Nonlinear Sub-Optimal Control for Polynomial Systems – Design and Stability

Many real world systems are inherently nonlinear. Therefore, the linear quadratic regulator theory is rarely efficient for these systems. In this paper, we propose the design of an optimal feedback control for polynomial systems in the indeterminate state variables. To deal with the case of a nonlinear infinite-horizon-cost-functional, we investigate the control based on the Lyapunov functions (LF) and by using the Kronecker product (KP) algebra. Then, we analyze the stability of the feedback and its domain of attraction (DA) in form of convex problems based on the linear matrix inequality (LMI) formalism. The practical sub-optimal control is evaluated through simulation results and comparative schemes.


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 statedependent-Riccati (SDR) equation, the nonlinear-matrixinequality-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 statefeedback 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.

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.

1. Definitions
Definition 1: For any vector  n x and any integer j ,  j j n x is the j -power of a vector x and is the non-redundant j -power of the vector x with   n j  standing for the binomial coefficient.We have  j , Mtar et al., 2009;Brewer, 1978).
Definition 2: Let   wx be any homogenous form of degree 2 j , then the SMR of   wx in any  n x is given by   j T j w x x Wx  (Chesi, 2005;Chesi, 2003).j x is considered a base vector of the homogenous function of degree j in x .
W is a suitable but non-unique symmetric matrix SMR, also known as Gram matrix.All matrices W can be linearly parameterized as

 
, where given by Using theorem T2.13 of (Brewer, 1978),   wx can be written using a new formulation given by RMR as . Then, similarly to the homogenous forms of even order shown above, we propose a complete RMR of , where Therefore it is called the mat notation.
Notation 2: M  stands for the Moore-Penrose pseudoinverse of any full rank matrix M .(Halmos, 1974).
Notation 4: For any vector  n x and integers p and  , we denote by

Problem Statement
Consider the nonlinear system given by

Gx
. By using the KP tensor, we write and then,   0 () vector field in the state vector x given by   & Benabdelkader, 2012a;Rotella & Tanguy, 1988).
For Q a symmetric non-negative definite matrix of qq  and R a symmetric positive definite (SPD) matrix of mm  , we propose the design of a state feedback which minimizes the continuous-time cost functional We denote by   Vx the optimal cost with an initial condition x at t (Goh, 1993;Borne et al., 1990) where   arg min u uJ   is the optimal control.The optimality conditions, corresponding to the problem ( 5) and ( 6), are given by (Borne et al., 1990) where   x Vx denotes the derivative of   Vx w.r.t. the state vector x ; i.e.  

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, vec and mat notations (Khayati & Benabdelkader, 2012a).Such a design is based on the determination of the cost function   Vx 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   , P is an SPD constant matrix of nn  and j P constant matrices of for 1 and 2 for 2 and 1 The expression of   Vx 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 mat notation, introduced in Section 2, we obtain the derivative of (13) w.r.t. x where () n j D is the square j -differential Kronecker matrix of jj nn  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 mat notation, introduced in Section 2, we obtain from the nonlinear SDR equation ( 9) where

Determination of p P
In this Section, the matrices p P , for 1, , pp  , will be computed from ( 17) by cancelling the coefficients of 1 p x  .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 1 P is obtained by cancelling the terms of 2 x , in (17).The operator

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 P , 2 P , …, p P computed in Section 5.It is based on an approximated cost   Vx given by ( 10).An analytical form of the state feedback can be obtained by using ( 8), ( 15), ( 16) and ( 18) (Khayati & Benabdelkader, 2012a)   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 P to 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 1 g  (where g is the order of the term

 
Gx in ( 5)), in particular, when g is non-zero.The stability of the proposed closed-loop feedback ( 5) and ( 25) will be discussed in the following section.

Stability of the Sub-Optimal State Feedback
To investigate the stability of the closed loop system, we consider   Vx, given by ( 10 Vx, along the trajectories of the closed loop system ( 5) and ( 25), is given by ,, F B C is stabilizable- detectable.Note that if a solution P of the ARE ( 19) exists, then it is the unique SPD matrix solution of the optimal control for the linearized system and   11 F B P  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 0 x  In the following, we assume Chesi, 2009;Chesi, 2003).The computation of the maximum  s.t.

B
, i.e. (5) and ( 25) is LAS, corresponds to the LEDA of the closed-loop dynamics and is given by    B where (Chesi, 2009)

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 , 2 P , …,   , we have , where ij V is given by ( 16).Using the nonredundant vector power series p x and the vector notations p X introduced in Section 2, without loss of generality, we assume that  t p , with 0 tg pp , and 0 for f even, respectively.We obtain where

S
is the SMR matrix of the terms of (Mtar et al., 2009), and 32) can be rewritten as follows The decision variables  are set by the concatenation of all free variables   where the factor    21) and ( 23), the sub-optimal state-feedback system (5) and ( 25) is GAS.  (38  holds, then the LMI constraint (37) holds 0  , then we select 1

Sub
  and we have  decreasing with  (i.e.  as 0    21) and ( 23), if the LMI ( 38) is feasible in 0   and  , then the sub-optimal state-feedback system (5) and ( 9) is GAS.
 of  , given by ( 30), is computed by , where  is a solution of the following eigen-value problem (EVP): max

 
subject to 10     and LMI (38).If arg max  of this EVP is negative, then the linear inequality constraint 10     corresponds to 1   as st pp  .Remark: The results discussed above can be proven using simply the theorem 1 of (Chesi, 2003) and the proposition 2 of (Chesi, 2005).

Example
As an example, we consider the design of a nonlinear aircraft control problem which has been exhaustively treated in literature (see e.g.(Banks & Mhana, 1992) x is the angle of attack in rad , 2 x the pitch angle in rad , 3 x the pitch rate in rad sec and u the control input provided by the tail deflection angle in rad (Banks & Mhana, 1992).Note that terms involving nonlinearities in u with 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  

H x x
 ,  Q  3 0.25 I and 1 R  .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 J is evaluated with different initial conditions in terms of angle of attack,  

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.

.
), as a Lyapunov candidate function.  Vx is a radially unbounded continuous function, and its derivative exists and is continuous.From (10The time derivative of the LF   ) in the vector  and the scalar  .If 0   s.t. the LMI 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   1 0 23 x  .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 3 which are relatively small) exhibits a significant added-value in terms of cost estimation and domain of attraction interval performances compared to the other methods.

Consider the form of degree 3 in three variables
, we write ).Thus, for P solution of the ARE (19), given  s.t.
p P computed from (

Table 1 .
Cost index PW J and cost errors (expressed in % of PW J ) PWJ