Volume 1, Issue 1, Year 2012 - Pages 26-34

DOI: 10.11159/ijecs.2012.004

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

Karim Khayati, Riadh Benabdelkader

Royal Military College of Canada, Department of Mechanical and Aerospace Engineering, PO Box 17000, Station Forces, Kingston, Ontario, Canada K7K 7B4

karim.khayati@rmc.ca; riadh.abdelkader@rmc.ca

**Abstract** - *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.*

** Keywords:** Polynomial Systems, Matrix KP, Nonlinear State-Feedback, Stability, Sub-Optimal Control

© Copyright 2012 Authors - This is an Open Access article published under the Creative Commons Attribution License terms. Unrestricted use, distribution, and reproduction in any medium are permitted, provided the original work is properly cited.

#### 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 *x*and is
the non-redundant *j*-power of the vector *x*with 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 2*j*,
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 *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 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 2*j*+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 *β*is 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 *V*is a vector of dimension
, then is the -matrix verifying . Therefore it is called
the *mat*notation.

*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 of*T*_{1}, *T*_{2}, , *T*_{p}, denoted by , with and (Halmos, 1974).

*Notation 4: *For any
vector and
integers *p*and *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 *x*and *y*column-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 *x*given by with (Khayati & Benabdelkader,
2012a; Rotella & Tanguy, 1988).

For *Q*a symmetric non-negative definite
matrix of and
*R*a 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 *x*at
*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, *vec*and *mat*notations (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 , *P*is an SPD constant matrix of and *P*_{j}constant matrices of . Note that *V*(*x*)can be expressed in a
compact form

where

And equivalently, by
using the Cholesky decomposition, *P*_{1}exists *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 *mat*notation, 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 *mat*notation,
introduced in Section 2, we obtain from the nonlinear SDR equation (9)

where

#### 5. Determination of*P*_{p}

_{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 *P*_{1}is 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*=*P*_{1}^{T}P_{1}is SPD, we use (14), (16),
(18) and the *mat*notation
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

*vec*and

*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 *p*and is regular for *p*even and singular for *p*odd (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*, *P*_{2}, , *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*, *P*_{2}, , *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, *P*_{2}, , *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 _{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

*g*+ 1(where

*g*is the order of the term

*G*(

*x*)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.

#### 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 *B*and C_{1}by 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 *α**s.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*, *P*_{2}, , 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

*x*of 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

*f*odd, and and for

*f*even, respectively. We obtain

where <is the SMR matrix of the terms of order 2*i*in , 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 *β*are 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 *P*solution
of the ARE (19), given *α**s.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 *v*be 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_{t}is 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 *P*solution of the ARE
(19), given *α**s.t.*
and *P*_{2}, , 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 *β*and 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 *P*solution of the ARE (19), given *α**s.t.* and *P*_{2}, , 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 *x*_{1}is the angle of attack
in rad, *x*_{2}the pitch angle in rad, *x*_{3}the pitch rate in rad/secand *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 ,*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
*J*is evaluated
with different initial conditions in terms of angle of attack, *x*_{1}(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.

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