Multiple Sliding Surface Control for Systems withMismatched Uncertainties

Mayuresh B Patil∗Divyesh Ginoya

College of Engineering, Pune

Email: mayur.mayuresh@gmail.com∗Email: dlginoya007@gmail.com

S B Phadke∗P D Shendge

College of Engineering, Pune

Email: sbp.instru@coep.ac.in∗Email: pds.instru@coep.ac.in

Abstract—This paper proposes a new method for slidingmode control of systems with mismatched uncertainties. Theproposed method employs a multiple sliding surface (MSS)approach combined with inertial delay control (IDC) forestimating unmatched disturbances. It has the advantage oftackling constant as well as time-variant and state dependentdisturbances/uncertainties. The proposed method is validated bysimulation and compared with traditional sliding mode control(SMC) and integral sliding mode control (I-SMC) method.

Index Terms—mismatched disturbance, sliding mode control(SMC), integral sliding mode control (I-SMC), multiple slidingsurface control (MSSC), inertial delay control (IDC)

I. INTRODUCTION

Sliding Mode Control (SMC) has been widely studied

and extensively employed in industrial applications as it’s

conceptually simple, and in particular its powerful ability to

reject disturbances and plant uncertainties [1]-[3]. Most of the

SMC results are satisfactory when systems satisfy matching

conditions i.e. the disturbance is present in the same channel as

that of the input. However, the matching condition assumption

is fairly restrictive and is not satisfied by many practical

systems such as under-actuated mechanical system, electro-

mechanical systems connected in series, flight control systems

[4] and others.

As the demand for attenuation of mismatched uncertain-

ties/disturbances has increased, the research is going on to

solve this problem and to develop different control strategies

[5]-[10]. These mismatched uncertainties affect system sta-

bility and cause deterioration in the output. Hence, one of

the approaches to handle this issue is to focus on system

stability using some classical control design tools such as

Riccati approach [10],[11], adaptive approach [12] etc. The

mismatched uncertainties considered by the authors stated

above must be bounded or vanishing. Again, this assumption

may not be satisfactory for many engineering systems may

suffer from these types of disturbances.

We can state another approach as the integral sliding mode

control (I-SMC). It introduces an integral term into sliding

surface, which makes the system initial states start from

the sliding mode and eliminates the reaching phase. Thus,

I-SMC enhances robustness against matched uncertainty of

the conventional SMC. Compared with SMC for mismatched

uncertainties, the I-SMC method is more useful due to its

simplicity and robustness and has been applied to many

practical systems [13]-[15]. Still, the drawbacks of integral

action come along with the results such as large overshoot

and long settling time.

Recently, many authors introduced a disturbance observer

(DOB) for SMC to eliminate the the effect of uncertain-

ties. Also Non-linear Disturbance Observer (NDOB) along

with SMC has been proved as a good strategy for systems

with mismatched uncertainties [16]. But the major limita-

tion of this strategy is that it can only handle bounded

slow varying/constant disturbances. Multiple sliding surface

control (MSSC) can be used for systems with mismatched

disturbances/uncertainties. For the estimation of uncertainty,

several techniques have been used by several authors. Function

approximation technique is one of them. It is used to transform

the uncertainties into a finite combination of orthonormal basis

function [17].

In this paper, multiple sliding surface control (MSSC)

method is proposed to handle such mismatched uncertainties

using inertial delay control (IDC) [18] - [20]. The key features

of this proposed method are :

1) It allows a sliding controller to be designed as if the

system has a reduced relative degree by defining one of

the states as a synthetic input to the reduced order plant.

2) The synthetic states are then controlled by a second

controller to make sure that the synthetic control will

track the profile defined by the second controller.

3) For estimation of these mismatched distur-

bances/uncertainties inertial delay control (IDC) is

proposed.

This paper is organised as follows : In section II control

using traditional SMC and I-SMC is derived. In Section III

multiple sliding surface control (MSSC) with inertial delay

control (IDC) is proposed. Section IV shows the simulation

results to support the proposed method.

978-1-4673-6190-3/13/ $31.00 c©2013 IEEE

II. EXISTING SMC METHODS AND THEIR DRAWBACKS

Consider the following third-order system with mismatched

disturbance, stated as

x1 = x2 + d1(t)

x2 = x3 + d2(t)

x3 = a(x) + b(x)u+ d3(t)

y = x1

(1)

where x1, x2 and x3 are states, u is the control input, d1(t),d2(t) are the disturbances, d3(t) is the lumped uncertainty

and y is the output. It is assumed that these disturbances are

bounded.

A. Traditional SMC

The sliding mode surface and the control input of traditional

SMC are designed as follows:

σ = x3 + c1x2 + c2x1 (2)

u = −b−1(x)[a(x) + c1x3 + c2x2 + ksgn(σ)] (3)

Combining (1), (2) and (3) we’ll get,

σ = −ksgn(σ) + (c2d1(t) + c1d2(t)) (4)

(4) shows that the states of the system (1) will reach

the sliding surface σ = 0 in finite time as long as the

switching gain in control input (3) is designed such that

k > c2||d1(t)|| + c1||d2(t)||. When we analyse the sliding

mode condition, as σ = 0 we’ll get

x1 = −c1x1 − c2x1 + d1 + d2 + c1d1 (5)

Remark 1: From (5) it can be stated that the states can’t

not be drawn to the desired point even if the control law in

(3) can force the states to reach the sliding surface in finite

time. This is the major drawback of traditional SMC and it

can’t be used alone for an unmatched system.

B. Integral SMC

Considering traditional SMC, the very less modified solu-

tion for these unmatched systems is integral SMC (I-SMC).

The sliding surface is designed as

σ = x3 + c1x2 + c2x1 + c3

∫

x1 (6)

The control input can be designed as

u = −b−1(x)[a(x) + c1x3 + c2x2 + c3x1 + ksgn(σ)] (7)

Considering (1), (6) and (7) we get,

σ = −ksgn(σ) + (c2d1(t) + c1d2(t)) (8)

Similar to the above method, the states of the system (1)

will reach the sliding surface in finite time but the condition

is k > c2||d1(t)||+ c1||d2(t)||. Considering the condition σ =

0, we get,

...x1 = −c1x1 − c2x1 − c3x1 + d1 + d2 + c1d1 (9)

Remark 2: From (9) it can be stated that the state can slide

to the desired point if the system has reached the sliding

surface in finite time and the disturbance has a constant

steady state value. If the disturbances are time-varying or state

dependable this method fails to achieve the desired output.

Also, as stated earlier in this paper, due to integral action

some adverse effects are seen in the control action such as

large overshoot and long settling time.

III. MULTIPLE SLIDING SURFACE CONTROL (MSSC)

DESIGN WITH INERTIAL DELAY CONTROL (IDC)

The goal of the controller is to make x1 track the desired

trajectory x1d. Applying MSS control to the system in (1), the

first sliding surface is defined as

s1 = x1 − x1d (10)

To minimize the initial control we modify our sliding

surface as-

s∗1= s1 − s1(0)e

−α1t (11)

where s1(0) is the initial value for s1.

s∗1= x2 + d1 − x1d + α1s1(0)e

−α1t

d1 = s∗1− x2 + x1d − α1s1(0)e

−α1t (12)

Using IDC, we can estimate d1. Let the estimation of d1 be

e1 and is defined as-

e1 =1

τ1s+ 1d1 (13)

A modified second sliding surface is defined as-

s∗2= s2 − s2(0)e

−α2t (14)

s∗2= x2 − x2d − s2(0)e

−α2t (15)

x2d is the synthetic input and is to design in such a way

that it’ll make s∗1s∗1< 0 and it will negate the effect of the

uncertainty d1. Solving (12), (13) and (15) together we’ll get

x2d and e1 as-

x2d = x1d − e1 − α1s1(0)e−α1t − k1s

∗

1(16)

e1 =1

τ1[s∗

1+

∫

(k1s∗

1− s∗

2− s2(0)e

−α2t] (17)

Moving to next step,

s∗2= x3 + d2 − x2d + α2s2(0)e

−α2t

s∗2= x3 + e2 + α2s2(0)e

−α2t

e2 = s∗2− x3 − α2s2(0)e

−α2t (18)

Using IDC, we can estimate e2. Let the estimation of e2 be

e2 and is defined as-

e2 =1

τ2s+ 1e2 (19)

A modified third sliding surface is defined as-

s∗3= s3 − s3(0)e

−α3t (20)

s∗3= x3 − x3d − s3(0)e

−α3t (21)

x3d is the synthetic input and is to design such that it’ll make

s∗2s∗2< 0 and it’ll negate the effect of lumped uncertanity e2.

Solving (18), (19) and (21) together we’ll get x3d and e2 as

x3d = −e2 − α2s2(0)e−α2t − k2s

∗

2(22)

e2 =1

τ2[s∗

2+

∫

(k2s∗

2− s∗

3− s3(0)e

−α3t)] (23)

Now solving to design control input u,

s∗3= a(x) + b(x)u+ d3 − x3d + α3s3(0)e

−α3t

s∗3= a(x) + b(x)u+ e3 + α3s3(0)e

−α3t

e3 = s∗3− a(x)− b(x)u− α3s3(0)e

−α3t (24)

Again using IDC, we can estimate e3. Let the estimation of

e3 be e3 and is defined as-

e3 =1

τ3s+ 1e3 (25)

Now, designing control input u such that it will drive s∗3

to

0 and will negate the effect of e3. Hence the desired control

input u is-

u = −b−1(x)[a(x) + α3s3(0)e−α3t + e3 + k3s

∗

3] (26)

and e3 is

e3 =1

τ3[s∗

3+

∫

k3s∗

3] (27)

The ultimate boundedness can be proved using Lyapunov

stability criterion.

IV. NUMERICAL SIMULATION

Consider the following system for simulation studies,

x1 = x2 + d1(t)

x2 = x3 + d2(t)

x3 = −2x1 − x2 + ex1 + u+ d3(t)

y = x1

(28)

Both traditional SMC and I-SMC methods are applied

and compared along with the proposed strategy on above

mentioned system. It is assumed that all above disturbances

are acting on the system from initial. These are considered

as uncertainties in the system. These methods are applied for

various types of disturbances stated below. At the end of the

discussion of each case, advantages of MSSC are stated.

A. Case 1 : Constant disturbances

Consider the initial states of system (28) as x(0) =[1 0 0]T .

Initially, the disturbances are of different values but are consid-

ered as constants. They are taken as- d1(t) = 0.25, d2(t) = 0.1and d3(t) = 0.2.

If we consider the output, it can be observed from Fig.1(a)

that the traditional SMC fails to drive the state to desired

position. In other words, steady state error is present. This

shows that traditional SMC method is sensitive to mismatched

disturbances.

TABLE I: control parameters for case 1 and case 2

CONTROLLERS PARAMETERS

SMC c1 = 2, c2 = 2, k = 3

I-SMC c1 = 2, c2 = 2, c3 = 3, k = 3

MSSC with IDC τ1 = τ2 = τ3 =0.01, k1 = k2 = k3 =1,α1 = α2 = α3 =0.1

It is also vary much visible from Fig.1(a) that I-SMC tries to

drive the state to desired equilibrium point but sacrifices more

time. This is relevant to the fact about integral action that it

increases the settling time. Also, it brought the overshoot in

the output and is another drawback of it.

From Fig.2(b) and 2(c), it can be seen that the chattering

in the control input is present due to signum function used in

the design procedure. It is another drawback of the traditional

SMC as well as I-SMC method.

In Fig. 1(a), both MSSC and I-SMC methods can finally

suppress the mismatched uncertainties, but I-SMC method

brings the state around 65 sec whereas MSSC method brings it

around 7 sec to desired equilibrium point. It shows that MSSC

has a much quicker convergence rate than that of the I-SMC

method. Also, Fig.2(a) shows that the control input designed

using MSSC method has no chattering. It provides smooth

control input.

B. Case 2 : Time varying disturbances

Here, we consider initial states of system (28) as well as the

control parameters same as Case 1. As stated earlier, our major

concern is about the fact that the disturbances/uncertainties

can’t be always constant. The state variable in one channel

can be a disturbance in the other channel as that of the

flexible joint or flexible link system. They can be time-variant

or state dependant. Our aim is to reduce the effect of such

disturbances/uncertainties present in the system. Here, the

disturbance in first and second channel is considered time

dependent which are d1(t) = 0.1sin(t) and d2(t) = 0.5sin(t)respectively. And the third channel has state dependent uncer-

tainty and let’s assume it as d3(t) = x1x2.

Fig.3(a) shows the comparison of output for the three

methods. It can be clearly seen that the traditional SMC as

well as I-SMC method has failed to drive the output to desired

point. Hence these two methods can’t give the desired results

for mismatched systems with disturbances apart from constant

or slow varying ones. Also, as seen in Fig.4(b) and 4(c), in

the control input chattering is present which is not desirable.

The MSSC method drives the output to the desired point

though the disturbances are not constant. Also, Fig.4(a) shows

that the control input designed using MSSC method has no

chattering. This is another advantage of this method.

0 2 4 6 8 10 12 14 16 18 20−1.5

−1

−0.5

0

0.5

1

1.5

2

time (sec)

statex1

SMC

I−SMC

MSSC

(a)

0 2 4 6 8 10 12 14 16 18 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

time (sec)

statex2

SMC

I−SMC

MSSC

(b)

0 2 4 6 8 10 12 14 16 18 20−2

−1

0

1

2

3

time (sec)

statex3

SMC

I−SMC

MSSC

(c)

Fig. 1: state variables in simulation studies of Case 1

0 2 4 6 8 10 12 14 16 18 20−6

−4

−2

0

2

4

time (sec)

controlu

MSSC

(a)

0 2 4 6 8 10 12 14 16 18 20−4

−2

0

2

4

6

time (sec)

controlu

SMC

(b)

0 2 4 6 8 10 12 14 16 18 20−8

−6

−4

−2

0

2

4

6

time (sec)

controlu

I−SMC

(c)

Fig. 2: control input showing chattering reduction for constant

disturbances

0 2 4 6 8 10 12 14 16 18 20−1

−0.5

0

0.5

1

1.5

time (sec)

statex1

SMCI-SMCMSSC

(a)

0 2 4 6 8 10 12 14 16 18 20−2

−1.5

−1

−0.5

0

0.5

1

time (sec)

statex2

SMCI-SMCMSSC

(b)

0 2 4 6 8 10 12 14 16 18 20−3

−2

−1

0

1

2

3

4

time (sec)

statex3

SMCI-SMCMSSC

(c)

Fig. 3: state variables in simulation studies of Case 2

0 2 4 6 8 10 12 14 16 18 20−8

−6

−4

−2

0

2

4

6

time (sec)

controlu

MSSC

(a)

0 2 4 6 8 10 12 14 16 18 20−6

−4

−2

0

2

4

6

time (sec)

controlu

SMC

(b)

0 2 4 6 8 10 12 14 16 18 20−8

−6

−4

−2

0

2

4

6

time (sec)

controlu

I-SMC

(c)

Fig. 4: control input showing chattering reduction for time

varying disturbances

C. Effect of Change in linear Gain

We can see the chattering in the control input when we

design it using traditional SMC and I-SMC. This is due to

the discontinuous switching gain present in the control input.

We can get less chattering if we reduce the switching gain,

but at the same time these two methods fail to reject these

disturbances effectively. While using MSSC with IDC, the

output reaches the desired point after some time (6 sec in

the Case 1 and around 9 sec in the Case 2). We can increase

the rate by increasing the value of linear gains (k1, k2 and k3)

or by decreasing time constants (τ1, τ2 and τ3).

V. CONCLUSION

In this paper, a new approach for controlling systems

affected by mismatched disturbances/uncertainties is proposed.

The proposed approach successfully counters the effect of

mismatched uncertainties and outperforms the traditional SMC

and I-SMC approaches. In case of traditional SMC and I-SMC

methods ,while handling these mismatched uncertainties, the

error in the output is distinctly large whereas MSSC with IDC

almost eliminates the error. This conclusion is validated by

simulation for the cases of constant and time varying as well

as state dependent disturbances. Unlike the traditional SMC

and I-SMC approaches, the proposed control is chatter free.

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