Design Method of ILQ Servo System with Tracking Robustness

ELSEVIER

Copyright © IFAC Robust Control Design Milan. Italy, 2003

IFAC PUBLICATIONS www.elsevier.comllocate/ifac

DESIGN METHOD OF ILQ SERVO SYSTEM WITH TRACKING ROBUSTNESS Sadaaki Kunimatsu * Takao Fujii *

*

Department of Systems and Human Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka, 560-8531 JAPAN

Abstract: A robustness problem of output trajectories is considered here in the design of servo systems. Our purpose is to design a robust servo system with tracking robustness in that its output trajectory remains in a neighborhood of the nominal trajectory when the dynamics of the system is perturbed. For this purpose we propose a control configuration of robust servo system with a conditional feedback. In this configuration the robust servo system is designed by the Inverse Linear Quadratic (ILQ) design method as proposed by the second author, and the conditional feedback is added to achieve tracking robustness of the output response. First, we state the configuration of this servo system, called Model Reference ILQ (MR-ILQ) servo system, and then provide a theoretical basis for parameter tuning. Second, we clarify the relation between these parameters and the closed loop performances such as robust stability and tracking robustness. Finally, we establish a new design method of ILQ servo system with tracking robustness. We also illustrate a numerical example to show the validity of our results. Copyright © 2003 IFAC

Keywords: tracking robustness, quadratic stabilization, conditional feedback, 11 analysis, ILQ design method

1. INTRODUCTION

composed of several control parameters with different functions.

Robustness of output trajectories is one of the most important performances that should be achieved in the design of robust servo systems. There are several design methods such as mixed H 2 / Hoc control (Khargonekar and Rotea, 1991), 11 synthesis (Zhou et al., 1995), £1 control (Khammash and Pearson, 1991) and so on that achieve robust stability and robust performance. Although these methods yield controllers with unified functions to control systems as a whole, each function can not be assigned to different control parameters. Our motivation here is to construct a robust servo system with robustness of transient responses against structured uncertainties, which we call tracking robustness, by providing a simple configuration of servo system

In this paper, we propose a design method for servo controllers achieving tracking robustness, by using the control configuration of robust servo system with a conditional feedback. The organization of this paper is as follows. In Section 2 and 3, we show the configuration of this servo system based on the Inverse Linear Quadratic (ILQ) design method as proposed by the second author (Fujii, 1987) and then provide a theoretical basis for quadratic stability from the viewpoint of the inverse problem. In Section 4, we clarify the relation between these parameters and the closed loop performances such as robust stability and tracking robustness, by using 11 analysis. In Section 5, we establish a new design method of

523

ILQ servo system with tracking robustness. In Section 6, we also illustrate a numerical example to show the validity of our results.

Augmenting the system (1) with the servo co troller (2) yields the following augmented syster

The notation is fairly standard in this paper. In particular Fu(M, N) denotes upper linear fractional transformation, b-diag denotes blockdiagonal and COllSiSm {Ad := [AT, ... , A;'f·

(31

x e = (A p + D e<1Ep )x e + (Be - e = Cexe

Ap

(la) (lb)

where V e := v - (u

p

:=

[Ea 0]

+ D e<1Eb)ve

(4a (4b

+ SET: Eax)

2.2 Problem Formulation Based on the preliminary consideration, we wil formulate two problems in this section. First, w, design an optimal servo system with a desired in put/output characteristics for the nominal systen (1) with <1 = 0, which can be achieved by use 0 the ILQ design method, as follows.

In this paper we consider a design problem of servo systems (Davison, 1976) such that the output responses yet) track generalized reference inputs r(t), where r(t) are the outputs of the following linear free system with arbitrary initial states.

Problem 2. Find a state feedback control V e = -Kex e for the augmented system (4) with <1 = ( such that 1) it ensures LQ optimality, namely, it minimize~ some quadratic performance index, and 2) the optimal servo system with the result· ing state feedback gains has a desired in· put/output characteristics.

xr(t) = Acxr(t), r(t) = Ccxr(t) Ac := block-diag{A r }, CC:= block-diag{erJ

l
The nominal controller obtained in Problem 2 can then be upgraded to a robust one with tracking robustness, by solving both problems of quadratic stabilization and tracking robustness simultaneously as follows.

(2c)

where the pair (A r , br ) has a controllable canonical form. The following is a well-known assumption for the existence of robust servo controllers.

Problem 3. Find an error regulation compensator Q e( s) achieving quadratic stabilization and tracking robustness for the augmented system (4).

Assumption 1. For all eigenvalues TJj (1 S j S q) of A r , the following condition holds. det [ A

of, E

x e = (A e + D e<1Ee )xe + (Be - e = Cexe

s

1

~]

Let us consider a feedback transformation u : SET: Eax using new inputs v, where S : defined as S := V1 J- 2 Vt based on the fo lowing singular value decomposition of E b : [U 1 U2 ] block-diag{J, o} [VI v2 f. Then the au~ mented system (3) can be rewritten by the folloVl ing equations with ET: E e = 0:

where A, B, C =: COhSiSm {cd, D, Ea, and E b are known constant matrices of appropriate sizes, respectively, and a triple (A, B, C) is a minimal realization with rank B = m and no right-half plane zeros. The uncertainty <1 is also an unknown constant matrix, where the maximum singular value of <1 is restricted by 11<111 1.

l
[

V -

We consider the n-dimensional, m-input and moutput system with structured uncertainties:

Gc(s) := (sI - Ac)- Bc

-:cC :c] , B e :=

Here X, Xc and u denotes the states and tt control inputs in steady states, respectively.

2.1 Preliminaries

xe(t) = Acxc(t) + Bc(r(t) - yet)) Bc := block-diag{br }

:= [

Ce := [C 0], De := [D T

Compared with the usual ILQ servo system (Fujii, 1993), the Model Reference ILQ(MR-ILQ) servo system proposed here has an additional structure with a conditional feedback QE(S) that consists of an internal model and a constant gain, in order to reduce the error between actual output responses and desired ones. The attractive feature of this servo system lies in its aim to attain tracking robustness by use of a very simple structure.

+ D<1Eb)U

(3;

where X e := [(x - x)T, (xc - xe)Tf, Ue := u U, e:= r - y

2. MR-ILQ SERVO SYSTEM

x(t) = (A + D<1Ea )x + (B yet) = Cx

+ D e<1Eb)ue

~TJj I ~] =I 0

For this purpose we construct a quadratically stabilizing controller with tracking robustness as

524

e = -Kex e + QE(S)(Y* - y) where y* are the outputs of a reference model G~r(s), that is y* = G~r(S)T. Based on this solution, a quadratically stabilizing controller with tracking robustness is derived for the original augmented system (3) by the following equation:

V

U

T

e = -(Ke + :=Er Ep)x e + Qe(s)(y* - y)

Fig. 2. ILQ servo system with 2 degrees of freedom

Finally the resulting configuration of the MR-ILQ servo system is shown in Fig. 1.

where G~r(s) := Q~(s)G~r(s)+Kg.Gc(s),k~ := K~ + QDC, kg. := Kg. + Qc. Therefore the quadratic stabilization part of the statement in Problem 3 becomes as follows.

:.......................... [Reference Model Part].: G *y r(s) 1 - : - - _ 9 + - - - - - - - - , L - : : " " - - l y.

Problem 4. Find a quadratically stabilizing controller k e := V-I EV [k~ - kg.] for the augmented system (4). .

.

-c.. [ILQ Part].;

:

Before discussing this problem in Section 3, we derive a preliminary result. Now that we have obtained K~ and Kg. satisfying (A.l) by solving Problem 2, and in addition QDCB = 0 by the definitions of Q~ and Dc, the following equations are satisfied.

Fig. 1. Configuration of MR-ILQ servo system Here K e := Ev [K~ - Kg.] , Ev := V-I EV (See Appendix A), QE(S) := EvQ~(s) and G=(s) is the transfer function of the system (1) after a feedback transformation u = v -:= Eax.

Er

k e = V-I EV [K~ - Kg.] -0

KFB = I

2.3 Problem Reformulation

First we choose, as a reference model G~r (s), the input/output characteristics designed by the ILQ design method in Problem 2 as follows.

Problem 5. Find an error regulation parameter Q~ that achieves tracking robustness while satisfying quadratic stability required in Problem 4.

(5)

where BCe := [0 Bt]T. Since a control system with simple structure is desirable, we next choose the error regulation compensator Q~(s) in Fig. 1 as follows. Q~(s) := Qc(sI - Ac)-l Bc

= D-l c

where {3( s) :=

diag

l~i~m

3. QUADRATIC STABILIZATION In this section, we state quadratic stabilization as a means of robust stabilization for the closed loop system. Here we consider a quadratic stabilization problem from the viewpoint of an inverse problem (Fujii, 1993) in analogy with the ILQ design method.

+ QD

3i(S)} {qifo(s)

{3'f sq + !3'f-l sq-l + ... + {3p,

Qc:= QpQ~, Q~:= b-diag

{[!3?,""

._ Q QO QO._ dia { QD·- P D' D '-l~i~~

{3'f:::: 0

l~i~m

0

{3'f-l

(6b)

Finally we state the remaining part of Problem 3 for tracking robustness. Briefly speaking, it is achieved by tuning qi (1 ::; i ::; m), as will be discussed in Section 4.

In the previous section we formulated two design problems of the MR-ILQ servo system. In this section we reformulate Problem 3 in more detail.

G~r(s) := Ce(sI - A e + BeKe)-l BCe

(6a)

As a preliminary, we define transformation matrices by using some n + qm x n + m(q - 1) matrix Il =: [ IlT Iln T spanning acornplement subspace of Im (Be) and K := K~Ill - Kg.Il2 as follows.

l}

if d i = 1 } if d i >1

Qp := D;;lQc;" Qc;,:= diag {q;} :::: 0 l~i~m

Te

and di (1 ::; i ::; m) is the relative degree for the i-th output of G(s), which is defined by di := min{klciAk-l B f- O}. The decoupling matrix Dc := COlI
T e-

l

:=

=..

B] [I -K

Ill [ Il2 0

0]

V-I

(7a) (7b)

Using this transformation matrix, we transform A e , Be, and K e satisfying (6) as follows.

525

robust stability problem with augmented perturbation L1 p (s) as shown in Fig. 3.

(8a) (8b) (8c)

Lemma 6. (Khargonekar et al., 1990) The state feedback controller with a gain Re is a quadratically stabilizing controller for the augmented system (4a) if and only if A e - BeRe is stable and I/u := IIG e(s)lIoo < 1 where Ge(s)

:=

-

-

Fig. 3. Robust performance vs robust stability

1

(Ee - EbKe)(sI - A e + BeKe )- D where w is related to Zl by w = L1z 1, Z2 := y* - y and P(s) is a transfer function defined by

Lemma 7. (Kunimatsu et al., 2003) Let G~(s) := limL'-+ooGe(s). Then G~(s) can be expressed as G~(s) = E(sI - A ll )-lTll D - iJ where fJ := - 0 -1 EbKFD, E:= El - EbV A 21 .

Zl] = P(s) [ Z2

[w] ,P(s) =: r

[Pll(S) P12(S)] P21 (S) P 22 (S)

and it is given as follows: Based on the above preliminaries, we can obtain the following theorem.

o o o

De Theorem 8. (Kunimatsu et al., 2003) The state feedback controller with a gain Re satisfying (6) is a quadratically stabilizing controller for the augmented system (4a) if All is stable and 1/00 := IIG~(s)lloo < 1. In addition stability of All is a necessary condition as well.

Beej 0

0 0

= Ge(s), P 12 (S) = (Ee - EbKe)(sI - A e + BeKe)-lBee, P21 (S) = -Ce(sI - A e + B eR e )-l De and P22 (S) = O. Our objective in this section is to make ,max := max IIAII$lI1Fu(P, L1) 11 00 as small as possible while satisfying robust stability, namely IIGe(s)lIoo < 1. As a preliminary to apply J-t analysis, we define P..,(s) by from which we see Pll(s)

4. TRACKING ROBUSTNESS

In this section, we consider a tracking robustness problem, in which our aim is to reduce the maximum value of error signals y* - y, namely .coo norm of it. Since L 1 norm is an upper bound of the induced norm by .coo signal, we think that the L 1 norm of the operator from r to y* - y is a better choice as a criterion of tracking robustness. Since direct estimation of L 1 norm is difficult (Khammash and Pearson, 1991), we pay attention to an indirect estimation of L 1 norm using the following inequality (Zhou et al., 1995):

P..,(s):= [

,-I

Pll(S) P21 (S)

,-I

P12 (S)] 3 P22 (S) , , > 0

Let J-tA p(P.., (jw)) be the structured singular value of P..,(jw) associated with L1 p (jw). Then J-tAp(P..,) is bounded as follows (Zhou et al., 1995).

J-tAp(P..,(jw))

(9)

::; JIIGe(jw)1I 2 + 2,-1 1/P12 (jw) 11 1/P21 (jw)1I

IIG(s)IIH ::; IIG(s)lIoo ::; IICeAtBIILl ::; 2nIlG(s)IIH where G(s) := C(sI - A)-l B E RHoo , A E Rnxn and II·IIH means Hankel norm. Due to this

Theorem 9. Let P(s) be stabilized by Kfj., Kg, Ev and Q~(s). Then a set of these controllers is a quadratically stabilizing controller with tracking robustness for the closed loop system in Fig. 1, that is, the inequality max IIAII9I1Fu(P, L1)1I00 < , holds for some performance bound, > 0, if the following condition is satisfied.

inequality, we can reduce L 1 norm indirectly by reducing H oo norm of the transfer function from r to y* -y, which justifies the use of H oo norm. In the following we consider the tracking robustness problem defined above as a robust performance problem.

IIGe(s)lI~ + 2,-1 1/P12(S) 1100 !/P21 (s)lIoo < 1 (10) Proof: Since maxIIAII9I1Fu(P,L1)1I00 < , and IIGe(s)lIoo < 1 if SUPWERJ-tAp(P..,(jW)) < 1 by Main Loop Theorem (Zhou et al., 1995), it is enough for this proof to show that sup wER J-tLj,p(P..,(jw)) < 1 if (10) holds. From (9), SUpwER J-tLj,p(P..,(jw)) ::; JIIGellZx, + 2,-1 1/P12 1100 1/P21 1100' Hence the desired result follows if (10) holds. 0

4.1 Analysis of Tracking Robustness by J-t Analysis Let L1 f (s) E RH oo with lIL1f(s)lIoo ::; 1 be a virtual perturbation. Using this Ll f (s) and L1 given in Section 2.1, we define the following augmented perturbation L1 p(s) := b-diag{ L1, L1 f (s)}. Then a robust performance problem is equivalent to a

526

by regarding the order of (3i (s) as the same order of 4>i(S).

Lemma 10. Let PIT(s) := limE......ooP21 (s). Then PIT(s) can be expressed as -

-

1

P2f(s) = C 1(s1 - A ll )- TllD = diag {

1$i$m

O'(s) } Dp(s)D 4>i(S) + qd3i (s)

(lla)

By this lemma, we should choose an appropriate qi (1 :S i :S m) in the range of qi E [0, qi).

(llb)

5.2 Design Procedure

Here 4>i(S) is given by (A.2), Dp(s) := co11
{L::~~';11B:(s)ciA~-1}

L::~=1(*) := 0- ;nd = Bf(s)sk + (f(s) with

The preceding analysis yields the following procedure of the MR-ILQ design method.

where

Bf(s) is defined by 'l/Ji(S) 'l/Ji(S) given in (A.2).

Step!: Obtain the singular value decomposition of Eb if E b i= O. Step2: Transform G(s) into G::o{s) using u = v~Er Eax if Ea i= O. Step3: Specify input/output characteristics G yr and obtain K~, Kg using the ILQ design method as stated in Theorem 15. Step4: Specify (3i(S) and obtain the resulting (1 :S i :S m) by Lemma 13. Step5: Specify qi E [0, qi) such that All is stable and V oo < 1, while trading off between robust stability and tracking robustness. Step6: Obtain E and specify E within E > E while trading off between the asymptotic degree to desired input/output characteristics, the degree of quadratic stability and quantity of control inputs. Step7: Repeat these procedure from Step3 until desired characteristics can be obtained.

Er

We state the following two relations between P21 (S) and PIT(s) without proof.

Lemma 11. 11P21(S)1100:S IIPIT(s)lloo+(I!(O"min(2), where O"min > (2 is the minimum singular value of E, while (1 and (2 are both positive scalars independent of 0"min.

q;

By Lemma 10 and 11, we can see that IIP21 (s)1I00 becomes smaller as qi with d i > 1 and 0" min increase. we also see that P12 (s) is independent of qi by its definition. By these facts and (10), we claim that the desired tracking robustness can be achieved since we can make IIFu(P, L1)1100 effectively small by increasing qi. Note that the i-th row with di = 1 of P 21 (s) hardly influence 11P21(S)1100 at all since that of PIT(s) is 0 by Lemma 10. Therefore we can deduce that the i-th output with d i = 1 is robust against structured uncertainties.

6. NUMERICAL EXAMPLE To show the validity of our results, we illustrate unit step responses for the following multiinput/multioutput system (Kawarabayashi and Fujii, 1990) where structured uncertainties do not satisfy the matching condition. A = col{[O, 0, OJ, [0, -2.5, 0], [5.5, -3, -OA]}, B = col{[2, 0], [0, 0.5]' [0, O]}, C = col{[O, 0, 1], [0, 1, O]}, D = col{[O, 1], [0, 0], [1, O]}, Ea = col{[O, 0, 2], [0, 0, O]}, Eb = col{[O, 0], [0.2, O]} where we see Ea = 0 and V = 12. We also see d1 = 2, d2 = 1 and Dc = col{c1AB, C2B} = col{[I1, -1.5], [0, 0.5]}, det(D c ) i= 0 We first specify input/output characteristics Gyr(s) = diag {1/(5s + 1)2, 1/(5s + I)} = G~r(s) Then we can determine Kg = D~1col{[0.04, 0], [0, 0.2]} and K~ = D~1col{[5.5, -3, Oj, [0, 1, O]}. We next specify Q~ = diag{l, 0.2}, Q~ = diag{I,O} and Qlj, = diag{q1' q2}, then Q~(s) = D~ 1diag {q1 (s + 1)/ s, 0.2q2/s}, PIT(s) = diag {25s/((5s + 1)2 + 25q1(S + 1)), O}. We now show some simulation results for E = 10012, L1 = 12 and [Qlj, = diag{1.8, O} or Qlj, = diag{l1, O}l in Fig. 4.

Corollary 12. Let Im(D) C Im(B) or det(CB) i= O. If the closed loop system is robustly stable, that is V q < 1, then tracking robustness can always be achieved, namely (10) is always satisfied. Proof: if Im(D) c Im(B) or det(CB) i= 0, then PIT(s) = o. Hence (10) is obviously satisfied for a large O"min if IIG e (s)lloo < 1 0

Er

5. MR-ILQ DESIGN METHOD 5.1 Specification of Closed Loop Poles

In this section, we state how to specify the eigenvalues of All, namely the roots of ~i(S) := 4>i(S)+ qd3i (s) given by (l1b). With the help of algebraic approach to the Root Locus method, the following lemma yields a range of qi (0 :S i :S m) such that ~i(S) is Hurwitz.

Lemma 13. (Barmish, 1994) Let H(a) be the Hurwitz matrix of a polynomial a(s). Then the maximum value of qi ::::: 0 such that ~i(S) is Hurwitz is q; = 1/>"~ax(-H-1(4>dH((3i)) (0 :S i :S m) where >"~ax(A) is the maximum real eigenvalue = +00 if the matrix A has of a matrix A, and no real eigenvalues. The H((3i) is similarly defined

Here the dashed line (- -) and the dash dot line (.) indicate the error output response with respect to the first output for q1 = 1.8 and q1 = 11, respectively, while the solid line (-) indicates the second output for q2 = o. From this figure, we can

q;

527

Kunimatsu, S., T. Fujii and T. Tsujino (2003). Quadratic stabilization of ILQ servo systems with generalized reference inputs. Trans. of the Soc. of Inst. and Contr. Eng. 39(3),307309. (in Japanese). Zhou, K., J. C. Doyle and K. Glover (1995). Robust and Optimal control. PRENTICE.

error output respons.

," ,

.. ,.j

I.:

: ~: ,'\ : ~.: ~: ..

:

, f

: I

O.04· f

:



J

\1

I

.

~

,

~ ,'\

I 0.02

: ......... _.-.... ,.

I.'

"

. ..;..

,

.

,

' .....

\~/.-\ . . "

:

-

Appendix A. ILQ DESIGN METHOD

-_."-.._.~:":._\~

The ILQ design method is an LQ design method proposed from the viewpoint of the inverse LQ problem as described below. (The configuration of the ILQ servo system has already been shown in the lower part of Fig. 1.)

ti.Jllelsec)

Fig. 4. Error output response (y - y*) see that the first error output for ql = 11 is smaller than that for ql = 1.8. We next see that the second output is robust against structured uncertainties for any choice of qz when dz = 1.

Lemma 14. (Fujii, 1987) The state feedback V e = -Kexe is the optimal controller with respect to the quadratic performance index J e := JoOO(x;Qexe +v; Reve ) dt for some Qe > 0, Re > 0 only if the following two conditions are satisfied.

7. CONCLUSION As an extension of the usual ILQ design method, we have proposed the MR-ILQ design method, which yields such a robust servo system that has a simple structure and can achieve quadratic stability and tracking robustness only by tuning Error Regulation Parameter Q~ and Gain Thning Parameter E essentially. In particular we have shown the fact that the i-th output is robust against structured uncertainties if its relative degree is 1, namely di = 1. We have confirmed that this fact is ture by the numerical example to show the validity of our result.

Ke

= V-I EV [K~ - Kg]

K~B = I

(A. la) (A.lb)

where K~, Kg are matrices of appropriate sizes. In the ILQ design method, we first asymptotically specify desired input/output characteristics Gyr(s) = Ce(sI - A e + BeKe)-1 Bee from reference inputs r to outputs y by appropriately designing K~ and Kg, and then guarantee LQ optimality by proper choice of E. We can first specify asymptotic input/output characteristics with decoupling structure as follows.

REFERENCES Barmish, B. R. (1994). New Tools for Robustness of Linear Systems. Macmillan Publishing Co. Davison, E. J. (1976). The robust control of a servomechanism problem for linear time invariant systems. IEEE Trans. AC21(1), 25-34. Fujii, T. (1987). A new approach to the LQ design from the viewpoint of the inverse regulator problem. IEEE Trans. AC32(1l), 995-1004. Fujii, T. (1993). Design of tracking systems with LQ optimality and quadratic stability. Proc. 12th IFAC World Congress pp. 435-442. Kawarabayashi, S. and T. Fujii (1990). Design of optimal servo-systems for engine test bed by ILQ method. Proc. of the 29th IEEE CDC pp. 1579-1583. Khammash, M. and J. B. Pearson (1991). Performance robustness of discrete-time systems with structured uncertainties. IEEE Trans. AC 36(4), 398-412. Khargonekar, P. P. and M. A. Rotea (1991). Mixed H z / H oo optimal control. IEEE Trans. AC 36(7), 824-837. Khargonekar, P. P., I. R. Petersen and K. Zhou (1990). Robust stabilization of uncertain linear systems. IEEE Trans. AC35(3), 356-361.

Theorem 15. (Fujii, 1993) The asymptotic transfer function Gc;.(s) := limE-+ooGyr(s) can be specified as

G~(s) = l::;.::;m diag {:i«S))} 'l'i S

E RH oo

(A.2a)

i(S) = 'l/Ji(s)a(s) + ri(s) ri(s) := r'f-l sq-l + ... + r; s + r? deg i(S) = di + q - 1 deg 'l/Ji(S) = di -1

(A.2b) (A.2c) (A.2d) (A.2e)

by determining K~ and Kg uniquely as follows. K~

= D-;l

Kg

=

col {ci'l/J(A s )}

l
D-;lblo~k-diag{[r? ... r'f-1n

(A.3a) (A.3b)

l
where d i and Dc-defined in Section 2.3 are the relative degree for the i-th output of G(s) and the decoupling matrix, respectively

Theorem 16. (Fujii, 1993) There always exits a diagonal E > 0 such that the K e satisfying (A.l) and (A.3) is the optimal controller with respect to the quadratic performance index J e for Re := VTE-1V and some Qe > O.

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