Robust stabilization of uncertain systems with persistent disturbance
and a class of non-linear actuators
CHIH-CHIN HSUy and I-KONG FONGy*
Linear uncertain systems perturbed by persistent disturbances and driven by actuators subject to band-bounded non-linearities are considered. A set of linear matrix inequality based sufficient conditions are derived for designing state feedback controllers which assure ultimate boundedness of closed-loop state trajectories. The effect of the non-linear actuators is identified to that of the considered persistent disturbances. The control purpose is to minimize the ultimate boundedness region to which the state trajectories are eventually confined, while tolerating disturbances of given or the largest magnitudes. Finally, an experimental system, an inverted pendulum on a Stewart platform making translational movement, is arranged to demonstrate the feasibility and effectiveness of the derived results.
1. Introduction
In practical control systems, the actuators or driving devices like power amplifiers always have inherent non-linearities due to technological or physical constraints. For example, all actuators have a limited operation range. Though many actuators are manufactured to have pretty good linear characteristics over their opera-tion ranges, non-linearities such as deadzone inevitably exist. If these non-linearities are not properly accounted for, they will deteriorate the overall performance or result in instability. Consequently, for many decades control problems with non-linear actuators have attracted considerable interest, and no less recently (Hanrion and Tarbouriech 1999, Henrion et al. 1999, Kapila et al. 1999 a,b, Fong and Hsu 2000).
Among all actuator non-linearities, the saturation is probably the most frequently encountered and studied (Bernstein and Michel 1995, Mahmoud 1997, Kapila and Pan 1999, Stoorvogel and Saberi 1999). As to still other non-linearities like deadzone, hysteresis, backlash, and quantization, commonly seen in hydraulic or elec-tromagnetic devices, general discussions are rare. Traditionally, describing functions are used to approx-imate the frequency domain effects of many kinds of non-linearity, yet the main purpose is to investigate the possible existence of periodic solutions such as limit cycles (Khalil 1992). To ensure system stability, more precise analysis is needed. Individual results usually cover specific types of nonlinearity or use approximations (Choi and Kim 1996, Pare and How 1998).
Besides actuator non-linearities, there are usually unwanted disturbance inputs produced by the environ-ments of control systems that must be taken care of. For example, effects of persistent wind on a flying vehicle need to be compensated by the flight control system. There are plenty of research works about the factor, but most assume that the disturbance has finite signal energy or a certain probabilistic distribution, and adopt the H1=H2 theories to solve the related control
problems. However, some disturbances can not be modelled by the above methods, such as persistent envi-ronmental temperature changes for a process control system. For these cases signal peak value is more appropriate for modelling the signal size, and the optimal ‘1=L1 methods (Vidyasager 1986, Dahleh and Pearson 1987) naturally arise, which tries to minimize the worst-case peak to peak gain of the closed-loop system from exogenous inputs to regulated signals. In the last decade, many authors (Dahleh and Shamma 1992, Diaz-Bobillo and Dahleh 1992, Shamma 1993, 1996, Dahleh and Diaz-Bobillo 1995) have developed various optimal ‘1=L1control methods,
though the proposed procedures for designing the optimal or sub-optimal controllers often need intensive computations, and the resultant controllers are either linear ones with high orders or nonlinear ones. To get simpler results, the optimal ‘1=L1 problems are
sometimes modified a little. In works like Abedor and Nagpal (1996) and Nguyen and Jabbari (1999) the considered signal norm is the supremum of the Euclidean norms of the signal vector at all time instants. This may facilitate the application of the linear matrix inequality (LMI) approach, and gives controller design methods with convenient computation tools.
In this paper, we consider both the effects from actuator non-linearities and external distur-bances. We study the band-bounded non-linearities
International Journal of ControlISSN 0020–7179 print/ISSN 1366–5820 online # 2004 Taylor & Francis Ltd http://www.tandf.co.uk/journals
DOI: 10.1080/0020717042000297207
Received July 2002. Accepted 10 September 2004. * Author for correspondence. e-mail: ikfong@cc.ee.ntu. edu.tw
yDepartment of Electrical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China.
(Hsu and Fong 2001 a), which, unlike the sector-bounded non-linearities, are able to include the dead-zone and hysteresis as special cases. Furthermore, we allow disturbance in different channels to have different bounds. As to the regulated state variables, we adopt the ultimate boundedness control problem formulation, and try to design state feedback controllers which ensure that state trajectories reach and stay within a compact ultimate boundedness region around the origin. For given disturbances, the ultimate boundedness region is minimized. If the problem does not specify the bounds on disturbances beforehand, then the size of the ultimate boundedness region and the bounds of the tolerable disturbance can be obtained simultaneously in some optimal sense. Roughly speaking, this is a variant of the minimization problem of the closed-loop system peak to peak gain. We utilize the LMI approach here, so the proposed state feedback controller design procedure is very straightforward. Finally, we build an experimetal system to illustrate the effectiveness of the proposed approach.
Some notations are defined first. Let R be the field of real numbers, Rn the real vector space of n-tuple vectors, and Rmn the real vector space of m n matrices. For any X, Y 2 Rnn, X Y means that X, Y are symmetric and X Y is positive semi-definite. Similar notations apply to symmetric positive/negative definite matrices. If X > 0, then maxðXÞ denotes its largest eigenvalue. The transpose of a real matrix X is denoted by XT. In is the n n identity matrix, and
ein is the ith column of In. In a symmetric block matrix,
for simplicity the symbol will be used to replace the submatrices that lie above the diagonal. Finally, the
notation Cofg represents the convex hull of the set in the argument.
2. Problem formulation
Consider the uncertain feedback system described by the mathematical model
_ x xðtÞ ¼ AðtÞxðtÞ þ BuðtÞuðtÞ þ BdðtÞdðtÞ, xð0Þ ¼ xo ~ u uðtÞ ¼ KxðtÞ uðtÞ ¼ ½1ðt, ~uu1ðtÞÞ mðt, ~uumðtÞÞT: 9 > = > ; ð1Þ
In (1) the first linear state equation describes the plant to be controlled, where xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the control input vector, and d ðtÞ 2 Rp is the exogenous disturbance input vector. The second equation represents the state feedback controller, where K 2 Rmn is the state feedback gain matrix, and
~ u
uðtÞ 2 Rm is the controller output vector. Finally, the third equation stands for the non-linear characteristics of the actuators, where ~uuiðtÞis the ith component of ~uuðtÞ,
and ið, Þ : ½0, 1Þ R ! R denotes the ith non-linear
characteristic, illustrated in figure 1. Note that the dash– dotted curve (or curves, as the relationship may be time-varying) of ið, Þ is only known to be confined to the
band between the two parallel dashed lines which are characterized with unit slope and bias i. Hence we
call it the band-bounded non-linearity (Hsu and Fong 2001 a). This kind of non-linearity can accommodate characteristics such as backlash, hysteresis, deadzone, and quantization, which are not included by the sector-bounded non-linearity (Khalil 1992). It may be interpreted as that at the ith channel, i ¼ 1, 2, . . . , m, there is a non-linear actuator accepting the ith controller
Figure 1. Actuator input–output characteristic with unity band gain.
output ðeimÞTKxðtÞ and producing the control input ðeimÞTKxðtÞ þuiðtÞ where i uiðtÞ i. In other words, the ith actuator has the input-output character-istic ið, Þ which is only known to be bounded by
~ u
uiiið, Þ ~uuiþi for all ~uui2 R.
Regarding this system, assume that the plant fAðtÞ, BuðtÞg is controllable, and there are two vectors
d, dd 2 Rp such that didiðtÞ ddi for all t and i ¼
1, 2, . . . , p, where di, diðtÞ, and ddi are the ith component
of d, d(t), and dd respectively. Note that there is no matching condition (Lunze 1998) assumption on the disturbance influence coefficient matrix Bd(t), so the
disturbance d(t) may enter the system in a quite unrestrictive way. The constant vectors d, dd may be given or not, and if they are not given, then we wish to determine how large each of their components may be. This assumption allows one to handle disturbances with only the worst case information (Boyd and Barratt 1991). This is important in many real-world applica-tions, since exogenous disturbance are often persistent, i.e. they continue acting on the system as long as the system is in operation.
As mentioned above, we see that u(t) in (2) may be written as uðtÞ ¼ ~uuðtÞ þuðtÞ, and the ith component uiðtÞ of uðtÞ satisfies iuiðtÞ i for all t 0.
This allows us to re-write (1) as _ x xðtÞ ¼AðtÞxðtÞ þ BuðtÞuuðtÞ~ þ½BuðtÞ BdðtÞ uðtÞ dðtÞ " # , xð0Þ ¼ x0 ~ u uðtÞ ¼KxðtÞ: 9 > > > > = > > > > ; ð2Þ
Now we simplify (2) by re-defining some notations: ~uuðtÞ is replaced by u(t), ½BuBd is replaced by Bw,
½uTðtÞ dTðtÞT is replaced by w(t), and the dimension m þ p of the vector ½uTðtÞ dTðtÞT is replaced by p, where p > m. After these modifications, we can consider the feedback control system modelled by
_ x xðtÞ ¼ AðtÞxðtÞ þ BuðtÞuðtÞ þ BwðtÞwðtÞ, xð0Þ ¼ x0 uðtÞ ¼ KxðtÞ: ) ð3Þ
where wðtÞ ¼ ½w1ðtÞ wpðtÞ and wiwiðtÞ wwi. Note
that here wwi¼ wi¼i for i ¼ 1, 2, . . . , m, and wwmþj,
wmþj, respectively, equal to ddj, dj for j 1. Clearly, by
doing the above re-formulation, we show that the effect of band-bounded non-linear actuators is the same as that of exogenous disturbances, and may be considered together. Consequently, the assumption about wiðtÞin (3) may be expressed in the convex form
wiðtÞ ¼ ½1 iðtÞ wwiþiðtÞwi ð4Þ
where 0 iðtÞ 1 for all t and i ¼ 1, 2, . . . , p. With PðtÞ ¼ diagð1ðtÞ, . . . , pðtÞÞ, we have a more compact
expression
wðtÞ ¼ ½IpPðtÞ ww þPðtÞw: ð5Þ
Let fP1, P2, . . . , P2pg be the set of 2pdistinct diagonal
p pmatrices whose diagonal elements are either 0 or 1, and let the convex hull (Horn and Johnson 1985) of the set be denoted by CofP1, P2, . . . , P2pg. With (5), the
open-loop system of (3) can be represented by _
x
xðtÞ ¼ AðtÞxðtÞ þ BuðtÞuðtÞ
þBwðtÞf½IpPðtÞ ww þPðtÞwg ð6Þ where PðtÞ 2 CofP1, P2, . . . , P2pg for all t.
The parameters of the system (11) are assumed to subject to the polytopic uncertainty
AðtÞ BuðtÞ BwðtÞ
½
2Cof½A1 Bu1 Bw1, . . . , ½Aq Buq Bwqg: ð7Þ
Facing the parameter uncertainty, actuator non-linearities, and persistent disturbances, we intend to find state feedback controllers to achieve ultimate boundedness control for the closed-loop system (Corless and Leitmann 1981, Garofalo et al. 1989), which basically means making every state trajectories of (1) enter a neighbourhood (the ultimate boundedness region) of the origin of the state-space eventually. Here, we consider the ellipsoidal neighbourhood Ec¼ fxTPx cg for some P > 0 and c > 0. Of course, it is the best if the ‘size’ (with any acceptable definition) of Eccan be guaranteed to the smallest while bounds j ddij
and jdij, i ¼ 1, 2, . . . , p, of d(t) in (1) are as large as possible if they are not given beforehand.
We end this section by presenting one lemma which will be useful subsequently.
Lemma 1 (Xie 1996): For any real matrix function F(t) of t such that FTðtÞF ðtÞ I for all t, and real constant matricesS1,S2, andS > 0 with matching dimensions, we
have
T1FðtÞ2þT2FTðtÞ1T11þT212, for all t:
3. The main results
The following theorem is our main result which pose LMIs for the polytopic uncertain system. Feasible solutions of the LMIs in the theorem give state feedback gains for the ultimate boundedness control purpose.
Theorem 1: Consider the uncertain system(3) perturbed by bounded disturbances(4) and polytopic uncertainty (7). For a given o >0, if there exist matrices Q 2 Rnn,
Y 2 Rmn, vectors ww 2 Rp, w 2 Rp, and real scalars 11 , ^ satisfying the LMI conditions
Q ^In>0, 1 > 11>0, 11þmaxðQÞ <2o ð8Þ eipðww wÞ > 0, 8i ¼1, 2, . . . , p ð9Þ j Tjl ^ Q 0 11In 2 6 4 3 7 5 < 0, 8j ¼1, 2, . . . , q and l ¼ 1, 2, . . . , 2p ð10Þ where j¼AjQ þ QATj þBujY þ Y T BTuj and jl¼
Bwjf½IpPlww þ Plwg, then uðtÞ ¼ KxðtÞ ¼ YQ 1
xðtÞ makes every state trajectories of system (3) converge to and stay within the ultimate boundedness region E1
1 ¼ fx 2 R nj
xTPx ¼ xTQ1x 11<1g eventually,
and the length of the major semi-axis of E1 1 is less
than 0.
Proof: For Ec¼ fx 2 RnjxTPx cgwith P > 0 to be
an ultimate boundedness region of the closed-loop system (3), it suffices to ensure that the time derivative of the Lyapunov function candidate vðxÞ ¼ xTPx is negative along all state trajectories of (3) outside Ec
for all considered w(t). Here we require dv½xðtÞ=dt to be negative outside a ball B ¼ fx 2 Rn j xTx g
which is contained by the set Ec for all considered
w(t). This can be implied by the S-procedure (Boyd et al. 1994), or by the existence of 1>0 such that
xTðtÞ½ATðtÞP þ PAðtÞ þ PBuðtÞK þ KTBTuðtÞPxðtÞ
þ fwwT½IpPðtÞ þ wTPðtÞgBTwðtÞPxðtÞ
þ xTðtÞPBwðtÞf½IpPðtÞ ww þPðtÞwg
1½ xTðtÞxðtÞ <0 ð11Þ
for all ½AðtÞ BuðtÞ BwðtÞ 2 Cof½A1Bu1Bw1, . . . , ½Aq Buq
Bwqg, PðtÞ 2 CofP1, P2, . . . , P2pg, and t 0. The
con-dition (11) is equivalent to xTðtÞ½ATjP þ PAjþPBujK þ K T BTujPxðtÞ þ f ww T ½IpPl þwTPlgBTwjPxðtÞ þ xTðtÞPBwjf½IpPlww þ Plwg 1½ xTðtÞxðtÞ <0 ð12Þ for all t 0, j ¼ 1, 2, . . . , q, and l ¼ 1, 2, . . . , 2p. In the matrix form with the new variable ^ ¼ 1 >0, the
condition (12) may be expressed as ATjP þ PAjþPBujK þ K T BTujP þ 1In fwwT½IpPl þwTPlgBTwjP ^ " # <0: ð13Þ
Multiplying (13) from the left- and right-hand sides by diagðP1, 1Þ, and letting Q ¼ P1>0, Y ¼ KQ, further transform (13) into QATj þAjQ þ BujY þ Y T BTujþ1Q 2 fwwT½IpPl þwTPlgBTwj ^ " # < 0: ð14Þ It follows from the Schur complement (Boyd et al. 1994) that (14) is equivalent to (10). The condition B¼
fx 2 RnjxTx g Ec¼ fx 2 Rn j xTPx cg is
satis-fied by setting c ¼ 11 and requiring accordingly
Q ^In>0. In addition, the major semi-axis length of
Ecequals ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 maxðQÞ
p
, which will be no greater than o provided (8) holds, since
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
1 maxðQÞ
p
12½11þ
maxðQÞ. As to (9), it is easy to see that magnitudes
for different disturbances should satisfy (9) under the assumptions. This concludes the proof. œ Closer inspection of the main LMIs (8), (9), and (10) reveals that the Theorem 1 is also applicable in the situation where the only information of fAðtÞ, BuðtÞ,
BwðtÞg is known, because even if ww, and w are viewed
as variables, (8), (9) and (10) are still LMIs. Naturally, it would the best if we could set an objective function for LMIs (8), (9) and (10) to form an optimization problem, enabling us to find the state feedback controller which results in the smallest ultimate boundedness region while tolerating the largest disturbance. Thus we form the following convex optimization problem
min 11þ kYk X p j¼1 ðwwiwiÞ subject to ð8Þ, ð9Þ, and ð10Þ 9 > = > ; ð15Þ
where the two-norm of Y is included in the objective function to indirectly minimize the controller gain K ¼ YQ1, while 11 is for reducing the size of E1
1 .
Finally, the last two terms reflect our desire to withstand disturbances of the largest magnitudes. If in the problem to be studied the disturbance magnitudes ww and w are given beforehand, then in Theorem 1, the main LMI (10) is with respect to less number of variables, and the term Ppi¼1ðwwiwiÞ may be taken out from the
objective function (15). Of course, proper weightings can also be multiplied to individual terms when deemed necessary.
4. Stabilization of an inverted pendulum on a Stewart platform
This section presents a practical application of the theory developed in } 3 to the stabilization of an inverted pendulum on a Stewart platform (Bhaskar and Mruthyunjaya 2000). A photograph of the experimental equipment is shown in figure 2. As clearly C.-C. Hsu and I.-K. Fong
displayed in the photograph, the inverted pendulum itself has only single degree-of-freedom, but is mounted on a Stewart platform, which is the most celebrated parallel manipulator capable of performing six degree-of-freedom movement (Hsu and Fong 2001 b). The inverted pendulum system includes a ballscrew table cart and a stick freely pivoted at the cart. To balance the stick, the cart is moved back and forth by using an electric motor to drive the ballscrew table. Besides, optical encoder sensors are mounted at appropriate places to measure the angle between the stick and the vertical line, as well as rotation angle of the ballscrew, which is then converted to the position of the cart. Velocity of the cart and the angular velocity of the stick are obtained indirectly by computing the finite derivatives of the sensor outputs. The goal here is to design and test controllers that will stabilize the inverted pendulum at its vertical position, despite the presence of actuator non-linearities, or disturbance and uncertainty caused by the translational movement of the Stewart platform.
4.1. Mathematical model of the experimental system A schematic diagram of the inverted pendulum system on the Stewart platform is shown in figure 3. Here, we attach frames fPg and fBg with origins OP
and OB to the centroids of the upper platform and the
lower base of the Stewart platform, respectively. The unit vectors of the frames fPg and fBg are fiP, jP, kPg
and fiB, jB, kBg, respectively. The relative position of the
origin of the frame fPg with respect to that of the frame fBg can be described by O!BOP¼xPiBþyPjBþzPkB.
Because in our experiment the upper platform will only move translationally, the three Roll–Pitch–Yaw
Euler angles often used to characterize the orientation of the upper platform will not be needed in this study. Therefore, we have iP¼iB, jP¼jB, and kP¼kB. The
ballscrew table is placed to align with the iPaxis on the
upper platform, and the centroids of both objects are assumed to coincide when the table is reset to the middle position.
In figure 3, the displacement of the table cart from the centroid OP is denoted by xc. Let rcartðtÞ be the
displacement vector of the cart from OP, then
rcartðtÞ ¼ xciP¼xciB: ð16Þ Let the position vector of the stick from OPto its centre
of mass be rstickðtÞ, then
rstickðtÞ ¼ ½xcþls sinð Þ iPþls cosð Þ kP
¼ ½xcþls sinð Þ iBþls cosð Þ kB ð17Þ
where lsdenotes the half length of the stick, and is the
angle between the vertical line and the stick. In the mechanical system, the table cart is constrained to move only in the iP direction on the track of the
ball-screw, so any forces acting in the jP and kP directions have no effect on the motion of the cart. If the effect of friction is neglected, then the differential equation describing the motion of the cart can be obtained by summing all forces in the iP direction. Thus we have
mc
d2
dt2rcartðtÞ ¼ ðf fhþmcxx€PÞiP¼ ðf fhþmcxx€PÞiB
ð18Þ where mc is the mass of the table cart, f is the
force applied to the cart in the iP direction, and fh is
the reaction force from the stick in the iP direction. Figure 2. Photograph of the inverted pendulum system on a Stewart platform.
Similarly, applying Newton’s Second Law of Motion to the stick, we get
msd
2
dt2rstickðtÞ ¼ ðfhþmsxx€PÞiPþ ðfvmsg þ ms€zzPÞkP
¼ ðfhþmsxx€PÞiBþ ðfvmsg þ ms€zzPÞkB
ð19Þ where ms is the mass of the stick, g is the gravitational
acceleration, and fvis the reaction force from the cart in
the jPdirection. From the rotational motion of the stick, one more equation of motion can be derived. Summing the moments about the stick’s centre of mass yields
fvlssinð Þ fhlscosð Þ ¼
1 3msl
2
s € ð20Þ
where ð1=3Þmsl2s is the stick’s moment of inertia about its
mass centre. By combining (18), (19) and (20), fhand fv
can be eliminated to yield
ðmcþmsÞxx€cþmsls cosð Þ € ¼ f þ ðmcþmsÞxx€P þmslssinð Þ _ 2 ð21Þ 3 4ms cosð Þ €xxcþmsls ¼€ 3 4msgsinð Þ 3 4ms€zzP sinð Þ þ3 4msxx€P cosð Þ: ð22Þ Equations (21) and (22) form a set of non-linear differ-ential equations that describe the motion of the inverted pendulum system on a Stewart platform mak-ing only translational movement. The set of equations may be linearized provided the stick is not allowed to fall too far from the vertical line, and to rotate too quickly. Under these conditions, sinusoidal terms in
the equations can be approximated by sinð Þ ffi and cosð Þ ffi 1, and the square term can be approximated by _ 2ffi0. Thus (21) and (22) are approximated by ðmcþmsÞxx€cþmsls ¼ f þ ðm€ cþmsÞxx€P, ð23Þ 3 4msxx€cþmsls ¼€ 3 4msg 3 4ms€zzP þ 3 4msxx€P: ð24Þ Let x1¼xc, x2¼xx_c, x3¼ , and x4¼ _ , then (23)
and (24) can be cast into the state equation
_ x x1 _ x x2 _ x x3 _ x x4 2 6 6 6 6 4 3 7 7 7 7 5¼ 0 1 0 0 0 0 3msg ðmsþ4mcÞ þ 3ms€zzP ðmsþ4mcÞ 0 0 0 0 1 0 0 3ðmsþmcÞg ðmsþ4mcÞls 3ðmsþmcÞ€zzP ðmsþ4mcÞls 0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 x1 x2 x3 x4 2 6 6 6 6 4 3 7 7 7 7 5 þ 0 4 ðmsþ4mcÞ 0 3 ðmsþ4mcÞls 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 f þ 0 € x xP 0 0 2 6 6 6 6 4 3 7 7 7 7 5: ð25Þ
In (25), it is seen that the bounded accelerations €zzP in the direction of kB and €xxP in the direction of iB generated by the motion of the Stewart platform introduce parameter uncertainty and exogenous distur-bance, respectively, to the state equation. As to the input term f, it is mentioned above that the cart is driven by an electric motor, actually an AC servo motor, con-nected to a ballscrew table, which transforms the motor torque into the driving force f. Hence, it
Figure 3. Schematic diagram of the inverted pendulum system on a Stewart platform.
is more convenient to use the torque generated by the AC servo motor instead of the force f as the control input variable. Let the screw drive patch be pd (m/rev). According to the principle of conservation
of mechanical energy, we have
f ¼ b2 pd
ð26Þ
where b is the efficiency coefficient and is usually
assumed to be between 0.8 and 0.9. Thus, equation (25) can be re-written as _ x x1 _ x x2 _ x x3 _ x x4 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ¼ 0 1 0 0 0 0 3msg ðmsþ4mcÞ þ 3ms€zzP ðmsþ4mcÞ 0 0 0 0 1 0 0 3ðmsþmcÞg ðmsþ4mcÞls 3ðmsþmcÞ€zzP ðmsþ4mcÞls 0 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 x1 x2 x3 x4 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 þ 0 4 ðmsþ4mcÞ ð2b pd Þ 0 3 ðmsþ4mcÞls ð2b pd Þ 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 þ 0 € x xP 0 0 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 : ð27Þ
State equation (27) is the linear model representing our plant, which will be driven by the actuator consisting of an AC servo motor and a power amplifier. The motor driver is operated in the torque mode, which means that the torque generated by the motor is supposed to be proportional to the input voltage to the driver. However, deadzone non-linearity exists between driver input and motor torque output due to mechanical friction and electrical factors. The width of the deadzone can be measured by applying small voltage to the motor driver incrementally until the ballscrew starts to move. The measured threshold voltage is 0.06 V, which corresponds to a torque control input of magnitude 0.0228 kg-m2=s2.
For this physical system, the cart is only allowed to move within a range of length 0.30 m due to finite ball-screw track length. This will restrict the magnitude of the state variable x1. Therefore, stabilizing state
feedback gains designed by conventional methods, such as the pole-placement method, may be affected by the non-linearities or may violate the constraint. Here, we will look for state feedback laws to stabilize the system despite the existence of actuator non-linearities, exogenous disturbance, and parameter uncertainty. Hence for this experimental system, it is expected to see that the stick is kept upright with some small swings, and the cart is restrained to move back
and forth inside a narrow neighbourhood of its reset position. To facilitate future reference, we list some parameters values of the inverted pendulum system in table 1.
4.2. Stabilization of the inverted pendulum
Comparing the system models (3) and (27), we see that x 2 R4, u ¼ , the motor torque output, and
AðtÞ ¼ 0 1 0 1 0 0 3msg ðmsþ4mcÞ 3ms€zzPðtÞ ðmsþ4mcÞ 0 0 0 0 1 0 0 3ðmsþ4mcÞg ðmsþ4mcÞls þ3ðmsþ4mcÞ€zzPðtÞ ðmsþ4mcÞls 0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 , BuðtÞ ¼ 0 4 ðmsþ4mcÞ 2b pd 0 3 ðmsþ4mcÞls 2b pd 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 BwðtÞ ¼ 0 0 4 ðmsþ4mcÞ 2b pd 1 0 0 3 ðmsþ4mcÞls 2b pd 0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 : ð28Þ
Note that the above Bw(t) has two columns, though
in (27) the only exogenous disturbance is €xxPðtÞ. This
is due to how we treat actuator non-linearities. The deadzone characteristic of the motor and driver is regarded as a kind of the band-bounded non-linearities. The effect of the deadzone on the control input, denoted as uðtÞ, is replaced by an equivalent disturbance input w1ðtÞ, which together with the
‘‘true’’ exogenous disturbance w2ðtÞ ¼ €xxPðtÞ form the
disturbance vector wðtÞ 2 R2. For w1ðtÞ, we have
jw1ðtÞj ww1¼ ¼0:0228 kg-m2=s2, while for w2ðtÞ we
have jw2ðtÞj ww2¼0:0395 m=s2, because horizontally,
Parameter Notation Value Unit
half stick length ls 0.16 m
stick mass ms 0.48 kg
cart mass mc 5.73 kg
ballscrew drive pitch pd 0.02 m/rev
ballscrew track length — 0.30 m
ballscrew drive efficiency b 0.9 —
Table 1. Various parameter values of the inverted pendulum system.
the upper platform of Stewart platform is commanded to make the simple harmonic motion xPðtÞ ¼ am sinð2t=TmÞ and yPðtÞ ¼0, where
am¼0:1 m and Tm¼10 s. If we let ww ¼ ½ ww1 ww2T, then
it is possible to write
wðtÞ ¼ ½I22 PðtÞ ww ð29Þ
where PðtÞ 2 CofP1, P2, P3, P4g, and fP1, P2, P3,
P4g is the set of four distinct diagonal 2 2 matrices
with either zero or unity diagonal elements. In addition to the horizontal movement, the upper platform of the Stewart platform is commanded to make a vertical simple harmonic motion zPðtÞ ¼0:8 þ am sinð2t=TmÞ
(m). Hence from the parameter values listed in table 1 and (28), we have AðtÞ BuðtÞ BwðtÞ ½ 2Co 0 1 0 0 0 0 0 0 0 0:606 0 48:33 48:33 1 0 0 0 1 0 0 0 0 0 48:96 0 226:56 226:56 0 2 6 6 6 6 4 3 7 7 7 7 5 8 > > > > < > > > > : , 0 1 0 0 0 0 0 0 0 0:6 0 48:33 48:33 1 0 0 0 1 0 0 0 0 0 48:57 0 226:56 226:56 0 2 6 6 6 6 4 3 7 7 7 7 5 9 > > > > = > > > > ; : ð30Þ Note that the units for the linear and angular displacements are (m) and (rad), respectively.
Up to now a mathematical model in consistent with the one formulated in a } 2 is derived. However, practically all state and control input variables in physi-cal systems have upper bounds on their magnitudes. In this particular physical system, through some test runs it is discovered that the short ballscrew track length of 0.30 m is the most restrictive, as it limits the traveling range of the cart. Hence, the following state constraints xðtÞ 2 Sðh, Þ ¼ fx 2 R4 j hTix i, i ¼ 1, 2g ð31Þ with h1¼ ½1 0 0 0T, h2¼ ½1 0 0 0T, and 1¼ 2 ¼
0:15 must be accommodated. To be able to do so, the set invariance concept (Blanchini 1999) is introduced here because in the literature (Boyd et al. 1994) it is successfully utilized in the quadratic stabilization problem subject to the same type of state constraints. More specifically, an ellipsoid positively invariant set P ¼ fx 2 Rnj xTPx <1g is considered here, and it is sufficient to require that P be contained by Sðh, Þ, or equivalently to augment the following LMI
hTiQhi 2i, i ¼1, 2 ð32Þ
where Q ¼ P1>0, to the LMIs in Theorem 1.
Corollary 1: Consider the polytopic uncertain system(3) with (30) that is subject to the state constraints (31) as well as the bounded disturbances(29). For a given o>0,
if there exist matrices and scalars Q 2 Rnn, Y 2 Rmn, 11 2 R, and ^ 2 R satisfying the LMIs (8), (10) and
(32), then uðtÞ ¼ KxðtÞ ¼ YQ1xðtÞ makes every state tra-jectories of closed-loop system originating from the set fxo2 RnjxToPxo¼xToQ1xo 1g converge to and stay
within the ultimate boundedness region E1 1 ¼
fx 2 RnjxTPx ¼ xTQ1x 11<1g eventually. Also,
the length of the major semi-axis of E1
1 is less than 0.
Based on the feasibility problem stated in the above corollary, a convex optimization problem
min $111þ$2kY k
subject to ð8Þ, ð10Þ, and ð32Þ )
ð33Þ
is formed. In a representative application case we set o¼20, $1¼50, and $2¼1. The resultant optimal
solution of (33) is K¼4:78 11:38 21:39 4:44, P¼ 64:97 16:21 41:73 3:6 16:21 20:06 35:19 4:72 41:73 35:19 87:73 8:26 3:6 4:72 8:26 1:18 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ^¼0:0075, 11 ¼0:2:
Figure 4 shows the closed-loop system responses corresponding to the initial conditions xo¼
½0:117 0 0:014 0:128T in the presence of the translational movement xPðtÞ ¼0:1 sinð2t=10Þ m and zPðtÞ ¼0:8 þ 0:1 sinð2t=10Þ m of the upper platform of the Stewart platform. It is seen that the ultimate boundedness control is achieved.
5. Conclusions and discussions
For a class of linear uncertain systems subject to persistent disturbances and driven by the band-bounded non-linear actuators, an LMI based robust state feedback controller design method is proposed to guarantee ultimate boundedness of state trajectories. The problem formulation is flexible in that the effect of band-bounded nonlinear actuators may be accommodated like that of exogenous disturbances, and that every disturbance may have theirs own bounds. Moreover, an inverted pendulum on a Stewart platform making translational movement is introduced, and its corresponding constrained control problem is discussed and solved by the proposed method. The experimental system can also be used as a good test bench for other controller design methods.
It would be a challenge to stabilize the inverted pendulum on the Stewart platform making 6-DOF movements, and it is conjectured that more information, such as the rates of three Euler angles of the Stewart platform, which can be measured indirectly by three gyros, will be needed. Because this problem is similar to the stabilization of missiles vertically launched on a ship, it worths further investigations.
Acknowledgement
This research is supported by the National Science Council of the Republic of China under Grant NSC 89-2213-E002-088.
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