IET Control Theory & Applications Research Article
State-input constrained asymptotic
null-controllability by a set-valued approach
ISSN 1751-8644
Received on 14th December 2014 Revised on 10th April 2015 Accepted on 25th May 2015 doi: 10.1049/iet-cta.2014.1333 www.ietdl.org
Lahoucine Boujallal, Khalid Kassara
Department of Mathematics, University Hassan II, P.O. Box 5366, Casablanca, Morocco E-mail: k.kassara@fsac.ac.ma
Abstract:This study presents a unified approach to investigate asymptotic null-controllability under possibly mixed con- straints on state and control, for standard ordinary differential equation control systems. Using tools from set-valued analysis and viability theory, initial data which can be steered to the origin at infinity are provided through a new type of Lyapunov functions. The corresponding controls are given via selections of adequately designed multifunctions, which are examined in both cases of convex constraints and the class of affine-control systems. Finally, numerical examples from classical mechanics are given to verify the theoretical results.
1 Introduction
We consider a constrained control system of the form,
˙
x=f(x,u), (x,u)∈K, (1) where f stands for a smooth function from Rn×Rp to Rn, for integersnandp; andKdenotes a closed subset ofRn×Rp.
System (1) is said to be asymptotically null-controllable from a subset of Rn, whenever for all x0∈ there exists a control
¯
u(·) yielding a trajectory ¯x(·) issued from x0, which satisfy the following statements,
(¯x(t),u(t))¯ ∈K, for allt≥0, (2a) and,
tlim→∞x(t)¯ =0. (2b)
On the basis of classical Lyapunov theory, several types of control Lyapunov functions (CLFs) have been introduced and/or used by most of the studies dealing with this problem, in the special case of no constraints (i.e.K=. Rn×Rp). These stand for positive definite continuous functions whose derivatives can be made negative by appropriate choices of controls.
Paving the way, [1] and then [2] – for non-autonomous sys- tems – have investigated existence for such functions in connection with asymptotic controllability, by introducing a local notion of relaxed controls. In [3], the author has proceeded by means of feedback linearisation and questions of regularity have been raised by Rifford [4], which examined a new type of CLF based on the Clarke’s generalised gradient.
The work by Coron [5] has led to the use of discontinuous feed- backs. While Camilliet al. [6] showed in an earlier paper that the CLFs may be constructed as viscosity solutions of a first-order partial differential equation that generalises Zubov’s equation.
Asymptotic controllability is also apart of the stabilisability problem, as the former is needed to be achieved around the equi- librium state, and local existence of CLFs can provide stabilising feedbacks on the condition that sensitivity with respect to initial data is satisfied.
Motivated by the ubiquitous appearance of constraints on state and/or control in real-world applications, numerous works have been concerned with asymptotic controllability under constraints.
Thus [7,8] consider constraints of the form,
K=. S×C, withS⊂Rn andC⊂Rp, (3)
then they pose and solve the problem of designing corresponding feedbacks in the framework of non-smooth analysis. The authors [9–11] have also investigated the problem, relative to several types of constraints. Techniques of input saturation have been also considered in a series of studies, among them we cite [12–14].
The following facts deserve to be quoted, as they motivate our study:
(a) None of the approaches mentioned above apply in the under- studied case of mixed constraints, i.e. subsetKis strictly in a more general form than (3).
(b) There is no approach that encompasses all the cases with regard to dynamicsf and constraints subsetK.
(c) An inevitable common problem one has to face, as inherently revealed by all the above studies, consists of whether the proposed control laws will produce a system solution.
Inspired by the studies done in [15, 16], we intend to offer a unified set-valued scheme in order to deal with constrained asymptotic controllability for possibly mixed constraints. Gener- ally, the contribution of this paper is to seek conditions under which, any arbitrary Lyapunov functionϕ (defined independently from dynamics f, in a new sense to be precised) can generate a domainϕ and a closed-convex valued multifunctionGϕ, in such a manner that:
constrained asymptotic null-controllability fromϕ holds via controls provided by selections of multifunctionGϕ. One advantage of setting the problem as a viability problem – we refer the reader to [17] as a basic reference on viability theory – is that it will be guaranteed that the controls derived, will produce global solutions of the associated non-linear system, overcoming the concern that is mentioned in item (a). This is due to the fact that, by construction, multifunction of regulationGϕinvolves both linear growth and tangential conditions for viability.
Also, it is noteworthy that, whenever constraints subset K is convex, the mapGϕ is always closed-convex valued – even when the system is not affine dependent in the control. As a result, it may have continuous selections, and universal formulas can be exhib- ited. Furthermore, despite its discontinuity, the minimal selection may be used to provide controls that solve the problem, whenever the mapGϕ is lower semi-continuous.
The remaining of the paper is structured as follows: In Section2, we provide some notations and preliminary lemmas. Section 3is devoted to present the main elements of our set-valued approach,
then Section4to state and prove the main results for convex con- straints. In Section 5, we investigate the class of control-affine systems, then Section6is concerned with treating some examples from classical mechanics. Finally, we conclude in Section 7 by comments and discussions.
2 Definitions and preliminary results
Through this paper, the Euclidean norm is denoted| |, and,is the usual inner product. For a vector z we denote by zi its ith component. LetT be a linear operator then its adjoint operator is denoted byT . Also, we consider the notation
∇xf =. ∂fi
∂xj
ij
and∇uf =. ∂fi
∂uj
ij
.
LetKbe a non-empty subset of an Euclidean space. The contingent cone to subsetK at pointx∈K is defined by
TK(x)=
y |lim inf
ε↓0
d(x+εy,K)
ε =0
,
whered(y,K)=. inf{|z−y| |z∈K}. WhenK is closed and con- vex, the operator of best approximation onK, is denoted byπK(·).
It is given by
|x−πK(x)| =d(x,K), for allx∈Rn.
Next, we provide a result by Aubin and Frankowska [18], which is crucial for building our control laws.
Lemma 1:Let ψ:RN →Rbe a differentiable mapping, for an integer N. Given a closed convex subsetK of RN and letD=. {z∈K |ψ (z)≤0} andz0∈D. Suppose there existsy0∈TK(z0) such that dψ (z0)y0<0, then
y∈TD(z0)⇐⇒
y∈TK(z0) and
dψ (z0)y≤0 ifψ (z0)=0,
(4)
where dψ (·)denotes the differential operator ofψ.
Let g:RN →RN, then subset D is said to be locally viable under system
˙ z=g(z),
z(0)=z0, (5)
if for allz0∈Dthere exist¯t>0 and a solution to system (5),z(¯ ·) on[0,¯t)which is viable inD[i.e. satisfies¯z(t)∈Dfor allt]. Such a property can be characterised [17] in terms of contingent cones as follows.
Lemma 2:Assume that functiongis continuous on the closed sub- setD. Then,Dis locally viable under system (5) if and only if the following tangential condition holds
g(z)∈TD(z), for eachz∈D. (6) According to [17], the viable solutions provided by condition (6) are global whenever dynamics g has linear growth on subsetD, i.e. there exists a numberc>0 such that
|g(x)| ≤c(1+ |x|), for allx∈D.
Nevertheless, such a condition is not required when functiongis bounded on subset D, or in particular when the latter subset is bounded.
By selection of a multifunctionQit is meant a functiongsuch that g(x)∈Q(x) for allx. Michael selection theorem [19] which will be repeatedly used in this paper, can be stated as follows.
Lemma 3: If a multifunctionQislscand has closed convex values, then for all(x0,y0)such thaty0∈Q(x0), there exists a continuous selectionσ ofQwhich satisfiesσ (x0)=y0.
The short-term lsc in Lemma3means that multifunction Qis lower semi-continuous, i.e. for every x∈Dand any sequence of elementsxk ofDconverging tox, it holds that
for each y∈Q(x), there exist a sequence yk∈Q(xk) that converges toy.
Ultimately, in a way that fits our context, the Lyapunov functions we will consider in this paper, consist of C1 real-valued functions ϕdefined onRn×Rnand satisfying
h:[0∞)→, differentiable, and
ϕ(h(t),h(t))˙ ≤0 for allt≥0, =⇒ h(t)→0 fort→ ∞, (7) wherestands for a subset ofRn. Note such functionsϕdepend only upon subset, we thereby call them -Lyapunov functions.
For instance, the notion of Lyapunov pairs(V,W)considered by Clarkeet al.[7], coincide with our terminology by taking function ϕas follows
ϕ(x,y)= ∇V. (x),y +W(x)for allx,y, for smooth non-negative functionsV andW.
3 Construction of the set-valued method
This section is devoted to present the elements of our control strat- egy. Throughout, we assume that function f is continuous and has linear growth on subset K. Consider the map defined on K as follows
G(x,u)= {v. ∈Rp |(f(x,u),v)∈TK(x,u)}. (8) Let=. π1(K), whereπ1denotes the mapping(x,u)→x, andϕ be an-Lyapunov function in the sense of (7), then set
Dϕ= {(x,. u)∈K|ϕ(x,f(x,u))≤0}, (9) and define the map given for each (x,u)∈Dϕ by
Gϕ(x,u)= {. v∈Rp |(f(x,u),v)∈TDϕ(x,u)}. (10) It provides the regulation map in subset Dϕ, relative to the augmented control system
˙
x=f(x,u),
˙
u=v, (11)
wherevhas values inRpand stands for the control. Let
ϕ=. π1(Dϕ). (12)
Then, we are ready to show the following result.
Theorem 1:If the map Gϕ has a continuous selection which possesses linear growth, then system (1) is asymptotically null- controllable fromϕ.
Proof:For such a selectiong, it holds that
(f(x,u),g(x,u))∈TDϕ(x,u)for all(x,u)∈Dϕ. This sets us in the context of viability theory [17], which involves that system
˙
x=f(x,u), x(0)=x0,
˙
u=g(x,u), u(0)=u0 (13) has a solution(x,¯ u)¯ which is viable inDϕ, for each(x0,u0)∈Dϕ. This solution is global as bothf and g possess linear growth. It follows that x¯ has values in and satisfiesϕ(x(t),¯ x(t))˙¯ ≤0 for all t. Then (7) yields x(t)¯ →0 whent→ ∞ and thereby x¯ and
¯
u satisfy (2), leading to the asymptotic null-controllability from
subsetϕ.
Remark 1:Suppose that the map Gϕ(·) is convex valued. That occurs when subsetDϕ is convex, in which case, its contingent cone at every point is convex. Then a very pertinent question is whether its minimal selection gϕ can be used in the proof of Theorem 1 instead of continuous selection g. The answer is negative in general because this selection, as given by
gϕ(x,u)=. πGϕ(x,u)(0), (14) for all (x,u)∈Dϕ, is not continuous in general. Thanks to [17, Theorem 4.3.2], it follows that whenever mapGϕ islsc, then sys- tem (13), with g=. gϕ, still has a viable solution in subset Dϕ. As a consequence Theorem1remains still valid provided that the statement ‘the mapGϕhas a continuous selection which possesses linear growth’, is replaced by ‘the mapGϕ islscand its minimal selection possesses linear growth’. This fact may be of interest whenever slow controls are rather required.
To provide an expression of the feedback mapGϕ(·)of (10), we first proceed to compute the contingent coneTDϕ(·). Suppose there- after that both dynamicsf and Lyapunov function are differentiable and set for all(x,u)∈Dϕ
ψ (x,u)=. ϕ(x,f(x,u)). (15) It follows that subsetDϕ given by (9) can be written as
Dϕ= {(x,u)∈K|ψ (x,u)≤0}.
Note that the partial differentials ofψ are given onDϕ by
∇xψ (x,u)= ∇xϕ(x,f(x,u))+(∇xf(x,u))∇yϕ(x,f(x,u)), (16) and
∇uψ (x,u)=(∇uf(x,u))∇yϕ(x,f(x,u)). (17) Then, we set the constraints qualification condition
For all(x,u)∈Dϕ, such thatψ (x,u)=0, there exists(y,v)∈TK(x,u)such that:
∇xψ (x,u),y + ∇uψ (x,u),v<0.
(18)
Subsequently, we need to define the following functions and maps ϕ(x,u) .
= ∇xϕ(x,f(x,u)),f(x,u) +
∇yϕ(x,f(x,u)),∇xf(x,u)f(x,u)
, (19)
mϕ(x,u)=. (∇uf(x,u))∇yϕ(x,f(x,u)), (20) Cϕ(x,u) .
= {v∈G(x,u)|ϕ(x,u)+
mϕ(x,u),v
≤0}, (21) for each(x,u)∈K. In the following result, we provide an expres- sion of the feedback map.
Lemma 4:Assume that function f is differentiable on subset K.
Under condition (18), we get for all(x,u)∈Dϕ
Gϕ(x,u)=
G(x,u)ifψ (x,u) <0,
Cϕ(x,u)ifψ (x,u)=0. (22) Proof:Thanks to Lemma1and condition (18), the contingent cone of subsetDϕ ca be provided as follows
(z,w)∈TDϕ(x,u)⇐⇒
(z,w)∈TK(x,u) and, ifψ (x,u)=0 then
∇xψ (x,u),z + ∇uψ (x,u),w ≤0, for each(x,u)∈Dϕ. According to (14), we get
v∈Gϕ(x,u)⇐⇒
(f(x,u),v)∈TK(x,u) and, ifψ (x,u)=0 then
∇xψ (x,u),f(x,u) + ∇uψ (x,u),v ≤0.
By considering (19), (20), and (8), we easily can see that the last
expression is equivalent to (22).
Example 1:To illustrate by a mathematical example, consider a problem with mixed constraints
˙
x=asin(xu)+be−λu, with cstr. 0≤x≤u, (23) where a, b, and λ denote real numbers. We apply the previous results through the following scheme
(i) Set, f(x,u)=. asin(xu)+be−λu and K= {(x,. u)∈R2|0≤ x≤u}.
(ii) Compute the contingent cone of subset K. We use [18] to obtain
TK(x,u)=
R2, ifu>x>0,
{(y,v)|y≤v}, ifu=x>0, {(y,v)|y≥0}, ifu>0,x=0, {(y,v)|0≤y≤v}, ifu=x=0.
(iii) Note,=. [0, ∞)and find an-Lyapunov functionϕ, which satisfies qualification condition (18). Here, take for instance
ϕ(x,y)=. y+αx, for all(x,y)∈R2, (withα >0).
It actually stands for an-Lyapunov function (by using a simple Gronwall inequality) and condition (18) is shown to be satisfied wheneverλ >0 andb>0.
(iv) Compute functionsϕ and mϕ of (19) and (20), respectively.
They are given for all(x,u)∈K, by
ϕ(x,u)=(aucos(xu)+α)f andmϕ(x,u)=axcos(xu)−be−λu. (v) Due to (21), the mapCϕcan then be expressed as follows: (see equation at the bottom of the next page)
where we seek continuous selection v=g(x,u) in the form v= αu+β. As a result, since
−α2x−αλbue−λu≤0, for all(x,u)∈K, then, we can pickβ, as follows
β=axcos(xu)−λb e−λu.
In checking whether (f,v)∈TK(x,u)for all(x,u)∈Dϕ, it turns out that the following conditions are required to be held
a<0, b>0, λ≥1, and λ(λ−1) < α.
Whence, the set-valued maps Cϕ, and then Gϕ (as Cϕ⊂Gϕ), have a continuous selection, having linear growth, and which is
0 0.2 0.4 0.6 0.8 1 0
1 2 3 4 5 6 7
state control
Fig. 1 Simulation result of Example 1 with a= −2, b=0.9,α=1.7, and λ=2
given by
g(x,u)=αu+axcos(xu)−λbe−λu. (24) (vi) To get both state and control which achieves null- controllability from initial datax0∈ϕ, letu0 such that
f(x0,u0)+αx0≤0, and x0≥u0≥0, and solve augmented system, see Fig.1
˙
x=asin(xu)+be−λu, x(0)=x0,
˙
u=αu+axcos(xu)−λbe−λu, u(0)=u0.
In what follows, we address an instance where system (1) is asymp- totically null-controllable from the whole domain. For this end, we need to introduce the following map, for eachμ≥0
Cϕμ(x,u)= {. v∈G(x,u)|ϕ(x,u)+
mϕ(x,u),v
≤ −μ}, (25) for each(x,u)∈K.
Theorem 2:Assume that, for some μ >0, the mapCμϕ given by (25) has a continuous selection which has linear growth, then system (1) is asymptotically null-controllable from.
Proof:Letg be such a selection of the mapCϕμ. As qualification condition (18) is satisfied, this map has values included inCϕ⊂Gϕ. It follows that the selectiongis also a continuous selection ofGϕ, which has linear growth. By Theorem 1, system (1) is therefore asymptotically null-controllable from subsetϕ. The rest of the proof is devoted to show that system (1) is asymptotically null- controllable from subset\ϕ.
Let x0 belong to \ϕ. Therebyϕ(x0,f(x0,u)) >0, for allu such that (x0,u)∈K. Since g is a continuous selection of G, it follows that system
˙
x=f(x,u), x(0)=x0,
˙
u=g(x,u), u(0)=u0,
admits aK-viable solution(¯x,u)¯ (on horizon [0,∞)as bothf and gpossess linear growth), for allu0such that(x0,u0)∈K. Let such u0 be given, we readily have
d
dtϕ(¯x(t),f(¯x(t),u(t)))¯ =
∇xϕ(¯x(t),x(t)),˙¯ x(t)˙¯
+
∇yϕ(¯x(t),x(t)),˙¯ ∇xf(x(t),¯ u(t))¯ x(t)˙¯
+ ˙¯u(t),∇uf(x(t),¯ u(t))¯ ∇yϕ(x(t),¯ f(x(t),¯ u(t)))¯ , which yields, according to (19) and (20)
d
dtϕ(¯x(t),f(¯x(t),u(t)))¯ =ϕ(¯x(t),u(t))¯ + ˙¯u(t),mϕ(x(t),¯ u(t))¯ .
Whence
ϕ(¯x(tf),f(¯x(tf),u(t¯ f)))=ϕ(x0,f(x0,u0)) +
tf 0
[ϕ(¯x(t),u(t))¯ + g(¯x(t),u(t)),¯
mϕ(¯x(t),u(t))]¯ dt.
Sincegis also a continuous selection ofCϕμ, then ϕ(x(t¯ f),f(x(t¯ f),u(t¯ f)))≤ϕ(x0,f(x0,u0))−μtf. Thereby
(¯x(tf),u(t¯ f))∈Dϕ for tf ≥ ϕ(x0,f(x0,u0))
μ .
Let xf = ¯. x(tf) and uf = ¯. u(tf). Then xf ∈ϕ, and the use of Theorem1involves that system
˙
z=f(z,w), z(tf)=xf,
˙
w=g(z,w), w(tf)=uf, has a solution (¯z,w)¯ on horizon tf,∞
, which ranges in Kand satisfies
¯
z(t)→0 whent→ ∞.
Consequently, the control given byu¯ on [0,tf] andw¯ on (tf,∞), achieves asymptotic null-controllability fromx0.
v∈Cϕ(x,u) ⇐⇒
(f,v)∈TK(x,u), and,
−α(α+ucos(xu))x+v(axcos(xu)−be−λu)≤0, iff +αx=0.
⇐⇒
(f,v)∈TK(x,u), and,
−α2x−αλbue−λu+β(axcos(xu)−λbe−λu)≤0, iff+αx=0,
4 Case of convex constraints
It turns out from the previous section that, under qualification con- dition (18), the property of asymptotic null-controllability mainly relies on whether the map Cμϕ(·) [for some μ≥0, as given by (25)] admits a continuous selection which has linear growth. This actually may hold when it has convex values, providing the oppor- tunity to use Michael selection theorem. For that purpose, in all that follows, we assume that
dynamicsf isC1and has linear growth; and subsetKis convex.
For the sake of conciseness, we begin by highlighting both condi- tions that have to satisfy the -Lyapunov functions in order that constrained asymptotic null-controllability be held.
Definition 1:Let μ≥0, we say that -Lyapunov function ϕ belongs to subset μ whenever there exists c>0 such that for all (x,u)∈Dϕ, there exists v which satisfies the following statements
(f(x,u),v)∈int(TK(x,u)), (26a) ϕ(x,u)+
v,mϕ(x,u)
<−μ, (26b)
and |v|<c(|x| + |u| +1), (26c) where functions ϕ and mϕ, are, respectively, given by (19) and (20).
Then, we can state the following technical result.
Lemma 5:Let μ≥0 and assume that ϕ∈μ, then both maps G,Gϕ, and Cϕμ, as given, respectively, by (8), (22), and (25) are lsc. (noteCϕ0=Cϕ).
Proof:In order to show that maps G and Cμϕ arelsc, we rewrite them in the context of [18, Proposition 1.5.2] in the form
{v∈ ¯F(x,u) | ¯f(x,u,v)∈ ¯G(x,u)}, where for each(x,u)∈Dϕ, we have for the mapG
F¯(x,u)=Rp, f¯(x,u,v)=(f(x,u),v), andG(x,¯ u)=TK(x,u), and for the mapCμϕ
F(x,¯ u)=G(x,u), f¯(x,u,v)=
mϕ(x,u),v , and G(x,¯ u)=
−∞,−ϕ(x,u)−μ .
Thereafter, we use (26) to easily check each of the hypotheses of the cited proposition, which are listed below:
1. The mapF¯ islscwith convex values.
2. f¯ is continuous.
3. For all(x,u)∈Dϕ, the mappingv→ ¯f(x,u,v)is affine.
4. For all(x,u),G(x,¯ u)is convex and its interior is non-empty.
5. The graph of the map(x,u)∈Dϕ→int(G(x,¯ u))is open.
6. For all(x,u)∈Dϕ, there existsv∈ ¯F(x,u)such thatf¯(x,u,v)∈ int(G(x,¯ u)).
Next we prove that the mapGϕ islsc. Let(xn,un)nbe a sequence ofDϕ that converge to(x,u)∈Dϕ andv∈Gϕ(x,u). We have to
seek a sequence(vn)nthat satisfies
vn∈Gϕ(xn,un)for eachn,
and vn→v. (27)
Assume thatψ (x,u) <0. Since the function ψ is continuous and (xn,un)→(x,u)we can consider the smallest integern0such that
ψ (xn,un) <0 for alln≥n0. Then, the sequence defined by
vn=
v ifn≥n0, wnifn<n0, where
wn∈Cϕ(xn,un)for alln<n0,
merely satisfies (27) due to the fact that ψ (xn,un)=0 whenever n<n0. Now suppose thatψ (x,u)=0, thenv∈Cϕ(x,u). Since the mapCϕ(·)islsc, it follows that there exists a sequence(vn)nsuch thatvn∈Cϕ(xn,un)for eachnandvn→v. Thanks to (22) we get, vn∈Gϕ(xn,un)for alln, as required in (27).
Next, we state and prove the following result.
Theorem 3:Let μ≥0 and ϕ belong to μ. If μ=0 then sys- tem (1) is asymptotically null-controllable from ϕ, or else it is asymptotically null-controllable from the whole domain. Proof:Suppose thatμ=0 and letϕ∈0. It follows that the well defined map
F(x,u)= {. v∈Gϕ(x,u)| |v| ≤c(|x| + |u| +1)}, for all(x,u)∈Dϕ, is lsc. This is due to (26c) and lower semi- continuity of the map Gϕ, as proved in Lemma5.
Furthermore, the map F has closed convex values. Thus, Lemma3implies thatF has a continuous selection, which there- fore is a continuous selection of the feedback mapGϕthat possesses linear growth. We are now able to use Theorem1to conclude that system (1) is asymptotically null-controllable fromϕ.
Now assume that μ >0 and consider the map defined on Dϕ by
R(x,u)=.
v∈Cμϕ(x,u)| |v| ≤c(|x| + |u| +1) , whereCμϕ is given by (25). As the latter map islsc(by Lemma5), condition in (26c) implies that the mapRis alsolsc. As a result, Michael selection theorem (stated in Lemma3) yields a continuous selection of the map R, which stands for a continuous selection of the map Cϕμ that has linear growth. The proof ends by using
Theorem2.
To be more precise, by Michael selection theorem, whenever v0∈Fϕ(x0,u0)there exists a continuous selection g of Fϕ such thatv0=g(x0,u0). This implies that, initially both control and its velocity can be freely chosen.
One important fact provided by the convex constraints setting, consists of the ability to use the minimal selection of the closed convex valued map Gϕ. This is given, thanks to Lemma4, for all (x,u)∈Dϕ, by the expression
gϕ(x,u)=. πGϕ(x,u)(0)=
πG(x,u)(0)ifψ (x,u) <0,
πCϕ(x,u)(0)ifψ (x,u)=0, (28) where maps G and Cϕ, are given, respectively, by (8) and (21).
Although mappinggϕis discontinuous, it may lead to a slow con- trol law which steers constrained system (1) to the origin, as we prove in the following result.
0 1 2 3 4 5 6 7 8
−14
−12
−10
−8
−6
−4
−2 0 2 4
Time x1
x2 u1 u2
0 1 2 3 4 5 6 7 8
−25
−20
−15
−10
−5 0
Time
constraint
a b
Fig. 2 Simulation results for Example 2 withα=(0.48, 0.48),β=(1,1), x0=(3, 1), and u0=(−2, 3) aTrajectories of state and control
bEvolution of constraint:a,u −ρb,x
Theorem 4:Letϕbelong to0, then for all(x0,u0)∈Dϕ, system
˙
x=f(x,u), x(0)=x0,
˙
u=gϕ(x,u), u(0)=u0, (29) has a solution (x,¯ u)¯ :[0,∞)→K which satisfies x(t)¯ →0 at infinity.
Proof:This is due to lower semi-continuity of the map Gϕ, then by using [17, Theorem 4.3.2], system (29) has a solution (x,¯ u)¯ over a bounded horizon. Since gϕ has linear growth (thanks to (26c)), then the couple(f,gϕ)has linear growth too. Therefore, the solution can be extended to an infinite horizon, and convergence to zero ofx¯ at infinity results from the fact thatϕis an-Lyapunov
function.
5 Example from the class of control-affine systems
Consider control system
˙
x=f0(x)+ p
j=1
ujfj(x) (30a)
with mixed constraints
x≥0 and u,a ≤ρx,b, (30b) where p≥1 and fj,j=0,. . .,p denote C1 vector fields defined on Euclidean space Rn and both having linear growth. The cou- ple(a,b)belongs toRp×Rn, witha=0, andρstands for a real number. Letf =. f0+p
j=1ujfj and Kbe, respectively, the asso- ciated dynamics and constraints subset, then subset of (12) is equal toRn+, and thereby a convenient-Lyapunov function can be given by
ϕ(x,y)= x,. α + y,β, for all(x,y)∈Rn×Rn, whereαand β denote vectors ofRn whose both coordinates are positive.
Indeed, if aC1 functionh:R+→satisfies :ϕ(h(t),h(t))≤ 0, for allt, then the real non-negative functionν .
=β1h1+ · · · + βnhnis such that
˙ ν(t)+inf
i
αi
βi
ν(t)≤0, for allt≥0.
As a result ν(t)→0 at infinity. Thereby both functionshi do so for all i=1,. . .,n. Now, let us express the function ψ of (15), we get
ψ (x,u)= x,α + f0(x),β)+ p j=1
ujfj(x),β,
for all(x,u)∈K, (31)
and let Dϕ= {(x,u)|ψ (x,u)≤0}andϕ=π1(Dϕ). The partial differentials of function ψ are given by
∇xψ = ∇f0,β + p
j=1
uj∇fj,β +α,
∇uψ =
f1,β,. . .,fp,β .
This yields the functionsϕ andmϕ of (19) and (20) ϕ= ∇xψf0+
p j=1
uj∇xψfj, andmϕ=
f1,β,. . .,fp,β . (32) It is noteworthy here that the control-affine structure of the system involves that function ϕ is affine in control u and function mϕ depends only from statex.
Using [18, Chapter 4] on tangent cones, we get the contingent cone of subsetK, as follows
TK(x,u)=
(y,v)|y∈ n i=1
TR+(xi)andv,a ≤ρy,b
, (33) for all(x,u)∈K, where
TR+(r)=
R ifr<0, R+ ifr=0,
for allr∈R+. Next, we need to consider the conditions x∈ϕ andxi=0 =⇒ f0(x)i≥0 and
fj(x)i=0, for allj=1,. . .,p. (34) Then, we have the following result.
Theorem 5:In addition to condition of (34), assume that for each (x,u)∈Dϕ, the linear inequalities
mϕ(x),v ≤ −ϕ(x,u), anda,v ≤ρb,f(x,u), (35) lead to a continuous selectionv=g(x,u), which has linear growth, then constrained system given by (30) is asymptotically null- controllable from subsetϕ.
Proof:Equation (34) implies thatf(x,u)∈TRn+(x,u)for all(x,u)∈ Dϕ. Thereby, due to (33) and (35), we get
(f(x,u),g(x,u))∈TK(x,u)and
mϕ(x),v +ϕ(x,u)≤0, for all(x,u)∈Dϕ.
As a result function g stands for a continuous selection of the map Cϕ of (21). It therefore achieves null-controllability from
subsetϕ.
Remark 2:It is of interest to notice that the minimal selection gϕ
that is highlighted by Theorem4can be provided here by solving the following quadratic programs, for each(x,u)∈Dϕ
g1(x,u)=. min
|v|2| a,v ≤ρf(x,u),b
, forψ (x,u) <0, and
g2(x,u)=. min
|v|2| a,v ≤ρf(x,u),b, and mϕ(x),v ≤ −ϕ(x,u)
, forψ (x,u)=0.
Then mappinggϕcan then be given by the following expression gϕ(x,u)=
g1(x,u)ifψ (x,u) <0, g2(x,u)ifψ (x,u)=0.
Example 2:Consider control system
˙
x1=x2+x1u1,
˙
x2=exp(−x1)+x2u2, (36) with constraints given by (30b), where a=(1, 1),b=(0.5, 0.3), andρ=1. We then see that this system can be examined in the context of Corollary1, where the fieldsfj are given by
f0(x)=(x2, exp(−x1)), f1(x)=(x1, 0), and f2(x)=(0,x2). The functionsϕ andmϕ of (19) and (22), respectively, are given for all(x,u)∈Kby
ϕ(x,u)=(.48+u1−exp(−x1)) (x2+x1u1) +(1.48+u2)(exp(−x1)+x2u2) and
mϕ(x,u)=(x1,x2). See Fig.2for simulation results.
Fig. 3 Mechanical system under consideration
0 5 10 15 20 25 30 35 40
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4 0.5
Time displacement velocity
Fig. 4 Evolution of the state without control (u=0), for the same initial data as in Fig.5
Corollary 1:In addition to condition of (34), assume that p
j=1
ajfj,β>0 for allx∈\ {0}. (37)
Letλ:Dϕ→Rbe a continuous function which has linear growth and satisfies
λ(x,u)≤min
ρ f(x,u),b p
j=1ajfj,β,− ϕ
p
j=1|fj,β|2
, for all(x,u)∈Dϕ,x=0. (38) Thenλmϕ stands for a continuous selection which achieves null- controllability from subset ϕ.
Proof: It easily can be verified that the selection g=. λmϕ satis- fies (35) for all(x,u)∈Dϕ. Since both fieldsfj,j=1,. . .,phave linear growth, then function mϕ of (32), has linear growth too.
Thereby continuous selection λmϕ has linear growth, as required
by Theorem5.
0 5 10 15 20 25 30 35 40
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4
Time
displacement velocity
0 5 10 15 20 25 30 35 40
−0.15
−0.1
−0.05 0 0.05 0.1 0.15 0.2 0.25
Time
The control u
0 5 10 15 20 25 30 35 40
−1.05
−1.04
−1.03
−1.02
−1.01
−1
−0.99
−0.98
−0.97
−0.96
−0.95
Time x12+x22+u2−1
a b
c
Fig. 5 Simulation result of system (42) withα=0.1584 and A=1, using an optimisation tool for computing the minimal selection aEvolution of state with(q0,q˙0)=(0.3, 0.2)
bEvolution of the control withu0=0.1 cEvolution of the mixed constraint (43)
6 Application to Lagrange equation
Lagrange equation [20] consists of a unified setting, providing the equations of motion for a wide range of mechanical systems, described by a set of generalised coordinates. It generally [21]
has the form of the following system of second-order differential equations
μ(q)¨q=γ (q,q)˙ +δ(q,q)u,˙ (39a) which we assume to be subject to mixed state-input constraints given through inequalities as follows
ζk(q,q,˙ u)≤0, for allk=1,. . .,l, (39b) wherelstands for an integer, and functionsζkdenote continuously differentiable mappings fromRn×Rn×Rp toR.
In (39a), function μ maps vectors ofRn into positive-definite matrices ofL(Rn)andγ denotes aC1differentiable function from Rn×RntoRn. Whileδtakes values inn×pmatrices. The control uhas values belonging to∈Rp.
To set constrained system (39) in the framework of this paper, let
x=(x1,x2) =. (q,q)˙ and =. π1(K).
Asμ(·)is invertible (because it is positive-definite), we get
˙ x1=x2,
˙
x2=μ(x1)−1γ (x1,x2)+μ(x1)−1δ(x1,x2)u.
We thus recover the dynamics of a control-affine system as given in (30), by setting
f0(x)=. (x2,μ(x1)−1γ (x1,x2)), and
fj(x)=. (0,cj(μ(x1)−1γ (x1,x2))), forj=1,. . .,p wherecj(·)denotes thejth column of a matrix. While the associated constraints subsetKis given by
K=
(x,u)∈Rn×Rn×Rp|ζk(x,u)≤0, for allk=1,. . .,l
. (40)
Its contingent cone at points (x,u)∈Kthat satisfy qualification condition (4) can be expressed by formula
TK(x,u)= {(y,v)∈Rn×Rn×Rp| ∇xζk(x,u),y
+ ∇uζk(x,u),v ≤0, for allk∈I(x,u)}, (41a) where
I(x,u) .
= {k|ζk(x,u)=0} (41b) denotes the subset of active constraints. Below we treat two examples by using the results of the previous sections.
Example 3:As an application, we consider the system illustrated by Fig. 3, which represents a bead of mass m, sliding with- out friction along a wire having the shape of a parabola y= Ax2, for a positive number A, and subject to the earth’s gravi- tational field g. Then, the Lagrange equation of motion is given as follows
(1+4A2q2)q¨= −4A2qq˙2−2gAq+u. (42) We assume that the mixed state-control constraint
q2+ ˙q2+u2≤1 (43) is imposed on the system. The termuinvolves a force applied to the bead, which we seek in feedback form in such a manner that the origin is reached at infinity. Obviously, (42) belongs to the class of Lagrange equations (39a) with
μ(q) .
=1+4A2q2, δ≡1, and
γ (q,q)˙ = −4A. 2q˙q2−2gAq.
Let x=(x1,x2) =. (q,q)˙ , then we can rewrite system (42) as follows
˙ x1=x2,
˙
x2= −4A2x1x22−2gAx1+u 1+4A2x21 .
From (43), we get, K= ¯B(0, 1) and = [−1 1] × [−1 1]. Note the contingent subset ofKis given by
TK(x,u)= {(y,v)∈R2×R| x,y +uv≤0}, for|x|2+ |u|2=1.
Let
ϕ(x,y)= x,y +α|x|2, for allx,y.
This defines an-Lyapunov function on spaceR3. The functions ϕ,mϕ, are, respectively, given by
ϕ(x,u)=(x2+2αx1)x2
+(2αx2+x1)
−4A2x1x22−2gAx1+u 1+4A2x12
+
−8A2x1(−4A2x1x22−2gAx1+u)
(1+4A2x21)2 +−4A2x22−2gA 1+4A2x12
x22
+
−4A2x1x22−2gAx1+u
1+4A2x21 − 16A2x1x22 1+4A2x12
×
−4A2x1x22−2gAx1+u 1+4A2x21
0 5 10 15 20 25 30 35 40
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
Time pendulum angles angular velocity
Fig. 6 Evolution of the state without control (u=0), for the same initial data as in Fig. 7. Note although it is bounded it does not converge at infinity
and
mϕ(x,u)= x2
1+4A2x12,
for each(x,u)∈K. Then, we are ready to achieve asymptotic null- controllability fromϕ. See Figs.4and5for simulation results.
Example 4:Consider the following Lagrange equation
¨
q= −ω2sin(q)−ucos(q). (44) It describes [20] the motion of a simple pendulum, which is free to oscillate in a vertical plane, subject to earth gravity. The term ω denotes the usual angular frequency. Controlu represents the acceleration of the suspension point of the pendulum, which we suppose moving on an horizontal straight line. We assume that the mixed state-input constraint
−log(q2+1)− ˙q2+u2≤0 (45) is imposed on the system. Let
x=(x1,x2) =. (q,q)˙ . Then, we get
˙
x1=x2=f1(x,u),
˙
x2= −ω2sin(x1)−ucos(x1)=f2(x,u), and
K=
(x1,x2,u)∈R3| −log(x21+1)−x22+u2≤0
. It follows that
(f(x,u),v)∈TK(x,u) ⇐⇒ − x1x2
1+x21
−x2
−ω2sin(x1)−ucos(x1)
+uv≤0,
for all(x,u)∈K. Let, forα >0
ϕ(x,y)= x,y +α|x|2, for allx,y,
0 5 10 15 20 25 30 35 40
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4 0.5
Time pendulum angles angular velocity
0 5 10 15 20 25 30 35 40
−0.2
−0.15
−0.1
−0.05 0 0.05 0.1 0.15
Time
The control u
0 5 10 15 20 25 30 35 40
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05 0
Time
−log(x12+1)−x22+u2
a b
c
Fig. 7 Simulation result of system (45) forα=0.0747 andω=1, using an optimisation tool for computing the minimal selection aState trajectories of system (44) with(q0,q˙0)=(0.5,−0.2)
bEvolution of the control withu0= −0.1 cEvolution of the mixed constraint (45)
which defines an -Lyapunov function. Then functions ϕ, and mϕ, are, respectively, given by
ϕ(x,u)=
x2+2αx1+x2
−ω2cos(x1)+usin(x1)
x2 +
−ω2sin(x1)−ucos(x1)+2αx2+x1
×
−ω2sin(x1)−ucos(x1)
, mϕ(x,u)= −x2cos(x1).
See Figs.6 and 7which illustrate the state of the system in the cases (a) with control, and (b) without control.
7 Concluding remarks and discussions
The main facts to point out about the set-valued approach devel- oped in this work, can be listed as follows:
1. It may easily incorporate any type of constraints on both state and control.
2. One of the serious problems one may face in handling non- linear control design, in general, consists of whether the proposed controls produce a solution to the excited non-linear system. Here, this is bypassed by using both tangential condition of viability theory and linear growth condition.
3. Systems under consideration, can be highly non-linear, and may not be restricted to the class of affine-control dependent sys- tems, as usually supposed in the classical CLF method. Illustrative Example1actually emphasises this fact.
4. The approach is global (non-local), it may deliver all the initial states from which the system is asymptotically null-controllable (not only the ones near the origin), going up to the whole state constraints domain.
5. Nevertheless, the following difficulties can be emphasised:
(a) The method works only for smooth control systems, with a C1 dynamics having a linear growth. A non-example, in the case of input constraints, would be the dynamics involving stan- dard input saturation function [13, 14]: this function is not differentiable.
(b) It needs computing contingent cones, generally known as a hard problem when unusual constraints are assumed. The cases of constraints described by convex subsets with smooth equalities and/or inequalities are thoroughly studied, we refer the reader to [18], one of the basic references on that subject.
8 Acknowledgments
The authors are very grateful to the reviewers for their comments and suggestions that improve the manuscript.
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