Sage Journals: Discover world-class research

Abstract

Reliability evaluation plays a more and more important role in nuclear engineering, aviation industry, and so on. Due to the special integrity of a control system, traditional reliability methods, such as the reliability block diagram and fault tree analysis, are not proper to make a reasonable and accurate reliability evaluation on a control system. However, a systematic methodology to predict performance reliability of a control system is rarely reported. In this article, a methodology of performance reliability based on the performance requirement for an automatic control system, not just limited to the stability reliability or probabilistic robustness evaluation, is first proposed. A solution to evaluate control performance reliability is introduced with analytical approximation approach called first-order second-moment method. Also, a robust iterative algorithm is adopted to overcome the bifurcation and chaos problems encountered when searching the most probable point for highly nonlinear limit state function. An elaborate example based on a servo motor control system is presented to calculate performance reliability of the settling time of the control system with first-order second-moment method. Its numerical approximation is demonstrated to be accurate through Monte Carlo simulation.

Keywords

Performance reliability control system first-order second-moment method robust iterative algorithm most probable point

Introduction

Lack of a systematic methodology for control performance reliability evaluation for automatic control systems

Demand of evaluating the reliability of the control performance of a control system grows gradually and becomes more and more urgent along with the increasing requirement of the reliability and safety in many important civil engineering fields, such as nuclear engineering and aviation industry. Such evaluation of control system reliability also guides the parameter design of a control system and helps making decisions on the selection of several alternative control strategies. Moreover, since the essence and motivation of control technology ultimately stems from the requirement of system reliability,¹ methodology of control system reliability prediction should be set up and developed to assess its control performance quality for control system with uncertainties of either model parameter or external excitation. Although the robust control technique is well developed to handle the instability problem of control systems with great uncertainties, such a technique is undoubtedly a resource-consuming and tough task since it considers all of the probable uncertainties of a control system. Actually, situations with extremely low probability can usually and wisely to be ignored while high reliability can still be guaranteed. Hence, evaluation of control system reliability should be performed to indicate the extent at which the maximum range of uncertainties can be neglected while a required reliability can still be reached.

However, the traditional reliability analysis which holds the idea that separate two-state failure analyses of components distribute to the deduction of system reliability through proper reliability modeling, such as fault tree analysis, reliability block diagram, Markov chain modeling² is no longer appropriate and efficient to describe and analyze the control system reliability. It is because that the performance of the whole control system is not simply or merely depending on the functioning conditions of separated components, but on the overall synergistic effect of components in a control system. Failures or function deterioration of one component may not necessarily lead to the failures of a whole control system, since the parameter adjustment, robust control strategy, or controller reconfiguration can maintain a control system functioning normally. Therefore, it is hard and impractical to get clear definitions of failure criterion for every component in a control system, which hinders the application of traditional reliability modeling to evaluate the reliability of control system. Therefore, there is a demand of establishing a new methodology to evaluate the control system reliability which can describe the control performance reliability as a whole.

A well-suited method to model control system reliability is the stress–strength interference theory which is the core theory of structural reliability analysis that is widely applied to solve the reliability prediction of mechanical structures. Probabilistic stability measures for controlled structures was first introduced and proposed to solve the reliability prediction for a structural control system with probabilistic model uncertainties with a similar analysis method used in structural reliability.^3,4 These approaches apply Routh–Hurwitz stability method to judge whether a linear control system is stable and hence utilize this method to establish failure criteria for the whole control system stability, which, in other words, is named as limit state functions in structural reliability. Then, the reliability evaluation of control stability can then be transformed as structural reliability prediction that gets fully developed through several decades.⁵ As the research of the stochastic robustness evaluation of control system with probabilistic model uncertainties is progressively ongoing, researchers started to focus on reliability prediction of the control stability for those systems under stochastic excitation⁶ or under both model and excitation uncertainties. Methodology of stability reliability calculation for a controlled structure has gotten a remarkable progress, but these researches are limited to the reliability analysis of stability of controlled structures. It is rarely reported on the evaluation approach either for general auto-control systems or for other important performances such as the settling time, percent overshoot, steady-state error, system bandwidth, gain margin, and phase margin. Moreover, it is necessary to set up a systematic methodology for performance reliability evaluation to guide the quality control and management for a control system design and manufacture since the performance requirements vary in different control systems. Therefore, this article will present a systematic methodology for reliability evaluations of various control performances for general automatic control systems.

Obstacles of control system reliability calculation

Since methods of evaluating control system reliability can be solved in a way which is similar to the one of structural reliability, the main obstacle to calculate control performance reliability lies at the estimation of the probability of failure by a multi-dimensional integral on the failure domain defined by the limit state functions. There are three main approaches to remove this tough obstacle: numerical calculation, simulation with Monte Carlo simulation (MCSim),⁷ and approximation solutions, in which first-order reliability method (FORM) is the most important and popular one.^8,9 However, direct numerical calculation of multi-dimensional integral in a complex integral domain takes more calculation time and makes larger accumulative error as the dimension of integral increases,¹⁰ and simulation methods, even those with important sampling schemes, are time-consuming and impractical to problems with low probability of failure and implicit limit state functions.¹¹ Hence, FORM gains its popularity and widespread application in practice for its computation efficiency and accuracy despite of its drawbacks that approximation error grows as the nonlinearity of limit state functions become higher, and this method gives no indications on whether the estimation is conservative or aggressive without knowing the concavity and convexity of limit state function.¹² A correction factor taking into account the curvatures of the limit state functions can be multiplied to the approximated reliability value in order to decrease the error brought by the its serious nonlinearity.¹³ Moreover, in order to overcome the hurdles in FORM approximation for highly nonlinear limit state function, an innovative approach based on universal generating function avoids the approximation with linear hyperplane and applies maximum entropy principle to obtain the optimally fitted probability density function (PDF) of limit state function.¹⁴ Such a method can obtain reliability approximation at any precision but is seriously limited to applications with no more than one limit state function.

For the reliability prediction of control systems with more than one requirement, the reliability calculation would encounter another obstacle: multi-dimensional integration with complex failure domain defined by multiple limit state functions. Similarly, such integration can be solved by FORM but requires more effort to deal with the complex integral domain. Among solutions to this question, first-order multinormal (FOMN) approximations reduce the order of integral dimension and hence obtain the approximation by a product of several single integrations through a recursive scheme using condition probability formula and linearization.¹⁵ Another method called product of conditional marginal (PCM) is also based on a simple computational procedure involving a product of one-dimensional normal integrals and demonstrated to be more accurate and sufficient than FOMN since PCM does not involve heavy computation of linearization.¹⁶ For those situations when the computation efficiency is more important than the calculation accuracy, probability network evaluation technique (PNET) may be the best alternative since it reduces the multi-dimensional normal distribution to a nonsingular distribution of independent variables which is easy to estimate its reliability.^17,18

In this article, some potential obstacles of reliability evaluation for control systems are first presented and alternative solutions to solve these obstacles are proposed in this section. Then, based on these summaries of obstacles and solutions, a systematic methodology of performance reliability of automatic control systems is thoroughly introduced in section “Obstacles of control system reliability calculation.” This methodology can describe and analyze the reliability of control as a whole in a similar way as in analyzing the structural reliability. Detailed algorithms of FORM and its iterative algorithm to calculate control system reliability are presented in section “Principle of FORM and its iterative algorithm.” A case study is elaborated to apply the proposed method to evaluate the reliability of settling time of a servo motor control system in section “A case study of a servo motor control system.” The estimation of reliability would be compared with the numerical results from MCSim to show the validity of proposed methodology of control system reliability. Finally, the last section concludes the article.

Methodology of performance reliability evaluation of a control system

Things are born with uncertainties, and a control system is no exception. Generally, there are two main types of uncertainties for a control system: model inconsistency between the mathematically described model and the practical system and uncertainty of external excitations. In this article, the former will be discussed. Model uncertainty mainly comes from incomplete knowledge on the physical system, deviations from calculation approximation, and stochastic manufacturing errors. All of these may result in the uncertainty of a control system behavior. It is theoretically demonstrated that uncontrolled model uncertainties of a control system may lead to system instability, steady-state error, and several important response performances.¹⁹ In order to evaluate the probability that a given control system to work normally with specific uncertainties, it is necessary to define an index of reliability for such a system capacity.

Definition

As a control system, its performance reliability is a general designation of its probability to satisfy certain specific performance

R = \Pr (g (x) \geq 0)

(1)

where $g (x) \geq 0$ is mathematical expression of the specific performance requirement which is called as limit state function in general, and $x = (x_{1}, x_{2}, \dots, x_{n}) \in R^{n}$ is a vector of stochastic system parameters which is a set of system uncertainties. Assuming that the stochastic parameter vector obeys an n-dimensional PDF $f_{x} (x)$ which can be obtained through statistical experiments, then the performance reliability is essentially the integration of PDF on the domain satisfying the performance requirements $g (x) \geq 0$

R = \int_{g (x) \geq 0} \dots \int f_{x} (x) d x

(2)

Because that performance requirements vary in different control systems, there are various kinds of performance reliability, such as stability reliability, settling time reliability, steady-state error reliability, and gains/phase margin reliability.

Since complex systems with high-order transfer function could be approximated by a second-order system using some order-reducing methods, here, a detailed methodology of control performance reliability would be given based on linear second-order control system.

In the control system theory, compared with state–space representation, it is much easier to explore the relationship between system performance and model parameters using transfer functions representation. Therefore, the performance reliability of a linear control system adopts transfer function method to describe control systems. The typical transfer function of a second-order control system can be written in the following form

G (s) = \frac{K}{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}}

(3)

where $K$ is the proportionality coefficient, $ζ$ is the dimensionless damping ratio, and $ω_{n}$ is the natural frequency. Its control performance can be completely determined by the last two parameters. Therefore, the performance reliability of second-order control system can be expressed with these two parameters. Some definitions would be listed as follows.

Definition 1.1

Stability reliability of a control system is defined as its probability to keep stable. Stability reliability is the most fundamental and important reliability index for a control system. To a second-order control system described with the transfer function in equation (3), its stability depends on the value of damping ratio $ζ$ , since the natural frequency $ω_{n}$ is always greater than 0 under most circumstances. The stability condition of second-order control system is $ζ > 0$ , and hence, its stability reliability can be written as follows

R = \Pr (ζ > 0) = \int_{0}^{+ \infty} f_{ζ} (ζ) d ζ

(4)

Usually, the PDF $f_{ζ} (ζ)$ of $ζ$ is not given directly in practical application but can be deducted from the available PDF of the basic system parameters $x = (x_{1}, x_{2}, \dots, x_{n}) \in R^{n}$ through the relationship function $ζ = ζ (x)$ . However, the deduction becomes cumbersome when the relationship function $ζ (x)$ has a complicated form. An alternative way to avoid this troublesome deduction is transformed the direct integration in equation (4) to indirect multiple integration in the system–parameter space in the domain satisfying the conditions $ζ > 0$

R = \Pr (ζ (x) > 0) = \int_{ζ (x) > 0} \dots \int f_{x} (x) d x

(5)

The calculation of the multiple integration above is also very complicated, but its approximation can be easily obtained through the FORM which will be introduced later in this article.

Definition 1.2

Settling time reliability of a control system is defined as the probability to settle within a certain percentage of input amplitude in a required time. For a second-order control system, its settling time for which the response remains within 2% of the final value can be calculated through

T_{s} = \frac{4}{ζ ω_{n}}

(6)

In some situations, when the efficiency is put in the first place or the control system works with other systems, the settling time is often required to be set within a certain value $t_{s}$ . When the settling time is out of this specific requirement, the expected output cannot be transferred to other coordinated systems and thus will cause the failure of the whole system. Hence, the failure of a control system can be defined as the probability of settling time more than a required value $t_{s}$ and correspondent reliability is named as settling time reliability

\begin{array}{l} R = \Pr (\frac{4}{ζ ω_{n}} > t_{s}) = \Pr (4 - t_{s} ζ ω_{n} > 0) \\ = \int \int_{4 - t_{s} ζ ω_{n} > 0} f_{ζ, ω_{n}} (ζ, ω_{n}) d ζ d ω_{n} \end{array}

(7)

Also, $ζ$ and $ω_{n}$ can be expressed as a function of system parameters: $ζ = ζ (x)$ and $ω_{n} = ω_{n} (x)$ , and hence, the settling time reliability can be transformed as follows

R = \Pr (4 - ζ (x) ω_{n} (x) > 0) = \int_{4 - t_{s} ζ (x) ω_{n} (x) > 0} \dots \int f_{x} (x) d x

(8)

Similarly, reliability of other performance can be defined in the same way, and its calculations can be ultimately transformed as a multiple integration in equation (2). It can be concluded that the integral objects are almost the same and the difference lies in the variety of a limit state function $g (x) \geq 0$ .

Furthermore, to control the systems which have to meet more than one requirement, the limit state function can be expressed as a combination of all requirements

R = \Pr (⋂_{i} g_{i} (x) > 0) = \int_{⋂_{i} g_{i} (x) > 0} \dots \int f_{x} (x) d x

(9)

Approximation of equation (9) is far more difficult than the one in equation (2) and method to solve it will be discussed in the following section.

Principle of FORM and its iterative algorithm

Principle of FORM

In order to solve the integration in equation (2), an efficient method to make an accurate approximation of this integral is FORM which is widely used in structural reliability analysis. Such a method utilizes first two moments of the multi-dimensional PDF $f_{x} (x)$ of uncertainties $x$ : its mean $μ_{X}$ and its standard deviation $Σ_{X}$ . Before its approximation for the integration, a proper transformation must be carried out first to transform the given random space $x$ to a normal random space $y'$ . There are always various approaches to complete such a transformation: one is Rosenblatt transformation method²⁰ and the other orthogonal transformation of normal random variables.²¹ Through these approaches, the original random space $x$ is transformed to a normal random space $y'$ and so does the limit state function

x, g (x) \to y', h' (y')

(10)

where $y'$ is the normal vector. Obtaining the normal variables $y'$ , a standard normal $y$ can be easily acquired through the Hasofer–Lind transformation

Y_{i} = \frac{{Y'}_{i} - μ_{{Y'}_{i}}}{σ_{{Y'}_{i}}}

(11)

And the correspondent transformed limit state function is $h (y)$ . So far, a transformation from an arbitrary random space $x$ to a standard normal space $y$ is accomplished. It is demonstrated that the integration in the failure domain remains the same before and after this transformation.³ That is why the approximation of the integration can be handled in the new random space $y$ . Moreover, such a transformation is useful because in the standard normal space, special properties of the standard normal distribution can be directly applied so that the approximation for the integration of equation (2) can be simplified greatly.

Since direct integration of equation (2) in the failure domain with a nonlinear boundary $h (y)$ is computationally intractable and impractical, FORM tries to solve this problem using a linear approximation to replace the nonlinear boundary through a first-order Taylor expansion in the limit state function, so that the integration can be easily calculated in a domain determined by a linear boundary. Now that a linear approximation is adopted to replace the nonlinear boundary using first-order Taylor expansion, the problem becomes that at which point the Taylor expansion should be carried out. On this problem, there is a widely acknowledged view that the expanded point is the point closest to the origin which is also called the most probable point (MPP) because the standard normal PDF decays exponentially with the square of the distance from the origin. A better understanding of FORM can be illustrated in bi-variable standard normal space (see Figure 1). The integration in a nonlinearly bounded region is approximated with the one in a linearly bounded region. Actually, the calculation of integration in the domain defined by the linearly approximated limit state function can be largely simplified by rotating the coordinates to the direction parallel to the linearly approximated boundary. This approximation may introduce approximation error which grows as the nonlinearity of limit state functions becomes higher.¹² The space defined by the new rotated coordinate system $V - W$ in Figure 1 is also a standard normal one. To make a better and clearer explanation, here, the $V$ axis that is parallel to the line connecting the MPP and the origin is named as the optimal integral axis. Then, this bi-variable normal integration can be approximated through the following

\begin{array}{l} \int \int_{h (y_{1}, y_{2}) > 0} f_{Y_{1} Y_{2}} (y_{1}, y_{2}) d y_{1} d y_{2} \\ \approx \int \int_{h' (y_{1}, y_{2}) > 0} f_{Y_{1} Y_{2}} (y_{1}, y_{2}) d y_{1} d y_{2} \\ = \int_{- \infty}^{+ \infty} \int_{- β}^{+ \infty} f_{V W} (v, w) d v d w = 1 - Φ (- β) \end{array}

(12)

where $f_{Y_{1} Y_{2}} (y_{1}, y_{2})$ is a joint PDF of $y_{1}$ and $y_{2}$ , which in this case is bi-variable normal distribution, $f_{VW} (v, w)$ is a joint PDF of $v$ and $w$ and also obeys bi-variable normal distribution, $β$ is the distance between the MPP and the origin, and $Φ (\cdot)$ is standard normal cumulative distribution function. Here, it can be seen that the approximation of integration is simplified as a single integration along the optimal integral axis $V$ .

Figure 1.

An illustration of FORM in bi-variable normal space.

This conclusion can be applied to multiple integration in equation (2) with no rectifications

R = \int_{g (x) \geq 0} \dots \int f_{x} (x) d x \approx 1 - Φ (- β)

(13)

where $β$ is also the distance between the MPP and the origin in n-dimensional space, named as reliability index.

However, the approximation for the integration with multiple limit state functions just as equation (9) is a little bit complicated

\begin{array}{l} R = \int_{\cap_{i}^{m} g_{i} (x) > 0} \dots \int f_{x} (x) d x \\ \approx 1 - \int_{- \infty}^{- β_{1}} \dots \int_{- \infty}^{- β_{m}} f_{V} (v, R) d v = 1 - Φ_{m} (- β, R) \end{array}

(14)

where $β = (β_{1}, β_{2}, \dots, β_{m})$ is a set of all the correspondent reliability index for each limit state function $g_{i} (x) > 0$ , $V = (V_{1}, V_{2}, \dots, V_{2})$ is a set of all the correspondent optimal integral axis for each $β_{i}$ , $R = (R_{ij})_{m \times m}$ is a correlation matrix whose element $R_{ij}$ is the product of the slope $α$ of axis $V_{i}$ and $V_{j}$ in the original coordinates: $R_{ij} = α_{i} α_{j}$ , and $Φ_{m} (\cdot)$ is a m-dimensional normal cumulative distribution function.

Methods to approximate the multi-dimensional integration in equation (14) include FOMN approximations, PCM, PNET, and so on.

Iterative algorithm to search MPP

Searching the MPP, the optimal point on the limit state surface that is closest to the origin, is the key barrier in evaluating reliability with FORM. Among many mathematical optimization schemes, such as the gradient projection method, the augmented Lagrangian method and the sequential quadratic programming method, Hasofer–Lind and Rackwitz–Fiessler (HL-RF) iterative algorithm is highly recommended and widely used. However, when limit state surface is complicated and highly nonlinear in standard normal space, HL-RF iterative algorithm would encounter the bifurcation and chaos problem and fail to converge.²² Several modified iterative algorithms are engaged in solving this problem by altering the infinite search step length into a finite one using various step length selection rules.^23,24 A robust iterative algorithm is proposed to introduce a new step length to control the convergence of the sequence and demonstrates its efficiency and simplicity to find MPP in FORM.²⁵ This iterative algorithm will be introduced as follows.

Suppose $y^{(k)} = (y_{1}^{(k)}, y_{2}^{(k)}, \dots y_{n}^{(k)})$ is the kth iteration point and its distance to the origin is $β^{(k)}$ and the correspondent slope is $α^{(k)}$ satisfying the following expression

y^{(k)} = α^{(k)} β^{(k)}

(15)

Bring in a point $y_{λ}^{(k + 1)}$ which moves a step with length $λ > 0$ along the negative gradient direction from the point $y^{(k)}$ (see Figure 2). It is formulated as follows

y_{λ}^{(k + 1)} = y^{(k)} - λ {\frac{\partial h}{\partial y} |}_{y^{(k)}}

(16)

Figure 2.

The iterative procedure for searching MPP.

Then, its slope is as follows

α^{(k + 1)} = \frac{y_{λ}^{(k + 1)}}{‖ y_{λ}^{(k + 1)} ‖}

(17)

It is not difficult to judge that the line connecting $y_{λ}^{(k + 1)}$ and the origin inevitably intersects the limit state surface at point $y^{(k + 1)}$ . Generally, $y^{(k + 1)}$ can be obtained through solving combination of the limit state equation and the line equation $O Y^{(k + 1)}$ . However, this computing procedure is a complicated iterative procedure. An alternative procedure is designed through Taylor expansion to the limit state equation at point $y^{(k)}$

h (y^{(k)}) + {\frac{\partial h}{\partial y} |}_{y^{(k)}} (y^{(k + 1)} - y^{(k)}) = 0

(18)

Then

y^{(k + 1)} = - \frac{h (y^{(k)})}{{\frac{\partial h}{\partial y} |}_{y^{(k)}}} + y^{(k)}

(19)

and correspondent distance $β^{(k + 1)}$ from origin can be obtained through equation (15). In order to make this iterative algorithm convergent, it is necessary to satisfy the condition

‖ y^{(k + 1)} - y^{(k)} ‖ \leq ‖ y^{(k)} - y^{(k - 1)} ‖

(20)

If it cannot be satisfied, the step length $λ$ should be modified to a smaller one as follows

λ = \frac{λ}{c}

(21)

where $c$ is a division factor whose desired value is 1.2–1.5.

So far, the iterative algorithm is completely presented. Its efficiency and validity can be figured out in the following case study.

A case study of a servo motor control system

Control model of a servo motor control system and its simplification

The widely used servo motor control system with three-loop control is chosen as the case study in this article. The task of the servo control system is to achieve a precise angle outputs according to input signals.

A basic control model of direct current (DC) motor with no control modules can be illustrated by the transfer function block diagram shown in Figure 3. There are mainly four parts in this block diagram: voltage equations in armature, electromagnetic force effect on rotor, dynamic equation on motor spindle, and induced electromotive force effect on the field coil. Three variables in this model, armature current $I_{a}$ , motor speed $Ω$ , and angle $Θ$ , would be utilized to achieve precise servo control.

Figure 3.

Transfer function block diagram of DC motor.

The first-loop armature current control is achieved with two-parameter proportional–integral (PI) controller (see Figure 4). It is noted that the mechanic damping ratio is neglected in control model since it is rather small. Through proper parameter selections, $τ_{i} = T_{a}$ and $α = R_{a}$ , its transfer function can be deduced to a simple form as a pure integrator

Ω (s) = \frac{1 / K_{e}}{T_{m} s} U_{i} (s) - \frac{1}{Js} T_{d} (s) = \frac{1}{Js} (\frac{K_{t}}{R_{a}} U_{i} (s) - T_{d} (s))

(22)

where $T_{m} = R_{a} J / K_{e} K_{t}$ is designated as the electromechanical time constant.

Figure 4.

DC motor control system with armature current control.

To show the benefit of this current control, electromagnetic torque is deduced as follows

T_{em} (s) = K_{t} I_{a} (s) = \frac{K_{t}}{R_{a}} U_{i} (s)

(23)

It can be found that the armature current is proportional to the voltage input and will not jump up sharply in spite of the influence of the back electromotive force caused by the overload. Therefore, the armature current control is actually an over-current protection for the armature.

The second-loop motor speed control is designed with a proportional controller on the foundation of the armature current control (see Figure 5). The transfer function can be easily obtained

Ω (s) = \frac{1 / K_{f}}{{T'}_{m} s + 1} (U_{v} (s) - \frac{R_{a}}{K_{v} K_{t}} T_{d} (s))

(24)

wherein

T'_{m} = \frac{T_{m}}{K_{v 0}} and K_{v 0} = \frac{K_{v} K_{f}}{K_{e}}

(25)

Figure 5.

Motor speed control design based on current control.

Finally, the third-loop motor angle control is accomplished with a simple proportional controller (see Figure 6). And the correspondent transfer function is as follows

G (s) = \frac{Θ_{o} (s)}{Θ_{i} (s)} = \frac{K_{p} / K_{f}}{{T'}_{m} s^{2} + s + K_{p} / K_{f}}

(26)

Figure 6.

Motor angle control design based on two-loop control of current and speed.

Therefore, it can be concluded that the servo motor control system with three-loop control is essentially a second-order control system. This system can achieve zero steady-state error for step input. Its natural frequency and damping ratio is, respectively, as follows

ω_{n} = \sqrt{\frac{K_{p}}{K_{t} {T'}_{m}}}, ζ = \frac{1}{2} \sqrt{\frac{K_{t}}{K_{p} {T'}_{m}}}

(27)

Reliability calculation

In some applications, the settling time is required to be less than a specific value in order to coordinate well with other machines. For the control system discussed above, the settling time can be calculated through equations (6), (25), and (27)

t_{s} = \frac{8 K_{v} K_{f} R_{a} J}{K_{e}^{2} K_{t}}

(28)

Assuming that the controllers are realized through analog circuit, therefore, all of the control parameters contain a certain range of uncertainty due to manufacturing error. To achieve the settling time less than $T_{s}$ , the values and correspondent deviations of all parameters in equation (28) are designed as listed in Table 1. According to equation (2), to calculate its settling time reliability, the limit state function must be obtained through equation (28)

g (P) = 8 K_{v} K_{f} R_{a} J - K_{e}^{2} K_{t} T_{s} < 0

(29)

where $P$ is a vector containing design parameters $P = (R_{a}, J, K_{e}, K_{t}, K_{v}, K_{f})$ .

Table 1.

Mean values and deviations of control parameters.

Design parameters	Mean	Deviation
Resistance in armature circuit $(R_{a} / Ω)$	3	0.5
Rotary inertia of motor spindle $(J / kg m^{2})$	1.8	0.5
Back EMF coefficient $(K_{e} / V s / rad)$	7.1	0.1
EMT coefficient $(K_{t} / N m / A)$	7.1	0.1
Gain in speed controller $(K_{v} / V / V)$	2500	100
Motor feedback coefficients $(K_{f} / V s / rad)$	0.24	0.01

EMF: electromotive force; EMT: electromotive torque.

Given the limit state function, the reliability evaluation can be obtained through the FORM proposed in section “Principle of FORM and its iterative algorithm.” Results are shown in the next section.

Results and discussion

Validity of iterative algorithm to search MPP

Because wrong solutions to MPP are frequently obtained using a traditional HL-RF iterative algorithm, it is necessary to validate that this shortcoming does not exist in the improved iterative algorithm. Therefore, in this section, the calculated MPP and the limit state equation are projected on several two-dimensional planes to illustrate that this MPP is the closest point to the origin. Through these illustrations, the efficiency and accuracy of the proposed iterative algorithm is roughly demonstrated.

For the problem of settling time reliability evaluation in section “A case study of a servo motor control system,” the reliability index $β$ and its MPP $y^{*}$ in the standard normal space can be calculated through the proposed iterative algorithm in section “Principle of FORM and its iterative algorithm” as $β = 0.3096$ , $y^{*} = (0.1606, 0.2566, 0.0279, 0.0412, 0.0139, 0.0395)$ . Hence, its settling time reliability approximation is $R = 1 - Φ (β) = 0.6216$ . However, the traditional HL-RF iterative algorithm does not get any converged solution.

To demonstrate the validity of the MPP, limit state functional curves with four of six standard normal variables which have the values of MPP are depicted on space of $R_{a} - J$ , $R_{a} - K_{e}$ , and $K_{e} - K_{t}$ , respectively (see Figure 7(a)–(c)), and illustrations of other spaces are not presented in this article for simplicity. From these figures, it is apparent to see that lines connecting the MPP and the origin are rightly perpendicular to the planar limit state functional curve, which demonstrates that the calculated MPP is just the nearest point from the origin. Therefore, the proposed iterative algorithm is efficient to search MPP with high accuracy even in multi-dimensional space.

Figure 7.

Limit state functional curves projected into (a), (b), (c) two-dimensional plane and (d) three-dimensional space.

Moreover, except for the planar limit state functional curve in $R_{a} - J$ space, other limit state functional curve is nearly linear (see Figure 7(a)–(c)). And, three-dimensional limit state functional surface is depicted in the $J - K_{e} - R_{a}$ space (see Figure 7(d)). In this figure, it can be found that limit state functional surface is slightly nonlinear on the intersection of $R_{a} - J$ coordinate plane. Therefore, the approximation to the reliability evaluation in this case study with FORM still holds a certain accuracy. Its accuracy is verified by the comparison with the relatively exact calculation through MCSim. The reliability in the case study calculated through MCSim is approximately 0.6466. So, the evaluated error of FORM is 3.87% which shows a good approximation of FORM.

Accuracy of approximation with FORM

In order to verify the accuracy of approximation with FORM, some design parameters are altered to obtained different approximated results, and this series of results are compared with those calculated through MCSim. In this case study, the deviation rotary inertia of motor spindle $J$ is chosen as a variable and several results are calculated with FORM with variation of $J$ . The numerical results calculated through both FORM and MCSim and the estimated errors of FORM approximation are plotted in the same graph (see Figure 8). The broken line designated with squares illustrates that the approximated error varies between 1% and 5% with variation of parameter $J$ as horizontal ordinate, meaning that the approximation method, FORM, maintains a good accuracy with variation of design parameters. Also, from the broken line designated with triangles and circles, the reliability approximated with FORM (designated as Rform in Figure 8) is always conservative as compared with those calculated through MCSim (designated as Rmcs in Figure 8). Moreover, the iterative time of FORM to obtain the desired solution is about seven steps, which is far less than that through MCSim that costs 10,000 cycles to evaluate the reliability at a high precision. Therefore, it can be concluded that the solution to the case study with FORM is of good accuracy and great efficiency.

Figure 8.

Reliability evaluation through FORM and MCSim and the approximated error.

Reliability sensitivity analysis of the parameter deviations

Reliability sensitivity is a useful index to indicate the influence of the parameters to the whole system reliability. It is defined as the following expression

α_{i} = \frac{\partial β}{\partial y_{i}}

(30)

where $α_{i}$ is the reliability sensitivity of the parameter $y_{i}$ . In designing a control system, the reliability sensitivity of each design parameter will give a great enlightenment on the deviation control of these design parameters. Improving the most sensitive parameters will increase the system reliability significantly. In this case study, the sensitivity of the deviation of each parameter is investigated and listed in Table 2. From Table 2, the motor feedback coefficients $K_{f}$ , rotary inertia of motor spindle $J$ , and resistance in armature circuit $R_{a}$ play important roles in servo motor control system, and hence, strict quality control should be paid on those parameters of motor in order to guarantee high settling time reliability.

Table 2.

Reliability sensitivity of parameter deviations.

Design parameter deviation	Sensitivity
$K_{f}$	0.5214
$J$	0.4496
$R_{a}$	0.1655
$K_{e}$	0.02389
$K_{t}$	5.9639e−3
$K_{v}$	4.8052e−5

Conclusion

In this article, the methodology to evaluate the control performance reliability is first proposed, especially for the second-order control system. In order to make an accurate approximation of the multi-dimensional integration which is the main obstacle lying in control performance reliability, FORM is introduced and a robust iterative algorithm is adopted to search the MPP which is a tough task in the procedure of FORM. This method and its algorithm are demonstrated to be accurate and efficient through a case study of settling time reliability evaluation of a servo motor control system. In the case study, a servo control system is designed with three-loop control, and the deduction and simplification of its transfer function are elaborated in this article. Then, a method to evaluate the performance of the control system is applied to this case and an accurate reliability estimation is obtained with an estimated error of 3.87%. Moreover, based on this method, reliability sensitivity is carried out to show the key components in the control system which provides useful guides on the quality control of the system parameters.

Further research aims at the following two aspects. One of them is that a more accurate approximation method can be developed either through second-order approximation to the limit state function or by multiplying a correction factor to error compensation of nonlinearity of limit state function. The other one is that an optimal reliability design method can be achieved using the reliability sensitivity based on the methodology of control performance reliability evaluation proposed in this article.

Footnotes

Academic Editor: Elsa de Sa Caetano

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors received the support of the Fundamental Research Funds for the National Natural Science Foundation of China (Grant no. U1361127), the National Key Research and Development Program of China (Grant no. 2016YFC0600907).

References

Taflanidis

Scruggs

Beck

. Reliability-based performance objectives and probabilistic robustness in structural control applications. J Eng Mech 2008; 134: 291–301.

Rausand

Høyland

. System reliability theory: models, statistical methods, and applications. New York: Wiley, 2003.

Spencer

Jr Sain

Kantor

. Probabilistic stability measures for controlled structures subject to real parameter uncertainties. Smar Mat St 1992; 1: 294–305.

Spencer

Sain

Won

. Reliability-based measures of structural control robustness. Struct Saf 1994; 15: 111–129.

Melchers

. Structural reliability: analysis and prediction. London: Wiley, 1987.

Field

Jr Bergman

. Reliability-based approach to linear covariance control design. J Eng Mech 1998; 124: 193–199.

Patelli

Pradlwarter

Schuëller

. On multinormal integrals by importance sampling for parallel system reliability. Struct Saf 2011; 33: 1–7.

Cornell

. Structural safety specifications based on second-moment reliability analysis, Symposium IABSE, London, 1969.

Schueller

. Structural reliability-recent advances. Struct Saf Reliab 1998; 1: 3–35.

10.

Pandey

. An effective approximation to evaluate multinormal integrals. Struct Saf 1998; 20: 51–67.

11.

Harbitz

. Efficient and accurate probability of failure calculation by use of the importance sampling technique. In: Proceedings of the 4th international conference on applications of statistics and probability in soil and structural engineering (ICASP4), Pitagora Editrice Bologna, 1983, pp.825–836.

12.

Dolinski

. First-order second-moment approximation in reliability of structural systems: critical review and alternative approach. Struct Saf 1983; 1: 211–231.

13.

Hohenbichler

Gollwitzer

Kruse

. New light on first-and second-order reliability methods. Struct Saf 1987; 4: 267–284.

14.

Zhou

Huang

H-Z

Xiao

N-C

. Reliability analysis method based on universal generating function. J Mach Des 2011; 28: 93–96.

15.

Gollwitzer

Rackwitz

. An efficient numerical solution to the multinormal integral. Probabilist Eng Mech 1988; 3: 98–101.

16.

Yuan

Pandey

. Analysis of approximations for multinormal integration in system reliability computation. Struct Saf 2006; 28: 361–377.

17.

Ditlevsen

Bjerager

. Methods of structural system reliability. Struct Saf 1986; 3: 195–229.

18.

Zhou

. PNET-based reliability sensitivity analysis of structure systems. PhD Dissertation, Northeastern University, Shenyang, China, 2008.

19.

Zhao

. Multivariate robust control system. Harbin, China: Harbin Institute of Technology Press, 2011.

20.

Rosenblatt

. Remarks on a multivariate transformation. Ann Math Stat 1952; 23: 470–472.

21.

Rackwitz

Fiessler

. Structural reliability under combined random load sequence. Comput Struct 1978; 9: 489–494.

22.

Yang

Lin

. Chaotic dynamics analysis of FORM in structural reliability. Chin J Theor Appl Mech 2005; 37: 799–804.

23.

Ding

. A search algorithm for structural reliability index. Chin J Comput Mech 2005; 22: 787–791.

24.

Junjie

Yong

Jia

. Equal step iteration method for reliability index and its proof of convergence. J Huazhong Univ Sci Technol 2008; 36: 129–132.

25.

Gong

. A robust iterative algorithm for structural reliability analysis. Struct Multidiscip O 2011; 43: 519–527.