www.ietdl.org Published in IET Control Theory and Applications Received on 22nd January 2011 Revised on 29th July 2012 doi: 10.1049/iet-cta.2011.0057
ISSN 1751-8644
Dynamic observer-based robust control and fault detection for linear systems X.-J. Li1 G.-H.Yang2,3 1 College
of Sciences, Northeastern University, Shenyang, Liaoning 110004, People’s Republic of China of Information Science and Engineering, Northeastern University, Shenyang, Liaoning 110004, People’s Republic of China 3 State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, People’s Republic of China E-mail:
[email protected] 2 College
Abstract: This study is concerned with the design of dynamic observer-based robust controller that also facilitates the acquisition of information used for fault detection (FD) purpose in feedback control systems. Through introducing a weighting matrix, the combination of observer states is utilised to generate a residual signal to detect faults. The first technical contribution is to construct a new linearising change-of-variables that is able to convert the dynamic observer-based controller design problem into linear matrix inequality-based optimisation problem. The second one is to show that the proposed dynamic observer-based controller can achieve a better H∞ performance compared with the existing static (Luenberger) observerbased controller design approaches. Finally, via the simple residual structure, a convex fault detector design condition with some parameter matrices fixed is developed for guaranteeing the H− performance used to measure the fault sensitivity. An F-18 aircraft model is given to show the satisfactory FD performance and control performance.
Nomenclature For a matrix A, AT and A⊥ denote its transpose and orthogonal complement, respectively; He(A) := A + AT ; the symbol ∗ in a matrix represents the symmetric entries; I denotes the identity matrix with an appropriate dimension; σmax G and σmin G denote the maximum and minimum singular values of the transfer matrix G, respectively; a block diagonal matrix with matrices X1 , X2 , . . . , Xn on its main diagonal is denoted as diag(X1 , X2 , . . . Xn ).
1
Introduction
A state observer reconstructs the states of a dynamic system and has important applications in many aspects such as realisation of feedback control, system supervision, and fault diagnosis. Especially, the observer-based controller design theories have been greatly developed in the past two decades. To mention a few, Choi and Chung [1] presented a robust observer-based H∞ control design method for linear time-delay systems with norm-bounded time-varying uncertainty. The observer-based controller designs for linear systems with state perturbations and structural parameter uncertainties were addressed in [2, 3]. Ref. [4] studied fuzzy observer design and H∞ controller design for Takagi-Sugeno (T-S) fuzzy systems. Ref. [5] investigated the observer-based output feedback sliding mode control for Itô stochastic timedelay systems. Ref. [6] presented a pole assignment method IET Control Theory Appl., 2012, Vol. 6, Iss. 17, pp. 2657–2666 doi: 10.1049/iet-cta.2011.0057
to study the observer-based stabilisation of switching linear systems. In all of these works, a common feature is that a constant observer gain is designed to achieve different control specifications. The dynamic observer structure is developed in [7, 8], where the constant observer gain is replaced with a dynamic filter, and the dynamic observer-based control problem has been further investigated in [7]. As only the stability problem of the closed-loop system is considered, the controller gain and the dynamic observer gains in [7] can be designed independently by the separation principle. This paper aims firstly to the dynamic observer-based H∞ controller design, where the separation principle used in [7] does not hold any longer. In order to overcome this difficulty, a new form of change-of-variables to linearise the controller and dynamic observer synthesis conditions will be developed, and the comparison with the existing results shows that the proposed dynamic observer-based controller can achieve a better H∞ performance. On the other hand, the robust FD problem of linear time-invariant (LTI) systems has been considered by many authors. To measure the fault sensitivity, the definition of H− index has been introduced in the full frequency domain [9]. With such defined performance index, the fault detection (FD) observer design in [10] has been formulated as a multiple objective optimisation problem, and there has also appeared a number of results using the H∞ /H− techniques to design fault detector [11–15]. However, faults usually occur in finite-frequency ranges; 2657 © The Institution of Engineering and Technology 2012
www.ietdl.org therefore, some important finite-frequency fault detector design approaches have been further considered in [16–19] by using Generalised Kalman-Yakubovi˘c-Popov (KYP) Lemma [20], and the finite frequency H∞ filter design has also been investigated in [21–24]. Note that, most existing residual generators are designed to be output estimation errors [9, 15, 16, 18, 25, 26], and then the detection of incipient faults in close-loop systems may become difficult [27], in addition, the FD filter design conditions for guaranteeing the H− performance (high-gain performance) are non-convex [13, 15, 19, 28]. In fact, when a fault occurs, an effort made by the controller attempts to minimise the effect of faults on the plant input–output relationship, and such an effort should be reflected in the observer states [29], and therefore, by introducing a weighting matrix, the combination of observer states is utilised to generate a residual to achieve the detection objective in this paper. It should be remarked that this residual structure has been used in [19, 29]. However, Wu [29] did not consider the H− performance and [19] only provided the bilinear matrix inequality (BMI)based FD filter design conditions for guaranteeing the H− performance. Generally, the BMI problems are NP-hard and inefficient. In the next section, by exploring the proposed residual generator of [19, 29], a convex FD system design condition with some parameter matrices fixed is developed for guaranteeing the H− performance. The main contributions of this paper are summarised as follows. First, by introducing a new linearising changeof-variables, the dynamic observer and controller design conditions can be linearised for the case that the separation principle does not hold. Second, it is shown that the proposed dynamic observer-based controller can achieve a better H∞ performance compared with the existing static observer-based controller design approaches. Finally, a convex fault detector design condition with some parameter matrices fixed is developed for guaranteeing the H− performance, which is an improvement for the known non-convex detector design approaches [13, 15, 19, 28]. This paper is organised as follows: in Section 2, the control objectives and design objectives of this work are presented. In Section 3, controller and fault detector design conditions are developed. An example taken from the existing literature is given in Section 4, which is followed by a conclusion in Section 5.
2
In this paper, the linear system to be considered has the following form
(2)
yˆ (t) = C1 xˆ (t)
(3)
ξ˙ (t) = AL ξ(t) + BL (y(t) − yˆ (t))
(4)
η(t) = CL ξ(t) + DL (y(t) − yˆ (t))
(5)
u(t) = K xˆ (t)
(6)
where AL , BL , CL , DL and K are real matrices of appropriate dimensions to be designed, η(t) ∈ Rn is the correction signal, and ξ(t) ∈ Rn is the auxiliary state. Remark 1: When AL = 0, BL = 0, CL = 0 and DL = H , the dynamic observer (2)–(5) is equivalent to the Luenberger observer. Thus, it can be regarded as an extension of the Luenberger observer [7, 25, 30]. In the next section, an example is given to illustrate that the dynamic observerbased controller can achieve better H∞ performance compared with the known Luenberger observer-based controller design approaches. Remark 2: The dynamic observer-based controller architecture (2)–(6) has been provided in [7], where only the stability problem is considered. An important feature of [7] is that the controller parameter K and the observer parameters AL , BL , CL , DL can be designed independently by separation principle. However, it is well known that the separation property does not hold any longer for the H∞ controller design. This paper will address this difficulty through constructing a new form of change-of-variables to linearise the controller and observer synthesis conditions. The detection objective is to generate a residual signal in the closed-loop system to detect faults. It has been pointed out that when the faults occur, in order to attenuate the effect of faults on the performance output, the effort made by the dynamic observer-based controller will be reflected in the observer states. Therefore the combination of observer states is utilised to generate a residual
y(t) = C1 x(t) (1)
where x(t) ∈ Rn is the state space vector, y(t) ∈ Rny and u(t) ∈ Rnu are the measurement output and control input, d(t) ∈ Rnd is the unknown input vector with bounded energy, z(t) ∈ Rnz is the performance output to be controlled, and f (t) ∈ Rnf denotes the actuator fault signal. A, B, Bd , Bf , C1 , C2 , D and Dd are known matrices with appropriate dimensions. The following assumptions on the system model are standard: (A, B) is stabilisable and (A, C1 ) is detectable.
(7)
Remark 3: In [9, 15, 16, 25, 26, 31], the generated residual is designed to be the output estimation error r(t) = y(t) − yˆ (t)
x˙ (t) = Ax(t) + Bu(t) + Bd d(t) + Bf f (t)
2658 © The Institution of Engineering and Technology 2012
x˙ˆ (t) = Aˆx(t) + Bu(t) + η(t)
r(t) = V xˆ (t)
Problem statement
z(t) = C2 x(t) + Du(t) + Dd d(t)
The control objectives are to minimise the effects of disturbances and faults on the performance output z(t). Then a dynamic observer-based controller is chosen to achieve the desired control performance and its architecture is described as [7]
(8)
It will be shown in the simulation section that the proposed residual in (7) can be used to detect the actuator faults in the closed-loop systems; on the contrary, the residual given in (8) is difficult to be applied in the systems under feedback control. Denoting e(t) = x(t) − xˆ (t), x˜ = [xT , eT , ξ T ]T , we obtain the following augmented error system ¯ x(t) + B¯f f (t) + B¯d d(t) x˙˜ (t) = A˜ r(t) = C¯1 x˜ (t)
(10)
z(t) = C¯2 x˜ (t) + Dd d(t)
(11)
(9)
IET Control Theory Appl., 2012, Vol. 6, Iss. 17, pp. 2657–2666 doi: 10.1049/iet-cta.2011.0057
www.ietdl.org 2.1
where ⎡
⎤
A + BK −BK 0 0 A − DL C1 −CL ⎥ ⎦ AL 0 B L C1 ⎡ ⎤ ⎡ ⎤ Bd Bf B¯f = ⎣Bf ⎦ , B¯d = ⎣Bd ⎦ 0 0 ¯ C1 = V −V 0 , C¯2 = C2 + DK
Preliminaries
The following useful lemmas are essential for the later developments.
⎢ A¯ = ⎣
Lemma 1 (elimination lemma [32]): Let , , be given. There exists a matrix F satisfying
−DK
0
F + (F )T + < 0 (12)
if and only if the following two conditions hold
Consider the system described by (9)–(11), the design problem under consideration is to find a controller and a detector such that follows. 1. For control objectives, minimising the effects of the external disturbances and faults on the performance output z(t). Then, for prescribed suitable levels β1 , β2 , the following small-gain specifications can be formulated ∀ | ω |≤ ω¯ 1
(13)
∀ω ∈ (−∞, +∞)
(14)
sup σmax (Gzf (jω)) < β1
T
⊥ ⊥ < 0 ⊥
T T
¯ −1 B¯f Gzf = C¯2 (jωI − A) ¯ −1 B¯ d + Dd Gzd = C¯2 (jωI − A)
∀ | ω |≤ ω¯ 1
(15)
∀ω ∈ (−∞, +∞)
(16)
inf σmin (Grf (jω)) > γ1 ω
sup σmax (Grd (jω)) < γ2 ω
where ¯ −1 B¯ f Grf = C¯1 (jωI − A) ¯ −1 B¯d Grd = C¯1 (jωI − A) denote the transfer functions from f (t) and d(t) to r(t), respectively.
(18)
G(jω)T G(jω) I < 0, I
∀ω ∈ [− , ]
holds, where G(jω) = C(jωI − A)−1 B + D. (ii) There exist matrices P = P T , Q > 0 such that
denote the transfer functions from f (t) and d(t) to z(t), respectively. 2. For detection objectives, minimising the effects of the external disturbances on the residual output r(t), while increasing the sensitivity of faults to the residual output r(t). Then, for prescribed suitable levels γ1 , γ2 , the following high-gain specification and small-gain specification should be satisfied
