arxiv:1812.11538v2 [cs.sy] 9 may 2020 · that distinguishes the control of nonholonomic systems...

2010; 00:1–15Published online in Wiley InterScience (www.interscience.wiley.com).

Smooth, time-invariant regulation of nonholonomic systems viaenergy pumping-and-damping†

Bowen Yi1,2, Romeo Ortega2,3, Weidong Zhang1∗

1. Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China2. Laboratoire des Signaux et Systémes, CNRS-CentraleSupélec, Gif-sur-Yvette 91192, France

3. Department of Control Systems and Informatics, ITMO University, St. Petersburg 197101, Russia

SUMMARY

In this paper we propose an energy pumping-and-damping technique to regulate nonholonomic systemsdescribed by kinematic models. The controller design follows the widely popular interconnection anddamping assignment passivity-based methodology, with the free matrices partially structured. Twoasymptotic regulation objectives are considered: drive to zero the state or drive the systems total energy toa desired constant value. In both cases, the control laws are smooth, time-invariant, state-feedbacks. For thenonholonomic integrator we give an almost global solution for both problems, with the objectives ensuredfor all system initial conditions starting outside a set that has zero Lebesgue measure and is nowhere dense.For the general case of higher-order nonholonomic systems in chained form, a local stability result is given.Simulation results comparing the performance of the proposed controller with other existing designs are alsoprovided. Copyright c© 2010 John Wiley & Sons, Ltd.

Received . . .

KEY WORDS: nonholonomic systems, passivity-based control, interconnection and damping assign-ment, energy pumping-and-damping

1. INTRODUCTION

The study of mechanical system subject to nonholonomic constraints has been carried-out withinthe realm of analytical mechanics [4, 5]. The complexity and highly nonlinear dynamics ofnonholonomic mechanical systems make the motion control problem challenging [4]. A key featurethat distinguishes the control of nonholonomic systems from that of holonomic systems is that inthe former, it is not possible to render asymptotically stable an isolated equilibrium with a smooth(or even continuous), time-invariant (static or dynamic), state-feedback control law. The best onecan achieve with smooth control laws is to stabilise an equilibrium manifold [20]. This obstaclestems from Brockett’s necessary condition for asymptotic stabilization [6]—see also [4]. In viewof the aforementioned limitation, time-varying feedback [26, 15], discontinuous feedback [1, 12],switching control methods [18] and hybrid systems approaches [14], have been considered in thecontrol literature. In this paper we are interested in investigating the possibilities of regulatingnonholonomic systems via smooth, time-invariant state-feedback.

†the National Natural Science Foundation of China (61473183, U1509211, 61627810), National Key R&D Program ofChina (SQ2017YFGH001005), China Scholarship Council, and by the Government of the Russian Federation (074U01),the Ministry of Education and Science of Russian Federation (14.Z50.31.0031, goszadanie no. 8.8885.2017/8.9).∗Correspondence to: Shanghai Jiao Tong University, Shanghai 200240, China ([email protected])

Copyright c© 2010 John Wiley & Sons, Ltd.

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2 B. YI et al.

In the 1998 paper [10] a radically new approach to regulate the behaviour of nonholonomicsystems was proposed. The work was inspired by the classical field-oriented control (FOC) ofinduction motors, which was introduced in the drives community in 1972 [3], and is now thede facto standard in all high-performance applications of electric drives—see [19] for a moderncontrol-oriented explanation of the method. The basic idea of FOC is to regulate, with a smooth,time invariant, state-feedback law, the speed (or the torque) of the motor by inducing an oscillation,with the desired frequency and amplitude, to the motors magnetic flux, that is a two-dimensionalvector. From the physical viewpoint this is tantamount to controlling the mechanical energy via theregulation of the magnetic energy. As shown in [10], applying this procedure to the nonholonomicintegrator allows us to drive the state to an arbitrarily small neighborhood of the origin as well assolving trajectory tracking problems. Unfortunately, when the objective is to drive the state to zero,the control law includes the division by a state-dependent signal—rendering the controller not-globally defined. Although this signal is bounded away from zero along trajectories, in the face ofnoise or parameter uncertainty, it may cross through zero, putting a question mark on the robustnessof the design. It should be mentioned that the results of [10] were later adopted in [7] and are theinspiration for the transverse function approach pursued in [21, 22].

In [28] it is shown that FOC can be interpreted as an Interconnection and Damping AssignmentPassivity-based controller (IDA-PBC) [25] that assigns a port-Hamiltonian (pH) structure tothe closed-loop. The corresponding energy function has the shape of a “Mexican sombrero”,whose minimum is achieved in the periodic orbit that we want to reach, e.g., H`(x`) = β`, withx` part of the state coordinates, whose energy function is H`(x`), and β` is a positive, tuningconstant—see Figure 1. The same approach was proposed in [13] to induce an oscillation in theBall-and-Beam system and in [9] in walking robot applications. To assign the Mexican sombreroshape the energy function contains a term of the form (H`(x`)− β`)2, whose gradient can betransferred to the dissipation matrix of the pH system, giving then an interpretation of “EnergyPumping-and-Damping” (EPD). That is, a controller that injects or extracts energy from the systemdepending on the location of the state with respect to the desired oscillating trajectory, see Fig 3.This point of view was adopted in [2] to design a controller that swings up—without switching—thecart-pendulum system. In the sequel, we will refer to this controller design technique as EPDIDA-PBC, that is, a variation of IDA-PBC where the (otherwise free) dissipation matrix is partiallystructured. EPD IDA-PBC has been used in [28] to solve the more general orbital stabilizationproblem, where we made the important observation that, by setting β` = 0, we can achieveregulation to zero of the state.

The main objective of this paper is to show that an EPD IDA-PBC formulation of the schemeproposed in [10] provides a suitable framework for the solution of the following problems:

• Find a globally defined, smooth, time-invariant state-feedback that achieves either one of thefollowing asymptotic regulation objectives for nonholonomic systems: drive to zero the stateor drive the systems total energy to a desired constant value.

The objectives should be ensured for initial conditions starting sufficiently close to the desiredobjective but outside a set which has zero Lebesgue measure and is nowhere dense.† Followingstandard practice, the qualifier “almost” will be used to underscore the latter feature.

For the nonholonomic integrator we give an almost global solution for both problems—that is,all trajectories starting outside a zero-measure set converge to their desired value. For the generalcase of higher-order nonholonomic systems in chained form, it is shown that the EPD IDA-PBCmatching equation is always solvable, and a local result is given.

The reminder of the paper is organized as follows. In Section 2 we introduce the problemformulation and the EPD IDA-PBC method to achieve almost global regulation of nonholonomicsystems in its general form. In Section 3 we give the constructive solutions for the nonholonomic

†Clearly, to comply with Brockett’s necessary condition, in the case of regulation to zero this set should contain theorigin.

Copyright c© 2010 John Wiley & Sons, Ltd. (2010)Prepared using acsauth.cls

SMOOTH, TIME-INVARIANT REGULATION OF NONHOLONOMIC SYSTEMS 3

Hd

desired curve

Figure 1. Shape of the energy function assigned by the FOC, with the desired periodic orbit in red

integrator, which are extended to high-order nonholonomic systems in chained form in Section 4.The paper is wrapped-up with simulations results in Section 5 and concluding remarks in Section 6.

Notation. In is the n× n identity matrix. For x ∈ Rn, W ∈ Rn×n, W =W> > 0, we denote theEuclidean norm |x|2 := x>x, and the weighted-norm ‖x‖2W := x>Wx. All mappings are assumedsmooth. Given a function H : Rn → R we define the differential operator ∇H(x) :=

(∂H∂x

)>.

2. REGULATION OF NONHOLONOMIC SYSTEMS VIA EPD IDA-PBC

In this paper, we adopt the driftless system representation of the nonholonomic system

ẋ = S(x)u, (1)

with x ∈ Rn the generalized position, u ∈ Rm the velocity vector, which is the control input, n > m,and the mapping S : Rn → Rn×m. The corresponding constraint is

A>(x)ẋ = 0 (2)

with A : Rn → Rn×(n−m) full-rank. It is assumed that the system is completely nonholonomic,hence controllable. We refer the reader to [4] for further details on nonholonomic systems.

The proposition below shows that the problem of regulation of the system (1) can be recast as anEPD IDA-PBC design. Following the “FOC approach” advocated in [10]—see also [28]—the ideais to decompose the state of the system into two components as‡[

x`x0

]= x, x` ∈ Rn` , x0 ∈ Rn0 (3)

with n = n0 + n` and to find a smooth state-feedback that transforms the closed-loop dynamics intoa pH system of the form[

ẋ`ẋ0

]=

[J`(x)−R`(x) 0

0 J0(x)−R0(x)

]∇H(x), (4)

where the total energy function is given by

H(x) := H`(x`) +H0(x0),

with H0 : Rn0 → R, H` : Rn` → R and the interconnection and damping matrices

J0 : Rn → Rn0×n0 , J` : Rn → Rn`×n` , R0 : Rn → Rn0×n0 , R` : Rn → Rn`×n` (5)

‡See Remark 1 for the case of an arbitrary state partition.


4 B. YI et al.

satisfying

J0(x) = −J>0 (x), J`(x) = −J>` (x), R0(x) = R>0 (x) ≥ 0. (6)

The control objectives are to ensure that

limt→∞

H`(x`(t)) = β` > 0, limt→∞

x0(t) = 0, (7)

orlimt→∞

x(t) = 0. (8)

As seen in the proposition below these objectives are achieved making the trajectory converge tothe curveH`(x`(t)) = β` via the EPD principle, where β` > 0 in the first case and β` = 0 to regulatethe state to zero. The EPD principle imposes the following constraint on R`(x):

[R`(x) +R>` (x)]H

s` (x`) ≥ 0, (9)

where we defined the (shifted) energy function

Hs` (x`) := H`(x`)− β`. (10)

To streamline the presentation of the result we partition the matrix A(x) as

A(x) =

[A`(x)A0(x)

], (11)

with A` : Rn → Rn`×(n−m) and A0 : Rn → Rn0×(n−m).

Proposition 1Consider the system (1) and the state partition (3). Fix β` ≥ 0. Assume there exist energy functionsH0(x0), H`(x`) and interconnection and damping matrices (5), (6) verifying the followingconditions.

C1. The matching PDE

A>` (x)[J`(x)−R`(x)

]∇H`(x`) +A>0 (x)

[J0(x)−R0(x)

]∇H0(x0) = 0. (12)

C2. The EPD condition (9).C3. The minimum condition

∇H(x)|x=0 = 0, ∇2H(x) > 0, (13)

and the origin is isolated.C4. Define the function

Q(x) := ‖∇H0(x0)‖2R0(x) +1

2Hs` (x`)‖∇H`(x`)‖2R`(x)+R>` (x). (14)

For the system (4), there exists a function h : Rn → R such that the following detectability-like implication holds[

limt→∞

Q(x(t)) = 0 and x(0) /∈ I]⇒ (7) [or (8), respectively], (15)

with I := {x ∈ Rn | h(x) = 0} the inadmissible initial condition set.

Assume the initial conditions of the system are outside the set I. Then, the control law

u = [S>(x)S(x)]−1S>(x)

[(J`(x)−R`(x)

)∇H`(x`)(

J0(x)−R0(x))∇H0(x0)

]. (16)

ensures (7) when β` > 0 or (8) when β` = 0.



ProofSome simple calculations show that the closed-loop dynamics takes the pH form (4). Define thefunction

V (x) =1

2|Hs` (x`)|2 +H0(x0), (17)

the derivative of which isV̇ = −Q(x) ≤ 0, (18)

where the upperbound is obtained using (9). Invoking the properties of the function V (x), weconclude Lyapunov stability of the closed-loop dynamics with respect to the energy level set{x ∈ Rn|H`(x`) = β`, x0 = 0} for β` > 0 (or the origin if β` = 0, respectively).

According to Barbalat’s Lemma, together with (18), we have

limt→∞

Q(x(t)) = 0

for any intial conditions. Since x(0) /∈ I, and using the convergence implication (15) directly, wehave that (7) holds when β` > 0. If β` = 0 we conclude (8) recalling the minimum condition C3.222

Proposition 1 for β` = 0 does not contradict Brockett’s necessary condition. Indeed, in theproposed design we only guarantee that the origin of the closed-loop system is Lyapunov stablebut not asymptotically stable. More precisely, we establish the following implication[

|x(0)| < δ, x(0) /∈ I]⇒ lim

t→∞x(t) = 0,

which differs from the usual attractivity condition |x(0)| < δ ⇒ limt→∞ x(t) = 0. See Fig. 2.Interestingly, we have the following lemma, whose proof is given in Appendix A.

Lemma 1Consider the scenario of Proposition 1, with β` = 0. The following implication is true:

[C1−C4 ⇒ {0} ⊂ cl(I)],

where cl(·) denotes the closure of the set.

asymptotic stability asymptotic regulation

I

Figure 2. An illustration of the difference between asymptotic stability and the case studied in the paper

Remark 1To simplify the presentation we have assumed the direct partition of the state given in (3).Proposition 1 can be easily extended to the case where the partition is of the form[

x`x0

]= Px,

where P ∈ Rn×n is a permutation matrix.

Remark 2The EPD principle is codified in the inequality (9) and graphically illustrated in Fig. 3. Clearly,when β` = 0, the EPD IDA-PBC becomes the standard IDA-PBC with damping injection ensuredby (9).


6 B. YI et al.

H`(x`)

damping R`(x) + R>` (x) ¸ 0

pumping R`(x) + R>` (x) · 0

H`(x`) = ¯`

Figure 3. An interpretation to the EPD control method

Remark 3It should be pointed out that when β` > 0 the origin x = 0 is an unstable equilibrium point of theclosed-loop system. Indeed, the minimum condition (13) ensures that H(x) is a (locally) positivedefinite function whose derivative is given as

Ḣ = −‖∇H0(x0)‖2R0(x) − ‖∇H`(x`)‖2R`(x)+R>` (x)

.

On the other hand, in a small (relative to β`) neighborhood of x` = 0, the EPD condition (9)imposes that R`(x) +R>` (x) < 0. Hence, there exists a neighborhood of x = 0 where Ḣ > 0 that—according to Lyapunov’s first instability theorem [16]—implies that the origin is unstable.

Remark 4Standard IDA-PBC has been applied in [8, 20] to stabilise a manifold containing the desiredequilibrium point, in the latter publication including disturbance rejection. In [11, 12] switchedor non-smooth versions of IDA-PBC that ensure convergence to the desired equilibrium point areproposed.

Remark 5As indicated in the Introduction, the (mathemathically elegant) transverse function method of[21, 22] follows the same approach adopted here—which was originally inspired by [10].§ Thismethod can be used for the tracking problem of controllable driftless systems invariant on a Liegroup.

3. NONHOLONOMIC INTEGRATOR

In this section, we consider the benchmark example of the nonholonomic integrator described inchained form by

ẋ1 = u1

ẋ2 = u2

ẋ3 = x2u1. (19)

§The first use of the FOC approach for the control of nonholonomic systems is, erroneously credited to [7] in [21, 22].In view of the tangential reference to [10] made in [7], this is probably inadvertendly.



The system can be represented in the form (1), (2) with the definitions

S(x) =

1 00 1x2 0

, A(x) =x20−1

.The proposition below solves, via direct application of Proposition 1, the problems of regulation

of the energy or driving the state to zero for the system (19).

Proposition 2Consider the nonholonomic system (19) with the state partition x` = col(x1, x2) and x0 = x3. Fixβ` ≥ 0.

P1. The functionsH`(x`) =

1

2(x21 + x

22), H0(x0) =

1

2x23,

together with the mappings

J0 = 0, R0(x) = x22, J`(x) =

[0 −x3x3 0

], R`(x) =

[0 00 γHs` (x`)

],

with γ > 0 and Hs` (x`) defined in (10), verify conditions C1-C3 of Proposition 1.P2. The smooth, time-invariant control law (16) takes the form¶

u = uES(x) + uEPD(x) (20)

with

uES(x) =

[−x2x3x1x3

]uEPD(x) =

[0

−γHs` (x1, x2)x2

].

P3. The function Q(x), defined in (14), is given as

Q(x) = −x22[γ(x21 + x

22 − β`)2 + x23

]. (21)

P4. The control law (20) ensures (7) when β` > 0 or (8) when β` = 0 and γ = 1 with the set ofinadmissible initial conditions in C4 defined via the function

h(x) = (x21 + x22)x

23. (22)

ProofThe proof of claims P1-P3 follows via direct calculations, noting that the closed-loop system takesthe pH form

ẋ =

0 −x3 0x3 −γHs` (x`) 00 0 −x22

[∇H`∇H0

]. (23)

To apply Proposition 1 we need to prove the detectability-like condition (15) with the functionh(x) given in (22). First, we note that {x ∈ R3|x1 = x2 = 0 or x2 = x3 = 0} ⊂ I is an equilibriumset for the closed-loop system (23) that does not match the control objectives—therefore it has to beruled out. We will now prove that I is the set of inadmissible initial conditions.

¶We have splitted the control law into uES(x) and uEPD(x) to underscore the role of energy-shaping and EPD terms,respectively.


8 B. YI et al.

Starting outside I, La Salle’s Invariance Principle ensures that all trajectories will converge tothe largest invariant set contained in the set {x ∈ Rn | Q(x) = 0}. Thus, we have the systems stateconverges to the set {x ∈ R3|H`(x`) = β`, x3 = 0} or the largest invariant set in {x ∈ R3|x2 = 0}.Since the former is exactly the desired task, it is clear that we only need to prove the implication(15) for the case limt→∞ x2(t) = 0. Towards this end, we first note that

Ḣs` = Ḣ`

= (∇H`)>[0 −x3x3 −γHs`

]∇H`

= −γx22Hs` .

(24)

Similarly, we have from (23) thatẋ3 = −x22x3.

From these two equations we conclude that

limt→∞

x3(t) = 0 ⇔ limt→∞

Hs` (x`(t)) = 0, ⇔ x2(t) /∈ L2. (25)

Now, solving (24) we get

H`(x`(t)) = e−γ

∫ t0x22(s)dsH`(x`(0)) + β`[1− e−γ

∫ t0x22(s)ds].

In view of the constraint on the initial conditions, i.e., H`(x`(0)) 6= 0, and the fact that β` ≥ 0, wehave that

H`(x`(t)) =1

2[x21(t) + x

22(t)] > 0, ∀t ∈ [0,∞). (26)

Moreover, if β` > 0 we also have that

limt→∞

H`(x`(t)) > 0. (27)

The equivalences (25) and the inequalities (26), (27) will be instrumental to complete the proof.To apply LaSalle Invariance Principle, let us calculate the largest invariant set in {x ∈ R3|x2 =

0}. The second equation in (23) is given by

ẋ2 = x1x3 −γ

2x2(x

21 + x

22 − β`)

from which we conclude that

x2 ≡ 0 ⇒ x1x3 ≡ 0.

It implies three cases: 1) x1 ≡ 0, 2) x3 ≡ 0, or 3) non-zero signals x1(t) and x3(t) are orthogonal,i.e., x1(t)x3(t) = 0. From the dynamics of x3, we get the monotonicity of x3 with respect to time,thus excluding the third case. It yields

limt→∞

x1(t) = 0 or limt→∞

x3(t) = 0.

We proceed now to prove that the latter implies (7). If the systems state converges to theinvariant set {x2 = x3 = 0}, we conclude that the trajectories of the closed-loop system verifylimt→∞ x3(t) = 0, but from (25) we have that this is possible if and only if limt→∞Hs` (x`(t)) = 0.Therefore, we only need to consider the case of convergence to the invariance set {x1 = x2 = 0},i.e., limt→∞(x1(t), x2(t)) = 0.

If β` > 0 the inequalities (26) and (27) rule-out the possibility of limt→∞ x1(t) =limt→∞ x2(t) = 0, completing the proof for this case.

Let us consider now the case β` = 0. In this case, we have that the function V (x)—defined in(17)—takes the form

V (x) =1

2H2` (x`) +H0(x0),



and its derivative is given byV̇ = −2x22V.

Consequently, recalling (25), we have that

limt→∞

V (x(t)) = 0 ⇔ limt→∞

H`(x`(t)) = 0. (28)

The proof is concluded noting that if limt→∞(x1(t), x2(t)) = 0, the trajectories of the closed-loopsystem verify limt→∞H`(x`(t)) = 0. 222

Remark 6The controller of Proposition 2 (with β` = 0 and γ = 1) solves the problem of almost globalregulation to zero of the nonholonomic integrator with a smooth, time-invariant state-feedback.To the best of our knowledge, such a problem was still open in literature.

Remark 7Although not necessary for the analysis of the asymptotic behavior in Proposition 2, we haveadded in the control a tunable parameter γ > 0 that, as shown in the simulations, enhances theperformance. For the case of regulation of the state to zero, this parameter is taken equal to one.However, it is possible to add this tuning gain in an alternative controller, which incorporates adynamic extension that makes the constant β` a function of time β`(t) that asymptotically convergesto zero.

Remark 8Due to its smoothness and time-invariance, it is reasonable to expect that the transient performanceof the proposed design is better than the one resulting from the application of time-varying[26, 15], discontinuous [1, 12] or switching [18] feedback laws. Specifically, for the formerthere is poor transient performance with the presence of oscillations, which is clearly caused byinjecting sinusoidal signals. For the discontinuous feedback, see [1] for instance, the sensitivity tomeasurement noise is due to the appearance of some state in the denominator. This fact is illustratedvia simulations in Section 5.

Remark 9Notice that the system (19) is diffeomorphic to the system considered in [10], that is,

ż1 = u1

ż2 = u2

ż3 = z1u2 − z2u1,

via the change of coordinates z 7→ (x1, x2, x1x2 − 2x3), and they are both particular cases of thedynamical model of the current-fed induction motor [10, 19].

4. NONHOLONOMIC SYSTEMS IN CHAINED FORM

Now we extend the results to the high dimensional nonholonomic systems with chained structure.That is, the n-dimensional system (1) with

S(x) =

1 00 1x2 0x3 0...

...xn−1 0

. (29)

It is well-known [23, 24] that arbitrary nonholonomic systems of order n ≤ 4 can always betransformed into the previous chained form. Hence, the class considered in this section covers alarge number of practical applications.


10 B. YI et al.

We have the following proposition whose proof is very similar to the proof of Proposition 2.Unfortunately, due to the complicated nature of the zero dynamics for the output Q(x), we can onlyprove a local convergence result for this general case.

Proposition 3Consider the nonholonomic system (1), (29) with the state partition x` = col(x1, x2, x4, . . . , xn)and x0 = x3. Fix β` ≥ 0.

S1. The functionsH0(x0) =

1

2x23, H`(x`) =

1

2|x`|2,

together with J0 = 0, R0(x) = x22 and the matrices

J`(x) =

0 −x3 0 . . . 0x3 0 x

23 . . . x3xn−1

0 0 0 . . . 00 −x23 0 . . . 0...

......

......

0 −x3xn−1 0 . . . 0

R`(x) =

0 0 0 . . . 00 γHs` (x`) 0 . . . 00 0 0 . . . 00 0 0 . . . 0...

......

......

0 0 0 . . . 0

where Hs` (x`) is defined in (10) and γ > 0, verify conditions C1-C3 of Proposition 1.

S2. The smooth, time-invariant control law (16) takes the form (20) with

uES(x) =

[−x2x3

x1x3 + x3

(x3x4 + . . .+ xn−1xn

)]

uEPD(x) =

[0

−γHs` (x`)x2

].

S3. The function Q(x), defined in (14), is given as

Q(x) = −x22(γ(Hs` (x`))

2 + x23). (30)

S4. There exists δmin > 0 such that for all δ ≤ δmin the control law (20) ensures (7) when β` > 0or (8) when β` = 0 and γ = 1, or convergence to the following invariant set

{x ∈ Rn | x2 = 0, x1 + x3x4 + . . .+ xn−1xn = 0, x3 6= 0, Hs` (x`) 6= 0},

provided the initial state starts in the set {x ∈ Rn |(Hs` (x`))2 + x23 ≤ δ}.

ProofThe proof of claims S1-S3 follows via direct calculations, noting that the closed-loop system takesthe pH form

ẋ =

0 −x3 0 0 . . . 0x3 −γHs` (x`) 0 x23 . . . x3xn−10 0 −x22 0 . . . 00 −x23 0 . . . . . . 0...

...... . . .

0 −x3xn−1 0 0 . . . 0

∇H(x). (31)



Similarly to the case of the nonholonomic integrator, we also have the key relationships

Ḣs` = −γx22Hs` , ẋ3 = −x22x3.

Hence, the equivalence (25) holds true. Also, in view of (30) we only need to study the case x2 ≡ 0,when we have from the closed-loop dynamics (31) that

x3(x1 + x3x4 + . . .+ xn−1xn) = 0.

Hence, x3 = 0 or x1 + x3x4 + . . .+ xn−1xn = 0. In the first case, we clearly have limt→∞ x3 =0, and using (25), we conclude that limt→∞Hs` (x`(t))) = 0, achieving the control objective.Therefore, we conclude the state will converge into the following set

{x ∈ Rn | Hs` (x`) = 0, x3 = 0} ∪ {x ∈ Rn | x2 = 0, x1 + x3x4 + . . .+ xn−1xn = 0}. (32)

completing the proof. 222

Remark 10As shown in the proposition above, the matching PDEs are always solvable satisfying all theassumptions of the EPD IDA-PBC design. Unfortunately, for n ≥ 4 the invariant set to which alltrajectories converge given in (32) contains, besides the target set, an additional set that complicatesthe convergence analysis. Thus, we can only guarantee local convergence. Moreover, simulationevidence proves that—starting far away from the desired equilibrium—the closed-loop systemtrajectories will not converge to their desired values, confirming the local nature of our result.

5. SIMULATIONS

The performance of the proposed controller is illustrated via simulations with Matlab/Simulink,which are summarized as follows.

E1 In Fig. 4 we give the simulation results of the energy regulation controller of Proposition 2,i.e., with β` > 0, the initial conditions x(0) = (3, 2, 2)> and γ = 5.

E2 In Fig. 5 we repeat the simulation above, but for state regulation, that is, β` = 0. We also givethe simulation results of the well-known Pomet’s method [26] with

u1 = −(x2 + x3 cos(t))x2 cos(t)− (x2x3 + x1)u2 = x3 sin(t)− (x2 + x1 cos(t)).

As expected, due to the periodic signal injection in the feedback law—which was designedfollowing the procedure proposed in [26]—large oscillations are observed in the lengthytransient stage. Clearly, the new design outperforms Pomet’s method with a significantlyimproved transient performance.

E3 To evaluate the robustness of the EPD IDA-PBC method, we repeat the experiment aboveadding (unavoidable) high-frequency noise in the measurable state.‖ Fig. 6 illustrates thatthe state now converge to a small neighborhood of the desired equilibrium point. Here, wecompare the new design with the famous (exponentially convergent) discontinuous design ofAstolfi [1]

u1 = −kx1

u2 = p2x2 + p3x3x1,

with k = 1, p2 = −5 and p2 = 9. As shown in the figure the state trajectories grow unboundedin finite time. Thus ad-hoc modifications are needed to deal with this problem in practice.

‖The measurement noise is generated by the block “Uniform Random Number”, where the signals are limited to[−0.1, 0.1] and the sample times are selected as 0.01.


12 B. YI et al.

E4 Simulations of the energy regulation controller for the case n = 4 were also carried-out. InFigs. 7a-7b, we fix γ = 0.5. The trajectory in Fig. 7a achieves the desired objective. However,Fig. 7b shows that it fails for a larger x4(0). If we fix x1(0) = 0.5, x2(0) = 1 and x3(0) = 0.1,after extensive simulations we can find the critical initial value for x4(0) to be between 0.9and 1—for x4(0) > 1 the state will converge to the undesired set and the desired objective isnot achieved. Increasing the parameter γ, as shown in Figs. 7c-7d, the controller will achievethe desired target again. Roughly speaking, in this simulation case a larger γ enlarges the“domain of attraction”—however, this pattern was not observed in other simulation scenarios.

(a) Trajectories in the state space

(b) x3(t) and partial energy H`(x`(t))

(c) Trajectories of x` and function H`(x`)

(d) Control inputs

Figure 4. Energy regulation (with β` = 0.5, 2) of the nonholonomic integrator

6. CONCLUDING REMARKS

We propose in this paper a variation of the well-known IDA-PBC design methodology, called EPDIDA-PBC, that is suitable for the problem of regulation of nonholonomic systems. Two asymptoticregulation objectives are considered: drive to zero the state or drive the systems total energy to adesired constant value. In both cases, the objectives are achieved excluding a set of inadmissibleinitial conditions. The main feature of this approach is that, in contrast with the existing methodsreported in the literature, it yields smooth, time-invariant state-feedbacks that, in principle, have abetter transient performance. This fact is illustrated via simulations.

We should also point out that in the state regulation case the zero equilibrium point is renderedstable in the sense of Lyapunov, but not asymptotically stable. We also prove in Lemma 1 that,under the conditions of Proposition 1, the set of inadmissible initial conditions contains the origin.On the other hand, as indicated in Remark 3, for the case of energy regulation, convergence to apoint ensuring the objective is achieved rendering the zero equilibrium unstable.

Current research is under way to sharpen our result for high-dimensional systems in chainedstructure. In particular, we are investigating alternative solutions to the matching equation (12) viathe proposition of different interconnection and damping matrices.




(b) State trajectories x(t) of Pomet’scontroller

(c) State trajectories x(t) of EPD IDA-PBC

(d) Control inputs

Figure 5. State regulation of nonholonomic integrator with EPD IDA-PBC and Pomet’s controller


(b) State trajectories x(t) of EPD IDA-PBC

(c) State trajectories x(t) of Astolfi’scontroller

(d) Control inputs

Figure 6. Robustness evaluation of EPD IDA-PBC and Astolfi’s controller for the nonholonomic integrator


14 B. YI et al.

(a) γ = 0.5, β` = 0.5, x(0) =(0.5, 1, 0.1, 0.5)>

(b) γ = 0.5, β` = 0.5, x(0) =(0.5, 1, 0.1, 2)>

(c) γ = 5, β` = 0.5, x(0) =(0.5, 1, 0.1, 1)>

(d) γ = 50, β` = 0.5, x(0) =(0.5, 1, 0.1, 1)>

Figure 7. EPD IDA-PBC of the nonholonomic system with chained structure and n = 4

ACKNOWLEDGEMENTS

The authors would like to express their gratitude to Prof Andrew Teel, for his careful reading of ourmanuscript, and many thoughtful comments that helped improve its clarity. The first author wouldlike to thank Chi Jin (University of Waterloo) for fruitful discussions.

A. PROOF OF LEMMA 1

The proof is established by contradiction, that is, showing that if zero is not in the closure ofI, then we contradict Brockett’s necessary condition for asymptotic stabilization of the origin ofnonholonomic systems [6].

In Proposition 1 we prove that, if conditions C1-C4 hold, for all x(0) /∈ I, we have that

limt→∞

x(t) = 0,

regardless of whether I contains the origin or not.Denote Bδ := {x ∈ Rn| |x| ≤ δ}. If cl(I) does not contain the origin, we can always find a (small)

constant δm > 0, such thatBδm ∩ I = ∅.

From the facts that ∇H(0) = 0 and ∇2H(x) > 0, as well as Ḣ ≤ 0, we have that the sublevelsets

E� := {x ∈ Rn| H(x) ≤ �}are invariant for any � ≥ 0. Given δm > 0, we can always find small �m > 0 such that

E�m ⊂ Bδm ,



consequently E�m ∩ I = ∅. Thus, we have proven the following implication

x(0) ∈ E�m ⇒

{x(t) ∈ E�m , ∀t ≥ 0limt→∞

x(t) = 0.

The invariance of the set E�m proves Lyapunov stability of the zero equilbrium, that together withthe second attractivity condition, implies asymptotic stability of the zero equilibrium of the closed-loop system. Since the controller of Proposition 1 is smooth and time invariant, this contradictsBrockett’s necessary condition, completing the proof. 222

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1 Introduction2 Regulation of Nonholonomic Systems via EPD IDA-PBC3 Nonholonomic Integrator4 Nonholonomic Systems in Chained Form5 Simulations6 Concluding remarksA Proof of Lemma ??

arxiv:1812.11538v2 [cs.sy] 9 may 2020 · that distinguishes the control of nonholonomic systems...

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