Model-based strategies for sensor fault accommodation in uncertain dynamic processes with multi-rate sampled measurements

Chemical Engineering Research and Design 1 4 2 ( 2 0 1 9 ) 204–213

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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Model-based strategies for sensor fault accommodation in uncertain dynamic processes with multi-rate sampled measurements James Allen, Shaoru Chen, Nael H. El-Farra ∗ University of California Davis, Department of Chemical Engineering, One Shields Avenue, Davis, CA 95616, United States

a r t i c l e

i n f o

a b s t r a c t

Article history:

This work presents model-based strategies for the accommodation of sensor faults in

Received 5 October 2018

dynamic processes with multi-rate sampled state measurements. The developed strategies

Received in revised form 30

account explicitly for both closed-loop stability and performance considerations. Initially,

November 2018

a model-based feedback control system, in which a model predictor compensates for the

Accepted 3 December 2018

unavailability of state measurements between sampling times, is designed. The stability

Available online 10 December 2018

and performance characteristics of the multi-rate sampled-data closed-loop system are analyzed and explicitly characterized in terms of the state sampling rates, the fault parameters,

Keywords:

and the various process, model and controller design parameters. The resulting character-

Sensor faults

izations provide insight into the robustness and margins of tolerable faults that can be

Fault accommodation

accommodated, and are used to develop both stability-based and performance-based fault

Sampled-data systems

accommodation schemes that aim to maintain closed-loop stability and minimize the degra-

Multi-rate sampling

dation of closed-loop performance in the presence of sensor faults. Two types of sensor faults

Chemical processes

are considered. These include faults that manifest themselves as improper sensor readings, as well as faults that cause drift in the sensor sampling rate. The first type of fault introduces errors in the model state updates at the sampling times, while the second type of fault alters the rate at which the sensors sample the process states. The developed methods are illustrated through a case study involving a cascade of two non-isothermal continuous-stirred tank reactors with plant-model mismatch and access to full state measurements that are sampled at different rates. The case study gives some insight into the effects of sensor faults on the stability and performance of the sampled-data state feedback control system, and how these effects can be mitigated through use of fault-tolerant control. © 2018 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1.

Introduction

The ability to compensate for control system faults and malfunctions is an important capability for modern-day process control systems. When left unaddressed, faults in the control system components, such as the measurement sensors and the control actuators, can lead to substantial degradation in the achievable closed-loop control quality, which can translate

∗

into economic losses through product quality deterioration and, in some cases, safety hazards due to process instabilities and loss of controllability. The increased emphasis placed on process safety and meeting stringent product quality requirements in industrial applications have motivated a significant and growing body of research work on the problem of faulttolerant control (see, e.g., (Frank and Ding, 1997; Simani et al., 2003; Blanke et al., 2003; Isermann, 2005; Zhang and Jiang,

Corresponding author. E-mail address: [email protected] (N.H. El-Farra). https://doi.org/10.1016/j.cherd.2018.12.003 0263-8762/© 2018 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Chemical Engineering Research and Design 1 4 2 ( 2 0 1 9 ) 204–213

2008; Mhaskar et al., 2013) for some results and references in this area). The fault-tolerant controller design problem is commonly addressed in the context of a conventional feedback control setting where measurements of the process output are assumed to be available continuously for the controller to utilize in compensating for the effects of the faults and driving the process state to the desired operating point. However, with real-time operation practices this is generally not the case, and measurements of the process state variables are usually limited in availability by the capabilities of the measurement sensors and/or the resources of the sensor-controller communication medium. Discretely sampled-data systems are the norm, rather than the exception, in industrial processes as continuous measurement of process data often proves logistically or technologically prohibitive. In sampled-data systems, the stability and performance characteristics of the closedloop system are critically dependent on the rate at which the sampled measurements are taken. The design of fault-tolerant control systems for sampleddata processes has been the subject of a number of prior works. In Sun and El-Farra (2011), a model-based framework for fault detection and control actuator reconfiguration was developed for processes with sampled and delayed measurements and control actuator faults. While reconfiguration of the control system is a viable option for the handling of faults, it is not always an ideal solution, especially in situations where the availability of component redundancy is either costly on inherently limited by process design considerations. The ability of the process to maintain a satisfactory level of operability in the faulty control configuration is a worthwhile pursuit as was demonstrated by Napasindayao and El-Farra (2013a) where a stability-based fault accommodation strategy was developed and applied to particulate processes subject to control actuator faults. These results where later generalized by Napasindayao and El-Farra (2015) to incorporate fault estimation capabilities through an optimization-based moving horizon parameter estimation scheme to aid in the implementation of the fault accommodation logic. In addition to closed-loop stability considerations, the development of performance-based fault accommodation strategies that aim to minimize performance deterioration following a fault event was pursued by Allen and El-Farra (2017); however, this was done in the context of control actuator faults. In addition to their focus on control actuator faults, a common consideration among these prior works is the assumption that all process states are sampled at the same rate. In many practical applications, limitations on the measurement capabilities of different sensors usually lead to different sampling rates, and in such instances a synchronized sampling mechanism may not be the best choice (Wang et al., 2004). The use of multi-rate sampling can also be triggered by the relative significance of the measurements collected. For example, for sensors placed at certain critical locations in the process where frequent monitoring and tight control are required, one expects that a fast sampling rate would be applied. For sensors located at other less critical locations, reduced sampling rates can be applied to optimize energy resource consumption and reduce cost. It is well-known at this stage that the use of multirate sampled measurements in feedback control systems has a significant impact on the stability and performance properties of the closed-loop system (see, e.g., (Tippett and Bao, 2014; Shao and Cinar, 2015; Pasanda and Montazeri, 2018; Ravi and

205

Kaisare, 2018; Jia et al., 2018) for some results on the analysis and control of multi-rate sampled-data systems). A more robust framework for fault-tolerant control should therefore account explicitly for the possibility of sampling each state at a different rate (see, e.g., (Dong et al., 2016; Gao et al., 2017; Zhang et al., 2018) for some results in this direction). Furthermore, compared with the focus on control actuator faults, the problem of handling sensor faults has received less attention. Sensor faults are quite common in practice and are often critical to the overall system performance, especially with the increased reliance on dense sensor deployment and sensor networks in many industrial applications. Sensor malfunctions that manifest themselves, for example, as improper sensor readings or cause deterioration in the sampling frequency not only limit our ability to monitor the process and diagnose the health status of process operations, but may also interfere directly with the control quality when these errors are introduced in the feedback controller implementation. Examples of previous works on sensor fault handling include previous studies on passive fault-tolerant control approaches for handling sensor data losses (Gani et al., 2008), stability-based compensation of abrupt changes in measurement sampling rates (Napasindayao and El-Farra, 2013b), and active sensor fault-tolerant control approaches based on sensor reconfiguration (Yao and El-Farra, 2014). A common theme in these studies has been the almost exclusive emphasis on closed-loop stability considerations. While maintaining closed-loop stability is of paramount importance, minimizing the deterioration in closed-loop performance following a fault event is also an important objective of fault-tolerant control. Furthermore, in situations when multiple stabilizing fault accommodation measures can possibly be taken by the control system, it is important to utilize performance considerations as a means of further discriminating between the feasible fault accommodation decisions. This necessitates the selection and characterization of an appropriate performance metric that takes into account the characteristics of the system. To date there has not been a rigorous assessment or characterization of the closed-loop stability and performance properties of multi-rate sampleddata processes in the context of sensor fault-tolerant control, nor has there been any thorough assessment of performancebased accommodation strategies for multi-rate sampled-data processes. These are important gaps that the current work aims to address. The measurement errors that can be caused by sensor faults may deteriorate the overall control quality and thus need to be accounted for explicitly in the control system design and operation. Motivated by these considerations, the objective of this work to develop analysis and accommodation tools for sensor fault-tolerant control of multi-rate sampled-data process systems. The focus is on providing insights into the development and implementation of sensor fault accommodation strategies from both stability and performance perspectives. The remainder of the paper is organized as follows. Section 2 introduces a motivating example that is used throughout the paper to illustrate the development and implementation of the proposed framework. A stabilizing model-based feedback controller is designed and its implementation using a multi-rate measurement sampling scheme is described. The stability of the closed-loop system is then analyzed in Section 3 leading to an explicit characterization of the closed-loop stability region in terms of the process parameters, the sever-

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ity of the sensor faults, and possible fault accommodation variables. The implications of the resulting characterization for the development of stability-based fault accommodation strategies (both passive and active) are highlighted. The limitations of stability-based fault accommodation strategies are touched upon prior to introducing a suitable performance metric in Section 4, which captures the settling time of the closed-loop performance output following an impulse disturbance event. The performance metric is characterized in terms of the same process and controller parameters that govern closed-loop stability, and the developed characterization is then used to devise performance-based fault accommodation strategies. Throughout the paper, simulation results based on the motivating example are presented to illustrate the advantages of fault accommodation from both the stability and performance perspectives. Finally, some concluding remarks are given in Section 5.

2.

Preliminaries

2.1.

Motivating example

In this section, we introduce a motivating example that will be used throughout the paper to illustrate the development and implementation of the sensor fault accommodation strategies. To this end, we consider a process composed of a series cascade of two non-isothermal continuous-stirred tank reactors with a recycle stream and three parallel irreversible elementary reactions. Following Sun and El-Farra (2008), the process dynamics can be captured by the following mass and energy balances:

Table 1 – Process parameters and steady state values Parameter

Value

F0 F1 F3 Fr V1 V2 R T0 T03 CsA0 CsA03 H1 H2 H3 k10 k20 k30 E1 E2 E3  Cp T1s CsA1 T2s CsA2 Q1s Q2s

4.998 m3 /h 39.996 m3 /h 30.0 m3 /h 34.998 m3 /h 1.0 m3 3.0 m3 8.314 kJ/kmol/K 300.0 K 300.0 K 4.0 kmol/m3 2.0 kmol/m3 5.0 × 104 kJ/kmol 5.2 × 104 kJ/kmol 5.4 × 104 kJ/kmol 3.0 × 106 h−1 3.0 × 105 h−1 3.0 × 105 h−1 5.0 × 104 kJ/kmol 7.53 × 104 kJ/kmol 7.53 × 104 kJ/kmol 1000.0 kg/m3 0.231 kJ/kg/K 457.9 K 1.77 kmol/m3 415.5 K 1.75 kmol/m3 0 kJ/h 0 kJ/h

feed stream concentrations, CA0 and CA03 as the manipulated inputs.

2.2.

 F0 Fr Q1 = (T0 − T1 ) + (T2 − T1 ) + Gi (T1 )CA1 + V1 V1 cp V1

Model-based controller synthesis

3

T˙ 1

The control problem is addressed on the basis of the linearization of the above process dynamics around the operating steady state:

i=1

 F0 Fr (CA0 − CA1 ) + (CA2 − CA1 ) − Ri (T1 )CA1 V1 V1 3

˙CA1 =

3 

i=1

˙T 2

F1 F3 = (T1 − T2 ) + (T03 − T2 ) + V2 V2

˙CA2

 F1 F3 = (CA1 − CA2 ) + (CA03 − CA2 ) − Ri (T2 )CA2 V2 V2

i=1

(1)

Q2 Gi (T2 )CA2 + cp V2 3

i=1

where Tj , Fj , Vj , CAj , and Qj denote the reactor temperature, stream flow rate, reactor volume, reactant concentration, and rate of heat input to the reactor, respectively, with j = 0 denoting the value with respect to the fresh inlet stream to reactor 1, j = 1 denoting reactor 1, j = 2 denoting reactor 2, j = 03 denoting the fresh inlet stream to reactor 2, and j = r denoting the recycle stream (note that not every variable contains every subscript). The expressions Ri (Tj ) = ki0 exp(−Ei /RTj ) and Gi (Tj ) = ((−Hi )/cp )Ri (Tj ), are expressed only for j = 1 and j = 2 with ki , Ei , and Hi , for i = 1, 2, 3, denoting the pre-exponential constants, activation energies, and enthalpies of the three reactions, respectively. The parameters  and cp denote the fluid density and heat capacity in the reactor. Using the process parameter values in Table 1, it can be shown that the process has unstable steady state at (T1s , CsA1 , T2s , CsA2 ) = an 3 (457.9 K, 1.77 kmol/m , 415.5 K, 1.75 kmol/m3 ). The control objective is to stabilize the process at this steady-state using the rates of heat transfer, Q1 and Q2 , as well as the fresh

˙ = Ax(t) + Bu(t) x(t)

(2)

where x represents the vector of process state variables given (in deviation variable form) by: x = [T1 − T1s

CA1 − CsA1

T2 − T2s

T

CA2 − CsAs ] ,

u is the vector of manipulated inputs given (in deviation variable form) by: u = [Q1 − Q1s

CA0 − CsA0

Q2 − Q2s

T

CA03 − CsA03 ] ,

A and B are constant state and input matrices, respectively, given by:

⎡ A

=

25.30

31.75

⎤

0

⎢ −78.03 −45.94 0 34.64 ⎥ ⎢ ⎥, ⎣ −2.84 1.42 14.70 0 ⎦ −22.45

⎡

B

4.97

=

⎢ ⎢ ⎢ ⎢ ⎣

−24.88

0

−5

0

0

0

0

2.82

0

0

0

0

0.347 × 10−5

0

0

0

0.945 × 10

13.47

⎤

⎥ ⎥ ⎥ 0 ⎥ ⎦ 5.71

(3)

Chemical Engineering Research and Design 1 4 2 ( 2 0 1 9 ) 204–213

207

It should be noted that the small values appearing in the input matrix entries B(1, 1) and B(3, 3) (which are associated with the rates of heat inputs) are due to a combination of the choice of process parameter values and the use of dimensionless state and input variables in deriving the state-space form of (2). Based on the linearized system of (2) and (3), a model-based state feedback controller of the following form is designed: u(t) ˙ xˆ (t)

= Kxˆ (t) (4) =

ˆ x(t) + Bu(t) ˆ Aˆ

Fig. 1 – A schematic representation of the sampling times and sampling periods for a multi-rate sampled-data system with two sensors.

where xˆ is the model state which provides an estimate of the ˆ and Bˆ are constant matrices that represent process state, A estimates of A and B, respectively, K is the feedback controller gain which is chosen to place the poles of the closed-loop model at ˛ · [−1 − 2 − 3 − 4], where ˛ is the pole location shift which will be used later as the fault accommodation parameter. The controller of (4) computes the control action using the model states when state measurements are unavailable from the sensors. At the sampling times, when the state measurements become available, the model states are updated using the actual states to provide the necessary feedback. In the case of a uniform sampling rate for all process states, all model states are updated at the same time. However, when different process states are sampled at different frequencies, not all model states are updated simultaneously. The next subsection describes the multi-rate measurement sampling mechanism and its implications for the implementation of the above model-based control strategy.

2.3. Controller implementation using multi-rate sampled measurements Referring to the example of (1), we consider that the reactant concentration and reactor temperature states can be measured at independent rates. We use the notation 1 to denote the sampling period for reactant concentration measurements (both CA1 and CA2 are assumed to have the same sampling period), and use the notation 2 to refer to the sampling period for the reactor temperature measurements (both T1 and T2 are assumed to have the same sampling period). To simplify the theoretical development and analysis of the multi-rate closed-loop system, we first recast the individual sampling periods as follows: i = ıi · , where  is selected such that each ıi is a positive integer. Following this recasting, the following characteristic time units can be identified: • Shortest time unit (STU):  s = gcd(ıi ) ∀ i ∈ {1, 2, . . ., m}, where gcd(·) denotes the greatest common divisor. • Basic time unit (BTU):  B = lcm(ıi ) ∀ i ∈ {1, 2, . . ., m}, where lcm(·) denotes the least common multiple. where m is the number of independent sampling rates in the multi-rate system (m = 2 for the reactor example). The STU is the shortest constant time period that the system could sample and capture all the state data being sampled by the sensors. The BTU is a new time period for the multi-rate system which

represents the time interval between two consecutive times at which all process states are sampled simultaneously. The inverse of the BTU captures the rate at which the sampling times of the different sensors are synchronized. Fig. 1 shows a schematic representation that illustrates this augmented multi-rate sampling scheme for the case of two sensors with different sampling periods. Here the two sensors, S1 and S2 , have sampling periods 1 = 0.2 and 2 = 0.3, leading to  = 0.1, ı1 = 2, ı2 = 3, and  s = 0.1,  B = 0.6. Here the notation jk is introduced to indicate a “potential” sampling time (this may or may not coincide with an actual sampling time), where j denotes the index of the STU inside the BTU which is denoted by the index k. Note that there are M =  B / s sub-intervals, each k =  k+1 of length  s , in each interval of length  B , resulting in M 0 k+1 (by convention this is denoted by 0 ). It can be seen that every sampling instance from S1 and S2 (marked by an “×” in Fig. 1) occurs at some jk although not every jk contains a sampling instance from the original system. Furthermore, it can be seen that at times 0k , k ∈ {0, 1, 2, . . . } all sensors in the multi-rate system sample simultaneously, thus creating a sampling scheme with a pseudo single-rate. Since this augmented scheme contains potential sampling times at which one or none of the actual sensors are sampling, we define a binary function ς(i, j) to denote whether the ith sensor is active at each of the potential sampling instances jk :

ς(i, j) =

1,

if j is divisible by ıi

0,

otherwise

(5)

It should be noted that although the sampling rates are independent, it is computationally advantageous to choose sampling rates that “mesh” well in the sense that they result in a smaller value for M. Based on the multi-rate sampling scheme described above, the model states that are updated at any potential sampling time jk are only those states corresponding to the measurement sensors transmitting their data at that time. In the absence of sensor faults, the model state update law takes the form: i

xˆ (jk ) = xi (jk ), ∀ ς(i, j) = 1 i ∈ {1, 2, . . ., m}, j ∈ {0, 1, 2, . . ., M − 1}

(6)

where xi is the vector of process states with the same ith sampling period.

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Chemical Engineering Research and Design 1 4 2 ( 2 0 1 9 ) 204–213

where

3. Analysis and accommodation of instability-inducing sensor faults

N = Is0 eFs RM−1

The objective of this section is to analyze the impact of sensor faults on the stability properties of the closed-loop system and use the results to devise stability-based fault accommodation strategies that preserve closed-loop stability in the presence of sensor faults. We consider two different kinds of sensor malfunctions. The first kind involves malfunctions in the form of errors in the state measurements, and the second kind includes faults that cause a change in the sampling frequency of the measurement sensor.

3.1.

j j−+1

Is

⎡

¯ i := diag{ ¯ i },  k

⎢

⎡ ⎢ Sj = ⎢ ⎣

=

¯ i xi ( k ), ∀ς(i, j) = 1  j

i

∈

{1, 2, . . ., m}, j ∈ {0, 1, 2, . . ., M − 1}

(7)

From the above representation, it is evident that this type of sensor faults introduce errors directly into the model state updates which translate into errors in the implementation of the model-based control action. To analyze the stability of the closed-loop system subject to the model-based controller of (4) and the model state update law of (7), we define T eT (t)] , where e(t) = the augmented state vector (t) = [xT (t) x(t) − xˆ (t) is the model estimation error, which represents the discrepancy between the actual and model states. The multirate sampled-data closed-loop system of (2)–(4) and (7), with m different sampling rates, can be formulated as: ˙ (t)

=

k ) F(t), t ∈ [jk , j+1

ei (jk )

=

¯ − I)xi ( k ), ∀ ς(i, j) = 1 ( j

i

∈

{1, 2, . . ., m}, j ∈ {0, 1, 2, . . ., M − 1}

(8)

i

where A + BK

−BK

ˆ + (B − B)K ˆ (A − A)

ˆ − (B − B)K ˆ A

O

⎤ ⎥

I4×4 ⎦ ,

=

Sj

⎡ j

⎢

I

O

j

Is = ⎣ 

ς(1, j)

O

⎤

I − Sj ⎥ ⎦

⎤ ⎥ ⎥ ⎦

ς(2, j) ς(1, j) O ⎡ ⎤ 1 − 1

j

for j ≥ 1

(11)

ς(2, j)

⎢ ⎥ ⎢ 1 − 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 1 − 1 ⎦ 1 − 2

xˆ (jk )



I4×4

R0 = ⎣ O

i ∈ {1, 2, . . ., m}

where the value of each diagonal element indicates the fault or health status of the ith sensor suite (the term sensor suite is used here to refer to a group of sensors that have the same sampling period). A healthy sensor (with no measurement ¯ i = 1, whereas any value errors) is indicated by a value of  k i ¯ k = / 1 indicates some error in the collected measurement. Based on this representation, the model state update law of (6) can be reformulated to account for sensor errors as follows: i

eFs ,

=1

Closed-loop stability analysis

To capture the effect of measurement sensor errors on the model state updates, we introduce the following diagonal matrix:

F=

Rj =



and we have assumed, for simplicity, that sensors of the ¯ 1 = diag{1 , 1 } and same kind have the same faults, i.e.,  2 ¯  = diag{2 , 2 }, where 1 and 2 represent the faults in the temperature and concentration sensors, respectively. This gives rise to the following model state update law: k xˆ (jk ) = (Sj − j )x(jk ) + (I − Sj )ˆx(j− )

(12)

k is the time instance immediately prior to  k . This where j− j form takes into account the fact that not all model states in xˆ are updated at every jk . By norm-bounding and analyzing the closed-loop system response of (10) and (11), it can be shown (see Yao and ElFarra (2012) for a similar proof) that the closed-loop system is guaranteed to be exponentially stable at the origin if the following condition is satisfied:



ˆ B, ˆ K, 1 , 2 , 1 , 2 ) < 1 max N(A, B, A,

(13)

where max (·) denotes the maximum eigenvalue magnitude of the matrix N. This stability condition imposes a constraint on the allowable ranges of several key parameters, including the plant-model mismatch, the feedback gain, the magnitudes of the various faults, and the sampling rates of the various sensors. As such this condition can be used to explicitly characterize the closed-loop stability region in terms of these parameters. An important implication of this is that it can be used to develop strategies for mitigating the potentially adverse effects of faults on closed-loop stability by varying a suitably chosen parameter (within its feasible range) to restore closed-loop stability. This point is demonstrated in the next subsection.

3.2.

Stability-based fault accommodation strategies

(9)

By solving the linear system of (8) and (9), it can be shown that: F(t− k ) j j

(t) = e

R Nk 0 ,

k t ∈ [jk , j+1 ),

∀ j ∈ {0, 1, . . ., M − 1}, k ∈ {0, 1, 2. . .}

(10)

In the discussion below, we note that if the notation  or  appears without a subscript then it is assumed that 1 = 2 =  and 1 = 2 = . This is done for ease of interpretation and to isolate trends when applicable. Furthermore, all simulation results include plant-model mismatch in the form of parametric uncertainty in the heat of reaction, i.e., H1model = (1 + ı)H1 with an uncertainty level of ı = 0.05.

Chemical Engineering Research and Design 1 4 2 ( 2 0 1 9 ) 204–213

Fig. 2 – A contour plot of max (N) as a function of the pole location shift parameter, ˛, and the sensor fault, , with  = 0.05 h. The boundary between the colored and uncolored regions is the unit contour line. Fig. 2 is a contour plot of max (N) as a function of the sensor fault size, , and the pole location shift parameter, ˛, which determines the feedback controller gain K (see Section 2). The uncolored region represents the area inside the unit contour line, and is therefore the closed-loop stability region. The colored part of the plot represents the region where the stability condition of (13) is not satisfied and therefore the closed-loop system is unstable. It can be seen from the plot that for a given pole shift location, ˛, there is a range of sensor fault values that can be tolerated while maintaining operation in the stable region. From a strict closed-loop stability standpoint, such faults do not require any active accommodation. The range of tolerable faults appears to increase as ˛ is increased (i.e., as the poles are shifted farther to the left in the left-half of the complex plane). It also appears that for severe faults with values of  significantly above or below 1, closed-loop stability is lost no matter what ˛ is. An important implication of the result in Fig. 2 is that it can be exploited to devise stability-based fault accommodation measures in the case where an estimate of the range of faults that the sensors are susceptible to is available, or when the fault value is known and such a fault places the operating

209

Fig. 4 – A contour plot of max (N) as a function of the temperature sampling period, 1 , and the concentration sampling period, 2 , with ˛ = 5 and  = 1. point in the unstable region. This point is demonstrated in the simulation scenario presented in Fig. 3. Fig. 3 shows the effect that a destabilizing sensor fault has on the closed loop system. The fault is represented by a step change in the value of  from  = 1 to  = 0.81 at t = 2 h, while ˛ = 1 and  = 0.05 h. It can be seen from Fig. 2 that the post-fault operating point lies within the unstable region. The plots in Fig. 3 show the closed-loop response of the reactor temperatures, T1 and T2 , as well as the manipulated heat transfer flow rates, Q1 and Q2 , when no fault accommodation takes place (see the dashed profiles) and when fault accommodation is implemented by changing the pole location shift parameter from ˛ = 1 to ˛ = 10 following the fault (see the solid profiles). In line with the predictions of Fig. 2, the chosen accommodation measure results in a stable post-fault operation as it restores the post-fault operating point back in the stable region. Fig. 4 is a contour plot of max (N) as a function of the sampling period of the temperature sensors, 1 , and the sampling period of the concentration sensors, 2 . The uncolored region represents the area inside the unit contour line, and is therefore the closed-loop stability region. The colored part of the plot signifies that the stability condition of (13) is not satisfied, and that the closed-loop system is unstable. It can be seen from the plot that the stability of the closed-loop system

Fig. 3 – Comparison between the closed-loop system responses subject to a stability-degrading sensor fault in the presence and absence of active fault accommodation measures. Top plots: Closed-loop temperature profiles. Bottom plots: Manipulated input profiles.

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Fig. 5 – A contour plot of max (M) as a function of the sampling period and the sensor fault magnitude, with ˛ = 1. depends more critically on the rate at which the temperature measurements are sampled than on the rate at which the concentration measurements are sampled. Specifically, notice that there exists a critical value for the sampling period of the temperature measurements above which the closed-loop system loses stability no matter how much faster the sampling of the concentration measurements is conducted. The inverse of this, however, is not true. For any concentration sampling period, one can always ensure closed-loop stability by sufficiently reducing the temperature sampling period. This type of comparative analysis of the relative influences of the different sampling rates on closed-loop stability could potentially aid in making process design decisions such as where to focus redundancy in sensor placement. For example, the predictions of Fig. 4 suggest that redundancy efforts in sensor placement should focus more on the temperature measurement sensors and less on the composition sensors given that variations in the temperature measurement sampling rate have a more profound influence on closed-loop stability. This is important in situations where the sensor faults manifest themselves in the form of malfunctions in the sampling mechanism leading, for example, to an unexpected increase in the sampling period. In the event that the temperature sampling period increases due to a malfunction in the temperature sensor, switching to a healthy backup ensures closed-loop stability. Beyond its implications for sensor redundancy, the analysis based on Fig. 4 can aid in developing explicit stability-based fault accommodation strategies. For example, an increase in the sampling period of the concentration measurements that shifts operation into the unstable region can be compensated for by a decrease in the sampling period of the temperature measurements as this would restore the operating point back within the stable region. The choice of the sampling period as an accommodation parameter is ultimately dependent on the particular characteristics of the measurement sensor and whether the sampling rate can actually be varied. Fig. 5 shows how the closed-loop stability region (the uncolored region) depends on the sampling period and the sensor fault size, when ˛ = 1. This plot provides insight into how robust a single sampling period is to a range of sensor fault values. This can be useful in situations where the sensor fault is known to fall within a certain range of values but the exact value of the sensor fault is unknown. In such cases, one can choose a suitable sampling period that places the operation for all potential values of the sensor fault in the stable regime, or let it be known that given the range of fault values stability cannot be guaranteed. The shape of the stability region in this

Fig. 6 – A contour plot of max (N) as a function of the pole location shift parameter, ˛, and the sampling period , with  = 1. case also has some useful implications for fault accommodation as it identifies how in some cases the sampling period may be varied to compensate for sensor faults that push the operating point slightly outside the stability region. Further insight into the stability characteristics of the sampled-data closed-loop system can be obtained from Fig. 6 which shows how the size and shape of the closed-loop stability region (the uncolored parts of the plot) depend on the sampling period and the pole location shift parameter (which captures how aggressive the feedback gain is). A striking feature of this plot is the existence, for any given pole placement, of two distinct (and disconnected) ranges for the sampling period that would result in stable closed-loop operation. While this trend could possibly be exploited to sample at a slower frequency (e.g., to reduce sensor operating costs) and still maintain closed-loop stability, it is unclear whether or not such an action would be advantageous, especially given the fact that the upper portion of the stability region is rather narrow which suggests that small perturbations in the sampling period due to sensor malfunctions could land the operating point inside the unstable region. This raises the question of how one can decide on a suitable fault accommodation measure when faced with multiple feasible measures that are all stabilizing. To address this problem, a performance metric is introduced and analyzed in the next section so as to further aid in the fault accommodation process.

4. Analysis and accommodation of performance-degrading faults Beyond maintaining closed-loop stability, minimizing the deterioration in closed-loop performance following a fault event is a primary objective of fault-tolerant control designs. Furthermore, in situations when multiple stabilizing fault accommodation measures can possibly be taken by the control system, it is useful to introduce a performance metric as a way of further discriminating between the feasible fault accommodation decisions. In this section, we first introduce a performance metric and characterize it with respect to the various control and operation parameters, and then we use the resulting characterization to present suitable performancebased fault accommodation strategies.

4.1.

Closed-loop performance characterization

Given the sampled-data nature of the control system considered, we select the extended H2 -norm (which we denote by

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GH2 ) as the performance metric of interest. The extended H2 -norm is a measure of the settling time of a prescribed performance output after an impulse disturbance is introduced into a prescribed process input. Characterizations of this kind of a performance metric were pursued in Montestruque and Antsaklis (2006) in the context of model-based networked control systems with a uniform update rate (see also (Sun et al., 2009)). In this section, we develop the necessary characterization appropriate for multi-rate sampled-data systems. To characterize extended H2 -norm, we first reconsider the state-space representation of the system in (4) to include a disturbance term and a performance output as follows: ˙ = Ax(t) + Bu(t) + Ed (t) x(t) z(t) = Jx(t)

(14)

where d is the disturbance input and z is the performance output. In the case of the motivating process example of (1), the performance output is chosen to be the outlet reactant concentration of the second reactor, CA2 . After some manipulations, and with the aid of the same notation used in Section 3.1, the sampled-data closed-loop system can be formulated as follows: ˙ (t) = F(t) + Hd(t), T T

T

e(jk )

=

j x(jk ) + (I

j ∈ {0, 1, 2, . . ., M − 1}

(15)

− Sj )e(jk ) −

z(t) = L(t) T

where H = [ET ET ] , L = [J O]. Similar to the closed-loop formulation of (8) and (9), the effects of sensor errors can be seen directly on the model estimation error reset at the various sampling times. By solving (15), it can be verified that the response of the closed-loop performance output to an impulse disturbance d = ı(t − t0 ) can be expressed as: F(t− k ) j j

R Nk H,

z(t) = Le

k t ∈ [jk , j+1 )

∞

z22 =

FT (t− k ) j

T

HT (Rj Nk ) e

LT Le

F(t− k ) j

(Rj Nk )Hdt

t0

⎡ ∞ M−1    = HT ⎣ k=0 j=0

⎤

k

j+1

k T k T F (t−j ) T

(Rj N ) e

F(t− k )

L Le

j

(Rj Nk )dt⎦ H

k j

⎡ ⎤ ∞ M−1   T = HT ⎣ (Rj Nk ) W0 (0, s )(Rj Nk )⎦ H j=0 k=0

⎡ ⎤ ∞ M−1   T k = HT ⎣ (NT ) (Rj ) W0 (0, s )Rj Nk ⎦ H

(17)

j=0 k=0

where



s

T

eF t LT LeFt dt

W0 (0, s ) = 0

⎡

⎛

⎛

GH2 = ⎣trace ⎝HT ⎝

M−1 

⎞ ⎞⎤1/2 Xj ⎠ H⎠⎦

(19)

j=0

T

NT Xj N − Xj + (Rj ) W0 (0, s )Rj = 0,

j ∈ {0, 1, 2, . . ., M − 1}

(20)

It can be seen from the way that the extended H2 -norm is computed above that it depends on the same set of parameters that influence closed-loop stability, including the magnitudes of the sensor faults, the feedback gain, the measurement sampling rates and the process-model mismatch. This can be exploited to characterize the performance landscape of the closed-loop system as a function of these parameters and identify suitable sensor fault accommodation strategies that account explicitly for performance considerations as is shown in the next subsection.

(16)

Based on this response, the 2-norm of the performance output can be computed as follows:



The extended H2 -norm, GH2 , of the multi-rate sampleddata system is then given by:

where Xj is the solution to the jth discrete Lyapunov equation given by:

k ), k ∈ {0, 1, 2, . . .} t ∈ [jk , j+1

(jk ) = [x(jk ) e(jk ) ] ,

Fig. 7 – A contour plot of the extended H2 -norm, GH2 , as a function of the pole location shift parameter, ˛, and the sensor fault size, , with  = 0.05 h.

(18)

4.2. Performance-based fault accommodation strategies In the following discussion, we note that the performance metric is plotted only in the region where the closed-loop stability condition of (13) is met, as the notion of performance is not meaningful when the system is unstable. Also, since the extended H2 -norm is a measure of the settling time of the closed-loop performance output response, lower values of GH2 indicate better performance. Fig. 7 shows a contour plot of the extended H2 -norm, GH2 , as a function of the pole location shift parameter, ˛, and the sensor fault size, , for  = 0.05 h. It can be seen that, for a given value of , the closed-loop performance improves as ˛ increases, i.e., as the poles are shifted farther to the left in the left-half of the complex plane. This is consistent with the intuition that faster settling times can be achieved when the feedback gain is made more aggressive. Another interesting trend that can be observed from the plot is that, for certain values of ˛, the contour lines appear to be nearly vertical over a certain range of fault values, thus indicating that the closedloop performance remains relatively unaffected by such faults. However, beyond this range of fault values, the closed-loop

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Fig. 8 – Comparison of the closed-loop system responses, subject to a performance-degrading sensor fault, in the presence and absence of fault accommodation. Top plots: Closed-loop reactor temperature profiles. Bottom plots: Manipulated input profiles. performance deteriorates considerably as the boundary of the closed-loop stability region is approached. Fig. 7 represents the performance corollary to Fig. 2, and shows the possible performance gains (in terms of reducing the settling time) that can be made through adjustment of the fault accommodation parameter in the case of a fault in the measurement sensors. These performance gains are illustrated in the simulation scenario considered in Fig. 8. Fig. 8 shows the effect of a performance-degrading sensor fault on the closed-loop system response. The fault is simulated by a step change in the value of  from  = 1 to  = 0.9 taking place at t = 2 hours, while ˛ = 1 and  = 0.05 h. It should be noted that for this fault, the closed-loop system remains stable if no accommodation were to take place since the postfault operating point lies well within the closed-loop stability region. Based on the performance landscape shown in Fig. 7, however, it can be seen that it is advantageous for the controller to take some accommodation measures following the fault by adjusting ˛. For example, it can be seen that by comparing the performance levels at (˛ = 10,  = 0.9) versus (˛ = 1,  = 0.9) in Fig. 7, that a reduction in the extended H2 -norm can be achieved through fault accommodation. The predicted performance improvement is further confirmed by the plots in Fig. 8 which show the closed-loop response of the reactor temperatures and manipulated heat transfer rates when no accommodation takes place (see the dashed profiles) and when fault accommodation is implemented by adjusting ˛, with ˛ = 1 → 10 post fault (see the solid profiles). Further insight into the performance landscape of the closed-loop system can be gleaned from Fig. 9 which depicts the performance variations as a function of the pole location and the sampling period. In this figure, one can seen that there are two distinct bands of potential operation regimes at varying sampling periods (as was noted in the earlier discussion of Fig. 6). With the additional knowledge that Fig. 9 provides in terms of closed-loop performance, it now becomes clear that even though the closed-loop system is stable in the upper band (see Fig. 6), the performance of the system deteriorates considerably relative to that corresponding to the smaller sampling periods (faster sampling rates). As a corollary to Fig. 4, Fig. 10 depicts the performance landscape of the sampled-data closed-loop system as characterized by the different sensor sampling rates. The stability

Fig. 9 – A contour plot of the extended H2 -norm, GH2 , as a function of the pole location shift parameter, ˛, and the sampling period, , with  = 1.

Fig. 10 – A contour plot of the extended H2 -norm, GH2 , as a function of the temperature measurement sampling period, 1 , and the concentration measurement sampling period, 2 , with ˛ = 5 and  = 1.

of the closed-loop system was shown to be highly dependent on the sampling period of the temperature measurements, 1 (as seen in Fig. 4) and it can now be seen in Fig. 10 that the performance of the closed-loop system within the stable region of operation is almost entirely dependent on the sampling rate of the concentration sensors, where reducing the concentration sampling period reduces the extended H2 -norm and improves the settling time following a disturbance.

Chemical Engineering Research and Design 1 4 2 ( 2 0 1 9 ) 204–213

5.

Concluding remarks

In this work, a framework for sensor fault accommodation in multi-rate sampled data systems was developed. The framework focused on the analysis of sensor faults and their effects on the stability and performance characteristics of the closedloop system, such as the range of allowable non-destabilizing faults for any given system configuration, the degree of uncertainty in process parameters for which the closed-loop system remains stable, and the performance of the system as a function of various faults and fault accommodation parameters. This was done by first characterizing the multi-rate sampled data nature of the system through use of an augmented sampling time scheme, after which the closed-loop stability properties of the multi-rate sampled-data system were analyzed, leading to an explicit characterization of the stability region of the system as a function of the relevant process and controller parameters. The effects of such variables as the sensor fault parameter, the pole shift location (the accommodation variable), and the sampling rates on closed-loop stability were explored and discussed. The benefits of introducing a performance metric were highlighted in the case of multiple allowable stabilizing configurations, and a suitable performance metric was introduced and characterized as a function of the multi-rate sampled-data system parameters. The focus of this study was to showcase the benefits of this explicit characterization of the closed-loop stability and performance of the system from a fault-tolerance standpoint, with an eye on real world implementation through use of multi-rate sampled-data systems. Here the results were applied specifically to sensor based-faults where the results of the explicit stability and performance characterizations could be used to calculate a priori the robustness margins for potential faults or variable drift of process variables in order for the system to remain operable in a satisfactory fashion, whether that be a stable system or some baseline measure for performance, these tools can aid in this type of a priori design. Future work will focus on the integration of fault detection and estimation capabilities into the proposed framework in order to complement and facilitate the implementation of the developed the fault accommodation strategies.

Acknowledgments Financial support by NSF, CBET-1438456, is gratefully acknowledged.

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