H∞ Control Application To Three-Inertia Benchmark Problem

Copyright a> IFAC Robust Control Design, Prague, Czech Republic. 2000

ROBUST SERVO SYSTEM DESIGN BY MIXED H2tH. CONTROL APPLICATION TO THREE-INERTIA BENCHMARK PROBLEM Yeon-Wook CHOE Department of Control & Instrumentation, Pukyong Nat 'I Univ., Pusan, Korea 608-739, E-mail: [email protected])

Abstract: The purpose of this paper is to propose an approach to design a robust servo controller based on the mixed Hi H_ theory, and conflrm its validity by applying this to a benchmark problem. First., the existing H_ servo problem is modi fled to a structure for the mixed H2/ ~ control problem by virtue of the internal model principle. By adopting this structure, we can divide speciflcations required in the robust servo system design into H2 and H_ performance criteria, respectively. We show that the proposed design approach is quite effective through an application to a three-inertia benchmark problem. Copyright @2000 IFAC Keywords: Robust performance, Servo system, LMI

I .INTRODUCTION Although the H_ control is a suitable technique to achieve the flrst two speciflcations, it is inevitable that the response should be rather conservative. Recently, it has been proved that., by introducing H2 speciflcation into the ~ design, we could simultaneously benefit from the H2 and H_ control design (Scherer, et aI., 1997). This approach is called a mixed HiH- control, and in consequence, a designer can arbitrary determine the trade-off between H2 (e.g. noise rejection) and ~ (e.g. robust stability) performance of the closed loop system.

In the past several years, the H2 and/or ~ control have attracted many researchers' attentions. Also, its effectiveness has been reported in various application flelds in these years. The H_ control theory provides us a quite powerful tool for shaping the loop gain in the frequency domain or obtaining the robust stability property. Therefore, it is quite useful to obtain satisfactory feedback properties such as low sensitivity. On the other hand, the H2 optimal control theory has been heavily studied since 1960's as the LQG optimal control problem. The H2 norm performance measure seems to be suitable for obtaining good command response.

The purpose of this paper is to propose an approach to design a robust servo controller based on the mixed HiH- control and confmn its effectiveness by application to a benchmark problem. The formulation of the mixed HiH- control problem for servo system is presented in Section 11. The main results are given in Section Ill, where the structure of generalized plant for robust tracking is proposed and, we also show that there exists a solution to the problem by virtue of an LMI approach. In Section IV, after designing a robust controller based on the proposed

For a servo system design. the following three speciflcations are of practical interests: (1) internal stability of the closed-loop system which must be guaranteed; (2) desired feedback characteristics such as robust stability, sensitivity reduction and disturbance attenuation; (3) desired transient and steady-state properties such as robust tracking to reference inputs.

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solution K to the H- servo problem.

method for the three-inertia benchmark problem. we analyze the results of simulation and check the validity of the proposed structure as a robust servo system design method.

[Theorem 1] Consider the feedback control system depicted in Fig.I. It is assumed that the plant G = ND- 1 and the reference signal generator

GR =D;I NR are described by using left and right coprime factorization, respectively. Then, the necessary and sufficient condition for the existence of a solution K to the H_ servo problem is that the following three conditions must be satisfied simultaneously.

2. PROBLEM FORMULATION 2.1 H_Servo Problem

Consider a unity feedback control system shown in Fig.l , where G(s) and K(s) denote a plant and controller, respectively. We assume that the class of a reference signal is described as

(Cl) K e O(G) , where Q represents the set of stabilized compensators to a plant G. (C2) (1- NK)D;le RH_ (C3)

where GR(s) is a transfer function which does not have stable poles, and ro is an unknown constant. The purpose of servo system design is to find a feedback compensator K(s) that satisfies the following three specifications: I) closed-loop internal stability, 2) desired feedback characteristics such as robust stability, sensitivity reduction and disturbance attenuation, and 3) robust tracking. Here, the robust tracking means that the output y(t) of the plant must track for any type of reference signal r(t) without steady-state error, i.e., lirn e(t) =0 , under plant

DK/a R e RH., wherea R denotes the largest invariant factor ofD R , DK is a denominator when a solution K is described by right coprime factorization andRH)s the set of all stable real rational functions.

Hence, H_ servo system must have the structure as depicted in Fig.2, where Kr internally stabilizes G/a R , and z and w denote the output of interest and an exogenous input, respectively. And a.~1 denotes a model related to the reference signal.

,-

P(S)

perturbation and/or a step type disturbance input d(t). d

,~ G. ~-f~ , K " + f-----.--.+

z ---"-----~

_ _- - w

y u y

Fig.1 Feedback Control System First, we consider the robust servo problem with Hnonn bound, called the H. servo problem and defined as follows (Hozumi, et al., 1997).

Fig. 2 The Original Problem

2.2 Mixed H/H_ Optimal Design Problem

[Definition] H_ servo problem: Consider a feedback control system depicted in Fig.l. For a given generalized plant P which includes the plant and weighting functions for loop shaping, and a reference signal r(t), fmd a controller K satisfying the following three specifications:

The basic block diagram used in this paper is given in Fig.3, in which the generalized plant P is given by the state-space equations

(S I) K internally stabilizes P

(2)

IIT....

(S2) (s)1!- < y (S3) K achieves the robust tracking property for the reference signal.

where se R" is state vector, u is the control input, w is an exogenous input (such as a disturbance input, sensor noise etc.), y is the measured output and z =[z. Zz f is a vector of output signal related to the

Ted (s) represents the transfer function from d to e.

There have been lots of researches related to the Hservo problem. Sugie et al. (1989) derived sufficient and necessary conditions for the existence of a

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the conventional H_ control theory. For the purpose of this, we, first, divide the control objectives into each H_ and H2 performance criterion, then describe the two criteria as one formation. In other words, a new structure for the mixed H/H_ control is required to deal with two criteria simultaneously. Here, we introduce the following interconnection for robust control system, on which a controller satisfying two criteria is designed.

performance of the control system (z.. is related to the H_ performance and ~ is related to the H2 performance ).

Zex>

-p

w

~

y

u

I Kr I

°1

1

Fig.3 Block diagram of Mixed H/H_ control Let Tzw be the closed transfer function from w to Z for the system P closed by the output-feedback control law u Kry . Our goal is to compute a dynamical output feedback controller KT

=

Fig. 4 The proposed generalized plant for Mixed H21H.. control

(3)

A.H_ Control Problem: In Fig.4, Wis) denotes a

that simultaneously meets H2 and H_ performance on the closed-loop behavior. The closed-loop system

weighting function related to the plant uncertainty (this case, additive uncertainty) and W.(s) is a sensitivity weighting function. We can summarize the robust tracking H_ control problem as follows :

Trw has the following

description (S 1) K,.(s) stabilizes P( s) .

(4)

(S2) IIT_(s)1I-

=11i.~::~:l < y

where TZ_I W (s)denotes the transfer function from w to Z_l and is related to robust stability requirement (for additive uncertainty)

The problem we concerned with can be summarized as minimizing the H2 norm of the channel

w

~ Z 2 (:

T2 ), while keeping the bound yon the ~ 11T._,w"-

norm of the channel w ~ z_(: T_), i.e.

=11(1 + GKfl KWa "-
(5)

K=a.~IKr The nominal performance condition is reflected by Since this problem can be reformulated as a convex optimization problem (Scherer, et al., 1997), the optimal solution under the given value of y can be obtained through LMI. As efficient interior-point algorithms are now available to solve the generic LMI problems, the mixed H/IL. problem can be solved without much difficulty in order to find the best trade-off between the H2 and ~ minimization.

11T._,w(st < y, where TZ2 _ W (s) denotes the transfer function from w to Z_2 . In this case, if (S2) is satisfied under the condition of r=Q.5, then robust performance can be guaranteed outright, since (S2) will satisfy the SISO robust performance test for additive uncertainty given by (Zhou, et al., 1996)

(6) 3. MAIN RESULTS The robust performance condition that was given in (6) is necessary and sufficient, and the left-hand side is actually the peak value of ~ (Packard and Doyle, 1993). In section 4, we will investigate this value as an index to robust performance when designing a

In order to find a robust servo controller K which satisfies good reference response (H2 performance) as well as desired feedback properties ~ performance), we adopt the mixed H21H.. control system rather than

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control system for a benchmark problem.

't d, (i =

B. H] control Problem : The traditional H2 optimization attempts to minimize the energy of the system output when the system is faced with white Gaussian noise input. So, in order to design a controller adept at handling noises, H2 optimization should be considered. That is, the H2 norm minimization of the transfer function Tt ,w s) from w

jj(i

1 - 3) = torque disturbance

=1- 3)[kgm2]= moment of inertia

d j (i = 1 - 3, a, b )(Nmslrad] = viscous-friction coefficient of motors, k j (i a, b )[Nmlrad] = torsional coefficient of connection part

=

to z] in Fig.4 is to be taken as a controller design problem, where p is a varying parameter. It is well known that the upper bound of

liT ,0'11:

is

defined by tr(CCI2WoC~2 )' where Wo solves the Lyapunov equation

Fig.5 Coupled three-inertia system

Problem 2: Position Control Problem : We want to design a controller by which the six design specifications given in (Hara, et al., 1995) are to be satisfied on condition that all of the 11 parameters are subject to change within the range of variation and there must exist hardware constraints.

(7)

Since Wo < W for any W satisfying

It is readily verified that

liT ,0'11: < v

4.1 Feedback Control/er Design by mixed H/ H_ control

if and only if

there exists W >0 satisfying (8) tr(C el2 WC~2) < v (Scherer, et al., 1997).

and In this problem, there are 11 physical parameters that are assumed to be changing within the given range of variation. If all the variations are reflected in the controller design, the obtained controller may be considerably conservative as well as complex. Therefore, we first find out principal parameters 0) and kJ which strongly affect the resonant frequency of the plant by plotting the frequency response curve, then a robust controller design dependent on these parameters is carried out by making use of the proposed structure for mixed H2~ control. The robustness on variations of the other parameters is evaluated through simulation.

C. Mixed HJH_ Control: The mixed H21H_ controller K, must satisfy both of the following criteria simultaneously

I~_wt
(9)

IIT ,wIl 2
(10)

The solution under the constraints of (9), (10) is dependent on the value of 'Y and v , and can be obtained using any available software such as Matlab LMI toolbox (Gahinet, et al., 1995).

The parameter variations ofjj and ko can be described by additive uncertainty such as

4. APPLICATION TO THE BENCHMARK PROBLEM

J') =J')• +WlJ 0h

(ll-a,b)

ko =ko• +Wk Ok.

We consider the benchmark problem given in (Hara, et al., 1995). The model treated in the problem is a coupled three-inertia system that reflects the dynamics of mechanical vibrations. A controller, by which robust performance (both in time and frequency-domain) condition must be satisfied, is required in order to solve the problem.

where

j3 o ,k

Qo

are nominal values, Wh. ' ~• are

constant values representing the range of variation, and (12)

The three-inertia benchmark problem is shown in Fig.5, where the meanings of each symbol are as follows:

Furthermore, if we express the reciprocal ofjjas (13)

8 j (i =1- 3) [radian]= angular displacement t= control torque [Nm]

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shown in Fig.7.

the conservativeness related to the additive uncertainty may be reduced up to a certain point by adopting (13) in place of(l1-a) in the plant dynamics. +

1 - - - -..-._ +

f-----'

INk.

Fig. 7 Structure for Mixed H21H_ controller design

4.2 Feedforward Controller Design

Since it is impossible to meet all the design specifications related to the output transient response only by a feedback controller, we adopt the structure of two-degree-of-freedom system to cope with this problem. That is, by adding a feedforward path, we try to improve the output time response. The feedforward controller is designed by using Model Matching Method (Sugie and Vidyasagar, 1989). The details are omitted.

Fig. 6 Representation for additive parameter variation If we define the input and output of variations as

then parameter variation from

Ws

to

z_, can

be

described as a structured perturbation using 4.3 Simulation Results

(15) After designing a robust controller based on Fig.7, we check the six design specifications through simulation. Matlab is used for computation.

The purpose of a controller is, if possible, to make z_, =[9 -9 9 -9 3 small in the

ut

presence of parameter variations and disturbance. This can be achieved by letting the closed-loop to transfer function T- 2 from wT = 'td 't]T d1

In Table 1, the parameter values used in the simulation are shown, where 'nominal' means an ideal case without parameter variations, 'CaseI' represents that the moments of inertia have their minimum values and the torsional and viscousfriction coefficients are varied maximally within the range of variation, 'Case2', on the contrary, represents that the moments of inertia have their maximum values and the torsional and viscousfriction coefficients are varied minimally, and the case that all of the parameters have their minimal value is represented by 'Case3'. The values of Wh' w:.• which express the magnitude

Z_2 have robust performance. Therefore, if it is

possible to design a controller by which we keep the H_-norm of the closed-loop transfer function T_ from w

=[Wj,

Wk.

't d,

't d,

t

to

Z =[Zj, Zk. 9 1 -92 9 2 -93 uf low, the design specifications can be satisfied. In other words, the output will track the reference signal and torsional vibrations between 6J and 61> 61 and 6j will be also suppressed under the condition of torque disturbance and parameter variations.

of variations on J3,k. are given as 0.004 (20% variation) and 8 (10% variation), respectively. And constants a.J31,J32 are determined as 10, 0.08, and 0.05, respectively through several trial and errors.

As a next step, by introducing H2 specifications into the H- design, we can allow to take noise transmission aspects into account. That is, a controller, by which H2-norm of the transfer function T2 from w (refer to Fig. 7) to Z2 is to be minimized under the condition of H- norm, can be designed by applying the mixed Hi H_ control to the structure

Table J Nominal and varied parameters Para-

meters

141

Nominal

Case 1

Case 2

Case 3

j, J2

jJ

0.001 0.001 0.002

0.0009 0.0009 0.001

0.0011 0.0011 0.003

0.0009 0.0011 0.003

920 80

1012 88

828

828

72

72

0.005 0.001 0.007 0.001 0.001 5000

0.055 0.0011 0.035 0.01 0.0011 5000

0.045 0.0009 0.0014 0.0002 0.0009 5000

0.045 0.0009 0.0014 0.0002 0.0009 4500

k.

~

d, d2 dJ d. ~ as

purpose of designing a robust controller, the design objectives such as better transient response and robust tracking are first defined in terms of H2 and H- optimization theory, then the generalized plant for the mixed H/H_ control is determined and solved by using a LMI algorithm. Practical computation to get a controller is now quite easy thanks to some excellent software such as Matlab. Gain PIal m ~ SoreitNty t.n:1icn 10' E'--'-~~~~-~~ ' .,.:.......~.-.~ . ~~~~-d

,

", ~.

n : ;

(1) Tracking ability - specification no . I

,':

The reference tracking ability is shown in Fig. 9, where 93 and t represent plant output and control input. From the results of simulation, we know that the design specifications were sufficiently satisfied in the presence of the parameter variations and disturbance.

;

I

I .

~, .rr · , I :iA. = , ' ' ~ 0.,r :\ \ ~

-:'O.5 ~

:

J

'~ '-,~

0: '

,

~--:------'

0..'

2 f~

0.'

i.

_ _ - --

: ,

-,--'=--_ _ _ - ' 0.2

0.'

0..2

Timeloet.J

0..'

It is thought that the difficulty in selecting weighting

functions, that is essential to the general H_ control theory, can be avoided to some degree if we use the proposed structure for controller design. While the LMI-based approach is computationally more involved for large problems, it has the merit of eliminating the regularity restrictions attached to the Riccati-based solution. For example, the problems caused by adding an integrator in the loop for servo system design can be easily handled through the generalized plant proposed here.

'~

,

!. ._~ ,:k~ , 0.

-,0.

!

0..2

Timolsec.J

Fig.9 Gain plots of complementary sensitivity function

_ __

0..0.15[1,

I

~

0.0.;:[( 0.1 ~· OO5 r

\q------

V ~. 01 ~ : --:---_::_:_---' 0 0.2 0.'

Time IMC.J

Tome IMC.J

Fig. 8 Step Responses to the reference input (2) Complementary Sensitivity Function Specification no . 4

REFERENCES The gain plots of the complementary sensitivity function are shown in Fig.lO. We know that, although the condition - the gain must be under 20[dB] over all the frequencies considered - is satisfied despite of variations, the other one - the gain must be under -20[dB] above 300[radlsec] frequency - cannot be met in any cases. Actually, since it was already known that specification no. 4 and the others had a reciprocal relationship each otherl6), it is impossible to meet all the specifications simultaneously.

Gahinet, P., A. Nemirovski, A.J. Laub and M. Chilali (1995). In: LMI Control Toolbox, The MATHWORKS Inc. Ham, S., etc (1995). A coupled three-inertia benchmark problem. In: Measurement and Control, vo1.34, no.5, pp. 403/409 (In Japanese) Hozumi, J., S. Hara and H. Fujioka (1997). Robust Servo Problem with Hoc norm Constraint. In Int. J. Control, vo!. 66, no. 6,pp 803-823 Packard, A. and J.C. Doyle (1993). The Complex Structured Singular Value. In: Automatica, 29-1 , pp.711109 Scherer, c., D. Gahinet and M. Chilali (1997). Multiobjective Output-Feedback Control via LMI Optimization. In IEEE trans. IEEE Trans. Auto. Contr., voI.AC-42, no.2 pp. 896/911 Sugie, T. and M. Vidyasagor 1989). Further results on the robust tracking problems in the twodegree-of-freedom control systems. In: Systems and Control Letters, 13, pp 1011108

5. CONCLUSION In this paper, we have proposed a generalized plant structure for the mixed HiH_ control theory in order to design a robust servo controller achieving good robust performance. And the effectiveness of the proposed structure was confirmed by applying to a coupled three-inertia benchmark problem. For the

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