Control of nonlinear non minimum phase s

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/220665252 Control of nonlinear non-minimum phase systems using dynamic sliding mode Article in International Journal of Systems Science · February 1999 DOI: 10.1080/002077299292533 · Source: DBLP CITATIONS READS 13 43 2 authors: Xiao-Yun Lu S.K Spurgeon University of California, Berkeley University College London 127 PUBLICATIONS 938 CITATIONS 321 PUBLICATIONS 5,188 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: A Robust Exact Differentiator Block for MATLAB/Simulink View project All content following this page was uploaded by Xiao-Yun Lu on 03 September 2014 The user has requested enhancement of the downloaded file All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately International Journal of Systems Science, 1999, volume 30, number 2, pages 183± 198 Control of nonlinear non-minimum phase systems using dynamic sliding mode Xiao-Y un Lu ² and Sarah K Spurgeon² A newly developed dynamic sliding mode control technique for multiple input systems is shown to be useful in the control of nonlinear, non-minimum phase systems where the zero dynamics have no ® nite escape time The system may not be dynamic feedback linearizable To achieve asymptotic performance, unbounded control may be necessary as determined by the zero dynamics As long as the growth rate of the zero dynamics is no more than exponential, ultimate bounded performance can be achieved with ® nite control e ort Lagrange stability analysis of the closed-loop system resulting from the proposed variable structure scheme is performed Essentially a thin layer is introduced around the sliding surface Outside the layer, the sliding mode controller is used; inside the layer, the controller is designed to asymptotically (exponentially) stabilize the dynamic compensator It is shown that there is a trade-o between control performance and control e ort The method is illustrated by the control of the Inverted Double Pendulum which is not dynamic-feedback linearizable and is non-minimum phase and thus constitutes a testing example for the proposed scheme Introduction The concept of zero dynamics is not essential in linear control systems design because, as long as the system is controllable, it can be stabilized with static state feedback However, it is generally recognized that zero dynamics play an important role in nonlinear controller design when the system is not exactly linearizable (Isidori 1995, Isidori and Byrnes 1990, Marino 1988) In nonlinear controller design, di erent types of zero dynamics arise from the stability analysis of di erent design methods The zero dynamics of Isidori (1995) and Isidori and Byrnes (1990) where static state feedback is employed are the dynamics of the uncontrolled states The zero dynamics adopted in Lu and Spurgeon (1995, 1996a, b) are the dynamics of the control, which is a generalization of the zero dynamics used by Fliess (1990) Recently, work by Lu and Spurgeon (1995, 1996a, b) proved that if the zero dynamics are uniformly asymptotically stable (minimum phase), then the overall Received 23 July 1997 Revised January 1998 Accepted January 1998 ² Control Systems Research, Department of Engineering, University of Leicester, Leicester LE1 7RH, UK e-mail: xyl@sun.engg.le ac.uk, tel: + 44-116-252 2567; sks@sun.engg.le.ac.uk, tel: + 44-116252 2531 0020± 7721/99 $12.00 Ñ closed-loop system resulting from a sliding mode design is uniformly asymptotically stable However, in many practical situations, a bounded control signal is acceptable There is no need for the control to be asymptotically stable It is thus possible to remove the restriction of uniform asymptotic stability of the zero dynamics relating to the control and still yield useful results Some e ort has been made to control nonlinear nonminimum phase systems The remarkable work in Hauser et al (1989) and Isidori and Byrnes (1990) may be considered as the ® rst step into the control of slightly non-minimum phase systems Here slightly nonminimum phase means that the linearization has no eigenvalues with strictly positive real parts Isidori and Byrnes (1990) use centre manifold theory for stability analysis while Hauser et al (1989) use a minimum phase system to approximate the non-minimum phase system These methods fail if the growth rate of the zero dynamics exceeds certain limits In the paper by Lu et al (1997) the control of a 2-dimensional non-minimum phase system is considered A generalized Lyapunov method is used in considering Lagrange stability of the closed-loop system where the zero dynamics have restricted growth rate However, this method is di cult to generalize if the number of controls is greater than and the zero dynamics have a high growth rate 1999 Taylor & Francis Ltd 184 X.-Y Lu and S K Spurgeon From the work in Isidori (1995), it is known that the zero dynamics essentially depends on the choice of output A nonlinear system may be linearized with respect to some outputs but not others Based on this idea, Gopalswamy and Hedrick (1993) use a modi® ed output to track the reference output while simultaneously rendering the zero dynamics acceptable The output tracking considered is to zero order only which makes the control task slightly easier The work in Tornabe (1992) uses direct dynamic state feedback to control non-minimum phase linear systems and linearizable nonlinear systems If a system is linearizable and locally weakly controllable, there is no zero dynamics Thus the system is minimum phase by de® nition The unicycle control problem in the paper by Sira-Ramirez (1993b) is another example of a non-minimum phase system Nonlinear systems which are linearizable via static or dynamic feedback and coordinate transformation are studied in great detail by Isidori (1995), Marino (1988) and Nijmeijer and van der Schaft (1990) However, the condition for linearizability is very restrictive and thus the class of nonlinear systems which can be linearized is very limited It is noted that a nonlinear system which is static (dynamic) output feedback linearizable must be static (dynamic) state feedback linearizable If a system is not dynamic feedback linearizable, asymptotic feedback linearization appears a very promising alternative approach (Lu and Spurgeon 1995, 1996a, 1998) This paper extends the sliding mode control method developed by Lu and Spurgeon (1996a, 1997) for minimum phase systems to non-minimum phase systems where the corresponding zero dynamics (i.e the dynamics of the control) have no ® nite escape time and a growth rate which is at most exponential It is not necessary for the system to be feedback linearizable The inverted double pendulum discussed by Fliess et al (1995) and Lu and Spurgeon (1996b) and the unicycle example of Sira-Ramirez (1993b) are pertinent examples It is demonstrated that for such systems to achieve asymptotic performance, unbounded control may be necessary (Sira-Ramirez 1993b) However, by using a variable structure control, Lagrange stability can be achieved for the closed-loop system (Lu and Spurgeon 1996b) More speci® cally, ultimately bounded regulation can be achieved for any given ² > with ® nite control by using variable structure control around the sliding surfaces This method may be considered as a type of approximate sliding mode method Outside a thin layer, the dynamic sliding mode controller described in the paper by Lu and Spurgeon (1996a) is used; within the layer, the controller is designed to exponentially stabilize the near zero dynamics which are the dynamics of the compensator when the outputs and their derivatives are su ciently small The design method is demonstrated using the two input Inverted Double Pendulum which is both non-minimum phase and not dynamic feedback linearizable The paper is structured as follows Section provides some background for dynamic sliding mode control and a bound estimate for a type of unstable system; section describes a dynamic sliding mode control design method; section is for partial stability analysis which is related to the performance of the closed-loop system; section gives Lagrange stability analysis of the closedloop system resulting from a variable structure control design; control of the Inverted Double Pendulum is used as an illustrative example in section The following notation will be used throughout Nd x x x R n , x x there holds De® nition 2.4: m; (2) All u " n1 b T T The di erential I± O system (1) is called De® nition 2.1: (1) p , , yp, , np T yp t , u^, t or equivalently by the Implicit Function Theorem and n y1 u " i , , ,i 1, , m, are C -functions; det ¶ ¶ u b u1 , ,u m , , umb is satis® ed with y Nd for all t for some d , d u > De® nition 2.2: is de® ned as 0, u^, t m 0, and for u Nd u u0 Clearly if the trajectories of (4) are exponentially bounded, the trajectories u t of (4) are de® ned for any initial conditions u Nd u u0 and t when " t Nd (no ® nite escape time) Remark 2: To control non-minimum phase behaviour with direct sliding mode methods (Lu and Spurgeon 1997), the differential I± O systems need to be restricted to the following form u p 0, , t u^ Equation (1) is called minimum phase if there exist d u > b b m , such that (3) is uniand u0 R b , b formly asymptotically (exponentially) stable for initial b condition u Nd u u0 , where u u1 , , u1 , b m Otherwise, it is non-minimum phase , um , , um Physically, this zero dynamics may be viewed as the level of control required to maintain the states near zero For non-minimum phase systems, it is necessary to extend this concept to the case when y Nd This leads to a new concept, that of near zero dynamics An advantage of these dynamics will be that the unstable behaviour of the uncontrolled states is transferred to the control It is this advantage which makes it possible to design a variable structure controller to curb the control variable whilst ensuring ultimately bounded regulation Consider (1) as a dynamics of u and n y y1, , y1 , , ym , , ymnm as time variant parameters The following concept arises Remark 2.1: T c is called the bound of exponential growth rate The zero dynamics corresponding to (1) u ,t M exp ct ut (3) Regularity Condition yn u y, u, t u a y, u, t u b u b y, u, t , where the highest order derivative of the control appears n T linearly, y n y1 , , ymnm and u a y, u, t : R u b y, u, t : R De® nition 1’: (a) p (b) n b R R b m m R R R R m The system (5) is proper if m; y, u, t and functions; u a n u b y, u, t are continuous matrix (c) Regularity Condition det u a y, u, t is satis® ed with y Nd for some d > and t > and for u Nd u u0 Without loss of generality, u0 can be chosen zero from now on From the regularity condition (6), it is obtained that 186 X.-Y Lu and S K Spurgeon ub u a y, u, t y n q y, y for y n u a y, u, t u b y, u, t , u, t Nd for some d > 0, t > and for u The near zero dynamics corresponding to (5) is de® ned as u q " t , u, t zÇ b zÇ b b zÇ b " qm t , z, t zb m zt b zt z t0 1: There exist constants Ma, Ma0 , Mb , Mb0, M a 0, such that for " t Nd d > , for all t and z Nd Assumption u a " t , z, t Ma z Ma0 u b " t , z, t Mb z Mb0 t , z, t M u a " 10 a If Assumption is satis® ed, there exists t0 > such that the near zero dynamics (8) is exponentially bounded for initial condition z Nd z , d z d u, for " t Nd and for t t0 c0 zt z t0 exp c t t0 c where c M a Mb 0, c0 M a d Mb0 If c 0, or equivalently Mb 0, for any c1 > 0, there exists T > such that Lemma 2.1: zt c0 exp c1 t , t0 t a c z c Mb z d M d z z dt 2z zÇ t0 c z t0 t0 z Ç n1 z Ç n1 z n1 z Ç 1m z m1 z m1 nm1 u 1 z , u^, t 11 z Ç m1 nm1 z Mb0 T z T From d z dt it is deduced that z Ç 11 Mb0 a d t t0 c0 M a Mb , c0 c0 t Without loss of generality, suppose that n1, , nm1 > and nm1 nm1 m2 1, m1 m2 m The system (5) may be expressed in the following generalized controller canonical form (GCCF) By Assumption and the structure of (9), there exist c, c0 such that M c0 d¿ ¿ The proof for the second assertion is trivial and thus omitted h Proof : zÇ c z c0 ¿ t0 exp c t ¿ d¿ z t0 c0 t t0 z t0 exp c t t0 c0 c0 t t0 exp c t t0 c c0 z t0 z t0 exp c t c c0 c0 z t0 exp c t t0 c c c0 z t0 exp c t t0 c t , z, t " t0 c0 From the Gronwall± Bellman Inequality m t z t0 zÇ c z and z2 q1 z zÇ with " t Nd for some d > b T Let z z , ,z m , z i ui , uÇ i , , ui i The system (8) has the following state space realization zÇ d z dt Nd u De® nition 2’: b d z dt z Ç m1 nm1 u m1 z Ç 1m z z Ç nm u m m where z and z i z 1i , , z ni z , , z m T i z , u^, t m z , u^, t yi , , yi ni ,i 1, , m 187 Control of nonlinear non-minimum phase systems using dynamic sliding mode xÇ 2.2 Sliding surface and sliding reachability condition Based on the above GCCF, the sliding surface should be chosen to yield the required performance of the reduced order system when the ideal sliding mode is reached In this paper, the following direct sliding surface is used ni si j aj i z ji , i 1, , m, zÇ 12 j i where are Hurwitz polynomials with j aj ¸ i ani 1, i 1, , m, which was ® rst proposed for SISO systems by Sira-Ramirez (1993a) and generalized to the MIMO case in the work of Lu and Spurgeon (1996a, 1997) The choice of sliding reachability condition should guarantee that the sliding surface is reached in ® nite time or asymptotically and that an expression for the control may be easily recovered For this purpose, the general sliding reachability condition sTÇs < in the papers by De Carlo et al.(1988) and Utkin (1992) is modi® ed as follows deđ ned by g is,i ầsi such that for i (a) g 13 1, , m i 0; (b) g i s is C (c) g i s is bounded for s 1, , m x t0 , w0, t 0; Nd ; sTg s sTKs The following discontinuous sliding reachability condition is strong Let ·ij R m m be a positive de® nite matrix and ·i m 0, i 1, , m Example: ·1 m g s sgn s1 : ·ij s : ·m m sgn ·i m si sgn sm It is clear that sT ·ij s m si i sT ·ij s t I Let I 0, The equilibrium point w of (14) is partially asymptotically stable if (a) it is partially stable with respect to x; (b) if ² > 0, d > and T > such that w0 satisfying w0 < d , x t0 , w0 , t a Theorem V:I V i VÇ 1: q R ,i x ² t > T Suppose there exists a function R such that, for some functions q R , 1, 2, 3, and every t, w I V V t, w t, w a a x ; x Then (i) for any d > and t0 , w0 I Nd V x t; t0 , z0 uniformly in t0, z0 when t ; R m , (ii) w is uniformly asymptotically stable with respect to x 2.4 Di erential equations with discontinuous right hand side Due to the presence of uncertainty which may be discontinuous and the use of a discontinuous control action in sliding mode design, di erential equations with discontinuous right hand side must be considered Such equations are extensively studied by Filippov (1964) and Paden and Sastry (1987) The main results of Filippov (1964) will be cited here Consider the system xầ f x, t 15 R R satisđ es the following conwhere f , : R dition: Condition B: There exists an open set Q R n , Q, such that f , is de® ned everywhere and is Lebesgue measurable almost everywhere in Q R For any T 0, n 2.3 Partial asymptotic stability Consider the pair of di erential equations ² De® nition 2.7: (b) (d) There exists a positive de® nite matrix K such that sTg ·, s The equilibrium point w of (14) is partially stable with respect to x if ² > 0, d > such that w0 satisfying w0 < d , (a) a -function of s if si 14 De® nition 2.6: A strong sliding reachability condition is De® nition 2.5: , g t, x, z , n R , g: I q q q R R R where f : I V V are continuous functions, V is a domain, i.e open connected subset, in R n and V , and I 0, is the maximal interval of existence of solutions for x0 , z0 Nd , d > f t, 0, g t, 0, 0 Let w x, z The following results relating to partial asymptotic stability are quoted from Rouche et al (1977) i n f t, x, z 188 X.-Y Lu and S K Spurgeon there exists a Lebesgue integrable function BT t such that f x, t BT t for almost all t 0, T Under Condition B, for any initial condition x t0 x0 Q, t0 0, T , there exists an absolutely continuous solution of (15) for t t0 , T Typical examples of such functions are sign function sgn and saturation function sat The systems considered in this paper automatically satisfy Condition B g s Çs From (19) the highest order derivatives of the control b T ub u1 , , umb m are obtained as ub p z , u, t u a y, u, t x n1 j Direct sliding mode control 3.1 Design method Step Choose sliding surfaces (12) Step Choose a strong sliding reachability condition, i.e sTg s sTKs Step Choose the sliding gain matrix K such that T BD0 K BD0 > 0, 0m2 m2 16 where D0 diag d1 , , dm1 with di 0, , T 0, of dimension ni for i 1, , m1 and dj of dimension for j m1 1, , m; A diag A1 , , Am1 with Ai the companion i ni matrix of the Hurwitz polynomial j aj j i ¸ with ani 1, A and B satisfy the Lyapunov equation T T A B BA In g s j ni Çs 11 j nm j 1 aj aj i j z z i j j m z u a y, u, t u y, u, t b 18 j ni j nm j aj aj z aj i m j 1 m i j z aj m 20 , i z j z m j 1 zÇ zÇ b z2 zb 1 Çzb p1 m Çzb zÇ b 1 z , z, t 21 m zÇ z2 m m m m z1 , , zb i zi m zb pm z , z, t , m ui , uÇ i , , ui i i b i , 1, , m z , ,z m z 22 T Equation (21) together with (11) yields a closed loop m m system of dimension i ni i b i which can be written as: j z aj i j z if the Regularity Condition is satis® ed Note that pi z , u, t is a continuous function if si because g i is C if si This dynamic feedback can be realized in canonical form by introducing the pseudo-state variables as i Now set n1 1 where u b aj j nm Step Di erentiating (12) with respect to time t along the trajectories of (11) and considering (5) leads to n1 y, u, t u b 17 m ni aj u a u b y, u, t u y, u, t b g s, 19 m j where s is as de® ned in (12), to determine the feedback control Equation (18) in s-coordinates becomes zÇ F z , z, t zÇ P F z , z, t P z , z, t z , z, t , , z n1 , u 1; ; z 2m , , z nm , u m T 1 m m T z2 , , zb , p1; ; z2 , , zb , pm z m m 23 where zÇ is replaced with P z , z, t 189 Control of nonlinear non-minimum phase systems using dynamic sliding mode S tability of direct sliding mode z Ç 1i z Ç ni i z Ç ni z i z i ni z Ç ni i z K BD aj u j i z j i si Cx, x z Az Çs zÇ P z , s, z, t z , u^, t g js > 0, 26 27 C T 2s s T 28 BD T x s ,s 2G x, s s, g 2G x s, Ks C Q BD g s 24 P z , s, z, t z2 , , zb , p1; ; z2 m , , zb m , pm T, vT1, vT2 0, " 0, If v2 If v2 2G where A and D are as de® ned in Step of the algorithm C H T 2G K 0, v1 HT K R l , v2 R m , v1 v2 vT1Cv1 > because C is positive de® nite vT1Cv1 " m 4.1 Some stability results Some stability results for a particular nonlinear system are presented here These will enable the stability of the system (23) to be analysed T T BD is positive de® nite Let v vT1, vT2 T and H BD G Consider " T 2G It follows that Vầ is negative deđ nite if and only if Ds T BA BD xT, sT Q for j m1 1, , m Putting them together, the closed-loop system (23) can be written equivalently as Ç BD < Bx, x > Cx, x VÇ is transformed as Çsj 1 2G C which is positive de® nite and radially unbounded Di erentiating (28) with respect to t along the trajectories of (25) 1, , m1; and the jth degenerate block zÇ T Proof: The proof is carried out in several stages Consider the following Lyapunov function candidate g is j sTKs and K R m m (2) The matrix K satisđ es V j ầsi for i sTg ·, s Then (25) is globally exponentially stable z , u^, t i ni 25 where A R l l is Hurwitz, G R m l , and D R l m (1) g s g s , , g m s T is a strong sliding reachability condition which satis® es ATB is transformed as z Ç 1i Ax Ds, Gx g s , xầ ầs where C is chosen positive deđ nite and A, B, and C satisfy the following L yapunov equation: u i i Let s s1, , sm and consider the following time invariant nonlinear system Lemma 1: This section formally analyses the partial asymptotic stability of the closed-loop system First, the closedloop system is transformed into a proper form for stability analysis Then a fundamental lemma concerning the stability of a particular nonlinear system is proved Finally, the partial stability of the closed-loop system can then be proved First note that (12) determines a linear coordinate transformation between z z , s , where z z 11 , , z n11 1, , z 1m , , z nmm Under this coordinate transformation, the typical ith non-degenerate block given by T > v1 Cv1 2vT1HTv2 vT2Kv2 2vT1HTv2 vT2 HC 1HT v2 Because C > 0, there exists a positive de® nite matrix P such that C P2 Then " T > v1 PPv1 Pv1 2vT1PP HTv2 HP T v2 , Pv1 vT2 HP P HT v2 HP T v2 190 X.-Y Lu and S K Spurgeon This proves that Vầ is negative deđ nite along the trajectories of (25) if (26) is satis® ed Because V is essentially time invariant, the discussion above implies that ¸min Q0 x V Vầ x ámax Q0 x ámin Q 2 , 29 , where x T xT, sT and Q0 diag B, 12 I (25) is then globally exponentially stable by the Lyapunov Theorem (Hahn 1967) If g i s is discontinuous at si 0, i 1, , m, from (29), ¸min Q V ámax Q0 Vầ x ámin Q t ámax Q0 exp ¸min Q0 , 30 x For the system (30) and arbitrarily ® xed > 0, choose L to have distinct eigenvalues with negative real parts Then there exists µ > such that the solution of (30) has the following estimate: t | t0 , t0, t , where h only depends on L and is independent of x t0 Proof: h>0 , µ | and Choose the matrix with w1 , , wn a set of n linearly independent eigenvectors of L Then G W L W is a diagonal matrix with G ii the eigenvalues of L Now the solution of àL xầ x may be expressed as xt x t0 exp x t0 I µL µL t t0 t t0 2! µL G t µ t0 G t µ t0 W xt x t0 W W µa h x t0 exp max Re ¸i i n G exp t µ t t0 t0 , L max ¸i i n L , µ | >| µa h x t0 exp exp | t t t0 t0 h as required This completes the proof 4.2 Partial exponential stability of closed-loop system Partial asymptotic stability implies that to achieve asymptotic performance, unbounded, but no ® nite escape time, control e ort might be necessary In the following theorem, the exponential boundedness for z t will be relaxed to no ® nite escape time, i.e Assumption is not necessarily satis® ed Theorem 4.1: Suppose (a) the near zero dynamics (7) have no ® nite escape time for z U0; (b) the sliding reachability condition is strong; (c) K R m m in the sliding reachability condition (De® nition 2.5) satis® es w1 , , wn W W and h x t0 | h exp x t0 W exp xt xt Lemma 2: x t0 t0 will lead to The following result veri® es that for a linear stable system, arbitrarily given decay rate can be achieved via the selection of one parameter Consider xt t0 t Thus h µL t µL I W 1x t0 W exp This completes the proof xầ àL where h W W which is independent of and x t0 Thus choose µ such that ¸min Q t ¸max Q0 exp W 1x t0 W W 2! a Thus V W 1x t W t t0 K BD0 T BD0 0m2 > 0, 31 m2 where A diag A1, , Am , Ai is the companion matrix i n of the Hurwitz polynomial j i aj ¸j with anii for i 1, , m D0 diag d1 , , dm1 , di 0, , 0, , dim di ni 1, i 1, , m1 A and B satisfy the L yapunov equation (17) Let x z , s Then the closed-loop system (17) is exponentially stable for z when z , z Nd U0 191 Control of nonlinear non-minimum phase systems using dynamic sliding mode First note that (24) is equivalent in stability with (23) due to the linear transformation (12) from z to z , s Thus there exists d > such that all the conditions above hold for z , s Nd 0, and z U0 Now consider (24) Note that the ® rst two equations are in the form of (25) with l n m, and D replaced by diag D0 , 0m2 m2 and G Now replacing D with diag D0 , 0m2 m2 and setting C I, G in (26) produces (31) Thus the results are obvious from Lemma 4.1 Proof: This result guarantees that exponential regulation may be achieved possibly with unbounded but no ® nite escape time control e ort The next stage will be to introduce the variable structure control to stabilize (bounded-input-bounded-state) the control e ort This will lead to Lagrange stability of the closed-loop system Remark 4.1: i If Assumption is satis® ed, z t , i.e the trajectories of the dynamic compensator, are exponentially bounded with growth rate c as in Lemma 2.1 The control varii ables z1 ui , i 1, , m, are reduced exponentially ²0 as and independently of each other when E z follows: i z2 zÇ z3 i zÇ b zÇ b i i i 1 and has i ,1 i ¸min ki b ¸ki m L i , where L i , i 1, , m, are the companion matrices of the Hurwitz polynomials in (32) Step Choose the initial conditions z Nd and z Nd z for (23) such that they are compatible, where d > is su ciently small and < d z d u where d u is the same as in the Regularity Condition (1) If E ²0, the closed-loop system (23) can be rewritten as follows: Ç z A0z Ç z u a z , z, t P0 z , z, t u b 33 z , z, t P z , z, t zÇ where P0 z , z, t p1 , , pm , z z n11 , , z nmm , m i A0 diag A0 , , A0 and A0 is of the following form T i A0 0 0 ni ni (2) If E < ²0 , Ç z A0z , Ç z u a z , z, t ầzb àL zầ where zb zb 1 u b z , , zb m , L diag m b µ j zÇ b 34 z , z, t , b µ a i j 1 zj i zj i m zj m j a j L 1, , L m and i zầ i b > c, ámin Lagran ge stability of the closed-loop system Step Let < ²0 < 1, E z , s If E > ²0, the dynamic feedback design from section is used If E ²0, a further controller is designed to asymptotically (exponentially) stabilize the dynamic compensator with an attenuation rate greater than the growth rate of the near zero dynamics Note that choosing E s , E z and E z , s are equivalent theoretically because of the choice of sliding surface and sliding reachability condition In practice, choosing E z is more direct and convenient for implementation This is equivalent to introducing a layer of thickness ²0 about the sliding surface Note that in the paper by Slotine and Coetsee (1986) a thin layer around the sliding surface is used to reduce chattering The controller within the layer is also variable structure in nature and is designed as follows i b where j i a j ¸ j is Hurwitz with a distinct roots, i 1, , m satisđ es b i zb b µ i j a m µ 32 i i j i zj i , j a j The Lagrange stability analysis is now carried out for the closed-loop system which is governed by (33) and (34) in a variable structure form The main idea here can be sketched as follows Suppose the time interval t0, 192 X.-Y Lu and S K Spurgeon is partitioned into the following two sequences of disjoint sub-intervals: t0 , t1 t1 , t2 , t2, t3 , , t2i , t2i , , 35 , t3, t4 , , t2i 1, t2i , , 36 where ti is a monotone strictly increasing sequence such that ti i t1 is the minimum time instant such that z t1 where s , µ and h are the same as in L emma 4.2 In particular, if µ is bounded and z t2i is a bounded sequence, then Mu i is bounded (3) The bound for z t in the time interval ²0 t t2i 1, t2i , D t t2i t2i < 1, with z t2i has the estimate z m t j ² Note that ti is the time instant at which the con® guration switches from one controller to the other In the time intervals (35), the sliding mode controller is used, i.e the closed-loop dynamics are governed by (33) In this situation, the z -trajectory will be exponentially stable with a prescribed decay rate while the z-trajectory will be exponentially bounded In the time intervals (36), the closed-loop dynamics are governed by (34) In this case, the z -trajectory will be bounded by a prescribed ² > while the z-trajectory will be exponentially reduced to a prescribed bound d z > Let D tmin and D tmax denote the minimum and maximum time steps in simulation or on-line signal measurement From now on, i denotes the ordinary number of the sub-interval pairs t2i 2, t2i t2i , t2i , i 1, 2, It is important to note that according to Condition B the closed-loop trajectories of (33) and (34) respectively are absolutely continuous The trajectories of the overall closed-loop system are continuous when variable structure control is adopted It is necessary to note that to estimate the trajectory bound x t for t tu , tạ of the dynamical system f x, t xầ it is su cient to know the initial condition x tu and the bound on the growth rate f x, t on the same interval First the following estimate is given Under Assumption 1, (1) for t t2i , t2i , i 1, 2, , the z t trajectory evolving as in (33) has the following estimate Lemma 5.1: c0 z t2i exp c t t2i c (2) for t t2i 1, t2i , i 1, 2, , the upper bound for u in (34) has the following estimate u zt µhMa µhMa0 Mu i z t2i z t2i exp Mb z t2i 2s 1D exp tmin s 1D tmin Mb0 37 nMu i D t t ²0 D The estimate in (1) directly comes from Lemma 2.1 For t t2i 1, t2i , the growth rate of z t can be estimated as follows From (33), Proof: u z , z, t zÇ b u a u b z , z, t while àL zầ z and by the choice of L , there exists s zt z t2i zÇ b Remark 1: nj nj n²0 µh exp s 1 > such that t t2i , , zt where h is the same as in Lemma 4.2 In particular z t2i exp u b z , z, t z t2i s t2i t2i Thus u u a z , z, t zầ b àh u a µh z , z, t Ma z µh Ma0 Ma z µh Ma z µh Ma z t2i µh Mu i Ma0 , µ z z t2i u b z z , z, t Mb z Mb0 µh Ma0 Mb z Mb0 µh Ma0 Mb z Mb0 exp Mb z t2i 2s 1D tmin exp s 1D tmin Mb0 Note that Mu i , is a monotone increasing function of z t2i when is đ xed; it is a monotone increasing function of µ when z t2i is ® xed The last assertion in (2) is obvious This proves (2) To prove (3), notice the canonical structure of (11) For j 1, , m, 193 Control of nonlinear non-minimum phase systems using dynamic sliding mode z Ç 1j z zÇ j j ni z z ầ nj u aj j (5) choose > big enough in (34) such that µ j j nj z , u^, t zÇ b u aj z , u^, t u bj z ²0 nj t2i , t2i Mu i D t ²0 j z j nj D t ²0 Mu i D t ²0 D j ²0 z ²0 ²0 D j t t2 ² 0D ²0 D tnj Mu i D tnj Then nj m z t c0 c d z z k j k j m n²0 j nj nj ² 0D t nMu i D t This proves (3) h From Assumption and Lemma 5.1, z t will not a ect the growth of z t However, z t , particularly the initial condition sequence z t2i , may a ect the bound of z t through the e ect on its growth rate Remark 5.2: Suppose that for any given ² > given, (1) the sliding reachability condition is chosen strong as in De® nition 2.5; (2) K in the sliding reachability condition satis® es (16); (3) Assumption is satis® ed; (4) D tmax is chosen to satisfy: 1, tmax ² m nj nj 2n j ² nMu , 38 where Mu L µh Ma L 22 exp µh L2 h 2s 1D c0 c d s 1D tmin s 1D tmin Mb0 c /s hL , d z Nd ; z t0 , z t2 and for Suppose ² d and let ²0 ² /2n Consider the closed-loop systems (33) and (34) The proof is by induction on each pair of sub-intervals t2i , t2i t2i , t2i , i 1, 2, , in which the bounds for z t and z t are estimated for z t0 , z t0 Nd Nd z ² for t We will prove that z t t1 and that d z is always achievable by proper choice of z t2i design parameters This also implies that z t is bounded Proof: S -1 Consider the ® rst pair of sub-intervals t t0 , t1 t1 , t2 , i For t t0 , t1 , z t has the following estimate z t z t0 exp s t z t0 exp s t1 40 t0 In particular z t1 t0 To render z t ²0 z t0 e s t1 t0 it is su cient that z t1 The time period needed to render following estimate t0 s ln z t0 ²0 ²0 z t ²0 zt From (41) z t0 has the 41 If Assumption is satis® ed, for t t0 , t1 the following estimate due to Lemma 2.1 ²0 39 , ¸min 1D tmin ln and t1 c0 c /s c and µ and h are the same as in L emma 4.2 exp d z z is uniformly ultimately bounded by ² for z t0 tmin Mb L exp Ma0 d z ²0 c /s h L1 Then there exists a variable structure feedback control as in Step such that Theorem 1: D d ln (b) z t L for t Nd Nd z t D L1 (a) t Inductively z ¸minD tmin ni The bound for z t in the time interval t has the following estimate z max c0 exp c t c , zt t0 has 42 194 X.-Y Lu and S K Spurgeon z t1 z t0 c0 exp c t1 c z t0 c0 c c0 c d z z t0 In summary, for t t0 , t1 t1, t2 if (44) and D tmax as in (46) then t0 c /s c /s d exp z t1 s exp h z t1 t s t1 t h z t1 exp h L exp , t1 43 D , i.e z t is exponentially decreasing in t1 , t2 To render h L exp z t2 s 1D tmin d z it is su cient to choose µ such that z t2 h L exp s 1D tmin h L exp àáminD tmin d z or ¸minD tmin h L1 ln 44 d z ²0 when t It is true that z t t1 , t2 except at the last time step Thus by (3) of Lemma 5.1, which gives the growth rate of z t in this sub-interval, it holds that m z t2 nj nj n²0 j tmax ² 0D nMu D D m j nj nj tmax ² 0D nMu D tmax tmax 1, tmax ² ² m j then z t nj nj ² 2n ²0 ²0 ² ,t z t2 ² d nMu t1, t2 In particular , t1 t3 t2 z t2 ln s s ²0 ln 2n 47 by (47) For t t3 , t4 zt h L exp s h L exp s 1D tmin t t3 Thus µ in Lemma 4.2 is chosen such that s hL exp s 1D tmin àáminD satis® es tmin d z 49 Or equivalently µ or equivalently D tmin hL exp d z t similar to (41) Under Assumption 1, the near zero dynamics (9) of z t is exponentially bounded according to Lemma 2.1 For t t2, t3 c0 zt z t2 exp c t t2 c c0 d z exp c t t2 c 48 c0 d z z t3 exp c t3 t2 c c0 c d z 2n /s L2 c tmax is chosen such that D z t2 tmax 45 Thus if L 1, Consider the interval t t2 , t3 t3, t4 ² It is noted that z t2 2n²0 The time period ² has the following t2 , t3 needed to render z t3 estimate s 1D tmin s 1D tmin z t1 s S -2 where h is the same as in Lemma 4.2 , and z t2 h L exp zt For t t1 , t2 , due to the control choice in (34), the following estimate is true for some s , c0 0, and c > zt ² z t2 L ²0 is chosen as in , , z t , ámin D tmin hL ln d z 50 Then 46 z t4 Similar to (46) if D tmax D tmax is chosen such that 1, ² z t ²0 ²0 d z ² ² 2n j ,t m nj nj t3, t4 nMu 51 195 Control of nonlinear non-minimum phase systems using dynamic sliding mode In summary, for t t2 , t3 t3, t4 if µ and D tmax are chosen as in (50) and (51) respectively, it holds that , ², z t ² z t4 zt L 2, z t3 L 2, z t4 d z i For t t2i 2, t2i t2i that the following estimate is true (ai ) zt L (b1 ) Mu i Mu (ci ) ² z t (d1 ) ² z t2i (e1 ) d z z t2i S -i , t2i , suppose L 2, Mu , Mu i z t2 i , ², z t2 i d z z t The inverted double pendul um The design procedure will now be illustrated using the two input Inverted Double Pendulum The system is non-minimum phase The model is quoted from the paper by Fliess et al (1995) and shown in ® gure Suppose the control variables u, v yÇ , Çz are the velocities of the suspension point y, z The equations of motion are p1 Then repeating the process carried out as in S -2, it is easy to prove that for t t2i , t2i t2i 1, t2i , the following estimates are true if µ and D tmax are chosen as in (50) and (51) respectively z t2i (2) As a special case, if z t O tr for some r > 0, µ > can be chosen arbitrarily (3) The bound for control has an explicit estimate L For ® xed L , there is a trade-o between s , ²0 , D tmin, D tmax and d z ; (4) L cannot be made arbitrarily small even when s (or equivalently µ) is chosen arbitrarily big I xÇ cos a x2 N1 yÇ cos x1 N1 zÇ sin x1, I xÇ cos x1 x2 N2 yÇ cos x2 N2 Çz sin x2, pÇ N1 g sin x1 N1 xÇ yÇ sin x1 N1 xÇ zÇ cos x1 , pÇ N2 g sin x2 N2 xÇ yÇ sin x2 N2 xÇ zÇ cos x2 , p2 yÇ ² I1 xÇ zÇ I2 xÇ u, v, 52 It is thus proved by induction that zt L 2, z t ² t , x2 Beam t2 and as a by-product Mu i Mu , i This proves (a) and (b) This completes the proof 3, 4, z h (1) If z t is polynomially bounded, i.e zt O tr for some r > 0, as in the case c in Lemma 2.1, it is also exponentially bounded when t T for some T > Then the argument above is still true with t0 T Similarly, it is thus concluded that all the cases when the zero dynamics have growth rate no more than exponential from some time point T > onwards, the Lagrange stability analysis can be carried out in the same way Of course, z t should have no ® nite escape time in the region concerned g v Remark 5.3: x1 Beam u O y Figure Inverted double pendulum 196 X.-Y Lu and S K Spurgeon where p1 and p2 are the generalized impulsions associated with the generalized coordinates x1 and x2 respectively, and g, I, I1, I2 , N1 and N2 are constant physical parameters de® ned by I1 M1 M2 l12 N1 M1 M2 l1, N2 M2 l2 , I , I2 M2 l1l2 , In the following discussion, l1 0, l2 5, M1 1, M2 0, g 81, h Note that u and u are smooth near the origin because , ¶ , ¶ M2 l2 , u u uÇ vÇ I xÈ cos x1 x2 N1 vÇ sin x1 N1 g sin x1 I xÇ xÇ xÇ sin x1 x2 N1 uÇ cos x1 , I Èx1 cos x1 x2 I2 xÈ N2 vÇ sin x2 N2 g sin x2 I xÇ xÇ xÇ sin x1 x2 N2 uÇ cos x2 The system is required to asymptotically track the constant trajectory x1 , xÇ , x2 , xÇ 0, 0, h, with h and < h < uÇ u I cos e1 e3 h gN2 sin e3 N2 uÇ cos e3 h N2 vÇ sin e3 I sin e1 e3 h e2 e2 N1 uÇ cos e1 I sin e1 h e4 e2 Çe3 e4, Çe4 u e3 N2 I cos2 h N1 I I1 gN2 sin e3 n2vÇ sin e3 h I sin e1 I cos e1 e3 N1 vÇ sin e1 I1I2 e3 uÇ e1 ,Çe1 ,e2 ,Çe2 d c3 vÇ e1 ,Çe1 ,e2 ,Çe2 d c4 e3 , h e2 e2 e4 e2, e4, / , s2 , Çs1 k11 s1 k12s2 k10 sgn s1 Çs2 k21 s2 k22s2 k20 sgn 54 where e4 N1uÇ cos e1 h e4 e2 a1 e1 a1 e3 where a1 , a1 , 2, are Hurwitz (e) Choose a sliding reachability condition with switching h gN1 sin e1 h I sin e1 I2 cos2 e1 e3 They are of the form uÇ c1 , vÇ c2 , where c1 , c2 are constants This means that, to achieve asymptotic regulation, u O t and/or v O t t is expected d , there exist constants From (53), if e1 , eÇ 1, e2 , eÇ c3 , c4 such that s1 s2 N1 vÇ sin e1 N2 uÇ cos e3 h N2 I1 sin h from which it is concluded that the near zero dynamics are also polynomially bounded for any initial condition uÇ , vÇ According to Remark 5.3, any µ > will work Choose µ in the controller design process (d) Choose sliding surfaces h / N2 I sin h cos h N1 I2 N2 I1 cos h h e4 0.9681 , gN2 I1 sin h e4 I2 g N1 sin e1 e 0,h 0.1 2354 e 0,h gN2 I sin h cos h vÇ (a) Solving out Èx1 and xÈ and de® ning the error states as x1 , xÇ 1, x2 h, xÇ e1 , e2, e3 , e4 yields the following generalized controller canonical form: ầe2 h which is satisđ ed globally Thus d u and d z > can be chosen arbitrarily (c) u i e, u^ e 0, i 1, 2, yields the following zero dynamics I1 xÈ e2, e3 (b) The Regularity Condition is checked as follows: where M1 and M2 (resp l1 and l2 ) are the masses (resp lengths) of beams and which are assumed to be homogeneous It is proved by Fliess et al (1995) that this system is non-¯ at, i.e not dynamic feedback linearizable with respect to the control variables u and v It is clear from the model that m1 m 2, n1 n2 2, n n1 n2 To apply the design method, p1 and p2 are eliminated to yield the following I± O form: Çe1 I2 cos2 e1 I1I2 e 0,h 0.1 , h 53 K kij and k10 k20 From the above choices, A1 From (17) it is obtained that 1 A2 and D I2 197 Control of nonlinear non-minimum phase systems using dynamic sliding mode Tracking Error e(t) 15 B u(t) 10 0 T BD BD K 5 10 t 15 10 20 50 s1, s2 50 100 150 200 10 t 15 20 10 t 15 20 0 Figure 10 t 15 20 Asymptotic tracking achieved with v = O t 16 0 16 T BD BD > is clearly satis® ed The dynamic feedback controller is then solved out using MATHEMATICA Using this controller, asymptotic tracking is achieved with bounded u but unbounded v O t as the simulation shows (® gure 2) This result may be predicted from the zero dynamics above In all the simulations, initial conditions are chosen as: x 0 2, 2, 23, 13 , u0,v0 2, 15 Now introduce a layer Let ²0 and E e1 e2 e3 e4 If E ²0 , the above dynamic sliding mode controller is used If E < ²0, let vÇ 4v to reduce the controller exponentially Approximate tracking is then achieved with ® nite control as shown in ® gure 15 Tracking Error e 10 5 0 10 t 12 14 16 18 20 10 12 14 16 18 20 10 12 14 16 18 20 u(t) 10 30 20 10 v(t) 10 Conclusion A novel dynamic sliding mode control method (Lu and Spurgeon 1996a, 1998) for asymptotic linearization of MIMO nonlinear minimum phase systems is extended to non-minimum phase systems Here the zero dynamics of interest are those of the control A layer of thickness ²0 < ² < is introduced about the sliding surface Outside the layer, the sliding mode controller for minimum phase systems developed by Lu and Spurgeon (1996a) is used If the growth rate of the zero dynamics is no more than exponential for some initial conditions and from some time point t T, the controller within the ²0 -layer may be designed to bring back the control exponentially with an attenuation rate greater than the growth rate of the near zero dynamics Thus ultimately bounded performance can be achieved with bounded control The method is applied to the control of the Inverted Double Pendulum which is both non-minimum phase and not dynamic feedback linearizable Acknowledgment This work was supported by UK EPSRC (grant reference GR/L21303) s1, s2 References DeC arlo , R A., Z ak , S H , and M atthews, G P., 1988, Variable 10 12 14 16 18 20 Figure Inverted Double Pendulum, a non-minimum phase system, ultimate bounded tracking is achieved with ® nite control structure control of nonlinear multivariable systems: a tutorial Proceedings of the IEEE, 76, 212± 232 F ilippov , A F., 1964, Di erential equations with discontinuous right hand side American Mathematical Society Translations, 42, 199± 231 198 Control of nonlinear non-minimum phase systems using dynamic sliding mode F liess, M., 1990, What the kalman state variable representation is good for Proceedings IEEE CDC, Honolulu, Hawaii, pp 1282± 1287 F liess, M., Levine, J., M artin, P., and 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systems Systems and Control Letters, 19, 193± 204 Utkin, V I., 1992, Sliding Modes in Control and Optimization (Berlin: Springer-Verlag) van der Scha ft , A J., 1989, A set of higher-order di erential equations in the inputs and outputs Systems and Control Letters, 12, 151± 160  ... sliding mode control technique for multiple input systems is shown to be useful in the control of nonlinear, non- minimum phase systems where the zero dynamics have no ® nite escape time The system... Conclusion A novel dynamic sliding mode control method (Lu and Spurgeon 1996a, 1998) for asymptotic linearization of MIMO nonlinear minimum phase systems is extended to non- minimum phase systems... 1967, Stability of Motion (New York: Springer-Verlag) Hau ser , J., Sastry , S. , and M eyer , G., 1989, Nonlinear controller design for ¯ ight control systems IFAC Nonlinear Control Systems Design,