υ3∈R2n+n1+n2+p0+q0+q, ϑ1 and ϑ2 are vectors consisting of any m1 elements of υ1 and υ2, ϑ3 is a vector consisting of any m1+m2 elements of υ3. Furthermore, the corresponding observer-based controller gain matrices are given by (26) and (27) and the guaranteed cost is given by (28).Proof.
The following equation denotes that(34)Ψυ3,ϑ3,x1,x∧=−υ3TΩx1,x∧+ε3x1,x∧Iυ3−ϑ3TΛ3ϑ3,ϑ3=ϑ31ϑ32…ϑ3,m1+m2T.
We define(35)sυ3,ϑ3,x1,x∧=υ3jTκjx1,x∧υ3j=υ3j2κjx1,x∧≥0.
From condition (33), we have(36)Ψυ3,ϑ3,x1,x∧=−υ3TΩx1,x∧+ε3x1,x∧Iυ3−∑j=1m1+m2υ3jTκjx1,x∧gjυ3j=−υ3TΩx1,x∧+ε3x1,x∧Iυ3−∑j=1m1+m2sυ3,ϑ3,x1,x∧gj≥0.
Then, by Lemmas3 and 4, for x1∈BR1 and x∧∈BR2, inequality (36) leads to −Ωx1,x∧+ε3x1,x∧I≥0, i.e. Ωx1,x∧<0. Similarly, the conditions in (33) lead to P1xJ>0 and P2xJ>0, respectively.
Finally, by Theorem1, Problem 1 is solvable with the corresponding observer-based controller gain matrices given by (26) and (27) and with the guaranteed cost given by (28).Remark 2.
In Theorem2, ε1xJ, ε2xJ, and ε3x1,x∧ are the given positive-definite polynomial functions to ensure the positive definiteness of P1xJ, P2xJ, and −Ωx1,x∧ in conditions (33), and all the coefficients corresponding to those positive-definite polynomial functions should be sufficiently small. Furthermore, the highest degree for every entry of polynomial matrices P1xJ and P2xJ should be 2d (d=0,1,2,…) to satisfy the conditions (33).
In order to further simplify the complexity of the control algorithm, we obtain the following corollary by restricting the polynomial matricesP1xJ and P2xJ in Theorem 2 to be constant matrices, and Y1x1,x∧ and Y2x1,x∧ to be polynomial matrix of substate x1.Corollary 1.
Supposex1∈BR1. For the given scalars γ>0 and ε0>0, the polynomial functions ε1>0, ε2>0 and ε3x1>0, there exists an observer-based output feedback controller of the form in (12) and (13) such that the closed-loop system in (14) satisfies the conditions i-iii in Problem 1, if there exist polynomial matrices P1∈R+n×n, P2∈R+n×n, X1x1∈Rx1p×n, X2x1∈Rx1n×n0, and SOS polynomial functions τix1, (i=1,2,…,m1) such that(37)υ1TP1−ε1Iυ1∈∑sos,υ2TP2−ε2Iυ2∈∑sos,−υ3TΩx1+ε3x1Iυ3−∑i=1m1τix1gi∈∑sos,Ωx1=Ω1x1∗∗Ω2x1−P10Ω30−P1Ω3=0I00000,Ω2x1=X1Tx1−B1Tx10−D1Tx1−D2Tx100−F1Tx1,Ω1x1=Ω11x1∗∗∗∗∗∗0Ω22x1∗∗∗∗∗C1x1P1+D1x1X1x10−I∗∗∗∗C2x1P1+D2x1X1x100−I∗∗∗ε0ETx1ε0ETx1P200−ε0I∗∗Fx1P1+F1x1X1x10000−ε0I∗B2Tx1B2Tx1P20000−γ2I,Ω11x1=SymAx1P1+B1x1X1x1,Ω22x1=SymP2Ax1−X2x1C0.
In this case, the corresponding observer and controller gain matrices are respectively given byKx1=X1x1P1−1 and Lx1=P2−1X2x1. Moreover, the guaranteed cost is given by J2∗=xT0P1−1x0+eT0P2e0.Remark 3.
In Corollary1, ε1, ε2, and ε3x1 are the given and sufficiently small constants which play the same role as the positive-definite polynomial functions ε1xJ, ε2xJ, and ε3x1,x∧ in Theorem 2, i.e., they are introduced to guarantee the positive definiteness of P1, P2, and −Ωx1 in conditions (37).Remark 4.
On the basis of Corollary1, for the given H∞ performance γ>0, a suboptimal guaranteed cost controller in the sense of minimizing the guaranteed cost J2∗ can be determined by the following optimization problem:(38)minα,s.t.37,−υ4TΩ0υ4∈∑sos,where υ4∈R2n+1, Ω0=−αxT0eT0P2x0−P10P2e00−P2.
In fact,−υ4TΩ0υ4∈∑sos implies J2∗=xT0P1−1x0+eT0P2e0≤α by Schur complement Lemma and Lemma 3. Therefore, minimizing the value of α will decrease the guaranteed cost J2∗, and may bring the true guaranteed cost closer to the optimal one.


## 4. Numerical Example

In this section, to illustrate the proposed robust nonlinear mixedH2/H∞ output-feedback control approach, the nonlinear mass-spring-damper system is considered. The dynamic model of this nonlinear system is given by(39)mX¨+gX,X˙+fX=ϕXu,where m is the mass; u is the control input; fX is nonlinear spring term; gX,X˙ is the nonlinear damper term; ϕX is the nonlinear input term. Assume that gX,X˙=c1X˙, fX=c2X+c3X3, ϕX=1+c4X2, and the parameters are chosen as c1=1, c2=0.01, c3=0.1, c4=0.13, m=1.Letx1=X (displacement) and x2=X˙ (velocity). By considering the disturbances and defining measured and controlled outputs, the state-space equation of the above second-order nonlinear system can be written as the following polynomial form:(40)x˙=01−0.01−0.1x12−1x+01+0.13x12u+00.010.50w,y=10x,z1=0.1000x+00.01u,z2=0.6000.3x+01u.Suppose the uncertainties existing in the stiffness coefficient of the spring, the damping coefficient of the damper and the input term satisfy the following structures:(41)Ex1=0.01000.1x1+0.01,Fx1=0000.1x1,F1x1=00.1.For the sake of simplicity in calculation, the state-region constraint is chosen asg1=0.3−x12; X1x1 and X2x1 are designed as polynomial matrices with degree 2; τ1x1 is designed as an SOS polynomial function with degree 4. Moreover, the other parameters are chosen as ε0=101, ε1=ε2=ε3x1=10−5, γ=0.9. By Corollary 1, a robust nonlinear mixed H2/H∞ output feedback controller of the form in (12) and (13) is obtained. The resulting P1, P2, X1x1, and X2x1 are given herein.(42)P1=1.0581−1.1987−1.19872.5315,P2=2.8256−0.6741−0.67410.2461,X2x1=2.3794x12+0.0522x1+2.3376−0.0832x12−0.0346x1+3.7705,X1x1=0.0204x12−0.0003x1−0.2518−0.0014x12+0.0001x1−0.5936.Furthermore, in order to show the advantages in control performance of the proposed nonlinear mixedH2/H∞ control approach, a traditional robust nonlinear H∞ output feedback controller is designed by setting C2x1=0 and D2x1=0 in Corollary 1 for comparison. The resulting P1, P2, X1x1 and X2x1 are given herein.(43)P1=2.6763−1.3236−1.32364.9149,P2=2.4254−0.7251−0.72510.4402,X2x1=2.5020x12+0.0068x1+1.3702−0.0722x12−0.0132x1+2.7122,X1x1=0.0710x12+0.0010x1−1.6806−0.0069x12+0.0007x1−1.0376.Then, starting from the initial conditionsx0=−0.51.5T and x∧0=−0.50.5T and subjected to the following disturbance:(44)w=sin0.1t+cost2T,0<t≤10,0,t>10.The simulation results are shown in Figures1–3 Furthermore, to demonstrate the state estimation accuracy of traditional H∞ output-feedback control strategy and the proposed H2/H∞ output-feedback control strategy in a quantitative way, the root mean square errors (RMSE) between the actual values and the estimated values of the system states are computed and shown in Table 2.Figure 1
Time responses of statesx1 and x2 using the nonlinear H∞ control approach and the proposed nonlinear mixed H2/H∞ control approach.Figure 2
Time responses of estimation errorsx1−x∧1 and x2−x∧2 using the nonlinear H∞ control approach and the proposed nonlinear mixed H2/H∞ control approach.Figure 3
Time responses of control inputu using the nonlinear H∞ control approach and the proposed nonlinear mixed H2/H∞ control approach and the actual truncated norms of a disturbance input.Table 2
RMSE in Figure2.
State variablesRMSE using proposed H2/H∞ controlRMSE using traditional H∞ controlx1, x∧10.01120.1851x2, x∧20.02630.2151The aims of the design controller are to achieve good robust performance and robust stability against the uncertainties and the disturbances. As can be seen from Figure1, the closed-loop system is stabilized at the origin for both H∞ control and the proposed mixed H2/H∞ control, and the influences of the uncertainties and disturbances on system gradually disappear with the time increasing. However, from Figures 1 and 2, we can see that, compared with the traditional H∞ controller, the nonlinear mixed H2/H∞ controller enhances the transient performance of the system, with a smoother transition and smaller overshoot. From Table 2, we conclude that the RMSEs of the proposed H2/H∞ output-feedback controller are less than those of the traditional H∞ output-feedback controller, which indicate a better state estimation level.Moreover, applying the nonlinearH∞ controller and the proposed nonlinear mixed H2/H∞ controller to system (38), we compute the truncated norms zt2/wt2 to show the actual disturbance attenuation level from disturbance w to output z over t∈0,10. From Figure 3, we can see the truncated norms for both the nonlinear H∞ control design and the proposed nonlinear mixed H2/H∞ control design are indeed less than the expected value γ=0.9. Nevertheless, the truncated L2-gain of the proposed nonlinear mixed H2/H∞ control design is less than the one of the traditional nonlinear H∞ control design, which indicates a better level of disturbance attenuation. Therefore, the nonlinear mixed H2/H∞ controller enhances the disturbance attenuation level. With the given initial states above, the guaranteed cost is J2∗=1.2281. As a result, the nonlinear mixed H2/H∞ controller not only guarantees the asymptotic stability of the closed-loop system, but also has better transient performance and robust performance.Finally, all simulations above are performed on an ordinary Core i7 CPU with base frequency of 3.4 GHz. The version of MATLAB is R2013a with SOSTOOLS 3.0.


## 5. Conclusions

In this paper, an SOS-based robust nonlinear mixedH2/H∞ output-feedback control design approach is proposed for a class of uncertain polynomial systems. This control design approach mainly shows that the problem above is formulated as an SOS convex programming problem, while the observer and the controller are designed simultaneously in a single-step way, thus reducing the difficulty in constructing Lyapunov function and implementing numerical computation caused by the non-convexity of nonlinear output-feedback control design. In the simulation, compared with the traditional H∞ controller, the nonlinear mixed H2/H∞ controller enhances the transient performance of the system with a smoother transition and smaller overshoot, and achieves a better disturbance attenuation level and state estimation level. Therefore, the proposed robust nonlinear mixed H2/H∞ control method not only guarantees the stability of the closed-loop system, but also has good transient performance and robust performance. Furthermore, input constraint is also a very interesting problem; on this topic, we will adopt the promising intelligent control methods, such as adaptive, fuzzy, and neural networks -based control methods, to address those input nonlinearity in the follow-up study.



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