1. Introduction
One of the fundamental control problems is tracking control systems, where output signal has to track a ref-erence signal or desired output without steady-state error. Therefore a great deal of interest has been di-rected to the design problem of servomechanisms for linear multivariable dynamical systems (e.g. Davison 1972, Furuta and Komiya 1972). If a controlled system has uncertainties such as unmodeled dynamics, un-known parameter variations and so on, then it is re-quired to achieve robust stability and tracking per-formance. Therefore design problems of robust track-ing control systems for uncertain dynamical systems have been extensively studied (e.g. Schmitendorf and Barmish 1986, Hopp and Schmitendorf 1990). It is usu-ally assumed that desired outputs are modelled by the outputs of some free time-invariant linear systems, whose future values are not available. Thus any servo-mechanism has to utilize instantaneous values of de-sired outputs, errors and so on for control purpose.
By the way, it is well known that when the future in-formation about reference signals and/or disturbances is available, the performance of transient responses will be greatly improved. This kind of control problem, in which information on future is utilized, is called the preview control problem (Tomizuka 1975) and a large
number of design method of preview control systems have been proposed (e.g. Katayama et al. 1985, Fujisaki and Narasaki 1997). Furthermore, some ∞preview control systems (e.g. Cohen and Shaked 1997) and ro-bust preview tracking controllers (e.g. Takaba 1998) have been derived. Cohen and Shaked (1997) consid-ered the problem of ∞preview tracking control for linear time-varying systems. Also, Takaba (1998) de-rived a design method of a state feedback controller with integral and preview actions in terms of linear matrix inequalities (LMIs) for discretetime systems with polytopic uncertainties.
On the other hand in most practical situations, com-plete state information for general multivariable dy-namical systems cannot be utilized. Thus in the case that the full state information of multivariable dynami-cal systems cannot be measured, some observerbased quadratic stabilizing controllers (e.g. Petersen 1985, Jabbari and Schmitendolf 1993), robust ∞controllers (e.g. Iwasaki and Skelton 1994, Park and Bien 1994 ) and robust output feedback control systems (e.g. Ben-ton 1999) have been presented. Furthermore, a design method of observer-based guaranteed cost controllers for uncertain linear dynamical systems has been sug-gested (Oya et al. 2004). However, so far the design problem of observer-based robust preview tracking controllers for uncertain discrete-time systems has lit-tle been considered as far as we know.
From this viewpoint in this paper, we deal with an observer-based robust tracking control problem for
un-Observer-Based Robust Tracking Control Scheme with Preview
Action for Uncertain Discrete-Time Systems
Hidetoshi OYA*
This paper deals with a design problem of an observer-based robust preview control system for uncertain discrete-time systems. In this approach, we adopt 2-stage design scheme and we derive an observer-based robust controller with integral and preview actions such that a disturbance attenuation level or a given performance index is satisfactorily small for allowable uncertainties. In this paper, we show that sufficient conditions for the existence of the observer-based robust preview controller are given in terms of linear matrix inequalities (LMIs). Finally, illustrative examples are in-cluded.
Key words: preview tracking control system, observer-based control, disturbance attenuation level, 2-stage design,
LMIs. Vol. 41, No. 1, 2007
*電気電子工学科 講師
certain discrete-time systems under the assumption that finite future values of reference signals or desired output are available at each time instant. In order to de-rive a simple design method of the observer-based ro-bust preview controller via LMIs framework, we adopt a similar way to the design approach derived by Oya et al. (2004). Namely, the proposed design method is roughly separated into two parts. Firstly, an observer gain matrix is designed and next, a control gain matrix is determined such that a disturbance attenuation level or an upper bound on a given performance index for an augmented dierence system consisting of a tracking error, an observer, an estimation error system and the reference signal is satisfactorily small for allowable un-certainties. In this paper, we show that sufficient condi-tions for the existence of the observerbased robust pre-view tracking controller for uncertain discrete-time systems are given in terms of LMIs.
This paper is organized as follows. In Sec. 2, we de-fine the class of uncertain discrete-time systems under consideration, and introduce an observer, an estimation error system, a tracking error system and an aug-mented system. Sec. 3 contains the main results. Fi-nally, numerical examples are presented to illustrate the results developed in this paper.
In the sequel, we use the following notation. The transpose of a matrix and the inverse of one are de-noted by T and 1 respectively. H
e{} means
Tand diag(
1, · · ·,N) denotes a block diagonal
matrix composed of matrices ifor i1, ···, N. Also, In
represents n-dimensional identity matrix. For real sym-metric matrices and , (resp. ) means that is positive (resp. nonnegative) definite ma-trix. E{ · } and Tr{ · } denote its expectation and its trace, respectively. Furthermore, 2[0,∞) is 2-space (i.e. the collection of all square integrable functions) and for a signal f(t)∈2[0,∞) , f(t)2denotes its 2
-norm.
2. Problem Formulation
We consider an uncertain discrete-time system with the following state space representation.
x(t1)A(q)x(t)B(q)u(t)
y(t)C(q)x(t) (1)
where x(t)∈n; u(t)∈mand y(t)∈lare the vectors
of the state, the control input and the measured output, respectively and the parameter q∈N
(q(q1, · · ·,qN)T) is a constant vector of uncertainties.
Also the matrices A(q), B(q) and C(q) in eq. (1) depend affinely on the parameters qkfor k1, ···, N. That is
(2)
where the matrices A, B and C denote the known nomi-nal values and the matrices Ak, Bkand Ckfor i1, ···, N
represent the structure of the uncertainties. In eq. (2), the unknown parameter qk for k1, ···, N ranges
be-tween known extremal values qk0 and qk0 (i.e.
qk∈[qk, qk]). This assumption means that the
param-eter q∈N belongs to the following parameter box
(Gahinet et al. 1996).
(3) We also assume that for ∀q ∈D the pairs (A(q), B(q)) and (C(q), A(q)) are controllable and observable, respec-tively and the following relation holds.
(4)
The assumption of eq.(4) guarantees that the system (A(q), B(q), C(q)) for ∀q ∈D has no zeros at z1 and it is necessary for the existence of robustly tracking con-troller with integral action.
Let r(t)∈l be the reference signal or the desired
output which is assumed to be previewable. That is, we assume that the h future values of the reference sig-nal r(t) (i.e. r(t1), ···, r(th)) are available at each time
t as well as the present and the past values of the
refer-ence signal and
r¯(t)∈2[0,∞) (5)
In eq. (5), r¯(t) is a difference vector given by
r¯(t)r(t1)r(t). Under the assumption of eq. (5), there
exists a constant value r∞such that
(6) Now in order to estimate the state x(t) for the uncer-tain system of eq. (1), we introduce the following full state observer (Hagino and Komoriya 1989).
xe(t1)Axe(t)Bu(t)Hr( y(t)Cxe(t)) (7) lim ( ) t→∞r t r∞ rank ( ( ( A I B C n l n θ θ θ ) ) ) 0 D{θ∈N|θ ∈[θ θ, ] , ,⋅⋅⋅ } k k k fork 1 N ( ) ( ) ( ) ( ) θ θ θ θ θ A B C A B C A B C k k N k k k 0 0 1 0
∑
where Hr∈
nlis the observer gain matrix. In addi-tion to the observer of eq. (7), we introduce the estima-tion error vector ze(t)
x(t)xe(t), then we see from eqs.
(1) and (7) that the estimation error satises the relation of eq. (8). In eq. (8), Ae(q), Be(q) and Ce(q) are the
matri-ces given by Ae(q)A(q)A, Be(q)B(q)B and
Ce(q)C(q)C, respectively.
Let x¯(t)x(t1)x(t) and u(t)u(t1)u(t) be the
difference state vector and the difference control input vector respectively. Additionally, we introduce the tracking error vector e(t)r(t)y(t) and difference
vec-tors x¯e(t)
xe(t1)xe(t), z¯e(t)
ze(t1)ze(t) and
w¯(t)w(t1)w(t). Since q ∈Nis assumed to be
con-stant, we see from eqs. (1), (7) and (8) that the following relation is satised.
e(t1)e(t)C(q)x¯e(t)C(q)z¯e(t)r¯(t) (9)
Note that it follows from w(t)∈[0, ∞) that the differ-ence vector w¯(t) is a signal of 2-space, i.e.
w¯(t)∈ [0, ∞). Also since h future values of the refer-ence signal r(t1), ···, r(th) are available at time t, we define the difference vector of the reference signal de-scribed by
r¯h(t)(r¯T(t) r¯T(t1) ··· r¯T(th))T (10)
From eq. (10), the difference vector r¯h(t) satisfies
r¯h(t1)Arhr¯h(t)Brhr¯(th1) (11)
where Arhand Brhare the matrices expressed as
(12)
Brh(0 0 ··· Il) T
Furthermore from the denition of the difference vector
r¯h(t), r¯(t) can be written as
r¯(t)Grr¯h(t) (13)
where Gris the matrix given by
Gr(Il 0 · · · 0) (14)
Now we introduce an augmented vector x (t)∈nNh
given by x (t)(eT(t) x¯ e T(t) r¯ h T(t) z¯ e T(t))T where N h nl(h2). Then we obtain x (t1)(q)x(t)(q)u¯(t) w(t) (15) In eq. (15), w (t)r¯(th1) and (q), (q) and are
the matrices described as
(16) 12(q)(C T(q) CT(q)H r T 0)T 21(q)(0 Ae(q)HrCe 0) 22(q)Ae(q)HrC(q) (q)(1 T 2 T(q))T 1(0 B T 0)T, 2(q)Be(q) (17) 1(0 B T 0)
Note that from eqs. (16) and (17), we find that the ro-bust stabilizability of the augmented system of eq. (15) does not depend on the preview length h, because all the eigenvalue of the matrix Arhare stable and r¯h(t) is
uncontrollable by the difference control u¯(t).
It is well known that the integral action of the con-troller is introduced by including the difference control in the performance index (Katayama 1985). There-forewe define the following performance index.
(18) where the weighting matrices er∈
NhNh,
z∈
nn
and r∈
nnare positive definite which can be ad-justed by designers and xer(t) is the vector given by xer(t) (eT(t) x¯ e T(t) r¯ h
T(t))T. It should be noted that the
physical interpretation of the performance index is to achieve the asymptotic tracking without excessive rate of change in the control input.
Using the augmented vector x (t) and the difference control u¯(t), we introduce the following vector.
z(t)x(t)u¯(t) (19) t e T e e eT z e T r x tr rx tr z t z t u t u t 0 ∞
∑
{ ( ) ( ) ( ) ( ) ( ) ( )} 11 0 0 0 0 ( ) ( ) ( ) θ θ θ I C A H C A l r r e rh Γ ( ) ( ) ( ) ( ) ( ) θ θ θ θ θ 11 12 21 22 A I O I O r l l h 0 0 0 0 0 O O where and are the matrices given by
(20)
(0 0 1/2r )T
Then the performance index of eq. (18) is expressed as (21) Now we consider the control given by
(22) where Keris the control gain matrix given by Ker
(Ke
Kxe Krh). From the definition of difference signals and
the augmented vector xer(t), we can obtain the control
input u(t) of eq. (23). Here, we have used the assump-tion that Krh
(Krh(0) Krh(1) · · · Krh(h)) and xe( j )0, u( j )0 for any j0.
(23) Substituting eq. (22) into eq. (15) yields
x (t1)K(q)x (t) w(t)
z(t)Kx (t)
(24) where K(q) and Kare the matrices such that
(25) KKer
From the above discussion, the design problem of the observer-based robust controller with integral and preview actions for the uncertain discrete-time system of eq. (1) is reduced to the design problem of the ob-server-based robust stabilizing controller for the aug-mented system of eq. (15). If the augaug-mented system of eq. (15) is robustly stabilized by a given controller, then by the definition of the augmented vector x (t), we have
e(t)→0 and x(t)x(t1)→0 as time t goes to infinity.
Note that the exogenous signal w (t) is not available for control.
3. Design of the Observer-Based Robust Preview Tracking Controller
In this section, on the basis of the design approach derived in the work of Oya et al. (2004), we consider to design the observer-based robust controller (i.e. the de-sign problem to determine the observer gain matrix Hr
and the control gain matrix Ker) such that the
aug-mented system of eq. (24) is robustly stable with dis-turbance attenuation level g0 for "q ∈D. Now, we firstly give a definition for the observer-based robust controller with disturbance attenuation level g0.
Definition 1 If the augmented system of eq. (24) is robustly stable (internally stable) and the relation
gw(t)2z(t)2 (26) holds for zero initial condition x (0)0, then the aug-mented system of eq. (24) is robustly stable with dis-turbance attenuation level g0.
3.1 Design of the Observer Gain Matrix From eq. (8), the estimation error satisfies the rela-tion of eq. (27).
In this paper, we consider to design the observer gain matrix Hrwhich stabilizes the following system
obtained by ignoring the u¯(t) and x¯e(t) in eq. (27).
lz(t1)(A(q)HrC(q))lz(t) (28)
Now for the uncertain system of eq. (28), we intro-duce the quadratic function H(lz, t)
lz
T(t)l
zlz(t) as
a Lyapunov function candidate where the matrix lz∈
nnis a symmetric positive definite. Then the first order difference D H(lz, t)
H(lz, t1) H(lz, t)
along the trajectory of the system of eq. (28) satisfies D H(lz, t) l z T(t)F lz(q)lz(t) (29) Flz(q)H(qF) T lzH(q)lz H(q)((q)HrC(q)) (30) Therefore if there exist the matrices Hrand lz
sat-isfying the condition Glz(q)0 for "q ∈D, then the
quadratic stability of the system of eq. (28) is ensured. Namely, the quadratic function H(lz, t) becomes a
K e e K K r r ( ) ( ) ( ) ( ) ( ) ( ) θ 11θθ 1θ 12 θθ 21 2 22 u tr K x tx e Ke e j K j r t j j t j h r e h ( ) ( ) ( ) ( ) ( ) 0 1 0
∑
∑
u t K x t K e t K x t K r t e e e x e r h r r e h ( ) ( ) ( ) ( ) ( ) t T t t t 0 2 2 ∞∑
ζ ( ) ( )ζ ζ( ) 0 0 0 0 0 0 0 0 0 0 1 2 1 2 e z r Lyapunov function for the system of eq. (28). However, by introducing a symmetric positive definite matrix lz∈
NhNhand a design parameter
H∈
nnwhich is a symmetric positive definite matrix, we now con-sider the condition of eq. (31) (see Remark 1).
The condition of eq. (31) can be written as eq. (32) at the top of the previous page. In eq. (32), lzis a matrix
satisfying lz
lzHrand zand lz
1are the positive definite matrices given by
lz 1 lz 1 lzlz 1 lz diag(Il,lz, In) (33) If there exist the matrices lz, lzand lzsatisfying
the condition of eq. (32), then quadratic stability of the system of eq. (28) is ensured, because the matrices lz
and H are positive definite, i.e. the quadratic function
H(lz, t) becomes a Lyapunov function.
We now define a set of the 2Nvertices of the
parame-ter box D of eq. (3) such as Dvex
{w∈N|w
k∈{qk,qk} for k1,···, N} (34)
Furthermore using Schur complement formula (Boyd et al. 1994), the design problem of the observer gain matrix Hris reduced to the problem of finding the
ma-trices lz0, lz0 and lz which satisfy the LMI
condition of eq. (35). Thus, if the solution of LMI of eq. (35) exists, then using the solution lz, lzand lz, the
observer gain matrix Hrcan be obtained as
Hrlz
1
lz (36)
3.2 Design of the Control Gain Matrix In the previous section, the observer gain matrix Hr
has been derived. Hence, we consider to design the con-trol gain matrix Ker. Firstly, we shall give a theorem for
robust stability with disturbance attenuation level g for the augmented system of eq. (24).
Theorem 1 The augmented system of eq. (24) is robustly stable with disturbance attenuation level g0 if there exist the control gain matrix Kerand symmetric
positive definite matrix x∈(nNh)(nNh)satisfying the inequality of eq. (37)
Proof: We introduce the quadratic function
K(x , t)
xT(t)
xx (t) as a Lyapunov function candi-date. By evaluating the first order difference of the function K(x , t), i.e. D K(x , t)
K(x , t1)
K(x , t) along the trajectory of the augmented system
of eq. (24) and considering the Hamiltonian (z, w) D K(x , t)z T(t)z(t)g2wT(t)w (t) (38) (31) (32) (35) (37) Φξ( )θ θ ε θ ε ( ) ( ) K K T K n m N K K I h 0 0 0 0 ξ γ θ n m N I h 0 0 2 0 for∀ ∈D λ λ λ λ λ λ λ λ λ θ θ θ θ θ θ θ θ θ z z z z z z z z z H T T T T T T A C C C A C C C ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 ∀ ∈ for Dvex Υ Φ Φ λ λ λ λ λ λ λ λ λ λ θ θ θ θ θ θ θ θ θ θ z z z z z z z z z z H C H C C C C e T e H T ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 1 1 1 λzC( )θ H θ 0 0 for ∀ ∈ Υλ θ Φλ θ λ θ θ θ θ θ z z z C H C C H C r T r H ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 for∀ ∈D
we get the relation of eq. (39). If the inequality condi-tion of eq. (37) is satisfied, then we easily see that the inequality condition of eq. (37) is equivalent to
(z, w)0 for "q ∈D (40) because the relation of eq. (39) can be written as
(41)
Thus we easily see from (1, 1)-block of the inequality of eq. (37) that the augmented system of eq. (24) is ro-bustly stable (internally stable). Moreover for zeroini-tial condition x (0)0, summing up the inequality of eq. (40) yields the following relation.
g2w(t)2 2z(t)
2
2 () (42)
From the above discussion, if the matrices Kerand x
satisfying the condition of eq. (37), then the augmented system of eq. (24) is robustly stable with disturbance attenuation level g. Therefore it follows that the result
of the theorem is true.
Theorem 1 provides a sufficient condition for the existence of the observer-based robust preview track-ing controller with disturbance attenuation level g0. The following theorem provides a design method of the observer-based robust preview tracking controller with disturbance attenuation level g.
Theorem 2 Consider the augmented system of eq. (24) and the controller which is composed of the control law of eq. (23) and the observer of eq. (27).
For a given constant g0, if there exist the solution er0, erand z0 of the LMI of eq. (43), then the
observer-based robust controller consisting of the con-trol gain matrix Kerwhich can be obtained as
Kererer
1 (44)
and the observer gain matrix He which is designed
such as Hrlz
1
lzby solving the LMI condition of
eq. (35) robustly stabilizes the augmented system of eq. (24) and achives the disturbance attenuation level g. In eq. (43), the matrices x, xand K(q)xare the ma-trices expressed as eqs. (45) and (46).
x diag(er,z)x 1 x (er 0) , er Kerer (45) (46)
Proof : By pre- and post-multiplying eq. (37) by the
matrix diag(x, InmNh) and using the matrices x∈
(nNh)(nNh)and
er∈
mNh, simple algebraic
manip-ulation gives the matrix inequality of eq. (48). Further-more by using the Schur complement formula (Boyd et al. 1994), the inequality of eq. (48) is reduced to the fol-lowing condition.
The matrix inequality condition of eq. (47) is LMI in er, z and er because the matrix K(q)x is
ex-pressed as eq. (48). Namely, the matrix inequality con-dition of eq. (46) is equivalent to the LMI concon-dition of eq. (43). Furtheremore, we easily see from eq. (45) that if there exists the solution of LMI of eq. (43), then the control gain matrix Keris obtained as eq. (44).
From the above discussion, the proof of Theorem 2
is completed.
In eq. (43) by setting g *g2, the inequality condi-tion of eq. (43) can be written as eq. (49). The condicondi-tion of eq. (49) is LMIs in er, z, erand g * because in eq.
(49), the parameter g * appears anely. The LMI
condi-K e z e z r r ( ) ( ) ( ) ( ) ( ) ( ) θ ξ 11θθ 1θ 12 θθ 21 2 22 ( , ) ( ) ( ) ( ) ( ) ( ) ζ ω ξ ω θ ξ ω ξ t t t t T Φ (z, w)xT(t)(T K(q)xK(q) T KKx)x(t)He{w T(t) T xK(q)x(t)} wT(t) T x w(t)g 2wT(t)w(t) (39) (43) Ux(q)0 for "q ∈D (47) (48) K T n m N K n m N I h I h ( )θ ε ( )θ ε γ θ ξ ξ ξ ξ ξ ξ 0 0 0 0 0 0 0 1 2 for∀ ∈D Υξ ξ ξ ξ ξ ξ ξ ξ ξ θ θ ε θ ε γ θ ( ) ( ) ( ) 0 0 0 0 0 0 0 2 I I n m N K T T T T T K n m N h h ∀ ∈ for Dvex
tion of eq. (49) also defines a convex solution set of (er,z,er,g *). Therefore various ecient convex
opti-mization algolithms can be used to test whether the LMI of eq. (49) is solvable and to generate particular solutions. Moreover, its solutions parametrize the set of the observer-based robust preview tracking controllers and the parametrized representation can be exploited to design the observer-based robust tracking controller with some additional requirements.
In this paper therefore, we seek to minimize g * sub-ject to the constraint of eq. (49) and the observer gain matrix designed as eq. (36). Namely, we consider to de-sign the control gain matrix which minimizes the pa-rameter g * and then we get the following theorem. Note that the observer-based robust controller minimiz-ing the parameter g * is not optimal but sub-optimal, because we consider the constrained convex optimiza-tion problem of eq. (50) under the observer gain matrix designed as Hrlz
1
lz(Oya et al. 2004).
Theorem 3 Consider the augmented system of eq. (24) and the controller which is composed of the control law of eq. (23) and the observer of eq. (27). Then the observer-based robust controller consisting of the control gain matrix Ker which is designed as Kererer
1and the observer gain matrix H
rwhich is
obtained as Hrlz
1
lzby solving the LMI of eq. (35)
robustly stabilizes the augmented system of eq. (24) with disturbance attenuation performance g (√g *).
3.3 Special Case: r(t+j )r(t+h) for jh+1 In the above, the reference signal r(t) satisfying
r¯(t)r(t+1)r(t)∈2[0,∞) is considered. Now we con-sider a special case that the reference signal r(t) satis-fies the relation r(tj)r(th) for jh1. This control
problem is also considered in the work of Takaba (1998).
From the definition of w (t), if the reference signal r(t) satisfies the relation r(tj)r(th) for jh1, then w (t)¤0. Namely, the augmented system of eq. (24) can be rewritten as
x (t1)K(q)x (t)
z(t)Kx (t)
(51)
Also from the condition of eq. (37), we get the condition K
T(q)
xK(q)K
T
Kx0 for ∀q ∈∆ (52) and for all initial condition x (0), we have the following relation, because w (t)¤0.
xT(0)
x(0) (53)
Furthermore, by introducing the complementary ma-trices x diag( er,z)x 1, x ( er 0) and er
Kerer, and using the Schur complement formula
(Boyd et al. 1994), we find that the condition of eq. (52) is equivalent to the condition of eq. (54).
In this case, we adopt a similar way to observer-based guaranteed cost control (Oya et al. 2004). Namely, we assume that the initial value x (0) is zero mean random vector satisfying E{x (0)xT(0)}I
n+Nh
and E{x (0)}0. Then the upper bound on the perform-ance index of eq. (53) is given as Tr{x}. Thus, we consider the constrained optimization problem.
However, the condition of eq. (54) is LMIs in er, z
and er. Thus we introduce a complementary variable
x∈(n+Nh)(n+Nh)satisfying
(49)
(50)
for "q∈Dvex (54)
(55)
Minimize subject to eq. (54), 0 and 0
ξ, er, z, er[ {ξ}] r Tr e z ξ ξ ξ ξ ξ ξ ξ ξ θ θ K T T T T K n m N I h ( ) ( ) 0 0 0
Minimize subject to eq. (49), and
γ*, , z, er[ *]γ γ* , r e z 0 0 0 ξ ξ ξ ξ ξ ξ ξ ξ θ ε θ ε γ θ 0 0 0 0 0 0 0 I I n m N K T T T T T K n m N h h ( ) ( ) * ∀ ∈ for Dvex
(56) Then the minimization problem of Tr{x} can be transformed into that of Tr{x}. Therefore, the design problem of the control gain matrix to minimize the upper bound on the performance index of eq. (53) is re-duced to the following constrained convex optimization problem, because the condition of eq. (56) is also LMIs in xand x.
Note that the observer-based robust controller minimiz-ing the upper bound on the performance index of eq. (53) is not optimal but sub-optimal, because we con-sider the optimization problem of eq. (57) under the ob-server gain matrix designed as Hrlz
1 lz.
As a result, the following theorem is obtained. Theorem 4 Consider the augmented system of eq. (51) and the controller which is composed of the control law of eq. (23) and the observer of eq. (27).
There exists the control gain matrix Kerminimizing
the upper bound on the performance index of eq. (53), if there exist the optimal solution x0, Wer, er0
and z0 of the constrained convex optimization
prob-lem of eq. (57). Also, using the solution of the LMI con-dition of eq. (35), the observer gain matrix Hr is
de-signed in advance such as Hrlz
1 lz.
If the solution x, Wer, erand zof the constrained
convex optimization problem is obtained, then the con-trol gain matrix is given by Kererer
1.
Remark 1 In this paper, the observer of eq. (7) is designed such that quadratic stability of the system of eq. (28) is ensured and the condition of eq. (31) is satis-fied, because in order to get the control gain matrix Ker
and the symmetric positive definite matrix x satisfy-ing the inequality of eq. (37) or eq. (54), (2, 2)-block of the matrix inequality condition K
T(q)
xK(q)
T
KKx0 for "q ∈D (i.e. (1, 1)-block of the condi-tion of eq. (37) or eq. (52)) mnust be negative definite. Namely, the observer gain matrix Hr has to be
deter-mined, making allowance for the inequality of eq. (58) at the top of the previous page and that is a necessary
condition for the existence of the control gain matrix
Ker and the symmetric positive definite x satisfying
the condition of eq. (37) or eq. (54).
Thus introducing a symmetric positive definite ma-trix lz∈
NhNh and a design parameter
H∈nn,
we consider the condition of eq. (31).
Remark 2 If the augmented system of eq. (24) is robustly stable with disturbance attenuation level g, then for w (t)∈2[0,∞), "x (0) and "q ∈D we have
(59) Remark 3 The result shown in Sec. 3.3 is equiva-lent to the existing result (Oya et al. 2005), i.e. the result derived by Oya et al. (2005) is included as a special case of the result of this paper. Namely, the result of this paper is an extended version of the existing result (Oya et al. 2005) and thus the resulting controllers can be applied to more practical reference signals.
4. Illustrative Examples
In this section, we illustrate the effectiveness of the proposed observer-based robust preview tracking con-troller by the following simple example.
Consider the uncertain discrete-time system with un-known parameters of eq. (60). In eq. (60), the parame-ters d1and d2are the uncertainties and are assumed to take the value within the interval [0.15, 0.15] and [0.10, 0.10], respectively.
(60) By choosing the design parameter HI2and solv-ing the LMI condition of eq. (28), we obtain the follow-ing observer gain matrix.
Hr(1.28941 0.81503)
T (61)
Next, we select the weighting matrices er,zand r
such that erI3, z4.0I2and r1.0 and let h5. By
applying Theorem 3 and solving the constrained convex optimization problem of eq. (50), we get the con-trol gain matrix Kergiven by eq. (62) and the parameter
x t x t u t y t x t ( ) . . . . ( ) . . ( ) ( ) ( . ) ( ) 1 1 0 1 0 0 5 0 85 1 0 1 0 1 0 0 1 2 δ δ lim ( ) t→∞ξ 0t ξ ξ I I n N n N h h 0 (57) (62) Ker K Ke xe Krh Krh Krh Krh Krh Krh ( ( ) ( ) ( ) ( ) ( ) ( )) ( . . . ) 0 1 2 3 4 5 0 04806 0 09330 0 05654 0 04649 0 00137 0 00304 0 00171 0 00070 0 00036
Minimize subject to eq. (54) and (56), 0 and 0
ξ, , , ξ
[ { }]
er z er r
g*29.69464 (i.e. the disturbance attenuation level is g5.44928).
Now we assume xe( j)0, u( j)0 and r( j)0 for any
j0 mentioned above. Also in this example, the
refer-ence signal r(t) is supposed to vary such that (63)
In order to examine the robustness of the proposed controller, we consider four cases such that
• Case 1): d10.15 and d20.1 • Case 2): d10.15 and d20.1 • Case 3): d10.15 and d20.1 • Case 4): d10.15 and d20.1
Namely, the unknown parameters d1and d2take the values of extremal point of the intervals [0.15, 0.15] and [0.1, 0.1 ], respectively.
The results of the simulation of this example are de-picted in Figure 1–5. We see from these figures that the proposed observer-based robust preview controller achieves robust tracking performance and robustly stabilizes the augmented system of eq. (24).
5. Conclusions
In this paper, a design method of an observer-based robust preview tracking control system for uncertain discrete-time systems under the assumption that finite future values of reference signals are available at each
r t t t ( ) . . 0 0 20 10 0 20 for for
Fig. 1. Time histories of the controlled output y(t)
Fig. 2. Time histories of the state x1(t)
Fig. 3. Time histories of the state x2(t)
Fig. 4. Time histories of the control input u(t)
Fig. 5. Time histories of incremental variation of the
time instant has been presented. The proposed ob-server-based robust preview tracking controller is eas-ily obtained through a constrained convex optimization problem, because adopting 2-stage design approach (Oya et al. 2004), the design problem of the observer-based robust tracking controller with preview action is reduced to the LMIs. Therefore, the proposed observer-baed robust controller with integral and preview ac-tions can be easily obtained by using commercially available software such as MATLAB’s LMI Control Toolbox and Scilab’s LMITOOL. In this paper, the ref-erence signal r¯(t)∈2[0,∞) is considered and the ob-server-based robust preview tracking controller is de-termined such that the augmented system is robustly stable with disturbance attenuation level g. Moreover, the special case that the reference signal satisfies the relation r(tj)r(th) for jh1 (i.e. w(t)
r¯(th1)¤0) are discussed. Namely, the existing result
(Oya et al. 2005) is included as a special case of the re-sult of this paper and therefore the rere-sulting controllers can be applied to more practical reference signals.
The future research subject is an extension of the proposed design method of an observer-based robust preview tracking control system to such a broad class of systems as uncertain time-delay systems, uncertain large-scale interconnected systems and so on. Further-more in future work, we will examine the controller de-sign algorithm for the minimization of the disturbance attenuation level g or the upper bound on the perform-ance index for special case mentioned in Sec. 3.3, be-cause the observer-based robust tracking controller de-rived by our design method is not optimal.
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