WP-13-1 = 3:20 Proceedings of the 29th Conference

on Declslon and Control Honolulu, Hawall . December 1990

Ripple-Free Model Reference Adaptive Control

H. Buller G. Honderd A.R.M. Soeterboek

Delft University of Technology, Depl. of Electrical Engineering, Control Lab., P.O.Box 5031, 2600 G R Delft, Netherlands, Phone: t 3 1 15 785114, Fax: i -31 15 626738

Abstract In most existing Model Reference Adaptive Control (MRAC) schemes, the plant-model output error is driven to zero and so y,(k) -+ y,(k) as t + CO. In discrete systems, this implies a can- cellation of the process zeros, which may lead to ‘ringing’ eflects and the inability to control nonminimum-phase plants. Redraw- ing the primary controller structure of a general MRAC scheme shows that the closed loop approaches a Minimum-Settling Time (MSTC)-like controller. This paper shows how this original MSTC behaviour can be changed to a Ripple-Free Response Control (RFRC), which ef- fectively leaves the process zeros unchanged. The plant output then tends to y,(k) + ( J ( Z - ~ ) / & > y,(k), in which h(2-I) is the estimate of the plant numerator polynomial and b = E,”,, b, is a correction factor for the DC gain of h(z - ’ ) . As the process zeros are untouched, an originally present ‘intersampling ripple’ disap- pears and nonminimum-phase plants can be controlled. However, the cost of these improvements is a slowing down or the process response by m samples (the degree of B(z-’)). Using modified signal and parameter vector definitions, it is shown that a stan- dard update scheme can yield the desired ripple-free behaviour while maintaining a stable adaptive law. Simulations confirm the ripple-free characteristics of the algo- rithm and practical application to a water level control problem shows that good results in a practical situation can be obt.ained.

Introduction Model reference adaptive control systems were originally devel- oped using continuous-time analysis, being stimulated by (among others) the presence of Lyapunov’s and Popov’s stability theories. It is therefore not surprising that the first steps towards discrcte- time MRAC were based on the existing continuous-lime designs. In (11 adaptive laws for the discrete counterpart of the most ba- sic form of MRAC are derived, in which all state variables of the process are available. However, because of the unclear link be- tween the state variables of the continuous-time physical process and the discrete-time reference model, most discrete-lime de- signs are developed for single-input, single-output processes. In 121 and [3] the continuous-time ‘Augmented Error Method’ is transformed to a discrete form using Lyapunov’s method. Sever- al others ((4],[5]) derived the adaplive laws using hyperstability theory, each contributing several changes and eiiliancements to the design. The development of these discrete-time counterparts of originally continuous-time designs needs a discrete form of existing stability theories. Lyapunov’s stability theory and Popov’s hyperstability theory both have a discrete counterpart that can be used in the design of discrete-time MRAC. IIowever, the inherent one-step delay in the adaptation in the discrete case prohibits direct ap- plication of these theories. A solution is found in the form of

a correction factor for the output error, linking the ‘a priori’ to the ‘a posteriori’ error ([l] and [ 6 ] ) . Furthermore, speaking in terms of hyperstability theory, the introduction of new adapta- tion schemes based on least-squares estimation methods violates the passivity property of the nonlinear part of the error equation. A stability analysis in this case leads to a ‘super-strictly positive real’ (super-SPR) demand on the linear part. A modification of Popov’s hyperstability theory was developed in 171 and (81, which implies that demands on the linear part and the nonlinear part may be exchanged. This means that the SPR demand on the linear part may be violated, provided that the nonlinear part satisfies a heavier passivity demand than originally, and vice ver- sa. The availability of this modified stability method enables a smooth introduction of more complex adaptation schemes (such as least-squares methods), maintaining the stability proof. In later discrete MRAC schemes, the design starts with the as- sumption that all process parameters are known. A basic con- troller can then be designed that makes the transfer of the con- trolled process equal to that of the reference model. Finally, the controller is augmented with a parameter estimation scheme (usually based on a least-squares method), that directly adjust- s the controller parameters. The design is made in such a way that the overall system stability is guaranteed, which implies that the parameter estimation part and the control part are analysed simultaneously. To be able to perform such an analysis, the pri- mary controller cannot be very complex. In 161 and [ g ] a general MRAC structure was developed that has many existing schemes as particular cases. Other unilications of discrete MRAC algo- rithms can be found in (101 and (11).

This paper is organized as follows. In the next section, the ‘In- dependent Tracking and Regulation’ (ITR) algorithm ([61, (91) is described in a compact way to lay a basis for the rest of the paper. In the following section, it is shown that ITR, as most existing MRAC schemes, is a pole and zero cancellation method which involves ‘ringing’ effects and has dimculties in coritrolling nonminimum-phase plants. By reconsidering the primary con- troller structure, a modification to a ‘Ripple-Free Response Con- troller’ (RFRC) (121 appears to be possible, which does not suffer from the mentioned difficulties. A new definition of the parame- ter and signal vectors is necessary. Asymptotic stability can be guaranteed, but the main problem appears to be the definition of the error signal of which this is so. Next, simulation results with the new adaptive scheme are presented, and results of a practical application to a water level control problem are shown. Finally, the last seclion draws some conclusions concerning the method.

Independent Tracking and Regulation This section presenls the ITR algorithm ( [ 6 ] , 191) because it can be considered a more or less ‘standard’ MRAC algorithm, and the design of the RFRC MRAC follows the same lines as ITR.

1004 CH2917-3/90/0000-1004$1 .OO @ 1990 IEEE

The mathematical details will not be covered here: they can be found in the mentioned literature.

Primary control ler s t r u c t u r e For the controller derivation, the following notation for the plant to be controlled will be used :

A( z - ' ) yp (k) = z - ~ B ( z- ' )u (k) where:

A(z - ' ) = 1 + ulz-' + . . . + u,z-"

B(z- ') = bo + biz-' + . . . + 6 , Y m

The process numerator and denominator orders (m and n re- spectively), and the time delay d are assumed to be known and equal to the reference model orders.

The control goals are twofold. First, in tracking, the transfer y , ( k ) / r ( k ) of the controlled process should be equal to the refer- ence model:

A , (z-')y,( k ) = Y d B , ( .-I).( k)

Second, in regulation (r(k) = constant), an initial disturbance of the output error e ( k ) = yp(k) - y,(k) should be eliminated with dynamics specified by:

C ( z - ' ) e ( k + d ) = 0 V k 2 0

In this equation, C(z- ' ) is an asymptotically stable polynomial of order n,. Now the name 'Independent Tracking and Regu- lation' becomes clear: the user can specify the desired tracking properties with A,(%-') and Bm(z-') , and the regulation objec- tives with C(z- ' ) . If the plant parameters a,, bi and d are all known, it can be shown (161) that the control objectives can be fulfilled by applying the controller structure of figure 1. It can be seen that the reference model without the time delay d , and the regulation filter C(z- ' ) are placed in cascade with the process. The coefficients of the polynomials Q(z- ' ) and R(z-') are the controller parameters. The control signal u(k) is calculated as:

1 ~ ( k ) = - [ C ( ~ - ' ) y , ( k + d ) - Q ( z - ' ) ~ ( k ) - R(z-')l/,(k)] ;

bo Q(z-1) = q1z-'+ . . . + q,pZ-nq

R(z- ' ) = ro + rlz-'+ . , . + r,,z-"'

The polynomials Q(z - ' ) and R(z-') are of appropriate orders n, and n,, respectively, that can be determined as a function of n,m,n, and d. Note that ideally, yp(k) = y,(k) and therefore C(z- ' ) will appear as l/C(z-') in the loop transfer function, thereby determining the regulation behaviour. The controller parameters in R and Q thus depend on C. Note that C can only slow down the regulation, and hence C should only be included if the result with C = 1 yields a 'wild' control signal U . The control signal u(k) is calculated as:

(1) 1 u ( k ) = - [ ~ ( ~ - l ) ~ , ( k + d ) - e:Wo(k) ] b0

with Bo and wo the parameter vector and signal vector, respec- t ively :

e,T = [st , . . . , qn,,ro,. . ., rn,]

U,' = [U(k - I),. . . ,U(k - TZ*),Yp(k),. . . ,Y(k - %)I Adding bo to Bo, and u(k) to w o ( k ) , leads to the vectors w ( k ) and U :

eT = (bo, e:)

J ( k ) = (u(k) , w m )

If the process parameters U;, bi and d are known, the controller parameter vector U can be determined explicitly, such that both the tracking and the regulation condition are satisfied (161).

Adapt ive ITR The second step in the ITR design is the extension with an adap- tive mechanism, which consists of a direct least-squares parame- ter estimator which recursively updates the (symmetric) adapta- tion gain matrix l':

0 < A1 I 1; 0 I A, < 2; r(0) > 0 This recursion formula, in combination with the parameter up- date law:

e ( k ) = O(k - 1) + l ' (k)w(k - d ) ~ ( k ) (4)

results in an asymptotically stable error signal ~ ( k ) , provided that the error equation has a form:

~ ( k ) = H(z- ' ) (0. - U ( k ) ) T w ( k - d ) (5 )

Here, U' is the exact parameter vector, for which r ( k ) = 0. H(z- ' ) must be 'super' SPR and determines the maximum value of & / X I (81. For simplicity, from now on H(z- ' ) is chosen 1. Us- ing the identity: C(z-')y,(k) = BT(k - d)w(k - d) ((61, compare (2) for the fixed-parameter case), ~ ( k ) can be found to be:

with:

For the adaptation, ~ ( k ) is calculated as:

E(k) = C(z-')y,(k) - U T ( k ) w ( k - d ) (8)

which follows from (6) and (7). The term E vanishes as adap- tation proceeds, the proof of which is quite complex ((61, (9)) . To calculate t ( k ) , 8(k) is required, but to calculate 8(k), ~ ( k ) is needed: this is perhaps the best illustration for the need of an a priori/ a posteriori correction. The error ~ ( k ) can be calculated by using known signals up to t = k - 1:

and so:

1005

C(z-')yp(k) - B"(k - l ) w ( k - d ) 1 + W T ( k - d ) r ( k ) w ( k - d )

c(k) =

R(z-1)

Figure 1: Primary ITR controller

(9)

The required error signal ~ ( k ) can thus be determined, making the equivalent error scheme consist of:

1. A linear part with transfer H(z-'), allowing the nonlinear part not to satisfy Popov's inequality to a certain amoun- t. Note that ~ ( k ) is known, and so the linear error transfer H ( z - ' ) equals 1 by default. Another H ( z - ' ) must be im- plemented purposely.

2. A nonlinear part containing the least-squares adaptation scheme, in which the maximum value of Xz depends on H ( z - ' ) in the linear part. If H ( 2 - I ) = 1 and A 1 = 1, Xz 5 2. Usually, Xz is chosen 1.

The described ITR method can be extended with several poly- nomials and filters to obtain existing MRAC schemes as special cases IS]. For the purpose of this paper, the lines in the derivation of ITR suffice as a basis for the next section.

Ripple-Free MRAC This section describes the design of a new MRAC scheme, making use of a 'ripple-free' or 'minimum-variance' control strategy. The characteristic of such a controller is that the process zeros are not cancelled, and thus ringing effects are avoided. In addition, nonminimum-phase plants can be controlled.

If 0 = e', the controller structure shown in figure 1 can be re- drawn as that in figure 2, in which M ( z - ' ) = z - ~ / C ( Z - ' ) . The closed-loop transfer is equal to M ( z - ' ) and resembles a mini- mum settling-time controller. A characteristic of this structure is that u(k) = (A/B)y,(k + d) , and hence process zeros appear as poles in the equation for u(k). After a change in y,(k -t d) , u(k + i) does not settle to a constant value but becomes an infi- nite series of control actions, because of the poles in the equation for u(k). On the other hand, yp(k) = ym(k) at every instant k . Note that for regulation, the loop transfer contains the factor l / C ( f ' ) , which smoothens the regulation behaviour. Because the process usually has a zero on the negative real axis in the z- plane, MSTC involves the presence of ringing and intersampling ripple because it compensates all process poles and zeros. To avoid zero cancellation, the cause of these problems, M can be modified to:

The parameter b = xzo b; is a compensation for the DC gain of B(z- ' ) . The closed-loop behaviour is known as 'ripple-free re- sponse control' (RFRC) [ 121 or 'minimum-variance control' [ 13). Because the process zeros are not cancelled, no ringing occurs. In RFRC, the control signal is u ( k ) = Ay,(k + d ) , and thus the control settles after n + 1 samples. The process zeros remain present in yp(k):

Because B(2-I) is of order m, it will take m samples before yp matches ym, and so the process response lags behind the model response. The new primary controller is depicted in figure 3. A transformation to a structure which is similar to figure 1 is on- ly possible in the non-adaptive case and is not considered here. The controller parameters in figure 3 are directly calculated from A(z-') and S(z-') , which in the adaptive case are replaced by their estimates A(z- ' ) and & ( . . - I ) . A(z- ') and E ( z - ' ) are cal- culated with the least-squares algorithm, as in ITR. Because in this case the process parameters are estimated rather than the controller parameters, other definitions for w ( k ) and B ( k ) must be used. The new parameter vector B(k) contains estimates of the process numerator and denominator coefficients bi and a,, and the corresponding signal vector w ( k ) contains series of mea- surements of the process input and output values:

Note that due to its definition, w ( k ) cannot be generated. This is not a problem because w ( k ) is needed neither in the derivation of the adaptive law, nor in the actual control algorithm. To show that the adaptation is asymptotically stable, the following type error equation is required:

In the underlying RFRC controller, signal and parameter vectors have changed. t ( k ) can be calculated as follows:

The first, term equals:

1006

model filter process

Figure 2: Alternative view of the primary controller of ITR

model filter process

Figure 9: Ripple-free primary controller structure

The required error signal for a stable adaptive law then becomes:

Substituting the adaptive law (4) yields the a priori/ a posteriori correction as in the ITR case:

y,(k) - B T ( k - l)w(k - d ) 1 + W T ( k - d ) r ( k ) w ( k - d )

c ( k ) =

Hence, the error signal t ( k ) in the ripple-free MRAC controller is calculated in a similar way as that in the original ITR controller (9); the only difference is the exclusion of C(z- ' ) in the calcula- tion of t(k), which is due to the different definition of w ( k ) and B ( k ) . C(z- ' ) does not influence the estimated parameters in the RFRC case, but is included in the explicit controller calculation.

An interesting aspect is the actual meaning of the error signal c(k). In the standard ITR case, t (k) was: ~ ( k ) = C(z-')(y,(k) - y m ( k ) ) + T ( k ) , in which ~ ( k ) = ( B ( k - d ) - B ( k ) ) T w ( k - d ) vanishes as adaptation converges. This implies that after the adaptation, yp(k) = y m ( k ) . In the RFRC controller, ~ ( k ) can be formulated as follows. The control signal is calculated as:

And SO the second term in (12) is equal to:

Combining equations (12), (13) and (15) yields:

t(k) = Y,(k) - { Y p ( k ) + W k ) - ACy,(k t d ) }

= i C y m ( k f d ) - bCu(k)

AB + d ) - 6-u(&)) A

Substituting (14) in (16) yields:

Hence, ~ ( k ) is a filtered version pf a modified plant-model er- ror. As adaptation converges, A 4 A and b -+ 6 if w ( k ) is persistently exciting, and so ~ ( k ) is the filtered difference be-

which indeed implies ripple-free behaviour. Hence, where in ITR ~ ( k ) --* 0 involved y p ( k ) 4 y,(k), in the new ripple-free con- troller y , ( / ~ ) --* ( B / b ) y , ( k ) .

tween by,(k) and B y m ( k ) . If ~ ( k ) -+ 0, Y J ~ ) -+ (Blb)y,(k),

Simulation Examples The new control algorithm has been tested in simulation on sev- eral processes. To be able to choose a forgetting factor X I < 1 in the algorithm, it is necessary to limit r(k) in some way. In all experiments, a 'limited trace' algorithm has been used, which leaves r(k) unchanged if the trace of Z'(k) is smaller than a spec- ified maximum value. If the trace is larger, r(k) is scaled to an acceptable value. This method is based on the observation that a small trace of r(k) should not be made larger than the least- squares algorithm seems fit, as is done in the 'constant trace' algorithm. The presence of ringing involves a larger richness of w ( k ) in the MSTC primary controller, resulting in a somewhat slower convergence in the RFRC case, as will be shown below. The forgetting factor A 1 = 0.98, and X2 = 1. The first simulations were performed with the process and model transfer functions, respectively:

8 s Z + s + 8

W,(S) = ~ poles a t s = -0.5 f 2.8~'

l6 poles a t s = -4 (2x1 wm(s) == s2 + 8 s + 1 6

Figure 4 shows the results with the original MSTC algorithm (C(z - ' ) = I), resulting in fast convergence but a ringing control signal, especially directly after starting the adaptation. The (hor- izontal) magnification shows what really happens to the process output y , (k ) : at the sampling instants y,(k) = y,(k), but in between an intersampling ripple is present. Figure 5 shows re-

1007

time (sec)

Figure 4: Response with MSTC-MRAC, C = 1

5 1 0 5 10 IS 20 2s

time (sec)

Figure 5: Response with RFRC MRAC

sults with the new RFRC algorithm. The adaptation converges somewhat slower, and there is a remaining delay of one sampling interval between yp(k) and y,(k). The control signal has howev- er become much smoother and the ringing effect, together with

Figure 6: RFRC MRAC applied to a nonminimum-phase plant

the intersampling ripple, has disappeared. In figure 6 the results with a nonminimum phase process:

-4s + 8 = s2 + 2s + 8

are shown, with the same reference model and controller para- meters. The MSTC scheme becomes unstable immediately after starting the adaptation and is not shown here. Obviously, the nonminimum-phase characteristics remain because the process zeros are not cancelled.

A Practical Application The RFRC algorithm has also been tested on a fluid level process, which is shown in figure 7 and can be described as follows. The

Figure 7: The water level process

lower part of the system consists of a large container filled with water. Out of this container water is pumped, via pipes and valves, to the water column. Three valves influence the water flows. Actuator valve S3 controls the water input to the column. This valve is connected, via an electropneumatic transducer, to the control computer (a pVAX I1 workstation), equiped with the real-time shell MUSIC (Multi-purpose SImulation and Control). Valve S2 is necessary for the pump to operate properly when S3 is closed. Valve S1 determines the outflow and thus makes it possible to disturb the flow of water out of the column. The water level in the column is measured with a pressure sensor at the bottom. The relations between the level in the water column, pressure of the sensor, outflow in linearized form and inflow are respectively:

I S ( S ) - US(5) N ( s ) = N ( 0 ) + H ( s ) = 1.9 -I- 0.067N(s)

US(S) = U, + I (UN(s) 3.6V (s)e-0.6ss

1 + 0.5s I S ( S ) =z -6+

In these formulas N is the water level in scale divisions, H is the output of the pressure sensor, US is the outflow, IS is the inflow and V is the voltage on the electropneumatic transducer conlrolling valve S3. V is limited between 0 and 10 Volts, causing an asymmetry in the response between the rising and dropping rate of the water level, depending on valve SI. The parameters U, and Ku, influencing the outflow, also depend on the setting of valve SI. For outflow settings of 30%, 40% and 50010, respectively, I(rr has a value of 0.02, 0.04 and 0.06. Neglecting offsets and higher-order effects, an approximation of the process by a first- order system:

-- H ( s ) Ke-"a V ( s ) ST + 1

results in a value of K of 12,6 and 4, and a value of T of 50,25 and 17 seconds, respectively. Figure 8 shows the tracking properties

1008

10 1

I 0 50 100 150 200 250

10

time (sec)

Figure 8: Tracking properties of RFRC MRAC

of the RFRC algorithm. The reference model has been chosen such that a t a nominal outflow of 40%, the outflow rate can just follow the reference model. It is seen that after initial startup overshoot, the response approaches the reference model response. Figure 9 shows the regulation results, again starting with wrong

I 50 100 150 200 250

time (sec)

Figure 9: Regulation properties of RFRC MRAC

parameter estimates. At t = 100 sec an input disturbance is applied by releasing a lot of water from the column, after which the outflow is changed from 40% to 30%. At 1 = 150 sec, the same is done, changing from 30% to 50%. It is seen that the regulation properties are satisfactory.

Conclusions It is possible to change the primary control structure of a MRAC scheme from the usual Minimum Settling Time-type algorithm to a Ripple-Free Response-like scheme. RFRC does not try to com- pensate process zeros, avoiding ringing effects (like intersampling ripple). The main issue in the derivation of the RFRC-MRAC scheme is the establishment of a stable adaptive law which guar- antees a ripple-free behaviour if the error signal tends to zero. In the original MRAC scheme, the error signal consisted of the desired error plus a transient term which tends to zero when adaptation proceeds. In the new MRAC scheme, the error con- sists of a filtered form of the desired RFRC error. Hence, if the error used in the adaptation tends to zero, the RFRC behaviour

is achieved. Note that a process zero on the unit circle cannot be dealt with, because this would yield b = 0 and hence l / b , used in the control law, cannot be calculated. The new scheme avoid- s ringing effects and can control nonminimum-phase processes. The cost of these advantages is a slowing down of the process response by a number of samples equal to the order of the nu- merator polynomial. Simulations confirm the RFRC-like charac- teristics and practical results on a water level system show the usability of the method.

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