Discrete time sliding mode controllers with relative degree one and two switching variables

Accepted Manuscript

Discrete Time Sliding Mode Controllers with Relative Degree One and Two Switching Variables Paweł Latosinski, Andrzej Bartoszewicz ´ PII: DOI: Reference:

S0016-0032(18)30478-2 https://doi.org/10.1016/j.jfranklin.2018.07.006 FI 3549

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

13 February 2018 21 May 2018 12 July 2018

Please cite this article as: Paweł Latosinski, Andrzej Bartoszewicz, Discrete Time Sliding Mode Con´ trollers with Relative Degree One and Two Switching Variables, Journal of the Franklin Institute (2018), doi: https://doi.org/10.1016/j.jfranklin.2018.07.006

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Discrete Time Sliding Mode Controllers with Relative Degree One and Two Switching Variables

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Pawel Latosi´ nski, Andrzej Bartoszewicz

Institute of Automatic Control, L´ od´z University of Technology 18/22 B. Stefanowskiego St., 90-924 L´ od´z, Poland

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Abstract

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In this paper, a new reaching law based sliding mode control strategy for discrete time systems is introduced. Contrary to most existing approaches, the new strategy uses a sliding variable with relative degree two. It is demonstrated that the new reaching law drives the sliding variable to a narrower quasi-sliding mode band than its relative degree one equivalent, while simultaneously ensuring the desired dynamic properties of the system. Furthermore, it is shown that the smaller quasi-sliding mode band width is reflected in reduced magnitude of all state variables in the sliding mode.

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Keywords: Sliding mode control, discrete-time systems, reaching law approach

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1. Introduction

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Continuous time sliding mode controllers [1–3] are well known to be computationally efficient and robust with respect to matched disturbance and parameter uncertainties [4]. These attractive properties have earned them considerable popularity in the control engineering community [5–8] and led to many further advances in the field. The most significant development in the context of this paper is the introduction of discrete time sliding mode controllers [9, 10]. The fact that such controllers only calculate a new value of the control signal at fixed intervals makes it impossible to attain ideal sliding motion and thus reject matched disturbance completely. However, these strategies still provide remarkable robustness by confining the system representative point to a vicinity of the sliding surface. Various authors have

Preprint submitted to Journal of the Franklin Institute

August 3, 2018

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proposed new discrete time sliding mode control strategies [11–17] with the aim of improving the dynamic response of the system. In this paper, a discrete time sliding mode control strategy will be developed by means of reaching law approach. Although this approach was first proposed for continuous time systems [7], its discrete time equivalent [12] has gained a significantly greater popularity in the control engineering community. In the reaching law approach, an a priori defined evolution of the sliding variable is applied to determine the control signal. Several authors have proposed new discrete time reaching laws [18–25] improving upon the original formula. Furthermore, some works without the classic assumption about matched disturbance have been published [26–30]. Most reaching law based strategies are designed with relative degree one sliding variables in mind. Typically, sliding variables with arbitrary relative degree are only applied in continuous time sliding mode control [31–33]. The use of such variables in the context of discrete time systems is not common, but a few recent works have considered arbitrary relative degree for discrete reaching laws [34–40]. It has been demonstrated that such an approach can lead to reduced state error in the sliding phase without sacrificing other favorable properties of the system. In our work, sliding mode controller design for discrete time systems will be considered. First, a reaching law based strategy with relative degree one sliding variable will be referred to and its properties will be briefly described. Then, the strategy will be extended to the case of relative degree two variables. It will be demonstrated that increasing the relative degree of the variable relaxes the restrictions placed on the design parameters. It will be further proven that such an approach allows one to obtain a smaller quasi-sliding mode band width and reduce state error in the sliding mode. The remainder of the paper is organized as follows. In section II, the considered class of systems is described and a reaching law based strategy for relative degree one variables is presented. In the same section, properties of the strategy are referred to and an additional property not present in existing literature is proven. In section III, relative degree two sliding variables are introduced and a modified reaching law for such variables is designed. Further in that section, properties of the proposed strategy are investigated and its advantages over the classic relative degree one approach are highlighted. These advantages are verified in Section IV by means of a simulation example. Finally, Section V gives concluding remarks.

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2. Relative degree one sliding variables In this paper, we consider a class of discrete time objects represented in the state space by the following equation (1)

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x(k + 1) = Ax(k) + b {u(k) + a ˜[x(k)] + d(k)} .

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Vector x ∈ Rn in this equation is the system state, A ∈ Rn×n is the state matrix, b ∈ Rn is the input distribution vector, u is the scalar control signal, d is the scalar disturbance and a ˜ is a nonlinear function representing modeling uncertainty. The disturbance and model uncertainties satisfy the so-called matching conditions, which means they affect the plant through the same channel as the control signal. For any k, they are also bounded in the following way dmin ≤ a ˜[x(k)] + d(k) ≤ dmax .

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Assumptions about matched and bounded perturbations are typical for robust control and enable several valuable properties of the system, which will be demonstrated later in this paper. Before presenting a reaching law based control strategy for the class of plants (1), we will formally define the relative degree of any constructed output of a discrete time system.

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Def. Constructed output (or variable) s is said to have relative degree r with reference to input u if and only if s(k + i) = f [x(k)] for any 0 ≤ i < r and s(k + r) = f [x(k), u(k)].

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In other words, a relative degree r output is only affected by the control signal from r time instants ago. The strategy proposed in this section will use a typical relative degree one sliding variable, which has the following form s1 (k) = cT 1 x(k),

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where c1 ∈ Rn is a vector selected to ensure stability of the system and ”1” subscripts denote the selected relative degree. The relative degree equal to one implies that the control signal affects the variable at the subsequent time instant, which means cT 1 b 6= 0 must be satisfied. Furthermore, in order to guarantee stability of the discrete time plant, c1 must be chosen in a way that places all poles of the closed loop system inside the unit circle. In this paper, vector c1 ensuring that all poles are placed at zero will be used. This is achieved by solving h

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−1 T n det In z − A + b(cT 1 b) c1 A = z

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d1 =

cT 1b (dmax + dmin ), 2

δ1 =

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for elements of vector c1 . Such a choice of this vector guarantees that the closed loop system state matrix is nilpotent. This in turn ensures that the effect of past disturbance and model uncertainty on the system state is always reduced to zero in finite time. Before presenting the reaching law based strategy, let us first define the mean effect of perturbations on the sliding variable and its maximum deviation from the mean as cT 1b (dmax − dmin ). 2

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In the next subsection, the discrete time reaching law for the relative degree one sliding variable (3) will be given.

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2.1. Reaching law based strategy In the reaching law approach, the evolution of the sliding variable is specified a priori and applied to design the controller. Since relative degree one variables are affected by the control signal from the previous time instant, the reaching law must define the evolution one step in advance. The reaching law used in this paper, originally introduced in [39], has the following form s1 (k + 1) =f [s1 (k)]s1 (k) − αsgn[s1 (k)] + cT a[x(k)] + cT 1 b˜ 1 bd(k) − d1 ,

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where d1 is defined by (5), function

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)

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|·| f (·) = min 1, β

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and α, β are positive design parameters. According to the conventional reaching law-based controller design procedure [12], the control signal obtained from reaching law (6) has the following form n

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−1 u(k) =(cT −cT 1 Ax(k) + f [s1 (k)]s1 (k) − αsgn[s1 (k)] − d1 . 1 b)

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It has already been demonstrated in [39] that reaching law (6) ensures switching type quasi-sliding motion of the system. Indeed, this reaching law guarantees an upper bounded convergence rate of the sliding variable to the vicinity of the switching surface and confines the system representative point to the vicinity in finite time. Furthermore, it ensures that the switching plane is crossed in every step in the sliding phase. These properties, which have been proven in [39], will be formally stated in the following three theorems. 4

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Theorem 1. If the reaching law for system (1) is chosen according to (6) with α > δ1 , then for any initial state the representative point will move towards the sliding surface and either arrive on it or cross it in finite time.

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Theorem 2. If the reaching law for system (1) is chosen according to (6) with α > δ1 and (α + δ1 )2 , α − δ1

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then the system representative point will keep crossing the sliding surface in every step after crossing it for the first time.

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Theorem 3. If the reaching law for system (1) is chosen according to (6) with α > δ1 and β selected as in (9), then after crossing the sliding surface for the first time, the state is confined to the band x : |cT 1 x| ≤ α + δ1

around the surface for all future time instants.

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In conclusion, the proposed reaching law ensures a finite time reaching phase and switching type motion in the sliding phase. These properties related to the evolution of the sliding variable have been proven in [39]. However, in this paper a new property not known previously will be stated and proven. This property will describe the maximum admissible state error in the sliding phase.

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2.2. State error estimation In this subsection, it will be demonstrated that the reaching law based strategy (6) ensures uniform ultimate boundedness of all state variables. This will be accomplished by proving that the absolute value of each variable is upper bounded in each step after a finite number of initial time instants. To that end, the following theorem will be formulated and proven. Theorem 4. If the reaching law for system (1) is chosen according to (6) with α > δ1 and β selected as in (9), then there exists a finite k1 such that for every k ≥ k1 the following inequality is satisfied

h n−1 ii X T −1 T p A − b(c b) c A b , 1 1 j

−1 |xj (k)| ≤ (cT 1 b) (α + δ1 ) ·

i=0

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for all state variables xj (j = 1, . . . , n), where pj = [0| .{z . . 0} 1 0| .{z . . 0}]. j−1

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Proof. Let k0 be the first sampling instant for which the system representative point has crossed the switching plane. Then, Theorem 3 implies that |s1 (k)| ≤ α + δ1 for all k ≥ k0 . Substituting the control signal (8) into (1), we get −1 x(k + 1) = Ax(k) + b˜ a[x(k)] + bd(k) + b(cT 1 b)

· {f [s1 (k)]s1 (k) − αsgn[s1 (k)] − d1 − cT 1 Ax(k)}

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−1 T T −1 T = A − b(cT a[x(k)] 1 b) c1 A x(k) + b(c1 b) {c1 b˜

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+ cT 1 bd(k) + f [s1 (k)]s1 (k) − αsgn[s1 (k)] − d1 }.

Furthermore, substitution of (6) into (13) yields h

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−1 T T −1 x(k + 1) = A − b(cT 1 b) c1 A x(k) + b(c1 b) s1 (k + 1).

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Denoting

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−1 T P1 = A − b(cT 1 b) c1 A ,

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we can express x(k) as

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−1 x(k) =P1 x(k − 1) + b(cT 1 b) s1 (k)

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=P1n x(k − n) +

n−1 X i=0

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−1 P1i b(cT 1 b) s1 (k − i)

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Relations (4) and (15) imply that P1n = 0. Consequently, since each i−th |s1 (k − i)| is bounded by α + δ1 for all k ≥ k0 + n, the absolute value of the j-th element of the state vector can now be expressed as

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|xj (k)| = |pj x(k)| = ≤

n−1 X i=0

T −1 (c1 b) s1 (k

n−1 X −1 pj P1i b(cT 1 b) s1 (k i=0

− i) · pj P1i b

≤

− i)

n−1 T −1 X (c1 b) (α + δ1 ) pj P1i b ,

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i=0

where pj is the vector specified in (12). The obtained upper bound, which holds for all k ≥ k1 = k0 + n, is consistent with the one given in relation (11). 6

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In conclusion, it has been demonstrated that in n time instants after the sliding variable crosses the switching plane for the first time, each state variable becomes upper bounded and it remains limited for all future steps. This demonstrates that constraints are imposed on all state variables. In the next section, relative degree two sliding variables will be considered and a new reaching law for such variables will be designed. It will be demonstrated that the new approach ensures a narrower quasi-sliding mode band and smaller state error compared to the relative degree one strategy (6). 3. Relative degree two sliding variables

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In this section, a sliding mode control strategy for sliding variables with relative degree two will be designed. For discrete time plants, such variables are only affected by control signal and disturbance from two time instants ago. A new reaching law taking relative degree two variables into account will be designed and its properties will be investigated. The proposed reaching law will be significantly different than the one proposed in [39] and will employ a modified design procedure. It will be demonstrated that, compared to (6), the new strategy reduces state error by relaxing restrictions on the choice of design parameters. The considered relative degree two sliding variable has the following form s2 (k) = cT 2 x(k),

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T where c2 is such a vector that cT 2 b = 0 and c2 Ab 6= 0. One can easily notice that the value of this variable at time k +1 can be explicitly calculated taking into account only the measured state vector x(k). Indeed, substituting the state equation (1) into relation (18), one obtains

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T s2 (k + 1) = cT ˜[x(k)] + d(k)} = cT 2 Ax(k) + c2 b {u(k) + a 2 Ax(k).

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In other words, the exact value of a relative degree two sliding variable can be obtained one step in advance. This allows one to use additional information about the evolution of this variable to design a sliding mode control strategy. Just like in the case of relative degree one variables, in this paper vector c2 will be selected to ensure that all poles of the closed loop system are placed in zero. In other words, it must satisfy h

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−1 T 2 det In z − A + b(cT = zn. 2 Ab) c2 A

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Such a choice of vector c2 guarantees that the closed loop system state matrix is nilpotent, which negates the effect of past disturbance and model uncertainty on the system state in finite time. A relative degree two variable is affected by disturbance and model uncertainties differently than the variable discussed in the previous section. For the considered variable, we define the mean effect of perturbations and its maximum admissible deviation from the mean as cT Ab δ2 = 2 (dmax − dmin ). 2

cT Ab (dmax + dmin ), d2 = 2 2

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Remark 1. Vectors c1 for relative degree one variables and c2 for relative degree two ones have both been selected to ensure a dead-beat response of the closed loop system. In other words, they both aim at driving the state vector of a nominal system to zero in exactly n = dim(A) steps. However, only one such strategy can exist for a given plant. Consequently, the closed loop system matrices for both cases are equal to each other, i.e. they satisfy T T −1 T 2 T −1 T A − b(cT 1 b) c1 A = A − b(c2 Ab) c2 A . This in turn implies c1 = γc2 A for an arbitrary γ 6= 0. If the vectors are scaled to ensure γ = 1, then one T gets cT 1 b = c2 Ab and relations (5) and (21) imply d1 = d2 and δ1 = δ2 .

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A new reaching law for the case of relative degree two variables will now be presented and proven to ensure better robustness of the system than its relative degree one equivalent.

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3.1. Reaching law for relative degree two sliding variables In this section a new reaching law for sliding variables with relative degree two will be proposed. Contrary to the method used in [39], the new method will be explicitly based on the relative degree one reaching law (6). Since relative degree two sliding variables are only affected by the control signal from two time instants ago, the reaching law must specify its evolution two steps in advance. The reaching law proposed in this paper for such variables is obtained by substituting the left hand side of (6) into its right hand side and can be expressed as s2 (k + 2) = f [s2 (k + 1)]f [s2 (k)]s2 (k) − αsgn[s2 (k + 1)]

− f [s2 (k + 1)]αsgn[s2 (k)] + cT a[x(k)] + cT 2 Ab˜ 2 Abd(k) − d2 ,

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where f is the same function as in (7) and α, β are positive design parameters. The proposed reaching law aims to achieve similar properties of the system 8

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as its relative degree one equivalent. Namely, it is designed to drive the representative point to the vicinity of the sliding surface in finite time while ensuring a limited sliding variable convergence rate. Just like in the case of relative degree one variables, the control signal is determined by substituting known system parameters into the left hand side of (22) and solving the obtained equation for u(k). This gives n

−1 2 u(k) =(cT − cT 2 Ab) 2 A x(k) + f [s2 (k + 1)]f [s2 (k)]s2 (k)

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− αsgn[s2 (k + 1)] − f [s2 (k + 1)]αsgn[s2 (k)] − d2 .

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It will now be demonstrated that the proposed reaching law ensures a finite time reaching phase, guarantees switching type motion in the sliding phase and confines the system representative point to a layer around the sliding surface. To that end, the following three theorems will be proven.

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Theorem 5. If the reaching law for system (1) is chosen according to (22) with α > δ2 , then for any initial state the representative point will move towards the sliding surface and either arrive on it or cross it in finite time. Proof. Let s2 (k) > 0 and s2 (k + 1) > 0. Since function f assumes values from 0 to 1 and (21) implies |cT a[x(k)] + cT 2 Ab˜ 2 Abd(k) − d2 | ≤ δ2 , (22) gives

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s2 (k + 2) =f [s2 (k + 1)]f [s2 (k)]s2 (k) − α{1 + f [s2 (k + 1)]}

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+ cT a[x(k)] + cT 2 Ab˜ 2 Abd(k) − d2 ≤s2 (k) − α + δ2 < s2 (k).

Likewise, let s2 (k) < 0 and s2 (k + 1) < 0. Then, reaching law (22) yields

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s2 (k + 2) =f [s2 (k + 1)]f [s2 (k)]s2 (k) + α{1 + f [s2 (k + 1)]} + cT a[x(k)]x + cT 2 Ab˜ 2 Abd(k) − d2 ≥s2 (k) + α − δ2 > s2 (k).

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Since α − δ2 is a strictly positive constant, relations (24) and (25) imply that for any initial conditions of the system, the representative point is guaranteed to arrive on the sliding surface or cross it in a finite number of steps. Next, it will be demonstrated that when the sliding hyperplane is crossed, it will be crossed again in each subsequent step and the system representative 9

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point will be confined to a specified band around the hyperplane. Therefore, the proposed strategy ensures switching type motion in the sliding phase. However, before these properties are demonstrated, an important fact will be stated in the following theorem. Theorem 6. If the reaching law for system (1) is chosen according to (22) with α > δ2 and β>

α(α + δ2 ) , α − δ2

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then for any initial condition of the system there exists a finite k1 such that |s2 (k1 )| ≤ α + δ2 and sgn[s2 (k1 + 1)] = −sgn[s2 (k1 )].

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Proof. The proof will be conducted for such an initial state that s(0) > 0, since the analysis for s2 (0) < 0 is almost identical. From Theorem 5 we know that there exists a finite k0 such that s2 (k0 ) > 0 and s2 (k0 + 1) < 0. Furthermore, since k0 is the first time instant for which the sliding hyperplane has been crossed, we have s(k0 − 1) > 0. Then, relations (21) and (22) imply that the maximum overshoot of s2 (k0 + 1) satisfies the inequality s2 (k0 + 1) =f [s2 (k0 )]f [s2 (k0 − 1)]s2 (k0 − 1) − α − αf [s2 (k0 )]

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+ cT a[x(k0 − 1)] + cT 2 Ab˜ 2 Abd(k0 − 1) − d2 ≥ − αf [s2 (k0 )] − α − δ2 .

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It will now be demonstrated that the sliding hyperplane is crossed again after being crossed for the first time. Relations (21) and (22) give

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s2 (k0 + 2) =f [s2 (k0 + 1)]f [s2 (k0 )]s2 (k0 ) + α − αf [s2 (k0 + 1)] + cT a[x(k0 )] + cT 2 Ab˜ 2 Abd(k0 ) − d2 ≥f [s2 (k0 + 1)]{f [s2 (k0 )]s2 (k0 ) − α} + α − δ2 .

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Since f [s2 (k0 )]s2 (k0 ) is an increasing function for s2 (k0 ) ≥ 0, it must only be proven that s2 (k0 + 2) is positive when s2 (k0 ) = 0. For such a value of s2 (k0 ), the maximum overshoot of s2 (k0 + 1) described in relation (27) equals −α − δ2 . Taking that fact into account, along with relations (7) and (26), inequality (28) further implies s2 (k0 + 2) > − α

(α + δ2 )(α − δ2 ) α + δ2 + α − δ2 > −α + α − δ2 = 0. (29) β α(α + δ2 ) 10

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Thus, the sliding hyperplane has been crossed again after being crossed for the first time. Next it will be shown that the property described in the theorem is always true in finite time. If s2 (k0 ) ≤ α + δ2 then the theorem is already proven for k1 = k0 . On the other hand, for s2 (k0 ) > α + δ2 it will be demonstrated that s2 (k0 + 2) is strictly smaller than s2 (k0 ). Indeed, since function f is upper bounded by 1, reaching law (22) gives s2 (k0 + 2) =f [s2 (k0 + 1)]{f [s2 (k0 )]s2 (k0 ) − α} + α

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+ cT a[x(k0 )] + cT 2 Ab˜ 2 Abd(k0 ) − d2 ≤f [s2 (k0 + 1)]{f [s2 (k0 )]s2 (k0 ) − α} + α + δ2 =s2 (k0 ) − s2 (k0 ){1 − f2 [s2 (k0 + 1)]f [s2 (k0 )]} − αf [s2 (k0 + 1)] + α + δ2 .

(30)

Since s2 (k0 ) > α + δ2 , from inequality (30) one gets

s2 (k0 + 2)
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(α + δ2 )2 =s2 (k0 ) − f [s2 (k0 + 1)] α − β

#

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Taking relation (26) into account, one further obtains "

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(α + δ2 )(α − δ2 ) s2 (k0 + 2)
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Thus, after crossing the sliding hyperplane again, the sliding variable will be strictly smaller than s2 (k0 ). Consequently, there exists a finite k1 for which s2 (k1 − 1) < 0 and 0 < s2 (k1 ) < α + δ2 . Then, relation (22) gives

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s2 (k1 + 1) ≤f [s2 (k1 )]f [s2 (k1 − 1)]s2 (k1 − 1) + f [s2 (k1 )]α − α + δ2 α + δ2 α − δ2 < α − α + δ2 < α − α + δ2 = 0, β α

(33)

which means that s2 (k1 + 1) is negative. The theorem is therefore true at time k1 . 11

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Next it will be shown that the proposed reaching law for relative degree two sliding variables confines the system representative point to a band around the switching plane in the sliding mode. To that end, the following theorem will be formulated and proven. Theorem 7. If the reaching law for system (1) is chosen according to (22) with α > δ2 and β selected as in (26), the sliding variable satisfies |s2 (k)| ≤ α + δ2 and sgn[s2 (k + 1)] = −sgn[s2 (k)], then the system representative point will remain confined to the quasi-sliding mode band n

x : |cT 2 x| ≤ α + δ2

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around the sliding plane at time k + 2 and will cross the sliding hyperplane between time instants k + 1 and k + 2.

Proof. Let 0 < s2 (k) ≤ α + δ2 and s2 (k + 1) < 0. Relation (29) implies that s2 (k + 2) > 0, which means that the sliding hyperplane will be crossed in the next step. Thus, in order to satisfy (34), only s2 (k + 2) ≤ α + δ2 needs to be proven. Reaching law (22) gives

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s2 (k + 2) =f [s2 (k + 1)]f [s2 (k)]s2 (k) + α − f [s2 (k + 1)]α + cT a[x(k)] + cT 2 Ab˜ 2 Abd(k) − d2 .

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a[x(k)] + cT Since |cT 2 Abd(k) − d2 | ≤ δ2 , relation (35) implies 2 Ab˜ s2 (k + 2) ≤f [s2 (k + 1)] {f [s2 (k)]s2 (k) − α} + α + δ2 .

(36)

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It will now be demonstrated that expression f [s2 (k)]s2 (k) − α is negative. Indeed, since s2 (k) ≤ α + δ2 , from relation (7) one obtains f [s2 (k)]s2 (k) − α ≤

(α + δ2 )2 − α. β

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Then, substitution of the known parameter restriction (26) into (37) yields f [s2 (k)]s2 (k) − α <

δ2 α2 − δ22 − α = − 2 < 0. α α

(38)

Furthermore, substitution of (38) into (36) gives δ2 s2 (k + 2) ≤ f [s2 (k + 1)] − 2 α 12

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+ α + δ2 < α + δ2 .

(39)

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Likewise, repeating (35)−(39) for the case of 0 > s2 (k) ≥ α + δ2 and s2 (k + 1) > 0, one obtains 0 > s2 (k + 2) > −α − δ2 . In conclusion, for any values of the sliding variable satisfying assumptions of the theorem, the representative point at time k + 2 will be confined to the band (34).

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Theorem 7 implies that once conditions |s2 (k0 )| ≤ α + δ2 , |s2 (k0 + 1)| ≤ α + δ2 and sgn[s2 (k0 )] = −sgn[s2 (k0 + 1)] are satisfied for a certain k0 , then the system representative point will belong to the band (34) for all future steps and will continue switching type quasi-sliding motion around the sliding surface. The existence of such a state is guaranteed by Theorem 6 since, as seen in relation (32), the sliding variable always decreases as long as it exceeds α + δ2 . Furthermore, assumption (26) related to design parameters in those theorems is more lenient than its equivalent (9) in Theorems 2 and 3 referring to the case of relative degree one sliding variable. Indeed, solving relations (9) and (26) for α, one obtains the lower bounds of this parameter with respect to β, which can be seen in Figure 1. Values of both design parameters are scaled by δi for i = 1, 2. The blue line represents the lower bound for relative degree one strategy (6), while the red line shows the bound for the proposed relative degree two approach (22). It can be clearly seen that the lower bound placed on parameter α is significantly lower for the case of relative degree two variable depicted by the red line. This is an advantage of strategies for relative degree two variables as the quasi-sliding mode band width for both relative degree one and two strategies is a linear function of the design parameter α, which means it is smaller for the case of relative degree two variables. In the next section it will be demonstrated that achieving a smaller quasi-sliding mode band width results in reduced state error.

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3.2. State error for relative degree two variables It will now be demonstrated that the proposed reaching law (22) for relative degree two sliding variables guarantees uniform ultimate boundedness of each state variable. The obtained bound will be proportional to the quasisliding mode band width and as a result, it will prove to be narrower than the one obtained in Theorem 4 for relative degree one variables. The following theorem will demonstrate this property. Theorem 8. If the reaching law for system (1) is chosen according to (22) with α > δ2 and β is selected as in (26), then there exists a finite k2 such

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Figure 1: Lower bound of α in relation to β

that for every k ≥ k2 the following inequality |xj (k)|

h n−1 i T X −1 T −1 T 2 i ≤ (c2 Ab) (α + δ2 ) · pj A − b(c2 Ab) c2 A b .

(40)

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is satisfied for all j = 1, . . . , n, where pj is defined by (12).

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Proof. The analysis is similar to the proof of Theorem 4. Let k0 be the first time instant for which assumptions of Theorem 7 are satisfied. Substitution of the control signal (23) into (1) gives −1 x(k + 1) =Ax(k) + b˜ a[x(k)] + bd(k) + b(cT 2 Ab)

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2 · {−cT 2 A x(k) + f [s2 (k + 1)]f [s2 (k)]s2 (k) − αsgn[s2 (k + 1)] − f [s2 (k + 1)]αsgn[s2 (k)] − d2 }

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−1 T 2 T −1 = A − b(cT 2 Ab) c2 A x(k) + b(c2 Ab)

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· {cT a[x(k)] + cT 2 Ab˜ 2 Abd(k) + f [s2 (k + 1)]f [s2 (k)]s2 (k) − αsgn[s2 (k + 1)] − f [s2 (k + 1)]αsgn[s2 (k)] − d2 }.

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Then, the following notation is introduced h

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−1 T 2 P2 = A − b(cT 2 Ab) c2 A .

Substitution of (22) and (42) into (41) yields

−1 x(k + 1) =P2 x(k) + b(cT 2 Ab) s2 (k + 2).

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(43)

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−1 x(k) =P2 x(k − 1) + b(cT 2 Ab) s2 (k + 1)

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Since vector c2 is selected to place all poles of the closed loop system in zero, relation (42) implies P2n = 0. Taking that fact and relation (43) into account, one can express vector x(k) as (44)

−1 P2i b(cT 2 Ab) s2 (k − i + 1).

Considering relation (44), we conclude that for all k ≥ k2 = k0 + n the j-th state variable is bounded in the following way

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|xj (k)| =|pj x(k)| ≤

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where pj is defined as in (12). The obtained bound is consistent with the one presented in (40).

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It has been demonstrated that the state error is always bounded in n steps after conditions of Theorem 7 are satisfied and will remain bounded for all time instants afterwards. Furthermore, if vectors c1 and c2 are selected ex−1 T T −1 T 2 actly as stated in Remark 1, then A − b(cT 1 b) c1 A = A − b(c2 Ab) c2 A , T T c1 b = c2 Ab and δ1 = δ2 . Consequently, the obtained bound becomes identical to the one given in Theorem 4 for sliding variables with relative degree one. Therefore, due to the relaxed parameter selection for the case of relative degree two variables, relation (40) determines smaller range of xj than relation (11).

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Remark 2. Although the upper bounds placed on each state variable in Theorems 4 and 8 are always true, the obtained bounds (11) and (40) can prove to be too conservative in certain special cases. Consider the case where all elements pj P1i b for i = 1, . . . , n have the same sign. Then, since the sliding variable changes its sign in each step in the sliding phase, only every second element of the sum (17) contributes towards the upper bound. Consequently, the bound becomes |xj (k)| ≤

  n  X T −1 (c2 Ab) (α + δ2 ) · max pj P2i b ,   i=0 i: even

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4. Simulation results

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Properties of the proposed strategy will now be verified by means of a simulation example. Reaching law based strategies (8) and (23) will be applied to the system (1) with 

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20 1 1 1/2 1/6       A =  0 1 1  , b =  1/2  , x(0) =  0  0 0 1 1 0

(47)

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and their performance will be compared. The plant is subject to disturbance d(k) = (−1)b(k/15)+1c with dmax = −dmin = 1. First, vectors c1 and c2 for both strategies are selected according to (4) and (20), respectively. This gives cT 1 =[1 1 1/3],

cT 2 = [1 0

− 1/6].

(48)

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Relations(5), (21) and (48) give d1 = d2 = 0 and δ1 = δ2 = 1. Parameters of reaching law (6) are selected to satisfy the assumptions of Theorems 1-4. To ensure a good balance between convergence rate and quasi-sliding mode band width, we choose α = 2.01 and β = 9. Likewise, parameters of reaching law (22) must satisfy assumptions of Theorems 5-8. Parameter β is once again set to 9. Then, the smallest possible value of α is selected, which equals 1.36. Results of the comparison are illustrated in the following three figures. Figure 2 shows the evolution of sliding variables s1 (k) and s2 (k), Figure 3 illustrates the control signal for both cases and Figure 4 depicts the first state variable. In all three figures the solid red line corresponds to the novel relative degree two reaching law (22) proposed in this paper, while the dashed blue line represents the relative degree one strategy (6). It can be seen from Figure 2 that the strategy for relative degree two sliding variables results in a narrower quasi-sliding mode band than the classic relative degree one approach. The relative degree one variable is limited by ±3.01 in the sliding mode, as depicted by the blue dotted lines. Meanwhile, the relative degree two variable is bounded by ±2.36, as depicted by red dotted lines. These results are consistent with relations (10) and (34), respectively. Figure 3 demonstrates that the relative degree two strategy requires less control effort in the sliding mode than the classic approach. Finally, Figure 4 illustrates that the system output x1 assumes strictly smaller values for the relative degree two approach than for the relative degree one 16

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Figure 3: Control signals

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strategy. Indeed, in the relative degree one case it is limited by ±2.0067 (blue dotted lines) and for the relative degree two variable it is bounded by 1.5733 (red dotted lines). These values are compliant with the comments presented in Remark 2. 5. Conclusions

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In this paper, sliding variables with relative degree higher than one have been considered. An existing reaching law based sliding mode control strategy has been extended to the case of relative degree two variables. It has been demonstrated that both strategies ensure similar desirable properties of the system. Namely, they ensure a finite time reaching phase, switching type motion in the sliding phase and they confine the representative point to a specific quasi-sliding mode band. However, since design parameter restrictions are relaxed in the case of relative degree two variables, the newly proposed strategy allows one to achieve a narrower quasi-sliding mode band than the classic relative degree one approach. It has been further proven that reduced QSMB width for the case of relative degree two variables is reflected in lower state error in comparison to the the existing method. References

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[1] S. V. Emelyanov, Variable Structure Control Systems, Nauka, Moscow, 1967.

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[2] Y. Itkis, Control Systems of Variable Structure, Nauka, Moscow, 1976.

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[3] V. Utkin, Variable structure systems with sliding modes, IEEE Transactions on Automatic Control 22 (2) (1977) 212–222. [4] B. Draˇzenovi´c, The invariance conditions in variable structure systems, Automatica 5 (3) (1969) 287–295.

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