#### Abstract

This paper considers the robust nonlinear mixed *H*_{2}/*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.

#### 1. Introduction

The mixed *H*_{2}/*H*_{∞} control design has attracted considerable attention in recent years, because it combines the merits of both the *H*_{2} guaranteed cost control and the *H*_{∞} robust control. Mixed *H*_{2}/*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 *H*_{2} 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 *H*_{2}/*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 *H*_{2}/*H*_{∞} output-feedback control design problem using SOS technique. This control design approach presents that the nonlinear mixed *H*_{2}/*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 Section 2, the problem description is introduced. In Section 3, the sufficient conditions to robust nonlinear mixed *H*_{2}/*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 Table 1.

#### 2. Problem Description

Consider the following uncertain polynomial system:where is the state vector with the measurable substate and the immeasurable substate ; is the control input vector; is the disturbance input vector; is the measured output vector; , , and are the known polynomial matrices; and denote the time-varying parameter uncertainties and satisfy the following assumption.

*Assumption 1. *where , , and are known polynomial matrices with appropriate dimensions that characterise the structures of the uncertainties; is an unknown time-varying matrix satisfying .

Define the controlled output equations aswhere , , , and () are known polynomial matrices with appropriate dimensions.

Respectively introduce the *H*_{∞} and *H*_{2} performance measures asandwhere is a positive scalar representing a desired *H*_{∞} disturbance attenuation level.

The robust nonlinear mixed *H*_{2}/*H*_{∞} output-feedback control problem to be addressed in this paper is specifically described as follows:

*Problem 1. *(Robust nonlinear mixed *H*_{2}/*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 with (ii)The *H*_{2} performance guarantees under , where is a positive constant representing the guaranteed cost(iii)The -gain from the disturbance input to the controlled output is less than a given positive scalar , i.e., under zero initial condition.

*Remark 1. *In fact, the terms *H*_{2} 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 -gain, respectively. In this paper, we refer to the proposed problem as *H*_{2}/*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 is an SOS polynomial, if there exist polynomials such thatTherefore, if a polynomial can be decomposed into the form in (7), i.e, , we have for all . 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 and be real matrices of appropriate dimensions. For any matrix satisfying and a positive scalar , we have*

Lemma 2. *(see [30]). For any and any positive definite matrix , the following inequality holds*

Lemma 3. *(see [8]). Let be a symmetric polynomial matrix of degree in and be a column vector whose entries are all monomials in with degree no greater than , and consider the following conditions:*(i)*, *(ii)*, where *(iii)*There exists a positive semidefinite matrix satisfying , where denotes the Kronecker product.**Then, .*

Lemma 4. *(Generalized S-procedure) (see [17]). Given , if there exist such thatthen*

#### 3. Robust Nonlinear Mixed *H*_{2}/*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:where is the estimate of , , and 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:where

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 - in Problem 1 for all admissible uncertainties under Assumption 1.

On the basis of Lyapunov stability theory, a solvable criterion for Problem 1 is first established as follows.

Lemma 5. *Consider the uncertain system in (1), (3) and (4). For the given scalars and , if there exist polynomial matrices , ,and , such that the closed-loop system in (14) satisfies the following inequality:where , denotes the -th entry of , represents the row indices of whose corresponding row is equal to zero; is the -th row of .Then, Problem 1 is solvable with the controller of the form in (12) and (13), and the guaranteed cost is given by*

*Proof. *Choose Lyapunov function for the closed-loop system in (14) asDifferentiating along the trajectories of (14) and using Lemma 1 yieldFrom condition (16) and Schur complement Lemma, we haveWhen , from (20), we obtainwhich implies the asymptotic stability of the closed-loop system in (14).

Furthermore, by integrating both sides of the inequality (21) from to , we obtainWhen , , we haveFrom inequality (20), we obtainwhich implies that the -gain from the disturbance input to the controlled output of the closed-loop system in (14) is less than .

Obviously, the condition in (16) is difficult to solve, because the cross couplings among matrix variables , and 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 as a positive-definite diagonal matrix.

Theorem 1. *Consider the uncertain system in (1) (3) and (4). For the given scalars and , 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 - in Problem 1, if there exist polynomial matrices , , , and , such that the following inequality holds**In this case, the corresponding observer and controller gain matrices are respectively given byand**Moreover, the guaranteed cost is given by*

*Proof. *Choosing , from (16), we haveMultiplying both sides of (29) by and denoting , yieldUsing Lemma 2 and choosing , we haveThen, by Schur complement Lemma, (30) and (31) lead to the condition (25).

In Theorem 1, 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:where and are the polynomial functions of and , 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. *Suppose and . For the given scalars and , the polynomial functions , and , 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 - in Problem 1, if there exist polynomial matrices , , , and SOS polynomial functions , , such thatwhere , , , and are vectors consisting of any elements of and , is a vector consisting of any elements of . 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 thatWe defineFrom condition (33), we haveThen, by Lemmas 3 and 4, for and , inequality (36) leads to , i.e. . Similarly, the conditions in (33) lead to and , respectively.

Finally, by Theorem 1, 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 Theorem 2, , , and are the given positive-definite polynomial functions to ensure the positive definiteness of , , and 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 and should be () 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 matrices and in Theorem 2 to be constant matrices, and and to be polynomial matrix of substate .

Corollary 1. *Suppose . For the given scalars and , the polynomial functions , and , 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 - in Problem 1, if there exist polynomial matrices , , , , and SOS polynomial functions , () such that**In this case, the corresponding observer and controller gain matrices are respectively given by and . Moreover, the guaranteed cost is given by .*

*Remark 3. *In Corollary 1, , , and are the given and sufficiently small constants which play the same role as the positive-definite polynomial functions , , and in Theorem 2, i.e., they are introduced to guarantee the positive definiteness of , , and in conditions (37).

*Remark 4. *On the basis of Corollary 1, for the given *H*_{∞} performance , a suboptimal guaranteed cost controller in the sense of minimizing the guaranteed cost can be determined by the following optimization problem:where , .

In fact, implies by Schur complement Lemma and Lemma 3. Therefore, minimizing the value of will decrease the guaranteed cost , and may bring the true guaranteed cost closer to the optimal one.

#### 4. Numerical Example

In this section, to illustrate the proposed robust nonlinear mixed *H*_{2}/*H*_{∞} output-feedback control approach, the nonlinear mass-spring-damper system is considered. The dynamic model of this nonlinear system is given bywhere is the mass; is the control input; is nonlinear spring term; is the nonlinear damper term; is the nonlinear input term. Assume that , , , and the parameters are chosen as , , , , .

Let (displacement) and (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:

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:

For the sake of simplicity in calculation, the state-region constraint is chosen as ; and are designed as polynomial matrices with degree 2; is designed as an SOS polynomial function with degree 4. Moreover, the other parameters are chosen as , , . By Corollary 1, a robust nonlinear mixed *H*_{2}/*H*_{∞} output feedback controller of the form in (12) and (13) is obtained. The resulting , , , and are given herein.

Furthermore, in order to show the advantages in control performance of the proposed nonlinear mixed *H*_{2}/*H*_{∞} control approach, a traditional robust nonlinear *H*_{∞} output feedback controller is designed by setting and in Corollary 1 for comparison. The resulting , , and are given herein.

Then, starting from the initial conditions and and subjected to the following disturbance:

The simulation results are shown in Figures 1–3 Furthermore, to demonstrate the state estimation accuracy of traditional H_{∞} output-feedback control strategy and the proposed *H*_{2}/*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.

The 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 Figure 1, the closed-loop system is stabilized at the origin for both *H*_{∞} control and the proposed mixed *H*_{2}/*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 *H*_{2}/*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 *H*_{2}/*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 nonlinear *H*_{∞} controller and the proposed nonlinear mixed *H*_{2}/*H*_{∞} controller to system (38), we compute the truncated norms to show the actual disturbance attenuation level from disturbance to output over . From Figure 3, we can see the truncated norms for both the nonlinear *H*_{∞} control design and the proposed nonlinear mixed H_{2}/*H*_{∞} control design are indeed less than the expected value . Nevertheless, the truncated -gain of the proposed nonlinear mixed H_{2}/*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 *H*_{2}/*H*_{∞} controller enhances the disturbance attenuation level. With the given initial states above, the guaranteed cost is . As a result, the nonlinear mixed *H*_{2}/*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 mixed *H*_{2}/*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 *H*_{2}/*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 *H*_{2}/*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.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest.

#### Acknowledgments

The authors would like to thank Zhejiang Provincial Natural Science Foundation of China (Grant nos. LQ20F030001 and LGG20F020008) for supporting this research.