August 2015 

NASA/TM-2015-218795 

A Case Study on the Application of a Structured 

Experimental Method for Optimal Parameter 

Design of a Complex Control System 

Wilfredo Torres-Pomales 

Langley Research Center, Hampton, Virginia 
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August 2015 

NASA/TM-2015-218795

A Case Study on the Application of a Structured 

Experimental Method for Optimal Parameter 

Design of a Complex Control System 

Wilfredo Torres-Pomales 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA STI Program / Mail Stop 148 

NASA Langley Research Center 

Hampton, VA 23681-2199 

Fax: 757-864-6500 

Acknowledgments 

I would like to express my gratitude to the reviewers, Mr. Sixto Vazquez and Dr. Kenneth Eure of NASA 

Langley Research Center, and to Professor Resit Unal from Old Dominion University who introduced me 

to the field of Robust Design and Reliability Engineering. 
iii

Abstract 

This report documents a case study on the application of 

Reliability Engineering techniques to achieve an optimal balance 

between performance and robustness by tuning the functional 

parameters of a complex non-linear control system. For complex 

systems with intricate and non-linear patterns of interaction 

between system components, analytical derivation of a 

mathematical model of system performance and robustness in 

terms of functional parameters may not be feasible or costeffective. The demonstrated approach is simple, structured, 

effective, repeatable, and cost and time efficient. This general 

approach is suitable for a wide range of systems. 
iv

Table of Contents 

1. Introduction ................................................................................................................................................ 1

1.1. Background Systems Concepts............................................................................................................ 1

1.2. Overview of Content ........................................................................................................................... 2

2. System Under Study ................................................................................................................................... 2

2.1. Plant ..................................................................................................................................................... 2

2.1.1. Ideal Plant Model.......................................................................................................................... 2

2.1.2. Laboratory Plant Model ................................................................................................................ 3

2.2. Controller ............................................................................................................................................. 4

2.2.1. Pendulum Controller..................................................................................................................... 5

2.2.2. Arm Controller ............................................................................................................................. 6

3. System Tuning Problem ............................................................................................................................. 6

4. Reliability Engineering Concepts ............................................................................................................... 7

4.1. Design of Experiments ........................................................................................................................ 7

4.2. Response Modeling ............................................................................................................................. 8

4.3. Optimal and Robust Parameter Design ................................................................................................ 9

5. System Tuning Approach ..........................................................................................................................10

5.1. Control and Noise Factors ..................................................................................................................11

5.2. Stimulus Tracking Signal ...................................................................................................................12

5.3. Response Metrics ................................................................................................................................12

5.4. Experiment Design .............................................................................................................................13

6. Results .......................................................................................................................................................13

6.1. Experimental Response Measures ......................................................................................................13

6.2. Response Model .................................................................................................................................13

6.3. Optimal Parameter Design ..................................................................................................................15

6.4. Verification .........................................................................................................................................15

7. Final Remarks ............................................................................................................................................18

8. References .................................................................................................................................................19

Appendix A. MATLAB Script for Simulation Experiment ...........................................................................20
1

1. Introduction 

This report documents a case study on the application of Reliability Engineering techniques to achieve 

an optimal balance between performance and robustness by tuning the functional parameters of a complex 

non-linear control system. The system of interest is the nonlinear pendulum control system reported by 

Torres-Pomales and Gonzalez [2]. The following subsection examines fundamental systems concepts and 

establishes a general context for the case study. 

1.1. Background Systems Concepts 

An engineered system is an entity with a purpose that is achieved by the interactions (i.e., relations) of 

its internal components (i.e., sub-entities) with each other and with the external environment (i.e., 

surroundings). The concept of system is circular and recursive such that, in general, both the environment 

and the components of a system are themselves systems. A system of interest (SOI) (or system in focus) is 

always contained within a larger super-system (or meta-system) consisting of the SOI and its environment. 

The purpose (i.e., a goal) of an engineered system is defined in terms of the desired (i.e., intended) 

effect on its environment achieved by the flow of matter, energy, or information between the system and 

its environment. The system purpose is the basis for the specification of the desired functional and quality 

attributes (i.e., properties) of the system. The purpose of system components is to contribute to 

accomplishing the larger goal of the containing system. Thus, in general, there is a dependence (i.e., a 

relation) between a system and its environment. Avizienis et al. define system dependability as “the ability 

to deliver service that can justifiably be trusted” and also “the ability to avoid service failures that are more 

frequent and more severe than is acceptable” [1]. Dependability is a function of criticality, which is a 

measure of the importance (i.e., degree or strength of influence) of a factor in achieving a larger goal. The 

failure or variability in performance of a high criticality component can have a severe effect on the operation 

of the containing system. Dependability is also a function of trust (i.e., confidence) in the service delivered 

by a system. Assurance is the level of confidence in the properties of a system. This confidence is bounded 

by the various forms of uncertainty (both epistemic and aleatoric) that threaten the ability to achieve and 

ascertain the desired system properties. Uncertainties can originate internal to the system, or they can be 

external in nature and their effects propagate to the system (for example, at the inputs). Dependability is 

bounded by the ability to measure uncertainties and predict their effects on goals. In general, the ability to 

assess uncertainties and their effects diminishes as system complexity (i.e., number and variety of 

components, relations, and states) increases. The dependability of a component (or sub-system) can be 

seen as a measure of the level of associated risk (i.e., potential for unwanted outcome) with respect to the 

goals of the containing system. 

Goals, dependencies, and uncertainties are interrelated. In engineering a system, we want to achieve a 

balance between performance (i.e., utility or value with respect to purpose) and robustness (i.e., 

insensitivity to uncertainties and undesired influences) in achieving the goals. The relation between goals 

and uncertainties is primarily determined by the dependencies defined in the system architecture and is a 

function of system parameters. Once a system architecture has been specified, we want to determine the 

combination of parameter values that results in the optimal balance between performance and robustness. 

For complex systems with intricate and non-linear patterns of interaction between system components, 

analytical derivation of a mathematical model of system performance and robustness in terms of functional 

parameters may not be feasible or cost-effective. Even if a model can be derived, closed-form solutions 

may not be possible. Without effective analytical means to determine the best combination of system 

parameters, the alternative option is experimentation. However, trial-and-error can be time consuming and 

the results can be unpredictable. We want a methodical, effective, repeatable, and cost and time efficient 

approach to tune the parameters of a system. 
2

1.2. Overview of Content 

The next section describes the system of interest, including the plant and the controller. This is followed 

by the system tuning problem statement. An overview of the Reliability Engineering concepts applied in 

this study is given. This is followed by a description of the approach used to tune the system. Detailed 

results are presented. Conclusions and final remarks end the report. 

2. System Under Study 

Figure 1 is the block diagram of the control system. The state of the plant X is controlled by the motor 

torque Td. The system is designed to track the target arm position d while maintaining control of the 

pendulum above the horizontal. The swing-up problem was not part of this study. 

 

2.1. Plant 

The plant consists of a motor which generates a torque , a vertical shaft, a horizontal arm with position 

and speed denoted  and θ̇, and a freely rotating pendulum with position and speed denoted  and φ̇. For 

this case study, it is assumed that the pendulum is always above the horizontal position, i.e.,  ≤  90o. Two 

plant models were developed. The Ideal Plant Model does not include motor dynamics and assumes that 

the desired motor torque equals the applied torque (i.e., Td = ). The Laboratory Plant Model includes 

model dynamics, friction, and other hard nonlinearities. 

2.1.1. Ideal Plant Model 

The ideal plant model was derived using the Euler-Lagrange equation. Variable m denotes the 

pendulum mass, J1 is the moment of inertia of the shaft and arm relative to the shaft’s main axis, J2 denotes 

the moment of inertia of the pendulum with respect to its center of mass, L1 is the length of the arm, L2 is 

the length of the pendulum, and g is the gravitational constant. 

Lab Plant Model

Controller

Torque Td

State X

Target Arm 

Position d

Figure 1: High-Level Block Diagram of Control System
3

Aθ̈+ Bφ̈+ C = τ (1) 

Dθ̈+ Eφ̈ + F + G = 0 (2) 

where: 

A = J1 + mL1

2 + (1⁄4) mL22

sin2() E = J2 + (1⁄4) mL2

2

B = (1⁄2) mL1L2cos() F = - (

1

/8) mL2

2

sin(2)(θ̇)

2

C = - (1⁄2) mL1L2sin()(φ̇)

2 + (1⁄4) mL22

sin(2)(φ̇θ̇) G = - (1⁄2) mgL2sin()

D = B

2.1.2. Laboratory Plant Model 

The laboratory plant model was developed starting from the ideal model and adds motor dynamics, 

friction, and some hard nonlinearities. 

Aeθ̈+ Be φ̈+ Ce = Td (3) 

Deθ̈+ Eeφ̈+ Fe + Ge + He = 0 (4) 

where: 

Ae = A + Jm Ee = E

Be = B Fe = F

Ce = C + amθ̇ Ge = G

De = D He = (1⁄2) bpφ̇

Here Td denotes the torque developed by the motor, Jm denotes the rotational inertia of the motor, and 

am and bp are viscous friction terms. In addition, the Laboratory Plant Model includes torque saturation 

Td,max, motor angular speed saturation θ̇

d,max, static and dynamic motor friction Tstatic and Tdynamic, and input 

torque dead zone Td,dead. In addition, scaling factors were used to more closely match the response of the 

model to the physical plant. Table 1 lists the model parameter values. 
4

Table 1: Parameter Values for Laboratory Plant Model 

Parameter Value Parameter Value Parameter Value

J1 0.04699 kg-m2 am 0.01932 N-m-s dAe 1.20

J2 0.00780 kg-m2 bp 0.01 N-m-s dBe 1.12

Jm 0.03790 kg-m2 Td,max 9.8 N-m dCe 1.05

L1 0.36 m θ̇

d,max 28.27 rad/s dDe 1.12

L2 0.28 m Tstatic 0.10 N-m dEe 0.98

m 1.1564 kg Tdynamic 0.08 N-m dFe 1.00

g 9.8 m/s2 Td,dead 0.147 N-m dGe 1.08

dHe 5.50

2.2. Controller 

Figure 2 illustrates the structure of the system controller. This is a dual-loop architecture in which the 

inner loop is responsible for the state of the pendulum and the outer loop generates pendulum position 

commands based on the target arm trajectory. 

The system control laws were developed based on a Simplified Plant Model with scaling factors added 

to minimize the modeling error relative to the Laboratory Plant Model. The equations of the Simplified 

Plant Model are: 

∅̈= H̅a+ L̅aTd (5) 

θ̈= Q̅

a

+ R̅aTd (6) 

where: 

Target Arm 

Position d

Arm 

Controller 

Pendulum 

Controller 

Target 

Pendulum 

Position d Torque Td

State

, θ̇, , ∅̇

Figure 2: Block Diagram of System Controller
5

H̅a = -[Aa(Fa + Ga)]/(AaEa - BaDa) Ca = 0

L̅a = -Da/(AaEa - BaDa) Da = D

Q̅

a

= Ba(Fa + Ga)]/(AaEa - BaDa) Ea = J2 + (1⁄4) mL2

2

R̅a = Ea/(AaEa - BaDa) Fa = - (

1

/8) mL2

2

(θ̇)

2

Aa = J1 + mL1

2 Ga = - (1⁄2) mgL2()

Ba = (1⁄2) mL1L2

The scaling factors in Table 2 were applied to the Simplified Plant Model. 

Table 2: Scaling Factors for Simplified Plant Model 

Parameter Value Parameter Value

dAa 1.35 dEa 0.98

dBa 1.00 dFa 1.05

dCa 0.00 dGa 1.00

dDa 1.00 dLa 1.05

2.2.1. Pendulum Controller 

The pendulum control law used the sliding controller theory presented by Slotine and Li [2]. The 

sliding surface is: 

s = (

d

dt + λ)2

(∫ φ̃ dt

t

0

) = φ̃ + 2λφ̃

̇ + λ2 ∫ φ̃dt

t

0

where φ̃ = φ − φd. The desired motor torque is: 

Td = (- H̅a + V + W)/ L̅a (7) 

with: 

V = φ̈

d − 2λφ̃

̇ − λ

2φ̃  = 

W = -K sat(s/)

β = √L̅u L̅⁄ l
6

K = β[F̂+η] + (β-1)|û| F̂ = |H̅u- H̅l

|

û = −H̅a + V

Factors H̅u, H̅l, L̅u, and L̅l are upper (u) and lower (l) bounds for H̅ and L̅ of the Laboratory Plant Model. 

These factors are given by the corresponding factors in the Simplified Plant Model using the scaling factors 

in Table 3. The tuning parameters for the Pendulum Controller are , , and . 

Table 3: Scaling Factors for Error Bound Models

Parameter Value Parameter Value

dAu 1.25 dAl 1.60

dBu 1.00 dBl 1.00

dCu 0.00 dCl 0.00

dDu 1.00 dDl 1.00

dEu 0.90 dEl 0.98

dFu 1.30 dFl 0.85

dGu 1.30 dGl 0.80

dLu 0.90 dLl 0.90

2.2.2. Arm Controller 

The arm control law is a PID law (Proportional, Integral, Derivative) with a nonlinear scaling factor to 

map the desired arm angular acceleration to the desired pendulum position. 

φd = [−K1θ̃

̇ − K2θ̃ − K3 ∫ θ̃ dt t

0

] [(

1

2

⁄ )(L1⁄ ) L2 θ̇2 + (g/L1)] (8) 

where θ̃ = θ − θd. The tuning parameters for the Arm Controller are K1, K2, and K3. 

3. System Tuning Problem 

The purpose of the control system is to move the arm following the trajectory given by the Target Arm 

Position d while balancing the pendulum. Ideally, the arm would perfectly track the target position with 

no effects due to variations in the characteristics of the plant. In real life, it is not possible to mitigate 

completely the effects of delays and functional coupling between system elements. Thus, the goal is to 

achieve an optimal balance between arm tracking performance and robustness. The robustness attribute 
7

would account for plant modeling errors and variability in the characteristics of the physical plant. Note 

that swing-up of the pendulum is not within the scope of this arm-tracking problem. The applicable system 

initial conditions include those where the pendulum is above the horizontal position ( ≤ 90o) and the 

combined values of the plant state variables , φ̇, , and θ̇ are within the stable region of the control system. 

It is also assumed that deriving a closed-form analytical derivation of optimal parameter settings is not 

feasible. 

4. Reliability Engineering Concepts 

Statistical techniques from the field of Reliability Engineering enable learning about and 

characterization of products and processes by means of systematic experimentation and modeling. The 

following sections present brief overviews of these topics. 

4.1. Design of Experiments 

Design of Experiments (DOE) theory is applied in the development of strategies to investigate the 

effects of independent controlled variables on system performance measured at selected variables of interest 

[4] [5]. The system of interest (SOI) is modeled as a black box to study the relations between Control, 

Input Signal, and Noise Factors (i.e., variables, parameters) and the Response Variable, as illustrated in 

Figure 3. The Noise Factors represent sources of system variability. In this model, the value of the response 

variable is a function of the Control, Input Signal, and Noise variables: y = f(X, S, Z). We seek efficient 

experiments that yield insightful knowledge about the system with as few resources as possible. 

In a basic experiment, we study the relation between the Control Factors (denoted x1, x2, ..., xk) and 

the Response Variable (denoted y) while the Input Signal Factors are given and the Noise Factors are not 

controlled. An experiment consists of a series of test points in which the Control Factors are assigned 

particular values and the Response is measured. In a two-level experiment, the Control Factors are tested 

at two different values. For a quantitative variable, the levels are the extremes of the value range, denoted 

xmax and xmin. For a qualitative variable, which has discrete values, the two levels are different values in the 

variable’s value set. A Full Factorial (Experiment) Design consists of all possible combinations of the 

control variables. This is a complete and systematic study of the design space defined by the Control 

Factors. A two-level full factorial design consists of 2ktest points. A Fractional Factorial Design includes 

a subset of test points in a Full Factorial Design. There are 2k-p test points in a two-level fractional factorial 

design, which consists with k-p independently assigned control factors and p confounded factors that are 

Input 

Signal 

Factors S

Noise 

Factors Z

Control 

Factors X

Response 

Variable y 

System of 

Interest

Figure 3: High-Level Black-box Model of the System of Interest (SOI)
8

assigned values as function of the independent control factors. 

In designing experiments, there is a tradeoff between the quality of data and the cost needed to generate 

it. In general, we want to maximize the accuracy of the characterization made possible by the data (i.e., the 

benefit) while minimizing the number of test points (i.e., the cost). The simplest way to increase the quality 

of information generated by an experiment is to increase the number of levels of the control factors. A 

three-level fractional factorial design consists of 3k-p test points. The obvious problem with this approach 

is that the number of test points (and thus, the cost of the experiment) increases exponentially. A Central 

Composite Design (CCD) consisting of full or factorial experiment portion, a center portion, and an axial 

portion offers a more favorable tradeoff between quality and size of an experiment [6]. Figure 4 illustrates 

the structure of a two-factor CCD experiment. The values -1 and +1 are standardized representations of the 

variables values, xmin and xmax, respectively. In this figure, the four points at the corners of the box are the 

factorial portion of the experiment: (1, 1). Point (0, 0) is the central portion. The points at the ends 

of the central axes (0, ) and (, 0) are the axial portion. Notice that this is five-level experiment with 

the control factors taking values 0, 1, and . When  = 1, the experiment is called a Face-Centered 

Design and the factors take only three possible values: 0 and 1. The number of test points in a CCD 

experiment is 2k-p + 2k + 1. 

4.2. Response Modeling 

We want to use the results of an experiment to generate a mathematical model of the relation between 

the control factors x and the response variable y. Here, we consider only polynomial models. A two-level 

experiment that tests each factor only at two values contains enough information to build a linear model. 

In addition to the direct relation between each control factor and the response variable, we want to consider 

possible interactions between control factors. The general expression for a linear polynomial model with 

two-factor interaction terms is: 

y = bo + i bixi + ij bijxixj (9) 

where 1 ≤ i ≤ k, 1 ≤ j ≤ k, and i  j. If the variables are assigned values in the 1 as in Figure 4, the 

(-1, +1) (+1, +1)

(-1, -1) (+1, -1)

(0, 0)

(0, +)

(+, 0)

(0, -)

X (-, 0) 2

X1

Figure 4: Structure of a Central Composite Design (CCD)
9

magnitude of the coefficients bi is a measure of the strength of the effect (and hence, importance) of the 

corresponding factor xi or interaction xixj. Note that higher-order interactions are possible, but as a rule of 

thumb, the strength of interactions decreases as their order (i.e., number of interacting factors) increases. 

For a three-level experiment such as a CCD, the system response can be modeled with a second-order 

polynomial. 

y = bo + i bixi + ij, ij bijxixj + i biixi

2

 (10) 

This model can achieve better accuracy for systems with non-linear responses. 

The basic relation between modeling and experiment design is that high-order models can achieve 

better accuracy but require experiments with higher granularity and, thus, higher cost. We need models 

that have adequate fidelity for the intended use at a reasonable cost. Analysis of the application and careful 

judgment are required to determine the best combination of experiment design and response model. 

4.3. Optimal and Robust Parameter Design 

Once a model has been developed, an optimization tool can be used to determine the combination of 

control factors that results in the best system response. There are three categories of optimization problems: 

the-smaller-the-better, the-larger-the-better, and the-closer-to-nominal-the-better (i.e., as close as possible 

to a target). The quality of system performance is determined not only on achieving the desired target value 

under ideal conditions, but must also take into accounts the effects of variability in uncontrolled noise 

factors. To achieve a high quality system design, both the mean and the dispersion in system performance 

must be controlled. An optimal parameter design achieves the best balance between performance and 

robustness (i.e., insensitivity to variability). For this, we need a parameter optimization approach that can 

account for the mean value y̅ and deviation y of the system response. One approach to this is the use of 

Dual Response Surfaces, in which models are developed for the mean response and the standard deviation, 

and an optimizer is used to minimize the deviation with a constraint on the error of the mean response 

relative to the desired target [7]. Another approach described below is to optimize signal-to-noise ratios, 

which is the approach used in the case study reported here. 

To enable a robust parameter design, the experiment must generate data suitable for the characterization 

of the effects of the noise factors z (see Figure 3). Note that the categorization of factors as either control 

or noise can be a matter of judgment based on the goals of the parameter design activity. Factors with weak 

effects on the response variable could be classified as noise in order to simplify the design activity by 

focusing on the strongest control factors. Because of this, it is possible that some noise factors are 

controllable in an experiment and others may really be uncontrollable. For simulation-based experiments 

such as the one considered in the case study reported here, the set of controllable noise factors may be larger 

than for real physical experiments. Figure 5 illustrates the high-level structure of an experiment intended 

to characterization and optimization of system performance and robustness. The inner and outer arrays are 

independent experiments for control and noise factors as described in preceding sections. The complete 

outer array experiment is executed for each test point of control factors in the inner array. A response 

measure is made for every test point of control and noise factors, and robustness measures (such as mean 

response and deviation) are made by combining the results of a complete noise factors experiment for a 

particular control factors test point. 

The signal-to-noise ratio is a robustness measure intended to capture with a single metric both the 

deviation of the average response from the target response ym and the dispersion of the response. The mean 

square deviation of the response is given by: 
10

msd = (y̅ – ym)

2 + y2

 (11) 

The signal-to-noise ratios for the optimization problems of the-smaller-the-better, the-larger-the-better, 

and the-closer-to-nominal-the-better are given in Table 4. This robustness measure is computed for each 

test point of the inner array in Figure 5. At the completion of the experiment, a model can be built to relate 

the control factors to the signal-to-noise ratio SN. A control factor optimization based on the maximization 

of the signal-to-noise ratio gives the system configuration with the best balance of performance and 

robustness. 

Table 4: Formulas for Signal-to-Noise Ratio

Case Signal-to-Noise Ratio (SN)

the-smaller-the-better -10 log10[(1/r) i yi

2

]

the-larger-the-better -10 log10[(1/r) i (1/yi

2

)]

the-closer-to-nominal-the-better 10 log10[y̅

2/y2]

5. System Tuning Approach 

This section describes the elements of the approach followed to tune the parameters of the control 

system. This case study was intended as a proof of concept and demonstration, and as such, the tuning 

problem was scoped to be representative but without excessive complexity. MATLAB was used to model 

the system and perform simulation experiments. 

Inner 

Experiment 

Array 

Response 

Measures 

Array 

Outer 

Experiment 

Array 

Robustness 

Measures 

Array 

1

2

3

.

.

.

n

1 2 3 ... r

Figure 5: Structure of Robust Parameter Design Experiment
11

5.1. Control and Noise Factors 

Table 5 lists the chosen control and noise factors. The baseline values for the noise factors are the 

nominal values given in Table 1. The baseline values for the Arm Controller and the Pendulum Controller 

factors where determined by trial-and-error to result in a stable system (i.e., small state perturbations are 

mitigated asymptotically) with the region of convergence for the pendulum state given in Table 6 (i.e., if 

the initial pendulum state is in this region, the plant state will converge the zero state (, φ̇, , and θ̇) = (0, 

0, 0, 0), assuming d = 0. The ranges of the control and noise factors were specified as percentages of the 

baselines values. 

Table 5: Control and Noise Factors for the Control System

Category System 

Element Factor Baseline Value

Range

(%)

Control

Factors

Arm 

Controller

K1 2.50 50

K2 2.00 50

K3 0.10 50

Pendulum 

Controller

 20.00 50

 0.017 50

Noise

Factors

Laboratory 

Plant Model

L1 0.36 15

L2 0.28 15

m 1.1564 15

Table 6: Region of Convergence of Baseline Control System

Initial 

Pendulum 

Position  (

o

)

Range of 

Initial 

Pendulum 

Speeds ∅̇ (

o

/s)

Initial 

Pendulum 

Position  (

o

)

Range of 

Initial 

Pendulum 

Speeds ∅̇ (

o

/s)

+35 (-600, +30) -35 (-30, +600)

+25 (-500, +190) -25 (-190, +500)

+15 (-525, +340) -15 (-340, +525)

+5 (-525, +500) -5 (-500, +525)
12

5.2. Stimulus Tracking Signal 

The chosen Arm Position Target d, which corresponds to the Input Signal in Figure 3, was a ramp 

waveform from 00

to 900 with a 30o/s slope from seconds 2 to 5. This is illustrated in Figure 6. 

5.3. Response Metrics 

Each experimental test point consisted of a 30 second run in which the control and noise factors were 

held constant and the stimulus signal was applied. Each run was divided into two phases: a transient phase 

(denoted l) from 0 to 15 seconds, and a steady-state phase (denoted u) from 15 to 30 seconds. The goal of 

the control system is to minimize the arm tracking error, defined as e = θ̃ =  - d, while balancing the 

pendulum. The performance metrics are given in Table 7, where Tl = 15 seconds and Tu = 30 seconds. 

Note that an outer array experiment consists of m test points (see Figure 5), and the signal-to-noise ratio is 

computed based on the results of one outer array experiment (i.e., one inner array row). 

Figure 6: Arm Position Tracking Target
13

Table 7: System Response Performance Metrics

Experiment 

Segment Phase Description Label Formula

Test Point (i, j)

Transient

Mean Square 

Error (MSE) Pl (

1

Tl

) ∫ e

2 dt

Tl

o

Maximum Square 

Error el

2 max(e2

) for 0 ≤ t ≤ Tl

Steady 

State

Mean Square 

Error (MSE) Pu (

1

Tu- Tl

) ∫ e

2 dt

Tu

Tl

Maximum Square 

Error eu

2 max(e2

) for Tl ≤ t ≤ Tu

All Combined Error E w1Pl + w2el

2 + w3Pu + w4eu2

; wq = 0.25

Inner Array 

Row

Signal-to-Noise 

Ratio SN −10 log10 [(

1

m

) ∑ Ej

2

m

j=1 ]

5.4. Experiment Design 

The Inner Experiment Array for the Control Factors was a Central Composite Design for the five factors 

K1, K2, K3, , and , which are denoted X1, X2, X3, X4, and X5. The factorial portion of the Inner Array was 

a half-fractional factorial design with confounded factor X5 = X1X2X3X4. 

The Outer Experiment Array for the Noise Factors was a half-fractional factorial design for the three 

factors L1, L2, and m denoted N1, N2, and N3. The confounded noise factor was N3 = N1N2.

6. Results 

As stated above, the experiment was performed using MATLAB. This section describes the experiment 

results and the parameter design with optimal signal-to-noise ratio. 

6.1. Experimental Response Measures 

Table 8 shows the results of the experiment. The combined error measures for each test point are given 

in the columns labeled E1 to E4. The signal-to-noise ratio for each row of control factors is given in the 

S/N column. 

6.2. Response Model 

The system response was modeled with a quadratic polynomial as in equation (10). Rather than 

including all the two-factor terms, an effects analysis was performed to identify the strongest interactions. 
14

These were X1X2, X1X3, X1X4, X2X3, and X1X5. The full model for the signal-to-noise ratio is: 

SN = bo + b1X1 + b2X2 + b3X3 + b4X4 + b5X5 + b12X1X2 + b13X1X3 + b14X1X4 + b23X2X3 + b15X1X5 + b11X1

2

+ b22X2

2 + b33X3

3 + b44X44

 + b55X5

5

 

The model coefficients were determined using regression. The values are given in Table 9

Noise Factors

N1/L1 -1 1 -1 1

N2/L2 -1 -1 1 1

Control Factors N3/m 1 -1 -1 1

X1/k1 X2/k2 X3/k3 X4/lambda X5/epsilon E1 E2 E3 E4 S/N

1 -1 -1 -1 -1 1 164.759 211.915 169.640 202.773 -104.769

2 1 -1 -1 -1 -1 92.331 105.338 98.999 101.663 -92.042

3 -1 1 -1 -1 -1 140.695 174.424 142.577 172.215 -101.287

4 1 1 -1 -1 1 101.038 102.994 138.310 98.582 -94.268

5 -1 -1 1 -1 -1 164.944 240.843 168.811 230.420 -106.383

6 1 -1 1 -1 1 96.915 105.989 107.866 101.518 -92.725

7 -1 1 1 -1 1 145.174 171.730 152.304 162.395 -101.280

8 1 1 1 -1 -1 91.764 103.789 125.609 99.935 -93.272

9 -1 -1 -1 1 -1 155.422 203.201 153.378 206.819 -104.023

10 1 -1 -1 1 1 82.765 97.702 87.739 94.618 -90.194

11 -1 1 -1 1 1 128.789 159.214 131.805 154.813 -99.436

12 1 1 -1 1 -1 83.854 98.019 109.475 98.034 -91.652

13 -1 -1 1 1 1 153.453 216.126 154.486 207.743 -104.435

14 1 -1 1 1 -1 83.720 101.371 85.523 99.925 -90.649

15 -1 1 1 1 -1 128.261 159.511 130.653 174.621 -100.152

16 1 1 1 1 1 83.644 94.659 110.132 91.651 -91.184

17 0 0 0 0 0 97.714 118.274 100.694 114.897 -93.690

18 -1 0 0 0 0 137.437 174.931 139.790 170.091 -101.060

19 1 0 0 0 0 83.531 96.579 90.889 93.246 -90.258

20 0 -1 0 0 0 100.726 124.102 103.103 120.556 -94.476

21 0 1 0 0 0 95.844 114.538 99.449 111.058 -93.176

22 0 0 -1 0 0 97.679 118.160 100.654 114.810 -93.677

23 0 0 1 0 0 97.759 118.347 100.751 114.936 -93.700

24 0 0 0 -1 0 104.244 122.517 109.246 117.315 -94.645

25 0 0 0 1 0 96.731 118.287 99.177 115.201 -93.600

26 0 0 0 0 -1 96.856 118.396 99.271 121.655 -93.938

27 0 0 0 0 1 98.500 118.874 102.004 114.817 -93.805

Table 8: Experiment System Response Measures
15

Table 9: Coefficients for Model of Signal-to-Noise Ratio of System Response

Coefficient Value Coefficient Value Coefficient Value Coefficient Value

b0 -93.4849 b4 0.8526 b14 0.1848 b22 -0.3668

b1 5.3657 b5 0.0723 b23 0.2449 b33 -0.2291

b2 0.7772 b12 -1.3888 b15 -0.1675 b44 -0.6631

b3 -0.1351 b13 0.1913 b11 -2.2001 b55 -0.4123

6.3. Optimal Parameter Design 

The optimizing compiler in Excel was used to determine the maximum signal-to-noise ration within 

the range of the control factors using the model in the previous section. The optimal signal-to-noise ratio 

was SNoptimal = -89.62 with the parameter values given in Table 10. 

Table 10: Optimal Values of the System Control Factors

Control 

Factor Value Control Factor Value

K1 3.750  27.826

K1 1.035  0.016

K1 0.080

6.4. Verification 

A brief experiment was run to verify the optimization results. For this verification experiment, the 

control factors were set as in Table 10 and noise factors were cycled as in the outer array in Table 8. The 

resulting signal-to-noise ratio was SNverification = -90.082, which is a close match to the optimal value. Figure 

7 to Figure 10 show the arm position tracking response for each verification test point. The steady-state 

oscillations are probably due to time delays in the system and the torque dead zone of the motor. Figure 11

shows the time history of all the plant state variables for the verification run with Noise Factors set to (N1, 

N2, N3) = (+1, +1, +1). 
16

Figure 7: Verification Run for Noise Factors (N1, N2, N3) = (-1, -1, +1)

Figure 8: Verification Run for Noise Factors (N1, N2, N3) = (+1, -1, -1)
17

Figure 9: Verification Run for Noise Factors (N1, N2, N3) = (-1, +1, -1)

Figure 10: Verification Run for Noise Factors (N1, N2, N3) = (+1, +1, +1)
18

7. Final Remarks 

This case study has demonstrated the application of Reliability Engineering techniques for optimal 

tuning of a complex control system. These techniques pertain to experiment design and mathematical 

modeling of the effects of control and noise factors on system performance and robustness. The applied 

approach enables the generation of insight into the relation between goals, dependencies, and uncertainties 

in a system. This is especially useful when complexity limits our ability to derive closed-form analytical 

expressions of system performance. The examined system contained many complex features such a dualloop control architecture, plant and controller state (i.e., memory), and a variety of nonlinear relations 

including quadratic, sinusoidal, and thresholds. 

The demonstrated approach is simple, structured, effective, repeatable, and cost and time efficient. This 

general approach is suitable for a wide range of systems. The case study showed that the applied techniques 

are able to predict performance with high accuracy. Simulation-based experimentation in a flexible 

environment such as MATLAB enables quick turnaround in the generation, characterization, and tuning of 

virtual system prototypes. 

There are several ways to expand the scope and depth of the case study presented here. Future case 

studies could increase the number of plant and controller parameters in the sets of control and noise factors. 

Also, instead of a single ramp input signal, a set of input signals could be applied such as steps, square 

waves, and sinusoidal waveforms. In that case the input signals would be handled as additional noise factors 

and considered in the design of the experiment outer array. This would expand the set of conditions in the 

Figure 11: Plant State Variables for Verification Run with Noise Factors (N1, N2, N3) = (+1, +1, +1)
19

characterization of robustness. Additionally, in this case study, we chose to follow a trial-and-error 

approach to identify a stable system configuration that served as a baseline for the study of performance 

and robustness. In tuning a nonlinear system from a stability perspective, we are primarily interested in the 

size of the region of convergence. It should be possible to develop simple metrics for the size of the region 

of convergence. Given that, an experiment could be designed to explore the system design space and 

identify the proper tuning of parameters to achieve an optimal balance of stability, performance, and 

robustness. 

8. References 

[1] A. Avizienis, J. C. Laprie, B. Randell and C. Landwehr, "Basic Concepts and Taxonomy 

of Dependable and Secure Computing," IEEE Transactions on Dependable and Secure 

Computing, Vols. 1, No. 1, pp. 11 - 33, January - March 2004. 

[2] W. Torres-Pomales and O. R. Gonzalez, "Nonlinear Control of Swing-Up Inverted 

Pendulum," in Proceedings of the 1996 IEEE International Conference on Control 

Applications, Dearborn, MI, 1996. 

[3] J.-J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, Inc., 1991. 

[4] NIST, "NIST/SEMATECH e-Handbook of Statistical Methods," National Institute of 

Standards and Technology (NIST), 30 October 2013. [Online]. Available: 

http://www.itl.nist.gov/div898/handbook/index.htm. [Accessed May 2015].

[5] ReliaSoft, Experiment Design and Analysis Reference, ReliaSoft, 2015. 

[6] R. Unal, R. A. Lepsch, W. Engelund and D. O. Stanley, "Approximation model building 

and multidisciplinary design optimization using response surface methods," in AIAA, 

NASA, and ISSMO Symposium on Mutidisciplinary Analysis and Optimization, Reston, 

VA, 1996. 

[7] O. Yeniay, R. Unal and R. A. Lepsch, "Using dual response surfaces to reduce variability 

in launch vehicle design: A case study," Reliability Engineering & System Safery, vol. 91, 

no. 4, pp. 407 - 412, 2006. 
20

Appendix A. MATLAB Script for Simulation Experiment 

% Run Experiment on pendulum control system

% Notes:

% 1. Performance measure results are saved to file

%

echo on ; 

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Set filename for control and noise matrices

% => This one in MATLAB format

% => This one will be in Excel-loadable format

control_matrix_filename_TXT = 'control_matrix.dat' ; 

noise_matrix_filename_TXT = 'noise_matrix.dat' ; 

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Set filename for error matrix

% => This one in MATLAB format

% => This one will be in Excel-loadable format

E_matrix_filename_TXT = 'E_matrix.dat' ; 

Pl_matrix_filename_TXT = 'Pl_matrix.dat' ; 

Pu_matrix_filename_TXT = 'Pu_matrix.dat' ; 

e2l_matrix_filename_TXT = 'e2l_matrix.dat' ; 

e2u_matrix_filename_TXT = 'e2u_matrix.dat' ; 

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Set filename for performance measures

% => This one in MATLAB format

perf_measures_filename_MAT = 'perf_measures_MAT' ; 

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Set filename for signal-to-noise measures

% => This one will be in Excel-loadable format

% ==> Make entries TAB separated and include end-of-line character

% ==> Can also include headings (for rows, columns, table caption, etc.)

SN_measures_filename_TXT = 'SN_measures.dat' ; 

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Set model information

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Specify model name

model_name = 'system_controller_n_plant_enhanced_v3p0' ; 

 

% Run time (seconds)

t_final = 30.0 ;

 

% Transient interval duration (seconds)

t_Transient = 15.0 ;

 

% Set plant initial conditions

Initial_arm_angle_deg = [0 0.0] ;

Initial_arm_speed_deg_sec = [0 0.0] ;

Initial_pendulum_angle_deg = [0 0.0] ;

Initial_pendulum_speed_deg_sec = [0 0.0] ;
21

 

% Baseline controlled model parameters

k1_base = 2.50 ; % X1

k2_base = 2.00 ; % X2

k3_base = 0.10 ; % X3

lambda_base = 20.0 ; % X4

epsilon_base = 0.017 ; % X5 

eta_base = 1.0 ; % This one will remain constant

% Baseline noise (uncontrolled) model parameters

J1_kg_m2_base = 0.04699 ; % This one will remain constant

J2_kg_m2_base = 0.007800 ; % This one will remain constant

L1_m_base = 0.36 ; % N1

L2_m_base = 0.28 ; % N2

m_kg_base = 1.1564 ; % N3

% Range percentage for model parameters (%)

r_control = 50.0 ;

r_noise = 15.0 ; 

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Set experiment matrices

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Matrix for Controlled parameters

% X1, X2, X3, X4, X4, X5(= X1X2X3X4)

Control_matrix = ...

[ 

-1, -1, -1, -1, +1;

+1, -1, -1, -1, -1;

-1, +1, -1, -1, -1;

+1, +1, -1, -1, +1;

 

-1, -1, +1, -1, -1;

+1, -1, +1, -1, +1;

-1, +1, +1, -1, +1;

+1, +1, +1, -1, -1;

 

-1, -1, -1, +1, -1;

+1, -1, -1, +1, +1;

-1, +1, -1, +1, +1;

+1, +1, -1, +1, -1;

 

-1, -1, +1, +1, +1;

+1, -1, +1, +1, -1;

-1, +1, +1, +1, -1;

+1, +1, +1, +1, +1;

 

0, 0, 0, 0, 0;

 

-1, 0, 0, 0, 0;

+1, 0, 0, 0, 0;

0, -1, 0, 0, 0;

0, +1, 0, 0, 0;

0, 0, -1, 0, 0;

0, 0, +1, 0, 0;

0, 0, 0, -1, 0;

0, 0, 0, +1, 0;
22

0, 0, 0, 0, -1;

0, 0, 0, 0, +1 ] ;

 

% Matrix for Noise parameters

% N1, N2, N3(= N1N2) 

Noise_matrix = ...

[ 

-1, -1, +1; 

+1, -1, -1;

-1, +1, -1;

+1, +1, +1 ] ;

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Performance Measure matrices

weights = [0.25, 0.25, 0.25, 0.25] ;

num_metrics = 4 ;

num_ctrl_rows = size(Control_matrix, 1) ;

num_ctrl_columns = size(Control_matrix, 2) ;

num_noise_rows = size(Noise_matrix, 1) ;

num_noise_columns = size(Noise_matrix, 2) ;

 

Pl = zeros(num_ctrl_rows, num_noise_rows) ;

Pu = zeros(num_ctrl_rows, num_noise_rows) ;

e2l = zeros(num_ctrl_rows, num_noise_rows) ;

e2u = zeros(num_ctrl_rows, num_noise_rows) ; 

U = zeros(num_ctrl_rows, num_noise_rows, num_metrics) ;

E = zeros(num_ctrl_rows, num_noise_rows) ;

SN = zeros(num_ctrl_rows, 1) ;

 

%pause

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Loop over Control Matrix

for i = 1 : num_ctrl_rows

 % Factor values for current control point

 k1 = [0 (k1_base * ...

 (1 + Control_matrix(i,1)*r_control/100.0)) ] ;

 k2 = [0 (k2_base * ...

 (1 + Control_matrix(i,2)*r_control/100.0)) ] ;

 k3 = [0 (k3_base * ...

 (1 + Control_matrix(i,3)*r_control/100.0)) ] ;

 lambda = [0 (lambda_base * ...

 (1 + Control_matrix(i,4)*r_control/100.0)) ] ;

 epsilon = [0 (epsilon_base * ...

 (1 + Control_matrix(i,5)*r_control/100.0)) ] ;

 eta = [0 eta_base] ; % This one remains constant

 

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 % Loop over Noise matrix

 for j = 1 : num_noise_rows

 % Factor values for current noise point

 J1_kg_m2 = [0 J1_kg_m2_base] ; % This one remains constant

 J2_kg_m2 = [0 J2_kg_m2_base] ; % This one remains constant

 L1_m = [0 (L1_m_base * (1 + 

 Noise_matrix(j,1)*r_noise/100.0))] ;

 L2_m = [0 (L2_m_base * (1 + 

 Noise_matrix(j,2)*r_noise/100.0))] ;
23

 m_kg = [0 (m_kg_base * (1 + 

 Noise_matrix(j,3)*r_noise/100.0))] ;

 

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 % Run the model

 plant_state = sim(model_name, 'StopTime', 't_final') ;

 

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 % Get the plant state histories

 time = plant_state.get('time') ;

 arm_position_target_deg = 

 plant_state.get('arm_position_target_deg');

 arm_position_deg = plant_state.get('arm_position_deg') ;

 arm_speed_deg_sec = plant_state.get('arm_speed_deg_sec') ;

 pendulum_position_deg = 

 plant_state.get('pendulum_position_deg') ;

 pendulum_speed_deg_sec = 

 plant_state.get('pendulum_speed_deg_sec') ;

 

 %show_plant_state_v1

 

%pause 

 

 % Get number of time points in the simulation

 num_time_points = length(time) ; 

 

 % Scan full length of simulation run

 for k = 1 : num_time_points

 

 % For Lower/Transient time interval

 if (time(k) <= t_Transient)

 Pl(i,j) = Pl(i,j) + ...

 (arm_position_deg(k) – ...

arm_position_target_deg(k))^2 ;

 e2l(i,j) = max([e2l(i,j), ...

 (arm_position_deg(k) – 

 arm_position_target_deg(k))^2]) ;

 

 % For Upper/Steady-state time interval

 else

 Pu(i,j) = Pu(i,j) + ...

 (arm_position_deg(k) – 

arm_position_target_deg(k))^2 ;

 e2u(i,j) = max([e2u(i,j), ...

 (arm_position_deg(k) – 

 arm_position_target_deg(k))^2]) ; 

 end

 end

 

 % Finish up averaging sums of squares

 Pl(i,j) = Pl(i,j)/num_time_points ;

 Pu(i,j) = Pu(i,j)/num_time_points ;

 

 % Store basic measures

 U(i, j, 1) = Pl(i,j) ; 

 U(i, j, 2) = e2l(i,j) ;

 U(i, j, 3) = Pu(i,j) ;
24

 U(i, j, 4) = e2u(i,j) ; 

 

 % Computer composite measure

 for k = 1 : num_metrics

 E(i, j) = E(i,j) + weights(k)*U(i, j, k) ;

 end % k

 

 % Update signal-to-noise ratio with current composite measure

 SN(i) = SN(i) + E(i, j)*E(i, j) ;

 

%pause

 end % j

 

 % Compute Signal-to-Noise ratio for current control point

 SN(i) = -10*log(SN(i)/num_noise_rows) ; 

 

%pause 

 

end % i

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Save experiment control and noise matrices in Excel-loadable format

save(control_matrix_filename_TXT,'Control_matrix','-ascii', '-tabs') ;

save(noise_matrix_filename_TXT, 'Noise_matrix', '-ascii', '-tabs') ;

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Save performance matrices in MATLAB format

save(perf_measures_filename_MAT, ...

 'Pl', 'e2l', 'Pu', 'e2u', 'U', 'E', 'SN') ;

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Save performance matrices in Excel-loadable format

save(Pl_matrix_filename_TXT, 'Pl', '-ascii', '-tabs') ;

save(Pu_matrix_filename_TXT, 'Pu', '-ascii', '-tabs') ;

save(e2l_matrix_filename_TXT, 'e2l', '-ascii', '-tabs') ;

save(e2u_matrix_filename_TXT, 'e2u', '-ascii', '-tabs') ;

save(E_matrix_filename_TXT, 'E', '-ascii', '-tabs') ;

save(SN_measures_filename_TXT, 'SN', '-ascii', '-tabs') ;

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Finished

echo off ; 
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 4. TITLE AND SUBTITLE

A Case Study on the Application of a Structured Experimental Method for 

Optimal Parameter Design of a Complex Control System 

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Torres-Pomales, Wilfredo

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NASA Langley Research Center

Hampton, VA 23681-2199

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14. ABSTRACT

This report documents a case study on the application of Reliability Engineering techniques to achieve an optimal balance 

between performance and robustness by tuning the functional parameters of a complex non-linear control system. For complex 

systems with intricate and non-linear patterns of interaction between system components, analytical derivation of a 

mathematical model of system performance and robustness in terms of functional parameters may not be feasible or 

cost-effective. The demonstrated approach is simple, structured, effective, repeatable, and cost and time efficient. This general 

approach is suitable for a wide range of systems. 

15. SUBJECT TERMS

Control; Experiment; Optimization; Reliability; Robustness

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