Output Tracking with Discrete-Time Integral Sliding Mode Control

To solve thisproblem, focus was placed on output feedback based sliding mode control [1]-[6].. In [9] it was shown that in the case of delayed disturbance estimation a worst case accurac

Sliding Mode Control

Xu Jian-Xin and Khalid Abidi

Department of Electrical and Computer Engineering, National University of

Singapore, 4 Engineering Drive 3, Singapore 117576

{elexujx,kabidi}@nus.edu.sg

1 Introduction

Sliding mode control is a very popular robust control method owing to its ease

of design and robustness to “matched” disturbances However, full state mation is required in the controller design which is a drawback since in mostpractical applications only the output measurement is available To solve thisproblem, focus was placed on output feedback based sliding mode control [1]-[6] Two approaches arose: a design based on observers to construct the missingstates, [3],[4], the other design focused on using only the output measurement,[1],[2] Both approaches present certain strengths and limitations

infor-Computer implementation of control algorithms presents a great convenienceand has, hence, caused the research in the area of discrete-time control to in-tensify This also necessitated a rework in the sliding mode control strategy forsampled-data systems Most of the discrete-time sliding mode approaches arebased on the availability of full state information, [7]-[9] A few approaches didfocus on the output measurement, [5],[6] In [5],[6], the control design was based

on the assumption that the state matrix of a discrete-time system is ible This is true for sampled-data systems In this chapter we will focus onstate based approaches as well as expand upon the work of [5],[6] by focusing

invert-on arbitrary reference tracking of a linear time invariant system with matcheddisturbance

A delay in the state or disturbance estimation in sampled-data systems is aninevitable phenomenon and must be studied carefully In [9] it was shown that

in the case of delayed disturbance estimation a worst case accuracy of O(T )

can be guaranteed for deadbeat sliding mode control design and a worsted case

accuracy of O(T2) for integral sliding mode control While deadbeat control

is a desired phenomenon, it is undesirable in practical implementation due tothe over large control action required In [9] the integral sliding mode designavoided the deadbeat response by eliminating the poles at zero A similar effect

is possible in an integral sliding mode design for output tracking

This chapter considers the output tracking of a minimum-phase linear tem subject to matched time varying disturbance To accomplish the task of

sys-G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 247–268, 2008 springerlink.com  Springer-Verlag Berlin Heidelberg 2008c

arbitrary reference tracking three approaches will be considered: 1) State back, 2) Output Feedback, and 3) Output Feedback with a State Observer.

Feed-In each approach the objective is to drive the output tracking error to a tain neighborhood of the origin For this purpose a discrete-time integral slidingsurface (ISM) is proposed The proposed scheme allows full control of the closed-loop error dynamics and the elimination of the reaching phase The elimination

cer-of deadbeat response helps in avoiding the generation cer-of over large control puts It is also worth to highlight that the discrete-time integral sliding mode

in-control (ISMC) can achieve the O(T2) boundary for output tracking error even

in the presence of O(T ) accuracy in the state estimation.

where the state x∈  n, the output y∈  m, the control u∈  m, and the

dis-turbance f∈  m is assumed smooth and bounded The discretized counterpart

disturbance dk represents the influence accumulated from kT to (k + 1)T , in the

sequel it shall directly link to xk+1 = x((k + 1)T ) From the definition of Γ it

can be shown that

dynamics of the sampled-data system has m closed-loop poles assigned to desired

locations

In [9] it was shown that as a consequence of sampling, the disturbance inally matched in continuous-time will contain mismatched components in thesampled-data system This is summarized in the following relation, [9],

where vk = v(kT ), v(t) = dt d f (t), d k = O(T ), d k − d k −1 = O(T2), and dk −

2dk −1+ dk −2 = O(T3) Note that the magnitude of the mismatched part in the

disturbance dk is of the order O(T3)

3 Output Tracking ISM: State Feedback Approach

3.1 Controller Design

Consider the discrete-time integral sliding-surface defined below,

σ k= ek − e0+ ε k

ε k = ε k −1 + Ee k −1 (5)

where ek= rk − y k is the tracking error, rk is the reference trajectory, yk is the

output, σ k ,  k ∈  mare the sliding function and integral vectors, e0is the intial

error, and E ∈  m ×m is the design matrix The output tracking problem is to

force yk → r k

To proceed with the controller design, consider a forward expression of (5)

σ k+1= ek+1 − e0+ ε k+1

Substituting ε k+1 and (2) into the expression of the sliding surface in (6) and

equating σ k+1to zero leads to

σ k+1= ek+1 + Ee k − e0+ ε k= ek+1 − (I m − E)e k + σ k= 0 (7)

where I m is a unity matrix of dimension m If we substitute y k+1 = CΦx k +

CΓ u k + Cd k into (7) and solve for the equivalent control ueq k we have

ueq k = (CΓ ) −1[r

k+1 − Λe k − CΦx k − Cd k + σ k] (8)

where Λ = I m − E Under the assumptions made, the control cannot be

im-plemented in the same form as in (8) because of the lack of knowledge of the

disturbance dk To overcome this, the disturbance estimate will be used fore, the final controller structure is given by

There-uk = (CΓ ) −1

rk+1 − Λe k − CΦx k − C ˆd k −1 + σ k



(9)where ˆd is the disturbance estimate and in the case of full state availability the

following delay based estimation can be used, [7],

Since, the difference between ueq k and ukis the substitution of dk with ˆdk −1 the

sliding surface σ k+1will no longer be zero but rather a function of the difference

dk − ˆd k −1 as follows

σ k+1 = C(ˆdk −1 − d k) (12)Thus, we obtain the closed-loop state dynamics during sliding mode In order toconclude on the stability of (11) we propose the following Lemma

Lemma 1 The eigenvalues of 

Φ − Γ (CΓ ) −1 (CΦ − ΛC) are the eigenvalues

of Λ and the non-zero eigenvalues of [Φ − Γ (CΓ ) −1 CΦ].

Proof: See Appendix.

According to Lemma 1 the matrix

Φ − Γ (CΓ ) −1 (CΦ − ΛC) has m poles to

be placed at desired locations while the remaining n −m poles are the open-loop

zeros of the system Since, the system (2) is assumed to be minimum phase,

the fixed n − m poles are stable Therefore, stability of the closed-loop state

the open-loop zeros of the system are stable and, thus, it should be minimumphase.

3.3 Tracking Error Bound

In order to calculate the tracking error bound we must find the bound of δ k

Looking back at (16), δ k was given by

δ k=−C(d k − ˆd k −1 − d k −1+ ˆdk −2 ). (17)

From (10) ˆdk −1= dk −1, therefore, (17) becomes

δ k=−C(d k − d k −1 − d k −1+ dk −2 ). (18)which simplifies to

δ k=−C(d k − 2d k −1+ dk −2 ). (19)

In [9] it is shown that dk − 2d k −1 + dk −2 = O(T3) if the smoothness and

boundedness conditions on f (t) hold Therefore,

δ k=−C(d k − 2d k −1+ dk −2 ) = O(T3). (20)

According to [9] the ultimate error bound one k  will be one order higher than

the bound on δ k due to convolution and since the bound on δ k is O(T3) theultimate bound on e k  is O(T2) Thus, the ultimate bound on the trackingerror is

is a function of the output tracking error

In order to proceed we will first define the reference model

where xr,k ∈  n is the reference model state vector, yr,k ∈  mis the reference

model output vector, and rk ∈  mis the reference trajectory Due to the beat nature of the reference model its output is the desired reference trajectory

dead-rk and, therefore, tracking this reference model would lead to the desired sponse The only drawback is that the stability of the reference model requires

re-that the system (2) satisfy the minimum phase condition Now define D = CΦ −1,

consider the sliding surface

σ k = D [x r,k − x k ] + ε k

ε k = ε k −1 + ED [x r,k −1 − x k −1] (23)

where D eliminates the state dependency, σ k , ε k ∈  m are the sliding function

and integral vectors, and E ∈  m ×m is a design matrix of rank m The design

associated with D is adopted in [5],[6] Note that unlike the sliding surface (5)

which is based on the output yk, sliding surface (23) is based on the states

xk To proceed with the design consider a forward expression of the slidingsurface (23)

σ k+1 = D [x r,k+1 − x k+1 ] + ε k+1

ε k+1 = ε k + ED [x r,k − x k] (24)Substituting the sampled-data system (2) into (24)

σ k+1 = D [x r,k+1 − Φx k − Γ u k − dk] + ε k+1

ε k+1 = ε k + ED [x r,k − Φx k −1 − Γ u k −1 − d k −1] (25)

From the definition of D we have DΦx = y, therefore, we can eliminate x from

(25) resulting in the expression for the sliding surface

and the expression for the integral variable ε k+1

ε k+1 = ε k + E [Dx r,k − y k −1 − DΓ u k −1 − Dd k −1 ] (27)

Sliding mode condition occurs when σ k+1= 0, therefore, setting the right-hand

side of (26) to zero and solving for the equivalent control ueq k , we get

ueq k = (DΓ ) −1 [Dx

and ε k+1 is found from (27) Controller (28) is not practical as it requires a priori

knowledge of the disturbance Thus, the estimate of the disturbance will be used.However, note that the disturbance estimate used in the state feedback controllerdesigned in Section 3 requires full state information which is not available in thiscase Therefore, an observer that is based on output feedback will have to beused If we substitute the delayed disturbance estimate, ˆdk −1, instead of dk thefinal controller becomes

uk = (DΓ ) −1

Dx r,k+1 − y k − Dˆd k −1 + ε k+1



(29)and the expression for the integral variable (27) becomes

To get the state dynamics, yk will be replaced by DΦx k and [yk −1 + DΓ u k −1]

will be replaced by [Dx k − Dd k −1] and the result is simplified

We see from (35) that σ k+1 is a function of d and ˆ d It will be shown later

that the disturbance observer is not dependent on the state dynamics and, thus,

σ k+1 is not coupled to xk+1 Using a delayed form of (35), σ k = D(d k −1 −

finally substituting (22) into the r.h.s of (37) and using the fact that [I −

Γ (DΓ ) −1 D]Γ = 0 we obtain

Δx k+1=

Φ − Γ (DΓ ) −1 (DΦ − ΛD)Δx k − ζ k (38)

where Δx = x r − x From Lemma 1 the closed-loop poles of (38) are the

eigenvalues of Λ and the open-loop zeros of the system (Φ, Γ, D) Thus, m poles

of the closed-loop system can be selected by the proper choice of the matrix Λ while the remaining poles are stable only if the system (Φ, Γ, D) is minimum

phase

We have established the stability condition for the closed-loop system, but,have not yet established the tracking error bound For this we need to first discussthe disturbance estimate ˆdk upon which the tracking error bound depends Thenext section will address this

4.3 Disturbance Observer Design

In order to design the observer we first need to note that according to (4) thedisturbance can be written as

dk = Γ f k+1

2Γ v k T + O(T

3) = Γ η k + O(T3) (39)

where η k = fk+12vk T If η k can be estimated then the estimation error of dk

would be O(T3) which is acceptable in practical applications Define the observer

on an integral sliding surface

σ d,k = ed,k − e d,0 + ε d,k

ε d,k = ε d,k −1 + E ded,k −1 (41)

where ed,k = yk − y d,k , is the output estimation error, σ d , ε d ∈  m are the

sliding function and integral vectors, and E d is a design matrix

Since the sliding surface (41) is the same as (5), by following the derivation

procedure shown in Subsection 3.1, that is, letting σ d,k+1= 0, we obtain

ˆ

η k = (CΓ ) −1[y

k+1 − Λ ded,k − CΦx d,k + σ d,k]− u k (42)

where Λ d = I m − E d Expression (42) is the required disturbance estimate and

is similar in form to (8) Note, however, that (42) requires the future value of

the output y which is not possible therefore the delayed (42) is used

η k −1 = (CΓ ) −1[yk − Λ ded,k −1 − CΦx d,k −1 + σ d,k −1]− u k −1 . (43)

Since, only ˆη k −1 is available the observer model (40) will be delayed as well.

Substitution of (43) into the delayed form of (40) and following the same steps

of the derivation of (11) we obtain

If we set E d = I m − Λ d where Λ d is a diagonal matrix it is shown in Lemma 1

that the poles of the closed-loop system (45) are the eigenvalues of Λ d and the

non-zero eigenvalues of [Φ − Γ (CΓ )CΦ] In control applications, we can choose eigenvalues Λ d closer to origin comparing with the controller eigenvalues Λ Premultliplication of (45) with C yields

Φ − Γ (CΓ ) −1 (CΦ − Λ d C)

is stable, for k large enough



Φ − Γ (CΓ ) −1 (CΦ − Λ d C)k −1 → 0

and the disturbance estimate will converge to the actual disturbance As a result,

when k is large enough

dk − ˆd k −1 = Γ (η k − ˆη k −1 ) = Γ (η k − η k −1 ) = O(T2). (51)

4.4 Tracking Error Bound

Recall that the closed-loop system was in the form

than the bound on ζ due to convolution and since the bound on ζ k is O(T3) theultimate bound onΔx k  is O(T2) Thus, the ultimate bound on the trackingerror is

5 Output Tracking ISM: State Observer Approach

5.1 Controller Structure and Closed-Loop System

In this section we discuss the observer based approach for the unknown states.Recall that the state based ISM control was given by (8)

ueq k = (CΓ ) −1[r

k+1 − Λe k − CΦx k − Cd k + σ k ] (58)Under the assumptions made, the control cannot be implemented in the same

form as in (58) because of the lack of knowledge of the states xk as well as

the disturbance dk To overcome this, the state and disturbance estimates tained from the observers will be used Therefore, the final controller structure isgiven by

It will be shown later that the state estimation error κ k is not dependent

on the state dynamics xk and, thus, the dynamics of the sliding function is

not coupled to xk Therefore, the stability of (62) depends on the matrix

ek+1 = Λe k + ξ k (65)

where ξ k is given by

ξ k =−C(d k − ˆd k −1 − d k −1+ ˆdk −2 ) + CΦ(κ k − κ k −1 ). (66)Thus, as in the state feedback approach, the output tracking error depends on

the proper selection of the eigenvalues of Λ In order to better understand the

effect of κ k on the tracking error bound we will discuss the state observer in thenext section

5.2 State Observer

State estimation will be accomplished with the following state-observer

ˆ

xk+1 = Φˆxk + Γ u k + L(y k − ˆy k) + ˆdk −1 (67)ˆ

where ˆx, ˆy are the state and output estimates and L is a design matrix Observer

(67) is well-known, however, the term ˆdk −1 has been added to compensate for

the disturbance Since, only the delayed disturbance is available it is necessary

to investigate the effect it may have on the state estimation Subtracting (67)from (2) we get

˜

xk+1 = [Φ − LC]˜x k+ dk − ˆd k −1 (69)where ˜x = x− ˆx is the state estimation error The solution of (69) is given by

Since, dk − ˆd k −1 = O(T2) it was shown in [9] that the ultimate bound on ˜xk

as k → ∞ is O(T ) However, it will be shown that by virtue of the integral action in the ISM control, the O(T ) error introduced by the state observer will

be reduced to O(T2) in the overall closed-loop system

5.3 Tracking Error Bound

In order to calculate the tracking error bound we must find the bound of ξ k

Looking back at (66), ξ k was given by