Robust tracking method for uncertain MIMO systems

Available online at www.sciencedirect.com

Journal of the Franklin Institute 350 (2013) 437–451 www.elsevier.com/locate/jfranklin

Robust tracking method for uncertain MIMO systems of realistic trajectories Laura Celentanon Universita degli Studi di Napoli Federico II, Dipartimento di Informatica e Sistemistica, Via Claudio 21, 80125, Napoli, Italy Received 28 March 2012; received in revised form 22 October 2012; accepted 4 December 2012 Available online 12 December 2012

Abstract In this paper new results which allow to determine the performances of a given control system with integral action, with uncertain MIMO plant and with reference and disturbance having bounded derivative, are provided. Moreover, other useful theorems are stated to design a controller forcing an uncertain MIMO system to track a generic reference signal with bounded derivative in presence of a generic disturbance with bounded derivative, with prefixed maximum time constant and error. The used approach is based on the determination of a first-order majorant system of an appropriate representation of the control system by calculating the eigenvalues of suitable matrices only in correspondence of the extreme values of the uncertain parameters. The utility and the efficiency of the proposed methods are illustrated with two significant examples. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Robust tracking of realistic trajectories; Uncertain MIMO systems; Practical stabilization; Lyapunov approach

1. Introduction Many mechanical, electrical, electro-mechanical, thermic, chemical, biological and medical linear systems exist, subject to parametric uncertainties and with non standard references and disturbances, which need to be efficiently controlled, as the numerous manufacturing systems, the robotic and transport systems, etc. For the above mentioned systems, in an ever more dynamic and global society, it is required to design and construct robust controllers able to *

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0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.12.002

438

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track quickly, with precision, very varied and fast references (i.e., with high derivatives), despite the action of disturbances very different in nature. Numerous scientific papers are available in the literature (e.g., [1–19,21,22]), some of which are also very recent [16–18,21,22]. However the following practical limitations remain in several cases: 1) the considered classes of systems are often particular; 2) the considered references and disturbances are almost always standard waveforms (polynomial and/or sinusoidal ones); 3) the controllers are often not very robust and/or do not allow satisfying more than one specification; 4) the control signals are in some cases excessive and/or unfeasible because of the chattering. A crucial problem is to force a process or a plant to track sufficiently regular generic references, e.g., the generally continuous piecewise linear signals easily produced by using digital technologies. Hence, new theoretical results are needful for both the scientific and the engineering community to design control systems with non standard references and/or disturbances and/or with ever harder specifications. The paper states and presents new results which allow:  to determine the performance of a given control system with integral action, with uncertain MIMO process and with reference and disturbance having bounded derivative,  to design a controller forcing an uncertain MIMO system to track, with prefixed maximum time constant and error, a generic reference signal with bounded derivative in presence of a generic disturbance with bounded derivative too. In detail, for a generic uncertain MIMO control system with integral control action, lemmas and theorems are provided, which allow to easily determine a positive first-order majorant system or to design its parameters, whose inputs are the norm of the reference’s derivative and the norm of the disturbance’s derivative, and whose output is the norm of the error. In other words, some theorems are stated, which allow to analyze or to design a control system, with uncertain MIMO process, guaranteeing a fixed response time and a fixed tracking error for a given working velocity (i.e., the derivative of the reference signal). Finally, two significant examples of application and some comparisons, well showing the utility and the efficiency of the proposed results, are reported.

2. Problem formulation and preliminary results Consider the uncertain LTI MIMO plant described by _ ¼ AðpÞxðtÞ þ BðpÞuðtÞ þ EðpÞdðtÞ, xðtÞ

yðtÞ ¼ CðpÞxðtÞ þ DðpÞdðtÞ,

ð1Þ

where x(t)ARn is the state, u(t)ARr is the control signal, d(t)ARl is the disturbance, y(t)ARm is the output, A(p), B(p), E(p), C(p), D(p) are matrices of suitable dimensions which are multilinear functions of the vector parameter pAY. Suppose that Y¼ [p,pþ]CRm is an

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hyper-rectangle and that the following controllability condition An1 ðpÞB ð pÞ ¼ n

rank½ BðpÞ AðpÞBðpÞ   

ð2Þ

is satisfied for at least a p^ 2 Y. Remark 1. In the following, for the sake of simplicity, the dependency of x(t), u(t), d(t), y(t) on time and the dependency of A(p), B(p), E(p), C(p), D(p) on p will be omitted. The aim is to state results: 1) to estimate, 8pAY and for an assigned controller with Integral (I) action, the maximum time constant and the tracking error of a generic reference signal r with bounded derivative in presence of a generic disturbance d with bounded derivative; 2) to design a LTI controller forcing an uncertain MIMO system to track a generic reference signal r with bounded derivative in presence of a generic disturbance d with bounded derivative, 8pAY, with prefixed maximum time constant and error. Remark 2. It is well-known that the disturbances afflicting many real processes are not polynomial and/or cisoidal signals; as the environment temperature, the mechanical stresses of gravitational and/or of electro-mechanical and/or of fluid dynamic type, due to the interactions with the outside world, the side effects of drugs and/or foods, etc. Moreover, concerning the reference signals, note that, e.g., when a versatile and fast production system is required, the system must have high performances for different reference signals of type r(t) ¼ rl(Vt) both at the variation of rl, to change the manufacturing product, and at the variation of V, to vary the working velocity (even if the planners and the builders spend a great effort to make it as higher as possible). It is clear that this class of references is not always standard, and maxabsðr_ ðtÞÞ is proportional to the manufacturing velocity factor V. Often r(t) is a continuous piecewise linear signal, which can be easily produced by using digital technologies (see Fig. 1). 2 r

1 0 -1

0

2

4

0

2

4

6

8

10

6

8

10

r

2 0 -2

time[s]

Fig. 1. Possible reference signal r with its bounded derivative r_ .

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As mentioned in Section 1, the problems (1) and (2) have not suitably been solved in the literature [1–19,21,22] in a very general framework, even if they are very important from a theoretical and practical point of view. Among the several controllers available in the literature, for the sake of brevity, later on it will be considered only the well-known state feedback control scheme with an I action depicted in Fig. 2 (see also [20]). Concerning this suppose that there exists at least a p^ 2 Y such that, in addition to (2), also the following condition is satisfied   A B rank ¼ n þ m: ð3Þ C 0 From this control scheme it easily can be derived that x_ ¼ Ax þ Bu þ Ed,

u ¼ KI z þ KR x,

y ¼ Cx þ Dd

z_ ¼ ry ¼ rCxDd, which can be rewritten as: x_ ¼ Ac x þ Bc r þ Ec d, where:



Ac ¼

ð4Þ

y ¼ Cc x þ Dd;

   BKI 0 , , Bc ¼ 0 I    x 0 , x¼ : z

A þ BKR C

 Cc ¼ C

ð5Þ  Ec ¼

E D



ð6Þ

To solve the problems 1) and 2) the following important preliminary results are stated. Lemma 1. The control system (5) can be described also by: z_ ¼ Ac z þ Bc r_ þ Ec d_ ,

e ¼ Hc z,

or equivalently    I D Cc ðsIAc Þ1 Bc

 Hc ¼ 0

 I ,

  Ec ¼ Hc ðsIAc Þ1 Bc

ð7Þ

 Ec s;

ð8Þ

where e ¼ ry is the tracking error and I is the identity matrix of an appropriate order.

Fig. 2. State feedback control scheme with an I control action.

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Proof. By making the change of variables   w z¼ ¼ Ac x þ Bc r þ Ec d e

441

ð9Þ

and by using the first of (5) it turns out to be z_ ¼ Ac x_ þ Bc r_ þ Ec d_ ¼ Ac ðAc x þ Bc r þ Ec d Þ þ Bc r_ þ Ec d_ ¼ Ac z þ Bc r_   þEc d_ , e ¼ 0 I z, hence, (7) follows.  Eq. (8) derives by equalizing the transfer function of the system x_ ¼ Ac x þ Bc       r r with the one of the system (7). , e ¼ Cc x þ I D d d

ð10Þ Ec



Remark 3. Lemma 1 is very significant because it allows to evaluate or estimate, via Lyapunov approach, the tracking error e. Lemma 2. If the process (1) can be written in the form yv þ A1 yn1 þ    þ An y ¼ Bn u þ En d, i.e., in the form 2

0

I

0

6 0 0 I 6 6 6 ^ ^ x_ ¼ 6 ^ 6 0 0 0 4 An An1 An2   y ¼ I 0    0 x,

  &  

0

3

2

0

3

2

0

3

7 6 0 7 6 0 7 7 6 7 6 7 7 6 7 6 7 7x þ 6 ^ 7u þ 6 ^ 7d 7 6 7 6 7 6 0 7 6 0 7 I 7 5 4 5 4 5 Bn En A1 0 ^

ð11Þ

where xARn, n¼ m  n, uARr, dARl, Ai (p)ARm  m, i¼ 1,2,y,n, Bn(p)ARm  r, En(p)ARm  l, pAY, then the controllability condition (2)–(3) is satisfied 8pAY iff rank Bn ¼ m; moreover, the control system (5) can be described also by the equations: 2 3 3 2 3 2 0 I 0  0 En I 607 7 6 0 7 6 0 0 I    0 6 7 7 6 7 6 6 7 7 6 7 6 7w þ 6 ^ 7r_ þ 6 ^ 7d_ ^ ^ ^ & ^ w_ ¼ 6 6 7 7 6 7 6 6 7 7 6 7 6 0 0 0  I 405 5 4 0 5 4 Bv Knþ1 An Bv Kn An1 Bv Kn1    A1 Bv K1 0 0 ¼ Ac w þ Bc r_ þ E c d_   e ¼ I 0    0 w ¼ H c w, ð12Þ   where Kn Kn1    K1 ¼ KR , Knþ1 ¼ KI . Proof. The fact that the controllability condition (2)–(3) is satisfied 8pAY iff rank Bn ¼ m follows by making explicit the matrices     A B : ð13Þ B AB    An1 B , C 0

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Equations (12) follow from (7) by changing the sign of Bc, Ec, Hc and by making the change of variable 3 2 0 0    0 I 6 I 0  0 0 7 7 6 7 6 6 I  0 0 7 ð14Þ w¼6 0 7z: 7 6       &       5 4 0 0  I 0

Definition 1. Give the system w_ ¼ Fw þ Gv,

z ¼ Hw,

F 2 Rnn ,

G 2 Rnr ,

H 2 Rmn

ð15Þ

nn

and a positive definite (p.d.) symmetric matrix PAR . A first-order positive system pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r_ ¼ ar þ bd, u ¼ cr, where rðtÞ ¼ :wðtÞ:P ¼ wT ðtÞPwðtÞ and dZ:v(t):, such that u(t)Z:z(t):, is said here to be majorant system of the system (15). Lemma 3. Give a p.d. symmetric matrix PARn  n, a symmetric matrix QARn  n, a matrix GARn  r and a matrix HARm  n with rank m. Then 8rZ0 it is:   min wT Qw ¼ lmin QP1 r2

ð16Þ

w2CP,r

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi max wT PGur lmax ðG T PG Þ r:u:,

w2CP,r

where CP,r ¼ {w::w:P ¼ r}, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :v ¼ Hw:r lmax ðHP1 H T Þ r,

8u 2 Rr ,

8 w 2 CP,r :

ð17Þ

ð18Þ

Proof. Since P is p.d. there exists a symmetric nonsingular matrix S such that P ¼ S2. Hence, by posing x¼ Sw, it is   min wT Qw ¼ min xT S 1 QS 1 x ¼ lmin S 1 QS 1 xT x w2CP,r x2CI,r x2CI,r  1 1 1  2   ð19Þ ¼ lmin SS QS S r ¼ lmin QP1 r2 , and so (16). Similarly max wT PGu ¼ max xT S1 S 2 Gu ¼ :SGu::x:x2CI,r x2CI,r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ uT G T PGu rr lmax ðG T PG Þ r:u:,

w2CP,r

ð20Þ

and so (17). Finally, by taking into account that, if F is a real matrix m  n with rank m, the matrix FTF has nm null eigenvalues and m positive eigenvalues equal to those of FFT, it is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :v: ¼ wT H T Hw ¼ xT S 1 H T HS 1 xr lmax ðS 1 H T HS 1 Þ :x: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð21Þ ¼ lmax ðHS 1 S 1 H T Þ :Sw: ¼ lmax ðHP1 H T Þ r and so (18).

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P Lemma 4. Let Q ¼ i1 ,i2 ,:::,il 2f0,1g Qi1 i2 il pi11 pi22    pill 2 Rnn be a symmetric matrix multil n n linearly depending on the parameters ½p1 p2    pl T ¼ p 2 P, P¼ [p ,pþ]CR  , and  PAR  1 1 a symmetric p.d. matrix. Then the minimum (maximum) of lmin QP (lmax QP ) is p2P p2P attained at one of the 2l vertices of P. Q can be written as Q0 þ pi Q1 , pi 2 Proof. Note that, fixed iA[1, l], for pj, jai, constant,  1 þ ½p ,p : Moreover, by taking into account that l QP xT Qx, it turns out ¼ min min i i x2fx: xT Px ¼ 1g to be lmin ððQ0 þ pi Q1 ÞP1 Þ ¼

min

þ pi 2½p i , pi 

xT ðQ0 þ pi Q1 Þx:

min

þ T pi 2½p i , pi , x2fx: x Px ¼ 1g

Therefore, said p^ i and x^ the points of minimum of f ðpi ,xÞ ¼ xT ðQ0 þ pi Q1 Þxjx2fx:   min þ lmin ððQ0 þ pi Q1 ÞP1 Þ ¼ min þ x^ T Q0 x^ þ pi x^ T Q1 x^ pi 2½p i , pi 

¼

min

ð22Þ xT Px ¼ 1g it

is

pi 2½p i , pi 

^ 1 , c^ 0 þ pþ ^ 1 g: ðc^ 0 þ pi c^ 1 Þ ¼ minf^c0 þ p i c i c

þ pi 2½p i , pi 

ð23Þ

The proof easily follows from (23). Remark 4. Note that even if the dependency of the matrices A(p), B(p), E(p), C(p), D(p) on the parameters p is multilinear, which is very recurrent in practice (see Example 2), the dependency of the characteristic polynomial coefficients pc(l,p) on the parameters p is not multilinear, but, in general, is very complex. Hence studying stability by using the usual methods (Routh, Kharitonov, roots locus method,y) turns out to be very complex and with corresponding results conservative if the dependency of the coefficients pc(l,p) is approximated with linear relationship whose codomains contain the variations of the coefficients of pc(l,p) with respect to p. In the light of these considerations, Lemma 4, as it will be shown in the following, is very important because it allows to easily determine a majorant system of an uncertain system. Lemma 5. Suppose that the matrix Ac given by the first of Eq. (6) for p ¼ _ p has n real and distinct eigenvalues (or with unitary geometric multiplicity) li, i ¼ 1,y, n, and m ¼ ðn þ mnÞ=2 distinct pair of complex conjugate eigenvalues (or with unitary geometric multiplicity) lh7 ¼ ah7joh, h ¼ 1,. . .,m; moreover, let be ui, i¼ 1, y, n, and uh7 ¼ uah7jubh, h ¼ 1, y, m, the eigenvectors associated. Then, by denoting with Z* the conjugate transpose of the matrix of the eigenvectors Z ¼ ½u1    uu ua1 þ jub1 ua1 2jub1    uam þ jubm uam 2jubm  and with L ¼ diag(l1, y, lu, a1þjo1, a1jo1, y, amþjom, amjom) the diagonal matrix of the eigenvalues, the matrix 

P ¼ ZZ

 n 1

" ¼

u X i¼1

ui uTi

þ2

m X 

uah uTah

þ

ubh uTbh



#1 ð24Þ

h¼1

  is always p.d. and the matrix Q ¼  ATc P þ PAc turns out to be  1   Q ¼  Zn L þ Ln Z 1 " #1 m u X   1X 1 1 ¼ ui uT þ uah uTah þ ubh uTbh ; 2 i ¼ 1 li i a h¼1 h

ð25Þ

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444

moreover it holds   lmin QP1 ¼ 2

max



j ¼ 1,2,...,nþm

 

Real lj ðAc Þ ¼

2 : tmax ðAc Þ

ð26Þ

Proof. As, for hypothesis, the eigenvalues of Ac are distinct (or with unitary geometric multiplicity), the matrix of the eigenvectors Z is nonsingular. Hence the matrix ZZ* is p.d. and therefore also its inverse P is p.d. Moreover, since Z1AcZ ¼ L  1  1  1 Ac T P þ PAc ¼ Ac n P þ PAc ¼ Z n Ln Z n Z n Z 1 þ Z n Z 1 ZLZ 1  1   ¼ Zn L þ Ln Z1 , ð27Þ from which Eq. (25) derives. In order to prove Eq. (26), note that  1    1   QP1 ¼  Z n L þ Ln Z 1 ZZ n ¼  Zn L þ Ln Z n ; ð28Þ    1 n hence the eigenvalues of QP are  lj þ lj ¼ 2 Real lj , j ¼ 1,. . .,n, and then Eq. (26) holds. Remark 5. Given the system x_ ¼ Ac x , note that, since the matrix P in (24) is always p.d., pffiffiffiffiffiffiffiffiffiffiffi xT Px is always a norm of x. Moreover, by using the P given by (24), for (26) it holds always :xðtÞ:P ret=tmax ðAc Þ :xð0Þ:P also when Ac has not all negative real part eigenvalues. 3. Main results Now the first main result can be stated. Theorem 1. Let KI, KR be two matrices such that the matrix Ac in (6) for p ¼ _ p has negative real part distinct eigenvalues. Then the majorant system of (7) with respect to the norm :z:P with P¼ (ZZ*)1, where Z is the eigenvectors matrix of Ac for p ¼ _ p, is given by r_ ¼ ac r þ bc max:_r : þ ec max:d_ :,

:e: ¼ hc r,

ð29Þ

where:

    lmin QP1 , Q ¼  Ac T P þ PAc ac ¼ min p2Vp 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     bc ¼ lmax Bc T PBc , ec ¼ max lmax Ec T PEc p2Vp qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   hc ¼ lmax Hc P1 Hc T ,

ð30Þ

ð31Þ

Vp being the set of vertices of the hyper-rectangle Y. 2 Proof. By choosing as ‘‘Lyapunov function’’ the quadratic form V ¼ zT Pz ¼ :z:P ¼ r2 , for z belonging to a generic hyper-circumference CP,r ¼ {z:zTPz ¼ r2}, it turns out to be   _ 2rrr max zT Qz þ 2zT PBc r_ þ 2zT PEc d_ z2Cp,r ,p2Y

r min zT Qz þ z2Cp,r ,p2Y

max 2zT PBc r_ þ

z2Cp,r ,p2Y

max 2zT PEc d_ :

z2Cp,r ,p2Y

The proof easily follows from Lemma 3 and Lemma 4.

ð32Þ

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445

Remark 6. Note that if Y ¼ f^pg then the time constant of the majorant system t¼ 1/ac is positive and coincides, thanks to Lemma 5, with the maximum time constant of the control system. Moreover ‘‘at steady-state’’ the tracking error satisfies relationship :e:r

hc bc hc e c max:_r : þ max:d_ :: ac ac

ð33Þ

Remark 7. Clearly if the initial state of the control system is not null and/or r(t) and/or d(t) are discontinuous in zero, the tracking error e(t) has an additional term whose practical duration depends on the time constant t¼ 1/ac of the majorat system.

Remark 8. Note that from (7) of Lemma 1 it is eðtÞ ¼ Hc e

Ac t

Z

t

z0 þ

 Hc eAc t Bc r_ ðttÞ þ Ec d_ ðttÞÞ;

ð34Þ

0

from which it easily follows that :eðtÞ:rmax:Hc eAc t ::z0 : þ max p2Y

Z

p2Y

Z

t

þmax p2Y

t

:Hc eAc t Bc :dt  max:_r :

0

:Hc eAc t Ec :dt  max:d_ ::

ð35Þ

0

If z0 ¼ 0, from (35) it is also Z 1 Z :eðtÞ:rmax :Hc eAc t Bc :dt  max:_r : þ max p2Y

p2Y

0

1

:Hc eAc t Ec :dt  max:d_ :

0

¼ Gr max:_r : þ Gd max:d_ ::

ð36Þ

The use of these estimates (even if in general less conservative), compared with that obtained by using Theorem 1 is complex, especially when the system is uncertain. For the sake of brevity, a method to design a controller such that: t ¼ 1=ac r^t, gr ¼ hc bc =ac r^gr , gd ¼ hc ec =ac r^gd , 8pAY, with prefixed t^ , g^ r and g^ d , will be presented only in the case of a process model in the form (11), Bn independent of p, and l¼ m. In this case, from Lemma 5 the second main result follows. Theorem 2. Let pa ðlÞ ¼ bnþ1 anþ1 þ bn an l þ    þ b1 aln þ lnþ1 be the characteristic polynomial of the nþ1 order low-pass Butterworth filter with cutoff frequency on ¼ a and p lk ¼ ei2nþ1ðnþ2kÞ , k ¼ 1,2,. . .,n þ 1, the kth root of p1(l)¼ pa(l)9a ¼ 1.

If in (12) it is posed h

Knþ1

Kn



K1  ¼ Byv

h

bnþ1 anþ1 I

b n an I

i     b1 aI  0 An ðp^ Þ

   A1 ðp^ Þ



,

ð37Þ

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 1 where Byn ¼ Bn T Bn BTn and I is the m-order identity matrix, then a majorant system of the system (12) with respect to the norm :w:P, with P ¼ TaP1Ta, where: 2

I 6 1 6 l1 I V ¼ pffiffiffiffiffiffiffiffiffiffiffi 6 n þ16 4 ^

 1 P1 ¼ VV n ,

ln1 I

2

I 0 6 6 0 I=a Ta ¼ 6 6^ ^ 4 0 0

  & 

I l2 I ^

  &

ln2



3 I 7 ln I 7 7 ^ 7 5 lnn

3 0 7 0 7 7 , ^ 7 5 I=an

ð38Þ

turns out to be r_ ¼ aga r þ bc :_r : þ ec :d_ :,

:e: ¼ r,

ð39Þ

in which

    lmin Q1a P1 1 , Q1a ¼  A1a T P1 þ P1 A1a ga ¼ min p2Vp 2 3 2 0 I 0  0 7 6 0 0 I  0 7 6 7 6 7 6 ^ ^ ^ & ^ 7 A1a ¼ 6 7 6 0 0  I 7 6 0 7 6 ~v ~ v1 ~1 5 4 A A A b bnþ1 I b I b I    I n n1 1 an an1 a A~ i ¼ Ai Ai ðp^ Þ, i ¼ 1,2,. . .,n,

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bc ¼ P1 ð1,1Þ ¼ 1:414, 2:236, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ec ¼ bc max lmax En T En :

3:696,

6:236, . . ., n ¼ 1, 2, 3, 4, . . .

p2Vp

ð40Þ ð41Þ ð42Þ

Proof. Eq. (40),  after  tedious  steps, follows  from the first of (12), from (29), (30) and (38), from the fact that l QP1 ¼ l Ta1 QP1 Ta and Ta Ac Ta1 ¼ aA1a . The second of (39), (41) and (42) derive from (31) for Bc ¼ Bc , Ec ¼ E c , Hc ¼ H c and from (38). Corollary 1. For a-N the parameter ga of the majorant system (39) turns out to be n1 p ¼ maxðRealðli ÞÞ i 2n ¼ 0:7071, 0:5000, 0:3827, 0:3090,. . ., n ¼ 1,2,3,4, . . .

lim ga ¼ g^ ¼ cos

a-1

ð43Þ

Proof. It is easy to see that for a-N the matrix V is the limit of the eigenvectors matrix of the matrix A1a (see (40)); hence A1N ¼ VLV1, AT11 ¼ An11 ¼ V n1 Ln V n , where

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L ¼ diagfdiagfl1 ,    ,l1 g,    ,diagfln ,    ,ln gg. Therefore   lmin Q1 P1 ¼ lmin ðAT11 PA11 P1 Þ ¼ lmin ðV n1 Ln V n V n1 V 1 A11 VV n Þ 1 ¼ lmin ðV n1 Ln V n V n1 LV n Þ ¼ lmax ðLn þ LÞ

ð44Þ

and hence (43) holds. Remark 9. From Theorems 6 and 7 it follows that for a large enough, the time constant of the majorant system Eq. (39) turns out to be t¼

1 1 t1 ffi ffi aga maxðrealðeigðA1a ÞÞÞ a

t1 ¼ 1:414, 2:000, 2:613, 3:236, . . ., n ¼ 1,2,3,4,. . . ;

ð45Þ

moreover it is: h c bc bc g1 ¼ ffi gr ¼ ac aga a g1 ¼ 2:000, 4:472, 9:657, 20:180, . . ., n ¼ 1,2,3,4, . . . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   g1 max lmax En T En p2Vp : gd ffi a

ð46Þ

Remark 10. Note that, on the basis of Theorems 6 and 7, it follows that, to have a good tracking error, the parameter a can be increased, more precise sensors and more powerful actuators can be used or j_r j can be reduced by means of a temporal factor (i.e., the working velocity must be reduced) and, if it is possible, d_ must be partially compensated. Remark 11. The case in which Bu is dependent on the parameter p can be dealt with by iteratively applying Theorem 2 with Bu ðpÞ ¼ Bu ðp^ Þ and Theorem 1 by increasing, if necessary, the design parameter a. The possibility of designing, in a direct way, a control system which has prefixed tracking error and maximum time constant is a quite complex problem. The discussion in the case of a SISO process without zeros is presented in [20] and [23]. 4. Examples The following examples show the great utility and efficiency of the results stated in the previous sections. Example 1. Consider the process (electric DC motor)  x_ ¼

40

120

1

0:01



 xþ

40 0



 uþ

controlled with the control law   u ¼ 50z þ 0:2503 6:9925 x,

0 0:333

z_ ¼ ry:

 d,

 y¼ 0

 1 x

ð47Þ

ð48Þ

L. Celentano / Journal of the Franklin Institute 350 (2013) 437–451

448

1

e r

0.5 0 -0.5 -1

0

2

4 time[s]

6

8

Fig. 3. Time histories of r(t) and of e(t).

By using Lemmas 1 and 5 and Theorem 1 a majorant system of the control system turns out to be r_ ¼ 10r þ 283:6989j_r j þ 12:4992jd_ j, 2 r ¼ :z:P ,

3:0162

jej ¼ 0:0122r

60:2929

6 P ¼ 4 60:2929 402:4253

1:4061e3 10:0525e3

402:4253

3

7 10:0525e3 5: 80:4851e3

ð49Þ

Therefore after about 3t ¼ 3  100 ms it is jejr0:3463max j_r j þ 0:0153 maxjd_ j. In the hypothesis that r(t) is the piecewise linear signal of Fig. 3 and d(t) ¼ 0, being maxj_r j ¼ 0:9003 it turns out to be jejr0:3463max j_r j ¼ 0.3118, while from the simulation it is max:e: ¼ 0:1945: Example 2. Consider the mechanical system of Fig. 4.

It is described by the equation "

m1 þ m2

m2

m2

m2

#

"

ka1 y€ þ 0

# " 0 ke1 y_ þ ka2 0

#  0 1 y¼ ke2 0

Fig. 4. The considered mechanical system.

  1 1 uþ 1 0

 1 d, 1

ð50Þ

L. Celentano / Journal of the Franklin Institute 350 (2013) 437–451

449

1

e1 e2

0.5

r1

0

r2

-0.5 -1

0

1

2

3

4

5

6

7

8

9

10

time[s]

Fig. 5. Time histories of r(t) and possible time histories of e(t).

in which: " y¼

y1 y2

m1 ¼ 1,

#

"

,

u¼

u1

#

u2

" ,

d¼

d1 d2

# ,

m2 ¼ 0:5,

ke2 ¼ 0:5740%,

ka1 ¼ 1710%, ka2 ¼ 0:5720%,

_ jd 1 jr0:1, d_ 2 r0:05:

ke1 ¼ 1720%, ð51Þ

By posing p1 ¼ ka1, p2 ¼ ka2, p3 ¼ ke1, p4 ¼ ke2, the process (50) can be described by " # " #     2p1 2p2 2p3 2p4 1 0 1 0 y€ þ y_ þ y¼ uþ d: 2p1 4p2 2p3 4p4 1 2 1 2

ð52Þ

Considering p^ 1 ¼ p^ 3 ¼ 1, p^ 2 ¼ p^ 4 ¼ 0:5 and by applying Theorem 2 the majorant system of the control system turns out to be r_ ¼ aga r þ 2:236:_r : þ 5:117:d_ :, :e: ¼ r ga ¼ 0:3811, 0:2403, 0:4366, 0:4745, 0:4887, 0:4958 for a ¼ 1, 2, 5, 10, 20, 50: ð53Þ For a¼ 10 it is g1a ¼ 0.475ffi0.500 in accord with Corollary 1; moreover after about 3t¼ 3.211 ms it turns out to be that :e:r0:4712max:_r : þ 1:078max:d_ : . Under the hypothesis  that  the references  r1,r2 are the triangular signals (triangular waves) of Fig. 5 and that d1 ¼ sin t=10 , d2 ¼ sin t=20 it turns out to be 0:4712max:_r : þ 1:078max:d_ : ¼ 0:5420, while from the simulations for p1 ¼ 1, p2 ¼ 0.5, p3 ¼ 1, p4 ¼ 0.5, it is max:e: ¼ 0:2441. Remark 12. From (53) it easily follows that, if it is desired to decrease the tracking error e without reducing the velocity of the carts, the design parameter a must be increased. Remark 13. If p5 ¼ 1=m1 ¼ 1710% the process (50) can be described by the equation " y€ þ

p1 p5

p2 p5

p1 p5

2p2 þ p2 p5

#

" y_ þ

p3 p5

p4 p5

p3 p5

2p4 þ p4 p5

# y¼

"

p5

0

p5

2

#

" uþ

p5

0

p5

2

# d:

ð54Þ By applying Theorem 1 with p^ 1 ¼ p^ 3 ¼ p^ 5 ¼ 1, p^ 2 ¼ p^ 4 ¼ 0:5 the majorant system of the control system with the parameters of the controller given by (37) with Bv ¼ Bv ðp^ Þ

L. Celentano / Journal of the Franklin Institute 350 (2013) 437–451

450

turns out to be r_ ¼ aga r þ 2:236:_r : þ 5:267:d_ :, :e: ¼ r ga ¼ 0:3971, 0:1638, 0:3005, 0:3225, 0:3305, 0:3343 for a ¼ 1, 2, 5, 10, 20, 50:

ð55Þ

For a ¼ 10 the control performances are slightly decreased with respect to the case in which the mass m1 does not present any uncertainty. Remark 14. It can be easily verified that the polynomial of the closed-loop control system is pl ¼ l6þa1l5þyþa6 with a1, a2,y, a6 (nonlinear functions of the parameters p1, p2,y, p5) belonging to the intervals: a1 2 ½37:6638 42:8087,

a2 2 ½734:597 879:873,

a4 2 ½7:206e4

a5 2 ½3:628e5

8:970e4,

4:455e5,

a3 2 ½9:065e3 a6 2 ½9:091e5

11:142e3 11:110e5:

ð56Þ

By applying the Kharitonov’s theorem, it turns out to be that the maximum real part of the poles is 1.873, that is larger than the pole of the majorant system equal to 3.293. 5. Conclusion In this paper a robust tracking method for uncertain MIMO systems of realistic trajectories has been provided. New results which allow to determine the performances of a given control system with integral action, with uncertain MIMO plant and with reference and disturbance having bounded derivative, have been provided. Moreover, other useful lemmas and theorems have been stated to design a controller forcing an uncertain MIMO system to track a generic reference signal with bounded derivative in presence of a generic disturbance with bounded derivative, with prefixed maximum time constant and error. The proposed method is based on two key results. The first, by using a suitable change of variables, puts the controlled process in a form such that the tracking error depends on the derivative of the reference and on the derivative of the disturbance. The second result allows to determine a first-order majorant system of the control system by calculating the eigenvalues of suitable matrices only in correspondence of the extreme values of the uncertain parameters. The utility and the efficiency of the these results have been illustrated with two examples and a comparison. Future developments are going on in this direction with further interesting results (see also [20] and [23]).

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