Global stabilization of multiple integrators by a bounded feedback with constraints on its successive derivatives
In this paper, we address the global stabilization of chains of integrators by means of a bounded static feedback law whose first time derivatives are bounded. Our construction is based on the technique of nested saturations introduced by Teel. We show that the control amplitude and the maximum value of its first derivatives can be imposed below any prescribed values. Our results are illustrated by the stabilization of the third order integrator on the feedback and its first two derivatives.
Actuator constraints is an important practical issue in control applications since it is a possible source of instability or performance degradation. Global stabilization of linear time-invariant (LTI) systems with actuator saturations (or bounded inputs) can be achieved if and only the uncontrolled linear system has no eigenvalues with positive real part and is stabilizable [SSY].
Among those systems, chains of integrators have received specific attention. Saturation of a linear feedback is not globally stabilizing as soon as the integrator chain is of dimension greater than or equal to three [FULLER69, SY91]. In [Teel92] a globally stabilizing feedback is constructed using nested saturations for the multiple integrator. This construction has been extended to the general case in [SSY], in which a family of stabilizing feedback laws is proposed as a linear combination of saturation functions. In [Marchand2003b] and [Marchand2005], the issue of performance of these bounded feedbacks is investigated for multiple integrators and some improvements are achieved by using variable levels of saturation. A gain scheduled feedback was proposed in [Megretski96bibooutput] to ensure robustness to some classes of bounded disturbances. Global practical stabilization has been achieved in [Gayaka:2011bm] in the presence of bounded actuator disturbances using a backstepping procedure.
Technological considerations may not only lead to a limited amplitude of the applied control law, but also to a limited reactivity. This problem is known as rate saturation [lauvdal97] and corresponds to the situation when the signal delivered by the actuator cannot have too fast variations. This issue has been addressed for instance in [SilvaTarbouch03]-[Freeman:1998tp]. In [SilvaTarbouch03, Galeani], regional stability is ensured through LMI-based conditions. In [lauvdal97], a gain scheduling technique is used to ensure semi-global stabilization of integrator chains. In [saberi2012], semi-global stabilization is obtained via low-gain feedback or low-and-high-gain feedback. In [Freeman:1998tp], a backstepping procedure is proposed to globally stabilize a nonlinear system with a control law whose amplitude and first derivative are bounded independently of the initial state.
In this paper, we deepen the investigations on global stabilization of LTI systems subject to bounded actuation with rate constraints. We consider rate constraints that affect only the first derivative of the control signal, but also its successive first derivatives, where denotes an arbitrary positive integer. Focusing on chains of integrators of arbitrary dimension, we propose a static feedback law that globally stabilizes chains of integrators, and whose magnitude and first derivatives are below arbitrarily prescribed values at all times. Our control law is based on the nested saturations introduced in [Teel92]. We rely on specific saturation functions, which are linear in a neighborhood of the origin and constant for large values of their argument.
This paper is organized as follows. In Section 2, we provide definitions and state our main result. The proof of the main result is given in Section 3 based on several technical lemmas. In Section 4, we test the efficiency of the proposed control law via numerical simulations on the third order integrator, with a feedback whose magnitude and two first derivatives are bounded by prescribed values. We provide some conclusions and possible future extensions in Section 5.
Notations. The function is defined as . Given a set and a constant , we let . Given and , we say that a function is of class if its differentials up to order exist and are continuous, and we use to denote the -th order derivative of . By convention, . The factorial of is denoted by . We define . We use to denote the set of matrices with real coefficients. denotes the -th Jordan block, i.e. the matrix given by if and zero otherwise. For each , refers to the column vector with coordinates equal to zero except the -th one equal to one.
2 Statement of the main result
In this section we present our main result on the stabilization of the multiple integrators with a control law whose magnitude and first derivatives are bounded by prescribed constants. Given , the multiple integrator of length is given by
Letting , System (1) can be compactly written as
In order to make the objectives of this paper more precise, we start by introducing the notion of -bounded feedback law by for System (1), which will be used all along the document.
Given and , let denote a family of positive constants. We say that is a -bounded feedback law by for System (1) if, for every trajectory of the closed loop system , the time function defined by for all satisfies, for all ,
Based on this definition, we can restate our stabilization problem as follows. Given and a set of positive real numbers , our aim is to design a feedback law which is a -bounded feedback law by for System (1) such that the origin of the closed-loop system is globally asymptotically stable. The case corresponds to global stabilization with bounded state feedback and has been addressed in e.g. [Teel92, Marchand2003b, Marchand2005]. The case corresponds to global stabilization with bounded state feedback and limited rate, in the line of e.g. [SilvaTarbouch03, Galeani, lauvdal97, saberi2012, Freeman:1998tp]. Inspired by [Teel92], our design for an arbitrary order is based on a nested saturations feedback, where saturations belong to the following class of functions.
Given , is defined as the set of all functions of class , which are odd, and such that there exists positive constants , , and satisfying, for all ,
, when ,
, when ,
, when .
In the sequel, we associate with every the -tuple .
The constants , , , and will be extensively used throughout the paper. Figure 1 helps fixing the ideas. represents the saturation level, meaning the maximum value that can be reached by the saturation. denotes the linearity threshold: for all , the saturation behaves like a purely linear gain. is the value of this gain that is, the slope of the saturation in the linear region. represents the saturation threshold: for all , the function saturates and takes a single value (either or ). Notice that it necessarily holds that and the equality may only hold when . We also stress that the successive derivatives up to order of an element of are bounded. An example of such function is given in Section 4 for .
Based on these two definitions, we are now ready to present our main result, which establishes that global stabilization on any chain of integrators by bounded feedback with constrained first derivatives can always be achieved by a particular choice of nested saturations.
Given and , let be a family of positive constants. For every set of saturation functions , there exists vectors in , and positive constants such that the feedback law defined, for each , as
is a -bounded feedback law by for System (1), and the origin of the closed-loop system is globally asymptotically stable.
The proof of this result is given in Section 3. It provides above theorem we give below also provides an explicit choice of the gain vectors and constants .
In [SSY], a stabilizing feedback law was constructed using linear combinations of saturated functions. That feedback with saturation functions in cannot be a -bounded feedback for System (1). To see this, consider the multi-integrator of length , given by . Any stabilizing feedback using a linear combination of saturation functions in is given by , where the constants , , , and are chosen to insure stability of the closed-loop system according to [SSY]. Let for all . A straightforward computation yields . Now consider a solution with initial condition , and such that . We then have , whose norm is greater than for some positive constants . Thus grows unbounded as tends to infinity, which contradicts the definition of a -bounded feedback.
Our construction is developed for chains of integrator, but it may fails for a general linear system stabilizable by bounded inputs. Consider for instance the harmonic oscillator given by , and a bounded stabilizing law given by with for some integer . The time derivative of verifies , which grows unbounded as the state norm increases, thus contradicting the definition of -bounded feedback.
3 Proof of the main result
3.1 Technical lemma
We start by giving a lemma that provides an upper bound of composed functions by exploiting the saturation region of the functions in .
Given , let and be functions of class , be a saturation function in with constants (), and and be subsets of such that . Assume that
and there exists positive constants such that
Then the th-order derivative of , defined by , satisfies
where is a polynomial function of , and for each .
Proof of Lemma 1.
The proof relies on Faà Di Bruno’s formula, which we recall
Lemma 2 (Faà Di Bruno’s formula, [fdb], p. 96).
Given , let and . Then the -th order derivative of the composite function is given by
where is the Bell polynomial given by
where denotes the set of tuples of positive integers satisfying
Using Lemma 2, a straightforward computation yield
Since , (3) ensures that the set is contained in the saturation zone of . It follows that
From definition of and (7), we get that
3.2 Intermediate results
In this subsection we provide two propositions which will be used in the proof of Theorem 1. We start by introducing some necessary notation.
Given and , let be saturations in with respective constants , . We define, for each ,
We also let
Note that these quantities are well defined since the functions are all in .
We also make a linear change of coordinates , with , that puts System (1) into the form
with the convention . The matrix can be determined from
For this system, we define a nested saturations feedback law as
Let be a trajectory of the system
with . Notice that with the above functions, the closed loop system (18) can be rewritten as
For , we also let
with , and
Note that from the definitions of and , we have , and a straightforward computation yields
The following statement provides explicit bounds on the successive derivatives of each functions , for each and the time function given by .
Given and , let be saturation functions in with respective constants for each . With the notation introduced in this section and the Bell polynomials introduced in (8), every trajectory of the closed-loop system (18) satisfies, for each and each ,
where , , and are independent of initial conditions and are obtained recursively as follows: for ,
and, for each ,
Proof of Proposition 1.
Let be a trajectory of the closed loop system (18). The right-hand side of (18) being of class and globally Lipschitz, System (18) is forward complete and its trajectories are of class . Therefore the successive time derivatives of , , and are well defined.
for all . Since , the above estimate is still true on . Moreover, from (19) it holds that at all positive times. has been proven for each .
We now prove by induction on the statement . Since , the case is done. Assume that, for a given , the statement holds for all . From Lemma 1 (with , , , , , , , , , , and (22)), we can establish that holds. Thus holds for all .
Assume that for a given , statements , and hold for all and all . Let . From (15), a straightforward computation yields
From , , we obtain that
Thus the statement is proven for all .
We now prove by induction on the statement . As before, since , the case for is done. Assume that for a given , the statement holds for all . From Lemma 1 (with , , , , , , , , , , and (22)), we can establish that holds. is thus satisfied for all .
We next provide sufficient conditions on the parameters of the saturation functions in guaranteeing global asymptotic stability of the closed-loop system (18).
Given and , let be saturation functions in with respective constants for each and assume that, for all ,
Then the origin of the closed-loop system (18) is globally asymptotically stable.
Actually the above proposition is almost the same as the one given in [Teel92], except that we allow the first level of saturation to have a slope different from .
Proof of Proposition 2.
To that end, we consider the Lyapunov function candidate . Its derivative along the trajectories of (18) reads
From (29b), we can obtain that for all
We next show that there exists a time such that , for all . To prove that we have the following alternatives : either for every , and we are done, or there exist such that . In that case there exists such that (otherwise thanks to (30), as which is impossible). Due to (30), we have in a right open neighbourhood of . Suppose that there exists a positive time such that . Then by continuity, there must exists such that , and for all . However, it then follows from (30) that for a left open neighbourhood of we have . This is a contradiction with the fact that on a right open neighbourhood of w have . Therefore, for every , one has and the claim is proved.
It follows from (29b) that
Therefore operates in its linear region after time . Similarly, we now consider , whose derivative along the trajectories of (18) satisfies
Reasoning as before and invoking (29b), there exists a time such that and operates in its linear region for all .
By repeating this procedure, we construct a time such that for all times greater than the whole feedback law becomes linear. That is
for all . System (18) becomes simply linear and its local exponential stability follows readily. Thus the origin of System (18) is globally asymptotically stable, which concludes the proof of Proposition 2. ∎
3.3 Proof of Theorem 1
Given and , let be saturation functions in with constants for each , and let be a family of positive constants. We let