Robust Stabilization Approach and 𝐻_∞ Performance via Static Output Feedback for a Class of Nonlinear Systems

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 𝐻_∞ 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 𝜇. 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 𝐻_∞ settings. A new 𝐻_∞ 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 𝐻_∞ synthesis via SOF is presented. Section 4 presents numerical examples for robust stabilization and 𝐻_∞ synthesis to illustrate the potential of the proposed conditions.

Notation 1. For conciseness the following notations are used: sym(𝐴)=𝐴+𝐴^𝑇,diag(𝐴,𝐵)=[^𝐴0_0𝐵], [^𝐴𝐵_•𝐶]=[_𝐵^𝐴𝐵_𝑇𝐶], and 𝑃=𝑃^𝑇>0 is a symmetric and positive definite.

2. Preliminaries

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

𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝑓(𝑘,𝑥), (2.1) where 𝑥(𝑘)∈ℜ^𝑛 is the state vector of the system. 𝐴∈ℜ^𝑛×𝑛 is a constant matrix and 𝑓(𝑘,𝑥) a nonlinear function in both arguments 𝑘 and 𝑥 satisfying 𝑓(𝑘,0)=0. This means that the origin is an equilibrium point of the system.

The nonlinear function 𝑓 is bounded by the following quadratic constraints:

𝑓^𝑇(𝑘,𝑥)𝑓(𝑘,𝑥)≤𝛼²𝑥^𝑇𝑀^𝑇𝑀𝑥, (2.2) where 𝛼>0 is the bounding parameter of the nonlinear function 𝑓 and 𝑀 is a constant matrix of appropriate dimensions.

The parameter 𝛼 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

𝑥𝑓^𝑇−𝛼²𝑀^𝑇𝑥𝑓𝑀00𝐼≤0. (2.3)

Remark 2.1. The nonlinear function 𝑓(𝑘,𝑥), 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 𝛼>0 if the equilibrium 𝑥=0 is globally asymptotically stable for all 𝑓(𝑘,𝑥) 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 Ω₀(𝑥) and Ω₁(𝑥) be two arbitrary quadratic forms over ℜ^𝑛, then Ω₀(𝑥)<0 for all 𝑥∈ℜ^𝑛−{0} satisfying Ω₁(𝑥)≤0 if and only if there exist 𝜏≥0: Ω₀(𝑥)−𝜏Ω₁(𝑥)<0,∀𝑥∈ℜ^𝑛−{0}. (2.4)

Proof. See [1].

Lemma 2.4 (Projection lemma [1]). Given a symmetric matrix 𝜓∈ℜ^𝑛×𝑛, and two matrices 𝑃, 𝑄 of column dimensions n, there exists X such that the following LMI holds: 𝜓+sym(𝑃^𝑇𝑋^𝑇𝑄)<0, (2.5) if and only if the projection inequalities with respect to X are satisfied: 𝒩_𝑃𝜓𝒩^𝑇_𝑝<0,𝒩^𝑇_𝑄𝜓𝒩_𝑄<0, (2.6) where 𝒩_𝑃 and 𝒩_𝑄 denote arbitrary bases of the nullspaces of 𝑃 and 𝑄, respectively.

Proof. See [1].

Lemma 2.5. Let Φ a symmetric matrix and 𝑁, 𝐽 be matrices of appropriate dimensions. The following statements are equivalent:(i)Φ<0andΦ+NJ^𝑇+JN^𝑇<0,(ii)there exists a matrix 𝐺 such that𝐽Φ𝐽+𝑁𝐺_𝑇+𝐺^𝑇𝑁^𝑇−𝐺−𝐺^𝑇<0. (2.7)

Proof. The proof is obtained remarking that (2.7) can be developed as follows: 𝐽Φ𝐽+𝑁𝐺_𝑇+𝐺^𝑇𝑁^𝑇−𝐺−𝐺^𝑇=𝐽Φ𝐽_𝑇0+sym{0𝐼𝐺^𝑇𝑁^𝑇−𝐼}<0, (2.8) 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 𝛼>0 if there exist a positive definite symmetric matrix 𝑄 and a positive scalar 𝛽=1/√𝛼 such that: 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒𝛽−𝑄0𝑄𝐴^𝑇𝑄𝑀^𝑇•−𝐼𝐼0••−𝑄0•••−𝛽𝐼<0. (2.9)

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 𝛼>0 if there exist a positive definite symmetric matrix Q , a matrix G of appropriate dimensions, and a positive scalar 𝛽=1/√𝛼, such that the following optimization problem: 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒𝛽−𝑄0𝜇𝑄0𝑄•−𝐼𝐼00••−𝑄0(𝐴−𝜇𝐼)𝐺•••−𝛽𝐼𝑀𝐺••••−𝐺−𝐺^𝑇<0, (2.10) is feasible for any prescribed scalar 𝜇∈]−11[.

Proof. Inequality (2.9) can be expressed as follows: −𝑄0𝑄𝐴^𝑇𝑄𝑀^𝑇=•−𝐼𝐼0••−𝑄0•••−𝛽𝐼−𝑄0𝜇𝑄0•−𝐼𝐼0••−𝑄0•••−𝛽𝐼_𝜙+sym00𝑀𝐴−𝜇𝐼_𝑁𝑄000_𝐽<0. (2.11) It is not difficult to proof that 𝜙<0 for any 𝜇∈]−11[. Now writing:(i)𝑁=[⁰⁰_{𝑀𝐴−𝜇𝐼}],(ii)𝐽=[𝑄000]^𝑇,and by lemma 2, there exists a matrix 𝐺 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:

𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝑓(𝑘,𝑥,𝑢)+𝐵𝑢(𝑘),𝑦(𝑘)=𝐶𝑥(𝑘), (2.12) where 𝑢(𝑘)∈ℜ^𝑚 is the control input,𝑦(𝑘)∈ℜ^𝑝 is the measured output, and 𝐵∈ℜ^𝑛×𝑚 and C∈ℜ^𝑝×𝑛 are constant matrices. We assume that the pair (𝐴,𝐵) is stabilizable and 𝐶 is full rank matrices. Also 𝑓(𝑘,𝑥,𝑢) is a nonlinear function which satisfies the following quadratic constraints:

𝑓^𝑇(𝑘,𝑥,𝑢)𝑓(𝑘,𝑥,𝑢)≤𝛼²(𝑥^𝑇𝐹^𝑇𝐹𝑥+𝑢^𝑇𝐻^𝑇𝐻𝑢), (2.13) where 𝛼>0 is the bounding parameter of the function 𝑓 and 𝐹 and 𝐻 are constant matrices of appropriate dimensions.

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

𝑢(𝑘)=𝐾𝑦(𝑘), (2.14) where 𝐾∈ℜ^𝑚×𝑝.

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

𝑥(𝑘+1)=(𝐴+𝐵𝐾𝐶)𝑥(𝑘)+𝑓(𝑘,𝑥,𝑢), (2.15) and function 𝑓 satisfies:

𝑓^𝑇(𝑘,𝑥,𝑢)𝑓(𝑘,𝑥,𝑢)≤𝛼²𝑥^𝑇(𝐹^𝑇𝐹+(𝐻𝐾𝐶)^𝑇𝐻𝐾𝐶)𝑥. (2.16)

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

𝑥𝑓^𝑇−𝛼²(𝐹^𝑇𝐹+(𝐻𝐾𝐶)^𝑇𝑥𝑓𝐻𝐾𝐶)00𝐼≤0. (2.17)

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 𝛼.

Theorem 2.10. System (2.12) is asymptotically stable by static output feedback with degree 𝛼>0 if there exist a positive definite symmetric matrix 𝑄, a matrix 𝑅∈^𝑚×𝑝, and a positive scalar 𝛽=1/√𝛼 such that the following optimization problem is solvable: 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒𝛽−𝑄0(𝐴𝑄+𝐵𝑅𝐶)^𝑇𝑄𝐹^𝑇(𝐻𝑅𝐶)^𝑇•−𝐼𝐼00••−𝑄00•••−𝛽𝐼0••••−𝛽𝐼<0, (2.18) where 𝑄𝑄=𝑉¹00𝑄₂𝑉^𝑇. (2.19) The static output feedback gain is given by 𝐾=𝑅𝑈𝐶₀𝑄₁⁻¹𝐶₀⁻¹𝑈^𝑇, (2.20) with 𝑈∈ℜ^𝑝×𝑝 and 𝑉∈ℜ^𝑛×𝑛 are unitary matrices, and 𝐶₀∈ℜ^𝑝×𝑝 matrix which are obtained by using the singular value decomposition of the matrix C : 𝐶=𝑈𝐶₀0𝑉^𝑇. (2.21)

Proof. According to the Theorem 2.6, system (2.15) is robustly stable if there exist a positive definite symmetric matrix 𝑄 and a positive scalar 𝛽=1/√𝛼 such that the following optimization problem is solvable: 𝐴𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒𝛽−𝑄0𝑄^𝑇𝑄𝑀^𝑇•−𝐼𝐼0••−𝑄0•••−𝛽𝐼<0. (2.22) Now we defin(i)𝐴=𝐴+𝐵𝐾𝐶,(ii)𝑀=(^𝐹_𝐻𝐾𝐶),(iii)𝑄 is replaced by (2.19) with: 𝑄₁∈ℜ^𝑝×𝑝, 𝑄₂∈ℜ^{(𝑛−𝑝)×(𝑛−𝑝)}.
Equation (2.22) becomes: 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒𝛽−𝑄0𝑄(𝐴+𝐵𝐾𝐶)^𝑇𝑄𝐹^𝑇𝑄𝐶^𝑇𝐾^𝑇𝐻^𝑇•−𝐼𝐼00••−𝑄00•••−𝛽𝐼0••••−𝛽𝐼<0. (2.23) Unfortunately, (2.23) is not convex in 𝐾 and 𝑄 and cannot be solved by the LMI tools. We can introduce some transformations to simplify the 𝐾𝐶𝑄 term of the inequality (2.23) by using (2.19) and (2.21) as follows: 𝐾𝐶𝑄=𝐾𝑈𝐶₀0𝑉^𝑇𝑉𝑄₁00𝑄₂𝑉^𝑇=𝐾𝑈𝐶₀𝑄₁𝐶₀⁻¹𝑈⁻¹_𝑅𝑈𝐶₀0𝑉^𝑇,_𝐶 (2.24) where 𝑅∈ℜ^𝑚×𝑝.
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 𝑄 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 𝛼>0, for an arbitrary prescribed number 𝜇 in ]−11[ if there exist a positive definite symmetric matrix 𝑄, matrices 𝑅∈ℜ^𝑚×𝑝, 𝐺∈ℜ^𝑛×𝑛, and a positive scalar 𝛽=1/√𝛼 such that the following optimization problem is solvable: 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒𝛽−𝑄0𝜇𝑄00𝑄•−𝐼𝐼000••−𝑄00𝐴𝐺+𝐵𝑅𝐶−𝜇𝐺•••−𝛽𝐼0𝐹𝐺••••−𝛽𝐼𝐻𝑅𝐶•••••−𝐺−𝐺^𝑇<0, (2.25) where 𝐺𝐺=𝑉¹0𝐺₂𝐺₃𝑉^𝑇. (2.26) The static output feedback gain is given by 𝐾=𝑅𝑈𝐶₀𝐺₁⁻¹𝐶₀⁻¹𝑈^𝑇, (2.27) with 𝑈,𝑉, and 𝐶₀ 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 𝑄, a matrix 𝐺 of appropriate dimensions, and a positive scalar 𝛽=1/√𝛼 such that 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒𝛽−𝑄0𝜇𝑄0𝑄•−𝐼𝐼00••−𝑄0(𝐴−𝜇𝐼)𝐺•••−𝛽𝐼𝑀𝐺••••−𝐺−𝐺^𝑇<0, (2.28) where the following hold: (i)𝐴=𝐴+𝐵𝐾𝐶,(ii)𝑀=(^𝐹_𝐻𝐾𝐶),(iii)𝐺 replaced by (2.26) where 𝐺₁∈ℜ^𝑝×𝑝, 𝐺₂∈ℜ^{(𝑛−𝑝)×𝑝}, 𝐺₃∈ℜ^{(𝑛−𝑝)×(𝑛−𝑝)},with 𝑈,𝑉, and 𝐶₀ are given by (2.21).
Then, inequality (2.28) becomes −𝑄0𝜇𝑄00𝑄•−𝐼𝐼000••−𝑄00𝐴𝐺+𝐵𝐾𝐶𝐺−𝜇𝐺•••−𝛽𝐼0𝐹𝐺••••−𝛽𝐼𝐻𝐾𝐶𝐺•••••−𝐺−𝐺^𝑇<0. (2.29) Unfortunately, (2.29) is not convex and cannot be solved by the LMI tools.
For this reason, we introduce some transformations to simplify the 𝐾𝐶𝐺 term of the inequality (2.29) by using (2.26) and (2.21) as follows: 𝐾𝐶𝐺=𝐾𝑈𝐶₀0𝑉^𝑇𝑉𝐺₁0𝐺₂𝐺₃𝑉^𝑇=𝐾𝑈𝐶₀𝐺₁𝐶₀⁻¹𝑈⁻¹_𝑅𝑈𝐶₀0𝑉^𝑇_𝐶, (2.30) where 𝑅∈ℜ^𝑚×𝑝.
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 𝐺=𝑄 and𝜇=0, we obtain the optimization problem (2.18) with 𝑄 satisfying (2.19). Therefore, if (2.25) is feasible, then (2.19) is feasible too.

3. Nonlinear Discrete-Time 𝐇_∞ Norm Characterization

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

𝐱(𝐤+1)=𝐀𝐱(𝐤)+𝐁𝐰(𝐤)+𝐟(𝐤,𝐱),𝐳(𝐤)=𝐂𝐱(𝐤)+𝐃𝐰(𝐤), (3.1) where 𝐰(𝐤)∈𝑅^𝐪 is the exogenous disturbance and 𝐳(𝐤)∈𝑅^𝐫 is the controlled output. 𝐀,𝐁,𝐂, and 𝐃 are known constant matrices of appropriate dimensions. 𝐟(𝐤,𝐱) is a nonlinear function satisfying the following quadratic constraints:

𝐱𝐰𝐟^𝐓−𝜶²𝐌^𝐓𝐱𝐰𝐟𝐌0000000𝐈≤0. (3.2)

We note 𝑇_𝑤𝑧 the transfer matrix from input 𝑤 to output 𝑧 when 𝑓(𝑘,𝑥)=0 is expressed by

𝐓_𝐰𝐳=𝐂(𝐬𝐈−𝐀)⁻¹𝐁+𝐃. (3.3)

Theorem 3.1. System (3.1) is robustly stable with ‖𝑇_𝑤𝑧‖_∞<𝛾 for a prescribed constant value 𝛾>0, if there exist a positive definite symmetric matrix 𝑄 and positive scalars 𝜏>0 and 𝛽=1/√𝛼 such that the following optimization problem is feasible: min𝛽−𝑄00𝑄𝐴^𝑇𝑄𝑀^𝑇𝑄𝐶^𝑇•−𝜏𝛾²𝐼0𝜏𝐵^𝑇0𝜏𝐷^𝑇••−𝐼𝐼00•••−𝑄00••••−𝛽𝐼0•••••−𝜏𝐼<0. (3.4)

Proof. Define the Lyapunov function: 𝑉(𝑥)=𝑥^𝑇(𝑘)𝑃𝑥(𝑘), (3.5) where 𝑃=𝑃^𝑇>0.
By using the dissipative theory, we show that 𝑉(𝑘+1)−𝑉(𝑘)+𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘)<0, (3.6) where 𝛾>0 is a prescribed scalar such that ‖𝑇_𝑤𝑧‖_∞<𝛾_∞<𝛾, (3.7) where 𝛾_∞ is the corresponding bounding norm bound when 𝑓=0.
Evaluating (3.6) leads to 𝑥^𝑇𝑤^𝑇𝑓^𝑇𝐴^𝑇𝑃𝐴−𝑃+𝐶^𝑇𝐶𝐴^𝑇𝑃𝐵+𝐶^𝑇𝐷𝐴^𝑇𝑃𝐵^𝑇𝑃𝐴+𝐷^𝑇𝐶𝐵^𝑇𝑃𝐵+𝐷^𝑇𝐷−𝛾²𝐼𝐵^𝑇𝑃𝑥𝑃𝐴𝑃𝐵𝑃^𝑇𝑤^𝑇𝑓^𝑇𝑇<0. (3.8) Now applying the S-procedure Lemma to (3.8) with (3.2), there exits 𝜏>0 such that 𝐴^𝑇𝑃𝐴−𝑃+𝜏⁻¹𝐶^𝑇𝐶+𝛼²𝑀^𝑇𝑀𝐴^𝑇𝑃𝐵+𝜏⁻¹𝐶^𝑇𝐷𝐴^𝑇𝑃𝐵^𝑇𝑃𝐴+𝜏⁻¹𝐷^𝑇𝐶𝐵^𝑇𝑃𝐵+𝜏⁻¹𝐷^𝑇𝐷−𝜏⁻¹𝛾²𝐼𝐵^𝑇𝑃𝑃𝐴𝑃𝐵𝑃−𝐼<0, (3.9) where 𝑃=𝑃/𝜏.
By Schur complement, (3.9) is equivalent to −𝑃00𝐴^𝑇𝑀^𝑇𝐶^𝑇•−𝜏⁻¹𝛾²𝐼0𝐵^𝑇0𝐷^𝑇••−𝐼𝐼00•••−𝑃⁻¹00••••−𝛽𝐼0•••••−𝜏𝐼<0, (3.10) where 𝛽=𝛼⁻².
Multiplying by diag(𝑄,𝜏,𝐼,𝐼,𝐼) with 𝑄=𝑃⁻¹ 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 𝐻_∞ norm of.

Theorem 3.2. System (3.1) is robustly stable with ‖𝑇_𝑤𝑧‖_∞<𝛾 for a prescribed constant value 𝛾>0, if there exist a positive definite symmetric matrix 𝑄, a matrix 𝐺 of appropriate dimensions, and positive scalars 𝜏>0 and 𝛽=1/√𝛼 such that for any prescribed scalar 𝜇 in ]⁻¹¹[ , the following optimization problem is feasible: min𝛽−𝑄00𝜇𝑄00𝑄•−𝜏𝛾²𝐼0𝜏𝐵^𝑇0𝜏𝐷^𝑇0••−𝐼𝐼000•••−𝑄00(𝐴−𝜇𝐼)𝐺••••−𝛽𝐼0𝑀𝐺•••••−𝜏𝐼𝐶𝐺••••••−𝐺−𝐺^𝑇<0. (3.11)

Proof. Inequality (3.4) can be written as follows: −𝑄00𝑄𝐴^𝑇𝑄𝑀^𝑇Q𝐶^𝑇•−𝜏𝛾²𝐼0𝜏𝐵^𝑇0𝜏𝐷^𝑇=••−𝐼𝐼00•••−𝑄00••••−𝛽𝐼0•••••−𝜏𝐼−𝑄00𝜇𝑄00•−𝜏𝛾²𝐼0𝜏𝐵^𝑇0𝜏𝐷^𝑇+••−𝐼𝐼00•••−𝑄00••••−𝛽𝐼0•••••−𝜏𝐼sym(000𝑀𝐶𝐴−𝜇𝐼𝑄00000)<0. (3.12) By lemma 2, denoting 𝜙=−𝑄00𝜇𝑄00•−𝜏𝛾²𝐼0𝜏𝐵^𝑇0𝜏𝐷^𝑇••−𝐼𝐼00•••−𝑄00••••−𝛽𝐼0•••••−𝜏𝐼<0,forany[000𝑀𝐶,𝜇∈]−11𝑁=𝐴−𝜇𝐼,𝐽=𝑄00000 (3.13) there exists a matrix 𝐺 of appropriate dimensions such that (3.11) holds where 𝛾,𝛽, and 𝜏 are scalars previously defined.

3.1. Static Output Feedback 𝐻_∞ Synthesis

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

𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵_𝑢𝑢(𝑘)+𝐵_𝑤𝑤(𝑘)+𝑓(𝑘,𝑥,𝑢),𝑧(𝑘)=𝐶_𝑧𝑥(𝑘)+𝐷_𝑧𝑢𝑢(𝑘)+𝐷_𝑧𝑤𝑤(𝑘),𝑦(𝑘)=𝐶_𝑦𝑥(𝑘), (3.14) where 𝑢(𝑘)∈ℜ^𝑚 is the control input, 𝑤(𝑘)∈ℜ^𝑞 is the exogenous disturbance, 𝑧(𝑘)∈ℜ^𝑟 is the controlled output, and 𝑦(𝑘)∈ℜ^𝑝 is the measured output. Also, 𝐵_𝑢,𝐵_𝑤,𝐶_𝑧,𝐷_𝑧𝑢,𝐷_𝑧𝑤 and 𝐶_𝑦 are known constant matrices of appropriate dimensions.

The system closed by SOF is written as

𝑥(𝑘+1)=𝐴_𝑐𝑙𝑥(𝑘)+𝐵_𝑐𝑙𝑤(𝑘)+𝑓(𝑘,𝑥,𝑢),𝑧(𝑘)=𝐶_𝑐𝑙𝑥(𝑘)+𝐷_𝑐𝑙𝑤(𝑘), (3.15) where the following hold:

𝐴_𝑐𝑙=𝐴+𝐵_𝑢𝐾𝐶_𝑦,𝐵_𝑐𝑙=𝐵_𝑤,𝐶_𝑐𝑙=𝐶_𝑧+𝐷_𝑧𝑢𝐾𝐶_𝑦,𝐷_𝑐𝑙=𝐷_𝑧𝑤. (3.16)

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

Theorem 3.3. System (3.14) is robustly static output feedback stabilizable with ‖𝑇_𝑤𝑧‖_∞<𝛾 for a prescribed constant value 𝛾>0 if there exist a positive definite symmetric matrix 𝑄, a matrix 𝑅 of appropriate dimensions, and positive scalars 𝜏>0 and 𝛽=1/√𝛼 such that the following optimization problem is feasible: min𝛽−𝑄00𝑄𝐴^𝑇+(𝐵_𝑢𝑅𝐶_𝑦)^𝑇𝑄𝐹^𝑇(𝐻𝑅𝐶_𝑦)^𝑇𝑄𝐶^𝑇_𝑧+(𝐷_𝑧𝑢𝑅𝐶_𝑦)^𝑇•−𝜏𝛾²𝐼0𝜏𝐵^𝑇_𝑤00𝜏𝐷^𝑇_𝑧𝑤••−𝐼𝐼000•••−𝑄000••••−𝛽𝐼00•••••−𝛽𝐼0••••••−𝜏𝐼<0, (3.17) where 𝑄𝑄=𝑉¹00𝑄₂𝑉^𝑇. (3.18) The static output feedback gain is then given by 𝐾=𝑅𝑈𝐶₀𝑄₁⁻¹𝐶₀⁻¹𝑈^𝑇, (3.19) where 𝑈,𝑉, and 𝐶₀ are given in (2.21).

Proof. According to Theorem 3.1, system (3.14) is robustly stable with ‖𝑇_𝑤𝑧‖_∞<𝛾 for a prescribed constant value 𝛾>0 if there exist a positive definite symmetric matrix 𝑄 and positive scalars 𝜏>0 and 𝛽=1/√𝛼 such that the following optimization problem is feasible: min𝛽−𝑄00𝑄𝐴^𝑇_𝑐𝑙𝑄𝑀^𝑇𝑄𝐶^𝑇_𝑐𝑙•−𝜏𝛾²𝐼0𝜏𝐵^𝑇_𝑐𝑙0𝜏𝐷^𝑇_𝑐𝑙••−𝐼𝐼00•••−𝑄00••••−𝛽𝐼0•••••−𝜏𝐼<0. (3.20) Now:(i)𝐴_𝑐𝑙,𝐵_𝑐𝑙,𝐶_𝑐𝑙 and 𝐷_𝑐𝑙 are replaced by (3.16),(ii)𝑀=(^𝐹_{𝐻𝐾𝐶𝑦}),(iii)𝑄 is replaced by (3.18) where : 𝑄₁∈ℜ^𝑝×𝑝, 𝑄₂∈ℜ^{(𝑛−𝑝)×(𝑛−𝑝)}. After same direct developments the result follows.

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

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

Theorem 3.4. System (3.14) is robustly SOF stabilisable with ‖𝑇_𝑤𝑧‖_∞<𝛾 for a prescribed constant value 𝛾>0 if there exist a positive definite symmetric matrix 𝑄, matrices 𝐺 and 𝑅 of appropriate dimensions, and positive scalars 𝜏>0, and 𝛽=1/√𝛼 such that the following optimization problem is feasible for any prescribed scalar 𝜇 in ]−11[: −𝑄00𝜇𝑄000𝑄•−𝜏𝛾²𝐼0𝜏𝐵^𝑇_𝑤00𝜏𝐷^𝑇_𝑧𝑤0••−𝐼𝐼0000•••−𝑄000𝐴𝐺−𝜇𝐺+𝐵_𝑢𝑅𝐶_𝑦••••−𝛽𝐼00𝐹𝐺•••••−𝛽𝐼0𝐻𝑅𝐶_𝑦••••••−𝜏𝐼𝐶_𝑧𝐺+𝐷_𝑧𝑢𝑅𝐶_𝑦•••••••−𝐺−𝐺^𝑇<0, (3.21) where 𝐺𝐺=𝑉¹0𝐺₂𝐺₃𝑉^𝑇. (3.22) The static output feedback gain is given by 𝐾=𝑅𝑈𝐶₀𝐺₁⁻¹𝐶₀⁻¹𝑈^𝑇, (3.23) with 𝑅∈ℜ^𝑚×𝑝, 𝑈,𝑉,and 𝐶₀ are given in (2.21).

Proof. According to the Theorem 3.3, the system (3.14) is robustly stable with ‖𝑇_𝑤𝑧‖_∞<𝛾 for a prescribed degree 𝛾>0 if there exist a positive definite symmetric matrix 𝑄, a matrix 𝐺 of appropriate dimensions, and positive scalars 𝜏>0 and 𝛽=1/√𝛼, for any prescribed scalar 𝜇 in ]−11[, such that the following optimization problem is feasible: min𝛽−𝑄00𝜇𝑄00𝑄•−𝜏𝛾²𝐼0𝜏𝐵^𝑇_𝑐𝑙0𝜏𝐷^𝑇_𝑐𝑙0••−𝐼𝐼000•••−𝑄00(𝐴_𝑐𝑙−𝜇𝐼)𝐺••••−𝛽𝐼0𝑀𝐺•••••−𝜏𝐼𝐶_𝑐𝑙𝐺••••••−𝐺−𝐺^𝑇<0, (3.24) where(i)𝐴_𝑐𝑙,𝐵_𝑐𝑙,𝐶_𝑐𝑙, and 𝐷_𝑐𝑙 are replaced by (3.16),(ii)𝑀=(^𝐹_{𝐻𝐾𝐶𝑦}),(iii)𝐺 replaced by (3.22), where 𝐺₁∈ℜ^𝑝×𝑝, 𝐺₂∈ℜ^{(𝑛−𝑝)×𝑝}. The optimization problem (3.24) is expressed as follows: min𝛽<0−𝑄00𝜇𝑄000𝑄•−𝜏𝛾²𝐼0𝜏𝐵^𝑇_𝑤00𝜏𝐷^𝑇_𝑧𝑤0••−𝐼𝐼0000•••−𝑄000𝐴𝐺+𝐵_𝑢𝐾𝐶_𝑦𝐺−𝜇𝐺••••−𝛽𝐼00𝐹𝐺•••••−𝛽𝐼0𝐻𝐾𝐶_𝑦𝐺••••••−𝜏𝐼𝐶_𝑧𝐺+𝐷_𝑧𝑢𝐾𝐶_𝑦•••••••−𝐺−𝐺^𝑇<0. (3.25) For this reason, we express differently the 𝐾𝐶_𝑦𝐺 term of the inequality (3.25), by using (3.22) and (2.21), in the same way as in (2.30). Consequently, we obtain 𝐾𝐶_𝑦𝐺=𝑅𝐶_𝑦. (3.26) 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 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵(𝑢(𝑘)+𝛼sin𝑦(𝑡)),𝑦(𝑘)=𝐶𝑥(𝑘), (4.1) where 𝑥(𝑘)=[𝑥₁(𝑘)𝑥₂(𝑘)]^𝑇, 𝑥₁(𝑘) and 𝑥₂(𝑘) being, respectively, the angular displacement and velocity; 𝑢(𝑘) represents the field current of the DC motor; the matrices 𝐴∈ℜ^2×2,𝐵∈ℜ^2×1, and 𝐶∈ℜ^1×2are given by: 𝐴=.10.095200.9048;𝐵=0.0480.952;𝐶=10 (4.2) The nonlinear function f is given by the following expression: 𝑓(𝑘,𝑥)=𝛼𝐵sin𝑦(𝑘)=𝛼𝐵sin𝑥₁(𝑘). (4.3) This function satisfies 𝑓^𝑇(𝑘,𝑥)𝑓(𝑘,𝑥)=𝛼²𝐵^𝑇𝐵sin²𝑥₁(𝑘)=0.9𝛼²sin²𝑥₁(𝑘)≤𝛼²𝑥^𝑇0.950000.95000𝑥. (4.4) In this case, we take 𝐹=[^0.950₀₀];𝐻=0.
The nonlinear system is unstable. The matrix 𝐴 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 𝛼=𝛼_max.We can see a drastic improvement obtained by Theorem 2.12.

Table 1 

Numerical Evaluation for Example 4.1.

Figure 1 

Inverted pendulum scheme.

Figure 2 

Output 𝑦=𝑥1.

Figure 3 

Output 𝑦=𝑥1.

Example 4.2. We consider now the nonlinear discrete-time system: 𝑥(𝑘+1)=−0.07250.19571.59310.03010.04040.60840.3764−0.16350.9024𝑥(𝑘)+1.21230.28950.61740.06510.93790.5110𝑢(𝑘)+𝑓(𝑘,𝑥(𝑘),𝑢(𝑘)),𝑦(𝑘)=0.9313−0.7534−0.0335120.5𝑥(𝑘), (4.5) with the nonlinear function 𝑓 satisfying the constraint (2.17), where 𝐹=𝐼₃ and 𝐻=𝐼₂. 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].

Table 2 

Numerical Evaluation for Example 4.2.

Example 4.3. We consider the nonlinear discrete-time system (3.14) with 𝐴=0.81890.08630.09000.08130.25241.00330.03130.2004−0.05450.01020.7901−0.2580−0.1918−0.10340.16020.8604,𝐵_𝑢=0.00450.00440.10010.01000.0003−0.0136−0.00510.0926,𝐵_𝑤=,𝐶0.0953000.0145000.086200−0.001100_𝑧=10−1000000000,𝐷_𝑧𝑢=001001,𝐶_𝑦=10000010,𝐷_𝑦𝑤=0, (4.6) with the nonlinear function 𝑓 satisfying the constraint (3.2), where 𝐹=𝐼₄ and 𝐻=𝐼₄. In Table 3, we present numerical result for 𝐻_∞ performance via SOF by applying Theorem 3.3 and for different values of 𝛾 which satisfy (3.7).In Table 4, we present numerical result for 𝐻_∞ performance via SOF by applying Theorem 3.4 and for different values of 𝛾 which satisfy (3.7).

Table 3 

Numerical Evaluation for Example4.3 by Theorem 3.3.

Table 4 

Numerical Evaluation for Example4.3 by Theorem 3.4.

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 𝐻_∞ 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.