Global stabilization of linear systems with bounds on the feedback and its successive derivatives
Abstract
We address the global stabilization of linear timeinvariant (LTI) systems when the magnitude of the control input and its successive time derivatives, up to an order , are bounded by prescribed values. We propose a static state feedback that solves this problem for any admissible LTI systems, namely for stabilizable systems whose internal dynamics has no eigenvalue with positive real part. This generalizes previous work done for singleinput chains of integrators and rotating dynamics.
1 Introduction
The study of control systems subject to input constraints is motivated by the fact that signals delivered by physical actuators may be limited in amplitude, and may not evolve arbitrarily fast. An a priori bound on the amplitude of the control signal is usually referred to as input saturation whereas a bound on the variation of control signal is referred to as rate saturation (e.g [saberi2012]).
Stabilization of linear timeinvariant systems (LTI for short) with input saturation has been widely studied in the literature. Such a system is given by
where , belongs to a bounded subset of , is an matrix and is an one. Global stabilization of can be achieved if and only if the LTI system is asymptotically null controllable with bounded controls, i.e., it can be stabilized in the absence of input constraint and the eigenvalues of have non positive real parts. Saturating a linear feedback law may fail at globally stabilizing as it was observed first in [FULLER69] and then [SY91] for the special case of integrator chains (i.e., when is the th Jordan block and ). As shown for instance in [OptCRyan], optimal control can be used to define a globally stabilizing feedback for but, when the dimension is greater than , deriving a closed form for this stabilizer becomes extremely difficult. The first globally stabilizing feedback with rather simple closed form (nested saturations) was provided in [Teel92] for chains of integrators and then in [SSY] for the general case. In [Lin95control], a global feedback stabilizer for was built by relying on control Lyapunov functions arising from a mere existence result. Other globally stabilizing feedback laws for have been proposed with an additional property of robustness with respect to perturbations. In [Saberi:2002ux], using lowandhigh gain techniques, a robust stabilizer was proposed to ensure semiglobal stability, meaning that the control gains can be tuned in such a way that the basin of attraction contains any prescribed compact subset of . This restriction has been removed in [saberi2000], where the authors provided a global feedback stabilizer for which is robust with respect to perturbations, based on an earlier idea due to Megretsky [Megretski96bibooutput]. Nonetheless, the feedback laws of [saberi2000] and [Megretski96bibooutput] require to solve a nonlinear optimization problem at every point , which makes its practical implementation questionable. In [chitour2015], an easily implementable global feedback stabilizer for which is robust with respect to perturbations was proposed but it only covers the multiple integrator case and it is discontinuous since it is based on sliding mode techniques. Robust stabilization of was also addressed in [AZCHCHGR15] by relying on the control Lyapunov techniques developed in [Lin95control].
In contrast to stabilization of LTI systems subject to input saturation, there are much less results available in the literature regarding global stabilization under rate saturation, i.e., when the first time derivative of the control signal is also a priori bounded. In [Freeman:1998tp], the authors rely on a backstepping procedure to build a bounded globally stabilizing feedback with a bounded rate, but the methodology does not allow to a priori impose a prescribed rate. In [SFbound], a dynamic feedback law inspired from [Megretski96bibooutput] is constructed and can even be generalized to take into account constraints on higher time derivatives of the control signal. However, as mentioned previously, the numerical efficiency of such feedbacks is definitely questionable. A rather involved global feedback stabilizer for achieving amplitude and rate saturations was also obtained in [SoSuAL] for continuous time affine systems with a stable free dynamics. This corresponds in our setting to requiring that the matrix is stable, i.e., (up to similarity). Finally, let us mention the references [lauvdal97], [lin1997semi] for semiglobal stabilization results and [SilvaTarbouch03] for local stabilization results using LMIs and antiwindup design. One should also mention [teel1996nonlinear] where a nonlinear small gain theorem is given for the behaviour analysis of control systems with saturation.
The results presented here encompass input and rate saturations as special cases. More precisely, given an integer , we construct a globally stabilizing feedback for such that the control signal and its first time derivatives, are bounded by arbitrary prescribed positive values, along all trajectories of the closedloop system. This problem has already been solved by the authors in [LCC1] for the multiple integrator and skewsymmetric cases. The solution given in that paper for the multiple integrator case consisted in considering appropriate nested saturation feedbacks. We also indicated in [LCC1] that these feedbacks fail at ensuring global stability in the skewsymmetric case and we then provided an ad hoc feedback law for this specific case. Here, we solve the general case with a unified strategy.
The paper should be seen as a first theoretical step towards the global stabilization of an LTI system when the input signal is delivered by a dynamical actuator that limits the control action in terms of magnitude and first time derivatives. Further developments are needed to explicitly take into account the dynamics of such an actuator. Possible extensions of this work may also address the question of global stabilization by smooth feedback laws (i.e., with respect to time) when all successive derivatives need to be bounded by prescribed values.
The paper is organized as follows. In Section 2, we precisely state the problem we want to tackle, the needed definitions as well as the main results we obtain, namely Theorem 1 for the single input case and Theorem 2 for the multiple input case. Section 3 contains the proof of the main results. In section 3.1.1 we show that the proof of Theorem 1 is a consequence of two propositions. The first one (cf. Proposition 1), we show that the feedback proposed in Theorem 1 is indeed a globally stabilizing feedback for . We actually prove a stronger result dealing with robustness properties of this feedback, as it is required in [Teel92] and [SSY]. The second proposition (cf. Proposition 2) specifically deals with bounding the first derivatives of the control signal by relying on delicate estimates. Section 3.2.1 contains the proof of Theorem 2 which is a consequence of Proposition 1 and Proposition 3, the latter providing estimates on the successive time derivatives of the control signal. We close the paper by an Appendix, where we gather several technical results used throughout the paper.
Notations :
We use and to denote the sets of real numbers and the set of non negative integers respectively. Given a set and a constant , we let . Given , we define . For a given set , the boundary of is denoted by . The factorial of is denoted by and the binomial coefficient is denoted .
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 differential of . By convention, .
Given , denotes the set of matrices with real coefficients. The transpose of a matrix is denoted by . The identity matrix of dimension is denoted by . We say that an eigenvalue of is critical if it has zero real part and we set where is the number of conjugate pairs of nonzero purely imaginary eigenvalues of (counting multiplicity), and is the multiplicity of the zero eigenvalue of . We define , and .
We use to denote the Euclidean norm of an arbitrary vector . Given and , we say that is eventually bounded by , and we write , if there exists such that for all .
2 Problem statement and main results
Given and , consider the LTI system defined by
(1) 
where , , , and . Assume that the pair is stabilizable and that all the eigenvalues of have non positive real parts. Recall that these assumptions on are necessary and sufficient for the existence of a bounded continuous state feedback which globally asymptotically stabilizes the origin of (1): see [SSY].
Given an integer and a tuple of positive real numbers , we want to derive a feedback law whose magnitude and first time derivatives are bounded by , .
Definition 1 (feedback law bounded by ).
Given , and , let be a tuple of positive real numbers. We say that is a feedback law bounded by for system (1) if it is of class and, for every trajectory of the closedloop system , the control signal , satisfies for all . The function is said to be a feedback law bounded for system (1), if there exist tuple of positive real numbers such that is a feedback law bounded by for system (1).
Based on this definition, we can write our stabilization problem of Bounded Higher Derivatives as follows.
Problem (BHD).
Given and a tuple of positive real numbers , design a feedback law such that the origin of the closedloop system is globally asymptotically stable (GAS for short) and the feedback is a feedback law bounded by for system (1).
Our construction to solve Problem (BHD) will often use the property of Small Input Small State with linear gain ( for short) developed in [SSY]. We recall below its definition
Definition 2 (, [Ssy]).
Given and , the control system , with and , is said to be if, for all and all bounded measurable signal eventually bounded by , every solution of is eventually bounded by . A system is said to be if it is for some . An inputfree system is called , if the control system is .
Remark 1.
It follows readily from this definition that if is , then all solutions converge to the origin. Note, however, that the property does not necessarily ensure GAS in the absence of input, as it does not imply stability of its origin.
When a feedback law ensures both global asymptotic stability and , we refer to is an stabilizing feedback.
Definition 3 (stabilizing feedback).
Given a control system with and , we say that a feedback law is stabilizing if the origin of the closedloop system is globally asymptotically stable. If, in addition, this closedloop system is , then we say that is stabilizing.
As mentioned before the feedback law given in [LCC1], which solves Problem (BHD) for the special case of multiple integrators, simply made use of nested saturations with carefully chosen saturation functions. We recall next why this feedback construction cannot work in general. For that purpose it is enough to consider the 2D simple oscillator case which is the control system given by , with , and . This system is one of the two basic systems to be stabilized by means of a bounded feedback, as explained in [SSY]. One must then consider a stabilizing feedback law , where is a fixed vector in and is a saturation function, i.e., a bounded, continuously differentiable function satisfying for and . Note that is chosen so that the linearized system at is Hurwitz. In particular it implies that . Pick now the following sequence of initial conditions . A straightforward computation yields that the first time derivative of the control along each trajectory satisfies , which grows unbounded as tends to infinity. Therefore this feedback can not be a bounded feedback.
In order to solve Problem (BHD) for the oscillator, we showed in [LCC1] that a feedback law of the type with and does the job and it also solves Problem (BHD) in case the matrix in (1) is stable. However, we are not able to show whether stabilizes or not the system in the case where . It turns out that the previous issue is as difficult as asking if a saturated linear feedback stabilizes or not the abovementioned 4D case, which is an open problem. It is therefore not immediate how to address the general case. This is why Theorem 1 is a non trivial extension of the solution of Problem (BHD) provided for the twodimensional oscillator.
2.1 Single input case
For the case of single input systems the solution of Problem (PHB) is given by the following statement.
Theorem 1 (Single input).
Given , consider a single input system where , and . Assume that has no eigenvalue with positive real part and that the pair is stabilizable. Then, given any and any tuple of positive real numbers, there exist vectors and matrices , , such that the feedback law defined as
(2) 
is a feedback law bounded by and stabilizing for system .
In view of Definition 3, the feedback law (2) globally asymptotically stabilizes the origin of (1), and thus solves Problem (BHD). We stress that, even though the exact computation of the control gains is quite involved (see proof in Section 3), the structure of the proposed feedback law (2) is rather simple. It should also be noted that, unlike the results developed in [LCC1], this feedback law applies to any admissible singleinput systems in a unified manner.
2.2 Multiple input case
To give the main result for LTI system with multiple input we need this following definition.
Definition 4 (Reduced controllability form).
Given and , a LTI system is said to be in reduced controllability form if it reads
(3) 
where, for some tuple in with , is Hurwitz, for every all the eigenvalues of are critical, and the pairs are controllable.
From Lemma in [SSY], it is then clear that without loss of generality, in our case, we can consider that system (1) is already given in the reduced controllability form. We can now establish the solution of Problem (BHD) for the multiple input case.
Theorem 2 (Multiple input).
This statement provides a unified control law solving Problem (BHD) for all admissible LTI systems. It allows in particular multiinput systems, which was not covered in [LCC1].
3 Proof of the main results
3.1 Proof of Theorem 1
In this section, we prove Theorem 1. For that purpose, we first reduce the argument to establishing of Propositions 1 and 2 given below. The first one indicates that the feedback given in Theorem 1 is stabilizing for in the case of single input. The second proposition provides an estimate of the successive time derivatives of the control signal.
3.1.1 Reduction of the proof of Theorem 1 to the proofs of Propositions 1 and 2
Let , and be a tuple of positive real numbers. Define . Consider a single input linear system where , and are and matrices respectively. We assume that the pair is stabilizable and that all the eigenvalues of have non positive real parts. As observed in [SSY], it is sufficient to consider the case where the pair is controllable and all eigenvalues of are critical. Indeed, since is stabilizable there exists a linear change of coordinates transforming and into and , where is Hurwitz, the eigenvalues of are critical and the pair is controllable. Then, it is immediate to see that we only have to treat the case where has only critical eigenvalues. From now on, we therefore assume that has only eigenvalues with zero real parts, and that the pair is controllable.
Our construction uses the following linear change of coordinates given by [SSY, Lemma 5.2]. This decomposition puts the original system in a triangular form made of onedimensional integrators and twodimensional oscillators.
Lemma 1 (Lemma in [Ssy]).
Let , , , be a controllable single input linear system. Assume that all the eigenvalues of are critical. Let be the nonzero eigenvalues of . Let be a family of positive numbers. Define
(6) 
Then there is a linear change of coordinates that puts in the form
(7)  
where for , and for .
With no loss of generality, we prove Theorem 1 for system (7), where the positive constants will be fixed later. Let be a positive constant. We rely on a candidate feedback under the form
(8) 
with
(9) 
It therefore remains to choose the positive constants such that the feedback law (8) is a feedback law bounded by , and stabilizing for system (7). For that aim, we rely on the next two propositions, respectively proven in Sections 3.1.2 and 3.1.3.
Proposition 1.
Proposition 2.
Let , , , be a controllable single input linear system. Assume that all the eigenvalues of are critical. Let be the nonzero eigenvalues of . Let , , be positive constants in . Then, there exist a positive constant , and continuous functions , , such that for any trajectory of the closedloop system (7) with the feedback law (8), the control signal defined by for all satisfies, for all ,
Pick in such a way that
Choose recursively , , such that
where the functions appearing above are defined in Proposition 2. By Proposition 1, the feedback law (8) is stabilizing for system (7). Moreover, as a consequence of Proposition 2, for any trajectory of the closedloop system (7) with the feedback law (8), the control signal defined by for all satisfies for all . Thus, the feedback law (8) is a feedback law bounded by for system (7). Since there is a linear change of coordinate () that puts (7) into the original form , the feedback law defined given in (2) can be picked as
and it is a feedback law bounded by , and stabilizing for (1). To sum up, the proof of Theorem 1 boils down to establishing Propositions 1 and 2.
3.1.2 Proof of Proposition 1
Proposition 1 is proved by induction on . More precisely, we show that the following property holds true for every positive integer .
In order to start the argument, we give intermediate results whose proofs are given in Appendix and which will be used for the initialization step of the induction and the inductive step. The first statement establishes for the onedimensional integrator.
Lemma 2.
Let . For every , the scalar system given by
(10) 
is , its origin is and its linearisation around zero is .
The next lemma guarantees that the twodimensional oscillator is .
Lemma 3.
For every , there exist such that for any the twodimensional system given by
(11) 
is , its origin is and its linearisation around zero is .
We now start the inductive proof of . For , we have to consider two cases. Either and corresponding to the simple integrator
(12) 
or and corresponding to the simple oscillator
(13) 
for some . In both cases, can be readily deduced by invoking Lemma 2 and 3 respectively. Given , assume that holds. In order to establish , it is sufficient to consider the following two cases:

, i.e, all the eigenvalues of are zero (multiple integrator);

, i.e some eigenvalues of have non zero imaginary part (multiple integrator with rotating modes).
In both cases we reduce our problem to the choice of only one constant using the inductive hypothesis.
case i)
Let be a set of positive numbers to be chosen later. Consider the multiple integrator given by
where for . Let . We then can rewrite this system as
for some matrices and of appropriate dimensions. From the inductive hypothesis, there exist functions for such that for any set of positive constants satisfying satisfying and , for each , the feedback law defined by
is stabilizing for . Choose satisfying the above conditions. The feedback law (8) is then given by
Since for all (see (1) and (9)), the closedloop system can be rewritten as
(14) 
with
(15)  
(16)  
(17) 
We now move to the other case where the dynamics involves multiple integrators with rotating modes.
case ii)
Let be a set of positive constants to be chosen later. Let , and be such that . Let be a set of non zero real numbers. Consider the following linear control system
where for , and for . Let . We then can rewrite this system as follows
From the inductive hypothesis, there exist functions for such that for any set of positive constant satisfying and , for each , the feedback law defined by
(18) 
is stabilizing for . Choose satisfying the above conditions. The feedback law (8) is then given by
By noticing that for all (see (1) and (9)), the closedloop system can be rewritten as
(19) 
with
(20)  
(21)  
(22) 
In both cases, it remains to show that there exists a function such that if then the closedloop systems (3.1.2) and (3.1.2) are , globally asymptotically stable with respect to the origin, and theirs respective linearizations at zero are asymptotically stable. It is sufficient to prove that the closedloop systems are and their linearization at zero are asymptotically stable. Indeed, from Remark 1, the property guarantees the convergence of any solution of the closedloop with no input. If moreover the linearized system is asymptotically stable, then the globally asymptotic stability of zero follows readily.