Automatica, Vol. 28. No. 1, pp. 145-151, 1992 Printed in Great Britain. 0005-1098/92 $5.00 + 0.00 Pergamon Press p|c ~) 1991 International Federation of Automatic Control Brief Paper Robust Output Tracking Control of Nonlinear MIMO Systems via Sliding Mode Technique* HAKAN ELMALI~" and NEJAT OLGAC~':~ Key Words--Automatic control; variable structure systems; nonlinear control systems; nonlinear systems; robust control; stability; smoothing; linearization techniques. venue, a powerful algorithm following Chang (1990) is introduced: second order sliding mode control. I/O linearization of a nonlinear system relies on the exact cancellation of the nonlinear terms. In the presence of modeling uncertainties and disturbances, however, this linearization may not be possible. But, for a certain class of disturbances which obey the so called "matching conditions", I/O linearization is guaranteed. Consequently the robust control strategy (SMC) can be performed effectively. It is important to note that there may be a remainder of the state dynamics which does not appear in the I/O linearized structure. This portion of the dynamics (also named zero dynamics) should behave bounded if not asymptotically stable. This point is discussed in Section 2. A characteristic component of SMC applications is s(x,t) =0, which represents the sliding hyperplanes. The second order SMC was originally conceived to introduce a low-pass filter for the s dynamics (i.e. the time variation of s) and a high-pass (or band-pass) filter for the output tracking error dynamics in series. This form of the second-order s dynamics rejects unwanted high-frequency signals resulting from the disturbances and uncertainties. The bandwidth of the s dynamics should be designed low to minimize the high frequency excitations to error dynamics. Therefore, the selection of SMC parameters for the s dynamics and the error dynamics are crucial to obtain the desired tracking performance. The difficulties associated with this philosophy are elaborated upon in Section 3. The novelty in this paper is at the junction of the two methodologies; I/O linearization and SMC, for nonlinear MIMO systems. Parallelism between these two operations is pointed out in Section 4. Computer simulations have proven very successful for the attitude control of a spacecraft on a circular orbit. In Section 5, several cases of parameter selection are presented. Abstract--The robust output tracking control problem of general nonlinear multi-input multi-output (MIMO) systems is discussed. The robustness against parameter uncertainties and unknown disturbances is considered. A second order sliding mode control (SMC) technique is used to establish the desired tracking. Input/output (I/O) linearization, relative degree, minimum phase and matching condition concepts are reviewed. Some earlier SMC strategies which are restricted to the systems in canonical form are extended to a much broader class of nonlinear dynamics. It is also shown that for unperturbed dynamics, the sliding phase of the SMC applications have a direct correspondence to the I/O iinearization operations. Interesting parametric flexibilities emanate within the formation of the second order SMC, designating the "s dynamics" and the "error dynamics" segments as frequency domain filters. However, a critical impasse is posed in the off-line selections of the design parameters. A set of example cases is presented for a spacecraft attitude control problem. These examples manifest that the proposed control strategy is tunable to a desired response despite the disturbances and uncertainties. 1. Introduction GENERALCLASSo f nonlinear control systems is considered in this paper. Towards a robust output tracking control of this system, an I/O linearization is performed, first. Then a second order SMC is applied on the I/O iinearized system. The relation between I/O linearization and SMC technique is discussed. It is shown that much broader class of nonlinear dynamics can be treated with this strategy, compared to the earlier applications of other investigators on the systems in canonical forms. The recent developments on the I]O linearization methods Isidori (1985), Sastry and Isidori (1989) and Byrnes and Isidori (1987) have brought a profound insight to the subtleties in the above mentioned procedures. We base this study upon these clarifications in I/O linearization. After the nonlinear transformation which converts any nonlinear system into a linear format, we come to the second level of endeavor: guaranteeing the robustness of the control strategy against modeling uncertainties and disturbances. At this m 2. I / 0 linearization of MlM O nonlinear systems with uncertainties A MIMO nonlinear control system is taken as: m i = t(x) + At(x, t) + ~ [gi(x) + Agi(x)lui i~l Yi=hi(x), i=l ..... m (la) where x(.):~+--~fi~_~n is the plant state vector, are system input and output vectors, respectively, f(-), g~(.) : ~ n - ~ ~ ' , i = 1. . . . . m, are smooth vector fields and h i ( . ) : ~ n - - ~ , i = 1. . . . . m, are smooth functions. At(x, t) and Ag~(x), i = 1. . . . . m, represent the disturbances and modeling uncertainties. For convenience the above equation will be rewritten in a condensed form: *Received 7 November 1990; revised 16 May 1991; received in final form 1 June 1991. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor V. I. Utkin under the direction of Editor H. Kwakernaak. ~-University of Connecticut, Mechanical Engineering Department, Storrs, CT 06269, U.S.A. ~Author to whom all correspondence should be addressed. u(.),y(.):~+--~'~ i = t(x) + At(x, t) + [G(x) + At;(x)]u y = h(x) 145 (lb) 146 Brief Paper a0((~ , q) = L ~ L ~ ' - ' h , ( T where Um) Ym) U = c o l (U 1 . . . . . y : c o l (Yl . . . . . G(x) = [g,(x) . . . . . Aaq(~, q) = L,x~L~f ~ lhi(T-'(~, q)) Abi([, q) Ag~(x)]. Throughout this paper, the boldface lower-case letters indicate vectors and the upper-case, matrices. The goal of the control problem is to guide the output y(x) along a desired trajectory ya(t) for the given system in equation (1). The control strategy utilized should be robust enough to handle the modeling uncertainties and unknown disturbances (Af and AG). The upper bounds of these variations are: IA~(x, t)l -< a'i(x, t) IAgi~(x)l<-/f0(x) i = l . . . . . = A(x) . m, k = l ..... LL~,L;" (3) Af(x, t) and Ags(x ) ~ K e r [dh~, dLthi, dL~h, . . . . . dL~Z- 2hi] (4) for i, j = 1 , . . . , m. This is called the "matching condition". It assures that the Af and AG terms do not appear in derivatives of y~ of order less than r~. If the perturbation vector fields satisfy (4), then there is a diffeomorphic coordinate transformation (~, q ) = T(x), which transforms system (1) to the following form: $,, = ~ . . ~i l = ~fi ~, = hi(I, q) + Ab,(~, + Aais([,q)lu/ ili = qi([, rl ) y,=~it for for q) + for ' ~ [aij(~ , rl) ]=l i = 1. . . . . i = ri + l . . . . . i=1 ..... m (5) q)) ~")col l(fl) <'o , (~)<'~> ..... b = col [bl, b 2. . . . . (~) AA=[Aa/j] (~7) ~'"~] b,~] Ab = col [Abi, Ab 2. . . . . (7) Ab,,] i,j=l ..... m. Superscripts in parenthesis indicate the order of time derivatives Since the inverse of the matrix A(~, q) is well defined, the apparent selection of the control: U = A-I(v- b) (8) linearizes the nominal part of equation (6). ~(') = v + Ab + A A A - I ( v - b) r~=q (9) where v is the "synthetic input". Note that the control law (8) makes the state vector r1 completely unobservable at the output. However, ~ can be considered as an external input to q dynamics. Consequently, it is desired that with this input, the dynamics: ¢~: q([, q) (I0) is bounded-input bounded-state (BIBS) stable. As discussed in Bymes and Isidori (1987), this is guaranteed if the zero dynamics, ¢1 = q(0, q) (11) is exponentially stable• Such systems are called hyperbolically minimum phase. The consideration in this paper is for this type of system only. 3. R o b u s t s e c o n d order sliding m o d e control To achieve zero tracking error despite the uncertainties and disturbances in system (9), we propose a second order SMC strategy in reference to Chang (1990). It starts with placing an additional zero, z0, in the error dynamics. ~s+ ZosSs= csie~i '), ~ cs,s= 1, i=O % is Hurwitz for each .= 1. . . . . ]=1 ..... ej=y~-y~a=~l-Y/a ~ = col [ ~ , ~ . . . . . ~'~ = COl [~1' ~2 . . . . . m r=~ri i=l ~;,l ~ ~r, ~ + Zog = e~') + CE = v + Ab + A A A - ~ ( v - b) - year)+ CE m for F__ v#-I i=l,...,m ~n--r 2 l ~"] ~ ~ " ~.--r] ~ (12) m where ~' = col [ ~ , f~ . . . . . m where ~ = 0 represents the jth sliding hyperplane, e~ is the tracking error and Yja is the jth component of the desired output and e~- ° is the indefinite time integral of the error e/. For simplicity, we select Zoi = Zo for/" = 1. . . . . m. The relation between s dynamics and the synthetic input v can be obtained by taking the time derivative of equation (12). Writing in vector notation: n where m where r~ - 2, and for all is nonsingular at x ~ xo. The system is said to have a strong (vector) relative degree in an open set @ if it has the same (vector) relative degree at every point xo ~ ~ . In this paper, we assume that the system (1) has a strong relative degree in its domain of definition. In the above equations, L ~ denotes the kth successive Lie derivative. Hermann and Krener (1977) have shown that equation (3) is the necessary and sufficient condition for the I / O linearization of the system (1). It is straightforward to show that, if the system has a (vector) relative degree (rl . . . . . rm), then the relative degree is unchanged by the addition of perturbations if: • m ~=q m [L.,L[~-'hl(X) "'" L,.L['-'h,(x)] . . . . . . . . . lhm(X ) ' ' " L~,L[" l h m ( X ) _ j La,L;'-'h,(T-'([, ~(') = b + A b + ( A + A A ) u L,~L~h,(x) = 0 fori=l ..... m, j = l ..... x in a neighborhood of xo. (ii) The m × m matrix i , j = 1. . . . . ~ e combined f o ~ of the equation (5) for all i, j = 1 . . . . . is rewritten suppressing the arguments (~, q): (2) n j=l ..... where o~(x, t) and /3i~(x) are the known upper bounds of unknown disturbances and control gain uncertainties, respectively. We first pursue the I/O linearization process (Isidori, 1985; Byrnes and Isidori, 1987) for a MIMO dynamics. The system (1) is said to have a (vector) relative degree [r 1. . . . . r,~] at xo, if: (i) for b,((~, q) = L ; ' h i ( T - ' ( ~ , q)) g,~(x)] AG(x) = [Agt(x) . . . . . 1(~, I~) s = col Is 1 , s 2 . . . . . e : CO! [ e l , e 2 . . . . . ,:, ..... . s,.] em] y~a") = col [(y~t) <'~), (y,])~'~). . . . . (yT)~r~)]. (13) Brief Paper s dynamics 147 error dynamics I[ I AAA-I(%-lpr-1 + "" +cO) * 1~ F~G. 1. Block diagram representation of the SMC strategy. A Lyapunov function is selected as: V = ½ ( ~ T ~ ~. s T ~ s ) , ~ = Oiag (w~2,j), (14) ./=1,...,m. For simplicity w,,) = co, for ] = 1. . . . . m is taken. Negative ~midefiniteness condition of the Lyapunov stability criterion, d V / d t ~ O, can be written as: ~(~ + ~s) ~ 0 05) Note that to obtain CE, the successive derivatives y~O, i - < r j - 1, j = 1. . . . . m should be measurable. However, this requirement may not always be satisfied. In that case a substitute requirement on the measurability of the full state x is adopted. Knowing the form of h(x) and satisfying the matching conditions of equation (4), the successive derivatives y~O can be evaluated as needed. Additionally, in equation (23) sgn (dsl/dt) has a discontinuity at dsl/dt = 0 . To avoid the discontinuity, a saturation function is replaced instead of sgn (dsi/dt) following SIotine (1984), namely which constitutes the attractivity condition for the SMC towards d s / d t = s = 0 . Using equations (12) and (13), equation (15) can be expressed as: ~[v + ~b + ~AA-I(v i g~ e ] (24) - b) - y~') + C ~ - z,~ + ~ ] s ] ~ 0. 06) _ A form of the "synthetic input" v is p r o ~ s c d as follows: v=t-k s~(~), k~0 = y~) - c ~ + z ~ - ~ ] s - k s ~ (~) 07) where s ~ ( ~ ) = c o i [ s g n ( ~ i ) ] , i = l . . . . . m, and ~ is the segment of the synthetic input which engenders the equality case of equation (15) for undisturbed system (i.e. Ab = 0 and ~ A = 0). For the perturbed system (i.e. A b ¢ 0, A A ¢ 0), substituting equation (17) in equation (16) and using equation (13), the attractivity condition can bc rewritten as: ~T[--k(l + A A A - ~ ) sgn (~) + A A A - a ( ~ - b) + Ab] ~ 0. e is the boundary layer thickness in dsi/dt space. During transient phase of the dynamics, dsi/dt may move in and out of this boundary layer. However, once Lyapunov function V < e2/2 is reached, dsi/dt remains entrapped in this layer. This, in turn, fulfills the objectives of the output tracking control problem. The s dynamics is obtained by substituting equation 0 7 ) into equation (13). ~ + k(l + A A A -~) (18) Equation (18) is reiterated using vector norms: =Ab+AAA + I I a A A - I ( ~ - b)ll + Ilabll] D(s) = Ab + AAA-l(y~dr) - b - CE) 09) Note that ~ T ~ (~) ~ ii~l~ and consequently - k ~ T s ~ (~) ~ - k II~ll. In order to adjust the rate V wc propose: - k + k I I a A A -1 s ~ (~)ll + l l a A A - ' ( ~ - b)ll + Ilabll ~ - ~ I I a A A - ~ ( t - b)ll + IIAbll + 1 - IIAAA -~ sgn (~)ll ~ , where IIAAA -~ s ~ (~)ll < 1 is assumed. (21) Note that llabll ~ II~(x)a(x, t)ll I I A ~ I I ~ II~(x)~(x)xll where P(x) (25a) (22) is an m x n matrix with rows P j ( x ) = . 1 = 1. . . . . m and ~ = 1~7~- ~(x,t) = ~ 1 [ai(X, t)] and ~(x) = [~i/(x)], i = 1 . . . . . n, ] = 1 . . . . . m.. ~ e gain k is updated at each control instant according to equation (21). ~ c overall expression for the control u in x domain becomes: row(dL~r / - I hi) u = A-l[y~ ~) - CE + z~ - w]s - k s ~ (~) - L~h]. (23) (25b) where the state dependent differential operator D is defined as~ 3- D ( ' ) = dt 2 (2O) where ~ > 0. Equation (20) leads to the acceptable values of k as: k~ ~(ytdr)-b-CE). When ~ is within target manifold, the smoothing argument of equation (24) converts equation (25a) into: 9 ~ ~1~11[ - k + k I I ~ A A - ' s ~ (~)11 ~0. sgn (~) - A A A - ~ z ~ + (1 + AAA-1)to2~s + (I + A A A 1),o~(-). (26) The overall control strategy( is represented by the block diagram in Fig. 1, where D - (.) represents the inverse of the state dependent differential operator of equation (26) and p = d/dt. It is d e a r from Fig. 1 that the loop closure is due to a nonzero AA, which is a result of the modeling uncertainty in the control gain G. AA is a function of the state x. The analysis of the dynamic nature of this control loop is not possible by observing the s dynamics and error dynamics without considering the contributions of the feedback elements. On the other hand, the lack of tools in the nonlinear systems theory creates aI~"dilemma. A systematic . . way of selecting the parameteric quanhtles zo, o~n, e, c~s does not exist. For instance, time and frequency domain characteristics of the system cannot be designed as in the * These are symbolic descriptions of equation (12) and feedback portion of equation (13). 148 Brief Paper case for linear systems. This point brings us to a severe problem of design parameter selections. In his recent paper, Chang (1990) approaches this design problem considering the s and error dynamics in series without taking the loop closure elements into account. His treatment is based on the analysis of frequency domain and cut-off frequency bounds. But due to the neglected feedback terms, results do not reflect the true characteristics of the control loop. 4. The parallelism between S M C and I / 0 linearization in ideal sliding Let us consider the original unperturbed dynamics of equation (lb) motion are: i = f(x) + G ( x ) u + T a C3toz - S3to3 C~ too S3to~ + C3~o3 S2 to, + ~ (s~to~ - c~to9 t(x) ~ (to2to3 -- 3 ~ ¢ 2 ¢ 3 ) - i=f(x)+G(x)u (27) y=h(x). On these dynamics, we now attempt to apply the SMC directly without an intermediary step of I / O linearization. We assume that the initial conditions are on the designed "sliding hype~lanes". We select an error dynamics, which indicates the form of the sliding hyperplanes, as in equation (12). Taking their time derivatives and writing them in the vector notation, we arrive at: ~ + Zo~ = e (n + C E = y(') - y(d") + CE. ~) ~(~) = (28) or ~ = s = O. (29) 0 0 0 0 0 0 0 0 t: For the ideal sliding dynamics, ds/dt = 0 is enforced. On the other hand, the Lyapunov function of equation (14) should attend its minimum during sliding, i.e. V = ½(i~g + to~sXs) = 0 - 3 ~ ¢ ~ ¢:) ( ~ 0 TM 1 ~. •0 ~ 0 The ideal sliding condition requires 0 1~=0 or ~(]+to2~s)=0. (30) ~ = 0 is self-evident during ideal sliding behavior. From equation (28) y(O(x, u) = y(a')(t) - CE(x). ~, Td=~ ~-1 ' /',~ f -S~C2 1 ~: ~ c,s~ + s,s~c~~ LC1C3 - S,S~S4J (31) ~..~ To satisfy equation (31), we select v = y(a~)(t) - C E ( x ) which leads to yt~)(x, u) = v. (32) Equation (32) is an exact duplicate of equation (9) without the perturbations. This shows that for the unperturbed system, the ideal sliding behavior represents the I / O iinearized structure, which proves the parallelism between SMC and I/O linearization procedure. The equivalent control in SMC is the same as the I/O iinearizing control of equation (8). Therefore, if it is known that the nonlinear system (1) is hyperbolically minimum phase, then the two steps (I/O linearization and SMC) can be combined into one and the same control can be obtained applying second order SMC directly to the nonlinear system. 5. Examples The above developed control strategy is applied to the attitude control problem of a spacecraft. Typically dynamics and associated characteristics are taken from Singh and Iyer (1989). If the spacecraft is on a circular orbit in an inverse square gravitational field, and the attitude of the space vehicle has no effect on the orbit, then the equations of where x = (0 x, t~x) x is the state vector, u = (T~, T2, T3)v is the control torque vector and T a is the external disturbance vector, t ~ = ( t o , , to2, to3)x is the instantaneous angular velocity vector with respect to the fixed inertial space, C i, Si represents cos (Oi), sin (0i), respectively. 0 = (0,, 02, 03) T are ordered roll, pitch and yaw angles about the body fixed coordinates, too is the orbital angular speed, I~, i = 1, 2, 3 are the moments of inertia about the body fixed axes. The objective of this control strategy is to track yd = (O,d, 02a, 03a) x in the presence of modeling uncertainties and external disturbances. It is selected as: Ya = [1 - e °353t(sin 0.353t + cos 0.353t)]r where r = [180,45,90]Xdeg. The values in this variation correspond to a damping ratio of 0.707 and a natural frequency of 0.5. The modeling uncertainties appear in the form of li + AI~, i = 1, 2, 3. The numerical values of the inertias for this simulation are ! = (8746, 888.2, 97.6) x kg m 2. The variations in the inertias are taken as: "',1 f "'q AI2~ = 0.511 + sin (0.1t)]~ v212~ AI~J " ~vfl3J where v, = 0.1, v 2 = 0.2, v 3 = 0.3. The external disturbances are generated by superposing a harmonic time function and Brief Paper 149 ~,~., i" ~ ' ~ , ~o -~ ,, ~ ,~,~ ,o~[,,~ ~.~,~, ~ 2 .~ 0 1 I 10 20 Time (sec) FIG. 2. d s / d t variations (case I). ~ , f,~ix,/"l ~(, 5 \ \_ ,-~-\~.~,,-,_.,~__~_ .,. /--~_--~'----- ............ -5 -7' I IO o ~ ~ Time (~e¢) FIG. 3. Tracking errors (case I). I100 / T I g ~ ~ ~ a ~ I I0 Tlma (~ec) FIG. 4. Control torques (case I). I 20 150 Brief Paper 20 j~,~ 16 12 s3 ~- o 0 0 O~ ~, - ~ " -4 ~ .~ - ¢ ~ "Tt % - , & ~-~. 2 -8 -~2 T I O I 20 IO Time (sec) FIG, 5. dsldt variations (case II). ~_~e~ I j| ~,l-~-~-- e 2 A ~ V " ~ ' ~ - ~ * ~ ~%~ w _ -I -2 ~ ,,,.~. _ ,,,"~ ~ ~ * ~ A _ J - ~ / ' ~ . ~ ~ ~ ~ ~ ~."~ ~ .~-~ ~ / ~ , ~ . ~ = .-.,~/~.-.~ f,'~ ~ J J ~ ~ e f -~ 0 L I0 L 20 Time (sec) FIG. 6. Tracking error (case II). 2300I/i 1700 T~ I100 Z 500 T -I00 ~, -700 -1300 ~--..~ ~ .-.- ~T~ 0 ~ tO Time (sec) FIG. 7. Control torques (case II). ~ 20 Brief random dither for the simulation. They are chosen as: Td~ = 40 sin (2~t) + T~.~a,oo,~ Td2 = 40 COS(2~tt) + T~.~,Oo~. Td3 = 20 sin (2~rt) + T3,~.ao,~ The random dither has a mean of (0, 0, 0) T Nm and standard deviation (10, 10, 10)T Nm. Corresponding terms to the equations (1) and (22) are: At(x, t) = (r(x)[,+,,, - t(x)l,) + T~ a{;(x) : ( { ; ( x ) b + , , , - {;(")l,) B 13(x) = 0 0 0 0 0 0 0 0 0 0 0 vl 1,(1 - Vl) 0 v~ 12(1 - 0 0 v3) 0 0 ~V2 + Vl)12 -~" I(V3 -- Vl)[ /3 ((020) 3 -- v,)ll (v~ + v:)& + I(v~ - v2)l 1~ v:)& (1 (vl + and c~ selection correspond to error dynamics poles of ( - 1 , - 1 ) . The tracking accuracy for 0 l, 02 and 03 (Fig. 3) are not desirable for this selection. The corresponding control torques T~, T~ and /~ are shown in Fig. 4. Case II. z o = l , to,,=2, c0=4, c~=4, e = 0 . 1 , # = 0 . In this selection, although the bandwidth of s dynamics is increased by 2 it is still small enough to filter the disturbances (TdS) at 2~trad/sec frequency. In comparing the set of response figures (Figs 5-7) with the previous ones, one can conclude that the ds/dt behavior of the system did not change--this is expected because both of these example cases treat the bandwidths below the excitation frequency, and the tracking error is much better in the second case. Because the poles of error dynamics are shifted to ( - 2 , - 2 ) making the error response faster and zo is reduced to (1) lessening the effect of the s dynamics oscillations on the error dynamics. These "educated" causal relations suffer from the lack of a positive tool as mentioned in the text. Nevertheless, the influence of the parametric variations on the tracking performance is quite apparent. Conclusions A combination of I/O feedback linearization of nonlinear systems and a second order sliding mode control strategy is performed. The examples reflect that the method is a very strong tool for output tracking problems of nonlinear systems in the influence of bounded disturbances and parameter uncertainties. The attitude angles of a spacecraft were controlled as an example via the reaction jets. The results for varying parametric selections are shown for performance improvements. Apparent further enhancements in response characteristics are currently studied. v3 0 (1 - 151 0 v~) &(1 - o(~, t) = Paper 30)02~2~3) (0)30)~ - 30)oZ~3~) va)lt + I(v2 - v3)112 ( ~ , ~ z (1 - v 3 ) 8 Acknowledgements--The authors wish to thank University of Connecticut Research Foundation for its financial support. They also express their gratitude to Professor Karl Hedrick and Professor L. W. Chang for their valuable input and discussion. - 3~;,~2) P(x) = - C 3 0 ) 2 q- 530) 3 52(53092 -~- C30)3) c~ c2 S2C 3 1 c2 S2 ~- ' The control design parameters are z o, ton, e, Co, cl, #. As explained above there is no systematic way of selecting these parameters. However, some educated suggestions are made for the following example cases to demonstrate the fact that the system behavior can be tuned. For the integrations a fourth order Runge-Kutta algorithm with a time step of 0.01 sec is used. The control loop closure period is taken as 0.04 sec. The initial attitude error is taken as 0 o = [ - 3 , 3, 5]T degrees. Case I. Zo=10, 0),,=l, c o = l , Cl=2, e = 0 . 3 , W=0. Since k selection in equation (21) is very conservative, # is chosen as 0. ta,, is selected small to limit the natural frequency of s dynamics (equation 26). In return, the s dynamics will be acting as a low pass filter with small bandwidth. Note that there is also a (I + AAA -1) factor to be considered with ton2 in equation (25), which makes the above conjecture very difficult to prove. Nevertheless, the ds/dt variations given in Fig. 2 reflect what was foreseen, co References Behtash, S. (190). Robust output tracking for nonlinear systems. Int. J. Control, 51, 1381-1407. Byrnes, C. and A. Isidori (1988). Local stabilization of minimum phase nonlinear systems. Syst. Control Lett. 11, 9-17. Chang, L. W. (1990). A MIMO sliding control with a second order sliding condition. ASME WAM, paper no. 90-WA/DSC-5, Dallas, Texas. Fernandez, B. R. and K. Hedrick (1987). Control of multivariable nonlinear systems by the sliding mode method. Int. J. Control, 46, 1019-1040. Hermann, R. and A. J. Krener (1977). Nonlinear controllability and observability. 1EEE Trans. Aut. Control, AC-22, 728-741. Isidori, A. (1985), Nonlinear Control Systems, An Introduction. Springer, New York. Sastry, S. S. and A. Isidori (1989). Adaptive control of linearizable system. I E E E Trans. Aut. Control, AC-34, 1123-1131. Singh, S. N. and A. Iyer (1989). Nonlinear decoupling sliding mode control and attitude control of spacecraft. IEEE Trans. Aerospace Electron. Syst., AES-25, 621-633. Slotine, J. J. (1984). Sliding controller design for nonlinear systems. Int. J. Control, 411, 421-434.