# Robust Stabilization Approach andH∞ Performance via Static Output Feedback for a Class of Nonlinear Systems

Authors: Neila Bedioui; Salah Salhi; Mekki Ksouri
Journal: Mathematical Problems in Engineering
 (2009)

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## 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 anH∞ criterion is also investigated. Numerical examples are provided to illustrate the effectiveness of the proposed conditions.


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## Body

## 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 ofH∞ 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. Section2 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+AT,diag(A,B)=[A00B], [AB•C]=[ABBTC], and P=PT>0 is a symmetric and positive definite.


## 2. Preliminaries

In this section, we consider nonlinear discrete-time system with the following state-space representation:(2.1)x(k+1)=Ax(k)+f(k,x),
where x(k)∈ℜn is the state vector of the system. A∈ℜn×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 functionf is bounded by the following quadratic constraints:(2.2)fT(k,x)f(k,x)≤α2xTMTMx,
where α>0 is the  bounding parameter of the nonlinear  function f and M is a constant matrix of appropriate dimensions.The parameterα 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(2.3)[xf]T[-α2MTM00I][xf]≤0.Remark 2.1.
The nonlinear functionf(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]).
LetΩ0(x) and Ω1(x) be two arbitrary quadratic forms over ℜn, then Ω0(x)<0 for all x∈ℜn-{0} satisfying Ω1(x)≤0 if and only if there exist τ≥0:
(2.4)Ω0(x)-τΩ1(x)<0,∀x∈ℜn-{0}.Proof.
See [1].Lemma 2.4 (Projection lemma [1]).
Given a symmetric matrixψ∈ℜn×n, and two matrices P, Q of column dimensions n, there exists X such that the following LMI holds:
(2.5)ψ+sym(PTXTQ)<0,
if and only if the projection inequalities with respect to X  are satisfied:
(2.6)𝒩Pψ𝒩pT<0,𝒩QTψ𝒩Q<0,
where 𝒩P and 𝒩Q denote arbitrary bases of the nullspaces of P and Q, respectively.Proof.
See [1].Lemma 2.5.
LetΦ a symmetric matrix and N, J be matrices of appropriate dimensions. The following statements are equivalent:(i)
Φ<0andΦ+NJT+JNT<0,(ii)
there exists a matrixG such that(2.7)[ΦJ+NGJT+GTNT-G-GT]<0.Proof.
The proof is obtained remarking that (2.7) can be developed as follows:
(2.8)[ΦJ+NGJT+GTNT-G-GT]=[ΦJJT0]+sym{[0I]GT[NT-I]}<0,
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:
(2.9){minimizeβ[-Q0QATQMT•-II0••-Q0•••-βI]<0.Proof.
See [13].Remark 2.7.
The stability condition given by Theorem2.6 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:
(2.10){minimizeβ[-Q0μQ0Q•-II00••-Q0(A-μI)G•••-βIMG••••-G-GT]<0,
is feasible for any prescribed scalar μ∈]-11[.Proof.
Inequality (2.9) can be expressed as follows:
(2.11)[-Q0QATQMT•-II0••-Q0•••-βI]=[-Q0μQ0•-II0••-Q0•••-βI]︸ϕ+sym([00A-μIM]︸N[Q000]︸J)<0.
It is not difficult to proof that ϕ<0 for any μ∈]-11[.  Now writing:(i)
N=[00A-μIM],(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:(2.12)x(k+1)=Ax(k)+f(k,x,u)+Bu(k),y(k)=Cx(k),
where u(k)∈ℜm is the control input,y(k)∈ℜp is the measured output, and B∈ℜn×m and C∈ℜp×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:(2.13)fT(k,x,u)f(k,x,u)≤α2(xTFTFx+uTHTHu),
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(2.14)u(k)=Ky(k),
where K∈ℜm×p.The closed loop system is given by the following state space representation:(2.15)x(k+1)=(A+BKC)x(k)+f(k,x,u),
and function f satisfies:(2.16)fT(k,x,u)f(k,x,u)≤α2xT(FTF+(HKC)THKC)x.Note that in this case, the constraint (2.16) is equivalent to(2.17)[xf]T[-α2(FTF+(HKC)THKC)00I][xf]≤0.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:
(2.18){minimizeβ[-Q0(AQ+BRC)TQFT(HRC)T•-II00••-Q00•••-βI0••••-βI]<0,
where
(2.19)Q=V[Q100Q2]VT.
The static output feedback gain is given by
(2.20)K=RUC0Q1-1C0-1UT,
with   U∈ℜp×p and V∈ℜn×n  are unitary matrices, and C0∈ℜp×p  matrix which are obtained by using the singular value decomposition of the matrix C :
(2.21)C=U[C00]VT.Proof.
According to the Theorem2.6, 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:
(2.22){minimizeβ[-Q0QÃTQMT•-II0••-Q0•••-βI]<0.
Now we defin(i)
Ã=A+BKC,(ii)
M=(FHKC),(iii)
Q is replaced by (2.19) with: Q1∈ℜp×p,  Q2∈ℜ(n-p)×(n-p).
Equation (2.22) becomes:
(2.23){minimizeβ[-Q0Q(A+BKC)TQFTQCTKTHT•-II00••-Q00•••-βI0••••-βI]<0.
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:
(2.24)KCQ=KU[C00]VTV[Q100Q2]VT=KUC0Q1C0-1U-1︸RU[C00]VT︸,C
where  R∈ℜm×p.
With this transformation we obtain the optimization problem given in Theorem2.10.Remark 2.11.
The optimization problem given by Theorem2.10 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∈ℜm×p, G∈ℜn×n, and a positive scalar  β=1/α  such that the following optimization problem is solvable:
(2.25){minimizeβ[-Q0μQ00Q•-II000••-Q00AG+BRC-μG•••-βI0FG••••-βIHRC•••••-G-GT]<0,
where
(2.26)G=V[G10G2G3]VT.
The static output feedback gain is given by
(2.27)K=RUC0G1-1C0-1UT,
with  U,V, and C0 are given in (2.21).Proof.
According to Theorem2.8, 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
(2.28){minimizeβ[-Q0μQ0Q•-II00••-Q0(Ã-μI)G•••-βIMG••••-G-GT]<0,
where the following hold: (i)
Ã=A+BKC,(ii)
M=(FHKC),(iii)
G replaced by (2.26)  where   G1∈ℜp×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
(2.29)[-Q0μQ00Q•-II000••-Q00AG+BKCG-μG•••-βI0FG••••-βIHKCG•••••-G-GT]<0.
Unfortunately, (2.29) is not convex  and cannot be solved by the LMI tools.
For this reason, we introduce some   transformations to simplify theKCG term of the inequality (2.29) by using (2.26) and (2.21) as follows:
(2.30)KCG=KU[C00]VTV[G10G2G3]VT=KUC0G1C0-1U-1︸RU[C00]VT︸C,
where  R∈ℜm×p.
With this transformation, we obtain the optimization problem given by Theorem2.12.The following lemma gives a connection of the results of Theorem2.10 with the one of Theorem 2.12.Lemma 2.13.
If the SOF stabilization problem is solvable by Theorem2.10, 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.


## 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:
(2.9){minimizeβ[-Q0QATQMT•-II0••-Q0•••-βI]<0.Proof.
See [13].Remark 2.7.
The stability condition given by Theorem2.6 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:
(2.10){minimizeβ[-Q0μQ0Q•-II00••-Q0(A-μI)G•••-βIMG••••-G-GT]<0,
is feasible for any prescribed scalar μ∈]-11[.Proof.
Inequality (2.9) can be expressed as follows:
(2.11)[-Q0QATQMT•-II0••-Q0•••-βI]=[-Q0μQ0•-II0••-Q0•••-βI]︸ϕ+sym([00A-μIM]︸N[Q000]︸J)<0.
It is not difficult to proof that ϕ<0 for any μ∈]-11[.  Now writing:(i)
N=[00A-μIM],(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:(2.12)x(k+1)=Ax(k)+f(k,x,u)+Bu(k),y(k)=Cx(k),
where u(k)∈ℜm is the control input,y(k)∈ℜp is the measured output, and B∈ℜn×m and C∈ℜp×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:(2.13)fT(k,x,u)f(k,x,u)≤α2(xTFTFx+uTHTHu),
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(2.14)u(k)=Ky(k),
where K∈ℜm×p.The closed loop system is given by the following state space representation:(2.15)x(k+1)=(A+BKC)x(k)+f(k,x,u),
and function f satisfies:(2.16)fT(k,x,u)f(k,x,u)≤α2xT(FTF+(HKC)THKC)x.Note that in this case, the constraint (2.16) is equivalent to(2.17)[xf]T[-α2(FTF+(HKC)THKC)00I][xf]≤0.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:
(2.18){minimizeβ[-Q0(AQ+BRC)TQFT(HRC)T•-II00••-Q00•••-βI0••••-βI]<0,
where
(2.19)Q=V[Q100Q2]VT.
The static output feedback gain is given by
(2.20)K=RUC0Q1-1C0-1UT,
with   U∈ℜp×p and V∈ℜn×n  are unitary matrices, and C0∈ℜp×p  matrix which are obtained by using the singular value decomposition of the matrix C :
(2.21)C=U[C00]VT.Proof.
According to the Theorem2.6, 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:
(2.22){minimizeβ[-Q0QÃTQMT•-II0••-Q0•••-βI]<0.
Now we defin(i)
Ã=A+BKC,(ii)
M=(FHKC),(iii)
Q is replaced by (2.19) with: Q1∈ℜp×p,  Q2∈ℜ(n-p)×(n-p).
Equation (2.22) becomes:
(2.23){minimizeβ[-Q0Q(A+BKC)TQFTQCTKTHT•-II00••-Q00•••-βI0••••-βI]<0.
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:
(2.24)KCQ=KU[C00]VTV[Q100Q2]VT=KUC0Q1C0-1U-1︸RU[C00]VT︸,C
where  R∈ℜm×p.
With this transformation we obtain the optimization problem given in Theorem2.10.Remark 2.11.
The optimization problem given by Theorem2.10 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∈ℜm×p, G∈ℜn×n, and a positive scalar  β=1/α  such that the following optimization problem is solvable:
(2.25){minimizeβ[-Q0μQ00Q•-II000••-Q00AG+BRC-μG•••-βI0FG••••-βIHRC•••••-G-GT]<0,
where
(2.26)G=V[G10G2G3]VT.
The static output feedback gain is given by
(2.27)K=RUC0G1-1C0-1UT,
with  U,V, and C0 are given in (2.21).Proof.
According to Theorem2.8, 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
(2.28){minimizeβ[-Q0μQ0Q•-II00••-Q0(Ã-μI)G•••-βIMG••••-G-GT]<0,
where the following hold: (i)
Ã=A+BKC,(ii)
M=(FHKC),(iii)
G replaced by (2.26)  where   G1∈ℜp×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
(2.29)[-Q0μQ00Q•-II000••-Q00AG+BKCG-μG•••-βI0FG••••-βIHKCG•••••-G-GT]<0.
Unfortunately, (2.29) is not convex  and cannot be solved by the LMI tools.
For this reason, we introduce some   transformations to simplify theKCG term of the inequality (2.29) by using (2.26) and (2.21) as follows:
(2.30)KCG=KU[C00]VTV[G10G2G3]VT=KUC0G1C0-1U-1︸RU[C00]VT︸C,
where  R∈ℜm×p.
With this transformation, we obtain the optimization problem given by Theorem2.12.The following lemma gives a connection of the results of Theorem2.10 with the one of Theorem 2.12.Lemma 2.13.
If the SOF stabilization problem is solvable by Theorem2.10, 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-TimeH∞ Norm Characterization

Stability is the minimum requirement and in practice a performance level has to be guaranteed. Performance objectives can be achieved viaH∞ norm optimization. In this section, we study the  H∞ control problem for the following   nonlinear discrete-time system:(3.1)x(k+1)=Ax(k)+Bw(k)+f(k,x),z(k)=Cx(k)+Dw(k),
where w(k)∈Rq  is the exogenous disturbance and  z(k)∈Rr 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:(3.2)[xwf]T[-α2MTM0000000I][xwf]≤0.We noteTwz the transfer matrix from input w to output z when f(k,x)=0 is expressed by(3.3)Twz=C(sI-A)-1B+D.Theorem 3.1.
System (3.1) is robustly stable with ∥Twz∥∞<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:
(3.4){minβ[-Q00QATQMTQCT•-τγ2I0τBT0τDT••-II00•••-Q00••••-βI0•••••-τI]<0.Proof.
Define the Lyapunov function:(3.5)V(x)=xT(k)Px(k),
where  P=PT>0.
By using the dissipative theory, we show that(3.6)V(k+1)-V(k)+zT(k)z(k)-γ2wT(k)w(k)<0,
where  γ>0 is a prescribed scalar such that
(3.7)∥Twz∥∞<γ∞<γ,
where  γ∞ is the corresponding  bounding norm bound  when f=0.
Evaluating (3.6) leads to
(3.8)[xTwTfT][ATPA-P+CTCATPB+CTDATPBTPA+DTCBTPB+DTD-γ2IBTPPAPBP][xTwTfT]T<0.
Now applying the S-procedure Lemma to (3.8) with (3.2),  there exits τ>0 such that
(3.9)[ATP̃A-P̃+τ-1CTC+α2MTMATP̃B+τ-1CTDATP̃BTP̃A+τ-1DTCBTP̃B+τ-1DTD-τ-1γ2IBTP̃P̃AP̃BP̃-I]<0,
where P̃=P/τ.
By Schur complement, (3.9) is equivalent to
(3.10)[-P00ATMTCT•-τ-1γ2I0BT0DT••-II00•••-P-100••••-βI0•••••-τI]<0,
where β=α-2.
Multiplying bydiag(Q,τ,I,I,I)  with Q=P̃-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  theH∞ norm of.Theorem 3.2.
System (3.1) is robustly stable with  ∥Twz∥∞<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:
(3.11){minβ[-Q00μQ00Q•-τγ2I0τBT0τDT0••-II000•••-Q00(A-μI)G••••-βI0MG•••••-τICG••••••-G-GT]<0.Proof.
Inequality (3.4) can be written as follows:
(3.12)[-Q00QATQMTQCT•-τγ2I0τBT0τDT••-II00•••-Q00••••-βI0•••••-τI]=[-Q00μQ00•-τγ2I0τBT0τDT••-II00•••-Q00••••-βI0•••••-τI]+sym([000A-μIMC][Q00000])<0.
By   lemma 2, denoting
(3.13)ϕ=[-Q00μQ00•-τγ2I0τBT0τDT••-II00•••-Q00••••-βI0•••••-τI]<0,for anyμ∈]-11[N=[000A-μIMC],J=[Q00000],
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 FeedbackH∞ Synthesis

In this section, we consider the static output feedback stabilization problem for the following nonlinear discrete   system (3.14):(3.14)x(k+1)=Ax(k)+Buu(k)+Bww(k)+f(k,x,u),z(k)=Czx(k)+Dzuu(k)+Dzww(k),y(k)=Cyx(k),
where u(k)∈ℜm is the control input,  w(k)∈ℜq  is the exogenous disturbance,  z(k)∈ℜr is the controlled output, and y(k)∈ℜp is the measured output. Also, Bu,Bw,Cz,Dzu,Dzw and Cy are known constant matrices of appropriate dimensions.The system closed   by SOF is written as(3.15)x(k+1)=Aclx(k)+Bclw(k)+f(k,x,u),z(k)=Cclx(k)+Dclw(k),
where the following hold:(3.16)Acl=A+BuKCy,Bcl=Bw,Ccl=Cz+DzuKCy,Dcl=Dzw.The objective of this section is to design static output feedbackH∞ controllers for nonlinear discrete time system (3.14).Theorem 3.3.
System (3.14) is robustly static output feedback stabilizable with  ∥Twz∥∞<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:
(3.17){minβ[-Q00QAT+(BuRCy)TQFT(HRCy)TQCzT+(DzuRCy)T•-τγ2I0τBwT00τDzwT••-II000•••-Q000••••-βI00•••••-βI0••••••-τI]<0,
where
(3.18)Q=V[Q100Q2]VT.
The static output feedback gain is then  given by
(3.19)K=RUC0Q1-1C0-1UT,
where  U,V, and C0 are given in (2.21).Proof.
According to Theorem3.1, system (3.14) is robustly stable with ∥Twz∥∞<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:
(3.20){minβ[-Q00QAclTQMTQCclT•-τγ2I0τBclT0τDclT••-II00•••-Q00••••-βI0•••••-τI]<0.
Now:(i)
Acl,Bcl,Ccl and Dcl are replaced by (3.16),(ii)
M=(FHKCy),(iii)
Q is replaced by (3.18)  where : Q1∈ℜp×p, Q2∈ℜ(n-p)×(n-p).   After same direct developments the result follows.


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

In this paragraph, anH∞ robust control for nonlinear systems (3.14) improving the previous approach is proposed.Theorem 3.4.
System (3.14) is robustly SOF stabilisable with ∥Twz∥∞<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[:
(3.21)[-Q00μQ000Q•-τγ2I0τBwT00τDzwT0••-II0000•••-Q000AG-μG+BuRCy••••-βI00FG•••••-βI0HRCy••••••-τICzG+DzuRCy•••••••-G-GT]<0,
where
(3.22)G=V[G10G2G3]VT.
The static output feedback gain is given by
(3.23)K=RUC0G1-1C0-1UT,
with R∈ℜm×p, U,V,and C0 are given in (2.21).Proof.
According to the  Theorem3.3, the system (3.14) is robustly stable with  ∥Twz∥∞<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:
(3.24){minβ[-Q00μQ00Q•-τγ2I0τBclT0τDclT0••-II000•••-Q00(Acl-μI)G••••-βI0MG•••••-τICclG••••••-G-GT]<0,
where(i)
Acl,Bcl,Ccl, and Dcl are replaced by (3.16),(ii)
M=(FHKCy),(iii)
G replaced by (3.22), where G1∈ℜp×p,  G2∈ℜ(n-p)×p.The optimization problem (3.24) is expressed as follows:
(3.25){minβ<0[-Q00μQ000Q•-τγ2I0τBwT00τDzwT0••-II0000•••-Q000AG+BuKCyG-μG••••-βI00FG•••••-βI0HKCyG••••••-τICzG+DzuKCy•••••••-G-GT]<0.
For this reason,   we express differently the  KCyG term of the inequality (3.25), by using (3.22) and (2.21), in the same way as in (2.30). Consequently, we obtain
(3.26)KCyG=RCy.
With this transformation, we obtain the optimization problem given in Theorem 3.4.


## 3.1. Static Output FeedbackH∞ Synthesis

In this section, we consider the static output feedback stabilization problem for the following nonlinear discrete   system (3.14):(3.14)x(k+1)=Ax(k)+Buu(k)+Bww(k)+f(k,x,u),z(k)=Czx(k)+Dzuu(k)+Dzww(k),y(k)=Cyx(k),
where u(k)∈ℜm is the control input,  w(k)∈ℜq  is the exogenous disturbance,  z(k)∈ℜr is the controlled output, and y(k)∈ℜp is the measured output. Also, Bu,Bw,Cz,Dzu,Dzw and Cy are known constant matrices of appropriate dimensions.The system closed   by SOF is written as(3.15)x(k+1)=Aclx(k)+Bclw(k)+f(k,x,u),z(k)=Cclx(k)+Dclw(k),
where the following hold:(3.16)Acl=A+BuKCy,Bcl=Bw,Ccl=Cz+DzuKCy,Dcl=Dzw.The objective of this section is to design static output feedbackH∞ controllers for nonlinear discrete time system (3.14).Theorem 3.3.
System (3.14) is robustly static output feedback stabilizable with  ∥Twz∥∞<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:
(3.17){minβ[-Q00QAT+(BuRCy)TQFT(HRCy)TQCzT+(DzuRCy)T•-τγ2I0τBwT00τDzwT••-II000•••-Q000••••-βI00•••••-βI0••••••-τI]<0,
where
(3.18)Q=V[Q100Q2]VT.
The static output feedback gain is then  given by
(3.19)K=RUC0Q1-1C0-1UT,
where  U,V, and C0 are given in (2.21).Proof.
According to Theorem3.1, system (3.14) is robustly stable with ∥Twz∥∞<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:
(3.20){minβ[-Q00QAclTQMTQCclT•-τγ2I0τBclT0τDclT••-II00•••-Q00••••-βI0•••••-τI]<0.
Now:(i)
Acl,Bcl,Ccl and Dcl are replaced by (3.16),(ii)
M=(FHKCy),(iii)
Q is replaced by (3.18)  where : Q1∈ℜp×p, Q2∈ℜ(n-p)×(n-p).   After same direct developments the result follows.


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

In this paragraph, anH∞ robust control for nonlinear systems (3.14) improving the previous approach is proposed.Theorem 3.4.
System (3.14) is robustly SOF stabilisable with ∥Twz∥∞<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[:
(3.21)[-Q00μQ000Q•-τγ2I0τBwT00τDzwT0••-II0000•••-Q000AG-μG+BuRCy••••-βI00FG•••••-βI0HRCy••••••-τICzG+DzuRCy•••••••-G-GT]<0,
where
(3.22)G=V[G10G2G3]VT.
The static output feedback gain is given by
(3.23)K=RUC0G1-1C0-1UT,
with R∈ℜm×p, U,V,and C0 are given in (2.21).Proof.
According to the  Theorem3.3, the system (3.14) is robustly stable with  ∥Twz∥∞<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:
(3.24){minβ[-Q00μQ00Q•-τγ2I0τBclT0τDclT0••-II000•••-Q00(Acl-μI)G••••-βI0MG•••••-τICclG••••••-G-GT]<0,
where(i)
Acl,Bcl,Ccl, and Dcl are replaced by (3.16),(ii)
M=(FHKCy),(iii)
G replaced by (3.22), where G1∈ℜp×p,  G2∈ℜ(n-p)×p.The optimization problem (3.24) is expressed as follows:
(3.25){minβ<0[-Q00μQ000Q•-τγ2I0τBwT00τDzwT0••-II0000•••-Q000AG+BuKCyG-μG••••-βI00FG•••••-βI0HKCyG••••••-τICzG+DzuKCy•••••••-G-GT]<0.
For this reason,   we express differently the  KCyG term of the inequality (3.25), by using (3.22) and (2.21), in the same way as in (2.30). Consequently, we obtain
(3.26)KCyG=RCy.
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 Theorem2.12 is compared to the methods presented in [11].Example 4.1 (see [5]).
We consider the nonlinear discrete-time system of inverted pendulum (see Figure1).The system can be described by
(4.1)x(k+1)=Ax(k)+B(u(k)+αsiny(t)),y(k)=Cx(k),
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∈ℜ2×2,B∈ℜ2×1, and C∈ℜ1×2are given by:
(4.2)A=[10.095200.9048];B=[0.0480.952];C=[10].
The nonlinear function  f is given by the following expression:
(4.3)f(k,x)=αBsiny(k)=αBsinx1(k).
This function satisfies
(4.4)fT(k,x)f(k,x)=α2BTBsin2x1(k)=0.9α2sin2x1(k)≤α2xT[0.95000][0.95000]x.
In this case, we take  F=[0.95000];H=0.
The nonlinear system is unstable. The matrixA 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=αmax.We  can see a drastic improvement obtained by Theorem 2.12.Table 1
Numerical Evaluation for Example4.1.
ApproachαmaxKTheorem2.10  condition0.8254·10-4-1.9987Theorem2.12 condition for μ=0.70.06688-0.3101Figure 1
Inverted pendulum scheme.Figure 2
Outputy=x1.Figure 3
Outputy=x1.Example 4.2.
We consider now the nonlinear discrete-time system:(4.5)x(k+1)=[-0.07250.19571.59310.03010.04040.60840.3764-0.16350.9024]x(k)+[1.21230.28950.61740.06510.93790.5110]u(k)+f(k,x(k),u(k)),y(k)=[0.9313-0.7534-0.0335120.5]x(k),
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 Example4.2.
ApproachαmaxK[11]0.1954[-0.1329-0.1268-0.4309-0.0405]New approach, (2.25), Theorem 2.12 for μ=00.3084[-0.0019-0.2766-1.15420.1067]New approach, (2.25), Theorem 2.12 for μ=-0.350.3329[0.0357-0.3141-0.98120.0924]Example 4.3.
We consider the nonlinear discrete-time system (3.14) with
(4.6)A=[0.81890.08630.09000.08130.25241.00330.03130.2004-0.05450.01020.7901-0.2580-0.1918-0.10340.16020.8604],Bu=[0.00450.00440.10010.01000.0003-0.0136-0.00510.0926],Bw=[0.0953000.0145000.086200-0.001100],Cz=[10-1000000000],Dzu=[001001],Cy=[10000010],Dyw=0,
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.
ApproachαmaxKApproach, (3.17), Theorem 3.3 for  γ∞=1.9918,γ=2.23610.0032[-2.0416-0.17960.6545-1.0362]Approach, (3.17), Theorem 3.3 for  γ∞=1.9918,γ=50.0288[-2.1323-0.08081.1624-1.2381]Table 4
Numerical Evaluation for Example4.3 by Theorem 3.4.
ApproachαmaxKApproach, (3.21), Theorem 3.4 for  γ∞=1.9918,γ=2.2361,μ=0.220.3643[-0.2993-0.1328-0.0685-0.0743]Approach, (3.21), Theorem 3.4 for  γ∞=1.9918,γ=5,μ=0.24000.4087[-0.3376-0.1185-0.0660-0.0737]


## 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 incorporateH∞ 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.



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*Source: 486470-2009-07-14.xml*