Analysis of Closed-Loop Performance and Frequency-Domain Design of Compensating Filters for Sliding Mode Control Systems

The development of the non-reduced order model becomes feasible owing to the locus of a perturbed relay system LPRS method [12] that involves the concept of the so-called equivalent gain

Frequency-Domain Design of Compensating

Filters for Sliding Mode Control Systems

Igor Boiko

Honeywell and University of Calgary, 2500 University Dr NW, Calgary, Alberta, T2N 1N4, Canada

i.boiko@ieee.org

1 Introduction

It is known that the presence of parasitic dynamics in a sliding mode (SM) sys-tem causes high-frequency vibrations (oscillations) or chattering [1]-[3] There are a number of papers devoted to chattering analysis and reduction [4]-[10] Chattering has been viewed as the only manifestation of the parasitic dynamics presence in a SM system The averaged motions in the SM system have always been considered the same as the motions in the so-called reduced-order model [11] The reduced-order model is obtained from the original equations of the sys-tem under the assumption of the ideal SM in the syssys-tem Under this assumption, the averaged control in the reduced-order model becomes the equivalent control [11] This approach is well known and a few techniques are developed in details However, the practice of the SM control systems design shows that the real SM system cannot ensure ideal disturbance rejection Therefore, if the difference be-tween the real SM and the ideal SM is attributed to the presence of parasitic dynamics in the former then the parasitic dynamics must affect the averaged mo-tions Yet, the effect of the parasitic dynamics on the closed-loop performance can be discovered only if a non-reduced-order model of averaged motions is used Besides improving the accuracy, the non-reduced-order model would provide the capability of accounting for the effects of ideal disturbance rejection and non-ideal input tracking The development of the non-reduced order model becomes feasible owing to the locus of a perturbed relay system (LPRS) method [12] that

involves the concept of the so-called equivalent gain of the relay, which describes

the propagation of the averaged motions through the system with self-excited oscillations Further, the problems of designing a predetermined frequency of chattering and of the closed-loop performance enhancement may be posed The solution of those two problems may have a significant practical impact, as it is chattering that prevents the SM principle from a wider practical use, and it is performance deterioration not accounted for during the design that creates the situation of “higher expectations” from the SM principle

In the present chapter, analysis of chattering and of the closed-loop perfor-mance is carried out in the frequency domain via the LPRS method Further,

G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 51–70, 2008 springerlink.com  Springer-Verlag Berlin Heidelberg 2008c

Fig 1 Relay feedback system

it is proposed that the effects of the non-ideal closed-loop performance can be mitigated via the introduction of a linear compensator The methodology of the compensating filter design is given

The chapter is organized as follows At first, the approach to the analysis of averaged motions in a SM control system is outlined, and the concept of the

equivalent gain of the relay with respect to the averaged signals is introduced.

After that the basics of the LPRS method are given In the following section, the non-reduced-order model of the averaged motions is presented Then, a mo-tivating example illustrating the problem is considered In the following section, the compensation mechanism is analyzed Finally, an illustrative example of the compensating filter design is given

2 Averaged Motions in a Sliding Mode System

It is known that the SM control is essentially a relay control with respect to the sliding variable Therefore, the SM system can be analyzed as a relay system If

no parasitic dynamics and switching imperfections were present in the system the ideal SM would occur It would feature infinite frequency of switching of the relay and infinitely small amplitude of the oscillations at the system output However, the inevitable presence of parasitic dynamics (in the form of the actu-ator and sensor dynamics) in series with the plant and switching imperfections

of the relay result in the finite frequency of switching and finite amplitude of the oscillations, which is usually referred to as chattering Chattering was the subject of analysis of a number of publications [1]-[4], [11] and was analyzed as a periodic motion caused by the presence of fast actuator dynamics To obtain the model of averaged motions in a SM control system that has parasitic dynamics and switching imperfections, let us analyze the relay feedback system under a constant load (disturbance) or a constant input applied (Fig.1) At first obtain the model that relates the averaged values of the variables when a constant in-put is applied to the system Later we can extend the results obtained for the constant input (and constant averaged values) to the case of slow inputs Let the SM system be described by the following equations that would

com-prise both: the principal dynamics and the parasitic dynamics, which are as-sumed to be type 0 servo system (not having integrators):

˙

x = Ax + Bu

u(t) =



c if σ(t) = f0− y(t) ≥ b or σ(t) > −b, u(t−) = c

−c if σ(t) = f0− y(t) ≤ −b or σ(t) < b, u(t−) = −c (2)

where A∈ R n ×n , B ∈ R n ×1 , C ∈ R1×n are matrices, A is nonsingular, c and b

are the amplitude and the hysteresis value of the relay nonlinearity respectively,

f0is a constant input, u ∈ R1is the control, x ∈ R n is the state vector, y ∈ R1

is the system output, σ ∈ R1 is the sliding variable (error signal), u(t −) is the

control at time instant immediately preceding time t Let us call the part of the

system described by equations (1) the linear part Alternatively the linear part

can be given by the transfer function W l (s) = C(Is − A) −1B The parasitic

dynamics can be present in both: the linear part (1) or/and in the nonlinearity (2) as a non-zero hysteresis value

Assume that: (A) In the autonomous mode (f (t) ≡ 0) the system exhibits a sym-metric periodic motion (B) In the case of a constant input f (t) ≡ f0, the switches

of the relay become unequally-spaced and the oscillations of the output y(t ) and

of the error signal σ(t) become asymmetric (Fig 2) The degree of asymmetry de-pends on f0(a pulse-width modulation effect); (C) All external signals f (t) applied

to the system are slow in comparison with the self-excited oscillations (chattering).

We shall consider as comparatively slow signals the signals that meet the following condition: the external input can be considered constant on the period of the oscil-lation without significant loss of accuracy of the osciloscil-lation estimation In spite of

this being not a rigorous definition, it outlines the framework of the subsequent analysis First limit our analysis to the case of the input being a constant value

f (t) ≡ f0 Under that assumption each signal has a periodic and a constant term:

u(t) = u0+ u p (t), y(t) = y0+ y p (t), σ(t) = σ0+ σ p (t), where subscript ”0” refers

to the constant term (the mean value on the period), and subscript ”p” refers to

the periodic term (having zero mean)

If we quasi-statically vary the input (slowly enough, so that the input value can be considered constant on the period of the oscillation – as per assumption C) from a certain negative value to a positive value and measure the values of

the constant term of the control u0(mean control) and the constant term of the

error signal σ0 (mean error) we can determine the constant term of the control

signal as a function of the constant term of the error signal: u0= u0(σ0) Two examples of this function are given in Fig 3 (for the first-order plus dead time plant; 1 and 2 are those functions for two different values of dead time) Let us

call this function the bias function It is worth noting that despite the fact that the original nonlinearity is a discontinuous function the bias function is smooth

and even close to the linear function in a relatively large range The described

effect is known as the chatter smoothing phenomenon [13] The derivative of this function (mean control) with respect to the mean error taken in the point σ0= 0

(corresponding to zero constant input) provides the so-called equivalent gain of the relay k n [13] The equivalent gain concept can be used for building the model

of the SM system for averaged values of the variables

k n = du0/dσ0| σ0=0= lim

f →0 (u0/σ0). (3)

Fig 2 Asymmetric oscillations at unequally spaced switches

Fig 3 Bias functions (negative part is symmetric)

Once the equivalent gain k n is found via analysis of the relay system having

constant input f (t) ≡ f0, we can extend the equivalent gain concept to the case

of slow inputs (as per assumption C) Therefore, the main point of the

input-output analysis is finding the equivalent gain value, and all subsequent analysis

of the forced motions can be carried out exactly like for a linear system with the

relay replaced by the equivalent gain.

The concept of the equivalent gain is used within the describing function method Yet, only approximate value of the equivalent gain can be obtained due to the

ap-proximate nature of the method The LPRS method [12] can provide an exact value

of this characteristic The basics of the LPRS method are presented below

3 The Locus of a Perturbed Relay System

Let us consider at first the solution of this problem via the describing function

(DF) analysis [14] The DF of the relay function and the equivalent gain can be

given as follows:

N (a) = 4c

πa



1−



b a

2

− j 4cb

k n(DF )= ∂u0

∂σ0





σ0=0

= 2c

πa

1



1−b a

where a is the amplitude of the symmetric oscillations The oscillations in

the relay feedback system can be found from the harmonic balance equation:

W l (jΩ) = −1/N(a), which can be transformed into the following form via the

replacement of N (a) with expression (4), accounting for (5) and using the switch-ing condition y (DF )(0) = −b, where t = 0 is the time of the relay switch from

“−c” to “+c”:

W l (jΩ) = −1

2

1

k n(DF ) + j

π

4c y (DF ) (t)





t=0

It follows from (4)-(6) that the frequency of the oscillations and the equivalent

gain in the system (1), (2) can be varied by changing the hysteresis value 2b of the relay Therefore, the following two mappings can be considered: M1: b → Ω,

M2: b → k n Assume that M1 has an inverse mapping (it follows from (4)-(6) for the DF analysis and is proved below via deriving an analytical formula of

that mapping) M −1

1 : Ω → b Applying the chain rule consider the mapping

M2



M −1

1

: Ω → b → k n Now let us define a certain function J exactly as the

expression in the right-hand side of formula (6) but require from this function

that the values of the equivalent gain and the output at zero time should be exact

values Applying mapping M2



M −1

1

: ω → b → k n , ω ∈ [0; ∞), in which we

treat frequency ω as an independent parameter, write the following definition of

this function:

J (ω) = −1

2

1

k n + j

π

where k n = M2



M −1

1 (ω)

, y(t) | t=0 = M −1

1 (ω) Thus, J (ω) comprises the two

mappings and is defined as a characteristic of the response of the linear part

to the unequally spaced pulse input u(t) subject to f0 → 0 as the frequency

ω is varied The real part of J (ω) contains information about gain k n, and the

imaginary part of J (ω) comprises the condition of the switching of the relay

and, consequently, contains information about the frequency of the oscillations

If we derive the function that satisfies the above requirements we will be able to

obtain the exact values of the frequency of the oscillations and of the equivalent

gain.

Let us call function J (ω) defined above as well as its plot on the complex plane (with the frequency ω varied) the locus of a perturbed relay system (LPRS) The word “perturbed” refers to an infinitesimally small perturbation f0applied to the system Assuming that the LPRS of a given system is available we can determine

the frequency of the oscillations and the equivalent gain k nas illustrated in Fig.4 The point of intersection of the LPRS and of the straight line, which lies at the

distance πb/(4c) below (if b>0) or above (if b < 0) the horizontal axis and is

Fig 4 The LPRS and oscillations analysis

parallel to it (line “−πb/4c“) offers computing the frequency of the oscillations

and the equivalent gain k n of the relay

According to (7), the frequency Ω of the oscillations can be computed via

solving equation:

(i.e y(0) = −b is the condition of the relay switch) and the gain k n can be computed as:

that is a result of the definition of the LPRS The LPRS, therefore, can be seen

as a characteristic similar to the frequency response W (jω) (Nyquist plot) of

the plant with the difference that the Nyquist plot is a response to the harmonic signal but the LPRS is a response to the square wave Clearly, the methodology

of analysis of the periodic motions with the use of the LPRS is similar to the one with the use of the DF In such an analysis, the LPRS serves the same purpose as the Nyquist plot, and the line “−πb/4c“ serves the same purpose as

the negative reciprocal of the DF (−N −1 (a)).

Obviously, formula (7) cannot be used for the outlined analysis It is a defini-tion In [12], a formula of the LPRS that involves only parameters of the linear part was derived as follows:

J (ω) = −0.5C[A −1+2π

ω(I− e 2π

ωA)−1 e π

ωA ]B+

+j π4C(I + e ω πA)−1(I− e π

Let us derive another formula of J (ω) for the case of the linear part given by a

transfer function, which will have the format of infinite series and be instrumental

at analyzing some effects in the SM systems Suppose, the system is a type 0

servo system (the transfer function of the linear part does not have integrators).

Write the Fourier series expansion of the signal u(t ) depicted in Fig 2:

u(t) = u0+4c π ∞

k=1

sin(πkθ1/(θ1+ θ2))/k ×

×{cos(kωθ1/2) cos(kωt) + sin(kωθ1/2) sin(kωt) }

where u0= c(θ1− θ2)/(θ1+ θ2), and ω = 2π/(θ1+ θ2)

Therefore, y(t) as a response of the linear part with the transfer function

W l (s) can be written as:

y(t) = y0+4c π ∞

k=1

sin(πkθ1/(θ1+ θ2))/k × {cos(kωθ1/2) cos[kωt + ϕ1(kω)]+ + sin(kωθ1/2) sin[kωt + ϕ1(kω)] } A l (ωk)

(11)

where ϕ l (kω) = argW l (jkω),

A l (kω) = |W l (jkω) |,

y0= u0|W l (j0) |

The conditions of the switches of the relay can be written as:



f0− y(0) = b

where y(0) and y(θ1) can be obtained from (11) if we set t = 0 and t = θ1 respectively:

y(0) = y0+4c π ∞

k=1

[0.5 sin(2πkθ1/(θ1+ θ2))Re W l (jkω) +sin2(πkθ1/(θ1+ θ2))Im W l (jkω)]/k

(13)

y(θ1) = y0+4c

π

∞ k=1

[0.5 sin(2πkθ1/(θ1+ θ2))Re W l (jkω)

−sin2(πkθ1/(θ1+ θ2))Im W l (jkω)]/k

(14)

Differentiating (12) with respect to f0(and taking into account (13) and (14))

we obtain the formulas containing the derivatives in the point θ1 = θ2 = θ =

π/ω:

c |W l(0)|

2θ

dθ1

df0 − dθ2

df0

+c θ

dθ1

df0 − dθ2

df0

∞

k=1

cos(πk)ReW l (ω k)−

− 2c

θ2

dθ1

df0 +dθ2

df0

∞

k=1

sin2(πk/2) dImW l (ω k)

c |W l(0)|

2θ

dθ1

df0 − dθ2

df0

+c θ

dθ1

df0 − dθ2

df0

∞

k=1

cos(πk)ReW l (ω k)

+θ 2c2

dθ1

df0 +dθ2

df0

∞

k=1

sin2(πk/2) dImW l (ω k)

where ω k = πk/θ Having solved the set of equations (15), (16) for d(θ1−θ2)/df0

and d(θ + θ )/df we obtain:

df0





f0=0= 0 which corresponds to the derivative of the period of the oscillations, and:

d(θ1− θ2)

df0





f0 =0

c



|W l(0)| + 2 ∞

k=1

cos(πk) ReW l (ω k)

Considering the formula of the closed-loop system transfer function we can write:

d(θ1− θ2)

df0





f0 =0

Solving together equations (17) and (18) for k n we obtain the following ex-pression:

2 ∞ k=1

(−1) k Re W l (kπ/θ)

(19)

Taking into account formula (19) and the definition of the LPRS (7) we obtain

the final form of the expression for Re J (ω) Similarly, having solved the set of equations (12) where θ1 = θ2 = θ and y(0) and y(θ1) have the form (13) and

(14) respectively, we obtain the final formula of Im J (ω) Having put the real

and the imaginary parts together, we can obtain the final formula of the LPRS

J (ω) for type 0 servo systems:

J (ω) =

∞ k=1

(−1) k+1 Re W l (kω) + j

∞ k=1

Im W l [(2k − 1)ω]

It is easy to show that series (20) converges for every strictly proper transfer

function (see Theorem 1 below) The LPRS J (ω) offers solutions of the periodic

and input-output problems for a relay feedback system – as illustrated by Fig

4, and does not require involvement of the filtering hypothesis Let us formulate that as the following theorem, the proof of which is given as the formulas above

Theorem 1 For the existence of a periodic motion of frequency Ω with

in-finitesimally small asymmetry of switching in u(t) caused by inin-finitesimally small constant input f0in the relay feedback system (1), (2), it is necessary that

equa-tion (8) should hold The ratio of infinitesimally small averaged control u0 to

infinitesimally small averaged error σ0 (the equivalent gain of the relay) will be determined by formula (9) The LPRS in (8) and (9) can be computed per (20)

4 Analysis of Chattering

Usually two main types of chattering are considered in continuous-time SM con-trol The first one is the high-frequency oscillations due to the existence of the discontinuity in the system loop along with the presence of parasitic dynamics

or switching imperfections, and the other one is due to the effect of an external noise In the latter case chattering will not be a periodic motion, as the source

of chattering is an external signal, which is not periodic However, the first type

of chattering is usually considered the main one as being an inherent feature

of SM control systems It is perceived as a motion, which oscillates about the sliding manifold [1] In the present chapter only this type of chattering is consid-ered Let us obtain the conditions for the existence of chattering considered as a self-excited periodic motion in a SM system Note that the SM system is a relay feedback system with respect to the so-called sliding variable, which is a function

of the system states As a result, the dynamics of a SM system not perturbed

by parasitic dynamics is always of relative degree one Also, the hysteresis value

of the relay of a SM system is always zero However, if parasitic dynamics are introduced into the system model the relative degree becomes higher than one.

Let us consider the configuration of the LPRS and in particular the location of its high-frequency segment in dependence on the relative degree of the

dynam-ics Let the transfer function W l (s) be given as a quotient of two polynomials of degrees m and n respectively:

W l (s) = B m (s)

A n (s) = b m s m +b m−1 s m−1 + +b

1s+b0

a n s n +a n−1 s n−1 + +a1s+a0 (21)

The relative degree of the transfer function W l (s) is (n − m) Then the

fol-lowing statements hold

Lemma 1 If W l (s) is strictly proper (n > m) then there exists ω ∗corresponding

to any given > 0 such that for every ω ≥ ω ∗ the following inequalities hold:

|ReW l (jω) − Re(b m /(a n (jω) n −m))| ≤ (ω ∗ /ω) n −m , (22)

|ImW l (jω) − Im(b m /(a n (jω) n −m))| ≤ (ω ∗ /ω) n −m (23)

This lemma simply means that at any frequency ω ≥ ω ∗:

W l (s) ≈ b m

a n s n−m

Lemma 2 (monotonicity of high-frequency segment of the LPRS).

If ReW l (jω) and ImW l (jω) are monotone functions of frequency ω and

|ReW l (jω) | and |ImW l (jω) | are decreasing functions of frequency ω for every

ω ≥ ω ∗∗ then within the range ω ≥ ω ∗∗ the real and imaginary parts of the

LPRS J (ω) corresponding to that transfer function are monotone functions of frequency ω.

Proof Since |ReW l (jω) | and |ImW l (jω) | are said to be monotone decreasing

functions of ω within the range ω ∈ [ω ∗∗;∞) their derivatives are negative.

Therefore, functions |ReW l (jkω) | and |ImW l (jkω) |, where k = 1, 2, , ∞, will

also have negative derivatives As a result, the derivatives of the following series are negative (being sums of negative addends):

d ∞ k=1

|Im W1[(2k−1)ω]|

2k −1

< 0 ,

d ∞ k=1

|ReW l (2kω) |

dω < 0 ,

d ∞ k=1 |ReW l [(2k −1)ω]|

From the first inequality and formula (20), it directly follows that the absolute

value of the imaginary part of the LPRS J (ω) is a monotone decreasing function

of frequency ω, as the sign of its derivative is negative For the proof of the monotonicity of ReJ (ω), let us note that the second derivatives of functions

|ReW l (jkω) | and |ImW l (jkω) |, where k = 1, 2, , ∞, are positive (otherwise,

considering the fact that those functions are monotone, arrival of W l (jω) at the origin at ω → ∞ would not be ensured) Now group the terms in the series (20)

for the real part by two and find the sign of the following derivative:

d[ |ReW l [(2k −1)ω]|−|ReW l [2kω] |]

dω , k = 1, 2, , ∞,

which is negative, because the first derivatives of |ReW l [(2k − 1)ω]| and

|ReW l [2kω] | are negative and the second derivatives are positive, and the terms

with higher ω have a negative derivative of smaller magnitude Therefore, the ab-solute value of the real part of the LPRS J (ω) is a monotone decreasing function

Taking account of the above lemmas address the following statement

Theorem 2 If the transfer function W l (s) is a quotient of two polynomials

B m (s) and A n (s) of degrees m and n respectively (21) then the high-frequency segment (where the above Lemma 1 holds) of the LPRS J (ω) corresponding to the transfer function W l (s) is located in the same quadrant of the complex plane where the high-frequency segment of the Nyquist plot of W l (s) is located with either the real axis (if the relative degree (n − m) is even) or the imaginary axis

(if the relative degree (n − m) is odd) being an asymptote of the LPRS.

Prove the above theorem for an arbitrary relative degree r = n − m ≥ 1 Let

us note that the following infinite series have finite sums for any positive integer

r ≥ 1:

S1(1) = ln2

S1(r) = 1 − 1

2r + 1

3r − 1

4r + =



1− 1

2r −1



ς(r)

S2(r) = 1 + 1

3r+1 + 1

5r+1 + 1

7r+1 + =



1− 1

2r+1



ς(r + 1)

where ς(r) is Riemann Zeta Function [19].

It can be shown that despite the assumption about the plant being type 0

servo system formula (20) remains valid for integrating plants too If applied

to the transfer functions of multiple integrators, this formula allows for deriving

the the LPRS of r-th order multiple integrator as follows:

J r −int (ω) =

 (−1) r/2 1

ω r S1(r) + j0 if r is even

0 + j ( −1) (r+1)/2 1 S2(r) if r is odd

Taking into account formula (19) and the definition of the LPRS (7) we obtain

the final form of the expression for Re J (ω) Similarly, having solved the set of equations (12) where... θ2 = θ and y(0) and y(θ1) have the form (13) and

(14) respectively, we obtain the final formula of Im J (ω) Having put the real

and the imaginary...

4, and does not require involvement of the filtering hypothesis Let us formulate that as the following theorem, the proof of which is given as the formulas above

Theorem For the