Nonlinear disturbance observer-based control for multi-input multi-output nonlinear systems subject to mismatching condition

Loughborough University Institutional Repository Nonlinear disturbance observer-based control or multi-input multi-output nonlinear systems subject to mismatching condition This item was submitted to Loughborough University's Institutional Repository by the/an author Citation: YANG, J, LI, S and CHEN, W-H, 212 Nonlinear disturbance observer-based control or multi-input multi-output nonlinear systems subject to mismatching condition International Journal o Control, 85 (8), pp 171-182 Additional Inormation: This article was published in the serial International Journal o Control [ c Taylor and Francis] and the deinitive version is available at: http://wwwtandonlinecom/doi/abs/118/2717921267552 Metadata Record: https://dspacelboroacuk/2134/1127 Version: Accepted or publication Publisher: c Taylor and Francis Please cite the published version

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International Journal o Control Vol, No, November 211, 1 17 Nonlinear disturbance observer based control or multi-input multi-output nonlinear systems subject to mismatching condition Jun Yang a, Shihua Li a and Wen-Hua Chen b a School o Automation, Southeast University, Key laboratory o Measurement and Control o CSE, Ministry o Education, Nanjing 2196, PR China; b Department o Aeronautical and Automotive Engineering, Loughborough University, Leicestershire, LE11 3TU, UK (Received 23 November 211) For a multi-input multi-output (MIMO) nonlinear system, the existing disturbance observer based control (DOBC) only provides solutions to those whose disturbance relative degree is higher than or equal to its input relative degree By designing a novel disturbance compensation gain matrix, a generalized nonlinear disturbance observer based control method is proposed in this article to solve the disturbance attenuation problem o the MIMO nonlinear system with arbitrary disturbance relative degree It is shown that the disturbances are able to be removed rom the output channels by the proposed method with appropriately chosen control parameters The property o nominal perormance recovery, which is the major merit o the DOBCs, is retained with the proposed method The easibility and eectiveness o the proposed method are demonstrated by simulation studies o both the numerical and application examples Keywords: disturbance attenuation; nonlinear disturbance observer; arbitrary disturbance relative degree; mismatching condition; multi-input multi-output nonlinear system 1 Introduction Developing eicient control approaches to counteract the adverse eects caused by model uncertainties and external disturbances is an active topic in both the control theory and application community Many elegant solutions, such as H 2 /H control, sliding-model control, adaptive control, backstepping control, and so on, have been widely investigated in the literature Although these methods have gained extensive applications and been proved to be eicient, they mainly ocus on the stability(or robust stability) o uncertain systems (Back and Shim 29), and in general the robustness is achieved at a price o sacriicing the nominal perormance (Back and Shim 28, 29, Yang, Chen, and Li 211) As a practical alternative approach, disturbance observer based control (DOBC) has been proved to be eective in compensating the eects o unknown external disturbances and model uncertainties in control systems and received a great deal o attention in control society, or example, the last decade s development o DOBC can be seen rom Chen, Komada, and Fukuda (2), Chen et al (2), Chen (23), Guo and Chen (25), Xia et al (27), Yoo, Yau, and Gao (27), Aldeen and Sharma (28), Back and Shim (28, 29), Fujisaki and Beekadu (29), Han (29), Shim and Jo (29), Wei and Guo (29, 21), Xia et al (211), Yang, Chen, and Li (211), and the reerences therein The major merit o the DOBC is that the robustness o the closed-loop system is obtained without sacriicing its nominal control perormance (Back and Shim 28) Corresponding author Email: lsh@seueducn ISSN: 2-7179 print/issn 1366-582 online c 211 Taylor & Francis DOI: 118/2717YYxxxxxxxx http://wwwinormaworldcom

2 J Yang S Li and W-H Chen In spite o the above excellent eature, most o the existing DOBCs are only available or systems whose disturbance relative degree is higher than or equal to its input relative degree (Chen et al 1999), which is also reerred to as the case that the disturbance satisies the so-called matching condition (Barmish and Leitmann 1982) Here the matching condition implies that the disturbance act on the system in the same channel as that o the control input The cases o mismatched and matched disturbance are illustrated by the ollowing typical examples (Marino, Respondek, and VD Schat 1989), ẋ 1 = arctan x 2 + w, ẋ 2 = u, y = x 1, ẋ 1 = arctan x 2, ẋ 2 = u + w, y = x 1, (1a) (1b) The disturbance in system (1a) is clearly mismatched one as it appears in dierent channel rom that o the control input, or equivalently, the disturbance relative degree ν = 1 is lower than its input relative degree σ = 2 The existing DOBC law u = K(x) ŵ (where ŵ is the disturbance estimation) is able to solve the disturbance attenuation problem o system (1b) where the disturbance is matched one, but clearly can not eectively compensate the disturbance in system (1a) It is worth noting that the mismatched disturbances and uncertainties widely exist in practical engineering and applications, such as light control systems (Chen 23), and motion control systems (Mohamed 27, Yang et al 211) Although the disturbance attenuation problems o multi-input multi-output (MIMO) nonlinear systems with mismatched disturbances have been concerned by Wei and Guo (29, 21), the problem is not solved entirely by DOBC since only the matched disturbances therein are compensated by DOBC but the mismatched uncertainty attenuation has to resort to other advanced eedback control methods, such as H control (Wei and Guo 21) and sliding-mode control (Wei and Guo 29) It is also reported that some constrains on the mismatched uncertainties (such as with bounded H 2 norm) are required by Wei and Guo (29, 21) Recently, several researchers have engaged in the ield o attenuating mismatched disturbances via DOBC due to its signiicance In She et al (28), Maeder, and Morari (21), Yang et al (211), mismatched disturbance attenuation approaches were proposed or linear systems A generalized linear extended state observer based control method was proposed in Li et al (212) to attenuate the mismatched disturbance o a class o single-input single-output (SISO) nonlinear system By designing a novel disturbance compensation gain, a nonlinear DOBC method was proposed or a special class o SISO nonlinear systems in Yang, Chen, and Li (211) Yang, Li, and Yu (212) proposed a disturbance observer based sliding-mode control method to compensate the mismatched uncertainties o SISO nonlinear systems However, disturbance attenuation o multi-input multi-output (MIMO) nonlinear systems subject to mismatching condition is still an unsolved problem in the DOBC community In this article, a generalized nonlinear disturbance observer based control () is proposed to compensate the eects caused by mismatched disturbances o MIMO nonlinear systems By properly designing a disturbance compensation gain matrix, it is proved that the mismatched disturbances could be eliminated rom the output channels The major merit o proposed method is that the promising disturbance attenuation perormance is achieved without sacriicing its nominal control perormance o the closed-loop system, which is reerred to as nominal perormance recovery The eectiveness o the proposed method is validated by simulation studies o both the numerical and application example

International Journal o Control 3 2 Problem ormulation and preliminaries 21 Problem ormulation Consider a MIMO nonlinear system depicted by { ẋ = (x) + g(x)u + p(x)w, y = h(x), (2) where x = [x 1,, x n ] T R n, u = [u 1,, u m ] T R m, w = [w 1,, w n ] T R n and y = [y 1,, y m ] T R m denote the state, input, disturbance and output vectors, respectively (x), g(x) = [g 1 (x),, g m (x)], p(x) = [p 1 (x),, p n (x)], and h(x) = [h 1 (x),, h m (x)] T are smooth vector or matrix ields on R n Without loss o generality, it is supposed that the equilibrium o system (2) in the absence o disturbances is x = The standard Lie derivative notation is used in this paper, stated as ollows (Isidori 1995) Deinition 1 (Isidori 1995): The vector relative degree rom the control inputs to the outputs o system (2) is (σ 1,, σ m ) at the equilibrium x i L gj L k h i(x) = (1 j m, 1 i m) or all k < σ i 1, and or all x in a neighborhood o x, and the m m matrix A(x) = L g1 L σ1 1 h 1 L g2 L σ1 1 h 1 L gm L σ1 1 h 1 L g1 L σ2 1 h 2 L g2 L σ2 1 h 2 L gm L σ2 1 L g1 L σm 1 h m L g2 L σm 1 h 2 h m L gm L σm 1 h m, (3) nonsingular at x = x For simplicity, it is reerred to as the input relative degree (IRD) Similarly, the disturbance relative degree (DRD) at x is deined as (ν 1,, ν m ) Remark 1 (Chen et al 1999): The existing nonlinear disturbance observer based control () is only available or nonlinear system (2) whose DRDs are higher than or equal to its IRDs, ie, ν i σ j or all 1 i m and 1 j m Remark 2: When it comes to the case that some DRD o nonlinear system (2) is strictly lower than some IRD, ie, ν i < σ j or some i, j {1,, m}, the disturbances will inevitably aect the states regardless o what eedback-based control method is employed In this case, the most sensible design goal would be to ind a control method such that the disturbances do not aect the interested output variables (at least in steady-state) As pointed in Remark 1, the existing is no longer available or this case, which motivates urther research on generalized or system (2) with arbitrary DRDs 22 Nonlinear disturbance observer The nonlinear disturbance observer (NDOB) proposed by Chen et al (1999) provides an adequate way to estimate the disturbances in system (2), given by { żw = l(x)[p(x)(λ(x) + z w ) + (x) + g(x)u], ŵ = z w + λ(x), (4) where ŵ = [ŵ 1,, ŵ n ] T and z w are the estimation vector o the disturbance vector w and the internal state vector o the nonlinear observer, respectively λ(x) is a nonlinear unction to be designed The observer gain l(x) is designed as l(x) = λ(x) x

4 J Yang S Li and W-H Chen The disturbance estimation error o (4) or system (2) is obtained by ė w (t) = l(x)p(x)e w (t) + ẇ, (5) where the estimation error is deined as e w = w ŵ Assumption 1: The derivatives o the disturbances in system (2) are bounded, ie, ẇ(t) < This is a general assumption made or the disturbance estimation All continuous and bounded disturbances satisies this assumption Lemma 1 (Chen and Guo 24): Suppose that Assumption 1 is satisied The disturbance estimation error system (5) is locally input-to-state stable (ISS) i the observer gain l(x) is chosen such that ė w (t) + l(x)p(x)e w (t) =, (6) is asymptotically stable Assumption 2: The disturbances in (2) are bounded and satisy lim t ẇ(t) = This assumption is made or steady-state analysis o disturbance estimation and compensation Lemma 2: Suppose that Assumption 2 is satisied The disturbance estimation ŵ o NDOB (4) can asymptotically estimate the disturbance w in system (2) i the observer gain l(x) is chosen such that (6) is asymptotically stable The proo o this lemma can be easily derived by combining the result o Lemma 1 with the ISS deinition in Khalil (1996) 3 Generalized nonlinear disturbance observer based control In order to present our main result, the disturbance decoupling or MIMO nonlinear systems with arbitrary DRDs is solved irstly 31 Disturbance decoupling Disturbance decoupling control, which could attenuate the disturbances rom the output channels, is an eective disturbance attenuation method The disturbance decoupling control approach has been developed or linear systems (Wonham 1985) and also studied or nonlinear systems (Isidori 1995) However, the existing disturbance decoupling methods are only available or systems under matched disturbances In this part, a novel disturbance decoupling control approach is proposed to attenuate the mismatched disturbances o the MIMO nonlinear systems Assumption 3: The distribution G = span{g 1,, g m } o the MIMO nonlinear system (2) is involutive This is a necessary condition or state eedback linearisation o MIMO nonlinear systems (Isidori 1995) Disturbance decoupling problem or a MIMO nonlinear system with arbitrary DRDs is solved by Theorem 31 Theorem 31 : Consider a MIMO nonlinear system (2) with disturbances have arbitrary DRDs satisying the condition in Assumptions 2-3 A disturbance decoupling control law which can compensate the disturbances in the output channels o (2) in steady state is given by u = A 1 (x) [ b(x) + v + Γ(x)w], (7)

where A(x) is the same as that in Eq (3), International Journal o Control 5 γ 11 (x) γ 12 (x) γ 1n (x) b 1 (x) v 1 γ 21 (x) γ 22 (x) γ 2n (x) Γ(x) =, b(x) = b 2 (x), v = v 2 γ m1 (x) γ m2 (x) γ mn (x) b m (x) v m with γ ij (x) = σi 2 k= c i k+1 L p j L k h i L pj L σi 1 h i, (i = 1, 2,, m; j = 1, 2,, n), σ b i (x) = L σi h i 1 i, v i = c i k Lk h i, (i = 1, 2,, m), k= where parameters c i k (i = 1, 2,, m; k =, 1,, σ i 1) have to be designed such that the polynomials p i (s) = c i + c i 1s + + c i σ i 1s σi 1 + s σi, (8) are Hurwitz stable Proo: The partially eedback linearisation o the nonlinear system (2) in the absence o disturbances is addressed irstly or the case o σ = σ 1 + σ 2 + + σ m < n since it is a general case (the case σ = n is a special case and very straightorward)(isidori 1995) Under the condition o Assumption 3, a new group o coordinate transormation which can eedback linearise system (2) is deined as (Isidori 1995) Φ(x) = [ ξ η ], (9) where ξ = [ ξ 1 T,, ξ mt ] T with ξ1 i ξ i ξ i 2 = = ξ i σ i h i (x) L h i (x) L σi 1, (1) h i (x) or 1 i m, and η = [η σ+1 (x),, η n (x)] T is selected such that the mapping Φ(x) has a jacobian matrix which is nonsingular and L gj η i (x) = or all σ + 1 i n and 1 j m The description o the system (2) under the new coordinates Φ(x) is then expressed as m + 1

6 J Yang S Li and W-H Chen subsystems, and the ith one o the irst m subsystems can be represented as Π i : ξ 1 i = ξi 2 + n L pk h i w k, k=1 ξ 2 i = ξi 3 + n L pk L h i w k, ξ i σ i k=1 = L σi h i(x) + m y i = ξ i 1, k=1 L gk L σi 1 h i u k + n L pk L σi 1 h i w k, k=1 (11) or all 1 i m The last subsystem has the ollowing orm Π m+1 : η = q(ξ, η) + r(ξ, η)w, (12) Let ξ = [ ξ 1 σ 1,, ξ m σ m ] T, by collecting the last state equation in (11) or all 1 i m to ormulate a new vector, it is obtained that ξ = b(x) + A(x)u + D(x)w, (13) where D(x) = L p1 L σ1 1 h 1 L p2 L σ1 1 h 1 L pn L σ1 1 h 1 L p1 L σ2 1 h 2 L p2 L σ2 1 h 2 L pn L σ2 1 h 2 L p1 L σm 1 h m L p2 L σm 1 h m L pn L σm 1 h m Substituting the control law (7) into Eq (13), yields ξ = v + Γ(x)w + D(x)w, (14) or an equivalent expression o n ξ σ i i = v i + (γ ik + L pk L σi 1 h i )w k (15) Combining Eq (15) with Eq (11), the subsystem Π i can be rewritten as k=1 Π i : { ξi = A i ξ i + D i (x)w, y i = C i ξ i, (16) or 1 i m, where 1 1 A i =, 1 c i ci 1 ci 2 ci σ i 1

International Journal o Control 7 D i (x) = [ d i 1,, d i n], d i j = σi 2 k= L pj h i L pj L h i, 1 j n, c i k+1 L p j L k h i C i = [1,,, ] 1 σi By simple matrix calculations, it can be veriied that or all x R n To this end, it ollows rom Eqs (16) and (17) that C i (A i ) 1 D i (x) =, (17) [ ] y i = C i (A i ) 1 ξi D i (x)w = C i (A i ) 1 ξi (18) It can be concluded rom Eq (18) that the disturbances are compensated rom the output channel o system (2) with arbitrary DRDs in a static way Particularly, i the system has a steady state, it can be obtained that the steady state value o the output y i satisies y is = 32 Generalized nonlinear disturbance observer based control 321 Control law design The proposed generalized law is constructed by replacing the disturbance vector w in the disturbance decoupling control law (7) with the disturbance estimation vector ŵ o the NDOB (4) Theorem 32 : Consider a MIMO nonlinear system (2) with disturbances have arbitrary DRDs satisying Assumptions 2-3, and also suppose that the observer gain l(x) o the NDOB (4) is selected such that system (6) is asymptotically stable A generalized law which compensates the disturbances in the output channels o system (2) is given by u = A 1 (x)[ b(x) + v + Γ(x)ŵ], (19) where ŵ is the disturbance estimation vector by the NDOB (4), A(x), b(x), v, and Γ(x) are the same as those deined in Eqs (3) and (7) Proo: Under the new group o coordinate transormation Φ(x) in (9), the nonlinear system (2) has been reormulated as those in Eqs (11)-(13) Substituting the control law (19) into Eq (13), gives or equivalently expressed as ξ i σ i = v i + ξ = v + Γ(x)ŵ + D(x)w, (2) n (γ ik + L pk L σi 1 h i )w k + k=1 n γ ik (ŵ k w k ), (21) k=1

8 J Yang S Li and W-H Chen or 1 i m Combining Eq (21) with Eq (11), the subsystem Π i is rewritten as Π i : { ξi = A i ξ i + D i (x)w Γ i (x)e w, y i = C i ξ i, (22) or 1 i m, where A i, D i (x), C i have been given in Eq (16), and Γ i (x) = γ i1 (x) γ i2 (x) γ in (x) It ollows rom Eqs (22) and (17) that y i = C i (A i ) 1 [ ξi D i (x)w + Γ i (x)e w ] = C i (A i ) 1 [ ξi + Γ i (x)e w ] (23) Eq (23) implies that the disturbances are decoupled rom the output channel o the nonlinear system (2) in an asymptotical way It is derived rom Eq (23) that ( y i C i (A i ) 1 ξ ) i + Γ i (x) e w ( C i (A i ) 1 ξ ) i +L 1 e w, (24) where L 1 is a constant such that Γ i (x) L 1 over the bounded area o x Since Lemma 2 shows that lim e w (t) =, the ollowing condition which represent the steady state o the output t can be obtained by taking limits on both sides o Eq (24), lim y i(t) C i (A i ) 1 t ( lim ξ ) i (t) +L 1 lim e w (t) t t = (25) This completes the proo Remark 3: I the disturbances in (2) are matched ones, the result in Theorem 31 reduces to the existing DOBC Particularly, g(x) = p(x) implies that Γ(x) = A(x), and the resultant control law (19) becomes u = A 1 (x) [ b(x) + v] ŵ, which is the same as that o the existing method (Chen et al 1999) Remark 4: In the absence o disturbances, the disturbance estimate by (4) satisies ŵ(t) i the initial state o the NDOB (4) is set as z w () = λ(x()) In this case, the control perormance under the proposed control law (19) recovers to that under the baseline eedback control law, which implies that the property o nominal perormance recovery is obtained by the proposed 322 Stability o the closed-loop system Assumption 4: The zero dynamics o the nonlinear system (2) in the absence o disturbances, ie, ẋ = (x) + g(x)u, under the baseline eedback control law u = A 1 (x)[ b(x) + v] are locally asymptotically stable at x This is a necessary condition or stability analysis o the closed-loop MIMO nonlinear system under the baseline controller (Isidori 1995)

Lemma 3 (Isidori 1995): Consider a system International Journal o Control 9 { ξ = Āξ + p(ξ, η), η = q(ξ, η), (26) and suppose that p(, η) = or all η near and p ξ (, ) = I η = (, η) has an asymptotically stable equilibrium at η = and the eigenvalues o Ā all have negative real part, then the system (26) has an asymptotically stable equilibrium at (ξ, η) = (, ) The stability o the resultant closed-loop system is established by Theorem 33 Theorem 33 : The nonlinear system (2) under the proposed law (19) is locally ISS around x i the ollowing conditions are satisied: (i) Assumptions 1, 3-4 are satisied; (ii) the control parameters c i k in the control law (19) are selected such that the polynomials pi (s) in (8) are Hurwitz stable; (iii) the observer gain l(x) is selected such that the system (6) is asymptotically stable; (iv) the disturbance compensation gain is selected such that g(x)a 1 (x)γ(x) + p(x) is continuously dierentiable around x Proo: Taking into account the observer dynamics (5) and substituting the proposed law (19) into the system (2), the augmented closed-loop system is obtained, described by { ẋ = F (x, ew, w) ė w = G(e w, ẇ) (27) where F (x, e w, w) = (x) + g(x)a 1 (x)[ b(x) + v Γ(x)e w ] +[g(x)a 1 (x)γ(x) + p(x)]w, (28) G(e w, ẇ) = l(x)p(x)e w + ẇ (29) Under the new coordinate transormations (ξ T, η T ) T, the closed-loop system consisting o the system ẋ = (x) + g(x)u and the baseline eedback control law u = A 1 (x)[ b(x) + v] is given by { ξ = Aξ, η = q(ξ, η), (3) where A = diag{a 1,, A m } It is easy to veriy that the closed-loop system (3) has the orm o system (26) The Assumption 4 and the condition (ii) imply that the conditions o Lemma 3 are satisied or system (3), and it can be concluded rom the lemma that the system (3) (or equivalently the system ẋ = F (x,, )) has an asymptotically stable equilibrium at (ξ, η) = (, ) (or x = ) Let X = [x T, e T w] T, the augmented closed-loop system (27) is rewritten as Ẋ = F (X) + G(X)w, (31) where F (X) = [ ] [ F (x, ew, ) g(x)a, G(X) = 1 ] [ ] (x)γ(x) + p(x) ẇ, w = G(e w, ) I n n w

1 J Yang S Li and W-H Chen 14 12 3 25 (a) Output,y 1 1 8 6 4 Reerence 2 2 4 6 8 1 12 14 16 18 2 Time,sec (b) Output,y 2 2 15 1 Reerence 5 2 4 6 8 1 12 14 16 18 2 Time,sec Figure 1 Output response curves o system (32) in the presence o unknown external disturbances under the control laws (19) (solid line), (33) (dotted line) and (34) (dashed line) The reerence signals are denoted by dash-dotted line According to the theorem o the asymptotic stability o cascade-connected systems (see Corollary 132 in Isidori (1999)), the system Ẋ = F (X) is locally asymptotically stable at X = Combining the above result with condition (iv), it ollows rom Lemma 54 in Khalil (1996) that the augmented closed-loop system (31) is locally ISS, which completes the proo o this theorem 4 Simulation stuides 41 A numerical example Consider a MIMO nonlinear system depicted by ẋ 1 = x 1 + x 1 x 2 + x 3 + w 1, ẋ 2 = sin x 1 + x 2 2 + x 4 + w 2, ẋ 3 = x 4 + u 1, ẋ 4 = u 2, ẏ 1 = x 1, y 2 = x 2 (32) The IRDs and DRDs o system (32) are calculated as (σ 1, σ 2 ) = (2, 2), and (ν 1, ν 2 ) = (1, 1), respectively This is the case where the DRDs o the system are strictly lower than the IRDs, and the conditions in Remark 1 are not satisied The law or system (32) is constructed by (19) with some derivative calculations The control parameters are selected as c 1 = 2, c1 1 = 9, c2 = 24, c2 1 = 1 The gain matrix in the NDOB (4) is selected as λ(x) = [5x 1, 5x 2 ] T To evaluate the eectiveness o the proposed method, both the baseline control and integral control methods are employed in the simulation studies or the purpose o comparison The baseline controller is given as The integral control is designed as u = A 1 (x)[ b(x) + v] (33) u = A 1 (x)[ b(x) + v + u int ], (34) where u int = [ c 1 i (y1 r 1 )dt, c 2 i (y2 r 2 )dt ] T, with r1 and r 2 the reerence signals, c 1 i and c 2 i are the integral coeicients, which are chosen as c1 i = 25 and c2 i = 3 or simulation studies The reerence signals are selected as r 1 = 1 and r 2 = 2, respectively The unknown external

International Journal o Control 11 8 6 6 3 4 Input,u 1 2 Input,u 2 3 6 (a) 2 4 2 4 6 8 1 12 14 16 18 2 Time,sec (b) 9 12 2 4 6 8 1 12 14 16 18 2 Time,sec Figure 2 Input proiles o system (32) in the presence o unknown external disturbances under the control laws (19) (solid line), (33) (dotted line) and (34) (dashed line) disturbances, w 1 (t) =, w 2 (t) =, or t < 4, w 1 (t) = 2, w 2 (t) = 3, or 4 t < 8, (35) w 1 (t) = 6 sin( π 2 t) + 2, w 2(t) = 8 sin(πt + π 2 ) 3, or 8 t 2, are considered to be imposed on system (32) The output response curves o system (32) subject to disturbances (35) under the three controllers are shown in Fig 1 The corresponding input proiles are shown by Fig 2 It is observed rom Fig 1 that the baseline controller achieves excellent tracking perormance, but quite poor disturbance attenuation perormance The integral control method improves the disturbance rejection perormance to some extent, however, it brings certain undesired control perormance, such as unsatisactory overshoots This implies that the disturbance rejection perormance o integral control is achieved at the price o sacriicing its nominal control perormance A brie observation o Fig 1 shows that the proposed method exhibits promising disturbance attenuation and reerence tracking perormance It is also observed rom Figs 1 and 2 that the response curves under the are overlapped with those under the baseline control method during the irst 4 seconds when there are no disturbances imposed on the system, which is a clear evidence that the property o nominal perormance recovery is retained or the method 42 Application to a permanent magnet synchronous motor The mathematical model o the permanent magnet synchronous motor (PMSM) in the rotor reerence rame is depicted by Krishnan (21) dω dt = K t J i q B J ω + d ω, di d dt = R s i d + n p ωi q + 1 u d + d id, L d L d di q dt = R s i q n p ωi d n pφ v ω + 1 u q + d iq, L q L q L q where ω the rotor speed, i d and i q the d-axis and q-axis stator currents, u d and u q the d-axis and q-axis stator voltages, R s, L d, and L q the stator resistance, d-axis and q-axis inductances, n p the number o pole pairs, φ v the rotor lux, J the moment o inertia, B the viscous riction coeicient, and K t = 3n p φ v /2, respectively d ω, d id, and d iq denote the unknown disturbances (36)

12 J Yang S Li and W-H Chen which lump all the inluence o parameter variations and external disturbances, given by d ω = 3n p φ v i q B 2J J ω J dω J dt 1 J T L, d id = L dn p L d ωi q R s i d L d di d L d L d dt, d iq = L qn p L q ωi d R s i q L q di q L q L q dt n p φ v ω, L q where T L the load torque, and R s = R st R s, L d = L dt L d, L q = L qt L q, φ v = φ vt φ v, B = B t B, J = J t J, R st, L dt, L qt, φ vt, B t, and J t denote the actual values o the above mentioned parameters Let x = [ω, i d, i q ] T the state vector, u = [u d, u q ] T the input vector, w = [ d ω, d id, d iq ] T the disturbance vector, and y = [ω, i d ] T the output vector, the mathematical model o the PMSM (36) can be represented by the compact orm in (2), with K t J i q B J ω (x) = R s i d + n p ωi q L, g(x) = d R s i q n p ωi d n pφ v ω L q L q 1 L d, p(x) = I 3 3, h(x) = 1 L q [ ] [ ] h1 (x) ω = h 2 (x) id According to the deinition o relative degrees or nonlinear systems Isidori (1995), the input relative degrees and disturbance relative degrees o the PMSM system are calculated as (σ 1, σ 2 ) = (2, 1), and (ν 1, ν 2 ) = (1, 1), respectively This is the case where the disturbances are mismatched ones The parameters o the PMSM under consideration are given as ollows: rated power P = 75 W, rated voltage U = 2 V, rated current I N = 471 A, number o pole paries n p = 4, armature resistance R s = 174 Ω, stator inductances L d = L q = 4 H, viscous damping B = 743 1 5 N m s/rad, moment o inertia J = 174 1 4 Kg m 2, rated speed 3 rpm, rotor lux φ v = 1167 wb, rated torque T N = 2 N m In order to show the eectiveness o the proposed method in controlling the PMSM, both the baseline and integral controllers are employed in the simulation studies or perormance comparison analysis The control parameters o the proposed method or the PMSM system are designed as c 1 = 44, c1 1 = 38, c2 = 7 The disturbance observer gain matrix is designed as λ(x) = [5x 1, 5x 2, 5x 3 ] The baseline controller is given by (33), where the control parameters are designed the same as the proposed method The integral controller is given by (34), where the integral coeicients are chosen as c 1 i = 16 and c2 i = 12, and the rest control parameters are selected the same as the proposed method 421 Tracking perormance under unknown load torque variation The speed tracking perormance is tested under unknown load torque variations in this part The unknown load torque T L = 2 N m is supposed to add on the PMSM at t = 2 sec Response

International Journal o Control 13 14 1 (a) Speed ω, rpm 12 1 8 6 4 2 Reerence 5 1 15 2 25 3 35 4 Time, sec 25 2 (b) d axis current i d, A 5 5 Reerence 1 5 1 15 2 25 3 35 4 Time, sec q axis current i q, A 15 1 5 (c) 5 5 1 15 2 25 3 35 4 Time, sec Figure 3 Variable response curves o the PMSM in the presence o unknown load toque variation under the control laws (19) (solid line), (33) (dotted line) and (34) (dashed line) The reerence signals are denoted by dash-dotted line curves o the rotor speed, d-axis and q-axis currents are shown in Fig 3 As shown by Fig 3, compared with the integral and baseline control methods, the proposed method obtains ine tracking perormance in the presence o unknown external load torque variations It is obtained rom Fig 3 that the response curves o the proposed method are overlaped with the baseline controller during the irst 2 sec when there is no disturbance, which implies the nominal perormance recovery o the proposed method It is also noticed that the integral control leads to undesirable transient control perormance although it can realize oset-ree tracking in the presence o load torque variation 422 Tracking perormance under uncertain parameters To show the robustness o the proposed control approach, the parameters are supposed to have variations rom their nominal operation values Two cases o parameter variations are considered here Case I: the stator resistance, the stator inductances, the rotor lux, the moment o inertia and the viscous riction coeicient are supposed to have perturbations R s = 2R s, L q = 2L q, L d = 2L d, φ v = 2φ v, J = 2J and B = 2B respectively The response curves o the PMSM under the proposed method in the presence o parameter variations in Case I is shown in Fig 4 It can be observed rom Fig 4 that the proposed method gains ine robustness perormance in such case o parameter perturbations Case II: the stator resistance, the stator inductances, the rotor lux, the moment o inertia and the viscous riction coeicient are supposed to have perturbations R s = 2R s, L q = 2L q, L d = 2L d, φ v = 2φ v, J = 2J and B = 2B respectively The response curves o the PMSM under the proposed method in the presence o parameter variations in Case II is shown in Fig 5

14 J Yang S Li and W-H Chen 12 1 (a) Speed ω, rpm 1 8 6 4 2 Reerence 5 1 15 2 25 3 35 4 Time, sec 25 2 (b) d axis current i d, A 5 5 Reerence 1 5 1 15 2 25 3 35 4 Time, sec q axis current i q, A 15 1 5 (c) 5 5 1 15 2 25 3 35 4 Time, sec Figure 4 Variable response curves o the PMSM in the presence o the irst case o parameter variations (Case I) under the control laws (19) (solid line), (33) (dotted line) and (34) (dashed line) The reerence signals are denoted by dash-dotted line The prominent robustness o the proposed method in the second case o parameter variations can be seen rom Fig 5 5 Conclusion The disturbance attenuation problem o a MIMO nonlinear system with arbitrary DRD via a DOBC approach, which has been recognized as a longstanding unsolved problem in the ield o DOBC, has been solved in this article by designing a new disturbance compensation gain matrix in the control law It has been proved that the disturbances in this case could be eliminated rom the output channels by the newly proposed with properly chosen control parameters The proposed method has retained the major eature o the DOBC method, ie, the ability o recovering its nominal control perormance, which has been veriied by simulation studies o both numerical and application examples Acknowledgements The work was supported in part by National 863 Project o the Twelth Five-Year Plan o China under Grant 211AA4A16, National Natural Science Foundation o China under Grant 617413, and Program or New Century Excellent Talents in University under Grant NCET- 1-328

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