Discretization of a non-linear, exponentially stabilizing control law using an L p -gain approach

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1 Discretization of a non-linear, eponentially stabilizing control law using an L p -gain approach Guido Herrmann, Sarah K. Spurgeon and Christopher Edwards Control Systems Research Group, University of Leicester, LE 7RH, U.K. gh7@sun.engg.le.ac.uk, eon@le.ac.uk, ce@sun.engg.le.ac.uk Abstract his paper considers the input-output stability of eponentially stable, non-linear systems with sampled-data output. Results for linear systems are generalized showing that the L p -gain with respect to the sampled-data output eists and converges to the L p -gain associated with the continuoustime output when the sampling period approaches +. he results are applied to a non-linear control configuration and compared to a Lyapunov function analysis based approach previously developed for a non-linear sliding-mode like control. In contrast to the Lyapunov function technique, the sampling-time constraint vanishes for a stable plant if no control is used. Introduction Analysis of discretized continuous-time control systems used for sampled-data control has been of interest for a long period. his is because controllers are often developed in continuous time but need to be applied via sampled-data technology. An etensive frame-work of theory has been developed for linear control of linear systems [, ]. Linear time-invariant (LI) systems are particularly useful for the application of H and H techniques []. In contrast, nonlinear systems often rely on Lyapunov function based analysis techniques [, 4]. he approach of [, 5] has been useful for determining reasonably low sampling-frequencies, which are sufficient for stable, robust closed loop control using a discretized non-linear control. he technique has been applied successfully to a sliding-mode based control. An important result for the discretization of LI systems is that any stable linear system (A B C) followed by a sample-and-hold process has finite L p -induced norm for constant sampling time >. Furthermore, the L p -norm of the discretized output signal approaches the L p -norm of the continuous output signal of (A B C) as! + []. his result has also been shown for linear time-varying systems []. It is of interest to etend this result to non-linear systems, since the L p -gain has also proved to be useful for non-linear systems. his paper presents an L p -gain approach for sampled-data outputs of strictly proper, affine, non-linear systems where the continuous outputs are Lipschitz functions of the states only. hese results are then used to analyse the closed-loop properties of a non-linear sampled-data control. A particular upper bound ~ for the sampling-time of the discretized non-linear control is derived so that asymptotic stability of the closed loop system follows for ~. his result is then compared to the Lyapunov function analysis based results of [, 6] for a sliding mode based non-linear control law. It will be seen that both approaches have their advantages and disadvantages from G. Herrmann is supported by a grant of the European Commission (MR-grant, project number: FMBIC9846). a theoretical and practical point of view. In the Lyapunov function based approach, an upper bound for the norm difference between the time-sampled state (t i ) and the value of the continuous signal (t) is found for each particular finite sampling interval [t i t i+ ]: sup (k(t );(t i )k)kk(t)k K t [t i t i+ ]: t [t i t i+] () he upper bound Kk(t)k decreases with decreasing sampling time. A problem of this approach [, 4, 5] is that the relation () can only be derived if an upper bound on the sampling-time < is assumed. For () being valid, this upper bound remains finite for any control even if the non-linear system is stable and the applied control is zero. hus, a minimum sampling frequency is always indirectly imposed. his minimum sampling frequency is not necessary with the L p -induced norm approach. Preliminaries: L p -gain of sampled-data LI systems Suppose a linear, finite dimensional, strictly proper system G, an input-output map of u! y, is given by the triple (A B C), wherea R nn is a stable matri. A sampleand-hold element may be introduced, measuring the output signal y(t) at well-defined time instants t i (i = :::) and holding the output of the element constant over the interval (t i t i+ ] for constant sampling time > : 8 t (t i t i+ ]: y (t y)=y(t i ) y (t = y)= t i =i: o symbolize the change in methodology from continuous to discrete-time, the superscript (=discrete) has been introduced. It was proved by [, heorem 9..] that the input-output map u! (y ; y) has finite L p -gain, (p [ ]). hel p -gain of the input-output map u! (y ;y) is bounded from above by: ;I (A B C) def = Z f H (t)dt + () () where the function f H and the term () satisfy: f H (t) def = sup Ce At B;Ce A(t;a) B () def = sup a( ) t[ ) Ce At B and kkis the induced Euclidean norm. his implies for the L p -norm on the in/finite horizon (): ku(t)k p = Z ku(t)k p dt! p of a Lebesgue-integrable input signal u(t) L p [ ]: ky (t);y(t)k p ;I ku(t)k p +(A B C () ) () he bias term (A B C () ) is bounded above by:

2 Z (A B C () ) sup a( ) ( Ce At (I ;e ;Aa )() p )dt! p : It R can be seen that the upper limit of the L p -gain ( f H (t)dt + ()) converges to for! using the following upper estimates of f H and the term () f H (t)kbkkck e ta e kak ; ()kbkkcke kak : Note that R e ta dt is finite for stable A. A similar argument also shows that lim!+ =. L p -gain of non-linear systems with sampled output Assume for a system with bounded input gain g ~ G _(t)=f()+g()u R n f: R n! R n g: R n! R n~m y = h() h : R n! R p h()=: (4) that the unperturbed system is eponentially stable. Further, suppose f () and g() are continuously differentiable in. he output measurements y = h() and the system function f () satisfy the following Lipschitz properties: kh( );h( )kk h k ; k kf( );f( )kk f k ; k: (5) Assuming eponential stability for u =,theremusteist by the converse stability theorem [7, heorem.] a () such that for c c c c 4 > : f();c kk c 4kk c kk ()c kk : (6) Combining now L p -stability characteristics of eponentially stable systems with the L p -gain of linear sampled systems, it can be shown that stable affine systems (4) have finite L p -gain with respect to an output h (t ) ; h() and that this gain approaches for! + : heorem An eponentially stable system (4) with sampled data output h (t ): 8 t (t i t i+ ]: h (t ) =h((t i )) h (t ) =: satisfying (5-6) has finite L p -gain (p [ ]) in the space L p [ ]( ). he gain converges to the value of the L p -gain with respect to the continuous output h() as! +. Further, it can be shown that for the output signal h (t ) ; h() the L p -induced norm has the limit as! +. Proof First consider an auiliary output ~y which enables the L p -stability result for sampled-data LI systems to be used. Using a stable matri F R nn, the output function ~y for the system (4) is: ~y() =f () ; F he auiliary output ~y is globally Lipschitz in since f() is Lipschitz. Further, due to eponential stability, the equality ~y() = holds. hus, the input-output map u! ~y has finite L p -gain using [7, heorem 6.]: k~yk p u!~y ku(t)k p + u!~y u!~y u!~y (7) Rewriting the system equation (4) using ~y() yields: _(t) =F +(~y() +g()u): It is now possible to interpret the function (~y() +g()u) as the input signal for a linear system (F I I). Sinceh() is globally Lipschitz with Lipschitz constant K h and g is bounded by ~ G, the relation () for discretized linear systems implies: kh (s ) ; h()k p K h ;I (F I I)k~yk p +K h ~G ;I (F I I) kuk p + K h (F I I () )(8) he term k~yk p in (8) can now be replaced by the epression in (7) implying that (h (t ) ; h()) has bounded L p -norm for L p -norm bounded input u. he norm vanishes for! + since ;I and converge to in this case. he latter and the eistence of a finite L p -gain with respect to the map u! h() [7, heorem 6.] implies the remaining part of the claim using the triangle inequality. 4 Discretizing a non-linear, continuous-time control he procedure for controller discretization is introduced using a generic eample. Certain assumptions are made on the system and an eponentially stabilizing control: Assumption : Assume for a system with bounded input gain kgk G, G > _(t)=f()+g()u R n f : R n! R n g : R n! R nm (9) an eponentially stabilizing continuous-time control u = u C ((t)) eists. Further, suppose f(), g() and u C ((t)) are continuously differentiable in. Assumption : he controlled system shall have certain Lipschitz properties: kf( );f( )kk f k ; k C u ( );u C ( ) Ku k ; k () Assumption : Suppose values of the continuous-time control are available via sampled measurements at well defined time instants u C ((t i )) = u C ((t))j t=ti t i = i i = :::. It is then possible to apply to the system a discretized control: 8 t (t i t i+ ]: u(t) =u (t ) =u C ((t i )) u(t )=: () he following theorem ensures stability of the discretized control using an L -gain approach: heorem here eists a ^ small enough such that for any < ^, the discretized control for the system (9) is asymptotically stable. he constraints on the sampling time vanishes, ~! +, for a control with Lipschitz constant K u! + and eponentially stable system (9). Note the contrast to [] where relations similar to () need to hold and therefore the upper bound on the sampling time remains finite for any control even if the control is zero and the system is open-loop stable. Proof Step I: It can be verified using the principle of complete induction that a continuous solution to the discrete controlled system eists []. Step II: Due to the eponential stability of u C ((t)), the equilibrium point is f() + g()u C () =. hus, there eists a constant K C > using the Lipschitz property of f(), the assumption on kgk in (9) and on u C () in () such that

3 kf( )+g( )u C ( );f( );g( )u C ( )kk C k ; k kf()+g()u C ()kk C kk: Further, there must eist a Lyapunov function [7, heorem.] such that: ; f((t))+g((t))u C ((t)) ;c k(t)k c > c 4 kk c kk ()c kk : () Step III: Investigate a system with state ^ and input perturbation vector ^u ; _^ = f(^(t))+g(^(t))u C (^(t)) f +g(^(t)) ^u () g For ^(t =)=(t =) and a particular choice of ^u, it will be seen that system () is equivalent to the closed loop with discretized control. Using Lemma of the Appendi, it follows that the system () with input perturbation ^u, output ~y ~y = f(^(t)) + g(^(t))u C (^(t)) ; F ^ (4) h and arbitrary but fied and stable matri F R nn has finite L -induced norm with upper bound = c 4(K C+kF k)g c. Choosing a scalar, <, it follows for = KC+kF p k c using Lemma : ;f(^(t))+g(^(t))u C (^(t)) + ^ ^ g(^)^u k^uk ; k~yk ;(;)c k^k : (5) hus, integrating inequality (5) and using (^) c k^k from () it follows that: ((t = ));((t =)) ;(;) c + Z k^u(s)k ds; Z ((s))ds c Z! k~y(s)k ds (6) Step I: he system dynamics () may be rewritten in terms of the signal ~y in (4) and an auiliary output ~u: _^ = F ^ +(~y + g(^(t))^u) ~u = u (t ^) ; u C (^(t)): (7) he signal u (t ^(t)) results as in () from a sample and hold process operating on the non-linear control u C (), which is now a function of ^(t). Considering (~y + g(^(t))^u) as the input signal, the dynamics in (7) have, as discussed in Section, a finite L -gain associated with the linear system (F I I). Using the bounds on kgk in (9), the Lipschitz constant for the continuous control u C in () (k~uk K u ^ (t ^) ; ^(t) ) and the relation jaj + jbj + jcj p q jaj + jbj + jcj, it follows that: k~u(t)k p p h K u ( ;I (F I I)) k~yk p +G k^uk p +((F I I () )) i (8) Step : Consider now inequality (8). Both sides may be squared and multiplied by the constant: ( ; K u ( ;I (F I I)) G ) which has to be positive. his is satisfied if the sampling time has been chosen small enough so that K ( u ;I (F I I)) G <. he resulting inequality can now be added to (6) and equivalently rewritten so that: ((t = )) ;((t = )) ;( ; ) c Z ((s))ds c + k^u(s)k p ;k~u(s)k p (;K ( u ;I (F I I)) G )! + K u ;I(F I I)) ( ; K ( u ;I (F I I)) G ) ; k~y(s)k p + K u (F I I ~() ) ( ; K ( u ;I(F I I)) G ) (9) he perturbation ^u may now be chosen to be ^u = ~u,which makes system () equivalent to the discrete controlled system (9). Hence, if the sampling frequency has been chosen small enough so that K ( u ;I (F I I)) G < p Ku ;I (F I I) p;k ( () u ;I (F I I)) G then it follows from (9) for = KC+kF p k : c! ((t = ))+ K u (F I I () ) (;K ( u ;I(F I I)) G ) ((t = )) +(;) c Z ((s))ds () c From the boundedness of ((t = )) and R ((s))ds, it follows that the system is asymptotically stable [8, heorem..4]. Observe that the right hand side of (8) has the value and the constraint on in () vanishes if the Lipschitz constant K u is zero. his would be satisfied if there is no control necessary to stabilize the system. 5 A sliding-mode like control law Linear, uncertain systems shall be considered _ = A + Bu + F(t ) () where R n, u R m and the known matri pair (A B) defines the nominal linear system, which is assumed to be controllable with B of full rank. he unknown function F(: :) : R R n! R n models parametric uncertainties and non-linearities of the system lying in the range space of B. Under these conditions, it is possible to define a linear transformation ~ such that the system () becomes: _z =z +A () _ =(+)z +(+)+B u (4) where ~z = ~ = z z R n;m R m (5)

4 and is a stable design matri. he sub-system () defines the null-space dynamics and the sub-system (4) represents the range-space dynamics. he matrices and stemming from the transformation of F(: :) are bounded (t ~z): RR n! R m(n;m) kkk (t ~z): RR n! R mm kkk he associated non-negative bounds K and K are assumed known and finite. It is assumed that F(: :) is Caratheodory to ensure eistence of solution (see [, 9]). It can be shown similarly to [6], that the sliding-mode like control law outlined below forces the system states ~z to satisfy (t) and the control is robust to matched uncertainty. his control has two parts: u(t) =u C L (z (t) (t)) + u C NL (z (t) (t)) (6) where u C L () and uc NL () are the linear and the non-linear control components. he linear controller part is u C L (z (t) (t)) def = ;B ; (z (t) +(; )(t)) (7) where is a stable design matri and P satisfies P + P = ;I m where I m is the (mm) identity matri. A Lyapunov function (t) for the analysis of the range-space dynamics is and (t) def = (t)p (t) (8) (t) def = z (t)p z (t) (9) with P + P = ;I (n;m) is appropriate for the nullspace dynamics. he non-linear control component u C def NL = ( ;(z(t) (t))b ; P(t) kp for [z (t)k+(kz(t)k+k(t)k) ] 6= for [z ] = R + () achieves robustness by counteracting the matched uncertainties and ensuring reachability of a cone shaped layer around the sliding mode. he epression (kz (t)k + k(t)k) results in a cone shaped layer around =,which is defined by the relation =!,where! R +! def = ma (P )= min (P ) ma (P ) min (P )( ; ); has to be positive, which can be ensured by imposing the constraint min (P ) ma (P ) > (z (t) (t)) is defined as: ;. he function (z (t) (t)) def = ( kp (t)k + kz (t)k): () where >. Note, that the non-linear control given in () and () is continuous and globally Lipschitz. he gains and in () have been defined so that they ensure robustness with respect to the matched disturbances and reachability of the cone shaped sliding mode layer: def =ma sup ma P ; +P ; () def =sup(kk)+! kp A P ; k+! kp ; kkp ; k () def =((; ) +) > : (4) However, it is also possible to adjust the parameter > for a compromise of using linear control for performance ( = ) orrobustness(! ). A proof of stability [, 6] makes it necessary to show that the null space Lyapunov function (t) in (9) is decreasing as soon as the states have entered the cone shaped layer, i.e. by imposing a quadratic stability constraint which will limit the choice of! with an upper bound. An implicit bound on! will be given in Sect. 5.. A theorem showing asymptotic stability of the control can be easily derived from results in [, 6]. 5. Discretizing the sliding-mode like control Since the sliding-mode like control u C L (z )+u C NL (z ) of (6-7) and () is globally Lipschitz, a time-discretized control u L (t z )+u NL (t z ) maybeappliedtothe system () instead of the continuous control (). he proof of stability for the discretized control using the L - gain approach may be sketched by following the different steps of the proof for heorem. Step I: he eistence of a solution of () has been proved by [] and the respective system (-4) has also a Caratheodory solution if ^u is of Caratheodory type. Step II: A global Lyapunov function can be provided by non-smooth theory using a ma-lyapunov function as described in [, 5]: (z (t) (t))= (z (t))+k ((t)) for! (k! +) (z (t)) for! (5) where k> has to be determined []. It has been shown in [] that this function is differentiable for almost all t since the ma-lyapunov function is Lipschitz in z and and the solution z (t) and (t) is absolutely continuous in t []. hus, the equality R t _ (z (s) (s))ds =((z (t) (t)) + const:) applies. his integrability feature of (z _ (t) (t)) allows the non-smooth ma-lyapunov function to be used as within the proof for heorem. Further, it can be shown similar to [, ] that the relation _ ; # ~,(#>) ~ holds for almost all t. his implies eponentially fast decay of (z (t) (t)) and eponential stability of the continuoustime controlled system. Step III: he auiliary output for (-4) may be introduced with respect to the sliding-mode-like control system: i h i h i A ~y=h + + ;F ( def = ~C h z i (6) ;(z(t) (t))p kp (t)k+(kz(t)k+k(t)k) for [z ] 6= for [z ] = where the scalar derived from (-) is bounded: ~ ma : hus, k Ck ~ is bounded and it is permissible to define F = : A ; ~map he output matri C ~ is partitioned as follows: h i ~C = C ~ ~ C C ~ R n(n;m) C ~ R nm : he definition of f and g for () are then implicit using the transformed system dynamics (-4) and the respective continuous-time control (6). his also determines the sub-states ^z and ^ and the input perturbation ^u R m as for (). In the following, the ma-lyapunov function

5 (^z (t) ^(t)) from (5) and the respective sub-functions in (8) and + k in (8-9) shall be investigated with respect to the newly introduced states ^z and ^. As a result, the L -gain of the system with input ^u and output ~y will be determined. Similar to [6], it can be shown for! ^ P (B u C NL (^z (t) ^(t)) + (^z (t) ^(t)) ) : hus, the uncertainty is compensated for! [6]. In this case, the derivative _ (^z (t)) + k _ ( ^(t)) satisfies for almost all t (and considering the perturbation ^u): _ (^z (t)) + k _ ( ^(t)) = 4 ^z (t) ^(t) ^u 5 4 ^z (t) ^(t) ^u ; ^#( (^z (t)+k ( ^(t))+ k^uk ; k~yk (7) 6 4 ;I+^#P A P C + ~ C ~ C + ~ ~ C P A C + ~ C ~ ;k I;^#P + ~ C ~ C kp B kb P ; I m 5 7 5(8) he scalars >, ^# > and have been introduced to ensure that the symmetric matri. For that reason, and ^# should not be chosen too large to ensure that the principal minors of are negative semi-definite for any value of ~ C. Similar arguments need to be made for k>. Further, the value of should be chosen as small as possible so that for any value of ~ C. he scalar will act as an upper bound on the L -gain from ^u to ~y. For!, the derivative _ is investigated; for almost all t: _ = ^z (t) ^ (t) ^z (t) ^ (t) + ; +! ;# ; (k! +) k~yk = 4 ;I + #P + C ~ C ~ (k! +) +P P A + C ~ C ~ (k! +) 5 A P + C ~ ~ C (k! +) ; P! + C ~ ~ C :(9) (k! +) where R. It can be shown that if # >,! > and are chosen small enough then there always eists (k! +) a so that. Assuming, ( ^(t)) ;! (^z (t)), it follows for almost all t: _ ; # ; (k! +) k~yk (4) hus, the sub-functions and + k of the ma- Lyapunov functions (^z (t) ^(t)) (5) have been now investigated with respect to their time derivative (7),(4) at those points where they define the ma-lyapunov function (5). It is then possible using non-smooth analysis to show that for almost all t the derivative (^z _ (t) ^(t)) satisfies: _(^z (t) ^(t)) ; min( ^# (k! +) #)(^z (t) ^(t)) + k^uk ; k~yk (4) he latter inequality actually is equivalent to (5). herefore, is an upper bound of the L -gain from ^u to ~y, while a quadratic stability margin (;min( ^# (k! +) #)(^z ^)) is considered. Step I & : he net steps follow the proof of heorem. he Lipschitz constant K u of the sliding-mode based control has been calculated in [], which can be used here. In particular, an inequality similar to () must follow for any and therefore asymptotic stability can be implied for small enough sampling time. 5. Eamples of discretized sliding-mode based controls he L -gain technique will be demonstrated and compared to the Lyapunov function analysis based discretization methodology of [5] using two eamples. First, consider a simple model of the inverted pendulum, which is formed by a light rod and a heavy mass attached to one end of the rod with the pivot of rotation at the other end. he system may be epressed in the form of () with h i h i A = B= =:5(sinc((t));) :5 ;: sin(t) = ;b+: sinc(t)= t if t 6= (4) if t = where is the angle of rotation, =[ ], _ b (:9 b :) the damping coefficient and u(t) the control torque. Furthermore, it is of interest to investigate the L -gain approach on a stable model in the nominal case when the uncertainty is set to zero and 6=, the non-linear control gain, is only used for reachability of the sliding-mode. hus, the system satisfies in this case: A = h ;:5 ;:5 i B = h i [ ]=: (4) he non-linear controllers for both systems are chosen to satisfy =:6 =. By demanding =(), the linear controller is used to compensate for the parametric uncertainty (;b +:) of. herefore assuming =, the value of ( < ) must be chosen as large as possible, so that the linear control is also used for reachability. Provided the matri (5) determining the slidingmode performance, the linear control matri (7) and the respective Lyapunov matrices P (9) and P (8) have been set, then the features of the sliding-mode cone and the parameter! can be determined using nominal system and controller parameters only: ; ; ma P + P ;jp A P ; j!jp j =:85 = For the single input, second order systems considered, this constraint is necessary and sufficient so that for any state satisfying!, _ ;:85 jz j for almost all t. his ensures consistency with the results in [5], comparability of the control of the nominal and the uncertain systems and implies stable performance for both eamples (9). he value of, <<, issetto:5 as a trade-off between controller performance and low sampling frequency as derived in the proof of heorem. he constant k from the ma-lyapunov function (5) has been adjusted via numerical methods so that the sampling frequency bound =^ for each controller is minimal. Note that the results (Figure ) for the L -norm based technique are more conservative than those calculated for the Lyapunov function analysis based approach. he sampling frequency which is sufficient to stabilize robustly the unstable inverted pendulum (4) is generally for the L -norm based methodology more than nine times higher than for the Lyapunov function based technique. A similar result applies to the stable system with no uncertainties (4) where

6 /τ Ω * =Σ /τ Ω * =Σ a) unstable pendulum (4) b) stable system (4) Figure : alues of sufficient sampling frequencies against changing controller poles = using the L -norm based approach ( - ) and the Lyapunov function analysis based approach ( -- ) the non-linear control is only used for reachability of the sliding-mode region. he reason for the more conservative results is connected to the conservative estimate of the L - gain with respect to the output ~y (-4), (6). Using the Matlab c -LMI toolbo, the matri C ~ C ~ was approimated by a polytopic set. Furthermore in the Lyapunov function based approach, the very large value of the Lipschitz constant of the non-linear control u C NL () could be compensated by the upper estimate of the sampling error sup t[ti t (k(t i+] );(t i )k) () [], which is not straightforward for the L -norm technique. Proof Introduce the auiliary system input gain ~g = g and the auiliary output h ~ = h using the positive scalar > from (44). Upper bounds for the left hand side of the Hamilton-Jacobian inequality of the input-to-output map u! y can be derived using the inequalities (5) and (6): f () + ~g()~g () = f() + g()g () + h ~ () h() ~ + h ()h() ;c kk + c ~ 4 G kk + K h kk! c 4 G ~ ( ) ; ( ) c K h c ~ 4 G + c ~ kk 4 G he term = 4 ( ; ) is non-positive for : his implies using [7, heorem 6.5], that the L - induced norm of the map u! y is smaller or equal and the inequality (44) holds. hus, the map u! y must have the same upper bound on the L induced norm. he second claim in (45) follows by carrying out the same steps including the quadratic stability margin. 6Conclusion his work has presented a theoretical approach to L - stability of discretized non-linear continuous-time control. In particular an approach was presented which shows that a sampled-data output of a strictly proper, eponentially stable, affine, non-linear system has finite L p -norm approaching the L p -norm of the continuous-time output for! +. his result has been used to show asymptotic stability of a discretized, continuous-time, non-linear control for small enough sampling time. An eample of a sliding-mode based control has shown that the methodology is theoretically viable but practically disadvantageous compared to a Lyapunov function analysis based technique. Appendi Lemma he eponentially stable system (4) satisfying (5-6) has finite L -induced gain and f () + g()u kuk ; kyk K hc 4 G ~ = K h = p (44) c c Hence, is an upper bound on the L -gain. Furthermore, a quadratic stability margin ;( ; )c kk is part of the upper bound for _ if the gain is increased to = using a scalar, < and f()+ g()u kuk ;kyk ;(;)c kk : (45) holds for = K h p c. References []. Chen and B. Francis. Optimal Sampled-Data Control Systems. Springer-erlag, London, 995. [] P. Iglesias. Input-output stability of sampled-data linear time-varying systems. IEEE rans. Aut. Cont., 4(9):647 65, 995. [] G. Herrmann, S. K. Spurgeon, and C. Edwards. A new approach to discretization applied to a continuous nonlinear, sliding-mode-like control using non-smooth analysis. Accepted in IMA J. of Math. Cont & Inf.. [4] S. Djenoune, A. El-Moudni, and N. Zerhouni. Stabilization and regulation of a class of nonlinear singularly perturbed system with discretized composite feedback. Int. J. of Syst. Sc., 9(4):49 44, 998. [5] G. Herrmann, S. K. Spurgeon, and C. Edwards. Discretisation of sliding mode based control schemes. In Proc. 8th CDC, Phoeni, pp , 999. [6] G. Herrmann, S. K. Spurgeon, and C. Edwards. A new non-linear, continuous-time control law using slidingmode control approaches. In Proc. 5th Int. Workshop on ariable Structure Syst., Florida, pp. 5 56, 998. [7] H. K. Khalil. Nonlinear Systems. Macmillan Publishing Company, New York, edition, 99. [8]. Lakshmikantham and S. Leela. Differential and Integral Inequalities, vol., Acad. Press, 969. [9] E. P. Ryan and M. Corless. Ultimate boundedness and asymptotic stability of a class of uncertain dynamical systems via continuous and discontinuous feedback control. IMA J. of Math. Cont & Inf., : 4, 984. [] G. Herrmann. Contributions to discrete-time nonlinear control using sliding mode control approaches. Internal nd Year PhD-Report, Leicester Univ., Eng. Dep., 999.