zyxwvutsrqponmlkjih zyx zyxwvutsrqponmlkj zyxwvutsrqp zyxwvut IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO. I 1, NOVEMBER 1996 + CL have equally many zeros in the right-half plane when p > 0. 3) The system ? c ( s ) / d ( s ) is stabilizable by a srdble controller. Proof ( 1 2 ) : By the extended version of the zero exclusion principle given above, and by using the fact that p o and p I are stable, we deduce that the polynomials X p o (s) (1- X)pl ( s j are stable for 0 5 X 5 1 iff 2) p o ( s ) p l ( - s ) + + 1691 Remarks: The equivalence between statements 1) and 2) is also shown in [ 6 ] . Systems n ( s ) / d ( s )resulting from 110 and p l may be nonproper. It may also be identically equal to infinity when d ( s ) 0. This special case happens only when PO(.) = l i p ~ ( s for ) some real X:, which is a trivial case. (We thank one of the reviewers for this remark.) The existence of a stable stabilizing controller for a system can be checked by the techniques described in [ l ] and [2].These conditions can thus be used as alternatives to the conditions given in [4] for testing the stability of convex combinations of polynomials. zyxwv + (1 - X ) p l ( j ~#) 0 0 < X < 1: Xp~(s+ ) (1 X)p~(s)has invariant Xpo(,j-i’) U, E R. (The family degree when 0 < X < 1 since P O . P I are stabie and have same sign.) This last condition can be expressed equivalently by po(j&) zyxwvutsrq zyxwvutsr + ppl(,jd) i.0 __- Multiplying both sides by equivalent condition 111( 0 < p 5w E R. j ~ =) p ~ ( - j w ) , we arrive at the + p o ( j ~ ) (p -~j ~ ) /1 ’# 0 0 < p , zu t R. A second application of the extended zero exclusion principle leads us to the conclusion. (2 + 3): From the extended zero exclusion principle, the condition that po(s)pl ( - s ) p have equally many zeros in the right-half plane when p > 0 is equivalent to + + U O ( . ~ U ) ~ I ( - ~ L O/L ) # 0 0 < p. w + REFERENCES 11 B. D. 0. Anderson and E. I. Jury, “A note on the Youla-Bongiorno-Lu condition,” Automutica, vol. 2, pp. 387-388, 1976. 121 V. Blondel and C. Lundvall, “A rational test for strong stabilization,” Automutica, vol. 31, pp. 1197-1198, 1995. 131 B. R. Barmish, New Toolsfor Robustness of Linear Systems. New York McMillan, 1994. 141 M. Fu and R. Barmish, “Maximal unidirectional perturbation bounds for stability of polynomials and matrices,” Sq’st. Contr. Lett., vol. 11, pp. 173-179, 1988. PI D. C. Youla, J. J. Bongiorno, and C. N. Lu, “Single-loop feedback stabilization of linear multivariable plants,” Automatica, vol. 10, pp, 159-173, 1974. 161 E. Zeheb, “Necessary and sufficient conditions for root clustering of polytopcs of polynomials in a simply connected domain,” IEEE Trans. Automat. Contr., vol. 34, pp. 986-990, 1989. zyxwvutsrq E R. The decompositions po(.s) = p o o ( - s 2 ) s p o l ( - s 2 ) and til (s) = p 1 ” ( - s 2 ) s p l l ( - s 2 ) lead to p”(,ju) = POO(U,’)+ j ~ , p ~ ~ and( u ~ ) pi ( - . i ~=) p10(d~)-juip11(~~).Therefore, anequivalent condition is given by + [PO0(LO2)PlO(U2) + + d2VOI (W”L)p11bJ2) On Variable Structure Output Feedback Controllers {L] +.iubol(L,‘2)Pio(U,2) - Poo(w’2)pll(-i’2j]# 0, 0 < p > 9 E R. C. M. Kwan By looking at the real and imaginary parts of this expression, we deduce the equivalent condition that Abstract-In a recent work by Zak and Hui [I], a sliding-mode controller for multi-inputlmulti-output(MIMO) systems using static output feedback was proposed. Very nice geometric conditions for how to Po”(LJ2)Plo(U2) U,2Pol(u2)pll(U,2)2 0 design sliding surfaces were given. However, there are two restrictive assumptions in it. One is that the uncertainties in the system must be whenever bounded by a known function of outputs which excludes some possible U1 E R. uncertainties in the A matrix if the system is described by the triple L d ~ 0 1 ( U , ” ) p 1 0 ( L J 2 ) -p”o(Ll:2)plI (U”] = 0, (A.4, C ) . The other one requires a matrix equality [l,(4.3)] to be held In other words, the polynomial n ( s ) = y o o ( , ~ ) p l o ( s ) + . ~ ~ , o l ( s j l ~ l l (which s ) may also be very difficult to sati3fy in many systems. In this paper, must take positive values whenever d ( s ) = s ~ i ~ l ( s ) p l o (-s j we propose a modification of the sliding mode controller for a class of poo(.s)pl~(s)]is equal to zero on the positive real axis. Since single-input/single-output(SISO) systems which can eliminate the abovementioned limitations and, under certain conditions, guarantee global p o o ( O ) p 1 0 ( 0 ) = p o ( O ) p l ( O ) > 0, this condition is satisfied if and closed-loop stability. Hence the range of applicability of the method in only if d ( s ) has an even number of zeros between each pair of positive 111 can be greatly broadened. + real zeros of n ( s ) .By the parity interlacing condition (see [SI), this is equivalent to the requirement that IrL(s)/d(s) is stabilizable by a stable controller. U Example: Let p o ( s ) = 10s“ s2 Gs 0.57 and p l ( s ) = Ius3 2 2 8s 1.57. It is shown in [3] that although both polynomials are stable, not all convex combinations of p o and p i are stable. We verify this result by direct application of Theorem 1. From P O (s) = - ( - s’ ) 0.5 7 s [ 10( - s’ ) 61 and pl ( s ) = - 2( -s 2 ) 1.57+.s[-10(-s2)+8] we constnict n ( s ) = 10.s’-14s2+s4.86s and d ( s ) = 1 0 0 -~138.~’+45.20~+0.8949. ~ The polynomial d ( s ) has a single zero (at 0.5991) between the pair (0, 0.6368) of zeros of n ( s ) , and the system r r / d is therefore not stabilizable by a stable controller. The convex combinations of 710 and 111 are thus not all stable. This can be verified by checking that 2/3p0 1 / 3 p l is unstable. + + + + + + + + ~ + + + 1. INTRODUCTION Sliding mode control (or variable structure systems control) is a popular robust control method among control engineers. It is simple to design and completely robust to “matched” uncertainties. Its importance and applications can be seen from two recent special issues in the Intevnational Journal of’ Control 121 and the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS [ 3 ] .One major drawback Of sliding control is that states have to be available. Since in many Manuscript reccived May 2, 1994; revised March 7, 1995 and July 1, 1995. The author is with Intelligent AutomaLion Inc., Rockville, MD 20850 USA (e-mail: ckwan@i-a-i.com). Publisher ltem Identifier S 0018-9286(96)06767-0. 001 8-9286/96$05.00 0 1996 IEEE zyxwvutsrqp IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 41, NO 11, NOVEMBER 1996 1692 practical applications, only system outputs or part of system states are available, this necessitates the development of sliding mode control methods using output feedback. Emelyanov et al. [4] used observers to reconstruct states. Zak and Hui [ I ] proposed a new static output feedback scheme without using observers. Very nice geometric conditions were given on the switching surface design. In this note, we concentrate on a single-inputlsingle-output (SISO) linear system of the form zyxwvutsrqpo zyxwvutsrqponm zyxwvutsrq zyxwvutsr zyxwvutsrq All :y = c x The second assumption is that to guarantee the sliding condition m E 0. Zak and Hui [ l ] required that (1) for some matrix 124. Even for SISO systems, (8) is difficult to achieve. If (8) cannot be satisfied, the authors in [ l , (4.6)] resorted to a bounded controller which can only guarantee local stability. In the next section, we shall propose a modification of the sliding mode controller which will eliminate the two limitations and, at the same time, guarantee the global closed-loop stability. It should be emphasized here that our controller is no longer static. Our simple dynamic controller can guarantee global stability under the following conditions. 1) Solvability of equation S = F C , i.e., Zak and Hui [ l , Th. with .rl E Rn-‘ , . r 2 E R..c = [x: X 2 I T , u E R , and :y t R. -4,, are matrices with appropriate dimensions. Assume the uncertainties A&, A&, Ah, d ( t ) are all bounded and satisfy the so-called “matching” conditions. i.e., 4.31, is needed. 2) Existence of S is needed [ I , Th. 4.11. 11. MAINRESULTS with D,’s unknown but bounded quantities. Hence (1) can be written From (7), we can express ~ ‘ in 2 terms of TI and 0 as as Note that SS is a scalar in SISO systems. Using (9), (2) can be recasted as zyxwvutsrqpon zyxwvutsr I.,+ zyxwvutsrqpo zyxwvutsrqp There are two major assumptions in the paper by Zak and Hui [l]. The first assumption of [ l ] is that The eigenvalues of (All AltS;’S1) in (loa) are {-XI. - X 2 . . . . , -XVL--1} which can be done by suitably choosing SI. S2 such that [l, Th. 4.11 is satisfied. To guarantee the sliding condition (T E 0. we differentiate the sliding variable in ( 5 ) and use (IO) to get ~ with p a known function of outputs. This is quite restrictive because it can be seen easily that (3) does not satisfy (4). In other words, AArl, AA22 may not be allowed to exist in the system. For example, let us consider x = [ a21 0 y=[l CI.22 + Aa2t [:‘]U O]C. The uncertainty is E = A a 2 2 x 2 . It is not possible, in this case, to Ax! r” 3s 2 fttftctior? 9f I . The sliding variable is defined as [I] where h!f with F a constant matrix. The choice of F is related to the design of state switching surface 0 =SX (6) or, with no loss of generality The matrix F should be selected to satisfy S = FC. Geometric conditions were given in [ 11 for how to choose F and S so that ( 1 2 - 1)prescribed distinct, nonzero, and real negative eigenvalues {-A1,-A2;... .-An-l} with A,’s>O can be assigned. N =si(411 AA12S,1S1) + s2(-421 A ~ ~ S ~ ’ S I ) + Szb(D1 - D2s,1sl) = S,.A,,S,’ + S a ( A a a +bDZ)SF1. - - Since (11) contains 2 1 , Zak and Hui used (8) to avoid the measurement of states. If (8) cannot be satisfied, a bounded controller is used which can only guarantee local stability. As we shall see in a moment, we can eliminate these limitations while, under certain conditions, guaranteeing global stability by a slight modification of the controller. First we need the following lemma. Lemma: Consider (loa). Let A,, be the minimum value of { A I , A ~ , . . . , A ~ - ~ } where {-A1,-A2:... -Xn-l} are the eigenvalues of ( A l l - A12ST1Sl) with A’s all positive real values. Then we have i) j ( e x p ( A l l - 2412S21Sl)t(( 5 k e x p ( - ~ , t ) for some k > ~ . ii) Ilrl(t)ll is bounded by ~ ( f for ) all time, where u l ( t ) is the zyxwvutsrqpo zyxwvutsr zyxwvutsrqponml zyxwvutsrq 1693 IbEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO. 11, NOVEMBER 1996 REFERENCES solution of + k11412s;-11( IoI, i ( t ) = -X,,u,(t) .i(O) 5 k ~ ~ . E L ( O j ~ ~ . [ I ] S. H. Zak and S. Hui, “On variable structure output feedback controllers for uncertain dynamic systems,” IEEE Trans. Automat. Contr., vol. 38, (12) no. IO, pp. 1509-1512, 1993. [21 A. Sabanovic, N. Sabanovic, and K. Ohnishi, “Sliding modes in power Proof: It is obvious that i) hoilds for some positive constant k . converters and motion control systems,” Special Issue on Sliding Mode To see ii), we solve (loa) to yield Control, Int. J, Contr., vol. 57, no. 5 , pp. 1237-1259, 1993. [3] J. Y. Hung, W. Gao, and J. C. Hung, “Variable structure control: (t)l)= I I C ( A 1 1 - A l a S 2 1 . w t I1 IIx1(0)lI A survey,” Special Issue on Sliding Mode Control, IEEE Trans. Ind. Electronics, vol. 40, no. 1, pp. 2-22,, 1993. + 1 1 e( A I I - A I z S; ’ -71) ( t- 7 )II 141 S. V. Emelyanov, S. K. Korovin, A. L. Nersisian, and Y. E. Nisenzon, “Output feedback stabilization of uncertain plants: A variable structure ’ (IA12S,11( IffIdT approach,” Int. J. Contr., vol. 55, no. 1, pp. 61-81, 1992. [ 5 ] E. Bailey and A. Arapostathis, “Simple sliding mode control scheme < kc-x“tllSL(o)II k*c-x-(‘-’) applied to robot manipulators,” Int. J. Contr., pp. 1197-1209, 1987. ]Iz, 1‘ zy zyxwvutsrqp zyxwvutsrqpo + - . IIAlzs,’ 1 1 101 5 e-X”i,a,(0) +. ‘ IIAlaS;l if ~ ( l (LT I’ &-Am“(-‘) 1 1 Iff1 d r = w ( t ) , 0 5) kll.cl(0)ll > 0 Discrete-Time Neural Net Controller for a Class of Nonlinear Dynamical Systems where ui ( t )satisfies (1 2). Thus, by comparing both sides of the above inequality, one can see that w(t) 5 I I z l ( t ) l I for all time if ~ ( 0 is) sufficiently large. S. Jagannathan and F. L. Lewis (-1.E.D. Abstract-A family of two-layer discrete-time neural net (NN) controllers is presented for the control of a class of mnth-order multi-input-multioutput (MIMO) dynamical system. No initial learning phase is needed so that control action is immediate; in other words, the neural network (NN) controller exhibits a learning-while-functioning-feature instead of a learning-then-control feature. A two-layer NN is used which is linear in the tunable weights. However, this is a far milder assumption than the adaptive control requirement of linearity in the parameters, since the universal approximation property of the NN means that any smooth U = (Sab)-’(l - D;”””)-’[-Hw(t) - Glal - 51 sgri ( a ) (13) nonlinear function can be reconstructed. The structure of the neural net controller is derived using a filtered error approach. It is indicated that where delta-rule-based tuning, when employed for closed-loop control, can yield unbounded N N weights if: 1) the net cannot exactly reconstruct a certain ( 1 4 4 H L required function, or 2) there are bounded unknown disturbances acting (14b) on the dynamical system. Certainty equivalence is not used, overcoming G 21 1 N 1 a major problem in discrete-time adalptive control. In this paper, new b 2 s11p[ISzhD,,(tjI]+ q , ‘t)>O. ( 1 4 ~ ) on-line tuning algorithms for discrete-time systems are derived which are Note that o = F g , and hence (13) does not require states. Substituting similar to f-modification for the case of continuous-time systems that include a modification to the learning rate parameter and a correction (13) into ( l l ) , one gets term to the standard delta rule. These improved weight-tuning algorithms guarantee tracking as well as bounded NN weights in nonideal situations so that persistency of excitation (PE) is not needed. The hound on Ilxl (0) 11 can be estimated by following the definition of each state. From their physical meanings, we can easily get a crude estimation of I(z1(0)11. Now let us go back to (1 1). Assume 0 < D F n < I D3 I < Dinax< 1 which is a quite reasonable assumption. The sliding controller is chosen as zyxw zyxwvuts Il-~~lI zyxwvutsr zyxwvutsr + Noting w ( t ) 2 I l z ~ ( f ) l from l the Lemma, the fact that 1 DJ 1 - Dinax,and using the inequalities in (14), (IS) then becomes a& 5 - r / l n l < 0. I. INTRODUCTION 2 (16) Hence o tends to zero in finite time [SI.The time to reach sliding mode is determined by q. The larger the 1 1 , the shorter the time will be to reach sliding. 111. CONCLUSIONS We proposed an output feedback sliding controller for a class of SISO systems. The proposed controller is a modification of the controller in [l]. Two major limitatlions of the scheme in [ I ] have been eliminated by using a simple dynamic controller. This implies that the applicability of the scheme by Zak and Hui can be greatly broadened. Another merit is that we can guarantee global stability under certain conditions [l, Ths. 4.1 and 4.31 by using output feedback only. Considerable research has been conducted in system identification or identification-based neural network (NN) control [SI, [IO], [12], [131 and little in the use of direct closed-loop NN controllers that yield guaranteed performance. On the other hand, some results presenting the relations between NN and direct adaptive control [2],as well as some notions on NN for robot control, are given in [4]. A direct continuous-time neural net robot controller was proposed in [4] which guarantees closed-loop tracking performance. However, little about Manuscript received August 15, 1993; revised April 30,1994 and November 22, 199.5.This work was supported in part by the National Science Foundation under Grant IRT-9216545. S. Jagannathan is with the Automated Analysis Corporation, Peoria IL 61602 USA (e-mail: saranj @cujo.tc.cat.com). F. L. Lewis is with the Automation and Robotics Research Institute, The University of Tcxas at Arlington, Fort Worth, TX 76118 USA. Publisher Item Identifier S 0018-9286(96)04373-5. 0018-9286/96$05,00 0 1996 IEEE