Comparison of LPV and nonlinear system theory: Arealization problemJuri Belikov ∗, Ülle Kotta, Maris TõnsoInstitute of Cybernetics, Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia
a r t i c l e i n f o
Article history:Received 17 October 2012Received in revised form17 October 2013Accepted 25 October 2013Available online 31 December 2013
The paper explores the utilization of the nonlinear realization theory to address the problem of trans-forming linear parameter-varying input–output (LPV-IO) equations into a state-space form with staticdependence on the so-called scheduling parameter. The necessary and sufficient solvability conditionsare given, and three additional subclasses of LPV-IO equations are suggested that are guaranteed to havea realization of the considered type.
The linear algebraic approach based on differential one-forms[1] has been successfully applied so far to solve a number of non-linear control problems, including controllability [2], reduction [3]and realization [4,5] of nonlinear i/o differential or differenceequations, accessibility and feedback linearization of state equa-tions [6], linear i/o equivalence [7] and transfer equivalence [4].Thus, working with differentials of nonlinear system equationsrather thanwith equations themselves has proved itself as a practi-cal and reliablemathematical tool. This approach is especiallywell-suited for checking (generic) necessary and sufficient solvabilityconditions of various control problems. However, in order to findthe control law to be implemented, or equivalent reduced systemequations, for example, one has at the last step to integrate certaindifferential one-forms to get back to the equation level. The inte-gration of (integrable in principle) differential one-forms is knownto be a difficult task, in general.
Another approach in control theory is based on employing thetools of linear parameter-varying (LPV) framework. In this case,the relations between the system inputs, outputs, and states areconsidered to be linear, but the model parameters are assumed tobe the functions of a time-varying signal, the so-called schedulingvariable p. The success of the LPV framework can be explained bythe fact that it has been developed with a clear orientation towardpractical applications. Moreover, its tools extend the well-known
linear time-invariant (LTI) approaches, being inherently simplercompared with their nonlinear counterparts [8]. The LPV approachis attractive in the sense that, compared to the nonlinear theory,it allows us to work with equations in their original form ratherthan with the so-called tangent linearized description of the sys-tem. The results have already been used, though many aspects ofthe framework require further development and improvement [8].
The goal of this paper is to study how nonlinear realization the-ory can contribute to the LPV theory. The choice of this problem ismotivated partly by the fact that, in both cases, the problem hasbeen studied via non-commutative polynomial methods and theobtained results are very similar; see [4,8,9] for the nonlinear andLPV cases, respectively. Note that in the LPV case, one is interestedto get state equations with static dependence on the schedulingvariable p. In this paper, a detailed analysis of two of such sub-classes of LPV models from [8] is justified by the known resultsfrom the nonlinear theory. In addition, three new subclasses ofLPV-IO models in a special form are suggested that have the stateequations without a dynamic dependence on p. Necessary and suf-ficient conditions are provided for LPV-IO equations to be realiz-able in a state-space form with only static dependence on p.
Finally, let us mention that while in this paper we focus on re-alization from the i/o representation of the system, given in termsof the higher order difference equation, there is a rich literature onrealization from i/o (response) maps (operators). This includes theearly papers [10–12] as well as the recent ones addressing the hy-brid systems [13] and systems with a certain algebraic structurelike polynomial or Nash systems [14] and the references therein.Whether this theory can contribute for LPV systems, is an openquestion that we do not pursue here.
J. Belikov et al. / Systems & Control Letters 64 (2014) 72–78 73
The paper is organized as follows. Section 2 recalls the algebraicframework of differential one-forms and presents the polynomialformalism that allows us to analyze nonlinear and LPV systems ina unified manner. In Section 3, the main notations and definitionsfrom the LPV framework are presented. The next section is devotedto the analysis of existing and new realizable subclasses togetherwith the extension of the tools developed within the nonlineartheory to the case of LPV systems. Concluding remarks are drawnin the last section.
2. Nonlinear realization theory
Hereinafter, if a time-dependent variable is denoted as ξ(t),then ξ [k] stands for the kth-step forward time shift ξ(t + k) andξ [−l] for the lth-step backward time shift ξ(t − l) with k, l ∈ Z+.Consider a nonlinear SISO discrete-time system, described by thei/o difference equation
y[n]= φ
y, y[1], . . . , y[n−1], u, u[1], . . . , u[s] , (1)
where u(t) and y(t) are the input and output signals of the system,respectively; φ is supposed to be a real-analytic function. More-over, we assume that s < n are non-negative integers. The solu-tion of the realization problem is given in the framework of thelinear algebraic approach [1]. The latter is focusing on generic sys-tem properties that hold on open and dense subsets of suitabledomains of definition, provided that they hold at some points ofsuch domains. The generic approach motivates the choice of ana-lytic functions in the system description (1), see [1, Chapter 1] formore details.
The assumption below is a standard assumption made in mostpapers and is not restrictive as it is the necessary condition forsystem accessibility.
Assumption 1. System (1) is submersive, if the map φ satisfiesgenerically the condition
∂φ
∂(y, u)≡ 0. (2)
Recall briefly the algebraic formalism from [15] that we use inthis paper. Denote by K the field of meromorphic functions, de-fined by system (1), in a finite number of (independent) variablesfrom the set C = {y, . . . , y[n−1], u[k], k ≥ 0}, and introduce theforward-shift operator σ : K → K . In particular, σ(y[n−1]) :=
φ(·), meaning that y[n] as a dependent variable has to be replacedbyφ(·) from (1). For the remaining elements ofC, the forward-shiftis defined in a standard manner, i.e., σy[α]
:= y[α+1], α = 0, . . . ,n − 2, σu[β]
:= u[β+1], β ≥ 0, where y[0]= y and u[0]
= u. More-over, the application of σ to ϕ ∈ K is defined by shifting the argu-ments of the function according to the rules described above, i.e.,
σϕ(y, . . . , y[n−1], u, . . . , u[l+1])
:= ϕ(y[1], . . . , φ(·), u[1], . . . , u[l+1]).
Under Assumption 1, there exists an inversive difference over-field1 of (K, σ ) such that σ , when extended to this overfield,becomes an automorphism [6,16]. Therefore, σ has an inverse op-erator σ−1, interpreted as a backward-shift operator. For construc-tion of backward-shift operator σ−1, see [6,17].
Consider the infinite set of symbols dC = {dy, dy[1], . . . ,dy[n−1], du[l], l ≥ 0} and denote by E the vector space over the fieldK spanned by the elements of dC, i.e., E = spanKdC. For F ∈ K ,define the operator d : K → E as follows dF :=
n−1i=0
∂F∂y[i]
dy[i]+k
j=0∂F∂u[j] du[j]. One says that dF is a total differential (or simply the
1 With a slight abuse of notation, for the field extension, we use the same symbol.
differential) of the function F . Note that any element in E is a vec-tor of the form ω =
i αidζi with dζi ∈ dC and αi ∈ K . Then the
operators σ : K → K and σ−1: K → K induce, respectively,
the operators σ : E → E and σ−1: E → E by
σ(ω) :=
i σ(αi)d(σ (ζi)) and σ−1(ω) :=
i σ−1(αi)d(σ−1(ζi)).
A sequence of subspaces H1 ⊃ · · · ⊃ Hk∗ ⊃ Hk∗+1 = Hk∗+2 =
· · · =: H∞ of E is defined as
H1 = spanK
dy, . . . , dy[n−1], du, . . . , du[s] ,
Hk+1 = {ω ∈ Hk | σ(ω) ∈ Hk} , k ≥ 1.(3)
Definition 1. The state-space description
x[1]= f (x, u)
y = h(x)(4)
is said to be realization of the i/o equation (1) if elimination of thestate variables x(t) ∈ Rn in (4) yields the i/o equation having thesame solution sets {u(t), y(t), t ≥ 0} (in the strong sense) as (1).
The system (4) is said to be realizable in the classical state-spaceform if it admits a realization in the sense of Definition 1. Accordingto [18], realization (4) is called generically single-experiment ob-servable, if the rank of the observability matrix generically equalsto n, i.e., if
rankK
∂(h(x), σ (h(x)), . . . , σ n−1(h(x)))
∂x
= n. (5)
Note that by the properties of the shift operator σ , σ k(h(x)) =
h(x[k]) in (5) for k = 1, . . . , n − 1, where x[k] has to be replacedvia (4) by a function depending on x, u, . . . , u[k−1].
We say thatω ∈ E is an exact one-form, ifω = dζ for some ζ ∈
K . A one-formω, forwhich dω = 0, is said to be closed. Every exactone-form is closed, but the converse holds only locally; see [1]. Asubspace is said to be integrable, if it has a basiswhich consists onlyof closed one-forms, and can be checked by the Frobenius theorembelow, where the symbol dω denotes the exterior derivative of theone-form ω and ∧ means the exterior or wedge product.
Theorem 1 ([19]). Let V = spanK{ω1, . . . , ωr} ⊂ E , where ω1,. . . , ωr are linearly independent over K . The subspaceV is integrableiff dωk ∧ ω1 ∧ · · · ∧ ωr = 0 for all k = 1, . . . , r.
Consider a polynomial in the form
λ(z) = λkzk + λk−1zk−1+ · · · + λ1z + λ0, (6)
where λl ∈ K, l = 0, . . . , k, and z is a formal variable (polynomialindeterminate). Note that λ(z) = 0, iff at least one of the functionsλl, for l = 0, . . . , k, is non-zero. In what follows, z will be inter-preted as the forward-shift operator.
Definition 2. The left skew polynomial ring induced by (K, σ ) isthe ringK[z; σ ] of polynomials in the form (6) with usual additionand multiplication, satisfying the relation
z · ξ = σ(ξ)z (7)
for any ξ ∈ K ⊂ K[z; σ ].
A skew polynomial λ(z) ∈ K[z; σ ] may be interpreted as anoperator λ(z) : E → E , satisfying
z idy := dy[i], z jdu := du[j], (8)
for i = 1, . . . , n − 1 and j ≥ 0. For λ(z) =k
l=0 λlzl, we define
λ(z)(γ dζ ) :=k
l=0 λl(zl· γ )dζ with γ ∈ K and dζ ∈ {dy, du}.
By differentiating the i/o equation (1) and using relations (8),we obtain the polynomial infinitesimal system description asa(z)dy + b(z)du = 0 (9)
with a(z) = zn −n−1
i=0 aiz i, b(z) = −s
j=0 bjzj and ai =
∂φ
∂y[i]∈
K, bj =∂φ
∂u[j] ∈ K .
74 J. Belikov et al. / Systems & Control Letters 64 (2014) 72–78
K[z; σ ] is defined as σ−1c (λ(z)) = σ−1(λ(z)− λ0).
Iterated k-fold application of operator σ−1c , denoted as σ−k
c ,obeys the following elementary property which is used in furthercalculations. Let r(z) =
ζ
i=0 rizi∈ K[z; σ ], and thenσ−l
c (r(z)) =ζ
i=l(σ−lri)z i−l.
Theorem 2 ([4]). The nonlinear system, described by the i/o differ-ence equation (1), has an observable state-space realization iff thesubspace Hs+2 = spanK{ω1, . . . , ωn}, where
ωl = σ−lc
a(z) b(z)
dydu
, l = 1, . . . , n, (10)
is integrable.
Remark 1. Note that for the realizable i/o model (1), the state co-ordinates xl, l = 1, . . . , n can be obtained by integrating the exactbasis vectors ofHs+2. However, the one-formsωl, found by formula(10), are not necessarily always exact. In such a case, one has to findfor Hs+2 a new (locally) exact basis, taking linear combinations ofωl over the field K .
Remark 2. Note that the realizability condition of the i/o equation(1) in the state-space form (4) depends on the number of shifts ofthe control variable in (1). For example, if the i/o equation dependsonly on u (and not on the higher order shifts), then the i/o equationhas always a state-space realization, since (Hs+2 =)H2 is alwaysintegrable. In general, one has to check integrability of Hs+2.
3. Realization of LPV i/o representation
Consider next the i/o representation of an LPV system [8]
(a(z) � p)y + (b(z) � p)u = 0, (11)
whereu, y, n, and s are as above; a(z) := zn+n−1
i=0 aiz i and b(z) :=sj=0 bjz
j, a(z), b(z) are polynomials with parameter-varying co-efficients. Moreover, p ∈ P is a scheduling variable, P is the so-called scheduling space2 and ξ �p = ξ(p, p[1], p[−1], . . .), where p[l]
means the application of the forward-shift operator to the schedul-ing variable, i.e., p[l]
:= z lp. It is usually assumed that p has to beconsidered as measurable and free signal of the control system. Incase p is a function of the inputs, outputs or states, the LPV systemis called a quasi-LPV system [8].
Another common way to describe LPV systems is by the use ofa state-space representation
x[1]= (A � p)x + (B � p)u
y = (C � p)x,(12)
where entries of matrices A, B, C,D are rational functions withdynamic dependence on the scheduling signal. Note that if the ma-trices depend only on the instantaneous values of p, then the de-pendence is called static.
Next, we recall the theorem from [8] that allows us to find thestate equations for the i/o equation (11). Note that the applicationof Theorem 3, in general, results in the state-space form (12) withdynamic dependence on p.
2 Note that in [8] a slightly more general class was considered in which thefeedthrough term for the case n = s was included in (11). We removed thisterm, introducing the constraint s < n. Moreover, we assumed that an = 1 inorder to compare the results for (11) with those for systems of the form (1). Theserestrictions will not affect the results of this paper. Moreover, in [8] the differentnotationswere used to represent the polynomials and the cut-and-shift operatorσc .
Theorem 3 ([8]). The state coordinates for the i/o equation (11) aredefined by
xn−l = σ−lc
a(z) b(z)
yu
, l = 1, . . . , n. (13)
4. Main results
Observe that the results of Theorems 2 and 3 are very similar.In both cases the result is given in terms of the cut-and-shift oper-ator, applied to the skew polynomial (vector). The main differenceis that whereas in the LPV case the formula (13) defines directlythe state coordinates, in the nonlinear case the formula only yieldsthe one-forms that have to be integrated (whenever possible, seeRemark 1) to get the state coordinates. Note also that integrabilityrestriction means that not all nonlinear i/o equations can be trans-formed into the state-space formwhereas all LPV-IOmodels can beconverted into LPV state-space form (12). Of course, this LPV state-space model, as shown in [8], depends in general, dynamically onthe scheduling parameter p, or said in other words, the matrices instate equations depend also on shifted values of the parameter p.Such state-space model is of high complexity and therefore, one isinterested to distinguish the LPV-IO model classes that will yieldstate equations with static dependence on p. A number of suchsubclasses was suggested in [8]. In the next subsections, we willshowhownonlinear realization theory can be applied to justify thesuggested subclasses and to develop three additional subclasses ofsuch i/o models.
4.1. Shifted form
Consider the special case of the LPV-IO model (11)
y[n]+
n−1i=0
an−ip[i] y[i]
=
sj=0
bn−jp[j] u[j], (14)
where the coefficients ai, bj have a special form of dynamic depen-dence3 on parameter p.
In [8,9], the precise procedure for deriving an LPV state-spacemodel of the form (12) from the i/o equation (14) was presentedyielding the equations
x[1]1 = x2 − a1(p)x1 + b1(p)u
...
x[1]n−1 = xn − an−1(p)x1 + bn−1(p)u
x[1]n = −an(p)x1 + bn(p)u
y = x1,
(15)
where bj = 0 for j = 1, . . . , n− s− 1 if s < n− 1, and coefficientsai, bi have static dependence on parameter p.
Now, recall from [20] the nonlinear i/o equation, being a specialcase of (1),
y[n]= f1(y[n−1], u[n−1])+ · · · + fn(y, u), (16)
where fi for i = 1, . . . , n are analytic functions, known in the lit-erature as the ANARXmodel [21]. It was proven in [20] that (16) is
3 Note that, again in analogywith (11),we intentionally removed the feedthroughterm b0(p[n])u[n] from (14) in order to compare with the result presented in [4].
J. Belikov et al. / Systems & Control Letters 64 (2014) 72–78 75
always realizable in the following classical state-space form
x[1]1 = x2 + f1(x1, u)...
x[1]n−1 = xn + fn−1(x1, u)
x[1]n = fn(x1, u)y = x1.
The ANARX structure is well-known and proved itself as a rea-sonable choice for solving modeling and control tasks of differentcomplexities. For example, it has been used in controlling a back-ward motion of a truck trailer [22] and modeling the motions ofsurgeon’s hand during medical surgery operation [23]. Moreover,note that a great number of models of real-life processes identi-fied from the i/o data, reported in the literature, assume the ANARXstructure.
It is interesting to observe that (14) becomes a special case of(16) by fixing p = y or p = u with fi(y, u) = −ai(y)y + bi(y)u orfi(y, u) = −ai(u)y + bi(u)u for i = 1, . . . , n, respectively. There-fore, from the point of view of the nonlinear realization theory, theresults obtained for model (14) are natural and predictable.
4.2. Augmented form
Consider the special case of the LPV-IO model (11)
y[n]+
n−1i=0
an−ip[n−1] y[i]
=
sj=0
bn−jp[n−1] u[j], (17)
where ai and bj have again a special form of dynamic dependence4on parameter p.
In [8], the LPV state-space representation was derived from thei/o equation (17) as follows
x[1]1 = −a1(p)x1 − · · · − an(p)xn
+ b2(p)xn+1 + · · · + bn(p)x2n−1 + b1(p)u...
x[1]n−1 = xn−2
x[1]n = xn−1
x[1]n+1 = u
x[1]n+2 = xn+1
...
x[1]2n = x2n−1
y = x1,
(18)
where bj = 0 for j = 1, . . . , n − s − 1 if s < n − 1. One may easilysee that (18) has coefficients with static dependence, but dim x =
2n, i.e., the realization is not minimal, regarding the number ofstate coordinates. The augmented form (17) is well-known in theidentification literature [24] and incorporates different cases, inparticular bilinear and quadratic models.
Now, consider the i/o equation (1). A similar approach to thatone described in [8] to construct the state-space representation of(17) is known in nonlinear realization theory and can be appliedto derive the so-called extended state equations. Taking the state
4 It is important to emphasize that the feedthrough term b0(p[n−1])u[n] wasalready removed.
coordinates as x = [y, y[−1], . . . , y[−n+1], u[−1], . . . , u[−s]], yields
x[1]1 = φ(x1, x2, . . . , x2n−1, u)...
x[1]n−1 = xn−2
x[1]n = xn−1
x[1]n+1 = u
x[1]n+2 = xn+1
...
x[1]2n = x2n−1
y = x1.
(19)
Again observe that like in the case with shifted form (18) be-comes a special case of (19) by fixing p to be either y or u.
4.3. New realizable subclasses of LPV models
Now, we present three new special subclasses of LPV i/o mod-els that can be converted into the LPV state equations not havingdynamic dependence on parameter p. For that purpose, we usenonlinear realization theory, developed in [4], and especially theresults from [17,25]. We intentionally omit the calculations here,because the results can be checked directly, using the state elim-ination algorithm from [1,26] adopted for discrete-time systems.Note that these subclasses cannot be directly deduced from the re-sults of Theorem 3; see more in Section 4.4.Subclass 1:y[n]
= u + b1p[n−1] u[n−1]
+an−2
p[n−1]
+ b2p[n−2] u[n−2]
+
n−3i=1
ai
p[n−2]
+ aip[n−1] u[i]. (20)
The state equations can be derived as
x[1]1 = x2 + a1(p)x3 + · · · + an−2(p)xn + b1(p)u
x[1]2 = x3 + a1(p)x4 + · · · + an−3(p)xn + b2(p)u
x[1]3 = x4...
x[1]n−1 = xnx[1]n = uy = x1.
Subclass 2:y[n]
= a1p[n−1] y[n−1]
+ b1p[n−1] u[n−1]
+
n−1i=1
i+1j=2
ajp[n−1] y[i−1]. (21)
The corresponding state-space representation can be found asfollowsx[1]1 = a1(p)x1 + · · · + an(p)xn + b1(p)u
x[1]2 = x1 + x3...
x[1]n−1 = x1 + xnx[1]n = x1y = x1.
76 J. Belikov et al. / Systems & Control Letters 64 (2014) 72–78
Subclass 3:
y[n]=
ki=0
n−k−1j=0
an−k−j,i+1p[j] y[i+j]
+
n−k−1i=0
bn−k−ip[i] u[i]. (22)
The state-space description is given by the following equations
Remark 3. Note that model (22) covers several important sub-classes. For instance, it reduces to the shifted form (14),when k = 0and to the controllability (Brunovsky) canonical form, presentedin [8], for k = n − 1.
Note that for all three subclasses (20)–(22), the state equa-tions have only static dependence on the parameter p, and the or-der of the state equation is the same as that of the i/o equation,i.e., dim x = n.
4.4. Application of nonlinear realization theory to LPV representations
In this subsection, we demonstrate how the nonlinear theorycan contribute to the solution of the realization problem in the LPVcontext. According to Theorem 3, an arbitrary LPV-IOmodel can berealized in the form (12), but, in general, the state equations de-pend dynamically on the scheduling parameter p. The followingexample demonstrates that only the shifted form (14) can be writ-ten in the state-space form with static dependence on p directlyfrom the application of Theorem 3.
Example 1. Consider the second-order LPV-IO representation
y[2]= a1(p[1])y[1]
+ a2(p)y[1]+ a3(p[1])y + a4(p)y
+ b1(p[1])u[1]+ b2(p)u[1]
+ b3(p[1])u + b4(p)u. (23)
Eq. (23) can be transformed to the i/o form (11) as followsz2 − (a1(p[1])+ a2(p))z − (a3(p[1])+ a4(p))
y
+−(b1(p[1])+ b2(p))z − (b3(p[1])+ b4(p))
u = 0. (24)
The application of Theorem 3 to (24) results in the state coor-dinates x1 = y and x2 = y[1]
It is easy to observe that the static dependence on p in (25) canbe guaranteed only in the case a2(p) = b2(p) = a3(p) = b3(p) =
0. Therefore, the direct application of the formulas from Theorem 3allows us to transform an arbitrary LPV i/o equation to the state-space formwith static dependence on p only for the particular sub-class of models known as the shifted form (14).
Our next goal is to find necessary and sufficient conditions forLPV-IO equations to admit a state-space realization with static de-pendence on p. For that purpose, we use the tools recalled in Sec-tion 2, adopted for the case of LPV models. This result (Theorem 4)
is based onnonlinear realization theory anddepends on an integra-bility condition. Since the extension of the algebraic setting fromSection 2 is rather straightforward, the detailed description is notprovided, and we briefly recall and adopt only the basic facts.
Note that we use a different symbol Kp to denote the field ofmeromorphic functions depending on the variables from the setCp = {y, . . . , y[n−1], u[k], k ≥ 0, p[κ], κ ≥ 0}. In a similar man-ner, one can consider the infinite set of symbols dCp = {dy, . . . ,dy[n−1], du[k], k ≥ 0, dp[κ], κ ≥ 0} and denote by Ep the vectorspace over the field Kp spanned by the elements of dCp. The dif-ferential, forward- and backward-shift operators can be defined inthe similar manner as in Section 2. Next, let ν := max{s, κ} anddefine a sequence of subspaces H∗
Since the algebraic frameworkwas presented for systems of theform (1), we rewrite equation (11) as
y[n]= −
n−1i=0
(ai � p)y[i]−
sj=0
(bj � p)u[j]. (26)
The LPV-IO representation (26) can be described via polynomi-als from the ring Kp[z; σ ], i.e., the polynomial ring, defined overthe field Kp. Apply the operator d to both sides of (26), yielding
dy[n]+
n−1i=0
(ai � p)dy[i]+
sj=0
(bj � p)du[j]+
n−1κ=0
∂ψ
∂p[κ]dp[κ]
= 0,
where ψ denotes the right-hand side of Eq. (26). The previousequation can be rewritten in the polynomial form as
a(z)dy + b(z)du + r(z)dp = 0, (27)
where a(z) = zn+n−1
i=0 aiz i, b(z) =s
j=0 bjzj, r(z) =
n−1κ=0 rκz
κ
and ai = ai � p ∈ P, bj = bj � p ∈ P, rκ =∂ψ
∂p[κ] ∈ Kp.
Theorem 4. The LPV-IO equation (11) has the state-space realizationwith static dependence on the scheduling parameter p, iff the subspaceH∗
ν+2 = spanKp{ω∗
1, . . . , ω∗n}, where
ω∗
l = σ−lc
a(z) b(z) r(z)
dydudp
, l = 1, . . . , n, (28)
is integrable.
Proof. The proof is a straightforward extension of the proof ofTheorem 11 from [15].
In order to illustrate the application of Theorem 4 to the real-ization problem of LPV-IO equations, the following two examplesare considered. The examples are chosen in a way to emphasizedifferent features5 of the computation procedure.
Example 2. Consider the following LPV-IO equation
y[3]= u[2]
+ p[1]u[1]+ y
that can be represented in the polynomial form (27) as a(z) =
z3 + 1, b(z) = −z2 − p[1]z, r(z) = −u[1]z.Note that n = 3, s = 2, κ = 1, and ν = max{2, 1} = 2. Next,
according to (28), the one-forms that define the differentials of thestate coordinates can be computed as follows ω∗
1 = z2dy − zdu −
5 different orders of the systems, shifts of the input signal and parameter p.
J. Belikov et al. / Systems & Control Letters 64 (2014) 72–78 77
pdu − udp, ω∗
2 = zdy − du, ω∗
3 = dy. Thus, H∗
4 = spanKp{dy[2]
−
du[1]− pdu− udp, dy[1]
− du, dy}, which is integrable. The choicex1 = y, x2 = y[1]
−u, x3 = y[2]−u[1]
−pu yields the LPV state-spacerepresentation of the form
x[1]1 = x2 + u
x[1]2 = pu + x3
x[1]3 = x1y = x1.
Next, we consider an example from [27] to perform the com-parative analysis of both realization algorithms.
Example 3. Consider the discrete-time LPVmodel of themotion ofa varying mass connected to a spring from [27]
p[1]y[2]− (p[1]
+ p)y[1]+ (T 2
d ks + p)y = T 2d u, (29)
where y is the position of the varying massm, u is the force actingon the mass, ks > 0 is the spring constant, Td > 0 is the discretiza-tion step size, and p := m. After differentiating (29), we obtain thepolynomial representation in the form (27), where
a(z) = p[1]z2 − (p[1]+ p)z + (T 2
d ks + p),
b(z) = T 2d ,
r(z) = (y[2]− y[1])z − y[1]
+ y.
From (29) one can get that n = 3, s = 1, κ = 2, and ν =
max{1, 2} = 2. By formula (28), the one-forms that define the dif-ferentials of the state coordinates can be computed as followsω∗
1 =
[pz−(p+p[−1])]dy+(y[1]−y)dp andω∗
2 = dy. Thus,H∗
ν+2 = H∗
3 =
spanKp{pdy[1]
− (p + p[−1])dy + (y[1]− y)dp, dy}. Since dy is the
basis vector of the subspace H∗
3 , ω∗
1 can be simplified, resulting inH∗
3 = spanKp{pdy[1]
+y[1]dp−pdy−ydp, dy}, which is integrable.The choice x1 = y and x2 = py[1]
− py yields the state equations
x[1]1 = x1 +
x2p
x[1]2 = x2 − Tdksx1 + T 2
d uy = x1.
(30)
However, in case we apply Theorem 3, the state coordinates arechosen as x1 = p[−1]y, x2 = py[1]
− (p+ p[−1])y, yielding the stateequations
x[1]1 =
1 +
pp[−1]
x1 + x2
x[1]2 = T 2
d u −Tdks + pp[−1]
x1
y =x1
p[−1]
(31)
which obviously depend dynamically on p.In [27] the representation (30) with static dependence on pwas
obtained as a result of a state transformation z = T (x, p), appliedto Eqs. (31). In general, there is no rule how to find a suitable statetransformation, whereas the application of Theorem 4 always re-sults in the state equations with static dependence on p, wheneverthe subspace H∗
ν+2 is integrable.Finally, according to [8], there exists an alternative approach to
address the realization problem, the basic idea of which consistsin elimination of dynamic dependence via state-transformations,whenever possible. In other words, starting from the state equa-tions (12) with dynamic dependence on p, we find, wheneverpossible, an equivalent form with a reduced number of shifted pa-rameters p. The overall procedure is very similar to that of [28]
for nonlinear systems, though the latter addresses the case of re-moving (or lowering) the input shifts. Of course, neither techniqueguarantees, in general, that the result depends statically on p or onthe control variable u, respectively.
4.5. On the application of LPV realization theory to nonlinear systems
That far we have demonstrated how nonlinear realization the-ory can contribute to LPV realization issues. Now,webriefly discussthe possible application of the tools from the LPV framework ontononlinear realization problems. In such a case, we rather focus onthe so-called quasi-LPV systems, when the scheduling signal p isa function of inputs and/or outputs, and so the assumption on in-dependence of p is violated. If we study realizability/realization ofsystem (1) in the framework of the LPV approach, we first have totransform Eq. (1) into the form (11), using the method describedin [9]. Note that, in the principle, there are two different possi-bilities to parameterize the right-hand side of Eq. (1). In order toillustrate the main idea, consider in detail the following simple ex-ample.
Example 4. Consider the second-order bilinear system
y[2]= uy[1]
+ u[1]y[1]. (32)
Let the parametrization to be p := y, and then the respectivequasi-LPV system has the form
y[2]= p[1]u + p[1]u[1]. (33)
According to Theorem 3, the state coordinates are x1 = y, x2 =
y[1]− pu yielding the state equations
x[1]1 = x2 + pu
x[1]2 = p[1]uy = x1
(34)
with dynamic dependence on p. However, recalling that p = x1, wehave p[1]
= x[1]1 = x2 + x1u transforming Eq. (34) into the form (4)
x[1]1 = x2 + ux1
x[1]2 = (x2 + ux1)uy = x1.
Thus, onemay see that if we take into considerations the case ofquasi-LPV systems, then it is sometimes possible to find the classi-cal state-space description for the particular nonlinear model, us-ing LPV tools, despite the fact that the original LPV model is notrealizable in the state-space form with static dependence on p.
5. Conclusions
In this paper, regarding the state-space realization problem,the LPV and nonlinear theories were compared. The main objec-tive was to investigate how the established nonlinear theory cancontribute to the LPV approach. Three new subclasses of LPV-IOmodels with only static dependence on the parameter p in stateequations were suggested. Moreover, the theorem that guaranteesrealizability of LPV-IO model in a state-space formwith only staticdependence on p is presented. It should be mentioned, however,that applicability of the suggested realization approach dependson the integration of certain integrable one-forms which is knownto be sometimes difficult task.
As for the possible future direction of the research initiated inthis paper, the minimal (in the sense of state dimension) realiza-tion and system reduction are in interest. Moreover, it is a generalbelief that the LPV approach may provide an easier way to address
78 J. Belikov et al. / Systems & Control Letters 64 (2014) 72–78
certain nonlinear control problems, once the nonlinear models arerepresented as quasi-LPVmodels. For example, in [29], the stabilityand controllability properties of a nonlinear system have been an-alyzed using the methods, developed for linear time-varying sys-tems. Similarly, onemay investigate how the developed LPV theorycan benefit for studying nonlinear systems. Despite the fact that inits current form, the LPV theory cannot directly help to solve thenonlinear realization problem, the authors do believe that the ad-ditional researchmay help tomodify the existing tools and developa new realization algorithm (merging nonlinear and LPV theories)without linearizing the system equations, at least for certain sub-classes of nonlinear systems.
Acknowledgments
The work was supported by the European Union through theEuropean Regional Development Fund, the target funding projectSF0140018s08 of Estonian Ministry of Education and Research.J. Belikov and M. Tõnso were additionally supported by the ESFgrant N8787.
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