Robust fault-tolerant control using an accurate emulator-based ...

International Journal of Control

ISSN: 0020-7179 (Print) 1366-5820 (Online) Journal homepage: http://www.tandfonline.com/loi/tcon20

Robust fault-tolerant control using an accurate emulator-based identification technique Rajamani Doraiswami & Lahouari Cheded To cite this article: Rajamani Doraiswami & Lahouari Cheded (2017): Robust fault-tolerant control using an accurate emulator-based identification technique, International Journal of Control, DOI: 10.1080/00207179.2017.1318452 To link to this article: http://dx.doi.org/10.1080/00207179.2017.1318452

Published online: 02 May 2017.

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Date: 03 May 2017, At: 10:32

INTERNATIONAL JOURNAL OF CONTROL,  https://doi.org/./..

Robust fault-tolerant control using an accurate emulator-based identification technique Rajamani Doraiswamia and Lahouari Chededb a Department of Electrical and Computer Engineering, University of New Brunswick, Fredericton, Canada; b Department of Systems Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

ABSTRACT

ARTICLE HISTORY

A robust fault-tolerant control scheme using an accurate and robustly identified model of a system operating in a closed loop is proposed. A robust identification of the system and the associated Kalman filter is proposed by the novel use of an emulator, which is a transfer function connected to the input, output or both, to mimic the likely operating scenarios, both normal and abnormal. From the emulator-generated data, a set of perturbed models are generated, which plays a crucial role in ensuring robustness of the identified model and subsequently that of the controller. The prediction error method is employed to identify the perturbed models and a robust optimal model is obtained as a best fit to the perturbed models. A robust Kalman filter-based state feedback controller is obtained from the parameterisation of all stabilising controllers generated using the emulator-perturbed models. It is shown theoretically that the proposed robust controller designed using the emulator-perturbed robustly identified model is superior to the conventional robust controller designed from the identified nominal model, and a significant improvement in robustness is confirmed using illustrative examples. A fault-tolerant control system is developed by exploiting the key properties of the freely available Kalman filter.

Received  October  Accepted  April 

1. Introduction Identification of the nominal model and the design of a controller for a physical system in the face of model uncertainty is a challenging problem. Robust control theory was developed to meet this challenge, and continues to dominate the field of control theory and its applications (Doraiswami & Cheded, 2010, 2012a; Doraiswami, Cheded, Khalid, Qadeer, & Khoki, 2010; Dullerud, 2000; Zhou, Doyle, & Glover, 1996). The approaches to modelling in the face of uncertainty are to include a perturbation term to the nominal model (identified nominal model) to capture the model uncertainty. The perturbation term, termed herewith as perturbation, induces perturbations in the nominal model to emulate the perturbed system model. The perturbation model generates a set of likely models in the neighbourhood of the nominal model and covers the behaviour of the system over both the low- and high-frequency regions. The measure of model uncertainty is chosen as a 2-norm or an infinity norm bound on the perturbation. The uncertainty model is adhoc and tends to be conservative as the system perturbations are unknown or partially known, and robustness to stability is an important consideration, and minimising the worst case will ensure stability of a robust controller. CONTACT Lahouari Cheded

[email protected]

©  Informa UK Limited, trading as Taylor & Francis Group

KEYWORDS

Emulators; identification experiments; accurate and robust model; robust and high performance controller; fault tolerant system

The cost function is the lowest when the model is nominal and the highest when the perturbation is maximum. High controller performance may not be achieved for a given operating regime, although stability is guaranteed for a wide operating range. A novel approach to the identifications and the subsequent controller design is proposed so that both the identified model as well as the controller designed using the identified model are robust in the face of model uncertainties instead of assigning the entire burden of robustness to the design of the controller. The approach here is similar to that outlined in (Doraiswami & Cheded, 2010, 2012a; Doraiswami et al., 2010), where the focus is not on the Kalman filter-based state-feedback controller, but rather on leveraging the power of emulators to accurately identify the system so as to reduce the design effort involved in the development of the robust controller. 1.1 Emulator-based identification The proposed robust identification scheme is motivated by the ideal case when all the perturbed systems and their input-output data are available. The ideal case of the availability of perturbed systems is met by generating the

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R. DORAISWAMI AND L. CHEDED

perturbed system by connecting an emulator to the accessible nominal system input or the output and in cascade with it during the off-line identification phase (Doraiswami & Cheded, 2013, 2015; Doraiswami, Diduch, & Stevenson, 2014). The cascade combination of the nominal system and the emulator mimics the perturbed system formed of the nominal system and the perturbation. The emulator is selected such that it can mimic the likely operating scenarios, both normal and abnormal. In the case when there is no prior information about the perturbation model, the emulator is chosen to be a unity-gain all-pass filter to induce phase variations in the cascade combination. If some information about the perturbation transfer function is known, then the emulator is chosen as a cascade combination of the known perturbation transfer function and an all-pass filter. A number of emulator-perturbed models are generated using the emulator-generated data covering all likely operating scenarios, and not merely the input-output data generated by the unperturbed nominal model. The prediction error method, which is a gold standard for system identification (Forsell & Ljung, 1999; Ljung, 1999), is used to identify the perturbed models. In recent years, the objective information criterion such as the AIC has been used to select an optimal model order, which is a trade-off between the parameter estimation error and the complexity of the model. An optimal model and its associated optimal Kalman filter are obtained as a best fit, in the sense of H∞ – or – norm, to the set of identified models. The optimal model thus obtained characterises the behaviour of the system over wider operating regions (in the neighbourhood of the operating point), whereas the conventional model characterises the behaviour merely at the nominal operating point only (that is, the conventional approach assumes that the model of the system remains unperturbed at every operating point). In Doraiswami et al. (2014), it is theoretically shown that the optimal model is more robust, that is, the difference between the maximum and minimum cost functions resulting from the variations in the emulator parameters is lower than that of the conventional approach which is based on performing a single experiment (that is, without using emulators). 1.2 Robust Kalman filter-based state feedback controller In order to meet the dual needs of fault tolerance and controller design, a Kalman filter-based scheme is used for both tasks. It is shown in Dullerud (2000) and Zhou et al. (1996) that the Kalman filter-based state feedback parameterises all stabilising controllers. The set of

emulator-perturbed models was used to determine the robust controller iteratively by exploiting properties of the algebraic Ricatti equation in achieving a trade-off between robustness and robust stability of the closedloop stability. It is shown theoretically that the proposed controller is ‘more’ robust than the one used in the conventional scheme based on the identified nominal unperturbed model (Dullerud, 2000; Zhou et al., 1996). The significant improvement in the robustness of both the performance and stability is confirmed using an illustrative example. The optimal Kalman filter plays an important role in monitoring the performance and health of a system, thanks to the key properties of its residual (Doraiswami & Cheded, 2010, 2012a; Doraiswami et al., 2010). Performance degradation or a fault may result from a mismatch between the true system model and the identified model, changes in its operating condition, malfunctioning of its components, variations in its noise and disturbance statistics, etc. In a high-performance control system, it is imperative that the closed-loop stability, performance, and the system’s heath condition be all monitored in order to initiate quick remedial actions when needed (Fekri1, Athans, & Pascoal, 2006; Ioannou & Sun, 2012). For instance, if the performance degradation stems from a system model variation, then either (1) the controller is adapted (redesigned) or (2) the system is reidentified, and the robust controller redesigned accordingly. If on the other hand, a fault is detected, it is first isolated, and then either the fault is accommodated or the system is shut down for repair. The fault-tolerance process includes detection of the fault, its isolation and accommodation. 1.3 Main contributions The proposed scheme unifies three related issues in the design of a control system, namely (1) how to maintain high performance and stability in the face of model uncertainties both in the identification and controller design phases, and (2) how to detect, isolate and accommodate an abnormal operating condition such as a fault. In this respect, the Kalman filter and emulators both play a key role in the system identification, controller design and fault-tolerant system development. The main contributions include the accurate emulator-based identification of a robust optimal model, the Kalman filter-based state-feedback controller and the fault-tolerance system development. It is shown theoretically as well as with evaluation on simulated and physical systems that the proposed scheme is significantly better compared to the conventional one with respect to both robust performance and robust stability.

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The paper is organised as follows. Section 2 gives the model of the system expressed in the state-space, transfer function and linear regression forms; its associated Kalman filter and its residual model as well as the key properties of the Kalman filter residual. Section 3 explains the role of emulators. A set of perturbed models is generated by connecting, in cascade, an emulator to the accessible input or output of the system so as to mimic the phase perturbation of the nominal model. The emulator is a Blashke product of first-order all-phase filters. Section 4 develops the proposed emulator-based robust identification of the emulator-perturbed model. An important requirement for reliable and accurate identification is that the residual (the error between the output and its estimate) must be a zero-mean white noise process. This ensures that the identified model has captured completely the static and dynamic behaviours of the model, and the leftover from the identification process is a mere information-less white noise process. Since the residual of the Kalman filter, satisfies the above property, namely that it is a zero-mean white noise process if and only if the identified model and the model embodied in the Kalman filter are identical to each other (Doraiswami & Cheded, 2012b). The residual of the Kalman filter associated with the system – instead of the coloured noise equation error of the system model – is minimised and in order to identify not only the system model but also the associated Kalman filter without the need for any prior knowledge of the statistics (e.g. mean and covariance) of the disturbance and the measurement noise. Section 5 focuses on the development of the proposed robust controller design to meet the requirement of robust performance and robust stability. An illustrative example giving the details of the design procedure is given. In Section 6, the performance of the proposed robust identification and a robust Kalman filterbased state feedback controller is evaluated on a physical process control system [6-8], as well as on simulated systems. The proposed scheme was extensively tested on simulated and physical laboratory-scale two-tank liquid level systems. These successful tests confirm the vital role played by emulator-based accurate identification techniques in the design of robust fault-tolerant controllers.

3

2.1. Model of the system .. State-space model The state-space model (A, B, C) is given by x(k + 1) = Ax(k) + Bu(k) + E w w(k) y(k) = Cx(k) + v (k)

(1)

where x(k) is an (nx1) state, n is the order given by x(k) = [ x1 (k) x2 (k) x3 (k) ... xn (k) ]T , u(k), w(k) and v (k) are scalars representing the control input, disturbance and measurement noise, respectively; y(k) is the measured output; A is an (nxn) matrix, B is an (nx1) input vector, E w is an (nx1) disturbance entry vector and C is a (1xn) associated with y(k). For simplicity, the direct transmission term, i.e. the ‘D-term’, is assumed to be zero in the statespace model of (1). It is assumed that the system (A, B, C) is controllable and observable. .. Frequency domain model The frequency-domain expression relating the input u(z) and output y(z) is given by y(z) = G(z)u(z) + ϑ (z)

(2)

= C(zI − A)−1 B is the transfer funcwhere G(z) = N(z) D(z) tion of the system; ϑ (z) is the output error ϑ (z) = C(zI − A)−1 E w w(z) + v (z). .. Linear regression model y(k) = ψT (k)θ + υ(k)

(3)

where θ is a 2n × 1 feature vector formed of the coefficients of the numerator and the denominator polynomials of the transfer function G(z); ψT (k) is a 1 × 2n regression vector: θ = [ a 1 a 1 . a n b 1 . b n ]T ψT (k) = [−y(k − 1) − y(k − 2) ... − y(k − n) (4) u(k − 1) u(k − 2) ...u(k − n)]

.. Nominal model Let (A0 , B0 , C0 ), G0 (z) and θ 0 be, respectively, the nominal models of (A, B, C), G(z) and θ. 2.2 Kalman filter

2. Problem formulation In this section, the model of the system and the associated Kalman filter are given. The key properties of the Kalman filter established in Doraiswami and Cheded (2012b) are re-stated here for convenience and ease of reference.

The Kalman filter model (A0 − K 0C0 , [K 0 B0 ], C0 ) is copy of the identified nominal model (A0 , B0 , C0 ) of the system (A, B, C) driven the residual. The Kalman filter associated with (A0 , B0 , C0 ) is xˆ0 (k + 1) = (A0 − K 0C0 ) xˆ0 (k) + B0 u(k) + K 0 y(k)

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ˆ y(k) = C0 xˆ0 (k) ˆ ek f (k) = y(k) − y(k)

(5)

ˆ ˆ where x(k) and y(k) are the estimates of x(k) and y(k); ek f is the Kalman filter residual. .. Residual model of the Kalman filter The residual model map relating the residual ek f (z) to system input u(z) and the system output y(z) is given in Doraiswami and Cheded (2012b): N0 (z) D0 (z) y(z) − u(z) ek f (z) = Dk f (z) Dk f (z)

from the system input and the system output, (2) monitoring the performance and the health of the system, and (3) implementing the Kalman filter-based state feedback controller and developing the fault-tolerant control system. If there is a performance degradation, then the controller is redesigned (adapted) or the system is reidentified and the controller redesigned accordingly. If a fault is detected, it is then isolated and either the fault is accommodated or the system is shut down for repair.

3. Emulator-generated perturbed models (6)

where G0 (z) = DN00 (z) is nominal transfer function of (z) (A0 , B0 , C0 ), Dk f (z) = |zI − A p0 + K 0C p0 |. .. Key properties of the Kalman filter

r The residual ek f (k) is a zero-mean white noise process if and only if there is no mismatch between the true model of the system (A, B, C) and its identified nominal model embodied in the Kalman filter (A0 , B0 , C0 ), that is if A = A0 , B = B0 and C = C0 ; equivalently G(z) = G0 (z); θ =θ 0 . r The Kalman filter estimates y(k) ˆ are the minimum variance estimate of y(k). r Thanks to the feedback (closed-loop) configuration of the Kalman filter with residual feedback, the Kalman filter provides the highest robustness against the effect of disturbance and model variations r If there is model-mismatch, (A, B, C) = (A0 , B0 , C0 ), G(z) = G0 (z) or θ = θ 0 , the residual ek f (k) will no longer be a zero-mean white noise process. There will be an additive term and an additive term, e f (k), referred to as a fault indicative term (Doraiswami & Cheded, 2012b):

The emulator-perturbed models play a crucial role in the robust identification of the system and the associated Kalman filter, in the design of robust controller, and in ensuring fault-tolerance. In the design phase, a set of emulator-perturbed models are generated to mimic the likely operating scenarios and a robust optimal model is then derived as the best fit to the set of perturbed models. In the robust controller design stage, the identified optimal Kalman filter-based state feedback gains are generated. A robust controller is selected by verifying which of the state feedback gains meets the requirements of performance and stability of the closed-loop system not only for the optimal model but also for all members of the set of the emulator-perturbed models. Each of the perturbed models mimics, in a way, the behaviour of the true perturbed plant. The perturbed model G(z) can be expressed as an additive or multiplicative combination of a model G p0 (z), and a perturbation term, denoted g(z) (Zhou et al., 1996): G (z) = G p0 (z) (1 +  (z))

(8)

The model perturbation term, denoted g(z) = (1 + (z)) , satisfies: g(z) ≤ 

(9)

e(k) = e f (k) + e0 (k) e f (k) = ψTf (k) θ

(7)

where θ = θ − θ 0 is the deviation in the feature vector; ψTf (z) = H(z)ψT (z) and e0 (z) = H(z)υ(z) are the filtered regression matrix ψTj (z) and the filtered equation D0 (z) error; H(z) = D(z)D is the filter. k f (z) Remarks 2.1: the residual model and the above-stated key properties are exploited in developing (1) the identification of the system and the associated Kalman filter

where (z) is a stable frequency-dependent perturbation of the nominal is model G p0 (z);  is the upper bound; (.) is a 2-norm or infinity norm; G p0 (z), termed unperturbed model, is a-priori known stable proper transfer function that generates all the perturbed models. A perturbed model G(z) is a member of the set of all perturbed model in the neighbourhood of an operating point:     G(z) ∈ G p0 (z)g(z)| g(z) ≤ 

(10)

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3.1. Emulators: mimicking perturbations The perturbation term g( jω) in (8) is unknown or partially known. During the identification phase, realistic perturbations are introduced in the nominal model by varying the emulator parameters to induce variations in the phase and the magnitude of the cascade combination of the subsystems, thereby satisfying the perturbed system model (8). In other words, the emulator plays the role of mimicking the perturbation term g( jω). An emulator, denoted E(z), is selected to be the Blashke product of first-order all-pass filters expressed as a product of n first-order all-pass filters, which induce phase changes in the nominal system: E(z) =

 m   γi + z−1 1 + γi z−1 i=1

(11)

where |γi | ≤ 1 to ensure stability, the emulator parameter is a n × 1 vector γ = [ γ1 γ2 . γm ]T . The order of the all pass filter is selected to equal that of nominal system model so that an mth emulator can induce all possible phase variations in the range, −mπ–mπ, so as to mimic those of g(z) (Doraiswami & Cheded, 2015; Doraiswami et al., 2014). The nominal model of the system, G0 (z), is G0 (z) = G p0 (z) E0 (z)

(12)

where E0 (z) is the nominal emulator with parameters {γi0 }. The definition of the nominal model as G0 (z), instead of G p0 (z), ensures that (1) the order of the nominal and the perturbed model G(z) is identical and (2) the numerator and denominator polynomials of the nominal and the perturbed models are continuous functions of the emulator parameters. Hence, the analysis of the perturbed models will be mathematically tractable. The following lemma establishes the condition on structure of the emulator transfer function so that it can mimic the static and the dynamic behaviour of the perturbed system G(z) given by (10). Lemma 3.1: there exists an emulator E(z) of an order m such that E(z)G p0 (z) mimics G(z) = G p0 (z)g(z), that is E(z)G p0 (z) = g (z) G p0 (z) for all g (z)

(13)

E (z) = g (z) for all g (z)

(14)

if

Remarks 3.1: the perturbation model g(z) and its upper bound  may not be known precisely especially those associated with high-frequency behaviour of the perturbed system. Although E(z) induces all possible phase

5

variations in the phase, condition (14) of Lemma 3.1 for all g(z) may not be met as the norm E(z) may not satisfy (9). Instead it satisfies the inequality: ˆ E (z) ≤ 

(15)

ˆ is an estimate of  given in (9). where  .. Emulator-generated models and data During the identification phase, the emulator E(z) is connected in cascade with G p0 (z) to induce variations in the phase and the magnitude of the cascade combination of the subsystems. A number of experiments are performed by varying the emulator parameters {γi } one at a time, two-at-a time and so on until all the m parameters are varied simultaneously. The experiments include the nominal system model. The emulator-generated data from all the experiments are acquired. For notational simplicity, the input u(z) is chosen to be identical for all experiments. The output of the lth experiment, denoted y (k) is y (z) = G (z) u(z) + ϑ  (z) ,  = 0, 1, 2, ..., nexp − 1 (16) where nexp is the number of experiments; G (z) = E  (z)G p0 (z); E  (z) is the perturbed emulator model at the lth experiment when its parameters are varied and ϑ  (z) is the output error. The emulator-generated model G (z) is a member of a set of all emulator-generated models including the unperturbed nominal model G0 (z): ˆ G (z) ∈ {G p0 (z)E(z)| E(z) ≤ }

(17)

4. Emulator-based identification An important requirement for reliable and accurate identification is that the residual (the error between the output and its estimate) must be a zero-mean white noise process. This ensures that the identified model has captured completely the static and dynamic behaviours of the model to be identified, and the leftover from the identification is an information-less white noise process. Since the residual of the Kalman filter satisfies a key property, namely, it is a zero-mean where noise process if and only if the identified model and the model embodied in the Kalman filter (Doraiswami & Cheded, 2012b) are identical, it is then the residual ek f (k) of the Kalman filter (6) associated with the system (1) that is minimised – instead of the coloured noise equation error υ(k) of the system model (3), in order to identify not only the system model but also the associated Kaman filter without the prior knowledge of the statistics of the disturbance and the measurement noise, e.g. their mean and the covariance.

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The prediction error method is used for identification as it can be shown that it meets the above requirements (Doraiswami & Cheded, 2014). A model order less than the true one will not capture the full system behaviour, while a model higher than the true one will capture both the system’s dynamics as well as the effects of the noise and other artefacts. In recent years, objective information criteria, such as the AIC, have been used to select an optimal model order, which is a trade-off between the parameter estimation error and the complexity of the model. 4.1 Robust optimal model A robust optimal model and the associated robust Kalman filter are identified using the prediction error method from the emulator-generated model given by (16). It is shown theoretically as well as with evaluation on simulated and physical systems that the proposed identification scheme is significantly better than the conventional one with respect to robustness of the identified model. The robust optimal model is identified as follows.

of identified models {Gˆ  (z)} from the emulator-generated data {u(k), y (k)}. Hence the robust optimal model is ˆ (15). robust in the face of all perturbations E(z) ≤  .. Performance of the robust optimal model The primary objective of robust identification is that the robust optimal model is ‘closest’ to the set of perturbed system models so that the robust performance of the controller designed using the robust optimal model is compromised. The performance of the robust identified model is evaluated using a measure of the mean deviation between Gopt (z) and the perturbed models when the ˆ of the norm of the emulator, E(z), and upper bound  upper bound  of the norm of the perturbation, g(z), differ from each other. The proposed emulator-based and the conventional nominal model schemes are also compared to each other. The outputs yopt (z), yˆ (z) and y0 (z) of Gopt (z), Gˆ  (z) and the identified nominal model Gˆ 0 (z) are given, respectively, by yopt (z) = Gopt (z)u(z)

(20)

y (z) = Gˆ  (z)u(z)

(21)

yˆ0 (z) = Gˆ 0 (z)u(z)

(22)

r A set of models {Gˆ  (z)} is identified from the inputˆ



output data {u(k), y (k)} where G (z) is the identified model of G (z) given in (17) that includes the identified unperturbed nominal model, termed conventional (or nominal) model, Gˆ 0 (z). r Select one of the identified models Gˆ l (z) from the set of identified models {Gˆ  (z)}. r Compute the H∞ or H2 norm of the deviation, denoted δ l , between the selected model Gˆ l (z), and the rest of the identified models {Gˆ  (z)},  = l: δ l = Gˆ l (z) − Gˆ  (z)  = 0, 1, 2, ..., nexp − 1;  = l

(18)

r The robust optimal model Gopt (z) the one which has

The mean deviation between Gopt (z) and {Gˆ  (z)} is computed using the H2 norm or H∞ norm. The mean deviations are functions of the estimate of the upper ˆ bound . Let the maximum deviation with the robust optimal model Gopt (z) and the nominal model G0 (z) be denoted conv given by Gmax opt and G0 ˆ Gmax opt = arg max Gopt − G (z) Gˆ  (z)

Gmax 0

Gˆ  (z)

the minimal deviation:   j = arg min δ l l

Gopt (z) = Gˆ j (z) 





(19)

Let (A , B , C ), (A , B , C ) and (A0 , B0 , C0 ) be the state-space models, respectively, of the identified perturbed model Gˆ  (z), the identified robust optimal model Gopt (z) and the identified conventional model Gˆ 0 (z); (A0 − K 0C0 , [B0 K 0 ], C0 ) and (A0 − K 0C0 , [Bc K 0 ], C0 ) are, respectively, the Kalman filters associated with (A0 , B0 , C0 ) and (A0 , B0 , C0 ). 0

0

0

Remarks 4.1: the identified robust optimal model Gopt (z) is a best fit in the sense of H∞ or H2 norm to the set

(23)

= arg max Gˆ 0 − Gˆ  (z)

The mean deviation using the H2 norm or H∞ , ˆ is denoted J(), ˆ = J()

1 nexp

nexp −1



Gopt − Gˆ  (z)

2

(24)

=0

Similarly the mean deviation for the conventional identification between the identified nominal model Gˆ 0 (z) and the set of identified perturbed models {Gˆ  (z)}, ˆ is denoted Jconv (), nexp −1 2 1  ˆ ˆ G0 − Gˆ  (z) Jconv () = nexp =0

(25)

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where Gˆ  (z) for  = 0 refers to the identified nominal model Gˆ 0 (z). Theorem 4.1: ˆ <  f or all  ˆ < Jconv i f  J () ≤ J() ˆ ≥  f or all  ˆ < Jconv i f  J () = J() max Gmax opt ≤ G0

(26)

Remarks 4.2: the overall performance of the identified model degrades when the estimated bound is smaller than the true one. In case when the estimated bound is larger than the true one, the order of the identified model is larger and the controller designed using the robust optimal identified model will also be of high order, which is not desirable. Hence the order of the identified model is selected using the Akaike Information Criterion to obtain an appropriate order. In the ideal case when the system does not suffer from any perturbations, then the conventional approach is better as there is no need to use emulators to perturb the system model.

Unperturbed model of the system G p0 (z) (8) is b1 z−1 + b2 z−1 1 + a1 z−1 + a2 z−2

(27)

where a1 = −1.6630, a2 = 0.81, b1 = 1 and b2 = 0. Emulator: the emulator E(z) is selected as a first-order all-pass filter given by E0 (z) =

γ10 + z−1 γ1 + z−1 ; E = (z) 1 + γ10 z−1 1 + γ1 z−1

(28)

where |γ1 | < 1 and γ10 = 0.9 to ensure stability of the emulator. System: the nominal system model is a first-order system: G0 (z) = G p0 (z)E0 (z) =

G (z) = E (z) G p0 (z) b1 γ1 z−1 + (b2 γ1 + b1 ) z−2 + b2 z−2 1 + (a1 + γ1 ) z−1 + (a1 γ1 + a2 ) z−2 + a2 γ1 z−3 (30)

The emulator-perturbed output (16) is y (z) = G (z) u(z)+ϑ  (z) ,  = 0.1, 2, ..., nexp (31) The robust optimal model is Nopt (z) Dopt (z) 0.9004z−1 − 0.3550z−2 +1.8331z−3 = 1 + 0.2129z−1 +0.0347z−2 + 0.7297z−3 (32)

Gopt (z) =

The identified nominal model using the conventional approach is Nˆ 0 (z) Gˆ 0 (z) = Dˆ 0 (z) 0.0159z−1 + 0.0022z−2 − 0.0108z−3 (33) = 1 − 0.7621z−1 − 0.6882z−2 + 0.7297z−3

4.2 Illustrated example: identification of robust optimal model

G p0 (z) =

.. Emulator-generated models and data In the identification phase, the nominal model is connected in cascade with the emulator. A number of emulator parameter-perturbed experiments were performed by only varying γ1 in the range 0.1–0.9 in steps of 0.1. The emulator-perturbed model G (z) (17) is given by

=

Proof: the emulator-perturbed models do not cover all ˆ <  and includes them the model perturbations if  ˆ ≥ . The conventional scheme disregards perall if   turbed models altogether.

7

0.9z−1 + z−2 (29) 1 − 0.7630z−1 − 0.6867z−2 − 0.7290z−3

The eigenvalues of the robust optimal and nominal models are, respectively, −0.5765, 0.8390 ± j0.3441 with absolute values 0.5765 and 0.9068; and −0.7852, 0.8307 ± j0.3439 with absolute values 0.7852 and 0.8991. The identified optimal robust Kalman filter and the conventional one are ⎡ ⎤ 1.1635 −0.4246 −0.8710 xˆ0 (k + 1) = ⎣1.0634 −0.5825 −0.4059⎦ xˆ0 (k) 0.0704 0.3522 −0.4513 ⎡ ⎤ ⎡ ⎤ 1 0.5727 + ⎣0⎦ u(k) + ⎣0.5496⎦ y(k) (34) 0 0.6294 Dk f (z) = 1 − 0.1296z−1 −0.2841z−2 +0.0814z−3 and eigenvalues −0.5884, 0.3590 ± j0.0974; ⎡

⎤ 0.3479 −0.1619 −0.6308 xˆconv (k + 1) = ⎣ 0.6598 −0.4237 0.0025 ⎦ xˆconv (k) −0.3858 0.5194 0.0028

8

R. DORAISWAMI AND L. CHEDED

A: perturbed outputs

C:perturbed phase

1

phase

outpits

0

0 -1

-4 -6

0

10

20

30

40

0

1

2

time

frequency

B: correlation

D:error norm

0.08

2

0.06

1.5

norm

correlation

-2

0.04 0.02

3

prop conv

1 0.5

0 500

1000

time lag

1500

0

0

5

10

15

experiments

Figure . Emulator-perturbed outputs and phases, correlation and errors.

⎡ ⎤ ⎡ ⎤ 1 0.6626 + ⎣0⎦ u(k) + ⎣0.4267⎦ y(k) 0 0.4839

of the equation error as can be deduced from subfigure (B). r The proposed emulator-based identified optimal robust model is significantly closer to all the perturbed plant models than does to the conventionally identified model whose identification merely relies on the nominal model, as shown in subfigure (D). In other words, the optimal robust model is more robust than the conventionally identified model in the face of model perturbations. The maximum deviation between the robust optimal model and the perturbed models is 0.9384, while that between the nominal one and the remaining perturbed models is 2.3648. Hence, the proposed emulator-based identification is 2.52 times ‘more robust’ than the conventional approach.

5. Robust controller design (35)

Dk f (z) = 1 + 0.0729z−1 −0.2855z−2 +0.1135z−3 and eigenvalues −0.7055, 0.3163 ± j0.2465. The maximum value of the norm of the deviations (23) between (1) the robust optimal model and the perturbed models Gmax opt is 0.9384 and (2) the nominal model and is 2.3648. The mean devithe perturbed models Gmax 0 ˆ and ation using H2 norm for the proposed scheme J() ˆ the conventional scheme Jconv () given by (24) and (25) were, respectively, 0.2181 and 0.5905. Subfigures (A) and (C) of Figure 1 show, respectively, the impulse responses {y (k)} (31), and the phases of the emulator-perturbed models {Gˆ  (z)}(30). Subfigure (B) shows the autocorrelation function of the Kalman filter residual, which is a zero-mean white noise process. Subfigure (D) shows the H2 norm of the deviationδ l (18). It is clear that the mean of the deviation of the proposed identified model is significantly smaller than that of the nominal identified model. Remarks 4.3:

r The emulator-perturbed models exhibit variations in the time-domain and frequency-response behaviours. The emulator induces variations in the step responses and the phases of the perturbed models as shown in subfigures (A) and (C). r The prediction error method meets the most important requirement of a reliable and accurate identification scheme, namely the autocorrelation of the residual is a zero-mean white noise process thanks to minimisation of the Kalman filter residual instead

5.1 Closed-loop system The closed-loop system is formed of the identified perturbed model (A , B , C ), and the optimal Kalman filter in the forward path, and a robust state-feedback controller implemented using the estimated states of the robust optimal Kalman filter (A0 − K 0C0 , [B0 K 0 ], C0 ), in the feedback path. The augmented system forward path is

     x (k + 1) x (k) 0 A = 0 K 0C A0 − K 0C0 xˆ0 (k) xˆ (k + 1)

 B u(k) (36) + B0

Note that the Kalman filter (A0 − K 0C0 , [B0 K 0 ], C0 ) is associated with the robust optimal model (A0 , B0 , C0 ) and not with the system model (A , B , C ). Assumptions 5.1: it is assumed that (A , B ) is controllable and (A , C ) is observable. 5.2 State feedback controller design .. Control objective The objective is to design a robust controller based on the identified robust optimal model so that the controller ensures robust performance in the face of perturbations. .. Parameterisation of all stabilising controllers The robust controller is designed from the identified robust optimal model, and not from the nominal model. The conventional approach of identifying the nominal

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model is commonly used. The Kalman filter-based statefeedback control strategy generates a class of all stabilising controllers if and only the system is both stabilisable and detectable (Dullerud, 2000; Zhou et al., 1996). Since the robust optimal model (A0 , B0 , C0 ) is stabilisable and detectable, a Kalman filter-based state feedback strategy is obtained using the separation principle governing the estimation of the states, and the implementation of the state feedback strategy (Goodwin, Graeb, & Salgado, 2001). First, the state feedback strategy is derived using the linear quadratic optimisation approach. Then the states of (A0 , B0 , C0 ) are substituted by those of the associated Kalman filter as (1) the states of the true system (A, B, C) may not be accessible and (2) the Kalman filter states enjoys the highest robustness against the effect of disturbance and model variations. A state-feedback controller is then obtained for the state-space model of the identified robust optimal model (A0 , B0 , C0 ). The state-feedback controller is computed using the algebraic Riccati equation as this approach is most useful in controller design, and plays an important role in H2 and H∞ optimal control (Zhou et al., 1996). The state feedback, when the states are accessible, ensures 6 dB gain margin and 60° phase margin. Since (A0 , B0 , C0 ) is controllable, there exists an optiˆ mal state-feedback controller u(k) which minimises the linear quadratic regulator cost function J(u) given by ∞   0 T x (k) Q f x0 (k) + R f u2 (k) J (u) =

(37)

k=0

uˆ (k) = arg min {J (u)}

(38)

u

The optimal cost function J 0 and the optimal feedback ˆ strategy u(k) are J 0 = min {J (u)}

9

The state feedback gains {F i0 } for the conventional model (A0 , B0 , C0 ) are similarly obtained. 5.3 Closed-loop augmented system The closed-loop system formed by substituting u(k) = ˆ r(k) − u(k) in the augmented model in forward path using (36) and (39):

x (k + 1) 0 xˆ (k + 1)



=

−B F A 0  0 K C A − K 0C 0

 B r(k) + B0

 −B0 F

x (k) xˆ p0 (k)



(41)

where the feedback gain F =F 0 for the optimal model and F =F i0 for the conventional model. The output of the closed-loop system y (k) and the control input u(k) are given by y (k) = C x(k) + v  (k) 0 u(k) = r(k) − F 0 xˆ (k)

(42)

The closed-loop system is formed of (1) the true system (A , B , C ) and the Kalman filter is associated with (A0 , B0 , C0 ) and not with the true system (A , B , C ). Hence, there is a mismatch between the perturbed system and the Kalman filter. The feedback controller (39) must ensure stability and performance of the closed-loop system for the class of all perturbed systems generated by the emulators. In other words, a single controller based on the optimal nominal Kalman filter basedˆ state feedback u(k) = −F 0 xˆ0 (k) must meet the stability and performance requirements without the need for continuous adaptation for the set of all perturbed systems {(A , B , C )} in the neighbourhood of the robust optimal model (A0 , B0 , C0 ).

u

0

uˆ (k) = −F 0 xˆ (k)

(39)

where x0 (k) is the state of (A0 , B0 , C0 ) and xˆ0 (k) is the estimate of the Kalman filter. The state feedback gain F 0 satisfies the following algebraic Riccati equation: P f = (A0 )T P f A0 − (A0 )T P f B0 T × (R f + (B0 ) P f B0 )−1 (B0 )T P f A0 + Q f (40) T F 0 = (R f + (B0 ) P f B0 )−1 (B0 )T P f A0 where Q f = QTf ≥ 0, R f > 0 P f = PTf ≥ 0. The algebraic Riccati equation is used in the design of controllers, while the Lyapunov function is used for the analysis of the control system (Zhou et al., 1996).

5.4. Robust controller A novel emulator-perturbed model approach to obtain the robust state feedback control strategy is proposed here and is similar in spirit to that proposed in (Gadewadiker, Lewis, & Abu-Khalaf, 2006; Gadewadiker, Lewis, & Subbarao, 2008; Zhou et al., 1996). (1) A set of control weights R f i , i = 0, 1, 2, ..., n f , are selected. (2) For each of the selected R f i , the state feedback gain F i and the dominant pole of the transition matrix A0 − B0 F i for the robust optimal model (A0 , B0 , C0 ) are obtained using the algebraic Riccati equation approach (40).

10

R. DORAISWAMI AND L. CHEDED

(3) The set of state feedback gains F i is used in the closed-loop augmented state transition matrix to transfer the augmented system formed of the optimal model (A0 , B0 , C0 ) and its associated optimal Kalman filter in the forward path (41) given by

A0 K 0C0

 −B0 F i , i = 0, 1, 2, ..., n f . A0 − K 0C0 − B0 F i (43)

(4) The steps (2) and (3) are repeated using the conventional model (A0 , B0 , C0 ) and its associated Kalman filter (A0 − K 0C0 , [Bc K 0 ], C0 ) in order to compare the performances of the proposed and the conventions approaches:

A0 K 0C0

−B0 F i0 A0 − K 0C0 − B0 F i0

 i = 0, 1, 2, ...n f (44)

.. Design of robust Kalman filter-based state feedback An iterative scheme for selecting a Kalman filter-based state feedback that can stabilise all the set of closed-loop systems is given in (41). The states of the perturbed model (A , B , C ) are not accessible and the Kalman filter is not associated with the perturbed model. In this case, the controller for the perturbed model becomes an output feedback problem and not a state feedback one (Gadewadiker et al., 2008; Gadewadiker et al., 2006) and hence an iterative scheme is employed. The iterative scheme is developed by exploiting the above-stated properties of the Ricatti equation. The performance measure is the linear quadratic regulator cost function, which is an extension of the cost function used for the optimal model (37) to the closed-loop system given by Jcli

∞    T xa (k) Q f xa (k) + R f i u2 (k) (u) =

(45)

k=0

where F i0 is the feedback gain computed for the conventional model (A0 , B0 , C0 ). (5) The optimal cost function, the stability margin using the eigenvalues of (43) and the control effort, namely the norm of the feedback gains, are determined. Remarks 5.1: The performance and the stability margin are measured using the cost function J i and the dominant eigenvalue, pidom , respectively. The lower the cost function J i , the higher is the performance, and the lower the dominant pole, the higher is the stability margin. The system is asymptotically stable if pidom < 1. The following are the properties of the algebraic Riccati equation approach.

r The weighting R f i is only varied for a chosen Q f

instead of varying both Q f and R f i as the state feedback depends only on the ratio of the Q f /R f i (Doraiswami et al., 2014). r The optimal cost function J i increases monotonically, while the dominant pole pidom and the normF i , decrease monotonically as R f i increases. In other words, the performance of the system, the stability margin, and the control effort decrease with the increase in the weight of the control input. The state feedback with a lower control weight ensures higher performance and stability at the expense of a higher control effort, and hence the design of state feedback is a trade-off between the desired performance, and the control effort.

where xa (k) = [ x (k) xˆ0 (k) ]T . The optimal cost function, denoted ηi , is similar to (39) and is given by   ηi = min Jcli (u) u

0

uˆi (k) = −F i xˆ (k)

for all i = 0, 1, 2, ..., l f ,  = 0, 1, 2, ..., nexp

(46) The performance of the closed-loop system Jcli with state feedback gains F i i = 1, 2, ..., l f and the stability of the closed-loop system (41) are now determined .The performance Jcli is compared with the optimal cost function, denoted η0 , that is associated with the robust optimal model when the state feedback gain is F 0 . The optimal cost ηi is deemed to be acceptable if ηi satisfies the inequality: η0 ≤ ηc ≤ η

(47)

where η is specified maximum value. .. Iterative scheme

r The iteration starts with using the initial feedback gain and parameters of the Ricatti equation F 0 ,Q f ,R f , P f given by (39) and (40). r Select (A , B , C ) r At the ith iteration, select the feedback gain F i , and determine the optimal cost ηcli . If ηcli does not satisfy the inequality (47), then move to the next iteration (i + 1)th by selecting F i+1 and repeat. r If the inequality (47) is satisfied, select (A+1 , B+1 , C+1 ) and repeat the steps.

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5.5

B:control effort

1.02

prop conv

0.9

0.74

0.98

0.6 0.96

0.8

2

6 8 weights C:performance: optimal

0.7

7

performanc e

0.6 0.66 3.5 0.64 3

0.5

0.4 5 10 15 control weight

5 10 15 control weight

Figure . State feedback: cost function and control effort.

.. Proposed robust controller: comparison ˆ = η0 , The optimal cost functions for the proposed J(u) ˆ = ηconv are comand that of the conventional Jconv (u) pared to each other. Lemma 5.4: η0 ≤ ηconv

0.4

6 5 4

(48)

Proof: The proposed controller is based on the identified robust optimal model Gopt (z) whereas the conventional robust controller Gˆ 0 (z) is obtained from the identified nominal model G0 (z) (Dullerud, 2000; Zhou et al., 1996). The maximum deviations of Gopt (z) and Gˆ 0 (z)are, max (23). From (26) of Therespectively, Gmax opt and G0 max orem 1, these deviations satisfy: Gmax opt ≤ G0 . Since the deviation of the conventional scheme is larger, then  η0 ≤ ηconv . 5.5 Illustrative example: robust controller design The optimal identified model (32) and the associated Kalman filter (34) of the example considered in Section 4.2 is stabilised by using a Kalman filter-based state feedback controller. .. Design of state feedback The steps (a)–(e) given in Section 5.4 on robust controller were all implemented for this example. Subfigures (A), (B), and (C) of Figure 2 compare the cost function, stability margin, and the control effort for the optimal and the conventional models as R f was varied in the range R f = 1 to R f = 15 using (37), (43), and (44).

0

5 10 15 control weight D: performance; conventional

8 1 2 3 4 5

7

3

0.62 5 10 15 control weight

10

8

0.68

4

4

performance

gain

pole

performanc e

0.72 4.5 0.7

prop conv

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0.76

5

A:stability margin

C:control effort

B: stability margin 0.78

gain

A: cost function

11

6 7 8 9 10

6 5 4 3

5 10 perturbations

15

5 10 perturbations

15

Figure . Performance with respect to model perturbations and feedback gains.

Remarks 5.2: It is interesting to note that, when the control weight R f increases. the cost function increases whereas both the stability margin and the control effort decrease. The performance deteriorates whereas the control effort decreases as R f increases. The performance and control effort profiles are similar for the optimal and the conventional models. There is a trade-off between the performance and control effort, which is exploited in developing the iterative scheme for designing the robust controller. .. Design of a robust state feedback The problem of selecting a Kalman filter-based state feedback that can stabilise all the set of perturbed plants is considered. The Kalman filter-based state feedback of the closed-loop system (41) and (42) using the augmented models for the optimal and the conventional schemes is given by (43) and (44). The set of perturbed systems {(A , B , C )} generated using the emulatorperturbed experiments given by (10) are identified using the emulator-generated data (16). Subfigures (A) and (B) of Figure 3 compare the stability margin and control effort for the optimal and conventional models as the control weight is varied from R f = 1 to R f i = 10. Note that that the closed-loop system of the perturbed plants and the conventional model is unstable when R f < 4 and stable for R f > 4, whereas with the optimal model, the closed-loop system is stable for all values of R f , thus showing the great impact that the optimality of the model has had on the stability of the closed-loop system. Subfigure (C) shows the performance of the closedloop system formed of (1) the perturbed models (A , B , C ),  = 0, 1, 2, ..., 17 (41) and (2) the optimal

12

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Kalman filter (41) and (43). The performance is measured using the cost function (45). The cost functions for the selected control weights R f = 1, 2, 3, 4, 5 are all shown in Subfigure (C). The performance associated with each of the control weight is indicated in the graph legend of this subfigure. Subfigure (D), similar to the subfigure (C), shows the performance of the closed-loop system with the conventional Kalman filter (41) and (44). As the closedloop system is stable for R f > 4, only, the performance is shown only for the selected control weights R f = 6, 7, 8, 9 and 10. Remarks 5.3: The closed-loop system is stable for all perturbations and for all selected control weights when the optimal Kalman filter-based state feedback is employed to stabilise the perturbed system. However the conventional Kalman filter-based state feedback stabilises the closed-loop when the control weight is selected to be high. The cost function is lower, hence the performance higher, for all model perturbations when the state feedback is implemented using the optimal Kalman filter (A0 − K 0C0 , [B0 K 0 ], C0 ). However, the performance with the conventional Kalman filter (A0 − K 0C0 , [Bc K 0 ], C0 ) is degraded as the control weights have to higher to ensure closed-loop stability. There is a trade-off between the performance and stability or the control effort.

5.6 Monitoring the status of the system The Kalman filter plays a crucial role in monitoring the status of the closed-loop system including its performance, stability and fault detection and isolation capabilities in view of its key property, namely that its residual is a zero-mean white noise process if and only if the true model and the model embedded in the Kalman filer are identical to each other, that is (A , B , C ) = (A0 , B0 , C0 ). Otherwise there will be an additional term, labelled a fault-indicating component, which indicates a model mismatch (7) (Doraiswami & Cheded, 2012a, 2013, 2015; Doraiswami et al., 2010, 2014). The fault-indicating component e f (k) is generally unknown and hence a composite hypothesis testing scheme is employed where e f (k) is replaced by its maximum likelihood estimate eˆ f (k) (Doraiswami et al., 2014). Depending upon the size of eˆ f (k), the status of the system is unfolded starting with (1) the performance degradation, (2) stability measure, (3) presence of fault and (4) isolation of the faulty subsystem. If there is degradation in the performance and stability measure, the system is re-identified and the controller redesigned accordingly to ensure high performance and stability over a

dc motor

pump inflow

RL controller

i

Qi

ω H1

H2

Qo

outflow leakage

Q

Figure . Two-tank liquid level system.

wide operating region. If a fault is detected, it is then isolated and either the fault is accommodated or the system is shut down for repair (Doraiswami & Cheded, 2012a; Doraiswami & Cheded, 2010; Doraiswami et al., 2010). The details of the monitoring scheme are not given here due to space constraint.

6. Evaluation on a physical process control system The proposed Kalman filtered-based controller was evaluated on a physical process control system formed of two tanks connected by a pipe. The first tank is filled by a DC motor-driven pump (Doraiswami & Cheded, 2013; Doraiswami & Cheded, 2015; Doraiswami et al., 2014). The controller maintains a desired fluid level in the second tank at a specified level, as shown in Figure 4. The system formed of the DC motor and a pump relating the input to the motor, u, and the flow Qi is given by Q˙ i = −am Qi + bm φ(u)

(49)

where am and bm are the parameters of the motor-pump subsystem and φ(u) is a dead-band and saturation-type of nonlinearity. The liquid level system is modelled by A1

dH1 = Qi − C12 ϕ (H1 − H2 ) − C ϕ (H1 ) dt

dH2 A2 = C12 ϕ (H1 − H2 ) − C0 ϕ (H2 ) dt

(50)

 where ϕ(.) = sign(.) 2g(.),Q = C ϕ(H1 ) is the leakage flow rate, Q0 = C0 ϕ(H2 ) is the output flow rate, H1 is the height of the liquid in tank 1, H2 the height of the liquid in tank 2, A1 and A2 the cross-sectional areas of the 2 tanks, g = 980cm/sec2 the gravitational constant, and C12 and Co are the discharge coefficients of the inter-tank and output valves, respectively.

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A: height

13

B: control input 1.6

200

180 1.4 160 1.2

140

control input

height

120 100

1

0.8

80 60

0.6

1 0.75 0.5 0.25

40

0.4

20

0

100

200

300

400

500

0

time

100

200

300

400

500

time

Figure . The closed loop system: emulator-based system input–output data.

The two-stage closed-loop identification scheme is employed to identify these subsystems. It consists of the following two stages [Doraiswami and Cheded, 2014].

r In Stage 1, the closed-loop system is identified using data formed of the reference input r(k), the error e(k), the controller input u(k), flow f (k) and the height h(k). r In Stage 2, the subsystems Geu , Guq and Gqh are then identified using the subsystem’s estimated input and output measurements obtained from the first stage: ˆ ]T = D−1 (z)N(z) r(z) (51) [ e(z) ˆ ˆ u(z) fˆ(z) h(z) 6.1 Emulator parameter perturbed identification The system is identified in a closed-loop configuration by performing a number of emulator parameter-perturbed experiments by perturbing the sensor, and the actuator gains. Figure 5 shows the step responses of the emulator parameter-perturbed closed-loop system input and the output data. The emulator parameters of the actuator and the sensor were varied: the emulator values were selected to be 1, 0.75, 0.5 and 0.25 times the nominal value.

6.2 Optimal nominal model The optimal state-space model (A0 , B0 , C0 ) and the associated Kalman gain K 0 are



 1.220 −0.2228 1 0.6040 0 0 0 A = ;B = ;K = ; 1 0 0 0.6046 Identification also yields the following four open-loop transfer functions: 0.4576z−1 u(z) = 0.0067 + ; Gˆ eu (z) = e(z) 1 − z−1 0.0104z−1 Q(z) = ; Guq (z) = u(z) 1 − 0.9968z−1 0.0104z−1 Q(z) = ; Guq (z) = u(z) 1 − 0.9968z−1 0.7856z−1 h(z) = Gqh (z) = Q(z) 1 − 1.0039z−1

(52)

Figure 6 shows the measurement data and their estimates. Subfigures (A), (B), (C) and (D) give the plots of the tracking error, control input, flow rate and height versus time, respectively. The two-tank level system is highly nonlinear as can be clearly seen, especially from the flow rate profile in subfigure (C). There is a saturation-type nonlinearity involved in this physical system.

14

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A: error

control

error

0

error estimate

-50

6.3 Parameterisation of stabilising controllers

B: control input

0

-100 -150

control estimate

-0.5

-1

-200 0

500

1000

0

500

1000 time D: height

0

200

-1

150

height

flow

C: flow rate

-2 flow estimate

-3 0

500

100 50 0

1000

The performance measures, the feedback gains and, step responses of the resulting closed-loop system are computed as shown in Section 5 on robust control. The closed-loop system formed of the optimal model (A0 , B0 , C0 ), its associated Kalman filter (A0p0 − K 0C0p0 , [K 0 B0p0 ], C0p0 ), (43) is

height estimate 0

500

1000 time

Figure . Step response of closed-loop system: r(k), e(k), u(k) f (k) and h(k).

Remarks 6.1: the system is marginally stable with a pole close to the unit circle with poles at 0.9989 and 0.2231. The eigenvalues of A0 − K 0C0 of the Kalman filter are 0.3645 and 0.2237.

⎤ 1.2220 −0.2228 −0.6905 0.5058 ⎢ 1 ⎥ x0 (k + 1) 0 0 0 ⎥ =⎢ 0 ⎣ 0 0.6332 0.5315 −0.3502 ⎦ xˆ (k + 1) 0 0.6337 1.0000 −0.6337 ⎡ ⎤ 1

0  ⎢ x (k) 0⎥ ⎥ r(k) × 0 +⎢ (53) ⎣ 1⎦ xˆ (k) 0 

⎡

The performance and the stability of the nominal closed-loop system (53) was verified in the face of model

A: performance measure

B: control effort

4.9

0.85 0.8

4.7 0.75

4.6

gain

c os t func tion

4.8

4.5 4.4

0.7 0.65

4.3 0.6 0.6

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C: perturbed closed loop

0.8

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D: robust closed loop output 1

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s tep res pons e

0.8 0.8 0.6 0.4

0.4 0.2

0.2 0

0.6

2

4

6

8 time

10

12

14

0

5

10

Figure . Stabiliser parameterisation: performance measure, feedback gains and step responses.

15

20 time

25

30

35

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uncertainty using the set of emulator-perturbed models {(A , B , C )} (36). Figure 7 shows the results of the performance of the identified optimal Kalman filter-based state feedback in the face of model perturbations as the control weights are varied. Subfigures (A), (B), (C) and (D) show, respectively, the measures of performance J i with respect to the selected weights {R f i }, the norm of the state feedback gains {F i } with respect to the selected weights {R f i }, and the step responses of the resulting closed-loop system when the emulator parameters were perturbed, and the nominal closed-loop system. The H∞ bound are η = 9.979 and η0 = 9.536.

15

approach, both of which constitute two important classes of engineering systems.

Acknowledgments The authors acknowledge the support of the department of Electrical and Computer Engineering, the University of New Brunswick, the National Science and Engineering Research Council (NSERC) of Canada, and the University of KFUPM, Saudi Arabia.

Disclosure statement No potential conflict of interest was reported by the authors.

References 7. Conclusion In order to ensure a robust performance and stability of a controller, which is designed using an identified system model, it is crucial that both the identification scheme as well as the controller design strategy be selected appropriately. A key point that was stressed and successfully addressed in this paper is that the accuracy of the physical parameter-perturbed identification method has produced a smaller modelling error which in turn allowed for a tighter norm bound and hence led to a control system that enjoys a stronger robustness in both stability and performance. The controller was designed using a Kalman filter-based state feedback and, as such, was, therefore, able to handle the model system uncertainty of the identified model and to provide tools for a trade-off between robust stability, robust performance and control input limitations. The identification and the control design were successfully evaluated on a number of simulated as well practical physical systems, including the laboratory-scale two-tank liquid level systems. It was shown theoretically as well through evaluation on simulated and physical systems that the proposed scheme is significantly better compared to the conventional one with respect to both robust performance and robust stability. This work has amply demonstrated the powerful use of emulators in providing an accurately identified system model and an optimal robust controller. Finally, the effectiveness of the combination of our emulator-based accurate identification technique coupled with robust control design technique, as demonstrated in this paper, can be readily extended to other controller design techniques or to any other design or development tasks that rely on accurately identified system models. Two open problems worth mentioning are the extension of this work to multi-input, multi-output systems, and to nonlinear systems using linear parameter-varying

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