A Comprehensive Analysis of Chattering in Second Order Sliding Mode Control Systems

The purpose of the present chapter is to present a systematic approach to thechattering analysis in control systems with second-order sliding mode controllers 2-SMC caused by the presenc

Second Order Sliding Mode Control Systems

Igor Boiko1, Leonid Fridman2, Alessandro Pisano3, and Elio Usai3

1 University of Calgary, 2500 University Dr N.W., Calgary, Alberta, Canadai.boiko@ieee.org

2 Department of Control, Engineering Faculty, National Autonomous University ofMexico (UNAM)

lfridman@servidor.unam.mx

3

Department of Electrical and Electronic Engineering (DIEE), University of

Cagliari, Piazza D’Armi, 09123 Cagliari (Italy)

{pisano,eusai}@diee.unica.it

Chattering is the most problematic issue in sliding mode control applications[30], [31], [14], [33], [36] Among the well known approaches based on smoothapproximations of the discontinuities [11],[29] and asymptotic state observers[10, 33], the use of the high order sliding mode control approach can attenuatethe chattering phenomenon significantly [15],[21], [2],[4], [5],[3],[22],[27]

There are different approaches to chattering analysis which take into accountdifferent causes of it: the presence of fast actuators and sensors [32], [17], [18],[19], [34], [6], time delays and/or hysteresis [35], [32], [33], quantization effects(see for example [23])

The purpose of the present chapter is to present a systematic approach to thechattering analysis in control systems with second-order sliding mode controllers

(2-SMC) caused by the presence of fast actuators We shall follow both the

time-domain approach, based on the state-space representation, and the domain approach

frequency-The estimation of the oscillation magnitude in conventional (i.e., first-order)SMC systems with fast actuators and sensors was developed in [32], [18], [19]via the combined use of the singularly-perturbed relay control systems theoryand Lyapunov techniques However, Lyapunov theory is not readily applicable toanalyze 2-SMC systems, for which new decomposition techniques are demanded.The Poincar´e maps were successfully used in [24], [13] to study the periodicoscillations in relay control systems In [17] a decomposition of Poincar´e mapswas proposed to analyze chattering in systems with first order sliding modes,which led to Pontryagin-Rodygin [25] like averaging theorems providing sufficientconditions for the existence and stability of fast periodic motions

The describing function (DF) technique [1] offers finding approximate values

of the frequency and the amplitude of periodic motions in systems with linear

G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 23–49, 2008 springerlink.com  Springer-Verlag Berlin Heidelberg 2008c

plants driven by the sliding mode controllers [37],[26] The Tsypkin locus [35]

provides an exact solution of the periodic problem, including finding exact values

of the amplitude and the frequency of the steady-state oscillation The mentioned frequency-domain methods were developed to analyze relay feedbacksystems and cannot be used directly for the analysis of 2-SMC systems In [7],[8]the DF method was adapted to analysis of the Twisting and the Super-twisting2-SMC algorithms [21] In [9], a DF based method of parameter adjustment ofthe generalized sub-optimal 2-SMC algorithm [3],[5] was proposed to ensure thedesired frequency and amplitude of the periodic chattering trajectories

above-In the present Chapter, a systematic approach to analysis of chattering in2-SMC systems is developed The presence of parasitic dynamics is considered

to be the main cause of chattering, and the corresponding effects are analyzed

by means of a few techniques The treatment is developed by considering the

”Generalized Sub-optimal” (G-SO) algorithm [3],[5] All the main results can beeasily generalized to the Twisting algorithm [21] with minor modifications in theproofs

For a class of nonlinear uncertain systems with nonlinear fast actuators:

1 It is proved that the approximability domain [38] of the 2-SMC G-SO

algorithms depends on the actuator time constant μ as O(μ2) and O(μ) for the

sliding 2-SMC variable and its derivative respectively

2 Sufficient conditions of the existence of asymptotically orbitally stable riodic solution are obtained in terms of Poincar´e maps

pe-For linear, possibly linearized, dynamics driven by 2-SMC G-SO algorithms,frequency-domain methods of analysis of the periodic solutions are developed,and, in particular:

3 The describing function method is adapted to perform an approximateanalysis of the periodic motions

4 The Tsypkin’s method is modified for the analysis of the systems driven

by 2-SMC G-SO algorithms This modification allows for finding exact values ofthe parameters of periodic motions, without requiring for the actuator dynamics

to be fast

The chapter is organized as follows: in Section 2, a class of nonlinear systemswith nonlinear fast actuators is introduced In Section 3 we show that the 2-SMCG-SO algorithm with suitably chosen parameters steers the system trajectories

in finite time towards an invariant vicinity of the second-order sliding set Wealso estimate the amplitude of chattering oscillations as a function of the actua-tor time constant In Section 4 sufficient conditions of the existence and stability

of fast periodic motions in a vicinity of the second-order sliding set are derivedvia the Poincar´e map approach In Section 5, frequency-domain approaches tochattering analysis are developed The describing function method is adapted insubsection 5.1 to carry out analysis of periodic motions in systems with linearplants In Subsection 5.2, the Tsypkin’s method is modified to obtain the pa-rameters of the periodic motion exactly Examples illustrating the application

of the proposed methodologies are spread over the chapter The proofs of theTheorems are given in the Appendix

2 2-SMC Systems with Dynamic Actuators

We shall consider a nonlinear single-input system:

˙

with the state vector x = [x1, x2, , x n] ∈ X ⊂ R n and the scalar “virtual”

control input z1 ∈ Z1 ⊂ R The plant input z1 is modifiable via the dynamicfast actuator

which defines the sliding manifold s1(x) = 0 assigning desired dynamic properties

(e.g stability) to the constrained sliding-mode dynamics Define

s2(x, z1) =∂s1(x)

∂x a(x, z1) : X × Z1→ S2 (4)and assume that the following conditions hold∀(x, z1)∈ X × Z1:

By virtue of the Inverse Function Theorem, one can explicitly define a vector

w ∈ W ⊂ R n −2 and a diffeomorfic state coordinate change x = Φ(s, w) : S ×

W → X , with s = [s1, s2]∈ S, which is one to one at any point where condition

˙s1= s2, ˙s2= f (w, s, z1), (10)

where g : W × S → R n −2 , f : W × S × Z1→ R, h : Z × U → R mare smooth

functions of their arguments such that f ∈ C2[ ¯W × ¯ S × ¯ Z1], g ∈ C2[ ¯W × ¯ S] and

h ∈ C2[ ¯Z × ¯ U ], where upper bar means the closure of domain This means that the “sliding variable” s1has a well–defined relative degree r = 2 with respect to the plant input variable z1 over the whole domain of analysis

We consider here the case when the actuator output z1 has the full relative

degree m, equal to the order of the actuator dynamics, with respect to the discontinuous control u.

Remark 3 The special form (9) for the internal dynamics can be always

achieved if the original dynamics (1) has affine dependence on z1 [20] We areconsidering in this paper the subclass of non-affine systems (1) for which such a

special choice of vector w can be found.

Conditions

Consider system (9)-(11) driven by the “Generalized Suboptimal” 2-SMC rithm [5]

where U and β are the constant controller parameters and s 1M i is the latest

“singular point” of s1, i.e., the value of s1 at the most recent time instant t M i

(i = 1, 2, ) such that ˙s1(t M i) = 0 Our analysis is semi-global in the sense

that the initial conditions w(0), s(0), z(0) are assumed to belong to the known, arbitrarily large, compact domains W0, S0, and Z0, respectively The solutions

of the system (9)-(12) are understood in the Filippov sense [16]

Remark 4 Since the relative degree between the sliding output s1 and the

discontinuous control u is m + 2, only sliding modes of order m + 2, occurring

onto the following sliding set1 [21], can take place

1 The successive total time derivatives of s1 must be evaluated along the trajectories

of system (9)-(11) in the usual way

The internal dynamics in the (m + 2)-th order sliding-mode is described by

with respect to u is m + 2.

Then, the system equilibrium point can be computed as the unique solution

w0, z0, u0(w0, z0) of the system of equations (15)-(20) The knowledge of theequilibrium point will be used later in the Chapter to define a local linearizationfor the system (9)-(11)

Assumption 1 The internal dynamics (9) and the actuator dynamics (11)

meet the following input-to-state stability properties for some positive constants

Assumption 3 Consider the actuator dynamics (11) with the constant input

u(t) = U , t ≥ t0 Then, ∀ε ∈ (0, 1) there is γ ∈ [γ m , γ M ] and N (ε) > 0 such that

|z1− γU| ≤ εγU ∀t ≥ t0+ N (ε)μ (25)Assumption 1 prescribes a linear growth ofw(t) and z(t) w.r.t.s(t) and

|u|, respectively Assumption 2 guarantees that the virtual plant control input

z1, with large enough magnitude, can set the sign of f (see Fig 1) The edge of constants ξ1,ξ2, H0, , G M is mainly a technical requirement With

knowl-sufficiently large U , and β ∈ [0.5, 1) sufficiently close to 1, stability can be sured regardless of ξ1, , G M Assumption 3 requires a “non-integrating” stable2

in-In this case the (m+2) th order sliding dynamics do not depend on the control, i.e.they do not depend on the definition of solutions in (m+2)-th order sliding mode

actuator dynamics whose settling time in the step response is O(μ) Note that γ and N are considered uncertain Assumption 3 is always satisfied in the special

case of a linear asymptotically stable actuator dynamics

Assume, only temporarily, that there exists a certain constant H such that

| ˜ H(s, w) | ≤ H, then conditions (23)-(24) reduce to the following:

z1≤ 0 ⇒ −H + G M z1≤ f(w, s1, s2, z1)≤ H + G m z1

z1> 0 ⇒ −H + G m z1≤ f(w, s1, s2, z1)≤ H + G M z1

(26)

which can be represented graphically as in Fig 1 The dashed lines limit the

“admissible region” for the uncertain function f

Fig 1 Bounding curves for the function f

Consider the following tuning rules:

Theorem 1 Consider system (9)-(11), satisfying Assumptions 1-3 and driven

by the Generalized Sub-Optimal controller (12), (27)-(28) Then, if H is ciently large and μ is sufficiently small, the closed loop system trajectories enter

suffi-in finite time the suffi-invariant domasuffi-in O μ defined by (29), where ρ1 and ρ2 are proper constants independent of μ.

Proof See the appendix.

Simulation example Consider system

The initial conditions and the controller parameters are s1(0) = 20, s2(0) = 5,

w(0) = 5, z1(0) = z2(0) = 0, U = 80, β = 0.8 Figure 2 shows the time evolution

of s1 and s2 when μ = 0.001 The amplitude of chattering was evaluated as

the maximum of |s1| and |s2| in the steady state, yielding |s1| ≤ 7E-4 and

|s2| ≤ 0.4 We performed a second test using μ = 0.01 The accuracy of s1

changed to 7E − 2, and the accuracy of s2changed to 4, in perfect accordance

with (29) Simulations show high-frequency periodic motions for s1, s2 and w.

Below those “chattering” trajectories are investigated in further detail

−30

−20

−10 0 10 20 30

Time [sec]

s 1 (t)

s

2 (t)

Fig 2 The steady-state evolution of s1 and s2

4 Poincar´ e Map Analysis

We are going to derive conditions for the existence of stable periodic motions,

in the vicinity O μ of the second order sliding set, in terms of the properties ofsome associated Poincar´e maps We will also give a constructive procedure tocompute the parameters of such periodic chattering motions Introducing the

new variables y1= μ −2 s1, y2= μ −1 s2, rewrite system (9)-(11) in the form

˙

μ ˙ y1= y2, μ ˙ y2= f (w, μ2y1, μy2, z1), (32)

Note that the Generalized Sub-Optimal algorithm (12) is endowed by the

homogeneity property u(μ2y1) = u(y1) Consider the Original System in the Fast Time (OSFT)

dw/dτ = μg(w, μ2y1, μy2), (34)

dy1/dτ = y2, dy2/dτ = f (w, μ2y1, μy2, z1), (35)

and the Fast Subsystem (FS) with the “frozen” slow dynamics (w ∈ W is

con-sidered here as a fixed parameter):

f (w, ¯ y+1(T (w), w), 0, ¯ z+1(T (w), w)) < 0 Now we can define for all w ∈ W the

of the of the domain f (w, y1, 0, z1) < 0 on the surface y2= 0 into itself, generated

by system FS (37)-(38) (the details of this mapping are described in Appendix B)

Let us suppose that the FS (37)-(38) has a nondegenerated isolated periodicsolution and the following conditions hold:

Condition 1.∀w ∈ W the FS (37)-(38) has an isolated T0(w)-periodic solution

(¯y10(τ , w), ¯ y20(τ , w), ¯ z0(τ , w)). (41)

Condition 2.∀w ∈ W the Poincar´e map Ξ(w, y0

1, z0) has an isolated fixed point(¯y ∗

1(w), z ∗ (w)) corresponding to the periodic solution (41).

Condition 3.∀w ∈ W the eigenvalues λ i (w) (i = 1, , m + 1) of the matrix

∂Ξ

∂(y1, z) (w, ¯ y

∗

are such that|λ i (w) | = 1.

Condition 4 The averaged systems

The following Theorem is demonstrated:

Theorem 2 Under conditions 1 − 4, system (31)-(33) has an isolated periodic solution near the cycle

(w0, ¯ y10(t/μ, w0), ¯ y20(t/μ, w0), ¯ z0(t/μ, w0)) (45)

with period μ(T0(w0) + O(μ)).

Proof See the Appendix.

Let ν j (w0) (j = 1, , n − 2) be the eigenvalues of the matrix dp

dw (w0) Supposethat the periodic solution of FS (37)-(38) is exponentially orbitally stable andthe equilibrium point of the averaged equations is exponentially stable, i.e :

Condition 5. |λ i (w) | < 1, (i = 1, , m + 1).

Condition 6 ν j (w0) are real negative, i.e., ν j (w0) < 0, ∀j = 1, , n − 2.

Theorem 3 Under conditions 1 − 6, the periodic solution (45) of the system (31)-(33) is orbitally asymptotically stable.

Proof See the Appendix.

Consider the following linear dynamics

We have shown that analysis of periodic solutions can be performed by referring

to the decomposition into fast and slow subsystem dynamics The FS dynamics

dy1/dτ = y2, dy2/dτ = z, dz/dτ = −z + u (47)generates the following Poincar´e map Ξ+(y1, z) = (Ξ1+(y1, z), Ξ2+(y1, z)) of the domain z < 0 on the surface y2= 0 into the domain z > 0 of the same surface

(see the Appendix for the detailed derivation):

p (y1, z) are the smallest positive roots of

the following equations:

(0, 0, 0) we can skip the computation of the map Ξ − (y1, z) and rewrite condition

for the periodicity of system (47) trajectory in the form:

Ξ1+(y01p , z0) =−y0

1p , Ξ2+(y 1p0 , z0) =−z0 (51)The fixed points are

y 1p0 ≈ 3.95, z0

and the switching times are

T+sw ≈ 2.01, T+p ≈ 3.93, T+0 = T+sw + T+p ≈ 5.94. (53)The Frechet derivatives entering the Jacobian matrix are given by

(54)

The eigenvalues of matrix J are eig(J) = [ −0.5182, 0.0141], i.e they are both

lying within the unit circle of the complex plane, which implies that the periodicsolution of the fast subsystem (47) is orbitally asymptotically stable

The averaged equations for the internal dynamics has the form ˙w = −w Now

from Theorems 1-3 it follows that: i) system (46) has an orbitally asymptotically

stable periodic solution lying in the O μ boundary layer (29) of the second-order

sliding set y1= y2= 0 ii) in the steady state the internal dynamics w variable features a O(μ) deviation from the equilibrium point w = 0 of the averaged

solution It is also expected from Theorem 2 that the period of oscillation is

O(μ) The period and amplitude of the periodic solutions of (46) can be easily inferred from (52) and (53) via proper μ-dependent scaling.

The above results have been checked by means of computer simulations The

initial conditions are w(0) = y1(0) = y2(0) = z(0) = 1 The value μ = 0.1 was

used in the first test It is expected, on the basis of the previous considerations,

that y1 exhibits a steady oscillation with amplitude μ2y0

1p ≈ 0.0395 and period 2μT+

p ≈ 1.18s The plots in Fig 3 highlight the convergence to the periodic solution starting from initial conditions outside from the attracting O μ domain

The Poincar´e map based analysis provides an exact but complicated approach.Therefore, the use and adaptation of frequency methods for the chattering analy-sis in control systems with the fast actuators driven by 2-SMC G-SO algorithmsseems expedient However, this approach applies to linear dynamics For thisreason in this section we assume that the plant plus actuator dynamics areeither linear or linearized in the conventional sense in a small vicinity of thedomain (29).

Subsection 5.1 discusses the problem of local linearization, and Subsection 5.1states formally the analysis problem Subsection 5.3 is devoted to the describingfunction approach to the analysis of the G-SO algorithm in the closed loop Thisapproach is approximate and requires the linearized system (actuator and plant)being a low-pass filter This assumption is equivalent to the hypothesis of theactuator being fast Thus, subsection 5.4 presents the modified Tsypkin locus[35] method, which does not require the filtering hypothesis, and, furthermore,

provides exact values of the frequency and the amplitude of the periodic

mo-tion as a solumo-tion of an algebraic equamo-tion and the use of an explicit formula,respectively

We have shown that controller (12) can provide for the appearance of a stable

sliding mode of order m + 2, and that the system (9)-(12), with full state vector

ξ = [w T , s T , z T ], has a fixed equilibrium point ξ0= (w0, 0, z0) (see Remark 4)

The constant equilibrium value for the actuator input is u = u0(w0, z0), then it

is reasonable to linearize the system (9)-(12) in the small neighborhood of the

point ξ0 by considering the constant control value u0in the terms depending on

it Simple computations yield the following linearized dynamics

˙ξ = Aξ + Bu,

must be evaluated in the considered equilibrium point ξ0and equilibrium control

value u0

Because of the system trajectories will converge to an O(μ) vicinity of the equilibrium, the accuracy of the linear approximation depends on the μ param- eter, the smaller μ the higher the accuracy The transfer function and harmonic

response of the linearized system (55)-(56), which is asymptotically stable byconstruction, can be computed straightforwardly

s1

β

p M

s1

β

Fig 4 Left: Closed-loop system with the G-SO algorithm Right: the control

charac-teristic in steady state

The term s 1M i appearing in the switching function of control (12) changes

step-wise at the time instants t M i (i=1,2, ) at which ˙s1(t M i) = 0 During the

periodic motion s 1M i is an alternating (ringing) series of positive and negative

values, i.e s p 1M,−s p

1M , s p 1M,−s p

1M (here the label “p” stands for periodic) The

control sign change would occur at the time when the plant output is equal

to ±βs p

1M Therefore, in the periodic motion, the control function (12) can berepresented by the hysteretic relay nonlinearity in Fig 4-right This representa-tion opens the way for the use of the frequency-domain methods developed foranalysis of relay feedback systems [1, 35] The main difference from the conven-tional application of the existing methods is that the hysteresis value is unknowna-priori

5.3 Describing-Function (DF) Analysis

DF analysis is a simple approach which can provide in most cases a sufficientlyaccurate estimate of the frequency and the amplitude of a possible periodicmotion The main difference between the considered case and a conventional

relay system is that the hysteresis value βs p 1M is actually unknown To solve this

problem we can consider that during a periodic motion the extreme values of the

output coincide, in magnitude, with its amplitude Therefore, s p 1M is actually theunknown amplitude of the periodic motion The DF of the relay with a negativehysteresis is given as follows [1]:

where b = βy M p is a half of the hysteresis, c = U is the relay amplitude and

A y = y M p is the amplitude of the harmonic input to the relay Then we can exploitthe given relationships between the hysteresis parameters and the oscillationparameters in order to obtain the following expression for the DF of the G-SOalgorithm:

As usual, the periodic solutions correspond to the points of intersection of the

W (jω) and −1/q(A y) loci, the latter being a straight line backing out of the

origin with a slope that depends only on parameter β, as depicted in Fig 5 Therefore, a periodic motion may occur if at some frequency ω = ω the phase characteristic of the actuator-plant transfer function W is equal to −1800− arcsin(β) If such a requirement is fulfilled, so that intersection between the two