Operator-based robust nonlinear control and fault detection for a Peltier actuated thermal process

Mathematical and Computer Modelling 57 (2013) 16–29

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Operator-based robust nonlinear control and fault detection for a Peltier actuated thermal process Shengjun Wen, Mingcong Deng ∗ Graduate School of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Nakacho, Koganei, Tokyo 184-8588, Japan

article

info

Article history: Received 20 December 2010 Received in revised form 30 May 2011 Accepted 7 June 2011 Keywords: Operator Robust nonlinear control Fault detection Uncertainty Peltier device

abstract In this paper, operator-based robust nonlinear control design scheme and fault detection technique using robust right coprime factorization approach are considered for a Peltier actuated thermal process. The Peltier actuated thermal process is a typical nonlinear control affine system, where the process temperature depends not only on input current, but also on the square of the input current. In addition, the real system is likely to contain a fault signal owing to various factors. As a result, it is difficult to stabilize the control system and to achieve high accurate control performance, and to detect the fault signal to guarantee the engineering safety when a fault occurs. In this paper, for pursuing robust stability and tracking performance of the nonlinear closed-loop control system, a design scheme of operator-based robust tracking control system is proposed by using robust right coprime factorization approach. Then, for the tracking operator control system, a fault detection design technique based on robust right coprime factorization approach is also investigated. Finally, simulation and experimental results are given to confirm the effectiveness of the proposed method. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction In system control engineering, most of the controlled systems are nonlinear dynamic systems, and its exact mathematical model cannot be derived. Besides, a real system is likely to contain a fault signal owing to various factors, including sensor fault, actuator fault and process fault [1–4]. For example, a Peltier actuated thermal process is a typical nonlinear control affine system, where the process temperature depends not only on the input current, but also on the square of the input current. And the temperature of the endothermic side and radiation side are unknown which are decided by the characteristics of the Peltier device and affected by the heat sink such that it is difficult to get the exact model [5]. In addition, aged electronic component may lead to an actuator fault. For this kind of system, using conventional designs to stabilize the control system and to achieve high accurate control performance is difficult. Especially, when a fault occurs, the control performance of the system may be reduced, even the control system is unstable. As a result, how to stabilize the control system, and to optimize the control performance and to detect the fault signal to guarantee the engineering safety are significant issues. How to apply existing linear methods for nonlinear processes has been studied by some researchers primarily because many effective analyses and design techniques exist for linear time-invariant systems. In general, to use linear control techniques, the nonlinear processes need to be approximated to obtain the linearization model, such as transfer functions [6]. It may lead to inaccuracy, complex implementation, consumption of excessive energy, and thereby increase the cost in effort to retain the robust stability. Therefore, nonlinear control design methodologies have been considered. Until now, there exist

∗

Corresponding author. Tel.: +81 42 388 7134; fax: +81 42 388 7134. E-mail addresses: [email protected] (S. Wen), [email protected] (M. Deng).

0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.06.021

S. Wen, M. Deng / Mathematical and Computer Modelling 57 (2013) 16–29

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Fig. 1. Peltier actuated control system.

some well-developed nonlinear control approaches for the nonlinear processes, such as gain scheduling method, feedback linearization method, Lyapunov redesign method, backstepping technique, sliding mode control theory and so on [6–8]. However, most of the techniques for nonlinear design assume that measurements of all states are available [8]. As a matter of fact, in practical control problems, usually not all states are measurable due to economical or technical reasons. Hence, the existing control design methods often cannot generate satisfactory performance. As for the fault detection problem, a large number of interesting design methods have been considered. In general, there exist two main research approaches. One is a mechanical method, that is, a number of sensors are used to depict the abnormal signal by detecting whether the present sensory data deviate from the standard value or not [1]. However, this approach has a large number of costs to detect all sensors’ data and to monitor them. Another is an analytical method by using measurable information of the process, such as knowledge-based detection method [4]. However, these known analytical method have its own advantages and disadvantages. For example, the main problem with the fault detection method which relies on the process’ mathematical model is that, for the Peltier actuated thermal process, detailed modeling is difficult [5]. To resolve the above problem, operator-based robust right coprime factorization approach is considered. Operator theory is an advanced control theory based on an idea that a signal in the input space is mapped to the output space, and some researches in regards to the theory have been conducted [9–15]. The robust stability of the perturbed nonlinear process was studied by Chen and Han [9]. Based on the design scheme, a robust tracking control system is proposed by Deng et al. [11], and the perturbed signal does not affect the process output. So, the operator-based control method can be applied to a broader class of nonlinear processes with perturbations. The advantage of the operator-based control is that the input–output time function model given by basic physical rules from the real system is adopted such that the approximation of the real system is avoided. It is also easy to ensure the stability of the nonlinear systems by using a Bezout identity. Especially, the robust stability against perturbations can be guaranteed under an inequality condition [11]. Moreover, it seems useful to establish signal-processing-based analytical approach to analyze and design the nonlinear fault detection system [2–4,8]. As a result, an operator-based robust right coprime factorization approach is applied to design robust nonlinear controller and detect fault signals for a Peltier actuated thermal process. In this paper, the thermal process is modeled as a right coprime factorization description. For obtaining robust stability and tracking performance of the nonlinear closed-loop control system, a design scheme of operator-based robust tracking control system is proposed by using robust right coprime factorization approach. Then, for the tracking operator control system, the fault signal on an actuator is analyzed by using three sorts of Bezout identities. Simulation and experimental results are represented to support the designed control system and fault detection system. The contents of this paper are written as follows. Firstly, a nonlinear control system based on Peltier actuated thermal process is described. And the model of the Peltier actuated thermal process is set up. Next, an operator-based robust tracking control system is designed by robust right coprime factorization method. Following this, a fault detection design scheme based on robust right coprime factorization approach is investigated. Further, simulation and experimental results are given to confirm the proposed method. Finally, we draw the conclusion on our current. 2. Peltier actuated control system and problem statement 2.1. Peltier actuated control system The control system on an aluminum plate thermal process with a Peltier actuated device is shown in Fig. 1 [5]. This control system is composed of a Peltier actuated part, serial communication (RS232C) interface part and computers. Peltier

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Fig. 2. Photo of Peltier actuated part.

Fig. 3. Configuration of Peltier actuated thermal process.

actuated part includes Peltier device, temperature sensor and aluminum plate. Peltier device is used to cool the aluminum plate, temperature sensor measures temperature of the aluminum plate. The serial communication interface is a kind of communication interface between the personal computer and the microcomputer. The controller is placed in the personal computer and the microcomputer is responsible for A/D convert and producing PWM control signal. Fig. 2 shows the photo of Peltier actuated part. Peltier device and temperature sensor are installed on both sides of aluminum plate respectively. On the heat radiation side of the Peltier device, a heat sink is installed to prevent it from heating too much. The configuration of the aluminum plate thermal process with a Peltier device is shown in Fig. 3, where S3 is a Peltier device and there exists a sensor for measuring temperature of the aluminum plate on the other side. 2.2. Modeling on Peltier actuated thermal process To obtain the mathematical model of the Peltier actuated thermal process, the following six thermal transfer mechanisms are considered. (1) Fourier’s law concerning thermal conduction q = −λ

dT

(1)

dx

where q: Heat flux (W/m2 ), λ: Thermal conductivity (W/mK), dT /dx: Temperature gradient (K/m). (2) Newton’s cooling law q = α(T0 − Tx )

(2)

where α : Heat transfer coefficient of air (W/m K), T0 : Environment temperature (K) and Tx : Object temperature (K). (3) Specific heat capacity equation 2

dQJ = mcdT

(3)

where dQJ : Variation of heat capacities (J), c: Specific heat coefficient (J/kg K), m: Mass (kg) and dT : Variation of temperature.

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Table 1 Parameters of the aluminum plate thermal process. 900 (J/kg K) 15 (W/m2 K) 238 (W/m K) 0.095 (m)

c

α λ d1

S1 S2 S3 S4

6.8 × 10−3 3.4 × 10−4 9.0 × 10−4 5.0 × 10−4

(m2 ) (m2 ) (m2 ) (m2 )

K Rp m Sp

0.63 (W/K) 4.2 () 0.195 (kg) 0.053 (V/K)

(4) Electrothermal amount by Peltier effect Q = Sp Ti

(4)

where Q : Thermal value (W), Sp : Seebeck coefficient (V/K), T : Temperature of the endothermic side (K) and i: Current (A). (5) Thermal conduction by temperature difference Q = K ∆T

(5)

where K : Peltier’s thermal conductivity (W/K) and ∆T : Temperature difference (K). (6) Joule heat by current Q = Rp i2

(6)

where Rp : Peltier’s resistance (Ω ). In the experimental system, middle part temperature of aluminum plate is Tx measured by sensor and temperature of two flanks S4 is T0 which equals to environment temperature. According to Fourier’s law, conductive heat from middle part to other two parts is Q1 = −2λS4 (T0 − Tx )/d1 , where S4 denotes the flank area and d1 the length of other two parts shown in Fig. 3. We calculate convective heat Q2 = −α(T0 − Tx )(2S1 + 2S2 − S3 ) from aluminum plate to environment by Newton’s cooling law, where S1 , S2 and S3 denote the corresponding area of aluminum plate in Fig. 3. Based on Eqs. (4)–(6), the total endothermic heat is taken away from aluminum plate by Peltier device is Q3 = Sp T1 i − K (Th − T1 ) −

1 2

Rp i2

where Th and T1 are the radiation side temperature and endothermic side temperature of Peltier device, respectively. In the above equation, Sp T1 i is the heat amount that transfers from the endothermic side to the radiation side by Peltier effect, K (Th − T1 ) shows the movement of heat by temperature difference in the two side of Peltier device, and Rp i2 /2 is Joule heat generated by the input current to Peltier device. Thus, the total heat transferred from aluminum plate is Q1 + Q2 + Q3 without considering thermal radiation (it is very small), which equals to specific heat consumption caused by variation of temperature. By specific heat capacity equation, we have specific heat consumption Q4 =

d(T0 − Tx )mc dt

.

As a result, a differential equation in regards to thermal transfer is obtained as follows [5]. d(T0 − Tx )mc dt

1

2λS4 (T0 − Tx )

2

d1

= Sp T1 i − K (Th − T1 ) − Rp i2 − α(T0 − Tx )(2S1 + 2S2 − S3 ) −

(7)

where parameters of the aluminum plate thermal process are given in Table 1. In Eq. (7), T0 − Tx is defined as process output y(t ) and the total endothermic amount Q3 is defined as control input ud (t ). Then, the model of the thermal process is reexpressed as the following form. y(t ) =

1 −At e cm



eAτ ud (τ )dτ

(8)

where A=

α(2S1 + 2S2 − S3 ) + cm

2λS4 d1

.

2.3. Problem statement and definitions of operator It is clear that the Peltier actuated thermal process is a typical nonlinear control affine system, where the output temperature depends not only on i, but also on i2 . In addition, T1 and Th are unknown which are decided by the characteristics of the Peltier device, and the temperature of radiation side Th is affected by the heat sink [5,13]. The relationship between T1 , Th and i is obtained shown in Eqs. (9) and (10) based on the result in [5], where the related parameters are given in Table 2.

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S. Wen, M. Deng / Mathematical and Computer Modelling 57 (2013) 16–29 Table 2 Calculation parameters of the radiation side and endothermic side temperature. k1 = 5.22 k7 = 6.96

k2 = 16.3 k8 = 15.316

k3 = 0.3 k9 = 0.1

k4 = 12.0 k10 = 6.86

k5 = 26.0 k11 = 15.1

k6 = 0.15 k12 = 0.1

Namely, there exist some unmodeled uncertainties. Moreover, the real engineering control system is likely to contain a fault signal owing to various factors, such as actuator fault caused by aged electronic component. T1 = (k1 + k7 )i + k2 e−k3 t − k8 e−k9 t

(Th − T1 ) = (k4 + k10 )i + k5 e

−k6 t

(9)

− k11 e

−k12 t

.

(10)

For the Peltier actuated thermal process, using conventional designs to achieve high accurate control performance will be difficult for the system with the appearance of i2 , uncertainties and faults. Here, operator-based robust right coprime factorization approach is considered. The advantage of the operator-based control is that the input–output time function model is adopted such that the approximation of the real system is avoided. It is easy to ensure the stability of the nonlinear systems by using a Bezout identity. Especially, the robust stability against uncertainties can be guaranteed under an inequality condition [11]. Also, it seems useful to analyze and design the nonlinear fault diagnosis system. Some basic definitions and notations about operator and right coprime factorization are introduced as follows. Let U and Y be linear spaces over the field of real numbers, and let Us and Ys be normed linear subspaces which is called the stable subspaces of U and Y , respectively. An operator is a mapping between the two spaces. For example, Q : U → Y is an operator mapping from U to Y . Assume that D (Q ) and R(Q ) are the domain and range of Q . If the operator Q : D (Q ) → Y satisfies the rule Q : ax1 + bx2 → aQ (x1 ) + bQ (x2 ), where x1 , x2 ∈ D (Q ) and a, b ∈ C , then Q is said to be linear, otherwise, it is said to be nonlinear. Let Z be the family of real-valued measurable functions defined on [0, ∞), which is a linear space. For each constant T ∈ [0, ∞), let PT be the Projection operator mapping from Z to another linear space, ZT , of measurable functions such that fT (t ) := PT (f )(t ) =



f (t ), 0,

t ≤T t >T

(11)

where fT (t ) ∈ ZT is called the truncation of f (t ). And let U e and Y e be two extended linear spaces, which are associated respectively with two given Banach spaces UB and YB of measurable functions defined on the time domain [0, ∞). Then, the operator Q : U → Y is said to be bounded input bounded output (BIBO) stable or, simply, stable if Q (Us ) ⊆ Ys and domain D (Q ) ⊆ U e and range R(Q ) ⊆ Y e [10]. In this paper, nonlinear cases are considered and stable operators always mean BIBO stable. Definition 1 ([10]). Let ϕ(U , Y ) be the set of stable operators from U to Y . Then ϕ(U , Y ) contains a subset defined by

U(U , Y ) = {L : L ∈ ϕ(U , Y )}, where L is invertible with L−1 ∈ ϕ(Y , U ). Elements of U(U , Y ) are called unimodular operators. Definition 2 ([10]). A nonlinear operator Q : G → Y e is called a generalized Lipschitz operator on G which is a subset of U e if there exists a constant B such that

  [Q (x)]T − [Q (˜x)]T  ≤ B∥xT − x˜ T ∥U Y

(12)

for all x, x˜ ∈ G and for all T ∈ [0, ∞). Note that the least constant B is given by the norm of Q with

∥Q ∥Lip := ∥Q (x0 )∥Y + sup

sup

T ∈[0,∞) x,˜x∈G xT ̸=x˜ T

  [Q (x)]T − [Q (˜x)]T  Y ∥xT − x˜ T ∥U

(13)

for any fixed x0 ∈ G. A nonlinear feedback control system with a given plant P is shown in Fig. 4. The plant operator P is said to have a right factorization composed of two operators N and D in the form P = ND−1 , where operators N and D−1 can be either linear or nonlinear. Suppose P is a nonlinear operator, for instance, both N and D are nonlinear operators in general. If, furthermore, the two operators N and D together satisfy Bezout identity SN + RD = L for some operators S and R, where L is unimodular operator, then the right factorization is said to be coprime [10]. For the given plant P, suppose that there is perturbations ∆P, where ∆P is bounded. The right factorization of the perturbed plant can be described as P + ∆P = (N + ∆N )D−1 . Right coprime factorization design of the perturbed plant is shown in Fig. 5. If the right coprime factorization of the perturbed plant P + ∆P satisfies SN + RD = L and ∥(S (N + ∆N ) − SN )L−1 ∥ < 1, then the perturbed feedback control system is stable, where ∥ · ∥ is Lipschitz operator norm and L is unimodular operator [11].

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Fig. 4. A nonlinear feedback control system.

Fig. 5. A nonlinear feedback control system with uncertainties.

For the Peltier actuated thermal process, the objective of the paper is to show how the operator-based robust right coprime factorization approach can be utilized to ensure robust stability and achieve satisfactory tracking performance. Also, how to detect the actuator fault signal in the tracking operator control system by using robust right coprime factorization approach is another objective. 3. Operator-based robust nonlinear control Consider the aluminum plate thermal process with Peltier device described by the right factorization P = ND−1 . From the above mentioned model, right factorization operators N and D−1 can be obtained and N and D are stable operators. The right factorization operators of the nominal process are shown as follows D(w)(t ) = cmw(t ) N (w)(t ) = e−At



(14) eAτ w(τ )dτ .

(15)

In real system, T1 and Th are unknown which are decided by the characteristics of the Peltier device and affected by the heat sink. That is, there is a bounded perturbation ∆P caused by unmodeled uncertainties and noise. We have right factorization of the perturbed process P + ∆P = (N + ∆N )D−1 , where ∆N concerning with the uncertainties and noise is bounded and N + ∆N is stable shown as follows.

(N + ∆N )(w)(t ) = (e−At + ∆1 )



eAτ w(τ )dτ

(16)

where ∆1 is regarded as perturbation concerning with the uncertainties and noise. For the Peltier actuated thermal process, operator-based tracking control system shown in Fig. 6 is considered. To control the thermal process, controllers S and R are designed to satisfy the Bezout identity SN + RD = I. Then, N and D are said to be right coprime factorization if there exist two stable operators S and R satisfying the Bezout identity, where R is invertible and I is an identity operator. In the same manner, under the perturbed condition, we assume that there exist S and R satisfying the perturbed Bezout ˜ Therefore, the perturbed process retains a right coprime factorization. If the conditions (17) identity S (N + ∆N ) + RD = L. and (18) are satisfied, then the stability of the perturbed nonlinear feedback control system is guaranteed. That is, these conditions imply that suitable design of controllers R and S will guarantee the robust stability of the controlled process. SN (w)(t ) + RD(w)(t ) = I (w)(t ) ∥(S (N + ∆N ) − SN )(w)(t )∥ < 1.

(17) (18)

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Fig. 6. Operator-based tracking control system.

Fig. 7. The equivalent diagram of Fig. 6.

Finally, the right coprime factorization controllers are R and S are designed shown in (19) and (20). R(ud )(t ) =

B cm

ud (t )

(19)

 dya (t ) + Aya (t ) . S (ya )(t ) = (1 − B) 

dt

(20)

It is worth to mention that the initial state should also be considered, that is, SN (w 0, t0) + RD(w 0, t0) = I (w 0, t0) should be satisfied. In this paper, we select the initial time t0 = 0 and w 0 = 0. Furthermore, based on the result in [14], an output tracking operator M can be designed, where M is a designed stable operator. Since robustly stable condition is satisfied, the effects from perturbation cannot be transmitted back to error signal e. This shows that Fig. 6 can be rewritten as Fig. 7, and the effects are removed by the presented robust control system. From Fig. 7, we design tracking operator M by considering the uncertain part ∆N. That is, based on the condition

(N + ∆N )L˜ −1 M (r )(t ) = I (r )(t )

(21)

the process output ya (t ) tracks to reference signal r (t ), where ya (t ) is observable. 4. Operator-based fault detection From the model of the Peltier actuated thermal process, we can see that the model is an obvious nonlinear control affine system. The output temperature depends not only on i, but also on i2 . It is different from the aluminum plate thermal system considered in [3,13]. In this system, the control signal (endothermic amount) is described as ud = Sp T1 i − K (Th − T1 )− Rp i2 /2. So, the fault detection will be difficult for the system with the appearance of i2 . In this paper, when fault occurs, if the Bezout identity (17) and the inequality condition (18) are still retained, then the fault system is stable. Therefore, a system of robust fault detection in regards to the tracking operator M is designed shown in Fig. 8. To detect fault signal, firstly three operators R0 , S0 and D represented in Fig. 8 are designed. In Fig. 8, three sorts of Bezout identities are satisfied shown as follows

(SN + RD)(w)(t ) = I (w)(t )

(22)

(S (N + ∆N ) + RD)(w)(t ) = L˜ (w)(t ) (S0 N + R0 D)(w)(t ) = I (w)(t ).

(24)

(23)

For the Peltier actuated thermal process, R0 and S0 are designed as S0 (ya )(t ) = (1 − K0 ) R0 (ud )(t ) =

K0 cm



ud (t ),

dy(t ) dt

+ Ay(t )

 (25) (26)

S. Wen, M. Deng / Mathematical and Computer Modelling 57 (2013) 16–29

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Fig. 8. Detailed fault detection system.

Fig. 9. Simulation result without fault.

where K0 is a gain for fault detection. For the case of ya being observable, sum of the output of S0 and that of R0 is a mapping from space W to U shown in Fig. 8. That is, the sum u0 is represented as follows u0 = R0 (ud )(t ) + S0 (ya )(t )

(27)

where ud = Sp T1 id − K (Th − T1 ) − /2 denote the real endothermic amount and id denote the real input current. Under a fault caused by input current, the real endothermic amount ud becomes Rp i2d

ud = R−1 (e)(t ) + uf

(28)

where R (e)(t ) = Sp T1 idf − K (Th − T1 ) − /2 is the computed endothermic amount, idf the computed input current and uf is the endothermic amount with respect to the fault signal if . The Bezout identity (24) is satisfied, which implies that signal w equals to u0 because the Bezout identity is a mapping from space W to U. For operator D, it means a mapping from signal w to ud , seeing from Fig. 8. Therefore, the predictive endothermic amount yd equals to the real endothermic amount ud after being affected by fault as long as w is equivalent to w0 , that is, yd = ud = Sp T1 id − K (Th − T1 ) − Rp i2d /2. However, connecting u0 to D directly is impossible due to the difference of spaces that u0 and w0 belong to. One way to solve this problem is to design a space-change operator L mapping from space U to W such that −1

L(R0 D + S0 N ) = I .

Rp i2df

(29)

Then, predictive value idp of the real input current is obtained. Eventually, the difference between the computed input current idf (before being affected by the fault) and the predictive input current idp (after being affected by the fault) is derived. In

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S. Wen, M. Deng / Mathematical and Computer Modelling 57 (2013) 16–29

Fig. 10. Original input current without fault.

Fig. 11. Experimental result without fault.

other words, the fault signal can be detected by Detected fault signal = abs (idf − idp ).

(30)

If there are no actuator faults, then the fault signal is 0. 5. Simulation and experimental results In this section, the considered control system design scheme is confirmed by simulation and experiment. The used parameters in the simulation and experiment are shown in Table 3. 5.1. Robust nonlinear control results Firstly, simulation result of robust nonlinear control is given. In the case of using Peltier device, the process output is temperature and the process input is the current flow. The initial temperature is 21.3 °C in simulation. Under the reference input r = 3.0, the desired temperature is 18.3 °C. Considering the real application, simulation is given by limiting the input

S. Wen, M. Deng / Mathematical and Computer Modelling 57 (2013) 16–29

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Fig. 12. Robust stability analysis for the non-fault system.

Fig. 13. Simulation result with fault.

Table 3 Parameters for simulation and experiment. Reference input Design parameter Gain for fault detection Simulation and experimental time Sampling time

r = 3.0 B = 0.1 K0 = 0.95 600 (s) 100 (ms)

current between 0.0 (A) and 2.2 (A). Simulation result on the aluminum plate is shown in Fig. 9, where the design parameter B = 0.1. From Fig. 9, the Peltier actuated temperature control is effective. In experimental system with Peltier device, where the desired temperature is also 18.3 °C. Peltier device is driven by PWM signal, namely, the control current is PWM signal shown in Fig. 10. For the sake of comparison with simulation result, we denote that by using effective value in Fig. 11, including the process input and output. The error of the temperature output is less than 0.1 °C. It is clear that the desired temperature control result has been obtained. The current in simulation

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S. Wen, M. Deng / Mathematical and Computer Modelling 57 (2013) 16–29

Fig. 14. Original input current with fault.

Fig. 15. Experimental result with fault.

(about 0.7 (A) after 100 s) and in experiment (about 1.5 (A) after 400 s) is different because the endothermic side temperature and radiation side temperature of the Peltier are affected by the heat sink and the radiation transfer heat of the aluminum is not considered. Moreover, according to the experimental data, the robust stability is also analyzed shown in Fig. 12. In Fig. 12, the result of the robust stability analysis is described by a time sequence and the ordinate axis denotes robust stability magnitude which is calculated by using the robust stability condition (18). To get the robust stability magnitude, S (N + ∆N )(w)(t ) and SN (w)(t ) denote the real measuring feedback signal and the calculating feedback signal based on the model, respectively. The norm is the generalized Lipschitz norm defined in Definition 2 and the initial value is 0. When the robust stability magnitude is less than 1, the robust stability condition (18) is satisfied, namely, the control system is stable. However, under the effects of the perturbations, there exists a small variation for the output temperature from Fig. 11. Differential operator in controller S leads that the robust stability magnitude looks similar to impulsive data. From Fig. 12, all robust stability magnitude is less than 1, which shows that the robust stability of the control system is guaranteed by using the proposed method.

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Fig. 16. Simulation result of fault detection.

Fig. 17. Experimental result of fault detection.

5.2. Fault detection results In the system, the actuator fault is considered, that is, there exist a fault for the actuator output. Here, assume that the real endothermic amount is fixed ud = 10 (W) when 300 s < t < 350 s. Both initial temperatures are set for 21.3 °C for simulation and experiment. When the reference inputs are r = 3, both desired temperatures are 18.3 °C. The input current is limited between 0.0 (A) and 2.2 (A). The fault detection operators R0 and S0 are designed according to Eqs. (25) and (26), where the gain for fault detection K0 = 0.95. Simulation and experimental results of the process input and output are shown in Figs. 13 and 15, respectively. Fig. 14 is original PWM signal of the input current. From Figs. 13 and 15, the process output tracks to the desired temperature before the fault signal happened. When 300 s < t < 350 s, the process input id is affected by the fault signal. According to the simulation result (see Fig. 13), the process input current (0.35 (A)) is smaller than the value before the fault happened such that the process temperature is increased. In Fig. 15, the process temperature also increases during the fault. But, the process input current (0.75 (A)) is not equal to that of simulation because the endothermic side temperature and radiation side temperature of the Peltier are affected by the heat sink and the radiation transfer heat of the aluminum is not considered. After the actuator fault is removed, the temperature is back to the desired temperature gradually.

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S. Wen, M. Deng / Mathematical and Computer Modelling 57 (2013) 16–29

Fig. 18. Robust stability analysis for the fault system.

Figs. 16 and 17 show the predictive endothermic amount yd , predictive input current idp and the comparison of real fault signal and detected fault signal. Before and after the fault, the real fault signal is zero. During the fault, for the real endothermic amount is assumed as ud = 10 (W) such that the real fault signal is idf − 0.35 (A) (for the simulation) and idf − 0.75 (A) (for the experiment), respectively. The Bezout identity (24) is satisfied, which implies that signal u0 equals to w . By the operators L and D, a map from u0 to ud is obtained. Therefore, yd equals to the real endothermic amount affected by fault, that is, yd = ud and idp is derived. Then, the fault signal can be detected by using abs(idf − idp ). These results show the effectiveness of the designed fault detection system. The robust stability is analyzed by using the experimental data like that without fault shown in Fig. 18. The result is expressed by a time sequence. From Fig. 18, it shows that the robust stability magnitude is less than 1 during the fault. Namely, the result shows that the control system with fault is also robust stable. 6. Conclusions In this paper, operator-based robust nonlinear control design scheme and fault detection technique for a Peltier actuated control system is presented by using robust right coprime factorization approach. The mathematical model of the controlled system is derived, which shows the Peltier actuated thermal process is a typical nonlinear process. To achieve high control accuracy and stabilize the control system, operator-based robust nonlinear control for the control system is presented by using robust right coprime factorization approach. Also, for checking the stable tracking system, a fault detection design technique based on robust right coprime factorization approach is investigated. Simulation and experimental results on nonlinear control and fault detection are given to confirm the proposed method. References [1] J. Korbicz, J.M. Koscielny, Z. Kowalczuk, W. Cholewa, Fault Diagnosis, Models, Artificial Intelligence, Applications, Springer, Berlin, 2004. [2] M. Deng, A. Inoue, K. Edahiro, Fault detection in a thermal process control system with input constraints using robust right coprime factorization approach, IMechE, Part I: Journal of Systems and Control Engineering 221 (2007) 819–831. [3] M. Deng, A. Inoue, K. Edahiro, Fault detection system design for actuator of a thermal process using operator based approach, ACTA Automatica Sinica 36 (2010) 421–426. [4] P.M. Frank, Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy—a survey and some new results, Automatica 26 (2008) 459–474. [5] M. Deng, A. Inoue, S. Goto, Operator based thermal control of an aluminum plate with a Peltier device, International Journal of Innovative Computing Information & Control 4 (2008) 3219–3229. [6] J. Hammer, Nonlinear systems stabilization and coprimeness, International Journal of Control 42 (1985) 1–20. [7] G. Bartolini, A. Pisano, E. Punta, E. Usai, A survey of applications of second-order sliding mode control to mechanical systems, International Journal of Control 76 (2003) 875–892. [8] T. Chai, Z. Geng, H. Yue, H. Wang, C. Su, A hybrid intelligent optimal control method for complex flotation process, International Journal of Systems Science 40 (2009) 945–960. [9] G. Chen, Z. Han, Robust right coprime factorization and robust stabilization of nonlinear feedback control systems, IEEE Transactions on Automatic Control 43 (1998) 1505–1510. [10] R.J.P. de Figueiredo, G. Chen, Nonlinear Feedback Control Systems – An Operator Theory Approach, Academic Press, San Diego, 1993. [11] M. Deng, A. Inoue, K. Ishikawa, Operator based nonlinear feedback control design using robust right coprime factorization, IEEE Transactions on Automatic Control 51 (2006) 645–648.

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