International Journal of Electrical and Computer Systems (IJECS)

ISSN: 1929-2716

1. Introduction

Numerous physical systems are very well known to be nonlinear by nature, but methods for analysing and synthesizing controllers for nonlinear systems are still not as well developed as their counterparts for linear models (Ekman, 2005). The investigation of new techniques for nonlinear problems such as the stability, the estimation and the control design remains a challenge until today (see e.g. (Zhu & Khayati, 2012; Zhu & Khayati, 2011; Won & Biswas, 2007; Khayati et al., 2006, Ekman, 2005)). In particular, to deal with the nonlinear optimal control problem, it has been stated in (Khayati, 2013) and references cited therein that a great variety of works shown in the literature used simple techniques, based on the local linearization, and more complex ones, such as (but not limited to) the state-dependent-Riccati (SDR) equation, the nonlinear-matrix-inequality- and frozen-Riccati-equation-based methods (Won & Biswas, 2007; Huang & Lu, 1996; Banks & Mhana, 1992). These methods could work well in some applications but rigorous theoretical proofs were lacking (Won & Biswas, 2007). The related grey area nevertheless covers the stability analysis of these closed loop controllers and also their implementation (complexity of the algorithms) within a large set of plants. These concerns have been discussed in separate works with a lot of compromises to achieve their goals (Won & Biswas, 2007; Ekman, 2005; Banks & Mhana, 1992).

Recently, the KP algebra has shown an important role in research activities dealing with control analysis and design (Mtar et al., 2009; Bouzaouche & Braik, 2006; Rotella & Tanguy, 1988). In these works, polynomial modelling structures represent the nonlinearities using the matrix KP and the vector power algebra (Steeb, 1997; Brewer, 1978). This modelling resembles the classical linearization, but with a difference. In fact, the order of truncation of the decomposition is high enough to represent closely and fairly the actual dynamics of the system.

In this paper, the optimal control for affine input nonlinear systems (i.e. linear w.r.t. the input but nonlinear in terms of the states (Rotella & Tanguy, 1988)) is considered. Such a large class contains well-known examples in control theory and many physical systems (e.g. mass-spring systems with softening/hardening springs, artificial pneumatic muscles, flight engine setups, etc.) (Chesi, 2009; Ekman, 2005; Banks & Mhana, 1992). The controller is developed using the well-known optimality conditions (Goh 1993; Borne et al., 1990; Rotella & Tanguy, 1988) by converting the nonlinear SDR equation into a set of algebraic equations using the KP algebra (Steeb, 1997; Rotella & Tanguy, 1988). The proposed method is using the same technique developed in (Rotella & Tanguy, 1988), but with a main difference of considering a given quadratic form for the cost index functional allowing the analysis of the stability of the optimal state-feedback (Goh, 1993). In fact, this analysis will show cases where the overall system will be globally asymptotically stable (GAS), or will estimate alternatively its DA and how much this domain can be large when the system is locally asymptotically stable (LAS) eventually. The stability and DA estimate features will be cast as convex problems that will be solved using LMI frameworks (Chesi, 2009; Chesi, 2005). Indeed, we will propose a technique that ensures the computation of the largest estimation of the domain of attraction (LEDA) using both the well-known complete square matrix representation (SMR) (Chesi, 2009; Chesi, 2003) and a new formalism of a complete rectangular matrix representation (RMR).

We will proceed as follows. In Section 2, we introduce a set of useful notations, definitions and properties regarding the matrix KP algebra, the vector power series and the SMR/RMR formulations. Section 3 is devoted to the problem statement of the nonlinear dynamics, the nonlinear quadratic cost functional to be optimized and the related optimality conditions. In Section 4, we introduce an LF-based optimal cost index that will be used in the transformation of the polynomial SDR equation. Then, Section 5 deals with the computation of a closely acceptable solution to this nonlinear equation in the unknown constant matrices, while in Section 6, an analytic and practical form of the state-feedback sub-optimal control is developed. Section 7 introduces the stability issue of the designed sub-optimal closed-loop. Moreover, in Section 8, we discuss the computation of the LEDA of this closed loop system. Finally, to illustrate the proposed technique, numerical and comparative results are presented in Section 9, while Section 10 concludes this work.

2. Useful Notations, Definitions and Proprieties

Notations and properties of matrices, vectors, dot product and KP tensors used in this paper are exhaustively discussed in the literature; e.g. (Schott, 2001; Steeb, 1997; Brewer, 1978). The proofs of the new lemmas introduced in this Section are based on theorems introduced in these references. Due to lack of space, all these theorems as well as the proofs of the lemmas shown below are omitted.

2. 1. Definitions

Definition 1: For any vector and any integer j, is the j-power of a vector xand is the non-redundant j-power of the vector xwith standing for the binomial coefficient. We have , s.t. (Mtar et al., 2009; Brewer, 1978).

Definition 2: Let w(x)be any homogenous form of degree 2j, then the SMR of w(x)in any is given by (Chesi, 2005; Chesi, 2003). is considered a base vector of the homogenous function of degree jin x. Wis a suitable but non-unique symmetric matrix SMR, also known as Gram matrix. All matrices Wcan be linearly parameterized as , where is a free vector with . is a linear parameterization of the set . We refer to W(β)as the complete SMR of w(x).

Definition 3: Let w(x)any form of degree 2j+1in given by , where . Using theorem T2.13 of (Brewer, 1978), w(x)can be written using a new formulation given by RMR as , with and . Then, similarly to the homogenous forms of even order shown above, we propose a complete RMR of w(x)as , where βis a vector of free parameters. is a linear parameterization of the set . We refer to as the complete RMR of w(x). The following two examples illustrate this new formulation.

Example 1:Consider the form of degree 3in two variables . Noting and , we obtain, for , M+L(β)=.

Example 2: Consider the form of degree 3in three variables . Noting and , we obtain, for , M+L(β)=.

2. 2. Notations

Notation 1: If Vis a vector of dimension , then is the -matrix verifying . Therefore it is called the matnotation.

Notation 2: M⁺stands for the Moore-Penrose pseudo-inverse of any full rank matrix M.

Notation 3: Given ,for any integer , we denote by and . We have where is the direct sum ofT₁, T₂, , T_p, denoted by , with and (Halmos, 1974).

Notation 4: For any vector and integers pand μ, we denote by .

2. 3. Lemmata

Lemma 1: and (Khayati & Benabdelkader, 2012a),

where is given by and therefore called the j-differential Kronecker matrix. I_n(resp. ) denotes the identity matrix of (resp. ), the permutation matrix of (Rotella & Tanguy, 1988; Brewer, 1978). Equivalently, can be derived from

Lemma 2: For xand ycolumn-vectors of and respectively and for any matrix , we have (Khayati & Benabdelkader, 2012a)

Lemma 3: Consider a matrix . Let be a partition of A, i.e. , . We have (Khayati & Benabdelkader, 2012a)

3. Problem Statement

Consider the nonlinear system given by

where designates the time, the state vector, the input vector. and for are analytic vector fields from into expressed as polynomials in x. Note that . By using the KP tensor, we write , and then, , with , and . Let be a vector field in the state vector xgiven by with (Khayati & Benabdelkader, 2012a; Rotella & Tanguy, 1988).

For Qa symmetric non-negative definite matrix of and Ra symmetric positive definite (SPD) matrix of , we propose the design of a state feedback which minimizes the continuous-time cost functional

We denote by V(x)the optimal cost with an initial condition xat t(Goh, 1993; Borne et al., 1990)

where is the optimal control. The optimality conditions, corresponding to the problem (5) and (6), are given by (Borne et al., 1990)

where denotes the derivative of V(x)w.r.t. the state vector x; i.e. .

4. Quadratic Cost Function Representation

Based on the optimality conditions discussed in (Borne et al., 1990; Rotella & Tanguy, 1988), we build the following procedure to obtain a suboptimal state feedback in a polynomial form using the KP tensor, vecand matnotations (Khayati & Benabdelkader, 2012a). Such a design is based on the determination of the cost function V(x)in a quadratic form. In fact, this function would be expected to satisfy the conditions of any Lyapunov candidate function (Goh, 1993). We propose (Khayati & Benabdelkader, 2012a)

with , Pis an SPD constant matrix of and P_jconstant matrices of . Note that V(x)can be expressed in a compact form

where

And equivalently, by using the Cholesky decomposition, P₁exists s.t. , then the cost function V(x)can be rewritten in a summation form as

with

The expression of V(x)given by (13) and (14) will be advantageous to solve the nonlinear SDR (9). Using theorems T2.3 and T4.3 in (Brewer, 1978) and applying lemmas 1, 2 and 3 and the matnotation, introduced in Section 2, we obtain the derivative of (13) w.r.t. x

with

where is the square j-differential Kronecker matrix of introduced in lemma 1 (see Section 2). Using the KP tensor, the theorem T2.13 of (Brewer, 1978), the lemmas 2 and 3, and the matnotation, introduced in Section 2, we obtain from the nonlinear SDR equation (9)

where

5. Determination ofP_p

In this Section, the matrices P_p, for , will be computed from (17) by cancelling the coefficients of . The details of such steps, based on the KP notations and theorems introduced in (Steeb, 1997; Brewer, 1978) as well as the lemmas 1, 2 and 3 shown in Section 2, are omitted due to lack of space.

First, the matrix P₁is obtained by cancelling the terms of , in (17). The operator is linear on matrices of the same dimensions. Noting that the first differential Kronecker matrix is given by and that P=P₁^TP₁is SPD, we use (14), (16), (18) and the matnotation to obtain the classical algebraic Riccati equation (ARE)

And thence, for a given , the calculation of P_p, , is obtained from (17) by cancelling the coefficients of . Using vecand mat notations, theorems T1.5, T1.6, T3.2, T3.4 of (Brewer, 1978) and the iterative form of the differential Kronecker matrix (2), we combine (14), (16) and (18) to obtain

where and . Note that is a Hurwitz matrix, then is regular for all . is a singular matrix for all nonzero integers pand is regular for peven and singular for podd (Khayati & Benabdelkader, 2012a; Rotella & Tanguy, 1988). Using the non-redundant vector power notation (Bouzaouche & Braiek, 2006), and the theorem T3.4 of (Brewer, 1978), we write where is the transformation matrix defined in Section 2 (Bouzaouche & Braiek, 2006). Two cases arise depending on p:

Case Ι pis even: Let be a full rank rectangular matrix. We obtain

If P, P₂, , P_p-1are known, can be calculated as a solution of the linear equation (21). Thus, is deduced. In fact, by using the Moore-Penrose pseudo-inverse of , we obtain

Case ΙΙ pis odd: Eq. (17) is rewritten using the non-redundant power series. Then, the coefficients of are given in (20), but multiplied by on the left hand side. Thus, this linear equation becomes

where > is a full rank rect-angular matrix and . If P, P₂, , P_p-1are known, can be calculated as a solution of the linear equation system (25). Thus, is deduced. In fact, by using the Moore Penrose pseudo-inverse of , denoted by , we obtain

6. Implementation of the State Feedback

Consider the nonlinear dynamics (5). The optimal control minimizing the functional cost (6) is obtained by the optimality conditions (8) and (9). We propose the design of a practical sub-optimal control using the matrices i>P, P₂, , P_p-1computed in Section 5. It is based on an approximated optimal cost V(x)given by (10). An analytical form of the state feedback can be obtained by using (8), (15), (16) and (18) (Khayati & Benabdelkader, 2012a)

with and

The KP tensor is used here to design a systematic computation of a sub-optimal state-feedback. The proposed nonlinear feedback (25) with (26) would not necessarily be implemented with a great number of computed matrices P_pto be so different from the linear control approximation, a priori. According to (Rotella & Tanguy, 1988), it can be concluded that the state-feedback obtained with only P(i.e., only the first order of the SDR equation) is more efficient than the solution issued from the linearized system. In fact, by computing only P, we may obtain a polynomial sub-optimal control of order g + 1(where gis the order of the term G(x)in (5)), in particular, when gis non-zero. The stability of the proposed closed-loop feedback (5) and (25) will be discussed in the following section.

7. Stability of the Sub-Optimal State Feedback

To investigate the stability of the closed loop system, we consider V(x), given by (10), as a Lyapunov candidate function. V(x)is a radially unbounded continuous function, and its derivative exists and is continuous. From (10), if

holds, then the Lyapunov candidate function V(x)is positive definite; that is , . Note that (27) is equivalent to . The time derivative of the LF V(x), along the trajectories of the closed loop system (5) and (25), is given by

Let us define Band C₁by and , respectively. We assume the triplet is stabilizable-detectable. Note that if a solution Pof the ARE (19) exists, then it is the unique SPD matrix solution of the optimal control for the linearized system and is a Hurwitz matrix (Rotella & Tunguy, 1988). Thus, the linearized system is asymptotically stable. Moreover, the nonlinear closed loop system (5) and (25) is LAS and s.t..

In the following, we assume and consider the closed ball . Given αs.t. ; i.e. for all nonzero , is an estimate of the DA if (Chesi, 2009; Chesi, 2003). The computation of the maximum s.t. , i.e. (5) and (25) is LAS, corresponds to the LEDA of the closed-loop dynamics and is given by where (Chesi, 2009)

8. LEDA Computation of the Closed Loop System

In this section, we present the mechanism to evaluate the LEDA of the obtained sub-optimal closed-loop system. Let be a given sphere. The problem (29) turns out that (Chesi, 2003)

We assume that P, P₂, , are obtained from (19), (21) and (23). The terms and are polynomials in x of degrees and , respectively. For any , we have

with , where V_ij is given by (16). Using the non-redundant vector power series and the vector notations introduced in Section 2, without loss of generality, we assume that , with , and s.t. . The terms and are polynomials in even and odd vector powers in xof orders and , respectively. and are integers s.t. and . Then, we use the SMR and RMR notations introduced in Section 2 to set the time derivative of the LF, , in a quadratic form. If we denote by and for fodd, and and for feven, respectively. We obtain

where <is the SMR matrix of the terms of order 2iin , a free vec-tor with and >stands for binomial coefficients (Mtar et al., 2009), and is the RMR of terms of order in (see Section 2). (32) can be rewritten as follows

where and

The decision variables βare set by the concatenation of all free variables and , . Two cases arise depending on the size of the values of and in (33).

8. 1. Case of

Using the transformation introduced in Section 2, we have if the LMI

holds in the free decision variable β. Thus, for Psolution of the ARE (19), given αs.t. and computed from (21) and (23), if the LMI (35) problem is feasible in β, then the sub-optimal state-feedback (5) and (25) is GAS.

8. 2. Case of

Let vbe the least common multiple of and , i.e. s.t. . Consider the well-posed vectors and introduced in Section 2. Noting , , then we have , and . Thus, (33) is equivalent to

with . is the pseudo-inverse of introduced in Section 2, , . Noting that T_tis SPD, let be the Cholesky decomposition. Then, from (36), , is equivalent to

where the factor depends on . If , then and monotically increasing with . If , then and monotically decreasing with . The following results hold.

Sub-case : , s.t. the LMI (37) holds. Thus, for Psolution of the ARE (19), given αs.t. and P₂, , computed from (21) and (23), the sub-optimal state-feedback system (5) and (25) is GAS.

Sub-case : Given , consider the LMI

in the vector βand the scalar . If s.t. the LMI (38) holds, then the LMI constraint (37) holds , then we select and we have decreasing with (i.e. as ). Thus, for Psolution of the ARE (19), given αs.t. and P₂, , computed from (21) and (23), if the LMI (38) is feasible in and β, then the sub-optimal state-feedback system (5) and (9) is GAS.

Sub-case and s.t. : Alower bound γ, given by (30), is computed by , where is a solution of the following eigen-value problem (EVP): subject to and LMI (38). If of this EVP is negative, then the linear inequality constraint corresponds to as .

Remark: The results discussed above can be proven using simply the theorem 1 of (Chesi, 2003) and the proposition 2 of (Chesi, 2005).

9. Example

As an example, we consider the design of a nonlinear aircraft flight control problem which has been exhaustively treated in literature (see e.g. (Banks & Mhana, 1992)) and defined by

where x₁is the angle of attack in rad, x₂the pitch angle in rad, x₃the pitch rate in rad/secand uthe control input provided by the tail deflection angle in rad(Banks & Mhana, 1992). Note that terms involving nonlinearities in uwith small effect on the dynamics are eliminated, as the approaches discussed here cannot account for nonlinear control terms, but are taken into consideration in the simulations. The performance index uses ,Q=and R=1. The simulations have been applied for the proposed LF-based technique as well as the linear control Lin where the dynamics is linearized about the origin, the KP-based design introduced in (Rotella & Tanguy, 1988) and the SDR-equation-pointwise-based (referred to as PW) technique (Banks & Mhana, 1992). The sub-optimal cost Jis evaluated with different initial conditions in terms of angle of attack, x₁(0), and same for the different methods. Table 1 shows the cost performance errors in %. The LF- (of orders 2 and 3), KP- (of orders 2 and 3) and Lin-based design costs are compared to the PW-technique one. A positive value corresponds to an improvement (i.e., a lower cost) with the given method compared to the PW cost, meanwhile a negative value corresponds to a higher cost. Figures 1-3 show the control variable, the angle of attack and the pitch angle, respectively, obtained with the initial condition . Due to lack of space the pitch rate figure is omitted. Curves of LF-based design, with orders of truncation 2 and 3, overlap almost during all the time showing very similar results in terms of transient behaviour and stability. Furthermore, the proposed design (with both orders 2 and 3which are relatively small) exhibits a significant added-value in terms of cost estimation and domain of attraction interval performances compared to the other methods.

Table 1. Cost index and cost errors (expressed in % of )
, , , , .

	0.0016	20.2	18.6	-0.6	-0.8	0.0
	0.0071	23.8	22.8	-1.6	-2.6	-0.2
	0.0196	30.9	30.3	-3.7	-6.8	-0.7
	0.0519	46.3	45.7	-13.3	-31.7	-4.3
	0.1056	48.3	46.3	Unstab.	Unstab.	Unstab.
	0.4081	71.4	65.6	Unstab.	Unstab.	Unstab.
	1.6170	58.5	50.9	Unstab.	Unstab.	Unstab.

10. Conclusions

A new nonlinear optimal control design for polynomial systems subject to nonlinear cost objectives is proposed. We develop a systematic and practical LF-based sub-optimal control approach using the KP notations. The analysis of the stability of the closed loop system is then discussed using LMI frameworks. The problem of the LEDA computation is cast as a convex EVP design. This method is expected to ensure a best compromise between the feasibility of the implemented scheme and the stability analysis of the overall system. An example showing simulations and comparative results successfully demonstrates the effectiveness of this technique. Furthermore, a modified version of this nonlinear optimal control will be presented to relax the conditions within the computation of the Lyapunov function matrices of high order, and also, improving the formulation of the stability feature (Khayati, 2013). Nevertheless, all those changes will be proposed by following the same overall procedure discussed in this paper.

References