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A TIME DELAY CONTROLLER FOR SYSTEMS WITH UNKNOWN DYNAMICS


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A TIME DELAY CONTROLLER FOR SYSTEMS WITH UNKNOWN DYNAMICS

Kamal Youcef-Toumi, Assistant Professor Osamu Ito, Giuate Student Departe of Mechanical Engineering Laboat

ory for Manuacturing and Productivity Massachusetts Instie of Technology Cambridge, MA 02139 U.S.A. estimation of specific parameters, discontinuous control, ABSTRACr nor repetative actions. Rather, it depends on the direct estimation of the effect of uncertainties. This is This paper focuses on the control of systems with unknown accomplished using ime delay. The gathered information is dynamis and deals with the cla of systems described by used to cancel the unknown dynamics and the unexpected x = f(x,t)+h(x,t)+B(x,t)u+d(t) where h(x,t) and d(t) are disturbances simultaneously. Then, the controller inserts the unknown dynamics and unexpected disturbances, desired dynamics into the plant In other words, the TDC respectively. A new control method, 'rme Delay Control uses past observation of the systen's response and the sed for such systems. Under the assumption (IDC), control inputs to directly modify the control actions rather of accessibility to all the state variables and their than adjusting the controller gains. This algorithm can deal derivatives, the TDC is characterized by a simple with large unpredictable system parameter variations and estimation technique of the effect of the uncertainties. This disturbances be unbounded. Yet, the system's performance is accomplished using time delay. The control system's is very satisfactory. structure, stability and design issues are discussed for linear In Section 2, the control problem is defined and the time-invariant and single-input- single-output systems. Time Delay Control algorithm is presented. Section 3 Finally, the control performance was evaluated through discusses the system's structure, stability analysis and both simulations and experiments. The theoretical and design issues for linear dme-invariant and single-inputexperimental results indicate that this control method shows single-output systems. Section 4 is an evaluation part of the excellent robustness properties to unknown dynamics and paper. Effectiveness of the TOC system is evaluated disturbances. through both simulations and experiments. Finaly, in 1. INRODUCTION Section 5, the results are suarized and the directions of the future research ae suggested. Most of the well-developed control theory, either in 2. DERIVATION OF THE CONTROL LAW the frequency domain or in the time domain, deals with systems whose mathematical rpsentations are completely 2.1 Error Dynamics and Structural Constraint known. However, in many practical situations, the parameters of the system are ither poorly known or operate The nonlinear systems considered in this paper are in environments where unpredictable large system descibed by the following dynamic quations. parameter variations and unexpected disturbances are possible. Underwater vehicles, robot manipulators, and i = f(x,t)+h(x,t)+B(x,t)u+d(t) (2.1) autonomous systems are a few examples. In such situations, the usual fixed-gain controller will be inadequate to achieve where x is an nxl plant state vector, u is an rxl control satisfactory perfrmance in the entire range over which the vector, B(x,t) is an nxr control disribution matrix with rank characteristics of the system may vary [81 . r, f(x,t) and h(x,t) are nxl nonlinear vectors representing Several advanced control techniques have been respectively known and unknown part of the plant developed for such systems. One of the primary methods is dynamics, and d(t) is an unknown disturbance vector. The Adaptive Control [3,6,10,121. In Adaptive Contrl, the variable t represents time. This expression is common in structure of the controller is selected a priori, usually PD or many applications. For example, in robot manipulators, PH) type controller. The controller gains are then updated h(x,t) can correspond to nonlinear torques caused by that of so using a recursively estimated parameters the plant Coriols and centriugal effects or nonlinearities such as dry the plant output closely follows the desired response. As friction at each joint. The vector d(t) represents any kdnd of stated in [5,7] , this method considers slowly varying disturbance such as extemal torques. The objective of this bounded and/or parameters, linear dynamic equations research is to be able to control such systems and guarantee uncertainty. performance despite the presence of large dynamic vmriatons Sliding Mode Control [11,14,18] is another in h(x,t) and large unexpected disturbances in d(t). In this powerful method which can deal with nonlinear systems. it is assumed that all the state variables and their paper, Based on Lyapunov's method, the control scheme is derivatives are accessible. Also, this paper will address characterized by a discontinuous function with high issues when the control distribution matrix B is known. frequency chattering. The plant parameter variations and Let us defne the reference model to be followed as a disturbaces are assumed bounded. linear time-invariant system given by Learning Contl [1,13] is an approach which is based on trial and error. Each time the system performs the Xm = Amxm+Bmr (2.2) same task, data is collected and used to update the control action. By repeating this process several mes, betterment where xm is an nxl model state vector, Am is an nxn in performance is obtained. Therefore, this approach is constant stable system matrix, Bm is an nxr constant restricted to repetative tasks only. command distribution matrix, and r is an rxl command This paper proposes another control method, Time vector. The error vector, e, is defined as the difference Delay Control (TDC) [15], which depends on neither between the plant and the model stae vectors:

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e = xm-x (2.3) The control objective is to force the eror to vanish with a desire dynamois: 6 = Aee (2.4) where Ae is an nxn ernro system matix, which defines some desirable dynamics. By combining Eqs. (2.1) through (2.3), one obtains an equation that governs the error dynamics. e = Ame+(-f-h-d+Amx+Bmr-Bu} (2.5) If it is possible to determine a control u of Eq. (2.5) such that the following equation is always met (2.6) -f-h-d+Amx+Bmr-Bu = Ke then, substitution of Eq.(2.6) into Eq.(2.5) leads to 6 = (Am+K)e = Ace (2.7) where K is an nxn error feedback matrix. In the above equation, the error system matrix Ae can be arbitrily determined through proper choice of the er feedbackin matrix K. The control signal u, that satisfies Eq.(2.6), must then be selected in order to obtain a desired error dynamics. However, Eq.(2.6) cannot always be satised because the number of controls is generally smaller than the number of the states. Thus, a best apoximat solution of the equation is adopted to detmine the control u: u = B+(-f-h-d+Anx+Bmr-Ke) (2.8) where B+ = (BTB)-IBT and is known as a pseudo-inverse matrix. Note that the matrix BTB is an nxr nonsingular matrix since B is of rank r. The condition for which Eq.(2.8) exactly satisfies Eq.(2.6) must be determined, sne this is not always the case. To find this condition, Eq.(2.8) is substituted into Eq.(2.1) to obtain k = f+h+BBt(-f-h-d+Amx+Bmr-Ke)+d (2.9) and tugh some algebiaic pulation, given in Appendix L the err dynamics is given by E = (Am+K)e + (1- BB+)(-f-h-d+Amx+Bmr-KeC (2.10) In order to obtain the desired error dynamics given by Eq.(2.7), the following s ral constraint must then be met so ta the earrx vanishes as time goes to infinity. (2.11) (1-BB+) (-f-h-d+Amx+Bmr-Ke) = 0 constaint of If B is nn and B-1 exists, thees Eq.(2.1 1) is always satisfied since I-BB+ = I-BB-1 is a zero matix. If not, the choice of the reference model and the error feedback gain matrix is somehow resticted since the above constraint equation contains Am, Bm and K. Moreover, some elements of unknown dynamics vector h(x,t) and unexpected disturbance vector d(t) should be known in order to ensure the system to satisfy the above constraint It is shown in Appendix H that this condition is always satisfied for systems expressed in caonical form. It must be noted that I-BB+ has rank of n-r, and thus Eq.(2.11) effectively consists of n-r constrint equations instead of n constraints. On the other hand we would like to completely control n states using r controls. So, the structural constraint simply indicates that r inputs can really control only r states and the rest n-r states should au dcally be contrlled under the cnint. 2.2 The Time Delay Estimatio and Control Actio A structural constaint for this scheme has been derived and a class of system that satisfies it has been identified. Now it is of interest to determine the control action u that will force the plant to follow the reference model in the face of unknown dynamics h(x,t) and unexpected disturbance d(t) which appear in the right hand side of Eq.(2.8). These two terms appear as a sum and their effect can be determined from the plant dynamic equation

the present time t is very close to that at time t-L in the past for a small ime delay L, (2.13) h(x,t)+d(t) h(x,t-L)+d(t-L) Now, combining Eqs.(2.12) and (2.13), the effect of h(x,t)+d(t) is esfimated by: li(x,t4(t) i(t-L)-f(x,t-L)-B(x,t-L)u(t-L) (2.14) A fundamental stepm this derivation is to delay time by an amount of L to estimate the system's unknown behavior, naly h(x,t)+d(t). The TDC contol law is then obtained by substituting Eq.(2. 14) into Eq.(2.8) and is given by u(t) = B+(t) (-f(t) - *(t-L) + f(t-L) + B(t-L) u(t-L) + Amx(t) + Bmr(t) - Ke(t)) (2.15) In the above equation, each term has the following meaning: (1) B+(t), a pseudo inverse matrix cancels the control matix

Thus this controller observes the ste derivatves and the inputs of the system at time t-L, one step into the pasts and determines the control action that should be commanded at timet. 2.3 Control Law for Systems in a Canonical Form As mentioned earlier, one class of systems that satisfy the structural constraint are those expressed in canonical form. Thus, the objective of this section is to derive the rime Delay Control law for this specific class of systems. Physical systems can often be described using this representation. Consider a nonlinear plant with n ssmtes and r inputs where each term of Eq.(2.3) can be partitioned as follows: X= N-; f(x,t)= Xs ] ; h(x,t)= 01
xr J

dynamics.

disturbance d(t), (3) the team, Amx+Bmr, inserts the desired dynamics of the reference modeL and (4) the error feedback term, -Ke, adjusts the error

(2) the term, -f(t)-k(t-L)+f(t-L)+B(t-L)u(t-L), attempts to cancel the udesired lnown nonlinear dynamics f(t), unknown nonlinear dynamics h(t) and the unexpected

B(t),

pfrx,t)
;

hr(x,t)
d(t)= [.O] (2.16)

B(x,(t1

where xq and 0 are (n-r)xl voctors, XS = Ixr+l,. . . T is also an (n-r)xl vector, xr, fr, hr and dr are rxl vecors, and Br is an n nonsingular matrix. The reference model and an erro feedback gain matix can also be partitoned in a similar fashion such dt the total controlled system satses the structural constint as of Eq.(.l 1) (see Appendix II). Am= ° 1 Bm= rJ K= 01

h(x,t)+d(t) = i(t)-f(x,t)-B(x,t)u (2.12) In order to obtain an estimate of the effect of the term h(x,t)+d(t), it is considered that the value of h(x,t)+d(t) at

Eq.(2A):

Amrl 2mr Kr 2*7 where, in this case, 0 is an (n-r)x(n-r), (n-r)xr or (n-r)xn mat, Iq is an (n-r)x(n-r) matrix, Amr is an rxn matrix, mr is an r matix, and Kr is an rxn matrix. Substituting Eqs.(2.16) and (2.17) into Eq.(2.15), the TDC control law becomes: u(t) = Br-1(t) (-fr(t) - r (t-L) + fr (t-L)+Br(t-L) u (t-L) + Amrx(t) + Bmrr(t) - Kre(t)) (2.18) Note that in this canonical system formulation, the above control law includes an inverse of the control distribution matrix Br instead of a pseudo-inverse matix as seen in the general TDC control law in the previous section. To this end, the fundamentals of this approach have been defined. These include a strucual constraint condition, a tm delay scheme and the ti delay control action. 3. ANALYSIS AND DESIGN FOR LTI-SISO SYSTEMS If the plant is a liner tim-invaiat (LT) and singleinput- single-output (SISO) system, discussion of a Time Delay Contrl system can be conideably simpler and more

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intuitive using classic frequency domain technique. In order to describe the main feature of this controller, this section will focus on the analysis and design for such systems. 3.1 Control System Structure Consider the following LTI-SISO plant with unknown system matrix A. * = Ax + bu y = xi (3.1) Tj where x [x, . . ,, , X 2j j Xni A = 0, 1, ,.. . .... ol 0, b.=0
,

b and al, a2, . . an are unknown constants, and b is a known constant. This formulation corresponds to a canonical form representation of Eq.(2.16) in the following
-aIV -a2V
-a3lp..

...........

-an J

.

manner

fr (x,t) = 0, hr (x,t) = -aixI -a2x2- -anxn, (3.2) The reference model and error feedback gai matrix K can be chosen based on Eq.(2.17) in order to satisfy the suctural constraint of Eq.(2.1 1). im=Amxm+bsr

dr(t)=0, B(x,t)=b

Ym=Xml where m[xmXl, Xm2, ..,xmn]T Am= O, 1, Q'

(3.3)

3.2 Stability Analysis It is also possible to discuss the stability issues of this controller using the above formulation. By combining Eqs.(3.6) through (3.8), one obtains

e 1 -Ls (3.15) the block diagram of Fig.1 can first be reduced to the one of Fig2 and then to the one of Fig.3. A typical root locus of the closed loop part of the system (inside the dotted line of Fig.3) is shown in Fig.4. If the delay time L is small enough, in other words, if the effective controller gainl/L is sufficiently large, one closed loop goes to negative infinity whereas all the other poles approach the fixed closed loop zeros. Therefore, pole/zero cancellation occurs, which makes the whole closed loop part to look like a first order sysem with a large bandwidth. The resultant block diagam is shown in Fig.5. Since the second block in Fig.5 can be regarded as almost unity, the whole system behaves as the reference model. From the above analysis, a Time Delay Control system can be interpreted as one where the command r is prefiltered by the reference model into Ym and then the error ym-y is forced to zero by the high gain integrator l/Ls with pole/zero cncellation.

Y(s)

bm

O,

O,

IV

l

, O
-

bm= °

K= 0,

L-amil.
ol
. . . . . .

...

o

-am2i -aml)3
..

..

amn]

bn

.........

0 0, o
.

...I
.

0 0
. .
. .

1(3.4)

L-kil -k2, *., -knJ Substituting Eqs.(3.2) through (3.4) into Eq.(2.18), the TDC control law can be obtained as follows. u(t) = u(t-L) + (1/b)f -*n(t-L) -anxn(t)-. ...-alxl(t) + bmr (t) + kn (xmn - Xn) + ..+kl(xml XI) ) (3.5) From the above Eqs.(3.1), (3.3) and (3.5), the following Laplace transfonmations are obtained for the plant, the model and the control action as G(s), Gm(s) and U(s), respectively. G(s) = Y(s)/U(s) = b/P(s) (3.6) where P(s)=sn+ansn-l+ ... +al Gm(s) = Ym(s)/R(s) = bm/Pm(s) (3.7) where Pm(S)=Sa4amnsn-l+. ...aml U(s) = { Pk(s)Ym(s) + bmR(s)
where Pk(s)=knsn-1+... +k2s+kl

.

.

.

.

R(s) Pm(S) (1e-Ls)P(s)+Pmk(s) Therefore, the characteristic equation of the closed loop is (3.17) Pm(S)( (1-.e-S)P(s)+Pmk(s) J = 0 Usually, the model's characteristic equation Pm(s) is chosen to be stable, thus one is to examine: (I -eU)P(s)+PMn(s) = 0 (3.18) or equivalendy, 1 + ( -e<Ls+(Pmk(s)fP(s)) ) = 0 (3.19)

-

____

Pm(s)+P(s)

(3.16)

-Pm(s)Y(s) )/{ b(l-e-Ls))

(3.8)

Pmk(5s)?=e-Inss+(amn+kn)sn-+ ... +(aml+kl)

system in frequency domain by analyzing the defined above LTI-SISO systems. Suppose that the desired error dynamics is governed be=Ame (3.9) which simplifies the algebra by making all the error feedback gains to be zero. k1=.... =kn=O (3.10) In addition, for a small time delay L, the following approximation holds.
(.1 in(t-L) =_ in(t) Now combining Eqs.(3.8), (3.10) and (3.11) leads to U(s) = ( bmR(s)-Pm(s)Y(s) )/( b(1-e-Ls) 1 (3.12) or using Eq.(3.7), U(s) = { Pm(s)Ym(s)-Pm(s)Y(s) 1/{ b(l-e-Ls) 1 (3.13) This equation can be reduced to U(s) = e-LsU(s) + { Pm(s)/b ) t Ym(s)-Y(s) (3.14) A block diagram for the whole system can then be obtained as shown in Fig.1. With the following approximation of

Now we can describe the main feature of the TDC

The stability condition of Eq.(3.18) or (3.19) can be solved by using existing techniques such as the Nyquist stability criterion, or Routh-Hurwitz criterion using the Pade approximation, and it gives the upper limit of the time delay L to maintain the stability. Strictly speaking, the above stabiWty test should be done for the entire possible range of the unknown polynominal P(s). However, a good result can usually be obtained using the "worst" P(s), which can be defined as the polynominal that corresponds to the "fastest" plant. Stability can also be interpreted intuitively using Eq.(3.18). If the time delay is infinitely small, L -.0, then the overall stability will depend only on Pms(s) and P(s). As seen in Eq.(3. 16), the stability condition depends on the time delay, the error feedback gains, and the characteristic equations of both the plant and the model. For example, for a first order system, the stability condition with a small dme delay becomes as follows using digital implementation technique [17]. (3.20) L < 1/( (am+k)a ) This stability condition states that a smaller time delay is required for a faster plant, a faster model and/or a larger error feedback gain. This general idea usually holds and intuitively makes sense. 3.3 Design Procedure As shown in Eq.(3.5), design parameters to be chosen are (1) error feedback gains kl through kn, which adjust error dynamics and (2) time delay L, which is related to model following performance. Substituting Eqs.(3.3) and (3.4) into Eq.(2.7), the error system matrix can be obtained and its characteristic equation can be calculated as (3.21) sn+(amn+kn)sn-l+ .. +(am1+kl) = 0 Using the above equation, error feedback gains, kl through kn, can be chosen to give the desired poles of the error dynamics. From Eqs. (3.6) through (3.8), the following transfer function is obtained and it is clear that we would like to have it as close to unity as possible.

Y(s)/Ym(s) = {PPm(s)+Pkc(s))/( (1 -e-LS)P(s)
+Pmk(s) 1

(3.22)

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A proper time delay L can be chosen by evaluating the above equaton, in order to meet model following specifications, which nt be defined in terms of gain, phase and fiequency range. Again, stricly speaking, the above ransfer function should be evaluated for the entire possible range of the unknown polynominal P(s) but a good result can usually be obtained using the "wom" P(s). A simple example which illustrates the design procedure can be found in [9]. 4. EVALUATION OF CONTROL SYSTEM PERFORMANCE 4.1 Simulations 4.1.1 First Order Nonlinear System Simulation was done for the folowing first order plant to demonstrate the effectiveness of the Time Delay Controller. (4.1) x = cos(x)+u+d where cos(x) is teated to be the unknown dynamics of the plant, and d = 0.5*1(t-0)5) is the unexpected disurbance with magnitude 0.5 and acts at t=0.5 se. Note that 1(t) represents a unit step function. The reference model to be followed was selected to be a first order system with a ime constant of 1 second. m=-xm+r (4.2) where r = 1(t). The desired error dynamics was defune to be L-- -aec (4.3) where ae = 1. The TDC control law is obtained simply by combining Eqs.(4.1) through (4.3) and (2.18). (4.4) u(t) = u(t.L) + ( -k(t-L)x+rkpe ) where k0 = ae-i. As seen in Fig.6.1, the response of the plant and that of the model match almost perfectly. In fact, the small error in Fig.6.2 confirms this result. As seen in Fig.6.3, the Time Delay ControUer completely cancells the step disturbance right after it is applied at inm t=.5 sec. Additional simulations were done for different erro feedback gain kp's. The relationship between the error dynamics and the error fedback gain is clearly seen in Fig.7. 1. The larger the gain is, the quicker the error dynamics is as predicted in Eq.(2.7). The error dynamics actually has the time constant of (I/ae) = 1, 1/2 or 115 second, which is shown in Fig.7.2. However, as seen in Fig.7.3, a larger control effort is of course necessary for

where
0 - cwn IL O _02 l0 with a natural frequency co and damping ratio 4 of 2.5 md/sec and 0.707, respectively. The input conmands rl and r2 were chosen to be rl Tr2=f(l2) *(1-cos(t*(p/4)) (rad) Ostc<4 sec 1 (rad), t 2 4 se For s licity, let the desre dynamics be

Anlr

W

0 %2

_0

0j,

Bm=r-

which corresponds to (4.8) Kr=0 From, Eqs. (4.5) throgh (4.8) and (2.18), the Time Delay Control law becomes:

= Ame

(4.7)

02 t * Hl-

[t|

-

2 .t-L +

,,1+ mt

1011
62

The folowing PD controller was derived and used to compare the results. Note that this PD otroller does not compensate for the Coriolis and c gal f

(4.9)

01

4.1.2 Higher Order Nonlinear System A robot manipulator, shown in Fig.8, was chosen as a nonlinear system to controL This system can be described by the following equations,

quicker convergence.

where
and

~[Iti = L-J + t :::IL) (4.5)
H= (12) * [2(5/3+cos02),

:3+cos02,

23+cose2,
h= (1/2) * H-i*

V3

.4

-jsin2*02*(201+62)
sinD2*612

effectiveness of the controller, the Coriolis and centrifugal force vector h is tated to be completely unknown. Th reference model was selected as a of second order systems described by
In
set

order to determine the

mi
d
0m2

: 1 I[

0

Omi

04

|

|

0

- |-|--

6mn2
-

|+ --+-

rji

dt

9m1
Orn2

Awr

0m 1
0m

Bmr

rz
(4.6)

(4.10) The fim simulation was done under the above formulation and with a delay of L=0.01 second. With the Thune Delay Controller, the manipator followed the desird model very well as shown in Figs.9. 1 In fact, the maximum position tracking error is considerably redued to only some 1/50 of that with the PD controller as shown in Fig.9.2. Even though the response chosen is slow, the nonlinea effects are significant with the PD controller [4] . The required torques for the both motors are satisfactory. Their magnitudes are not excessive and the signals are very smooth as seen in Figs.9.3. In many pracal siuations, the available acceleratio signal could be very noisy. Such a situation is simulated by applying white noise with a=0.03 (= about 10% of the maximum signal) to both signals. The acceleration measurement signals then became very noisy as shown in Figs. 10. However, the robot pulator still followed the reference model fairly well as seen in Figs.l 1.Ithrough 11.3. In fact, the position trackdng errors remained much less than that with the PD controller. This is because the plant acts as a low pass filtr and considerably reduces the high frequency components of the white measurement noise. Many physical plants, especially mechanical systems, can be considered as low pass filters simila to this case and then noise problem may not be significant in the TDC systems. If this is not the case, however, some efforts to eliminate the measurement noise are necessary in order to ensure good tracking performance. Also, this noise insensitivity property was confuired by simulation with white noise in tachometers [16]. 4.2 Experiments In order to see the effectiveness of the Time Delay Controller in a real system, the control law was used to position a servo motor system as shown in Fig.12. The inherent unknown dynamics here are viscous and dry friction. Also, additional uncertainty was simulated by an elastic spring that was physically a to the -motor load. Then, the dynamic equadons for this system are .0] d [00 = , (4.11) dt L{kJJ)*O -(bsyJ)*9 -dsSgn(J where 0 is motor rotation angle, tm is motor torque, and ks is an unknown spring constant, bs is an unknown viscous friction coefficient, ds is-unknown dry friction and J ( =

fX1)

=

jHt

Amr

t2J

02

+

Bmr.{rj)

907

4.0*10-4 kg-m-sec2 = 0.53 oz-in-sec2) is the total inertia. The reference model was chosen to be a second order system defined by d 70m = 1 O ml + ' 0 r 0
reference command r are 5 rad/sec, I and i/8 rad = 22.5 deg, respectively. Let the error dynamics to be e=Ame thus making error feedback gains to be zero. Then, combining Eqs.(4.11), (4.12) and (2.18), one obtains the TDC control law: 4ma')y. (tL) + J f -O(t-L) -o(dnq(t) -24OnO(t) +(O2nr(t) } (4.13) Since an angular acceleration signal was not available in the experimental hardware, the following approximation was

Time Delay Controller
r

-:k 2 -2lja [ bm [0| (4.12) i[m where the natural frequency (n, damping ratio ( and

$S~~~~~~~~~( )pm(de
.

b

m

y

e

P(g) m
b

u(t):

b
pla

y

L

Fig. I Block diagram of TDC for a LTI-SISO plat
Time DELlay Contrle

................... ... modelpln
............I..... e

.........

&(.4 )I/L (414) 0(t-L) _ {O(t)-8(t-L) The fmnal control law can be obtained by substituting Eq.(4.14) into Eq.(4.13). tm(t) = tm(t-L) + J ( j46(t)-(tL)/L (4.15) -0)2n((t) -2Cco)n6t) +02v(t) where the delay time L is 5 msec. In the first experiment, the following PD controller was used in order to see how large the unidentified viscous damping coefficient bs and dry friction ds are. Zm(t) = J ( -c%26(t) -2wnk§(t) +w,}r(t) (4.16) If the external spring is not attached and both bs and ds are small enough, the above controller wll sake the plant behave the same as the model. However, the asctal response showed a large steady state error as seen in Fig.13, which implies ta the unknown viscous damping coefficient and/or dry friction cannot be ignored and some compensation technique is necessary. In order to compensate for these undesired and unknown dynamics, the Time Delay Control law of Eq.(4.15) was used. This time, as shown in Fig. 14, the plant showed no steady state error, 0% overshoot and 0.8 sec 2% setting time, which means that the plant closely followed the model. An external spring was then attached to simulate additional unknown dynamics. Fig.15 (with a soft spring of ks=0.17 Nm/rad) and Fig.16 (with a strong spring of ks=0.23 Nmn/rad) show that the response remained almost the same despite the unexpexted extemal spring torque. Note that the controller does not depend on the unknown system parameters of k3, ds and bs. Thus, even if they change over time, the same performance will be guaranteed. Since the distinguishing feature of the Time Delay Controller is the angular acceleration term, the last experiment was done to see what would happen when the accceleration term is tken out of the control law tn(t) = ?m(t-L) + J ( -Xn20(t) -24w0b(t) +con2r(t) ) (4.17) The above PW controller was applied to the system without an external spring. As seen in Fig. 17, the system went into an oscillatory mode due to the high gain integrator part of zm(t) =rTm(t-L). This result can be interpreted as that the acceleration term, or the higher derivative term in general, in the Time Delay Controller stabilizes the system with a high gain integrator. Overall, the control signals of the TDC were very noisy due to the numerical differentiation in Eq. (4.14). However, the system showed very good performance and this indicates that use of higher derivative signals, with proper hardware, should not be very serious problem as also demonstrated in [2]. Again, this is because this servomotor worked as a lowpass filter and reduces the noise effect 5. CONCLUSION This paper discussed the control of systems with unknown dynamics. A new method, Time Delay Control (TDC),was proposed. The simple control algorithm does not require parameter identification, infinite frequency
.
0

used,

Fig 2 Reduced block diagram 1 of TC for a LSTl-SISO plant p (s) b L~~~~~aps )

r

b

y,

_

___

S

-E
Fig.3 Reduced block diagram 2of TDC for aLTI -SISO plant

t

hma.

Real

Fig.4 Root locus ol TX for a LTI-SISO plant model
r

b

m

y

m

i

y
p

____

P (s)
m

6us*l
Sao

Fig 5 Reduced blockdiagram '3 of TXC for a LTI-SISO plant

FP1 6.1 Response f the p-w and the mokl

1slc)

.....

-e

-0 912

@Si

/
*T
4'

switching or repetative action. It was shown that the choice

erro Fig 62 Resposof tXe

908

12

a'
6.4

*t* ml 1t (rat)

(1P)PD

"I *(TOMtSol

.4.~~~l
IF

e

-6.4
-0.4

0\5
F*.6.3 Rusponsea(t CMcomrolh
w

lra4)

Z2 u2{A:

-1. 2V
6

(mc{rSl~~~~~~~~~~~~~~~tm

e|

Ume(c)

eS

Fi&9.1 ReI= of ae&*
el ITDC)

o

d tM mdo

time Inc)}

65 .'84 __ __ ___ _ __(PD_J- _ _
2. 6 12

,

a

,

,

ts

tim.
e~ ( )\

(mc)

12
Fig.?. bip
-*

e2 (PD)
___v

d

*_

Sad on

0
,

e2
V.zFg9. I Fi.91

-

kV

g

ti__ 9.

,4..~19l
md 1

$-I).4 I4

_go F hee ll

.s

w.*

t

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Ti

I

(t1f2
-

I
a

r. ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
2 7.2 _ rim.F

"And

rZ S

_
Z.

Us. (we)

r-T.

lcp.o
175

-.3

-5

Fi.9.3 Ek_mlad fU tore ino

tw.

W.

time (NC)

S 4. -2

___ _ _

253
-3 , , _

al.

4-.

-

.

3

Fi.73 Lbsp.s t CM et bp

(r&&mA
unt mans and unit length

82.82

-. 4

time (WC)

-8. 4
*
1

-@

-21
.

K,

F 10ONiy
os

m. . o ckem-e m *
c f

time (ect

Motor 2 Mdotor I

2 2

/

sd

Kf

o

uit tac mass ant unitleangt

I2 @m2
(5rat?

1

2 (TDC), I

02

FigS two tegre.-o-(rntdoU robot mmnpulat*tc

"A

-5

N (PID)
W-

of reference model and error feedbackgain matrix is restricted depending on siz of a control distribution matrix. The simplified TDC control law was given for systems in a canonical form. Linear me-invariant (LTI) and single-input-singleoutput (SISO) plants were discussed. System's structure was explained as the combination of (I) a prefilter of the model, (2) a high gain integrator and (3) pole/zero cancellation. Stabilty oondition was given. In addition, a desig procedure was explined on how to choose proper error feedback gains and time delay. Performance of the control system was evaluated through both simulations and experiments. Through simulations for the first order plant, good model following and very quick disturbance rejection property were demonstrated. -It was also demonstrated that the error dynamics can be adjusted by choosing a proper error

experimental evaluation. ACKNOWLEDGEMENTS The authors greatfuly acknowledge a fellowship

feedback gain. The good model following property was confirmed through simulations for a more complicated noninear system - a robot nipulator. It turned out that measurement noise does not cause a serious problem because most physical plants act as low pass filters and considerably reduce the noise effect. Then, the validity of the controller was verified through the experments using a servomotor positioning system. The focus of the current research include stability analysis for nonlinear plants, discussion for cases with an unknown control distribution matrix, and more detailed

Fi 11s Re Ispo

2.

m n as UM motel the _M

4w.

6.

time (WC)

f

909

(rad)

2% settling time 0.8 se

(rad)

e2

41 *4 -0 S 12

4

Z'

4--

--F7~

-a

timesecI

*,.'
*

e

--"
/

2IP

e2 ETDC)

Iat
-

_~ ~ ~ ~ ~ ~ -a/$rad.
Time DeaY Cntdr
aA / rad. L ~~~~~~~~~r

4

Fitg 112 RlesponseOtte

r

timeI(xc)

21 nttling time
0.s
j

Fig.14 Experiment '

I

-_

Ita

Hart. ten tin. _0., ice.) Fig. I5 Ex peri ment '31 TDC witaaudt Wpins)

-

compensated spring torque

Fg,16 Experiment '4 1 TDC with a lard spring)

r

-

/l

rad

To

Bottm: 0f 0061 NE/div.) Botzrn time0o.ScJdlv.) Fig. 1 3 Experinetit ' I ( PD Corolr

Top tra

( 022 rWd4t. )

Top tr I ( 022 rd/iv. ) Bttom tra-,Tce i 0.061 Nn/div. Horat tre tite O S Jdv. ) Fig )7 Experiment ' 51 PIDCoavoler )

support from Matsushita Electric Industrial Co.,Ltd. This research was partly supported by the National Science Foundation under contract No. 8702839-MSM. APPENDIX I Algebraic Manipulation Leading to Eq. (2.10) Subtracting Eq.(2.9) from Eq.(2.2), nm-r = Amxm - BB+Amx + BB4Ke + {I-BB+) (-fh-d + Bmr) (I.1) Adding -AmX + Amx (=0) to the right hand side of Eq.(I. 1),
Xm-x Amxm-Amx + Amx - BB+Amx
=

obtained:
or e=

e = Ame + Ke +

(I-BB+) (-f-h-d+Amx+Bmr-Ke} (1.3)

(Am+ K)e +

(I-BB) [-f-h-d+Amx+Bmr-Ke) (2.10)

APPENDIX II Canonical System and the Structural Constraint From Eq.(2.16)

(BTB )-1 = [O BrT)
=

(.2) {I-BB+) -f-h-d+Amx+Bmr-Ke) By using the definition of the error in Eq.(2.3), Eq.(2. 10) is

+ BB+Ke + (I-BB+) (-f-hAd+Bmr) = Am(Xm-x) + Ke +

Then,

(BrTBr)-1

l B

B Br I (BrT)-l
[O

(II. 1)

I-BI+

=

I.B(BTB)-IBT [r 'O Br-I (BrT)LBr

BrTl

910

=-I-

[0

.

[O

I] (II.2)

JoI Q

From Eq.(2.16),

-f-h-d-Amx+Bmr-Ke

= -

(II.3)

-fr=hrrd-AmrX+Jmrr- Kre]

-

X-frhr-d-Amrz+Bmrr-Kre Substituting Eqs.(1.2) and (1I13) into Eq.(2.11), the structural constraint beconls: (I-BB+ { -f-h-d-AAmrx+BmrKe)

12. Tomik, and Horowit, R. "Model Reference Adaptive Control of Mechanical Manipulators", IFAC Adaptive Systems in Control and Signal Processing,San Francisco, CA 1983. 13. Uchiyama,M. "Formulation of High-Speed Motion Pattem of a Mechanical Arm by Trial",loI ais of Socit of Instrumrnt and Control Enginng of Jaan, Vol. 14, No.6, pp.706-712, December, 1978.

=:

0

I

(II4) 0l r 1 [J -fr-rd-Amrl+Bnrj

III1 1972

14. Utkn,V.I. "Equations of Sliding Mode in Discontinuous Systems", Autmation and Remote Cnrl

Thus, the structural constaint is always mct for systems expressed in canonical form.

15. Youcef-Toumi,K and Ito,O."On Model Reference Control Using Time Delay for Nolinear Plants with Unknown Dynamics", MITLezQwf LMP/RBT 86-06, June, 1986. 16. Youcef-Toumi,K and Ito,O. "ontroller Desig for Systems with Uniknown Dynamics" Eedi gfAm Control ConferMnce Minneapolis, MN, June,1987.

REFERENCES
1. Arimoto, S., Kawamura., S. and Miyazaki. F. "Can Mechanical Robots LeAn by Themselves ?" f fing cnd raonl i obodtics Reserch Kyoto, JAPAN, August, 1984.
2. Asaa, HL and Youcef-Touni, K. "Analysis and Design of a Direct-Drive Anm with a Five-Bar-Link Parallel Drive Mechanism", ASMEiJbnahtlQm of imi Sliam, MeasuMremnt and Cgol. Vol106, No.3, pp.22-230, Sept, 1984.

17. Youcef-ToumiK and Ito,O."Model Reference Control Using Tmhe Delay for Nlinear Plants with Unknownm Dyamcs, ?} inL of Intamudmal Federal Republic of Germany, July,1987. 18. Young,K-KD. "Controller Design fora Manipulator Using Theary of Variable Sture Systems IEEE Transton, SMC-8-2, pp.101-109,1978.

3. Astlm K.L and Wittenmark, B. "On Self-Tuning Regulators", A Au Vol.9, pp.185-199, 1973. 4. Brady, NC et a] Robot Moion: Plan and CQntroL M.I.T. Press, 1983. 5. Craig, J.J., Hsu, P. and Sastry, S.S. "Adaptive Control of Mechnical Maiphtors", P id MU 1nten%j=dalmf=eon Egbg9gs adArao April 7-10, 1986. 6. Dubowsky, S. and DesForges, D.T. "The Application of Model Referenced Adaptive Contrl to Robotic Manipulators", ASME JoraI of D c Systms. Measurement and Control, 101:193-200, 1979. 7. Hsia, T.C. "Adaptive Control of Robot Manipulators - A Review" Proding of theEEB In=raional Conference on Robotsd Autoaon. April 7-10, 1986. 8. Ih, C.C. and Wang, S.J. "Dynamic Modellng and Adaptive Control for Space Stations" JPL Pukl 85-57, &iiz

July, 1985.

9. Ito, 0. "Mo&l Reference Control Using rme Delay for Nonlinear Plants with Unknown Dynamics" M. Theis. Dept of Mechnical Egineen n, M.I.T.. May,1987

10. Kalman, RE, "Design of Self-optmizing Control", ASME Transaction. Vol.80, No.2, pp.468-478, 1958. 11. Slotine, J.-J.E. and Sastry, S.S. 'Tracking Control of Nonlinear Systems Using Sliding Surfaces with Applications to Robot Manipulators", Inteaional Joual gf ConQpl. 38-2, 465492, 1983.

911

Ikj

z

im

na

hrw r

-

0 0

=

Y

$X~~~tedylState

a M/ 6~~~

ro
ral!
z

Irm ~

~

~

_.

II a

sttln timeXX_

S_

Dte lay Corltroter, Fig~l ExperimenL 12 Tim 21t settlng ti me _ .,sec
1

overs_

ry r g M sz

~sringtoyrq;ue,

v e ( 0.2 r d/d, Bottom ltrae "ai 0 066 Mm/d4W.iv, Hoti trace. ntte ( 0,5 sec.dgeiv Flg 15 EFxperiment , TDC with a soft 3pring 2X settling time

Top tace

SS1~~~~~~~~~~

n8rd

F.g 1 6 Experiment 114 TDC wit

a at,d spring

D

-'~~-


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