SLIDING SURFACE MATCHED CONDITION IN SLIDING MODE CONTROL

Transcription

1 Asian Journal of Control, Vol. 9, No. 3, pp. 0-0, September SLIDING SURFACE MACHED CONDIION IN SLIDING MODE CONROL Ji Xiang, Hongye Su, Jian Chu, and Wei Wei -Brief Paper- ABSRAC In this paper, the recently developed sliding surface matching condition (SSMC) is viewed from a new point that the mismatched uncertain term should satisfy the minimum-phase condition for the nominal system. With this new viewpoint, SSMC is extended to a more general class of system with nonlinear uncertainties. Finally, a numerical example demonstrates the results. KeyWords: Sliding mode control, sliding surface matching condition (SSMC), nonlinear uncertainties. I. INRODUCION Sliding mode control (SMC) approach, originating from the theory of Variable Structure System (VSS) [1], has been widely applied to systems in the presence of modeling uncertainties and external disturbances. By a discontinuous control action, the system is forced to move along a predefined sliding surface such that its behavior is only determined by the sliding surface parameters but not influenced by uncertainties or disturbances. his is the socalled completely robust [] feature of SMC, for which it is assumed that the uncertainties and disturbances satisfy the matched condition. However, the uncertainties of most real systems dissatisfy this assumption. o solve this problem, Kwan [3] used an adaptive control law to design a time varying sliding surface for a class of mismatched uncertain systems, where parameter uncertainties are assumed in some subset of the space expanded by the system matrix. Manuscript received December 14, 004; revised December 4, 005; accepted September 7, 006. Ji Xiang and Wei Wei are with the Department of System Science and Engineering, College of Electrical Engineering, Zhejiang University, P.R. China ( jxiang@ zju.edu.cn). Hongye Su and Jian Chu are with the National Laboratory of Industrial Control echnology, Institute of Advanced Process Control, Zhejiang University, P.R. China. his work was supported by a grant from the Natural Science Fund for distinguished scholars of Zhejiang Province (No.R105341) and Priority supported financially by the New Century 151 alent Project of Zhejiang Province. he Riccati equation approach in [4] and linear matrix inequality (LMI) technique in [5,6] are respectively applied for the robust linear sliding surface design. hese robust methods, however, are unable to avoid conservatism. Further, as the sliding mode dynamics are related to the uncertainties, they cannot be described precisely. Recently, Chan et al., [7] presented a new matched concept. he sliding surface was designed such that the structure of the mismatched parameter uncertainties satisfies one matrix equation which is called sliding surface matched condition (SSMC). his condition shows the relationship between sliding matrix and uncertainties structure. However, how to attain such a sliding matrix was not addressed. Combining the traditional matched condition with the SSMC, Choi [8] puts forward a new invariant condition in terms of LMIs, by which one can judge whether there is a sliding surface on which the sliding mode dynamic is insensitive to the uncertainties. However, in contrast to the well-known matched condition, which means that the suffered uncertainties are located in the span space of the input distribution matrix, it cannot be directly known what kind of system satisfies SSMC. In addition, both results in [7] and [8] are limited to structural parameter uncertainties. In our early paper [9], we developed results of Choi [8] such that the mismatched uncertainties were extended to some linear-like functions. In this paper, SSMC is further investigated and explained from a new viewpoint. First, we show that SSMC is essentially equivalent to a minimum-phase property. Next, the concept of SSMC is extended to a more general class of systems with uncertainties of nonlinear terms. Last,

2 Asian Journal of Control, Vol. 9, No. 3, September 007 a Lorenz system as numerical example is illustrated to demonstrate our results. he notations are standard throughout this paper. A, A, and A denote the transpose, the conjugate transpose and the spectral norm of matrix A, respectively. M denotes the orthogonal complement matrix of full column rank of matrix M. II. PREVIOUS RESULS AND PROBLEM SAEMEN he considered uncertain system has the following form: ( ) ( ) x() t = A+ DF() t E x() t + B u() t + f( x, t) (1) where x(t) R n is the state vector, u(t) R m is the input vector, f (x, t) is the lumped matched uncertainties and perturbations, and DF(t)E is the structured parameter uncertainty. he following assumptions are introduced. n A1) he matrix pair (A R n n, B R m ) is controllable and rank(b) = m; A) f (x, t) is unknown but bounded such that f (x, t) ρ (x, t), where ρ (x, t) is a known nonnegative scalar valued function; q A3) F(t) R q is Luenberger-measurable and satisfies F (t) F(t) δi with δ > 0 a given constant; n q q n A4) D R and E R are known constant matrices of full rank with q m. he sliding surface is defined as [10]: 1 σ = B X x = 0, () where X R n n is a symmetric positive definite matrix determined later. he generality of Definition () can refer to [9]. Introduce a transformation matrix as B = 1 B X ( = ) [ XB ( B XB ) B( B X B ) ]. (3) By using z = x, the dynamical function System (1) in σ the new coordinates is z A11() t A1() t z 0 = + 1 ( () t + (, t) ), u f x σ A1() t A () t σ B X B (4) where 1 A11( t) = B A+ DF() E XB ( B XB ), (5) 1 1 A1 () t = B A+ DF() E B( B X B ), (6) 1 1 A1( t) = B X A+ DF() E XB ( B XB ), (7) A () t = B X A+ DF() E B( B X B ). (8) When the system trajectory is restricted on the sliding surface, i.e., σ = σ = 0, the behavior of the closed-loop system is equivalent to the following reduced order system: 1 1 z = B AXB ( B XB ) z + B DF( t) EXB ( B XB ) z. (9) It is obvious that the sliding mode dynamics is completely immune to the parameter uncertainty if and only if for all possible F(t) the following equation holds: = B DF() t EXB 0. (10) his condition is equivalent to the following two cases: Case I, B D = 0 and Case II, EXB = 0. In this venue, the sliding mode dynamics is governed by the following function: 1 z B AXB B XB z (11) = ( ). It is well-known that the condition for Case I is the matched condition on which many significant works have been constructed. If the matrix pair (A, B) is controllable, then there exists a symmetric positive definite matrix X such that the system (11) is stable [10]. o the best of our knowledge, there are a few papers discussing Case II [7-9]. Since Case II is related to the sliding surface, it is called SSMC to distinguish from the matched condition. he problem of SSMC can be formulated as designing a sliding surface {σ = B 1 X x = 0} such that: P1) system (11) is stable; P) EXB = 0. It is obvious that SSMC is more complex than the matched condition. he controllability of matrix pair (A, B) is only a necessary condition for SSMC. Chan et al., [7] first proposed SSMC by: c 1, E = E B X (1) where E c R q m. he used sliding matrix has the form: 1 B X = [ E K ], (13) q (m where K R q) is the designed parameter. he form (13) obviously owns some conservation. Choi [8] utilized

3 Ji Xiang et al.: Sliding Surface Matched Condition in Sliding Mode Control 3 the LMI technique to deduce a sufficient and necessary condition for the existence of SSMC. Lemma 1 [8]. here exists a linear sliding surface () such that P1) and P) hold if and only if there exist symmetric matrices Y 1 and Y satisfying the following LMIs: 1Γ 0 X = ΓY + BY B > B ( AΓY Γ+ ΓY ΓA ) B > 0 (14) 1 1 where n n symmetric matrix Γ is defined as Γ = I E g E, E g is the Moore-Penrose inverse of E and can be calculated g 1 1 by E = ER ( ERER) ( ELEL) E L, and (E L, E R ) is any full-rank factor of E. Lemma 1 presents a very explicit formula to both judge SSMC and obtain the corresponding sliding surface, but one can not intuitively know the intrinsic property of system satisfying SSMC and the considered parameter uncertainty is constrained on the structural form, DF(t)E. he remainder of this paper will show the intrinsic property of SSMC, followed by the structure of parameter uncertainties being released to more general nonlinear functions. III. MAIN RESULS 3.1 he intrinsic property of sliding surface matched condition heorem 1. here exists a sliding surface () satisfying P1) and P) if and only if the following conditions hold. C1) he triplet {A, B, E} is minimum phase C) rank(eb) = rank(e). Proof: he proof is constructive and accomplished by two Lemmas. Lemma. C1) and C) hold if and only if there exists a symmetric positive definite matrix X, matrix G and positive scalar ε such that: AX + XA ε BB < 0, (15) EX = GB. (16) Lemma 3. there exists a symmetric positive definite matrix X and matrix G satisfying the matrix equation EX = GB, if and only if there are two symmetric matrices X 1 and X such that: 1. X = E X E + BX B (17) he proofs of Lemma and Lemma 3 are shown in the appendix. By Lemma 3, (15) and (16) become two linear matrix inequalities: E X E + BX B > (18) 1 0, 1 BX 1 AE ( XE + B ) + ( E XE + BXB ) A ε BB < 0. (19) By Finsler Lemma [11], (19) is equivalent to: 1 1 B ( AE X E + E X E A ) B < 0. (0) Note that Γ = E q Γ E with Γ E R n and X 1 is a designed parameter, the LMIs (18) and (0) are equivalent to (14). his completes the Proof. Remark 1. Compared with the mathematic formulas of Lemma 1, C1) and C) of heorem 1 disclose the intrinsic property of SSMC. By heorem 1, one can intuitively judge whether the considered system satisfies SSMC. Remark. he inequality (19) can be simplified by: ξ < 0, AE X E E X E A BB (1) for a suitable positive scalar ξ. hus the formula to solve the sliding surface matching condition is expressed by the following LMIs over variables (X 1, X, ξ): E X E + BX B > ξ < 0 AE X E E X E A BB () which is simpler than (14) in Lemma 1 due to not calculating Γ. 3. Nonlinear uncertainties In this section, we extend the above results to a class of more general system with nonlinear uncertainties. Consider the following uncertain system: ( ) x() t = Ax() t + Φ(, t x) + B u() t + f( x, t) y() t = Cx () t (3) n where the uncertainty Φ(t, x) R is a nonlinear function, q q n y R is a middle vector and C R is a constant matrix. he definitions of the remaining vectors and parameters are the same as the system (1). he assumptions A1) and A) are still valid. he following conditions are also assumed valid. A5) he triplet {A, B, C} is minimum phase, A6) rank(cb) = rank(c), A7) here exists a known positive function ϕ(x) such that Φ(t, x) y ϕ(x). heorem : Given uncertain system (3) with assumptions A1), A) and A5)-A7), then there exists a sliding surface σ = B X 1 x with the symmetric positive definite matrix X coming from the following LMIs over the variables (X 1, X, ξ):

4 4 Asian Journal of Control, Vol. 9, No. 3, September , X = C X C + BX B > ξ < 0, AC X C C X C A BB (4) such that the reduced-order sliding motion system is asymptotically stable and is completely immune to the uncertain term Φ(t, x). Proof. By heorem 1, the LMIs (4) are feasible for system (3) and the middle vector y(t) is enforced zero when the system is restricted on the sliding surface. From assumption A7), we know Φ(t, x) = 0 when y(t) = 0. So the sliding motion is not influenced by the uncertain term Φ(t, x) and is asymptotically stable. Remark 3. For the parameter uncertainties, the condition A7) is weaker than A3) and A4). he nonlinear function Φ(t, x) is permitted to be non-differentiable, piecewisecontinuous, or discontinuous. It is easy to see that DF(t)Ex(t) is a special case of Φ(t, x) with ϕ(x) = δ D, whereas it is infeasible to express Φ(t, x) by DF(t)Ex(t) due to the restriction of F (t)f(t) δi. 3.3 Synthesis control law By heorem, the predefined sliding surface is σ = B X 1 x. Letting G = CBX yields y = Gσ. Now the following result can be established. heorem 3. he trajectory of System (3) can reach and subsequently remain on the sliding surface σ = 0 after finite time by the following control law: u( t) = ( B X B) BX Ax G B X ϕ( x) σ (5) 1 σ ( B X B ρ ( x, t) +η) σ where η is a positive scalar. Proof. Consider the Lyapunov function V = 0.5σ σ. Its derivative along the sliding surface is: ( 1 1 = σ B X Ax + B X Φ t x V 1 1 (, ) + B X Bu() t + B X Bf(, t x). ) (6) Substituting control law (5) into (6) and using assumption A), A7) and y = Gσ yield: V η σ, (7) which implies the sliding surface can be reached in finite time and then remain. his completes the proof. IV. NUMERICAL EXAMPLE o show an example, consider the Lorenz system [1], the famous model representing the thermally forced nonconservative hydrodynamical systems, namely: x 1 = a( x x1) x = xx bx1 x, x 3 = x1 x cx3 (8) with a = 10.0, b = 8.3, and c = 8/3. Without control, the origin is an unstable equilibrium point and the Lorenz system will present the chaotic dynamics [13]. Our goal is to stabilize the system (8) to origin by adding one controller at the variable x 1, i.e., x 1 = ax ( x1) + ut ( ). he closed-loop system thus formed can be described in the form of (3) with: a a A= b 1 0, B = 0, Φ( t, x ) = x y c 0 x and yt () = x1 (9) Regarding the nonlinear function Φ(t, x) as the uncertain term, one can see that it is mismatched. It follows from (9) that C = [1 0 0], which is qualified for Assumptions A5) and A6). Please note that we can not find a norm bounded function F(t), and constant matrices D and E such that Φ(t, x) = DF(t)Ex. herefore, although Φ(t, x) can be also thought of as a well-structured function because of the existence of constant matrix C, it does not belong to the considered uncertainties in [7,8] and [10]. Let ϕ(x) = x + x 3, then Φ(t, x) y ϕ(t, x). By theorem, the sliding surface can be worked out to be σ = [1 0 0] x. (30) Meanwhile, G = 1.0 is also obtained. Application of heorem 3 yields the following control law with η = 0.1, σ u σ (31) +κ () t = ax ax3 ( x + x3 ) 0.1 σ where κ = 0.01 is used to smooth the unit vector to avoid chattering. In [10], we show the mismatched uncertain term Φ(x) is unrelated to chattering and the global asymptotically stability are not influenced by this manipulation avoiding chattering. he simulation results are shown in Fig. 1 with initial state x(0) = [10, 0, 0]. Before control law injection, the dynamics of autonomous system (8) is chaotic. he control input is executed after t = 10s. From Fig. 1, it can be seen that the trajectories of the system (8) quickly enter into the sliding surface x 1 = 0 and then remain on it after t =

5 Ji Xiang et al.: Sliding Surface Matched Condition in Sliding Mode Control 5 APPENDIX Lemma. C1) and C) hold if and only if there exists a symmetric positive definite matrix X, matrix G and positive scalar ε such that: AX + XA ε BB < 0, (15) EX = GB. (16) Proof. (Necessity) Assume that (C1) and (C) hold. (C1) means that: Fig. 1. he history of system (8) with initial state x(0) = [10, 0, 0] and with control law (31) injected at the time constant t = 10s. 10s. he closed-loop system asymptotically converges to the origin in spite of the existence of mismatched nonlinear uncertainties. sin A E rank = n+ q, 0 B for all Re( s ) > 0. (A.1) Without loss of generality, we assume that the input 0 matrix has the form of B =. Partition E as E = [E 1 I m q (n m) q m E ], where E 1 R and E R. It follows from (n m) m (C) that E is of full row rank. Define L 1 R and m m L R as follows: 1 1 = 1 L E ( E E ) E, (A.) V. CONCLUSIONS he intrinsic property of the recently developed SSMC that the mismatched uncertain term should satisfy the minimum-phase condition is disclosed in this paper. his results in the SSMC being able to be judged intuitively. he uncertainties satisfying SSMC are then extended to a class of more general nonlinear functions. Finally, a numerical example validates the proposed results. E L =. E he system matrix is also partitioned as: A11 A1 A =, A1 A (A.3) (A.4) (n m) (n m) m where A 11 R and A R m. By performing elementary column and row operations, we have: si n m A11 A1 E 1 sin A E si n m A11 E1 A1 sim A E B 0 A1 E 0 Im 0 sin m A11 + L1 A1 0 sin m A11 + L1 A1 0 E A1 0. A1 E EA 1 EE si n m + E A11 L 1 1 A1 A (A.5)

6 6 Asian Journal of Control, Vol. 9, No. 3, September 007 Noting (A.1), this implies the matrix pair ( A11 L1 A1, E A 1) is detectable. hus, there exist a (n m) (n m) symmetric positive definite matrix X 1 R and (n m) (m q) matrix K R such that: X ( A L A KE A ) R ( A L A KE A ) X < 0. (A.6) We define the symmetric positive definite matrix X n n as: X1 X1( L1 KE ) X =, ( L1 E K ) X1 X + ( L1 E K ) X1( L1 KE ) (A.7) (n m) (n m) m m where X 1 R and X R are symmetric positive definite matrices. By Schur complement Lemma [11], we know X > 0. For simple presentation, we denote X3 = ( L1 E K ) X 1 and X = X + ( L1 E K ) X1( L1 KE ). Set G = E1X3 + EX. It can be seen that the Eq. (16) holds. Now, we prove X also satisfies the inequality (15). Substituting X into the left side of (15), it can be obtained that: where Ω 1 Ω3 Ω = AX + XA ε BB =, (A.8) Ω3 Ω 1 = Ω X ( A L A KE A ) ( A L A KE A ) X, 3 = , (A.9) Ω A X A X X A X A (A.10) Ω = A X + A X + X A + X A εi m (A.11) For any ε 0 > 0, there always exists a positive scalar ε such that Ω < ε 0 I m < 0. Noting (A.6), Ω 1 < 0 holds. hen applying the Schur complement formula, we know that Ω < 0 if: < 1+ε < 0. Ω Ω Ω Ω Ω Ω Ω (A.1) However (A.1) obviously holds for large enough ε 0. his implies the inequality (15) holds. In the beginning we assume that input matrix B has a regular form. Since coordinate transforming doesn t influence the feasibility of (15) and (16), the necessity is proven. (Sufficiency) Assume that (15) and (16) hold. It is known that rank(eb) rank(e). From (16), we have EBG = E XE, which in turn implies rank(e) = rank(e XE) rank(eb). So rank(eb) = rank(e), i.e., (C) holds. Suppose there exists a non-zero complex vector v1 such that: v λin A E v1 = 0, B 0 v (A.13) where V 1 C n and V C q. hen the following equations can be derived: 1 =λ 1+, Av v Ev (A.14) Bv 1 = 0. (A.15) Pre- and post-multiplying (15) by we have: * * * * * v 1 and V 1, respectively, ( λ +λ ) vxv+ vexv+ vxev < 0. (A.16) Noting that EXv 1 = GB v 1 = 0, it can be followed that Re(λ) < 0, which implies that (C1) holds. Proof is complete. Lemma 3. there exists a symmetric positive definite matrix X and matrix G satisfying the matrix equation EX = GB, if and only if there are two symmetric matrices X 1 and X such that: 1. X = E X E + BX B (17) Proof. (Sufficiency) Set G = EBX. hen from (17), it can be directly derived that EX = GB. (Necessity) Without loss of generality, assume that the rank of matrix E is q. Define a n (n q + m) matrix H 1 as: then H 1 = E B, (A.17) E I E B H E 0 EB n q 1 =. (A.18) Note that rank(gb E ) = rank(exe ) = rank(e), so rank(eb) = rank(e) = q. hus, from (A.18), it follows that rank(h 1 ) = n. hen one can easily draw out (n m) columns from E n (n m) to construct the matrix H R such that the matrix [H B] is nonsingular. hus the symmetric positive definite matrix X can be expressed as: P1 P3H X = [ H B], (A.19) P 3 P B

7 Ji Xiang et al.: Sliding Surface Matched Condition in Sliding Mode Control 7 (n m) (n m) m m (n m) m where P 1 R, P R and P3 R. Substituting (A.19) into EX = GB and noting EH = 0, we have: 3 + =, EBP H EBP B GB (A.0) which is equivalent to H EBP3 EBP G = 0. (A.1) B Note that the matrix [H B] is nonsingular, the following equations can be derived: G = EBP (A.), 3 = 3. BP E H (A.3) where H 3 is an arbitrary (n q) (n m) matrix. he matrix H can be rewritten as: = 4. H E H (A.4) (n q) (n m) where H 4 R. Now, noting E H3ΨH3 E = BP3 ΨP3B and using (A.19), (A.3), and (A.4), we have: Ψ 3 X = E ( H PH + H H + H H H H ) E 3 Ψ 3 + BP ( + P P) B, (A.5) (n q) (n q) where Ψ R is an arbitrary symmetric matrix. Let X1 = H4PH 4 + H 3H4 + H4H3 H3ΨH 3 and X = BP ( + P3 ΨP3) B, then the Eq. (17) holds. his completes the proof. REFERENCES 1. Utkin, V.I., Equations for Sliding Mode in Discontinuous Systems, Autom. Remote Contr., Vol. 1, No. 1, pp (1971), Vol., No., pp (197).. Drazenovic, B., he Invariance Conditions in Variable Structure Systems, Automatica, Vol. 5, pp (1969). 3. Kwan, C., Sliding Mode Control of Linear Systems with Mismatched Uncertainties, Automatica, Vol. 31, pp (1995). 4. Kim, K.S., Y. Park, and S.H. Oh, Designing Robust Sliding Haperplanes for Parametric Uncertain Systems: A Riccati Approach, Automatica, Vol. 36, pp (000). 5. Choi, H.H., Variable Structure Control of Dynamical Systems with Mismatched Norm-Bounded Uncertainties: An LMI Approach, Int. J. Contr., Vol. 74, pp (001). 6. akahashi, R.H.C. and P.L. Peres, H Guaranteed Cost-Switching Surface Design for Sliding Modes with Nonmatching Disturbances, IEEE rans. Automat. Contr., Vol. 44, pp (1999). 7. Chan, M.-L., C.W. ao, and.-. Lee, Sliding Mode Controller for Linear Systems with Mismatched ime-varying Uncertainties, J. Franklin Inst., Vol. 337, pp (000). 8. Choi, H.H., An LMI-Based Switching Surface Design Method for a Class of Mismatched Uncertain Systems, IEEE rans. Automat. Contr., Vol. 48, pp (003). 9. Choi, H.H., A New Method for Variable Structure Control System Design: A Linear Matrix Inequality Approach, Automatica, Vol. 33, pp (1997). 10. Xiang, J., H. Su, and J. Chu, Sliding Mode Control Design for a Class of System with Mismatched Uncertainties, Acta Automatica Sinica, Vol. 31, No. 5, pp (005). 11. Boyd, S., L.El. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control heory. SIAM, Philadelphia, PA (1994). 1. Chen, G. and X. Dong, From Chaos to Order: Methodologies, Perspectives and Applications, Nonlinear Science, series A, Vol. 4, World Scientific Publishing Co. Pte. Ltd. (1998). 13. Lorenz, E.N., Deterministic Nonperiodic Flow, J. Atmos. Sci., Vol. 0, No., pp (1963). 14. Utkin, V.I., Sliding Modes in Control and Optimization, Springer-Verlag, London (199).