Output Tracking with Discrete-Time Integral Sliding Mode Control

Output Tracking with Discrete-Time Integral Sliding Mode Control Xu Jian-Xin and Khalid Abidi Department of Electrical and Computer Engineering, National University of Singapore, Engineering Drive 3, Singapore 117576 {elexujx,kabidi}@nus.edu.sg 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 information is required in the controller design which is a drawback since in most practical applications only the output measurement is available To solve this problem, focus was placed on output feedback based sliding mode control [1][6] Two approaches arose: a design based on observers to construct the missing states, [3],[4], the other design focused on using only the output measurement, [1],[2] Both approaches present certain strengths and limitations Computer implementation of control algorithms presents a great convenience and has, hence, caused the research in the area of discrete-time control to intensify This also necessitated a rework in the sliding mode control strategy for sampled-data systems Most of the discrete-time sliding mode approaches are based on the availability of full state information, [7]-[9] A few approaches did focus 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 invertible This is true for sampled-data systems In this chapter we will focus on state based approaches as well as expand upon the work of [5],[6] by focusing on arbitrary reference tracking of a linear time invariant system with matched disturbance A delay in the state or disturbance estimation in sampled-data systems is an inevitable 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(T ) for integral sliding mode control While deadbeat control is a desired phenomenon, it is undesirable in practical implementation due to the over large control action required In [9] the integral sliding mode design avoided 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 system subject to matched time varying disturbance To accomplish the task of G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 247–268, 2008 c Springer-Verlag Berlin Heidelberg 2008 springerlink.com 248 X Jian-Xin and K Abidi arbitrary reference tracking three approaches will be considered: 1) State Feedback, 2) Output Feedback, and 3) Output Feedback with a State Observer In each approach the objective is to drive the output tracking error to a certain neighborhood of the origin For this purpose a discrete-time integral sliding surface (ISM) is proposed The proposed scheme allows full control of the closedloop error dynamics and the elimination of the reaching phase The elimination of deadbeat response helps in avoiding the generation of over large control inputs It is also worth to highlight that the discrete-time integral sliding mode control (ISMC) can achieve the O(T ) boundary for output tracking error even in the presence of O(T ) accuracy in the state estimation Problem Formulation Consider the following minimum-phase continuous-time system with a nominal linear time invariant model and matched disturbance ˙ x(t) = Ax(t) + B(u(t) + f (t)) y(t) = Cx(t) (1) where the state x ∈ n , the output y ∈ m , the control u ∈ m , and the disturbance f ∈ m is assumed smooth and bounded The discretized counterpart of (1) can be given by xk+1 = Φxk + Γ uk + dk yk = Cxk , y0 = y(0) where (2) T Φ = eAT , Γ = eAτ dτ B T eAτ Bf ((k + 1)T − τ )dτ , dk = and T is the sampling period (Φ, Γ, C) are controllable and observable Here the 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 Γ = BT + ABT + · · · = BT + M T + O(T ) ⇒ BT = Γ − M T + O(T ) (3) 2! where M is a constant matrix because T is fixed From (3), it can be concluded that the magnitude of Γ is O(T ) The control objective is to design a discrete-time integral sliding manifold and a discrete-time SMC law that will stabilize the sampled-data system (2) and achieve as precisely as possible output tracking Meanwhile the closed-loop Output Tracking with Discrete-Time Integral Sliding Mode Control 249 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 originally matched in continuous-time will contain mismatched components in the sampled-data system This is summarized in the following relation, [9], T dk = eAτ Bf ((k + 1)T − τ )dτ = Γ fk + Γ vk T + O(T ) (4) d where vk = v(kT ), v(t) = dt f (t), dk = O(T ), dk − dk−1 = O(T ), and dk − 2dk−1 + dk−2 = O(T ) Note that the magnitude of the mismatched part in the disturbance dk is of the order O(T ) Output Tracking ISM: State Feedback Approach 3.1 Controller Design Consider the discrete-time integral sliding-surface defined below, σ k = e k − e + εk εk = εk−1 + Eek−1 (5) where ek = rk − yk is the tracking error, rk is the reference trajectory, yk is the output, σ k , k ∈ m are the sliding function and integral vectors, e0 is the intial error, and E ∈ m×m is the design matrix The output tracking problem is to force yk → rk To proceed with the controller design, consider a forward expression of (5) σ k+1 = ek+1 − e0 + εk+1 εk+1 = εk + Eek (6) Substituting εk+1 and (2) into the expression of the sliding surface in (6) and equating σ k+1 to zero leads to σ k+1 = ek+1 + Eek − e0 + εk = ek+1 − (Im − E)ek + σ k = (7) where Im is a unity matrix of dimension m If we substitute yk+1 = CΦxk + CΓ uk + Cdk into (7) and solve for the equivalent control ueq we have k ueq = (CΓ )−1 [rk+1 − Λek − CΦxk − Cdk + σ k ] k (8) where Λ = Im − E Under the assumptions made, the control cannot be implemented 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 Therefore, the final controller structure is given by ˆ uk = (CΓ )−1 rk+1 − Λek − CΦxk − C dk−1 + σ k (9) ˆ where d is the disturbance estimate and in the case of full state availability the following delay based estimation can be used, [7], 250 X Jian-Xin and K Abidi ˆ dk−1 = dk−1 = xk − Φxk−1 − Γ uk−1 (10) Once the controller is designed, we need to examine the closed-loop state and output stability 3.2 Stability Analysis In order to derive the closed-loop state dynamics we substitute uk defined by (9) into (2) and obtain the following ˆ xk+1 = Φ − Γ (CΓ )−1 (CΦ − ΛC) xk + dk − Γ (CΓ )−1 C dk−1 −1 −1 +Γ (CΓ ) (rk+1 − Λrk ) + Γ (CΓ ) σ k (11) ˆ Since, the difference between ueq and uk is the substitution of dk with dk−1 the k sliding surface σ k+1 will no longer be zero but rather a function of the difference ˆ dk − dk−1 as follows ˆ σ k+1 = C(dk−1 − dk ) (12) Thus, we obtain the closed-loop state dynamics during sliding mode In order to conclude on the stability of (11) we propose the following Lemma Lemma 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 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 dynamics is guaranteed Substitution of (11) into yk+1 yields the dynamics ˆ yk+1 = −Λek + rk+1 + Cdk − C dk−1 + σ k (13) ˆ Substituting the result σ k = C(dk−2 − dk−1 ) obtained from (12) into (13) ˆ ˆ yk+1 = −Λek + rk+1 + C(dk − dk−1 − dk−1 + dk−2 ) (14) This yields the tracking error dynamics, ek+1 = Λek + δ k (15) where δ k is given by ˆ ˆ δ k = −C(dk − dk−1 − dk−1 + dk−2 ) (16) Remark From (15) we see that the reference tracking dynamics depends on the choice of Λ which is a design matrix However, internal stability require that Output Tracking with Discrete-Time Integral Sliding Mode Control 251 the open-loop zeros of the system are stable and, thus, it should be minimum phase 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 ˆ ˆ δ k = −C(dk − dk−1 − dk−1 + dk−2 ) (17) ˆ From (10) dk−1 = dk−1 , therefore, (17) becomes δ k = −C(dk − dk−1 − dk−1 + dk−2 ) (18) δ k = −C(dk − 2dk−1 + dk−2 ) (19) which simplifies to In [9] it is shown that dk − 2dk−1 + dk−2 = O(T ) if the smoothness and boundedness conditions on f (t) hold Therefore, δ k = −C(dk − 2dk−1 + dk−2 ) = O(T ) (20) According to [9] the ultimate error bound on ek will be one order higher than the bound on δ k due to convolution and since the bound on δ k is O(T ) the ultimate bound on ek is O(T ) Thus, the ultimate bound on the tracking error is (21) ek = O(T ) Output Tracking ISM: Output Feedback Approach 4.1 Controller Design In most practical situations, full state measurement is not possible and only the output is measured Thus, it makes more sense to try to derive a controller that is a function of the output tracking error In order to proceed we will first define the reference model xr,k+1 = Φ − Γ (CΓ )−1 CΦ xr,k + Γ (CΓ )−1 rk+1 yr,k = Cxr,k = rk (22) where xr,k ∈ n is the reference model state vector, yr,k ∈ m is the reference model output vector, and rk ∈ m is the reference trajectory Due to the deadbeat nature of the reference model its output is the desired reference trajectory rk and, therefore, tracking this reference model would lead to the desired response The only drawback is that the stability of the reference model requires that the system (2) satisfy the minimum phase condition Now define D = CΦ−1 , consider the sliding surface 252 X Jian-Xin and K Abidi σ k = D [xr,k − xk ] + εk εk = εk−1 + ED [xr,k−1 − xk−1 ] (23) where D eliminates the state dependency, σ k , εk ∈ 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 sliding surface (23) σ k+1 = D [xr,k+1 − xk+1 ] + εk+1 (24) εk+1 = εk + ED [xr,k − xk ] Substituting the sampled-data system (2) into (24) σ k+1 = D [xr,k+1 − Φxk − Γ uk − dk ] + εk+1 εk+1 = εk + ED [xr,k − Φxk−1 − Γ uk−1 − dk−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 σ k+1 = Dxr,k+1 − yk − DΓ uk − Ddk + εk+1 (26) and the expression for the integral variable εk+1 εk+1 = εk + E [Dxr,k − yk−1 − DΓ uk−1 − Ddk−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 , we get k ueq = (DΓ )−1 [Dxr,k+1 − yk − Ddk + εk+1 ] k (28) and εk+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 controller designed in Section requires full state information which is not available in this case Therefore, an observer that is based on output feedback will have to be ˆ used If we substitute the delayed disturbance estimate, dk−1 , instead of dk the final controller becomes ˆ uk = (DΓ )−1 Dxr,k+1 − yk − Ddk−1 + εk+1 (29) and the expression for the integral variable (27) becomes ˆ εk+1 = εk + E Dxr,k − yk−1 − DΓ uk−1 − Ddk−1 (30) ˆ where the disturbance estimate dk based on output information will be given in subsection 4.3 Output Tracking with Discrete-Time Integral Sliding Mode Control 4.2 253 Stability Analysis With the controller design, the stability of the closed-loop system must be analyzed For this we will substitute (30) in (29) and substitute the resulting expression for uk into (2) leading to the closed-loop equation during sliding mode, xk+1 = Φxk − Γ (DΓ )−1 [yk + E (yk−1 + DΓ uk−1 )] + dk ˆ −Γ (DΓ )−1 [D + ED] dk−1 + Γ (DΓ )−1 [Dxr,k+1 + EDxr,k + εk ] (31) To get the state dynamics, yk will be replaced by DΦxk and [yk−1 + DΓ uk−1 ] will be replaced by [Dxk − Ddk−1 ] and the result is simplified ˆ xk+1 = Φ − Γ (DΓ )−1 (DΦ + ED) xk + dk − Γ (DΓ )−1 Ddk−1 ˆ +Γ (DΓ )−1 [Dxr,k+1 + EDxr,k + εk + ED(dk−1 − dk−1 )] (32) Solving εk in (23) in terms of xr,k , xk and σ k εk = σ k − D(xr,k − xk ) (33) and substituting into (32) the following closed-loop dynamics is obtained ˆ xk+1 = Φ − Γ (DΓ )−1 (DΦ − ΛD) xk + dk − Γ (DΓ )−1 Ddk ˆ +Γ (DΓ )−1 [Dxr,k+1 − ΛDxr,k + ED(dk−1 − dk−1 ) + σ k ] (34) where Λ = Im − E Further, it can be shown that the expression of σ k+1 in (24) is of the form ˆ ˆ σ k+1 = D(dk − dk−1 ) + ED(dk−1 − dk−1 ) (35) ˆ We see from (35) that σ k+1 is a function of d and 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(dk−1 − ˆ ˆ dk−2 ) + ED(dk−2 − dk−2 ) in (34) we obtain xk+1 = Φ − Γ (DΓ )−1 (DΦ − ΛD) xk +Γ (DΓ )−1 [Dxr,k+1 −ΛDxr,k ]+ζ k (36) where ˆ ˆ ζ k = dk − Γ (DΓ )−1 D(dk−1 + dk−1 − dk−2 ) −1 ˆ ˆ +Γ (DΓ ) ED(dk−1 − dk−1 + dk−2 − dk−2 ) Subtracting both sides of (36) from xr,k+1 yields xr,k+1 − xk+1 = − Φ − Γ (DΓ )−1 (DΦ − ΛD) xk +[I − Γ (DΓ )−1 D]xr,k+1 + Γ (DΓ )−1 ΛDxr,k − ζ k (37) 254 X Jian-Xin and K Abidi finally substituting (22) into the r.h.s of (37) and using the fact that [I − Γ (DΓ )−1 D]Γ = we obtain Δxk+1 = Φ − Γ (DΓ )−1 (DΦ − ΛD) Δxk − ζ k (38) where Δx = xr − x From Lemma 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 discuss ˆ the disturbance estimate dk upon which the tracking error bound depends The next section will address this 4.3 Disturbance Observer Design In order to design the observer we first need to note that according to (4) the disturbance can be written as dk = Γ fk + Γ vk T + O(T ) = Γ η k + O(T ) (39) where η k = fk + vk T If η k can be estimated then the estimation error of dk would be O(T ) which is acceptable in practical applications Define the observer ˆ xd,k+1 = Φxd,k + Γ uk + Γ ηk yd,k = Cxd,k (40) where xd ∈ n is the observer state vector, yd ∈ m is the observer output ˆ vector, η ∈ m is the disturbance estimate and will act as the “control input” ˆ ˆ to the observer, therefore the estimate dk = Γ η k Since the disturbance estimate will be used in the final control signal it must not be overly large, therefore, it is wise to avoid a deadbeat design For this reason we will use an observer based on an integral sliding surface σ d,k = ed,k − ed,0 + εd,k εd,k = εd,k−1 + Ed ed,k−1 (41) where ed,k = yk − yd,k , is the output estimation error, σ d , εd ∈ m are the sliding function and integral vectors, and Ed 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 [yk+1 − Λd ed,k − CΦxd,k + σ d,k ] − uk (42) where Λd = Im − Ed 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 yk+1 which is not possible therefore the delayed (42) is used Output Tracking with Discrete-Time Integral Sliding Mode Control ˆ η k−1 = (CΓ )−1 [yk − Λd ed,k−1 − CΦxd,k−1 + σ d,k−1 ] − uk−1 255 (43) ˆ Since, only η 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 xd,k = Φ − Γ (CΓ )−1 (CΦ − Λd C) xd,k−1 + Γ (CΓ )−1 [yk − Λd yk−1 ] (44) Subtracting (44) from a delayed form of the system (2) and substituting dk−1 = Γ ηk−1 we obtain Δxd,k = Φ − Γ (CΓ )−1 (CΦ − ΛC) Δxd,k−1 (45) where Δxd,k = xk − xd,k The solution of (45) is given by Δxd,k−1 = Φ − Γ (CΓ )−1 (CΦ − Λd C) k−1 Δxd,0 (46) If we set Ed = Im − Λd where Λd is a diagonal matrix it is shown in Lemma that the poles of the closed-loop system (45) are the eigenvalues of Λd and the non-zero eigenvalues of [Φ − Γ (CΓ )CΦ] In control applications, we can choose eigenvalues Λd closer to origin comparing with the controller eigenvalues Λ Premultliplication of (45) with C yields ed,k = Λd ed,k−1 (47) Thus, the performance of the observer tracking depends on the choice of Λd Finally, we need to look at the disturbance estimation by the observer For this we subtract a delayed (40) from a delayed (2) to obtain ˆ Δxd,k = ΦΔxd,k−1 + Γ (η k−1 − η k−1 ) (48) ˆ To obtain the η − η relationship, premultiplying both sides of (48) with C and substituting (47) yield ˆ ηk−1 = (CΓ )−1 (CΦ − Λd C)Δxd,k−1 + η k−1 (49) or ˆ η k−1 = (CΓ )−1 (CΦ − Λd C) Φ − Γ (CΓ )−1 (CΦ − Λd C) k−1 Δxd,0 + ηk−1 (50) Since Φ − Γ (CΓ )−1 (CΦ − Λd C) is stable, for k large enough Φ − Γ (CΓ )−1 (CΦ − Λd C) k−1 →0 and the disturbance estimate will converge to the actual disturbance As a result, when k is large enough ˆ ˆ dk − dk−1 = Γ (η k − ηk−1 ) = Γ (η k − η k−1 ) = O(T ) (51) 256 X Jian-Xin and K Abidi 4.4 Tracking Error Bound Recall that the closed-loop system was in the form Δxk+1 = Φ − Γ (DΓ )−1 (DΦ − Λd D) Δxk − ζ k (52) and the term ζ k was given by ˆ ˆ ζ k = dk − Γ (DΓ )−1 D(dk−1 + dk−1 − dk−2 ) ˆ ˆ + Γ (DΓ )−1 ED(dk−1 − dk−1 + dk−2 − dk−2 ) (53) ˆ ˆ Substituting dk = Γ ηk + O(T ) and dk−1 = Γ ηk−1 we have ˆ ˆ ζ k = Γ (η k − η k−1 − η k−1 + η k−2 ) −1 ˆ ˆ + Γ (DΓ ) EDΓ (η k−1 − η k−1 + η k−2 − η k−2 ) + O(T ) (54) ˆ ˆ From (50) η k−1 → η k−1 when k is sufficiently large, therefore η k−1 − η k−1 + ˆ η k−2 − η k−2 → and (53) renders ζ k = Γ (ηk − 2ηk−1 + η k−2 ) + O(T ) (55) Since the disturbance f (t) is assumed smooth and bounded it can be shown that the magnitude of η k − 2η k−1 + η k−2 is O(T ) Also since Γ = O(T ) ζ k = O(T ) · O(T ) + O(T ) = O(T ) (56) According to [9] the ultimate error bound on Δxk will be one order higher than the bound on ζ due to convolution and since the bound on ζ k is O(T ) the ultimate bound on Δxk is O(T ) Thus, the ultimate bound on the tracking error is ek ≤ C Δxk = O(T ) (57) 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 = (CΓ )−1 [rk+1 − Λek − CΦxk − Cdk + σ k ] 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 obtained from the observers will be used Therefore, the final controller structure is given by Output Tracking with Discrete-Time Integral Sliding Mode Control ˆ uk = (CΓ )−1 rk+1 − Λek − CΦˆ k − C dk−1 + σ k x 257 (59) In order to derive the closed-loop state dynamics we substitute uk defined by (59) into (2), x xk+1 = Φ + Γ (CΓ )−1 ΛC xk − Γ (CΓ )−1 CΦˆ k + Γ (CΓ )−1 (rk+1 − Λrk ) −1 −1 ˆ +Γ (CΓ ) σ k + dk − Γ (CΓ ) C dk−1 (60) To proceed further we assume that the state estimate is related to the actual state as follows ˆ xk = xk + κk (61) where κk is the estimation error If we substitute (61) in (60) we obtain xk+1 = Φ − Γ (CΓ )−1 (CΦ − ΛC) xk + Γ (CΓ )−1 (rk+1 − Λrk ) ˆ +Γ (CΓ )−1 σ k + dk − Γ (CΓ )−1 C dk−1 − Γ (CΓ )−1 CΦκk (62) The sliding surface can similarly be derived as ˆ σ k+1 = C(dk−1 − dk ) + CΦκk (63) It will be shown later that the state estimation error κk 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 Φ − Γ (CΓ )−1 (CΦ − ΛC) which is stable if the system (Φ, Γ, C) is minimum phase and the matrix Λ has stable eigenvalues Analogous to (14), the output dynamics is given by ˆ ˆ yk+1 = −Λek + rk+1 + C(dk − dk−1 − dk−1 − dk−2 ) − CΦ(κk − κk−1 ) (64) and the output tracking error is given by ek+1 = Λek + ξ k (65) where ξ k is given by ˆ ˆ ξ k = −C(dk − dk−1 − dk−1 + dk−2 ) + CΦ(κk − κk−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 κk on the tracking error bound we will discuss the state observer in the next section 5.2 State Observer State estimation will be accomplished with the following state-observer ˆ ˆ ˆ xk+1 = Φˆ k + Γ uk + L(yk − yk ) + dk−1 x ˆ ˆ yk = C xk (67) (68) 258 X Jian-Xin and K Abidi ˆ ˆ where x, y are the state and output estimates and L is a design matrix Observer ˆ (67) is well-known, however, the term dk−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 ˆ ˜ xk+1 = [Φ − LC]˜ k + dk − dk−1 x (69) ˜ ˆ where x = x − x is the state estimation error The solution of (69) is given by k−1 ˆ [Φ − LC]k−1−i (di − di−1 ) ˜ ˜ xk = [Φ − LC]k x0 + (70) i=0 therefore, the state estimate is given by k−1 ˆ [Φ − LC]k−1−i (di − di−1 ) ˆ ˜ xk = xk − [Φ − LC]k x0 − (71) i=0 which means that κk from (61) is given by k−1 ˆ [Φ − LC]k−1−i (di − di−1 ) ˜ κk = −[Φ − LC]k x0 − (72) i=0 ˆ ˜ Since, dk − dk−1 = O(T ) 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(T ) 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 ˆ ˆ ξ k = −C(dk − dk−1 − dk−1 + dk−2 ) + CΦ(κk − κk−1 ) (73) From (72), the difference κk − κk−1 is given by ˜ κk − κk−1 = [In − (Φ − LC)](Φ − LC)k−1 x0 k−1 ˆ [Φ − LC]k−1−i (di − di−1 ) − i=0 k−2 ˆ [Φ − LC]k−1−i (di − di−1 ) + i=0 which can be simplified to (74) Output Tracking with Discrete-Time Integral Sliding Mode Control 259 ˆ ˜ κk − κk−1 = [In − (Φ − LC)](Φ − LC)k−1 x0 − (dk − dk−1 ) (75) Since (Φ − LC)k → for k large enough, we finally have the following ˆ κk − κk−1 = −(dk − dk−1 ) (76) Substituting (76) into (73) yields ˆ ˆ ˆ ξ k = −C(dk − dk−1 − dk−1 + dk−2 ) − CΦ(dk − dk−1 ) (77) ˆ ˆ and substitute dk = Γ η k + O(T ) and dk−1 = Γ η k−1 we get ˆ ˆ ˆ ξk = −CΓ (η k − η k−1 − η k−1 + η k−2 ) − CΦΓ (η k−1 − η k−2 ) (78) Since we are trying to calculate the steady state error bound, using the property ˆ η k → η k and substituting in (78) ξk = −CΓ (ηk − 2ηk−1 + η k−2 ) − CΦΓ (η k−1 − ηk−2 ) + O(T ) (79) Note that for a system of relative degree greater than 1, CB = Therefore, we can have 1 ABT + A2 BT + · · · 2! 3! 1 = CABT + CA2 BT + · · · = O(T ) 2! 3! CΓ = C BT + (80) Similarly 2 A T + · · · )Γ = C(I + O(T ))Γ = CΓ + O(T ) = O.(T ) 2! (81) Also, since the disturbance f (t) is assumed smooth and bounded it can be shown that the magnitude of η k − ηk−1 is O(T ) and η k − 2ηk−1 + ηk−2 is O(T ) Thus, we obtain CΦΓ = C(I + AT + ξ k = O(T ) · O(T ) + O(T ) · O(T ) + O(T ) = O(T ) (82) According to [9] the ultimate error bound on ek will be one order higher than the bound on ξ k due to convolution and since the bound on ξk is O(T ) the ultimate bound on ek is O(T ) Thus, the ultimate bound on the tracking error is ek = O(T ) (83) Remark From the result we see that even though the state estimation error is O(T ) we can still obtain O(T ) output tracking by virtue of the integral action in the controller design 260 X Jian-Xin and K Abidi Application to Motion Control Problems 6.1 Non-smooth Disturbance Motion control problems are widely encountered in industrial implementation of control algorithms and, thus, it is necessary to explore the applicability of the developed control laws The basic component in a motion control system is the electrical motor which in the absence of driver dynamics is given by x(t) + kf − Mv y(t) = x(t) ˙ x(t) = kf M u(t) + M f (x, t) (84) where M is the inertia, kf v is the damping factor, kf is a torque constant, u(t) is the control input and f (x, t) is the disturbance In most practical situations, the output y(t) is available via measurement of as the angular displacement In motion systems, the disturbance f (x, t) is generally dominated by a friction force that is characterized by a discontinuity that occurs at the onset of motion and can be assumed to satisfy the smoothness condition once the system is in motion It is, thus, vital to examine the system performance around the time the discontinuity occurs Beginning with the disturbance observer (43), since during the occurrence of the discontinuity the approximation (4) does not hold (48) becomes ˆ Δxdk = ΦΔxdk−1 + dk−1 − Γ ηk−1 (85) ˆ and it can be easily verified that the estimate η k converges to (CΓ )−1 Cdk Next look at the term ξ k which, associated with the state observer approach, is given (79) ξk = −CΓ (ηk − 2ηk−1 + η k−2 ) − CΦΓ (η k−1 − ηk−2 ) + O(T ) (86) Since, dk cannot be approximated by (4) the correct form should be ξ k = −C(dk − 2dk−1 + dk−2 ) − CΦ(dk−1 − dk−2 ) (87) It can be reasonably assumed that the discontinuity occurs rarely, therefore, if we assume that the discontinuity occurs at the k th sampling point, then ξ k ∈ O(T ) rather than O(T ) as the difference dk − dk−1 will no longer be O(T ) and instead will be of the order of dk which is O(T ) The tracking error dynamics of the state observer approach is given by ek+1 = Λek + ξ k (88) which has the solution k−1 ek = Λ k e0 + Λi ξ k−i−1 i=0 (89) Output Tracking with Discrete-Time Integral Sliding Mode Control 261 If the discontinuity occurs at k = kd during a certain time interval then ξ kd ∈ O(T ), and ξ k ∈ O(T ) for all other sampling points Therefore the solution of (89) would lead to a worst case error bound of ek = O(T ) (90) To counter the effect of a non-smooth disturbance, a nonlinear switching term can be incorporated in the control law (59) 6.2 Experimental Investigation To verify the effectiveness of the discrete-time integral sliding control design, experiments have been carried out using a linear piezoelectric motor which has many promising applications in industries The piezoelectric motors are characterized by low speed and high torque, which are in contrast to the high speed and low torque properties of the conventional electromagnetic motors Moreover, piezoelectric motors are compact, light, operates quietly, and robust to external magnetic or radioactive fields Piezoelectric motors are mainly applied to high precision control problems as it can easily reach the precision scale of micrometers or even nano-meters This gives rise to extra difficulty in establishing an accurate mathematical model for piezoelectric motors: any tiny factors, nonlinear and unknown, will severely affect their characteristics and control performance The configuration of the whole control system is outlined in Fig.1 The driver and the motor can be modeled approximately as a second order system shown in (84) with the system matrices A= kf − Mv , B= kf M , C= 10 where M = 1kg, kf v = 144N and kf = 6N/V where V stands for volt This simple linear model does not contain any nonlinear and uncertain effects such as the frictional force in the mechanical part, high-order electrical dynamics of the driver, loading condition, etc., which are hard to model in practice In general, Fig System Block Diagram 262 X Jian-Xin and K Abidi open−loop zero 2.5 1.5 −4 10 −3 10 −2 −1 10 10 sampling−time [sec] 10 Fig Open-loop zero of (Φ, Γ, D) with respect to sampling-time open−loop zero −0.2 −0.4 −0.6 −0.8 −1 −4 10 −3 10 −2 −1 10 10 sampling−time [sec] 10 Fig Open-loop zero of (Φ, Γ, C) with respect to sampling-time producing a high precision model will require more efforts than performing a control task with the same level of precision As in all motion control problems, only position feedback is possible Thus, leaving us with either the output feedback approach or the state observer approach both of which we will explore separately 6.3 Output Feedback Approach In order for the output feedback approach to be applicable the system (Φ, Γ, D), where D = CΦ−1 , must be minimum phase For discrete-time applications it is well known that the minimum phase condition is dependent on the continuoustime system as well as the sampling-time used For this particular system the plot of the open-loop zero versus the sampling-time is shown in Fig.2 It can Output Tracking with Discrete-Time Integral Sliding Mode Control 263 40 35 30 y [mm] 25 20 15 10 ISMC PI Reference 0 0.2 0.4 0.6 0.8 t [sec] Fig Position trajectory and comparison of ISMC and PI controllers’ performance 0.05 ISMC PI 0.04 e [mm] 0.03 0.02 0.01 −0.01 0.5 1.5 t [sec] Fig Tracking error of ISMC and PI controllers be seen that the open-loop zero approaches as T → ∞ but is never less than which means this particular system is non-minimum phase no matter what the sampling-time used is Thus, this approach is inapplicable in this case, underscoring the restrictive nature of this approach 6.4 State Observer Approach For the state observer approach the system (Φ, Γ, C) is required to be minimum phase From Fig.3 we see that for a sampling-time between 0.1ms and 1s the open-loop zero has a relatively larger stability margin From Fig.3 a selection of sampling-time T = 1ms would provide a fast enough convergence while having a good enough tracking error Upon sampling at T = 1ms the resulting sampleddata system state and gain matrices are 264 X Jian-Xin and K Abidi 1.0000 0.0009 , 0.8659 Φ= Γ = 2.861 × 10−6 5.6 × 10−3 and the open-loop zero is −0.954 To proceed with the implementation three parameters need to be designed: the state observer gain L, the disturbance observer integrator gain matrix Ed , and the controller integrator gain E The state observer gain is selected such that the observer poles are (0.4, 0.4) This selection is to ensure quick convergence Next, the matrix Ed is designed Note that for this second order system Ed is a scalar To ensure the quick convergence of the disturbance observer, Ed is selected such that the observer pole is λd = 0.9 which corresponds to s = −105.4 in the continuous-time Since the remaining pole of the observer is the non-zero open-loop zero z = −0.954 corresponding to a pole with real part of s = −47.1 in the continuous-time, it is the dominant pole Finally, the controller pole is selected as λ = 0.958 which is found to be the best possible after some trials Thus, the design parameters are as follows L = 1.059 231.048 , Ed = − λd = 0.1, E = − λ = 0.042 The reference trajectory rk used is a sigmoid curve as shown in Fig.4 The ISM results are compared to that of a PI controller as seen in Fig.4 and Fig.5 From the results we see that the ISM controller has a better tracking performance compared to a PI controller The results in Fig.6 are for the control inputs for the PI and ISM Finally, an extra load of 2.5kg is added without modifying the controller parameters We see from the results that the change of load barely effects the ISM controller performance as seen in Fig.7 The control for this case is seen in Fig.8 We can observe from the figures that the control input needed to overcome the deadzone is increased from around 1.25V to around 1.5V when the load is added 2.5 ISMC PI u [V] 1.5 0.5 0 0.2 0.4 0.6 t [sec] 0.8 Fig Comparison of the control inputs of ISMC and PI controllers Output Tracking with Discrete-Time Integral Sliding Mode Control 265 0.015 ISMC 0.01 e [mm] 0.005 −0.005 −0.01 −0.015 0.5 1.5 t [sec] Fig Position trajectory with ISMC loaded with 2.5kg 2.5 ISMC u [V] 1.5 0.5 0 0.2 0.4 0.6 t [sec] 0.8 Fig Control input for ISMC with 2.5kg load Conclusion This chapter presented a form of the discrete-time integral sliding control design for sampled-data systems with output tracking Three approaches are investigated: 1) State Feedback, 2) Output Feedback, and 3) Output Feedback with a State Observer Proper disturbance and state observers were presented The closed-loop stability of the system was not dependent on either observer and is designed separately It was shown that for all the three approaches the maximum bound on the tracking error is O(T ) at steady state It was also shown that even though the state observer produced O(T ) estimation error, the ISM state observer approach could still produce O(T ) tracking error Experimental comparison with a PI controller proves the effectiveness of the proposed methods 266 X Jian-Xin and K Abidi References ˙ Zak, S.H., Hui, S.: On variable structure output feedback controllers for uncertain dynamical systems IEEE Transactions on Automatic Control 38, 1509–1512 (1993) El-Khazali, R., DeCarlo, R.: Output feedback variable structure control design Automatica 31, 805–816 (1995) Edwards, C., Spurgeon, S.K.: Robust output tracking using a sliding mode controller/observer scheme International Journal of Control 64, 967–983 (1996) Slotine, J.J.E., Hedrick, J.K., Misawa, E.A.: On sliding observers for nonlinear systems ASME Journal of Dynamic Systems, Measurement and Control 109, 245– 252 (1987) Lai, N.O., Edwards, C., Spurgeon, S.: Discrete output feedback sliding mode based tracking control In: Proc 43rd IEEE Conference on Decision and Control, Nassau, Bahamas (2004) Lai, N.O., Edwards, C., Spurgeon, S.: On discrete time output feedback min-max controller International Journal of Control 77, 554–561 (2004) Su, W.C., Drakunov, S., Ozguner, U.: An O(T ) boundary layer in sliding mode for sampled-data systems IEEE Transactions on Automatic Control 45, 482–485 (2000) Utkin, V.I.: Sliding mode control in discrete-time and difference systems In: Zinober, A.S.I (ed.) Variable Structure and Lyapunov Control Lecture Notes on Control and Information Sciences, vol 193, Springer, Berlin (1994) Abidi, K., Xu, J.X.: On the discrete-time integral sliding mode control In: Proc IEEE Workshop on Variable Structure Systems VSS 2006, Alghero, Italy (2006) 10 Utkin, V.I., Shi, J.: Integral sliding mode in systems operating under uncertainty conditions In: Proc Conference on Decision and Control CDC 1996, Kobe, Japan (1996) 11 Cao, W.J., Xu, J.X.: Eigenvalue Assignment in Full-Order Sliding Mode Using Integral Type Sliding Surface IEEE Transactions on Automatic Control 49, 1355– 1360 (2004) 12 Fridman, L., Castanos, F., M’Sirdi, N., Khraef, N.: Decomposition and Robustness Properties of Integral Sliding Mode Controllers In: Proc IEEE Workshop on Variable Structure Systems VSS 2004, Vilanova i la Geltru, Spain (2004) 13 Astrom, K.J., Wittenmark, B.: Computer-Controller Systems Prentice Hall, Upper Saddle River (1997) Appendix: Proof of Lemma If the matrices Φ, Γ and C are partitioned as shown Φ= Γ1 Φ11 Φ12 , C = C1 C2 , and Γ = Φ21 Φ22 Γ2 where (Φ11 , C1 , Γ1 ) ∈ m×m , (Φ12 , C2 ) ∈ m×n−m , (Φ21 , Γ2 ) ∈ n−m×m and Φ22 ∈ n−m×n−m The eigenvalues of Φ − Γ (CΓ )−1 (CΦ − ΛC) are found from det λIn − Φ + Γ (CΓ )−1 (CΦ − ΛC) = (91) Output Tracking with Discrete-Time Integral Sliding Mode Control ⎡ Φ11 ⎢ λI − Φ11 + Γ1 C Φ21 − ΛC1 det ⎢ ⎣ Φ11 −Φ21 + Γ2 C − ΛC1 Φ21 −Φ12 + Γ1 C λI − Φ22 + Γ2 267 ⎤ Φ12 − ΛC2 ⎥ Φ22 ⎥=0 ⎦ Φ12 C − ΛC2 Φ22 (92) where Γ1 = Γ1 (CΓ )−1 and Γ2 = Γ2 (CΓ )−1 If the top row is premultiplied with C1 and the bottom row is premultiplied with C2 and the results summed and used as the new top row, using the fact that C1 Γ1 + C2 Γ2 = CΓ the following is obtained ⎤ ⎡ (λIm − Λ)C2 (λIm − Λ)C1 ⎦=0 Φ11 Φ12 det ⎣ −Φ21 + Γ2 C − ΛC1 λI − Φ22 + Γ2 C − ΛC2 Φ21 Φ22 (93) factoring the term (λIm − Λ) and premultiplying the top row with Γ2 (CΓ )−1 Λ and adding to the bottom row leads to ⎤ ⎡ C2 C1 Φ11 Φ12 ⎦ = (94) det(λIm − Λ) det ⎣ −Φ21 + Γ2 C λIn−m − Φ22 + Γ2 C Φ21 Φ22 Thus, we can conclude that m eigenvalues of Φ − Γ (CΓ )−1 (CΦ − ΛC) are the eigenvalues of Λ Now, consider ⎤ ⎡ C2 C1 Φ11 Φ12 ⎦ = (95) det ⎣ −Φ21 + Γ2 C λIn−m − Φ22 + Γ2 C Φ21 Φ22 Using the following relations C2 Φ21 − C2 Γ2 C Φ11 Φ21 = −C1 Φ11 + C1 Γ1 C Φ11 Φ21 (96) C2 Φ22 − C2 Γ2 C Φ12 Φ22 = −C1 Φ12 + C1 Γ1 C Φ12 , Φ22 (97) and multiplying (95) with λ−m λm we obtain ⎤ ⎡ λC2 λC1 Φ11 Φ12 ⎦ = λ−m det ⎣ −Φ21 + Γ2 C λIn−m − Φ22 + Γ2 C Φ21 Φ22 (98) Premultiplying the bottom row with C2 and subtracting from the top row and using the result as the new top row we have ⎤ ⎡ Φ11 Φ12 C2 Φ22 − C2 Γ2 C ⎥ ⎢ λC1 + C2 Φ21 − C2 Γ2 C Φ21 Φ22 ⎥ = λ−m det ⎢ ⎣ Φ11 Φ12 ⎦ −Φ21 + Γ2 C λIn−m − Φ22 + Γ2 C Φ21 Φ22 (99) 268 X Jian-Xin and K Abidi Using relations (96) and (97) we finally obtain ⎡ Φ11 −C1 Φ12 + C1 Γ1 C ⎢ λC1 − C1 Φ11 + C1 Γ1 C Φ21 λ−m det ⎢ ⎣ Φ11 −Φ21 + Γ2 C λIn−m − Φ22 + Γ2 C Φ21 ⎤ Φ12 Φ22 ⎥ ⎥ = Φ12 ⎦ Φ22 (100) We can factor out the matrix C1 from the top row to obtain ⎡ Φ11 Φ12 −Φ12 + Γ1 C ⎢ λIm − Φ11 + Γ1 C Φ21 Φ22 −m λ det(C1 ) det ⎢ ⎣ Φ11 Φ12 −Φ21 + Γ2 C λIn−m − Φ22 + Γ2 C Φ21 Φ22 ⎤ ⎥ ⎥=0 ⎦ (101) which finally simplifies to λ−m det(C1 ) det [λI + Φ − Γ (CΓ )CΦ] = (102) It is well known that [Φ − Γ (CΓ )CΦ] has at least m zero eigenvalues which would be canceled out by λ−m and, thus, we finally conclude that the eigenvalues of Φ − Γ (CΓ )−1 (CΦ − ΛC) are the eigenvalues of Λ and the non-zero eigenvalues of [Φ − Γ (CΓ )CΦ]  ... estimate dk based on output information will be given in subsection 4.3 Output Tracking with Discrete-Time Integral Sliding Mode Control 4.2 253 Stability Analysis With the controller design,... PI controllers Output Tracking with Discrete-Time Integral Sliding Mode Control 265 0.015 ISMC 0.01 e [mm] 0.005 −0.005 −0.01 −0.015 0.5 1.5 t [sec] Fig Position trajectory with ISMC loaded with. .. system (2) and achieve as precisely as possible output tracking Meanwhile the closed-loop Output Tracking with Discrete-Time Integral Sliding Mode Control 249 dynamics of the sampled-data system has