Observer-based Guaranteed Cost Fault-tolerant Controller Design for Networked Control Systems

INTRODUCTION

Networked Control Systems are feedback control system wherein the control loops are closed through a real-time network (NCSs) (Zhang et al., 2001; Antsaklis and Baillieul, 2007; Hespanha et al., 2007). Components directly connected to network (including sensor and controller) are regarded as nodes of network. Compared with traditional point-to-point control, networked control systems have more advantages such as shared resources, long range manipulation, low cost and easy of system maintenance. Hence, it has good application prospect. Long range manipulation, long range teaching and experiment, wireless network robot and industrial Ethernet technology can all be ascribed to be control systems based on network. However, because of insertion of the communication network in the feedback control loop, problems of time-delay, data packet dropout and disordered time sequence arise, which can degrade performance of the systems and even breakdown the systems. So, it is significant to do research on the fault-tolerant control for the networked Control Systems.

When time-delay is longer than one sampling period, it is called long time-delay. At present, the research on long time-delays networked control systems is mainly focused on modeling, stability analysis and estimate and control the time-delay (Yang and Fang, 2009; Yang et al., 2006; Zhang and Yu, 2009), etc. However, little attention has been to the study on fault-tolerant control. A kind of networked control systems with random time-delays are modeled as a discrete-time jump linear system with Markov delay characteristics and the actuator failures of networked control systems are analyzed based on jump linear system theory and fault-tolerant control theory (Huo and Fang, 2006). The uncertainty of network-induced delays is converted to the uncertainty of the parameter matrix, the sufficient conditions for closed-loop networked control systems with uncertain disturbance possessing robust integrity against sensor or actuator failures are given and the robust fault-tolerant controller is designed by Li et al. (2007). For a class of networked control systems with time-varying delays, based on the integrity fault-tolerant control theory and the time-delay-dependent stability criteria, the sufficient conditions for systems with integrity against actuator failures are given and the robust fault-tolerant cont roller is designed by Guo et al. (2008). A procedure is proposed for controlling a system over a network using the concept of an NCS-Information-Packet which is an augmented vector comprising control moves and fault flags, then the problem of fault-tolerant control for networked control systems is studied (Klinkhieo et al., 2006). The conclusions of these references are based on state feedback. However, in actual systems, as the complexity of network, the all states of the system are not convenience enough to be obtained. Then the state observer is prone to be used. Therefore it is essential to do research on networked control systems with state observer. The problem of state feedback stabilization for a class of networked control systems based on observer state is investigated (Kong and Fang, 2006). Based on observer state, the fault detection of the system is studied (Bao et al., 2005; Zhu and Zhou, 2006; Lu et al., 2007). However, as so far, the research on fault-tolerant Control for networked control systems with long time-delays based on state observer is rare.

In this study, we aim at solving the problem of integrity against sensor failures for networked control systems with long time-delays. According to possible sensor failures, the state observer of the system is designed and an augmented mathematic model for the networked control systems based on state observer is developed. Then the cooperative design approach of the controller and the observer is given and the existence conditions of guaranteed cost fault-tolerant control law are testified by use of the Lyapunov stability theory combined with bilinear matrix inequalities. Furthermore, the designs for guaranteed cost fault-tolerant control law and the state observer are presented.

PROBLEM FORMULATION

Observer-based networked control systems studied in this study is shown in Fig. 1. Where, the sensor and controller are regarded as the same node. Network only exists between controller and actuator, then the system has control time-delay τ.

We assume:

•	The time-delay τ is time varying and τ ε (0, T), where τ Z+

From Huang et al. (2008), the time-delay τ can be represented by τ = dT+Δτ_k≤�T, 0≤Δτ_k = h/2+Δτ≤h, where, h/2 is the mean and constant time delay; -h/2≤Δτ≤h/2 is the uncertain and time-vaying delay, including the offset of clock synchronization. d≤� = [τ^wc/T], where, [Δ] is the least integer larger than Δ, τ^wc is the worst case time-delay and known.

Assume that the process to be monitored is an LTI system described by:

(1)

Fig. 1:	Structure diagram of the observer-based networked control systems

where, x(t) ε Rⁿ, v(t) ε R^t and y(t) ε R^P are state vector, input vector, measure output vector. A_f, B_f and C_f are known matrices of compatible dimensions.

Then the discrete-time NCS model with delays is obtained as follows:

(2)

where, u(k) ε R^l denote the control output vector.

According to Xie et al. (2009) , we obtain

(3)

(4)

and

(5)

where, B, D and E are constant matrices determined by eigenvalues and eigenvectors of matrix A_f and B_f, F(τ_k) is an uncertain matrix satisfying F^T(τ_k)F(τ_k)≤I marked by F.

According to the system Eq. 2, the observer is designed as follow:

(6)

where, L ε R^nxp and denote the state observer gain matrix and the state estimate, respectively.

Aiming at system Eq. 2, using state feedback control law based on observer as follows:

(7)

where, K ε R^lxn.

In order to formulate the possible sensor failure faults, the fault model must be established first. Considering possible sensor failure faults, we can introduce a switched matrix H to the system (2) and lay the matrix M before the measure output vector y(k), where, H = diag(h₁, h₂,...,h_m) and for I = 1,2,...,m

Then the observer equation becomes:

(8)

Define the estimation error:

(9)

From Eq. 2, 8 and 9, we can obtain:

(10)

Let an augmented vector:

(11)

From Eq. 2, 7 and 10, the augmented system based on observer can be represented by:

(12)

Where:

Associating with the system (12), we define the following cost function:

(13)

where, Q and R are given positive-definite symmetric matrices.

Definition 1: Consider the system (12), If there exist a control law based on the observer and a scalar J' such that the overall networked control systems is asymptotically stable with H ε Ω and the closed-loop value of the cost function (13) satisfies J≤J', then J' is said to be a guaranteed cost and u(k) is said to be a guaranteed cost fault-tolerant controller based on the observer for the system (12).

For the convenience of notations, (*) is denoted as an ellipsis for terms that are induced by symmetry in the rest of this study.

MAIN RESULTS

Lemma 1 Zhang et al. (2001) If A, P and Q, are finite-dimension constant matrices, then Q = Q^T, P = P^T>0 we have:

Lemma 2 Zhang et al. (2001) Given matrices W, M, N of appropriate dimensions and with W symmetric, then:

For all F(k) satisfying F^T(k)F(k)≤I, if and only if there exists a scalar ε>0 such that:

Lemma 3 Lu et al. (2007) Given vectors a, b and matrices ú, X, Y, Z of appropriate dimensions and with X, Z symmetric, then:

(14)

we have

Theorem 1: Consider the system (12), if there exist symmetry positive-definite matrices P₁, S₁, S₂, symmetry matrix X, matrices K, L, Y, P₂, P₃, such that:

(15)

(16)

then the system (12) is asymptotically stable and the associated cost function satisfies J≤J', where:

(17)

have been defined above, δ(l) = w(l+l)-w(l), = [K -K 0 0],

Proof: Defining the Lyapunov function:

where:

From

we can obtain

(18)

The differentiate of V₁(k) satisfies the relation:

Let ξ^T(k) = [w^T(k) δ^T(k)] and from Eq. 18, we have:

Where:

Then, for any matrices X, Y and S₂ satisfying Eq. 14, based on Lemma 3, we can obtain:

The differentiate of V₂(k) and V₂(k) satisfy the relation as follows, respectively:

Therefore:

Let θ^T (k) = [ξ^T(k) w^T(k-�_ca)], we have:

Where

From Eq. 15, we can obtain:

(19)

Obviously, ΔV(k)<0. That means when inequalities Eq. 15 and 16 is satisfied, the system (12) remains asymptotically stable.

Now we shall prove that cost function is less than or is equal to J', where J' is defined in Theorem 1. From Eq. 19, we have:

(20)

That means:

(21)

By summing the inequality Eq. 16 from 0 to ∞, then:

(22)

Because the system is asymptotic stability, then

So, from Eq. 22, we can obtain:

This completes the proof of theorem 1.

It is noted that the inequalities of theorem 1 are not linear matrix inequality. In order to obtain the controller gain L and the observer gain L, the inequalities of theorem 1 can be transformed to bilinear matrix inequality, which can be solved with MATLAB and PENBMI (Kocvara and Stingl, 2004).

Theorem 2: Consider the system (12), if there exist a constant scalar μ>0, a scalar ε>0, symmetry positive-definite matrices W₁, þ₁, þ₂, matrices K, L, W₂, W₃, W₃, X₁, X₂, X₃, such that:

(23)

(24)

then the system (12) is asymptotically stable and the associated guaranteed cost J' satisfies (17), where , have been defined above,

Proof: From (15), we can obtain:

In light of Lemma 2 and using the Lemma 1, the inequality Eq. 13 can be equivalently rewritten as:

(25)

Pre-and post-multiplying both sides of Eq. 25 with

and its transpose, we have:

(26)

From Chen et al. (2003), let

where, μ is a constant scalar. For

we have:

(27)

Let

and using the Lemma 1 repetitiously, from Eq. 26 and 27, we can obtain (23).

Similar to the proof process of Eq. 23, Pre-and post-multiplying both sides of Eq. 25 with

we can obtain system (24).

This completes the proof of theorem 2.

The upper bound system (17) of guaranteed cost depends on the initial values of the system (12), which is inconvenient for the parameterized design of controller and observer. In order to delete the dependence, the initial values of the system (12) are assumed arbitrarily but satisfy the following condition:

(28)

where, U is given matrix.

Then finding the minimum of this upper bound can be formulated into an generalized eigenvalue minimization problem subject to linear matrix equality constraints. So, we can obtain:

That means

where, α, β and γ are scalars.

For W₁>0, we have:

Then the guaranteed cost fault-tolerant controller and observer gain and the minimization guaranteed cost can be obtained by solving the following minimization problem with MATLAB and PENBMI:

SIMULATION EXAMPLE

Consider a system described by (1), where:

assume that the initial conditions are T = 0.1s, d = 3, Δτ_k ε [1,1.1], x(0) = [0.8 -0.5]^T, Q = I, R = 0.1I, U = I, μ = 1.

Fig. 2:	Zero-input response of state x1

Fig. 3:	Zero-input response of state x2

From (3) and (4), based on Zhang et al. (2001), we have:

By solving the minimization problem with MATLAB and PENBMI, we can obtain:

In cases of sensor normal and possible failures, the switched matrices L₀ = diag(1,1), L1 = diag(0, 1) and L₂ = diag(1, 0) indicate actuator normal and sensor 1, 2 failure, respectively. In the case of L₀, L₁, L₂, zero-input response of state x₁, x₂ are shown in Fig. 2 and 3. The curves of zero-input response state x₁, x₂ in Fig. 2 and 3 illustrate that the networked control systems against possible sensor failure faults is asymptotically stable. It reveals that the presented method makes the networked control systems possess integrity againstsensor failures and the minimization guaranteed cost is J' = 1.2576x10³.