Robust frequency shaping sliding mode control

A.Jafari Koshkouei and A.S.I.Zinober

Abstract: Frequency shaping, linked with linear quadratic optimal and sliding mode control, is a technique for controlling systems with uncertainties. The authors propose some new methods for designing the sliding surface and sliding mode control when the LQ weighting functions are not constant at all frequencies. Furthermore, they introduce conditions for which the spectrum of the original reduced system is a subset of the spectrum of the augmented system, and develop an iterative optimal construction procedure for the sliding mode.

1 Introduction

The problem of robust stabilisation of uncertain systems can be solved using the sliding mode control (SMC) approach, which yields complete rejection of external disturbances satisfying the matching conditions. The robustness properties are usually achieved by using discon- tinuous controls. SMC has been extensively studied in the last three decades [l-51.

The frequency shaping approach to linear quadratic (LQ) design has been proposed in recent years [4, 6-13]. For example, Moore and Mingori [12] discussed frequency-shaped LQ design and spectral factorisation. They proposed techniques for the construction of optimal controllers which preserve the robustness properties of standard LQ state feedback. Tharp et al. [13] discussed the parameterisation of LQ frequency weightings, the associated dynamic controller and a two-phase procedure for the design of controllers utilising frequency weighting. They developed a technique to retain the spectrum of the closed-loop design model, resulting from a conventional LQ problem, as a subset of eigenvalues of the closed-loop augmented system.

In [4], sliding mode control design using frequency domain techniques was presented, but considered only the case when the control weighting matrix is dependent on frequency. In this paper, all the possible cases for which the weighting functions may be frequency-dependent will be studied. The frequency shaping of sliding mode control and design compensators for the reduced order system are studied. Furthermore, the conditions for the poles of the original LQ reduced-order system to remain the poles of the reduced-order system with compensator are obtained.

The frequency-dependent weighting functions may have a penalty on the control at high frequencies, e.g. the frequency response may drop off slowly as l lw at high frequencies [8]. For the case of state feedback in LQ design, the controller has 60" phase margin and ( $ 0 0 )

gain margin. Using appropriate frequency weighting func-

0 IEE, 2000 IEE Proceedings online no. 20000378 DOI: 10.1049/ip-cta:20000378 Paper first received 21st May 1999 and in revised form 7th February 2000 The authors are with the Department of Applied Mathematics, The University of Sheffield, Shefficld S10 2TN, UK

312

tions yields augmented control systems with the optimal sliding mode. Frequency shaping is a way of dealing with plant uncertainties and links linear quadratic optimal control and sliding mode control.

We use p.d., p.d.s. and u.p.d.s. for positive definite, positive definite symmetric and unique positive definite symmetric respectively, while W* indicates the complex conjugate transpose of the complex matrix W

2 Thesystem

Consider the linear time-invariant system

i = A X + BZL

z = c x

where x E R" is the state variable, A E R" ", B E R" "' 1s '

full rank, U E R" is the input control, C E R"' " such that CB is nonsingular, and z E R"' is the output. We assume that (A,B) is completely controllable and m < n. Define the sliding surface as s = Cx = 0. The system is in the sliding mode when the state lies on the sliding surface after some finite time, i.e.

s = c x = o t > t , (4 with t,s the time when the sliding mode is reached. During the sliding mode the control does not directly affect the motion [ l , 31. Differentiating eqn. 2 and inserting eqn. 1 yields

Ueq = -(cB)-'CAx (3)

where zieq is the equivalent control of the system during the sliding mode. The dynamics of the equivalent system is

x = (A - B(CB)-'CA)x (4) since CB is nonsingular, and the dimension of the null space of C is n - m. Therefore the dimension of the state variable space in the sliding mode is n - m. The control

U = ueq - (cB)-'(Hs + Ksgn(s)) (5)

with semi-p.d. matrix H and a diagonal matrix with positive entries K, enforces the system to the sliding mode. The sliding mode condition is

sTLi = -sTHs - sTKsgn(s) < 0

IEE Proc.-Control Theory Appl., Vol. 147, No. 3, Muy 2000

N affects the reaching time of the sliding mode and often is taken to be zero.

Assume Tis an orthogonal matrix such that

= [ 4 where B, is an m x m nonsingular matrix [2]. Let y = Tx, then

and CTT = [ C , C,]. When the ideal sliding mode occurs,

s = C,y l ( t ) + C,y,(t) = 0 t >_ t, (10)

Yz@> = -Kh(f) (11)

Yl(t> = (All -A,,KlY1(f) (12)

and then

with K= CF'C, . Eqn. 8 becomes

which is known as the reduced-order system and A l l - A I 2 K has n - nz eigenvalues, i.e. in the sliding mode m eigenvalues of eqn. I are zero [2]. We desire to find K such that A , , - A,,K is a stable matrix.

3 Sliding mode using the LQ approach

A technique for designing the sliding surface using the linear quadratic (LQ) approach has been considered by Young et a1 [5]. The basic idea is thaty, is the input control of the subsystem in eqn. 8 and LQ methods can be used to find the optimal control, or more precisely the optimal sliding mode. Consider the singular quadratic cost func- tional

8"

J = I xTQxdt (13) * t ,

where Q is a p.d.s. matrix. From eqn. 6

TQT7'= [ Q I I Q12 ] Q12 Q22

Then eqn. I3 and y = Tx yield

J = / ~ ( Y ~ Q I I Y I + ~ Y T Q I ~ Y ~ +.~$Q22~2)dt (14)

Without loss of generality we can assume QI2 = 0, then eqn. 14 becomes

J = ~ ~ ( Y : Q I I Y I + Y; Q22~2)df (15)

In eqn. 8 we can consider y2 as an input of the subsystem. The optimal control of this subsystem is

where P is a u.p.d.s. matrix solution of the Riccati equation

A3;P + PA11 - PAl,Q,;'Ay,P = -Q,1 (17)

y2 +Kyl = 0 is the sliding surface for system (7) , and [ K ZJTx = Cx = 0 is the sliding surface for the system in eqn. 1.

4 Frequency shaping of the sliding mode

In this Section, methods are presented for finding the sliding surface when the weighting matrices are functions of frequency. The quadratic cost in eqn. 15 can be written in the frequency domain using Parseval's Theorem as

1 " 271. -02

J = - 0 , 7 ( i w ) ~ ~ l(iw)yl (iw) +j,2*(iw)~22(iCo)yZ(iO)))do

(18)

where the matrices Ql I (io) and Q22(ic11) are frequency- dependent Hermitian weighting matrices. They are p.d. matrices for all frequencies except a set of frequencies with zero measure, i.e. for almost every frequency the weighting functions are p.d. matrices. Assume the weighting func- tions are proper rational functions of w2 [SI. This assump- tion guarantees that the optimal sliding solution is causal. Note that any real function can be approximated by a rational function. There exist four cases:

(i) both Q , , and Q2, are constant for all frequencies (ii) Qz2 is a function of o2 and Q,, is constant for all frequencies (iii) Q , , is a function of w2 and is constant for all frequencies (iv) both e,, and QZ2 are functions of (02

Cases (iii) and (iv) have not been considered before. Case (i) was dcscribed in Section 3 and (ii) has been presented by Young and Ozguner [4] and Hashimoto and Konno [9]. It has been shown by Anderson et a1 [6] that, when (2) ,( iw) and Q22(iw) are the inverse of each other, then identical closed-loop poles and an optimal sliding mode are obtained.

Case (ii) is considered first, i.e. Q,, is constant for all frequencies and is a function of w2 [ 131. Assume that W,(s) is a spectral factor of Q2z, i.e.

Q,,(irr)) = W2*(iw)W2(iw) (19)

Then the quadratic cost in eqn. 18 can be replaced by

+ ( W, ( iw)y2 (icfi))' W, ( iw)y2 (in))) d o (20)

where

fi(s) = W2(9)Yz(S)

This implies that J is the output of a filter or dynamic system with transfer function W2(s) and inputy,. If W,(s) is considered to be a transfer function, then W,(s) represents the precompensator transfer function of the system

4,12 = A*>2x,2 + B,,I*Y2

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The optimal sliding surface for the augmented system, which is the original system with the dynamic compensator in eqn. 22, is now studied. Consider

X, = A,x, + Bey, (23)

A, = ["'* 1. x, = [ :], B, = [I::], 0 A l l

and R,=Di2D,,2. The quadratic cost in eqn. 18 is converted to

w

(x,'Q,x, + 2xSNeyz + Y~TR,Y& (24) = Its

Therefore, the sliding surface is

s = y2 + Kx, = 0, K = R,'(B,TPe + N:) (25) where P, is the u.p.d.s. matrix solution of

ATP, + P A , - (P,B, + N,)R,-'(BTP, + NT) = -Q, (26)

(27)

For K= [K, K2] eqn. 25 becomes

s = y2 + K,X,,2 + Kzyl = 0

which is a linear operator of the states and is the sliding surface of the extended system

. I j = A j + B u (28)

with

0 B+vz Y = [z], Yl i= [A: A , , : ; ;] , j = [i2]

0 A21 Let e= [K 4. The sliding control in eqn. 5 is

zi = -(%-'(e& + Ksgn(s)) (29) which guarantees the occurrence of the sliding mode.

Assume

71 = KIXW, S Y 2 (30)

Then the system

Xw2 A W : X W 2 + Bw2Y2

v = K P W 2 +Y2 (31)

is a filter for y2 which is obtained by designing the sliding surface in eqn. 27 [4]. Therefore there is a filter for y2 corresponding to the sliding surface.

5 Iterative constructive procedure for the optimal sliding surface

One way to obtain various sliding surfaces is to alter the weighting functions in the hnctional performance index in eqn. 18. The problem is how should the weighting func- tions be selected? The iterative method below enables one systematically to consider various sliding surfaces and choose a desired sliding surface. This method may be applied a finite number times and the sliding surface chosen by comparing the eigenvalues of the reduced- order system. An example, design (a) , is presented in

314

Section 9. Conventional weighting matrices are considered which yield a new compensator and augmented system. From eqns. 22 and 30

U 1 = c1xw2 +DIY2

where

with CO = C,, Do = Dw2 and KO, = K, . Suppose

Q,(s) = D I + Cl(sl - AW2)-IBW2

i.e. Q, is the transfer function of the system

Now consider eqn. 23 with weighting matrices

and R I , = DTD1. The quadratic cost in eqn. 18 is converted to

w

J = .I @:QI,X, + 2x;Nley2 + Y ~ T R I , Y ~ ) ~ ~ (33) (3

Therefore, the sliding surface is

~1 = y2 + Kl,x, 0, K,, = RT,'(B,TPl, + N,T) (34)

where P I , is the u.p.d.s. matrix solution of the appropriate Riccati equation. For K,, = [K,

( 3 5 )

Since the quadratic cost fimctionals in eqns. 24 and 33 are different, the sliding surfaces in eqns. 25 and 34 are not the same, i.e. a new compensator and a new sliding surface have been obtained. By proceeding iteratively for some positive integer number N> 1 we obtain

ICl2], eqn. 34 is

s1 =Yz + K 1 1 X W 2 +Kl2Yl = 0

UN c N x ~ 2 + DNY2

where

Suppose

i.e. QN is the transfer function of the system

Now consider eqn. 23 with weighting matrices

and RNe=DLDN. The quadratic cost in eqn. 18 is converted to

03

(CQN~X~ + 2x,'Kve~2 + Y ~ T R N , v ~ > ~ ~ (37) = .I

IEE Proc.-Control Theovy AppL, RI. 147, No. 3, May 2000

Therefore, the sliding surface is

s , ~ = y2 + Kjvexe = 0, K N e = Rid (B,7'PNe + N,&) (38)

where PNe is the u.p.d.s. matrix solution of the Riccati equation

AlpNe + P,J, - (PNeBe + NNe)R;d (BZPNe + NLe) = QN

(39)

(40)

For KN = [R;:] eqn. 38 is given by

sN =Y2 + KNIxw2 f K N 2 Y l =

Therefore, for all N there exists an augmented system, an optimal sliding surface and a filter for y , relating to the sliding surface.

6 Design of a postcompensator

Next Case (iii) is discussed. Here Q , (s) is a function of w2 but Q22(s) is constant for all w

Q l l ( i w ) = W,*(io)Wl(iw) (41) where Wl(ico) is the spectral factor Qll(s). Then eqn. 18 becomes

l o o 2n . -cc + Y;(i~)e2zVz(iw)1 d w (42)

J = - I ' {(WI (iWlYl (iw>>* Wl (iW)Yl (iw)

Assume of the system

(iw) = W,(io)y,(iw) and the transfer function

iw, = 4V1xw1 +BWIYl

kvl = CWIXWI +DWlYl (43)

8, = + i y 2 (44)

is Wl(s) . Consider

Since C;, CYI > 0 and 0 6 , DWI - D:il (C;, C J I

C%;, D,I 2 0, Q is a semi-p.d.s. matrix. The quadratic cost (42) is now

J = p Q h +Y2222Y2) dt (45)

and the sliding surface is

s =y2 +KF1 = 0 , K = QTiBTF (46)

i T p + - pjQ;;jTF -Q (47)

where P is the u.p.d.s. matrix solution of the ARE

Assume K = [K , K2] then the sliding surface is given by

Y2 +KlY, +K2XWl = 0 (48)

The eqns. 43 describe apostcompensator for the system in eqn. 8. The eqns. 27 and 48 indicate that the sliding surface can be considered as a linear operator corresponding to the pre- and postcompensator, respectively.

IEE Proc-Control Theory Appl., Vol. 147, No. 3, Muy 2000

7 Relationship between the LO reduced system and the augmented sliding system

The augmented system is a new system which is a combination of the original LQ reduced-order system and a compensator, eqn. 43. Generally, the poles of the reduced-order system are not the poles of the augmented system. Since allocation of the system poles is very important for control design, it may be desirable to design a compensator such that the poles of the LQ system remain the poles of the augmented system. In this way some of the properties of the LQ system are conserved. In this Section, conditions are obtained for the poles of the LQ reduced-order system to be preserved as the poles of the reduced-order augmented system. Note that the general case of decomposition of the weighting matrix will be considered; in particular when the weighting matrix is not strictly proper. Consider the performance index in eqn. 18. Assume

Qi I (io) = eo + Q I I (io) (49) where e,, is constant matrix. Suppose

CO

J2 = 1 O ~ T Q O Y I +~2TQ22~2)dt (50) * f,

where Q22 is constant for almost all frequencies. Then an optimal sliding surface for the original system is

~2 = -Q;:AT~PY I = -KY I (51) where P is the u.p.d.s. solution of the Riccati equation

ATlP + P A l , - PA12QTiA:;P = -Qo (52)

Note that, if matrix A , , has eigenvalues on the imaginary axis, then one must ensure that Q, # 0 [14].

For

Q l l = W:W1

the transfer function of the system

iWI = 4,q~W, + + ~ w l Y l

iij = C W l k , l + DWIYl (53)

5, A&, + Bey, (54)

is g 1 ( s ) . Consider the augmented system . A

where

Then the index in eqn. 18 is converted to

J = +Y2TQ22Y2)dt (55)

where

Hence the optimal sliding surface is

~2 = -QTiBZFeYe = -KePe (56) where Pe is the u.p.d.s. solution of the Riccati equation

+ PeAe - PeBeQFiBLPe = -ell (57) 315

The system in the sliding mode is governed by , . A

?;I = (A, - B,K,)J;I

= (AI I - A&) + kNJ1

(58) Theorem 7.1 Zf

c w , = D w q (59)

then theAeige;va1ues of A l l - A12K are also the eigenva- lues o f A , - Be&. Proof: Assume A l , d2, . . . , Anpi,, are the eigenvalues of A l l - AI2K (assumed distinct) and vI , v2,. . . , v,-,], the corresponding eigenvectors, respectively. Therefore

A , , - AI2K = V-IAV

where V = [ v l v 2 . . . v * - ~ ] and A=diag(d1,A2,. . ., A,-,). The Hamiltonian matrix for eqns. 54 and 55 is

Since the eigenvalues of 2, - B,K, are the eigenvalues of He, it is sufficient to prove that the eigenvalues of A,, - A 12K are the eigenvalues of H,. Assume

Then

H,V =

V

p=[,:]

= PA

3 16

Therefore d,,A2, . . . ,A, -m are also eigenvalues of

Corollary 7.1 Assume that the conditions of Theorem 7.1 are satisfied and P is the u.p.d.s. solution of ARE, eqn. 52. The z1.p.d.s. solution of ARE, eqn. 57 is

A, - B,K,. 0

where P is the u.p.d.s. matrix solution of the Riccati equation

Proof: Substituting P, into the left-hand side of eqn. 57 and using Theorem 7.1 and eqn. 52, yields

8 Design of two compensators for frequency- dependent weighting functions

Now consider Case (iv), i.e. both Q , I and Q22 are functions of 02. Let W,(s) be the spectral factor of Qee, C = 1,2. Then We(s), L = 1,2, are the transfer functions of the systems in eqns. 43 and 22, respectively. W, and W, are post- and precoinpensators (see Fig. 1). Then the quadratic cost in

.

Fig. 1 Structure ojgeneralised syteni in the sliding mode

IEE P~oc.-Contrt,l Theoiy Appl. , Vol. 147, No. 3, May 2000

(64)

(65)

are frequency-shaped state variables and inputs, respec- tively. Consider

. - 21 = A ? , + Bx2

Q1 are semi-p.d.s. matrices which guarantee the existence of an u.p.d.s. solution of the conventional ARE. The quadratic cost in eqn. 18 can be replaced by

J = J:(iiQl12* + 2xTQ12Y2 + Y T Q 2 2 Y 2 M (67)

which is minimised with respect to y2. The optimal sliding surface is

s = y2 + Kil = 0, K = Qyi(BTP + QT2) (68)

where is a p.d.s. matrix solution of the ARE

ATP + PA + ( F B + &2)Q,;'(ETF + Q y 2 ) = -6, I (69)

Let K = [ K , K2 K3]. Then the sliding surface is given by

Y2 + KIYI + K2xw, + K3xw, = 0 (70)

Note that in this case the sliding surface is a linear operator of states. Unlike both the previous cases, the sliding is no longer in the phase plane (y1,y2). When both e,, and Q22 are frequency-dependent, two compensators are added to the sliding system and eqn. 70 is the sliding surface for the augmented system combined with two compensators.

9 Example: two-link robot manipulator

Robot manipulators are controllable nonlinear mechanical systems. The task of a two-link robot is to move to a given final position as specified by two constant given joint angles. Each link joint has a motor for providing input torque, an encoder for measuring joint position and a tachometer measuring joint velocity. It is desired to

design the control or a sliding surface such that the joint positions 0,. and d)r tend to the desired positions H , , and

which are specified by a motion planning system. When a robot hand is required to moved along a specified path, there is a tracking problem.

Consider the robot manipulator with two-link arm, which moves in a horizontal plane. Thc nonlinear equa- tions governing its movements are

with

a = J2 + 2m21112 cos <b,.

h = Jl + J2 + 4m21: + 7 + 4m21212 cos d),. 1

g==- J2h - a2

I> = 2m211 l2 sin cbr

where li is the length of link i, J, is the inertia moment of link i about axis i, m, is the ma;s of link i, v, is the viscous friction constant for axis i , and I is the moment of inertia of the axis motor (see Fig. 2).

For simplicity, the 'small' cross terms in eqns. 71 and 72 will be ignored below, but could be included in a simula- tion. From eqns. 71 and 72

(73)

Let

0,. = xI 8,. =x2 4,. = xj 4,. = ~4

then

. .. . . .. X I = x 2 x2 =x, x3 =xq xq =.q

Fig. 2 Robot manipulator with two link urms

IEE Psoc.-Control Theory Appl., Vol. 147, No. 3, Muy 2000 3 17

and

Substituting parameter values gives

0 1.0000 0 0

0 -0.3320 0 0.0187

B =

0 0 1.0000

0.7830 0 -0.1914

O l 0

I 130.8 -308.3

O I 0

. -308.3 3155.4

(74)

and

T =

0 0 0 1

O O 1 (75) 1 0 -0.3906 0 0,9206

1 0 -0.9206 0 -0.39061

is the transformation matrix given by eqn, 6, and

1 0 0 -0.3906 -0.9206

0.9206 -0.3906

(a ) Consider the functional in eqn. 18 with Q , = I2 and Q22(s) = W;(s)W2(s). Assume

W,(s) = Dw2 + Cw2(s12 - 4,*)-'BW2 with

0.3 0.1 0 0

0 0.6 0.5 1 A , = [ 3. BW2 = [ ]

= [ 0.3 0.91, [ 0.28 0,901 D, =

0.3 0 0.30 -0.02 WZ

The augmented system is eqn, 23 and

1 15.2715 0.5819 -0.6878 3.1932

16.8784 0.4274 -0,8771 -1.2032 K = [

The eigenvalues of A, - B,K are -0.3537f 1.0688i, -0.1495, -3.4473; thus the augmented system in the sliding mode is stable. The locations of the resulting eigenvalues depend upon the choice of the proper weight- ing function Q22. Simulation results are shown in Figs. 3a and 3b. Note that, from eqn. 75, O r =yI1 and 4,. = y12.

318

stote behaviour of the ougmented system

( ' ( I ( '

0 2 4 6 8 10 12 14 16 18 20 time,s

sliding functions 1.0r

51 -0 .d I I I I I I I I

0 2 4 6 8 10 12 14 16 18 20 time,s

Fig. 3 Responses of(a) a State behaviour of the augmented system 11 Sliding functions

The iterative constructive method is now applied. At the first step consider

r 0*3000 0*9000 1 0.3000 0.30 -0.02

115.2715 0.5:19]' D l = !l:O :oo] 16.8784 0.4274

1 r 15,4455 17.1424

0.8339 1.2374 1.1684 0.2460

0 0.2460 1.8104 Nle =

l o 0 1

1 r518.2784 16.3706 0 0 I 16.y06 1.3313 0 0

0 1 0 Qle =

L o 0 o i J

The gain matrix is

5.8304 0.0379 -0.4473 0.8252

12.5602 0.6850 -0.5926 -0.4663 [

[

[

K1, =

The eigenvalues of A , - B,K,, are -0.1819f0.86911', -0.1711, - 0.9311.

The second step yields the gain matrix

I 1

31.1239 0.2781 -3.3756 15.7457

26.4561 2.0299 -0.8349 -5.6859 K2, =

The eigenvalues of A, - B,K,, are -0.1031f0.77891', -0,1749, -0.6789. The third step yields the gain matrix

3.0334 0.1311 -0.2920 0.4816

11,7485 0.6377 -0.4192 -0.2967 &e =

The eigenvalues of A, - B,K3, are -0,1074f0.89023',

Therefore, various sliding surfaces and augmented systems with different poles have been found. One can then select a sliding surface with suitable eigenvalues taking into account the actual system requirements.

-0.1232, - 0,5245.

IEE Proc -Control Theory Appl, Vol. 147, No. 3, May 2000

(6) The following illustrates Case (iii) when Q l l is a function of frequency but e,, is constant. Assume Q, =0.05. Then the u.p.d.s. solution of ARE in eqn. 52 is P = 0.22361,. Therefore

1 -0.0873 0.2058

-0.2058 -0.0873 K = [

and A , , - Al,K has the eigenvalue -0.2236 (repeated twice). Consider

and Eqn. 53 with

-0.2236 0

0.4000 0.2764 0.4 0.5

1.5 4.50

1.5 0.45

Taking Q2, =I2 and

! 4.5500 7.4250 4.5000 7.4250

7.4250 20.5025 7.4250 20.4525

4.5000 7.4250 4.5000 7.4250

7.4250 20.4525 7.4250 20.4525

yields

r 0,6564 3.8063 0.7437 3.60051 K, =

1-2.3073 -3.2503 -2,1014 -3,16291

The eigenvalues o f a , - B,K, are -4.8346, - 1.3068 and - 0.2236 (repeated). The repeated eigenvalues - 0.2236 are the eigenvalues of A , , - A , , K . Since A l l = 0, the eigenvalues of the LQ closed-loop A l l - A I 2 K are zero if Q, = 0. On the other hand, we desire to find K, so that the eigenvalues of the closed-loop A , , - A,,K remain the eigenvalues of the closed-loop A, - B,K,. Qo should be selected as a nonzero matrix. Let us select Q, = 0.05. Then A , , - A, ,K is a stable matrix and the system stability is given by A , - B,K,. Simulation results are shown in Figs, 4a and 4b.

stote behaviour of the augmented system

-0.05 0 5

time,s sliding functions

1.0

I

0 5 10 15 time,s

Fig. 4 Responses qf (b) a State behaviour o f the augmented system 6 Sliding functions

IEE Proc.-Control Theory Appl.. Vol. 147. No. 3, May 2000

(c) Now consider Case ( i v ) in which both weighting matrices are frequency-dependent. Suppose

[ 1 3 1 ,

[ 0.95 3.001 c = D,, =

1 0 1.00 -0.05 W I

0 2.0 AW2 = [ 1, BW2 = [ '"],

2.0 0.5 7.0 0.4

1.0 0 0.7 0

-0.6 1.0 -0.6 0.7

Then

K =

0.0128 -2.7951 -0.6191 -5.21 15 1.7316 0.2719 1 3.8636 1.6441 2.7856 21.5820 -0.2407 1.4231 c The sliding surface is given by Eqn. 70. Simulation results are shown in Figs 5a and 5b. The transient responses are much faster for this case

So two compensators corresponding to the weighting matrices Q, , and Q22 have becn designed. The order of the augmented system is much higher than the LQ system.

Cases (ii), (iii) and (iv) can be applied to practical problems as illustrated by the abovc example. In all cases the sliding function has the form of a dynamic compensa- tor. The poles of Q22 in Case (ii) and the zeros of Q, , in Case (iii) correspond to the compensator poles and zeros, respectively, and similarly for Case (iv). All these compen- sators influence directly the reduced order closed-loop transfer function.

The frequency shaping of the sliding mode can provide additional flexibility for the design of the sliding mode using LQ techniques, Suitable design of the sliding surface may depend on the (i) model validity, (ii) actuator / sensor characteristics, (iii) sliding function objectives and (iv) disturbance spectrum [8].

In this example Q,, and Q22 are proper weighting functions. There may be cancellations between the poles of one compensator and the transmission zeros of the other compensator system, when Q,, and Q2* are not proper functions. A modified method for this case has been presented in [5].

stote behoviour o f the augmented system L 2 0

-2 -1, - 6

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 L.0 L.5 5.0 time,s

sliding functions 15

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 L.0 L.5 5.0 time,s

Fig. 5 Responses of (c)

a State behaviour o f thc augmented system h Sliding fiinctioiis

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10 Conclusions 11 References

Frequency shaping control design, linked with linear quadratic optimal and sliding mode control, is a technique for controlling systems with uncertainties. In this paper a new method for designing the control and the sliding surface has been proposed when the LQ weighting func- tions are not constant for all frequencies. By using this method, pre- and postcompensators can be designed. The resulting augmented system is a combination of the origi- nal system and the compensators. The order of the augmented system depends upon the dimension of the original system state and the weighting functions. The sliding mode can be expressed as a linear operator of states, i.e. a dynamical system. Additionally, conditions have been obtained to retain the spectrum of the original LQ reduced order system as a subset of the spectrum of the augmented system. This is important if compensators for the system are required such that the eigenvalues of the LQ system are the eigenvalues of the augmented system. Furthermore, an iterative constructive procedure has been developed to obtain the optimal sliding mode. This method enables one to find various sliding surfaces and, by comparing the eigenvalue locations in the left-hand half- plane, a sliding surface can then be selected to suit.

By using the H , approach as in [9], the sliding gain matrix could be found for Cases (iii) and (iv) of Section 4. Other H, methods to obtain the feedback gain matrix could be adapted for the augmented systems discussed in this paper.

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approach’, Int. J Control, 1983, 38, pp. 657-671 UTKIN, VI.: ‘Sliding modes in control and optimization’ (Springer- Verlag, Berlin, 19921, YOUNG, K.D., and O Z G m E R , U,: ‘Frequency shaping compensator design for sliding mode’, Int. 4 Control, 1993, 57, pp. 1005-1019 YOUNG, K.D., KOKOTOVIC, P.V, and UTKIN, VI,: ‘A singular perturbation analysis of high-gain feedback systems’, IEEE Truns. Autom. Control, 1977, 22, pp. 931-937 ANDERSON, B.D.O., MOORE, J.B., and MINGORI, D.L.: ‘Relations betwccn frequency-dependent control and state weighting in LQG problems’, Optimal Control Appl. Methods, 1987, 8, pp. 109-127 FRANCIS, B.A.: ‘A course in H , control theoiy’ (Springer-Verlag, Berlin, 1987) GUPTA, N.K.: ‘Frequency-shaped cost functionals: extension of linear quadratic Gaussian design methods’, J Gaid. Control, 1980, 3, pp.

HASHIMOTO, El., and KONNO, Y.: ‘Sliding surface design in the frequency domain’, in ‘Variable Structure and Lyapunov control’, ZINOBER, A.S.I. (Ed.) (Springer-Verlag, Berlin, 1994), pp. 75-84 KOSHKOUEI, A.J., and ZINOBER, A.S.I.: ‘Frequency shaping in the sliding mode and control design’, Control Industrial Processes Confer- ence, University of Hertfordshire, 1995, pp. 2 1-28; also in: ‘The Control of Industrial Processes: synthesizing the statistical and engineering approaches’, published by University of Hertfordshire, 1996, pp. 38-43 KOSHKOUEI, A.J., and ZINOBER, A.S.I.: ‘Frequency domain approach to sliding mode control’, European Control Conference, Brussels, 1997, [850] MOORE, J.B., and MINGORI, D.L.: ‘Robust frequency shaped LQ control’, Automafica, 1987, 23, pp. 641-646 THARP, H.S., MEDANIC, J.V, and PERKINS, W.R.: ‘Parametenza- tion of frequency weighting for a two-stage linear quadratic regulator based design’, Automatica, 1988, 24, pp. 415418 KOSHKOUEI, A.J., and ZINOBER, A.S.I.: ‘Comments on linear quadratic regulators with eigenvalue placement in a vertical strip’, IEEE Truns. Autom. Control, 1999, 44, pp. 1417-1419

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