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A control scheme for a class of uncertain linear plants
Chemical Engineering Science 54 (1999) 51—60
A control scheme for a class of uncertain linear plants Jose Alvarez-Ramı´ rez*, Rodolfo Sua´rez Divisio´ n de Ciencias Ba´ sicas e Ingenierı´ a, Universidad Auto´ noma Metropolitana-Iztapalapa, Apartado Postal 55-534, Me´ xico D.F., 09000 Me´ xico Received 6 September 1996; accepted 26 June 1998
Abstract An observer-based controller for a class of uncertain minimum-phase linear systems is presented. The main idea is to lump the uncertainties into a term, which is interpreted as a new uncertain state. In this way, the original system is posed in an extended state space where the dynamics of the new uncertain state can be reconstructed from measurements of the output. The design is simple. If the order of the system is n, first we assume that the output, its first n time derivatives and the uncertainties (new state) are available for feedback and design a state feedback controller in appropriate coordinates. Then we use a high-gain observer to estimate the derivatives of the output and the new uncertain state. On the contrary to traditional adaptive schemes, it results in a linear closed-loop system. Besides, a persistence of excitation condition is not necessary to assure closed-loop stability. Under signal tracking conditions, the controller yields global practical tracking, such that the tracking error can be made as small as desired by adjusting certain observer and controller parameters. The performance of the controller is evaluated by means of a simulation example. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Linear plants; High-gain observers; Robust control; Signal tracking
1. Introduction and problem statement In this work we present an approach to the design of robust controllers for linear systems. We depart from a state-space description of the plant. While in most traditional adaptive schemes (Astro¨m and Wittenmark, 1989) unknown parameters are considered separately from states, which leads to estimate separately parameters and states, here we consider products of parameters and states as uncertain terms which are lumped into a total uncertain term. In this way, our control design approach resembles the modeling error compensation due to Sun et al. (1994) and the time-delay compensation techniques due to Youcef-Toumi and Ito (1990). An important property of the lumped uncertain term is that it can be estimated by means of a high-gain observer. In this way, uncertain terms are interpreted as a new state of the system. The main advantage of the proposed control scheme over traditional (Astro¨m and
*Corresponding author. Fax: #52-5-7244900; e-mail: jjar@ xanum.uam.mx.
Wittenmark, 1989) and nonlinear (Kristic et al., 1992, 1994) adaptive designs is that it leads to closed-loop linear equations whose stability analysis can be addressed with standard techniques. To the contrary, traditional adaptive algorithms are typically highly nonlinear and this has meant that the associated theory is nonlinear. Another advantage of the proposed design procedure is that the dynamic order of the resulting controller depends only on the order of the plant, and not on the number of unknown parameters. In fact, the order of the controller is not higher than 2n. This means that the dynamic order of the resulting controller is lower than in traditional adaptive schemes (Astro¨m and Wittenmark, 1989) and nonlinear designs of adaptive controllers (Kristic et al., 1992, 1994). The proposed control scheme yields practical reference tracking. That is, the tracking error can be made as small as desired by adjusting certain controller parameters. We do not see this as a serious drawback since the performance of the practical controller is limited by various factors, like sampling rates and noise measurements. Our goal in this work is to develop a control strategy in a simpler setting that reveals its essential features,
0009-2509/99/$ — see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 2 1 4 - 0
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J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
thereby laying the groundwork for the more difficult cases, such as linearizable nonlinear systems. Problem statement. Consider linear plants of the following form: B(s) y(s)" u(s) A(s) b sK# 2 #b s#b u(s). " K> (1) sL#a sL\# 2 #a s#a L Many chemical plants, such as distillation columns and heat exchangers, can be modeled as in Eq. (1) (Morari and Zafiriou, 1989). We make the following assumptions on the plant. Assumption 1. The plant is minimum phase, i.e. the polynomial B(s)"b sK# 2 #b s#b is K> Hurwitz, and the plant order n, the relative degree o"n!m, and high-frequency gain b are known (the K> case where only the sign of b is known will be adK> dressed in a forthcoming paper). Assumption 2. The reference signal r(t) and its first n derivatives are known, bounded and continuous. Assumption 3. Only the output y is available for feedback.
The objective of this paper is to design an output feedback controller which guarantees boundedness of all signals of the closed-loop system and tracking of a given reference signal r(t) and the order of the resulting compensator be lower than the order of traditional (MRAC) (Astro¨m and Wittenmark, 1989) and nonlinear (Kristic et al., 1992) adaptive compensators. The paper is organized as follows. We first select a state-space description of the plant in Section 2, and then present our design procedure in Section 3 followed by a proof of stability and tracking in Section 4. Finally, we illustrate our procedure in an unstable second-order plant in Section 5.
2. Preliminaries The material in this section is straightforward and exists in the literature (see Astro¨m and Wittenmark, 1989; Kristic et al., 1992, 1994). However, for the sake of completeness in presentation a brief description of state-space representations of plant (1) is made. We start by designing a feedback controller assuming that the parameters of the plant are known and all internal states of the plant are available for feedback. This is not a reasonable assumption. We use it, however, as an intermediate assumption toward the final controller designed in the next section which will be a controller depending only on
the measured output and estimated (nonmeasured) states. In time domain, plant (1) can be described by the nth-order differential equation yL#a yL\# 2 #a y#a y L "b uK# 2 #b u#b u K> where yH"dHy/dtH and uG"dGu/dtG. We represent plant (1) by augmenting a series of m integrators at the input side of the system. We denote the states of these integrators by v "u, v "u, up to v "uK\. By taking K h "y, h "y, up to h "yL\ (internal states of the L plant), we can represent the augmented system by the (n#m)-dimensional linear state-space model [see for instance Khalil (1996) and references therein] h "h , j"1, 2 , n!1 H H> L K h "! a h # b v #b V L H H G G K> H G v "v , i"1, 2 m!1 G G>
(2)
(3)
v "V K where V31 is a new input. The state representation (2), (3) is not minimal since the states v"(v , 2 , v ) are not K observable from measurements of y"h 31. However, this is not an obstacle to stabilize the system (2), (3). The solution to this stabilization problem is well known and relies on standard techniques for linear and nonlinear systems. The state-representation (2), (3) allows us to use a time-varying change of variables to transform the system equation into an error space where the goal of the control design will be to drive the state of the system to zero. Let e"h!r31 be the output tracking error, and introduce the change of variables x "yH\!rH\" H h !rH\, j"1, 2 , n (tracking error vector). We reH write the h-subsystem (2) as e "x xR "x , j"1, 2 , n!1 (4) H H> L K xR "! a (x #rH\)# b v #rL#b V. L H H G G K> H G The series of integrators at the input side makes the derivatives of the input available for feedback. Consider the following dynamic linear state-feedback: u"v
vR "v , i"1, 2 m!1 G> L K vR " a (x #rH\)! b v G H H G G H G L # K x !rL b H H K> H
(5)
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
where the inner-loop gains K ’s are chosen in such a way H that the polynomial sL!K sL\!2 K s !K "0 be L Hurwitz. A possible procedure to choose the K ’s is to use H a reference model %L (s!j )"0, where the j ’s are the H H H desired closed-loop poles, such that sL! K sL\!2 L K s!K "%L (s!j ). H H Notice that v "v , i"1, 2 , m, so that the states G G v"(v , 2 , v ) are always available for feedback. In ad K dition,
L L V" a (x #rH\)! b v H H G G H G L # K x !rL b . H H K> H Let z"(x, v)31L>K. Under the state-feedback controller (5), the closed-loop system becomes zR "A z# (R), where R is a vector of signals compris? ing r(t) and their derivatives up to order n, and
A 0 . (6) A" A ? D A A is the Hurwitz companion matrix of the K ’s, A is the A G companion matrix of the b ’s, which is also Hurwitz G (minimum-phase assumption), and D is an m;n matrix involving the a ’s and the K ’s. The term (R) is the H H vector (0, 2 , 0, l(R))2, where
L l(R)" a rH\!rL b . (7) K> H H Observe that the closed-loop matrix (6) is in cascade form. It is not hard to see that the tracking error x(t) converges asymptotically to zero, while the compensator states v(t)"v(t) remain bounded for all t*0. Recall that the parameters a ’s, b ’s (except b ) are H G K> unknown and the state x31L is not available for feedback, so that the feedback controller (5) cannot be applied just as it is. The purpose of this paper is to look for controllers driven by the tacking error e"h!r31 which guarantee signal tracking while retaining the main features of the state feedback controller (5), such as linear closed-loop behavior and cascaded structure as in Eq. (6). That is, we look for a certainty equivalence design, which lead to a would-be linear controller (Astro¨m and Wittenmark, 1989) of the form (5).
3. An output feedback scheme In this section, we will derive a linear control scheme driven by the output tracking error e31. The main idea is to choose a state-space representation where the uncertainties in Eq. (4) (namely ! L a (x #rH\)# H H H K b v ) are lumped into a term g31 which is seen as G G G
53
a nonmeasured state. In this way, plant (1) is represented as a linear system embedded into a (n#m#1)-dimensional space together with an algebraic constraint. Consider the following dynamical system: e"x
G xR "x , j"1, 2 , n!1 H H> xR "g#rL#b V L K>
(8a)
L\ L K\ gR "! a x ! a rH# b v H H> H G G> H H G !a (g#rL#b V)#b V L K> K u"v v "v , i"1, 2 , m!1 (8b) G G> v "V K which is an (n#m#1)-dimensional linear system. It is not hard to prove that the function L K u(x, g, v, R)"g# a (x #rH\)! b v (9) H H G G H G is a first-integral (Wiggins, 1990) of system (8). That is, along the trajectories of system (8) we have that du/dt"0, for all t31, and any V31. The state-space representation (8) is externally equivalent to system (2), (3), as long as initial conditions satisfy u(x , g , v , 1 )"0. If we are able to control system (8) without making use of the constraint u(x , g , v , R )"0, we would be able to control system (2), (3). A geometric interpretation of the state-space representation (8) is the following. If the initial condition (x , g , v , R ) for system (8) satisfies u(x , g , v , R )"0, then, the dynamics of system (8) are equiva lent to the dynamics of the system (2), (3). That is, if (x(t), v(t)) is the solution to system (2), (3) with initial conditions (x , v , R ), and (x(t), g(t), v(t)) is the solu tion to system (8) with initial conditions (x , g , v , R ), then the equality u(x , g , v , R )"0 and the condition du/dt"0 imply that (x(t), v(t)) "%(x(t), g(t), v(t)) , where % denotes projection onto (x, v)-coordi nates. Geometrically, condition u(x , g , v , R )"0 means that if initial condition of uncertainty is located on the manifold u(x, g, v, R)"0, the trajectories of system (8) remain there and the corresponding trajectory of uncertainties lumped into g31 can be recovered just by projecting on the corresponding coordinates frame. It must be pointed out that for specific applications, it is not necessary to represent plant (1) as in Eq. (8). Such state-space representation is only a tool to obtain a control scheme and analyze the stability properties of the resulting closed-loop system. The role of the g31 in Eq. (8b) is to represent the uncertainty term ! L a (x #rH\)# K b v as H H H G G G
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a new state. In this way, as long as u(x , g , v , R )"0, g(t)"! L a (x (t)#rH\(t))# K b v (t) for all H H H G G G t'0. The next property establishes that the uncertainty g can be reconstructed from measurements of the output e"y!r31. Proposition 1. ¹he states (x, g)31L;1 can be reconstructed from measurements of the output e31 and the input V31. The above result is an important property because it establishes that the dynamics of the uncertain term g"! L a (x #rH\)# K b v can be reconH H H G G G structed from the availability of e and V. Resembling the structure of compensator (5), at this point we propose the following (n#m#1)-dimensional compensator driven by the output e: xNR "xN !¸Ha (e!xN ), j"1, 2 , n!1 H H> H xNR "gN #rL#b V!¸La (e!xN ) (10a) L K> L g"!¸L>a (e!xN ) L> vR "v , i"1, 2 , m!1 G G> L vR "V" !g# K xN !rL b (10b) K H H K> H u"v where xN 31 and g31 are estimates of x31L and g31, respectively, the K ’s are chosen as in Eq. (5), ¸'0 is an H adjustable parameter, and the constants a ’s are chosen in H such a way that the polynomial sL\!a sL!2 !a s!a "0 be Hurwitz. The dynamical system L L> (10) is a linear controller comprising a high-gain observer (Esfandiari and Khalil, 1992) for the states (x, g) and a compensator as the one described by system (5). Notice that since the parameters a ’s and b ’s are assumed to be H G unknown, the right-hand side of the Eq. (8b) is also unknown. Thus, such term was neglected in the construction of an observer for the state g. The order of the dynamic controller (10) is n#m#1. Kristic et al. (1992, 1994) designed a controller comprising a Kreisselmeier state estimator (Kreisselmeier, 1977) of order 2n, and a parameter estimator of order m#n (the number of unknown parameters), finally giving a (3n#m)-order compensator. This means that the dynamic order of the resulting controller (10) is lower than Kristic et al.’s adaptive scheme and traditional adaptive schemes. The generalization of the basic control design for minimum-phase, multi-input—multi-output plants where the high-frequency gains are known, is straightforward departing from an state-space representation analogous to the linear system (2). Some remarks on this extension are made in the conclusions section.
4. Stability analysis Let e be a vector in 1L> whose components are defined by e "¸L>\H(x !xN ), j"1, 2 , n H H H
(11)
e "g!g . L> Using the fact that g"! L a (x #rH\)# H H H K b v , and recalling that v "v , i"1, 2 , m, the dyG G G G G namics of the closed-loop system (2), (3), (10) are completely described by xR "A x# (e) A
(12a)
v "Dx#A v# (e)#t (R) eR "¸A e# (x, v, e)#t (R) (12b) where the matrices D, A and A are given as in A eq. (6), A is the (n#1)-dimensional Hurwitz companion matrix of the a ’s, and , , , t and t are linear H functions of its arguments, which are given in Appendix A. From the expressions for , , , t and t in Appendix A, and the assumption that r(t) and its derivatives are bounded, we obtain the existence of six positive numbers b , k"1, 2 , 6 (not depending on ¸) such that I the following inequalities hold for all (x, v)231K>L, e31L>, and ¸*1: " (e)")b "e", " (e)")b "e"
(13)
" (x, v, e)")b "z"#b "e" "t (R)")b , "t (R)")b .
(14)
Eqs. (12a) and (12b) become an (2n#m#1)-dimensional linear system. In principle, the stability analysis of such a system is easier than in traditional adaptive schemes. In fact, in adaptive controllers the parameter adaptation scheme induces nonlinear (Sastry and Bodson, 1989) or linear time-varying (Krause and Kumar, 1986) closed-loop equation whose stability analysis involve more elaborated stability theories. 4.1. Stability under regulation conditions Without loss of generality, we assume that r(t)"0. In this case R"0, From the expressions for t and t in the Appendix A, one can see that t (0)"0 and t (0)"0. Recall that matrix A in Eq. (6) and matrix ? A are Hurwitz, so that the Lyapunov equations A2 P #P A "!I and A2P #P A "!I ? ? ? ? L>K L> have a positive-definite solution. Let z"(x, v)31L>K, and consider the quadratic function ¼(z, e)" z2P z#e2P e, and take its time derivative along the ?
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J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
trajectories of the following linear system: xR "A x# (e) A v "Dx#A v# (e) eR "¸A e# (z, e). Using Eq. (13), we get
(15)
¼ Q "!"z"!¸"e"#z2P (e)#e2P (z, e) ? )"z"!(¸!j (P )b )"e"
#(j (P )b #j (P )b ) "z" "e" (16)
?
where "( , )2, b "max(b , b ) and j (P)
represents the maximum eigenvalue of the matrix P. Observe that the right-hand side in Eq. (16) is a quadratic form, which is negative for all ¸ in (¸*, R), where ¸*"j (P )b #(j (P )b
? #j (P )b )/4. (17)
This implies the global asymptotic stability of the linear system (15). From the results in Appendix A, it can be obtained that the estimates b ""K", b ""K/b ", b " K> "b"#"a"#"a K"#"b a"#"b a" and b ""a K"# "b K", L K K L K where a"(a , 2 , a ) and b"(b , 2 , b ). Given an up L K per bound of the parameters a31L and b31K, one can estimate an upper bound for the parameters b ’s. By G knowing an upper bound for the b ’s, an estimate of G ¸* can be obtained. 4.2. Stability under signal tracking conditions Based on the above result, we conclude that system (12) is an asymptotically stable linear system with a bounded additive perturbation t(t). Next, we will show that there exists a compact set !1L>K> (independent of ¸) which absorbs all trajectories of system (12). Proposition 2. ¸et L* as in Eq. (17). For all L in (L*, R), the trajectories of system (12) are uniformly bounded, and converge to a compact set !L1> > independent of L. Proof. See Appendix B.
)
The parameter ¸ in system (12) can be seen as the rate of estimation of uncertainties g31 and unmeasured states x31L. The result in Proposition 1 states that if the rate of estimation ¸ is larger than a certain value ¸*, where ¸* depends on the magnitude of the uncertain parameters a ’s, b ’s (except b ), then the trajectories of H G K> the closed-loop system (12) are uniformly bounded. This means that it is possible to find a compact set !L1L>K> such that all trajectories of Eq. (12) converge to it and remain there. By making a more detailed analysis of the closed-loop system (12) one can show that,
for sufficiently large ¸, there exists a family of sets ! (¸) L with the property that ! (¸ )-! (¸ ) for ¸ '¸ . That L L is, the larger the value of ¸, the smaller the absorbing set for the trajectories of system (12). However, to prove the results that follow, we will require only the existence of the set !. Results in Proposition 2 are useful to prove the following global practical stabilization result. Theorem 1. For any given initial condition (z , e )" (x , v , e ), and sufficiently large ¸, the asymptotic tracking error "x(tPR)" is on the order of L\, for L in (L*, R). Proof. Since eR "¸A e# (z, e)#t (t), we get R e(t)"exp (¸tA )e(0)# exp (¸(t!s)A ) ;( (z, e)#t (s)) ds. Performing the norm operator to the above equation, and making use of inequalities (13) and the fact that the matrix A is Hurwitz, we obtain "e")p "e "exp (!j ¸t)#p exp (!j ¸t) R ; exp (j ¸s)(b "e"#b "z"#b ) ds where j and p are positive constant depending only on the Hurwitz matrix A . After some algebraical manipula tions, we get R exp (j ¸t) "e")p "e "#p b exp (j ¸s) "e" ds R #p exp (j ¸s) (b "z"#b ) ds. The result in Proposition 2 implies that "z(t)")b , where b depends only on the initial conditions (z , e )31K;1L>. Then exp (j ¸t) "e")p "e "#(p b /j ¸) R ;[exp (j ¸t)!1]#p b exp (j ¸s)"e" ds where b "b b #b . Using the Gronwall’s inequality, we obtain "e(t)")"e" # (t) (see Appendix C), where QQ
(t) is a bounded continuous function with (R)"0 which is given by
(t)"k exp [!(j ¸!p b )t] p "e "j ¸!p"e "b !p b k" j ¸!p b and "e" given by QQ p b "e" " QQ j ¸!p b
(19)
(20)
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J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
so that lim "e(t)")"e" . For sufficiently large ¸, we R QQ have that j ¸!p b '0. On the other hand, "e" is of QQ the order of ¸\. Analogously to the differential equation governing the dynamics of the error e(t), we can integrate the equation xR "A x# (e) A R x(t)"exp(A t)x #exp (A t) exp (!A s) (e(s)) ds. A A A Using the triangle inequality, we obtain (recall that A is A Hurwitz)
"x(t)")p exp (!j t)"x "#p exp (!j t) A A A A R ; exp (j s) " (e(s))"ds A )p exp (!j t)"x "#b p exp (!j t) A A A A R ; exp (j s)"e(s)"ds A where p and j are two positive constants depending A A only on the matrix A . A By recalling that "e(t)")"e" # (t), we have that QQ "x(t)")p exp (!j t)"x "#b p exp (!j t) A A A A R ; exp (j s) ("e" # (s)) ds A QQ b p exp (!j t) A "p exp (!j t)"x "# A A A j A ;[exp(!j t)!1] "e" A QQ R #b p exp (!j t) exp (j s) (s) ds. A A A Using Eq. (19) to integrate the last term of the inequality above, we obtain
b p "x(t)")p exp (!j t)"x "# A [1!exp (!j t)]"e" A QQ A A j A b pk # A [exp (!¸ t)!exp (!j t)] A j !¸ A where ¸ "j ¸!p b . Then, we conclude that lim "x(t)")p b "e" /j . Since "e" is on the order of R A QQ A QQ ¸\, we have that lim "x(t)" is also on the order R of ¸\. )
The time function t (t) represents the estimated transiV ent behavior of the tracking error vector x31L. The first two terms in t (t) are on the order of O(¸). From the V expression for k [Eq. (19)] and recalling that "e " is of the order O(¸L\), we have that, for large values of ¸, t (t) V can be approximated by t (t)"%(¸L)t* (t) V ? V where t* (t)"exp (!¸ t)!exp (!j t) is a bounded V A time function and %(¸L) is a monotonic increasing function of ¸L\. The time function t (t) represents the V ? effects of the observer initial conditions xN 31L, g 31 in the tracking error x31L. For a given ¸, %(¸L) is a constant representing the amplitude of the overshoot in the estimation error e31L>. In this way, the larger the value of the observer gain ¸, the larger the overshoot in the estimated states xN 31L, g 31. This dynamic effect limits the achievable closed-loop performance since overshooting in e31L> is transmitted to the feedback loop. On the other hand, we have that the steady-state tracking error "x" "p b "e "/j is smaller for larger values of j . QQ A QQ A A We know that j is of the order of min+"Rej (A )", A I A (Hirsch and Smale, 1974), so that j depends on the A eigenvalues of the matrix A , and consequently on the A inner-loop gain vector K31L. Therefore, the steady-state tracking error "x" can be made smaller if the gain vector QQ K is chosen such that the eigenvalues of the matrix A are A located deeper in the open left-half complex plane. In general, large values of j is more or less related to A high-gain control, which in some cases may be undesirable. In practical situations, a compromise between the observer gain ¸ and the inner-loop gain vector K must be addressed in order to obtain small tracking error and good control performance during transients. 5. Example In this section, we illustrate the performance of the controller (10) via a second-order linear example from A stro¨m and Wittenmark (1989, pp. 147—151). We chose such an example from the adaptive control literature to have a reference against which we can judge the performance of the proposed controller. Consider a plant represented by the transfer function
Notice that the last expression in the proof of Theorem 1 can be written as
b y(s)" u(s) s(s#a)
"x(t)")"x" #t (t) QQ V where "x" "p b "e "/j and QQ A QQ A b p t (t)"p exp (!j t)"x "! A exp (!j t)"e" V A A A QQ j A b pk # A [exp (!¸ t)!exp (!j t)]. A j !¸ A
where a is unknown and b is known constant. We assume b"1. The poles of plant (21) are +0, !a,, so that for a(0 the plant is unstable. In this case, we do not add integrators at the system’s input; hence, V"u. The state-space representation (2), (3) of the plant is zR "z , zR "!az #bu, y"z . We want to design a controller such that the output of the system y(t) tracks the output of a reference model r(t) for any command signal r(t).
(21)
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
57
Fig. 1. Dynamics of the current uncertainty g(t) and its estimate g (t), which is provided by controller (22) with ¸"25. The estimate g (t) converges to the current value g(t), which allows to stabilize the unstable second-order plant (21).
Fig. 3. Simulations results for the second-order plant (21) and controller (22) with ¸"40, and initial conditions y(0)"yR (0)"0, xN (0)" xN (0)"1 and xN (0)"0.
Fig. 2. Dynamic behavior of the controlled output y(t) with the feedback controller (22) and regulation conditions r(t)"0.
Fig. 4. Output tracking error for several values of the observer parameter ¸. Notice that, after one period of the reference signal, the output tracking error achieves its steady-state dynamics.
A traditional model reference adaptive controller (MRAC) design is given by A stro¨m and Wittenmark (Example 4.8). The resulting controller (10) is described by xN "xN !¸a (e!xN ) xN "g #r#u!¸a (e!xN ) (22) g "!¸a (e!xN ) u"!r!g #K xN #K xN where e"y!r. The reference model is taken as G (s)"r(s)/r(s)"u/(s#2fus#u), hence K " K !u and K "!2fu. For simulations, we choose a "!3, a "!3, a "!1, such that the eigenvalues of matrix A are all located at !1. We consider first the case r(t)"0 (regulation), such that the uncertain state is given by g"!az . Fig. 1 presents the dynamics of the current value of g(t) and its estimate g (t), which is provided by controller (22) with ¸"25. The estimate g (t) converges to the current value g(t), which allows to stabilize the unstable second-order plant (21). The behavior of y(t) is presented in Fig. 2 for
three different initial conditions. Observe that y(t) is driven to zero without serious deterioration of the transient performance. Suppose now that r(t) is the output of the reference model G (s)"r(s)/r(s)"u/(s#2fus#u). Note that K r"!2fur!ur#ur, and r is available by integration of the reference model. Fig. 3 shows simulations results for ¸"40, and initial conditions y(0)"yR (0)"0, xN (0)"xN (0)"1 and xN (0)"0. The command signal r(t) is a square waveform, which is continuous almost everywhere. The effectiveness of controller (10) is demonstrated by the fact that the tracking error is not higher than 1% after few periods of the reference signal. To illustrate the fact that the asymptotic tracking error is reduced when the gain ¸ is increased, Fig. 4 shows the tracking error y(t)!r(t) for three different values of that gain: ¸"20, ¸"40, and ¸"60. For ¸'40, the peak in the relative tracking error as about 1% [recall that r(t) has unitary amplitude]. The large tracking error between 0 and 4 units of time are due to the peaking induced by the initial conditions of the compensator, which all were chosen as zero. Fig. 5 presents the control effort to
58
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
Fig. 5. Input signal corresponding to the signal tracking case shown in Fig. 3.
Fig. 7. Computed control input u(t) when 25% error in the estimation of the high-frequency gain b is assumed.
6. Conclusions
Fig. 6. Dynamics of the controlled output y(t) when 25% error in the estimation of the high-frequency gain b is assumed.
achieve the performance presented in Fig. 3. A comparison of our results with those in A stro¨m and Wittenmark (1989) shows that the performance achieved by our control design is as good as the one achieved by traditional adaptive schemes. At least in this example, it has been possible to design our controller to produce performance comparable to what is achieved by other designs. However, the main advantage of our design is that it uses a third-order compensator, while MRAC designs use a ninth-order filter in addition to the parameter adaptation (tenth-order compensator). Finally, to illustrate that the proposed controller (22) can also handle uncertainties in the high-frequency gain b, assume that the estimated value of the parameter b is bK "1.25. Fig. 6 presents the dynamics of the controlled output y(t) for the case r(t)"0 (regulation) and two initial conditions for y and yR . Notice that controller (22) takes the output y(t) to zero without excessive control effort (see Fig. 7) despite 25% uncertainty in the parameter b. It must be pointed out that if bK is decreased, instability of the closed-loop system is most likely to occur. As in traditional adaptive control approaches, to avoid instabilities due to uncertainties in the high-frequency gain b, bª must be taken as an upper bound (i.e. bª * b *b).
We presented a new design approach to design robust controllers for a class of linear plants. Our approach properly combines results from nominal stabilization and robust state observation. Our controller is simply a state feedback design with a linear observer. The dynamic order of the resulting controller is lower than in traditional adaptive schemes. From a stability analysis, we concluded that the steady-state tracking error will be of order O(¸\). Since ¸ is a design parameter, we can make the steady-state error arbitrarily small by adjusting the observer gain ¸. Our simulations show that the performance of the new controller is comparable to the performance of traditional controllers. As regards a possible generalization of the proposed design theory to a wider class of linear plants, one can observe that considering multi-input—multi-output minimum-phase plants with known high-frequency gains does not affect the subsequent developments, aside from obvious vectorial notations. If the (square, m-dimensional) MIMO plant is given by y(s)"G(s) u(s), the high-frequency gain matrix is G(0), which is assumed to be known and invertible. In this way, an extension of the proposed controller to MIMO plants would involve G(0)\ and m state observers like system (21). For instance, a simple (one-time constant), two-inputs—two-outputs model for binary distillation columns is given as y(s)"G(s) u(s) (Morari and Zafiriou, 1989), where y(s)"(x (s), x (s))2 are distillate and bottom " (deviation) compositions, respectively, u(s)"(R(s), »(s))2 are liquid and vapor (deviation) flowrates, respectively, and
1 g g . G(s)" (23) (s#q\) g g Plant (23) is minimum phase. A state-space realization of Eq. (23) is given by xR "!x /q#g R#g » " " xR "!x /q#g R#g ».
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
By assuming that the g ’s, the generalization of the proGH posed controller for plant (23) would involve the inverse of the high-frequency gain matrix G(0). In most practical cases the matrix G(0) is also uncertain (Morari and Zafiriou, 1989). At least for binary distillation columns, preliminary numerical simulations show that the proposed scheme is able to yield acceptable closed-loop response under set-point tracking and external disturbances despite of uncertainties in G(0). The stability of the proposed control scheme under uncertainties in G(0) for both single-input and multi-input plants requires more in depth research. The control of these cases (including the case of plants where the high-frequency gain b is also an uncertain parameter) are under progress K> and results and applications to chemical processes will be presented in a forthcoming paper. Appendix A. Computation of , and
Using the fact that g"! L a (x #rH\) H H H H # L b v , it is not hard to find the following expresG G G sions:
L 2
(e)" 0, 2 , 0, ! K ¸\L\>He #e H H L> H 2 L
(e)" 0, 2 , 0, ! K ¸\L\>He #e b H H L> K> H
(x, v, e)"(0, 2 , 0,
)2 L> where K\ L\
"# b v ! a x !a L> G G> H H> L G H L ; e # K (x !¸\L\>He ) L> H H H H L #b e # K (x !¸\L\>He ) K L> H H H H L K ! ax# bv . H H G G H G Notice that in all above expressions, e is at most of order ¸I, k)1. The expressions for t and t are given as follows: t (R)"(0, 2 , 0, L(R))2, where L(R) is given as in Eq. (7), and t (R)"(0, 2 , 0, t )2, where K
L L t " a rH#b !rL ! a rH\ . K H K H H H Appendix B. Proof of proposition 2 Consider the quadratic function ¼(z, e)"z2P z# ? e2P e and take its time derivative along the trajectories
59
of system (12). From Eq. (13), we obtain ¼ Q )!"z"!(¸!j (P )b )"e"#(j (P )b
? #j (P )b )"z""e".
#j (P )b "z"#j (P )b "e".
?
To simplify the presentation, let us define the following constants: c (¸)"¸!j (P )b , c "j (P )b #
? j (P )b , c "j (P )b and c "j (P )b . From
?
eq. (17), it can be seen that c (¸)'0, for all ¸*¸*. The right-hand side of the last expression can be written as ¼ Q )!"z"!(¸!j (P )b )"e"#(j (P )b
? #j (P )b )"z""e".
#j (P )b "z"#j (P )b "e".
?
To simplify the presentation, let us define the following constants: c (¸)"¸!j (P )b , c "j (P )b #
? j (P )b , c "j (P )b and c "j (P )b . From
?
eq. (17), it can be seen that c (¸)'0, for all ¸*¸*. The right-hand side of the last expression can be written as E("z", "e"; ¸)"!"z"!c (¸)"e"#c "z""e"#c "z"#c "e". We will show that for ¸'¸*, E("z", "e"; ¸) is negative for q except in a neighborhood of the origin. In turn, the above would imply that trajectories of system (12) are uniformly bounded. The quadratic function E("z", "e"; ¸) is defined on the closed first quadrant q "+"z", "e":"z"*0, "e"*0,. For m3[0, R], consider the set o(m)"+"z", "e" : "e""m"z", "e"O0, "z"O0,5q . If we evaluate E("z", "e"; ¸) in o(m), we obtain E("z", m"z"; ¸)""z"+(!1!c (¸)m #c m) "z"#(c #c m),. Since "z"'0, the sign of E("z", "e"; ¸) along the ray o(m) is given by the sign of the term into the brackets. It is easy to see that such a term is negative for all "z"'z*(m), where c #c m z*(m)" . (B.1) c (¸)m!c m#1 From the relation "e""m"z", the corresponding value for e*(m) is given by c m#c m e*(m)" . (B.2) c (¸)m!c m#1 The condition ¸'¸*, where ¸* is given as in eq. (17), implies that the second-order polynomial in the denominator is positive for all m3[0, R]. In fact, the polynomial c (¸) m!c m#1 has no roots in [0, R], for c!4c (¸)(0, which is exactly condition (17). So that z*(m)*0 and e*(m)*0, for all m in [0, R]. On the other hand, (z*(0), e*(0))"(c , 0) and (z*(R), e*(R))" (0, c /c (¸)). Thus, the continuity of Eqs. (B.1), and (B.2)
60
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
Inequality (18) has the form of inequality (C.1), so that Lemma 1 can be used with w (t)"exp(j ¸t)"e" w (t)"p b c(t)"p "e "#(p b /j ¸) [exp(j ¸t)!1]. After straightforward algebraic manipulations, we obtain
p b p "e "j ¸!pe "b !p b # "e") j ¸!p b j ¸!p b ;exp [!(j ¸!p b ) t]. From where the following definitions are taken:
Fig. B.1. Schematic diagram of the absorbing set c*(m).
with respect to m implies that, for a given ¸'¸*, the pair (z*(m), e*(m)) defines a continuous, finite valued curve c(m; ¸) in the first quadrant q (see Fig. B1) with the property that E("z", "e"; ¸) (0, for all ("z", "e")3q 3c>(¸), where c>(m, ¸)" +("z", "e")3q : c(m, ¸))0,, where c(m, ¸))0 indicates that the curve c(m, ¸) defined by the pair (z*(m), e*(m)), m3[0, R], is semi-negative valued. From the fact that *e*(m, ¸)/*¸)0 and *z*(m, ¸)/ *¸)0, we conclude that c*(m):"c>(m, ¸*) -c>(m, ¸), for all ¸ in (¸*, R). Therefore, E("z", "e"; ¸)(0 in the set ("z", "e")3q 3c*(m), for all ¸ in (¸*, R). Since c*(m) is a compact subset of 1L>K>, there exists a positive number c such that c*(m)L!"+z, e: ¼(z, e))c, (see Fig. B1). Since E("z", "e"; ¸)(0 implies that ¼ Q (0, we have that ¼ Q is negative in the subset q 3!. From Lyapunov arguments we conclude that all trajectories of system (12) are uniformly bounded and ! is an absorbing compact set, for all ¸ in (¸*, R). )
Appendix C. Computation of Eqs. (19) and (20) To compute Eqs. (19) and (20), the following result is required. Lemma A.1. (Gronwall’s inequality, Coddington and ¸evison, 1955). If w , w and c*0 on [0, t], c is differenti able, and
w (t))c(t)#
R
w (s)w (s) ds
(C.1)
then w (t))c(0) exp
R
w (s) ds#
R
c (s) exp
R
Q
(t)"i exp[!(j ¸!p b )t] p "e "j ¸!p"e "b !p b i" j ¸!p b p b "e" " . QQ j ¸!p b Notice that, if j ¸!p b '0, '(t) is bounded by i for all t*0.
w (q) dq ds.
References A stro¨m, K.J., & Wittenmark, B. (1989). Adaptive control. New York, USA: Adison-Wesley. Coddington, E.A., & Levison, N. (1955). ¹heory of ordinary differential equations, New York, USA: McGraw-Hill. Esfandiari, F., & Kahlil, H.K. (1992). Output feedback stabilization of fully linearizable systems. Int. J. Control, 56, 1007—1037. Hirsch, M.W., & Smale, S. (1974). Differential equations, dynamical systems, and linear algebra, New York, USA: Academic Press. Khalil, H. K. (1996). Adaptive output feedback control of nonlinear systems represented by input—output models. IEEE ¹rans. Automat. Control, 41, 177—188. Krause, J.M., & Kumar, K.S.P. (1986). An alternate stability analysis framework for adaptive control. Systems and Control ¸ett., 7, 19—24. Kreisselmeier, G. (1977). Adaptive observers with exponential rate of convergence, IEEE ¹rans. Automat. Control, 22, 2—8. Kristic, M., Kanellakopoulus, I., & Kokotovic, P.V. (1992). Nonlinear design of adaptive controllers for linear systems. IEEE ¹rans. Automat. Control, 39, 738—752. Kristic, M., Kanellakopoulus, I., & Kokotovic, P.V. (1994). Passivity and parametric robustness of a new class of adaptive systems. Automatica, 30, 1703—1716. Morari, M., & Zafiriou, E. (1989). Robust process control. Prentice-Hall, NJ, USA: Englewood Cliffs. Sastry, S.S., & Bodson, M. (1989). Adaptive control: stability. convergence and robustness. Englewood Cliffs, NJ, USA: Prentice-Hall. Sun, J., Olbrot, A.W., & Polis, M.P. (1994). Robust robust performance using model reference control and modeling error compensation. IEEE ¹rans. Automat. Control, 39, 630—635. Wiggins, S. (1990). Introduction to applied nonlinear dynamical systems and chaos. New York, USA: Springer. Youcef-Toumi, K., & Ito, O. (1990). A time-delay controller design for systems with unknown dynamics. ASME J. Dyn Measurement Control, 112, 133—142.
A control scheme for a class of uncertain linear plants Jose Alvarez-Ramı´ rez*, Rodolfo Sua´rez Divisio´ n de Ciencias Ba´ sicas e Ingenierı´ a, Universidad Auto´ noma Metropolitana-Iztapalapa, Apartado Postal 55-534, Me´ xico D.F., 09000 Me´ xico Received 6 September 1996; accepted 26 June 1998
Abstract An observer-based controller for a class of uncertain minimum-phase linear systems is presented. The main idea is to lump the uncertainties into a term, which is interpreted as a new uncertain state. In this way, the original system is posed in an extended state space where the dynamics of the new uncertain state can be reconstructed from measurements of the output. The design is simple. If the order of the system is n, first we assume that the output, its first n time derivatives and the uncertainties (new state) are available for feedback and design a state feedback controller in appropriate coordinates. Then we use a high-gain observer to estimate the derivatives of the output and the new uncertain state. On the contrary to traditional adaptive schemes, it results in a linear closed-loop system. Besides, a persistence of excitation condition is not necessary to assure closed-loop stability. Under signal tracking conditions, the controller yields global practical tracking, such that the tracking error can be made as small as desired by adjusting certain observer and controller parameters. The performance of the controller is evaluated by means of a simulation example. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Linear plants; High-gain observers; Robust control; Signal tracking
1. Introduction and problem statement In this work we present an approach to the design of robust controllers for linear systems. We depart from a state-space description of the plant. While in most traditional adaptive schemes (Astro¨m and Wittenmark, 1989) unknown parameters are considered separately from states, which leads to estimate separately parameters and states, here we consider products of parameters and states as uncertain terms which are lumped into a total uncertain term. In this way, our control design approach resembles the modeling error compensation due to Sun et al. (1994) and the time-delay compensation techniques due to Youcef-Toumi and Ito (1990). An important property of the lumped uncertain term is that it can be estimated by means of a high-gain observer. In this way, uncertain terms are interpreted as a new state of the system. The main advantage of the proposed control scheme over traditional (Astro¨m and
*Corresponding author. Fax: #52-5-7244900; e-mail: jjar@ xanum.uam.mx.
Wittenmark, 1989) and nonlinear (Kristic et al., 1992, 1994) adaptive designs is that it leads to closed-loop linear equations whose stability analysis can be addressed with standard techniques. To the contrary, traditional adaptive algorithms are typically highly nonlinear and this has meant that the associated theory is nonlinear. Another advantage of the proposed design procedure is that the dynamic order of the resulting controller depends only on the order of the plant, and not on the number of unknown parameters. In fact, the order of the controller is not higher than 2n. This means that the dynamic order of the resulting controller is lower than in traditional adaptive schemes (Astro¨m and Wittenmark, 1989) and nonlinear designs of adaptive controllers (Kristic et al., 1992, 1994). The proposed control scheme yields practical reference tracking. That is, the tracking error can be made as small as desired by adjusting certain controller parameters. We do not see this as a serious drawback since the performance of the practical controller is limited by various factors, like sampling rates and noise measurements. Our goal in this work is to develop a control strategy in a simpler setting that reveals its essential features,
0009-2509/99/$ — see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 2 1 4 - 0
52
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
thereby laying the groundwork for the more difficult cases, such as linearizable nonlinear systems. Problem statement. Consider linear plants of the following form: B(s) y(s)" u(s) A(s) b sK# 2 #b s#b u(s). " K> (1) sL#a sL\# 2 #a s#a L Many chemical plants, such as distillation columns and heat exchangers, can be modeled as in Eq. (1) (Morari and Zafiriou, 1989). We make the following assumptions on the plant. Assumption 1. The plant is minimum phase, i.e. the polynomial B(s)"b sK# 2 #b s#b is K> Hurwitz, and the plant order n, the relative degree o"n!m, and high-frequency gain b are known (the K> case where only the sign of b is known will be adK> dressed in a forthcoming paper). Assumption 2. The reference signal r(t) and its first n derivatives are known, bounded and continuous. Assumption 3. Only the output y is available for feedback.
The objective of this paper is to design an output feedback controller which guarantees boundedness of all signals of the closed-loop system and tracking of a given reference signal r(t) and the order of the resulting compensator be lower than the order of traditional (MRAC) (Astro¨m and Wittenmark, 1989) and nonlinear (Kristic et al., 1992) adaptive compensators. The paper is organized as follows. We first select a state-space description of the plant in Section 2, and then present our design procedure in Section 3 followed by a proof of stability and tracking in Section 4. Finally, we illustrate our procedure in an unstable second-order plant in Section 5.
2. Preliminaries The material in this section is straightforward and exists in the literature (see Astro¨m and Wittenmark, 1989; Kristic et al., 1992, 1994). However, for the sake of completeness in presentation a brief description of state-space representations of plant (1) is made. We start by designing a feedback controller assuming that the parameters of the plant are known and all internal states of the plant are available for feedback. This is not a reasonable assumption. We use it, however, as an intermediate assumption toward the final controller designed in the next section which will be a controller depending only on
the measured output and estimated (nonmeasured) states. In time domain, plant (1) can be described by the nth-order differential equation yL#a yL\# 2 #a y#a y L "b uK# 2 #b u#b u K> where yH"dHy/dtH and uG"dGu/dtG. We represent plant (1) by augmenting a series of m integrators at the input side of the system. We denote the states of these integrators by v "u, v "u, up to v "uK\. By taking K h "y, h "y, up to h "yL\ (internal states of the L plant), we can represent the augmented system by the (n#m)-dimensional linear state-space model [see for instance Khalil (1996) and references therein] h "h , j"1, 2 , n!1 H H> L K h "! a h # b v #b V L H H G G K> H G v "v , i"1, 2 m!1 G G>
(2)
(3)
v "V K where V31 is a new input. The state representation (2), (3) is not minimal since the states v"(v , 2 , v ) are not K observable from measurements of y"h 31. However, this is not an obstacle to stabilize the system (2), (3). The solution to this stabilization problem is well known and relies on standard techniques for linear and nonlinear systems. The state-representation (2), (3) allows us to use a time-varying change of variables to transform the system equation into an error space where the goal of the control design will be to drive the state of the system to zero. Let e"h!r31 be the output tracking error, and introduce the change of variables x "yH\!rH\" H h !rH\, j"1, 2 , n (tracking error vector). We reH write the h-subsystem (2) as e "x xR "x , j"1, 2 , n!1 (4) H H> L K xR "! a (x #rH\)# b v #rL#b V. L H H G G K> H G The series of integrators at the input side makes the derivatives of the input available for feedback. Consider the following dynamic linear state-feedback: u"v
vR "v , i"1, 2 m!1 G> L K vR " a (x #rH\)! b v G H H G G H G L # K x !rL b H H K> H
(5)
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
where the inner-loop gains K ’s are chosen in such a way H that the polynomial sL!K sL\!2 K s !K "0 be L Hurwitz. A possible procedure to choose the K ’s is to use H a reference model %L (s!j )"0, where the j ’s are the H H H desired closed-loop poles, such that sL! K sL\!2 L K s!K "%L (s!j ). H H Notice that v "v , i"1, 2 , m, so that the states G G v"(v , 2 , v ) are always available for feedback. In ad K dition,
L L V" a (x #rH\)! b v H H G G H G L # K x !rL b . H H K> H Let z"(x, v)31L>K. Under the state-feedback controller (5), the closed-loop system becomes zR "A z# (R), where R is a vector of signals compris? ing r(t) and their derivatives up to order n, and
A 0 . (6) A" A ? D A A is the Hurwitz companion matrix of the K ’s, A is the A G companion matrix of the b ’s, which is also Hurwitz G (minimum-phase assumption), and D is an m;n matrix involving the a ’s and the K ’s. The term (R) is the H H vector (0, 2 , 0, l(R))2, where
L l(R)" a rH\!rL b . (7) K> H H Observe that the closed-loop matrix (6) is in cascade form. It is not hard to see that the tracking error x(t) converges asymptotically to zero, while the compensator states v(t)"v(t) remain bounded for all t*0. Recall that the parameters a ’s, b ’s (except b ) are H G K> unknown and the state x31L is not available for feedback, so that the feedback controller (5) cannot be applied just as it is. The purpose of this paper is to look for controllers driven by the tacking error e"h!r31 which guarantee signal tracking while retaining the main features of the state feedback controller (5), such as linear closed-loop behavior and cascaded structure as in Eq. (6). That is, we look for a certainty equivalence design, which lead to a would-be linear controller (Astro¨m and Wittenmark, 1989) of the form (5).
3. An output feedback scheme In this section, we will derive a linear control scheme driven by the output tracking error e31. The main idea is to choose a state-space representation where the uncertainties in Eq. (4) (namely ! L a (x #rH\)# H H H K b v ) are lumped into a term g31 which is seen as G G G
53
a nonmeasured state. In this way, plant (1) is represented as a linear system embedded into a (n#m#1)-dimensional space together with an algebraic constraint. Consider the following dynamical system: e"x
G xR "x , j"1, 2 , n!1 H H> xR "g#rL#b V L K>
(8a)
L\ L K\ gR "! a x ! a rH# b v H H> H G G> H H G !a (g#rL#b V)#b V L K> K u"v v "v , i"1, 2 , m!1 (8b) G G> v "V K which is an (n#m#1)-dimensional linear system. It is not hard to prove that the function L K u(x, g, v, R)"g# a (x #rH\)! b v (9) H H G G H G is a first-integral (Wiggins, 1990) of system (8). That is, along the trajectories of system (8) we have that du/dt"0, for all t31, and any V31. The state-space representation (8) is externally equivalent to system (2), (3), as long as initial conditions satisfy u(x , g , v , 1 )"0. If we are able to control system (8) without making use of the constraint u(x , g , v , R )"0, we would be able to control system (2), (3). A geometric interpretation of the state-space representation (8) is the following. If the initial condition (x , g , v , R ) for system (8) satisfies u(x , g , v , R )"0, then, the dynamics of system (8) are equiva lent to the dynamics of the system (2), (3). That is, if (x(t), v(t)) is the solution to system (2), (3) with initial conditions (x , v , R ), and (x(t), g(t), v(t)) is the solu tion to system (8) with initial conditions (x , g , v , R ), then the equality u(x , g , v , R )"0 and the condition du/dt"0 imply that (x(t), v(t)) "%(x(t), g(t), v(t)) , where % denotes projection onto (x, v)-coordi nates. Geometrically, condition u(x , g , v , R )"0 means that if initial condition of uncertainty is located on the manifold u(x, g, v, R)"0, the trajectories of system (8) remain there and the corresponding trajectory of uncertainties lumped into g31 can be recovered just by projecting on the corresponding coordinates frame. It must be pointed out that for specific applications, it is not necessary to represent plant (1) as in Eq. (8). Such state-space representation is only a tool to obtain a control scheme and analyze the stability properties of the resulting closed-loop system. The role of the g31 in Eq. (8b) is to represent the uncertainty term ! L a (x #rH\)# K b v as H H H G G G
54
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
a new state. In this way, as long as u(x , g , v , R )"0, g(t)"! L a (x (t)#rH\(t))# K b v (t) for all H H H G G G t'0. The next property establishes that the uncertainty g can be reconstructed from measurements of the output e"y!r31. Proposition 1. ¹he states (x, g)31L;1 can be reconstructed from measurements of the output e31 and the input V31. The above result is an important property because it establishes that the dynamics of the uncertain term g"! L a (x #rH\)# K b v can be reconH H H G G G structed from the availability of e and V. Resembling the structure of compensator (5), at this point we propose the following (n#m#1)-dimensional compensator driven by the output e: xNR "xN !¸Ha (e!xN ), j"1, 2 , n!1 H H> H xNR "gN #rL#b V!¸La (e!xN ) (10a) L K> L g"!¸L>a (e!xN ) L> vR "v , i"1, 2 , m!1 G G> L vR "V" !g# K xN !rL b (10b) K H H K> H u"v where xN 31 and g31 are estimates of x31L and g31, respectively, the K ’s are chosen as in Eq. (5), ¸'0 is an H adjustable parameter, and the constants a ’s are chosen in H such a way that the polynomial sL\!a sL!2 !a s!a "0 be Hurwitz. The dynamical system L L> (10) is a linear controller comprising a high-gain observer (Esfandiari and Khalil, 1992) for the states (x, g) and a compensator as the one described by system (5). Notice that since the parameters a ’s and b ’s are assumed to be H G unknown, the right-hand side of the Eq. (8b) is also unknown. Thus, such term was neglected in the construction of an observer for the state g. The order of the dynamic controller (10) is n#m#1. Kristic et al. (1992, 1994) designed a controller comprising a Kreisselmeier state estimator (Kreisselmeier, 1977) of order 2n, and a parameter estimator of order m#n (the number of unknown parameters), finally giving a (3n#m)-order compensator. This means that the dynamic order of the resulting controller (10) is lower than Kristic et al.’s adaptive scheme and traditional adaptive schemes. The generalization of the basic control design for minimum-phase, multi-input—multi-output plants where the high-frequency gains are known, is straightforward departing from an state-space representation analogous to the linear system (2). Some remarks on this extension are made in the conclusions section.
4. Stability analysis Let e be a vector in 1L> whose components are defined by e "¸L>\H(x !xN ), j"1, 2 , n H H H
(11)
e "g!g . L> Using the fact that g"! L a (x #rH\)# H H H K b v , and recalling that v "v , i"1, 2 , m, the dyG G G G G namics of the closed-loop system (2), (3), (10) are completely described by xR "A x# (e) A
(12a)
v "Dx#A v# (e)#t (R) eR "¸A e# (x, v, e)#t (R) (12b) where the matrices D, A and A are given as in A eq. (6), A is the (n#1)-dimensional Hurwitz companion matrix of the a ’s, and , , , t and t are linear H functions of its arguments, which are given in Appendix A. From the expressions for , , , t and t in Appendix A, and the assumption that r(t) and its derivatives are bounded, we obtain the existence of six positive numbers b , k"1, 2 , 6 (not depending on ¸) such that I the following inequalities hold for all (x, v)231K>L, e31L>, and ¸*1: " (e)")b "e", " (e)")b "e"
(13)
" (x, v, e)")b "z"#b "e" "t (R)")b , "t (R)")b .
(14)
Eqs. (12a) and (12b) become an (2n#m#1)-dimensional linear system. In principle, the stability analysis of such a system is easier than in traditional adaptive schemes. In fact, in adaptive controllers the parameter adaptation scheme induces nonlinear (Sastry and Bodson, 1989) or linear time-varying (Krause and Kumar, 1986) closed-loop equation whose stability analysis involve more elaborated stability theories. 4.1. Stability under regulation conditions Without loss of generality, we assume that r(t)"0. In this case R"0, From the expressions for t and t in the Appendix A, one can see that t (0)"0 and t (0)"0. Recall that matrix A in Eq. (6) and matrix ? A are Hurwitz, so that the Lyapunov equations A2 P #P A "!I and A2P #P A "!I ? ? ? ? L>K L> have a positive-definite solution. Let z"(x, v)31L>K, and consider the quadratic function ¼(z, e)" z2P z#e2P e, and take its time derivative along the ?
55
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
trajectories of the following linear system: xR "A x# (e) A v "Dx#A v# (e) eR "¸A e# (z, e). Using Eq. (13), we get
(15)
¼ Q "!"z"!¸"e"#z2P (e)#e2P (z, e) ? )"z"!(¸!j (P )b )"e"
#(j (P )b #j (P )b ) "z" "e" (16)
?
where "( , )2, b "max(b , b ) and j (P)
represents the maximum eigenvalue of the matrix P. Observe that the right-hand side in Eq. (16) is a quadratic form, which is negative for all ¸ in (¸*, R), where ¸*"j (P )b #(j (P )b
? #j (P )b )/4. (17)
This implies the global asymptotic stability of the linear system (15). From the results in Appendix A, it can be obtained that the estimates b ""K", b ""K/b ", b " K> "b"#"a"#"a K"#"b a"#"b a" and b ""a K"# "b K", L K K L K where a"(a , 2 , a ) and b"(b , 2 , b ). Given an up L K per bound of the parameters a31L and b31K, one can estimate an upper bound for the parameters b ’s. By G knowing an upper bound for the b ’s, an estimate of G ¸* can be obtained. 4.2. Stability under signal tracking conditions Based on the above result, we conclude that system (12) is an asymptotically stable linear system with a bounded additive perturbation t(t). Next, we will show that there exists a compact set !1L>K> (independent of ¸) which absorbs all trajectories of system (12). Proposition 2. ¸et L* as in Eq. (17). For all L in (L*, R), the trajectories of system (12) are uniformly bounded, and converge to a compact set !L1> > independent of L. Proof. See Appendix B.
)
The parameter ¸ in system (12) can be seen as the rate of estimation of uncertainties g31 and unmeasured states x31L. The result in Proposition 1 states that if the rate of estimation ¸ is larger than a certain value ¸*, where ¸* depends on the magnitude of the uncertain parameters a ’s, b ’s (except b ), then the trajectories of H G K> the closed-loop system (12) are uniformly bounded. This means that it is possible to find a compact set !L1L>K> such that all trajectories of Eq. (12) converge to it and remain there. By making a more detailed analysis of the closed-loop system (12) one can show that,
for sufficiently large ¸, there exists a family of sets ! (¸) L with the property that ! (¸ )-! (¸ ) for ¸ '¸ . That L L is, the larger the value of ¸, the smaller the absorbing set for the trajectories of system (12). However, to prove the results that follow, we will require only the existence of the set !. Results in Proposition 2 are useful to prove the following global practical stabilization result. Theorem 1. For any given initial condition (z , e )" (x , v , e ), and sufficiently large ¸, the asymptotic tracking error "x(tPR)" is on the order of L\, for L in (L*, R). Proof. Since eR "¸A e# (z, e)#t (t), we get R e(t)"exp (¸tA )e(0)# exp (¸(t!s)A ) ;( (z, e)#t (s)) ds. Performing the norm operator to the above equation, and making use of inequalities (13) and the fact that the matrix A is Hurwitz, we obtain "e")p "e "exp (!j ¸t)#p exp (!j ¸t) R ; exp (j ¸s)(b "e"#b "z"#b ) ds where j and p are positive constant depending only on the Hurwitz matrix A . After some algebraical manipula tions, we get R exp (j ¸t) "e")p "e "#p b exp (j ¸s) "e" ds R #p exp (j ¸s) (b "z"#b ) ds. The result in Proposition 2 implies that "z(t)")b , where b depends only on the initial conditions (z , e )31K;1L>. Then exp (j ¸t) "e")p "e "#(p b /j ¸) R ;[exp (j ¸t)!1]#p b exp (j ¸s)"e" ds where b "b b #b . Using the Gronwall’s inequality, we obtain "e(t)")"e" # (t) (see Appendix C), where QQ
(t) is a bounded continuous function with (R)"0 which is given by
(t)"k exp [!(j ¸!p b )t] p "e "j ¸!p"e "b !p b k" j ¸!p b and "e" given by QQ p b "e" " QQ j ¸!p b
(19)
(20)
56
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
so that lim "e(t)")"e" . For sufficiently large ¸, we R QQ have that j ¸!p b '0. On the other hand, "e" is of QQ the order of ¸\. Analogously to the differential equation governing the dynamics of the error e(t), we can integrate the equation xR "A x# (e) A R x(t)"exp(A t)x #exp (A t) exp (!A s) (e(s)) ds. A A A Using the triangle inequality, we obtain (recall that A is A Hurwitz)
"x(t)")p exp (!j t)"x "#p exp (!j t) A A A A R ; exp (j s) " (e(s))"ds A )p exp (!j t)"x "#b p exp (!j t) A A A A R ; exp (j s)"e(s)"ds A where p and j are two positive constants depending A A only on the matrix A . A By recalling that "e(t)")"e" # (t), we have that QQ "x(t)")p exp (!j t)"x "#b p exp (!j t) A A A A R ; exp (j s) ("e" # (s)) ds A QQ b p exp (!j t) A "p exp (!j t)"x "# A A A j A ;[exp(!j t)!1] "e" A QQ R #b p exp (!j t) exp (j s) (s) ds. A A A Using Eq. (19) to integrate the last term of the inequality above, we obtain
b p "x(t)")p exp (!j t)"x "# A [1!exp (!j t)]"e" A QQ A A j A b pk # A [exp (!¸ t)!exp (!j t)] A j !¸ A where ¸ "j ¸!p b . Then, we conclude that lim "x(t)")p b "e" /j . Since "e" is on the order of R A QQ A QQ ¸\, we have that lim "x(t)" is also on the order R of ¸\. )
The time function t (t) represents the estimated transiV ent behavior of the tracking error vector x31L. The first two terms in t (t) are on the order of O(¸). From the V expression for k [Eq. (19)] and recalling that "e " is of the order O(¸L\), we have that, for large values of ¸, t (t) V can be approximated by t (t)"%(¸L)t* (t) V ? V where t* (t)"exp (!¸ t)!exp (!j t) is a bounded V A time function and %(¸L) is a monotonic increasing function of ¸L\. The time function t (t) represents the V ? effects of the observer initial conditions xN 31L, g 31 in the tracking error x31L. For a given ¸, %(¸L) is a constant representing the amplitude of the overshoot in the estimation error e31L>. In this way, the larger the value of the observer gain ¸, the larger the overshoot in the estimated states xN 31L, g 31. This dynamic effect limits the achievable closed-loop performance since overshooting in e31L> is transmitted to the feedback loop. On the other hand, we have that the steady-state tracking error "x" "p b "e "/j is smaller for larger values of j . QQ A QQ A A We know that j is of the order of min+"Rej (A )", A I A (Hirsch and Smale, 1974), so that j depends on the A eigenvalues of the matrix A , and consequently on the A inner-loop gain vector K31L. Therefore, the steady-state tracking error "x" can be made smaller if the gain vector QQ K is chosen such that the eigenvalues of the matrix A are A located deeper in the open left-half complex plane. In general, large values of j is more or less related to A high-gain control, which in some cases may be undesirable. In practical situations, a compromise between the observer gain ¸ and the inner-loop gain vector K must be addressed in order to obtain small tracking error and good control performance during transients. 5. Example In this section, we illustrate the performance of the controller (10) via a second-order linear example from A stro¨m and Wittenmark (1989, pp. 147—151). We chose such an example from the adaptive control literature to have a reference against which we can judge the performance of the proposed controller. Consider a plant represented by the transfer function
Notice that the last expression in the proof of Theorem 1 can be written as
b y(s)" u(s) s(s#a)
"x(t)")"x" #t (t) QQ V where "x" "p b "e "/j and QQ A QQ A b p t (t)"p exp (!j t)"x "! A exp (!j t)"e" V A A A QQ j A b pk # A [exp (!¸ t)!exp (!j t)]. A j !¸ A
where a is unknown and b is known constant. We assume b"1. The poles of plant (21) are +0, !a,, so that for a(0 the plant is unstable. In this case, we do not add integrators at the system’s input; hence, V"u. The state-space representation (2), (3) of the plant is zR "z , zR "!az #bu, y"z . We want to design a controller such that the output of the system y(t) tracks the output of a reference model r(t) for any command signal r(t).
(21)
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
57
Fig. 1. Dynamics of the current uncertainty g(t) and its estimate g (t), which is provided by controller (22) with ¸"25. The estimate g (t) converges to the current value g(t), which allows to stabilize the unstable second-order plant (21).
Fig. 3. Simulations results for the second-order plant (21) and controller (22) with ¸"40, and initial conditions y(0)"yR (0)"0, xN (0)" xN (0)"1 and xN (0)"0.
Fig. 2. Dynamic behavior of the controlled output y(t) with the feedback controller (22) and regulation conditions r(t)"0.
Fig. 4. Output tracking error for several values of the observer parameter ¸. Notice that, after one period of the reference signal, the output tracking error achieves its steady-state dynamics.
A traditional model reference adaptive controller (MRAC) design is given by A stro¨m and Wittenmark (Example 4.8). The resulting controller (10) is described by xN "xN !¸a (e!xN ) xN "g #r#u!¸a (e!xN ) (22) g "!¸a (e!xN ) u"!r!g #K xN #K xN where e"y!r. The reference model is taken as G (s)"r(s)/r(s)"u/(s#2fus#u), hence K " K !u and K "!2fu. For simulations, we choose a "!3, a "!3, a "!1, such that the eigenvalues of matrix A are all located at !1. We consider first the case r(t)"0 (regulation), such that the uncertain state is given by g"!az . Fig. 1 presents the dynamics of the current value of g(t) and its estimate g (t), which is provided by controller (22) with ¸"25. The estimate g (t) converges to the current value g(t), which allows to stabilize the unstable second-order plant (21). The behavior of y(t) is presented in Fig. 2 for
three different initial conditions. Observe that y(t) is driven to zero without serious deterioration of the transient performance. Suppose now that r(t) is the output of the reference model G (s)"r(s)/r(s)"u/(s#2fus#u). Note that K r"!2fur!ur#ur, and r is available by integration of the reference model. Fig. 3 shows simulations results for ¸"40, and initial conditions y(0)"yR (0)"0, xN (0)"xN (0)"1 and xN (0)"0. The command signal r(t) is a square waveform, which is continuous almost everywhere. The effectiveness of controller (10) is demonstrated by the fact that the tracking error is not higher than 1% after few periods of the reference signal. To illustrate the fact that the asymptotic tracking error is reduced when the gain ¸ is increased, Fig. 4 shows the tracking error y(t)!r(t) for three different values of that gain: ¸"20, ¸"40, and ¸"60. For ¸'40, the peak in the relative tracking error as about 1% [recall that r(t) has unitary amplitude]. The large tracking error between 0 and 4 units of time are due to the peaking induced by the initial conditions of the compensator, which all were chosen as zero. Fig. 5 presents the control effort to
58
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
Fig. 5. Input signal corresponding to the signal tracking case shown in Fig. 3.
Fig. 7. Computed control input u(t) when 25% error in the estimation of the high-frequency gain b is assumed.
6. Conclusions
Fig. 6. Dynamics of the controlled output y(t) when 25% error in the estimation of the high-frequency gain b is assumed.
achieve the performance presented in Fig. 3. A comparison of our results with those in A stro¨m and Wittenmark (1989) shows that the performance achieved by our control design is as good as the one achieved by traditional adaptive schemes. At least in this example, it has been possible to design our controller to produce performance comparable to what is achieved by other designs. However, the main advantage of our design is that it uses a third-order compensator, while MRAC designs use a ninth-order filter in addition to the parameter adaptation (tenth-order compensator). Finally, to illustrate that the proposed controller (22) can also handle uncertainties in the high-frequency gain b, assume that the estimated value of the parameter b is bK "1.25. Fig. 6 presents the dynamics of the controlled output y(t) for the case r(t)"0 (regulation) and two initial conditions for y and yR . Notice that controller (22) takes the output y(t) to zero without excessive control effort (see Fig. 7) despite 25% uncertainty in the parameter b. It must be pointed out that if bK is decreased, instability of the closed-loop system is most likely to occur. As in traditional adaptive control approaches, to avoid instabilities due to uncertainties in the high-frequency gain b, bª must be taken as an upper bound (i.e. bª * b *b).
We presented a new design approach to design robust controllers for a class of linear plants. Our approach properly combines results from nominal stabilization and robust state observation. Our controller is simply a state feedback design with a linear observer. The dynamic order of the resulting controller is lower than in traditional adaptive schemes. From a stability analysis, we concluded that the steady-state tracking error will be of order O(¸\). Since ¸ is a design parameter, we can make the steady-state error arbitrarily small by adjusting the observer gain ¸. Our simulations show that the performance of the new controller is comparable to the performance of traditional controllers. As regards a possible generalization of the proposed design theory to a wider class of linear plants, one can observe that considering multi-input—multi-output minimum-phase plants with known high-frequency gains does not affect the subsequent developments, aside from obvious vectorial notations. If the (square, m-dimensional) MIMO plant is given by y(s)"G(s) u(s), the high-frequency gain matrix is G(0), which is assumed to be known and invertible. In this way, an extension of the proposed controller to MIMO plants would involve G(0)\ and m state observers like system (21). For instance, a simple (one-time constant), two-inputs—two-outputs model for binary distillation columns is given as y(s)"G(s) u(s) (Morari and Zafiriou, 1989), where y(s)"(x (s), x (s))2 are distillate and bottom " (deviation) compositions, respectively, u(s)"(R(s), »(s))2 are liquid and vapor (deviation) flowrates, respectively, and
1 g g . G(s)" (23) (s#q\) g g Plant (23) is minimum phase. A state-space realization of Eq. (23) is given by xR "!x /q#g R#g » " " xR "!x /q#g R#g ».
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
By assuming that the g ’s, the generalization of the proGH posed controller for plant (23) would involve the inverse of the high-frequency gain matrix G(0). In most practical cases the matrix G(0) is also uncertain (Morari and Zafiriou, 1989). At least for binary distillation columns, preliminary numerical simulations show that the proposed scheme is able to yield acceptable closed-loop response under set-point tracking and external disturbances despite of uncertainties in G(0). The stability of the proposed control scheme under uncertainties in G(0) for both single-input and multi-input plants requires more in depth research. The control of these cases (including the case of plants where the high-frequency gain b is also an uncertain parameter) are under progress K> and results and applications to chemical processes will be presented in a forthcoming paper. Appendix A. Computation of , and
Using the fact that g"! L a (x #rH\) H H H H # L b v , it is not hard to find the following expresG G G sions:
L 2
(e)" 0, 2 , 0, ! K ¸\L\>He #e H H L> H 2 L
(e)" 0, 2 , 0, ! K ¸\L\>He #e b H H L> K> H
(x, v, e)"(0, 2 , 0,
)2 L> where K\ L\
"# b v ! a x !a L> G G> H H> L G H L ; e # K (x !¸\L\>He ) L> H H H H L #b e # K (x !¸\L\>He ) K L> H H H H L K ! ax# bv . H H G G H G Notice that in all above expressions, e is at most of order ¸I, k)1. The expressions for t and t are given as follows: t (R)"(0, 2 , 0, L(R))2, where L(R) is given as in Eq. (7), and t (R)"(0, 2 , 0, t )2, where K
L L t " a rH#b !rL ! a rH\ . K H K H H H Appendix B. Proof of proposition 2 Consider the quadratic function ¼(z, e)"z2P z# ? e2P e and take its time derivative along the trajectories
59
of system (12). From Eq. (13), we obtain ¼ Q )!"z"!(¸!j (P )b )"e"#(j (P )b
? #j (P )b )"z""e".
#j (P )b "z"#j (P )b "e".
?
To simplify the presentation, let us define the following constants: c (¸)"¸!j (P )b , c "j (P )b #
? j (P )b , c "j (P )b and c "j (P )b . From
?
eq. (17), it can be seen that c (¸)'0, for all ¸*¸*. The right-hand side of the last expression can be written as ¼ Q )!"z"!(¸!j (P )b )"e"#(j (P )b
? #j (P )b )"z""e".
#j (P )b "z"#j (P )b "e".
?
To simplify the presentation, let us define the following constants: c (¸)"¸!j (P )b , c "j (P )b #
? j (P )b , c "j (P )b and c "j (P )b . From
?
eq. (17), it can be seen that c (¸)'0, for all ¸*¸*. The right-hand side of the last expression can be written as E("z", "e"; ¸)"!"z"!c (¸)"e"#c "z""e"#c "z"#c "e". We will show that for ¸'¸*, E("z", "e"; ¸) is negative for q except in a neighborhood of the origin. In turn, the above would imply that trajectories of system (12) are uniformly bounded. The quadratic function E("z", "e"; ¸) is defined on the closed first quadrant q "+"z", "e":"z"*0, "e"*0,. For m3[0, R], consider the set o(m)"+"z", "e" : "e""m"z", "e"O0, "z"O0,5q . If we evaluate E("z", "e"; ¸) in o(m), we obtain E("z", m"z"; ¸)""z"+(!1!c (¸)m #c m) "z"#(c #c m),. Since "z"'0, the sign of E("z", "e"; ¸) along the ray o(m) is given by the sign of the term into the brackets. It is easy to see that such a term is negative for all "z"'z*(m), where c #c m z*(m)" . (B.1) c (¸)m!c m#1 From the relation "e""m"z", the corresponding value for e*(m) is given by c m#c m e*(m)" . (B.2) c (¸)m!c m#1 The condition ¸'¸*, where ¸* is given as in eq. (17), implies that the second-order polynomial in the denominator is positive for all m3[0, R]. In fact, the polynomial c (¸) m!c m#1 has no roots in [0, R], for c!4c (¸)(0, which is exactly condition (17). So that z*(m)*0 and e*(m)*0, for all m in [0, R]. On the other hand, (z*(0), e*(0))"(c , 0) and (z*(R), e*(R))" (0, c /c (¸)). Thus, the continuity of Eqs. (B.1), and (B.2)
60
J. Alvarez-Ramirez, R. Sua´ rez/Chemical Engineering Science 54 (1999) 51—60
Inequality (18) has the form of inequality (C.1), so that Lemma 1 can be used with w (t)"exp(j ¸t)"e" w (t)"p b c(t)"p "e "#(p b /j ¸) [exp(j ¸t)!1]. After straightforward algebraic manipulations, we obtain
p b p "e "j ¸!pe "b !p b # "e") j ¸!p b j ¸!p b ;exp [!(j ¸!p b ) t]. From where the following definitions are taken:
Fig. B.1. Schematic diagram of the absorbing set c*(m).
with respect to m implies that, for a given ¸'¸*, the pair (z*(m), e*(m)) defines a continuous, finite valued curve c(m; ¸) in the first quadrant q (see Fig. B1) with the property that E("z", "e"; ¸) (0, for all ("z", "e")3q 3c>(¸), where c>(m, ¸)" +("z", "e")3q : c(m, ¸))0,, where c(m, ¸))0 indicates that the curve c(m, ¸) defined by the pair (z*(m), e*(m)), m3[0, R], is semi-negative valued. From the fact that *e*(m, ¸)/*¸)0 and *z*(m, ¸)/ *¸)0, we conclude that c*(m):"c>(m, ¸*) -c>(m, ¸), for all ¸ in (¸*, R). Therefore, E("z", "e"; ¸)(0 in the set ("z", "e")3q 3c*(m), for all ¸ in (¸*, R). Since c*(m) is a compact subset of 1L>K>, there exists a positive number c such that c*(m)L!"+z, e: ¼(z, e))c, (see Fig. B1). Since E("z", "e"; ¸)(0 implies that ¼ Q (0, we have that ¼ Q is negative in the subset q 3!. From Lyapunov arguments we conclude that all trajectories of system (12) are uniformly bounded and ! is an absorbing compact set, for all ¸ in (¸*, R). )
Appendix C. Computation of Eqs. (19) and (20) To compute Eqs. (19) and (20), the following result is required. Lemma A.1. (Gronwall’s inequality, Coddington and ¸evison, 1955). If w , w and c*0 on [0, t], c is differenti able, and
w (t))c(t)#
R
w (s)w (s) ds
(C.1)
then w (t))c(0) exp
R
w (s) ds#
R
c (s) exp
R
Q
(t)"i exp[!(j ¸!p b )t] p "e "j ¸!p"e "b !p b i" j ¸!p b p b "e" " . QQ j ¸!p b Notice that, if j ¸!p b '0, '(t) is bounded by i for all t*0.
w (q) dq ds.
References A stro¨m, K.J., & Wittenmark, B. (1989). Adaptive control. New York, USA: Adison-Wesley. Coddington, E.A., & Levison, N. (1955). ¹heory of ordinary differential equations, New York, USA: McGraw-Hill. Esfandiari, F., & Kahlil, H.K. (1992). Output feedback stabilization of fully linearizable systems. Int. J. Control, 56, 1007—1037. Hirsch, M.W., & Smale, S. (1974). Differential equations, dynamical systems, and linear algebra, New York, USA: Academic Press. Khalil, H. K. (1996). Adaptive output feedback control of nonlinear systems represented by input—output models. IEEE ¹rans. Automat. Control, 41, 177—188. Krause, J.M., & Kumar, K.S.P. (1986). An alternate stability analysis framework for adaptive control. Systems and Control ¸ett., 7, 19—24. Kreisselmeier, G. (1977). Adaptive observers with exponential rate of convergence, IEEE ¹rans. Automat. Control, 22, 2—8. Kristic, M., Kanellakopoulus, I., & Kokotovic, P.V. (1992). Nonlinear design of adaptive controllers for linear systems. IEEE ¹rans. Automat. Control, 39, 738—752. Kristic, M., Kanellakopoulus, I., & Kokotovic, P.V. (1994). Passivity and parametric robustness of a new class of adaptive systems. Automatica, 30, 1703—1716. Morari, M., & Zafiriou, E. (1989). Robust process control. Prentice-Hall, NJ, USA: Englewood Cliffs. Sastry, S.S., & Bodson, M. (1989). Adaptive control: stability. convergence and robustness. Englewood Cliffs, NJ, USA: Prentice-Hall. Sun, J., Olbrot, A.W., & Polis, M.P. (1994). Robust robust performance using model reference control and modeling error compensation. IEEE ¹rans. Automat. Control, 39, 630—635. Wiggins, S. (1990). Introduction to applied nonlinear dynamical systems and chaos. New York, USA: Springer. Youcef-Toumi, K., & Ito, O. (1990). A time-delay controller design for systems with unknown dynamics. ASME J. Dyn Measurement Control, 112, 133—142.