Improved robustness and performance of discrete time sliding mode control systems

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Research Article

Improved robustness and performance of discrete time sliding mode control systems$ Sohom Chakrabarty a,n, Andrzej Bartoszewicz b a b

Department of Electrical Engineering, Indian Institute of Technology Roorkee, India Institute of Automatic Control, Technical University of Lodz, Poland

art ic l e i nf o

a b s t r a c t

Article history: Received 7 October 2015 Received in revised form 10 June 2016 Accepted 5 August 2016 This paper was recommended for publication by Dr. Oscar Camacho.

This paper presents a theoretical analysis along with simulations to show that increased robustness can be achieved for discrete time sliding mode control systems by choosing the sliding variable, or the output, to be of relative degree two instead of relative degree one. In other words it successfully reduces the ultimate bound of the sliding variable compared to the ultimate bound for standard discrete time sliding mode control systems. It is also found out that for such a selection of relative degree two output of the discrete time system, the reduced order system during sliding becomes finite time stable in absence of disturbance. With disturbance, it becomes finite time ultimately bounded. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Discrete time sliding mode control Relative degree Ultimate band Robustness Finite time convergence

1. Introduction Sliding mode control (SMC) has made its place in literature as a control which enables complete disturbance rejection in finite time, when the disturbance is bounded in nature and appears in the input channel. It works by applying a switching control, which brings the states of the system to a sliding surface in finite time, and once this sliding motion is achieved, the system is no longer affected by the disturbance. Though this idea is theoretically fascinating, it has practical limitations. The switching control needs actuators to switch at infinite frequency, which is not possible in the real world. Also, measurements by sensors and the control computation are done only after a specific time period. To remove this gap between theory and practice, discrete time sliding mode control (DSMC) theory was developed in [1–9]. Also many physical systems have inherently discrete time dynamics. It was seen that for these systems the states can no longer hit the sliding surface and stay there in presence of a disturbance, but converge inside an ultimate band around the surface in finite time. Hence the robustness of the system gets defined by the width of this ultimate band. ☆

Fully documented templates are available in the elsarticle package on CTAN. Corresponding author. E-mail addresses: [email protected] (S. Chakrabarty), [email protected] (A. Bartoszewicz). n

The discrete sliding mode control has traditionally been developed by taking outputs (or sliding variables) of relative degree one, i.e., the delay between the output and the control input is unit time step. This has given rise to proposals of various reaching laws of the form sðk þ 1Þ ¼ f ðsðkÞÞ, where s(k) is the sliding variable at the kth time step. Since s(k) is chosen as a relative degree one output, these reaching laws enable the calculation of control as sðk þ 1Þ, when calculated from the system dynamics, contains u(k). The most well-known of these reaching laws are laid down in [5,3,2]. Of the above, the first paper deals with a switching reaching law and the other two with nonswitching reaching laws. Even to this day, proposals of reaching laws are being laid down in literature, which exhibit different properties favourable to the design of control for a particular type of system. Some of these reaching laws are found in [10–17]. Very recent contributions highlighting the usefulness of DSMC can be found in [18,19]. However, all the above work proposes a sliding variable of relative degree one and finds out the ultimate band in each case. In the work presented in this paper, the authors aim to show that when the sliding variable is chosen with relative degree two, we get reduced width of the ultimate band and finite time stability during sliding in absence of disturbance. This is an important achievement, as this increases the robustness of the system as a whole, since robustness is directly related to the width of this ultimate band.

http://dx.doi.org/10.1016/j.isatra.2016.08.006 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Chakrabarty S, Bartoszewicz A. Improved robustness and performance of discrete time sliding mode control systems. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.08.006i

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2

A similar work has been recently presented in [20] where the reaching law given in [3] is used with an output of relative degree r ¼n, where n is the order of the system. The work is restrictive in the sense that r must be equal to n in order to obtain the properties of improved robustness and finite time stability. However, it has shown the path to the work in this paper where r ¼2 can bring about the desired properties of improved robustness and finite time stability for an n Z2 order system. The reaching laws that are used in this paper are the ones given in [5,2]. The switching reaching law in [5] contains a proportional and a switching term along with a term containing a disturbance d (k) (see Section 4). The non-switching reaching law in [2] contains a term which becomes zero in finite time in addition to this d(k) (see Section 3). This disturbance term d(k) is calculated as a direct functional relationship with the disturbance f(k) in the actual system, such that the control input becomes devoid of any disturbance terms. In this paper, we shall see that for relative degree two output, the disturbance term d(k) does not exhibit the same functional relationship with f(k) as in the case of relative degree one output. It is easy to conclude that the bound of this term d(k), which is denoted by dm, will also change because of this. Since dm has a direct influence on the ultimate bound, the ultimate bound will also vary accordingly. This observation laid to the work in this paper, which is described in detail in the following sections. In the remainder of the work, the terms ‘output’ and ‘sliding variable’ will be used interchangeably. This is because traditionally relative degree is defined for the output of a system, and sliding variable is a constructed output, if not a measured one. The next section introduces the idea of relative degree of an output of a discrete linear time invariant (LTI) system, which is the system considered in this work. The property of finite time convergence of all states in the sliding mode for relative degree two output is also shown in this section. Sections 3 and 4 discuss the conditions required to achieve reduction in the width of the ultimate bound for a non-switching and a switching reaching law respectively. Section 5 shows simulation examples for both the non-switching and switching reaching laws, comparing the results with selection of relative degree one output. The conclusions of the paper are collected in Section 6. Notations used in this paper: (1) jj jj is the standard Euclidean 2-norm of a vector or a matrix, (2) Ker(L) is the set of all vectors x A X which the linear transformation L : x↦y takes to y ¼ 0 A Y, X and Y being vector spaces, (3) det(A) denotes the determinant of the square matrix A.

2. Outputs with relative degree 1 and 2 Let us consider a discrete time LTI system x1 ðk þ 1Þ ¼ A11 x1 ðkÞ þ A12 x2 ðkÞ x2 ðk þ 1Þ ¼ A21 x1 ðkÞ þ A22 x2 ðkÞ þ B2 uðkÞ þ B2 f ðkÞ ðn  mÞ

ð1Þ

and x2 ðkÞ A R are the n states and uðkÞ A R is where x1 ðkÞ A R the input. The disturbance f ðkÞ A Rm is assumed to be bounded as j j f ðkÞj j r f m . The controls designed in the sequel guarantees bounded stability of the system in presence of such matched disturbance. The problem of unmatched disturbance has not been studied in this work. A12 A Rðn  mÞm , The above makes A11 A Rðn  mÞðn  mÞ , A21 A Rmðn  mÞ , A22 A Rmm and B2 A Rmm . We assume detðB2 Þ a 0. Obviously, written in the standard form of discrete LTI systems, we shall have " # A11 A12 A¼ A21 A22 m

m

" B¼

0

# ð2Þ

B2

The working definition of relative degree of an output for a discrete time system is given below. Definition 1. The relative degree r of an output s(k) of a discrete time system is the order of delay of the output in which the input u(k) first appears. Hence, if the relative degree is r, then sðk þ iÞ ¼ F i ðxðkÞÞ 8 i or and sðk þ iÞ ¼ F i ðxðkÞ; uðkÞ; uðk þ1Þ; …; uðk þ i rÞÞ 8 iZ r. 2.1. Asymptotic stability with relative degree 1 output For the above system, a relative degree one output is proposed as s1 ðkÞ ¼ C T1 xðkÞ ¼ Cx1 ðkÞ þ I m x2 ðkÞ

ð3Þ

mðn  mÞ

where C A R  C T1 B ¼ C

. Then " #  0 Im ¼ B2 B2

ð4Þ

and we can calculate the control u(k) from s1 ðk þ 1Þ ¼ C T1 AxðkÞ þ C T1 BuðkÞ þ C T1 Bf ðkÞ

ð5Þ

as obtained from the system dynamics, since B2 is non-singular. Design of C is done considering closed loop performance of the nominal system, i.e., when f ðkÞ ¼ 0. Then the output hits zero in finite time and for relative degree one systems, we get x2 ðkÞ ¼  Cx1 ðkÞ. Hence, the closed loop system becomes x1 ðk þ1Þ ¼ ðA11  A12 CÞx1 ðkÞ

ð6Þ

which is traditionally made asymptotically stable by choosing maxðj λ1 j Þ o 1

ð7Þ

where λ1 is an eigenvalue of ðA11  A12 CÞ. Since x2 ðkÞ is algebraically related to x1 ðkÞ, it also settles down to zero asymptotically. 2.2. Finite time stability with relative degree 2 output For the system (1), a relative degree two output will be s2 ðkÞ ¼ C T2 xðkÞ ¼ Cx1 ðkÞ

ð8Þ

where C A Rmðn  mÞ can be chosen same as in (3) or different, but satisfying the conditions in Theorem 1 below. Now " #   0 T C 0 C2 B ¼ ¼0 ð9Þ B2 implying s2 ðk þ 1Þ ¼ C T2 AxðkÞ þ C T2 BuðkÞ þ C T2 Bf ðkÞ ¼ C T2 AxðkÞ

ð10Þ

as calculated from the system dynamics (1) does not contain the control input u(k). Then we need to assume " #" # 0   A11 A12 C T2 AB ¼ C 0 A21 A22 B2 " #   0 ¼ CA11 CA12 B2 ¼ CA12 B2

ð11Þ

to be non-singular in order that the output is of relative degree 2,

Please cite this article as: Chakrabarty S, Bartoszewicz A. Improved robustness and performance of discrete time sliding mode control systems. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.08.006i

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3.2. Ultimate bound for relative degree two output

thus obtaining s2 ðk þ 2Þ ¼

3

C T2 A2 xðkÞ þ C T2 ABuðkÞ þC T2 ABf ðkÞ

ð12Þ

This now contains the control input u(k). Theorem 1. If KerðCÞ ¼ 0 and detðCA12 Þ a 0, then the output s2 ðkÞ with relative degree two as designed in (8) ensures finite time stability of the states of the system during sliding, in absence of the disturbance f(k). Proof. During sliding, s2 ðkÞ ¼ Cx1 ðkÞ ¼ 0. If KerðCÞ ¼ 0, it follows that x1 ðkÞ ¼ 0 during sliding. Now, we also have s2 ðk þ 1Þ ¼ CA11 x1 ðkÞ þ CA12 x2 ðkÞ ¼ 0 which implies x2 ðkÞ ¼  ðCA12 Þ  1 CA11 x1 ðkÞ during sliding. As x1 ðkÞ ¼ 0, it follows that x2 ðkÞ ¼ 0 as well, since CA12 is non-singular. Hence all the states become zero at the same time instant. □ Note that KerðCÞ ¼ 0 is only a sufficient condition and not a necessary one in order to achieve finite time stability. It is also clear that the above theorem points to an important achievement in the closed loop reduced order dynamics compared to the choice of relative degree one output. Of course, if there is a disturbance, then the finite time stability would be changed to finite time bounded stability, i.e., the states of the system will enter an ultimate bound in a finite time and stay there. Remark 1. In simulations, the parameter C is chosen the same for both relative degree one and two outputs for comparison purposes. However, selection of the parameter C for relative degree two output does not in any way require design of the same parameter for relative degree one output. The property of finite time stability is inherent to the relative degree two output systems when C is selected as per the conditions in Theorem 1, which are easy to satisfy.

It is already seen that s2 ðk þ 1Þ does not contain the input as well as the matched disturbance. Hence we find s2 ðk þ2Þ ¼ C T2 xðk þ 2Þ ¼ C T2 A2 xðkÞ þ C T2 ABuðkÞ þ C T2 ABf ðkÞ

ð17Þ

So we need to find out s2 ðk þ 2Þ from (13). We take the nominal part of the reaching law (without d(k)) and find out s2 ðk þ 2Þ. Then we add d2 ðkÞ to take care of the matched disturbance. This gives us s2 ðk þ2Þ ¼ sd ðk þ 2Þ þ d2 ðkÞ 8 n > < k  k sð0Þ for k o kn n sd ðkÞ ¼ k > n :0 for k Z k

ð18Þ

This requires d2 ðkÞ ¼ C T2 ABf ðkÞ in (18) so that the control     1 h i C T2 A2 xðkÞ  sd ðk þ 2Þ u2 ðkÞ ¼  C T2 AB

ð19Þ

becomes devoid of any uncertain terms. Therefore, we get the norm bound of d2 ðkÞ in this case to be d2m ¼ jj C T2 ABjj f m r jj CA12 jj jj B2 jj f m ¼ jj CA12 jj d1m which is then the ultimate bound

ð20Þ

δ2 as well.

Theorem 2. If in addition to the conditions as in Theorem 1, C also satisfies

σ max ðCA12 Þ o 1

ð21Þ

then the ultimate bound δ2 for the relative degree two output with reaching law (18) is lesser than the ultimate bound δ1 for the relative degree one output with reaching law (13), whether or not the parameter C is chosen same for both relative degree cases. Proof. The property is straightforward to see from (20).

□

3. Non-switching reaching law In [2], a reaching law for discrete time SMC systems has been introduced, which is

4. Switching reaching law

sðk þ 1Þ ¼ sd ðk þ 1Þ þ dðkÞ 8 n > k k n < sð0Þ for k o k n sd ðkÞ ¼ k > n :0 for k Z k

In [5], Gao et al. proposed a reaching law for discrete time SMC systems, which is of the form ð13Þ

and d(k) is an uncertainty derived from the system uncertainty f (k). It is evident that this reaching law brings the sliding variable s n (k) in finite time inside an ultimate bound dm 8 k Z k , where jj dðkÞjj rdm . It is beneficial for the understanding of the reader to denote the sliding variable s(k) and the uncertainty d(k) as s1 ðkÞ, d1 ðkÞ for relative degree one output and s2 ðkÞ, d2 ðkÞ for relative degree two output.

sðk þ 1Þ ¼ αsðkÞ  β sgnðsðkÞÞ þdðkÞ

ð22Þ

where α A ð0; 1Þ and β 4 dm are real constants, d(k) is the uncertainty derived from the system uncertainty f(k). At present, there are two ways to analyze Gao's reaching law, one provided in [21] and the other in [22]. In this paper, we shall work with the established and well-known analysis in [21]. As before, we shall denote the sliding variable s(k) and the uncertainty d(k) as s1 ðkÞ, d1 ðkÞ for relative degree one output and s2 ðkÞ, d2 ðkÞ for relative degree two output. 4.1. Ultimate bound for relative degree one output

3.1. Ultimate bound for relative degree one output It is easy to see that It is easy to see that s1 ðk þ 1Þ ¼

C T1 xðk þ1Þ ¼

s1 ðk þ1Þ ¼ C T1 xðk þ 1Þ ¼ C T1 AxðkÞ þ C T1 BuðkÞ þ C T1 Bf ðkÞ

C T1 AxðkÞ þ C T1 BuðkÞ þC T1 Bf ðkÞ

ð14Þ

This requires dðkÞ ¼ d1 ðkÞ ¼ C T1 Bf ðkÞ in (13) so that the control    1 h  i C T1 A xðkÞ  sd ðk þ 1Þ ð15Þ u1 ðkÞ ¼  C T1 B becomes devoid of any uncertain terms. Therefore, we get the norm bound of d1 ðkÞ in this case to be d1m ¼

jj C T1 Bjj f m

¼ jj B2 jj f m

which is then the ultimate bound

ð16Þ

δ1 as well.

ð23Þ

C T1 Bf ðkÞ

in (22) so that the control This requires dðkÞ ¼ d1 ðkÞ ¼    1 h   i C T1 A xðkÞ  αC T1 xðkÞ þ β 1 sgn C T1 xðkÞ ð24Þ u1 ðkÞ ¼  C T1 B becomes devoid of any uncertain terms. Therefore, we get the norm bound of d1 ðkÞ in this case to be d1m ¼ jj C T1 Bjj f m ¼ jj B2 jj f m which is the same as (16) in Section 3. Notice that we changed β to β1 to correspond to relative degree one. As we shall see in the

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4

sequel, for relative degree two, the conditions which guide the selection of β change. Hence the distinction is made. As per the analysis in [21] of the reaching law (22), we need ð1 þ αÞ β1 4 ð1  αÞd1m for crossing-recrossing s1 ðkÞ ¼ 0 at each successive step after crossing it for the first time. The ultimate bound is then calculated as

δ1 ¼ β1 þ d1m 4

2d1m 1α

Lemma 2. If β 2 4 1d2mα and sgnðs2 ðk þ 1ÞÞ ¼ sgnðs2 ðkÞÞ, sgnðs2 ðk þ2ÞÞ ¼ sgnðs2 ðkÞÞ.

then

Proof. With sgnðs2 ðk þ 1ÞÞ ¼  sgnðs2 ðkÞÞ, from (27) we get s2 ðk þ 2Þ ¼ α2 s2 ðkÞ  αβ 2 sgnðs2 ðkÞÞ  β 2 sgnðs2 ðk þ 1ÞÞ þ d2 ðkÞ ¼ α2 s2 ðkÞ  αβ 2 sgnðs2 ðkÞÞ þ β2 sgnðs2 ðkÞÞ þ d2 ðkÞ ¼ α2 s2 ðkÞ þ ð1  αÞβ2 sgnðs2 ðkÞÞ þ d2 ðkÞ

ð25Þ

ð31Þ

then for any j d2 ðkÞj o d2m , we get sgnðs2 ðk þ Since β 2ÞÞ ¼ sgnðs2 ðkÞÞ. □ d2m 2 4 1  α,

4.2. Ultimate bound for relative degree two output It is already seen that s2 ðk þ 1Þ does not contain the input, and hence we find s2 ðk þ 2Þ ¼ C T2 xðk þ 2Þ ¼ C T2 A2 xðkÞ þ C T2 ABuðkÞ þ C T2 ABf ðkÞ

ð26Þ

So we need to find out s2 ðk þ 2Þ from (22). We take the nominal part of the reaching law (without d(k)) and find out s2 ðk þ 2Þ. Then we add d2 ðkÞ to take care of the matched disturbance. This gives us s2 ðk þ 2Þ ¼ α2 s2 ðkÞ  αβ2 sgnðs2 ðkÞÞ  β2 sgnðs2 ðk þ 1ÞÞ þ d2 ðkÞ

ð27Þ

C T2 ABf ðkÞ

in (27) so that the control This requires d2 ðkÞ ¼ h    u2 ðkÞ ¼ ðC T2 ABÞ  1 C T2 A2 xðkÞ  α2 C T2 xðkÞ þ αβ2 sgn C T2 xðkÞ  i þ β2 sgn C T2 AxðkÞ

ð28Þ

becomes devoid of any uncertain terms. Therefore, we get the norm bound of d2 ðkÞ in this case to be d2m ¼ jj C T2 ABjj f m r jj CA12 jj jj B2 jj f m ¼ jj CA12 jj d1m which is same as (20) in Section 3. The task is now to determine the ultimate bound δ2 and the conditions on β2 that needs to be satisfied. These are evaluated by keeping in mind that the condition of crossing-recrossing about s2 ðkÞ ¼ 0 needs to be satisfied as per the assumption in [5]. For simplicity in understanding, we perform the analysis assuming s2 ðkÞ A R. For a higher dimensional output s2 ðkÞ, the same analysis holds for each of the elements of the vector. Let us consider the sliding variable s2 ðkÞ at two consecutive time instants. In other words, we take into account s2 ðkÞ and s2 ðk þ 1Þ, where k is any non-negative integer. Then one can either have sgnðs2 ðk þ 1ÞÞ ¼ sgnðs2 ðkÞÞ or sgnðs2 ðk þ1ÞÞ ¼  sgnðs2 ðkÞÞ. Lemma 1. If β 2 4 1dþ2mα and sgnðs2 ðk þ1ÞÞ ¼ sgnðs2 ðkÞÞ, then j s2 ðk þ 2Þj is strictly smaller than j s2 ðkÞj or crosses the hyperplane s2 ðkÞ ¼ 0. Proof. For sgnðs2 ðk þ 1ÞÞ ¼ sgnðs2 ðkÞÞ ¼ 1, from (27) we get s2 ðk þ 2Þ r α2 s2 ðkÞ  ð1 þ αÞβ2 þ d2m o s2 ðkÞ

ð29Þ

since β For sgnðs2 ðk þ 1ÞÞ ¼ sgnðs2 ðkÞÞ ¼  1, from (27) we get d2m 2 4 1 þ α.

s2 ðk þ 2Þ Z α2 s2 ðkÞ þ ð1 þ αÞβ2  d2m 4 s2 ðkÞ

ð30Þ

As k is an arbitrary non-negative integer, the above lemma implies that β2 4 1d2mα is necessary and sufficient condition for crossing re-crossing the sliding hyperplane s2 ðkÞ ¼ 0 at each successive step after crossing it for the first time. Furthermore, as the condition on β2 in Lemma 1 is always satisfied when the condition on β2 in Lemma 2 holds, we conclude that the latter is necessary and sufficient for generating the quasi-sliding mode in the sense of Gao et al. [5]. Indeed if β 2 4 1d2mα is satisfied, then the sliding variable crosses the sliding hyperplane in a finite time and then recrosses it again in every consecutive step. However, one may notice that the sequence fj sðkÞj g may not necessarily approach the sliding hyperplane monotonically, but the sequence of every alternate sample of fj sðkÞj g does. Ultimately the quasi-sliding mode is achieved when fsðkÞg starts crossing-recrossing the sliding hyperplane at each time step. We are now in a position to find out the ultimate bound δ2 for the sliding variable s2 ðkÞ, which is a measure of the robustness of the system concerned. The ultimate bound must be equal to the largest steady state value of the sliding variable for the maximum disturbance j d2 ðkÞj ¼ d2m . This is obtained from (27) putting s2 ðkÞ ¼ δ2 , which gives the largest value of s2 ðk þ 2Þ as

δ2 ¼ α2 δ2  αβ2 þ β2 þd2m

ð32Þ

This makes

δ2 ¼

ð1  αÞβ2 þ d2m 2d2m 4 ð1  α2 Þ ð1  α2 Þ

ð33Þ

since β 2 4 1d2mα. Theorem 3. If in addition to the conditions as in Theorem 1, C also satisfies

σ max ðCA12 Þ o 1 þ α

ð34Þ

then the ultimate bound δ2 for the relative degree two output with reaching law (27) is lesser than the ultimate bound δ1 for the relative degree one output with reaching law (22), whether or not the parameter C is chosen same for both relative degree cases. Proof. Let us consider a ρ 4 1. Then the inequalities in (25) and (33) can be written as equalities multiplying the RHS with this ρ. This gives us 2d1m ð1  αÞ 2d δ2 ¼ ρ 2m2 ð1  α Þ

From the above two inequalities, it is straightforward to conclude that j s2 ðk þ 2Þj o j s2 ðkÞj or sgnðs2 ðk þ 2ÞÞ ¼ sgnðs2 ðk þ 1ÞÞ ¼ □ sgnðs2 ðkÞÞ.

δ1 ¼ ρ

Geometrically, Lemma 1 can be interpreted as follows. If the states x(k) and xðk þ 1Þ are on the same side of the sliding hyperplane, then either xðk þ 2Þ is at the same side of the hyperplane and closer to it than x(k) or xðk þ 2Þ is on the other side of the hyperplane. As k is an arbitrary non-negative integer, the above lemma demonstrates that there exists such a finite k0 4 0 that for any i ok0 , we have sgn½s2 ðiÞ ¼ sgn½s2 ð0Þ and sgn½s2 ðk0 Þ ¼  sgn½s2 ð0Þ. In other words, the analysis shows that at some finite time instant k0, the sliding variable s2 ðkÞ will change its sign.

Hence taking into account d2m r jj CA12 jj d1m , we get

ð35Þ

δ2 2d2m =2d1m jj CA12 jj r ¼ ð1 þ αÞ ð1 þ αÞ δ1 Hence δ2 o δ1 if condition (34) is satisfied.

ð36Þ □

Here ρ is selected the same for both the ultimate bounds δ1 and δ2 . It can be considered as a selection parameter for δ1 which is kept same for the selection of δ2 , for fair comparison between the two ultimate bounds.

Please cite this article as: Chakrabarty S, Bartoszewicz A. Improved robustness and performance of discrete time sliding mode control systems. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.08.006i

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Fig. 1. Sliding variable for non-switching reaching law.

Fig. 2. State variables and control input for non-switching reaching law.

Please cite this article as: Chakrabarty S, Bartoszewicz A. Improved robustness and performance of discrete time sliding mode control systems. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.08.006i

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6

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Fig. 3. Sliding variable for switching reaching law.

Fig. 4. State variables and control input for switching reaching law.

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Remark 2. Compared to Theorem 2, the condition on C in Theorem 3 is more relaxed. Hence, with the switching reaching law (27), we can decrease the ultimate bound for relative degree two output with a less strict condition than required with the nonswitching reaching law (18).

5. Simulation example Simulation examples for a second order discrete LTI system with outputs of both relative degrees 1 and 2 are provided to compare performance. We consider an inherently unstable discrete-time second order system without any uncertainty and initial states ½  3 5T :     1 1:2 0 xðk þ1Þ ¼ xðkÞ þ ðuðkÞ þ f ðkÞÞ ð37Þ 3  0:5 1 where f(k) is a disturbance assuming value þ 0:1 for the first half of the simulation cycle and  0.1 for the remaining half. The disturbance is chosen at these extremities as it brings out the worst behavior of the system. The comparison between choices of relative degree one and two outputs is fair in such a scenario.

7

terms of improved robustness of the output against disturbances, where it is shown that the ultimate bound inside which the output converges in finite time can be made less by making the surface parameter satisfy some design conditions. It is also shown that for the selection of relative degree two output, we obtain a finite time stable reduced order system in absence of disturbance instead of asymptotically stable as in the relative degree one case. Hence, the overall performance of the discrete time sliding mode control system is improved by increasing the relative degree of the output.

Acknowledgment This work has been performed in the framework of the project ‘Optimal sliding mode control of time delay systems’ financed by the National Science Centre of Poland – decision number DEC 2011/01/ B/ST7/02582. The authors take this opportunity to thank The Foundation of Polish Science for extending support to the research as presented in this article under the grant Mistrz Program.

References 5.1. Non-switching reaching law n

Here, the reaching law of [2] with k ¼ 5 has been used for simulations. The surface parameter is selected as C ¼0.5, which satisfies the conditions required in Theorem 2. The ultimate bounds for the relative degree one and two outputs are obtained as δ1 ¼ 0:1 and δ2 ¼ 0:06 respectively. The plots of the output s(k) along with a zoomed view of it to show the ultimate bounds are given in Fig. 1. The plots of the state variables and the control input are given in Fig. 2. The plots corresponding to relative degree one output are shown with a dotted line whereas those with relative degree two output are shown with a smooth line. It can be seen from Fig. 2 that with relative degree two output, we have reduced the state errors when compared to that for relative degree one output. Also the control effort is lesser for the relative degree two case. 5.2. Switching reaching law Here, the reaching law of [5] has been used for simulations. The surface parameter is selected as C ¼0.9, which satisfies the conditions required in Theorem 3 with α ¼ 0:4. We select ρ ¼ 1:01 which gives the ultimate bounds as δ1 ¼ 0:3367 and δ2 ¼ 0:2597. For these values of the ultimate bound, β 1 ¼ 0:2367 and β2 ¼ 0:1836 are calculated. The plots of the output s(k) along with a zoomed view of it to show the ultimate bounds are given in Fig. 3. The plots of the state variables and the control input are given in Fig. 4. The plots corresponding to relative degree one output are shown with a dotted line whereas those with relative degree two output are shown with a smooth line. It can be seen from Fig. 4 that with relative degree two output, we have reduced the state errors when compared to that for relative degree one output. Also the control effort is lesser for the relative degree two case.

6. Conclusions In this paper, the authors have shown that choosing an output (sliding variable) of relative degree two instead of one, which is traditionally chosen, one gets a better performance of the discrete time sliding mode control system. The performance is better in

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Please cite this article as: Chakrabarty S, Bartoszewicz A. Improved robustness and performance of discrete time sliding mode control systems. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.08.006i