#### Abstract

This paper deals with the stability and stabilization problems for a class of discrete-time nonlinear systems. The systems are composed of a linear constant part perturbated by an additive nonlinear function which satisfies a quadratic constraint. A new approach to design a static output feedback controller is proposed. A sufficient condition, formulated as an LMI optimization convex problem, is developed. In fact, the approach is based on a family of LMI parameterized by a scalar, offering an additional degree of freedom. The problem of performance taking into account an criterion is also investigated. Numerical examples are provided to illustrate the effectiveness of the proposed conditions.

#### 1. Introduction

Modeling a real process is generally a complex and difficult task. Even if in numerous cases, a linear model can capture the main dynamical characteristics of a process. In some situations, it is necessary to take into account model uncertainties in order to design an efficient control law.

There exists an extensive literature dealing with this problem which is in fact the main problem in robust control design [1–8].

Among the numerous solutions allowing taking into account model uncertainties, a way which has been frequently investigated in literature consists in adding to the linear part of the model a nonlinear one which captures model uncertainties and frequently referred in literature as nonlinear systems with separated nonlinearity.

Nonlinear systems with separated nonlinearity are a class of nonlinear systems composed of a linear constant part to which another nonlinear function part is added. This function depends on both time and state and satisfies a quadratic constraint [9–16]. This class of nonlinear system can be considered as a generalized model for linear systems with parametric uncertainties where uncertainties can be norm bounded [2, 17] or polytopic [9, 10, 18, 19].

Many papers have investigated robust stability, analysis, and synthesis using essentially Lyapunov theory which has proved to be efficient in this context. Recent proposed approaches [9–13] are based on convex optimization problems involving linear matrix inequality (LMI) where the objective is to maximize the bounds on the nonlinearity that systems can tolerate without unstabilities. In particular, sufficient conditions where developed in the context of static state feedback and static or dynamic output feedback controllers [9–13, 15].

Even if the static output feedback stabilization (SOF) problem is considered as NP-hard [20] and still one of the most important open questions in the control theory, it concentrates the efforts of many researchers. SOF gains, which stabilize the system, are not easy to find due to the nonconvexity of the SOF formulation. In some papers the design of SOF controllers for a class of discrete-time nonlinear systems is proposed [11, 15, 17]. In this paper, we propose a new approach for robust static output feedback stabilization of a class of discrete-time nonlinear systems using LMI techniques. In fact, our approach is based on the introduction of a relaxation scheme to the SOF problem similar to [15, 21–23]. Our major objective is to maximize the admissible bounds on the nonlinearity guaranteeing the stability of system, with a prescribed degree . The main contribution is the possibility of decoupling the Lyapunov matrix to the SOF gain leading to less restrictive conditions.

The problem of performance is also treated in the context of settings. A new norm characterization is proposed for this class of nonlinear discrete time systems in terms of LMI formulation.

The paper is organized as follows. Section 2 presents robust stability condition for the class of nonlinear discrete time systems. Then, we develop our main results for robust stabilization by SOF. In Section 3, the problem of robust synthesis via SOF is presented. Section 4 presents numerical examples for robust stabilization and synthesis to illustrate the potential of the proposed conditions.

*Notation 1. *For conciseness the following notations are used: , , and is a symmetric and positive definite.

#### 2. Preliminaries

In this section, we consider nonlinear discrete-time system with the following state-space representation:

where is the state vector of the system. is a constant matrix and a nonlinear function in both arguments and satisfying . This means that the origin is an equilibrium point of the system.

The nonlinear function is bounded by the following quadratic constraints:

where is the bounding parameter of the nonlinear function and is a constant matrix of appropriate dimensions.

The parameter can be defined as a degree of robustness, because its maximization leads to an increase of robustness against uncertain perturbations. Note that constraint (2.2) is equivalent to

*Remark 2.1. * The nonlinear function , which satisfies the quadratic constraint (2.2), can be considered as parameter uncertainty [17].

In the sequel, we will use the following definition to present the concept of robust stability of the system (2.1), (2.2).

*Definition 2.2. *System (2.1) is robustly stable with degree if the equilibrium is globally asymptotically stable for all satisfying constraint (2.2).

In this section, we develop a method for studying robust stability of system (2.1). Before, we introduce some instrumentals tools which will be used in the proof of characterization of stability of system (2.1).

Lemma 2.3 (S-procedure lemma [1]). *Let and be two arbitrary quadratic forms over , then for all satisfying if and only if there exist :
*

*Proof. *See [1].

Lemma 2.4 (Projection lemma [1]). *Given a symmetric matrix , and two matrices , of column dimensions n, there exists X such that the following LMI holds:
**
if and only if the projection inequalities with respect to X are satisfied:
**
where and denote arbitrary bases of the nullspaces of and , respectively.*

*Proof. * [1].

Lemma 2.5. *Let a symmetric matrix and , be matrices of appropriate dimensions. The following statements are equivalent:*(i)(ii)*there exists a matrix such that*

*Proof. *The proof is obtained remarking that (2.7) can be developed as follows:
and by applying lemma 2.

##### 2.1. Stability Characterization

We first introduce the following theorem which gives a robust stability conditions for system (2.1). In fact, it is described by a convex optimization problem where we try to maximize the nonlinear bounding parameter without loss of the stability of system.

Theorem 2.6. *System (2.1) is robustly stable with degree if there exist a positive definite symmetric matrix and a positive scalar such that:
*

*Proof. *See [13].

*Remark 2.7. *The stability condition given by Theorem 2.6 is equivalent to the one introduced in [11] and [13].

A new condition for robust stability is proposed in the following theorem.

Theorem 2.8. *System (2.1) is stable with degree if there exist a positive definite symmetric matrix Q , a matrix G of appropriate dimensions, and a positive scalar , such that the following optimization problem:
**
is feasible for any prescribed scalar .*

*Proof. *Inequality (2.9) can be expressed as follows:
It is not difficult to proof that for any . Now writing:(i),(ii)and by lemma 2, there exists a matrix of appropriate dimensions such that inequality is satisfied.

*Remark 2.9. *The two optimization problems (2.9) and (2.10) are equivalent. In the case of stability analysis or state feedback control synthesis, no improvement is obtained by problem (2.10). The main advantage of problem (2.10) will appear when dealing with static output feedback. In that case, we will see that it theoretically improves the obtained results.

##### 2.2. Static Output Feedback Control

In this section, we investigate the static output feedback stabilization problem for nonlinear discrete systems.

We consider the nonlinear discrete-time system described as follows:

where is the control input, is the measured output, and and are constant matrices. We assume that the pair is stabilizable and is full rank matrices. Also is a nonlinear function which satisfies the following quadratic constraints:

where is the bounding parameter of the function and and are constant matrices of appropriate dimensions.

The objective is to find a static output feedback control law such as

where .

The closed loop system is given by the following state space representation:

and function satisfies:

Note that in this case, the constraint (2.16) is equivalent to

To establish a robust stabilization theory for system (2.12) with (2.13) by SOF, we give in the following theorem, an optimization problem which allows to stabilize the linear part of system (2.15) with (2.17) and at the same time to maximize the value of parameter .

Theorem 2.10. *System (2.12) is asymptotically stable by static output feedback with degree if there exist a positive definite symmetric matrix , a matrix , and a positive scalar such that the following optimization problem is solvable:
**
where
**
The static output feedback gain is given by
**
with and are unitary matrices, and matrix which are obtained by using the singular value decomposition of the matrix C :
*

*Proof. *According to the Theorem 2.6, system (2.15) is robustly stable if there exist a positive definite symmetric matrix and a positive scalar such that the following optimization problem is solvable:
Now we defin(i)(ii)(iii) is replaced by (2.19) with: ,

Equation (2.22) becomes:
Unfortunately, (2.23) is not convex in and and cannot be solved by the LMI tools. We can introduce some transformations to simplify the term of the inequality (2.23) by using (2.19) and (2.21) as follows:
where .

With this transformation we obtain the optimization problem given in Theorem 2.10.

*Remark 2.11. *The optimization problem given by Theorem 2.10 presents a sufficient condition for the robust stabilization by SOF of discrete-time nonlinear system (2.12). In fact to solve the BMI problem, we impose a diagonal structure to Lyapunov matrix as in [24] and we obtain a LMI convex problem.

##### 2.3. Main Results

In this section, we introduce a new approach for robust stabilization by SOF of nonlinear discrete time system (2.12). The following results present solutions to static output feedback problem in which an improved sufficient condition is presented. In fact, this approach is derived from Theorem 2.8.

Theorem 2.12. *The system (2.12) is robustly stable by static output feedback with degree , for an arbitrary prescribed number in if there exist a positive definite symmetric matrix , matrices , , and a positive scalar such that the following optimization problem is solvable:
**
where
**
The static output feedback gain is given by
**
with , and are given in (2.21).*

*Proof. * According to Theorem 2.8, the closed loop system (2.15) is robustly stable if there exist a positive definite symmetric matrix , a matrix of appropriate dimensions, and a positive scalar such that
where the following hold: (i),(ii),(iii) replaced by (2.26) where , , with , and are given by (2.21).

Then, inequality (2.28) becomes
Unfortunately, (2.29) is not convex and cannot be solved by the LMI tools.

For this reason, we introduce some transformations to simplify the term of the inequality (2.29) by using (2.26) and (2.21) as follows:
where .

With this transformation, we obtain the optimization problem given by Theorem 2.12.

The following lemma gives a connection of the results of Theorem 2.10 with the one of Theorem 2.12.

Lemma 2.13. *If the SOF stabilization problem is solvable by Theorem 2.10, then it is solvable by Theorem 2.12.*

*Proof. *If we consider the optimization problem (2.25) with and, we obtain the optimization problem (2.18) with satisfying (2.19). Therefore, if (2.25) is feasible, then (2.19) is feasible too.

#### 3. Nonlinear Discrete-Time Norm Characterization

Stability is the minimum requirement and in practice a performance level has to be guaranteed. Performance objectives can be achieved via norm optimization. In this section, we study the control problem for the following nonlinear discrete-time system:

where is the exogenous disturbance and is the controlled output. , and are known constant matrices of appropriate dimensions. is a nonlinear function satisfying the following quadratic constraints:

We note the transfer matrix from input to output when is expressed by

Theorem 3.1. *System (3.1) is robustly stable with for a prescribed constant value , if there exist a positive definite symmetric matrix and positive scalars and such that the following optimization problem is feasible:
*

*Proof. *Define the Lyapunov function:
where .

By using the dissipative theory, we show that
where is a prescribed scalar such that
where is the corresponding bounding norm bound when .

Evaluating (3.6) leads to
Now applying the S-procedure Lemma to (3.8) with (3.2), there exits such that
where .

By Schur complement, (3.9) is equivalent to
where .

Multiplying by with the both sides of (3.10), we obtain the optimization problem (3.4).

Now, we introduce the following theorem which can be seen as an alternate characterization of upper bounds of the norm of.

Theorem 3.2. *System (3.1) is robustly stable with for a prescribed constant value , if there exist a positive definite symmetric matrix , a matrix of appropriate dimensions, and positive scalars and such that for any prescribed scalar in , the following optimization problem is feasible:
*

*Proof. *Inequality (3.4) can be written as follows:
By lemma 2, denoting
there exists a matrix of appropriate dimensions such that (3.11) holds where , and are scalars previously defined.

##### 3.1. Static Output Feedback Synthesis

In this section, we consider the static output feedback stabilization problem for the following nonlinear discrete system (3.14):

where is the control input, is the exogenous disturbance, is the controlled output, and is the measured output. Also, and are known constant matrices of appropriate dimensions.

The system closed by SOF is written as

where the following hold:

The objective of this section is to design static output feedback controllers for nonlinear discrete time system (3.14).

Theorem 3.3. *System (3.14) is robustly static output feedback stabilizable with for a prescribed constant value if there exist a positive definite symmetric matrix a matrix of appropriate dimensions, and positive scalars and such that the following optimization problem is feasible:
**
where
**
The static output feedback gain is then given by
**
where , and are given in (2.21).*

*Proof. *According to Theorem 3.1, system (3.14) is robustly stable with for a prescribed constant value if there exist a positive definite symmetric matrix and positive scalars and such that the following optimization problem is feasible:
Now:(i) and are replaced by (3.16),(ii),(iii) is replaced by (3.18) where : , . After same direct developments the result follows.

##### 3.2. An Improved Approach of Static Output Feedback Synthesis for Robust Control

In this paragraph, an robust control for nonlinear systems (3.14) improving the previous approach is proposed.

Theorem 3.4. *System (3.14) is robustly SOF stabilisable with for a prescribed constant value if there exist a positive definite symmetric matrix , matrices and of appropriate dimensions, and positive scalars , and such that the following optimization problem is feasible for any prescribed scalar in :
**
where
**
The static output feedback gain is given by
**
with , and are given in (2.21).*

*Proof. *According to the Theorem 3.3, the system (3.14) is robustly stable with for a prescribed degree if there exist a positive definite symmetric matrix , a matrix of appropriate dimensions, and positive scalars and , for any prescribed scalar in , such that the following optimization problem is feasible:
where(i), and are replaced by (3.16),(ii),(iii) replaced by (3.22), where , . The optimization problem (3.24) is expressed as follows:
For this reason, we express differently the term of the inequality (3.25), by using (3.22) and (2.21), in the same way as in (2.30). Consequently, we obtain
With this transformation, we obtain the optimization problem given in Theorem 3.4.

#### 4. Numerical Examples

We present in this section two numerical examples to illustrate the proposed theory for SOF synthesis.

Our approach Theorem 2.12 is compared to the methods presented in [11].

*Example 4.1 (see [5]). *We consider the nonlinear discrete-time system of inverted pendulum (see Figure 1).The system can be described by
where , and being, respectively, the angular displacement and velocity; represents the field current of the DC motor; the matrices and are given by:
The nonlinear function *f * is given by the following expression:
This function satisfies
In this case, we take

The nonlinear system is unstable. The matrix is unstable which means that the linear part of systems is unstable. Therefore, we apply the approach given by Theorems 2.10 and 2.12 to stabilize system by SOF. We summarize in Table 1 the obtained results.Figures 2 and 3 show the output trajectory obtained respectively by applying Theorems 2.10 and 2.12 to system (3.26) with .We can see a drastic improvement obtained by Theorem 2.12.

*Example 4.2. *We consider now the nonlinear discrete-time system:
with the nonlinear function satisfying the constraint (2.17), where and . The results obtained byapproaches of Theorem 2.12 and in [11] are given in Table 2.The new approach (Theorem 2.12) improves the results obtained by [11].

*Example 4.3. *We consider the nonlinear discrete-time system (3.14) with
with the nonlinear function satisfying the constraint (3.2), where and . In Table 3, we present numerical result for performance via SOF by applying Theorem 3.3 and for different values of which satisfy (3.7).In Table 4, we present numerical result for performance via SOF by applying Theorem 3.4 and for different values of which satisfy (3.7).

#### 5. Conclusion

In this paper, the stabilization problem by static output feedback (SOF) for a particular class of nonlinear discrete time systems is investigated. A new sufficient condition is elaborated by using Lyapunov theory and formulated by LMI constraints. We obtain a convex optimization problem for maximizing the bound of the nonlinearity preserving the stability of the systems.

Finally, the proposed controller design method was extended to incorporate synthesis. An optimization problem, which is linear both in the admissible nonlinearity bound and the disturbance attenuation, is developed. Numerical comparisons with existing methods in literature illustrate the improvement obtained by our approaches. All of them can also be extended to the dynamic output feedback.

Other classes of nonlinear discrete time nonlinear systems exist and present some interesting characteristic from a practical point of view. For example, the ones where the linear part is affected by polytopic uncertainties. It would be interesting to extend the results of this paper to those classes. This will be exploited in a near future.