CC BY
17
Vol. 14, no. 1. 2018 ISSN2411-1473 sitito.cs.msu.ru
Когнитивно-информационные технологии в системах управления
Cognitive information technologies in control systems
УДК 517.977.5
DOI: 10.25559/SITITO.14.201801.091-100
SYNTHESIS OF FAULT ESTIMATION OBSERVER, BASED ON SPECTRAL MIMO H2
OPTIMIZATION
Evgeny I. Veremey, Yaroslav V. Knyazkin
Saint Petersburg State University, Saint Petersburg, Russia
Abstract
This paper is devoted to slowly varying additive fault detection with implementation of the observer-filter to be designed. Sensitivity of the observer to the external disturbance is to be minimized by the choice of its parameters. There are a lot of papers, devoted to fault detection issues, but such problems with initially given spectral features of external disturbances are not a subject of serious attention, in our opinion. External disturbances, which are considered in this research, can be presented as a sum of harmonic oscillations with given central frequency (sea wave disturbance) and a constant signal (ocean currents and wind). A suppression of the polyharmonical oscillations influence is considered as H2-optimization problem, which can be solved with application of the specific spectral approach in frequency domain. This approach is based on polynomial factorization that can improve computational effectiveness of the observer design procedures. This also guaranties nonuniqueness of the optimal solution that makes possible to provide integral action of the residual signal relatively to the external disturbance for the case when at least two sensors are used. The novel algorithm of adaptive fault estimation observer analytical synthesis is proposed and its effectiveness is demonstrated by the numerical example of the fault detection process with implementation of MATLAB package.
Keywords
Linear-quadratic functional; H2-optimazation; fault detection; optimal control; spectral approach; stability; integral action.
About the authors:
Evgeny I. Veremey, Doctor of Physical and Mathematical Sciences, Professor, Head of the Department of Computer Applications and Systems, Faculty of Applied mathematics and Control Processes, Saint-Petersburg State University (35 University Ave., Peterhof, St. Petersburg 198504, Russia); ORCID: http://orcid.org/0000-0001-8366-5654, e_veremey@mail.ru
Yaroslav V. Knyazkin, graduate student, Faculty of Applied mathematics and Control Processes, Saint-Petersburg State University (35 University Ave., Peterhof, St. Petersburg 198504, Russia); ORCID: http://orcid.org/0000-0001-8522-588X, yaroslavknyazkin@gmail.com
© Veremey E.I., Knyazkin Y.V., 2018
Том 14, № 1. 2018 ISSN2411-1473 sitito.cs.msu.ru
СИНТЕЗ ДЕТЕКТОРА ДИНАМИЧЕСКИХ СБОЕВ С ПРИМЕНЕНИЕМ СПЕКТРАЛЬНОГО
ПОДХОДА К MIMO H2 ОПТИМИЗАЦИИ
Е.И. Веремей, Я.В. Князькин
Санкт-Петербургский государственный университет, г. Санкт-Петербург, Россия
Аннотация
Данная работа посвящена задаче синтеза наблюдателей - фильтров для обнаружения воздействия медленно меняющихся аддитивных динамических сбоев. Требуется минимизировать чувствительность детектора к внешнему возмущению за счет выбора параметров наблюдателя. Несмотря на существование множества работ, посвященных обнаружению сбоев, подобным задачам при наличии известного спектрального состава внешнего возмущения, по нашему мнению, не было уделено достаточного внимания. Внешнее воздействие, рассматриваемое в данной работе, может быть представлено как сумма нескольких гармоник с известной центральной частотой (например, морское волнение) и постоянного сигнала (например, ветра или морского течения). Подавление полигармонического возмущения рассматривается как задача среднеквадратичной оптимизации, для решения которой привлекается специальный спектральный подход в частотной области, использующий параметризацию множества передаточных функций замкнутой системы. Этот подход основан на полиномиальной факторизации, что повышает вычислительную эффективность процедуры синтеза. Он также гарантирует неединственность оптимального решения, что дает возможность обеспечить дополнительные свойства, такие как астатизм разностного сигнала относительно внешнего возмущения, в случае если измеряется несколько координат состояния объекта. Предложен новый метод аналитического синтеза адаптивного детектора сбоев и его эффективность продемонстрирована на практическом примере - движении морского судна в горизонтальной плоскости с постоянной скоростью под действием ветра и морского волнения. Для проведения численного имитационного моделирования динамики судна используется среда МАТ1АВ.
Ключевые слова
Линейно-квадратичный функционал; Н2-оптимизация; обнаружение сбоев; спектральный подход;устойчивость; астатизм.
Introduction
Complexity of the controlled plants is permanently increasing that results in a significant probability of various malfunctions, that can cause degradation of the control effectiveness and, at worst, instability of the system. These circumstances make problems of safety and reliability crucial. Quick fault detection is necessary for well-timed correction of the control law to avoid catastrophic consequences. The designed fault detector must be simultaneously sensitive to the fault and irresponsive to external disturbance, such as sea waves one, wind, vibration etc., and the fault detection issues usually deal with trade-off between these properties. This problem has been paid
serious attention since the beginning of the 1970s, regarding both the theoretical context and the applicability to real systems [1-3]. Nowadays there are a lot of various approaches, based on such techniques as Principal Component Analysis (PCA) [4, 5], neural networks [6, 7], asymptotic observers [1, 2], parameter estimation [8, 9] etc. They can be split two main categories: model-free approaches (e.g. PCA), using any statistical criteria and model based ones, based on given mathematical description of the plant and usually including asymptotic observers.
Model based fault detection has been hot research area for the past decades (history of this issue is described in details in the monograph [1]).
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As a result, various effective approaches have been proposed for such systems as descriptor [10], Takagi-Sugeno [11], nonlinear [12], switched [12], stochastic [12, 13], Markovian jump [13, 14] ones etc., and there are a lot of papers and monographs devoted to this problem. However, some aspects of fault detection observer synthesis are not fully studied until now in our opinion. It is notable, that such issues with initially known spectral structure of the external disturbance are not fully studied until now and there are a lot of situations (e.g. marine ship motion process), where the external disturbance (sea waves one, vibration etc.) has given one, e.g. can be described as a random Gaussian process with the given spectral power density or as a sum of a few harmonics with known amplitudes and frequencies. This feature can be taken into account to improve effectiveness of the adaptive observers, providing their filtering properties.
The circumstances, mentioned above, motivate us to research devoted to fault estimation observer design for controlled plants affected by polyharmonical external disturbances. The stated in this paper H2 optimization problem can be solved with implementation of the special spectral approach in frequency domain, based on parameterization of the set of controllers, proposed in [14, 15]. Similar methods, based on polynomial factorization, have already been applied to H2 [17-20] and H« [21, 22] optimization problems, including ones devoted to fault estimation [23, 24]. They do not include very complicated procedures (e.g. solving of linear matrix inequalities (LMI)) that can improve computational effectiveness of the observer design procedures. The mentioned property is crucial for systems with real-time regime of operating, e.g. for onboard control systems. The optimal solution of the considered problem is not unique that makes possible to provide additional properties, such as integral action of the residual signal relatively to the external disturbance that is necessary in the considered example of the detector synthesis for a marine ship.
The rest of this paper is organized as follows. The next section introduces the equations of controlled plant, external disturbance, observer-filter and problem statement. Section 3 is devoted to description of spectral approach to solution of MIMO (Multiple Input and Multiple Output) H2
optimization problem with polyharmonic external disturbance, used as base for investigation, presented in this paper. Also we receive necessary and sufficient condition of H2 optimal suppression of the harmonic oscillations. Parameter computation process, based on modal synthesis, is described in Section 4. Finally, in Section 5, we describe overall results the investigation, merits and demerits of the proposed approach and mention some directions of the future research.
2. Problem statement
Let us introduce a linear time invariant plant
&= Ax + Bu + Ef + Hd (t),
a () (1) y = Cx,
where x e Rn is the state space vector, u e R"r is the control, d(t) is the scalar external disturbance,
f e Rnf is the slowly varying fault, i.e. f^« 0, and y e Rm is output measured signal. All components of the matrices A, B,C,E, H are known constants, the pairs {A, b} and {A, C} are controllable and observable respectively.
External disturbance d(t) for the system (1) is treated as sum of a polyharmonical function and a step one
d(t) = £ Adl sin(CT,t + 9, ) + 1(t)d0,
(2)
where Nd is the number of harmonics and Adi, a,, 9,- are their amplitudes, frequencies and phases respectively, d0 is a constant value and 1(t) is Heaviside step function. Let us suppose that the central frequency of d(t) is ra = ra0.
Adaptive fault estimation observer-filter has the following structure
Ax + Bu + v,
v(s) = L(s)(y - CX), r = C r (y - Cx) = Cr (x - X ),
(3)
where v(t) e Rn is the corrective signal, r(t) e R1 is the residual signal and L(s) is the transfer matrix. The residual signal is the fault message: if r(t) is
close to zero, then the plant operation is fault-free, else a fault occurred. Parameters of the transfer function L(s) characterize dynamics of the fault detection process and they are to be designed by the process of the optimal observer synthesis.
N
i=1
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Denote
ex = x - X, ey = Ce x = y - Cx,
then the error dynamics is written as &= Aev + Ef + Hd - v,
ey = Ce X =
v(s) = L(s)e v
(4)
(5)
r = C e = C e
r Crey CreX ■
Let us consider the following values, characterizing effectiveness of the filtration and sensitivity to fault:
Jo = Frd C/'®c)2, J =| K (0)|2, where (6) Frd (s), Ff (s) are the transfer functions from the external disturbance d(t) and fault f(t) to the residual signal r(t):
Frd = Cr (sI - A + L(s)C)-1 H, Fr/ = Cr (sI - A + L(s)C)-1 E.
(7)
Remark that J0 and J1 are functionals of L(s), and in such a way, we should design such L(s), that J0 (L) ^ min, J1 (L) ^ max, (8)
LeQL LeQL
where QL is set of L(s), providing stability of received closed-loop system (7) . In the framework of this research, the designed observer should be insensitive to the constant disturbance, i.e. the integral action of r(t) relatively to the signal d(t) should be guaranteed, i.e.
Frd (0) = 0. (9)
Firstly, we minimize the value J , characterizing suppression of polyharmonical disturbance and introduce auxiliary mean-square optimization problem. Consider the following auxiliary LTI plant:
Xf = A1x1 + B1u1 + H1d1, (10)
where x eR" , U eRm , A = AT , B = -CT , H = CT and dj e R is a harmonic disturbance with the frequency ra = ra 0. Let us accept that controller to be designed has the following tf-model U1 = LT (s)x1 = W(s)x1 = W(s)21 W1(s)x1, (11) where W1 (s), W2 (s) are the (m x n) and (m x m) polynomial matrix functions, and consider the following functional
1 T
J2(W) = lim - [ (xTRxj + k2uTQu ) dt
T^œ T ^
T
= lim — [ (eTe + k 2u[ Qui)dt,
T ^œ T »
(12)
where R = HHT , e = HTx1 , Q is symmetrical
positive definite matrix (e.g. unit one), k is positive value close to zero, and formulate an auxiliary mean-square optimization problem
J2(W) ^ min ,
WeQw
(13)
where Qw is the admissible set of the controllers (10). This set is such that all the roots of the characteristic polynomial
A(s) = (As (s))1-m det(WiBis - W2 (s)As (s)), where
(14)
B1s (s) = As (s)(sI 2 A1)21B1, As = det(A1)
are located in the open left-half complex plane. One can see that F d (s) = F^(s) and J
k^O
J
are
(16)
3. Spectral approach to MIMO H2 optimization
Firstly, we rewrite the functional (12) in frequency domain
1
J2 = _ J trace F (-s)RFx (s) + k2FT (-s)QFu, (s)]K (15)
j 0 K SA (s)ds,
where S^ (ra) = I •8(0-0)0) and F^ (s) , F^ (s) transfer functions from d1 to x1 and u1 Fx (s) = (sI - A1 - B1W)-1H1, Fu (s) = W(sI - A1 - B1W)-1H1, Dependency of the functional J2 from the controller W(s) is complicated and nonlinear, but it is quadratic in Fx (s) , Fu (s) and the problem (13)
can be solved by using of the approach, based on parameterization technique, presented in [15]. It is remarkable, that the method described below can be applied to the plants with multidimensional input, but it includes solution of the algebraic matrix Riccati equations that results in increasing of its computational complexity. Let us introduce the adjustable function-parameter ®(s).
O(s) = a(s)Fx (s) + P(s)Flli (s) , (17)
where a(s) , P(s) are (m x n) and (m x m)
polynomial matrixes. These parameters can be chosen in accordance to the approach, proposed in [15]. Let us solve the following matrix Riccati
о
о
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equation
SA1 + A[S--1SB1Q-iBT S + R = 0, (18)
k
and calculate them as follows
a(s) = ao ^ Q-1B[S, p(s) = po = I. (19) k
Then we denote
©(s) = As (s)P(s) + a(s)Bis (s),
Bis (s) - As (s)(sI - Ai)-1Bi, His (s) - As (s)(sI - Ai )-1 H i, P(s) - (sI - Ai),
(20)
consider presentation of the plant (10) in frequency domain:
(si - Aj )Fx - BjFu = H
(21)
and express F , Fu , using formulae (17), (21)
( F ^
X-
F
= M-
-11 H®
, M 0 =
si - a -Bj
. (22)
*0 Po
Now we apply Frobenius formula of block matrix inversion, taking into account the equality
(p(s) + a(s)P-iBi)-i = As (s)(As (s)p(s) + + a(s)Bi(s)) -1 = As©-i(s)( s), to calculate the inverse matrix M -i
M-1 =
^P-1 - Bis (s)©-1(s)a(s)P-1 Bis (s)©-1(s) - As (s)©-1(s)a(s)P-1 As (s)©-1(s)
then substitute the computed M-i to (22) and express of Fx (s), Fu (s) as functions of O(s):
Fx = Fx (0) = His (s)/As (s) + Bis (s)©-i(s)0(s),
i i ~ i ~ (23)
Fu, = (®) = As (s)©-i(s)0(s),
where O = (®-a0P-iHi) . Also we denote
Bi5= Dis (sI - Ai + Bia0)-iBi, Dis = det(sI - Ai + Bia0), to transform the transfer function ©-i(s) in simple form
©-i(s) = (a 0Bis + IAis )-i = Afs'I - A> 0(B^a 0 +
+ IAis)-i Bis = A^I - Ar> 0 (I + P-iBia 0 )-i P-iBi = = A-1I - A^a 0 (P + Bia 0 )-i Bi = A^I -
(24)
- A-1 a 0 DrX =
1
ADis
(As I-a oB15 ).
Now let us consider the expression from integrand in (14)
(25)
Fo*Fo = Fx; RFx1 + kXQFu ,
as function of the parameter O(s). Note that if we choose the parameters a(s), P(s) by the formulae (19), then
©-i(B*sRBis + k 2QA*As )©-1 = k 2Q, (this equality is proven in [15]), and (25) can be converted to the equivalent form
_Sfc___Sfc ^^ ^_Sfc _____
F0F0 - (Ti + O T2 )(Ti + T20) + T3 , where
Ti(s) = (WQ)-i©;iB*sRHis / As(s), T2(s) = kVQ ,(26)
T3 (s) = (H*sRHis) / As (s)As (-s) - Ti* (s)Ti (s).
One can see that the summand T3(s) does not depend on 0, so a minimum value of J2 is achieved if and only if
1 jm
i c * ^ ^ * ~
— I trace [(Ti + 0 T2)(Ti + T20)]Sd (s)ds , m J
J 0
reaches its minimum. In accordance to filtering property of the delta function
1 f* _* - * _* _ _ _
— rtrace [(T1 (s) + 0(s) T2 (s))(T1(s) + T2(s)0(s))]K
JKi
(27)
1
K SA (s)ds = — trace [(Ti* + 0*T*)(Ti + T2O)]/ ,
d jk J 0
i.e. J2 is function of ®(Jra0), and can be minimized by choice such ®(jra0) that the expression (27) is equal to zero:
Ti (jra 0 ) + T2 J 0 )0> J 0 ) - 0, or
®o Oo^) = T21 Oo)T1 Ooi) =
= -(k 2Q)-1©-1B*s RH1s / As (s)|
(28)
As a result, we substitute (28) to (23) and receive dynamics of the optimal closed loop system on the frequency ra0
Fx (/^ = His C^V As (/^ +
+ Bis (Jra0)©-i(jra0)®0(Jra0), (29)
Fu (>0) = As (jra0)©-i(j'ra0)®0(Jra0).
Finding of the matrices (29) does not complete the solution of the problem (13): we have to provide the desired behavior of the closed-loop system with the help of controller (11) similarly to [18].
Especially note that the controller provides the optimal dynamics of the closed-loop system (10), (11) on the frequency ra0 if and only if its transfer matrix W(s) satisfies the following equation
WC/®0)Fx,/0) = Fu,/0) (30)
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The proof of this statement can be proved similarly to the Theorem 4 from [18]. One can see, that (30) is equivalent to
L Oo)FXi Oo) = FUi Oo) . 4. Transfer matrix of the optimal observer
(31)
Fulfillment of the condition (31) from the previous section is not enough, because it guarantees only suppression of polyharmonical external disturbance (2), and it is also necessary to provide insensitivity to the constant one. Let us note that the designed observer should simultaneously suppress the external disturbance effect and be as sensitive to the fault action as possible and well known approaches, such as PIDcontroller or speed control law [19, 20] cannot be applied for solution of the issue, because they suppress any constant effect. One can see, that Fd (0) is function of the matrix L0 = L(0) and (9) results in
(- A + L°C)-1 H = e , (32) where e1 is a vector satisfying the condition Ce = 0, and its coordinates can be chosen in process of the observer synthesis. The expression (32) can be rewritten as the following system of linear equations
L0Ce 1 = Ae 1 + H .
Remark 1. It is necessary to provide the relation Fr/ (0) * 0, e.g. establishing the additional condition
(- A + L0C)-1 E = e2, Cre2 * 0.
Note that providing of the property (9) is not always possible and it is remarkable that its sufficient conditions can be written analytically in case of small dimension systems, including marine ships.
Example 1. Let us formulate such conditions for the problem considered in the next section: a transport marine ship moving on the horizontal plane with constant longitudinal speed presented by the model (1) with the following parameters:
f a
A =
"21 0
22 1
j3
0^ 0
0 /
H
(Ьхл h2 0
C =
Vector x e R3 consists of three components: drift angle, angular velocity and yaw angle; drift angle and yaw are measured. Let us denote as
coordinates of L0
Г ill ll2 ^
L0 = L(0) = l21 l22
V l3i l32 /
and consider components of (- A + L0C) 1H :
(- A + L0C)-1 H = -
1
det (- A + L0C)
'1122 '2 A 2 + a1 2^213 2 a 22'' 1132
a21h1l32 - a11h2l32 + h2l1 1l32 - h2l1 2^31 - h1l21l32 + h1l22l31
a 21^1 a11^2 + '2^11 '1121 a12^2131 + a 2 2^1131 ^ One can see, that the third component of the received vector depends only on the first column of L(s) and if we choose Cr = (0 0 1), then (9) is equivalent to the equation
'j a '21 ^2^ 1 '^21 a 2I 1+1 — 0 .(33) Now consider the functional J1(L) (6), characterizing sensitivity to the fault and demonstrate its dependency from the degree of stability of the closed-loop system. Let us implement the theorem of roots distribution [20, 25]: any polynomial A(s), deg A(s) = ~ with the degree of stability ast > 0 has the corresponding vector yeR~ , such as A(s) = A*(s,y), where A (s, y), if «is even;
A (s, y) =
(s + a , (y,ast))A (s,y), if n is odd;
n
A* (s, y)=n(s 2 + a1(y, a st) s + a0 (y, a st)),
i=1
a1(y, a st) =2a st+y L
(34)
ai(y, ast)=alt +y 2last +y 22, i =1 n*, (35) an•+1(y,ast)=yП*0 +ast , n* = [~/2] y = {yi1, yi2, y 21, y 22,-,y n*1, y n*2, y n^
Let us parameterize the characteristic polynomial A(s) (14) by the formulae (34), (35) and consider the value
n n
A(0, y* ) = na? (y*, ast ) = ]] (a* + yXt +y * )>a*\ i=1 i=1 The polynomial A(s) is denominator of the transfer function from fault to residual signal Frf (s), i.e. the value J is inversely proportional to ast and can be increased by its choice determining the trade-off between stability and sensitivity to fault signal.
a
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Finally, let us design stabilizing controller W(s) (11), satisfying the conditions (30), (9) using modal synthesis. Sincerely, it cannot be realized directly, because dependency of the characteristic polynomial (14) from the transfer matrix W(s) is nonlinear in case of m > 1. One of ways to avoid this difficulty is to find the columns lk (s), k = 1,K m, of L(s) = (lj (s) l2(s) K lm(s)), i.e. the rows of the controller W(s):
wk (s) = lTk (s) = Wik (s) / w2k (s) , k = 1,K m,
(s) , w2k(s) are polynomial rows and polynomials, consequently, designing m close-loop systems in the following way. Note that wk (s) can be constructed in any order, but suppose that they are designed from 1st to mth for simplicity. Denote Bis (s) = (b1s (s) b2s (s) A bms (s)) ,
Ao (s) = As (s) ,b% = b js (s) , j = 1Tm , < = H1s , define k = 1 and consider the plant (10) in frequency domain:
m
Ak_1 (s)x1 (s) = £bks-1 (s) u(s) + Hk;1 (s)d, (s) .(36)
i=k
On the kth step we design the feedback
uk (s) = wk (s)^(s) , wk(s) = W1k(s)/W2k(s) ,(37)
solving the equations
Ak_1 (s)w2k (s) _ W1k (s)bk_1 (s) = Ak (s), (38)
W1k 0'ra0)Fx 0'ra0) = K (jra0)(jra0)w2k O^ , (39)
where Ak (s) is a defined polynomial for the closed-loop system on this step, F^ (jra0) is the kth row of F (jra0) and taking into account the conditions of the integral action if Frd (0) depends on lk (0) = wk (0) . Let us note, that Ak_1(s) and coordinates of b^1 (s) can have common roots and they should be roots of (s). Then we substitute calculated feedback (37) to (36) and receive the following expression
m
(IAk_1(s) _ bk_1wk)x1(s) = £ bks~\s) u (s) + Hk;1(s)d1(s) ,
i=k+1
and rewrite it as
At (s)*(s) = S bs (s) u (s) + Hks (s)d1(s), where (40)
i=t+1
bts (s) = At (s)(IAt_i (s) - bt;1 (s)wt (s))-1bfs-1 (s) , (41)
Hts(s) = At(s)(IAt-1 (s) - bt;1 (s)wt(s))-1 Hfs- (s) . (42)
Then k increase and calculations (37)-(42) are repeated. Finally, we receive
L(s) = (wT (s) wk (s) K wk(s) ). Note that the polynomials Ak (s) , k = 1,K m can be parameterized in accordance to the formulae (34), (35), polynomial A(s) = ^(s) chosen on the final
step, is the characteristic polynomial (14) and it must be a Hurwitz one.
5. Example of Synthesis
Let us illustrate the practical implementation of the proposed approach by the example of the marine ship yaw motion. Assume that its model (1) has the following matrices:
^-0.0936 0.63
0A
A = 0.048 - 0.072 0 , B = E = 0.0160
v 0 1 0 y V 0 y
r0.0196A
( 0.41 A
H
C =
1 0
0 0
0.0076
V 0 V
and external disturbance d(t) can de expressed by the formula (2) as
d (t) = sin(ra 0t) + 0.1 sin(0.9ra 0t) + 0.1 sin(1. 1ra 0t), ra0 = 0.45 .
State vector x = (P ra <p)T e R3 consists of drift angle, angular velocity and yaw angle. Let us choose Cr =(0 0 1), Q = I, k=0.01. Then we compute transfer functions from (14), (20)
As (s) = s3 + 0.8106s2 + 0.0367s ,
B1s (s) =
f-s2 - 0.717s - 0.634s 0
- 0.048 - s - 0.0936 - s2 - 0.8106s - 0.0367
(
H1s (s) =
0.048 A
s + 0.0936 v s 2 + 0.8106s + 0.0367, solution of the matrix Riccati equation (18)
S =
f 0.0041 7.94-10"5 1.937-10"6^ 7.94-10"5 1.628-10"6 2.335-10"' 1.937-10"6 2.335-10"7
v 6.554-10\ parameters a0, ^ from (19)
f_40.919 _0.7940 _0.0194 A
a0 =i_0.0194 _0.0023 _0.0655 J, =1'
m
Том 14, № 1. 2018 ISSN2411-1473 sitito.cs.msu.ru
calculate transfer functions of the optimal closed loop system on the frequency ra0:
f0.0248 + 0.04j \ Fx 00) = -1.34 - 2.17 j v - 2.184 j f-0.0485 - 0.01191j ^^ =[ 0.0168 and, taking into account (32), receive
L(s) =
1
s4+3.58 s3 - 3.29 s2+0.73s - 0.708 7.43 s+9.38 s2 +1.5 s+1.9 4.95 s4 -1.1 s3 + s2 - 0.22 s +1.4-10-7 К
- 6.63s4 + 5.32-10-4s3 - 1.343s2 s4 + 4.342 s3 + 6.583 s2 + 6.453 s^
К 1.37 s +1.219 s
- 6.227s4 - 0.8s + 8.43 -10-6
Fig. 1 represents frequency responses Ard and
Ar/ of the transfer functions Frd and Frf of the
closed-loop system with L(s) computed in this
section. The curve Ard is close to 0 on zero
frequency and in the area of the central one ra0, i. e.
effect of the external disturbance d(t) is
successfully suppressed. Fig. 2 illustrates fault detection process: the constant part of the external disturbance intensifies at 100 s., but residual signal returns to zero, and the fault, occurred at 300 s. is successfully detected.
Fig.1. Frequency responses of the transfer functions Frd
Fig. 2. Fault detection process 6. Conclusion
A novel special approach in frequency domain to H2-optimal fault detection observers design is proposed and described in details in this paper. The presented method can be used for a wide spectrum of practical control applications concerned with plants affected by harmonical oscillations, such as various marine vehicles.
Proposed approach to ^-optimization is based on polynomial model presentation and the special parameterization technique for the transfer matrices of the closed-loop system. Its main feature is guaranteed no uniqueness of the problem solution that makes possible to design fault detection observers analytically and provide their additional properties, e.g. insensitivity to constant external disturbance effect.
Working capacity and effectiveness of the proposed method are illustrated by the numerical example: plane motion of a marine ship, considered as LTI plant of 3-th order.
Sincerely, the described approach has also some demerits. Firstly, it is the absence of universal conditions for integral action. Secondly, the proposed multistep method of modal synthesis can be intractable problem for high dimensional systems with many outputs. Overcoming of these negative features is the main direction of the future research. Thirdly, it is necessary to propose algorithm of adaptive threshold [1, 4] design. Finally, robust features or various delays should be taken into account in the sequel.
and F,
rf
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Vol. 14, no. 1. 2018 ISSN2411-1473 sitito.cs.msu.ru
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[23] Veremey E.I., Knyazkin Y.V. Spectral H2 fault estimation observer design based on allocation of the correction effect. Journal of Theoretical and Applied Information Technology. 2017; 95(12):2776-2782. Available at: http://www.jatit.org/volumes/Vol95No12/18Vol95No12.pdf (accessed 10.02.2018).
[24] Veremey E.I., Knyazkin Y.V. Spectral H-2 optimal correction of additive fault estimation observer. Proceedings of the 6th Seminar on Industrial Control Systems-Analysis, Modeling and Computation. EDP Sciences, 2016. DOI: https://doi.org/10.1051/itmconf/20160601005
[25] Veremey E.I., Smirnov M.N., Smirnova M.A. Synthesis of stabilizing control laws with uncertain disturbances for marine vessels. Proceedings of 2015 IEEE International Conference «Stability and Control Processes» in Memory of V.I. Zubov (SCP). 2015, p. 1-3. DOI: https://doi.org/10.1109/SCP.2015.7342219
Submitted 16.12.2017; Revised 10.02.2018; Published 30.03.2018.
СПИСОК ИСПОЛЬЗОВАННЫХ ИСТОЧНИКОВ
[1] Ding S.X. Model-Based Fault Diagnosis Techniques: Design Schemes, Algorithms and Tools. Springer Science & Business Media. 2012. DOI: https://doi.org/10.1007/978-1-4471-4799-2
[2] Chen J., Patton R.J. Robust model-based fault diagnosis for dynamic systems. Springer Science & Business Media, 2012. Vol. 3.1.
[3] Witczak M. Fault Diagnosis and Fault-Tolerant Control Strategies for Non-Linear Systems: Analytical and Soft Computing Approaches. Springer Science & Business Media, 2013. Vol. 266. DOI: https://doi.org/10.1007/978-3-319-03014-2
[4] Zanoli S.M. et al. Application of Fault Detection and Isolation Techniques on an Unmanned Surface Vehicle (USV) // IFAC Proceedings Volumes. 2012. Vol. 45, no. 27. Pp. 287-292. DOI: https://doi.org/10.3182/20120919-3-IT-2046.00049
[5] Wang Y., Ma X., Qian P. Wind turbine fault detection and identification through PCA-based optimal variable selection // IEEE
Том 14, № 1. 2018 ISSN2411-1473 sitito.cs.msu.ru
Transactions on Sustainable Energy. 2018. Pp. 1-9. DOI: https://doi.org/10.1109/TSTE.2018.2801625
[6] Janssens O. et al. Convolutional neural network based fault detection for rotating machinery // Journal of Sound and Vibration. 2016. Vol. 377. Pp. 331-345. DOI: https://doi.org/10.1016/j.jsv.2016.05.027
[7] Zhang W. et al. A deep convolutional neural network with new training methods for bearing fault diagnosis under noisy environment and different working load // Mechanical Systems and Signal Processing. 2018. Vol. 100. Pp. 439-453. DOI: https://doi.org/10.1016/j.ymssp.2017.06.022
[8] Isermann R. Process fault detection based on modeling and estimation methods — A survey // Automatica. 1984. Vol. 20, issue 4. Pp. 387-404. DOI: https://doi.org/10.1016/0005-1098(84)90098-0
[9] Mohanty R., Pradhan A.K Protection of Smart DC Microgrid with Ring Configuration using Parameter Estimation Approach // IEEE Transactions on Smart Grid. 2017. DOI: https://doi.org/10.1109/TSG.2017.2708743
[10] Wang Z., Shi P., Lim C.C. Н-/Ида fault detection observer in finite frequency domain for linear parameter-varying descriptor systems // Automatica. 2017. Vol. 86. Pp. 38-45. DOI: https://doi.org/10.1016/j.automatica.2017.08.021
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[13] Li H. et al. Fault-tolerant control of Markovian jump stochastic systems via the augmented sliding mode observer approach // Automatica. 2014. Vol. 50, issue 7. Pp. 1825-1834. DOI: https://doi.org/10.1016/j.automatica.2014.04.006
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[17] Veremey E.I. Efficient Spectral Approach to SISO Problems of H2-Optimal Synthesis // Applied Mathematical Sciences. 2015. Vol. 9, no. 79. Pp. 3897-3909. DOI: https://doi.org/10.12988/ams.2015.54335
[18] Veremey E.I. H2-Optimal Synthesis Problem with Nonunique Solution // Applied Mathematical Sciences. 2016. Vol. 10, no. 38. Pp. 1891-1905. DOI: https://doi.org/10.12988/ams.2016.63120
[19] Veremey E.I. Dynamical correction of control laws for marine ships' accurate steering // Journal of Marine Science and Application. 2014. Vol. 13, no. 2. Pp. 127-133. DOI: https://doi.org/10.1007/s11804-014-1250-1
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[22] Veremey E. Irregular Ида-optimization of control laws for marine autopilots. Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of VF Demyanov) (CNSA), 2017. IEEE, 2017. Pp. 1-4. DOI: https://doi.org/10.1109/CNSA.2017.7974028.
[23] Veremey E.I., Knyazkin Y.V. Spectral И2 fault estimation observer design based on allocation of the correction effect // Journal of Theoretical and Applied Information Technology. 2017. Vol. 95, no. 12. Pp. 2776-2782. URL: http://www.jatit.org/volumes/Vol95No12/18Vol95No12.pdf (дата обращения: 10.02.2018).
[24] Spectral И-2 optimal correction of additive fault estimation observer / E.I. Veremey, Y.V. Knyazkin // Proceedings of the 6th Seminar on Industrial Control Systems-Analysis, Modeling and Computation. EDP Sciences, 2016. DOI: https://doi.org/10.1051/itmconf/20160601005
[25] Synthesis of stabilizing control laws with uncertain disturbances for marine vessels / E.I. Veremey, M.N. Smirnov, M.A. Smirnova // Proceedings of 2015 IEEE International Conference «Stability and Control Processes» in Memory of V.I. Zubov (SCP). 2015. Pp. 1-3. DOI: https://doi.org/10.1109/SCP.2015.7342219
Поступила 16.12.2017; принята к публикации 10.02.2018; опубликована онлайн 30.03.2018.
Об авторах:
Веремей Евгений Игоревич, доктор физико-математических наук, профессор, заведующий кафедрой компьютерных технологий и систем факультета прикладной математики-процессов управления, Санкт-Петербургский государственный университет (198504, Россия, г. Санкт-Петербург, Петергоф, Университетский проспект, д. 35); ORCID: http://orcid.org/0000-0001-8366-5654, e_veremey@mail.ru
Князькин Ярослав Вячеславович, аспирант факультета прикладной математики-процессов управления, Санкт-Петербургский государственный университет (198504, Россия, г. Санкт-Петербург, Петергоф, Университетский проспект, д. 35); ORCID: http://orcid.org/0000-0001-8522-588X, yaroslavknyazkin@gmail.com
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