Robust controller design based on model

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Kybernetes 31,1

76

Robust controller design based on model reference approach Manuel A. Duarte-Mermoud Electrical Engineering Department, University of Chile, Santiago, Chile

Received June 2001

Ignacio Chang J. Electrical Engineering Department, Technical University of Panama, Panama Keywords Cybernetics, Control Abstract This paper presents a method for designing a ®xed controller, which is able to control in a stable fashion a family of linear time-invariant n-th order plants with arbitrary relative degree. This plant family is de®ned in terms of a nominal transfer function of rational type and bounded variations of the coef®cients of numerator and denominator polynomials. The design method is based on the algebraic relationship existing in the model reference adaptive control technique between the true plant parameters, the ideal controller parameters and the model reference parameters. While applying the proposed method, the resulting plant families are broader if compared with other techniques used to design robust controllers.

1. Introduction None of the real plants can be exactly modelled and there will always be uncertainties present in the resulting model of the plant. These uncertainties can be re¯ected in the plant parameters and usually they are quanti®ed by numeric intervals. Robust control attempts to design a ®xed controller for a speci®c model of the plant (considered representative in some sense) which is able to control in a stable fashion the plant even in the presence of uncertainties arising from a non suitable modelling process. That is one of the main reasons why robust control has become an important design tool in industrial Ê stroÈm et al., 1995). application (A Several researchers have related adaptive techniques with robust control mainly because of the advantages that can be reached (Hinrichsen and Martensson, 1990). As an example, there exist several robustness studies using multiple reference models (Ciliz and Narendra, 1994; Narendra and Balakrishnan, 1993) for a robot manipulator to improve the tracking capacities. A design methodology for robust controllers based on the m-synthesis is proposed in Prempain and Bergeon (1995). Another approach is due to Soh (1989), who chose from a set of controllers the most suitable according to an Kybernetes, Vol. 31 No. 1, 2002, pp. 76±95. q MCB UP Limited, 0368-492X DOI 10.1108/03684920210413773

The results reported in this paper have been funded by CONICYT through grant FONCECYT 1970351. The help provided by SENACYT from Panama is greatly appreciated by the second author.

optimisation process from a pole-placement viewpoint. Recently, Ozcelik and Robust controller Kaufman (1995) presented a methodology to design robust controllers for a design maximum range of plant parameter variations based on Kharitonov stability results, to determine the so called feedforward compensator. The aim of this paper is to propose a methodology to design robust controllers simple in essence so that a ®xed controller is able to guarantee the 77 stability of a properly de®ned plant family. Based on the relationships existing in MRAC between true plant parameters, ideal controller parameters and model reference parameters (Narendra and Annaswamy, 1989), variations in the model reference parameters (around their nominal values) are produced maintaining ®xed the ideal controller parameters and characterising the set of plants that can be controlled in a stable fashion with the unique controller. 2. Problem statement Let us consider the general model reference control scheme shown in Figure 1. Gm(s, E ) denotes a family of asymptotically stable reference models belonging to set M with parameters in set E, of the form Gm (s; E) = km

Z m (s; A) Rm (s; B)

M is the set of all possible model references de®ned by a rational transfer function which are asymptotically stable with zeroes in the open left half of the complex plane. Edenotes the hyperrectangle de®ned by E = {[a T b T ]T [ R n+ m =ai [ [ai ; a+i ]; bj [ [bj ; b+j ]; for i = 0; 1; . . .; m 1; j = 0; 1; . . .; n 1} Superscripts 2 and + denote minimum and maximum values of polynomial coef®cients ai and bj, respectively. km [ R is a known positive constant referred to as the high frequency gain of the model reference. Zm(s,A ) and Rm(s,B ) represent known monic and Hurwitz interval polynomials with parameters in hyperrectangles A and B, of degrees m and n (m , n ), respectively, de®ned as:

Figure 1. General model reference control scheme

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Z m (s; A) = s m + am 1 s m 1 + ´ ´ ´ + a1 s + a0 ; with aj [ [aj ; aj ]; j = m 1; . . .; 1; 0

78

R m (s; B) = sn + bn 1 s n 1 + ´ ´ ´ + b1 s + b0 ; with bi [ [b2i ; b+i ]; i = n 1; . . .; 1; 0 A and B are hyperrectangles de®ned as: £ ¤ A = {a [ R m =ai [ ai ; a+i ; i = 0; 1; . . .; m 1; with am = 1 £ ¤ B = {b [ R n =bi [ bi ; b+i ; i = 0; 1; . . .; n 1; with bn = 1

For notation purposes, the vectors a = [am 2 1, . . ., a1, a0]T [ R m, b = [bn 2 1, T . . ., b1, b0]T [ R n and r = b km a T b T c [ R n+ m+ 1 are de®ned. Let us characterise the plant family Gp(s, F ) belonging to set P with parameters in F by Gp (s; F) = kp

Z p (s; C) Rp (s; D)

P is the set of all possible plants de®ned by rational transfer functions with degree n and m and zeroes in the open left half of the complex plane. Kp [ R is a constant corresponding to the plant high frequency gain. Zp(s, C ) and Rp(s, D ) are monic interval polynomials with Zp(s, C ) Hurwitz, of degrees m and n (m , n ), respectively, de®ned as: £ ¤ Z p (s; C) = sm + cm 1 s m 1 + ´ ´ ´ + c1 s + c0 ; with cj [ cj ; cj + ; j = m 1; . . .1; 0

£ ¤ Rp (s; D) = sn + d n 1 s n 1 + ´ ´ ´ + d 1 s + d 0 ; with di [ di ; di + ; i = n 1; . . .; 1; 0

For notation purposes,£vectors ¤cT= [cm 2 1, . . ., c1, c0]T [ R m, d = [dn 2 1, . . ., d1, d0]T [ Rn and p = kp c T d T [ R n+ n+ 1 are de®ned. C, D and F are hyperrectangles de®ned similar to A, B and E, respectively, which are going to be determined in the design process. Let us consider a nominal plant and a nominal model reference de®ned as Gp (s) =

kp Z p (s) km Z m (s) and Gm (s) = Rp (s) R m (s)

Then, it is possible to ®nd the ideal controller parameters (Narendra and Robust controller Annaswamy, 1989) denoted by the vector [k ¤T u¤1 T u ¤0 u¤2 T ] [ R 2n for the design control scheme shown in Figure 2, with u¤1 T = [u ¤1;0 u ¤1;1 ´ ´ ´ u ¤1;n 2 ] [ R n 1 and u¤2 T = [u ¤2;0 u ¤2;1 ´ ´ ´ u¤ 2;n 2 ] [ R n 1 . The ideal controller parameter u ¤ , is such that the overall transfer function (plant together with the controller) equals the model reference transfer function. 79 Signals v1(t ), v2(t ) [ R n 2 1 are ®ltered versions of u(t ) and y(t ), respectively, de®ned as

v_1 (t) = Lv1 (t) + l u(t) v_2 (t) = Lv2 (t) + l yp (t) The pair (L, l ) is any controllable pair, with L [ R (n 2 1) £ (n 2 1) an asymptotically stable matrix and u ¤1 ; u ¤2 [ R n2 1 . Since (L, l ) can be chosen arbitrarily, the following controllable canonical structure is chosen 2 3 2 3 0 1 0 6 7 . 6 7 6 0 .. 7 6 ... 7 6 7 7 7 l = 6 L= 6 6 7 6 .. 7 6 7 6 . 1 7 405 4 5 1 l0 l1 ´ ´ ´ ln 2 This choice makes simple the relationships between parameters u ¤ , p and r. We denote the polynomials

u ¤1 (s) = u ¤1;n 2 s n 2 + u ¤1;n 3 s n 3 + ´ ´ ´ + u ¤1;1 s + u ¤1;0

Figure 2. Structure of the MRAC

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u ¤2 (s) = u ¤2;n 2 s n 2 + u ¤2;n 3 s n 3 + ´ ´ ´ + u ¤2;1 s + u ¤2;0 L(s) = s n 1 + ln 2 s n 2 + ´ ´ ´ + l1 s + l0

80

Equating the transfer function of the nominal plant together with the ideal controller and the nominal model reference we get k ¤ kp Z p (s)L(s) k Z (s) £ ¤ £ ¤= m m Rm (s) Rp (s) L(s) u ¤1 (s) kp Z p (s) u ¤2 (s) u ¤0 L(s)

(1)

Then, given the true plant parameters p [ R n+m+1 of nominal plant Gp(s ), and the nominal parameters r ¤ [ R n+ m+ 1 of the model reference Gm(s ), the ideal controller parameters u * [ R 2n can be obtained. Also, it is possible to obtain the family of plants Gp(s, F ) to be controlled in a stable way with the same ®xed ideal controller de®ned by u ¤ , assuming that some variations are allowed on the parameters of the model reference Gm(s, E ), represented by E, around the nominal model reference Gm(s ). This is what is done in the next section. 3. Design procedure for robust controller In this section the design procedure is explained considering two cases; unit relative degree and relative degree greater than one. The ideal controller parameters u ¤ are ®rst determined from the relationship existing between the true plant parameters p, ideal controller parameters u ¤ and model reference parameters r for the MRAC scheme (Narendra and Annasawmy, 1989). Keeping ®xed these parameters u ¤ and considering parameters variations in the model reference, the family of plants which is controlled in a stable form by the same ®xed controller de®ned by parameters u ¤ is determined. In order to guarantee the stability of the set controller-plant family, the Kharitonov Theorem is used (Barmish, 1994), which has been extended by several authors (Bose and Shi, 1987; Minnichelli et al., 1989). 3.1 Plants with unity relative degree (Chang, 1997; Chang and Duarte, 1998a) Let us assume that the relative degree of the plant is one, that is to say the plant order is nand m = n 2 1. In order that relationship (1) be satis®ed, values of k ¤ ; u¤ 0 [ R and u¤ 1 , u¤ 2 [ R n2 1 are chosen in a certain fashion. Let us choose k¤ =

km kp

Since the degree of polynomials L(s ) and Zm(s ) is n 2 freedom to impose that L(s) = Zm (s)

(2) 1, there is enough (3)

Using equations (2) and (3), relationship (1) becomes £ ¤ £ ¤ R p (s) L(s) 2 u¤1 (s) 2 kp u¤2 (s) + u¤0 L(s) = R m (s)

(4)

Robust controller design

To determine u ¤1 we impose that

Z p (s) = Z m (s) u ¤

1

(s)

(5)

since the coef®cients of Zm(s ) are arbitrary. Then we can write c0 + c1 s + ´ ´ ´ + cn 2 s n 2 + s n 1 = a0 + a1 s + ´ ´ ´ + an 2 s n 2 + s n 1 u ¤1;0 u ¤1;1 ´ ´ ´ u ¤1;n 2 s n 2

(6)

Equating the coef®cients we obtain the following relationships between the controller parameters u ¤1 [ R n2 1 and parameter vectors a and c [ R n2 1

u ¤1;0 = a0 c0 u ¤1;1 = a1 c1 (7)

.. .

u ¤1n 2 = an 2 cn 2 Replacing equation (5) in equation (4) we get: £ ¤ R p (s) kp u ¤2 (s) + u ¤0 Z m (s) = Rm (s)

(8)

Equating the coef®cients of the polynomials we get the following relationships between parameters u ¤0 [ R; u ¤2 [ R n2 1 and parameters b and d [ R n2 1

u ¤0 =

d n 1 bn 1 kp

(9)

and

u ¤2;0 =

d 0 b0 u ¤0 a0 kp

u ¤2;1 =

d 1 b1 u ¤0 kp kp

.. .

(10) u¤

u ¤2n 3 =

d n 3 bn 3 kp

u ¤2n 2 =

d n 2 bn 2 u ¤0 an 2 kp kp

a k 0 n 3 p

The set of equations (2), (7), (9) and (10) allow us to determine the ideal controller parameters u ¤ [ R 2n , for a nominal plant characterised in terms of

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82

parameters c and d [ R n 2 1, which are the coef®cients of polynomials Zp(s ) and Rp(s ), respectively, and for a nominal model reference de®ned in terms of parameters a and b [ R n2 1, which are the coef®cients of polynomials Zm(s ) and Rm(s ), respectively. Once the ideal controller parameters are obtained and kept constant a plant family of the form Gp(s, F ) which is controlled in a stable way with the same ®xed controller can be obtained. To do this we assume that the nominal model reference admits some parameter variations (in a stable fashion) generating a family of the form Gm(s, E ). From equations (5) and (8) and considering the existence of interval polynomials in the numerator (Zm(s, A )) and denominator (Rm(s, B )) for the nominal model reference transfer function Gm(s ) giving stable transfer functions, the following relationships can be stated: Z p (s; C) = Z m (s; A) u ¤1 (s) £ ¤ R p (s; D) = Rm (s; B) + kp u ¤2 (s) + u ¤0 Z m (s; A)

These expressions allow us to characterise a plant family given by Gp(s ) = kpZp(s, C )/Rp(s, D ), keeping constant the controller parameters u ¤ and providing some degrees of freedom for the model reference represented by Gm(s, E ) = kpZp(s, A )/Rp(s, B ). To verify that the resulting plant family together with the ®xed controller are globally stable the Kharitonov Theorem is used (Barmish, 1994) which states that an interval polynomial is stable if at most four corner polynomials obtained from the upper and lower bounds on parameters intervals are stable. Often, the resulting set controller-plant family is unstable since the controlled system obtained in this manner broaden the range from Rm(s, B ) to 0 Rm (s, B ), generating a new family of model references varying from Gm(s, E ) to 0 (s, E ) that might not belong to M. Therefore, a search procedure for the plant Gm family is established so that the system controller-plant family is stable for all members of the plant family. To clarify the above statement the following illustrative example is presented. Let us assume that the nominal plant Gp(s ) and the nominal model reference Gm(s ) are given by Gp (s) =

s 2 + 10s + 25 s 3 6s 2 + 11s + 6

Gm (s) =

s 2 + 4s + 3 s 3 + 10s 2 + 32s + 32

From equations (2), (7), (9) and (10) the ideal controller parameters turn out to be

u ¤ = [1; 22; 6; 16; 10; 43]T [ R 6 Let us consider model reference parameter variations characterised by Z m (s; A) = {s 2 + a1 s + a0 } with a1 [ [3:2; 4:8] and a0 [ [2:4; 3:6]

Rm (s; B) = {s 3 + b2 s 2 + b1 s + b0 }

(11a)


with b2 [ [8; 12], b1 [ [25.6; 38.4] and b0 [ [25.6; 38.4]. Equations (9) and (10) in this particular case become d 0 = b0 + kp u ¤0 a0 + kp u ¤2;0

d 1 = b1 + kp u ¤0 a1 + kp u ¤2;1

d 2 = b2 + kp u ¤0

Assuming the model reference parameter variations de®ned in equation (11a), the expression to obtain the variations of coef®cient d0 takes the following form: i £ + ¤ £ + ¤ h¡ ¤ ¢ ¡ ¤ ¢+ i h ¤ d 0 ; d 0 = b0 ; b0 + kp u 0 a0 ; kp u 0 a0 + kp u 2;0 ; kp u ¤2;0 (11b) Substituting the parameter values we get £ +¤ d 0 ; d 0 = [25:6; 38:4] + [ 16 £ 3:6; 16 £ 2:4] + [10; 10] £ +¤ d 0 ; d 0 = [ 22; 10] Similarly, the variations of the other parameters turn out to be £ +¤ £ ¤ d 1 ; d 1 = [ 8:2; 30:2] d 2 ; d +2 = [ 8; 4]

Now comes the question whether the set plant family-®xed controller equals the model reference family de®ned by equation (11a). To answer that question £ ¤ we obtain the interval for b0 given by equation (11b), denoting it by b00 ; b0 +0 and then we compare it with that obtained from equation (11a). Thus, h i £ i ¤ £ ¤ h + b0 0 ; b0 0 = d 0 ; d +0 + kp u ¤0 a0 ; kp u ¤0 a+0 kp u ¤2;0 ; kp u ¤2;0 (11c) h i 0 0+ b 0 ; b 0 = [6:4; 57:6]

Following the same above reasoning we obtain for the other parameters the following intervals, h i h i + + b0 1 ; b0 1 = [0; 64] b0 2 ; b0 2 = [8; 12] (11d)

Considering the initial interval given by equation (11a) and ®nal intervals given by equations (11c) and (11d), only that corresponding to parameter b2 coincides, whereas all others have been broadened. This clearly means that Rm(s, B ) has 0 suffered an enlargement to Rm (s, B ) with respect to the original interval de®nition. The aim of the above mentioned search procedure is to guarantee that the system controller-plant family be stable for all members of the plant family. From sensitivity analysis given in (Karnavas et al., 1993) it can be stated that variations in the zeroes of the model reference have more in¯uence in the pole

83

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placement of the controlled plant than variations in the poles of the model reference, as long as u ¤0 kp . 1. This can be seen from the sensitivity coef®cients resulting of the sensitivity analysis, which are given by di = 1 bi

84

di = kp u ¤0 ai

The proposed search procedure is stated as follows: Step 1. To reduce the interval polynomial Zm(s, A ), keeping constant the interval 0 (s, B ). Now it polynomial Rp(s, D ). The aim is to bring Rm(s, B ) to Rm is veri®ed if with this reduction of Zm(s, A ) stability of plantcontroller is achieved. Step 2. If after several reductions in Zm(s, A ) we reach the nominal value Zm(s ), without reaching stability, a small reduction in Rm(s, B ) is produced, keeping Zm(s, A ) in its nominal values. With this Rp(s, D ) reduces to R p0 (s, D ), and stability is analysed under these new conditions. If stability is not reached, Step 1 is repeated. Step 3. Steps 1 and 2 are subsequently repeated until a set of controller-plant family is obtained, which is controlled in a stable fashion for the same ®xed controller de®ned by u ¤ . Figure 3 shows schematically the above search procedure corresponding to the particular case of equation (corresponding to equations (9) and (10)) £

h i ¤ £ ¤ £ ¤ d i ; d +i = bi ; b+i + kp u ¤0 ai ; a+i + kp u ¤2;j ; u ¤2;j ; ; i = n 1; . . .1; 0

with u *2,j = 0, ; j = n 2

Figure 3. Search procedure to ®nd the plant family

1, u *2,j Þ

0, ; j = n 2

2, . . ., 1, 0.

3.2 Plants with relative degree greater than one (Chang, 1997; Chang and Robust controller Duarte, 1998b) design We have to point out that in the following study only parameter variations in the denominator of the nominal model reference are considered. This is because the relationships between nominal plant parameters, nominal model reference parameters and ideal controller parameters for plant of arbitrary relative 85 degree are much more complex if parameter variations in the numerator and denominator are allowed. This means that zeroes of the model reference are ®xed and so are those of the plant. The design procedure here is similar to that explained in the previous section, except that the relationships between true plant parameters, ideal controller parameters and model reference parameters are different. From equation (1) and since the relative degree is greater than one, the Bezout Identity plays an important role (Middleton and Goodwin, 1990). The arbitrary polynomial L(s ) has to be chosen containing Zm(s ) as a factor, i.e. L(s ) = Zm(s )L1(s ), L1(s ) being an arbitrary Hurwitz polynomial of degree n 2 m 2 1. Then, L1(s ) and L(s ) have the form L1 (s) = s n 2 + l1;n 3 s n 3 + ´ ´ ´ + l1;0 L(s) = Z m (s)L1 (s) = s n 1 + ln 2 s n 2 + ´ ´ ´ + l0 Choosing k¤ =

km kp

(12)

Equation (1) becomes £ ¤ £ ¤ Rp (s) L(s) u ¤1 (s) kp Z p (s) u ¤2 (s) + u ¤0 L(s) = Z p (s)Rm (s)L1 (s)

It can be shown from the MRAC that factor (L(s ) 2 u *1(s )) presents in the above equation can be expressed as a product containing Zp(s ). Thus, we can write L(s) = L(s) u ¤1 (s) = l (s)Z p (s)

(13)

with L(s) = s n 1 + Ln 2 s n 2 + ´ ´ ´ + L0 l (s) = s n m 1 + l n m 2 s n m 2 + ´ ´ ´ + l 0 L(s ) monic and P(s ) are of degree n 2 degree n 2 m 2 1.

1 and l (s ) is a monic polynomial of

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We de®ne P(s) = u ¤2 (s) + u ¤0 L(s)

(14)

Rp (s)l (s) = kp P(s) + L1 (s)Rm (s)

(15)

Thus, equation (1) becomes

86

From relationship (13) we get,

u ¤1;0 = l0 L0 u ¤1;1 = l1 L1 .. .

(16)

¤

u 1;n 2 = ln 2 Ln 2 u ¤1;n 3 = ln 3 Ln 3 Similarly, from relationship (14) we obtain

u ¤0 = P n 1 u ¤2;0 = P 0 u ¤0 l0 u ¤2;1 = P 1 u ¤1 l1 .. .

(17)

u ¤2;n 2 = P n 2 u ¤0 ln 2 Finally, equations (12), (16) and (17) allow us to determine the ideal controller parameters u ¤ , given the nominal plant parameters of Gp(s ) and nominal model reference parameters of Gm(s ). If we consider parameter variations in Rm(s ) of the type Rm(s, B ) and keeping constant parameter u ¤ , from equation (15) it is possible to obtain a plant family expressed as Rp(s, D ), which is de®ned by the following relationship R p (s; D)l (s) = kp P(s) + L1 (s)Rm (s; B) which implies the solution of the Bezout Identity. Once the plant family Rp(s, D ) is obtained the stability of the plant-controller is checked for all members of the plant family Rp(s, D ) together with the ®xed controller de®ned by parameters u ¤ . This is done by using the Kharitonov Theorem (Barmish, 1994). If stability is not achieved, the search procedure de®ned in Section 3.1 is used. The design procedure described in Sections 3.1 and 3.2 for n ¤ = 1 and ¤ n . 1, respectively, was programmed in a software called CRVMMR,

developed in MATLAB. The program is able to design robust controllers for Robust controller plants up to order ®ve with relative degrees up to four. It allows the ideal design controller parameters, the percentage of variation of model reference parameters (after the search procedure) and the plant parameter variations (absolute value and percentage) which de®ne the plant family stabilised by the ®xed controller de®ned by u ¤ to be displayed.

87

4. Design examples 4.1 Relative degree one In this section the design methodology proposed in this paper is applied to a third order plant. Let the third order plant with unity relative degree be de®ned by yp 0:5yp 2_ yp + yp = u + 3_ u + 2u (18) and the asymptotically stable model reference described by ym + 3ym + 3_ ym + ym = r + 4_ r + 4r

(19)

Initially, a parameter variation of 10 per cent in all coef®cients of the numerator of the model reference and 20, 25 and 30 per cent in the parameters of the denominator of the model reference are produced. km and kp parameters are assumed to be unity and kept constant. Applying the design procedure described in Section 3.1, the following results are obtained: The ®nal parameter variations for the model reference (after the search procedure) are shown in Table I. The ideal controller parameters denoted as: h iT T T u ¤ = k ¤ ; u ¤ 1 ; u ¤0 ; u ¤ 2 [ R 6 with u ¤1 ; u ¤2 [ R 2 are de®ned as

u ¤ = [1; 2; 1; 3:5; 14; 9]T [ R 6 The plant family to be controlled by the ®xed controller given by u ¤ is de®ned as kp Z p (s; C) Gp (s; F) = Rp (s; D) where Z p (s; C) = {s 2 + c1 s + c0 with c1 [ [2:9; 3:1] and c0 [ [1:9; 2:1]}

Numerator parameters Denominator Parameters

Percentage of initial variation

Percentage of ®nal variation

[10 10] [20 25 30]

[2.5 2.5] [20 25 30]

Table I. Variations of model reference parameters

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and R p (s; D) = {s 3 + d 2 s 2 + d 1 s + d 0 with d 2 [ [ 11; 0:1]; d 1 [ [ 3:1; 0:9] and d 0 [ [0:35; 1:65]}

88

Table II. Plant parameter variations

Figure 4. Scheme proposed by Soh (1989) for designing robust controllers

obtained by applying the Karithonov Theorem. Finally, plant parameter variations around their nominal values are shown in Table II. Remark If other initial parameter variations for the numerator coef®cients of the model reference, not necessarily equals, are de®ned (e.g. greater than 10 per cent) the resulting plant family not necessarily is broader in the sense that interval polynomials are larger. This is due to the fact that in applying the search procedure de®ned previously not only the model reference family can be reduced but also the plant family. For example if parameter variations of 20 and 30 per cent are used for the coef®cients of the numerator of the model reference (instead of 10 pr cent for all parameters used in the previous example), the design procedure provides plant parameters variations of 4.8 and 1.06 per cent, for the numerator coef®cients and 63.6, 43.1 and 120 per cent for the denominator coef®cients respectively. On the other hand, an increase in parameter variations of both numerator and denominator of the model reference could enlarge the range of parameter variations of the plant denominator. In what follows the proposed technique in this paper is compared with that proposed by Soh (1989) which designs a robust controller using an algorithmic procedure. The second order plant of unity relative degree proposed by Soh in his paper is analysed with both methods for the sake of comparison. Soh proposes the control scheme shown in Figure 4 such that the nominal vector of controller parameters, denoted by x*, guarantees the stability of the

Numerator parameters Denominator parameters

Nominal plant

Percentage of variation

[1 3 2] [1 2 0.5 2

[5 3.3] [65 55 120]

2 1]

closed-loop system for a plant family properly de®ned. To this extent vector x Robust controller is ®rst computed using the Bezout Identity and considering the desired location design of closed-loop poles in the open left half of the complex plane, a set of robust controllers is obtained. To choose the most suitable robust controller for the plant parameter variations an optimisation technique is used. Soh considers the second order plant de®ned by the transfer function

89

G(s; a; b) =

s2

s + b0 + a1 s + a0

(20)

where the parameter variations are de®ned as b0 [ [1; 1]

a1 = 2:2 + y1 + 2y2

a0 = 2:4 + 2y1 + y2

(21)

with yi [ [ 2 1; 1], for i = 1, 2. The closed-loop poles are chosen to be located at 2 1 ± j0.7 and 2 10.0. The way in which parameter variations on a1 and a0 are chosen satis®es the ªLumping Theoremº and it is necessary to compare with the design technique proposed in this paper. It can be noticed that the plant chosen by Soh is of relative degree one and uses the pole-placement technique, so that no parameter variations of the numerator are considered. The controller proposed by Soh (1989) has the form C(s) =

s + l0 p1 s + p2

(22)

Thus, the ®xed controller, after the optimisation procedure, becomes x* = [1.6643 10.1357 18.8942]T for the given yis and maximum admissible variations. The resulting plant family is given in Table III. Following the design procedure proposed in this paper for the second order plant chosen by Sho, and choosing the poles of the model reference in 2 3 and 2 4, we obtain the results shown in Table IV. From Tables III and IV it is observed that the percentage of parameter variations for plant transfer function obtained with the proposed method are considerably greater than those obtained by using the Soh method. Even though both methodologies are simple, in the Soh method it is necessary to perform a posterior analysis to determine the maximum uncertainty that parameters of plant denominator can admit, to then choose the

Parameters b0 a1 a0

Nominal plant

2

2

1 2.2 2.4

Range variation


± 2 5.2 to 0.8 2 154 to 0.6

± 136 125

Table III. Plant parameter variations obtained by Soh’s method

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suitable robust controller. To this extent, a non-simple optimisation procedure is used. In the method proposed here this is automatically obtained, the controller always being ®xed. In addition, variations of plant zeroes can also be determined by the method proposed in this paper, an aspect not considered by Soh’s method.

90 4.2 Relative degree two The method proposed in this paper was applied to one of the plants presented by Ozcelik and Kaufman (1995), based in turn on the problem studied by Masten and Cohen (1990) which was used to compare different adaptive control techniques. The plant and the model reference chosen are of relative degree two and they are de®ned by Plant: Gp (s) =

s2

K + a1 s + a0

(23)

s2

1 + 1:4s + 1

(24)

Model reference: Gm (s) =

The results obtained by applying the method proposed in this paper are summarised in Table V for three different cases. The design procedure considered that poles of the arbitrary polynomial L1(s ) of degree one, were

Table IV. Parameters Plant parameter variations obtained from b0 a1 the proposed method a0

Table V. Parameter variations obtained from the proposed method

Nominal ref mod.


1 7 12

± 90 95

1.Nominal mod ref. Nominal plant 2. Nominal mod ref. Nominal plant 3. Nominal mod ref. Nominal plant

Parameters K, a1, a0


1.0, 2.4, 3.0 0.5, 1.4, 1.0 1.0, 2.4, 3.0 1.75, 1.4, 1.0 1.0, 2.4, 3.0 3.0, 1.4, 1.0

[86; 90] [147; 305] [86; 90] [147; 305] [86; 90] [148; 305]


Nominal plant 2 2

1 2.2 2.4

± 286 475

Controller parameters [0.5; 2 [1.75; 2

10 2

1; 2 1; 2

[3.0; 3.4; 2

; 4.9 £ 10 2

3

286; 11.4 £ 105]

1.9; 6.3]

5

]


91

Figure 5. Robust control scheme proposed in this work for the second order example

located at 2 500 in the ®rst two cases and at 2 5000 in the third case, resulting in plant families of different sizes. The three cases were studied for different values of the plant high frequency gain kp; 0.5, 1.75 and 3.0. This is because the proposed methodology does not consider parameter variations of plant high frequency gain, unless the plant is of ®rst order. Table V shows the parameter variations for the plant family obtained by applying the proposed design procedure, together with the ideal parameter controller and model reference parameter variations. The control scheme is shown in Figure 5 for this particular case, where parameters k*, u*0 [ R and u*1, u*2 [ R are the ®xed controller parameters obtained from the nominal plant and nominal model reference. For the sake of comparison the design methodology developed in Ozcelik and Kaufman (1989) was also applied to the plant de®ned by equation (23). In Table VI, parameter variations obtained by applying the methodology presented by Ozcelik and Kaufman (1989) are presented. The corresponding robust controller is shown in Figure 6. The determination of the controller structure is based on the simpli®ed MRAC scheme proposed in Bar-Kana and Kaufmann (1985) and Kaufman and Neat (1993) which uses a feedforward compensator H(s ) in the nominal model reference and nominal plant, as shown in Figure 6. The procedure consists of determining the feedforward compensator H(s ) which in this case results to be H (s) =

0:5 0:025 + 2 s + 100 (s + 5)

(25)

In the scheme shown in Figure 6, ke(t ), kx(t ) and ku(t ) are adjustable gains which for this example turn out to be the following:

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Figure 6. Robust controller of Ozcelik and Kaufman (1995)

9

2

7

ke (t) = 9 £ 10 e(t) + 1 £ 10

Z

kx1 (t) = 2 £ 104 xm1 (t) + 2 £ 103 4

3

kx2 (t) = 2 £ 10 xm2 (t) + 2 £ 10

t

e(t)2 dt t0

Z Z

ku (t) = 18 £ 104 um (t) + 1:5 £ 103

t

xm1 dt t0 t

xm2 (t)dt t0

Z

t

um (t)dt t0

The control law is de®ned as up (t) = e(t)ke (t) + xm1 (t)kx1 (t) + xm2 (t)kx2 (t) + um (t)ku (t)

(27)

where xm1(t ) and xm2(t ) are the state variables of the model reference.

Table VI. Parameter variations using the method by Ozcelik and Kaufman (1989)

Parameter K a1 a0

Nominal mod. ref.

Nominal plant

1 1.4 1

1.75 1.4 1

Plant range

2

2

0.5±3.0 0.6 to 3.4 2.0 to 4.0

Percentage of variation 125 143 300

It can be observed that the design procedure proposed in this paper is much Robust controller simpler than that presented by Ozcelik and Kaufman (1995). For the proposed design method with the parameter variations chosen for the nominal model reference the range obtained for the plant family is larger than that obtained using the method of Ozcelik and Kaufman in all the three studied cases, i.e. for kp = 0.5, 1.75 and 3.0. If the nominal model reference parameters are chosen differently 93 even broader plant families could be obtained with the proposed method. Another important fact in the method of Ozcelik and Kaufman (1995) is that the resulting control signal up(t ) is rather complex and for the studied example the gains are quite high, giving potentially numerical problems in practical applications. In the proposed methodology the resulting control signal is much simpler and coef®cients are relatively small.

5. Conclusions and ®nal remarks A simple methodology has been derived for designing a robust controller obtained from a nominal model reference and a nominal plant, which is able to control in a stable fashion a plant family de®ned around the nominal plant. The method is based on the algebraic relationships existing in the MRAC relating the ideal controller parameters, the true plant parameters and the parameters of the model reference. These relationships allow us to determine a plant family de®ned in terms of interval polynomials, keeping constant the ideal controller and allowing some parameter variations in the model reference. Once the plant family is obtained, stability of the controller-plant family system is checked for all member of the plant family by using Kharitonov’s Theorem. A search procedure that always provides a ®xed controller, a model reference family and a plant family guaranteeing stability of the closed-loop system, has been also developed. The proposed methodology allows us to obtain wide ranges of parameter variations around the nominal plant. The range amplitude depends on the choice of the parameter variations for model reference around their nominal values, as well as the choice of the poles of the arbitrary and stable polynomial L1(s ). The farther from the origin the poles are, the wider the resulting range, since L1(s ) is a factor multiplying the polynomial denominator of the model reference family. To successfully apply the proposed methodology it is suggested to perform several trials with different sizes of the parameter variations of the model reference, as well as several trials with different pole locations of polynomial L1(s ) in order to obtain the widest plant family controlled with the same ®xed controller. Compared with other robust control designs, the proposed scheme is able to provide broader plant families controlled in a stable fashion by the ®xed controller. A mechanism of broaden the range of the parameter variations is also available in the method. The proposed algorithm has been programmed in MATLAB, allowing a quick interactive utilisation by the user through an easy to use interface, for

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robust controller design for plants up to order ®ve and relative degree up to four. Several future research directions arising from this work can be explored. The incorporation of parameter variations in the numerator of the model reference transfer function as well is one of the natural extensions of the method. Another interesting aspect to be studied later is to modify the design procedure to include not only stability but also transient performance.

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Ozcelik S., Kaufman H. (1995), ªRobust direct model reference adaptive controllersº. Proceedings of IEEE Conference on Decision and Control. New Orleans, USA, pp. 3955-60. Prempain E., Bergeon B. (1995), ªRobust tracking via the robust multiple reference modelº. Proceedings of European Control Conference. Rome, Italy, vol. 1, pp. 584-89. Soh, B.E. (1989), ªRobust pole-placement for uncertain interval systemsº, Proceedings of IEE, Part D, Vol. 136, pp. 301-6.


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