Gain scheduling control of nonlinear singularly perturbed time-varying ...
International Journal of Systems Science Vol. 36, No. 6, 15 May 2005, 357–364
Gain scheduling control of nonlinear singularly perturbed time-varying systems with derivative information HO-LIM CHOI and JONG-TAE LIM* Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Kusong-dong, Yusong-gu, Taejon, 305-701, Korea (Received 10 June 2004; in final form 11 March 2005) In this paper, we analyse the ultimate boundedness of nonlinear singularly perturbed time-varying systems and propose a control law using gain scheduling where the slow state and the exogenous signals are used as scheduling variables. In our control scheme, we have some flexibility in selecting the slow manifold of the system. Moreover, the derivative information can be properly engaged to manipulate the size of ultimate bound in tracking error of the controlled system. Keywords: Singularly perturbed time-varying systems; Ultimate boundedness; Gain scheduling; Slow manifold; Derivative information
1. Introduction Gain scheduling control methodology has been widely applied to many nonlinear systems. Since the introduction of the analytical framework of gain scheduling (Rugh 1991) and stability analysis (Lawrence and Rugh 1990), there has been considerable progress up to the present, e.g. potential hazard of gain scheduling and remedies (Shamma and Athans 1992), fast gain scheduling technique (Sureshbabu and Rugh 1995, Lee and Lim 1997) or trajectory planning (Costa and Oliveira 2002). In the gain scheduling approach, it is necessary to select the scheduling variables which are normally ‘slowly varying’ parameters. Then, the local controller is designed around the equilibrium manifold. More general notions on gain scheduling control and related research results are addressed in Rugh and Shamma (2000). In two-time-scale systems, there exist a slow state and a fast state. The singularly perturbed systems are the typical two-time-scale systems where the slow state plays a role of slowly varying parameters. The dynamics is degenerated into the reduced model
*Corresponding author. Email: [email protected]
and controlled by the slow state when the fast state is on the slow manifold. It has been shown that the behaviour of the system is constrained in O() bound of the slow manifold (Kokotovic et al. 1986, Retchkiman and Silva 1996, Khalil 2002). Thus, in the reduced model, the slow manifold works as a control input. For the control of singularly perturbed systems, there have been various results (Sharkey and O’Reilly 1988, Chen 2000, Shin and Lim 2002, Innocenti et al. 2003). One typical approach is the composite control scheme (Sharkey and O’Reilly 1988, Innocenti et al. 2003) where the slow control and fast control are designed separately. In those approaches, the slow manifold is not explicitly considered in the design process. In Shin and Lim (2002), the slow manifold is used as a design component, which can result in better performance depending on the initial states. In this paper, we analyse the ultimate boundedness of nonlinear singularly perturbed time-varying systems and propose a controller using the gain scheduling approach where the fast controller is the gain scheduling controller, and the slow state and the exogenous signals are used as scheduling variables. In our proposed scheme, the selection of the slow manifold is incorporated in the controller design process. This feature can lead to better transient response since the fast state
International Journal of Systems Science ISSN 0020–7721 print/ISSN 1464–5319 online 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.1080/00207720500111707
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can converge to the slow manifold more quickly. Also, the proposed controller utilizes the time-derivative information on scheduling parameters. The properly selected time-derivative information can result in smaller error bounds in reference tracking. Thus, the proposed scheme is the improvement over the existing control schemes because it can tackle both transient and steady-state response.
^ where A^ s ðx, z, wÞ ¼ ð@gðx, z, wÞÞ=@z, Ð 1 As ðx, wÞ ¼ As ðx, ^ ðx, wÞ, wÞ, and Rs ðx, s, wÞ ¼ 0 ½As ðx, ðx, wÞ þ s, wÞ A^ s ðx, ðx, wÞ, wÞd. From equations (2) and (3), we obtain the following system: x_ ¼ fc ðx, s, wÞ _s ¼ As ðx, wÞs þ Rs ðx, s, wÞs þ g1 ðx, w, w_ Þ þ g2 ðx, s, wÞ, ð4Þ
2. System analysis 2.1. System formulation Consider a system given by x_ ¼ f ðx, z, wðtÞÞ z_ ¼ gðx, z, wðtÞÞ,
ð1Þ
where x 2 Rn and z 2 Rm which are the slow state and the fast state, respectively. The parameter is a sufficiently small positive constant. The exogenous signal wðtÞ 2 Rp is continuously differentiable. Moreover, it is assumed that kwðtÞk and kw_ ðtÞk , t 0. The notations used in this paper are as follows: x, w, q, r denote the bounded subsets of x, w, q, r, respectively. xðwÞ denotes the frozen signal of x(w). i(A) denotes the eigenvalues of a matrix A. All norms are used as the Euclidean 2-norm unless specifically denoted. Assumption 1: Regarding the fast dynamics of the system (1), the following hold: (A1) (A2)
(A3)
g is twice continuously differentiable with respect to its arguments. For each constant x 2 x and w 2 w , there exists a continuously differentiable manifold z ¼ ðx, wÞ such that gðx, ðx, wÞ, wÞ ¼ 0. For each constant x 2 x and w 2 w , Re ½i ðð@g=@zÞðx, ðx, wÞ, wÞÞ s < 0, i ¼ 1, 2, . . . , m:
Set the change of variable s ¼ z ðx, wÞ. Then, due to (A2), ð1
d gðx, ðx, wÞ þ s, wÞd ¼ gðx, ðx, wÞ þ s, wÞ: ð2Þ 0 d
Also, using the results of Lee and Lim (1997) and Shin and Lim (2002), it can be shown that ð1
d gðx, ðx, wÞ þ s, wÞd ¼ As ðx, wÞs þ Rs ðx, s, wÞs, d 0 ð3Þ
where fc ðx, s, wÞ ¼ f ðx, s þ ðx, wÞ, wÞ, g1 ðx, w, w_ Þ ¼ ðð@ðx, wÞÞ=ð@wÞÞw_ , and g2 ðx, s, wÞ ¼ ðð@ðx, wÞÞ=ð@xÞÞ f ðx, s þ ðx, wÞ, wÞ From equation (4), we consider the slow dynamics part only. Letting s ¼ 0 and denoting f ðx, wÞ ¼ fc ðx, s, wÞjs¼0 , we have the reduced system: x_ ¼ f ðx, wÞ Assumption 2: hold: (B1) (B2) (B3)
ð5Þ
Regarding system (5), the following
f is twice continuously differentiable. For each constant w 2 w , there exists a smooth function x(w) such that 0 ¼ f ðxðwÞ, wÞ. For each constant w 2 w , Re½i ðð@f =@xÞðxðwÞ, wÞÞ e < 0, i ¼ 1, 2, . . . , n:
Let qðtÞ ¼ xðwðtÞÞ be the equilibrium manifold such that 0 ¼ f ðqðtÞ, wðtÞÞ. Then, we set e ¼ x q. Considering the original slow dynamics in equation (4), we have e_ ¼ x_ q_ ¼ f ðx, wÞ þ feðx, s, wÞ
@q w_ ; @w
ð6Þ
where feðx, s, wÞ ¼ fc ðx, s, wÞ f ðx, wÞ. Under Assumption 2, similar to equation (3), we have e_ ¼ Ae ðq, wÞe þ Re ðq, w, eÞe þ feðx, s, wÞ þ f1 ðw, w_ Þ,
ð7Þ
^ where A^ e ðx, Ð 1 wÞ ¼ ð@f ðx, wÞÞ=@x, Ae ðq, wÞ ¼ Ae ðq, wÞ, ^ ^ Re ðq, w, eÞ ¼ 0 ½Ae ðq þ e, wÞ Ae ðq, wÞd, and f1 ðw, w_ Þ ¼ ð@q=@wÞw_ . Using the relation x ¼ e þ q, we have e_ ¼ Ae ðq, wÞe þ Re ðq, w, eÞe þ feðe þ q, s, wÞ þ f1 ðw, w_ Þ _s ¼ As ðe þ q, wÞs þ Rs ðe þ q, s, wÞs þ g1 ðe þ q, w, w_ Þ þ g2 ðe þ q, s, wÞ:
ð8Þ
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2.2. System analysis: ultimate boundedness Regarding system (8), it is noted that Ae and As are Hurwitz for all q 2 q and w 2 w . Thus, we have two Lyapunov equations such as Ae ðq, wÞT Pe ðq, wÞ þ Pe ðq, wÞAe ðq, wÞ ¼ I and As ðq, wÞT Ps ðq, wÞ þ Ps ðq, wÞ As ðq, wÞ ¼ I, where Pe ðq, wÞ and Ps ðq, wÞ are symmetric positive definite real matrices. For notational convenience, the arguments of Ae ðq, wÞ, Pe ðq, wÞ, As ðq, wÞ, Ps ðq, wÞ and other functions are dropped whenever there is no confusion. As noted in Lawrence and Rugh (1990) and Lee and Lim (1997), there are several finite constants L1, L2, M 1 , M 2 , C 1 , and C 2 such that kPe k L1 , kPs k L2 , kP_ e k M 1 kw_ k, kP_ s k M 2 kw_ k, kRe k C 1 kek, and kRs k C 2 ksk. Also, using kw_ ðtÞk and a local Lipschitz condition, we have k feþ f1 k k1 ðksk þ Þ. Moreover, the following can be easily obtained: kg1 k k2 , kg2 k k3 kek þ k4 ksk where ki, i ¼ 2, . . . , 4, are finite positive constants. Now, we state the following analysis result. Recall that the ultimate boundedness of solutions of (8) means that there exists a constant c>0, independent of t0 0 such that for every a 2 ð0, cÞ, there exists ðaÞ > 0 such that keðt0 Þk a and ksðt0 Þk a ) keðtÞk and ksðtÞk , 8t t0 (Definition 4.6, Khalil 2002). Theorem 1: Suppose that Assumptions 1 and 2 hold. Then, regarding the system (8), there exist finite constants , , e0, and s0 such that for < , < , keð0Þk < e0 , and ksð0Þk < s0 , keðtÞk and ksðtÞk are ultimately bounded by eb and sb where eb ¼ ðð2L1 k1 Þ= ðð1 2L1 C1 M1 ÞÞÞ and sb ¼ ðð2L2 k2 Þ=ðð1 2L2 C2 2L2 k4 M2 ÞÞ, 0 < < 1. Proof: First, we define some constants for convenience: M1 ¼ M 1 , M2 ¼ M 2 , C1 ¼ C 1 supDe kek, and C2 ¼ C 2 supDs ksk, which leads to kP_ e k M1 , kP_ s k M2 , kRe k C1 , and kRs k C2 . Note that De and Ds are open balls, each containing e ¼ 0 and s ¼ 0. We set a Lyapunov function Vðe, s, tÞ ¼ V1 ðe, tÞ þ V2 ðs, tÞ where V1 ðe, tÞ ¼ eT Pe e and V2 ðs, tÞ ¼ sT Ps s. Then, along the trajectory of (8), V_ ¼ kek2 þ 2eT Pe fRe e þ feþ f1 g þ eT P_ e e 1 2 ksk2 þ sT Ps fRs s þ g1 þ g2 g þ sT P_ s s kek2 þ 2L1 C1 kek2 þ 2L1 k1 kek þ 2L1 k1 kekksk þ M1 kek2 1 2 ksk2 þ L2 C2 ksk2 þ 2L2 k2 ksk þ 2L2 k3 kekksk þ 2L2 k4 ksk2 þ M2 ksk2
kek
#T "
kek
#
¼ ksk
ksk
kekfð1 2L1 C1 M1 Þkek
2L1 k1 g 1 2L2 C2 ksk 2L2 k4 M2 ksk 2L2 k2 , ð9Þ where 2 ¼4
ð1 Þð1 2L1 C1 M1 Þ L1 k1 L2 k3
3 L1 k1 L2 k3 5 1 2L2 C2 2L2 k4 M2 ð1 Þ
with 0 < < 1. The matrix becomes positive definite when 1 2L1 C1 M1 > 0 and det > 0. Here, we have det ¼ ð1 Þ2 ð1 2L1 C1 M1 Þðð1=Þ ðð2L2 C2 Þ=ðÞÞ 2L2 k4 M2 ÞðL1 k1 þL2 k3 Þ2 , which leads to the condition ð1 Þ2 ð1 2L1 C1 M1 Þð1 2L2 C2 Þ : < ¼ ðL1 k1 þ L2 k3 Þ2 þ ð1 Þ2 ð1 2L1 C1 M1 Þð2L2 k4 þ M2 Þ Thus, V_ is negative definite when 1 2L1 C1 M1 > 0, 1 2L2 C2 > 0, and < for kek eb and ksk sb . Recalling the definitions of M1, M2, C1, C2, the condition 1 2L1 C1 M1 > 0 leads to keð0Þk < e0 and < , and the condition 1 2L2 C2 > 0 leads to ksð0Þk < s0 . Since the Lyapunov function Vðe, s, tÞ is in a quadratic form, we can follow the arguments of Lemma 2 of Lee and Lim (1997) regarding the ultimate bounds. Thus, it is concluded that the ultimate bounds œ of kek and ksk are eb and sb, respectively. Remark 1: Observe that as ! 0, each eb ! 0 and sb ! 0. Then, the asymptotic stability is achieved as claimed in (Kokotovic et al. 1986, Retchkiman and Silva 1996, Khalil 2002, Shin and Lim 2002). Thus, for better performance, it is necessary to reduce or, if possible, to remove the effects of perturbed terms (via the derivative information). Remark 2: For a sufficiently small < , it is noted that sb becomes small, which means that the fast state z reaches the slow manifold ðx, wÞ in a short time, and ðx, wÞ plays a role of a controller for the reduced system afterward. Thus, if the slow manifold of the closed-loop system can be designed with some flexibility, it can be utilized to yield a better transient response by reducing the convergence time of the fast state to the slow manifold.
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H.-L. Choi and J.-T. Lim In the second design phase, we design a gain scheduling controller for the control of fast dynamics. Linearization of the fast dynamics around z ¼ ðx, vÞ and u ¼ U where U ¼ Uðx, z, wÞjz¼ðx, vÞ for each constant v 2 v gives us the following:
3. Design of controller 3.1. Controller using gain scheduling with derivative information Consider a system given by
z_ ¼
x_ ¼ f ðx, z, wðtÞ, uÞ z_ ¼ gðx, z, wðtÞ, uÞ y ¼ hðx, wðtÞÞ,
ð10Þ
where x 2 Rn , z 2 Rm , u 2 Rl , and y 2 Rq , which are the slow state, the fast state, the control input, and the system output, respectively. The vector fields f and g are assumed to be twice continuously differentiable with respect to their arguments. Regarding the parameters and w(t), we assume the same condition as in section 2. Note that the system output only depends on the slow state and w(t), which seems to be restrictive. However, in many practical applications, this is indeed true, e.g. EMS systems, series DC motors, etc. The control goal is to force the output y(t) to asymptotically track a reference input rðtÞ 2 Rq while reducing the effect of exogenous signal w(t) and keeping all the closed-loop signals bounded. Assumption 3: There exists an isolated solution u ¼ Uðx, z, wÞ such that for each constant w 2 w , 0 ¼ gðx, z, w, Uðx, z, wÞÞ. Substituting u ¼ Uðx, z, wÞ into the slow dynamics of (10), we have the following reduced system:
_s ¼
@gðx, ðx, vÞ, U Þ @gðx, ðx, vÞ, U Þ sþ ug _ðx, vÞ, @z @u ð14Þ
where ug ¼ u U . Assumption 5:
The following
@gðx, ðx, vÞ, U Þ @gðx, ðx, vÞ, U Þ , @z @u
ð15Þ
is a controllable pair for each constant x 2 x and v 2 v . Then, the gain scheduling controller is given by ug ¼ Kp ðx, vÞs,
ð16Þ
where Kp ðx, vÞ is selected such that ð17Þ
ð11Þ
In the first design phase, we design a slow manifold ðx, w, rÞ which will work as a controller for (11). We denote v ¼ ½wT , rT T for notational convenience for the rest of paper and state the next assumption: Assumption 4: In an open neighbourhood x of the origin, there exists a smooth function xðvÞ for each constant v 2 v and r 2 r such that 0 ¼ f ðxðvÞ, vÞ and r ¼ hðxðvÞ, vÞ. Moreover, we can choose z ¼ ðx, vÞ for each constant v 2 v such that @f ðxðvÞ, vÞ e < 0, Re i @x
ð13Þ
From s ¼ z ðx, vÞ, we have
@gðx, ðx, vÞ, U Þ @gðx, ðx, vÞ, U Þ þ Kp ðx, vÞ @z @u
x_ ¼ f ðx, z, w, UÞ y ¼ hðx, wÞ:
@gðx, ðx, vÞ, U Þ ½z ðx, vÞ @z @gðx, ðx, vÞ, U Þ ½u U : þ @u
i ¼ 1, 2, . . . , n, ð12Þ
where f ðx, vÞ ¼ f ðx, ðx, vÞ, w, UÞ. From Assumption 4, letting qðtÞ ¼ xðvðtÞÞ and e ¼ x q, it is clear that limt!1 kyðtÞ rðtÞk ¼ 0 if limt!1 keðtÞk ¼ 0.
is a Hurwitz matrix for each constant x 2 x and v 2 v . In the third design phase, we add a derivative information term ud ¼ ðx, s, v, v_Þ to the controller. In particular, ud is selected such that ud becomes zero at each constant v 2 v . As noted in Remark 1, eb and sb are non-zero when 6¼ 0. From section 2, f1 and g1 are the -bounded terms. Thus, with the properly selected ud, we can cancel or reduce the effect of these terms. In overall, the proposed controller is summarized as u ¼ U þ ug þ ud :
ð18Þ
The proper selection of ud will be specified through Theorem 2 and Corollary 1. Remark 3: Regarding the proposed controller (21), the roles of each components are summarized as follows: The slow manifold ðx, vÞ and the gain scheduling controller ug are mainly responsible for the transient response where ug drives the fast state to ðx, vÞ.
Gain scheduling control of time-varying systems The derivative information controller ud is mainly responsible for the steady state response, i.e. the sizes of ultimate error bounds. In view of controller structure, it has a reference/disturbance feedforward compensation component and a proportional/derivative feedback component. Before we state the next result, we briefly explain the meanings of Assumptions 3–5: Assumption 3 resembles the existence condition of an isolated root of the fast dynamics. Assumption 4 states the existence of slow manifold such that the reduced system is stable. Assumption 5 states the controllability of the fast dynamics such that the gain scheduling controller can be designed, which derives the fast states to the designed slow manifold. Theorem 2: Suppose that Assumptions 3–5 hold. Then, regarding the closed-loop system (10), there exist finite constants , , e0, and s0 such that for < , < , keð0Þk < e0 , and ksð0Þk < s0 , keðtÞk and ksðtÞk are ultimately bounded by some finite constants eb and sb. Moreover, eb becomes zero if we can select ug and ud such that fgd ð@q=@vÞv_ ¼ 0 where fgd ¼ f ðx, z, w, uÞ ju¼U þug þud f ðx, z, w, uÞjz¼ðx, vÞ, u¼U . Proof: First, we obtain the following closed-loop system with the proposed controller x_ ¼ f ðx, z, w, U þ ug þ ud Þ z_ ¼ gðx, z, w, U þ ug þ ud Þ y ¼ hðx, wÞ:
@q e_ ¼ f ðx, vÞ þ feðx, s, v, ug Þ þ fbðx, s, v, ud Þ v_: @v
ð23Þ
From the design of slow manifold ðx, vÞ, we have f ðx, vÞ ¼ Ae e þ Re e,
ð24Þ
where Ae and Re are as specified in section 2. Substituting x ¼ e þ q, we obtain the following: e_ ¼ Ae e þ Re e þ feðe þ q, s, v, ug , ud Þ @q þ fbðe þ q, s, v, ud Þ v_ @v gðe þ q, s, v, ug , ud Þ _ðe þ q, vÞ: ð25Þ _s ¼ As s þ Rs s þ e As shown in Theorem 1, we set a Lyapunov function Vðe, s, tÞ ¼ eT Pe e þ sT Ps s. Through the similar algebraic manipulations, we obtain the ultimate bounds as follows: 2C1 k feð, ug , ud Þ þ fbð, ud Þ ð@q=@vÞv_k ð1 2L1 C1 M1 Þ 2L2 ke gð, ug , ud Þ _ðe þ q, vÞk , sb ¼ ð1 2L2 C2 M2 Þ
ð20Þ
Following the design of gain scheduling controller ug , we have ð21Þ
where As and Rs are as specified in section 2. Regarding the slow dynamics, we consider
where 0 < < 1. Finally, it is obvious that for the closed-loop system, Assumptions 1 and 2 are satisfied. Thus, the existence of , , e0, and s0 is assured from Theorem 1. œ From the analysis of Theorem 2, the following result can be obtained: Corollary 1: Assume that the slow dynamics is affine in u: x_ ¼ f ðx, z, w, uÞ ¼ f1 ðx, z, wÞ þ f2 ðx, z, wÞu and all the related conditions in Theorem 2 are satisfied. Considering ug ¼ Kp ðx, vÞs and ud ¼ Kd ðx, vÞv_, eb becomes zero if f2 ðx, z, wÞKp ðx, vÞs ¼ fe @q f2 ðx, z, wÞKd ðx, vÞ ¼ , @v
x_ ¼ f ðx, s þ ðx, vÞ, w, U þ ug þ ud Þ ¼ f ðx, vÞ þ fgd ðx, s, v, ug , ud Þ y ¼ hðx, wÞ
Also, fgd ðx, s, v, ug , ud Þ is divided into two parts: fgd ðx, s, v, ug , ud Þ ¼ feðx, s, v, ug , ud Þ þ fbðx, s, v, ud Þ, where fbðx, s, v, ud Þ denotes the part that fbðx, s, v, ud Þ ¼ 0 when ud ¼ 0. Then, from Assumption 4, we can set e ¼ x q, which results in
eb ¼
where e gðx, s, v, ug , ud Þ ¼ gðx, s þ ðx, vÞ, w, U þ ug þ ud Þ gðx, s þ ðx, vÞ, w, U þ ug Þ.
gðx, s þ ðx, vÞ, w, U þ ug Þ ¼ As s þ Rs s,
where f ðx, vÞ ¼ f ðx, s þ ðx, vÞ, w, U Þ and fgd ðx, s, v, ug , ud Þ ¼ f ðx, s þ ðx, vÞ, w, U þ ug þ ud Þ f ðx, vÞ:
ð19Þ
With s ¼ z ðx, vÞ, the fast dynamics becomes _s ¼ gðx, s þ ðx, vÞ, w, U þ ug þ ud Þ _ðx, vÞ ¼ gðx, s þ ðx, vÞ, w, U þ ug Þ þ e gðx, s, v, ug , ud Þ _ ðx, vÞ,
361
ð22Þ
where fe¼ f ðx, s þ ðs, vÞ, w, U Þ f ðx, ðx, vÞ, w, U Þ.
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Remark 4. In Theorem 2 and Corollary 1, even when there exists no ug and ud satisfying each conditions to make eb zero, each ug and ud can still be effectively designed to make the bounds of k feð, ug , ud Þ þ gð, ug , ud Þ _ðe þ q, vÞk smalfbð, ud Þ ð@q=@vÞv_k and ke ler, which result in a better tracking performance. In general, these two norm-bound conditions can be used in selecting the ud.
(Zero tracking error case) Consider the following nonlinear singularly perturbed system x_ ¼ 2x þ x2 þ 2z þ u þ wðtÞ z_ ¼ x x2 þ z þ u ð26Þ
where wðtÞ ¼ 0:1 sinðtÞ. Control goal:
Track rðtÞ ¼ 0:5 sin t while rejecting w(t).
First, we have u ¼ Uðx, z, wÞ ¼ x þ x2 z from the fast dynamics. Then, the reduced system is given by x_ ¼ 3x þ 2x2 þ z þ wðtÞ y ¼ x:
2
k31
Tðv, 0Þ ¼ 4 k31 þ1
3.2. Illustrative example
y ¼ x,
u ¼ 3r2 4r þ w þ k11 ðx rÞ þ k12 ðz 3r þ 2r2 þ wÞ þ k3 v_ 2r 7 ðK=Þ, k12 ¼ K 1, where k11 ¼ K ¼ ðð9 3 4rÞ=ðð1=Þ þ 4r 3ÞÞ, and k3 ¼ ½k31 , k32 . Moreover, in choosing the derivative information, the following matrix Tðv, 0Þ should be minimized by some proper values of k31 and k32
ð27Þ
Here, we have qðtÞ ¼ rðtÞ. Now, we choose the slow manifold z ¼ ðx, vÞ ¼ 3r 2r2 4rðx rÞ 2ðx rÞi w, i 0 to satisfy (12). Thus far, letting s ¼ z ðx, vÞ, we have f ðx, vÞ ¼ 2x þ x2 þ 2ðx, vÞ þ Ujz¼ðx, vÞ þ w, fe¼ 2s þ ug , and fb ¼ ud . By linearization of the fast dynamics, we have ð@g=@zÞ ¼ 1, and ð@g=@uÞ ¼ 1. Thus, Kp ðx, vÞ is selected such as Kp ðx, vÞ ¼ 2 to place the eigenvalue at 1, which leads to ug ¼ 2s. Moreover, with this choice, fe turns out to be zero. So far, the obtained controller is u ¼ U 2ðz ðx, vÞÞ þ ud where U ¼ x þ x2 ðx, vÞ. Now, in selecting the derivative information, we have kfeþ fb ð@q=@vÞv_k ¼ kud r_k and ke g _k ¼ kud _k. In our design, we choose ud ¼ r_ðtÞ to make eb ¼ 0. To verify the claims regarding the proposed method, we consider two cases: (1) To see the effect of slow manifold on the transient response, we select two slow manifolds such as 1 ðx, vÞ ¼ 3r 2r2 4rðx rÞ w and 2 ðx, vÞ ¼ 3r 2r2 4rðx rÞ 2ðx rÞ2 w. Note that in both approaches, the eigenvalues are placed at 3 equally. (2) To see the effect of derivative information on the steady state response, for comparison, we use the controller from Lee and Lim (1997) with the assumption that is known exactly. With the system eigenvalues at 3 and 3, we have
k32 1
3
5: k32 3 þ 4r
Hence, we choose k31 ¼ 10 and k32 ¼ ð3 4rÞ. Note that it is impossible to make kTðv, 0Þk ¼ 0, which means we cannot make limt!1 kyðtÞ rðtÞk ¼ 0 by the method in Lee and Lim (1997), even with derivative information. The simulation results of both comparisons are shown in figure 1. By two choices of slow manifolds coupled with gain scheduling controller, we show that the transient response is indeed much different depending on the choice of slow manifold as shown in figure 1(a)–(b). The effect of derivative controller design is shown through a direct comparison with the method of Lee and Lim (1997) in figure 1(c)–(d) where the proposed method achieves a better tracking performance. Another note is that there are existing control schemes using derivative information (Sureshbabu and Rugh 1995, Lee and Lim 1997). However, in these schemes, the full system information is needed including the value of . In our approach, we do not use the value of in controller design as long as < , which leads to simplicity and robustness by taking as an unknown small parameter. (Nonzero tracking error case) Consider the following nonlinear singularly perturbed system x_ ¼ 2x þ x2 z þ u þ wðtÞ z_ ¼ x x2 þ z þ u y ¼ x,
ð28Þ
where wðtÞ ¼ 0:1 sinðtÞ. This system is almost identical to the system (28) except for a slight difference in the slow dynamics. However, this difference actually makes it difficult to design a tracking controller. In fact, in this case, the design of derivative controller becomes more important. Since the design steps are similar, we only show the designed controller components: we have u ¼ U ðx, z, wÞ ¼ x þ x2 z; the selected slow manifold is z ¼ ðx, vÞ ¼ ð3=2Þr þ r2 þ 2rðx rÞ þ ðx rÞ2 þ ðw=2Þ to satisfy equation (12); the gain scheduling controller is ug ¼ 4s to place the eigenvalue at 3. Then, the remaining step is to design a derivative controller ud.
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Figure 1. Comparison when ¼ 1=20: Transient response: (a) plot of tracking error and (b) plot of control input; steady-state response: (c) plot of tracking error and (d) plot of control input.
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Thus far, with the designed controller components, we have fe ¼ s þ ug ¼ 5s, and fb¼ ud . Now, in selecting the derivative information, we have k feþ fb g _k ¼ kud _k. ð@q=@vÞv_k ¼ k 5s þ ud r_k and ke Unlike the previous case, this time we cannot make eb ¼ 0 by ud ¼ r_ due to 5s. We also need to reduce the effect caused by sb. By considering the structure of , we select ud ¼ 1:5ðw_ þ r_Þ. In figure 2, it is shown that the derivative controller indeed effectively improves the controller’s tracking performance.
4. Conclusions In this paper, we analyse the ultimate boundedness of nonlinear singularly perturbed time-varying systems and propose a control scheme in the context of gain scheduling. In our approach, there are mainly three design components—slow manifold, gain scheduling controller, and derivative information. With the properly chosen design components, the controller can result in better transient response and reference tracking performance over the existing control schemes.
Acknowledgement This work was supported by Satellite Technology Research Center.
References C.C. Chen, ‘‘Criterion for global exponential stabilisability of a class of nonlinear control systems via an integral manifold approach’’, IEE Proc. Contr. Theory Appl., 147, pp. 330–336, 2000. E.F. Costa and V.A. Oliveira, ‘‘Gain scheduled controllers for dynamic systems using sector nonlinearities’’, Automatica, 38, pp. 1247–1250, 2002. M. Innocenti, L. Greco and L. Pollini, ‘‘Sliding mode control for twotime scale systems: stability issues’’, Automatica, 39, pp. 273–280, 2003. H.K. Khalil, Nonlinear Systems, Englewood Cliffs, NJ: Prentice Hall, 2002. P.V. Kokotovic, H.K. Khalil and J. O’Reilly, Singular Perturbed Method in Control, London: Academic Press, 1986. G.A. Lawrence and W.J. Rugh, ‘‘On a stability theorem for nonlinear systems with slowly varying inputs’’, IEEE Trans. Automat. Contr., 35, pp. 860–864, 1990. S.-H. Lee and J.-T. Lim, ‘‘Fast gain scheduling on tracking problems using derivative information’’, Automatica, 33, pp. 2265–2268, 1997. Z. Retchkiman and G. Silva, ‘‘Stability analysis of singularly perturbed systems via vector Lyapunov methods’’, in Proc. 35th IEEE Conference on Decision and Control, Kobe, Japan, 1996, pp. 580–585. W.J. Rugh, ‘‘Analytical framework for gain scheduling’’, IEEE Contr. Sys. Mag., 11, pp. 79–84, 1991. W.J. Rugh and J.S. Shamma, ‘‘Research on gain scheduling’’, Automatica, 36, pp. 1401–1425, 2000. J.S. Shamma and M. Athans, ‘‘Gain scheduling: Potential hazard and possible remedies’’, IEEE Contr. Sys. Mag., 12, pp. 101–107, 1992. P.M. Sharkey and J. O’Reilly, ‘‘Composite control of nonlinear singularly perturbed systems: A geometric approach’’, Int. J. Control, 48, pp. 2491–2506, 1988. Y.-S. Shin and J.-T. Lim, ‘‘Control of nonlinear singularly perturbed systems using gain scheduling’’, IEICE Trans. Fundamentals, E85A, pp. 2175–2179, 2002. N. Sureshbabu and W.J. Rugh, ‘‘Output regulation with derivative information’’, IEEE Trans. Automat. Contr, 40, pp. 1755–1766, 1995.
Ho-Lim Choi received the BSE degree from the department of electrical engineering, The Univ. of Iowa, USA in 1996, and MS degree in 1999 and PhD degree in 2004, from KAIST (Korea Advanced Institute of Science and Technolgy), respectively. Currently, he is a BK21 post-doctoral research fellow at the department of electrical engineering, KAIST. His research interests are in the nonlinear control problems with emphasis on feedback linearization, gain scheduling, singular perturbation, output feedback, time-delay systems.
Jong-Tae Lim received the BSEE degree from Yonsei University, Seoul, Korea, in 1975, the MSEE degree from the Illinois Institute of Technology, Chicago, in 1983, and the PhD degree in Computer, Information and Control Engineering from the University of Michigan, Ann Arbor, in 1986. He is currently a professor in the Division of Electrical Engineering at the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology. His research interests are in the areas of system and control theory, communication networks, and discrete event systems. He is a member of IEEE, KIEE, and KITE.
Gain scheduling control of nonlinear singularly perturbed time-varying systems with derivative information HO-LIM CHOI and JONG-TAE LIM* Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Kusong-dong, Yusong-gu, Taejon, 305-701, Korea (Received 10 June 2004; in final form 11 March 2005) In this paper, we analyse the ultimate boundedness of nonlinear singularly perturbed time-varying systems and propose a control law using gain scheduling where the slow state and the exogenous signals are used as scheduling variables. In our control scheme, we have some flexibility in selecting the slow manifold of the system. Moreover, the derivative information can be properly engaged to manipulate the size of ultimate bound in tracking error of the controlled system. Keywords: Singularly perturbed time-varying systems; Ultimate boundedness; Gain scheduling; Slow manifold; Derivative information
1. Introduction Gain scheduling control methodology has been widely applied to many nonlinear systems. Since the introduction of the analytical framework of gain scheduling (Rugh 1991) and stability analysis (Lawrence and Rugh 1990), there has been considerable progress up to the present, e.g. potential hazard of gain scheduling and remedies (Shamma and Athans 1992), fast gain scheduling technique (Sureshbabu and Rugh 1995, Lee and Lim 1997) or trajectory planning (Costa and Oliveira 2002). In the gain scheduling approach, it is necessary to select the scheduling variables which are normally ‘slowly varying’ parameters. Then, the local controller is designed around the equilibrium manifold. More general notions on gain scheduling control and related research results are addressed in Rugh and Shamma (2000). In two-time-scale systems, there exist a slow state and a fast state. The singularly perturbed systems are the typical two-time-scale systems where the slow state plays a role of slowly varying parameters. The dynamics is degenerated into the reduced model
*Corresponding author. Email: [email protected]
and controlled by the slow state when the fast state is on the slow manifold. It has been shown that the behaviour of the system is constrained in O() bound of the slow manifold (Kokotovic et al. 1986, Retchkiman and Silva 1996, Khalil 2002). Thus, in the reduced model, the slow manifold works as a control input. For the control of singularly perturbed systems, there have been various results (Sharkey and O’Reilly 1988, Chen 2000, Shin and Lim 2002, Innocenti et al. 2003). One typical approach is the composite control scheme (Sharkey and O’Reilly 1988, Innocenti et al. 2003) where the slow control and fast control are designed separately. In those approaches, the slow manifold is not explicitly considered in the design process. In Shin and Lim (2002), the slow manifold is used as a design component, which can result in better performance depending on the initial states. In this paper, we analyse the ultimate boundedness of nonlinear singularly perturbed time-varying systems and propose a controller using the gain scheduling approach where the fast controller is the gain scheduling controller, and the slow state and the exogenous signals are used as scheduling variables. In our proposed scheme, the selection of the slow manifold is incorporated in the controller design process. This feature can lead to better transient response since the fast state
International Journal of Systems Science ISSN 0020–7721 print/ISSN 1464–5319 online 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.1080/00207720500111707
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can converge to the slow manifold more quickly. Also, the proposed controller utilizes the time-derivative information on scheduling parameters. The properly selected time-derivative information can result in smaller error bounds in reference tracking. Thus, the proposed scheme is the improvement over the existing control schemes because it can tackle both transient and steady-state response.
^ where A^ s ðx, z, wÞ ¼ ð@gðx, z, wÞÞ=@z, Ð 1 As ðx, wÞ ¼ As ðx, ^ ðx, wÞ, wÞ, and Rs ðx, s, wÞ ¼ 0 ½As ðx, ðx, wÞ þ s, wÞ A^ s ðx, ðx, wÞ, wÞd. From equations (2) and (3), we obtain the following system: x_ ¼ fc ðx, s, wÞ _s ¼ As ðx, wÞs þ Rs ðx, s, wÞs þ g1 ðx, w, w_ Þ þ g2 ðx, s, wÞ, ð4Þ
2. System analysis 2.1. System formulation Consider a system given by x_ ¼ f ðx, z, wðtÞÞ z_ ¼ gðx, z, wðtÞÞ,
ð1Þ
where x 2 Rn and z 2 Rm which are the slow state and the fast state, respectively. The parameter is a sufficiently small positive constant. The exogenous signal wðtÞ 2 Rp is continuously differentiable. Moreover, it is assumed that kwðtÞk and kw_ ðtÞk , t 0. The notations used in this paper are as follows: x, w, q, r denote the bounded subsets of x, w, q, r, respectively. xðwÞ denotes the frozen signal of x(w). i(A) denotes the eigenvalues of a matrix A. All norms are used as the Euclidean 2-norm unless specifically denoted. Assumption 1: Regarding the fast dynamics of the system (1), the following hold: (A1) (A2)
(A3)
g is twice continuously differentiable with respect to its arguments. For each constant x 2 x and w 2 w , there exists a continuously differentiable manifold z ¼ ðx, wÞ such that gðx, ðx, wÞ, wÞ ¼ 0. For each constant x 2 x and w 2 w , Re ½i ðð@g=@zÞðx, ðx, wÞ, wÞÞ s < 0, i ¼ 1, 2, . . . , m:
Set the change of variable s ¼ z ðx, wÞ. Then, due to (A2), ð1
d gðx, ðx, wÞ þ s, wÞd ¼ gðx, ðx, wÞ þ s, wÞ: ð2Þ 0 d
Also, using the results of Lee and Lim (1997) and Shin and Lim (2002), it can be shown that ð1
d gðx, ðx, wÞ þ s, wÞd ¼ As ðx, wÞs þ Rs ðx, s, wÞs, d 0 ð3Þ
where fc ðx, s, wÞ ¼ f ðx, s þ ðx, wÞ, wÞ, g1 ðx, w, w_ Þ ¼ ðð@ðx, wÞÞ=ð@wÞÞw_ , and g2 ðx, s, wÞ ¼ ðð@ðx, wÞÞ=ð@xÞÞ f ðx, s þ ðx, wÞ, wÞ From equation (4), we consider the slow dynamics part only. Letting s ¼ 0 and denoting f ðx, wÞ ¼ fc ðx, s, wÞjs¼0 , we have the reduced system: x_ ¼ f ðx, wÞ Assumption 2: hold: (B1) (B2) (B3)
ð5Þ
Regarding system (5), the following
f is twice continuously differentiable. For each constant w 2 w , there exists a smooth function x(w) such that 0 ¼ f ðxðwÞ, wÞ. For each constant w 2 w , Re½i ðð@f =@xÞðxðwÞ, wÞÞ e < 0, i ¼ 1, 2, . . . , n:
Let qðtÞ ¼ xðwðtÞÞ be the equilibrium manifold such that 0 ¼ f ðqðtÞ, wðtÞÞ. Then, we set e ¼ x q. Considering the original slow dynamics in equation (4), we have e_ ¼ x_ q_ ¼ f ðx, wÞ þ feðx, s, wÞ
@q w_ ; @w
ð6Þ
where feðx, s, wÞ ¼ fc ðx, s, wÞ f ðx, wÞ. Under Assumption 2, similar to equation (3), we have e_ ¼ Ae ðq, wÞe þ Re ðq, w, eÞe þ feðx, s, wÞ þ f1 ðw, w_ Þ,
ð7Þ
^ where A^ e ðx, Ð 1 wÞ ¼ ð@f ðx, wÞÞ=@x, Ae ðq, wÞ ¼ Ae ðq, wÞ, ^ ^ Re ðq, w, eÞ ¼ 0 ½Ae ðq þ e, wÞ Ae ðq, wÞd, and f1 ðw, w_ Þ ¼ ð@q=@wÞw_ . Using the relation x ¼ e þ q, we have e_ ¼ Ae ðq, wÞe þ Re ðq, w, eÞe þ feðe þ q, s, wÞ þ f1 ðw, w_ Þ _s ¼ As ðe þ q, wÞs þ Rs ðe þ q, s, wÞs þ g1 ðe þ q, w, w_ Þ þ g2 ðe þ q, s, wÞ:
ð8Þ
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2.2. System analysis: ultimate boundedness Regarding system (8), it is noted that Ae and As are Hurwitz for all q 2 q and w 2 w . Thus, we have two Lyapunov equations such as Ae ðq, wÞT Pe ðq, wÞ þ Pe ðq, wÞAe ðq, wÞ ¼ I and As ðq, wÞT Ps ðq, wÞ þ Ps ðq, wÞ As ðq, wÞ ¼ I, where Pe ðq, wÞ and Ps ðq, wÞ are symmetric positive definite real matrices. For notational convenience, the arguments of Ae ðq, wÞ, Pe ðq, wÞ, As ðq, wÞ, Ps ðq, wÞ and other functions are dropped whenever there is no confusion. As noted in Lawrence and Rugh (1990) and Lee and Lim (1997), there are several finite constants L1, L2, M 1 , M 2 , C 1 , and C 2 such that kPe k L1 , kPs k L2 , kP_ e k M 1 kw_ k, kP_ s k M 2 kw_ k, kRe k C 1 kek, and kRs k C 2 ksk. Also, using kw_ ðtÞk and a local Lipschitz condition, we have k feþ f1 k k1 ðksk þ Þ. Moreover, the following can be easily obtained: kg1 k k2 , kg2 k k3 kek þ k4 ksk where ki, i ¼ 2, . . . , 4, are finite positive constants. Now, we state the following analysis result. Recall that the ultimate boundedness of solutions of (8) means that there exists a constant c>0, independent of t0 0 such that for every a 2 ð0, cÞ, there exists ðaÞ > 0 such that keðt0 Þk a and ksðt0 Þk a ) keðtÞk and ksðtÞk , 8t t0 (Definition 4.6, Khalil 2002). Theorem 1: Suppose that Assumptions 1 and 2 hold. Then, regarding the system (8), there exist finite constants , , e0, and s0 such that for < , < , keð0Þk < e0 , and ksð0Þk < s0 , keðtÞk and ksðtÞk are ultimately bounded by eb and sb where eb ¼ ðð2L1 k1 Þ= ðð1 2L1 C1 M1 ÞÞÞ and sb ¼ ðð2L2 k2 Þ=ðð1 2L2 C2 2L2 k4 M2 ÞÞ, 0 < < 1. Proof: First, we define some constants for convenience: M1 ¼ M 1 , M2 ¼ M 2 , C1 ¼ C 1 supDe kek, and C2 ¼ C 2 supDs ksk, which leads to kP_ e k M1 , kP_ s k M2 , kRe k C1 , and kRs k C2 . Note that De and Ds are open balls, each containing e ¼ 0 and s ¼ 0. We set a Lyapunov function Vðe, s, tÞ ¼ V1 ðe, tÞ þ V2 ðs, tÞ where V1 ðe, tÞ ¼ eT Pe e and V2 ðs, tÞ ¼ sT Ps s. Then, along the trajectory of (8), V_ ¼ kek2 þ 2eT Pe fRe e þ feþ f1 g þ eT P_ e e 1 2 ksk2 þ sT Ps fRs s þ g1 þ g2 g þ sT P_ s s kek2 þ 2L1 C1 kek2 þ 2L1 k1 kek þ 2L1 k1 kekksk þ M1 kek2 1 2 ksk2 þ L2 C2 ksk2 þ 2L2 k2 ksk þ 2L2 k3 kekksk þ 2L2 k4 ksk2 þ M2 ksk2
kek
#T "
kek
#
¼ ksk
ksk
kekfð1 2L1 C1 M1 Þkek
2L1 k1 g 1 2L2 C2 ksk 2L2 k4 M2 ksk 2L2 k2 , ð9Þ where 2 ¼4
ð1 Þð1 2L1 C1 M1 Þ L1 k1 L2 k3
3 L1 k1 L2 k3 5 1 2L2 C2 2L2 k4 M2 ð1 Þ
with 0 < < 1. The matrix becomes positive definite when 1 2L1 C1 M1 > 0 and det > 0. Here, we have det ¼ ð1 Þ2 ð1 2L1 C1 M1 Þðð1=Þ ðð2L2 C2 Þ=ðÞÞ 2L2 k4 M2 ÞðL1 k1 þL2 k3 Þ2 , which leads to the condition ð1 Þ2 ð1 2L1 C1 M1 Þð1 2L2 C2 Þ : < ¼ ðL1 k1 þ L2 k3 Þ2 þ ð1 Þ2 ð1 2L1 C1 M1 Þð2L2 k4 þ M2 Þ Thus, V_ is negative definite when 1 2L1 C1 M1 > 0, 1 2L2 C2 > 0, and < for kek eb and ksk sb . Recalling the definitions of M1, M2, C1, C2, the condition 1 2L1 C1 M1 > 0 leads to keð0Þk < e0 and < , and the condition 1 2L2 C2 > 0 leads to ksð0Þk < s0 . Since the Lyapunov function Vðe, s, tÞ is in a quadratic form, we can follow the arguments of Lemma 2 of Lee and Lim (1997) regarding the ultimate bounds. Thus, it is concluded that the ultimate bounds œ of kek and ksk are eb and sb, respectively. Remark 1: Observe that as ! 0, each eb ! 0 and sb ! 0. Then, the asymptotic stability is achieved as claimed in (Kokotovic et al. 1986, Retchkiman and Silva 1996, Khalil 2002, Shin and Lim 2002). Thus, for better performance, it is necessary to reduce or, if possible, to remove the effects of perturbed terms (via the derivative information). Remark 2: For a sufficiently small < , it is noted that sb becomes small, which means that the fast state z reaches the slow manifold ðx, wÞ in a short time, and ðx, wÞ plays a role of a controller for the reduced system afterward. Thus, if the slow manifold of the closed-loop system can be designed with some flexibility, it can be utilized to yield a better transient response by reducing the convergence time of the fast state to the slow manifold.
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H.-L. Choi and J.-T. Lim In the second design phase, we design a gain scheduling controller for the control of fast dynamics. Linearization of the fast dynamics around z ¼ ðx, vÞ and u ¼ U where U ¼ Uðx, z, wÞjz¼ðx, vÞ for each constant v 2 v gives us the following:
3. Design of controller 3.1. Controller using gain scheduling with derivative information Consider a system given by
z_ ¼
x_ ¼ f ðx, z, wðtÞ, uÞ z_ ¼ gðx, z, wðtÞ, uÞ y ¼ hðx, wðtÞÞ,
ð10Þ
where x 2 Rn , z 2 Rm , u 2 Rl , and y 2 Rq , which are the slow state, the fast state, the control input, and the system output, respectively. The vector fields f and g are assumed to be twice continuously differentiable with respect to their arguments. Regarding the parameters and w(t), we assume the same condition as in section 2. Note that the system output only depends on the slow state and w(t), which seems to be restrictive. However, in many practical applications, this is indeed true, e.g. EMS systems, series DC motors, etc. The control goal is to force the output y(t) to asymptotically track a reference input rðtÞ 2 Rq while reducing the effect of exogenous signal w(t) and keeping all the closed-loop signals bounded. Assumption 3: There exists an isolated solution u ¼ Uðx, z, wÞ such that for each constant w 2 w , 0 ¼ gðx, z, w, Uðx, z, wÞÞ. Substituting u ¼ Uðx, z, wÞ into the slow dynamics of (10), we have the following reduced system:
_s ¼
@gðx, ðx, vÞ, U Þ @gðx, ðx, vÞ, U Þ sþ ug _ðx, vÞ, @z @u ð14Þ
where ug ¼ u U . Assumption 5:
The following
@gðx, ðx, vÞ, U Þ @gðx, ðx, vÞ, U Þ , @z @u
ð15Þ
is a controllable pair for each constant x 2 x and v 2 v . Then, the gain scheduling controller is given by ug ¼ Kp ðx, vÞs,
ð16Þ
where Kp ðx, vÞ is selected such that ð17Þ
ð11Þ
In the first design phase, we design a slow manifold ðx, w, rÞ which will work as a controller for (11). We denote v ¼ ½wT , rT T for notational convenience for the rest of paper and state the next assumption: Assumption 4: In an open neighbourhood x of the origin, there exists a smooth function xðvÞ for each constant v 2 v and r 2 r such that 0 ¼ f ðxðvÞ, vÞ and r ¼ hðxðvÞ, vÞ. Moreover, we can choose z ¼ ðx, vÞ for each constant v 2 v such that @f ðxðvÞ, vÞ e < 0, Re i @x
ð13Þ
From s ¼ z ðx, vÞ, we have
@gðx, ðx, vÞ, U Þ @gðx, ðx, vÞ, U Þ þ Kp ðx, vÞ @z @u
x_ ¼ f ðx, z, w, UÞ y ¼ hðx, wÞ:
@gðx, ðx, vÞ, U Þ ½z ðx, vÞ @z @gðx, ðx, vÞ, U Þ ½u U : þ @u
i ¼ 1, 2, . . . , n, ð12Þ
where f ðx, vÞ ¼ f ðx, ðx, vÞ, w, UÞ. From Assumption 4, letting qðtÞ ¼ xðvðtÞÞ and e ¼ x q, it is clear that limt!1 kyðtÞ rðtÞk ¼ 0 if limt!1 keðtÞk ¼ 0.
is a Hurwitz matrix for each constant x 2 x and v 2 v . In the third design phase, we add a derivative information term ud ¼ ðx, s, v, v_Þ to the controller. In particular, ud is selected such that ud becomes zero at each constant v 2 v . As noted in Remark 1, eb and sb are non-zero when 6¼ 0. From section 2, f1 and g1 are the -bounded terms. Thus, with the properly selected ud, we can cancel or reduce the effect of these terms. In overall, the proposed controller is summarized as u ¼ U þ ug þ ud :
ð18Þ
The proper selection of ud will be specified through Theorem 2 and Corollary 1. Remark 3: Regarding the proposed controller (21), the roles of each components are summarized as follows: The slow manifold ðx, vÞ and the gain scheduling controller ug are mainly responsible for the transient response where ug drives the fast state to ðx, vÞ.
Gain scheduling control of time-varying systems The derivative information controller ud is mainly responsible for the steady state response, i.e. the sizes of ultimate error bounds. In view of controller structure, it has a reference/disturbance feedforward compensation component and a proportional/derivative feedback component. Before we state the next result, we briefly explain the meanings of Assumptions 3–5: Assumption 3 resembles the existence condition of an isolated root of the fast dynamics. Assumption 4 states the existence of slow manifold such that the reduced system is stable. Assumption 5 states the controllability of the fast dynamics such that the gain scheduling controller can be designed, which derives the fast states to the designed slow manifold. Theorem 2: Suppose that Assumptions 3–5 hold. Then, regarding the closed-loop system (10), there exist finite constants , , e0, and s0 such that for < , < , keð0Þk < e0 , and ksð0Þk < s0 , keðtÞk and ksðtÞk are ultimately bounded by some finite constants eb and sb. Moreover, eb becomes zero if we can select ug and ud such that fgd ð@q=@vÞv_ ¼ 0 where fgd ¼ f ðx, z, w, uÞ ju¼U þug þud f ðx, z, w, uÞjz¼ðx, vÞ, u¼U . Proof: First, we obtain the following closed-loop system with the proposed controller x_ ¼ f ðx, z, w, U þ ug þ ud Þ z_ ¼ gðx, z, w, U þ ug þ ud Þ y ¼ hðx, wÞ:
@q e_ ¼ f ðx, vÞ þ feðx, s, v, ug Þ þ fbðx, s, v, ud Þ v_: @v
ð23Þ
From the design of slow manifold ðx, vÞ, we have f ðx, vÞ ¼ Ae e þ Re e,
ð24Þ
where Ae and Re are as specified in section 2. Substituting x ¼ e þ q, we obtain the following: e_ ¼ Ae e þ Re e þ feðe þ q, s, v, ug , ud Þ @q þ fbðe þ q, s, v, ud Þ v_ @v gðe þ q, s, v, ug , ud Þ _ðe þ q, vÞ: ð25Þ _s ¼ As s þ Rs s þ e As shown in Theorem 1, we set a Lyapunov function Vðe, s, tÞ ¼ eT Pe e þ sT Ps s. Through the similar algebraic manipulations, we obtain the ultimate bounds as follows: 2C1 k feð, ug , ud Þ þ fbð, ud Þ ð@q=@vÞv_k ð1 2L1 C1 M1 Þ 2L2 ke gð, ug , ud Þ _ðe þ q, vÞk , sb ¼ ð1 2L2 C2 M2 Þ
ð20Þ
Following the design of gain scheduling controller ug , we have ð21Þ
where As and Rs are as specified in section 2. Regarding the slow dynamics, we consider
where 0 < < 1. Finally, it is obvious that for the closed-loop system, Assumptions 1 and 2 are satisfied. Thus, the existence of , , e0, and s0 is assured from Theorem 1. œ From the analysis of Theorem 2, the following result can be obtained: Corollary 1: Assume that the slow dynamics is affine in u: x_ ¼ f ðx, z, w, uÞ ¼ f1 ðx, z, wÞ þ f2 ðx, z, wÞu and all the related conditions in Theorem 2 are satisfied. Considering ug ¼ Kp ðx, vÞs and ud ¼ Kd ðx, vÞv_, eb becomes zero if f2 ðx, z, wÞKp ðx, vÞs ¼ fe @q f2 ðx, z, wÞKd ðx, vÞ ¼ , @v
x_ ¼ f ðx, s þ ðx, vÞ, w, U þ ug þ ud Þ ¼ f ðx, vÞ þ fgd ðx, s, v, ug , ud Þ y ¼ hðx, wÞ
Also, fgd ðx, s, v, ug , ud Þ is divided into two parts: fgd ðx, s, v, ug , ud Þ ¼ feðx, s, v, ug , ud Þ þ fbðx, s, v, ud Þ, where fbðx, s, v, ud Þ denotes the part that fbðx, s, v, ud Þ ¼ 0 when ud ¼ 0. Then, from Assumption 4, we can set e ¼ x q, which results in
eb ¼
where e gðx, s, v, ug , ud Þ ¼ gðx, s þ ðx, vÞ, w, U þ ug þ ud Þ gðx, s þ ðx, vÞ, w, U þ ug Þ.
gðx, s þ ðx, vÞ, w, U þ ug Þ ¼ As s þ Rs s,
where f ðx, vÞ ¼ f ðx, s þ ðx, vÞ, w, U Þ and fgd ðx, s, v, ug , ud Þ ¼ f ðx, s þ ðx, vÞ, w, U þ ug þ ud Þ f ðx, vÞ:
ð19Þ
With s ¼ z ðx, vÞ, the fast dynamics becomes _s ¼ gðx, s þ ðx, vÞ, w, U þ ug þ ud Þ _ðx, vÞ ¼ gðx, s þ ðx, vÞ, w, U þ ug Þ þ e gðx, s, v, ug , ud Þ _ ðx, vÞ,
361
ð22Þ
where fe¼ f ðx, s þ ðs, vÞ, w, U Þ f ðx, ðx, vÞ, w, U Þ.
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Remark 4. In Theorem 2 and Corollary 1, even when there exists no ug and ud satisfying each conditions to make eb zero, each ug and ud can still be effectively designed to make the bounds of k feð, ug , ud Þ þ gð, ug , ud Þ _ðe þ q, vÞk smalfbð, ud Þ ð@q=@vÞv_k and ke ler, which result in a better tracking performance. In general, these two norm-bound conditions can be used in selecting the ud.
(Zero tracking error case) Consider the following nonlinear singularly perturbed system x_ ¼ 2x þ x2 þ 2z þ u þ wðtÞ z_ ¼ x x2 þ z þ u ð26Þ
where wðtÞ ¼ 0:1 sinðtÞ. Control goal:
Track rðtÞ ¼ 0:5 sin t while rejecting w(t).
First, we have u ¼ Uðx, z, wÞ ¼ x þ x2 z from the fast dynamics. Then, the reduced system is given by x_ ¼ 3x þ 2x2 þ z þ wðtÞ y ¼ x:
2
k31
Tðv, 0Þ ¼ 4 k31 þ1
3.2. Illustrative example
y ¼ x,
u ¼ 3r2 4r þ w þ k11 ðx rÞ þ k12 ðz 3r þ 2r2 þ wÞ þ k3 v_ 2r 7 ðK=Þ, k12 ¼ K 1, where k11 ¼ K ¼ ðð9 3 4rÞ=ðð1=Þ þ 4r 3ÞÞ, and k3 ¼ ½k31 , k32 . Moreover, in choosing the derivative information, the following matrix Tðv, 0Þ should be minimized by some proper values of k31 and k32
ð27Þ
Here, we have qðtÞ ¼ rðtÞ. Now, we choose the slow manifold z ¼ ðx, vÞ ¼ 3r 2r2 4rðx rÞ 2ðx rÞi w, i 0 to satisfy (12). Thus far, letting s ¼ z ðx, vÞ, we have f ðx, vÞ ¼ 2x þ x2 þ 2ðx, vÞ þ Ujz¼ðx, vÞ þ w, fe¼ 2s þ ug , and fb ¼ ud . By linearization of the fast dynamics, we have ð@g=@zÞ ¼ 1, and ð@g=@uÞ ¼ 1. Thus, Kp ðx, vÞ is selected such as Kp ðx, vÞ ¼ 2 to place the eigenvalue at 1, which leads to ug ¼ 2s. Moreover, with this choice, fe turns out to be zero. So far, the obtained controller is u ¼ U 2ðz ðx, vÞÞ þ ud where U ¼ x þ x2 ðx, vÞ. Now, in selecting the derivative information, we have kfeþ fb ð@q=@vÞv_k ¼ kud r_k and ke g _k ¼ kud _k. In our design, we choose ud ¼ r_ðtÞ to make eb ¼ 0. To verify the claims regarding the proposed method, we consider two cases: (1) To see the effect of slow manifold on the transient response, we select two slow manifolds such as 1 ðx, vÞ ¼ 3r 2r2 4rðx rÞ w and 2 ðx, vÞ ¼ 3r 2r2 4rðx rÞ 2ðx rÞ2 w. Note that in both approaches, the eigenvalues are placed at 3 equally. (2) To see the effect of derivative information on the steady state response, for comparison, we use the controller from Lee and Lim (1997) with the assumption that is known exactly. With the system eigenvalues at 3 and 3, we have
k32 1
3
5: k32 3 þ 4r
Hence, we choose k31 ¼ 10 and k32 ¼ ð3 4rÞ. Note that it is impossible to make kTðv, 0Þk ¼ 0, which means we cannot make limt!1 kyðtÞ rðtÞk ¼ 0 by the method in Lee and Lim (1997), even with derivative information. The simulation results of both comparisons are shown in figure 1. By two choices of slow manifolds coupled with gain scheduling controller, we show that the transient response is indeed much different depending on the choice of slow manifold as shown in figure 1(a)–(b). The effect of derivative controller design is shown through a direct comparison with the method of Lee and Lim (1997) in figure 1(c)–(d) where the proposed method achieves a better tracking performance. Another note is that there are existing control schemes using derivative information (Sureshbabu and Rugh 1995, Lee and Lim 1997). However, in these schemes, the full system information is needed including the value of . In our approach, we do not use the value of in controller design as long as < , which leads to simplicity and robustness by taking as an unknown small parameter. (Nonzero tracking error case) Consider the following nonlinear singularly perturbed system x_ ¼ 2x þ x2 z þ u þ wðtÞ z_ ¼ x x2 þ z þ u y ¼ x,
ð28Þ
where wðtÞ ¼ 0:1 sinðtÞ. This system is almost identical to the system (28) except for a slight difference in the slow dynamics. However, this difference actually makes it difficult to design a tracking controller. In fact, in this case, the design of derivative controller becomes more important. Since the design steps are similar, we only show the designed controller components: we have u ¼ U ðx, z, wÞ ¼ x þ x2 z; the selected slow manifold is z ¼ ðx, vÞ ¼ ð3=2Þr þ r2 þ 2rðx rÞ þ ðx rÞ2 þ ðw=2Þ to satisfy equation (12); the gain scheduling controller is ug ¼ 4s to place the eigenvalue at 3. Then, the remaining step is to design a derivative controller ud.
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Gain scheduling control of time-varying systems (a)
(b)
1.5
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φ1(x,w,r) φ2(x,w,r)
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5 time[sec]
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Conventional controller Proposed controller 4
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Figure 1. Comparison when ¼ 1=20: Transient response: (a) plot of tracking error and (b) plot of control input; steady-state response: (c) plot of tracking error and (d) plot of control input.
(a)
(b)
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4 Without ud With Ud
Without ud With Ud 3
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Figure 2.
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Comparison when ¼ 1=20. Steady-state response: (a) plot of tracking error and (b) plot of control input.
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Thus far, with the designed controller components, we have fe ¼ s þ ug ¼ 5s, and fb¼ ud . Now, in selecting the derivative information, we have k feþ fb g _k ¼ kud _k. ð@q=@vÞv_k ¼ k 5s þ ud r_k and ke Unlike the previous case, this time we cannot make eb ¼ 0 by ud ¼ r_ due to 5s. We also need to reduce the effect caused by sb. By considering the structure of , we select ud ¼ 1:5ðw_ þ r_Þ. In figure 2, it is shown that the derivative controller indeed effectively improves the controller’s tracking performance.
4. Conclusions In this paper, we analyse the ultimate boundedness of nonlinear singularly perturbed time-varying systems and propose a control scheme in the context of gain scheduling. In our approach, there are mainly three design components—slow manifold, gain scheduling controller, and derivative information. With the properly chosen design components, the controller can result in better transient response and reference tracking performance over the existing control schemes.
Acknowledgement This work was supported by Satellite Technology Research Center.
References C.C. Chen, ‘‘Criterion for global exponential stabilisability of a class of nonlinear control systems via an integral manifold approach’’, IEE Proc. Contr. Theory Appl., 147, pp. 330–336, 2000. E.F. Costa and V.A. Oliveira, ‘‘Gain scheduled controllers for dynamic systems using sector nonlinearities’’, Automatica, 38, pp. 1247–1250, 2002. M. Innocenti, L. Greco and L. Pollini, ‘‘Sliding mode control for twotime scale systems: stability issues’’, Automatica, 39, pp. 273–280, 2003. H.K. Khalil, Nonlinear Systems, Englewood Cliffs, NJ: Prentice Hall, 2002. P.V. Kokotovic, H.K. Khalil and J. O’Reilly, Singular Perturbed Method in Control, London: Academic Press, 1986. G.A. Lawrence and W.J. Rugh, ‘‘On a stability theorem for nonlinear systems with slowly varying inputs’’, IEEE Trans. Automat. Contr., 35, pp. 860–864, 1990. S.-H. Lee and J.-T. Lim, ‘‘Fast gain scheduling on tracking problems using derivative information’’, Automatica, 33, pp. 2265–2268, 1997. Z. Retchkiman and G. Silva, ‘‘Stability analysis of singularly perturbed systems via vector Lyapunov methods’’, in Proc. 35th IEEE Conference on Decision and Control, Kobe, Japan, 1996, pp. 580–585. W.J. Rugh, ‘‘Analytical framework for gain scheduling’’, IEEE Contr. Sys. Mag., 11, pp. 79–84, 1991. W.J. Rugh and J.S. Shamma, ‘‘Research on gain scheduling’’, Automatica, 36, pp. 1401–1425, 2000. J.S. Shamma and M. Athans, ‘‘Gain scheduling: Potential hazard and possible remedies’’, IEEE Contr. Sys. Mag., 12, pp. 101–107, 1992. P.M. Sharkey and J. O’Reilly, ‘‘Composite control of nonlinear singularly perturbed systems: A geometric approach’’, Int. J. Control, 48, pp. 2491–2506, 1988. Y.-S. Shin and J.-T. Lim, ‘‘Control of nonlinear singularly perturbed systems using gain scheduling’’, IEICE Trans. Fundamentals, E85A, pp. 2175–2179, 2002. N. Sureshbabu and W.J. Rugh, ‘‘Output regulation with derivative information’’, IEEE Trans. Automat. Contr, 40, pp. 1755–1766, 1995.
Ho-Lim Choi received the BSE degree from the department of electrical engineering, The Univ. of Iowa, USA in 1996, and MS degree in 1999 and PhD degree in 2004, from KAIST (Korea Advanced Institute of Science and Technolgy), respectively. Currently, he is a BK21 post-doctoral research fellow at the department of electrical engineering, KAIST. His research interests are in the nonlinear control problems with emphasis on feedback linearization, gain scheduling, singular perturbation, output feedback, time-delay systems.
Jong-Tae Lim received the BSEE degree from Yonsei University, Seoul, Korea, in 1975, the MSEE degree from the Illinois Institute of Technology, Chicago, in 1983, and the PhD degree in Computer, Information and Control Engineering from the University of Michigan, Ann Arbor, in 1986. He is currently a professor in the Division of Electrical Engineering at the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology. His research interests are in the areas of system and control theory, communication networks, and discrete event systems. He is a member of IEEE, KIEE, and KITE.
