In this paper some fixed-order controllers are designed via statistical methods for the benchmark problem originally presented at the 1990 American Control Conference. Based on some recent results by the authors, it is shown that the statistical approach is a valid method to design robust controllers. Two different controllers are proposed and their performance are compared with controllers with the same structure, designed using different techniques

Abdallah, Chaouki T.

Ariola, M.

Dorato, Peter

Koltchinskii, V.

English

University of New Mexico

University of New Mexico
UNM Digital Repository
Electrical & Computer Engineering Faculty
Publications Engineering Publications
12-1-1999
Statistical controller design for the linear
benchmark problem
Chaouki T. Abdallah
V. Koltchinskii
M. Ariola
Peter Dorato
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Recommended Citation
Abdallah, Chaouki T.; V. Koltchinskii; M. Ariola; and Peter Dorato. "Statistical controller design for the linear benchmark problem."
Proceedings of the 38th IEEE Conference on Decision and Control (1999): 507-509. doi:10.1109/CDC.1999.832832.
Proceedings of the 38* 
Conference on Decision & Control 
Phoenix, Arizona USA December 1999 
TuM04 13:20 
Statistical Controller Design for 
Problem 
V. Koltchinskii' M. Ariola2 
the Linear Benchmark 
C. T. Abdallah3, P. Dorato4 
Department of Mathematics and Statistics EECE Department 
The University of New Mexico The University of New Mexico 
Albuquerque, NM 87131 USA Napoli, ITALY Albuquerque, NM 87131 USA 
vlad@math.unm.edu ariola@unina.it {chaouki,peter}Qeece.unm.edu 
Dipartimento di Inforrnatica e Sistemistica 
Universita degli Studi di Napoli Federico I1 
Abstract 
In this paper some fixed-order controllers are designed 
via statistical methods for the Benchmark Problem 
originally presented at the 1990 American Control Con- 
ference. Based on some recent results by the authors, it 
is shown that the statistical approach is a valid method 
to design robust controllers. Two different controllers 
are proposed and their performance are compared with 
controllers with the same structure, designed using dif- 
ferent techniques. 
1 Introduction 
It has recently become clear that many control prob- 
lems are too difficult to admit analytic solutions. New 
results have also emerged to show that the computa- 
tional complexity of some "solved" control problems 
is prohibitive. In order to get around such problems, 
many authors have recently advanced the notion of 
probabilistic methods in control analysis and design. 
In control theory, some of the original (Monte Carlo) 
ideas have already been used, among the others, by Ray 
and Stengel [l], Tempo et al. [2], and by Khargonakar 
and Tikku [3], to solve robust analysis problems while 
Vidyasagar used learning theory to solve robust control 
problems [4], [5]. 
Unfortunately, and as acknowledged by the various au- 
thors, probabilistic methods, while more efficient than 
gridding techniques (which suffer from the curse of di- 
mensionality), still require a large number of samples 
in order to guarantee accurate designs. In [6] ,  we pro- 
posed a new control design algorithm which greatly re- 
duces the required number of plants sampled in order to 
achieve a certain performance level. In the current pa- 
per, we illustrate the use of our algorithm in designing 
'The research of V. Koltchinskii is partially supported by 
NSA Grant MDA904-99-1-0031 
2The research of M. Ariola is partially supported by the 
MURST 
3The research of C. T .  Abdallah is partially supported by Boe- 
ing Computer Services Grant 3-48181, and by NSF INT-9818312 
4The research of P. Dorato is partially supported by NASA 
contract NCCW-0087, at the NASA Center for Autonomous 
Control Engineering, University of New Mexico 
0-7803-5250-5/99/$10.00 0 1999 IEEE 507 
fixed-order robust controllers for the linear benchmark 
problem (see also [7]). 
2 Problem Formulation 
The benchmark problem was originally proposed in [SI. 
The plant consists of a two-mass/spring system with 
non-collocated sensor and actuator. The system can 
be represented in dimensionless state-space form as 
(see [si, [91) 
y = x z + v  
z = x2 
where x1 and x2 are the positions of the masses, c is the 
internal damping between the masses, f is a loop-gain 
uncertainty, U is the control input force, i- is a time 
constant for a first-order delay between the controller 
command U ,  and the actuator response U ,  w is the plant 
disturbance, y is the sensor measurement corrupted by 
the noise U, and z is the output to be controlled. 
Three design problems were proposed in [8]. The most 
demanding one requires: i) Closed-loop stability when 
the parameters ml, m2 and k are uncertain with mean 
value 1; ii) A 15 s settling time for unit disturbance 
impulse for the nominal plant ml = m2 = k = 1; 
iii) The minimization of the control effort and of the 
controller complexity. 
Many controllers were proposed for this problem. They 
are collected and analyzed in [9], where the authors, 
after evaluating the nominal performance, carry out 
a Stochastic Robustness Analysis in order to analyze 
the behavior when the plant parameters change. The 
six uncertain parameters are assumed to be uniform 
independent random variables in the following inter- 
vals: 0.5 < IC < 2, 0.5 < ml < 1.5, 0.5 < m2 < 1.5, 
0 < c < 0.1, 0.9 < f < 1.1 and 0.001 < T < 0.4. We 
shall denote by X E X R6 the vector of these uncer- 
tain parameters x = [ IC ml m2 c f i- 3'. 
Based on the specifications, our target is to design a 
fixed-structure controller such that: i) The nominal 
plant is stabilized; ii) The 15 s settling time specifica- 
tion is satisfied for the nominal plant; iii) The control 
effort does not exceed a one unit saturation limit in re- 
sponse to a unit w disturbance, for the nominal plant; 
iv) A certain cost function is minimized. This cost 
function accounts for the closed-loop stability and the 
performance in the presence of parameter variations. 
3 Statistical Design of Fixed-Order Controllers 
In this section, using a randomized algorithm which is 
described in detail and proven elsewhere [6], we shall 
design two fixed-order controllers. One of the charac- 
teristics of the algorithm is that the number of plants 
tested depends only on the accuracy and the confidence 
chosen but not on the complexity of the plant nor of 
the controller. 
Denoting by Y E Y IRm the vector of controller co- 
efficients, the two chosen controllers have the following 
structures 
(2) 
a1s2 + a2s + a3 
K2(s7 = s3 + b1s2 + b2s + b3 
The coefficients of the controllers are chosen to have 
uniform distributions. For the controller (1) these coef- 
ficients take values in the intervals a0 E [0.5, lo], a1 E 
whereas for the controller (2) they take values in a1 E 
[-2,-0.5], a2 E [-0.3,-0.11, bi E [1,5], b2 E [1,6], 
[-50,501, a2 E [-120,-40], a3 E [-40,-lo], bl E 
[70,170], b2 E [80,160], b3 E [loo, 1401. 
In order to use the randomized algorithm methodology, 
this problem has been reformulated in the following 
way (see also [6], [5]). Let us define a cost function 
W )  = m={$l(Y),$2(Y)) (3) 
where 
0 
1 otherwise 
if all requirements on the nominal plant are met 
and 
$2(Y) = E (C(X7 Y ) )  9 (4) 
where E indicates the expected value with respect to 
X ,  and 
if the random plant is not stabilized 
if both the control limit and the settling time 
specifications are not satisfied 
if either the control limit or the settling time 
specification is not satisfied 
otherwise 
508 
Our aim is to minimize the cost function (3) over Y .  
What we shall find is a suboptimal solution, a probably 
approximate near minimum of q ( Y )  with confidence 
1 - 6, level a and accuracy E (see [lo]), using the fol- 
lowing Procedure, which was derived in [6]. 
Procedure 
1. Let k = O  
2. Choose n controllers with random uniformly dis- 
tributed coefficients Yl, . . . ,Y, E Y ,  where (we 
indicate by 1.1 the floor operator) 
Evaluate for these controllers the function $1 (4) 
and discard those controllers for which $1 = 1. 
Let fi be the number of the remaining controllers. 
3. Choose m plants generating random parame- 
ters XI, . .. , X ,  E X with uniform distribution, 
where 
4. Evaluate the stopping variable 
where ri are Rademacher random variables, i.e. 
independent identically distributed random vari- 
ables taking values +1 and -1 with probability 
112 each. If y 5 €15, stop. The value of m is 
large enough to guarantee the required probabil- 
ity levels. If y > €15, let k = k + 1 and go back 
to step 3 
5. Choose the controller which minimizes the func- 
tion 
This is the suboptimal controller in the sense de- 
fined above. 
Remark 1 The proposed algorithm consists of two 
distinct parts: the estimate of the expected value in (4), 
which is given with an accuracy E and a confidence 
1 - 612, and the minimization procedure which is car- 
ried out with a confidence 1 - 612 and introduces the 
level a. As it can be seen from the Procedure, the 
number m of samples in X which are needed to achieve 
the estimate of the expected value (4), known as the 
sample complexity, is not known a priori but is itself 
a random variable. The upper bounds for this random 
sample complexity however, are of the same order of 
those that can be found in [4]. 
In both cases of our controllers, the procedure needed 
just one iteration to converge, i.e. k = 0. Therefore, 
for 6 = 0.05, a = 0.005 and E = 0.1, n evaluated to 
736 controllers and m evaluated to 50,753 plants. The 
suboptimal controllers are 
(5) 
(6) 
1 . 1 1 1 0 ~ ~  - 1.7393s - 0.2615 
K1(s) = s2 + 3.6814s + 2.9353 
K2(s )  = s3 + 92.1586~~ + 123.3358s + 131.8229 
31 .9432~~  - 76.6527s - 12.7876 
and the corresponding values of the cost function are 
!Po = 0.2683 for (5) and iP0 = 0.2062 for (6). As ex- 
pected, with a more complex controller we get a better 
result. 
4 Analysis of the Performance 
In this section we shall compare the performance of 
the two controllers (5)-(6) with those of 4 other con- 
trollers analyzed in [9] with the same structure as the 
ones proposed here. The transfer functions of these 4 
controllers can be found in [9]. 
The performance are quantified using a Monte Carlo 
evaluation. According to the distributions of the pa- 
rameters, 20,000 plants (see [9] for the choice of this 
number) are randomly generated and estimates of the 
following three metrics are calculated: 1. PI: Prob- 
ability of instability. This is the probability that a 
randomly generated plant is not stabilized. 2. P T ~ :  
Probability of exceeding the settling time. This is the 
probability that the 15 s settling time specification is 
not satisfied. 3. P,: Probability of exceeding the con- 
trol limit. This is the probability that the actuator 
displacement will exceed a one unit saturation limit in 
response to a unit w disturbance. 
Table 1: Robust Performance 
[ Controller 1 PI 1 PT, I P, I 
I 0.002 I 0.803 I 0.002 I Ki c I 0.041 0.874 1 0.041 
0.125 0.999 I 0.409 
The data for the controllers A, B, C and E are taken from [9] 
The settling time and the control limit specifications 
were considered to be violated also when the controller 
fails to stabilize a plant. As shown in Table 1, the 
two controllers designed with the statistical approach 
exhibit a better behavior in all the three cases. 
5 Conclusions 
In this paper, we have illustrated the use of sequential 
learning algorithms in designing robust controllers for 
linear systems. This approach has been developed in [6] 
and presents a significant extension of the results of [5]. 
The approach is not limited to linear problems, nor to 
the finite-dimensional case as one can apply similar de- 
sign steps to nonlinear systems and to delay-differential 
systems. In addition, discrete-time systems may be 
dealt with in exactly the same fashion. 
References 
[l] L. Ray and R. Stengel, “A Monte Carlo Approach 
to the Analysis of Control System Robustness,” Auto- 
matica, vol. 29, pp. 229-236, 1993. 
[2] R. Tempo, E. Bail and F. Dabbene, “Probabilis- 
tic Robustness Analysis: Explicit Bounds for the Min- 
imum Number of Samples,” Systems and Control Let- 
ters, vol. 30, pp. 237-242, 1997. 
[3] P. Khargonekar and A. Tikku, “Randomized Al- 
gorithms for Robust Control Analysis and Synthesis 
Have Polynomial Complexity,” in Proc. IEEE Conf. 
on Dec. and Control, (Kobe, Japan), pp. 3470-3475, 
1996. 
[4] M. Vidyasagar, A Theory of Learning and Gen- 
eralization: With Applications to Neural Networks and 
Control Systems. London: Springer-Verlag, 1997. 
[5] M. Vidyasagar, “Statistical Learning Theory and 
Randomized Algorithms for Control,” IEEE Control 
Systems Magazine, vol. 18, pp. 69-85, Dec. 1998. 
[6] V. Koltchinskii, C. T. Abdallah, M. Ariola, 
P. Dorato, and D. Panchenko, “Statistical Learning 
Control of Uncertain Systems: It is better than it 
seems,” Tech. Rep. EECE 99-001, UNM, 1999. Sub- 
mitted to Trans. Auto. Control, Feb. 1999. 
[7] C. T. Abdallah, M. Ariola, P. Dorato, and 
V. Koltchinskii, “Quantified Inequalities and Robust 
Control,” in Robustness in Identification and Control 
(A. Garulli, A. Tesi, and A. Vicino, eds.), pp. 373-390, 
Springer Verlag, London, 1999. 
181 B. Wie and D. S. Bernstein, “A Benchmark Prob- 
lem for Robust Control Design,” in Proceedings of the 
1990 American Control Conference, (San Diego, CA), 
pp. 961-962, May 1990. 
[9] R. Stengel and C. I. Marrison, “Robustness of 
Solutions to a Benchmark Control Problem,” Journal 
of Guidance, Control, and Dynamics, vol. 15, no. 5 ,  
[lo] M. Vidyasagar, “Statistical Learning Theory and 
Its Applications to Randomized Algorithms for Robust 
Controller Synthesis,” in ECC97, vol. Plenary Lectures 
and Mini-Courses, (Brussels, Belgium), pp. 162-190, 
1997. 
pp. 1060-1067,1992. 
509 


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Statistical controller design for the linear benchmark problem

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