Second-Order Sliding Sector for Variable Structure Control

The P R-sliding sector proposed in [5] for continuous-time systems was defined as a subset of the state space, inside which a norm of state decreases with zero control input.. The VS cont

Structure Control

Yaodong Pan1 and Katsuhisa Furuta2

1 The 21st COE Century Project Office, Tokyo Denki University, Ishisaka,

Hatoyama, Hiki-gun, Saitama 350-0394, Japan

neces-Sliding sectors have been proposed to replace the sliding mode for a chatteringfree VS controller and for the implementation of the VS controller in discrete-time systems [6][5] The first sliding sector proposed in [6] is a subset of thestate space where the closed-loop system is stable The VS control law with thesliding sector [6] ensures that the system moves into the sector in a finite time.The robustness of the VS control system with the sliding sector was proved in[7] by using the frequency domain criterion

The P R-sliding sector proposed in [5] for continuous-time systems was defined

as a subset of the state space, inside which a norm of state decreases with zero

control input The VS controller with the P R-sliding sector was designed such

that the system state moves from the outside to the inside of the sector with

a suitable VS control law, the control input is zero inside the sector, and thenorm of state keeps decreasing in the state space with specified negativity of its

derivative The VS control system with the P R-sliding sector is quadratically

stable and robust stable to bounded parameter uncertainty [5] The VS control

with the P R-sliding sector is called the lazy control because the VS control law is active only outside the P R-sliding sector, which may be useful to some practical

control systems

Theoretically the P R-sliding sector with the corresponding VS control law for

continuous-time systems is an invariant subset of the state space but with zero

G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 97–118, 2008 springerlink.com  Springer-Verlag Berlin Heidelberg 2008c

control input inside the P R-sliding sector, it can not ensure that the system state

remains inside the sector forever in a real control system with finite switchingfrequency In the most of case there exists a potential possibility for the system

state to move out of the sector The VS control law outside the P R-sliding sector

lets the system state move back into the sector as soon as it moves out of thesector Therefore it is necessary to switch the control input on the boundary

of the sector with infinite frequency to guarantee the invariance of the sector.Thus the invariance may be lost because of finite switching frequency when it

is implemented in a real control system although it does not affect the stability

of the VS control system with the P R-sliding sector It is proposed to use a

hysteresis function to design the VS control law (See Equation (20) of [5]), withwhich the system may stay inside the sector longer and the invariance is obtained

to a certain extent

As an effective method to deal with the chattering problem and to obtainhigher sliding accuracy, high-order sliding mode controllers [8][1][2][9] and high-order sliding mode observers [10][11] have been proposed and can be implemented

in many mechanical systems[12][13] The high-order sliding mode with s = ˙s =

· · · = s (r −1) ensure the sliding mode to be a high-order invariant subset in

the state space Similarly, a second order sliding sector has also been proposed[14] for continuous-time[15] and discrete-time[16] systems to attain the invariant

property The second-order sliding sector is a P R-sliding sector and an invariant

subset of the state space Therefore, it was named an invariant sliding sector in[14] Inside the invariant sliding sector, a norm of state decreases because it is a

P R-sliding sector And especially inside the sector, the state will not move out

of the sector with a suitable control law The VS controllers with the invariant

P R-sliding sector ensure that the system state moves from the outside to the

inside of the sector in a finite time, stays inside it forever after being moved into

it, and some Lyapunov function keeps decreasing in the state space with a VScontrol law The resultant VS control system thus is quadratically stable andwithout any chattering Such invariant sliding sector for continuous-time systemsremains invariant even if the switching frequency is finite In [17], Yu and Yu alsodiscussed the existence of an invariant sliding sector for a second order discrete-time system and gave conditions to guarantee the existence As comparisonwith those sliding sectors proposed in [6] and [5], the VS control system withthe invariant sliding sector is more realizable for practical implementations andmay result in a continuous control input

As the objective of this paper, based on the P R-sliding sector proposed in [5],

a second-order (i.e an invariant) P R-sliding sector for continuous-time systems

will be designed at first Then a quadratically stable VS control system with thesector will be proposed, where an internal and an outer sector are introduced tolet the VS control law be continuous on the system state and realizable Finallythe proposed VS control algorithm is implemented to an inverted pendulumcontrol system

This chapter is organized as follows Section 2 presents the P R-sliding sector [5], defines the second-order P R-sliding sector and describes the problem to be

considered in this chapter Section 3 designs the second-order P R-sliding sector

based on a normal switch function Section 4 proposes the VS controller with

the second-order P R-sliding sector Section 5 gives simulation results with the

inverted pendulum

2 Problem Description

In this chapter, a linear time-invariant continuous-time single input system withparameter uncertainties and external disturbances, described by the followingstate equation is taken into consideration

˙x(t) = Ax(t) + B(u(t) + d(x, t)) (1)

where x(t) ∈ R n and u(t) ∈ R1are the state and the input vectors, respectively,

A and B are constant matrices of appropriate dimensions, pair (A, B) is

control-lable, and d(x, t) represents parameter uncertainties and external disturbances.

It is assumed that d(x, t) is bounded and satisfies

d2(x, t) ≤ w(x), ∀x ∈ R n , ∀t ∈ [0, +∞), (2)

where w(x) = x T W x is a quadratic function function on x and W = W T ∈ R n ×n

is a known positive definite matrix

If the autonomous system ˙x(t) = Ax(t) of the above one in (1) is quadratically

stable, then there exists a positive definite symmetric matrix P and a positive semi-definite symmetric matrix R = C T C such that

˙

L(x) > −x T Rx for some elements x ∈ R n, and the other part satisfies the dition ˙L(x) ≤ −x T Rx for some other elements x ∈ R n The latter elements form

con-a specicon-al subset in which the Lycon-apunov function ccon-andidcon-ate L(x) decrecon-ase with

zero control input

We define a P -norm, denoted by ||x|| P as the square root of the Lyapunov

function candidate L(x) in (4), i.e.

Accordingly, we call this special subset a P R-sliding sector because the matrices

P and R together with the system parameter A determine the property of this

subset The definition of the P R-Sliding Sector found in [5] is given as follows.

Definition 1 The P R-Sliding Sector is defined on the state space R n as

where P and R are the matrices as described above, and s(x) and δ2(x) are a

linear and a quadratic functions, respectively.

Such P R-sliding sector is a subset of R n around a hyperplane s(x) = 0 and

is bounded by two surfaces s(x) = ±δ2(x) Inside the sector, the P -norm decreases Therefore a VS control law based on the P R-sliding sector can be designed such that the system moves into the sector where the P -norm decreases Unfortunately, the P R-sliding sector is not an invariant subset in the state space

when it is implemented in a real control system with finite sampling frequency

Similar to the definition of the second-order sliding mode control, i.e., s = ˙s = 0,

one more condition

is included to obtain the invariance of the sector besides the condition s2(x) ≤

δ2(x) used to define the sliding sector in (6).

Definition 2 The P R-sliding sector defined in (6) is said to be a Second-order

P R-Sliding Sector denoted by

2 Quadratic functions ξ2(x) and δ2(x) on x satisfy

whereN is the null set in R n The second-order P R-sliding sector defined above

is an invariant subset with the P -norm decreasing inside it because for any

system state inside the outer sector, the system will remain inside the outersector or move into the internal sector, i.e stay inside the second-order slidingsector, and for any state inside the internal sector, the system may move to theouter sector but will never move out of the sliding sector Therefore for some timemoment, if the system moves into the second-order sliding sector, the system willstay inside the sector since then The last condition dt d s2(x) ≤ d

dt δ2(x) in the

above definition shows the invariant property of a second-order sliding sector

By decomposing the sector to an internal and an outer sectors, a large controlinput to ensure the second-order property inside the outer sector can be avoidedwhile the invariance of the sector can be guaranteed

It has been pointed out in [5] that the P R-sliding sector exists with some

P and R for any controllable systems and can be designed by using the Riccati

equation In this chapter, the P R-sliding sector and the second-order P R-sliding

sector will be designed based on a normal switch function, which is used to realize

a VS control system with sliding mode The control objective with the

second-order P R-sliding sector is to let the system move into the sector in a finite time

and remain inside the sector since then with some suitable control rule and tostabilize the system quadratically

3 Second-Order Sliding Sector

3.1 Design of Switch Function

A switching function defined as

should be designed such that the reduced order system on the sliding mode:

¯

S2= 1.

An equivalent control input guaranteeing ˙s(x) = 0 is given by

u eq (t) = −(SB) −1 SAx(t) = −SAx(t), (SB = ¯ S ¯ B = 1). (19)With the control input

the system in Eq.(16) can be written as

˙¯

x(t) = ¯ A eq x(t) + ¯¯ B(v(t) + d(x, t)) (21)where the matrix ¯A eq is determined by

and v(t) is an alternative control input Taking a nonsingular transformation

as the following block forms:

I n −1 is the identity matrix of order (n − 1), O (n −1)×1 and O1×(n−1) are the

zero (n − 1)-column and row vectors, respectively.

It follows from Eq.(23) that

3.2 Design of Sliding Sector

With the switch function designed in the last subsection, a P R-sliding sector defined in Eq.(6) can be designed by choosing the switching function s(x) in

Eq.(27) as the linear function s(x) in Eq.(6) In this case, the problem is how

to determine the matrices P and R and also the control law to satisfy those conditions for a P R-sliding sector.

As the switch function is designed so that the reduced order system in Eq.(26)

is stable, there exists a positive definite symmetric matrix ˜P11 ∈ R (n −1)×(n−1)

such that the following Lyapunov equation holds for some positive definite metric matrix ˜Q11∈ R (n −1)×(n−1).

sym-− ˜ Q11= ˜A T11P˜11+ ˜P11A˜11 (28)Choose positive definite symmetric matrices ˜P and ˜ R as

where h is a large enough positive constant such that the matrix ¯ R is positive

definite Then a P R-sliding sector can be designed as

where ˜P and ˜ R are given by Eqs.(29) and (30), respectively, F1 and F2 are

transformation matrices defined in the last subsection, and Δ is chosen to be

Therefore inside the P R-sliding sector(i.e., s2(x) ≤ δ2

(x)), if the control law

is given by

u(t) = u eq + v(t) = u eq − kδ(x) · sgn (s(x)) , (37)

where δ(x) =



δ2(x) ,the equivalent control input u eq is given in Eq.(19) and

k (k > 1 + 1/ √ γ) is a large enough positive constant parameter, then we have

√ γ √ x T P x + √

x T W x)

≤ −x T

(t)Rx(t).

The following theorem concludes the above discussion

Theorem 1 The subset designed in Eq.(31) is a P R-sliding sector with

corre-sponding parameters discussed above, inside which the P -norm decreases with the control law given by Eq.(37) as

3.3 Second-Order Sliding Sector

Based on the P R-sliding sector designed in the last subsection, an internal and

an outer sectors for the second-order P R-sliding sector are designed as

Theorem 2 The P R-sliding sector designed in the last subsection with Δ = βP

and the internal and outer sectors determined above is a second-order P R-sliding sector as

where the equivalent control input u eq (t) is given in Eq.(19), k is a large enough

positive constant, and parameters are chosen to satisfy the following relations:

√ γ δ2(x))

< −x T (t)(R − h

√ α

√ γ βP )x(t)

< −1

2x

T (t)Rx(t), i.e., the P -norm decreases inside the internal sector.

Now let’s consider the outer sector In this case, αδ2(x) < s2(x) ≤ δ2(x) and

the control law is given by

not move out of the sector

At the same time, the derivative of the Lyapunov function candidate L(x) is

Therefore the designed P Rsliding sector is a secondorder one and the P

-norm decreases inside the sector

4 VS Controller with Second-Order Sliding Sector

With the second-order P R-sliding sector designed in Theorem 2, a VS

con-troller should be designed to move the system from the outside to the inside

of the second-order P R-sliding sector in a finite time while the P -norm keeps

decreasing

Theorem 3 With the internal sector (39), the outer sector (38), and the

second-order P R-sliding sector (40) designed in Theorem 2, the following tinuous VS control law

where the equivalent control input u eq (t) is given in Eq.(19), and α and k satisfy

the conditions given in Theorem 2.

Proof It is obvious that the proposed VS control law u(t) (48), which may

be denoted as u(t) := u(x(t)) := u(x) is continuous on x inside the internal

sector, inside the outer sector, and outside the second-order sector and is alsocontinuous in each switching instant happening in the boundaries of the second-

order P R-sliding sector (40) and the internal sector (39) satisfying s2(x) = δ2(x) and s2(x) = αδ2(x), respectively Thus the VS control law (48) is continuous in

the state space and no chattering will happen

It has been shown in the proof of Theorem 2 that the sector designed inTheorem 2 is a second-order one Thus the proof will be finished if it can be

shown that outside the second-order P R-sliding sector, the P -norm decreases

and the system moves into the sector in a finite time

Outside the second-order P R-sliding sector, the following inequality holds:

Now let’s consider the quadratic function V (x) defined in (46), then similar

it can be concluded that the system will move toward the sliding mode s(x) = 0,

will move into the sliding sector in a finite time although it may need infinite

time to converge to the sliding mode s(x) = 0 This ends the proof.

5 Simulation Results with Furuta Pendulum

5.1 Furuta Pendulum

Because of static friction inherently existing in all mechanical systems, it isdifficult to control the position, speed and torque of a motor at the slow speed

On the stabilization control of an inverted pendulum, it is much more difficult tokeep the pendulum strictly inverted at a predetermined motor position becausethe motor has to switch the rotation direction frequently to keep the balance.Hence it can be expected that the proposed VS control, which can take the time-varying friction into consideration, enhances the performance of the stabilizationcontrol for the inverted pendulum.

The apparatus is shown in Fig.1 The pendulum-link is equipped on the outputaxis of a direct drive motor, and the link can rotate around the pivot freely

Fig 1 Furuta pendulum

5.2 Derivation of Dynamic Equation

The dynamic equation of the pendulum can be obtained by using Lagrangemethod To calculate the energies, the coordinate values of each center of grav-ity(COG) are calculated The coordinate systems attached according to DH-

convention method are shown in Fig.2 θ1, θ2and τ are angles of the motor and

the pendulum, and the input torque respectively The DH-parameters are listed

in Tab.1 and the mechanical parameters and their meanings are described inTab.2