Identification of Linearized Models and Robust Control of Physical Systems

This chapter presents the design of a controller that ensures both the robust stability and robust performance of a physical plant using a linearized identified model . The structure of the plant and the statistics of the noise and disturbances affecting the plant are assumed to be unknown. As the design of the robust controller relies on the availability of a plant model, the mathematical model of the plant is first identified and the identified model, termed here the nominal model, is then employed in the controller design. As an effective design of the robust controller relies heavily on an accurately identified model of the plant, a reliable identification scheme is developed here to handle unknown model structures and statistics of the noise and disturbances. Using a mixed-sensitivity H∞ optimization framework, a robust controller is designed with the plant uncertainty modeled by additive perturbations in the numerator and denominator polynomials of the identified plant model. The proposed identification and robust controller design are evaluated extensively on simulated systems as well as on two laboratory-scale physical systems, namely the magnetic levitation and twotank liquid level systems. In order to appreciate the importance of the identification stage and the interplay between this stage and the robust controller design stage, let us first consider a model of an electro-mechanical system formed of a DC motor relating the input voltage to the armature and the output angular velocity. Based on the physical laws, it is a third-order closed-loop system formed of fast electrical and slow mechanical subsystems. It is very difficult to identify the fast dynamics of this system, and hence the identified model will be of a second-order while the true order remains to be three. Besides this error in the model order, there may also be errors in the estimated model parameters. Consider now the problem of designing a controller for this electro-mechanical system. A constant-gain controller based on the identified second-order model will be stable for all values of the gain as long the negative feedback is used. If, however, the constant gain controller is implemented on the physical system, the true closed-loop third-order system may not be stable for large values of the controller gain. This simple example clearly shows the disparity between the performance of the identified system and the real one and hence provides a strong motivation for designing a robust controller which factors uncertainties in the model.

2001).The proposed scheme is extensively tested on both simulated systems and physical laboratory-scale systems namely, a magnetic levitation and two-tank liquid level systems.
The key contribution herein is to demonstrate the efficacy of (a) the proposed model order selection criterion to reduce the uncertainty in the plant model structure, a criterion which is simple, verifiable and reliable (b) the two-stage closed-loop identification scheme which ensures quality of the identification performance, and (c) the mixed-sensitivity optimization technique in the H ∞ -framework to meet the control objectives of robust performance and robust stability without violating the physical constraints imposed by components such as actuators, and in the face of uncertainties that stem from the identified model employed in the design of the robust controller.It should be noted here that the identified model used in the design of the robust controller is the linearized model of the physical system at some operating point, termed the nominal model.The chapter is structured as follows.Section 2 discusses the stability and performance of a typical closed-loop system.In Section 3, the robust performance and robust stability problems are considered in the mixed-sensitivity H ∞ framework.Section 4 discusses the problem of designing a robust controller using the identified model with illustrated examples.Section 5 gives a detailed description of the complete identification scheme used to select the model order, identify the plant in a closed-loop configuration and in the presence of unknown noise and disturbances.Finally, in Section 6, evaluations of the designed robust controllers on two-laboratory scale systems are presented.

Stability and performance of a closed-loop system
An important objective of the control system to ensure that the output of the system tracks a given reference input signal in the face of both noise and disturbances affecting the system, and the plant model uncertainty.A further objective of the control system is to ensure that the performance of the system meets the desired time-domain and frequency-domain specifications such as the rise time, settling time, overshoot, bandwidth, and peak of the magnitude frequency response while respecting the constraints on the control input and other variables.An issue of paramount practical importance facing the control engineer is how to design a controller which will both stabilize the plant when its model is uncertain and ensure that its performance specifications are all met.Put succinctly, we seek a controller that will ensure both stability and performance robustness in the face of model uncertainties.To achieve this dual purpose, we need to first introduce some analytical tools as described next.

Key sensitivity functions
Consider the typical closed-loop system shown in Fig. 1 where 0 G is the nominal plant, 0 C the controller that stabilizes the nominal plant 0 G ; r and y the reference input, and output, respectively; i d and 0 d the disturbances at the plant input and plant output, respectively, and v the measurement or sensor noise.The nominal model, heretofore referred to as the identified model, represents a mathematical model of a physical plant obtained from physical reasoning and experimental data.Let w and z be, respectively, a (4x1) input vector comprising r, 0 d , i d and v , and a (3x1) output vector formed of the plant output y, control input u, and the tracking error e , as given below by: The four key closed-loop transfer functions which play a significant role in the stability and performance of a control system are the four sensitivity functions for the nominal plant and nominal controller.They are the system's sensitivity 0 S , the input-disturbance sensitivity 0 i S , the control sensitivity 0 u S and the complementary sensitivity 0 T , given by: 00 0 0 000 0 00 0 0 00 00 00 00 1 ,, , 11 1 The performance objective of a control system is to regulate the tracking error ery =− so that the steady-state tracking error is acceptable and its transient response meets the timeand frequency-domain specifications respecting the physical constraints on the control input so that, for example, the actuator does not get saturated.The output to be regulated, namely e and u, are given by: 00 00 () The transfer matrix relating w to z is then given by: 00 0 0

Stability and performance
One cannot reliably assert the stability of the closed-loop by merely analyzing only one of the four sensitivity functions such as the closed-loop transfer function 0 () Ts because there may be an implicit pole/zero cancellation process wherein the unstable poles of the plant (or the controller) may be cancelled by the zeros of the controller (or the plant).The cancellation of unstable poles may exhibit unbounded output response in the time domain.
In order to ensure that there is no unstable pole-zero cancellation, a more rigorous definition of stability, termed internal stability, needs to be defined.The closed-loop system is internally stable if and if all the eight transfer function elements of the transfer matrix of Equation ( 6) are stable.Since there are only four distinct sensitivity functions, 0 S , 0 i S , 0 u S and 0 T , the closed-loop system is therefore internally stable if and only if these four sensitivity functions 0 S , 0 i S , 0 u S and 0 T are all stable.Since all these sensitivity functions have a common denominator ( 00 1 GC + ), the characteristic polynomial 0 () s ϕ of the closedloop system is: sNs DsDs Ns where 00 (), () NsDs and 00 (), () NsDs are the numerator and the denominator polynomials of 0 () Gsand 0 () Cs, respectively.One may express internal stability in terms of the roots of the characteristic polynomial as follows.Lemma 1 (Goodwin, Graeb, and Salgado, 2001): The closed-loop system is internally stable if and only if the roots of 0 () s ϕ all lie in the open left-half of the s-plane.We will now focus on the performance of the closed-loop system by analyzing the closedloop transfer matrix given by Equation ( 6).We will focus on the tracking error e for performance, and the control input u for actuator saturation:

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The tracking error e is small if (a) 0 S is small in the frequency range where r and 0 d are large, (b) 0 u S is small in the frequency range where i d is large and (c) 0 T and is small in the frequency range where v is large.

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The control input u is small if (a) 0 u S is small in the frequency range where r , 0 d and v are large, and (b) 0 T is small in the frequency range where i d is large.Thus the performance requirement must respect the physical constraint that imposes on the control input to be small so that the actuator does not get saturated.

Robust stability and performance
Model uncertainty stems from the fact that it is very difficult to obtain a mathematical model that can capture completely the behavior of a physical system and which is relevant for the intended application.One may use physical laws to obtain the structure of a mathematical model of a physical system, with the parameters of this model obtained using system identification techniques.However, in practice, the structure as well as the parameters need to be identified from the input-output data as the structure derived from the physical laws may not capture adequately the behavior of the system or, in the extreme case, the physical laws may not be known.The "true" model is a more comprehensive model that contains features not captured by the identified model, and is relevant to the application at hand, such as controller design, fault diagnosis, and condition monitoring.The difference between the nominal and true model is termed as the modeling error which includes the following:

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The structure of the nominal model which differs from that of the true model as a result of our inability to identify features such as high-frequency behavior, fast subsystem dynamics, and approximation of infinite-dimensional system by a finite-dimensional ones.

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Errors in the estimates of the numerator and denominator coefficients, and in the estimate of the time delay

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The deliberate negligence of fast dynamics to simplify sub-systems' models.This will yield a system model that is simple, yet capable enough to capture the relevant features that would facilitate the intended design.

Co-prime factor-based uncertainty model
The numerator-denominator perturbation model considers the perturbation in the numerator and denominator polynomials separately, instead of lumping them together as a single perturbation of the overall transfer function.This perturbation model is useful in applications where an estimate of the model is obtained using system identification methods such as the best least-squares fit between the actual output and its estimate obtained from an assumed mathematical model.Further, an estimate of the perturbation on the numerator and denominator coefficients may be computed from the data matrix and the noise variance.Let 0 G and G be respectively the nominal and actual SISO rational transfer functions.The normalized co-prime factorization in this case is given by where 0 N and N are the numerator polynomials, and both 0 D and D the denominator polynomials.In terms of the nominal numerator and denominator polynomials, the transfer function G is given by: where N Δ and D RH ∞ Δ∈ are respectively the frequency-dependent perturbation in the numerator and denominator polynomials (Kwakernaak, 1993).Fig. 2 shows the closed-loop system driven by a reference input r with a perturbation in the numerator and denominator polynomials.The three relevant signals are expressed in equations (10-12).( )

Robust stability and performance
Since the reference input does not play any role in the stability robustness, it is set equal to zero and the robust stability model then becomes as given in Fig. 3

. Stability robustness model with zero reference input
The robust stability of the closed-loop system with plant model uncertainty is established using the small gain theorem.
Then the closed-loop system stability problem is well posed and the system is internally stable for all allowable numerator and denominator perturbations, i.e.: [ ] Proof: The SISO robust stability problem considered herein is a special case of the MIMO case proved in (Zhou, Doyle, & Glover, 1996).Thus to ensure a robustly-stable closed-loop system, the nominal sensitivity 0 S should be made small in frequency regions where the denominator uncertainty D Δ is large, and the nominal control input sensitivity 0 u S should be made small in frequency regions where the numerator uncertainty N Δ is large.
Our objective here is to design a controller 0 C such that robust performance and robust stability of the system are both achieved, that is, both the performance and stability hold for all allowable plant model perturbations [ ] for some 0 0 γ > .Besides these requirements, we need also to consider physical constraints on some components such as actuators, for example, that especially place some limitations on the control input.From Theorem 1 and Equation ( 6), it is clear that the requirements for robust stability, robust performance and control input limitations are inter-related, as explained next: • Robust performance for tracking with disturbance rejection as well as robust stability in the face of denominator perturbations require a small sensitivity function 0 S in the lowfrequency region and,

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Control input limitations and robust stability in the face of numerator perturbations require a small control input sensitivity function 0 u S in the relevant frequency region.
With a view to addressing these requirements, let us select the regulated outputs to be a frequency-weighted tracking error w e , and a weighted control input w u to meet respectively the requirements of performance, and control input limitations.W is chosen to be larger in magnitude than S W .For steady-state tracking with disturbance rejection, one may include in the weighting function S W an approximate but stable 'integrator' by choosing its pole close to zero for continuous-time systems or close to unity for discrete-time systems so as to avoid destabilizing the system (Zhou, Doyle, and Glover, 1996).Let rz T be the nominal transfer matrix (when the plant perturbation 0 0 Δ= ) relating the reference input to the frequency-weighted vector output w z , which is a function of 0 G and 0 C , be given by: ( ) It is shown in (McFarlane & Glover, 1990) that the minimization of rz T ∞ as given by Equation ( 18), guarantees not only robust stability but also robust performance for all

H ∞ controller design using the identified model
Consider the problem of designing a controller for an unknown plant G.We will assume however that the system G is linear and admits a rational polynomial model.A number of identification experiments are performed off-line under various operating regimes that includes assumptions on the model and its environment, such as : The length of the data record • The type of rich inputs • Noise statistics

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The plant operates in a closed-loop, thus making the plant input correlated with both the measurement noise and disturbances • Combinations of any the above Let ˆi G be the identified model from the th i experiment based on one or more of the above stated assumptions.Let ˆi C be the corresponding controller which stabilizes all the plants in the neighborhood of ˆi G within a ball of radius 1/ i γ .Given an estimate of the plant model ˆi G , the controller ˆi C is then designed using the mixed-sensitivity H ∞ optimization scheme , with both the identified model ˆi G and the controller ˆi C based on it, now effectively replacing the nominal plant 0 G and nominal controller 0 C , respectively.Let the controller ˆi C stabilize the identified plant ˆi G for all ˆ1/ ii γ ∞ Δ≤ where ˆi Δ is formed of the perturbations in the numerator and denominator of ˆi G .To illustrate the identification-based H ∞optimization scheme, let us consider the following example.Let the true order of the system G be 2 and assume the noise to be colored.Let ˆ:1 , 2 , 3 i Gi = be the estimates obtained assuming the model order to be 2, 3, and 4, respectively and let the noise be a zero-mean white noise process; 4 Ĝ is obtained assuming the model order to be 2, the noise to be colored but the input not to be rich enough; Let 5 Ĝ be an estimate based on correct assumptions regarding model order, noise statistics, richness of excitation of the input and other factors as pointed out above.
Clearly the true plant G may not be in the neighborhood of ˆi G , i.e. ˆi GS ∉ for all 5 i ≠ where The set ˆi S is a ball of radius ( 1/ i γ ) centered at ˆi G . to have a pole close to the unit circle to ensure an acceptable small steady-state error.The controller will have a pole at 0.99 approximating a stable integrator.The plant is identified for (a) different choices of model orders ranging from 1 to 10 when the true order is 2, and (b) different values of the standard deviation of the colored measurement noise v σ .Fig. 6 shows the step and the magnitude response of the sensitivity function.The closed-loop system is unstable when the selected order is 1 and for some realizations of the noise, and hence these cases are not included in the figures shown here.When the model order is selected to be less than the true order, in this case 1, and when the measurement noise's standard deviation v σ is large, the set of identified models does not contain the true model.
Consequently the closed-loop system will be unstable.

Comments:
The robust performance and the stability of the closed-loop system depend upon the accuracy of the identified model.One cannot simply rely on the robustness of the H ∞ controller to absorb the model uncertainties.The simulation results clearly show that the model error stems from an improper selection of the model order and the Signal-to-Noise Ratio (SNR) of the input-output data.The simulation results show that there is a need for an appropriate identification scheme to handle colored noise and model order selection to ensure a more robust performance and stability.

Identification of the plant
The physical system is in general complex, high-order and nonlinear and therefore an assumed linear mathematical model of such a system is at best an approximation of the 'true model'.Nevertheless a mathematical model linearized at a given operating point can be identified and the identified model successfully used in the design of the required controller, as explained below.Some key issues in the identification of a physical system include (a) the unknown statistics of the noise and disturbance affecting the input-output data (b) the proper selection of an appropriate structure of the mathematical model, especially its order and (c) the plants operating in a closed-loop configuration.
For the case (a) a two-stage identification scheme, originally proposed in (Doraiswami, 2005) is employed here.First a high-order model is selected so as to capture both the system dynamics and any artifacts (from noise or other sources).Then, in the second stage, lowerorder models are derived from the estimated high-order model using a frequency-weighted estimation scheme.To handle the model order selection, and the identification of the plant, especially an unstable one, approaches proposed in (Doraiswami, Cheded, and Khalid, 2010) and (Shahab and Doraiswami, 2009) are employed respectively.

Model order selection
For mathematical tractability, the well-known criteria based on information-theoretic criteria such as the famous Akaike Information Criterion (Stoica and Selen, 2004), when applied to a physical system, may require simplified assumptions such as long and uncorrelated data records, linearized models and a Gaussian probability distribution function (PDF) of the residuals.Because of these simplifying assumptions, the resulting criteria may not always give the correct model order.Generally, the estimated model order may be large due to the presence of artifacts arising from noise, nonlinearities, and pole-zero cancellation effects.
The proposed model order selection scheme consists of selecting only the set of models, which are identified using the scheme proposed in (Doraiswami, 2005), and for which all the poles are in the right-half plane (Doraiswami, Cheded, and Khalid, 2010).The remaining identified models are not selected as they consist of extraneous poles.
Proposed Criterion: The model order selection criterion hinges on the following Lemma established in (Doraiswami, Cheded, and Khalid, 2010).
Lemma: If the sampling frequency is chosen in the range 24 cs c f ff ≤ < , then the complexconjugate poles of the equivalent discrete-time equivalent of a continuous-time system will all lie on the right-half of the z-plane, whereas the real ones will all lie on the positive real line.This shows that the discrete-time poles lie on the right-half of the z-plane if the sampling rate ( s f ) is more than twice the Nyquist rate ( 2 c f ).Thus, to ensure that the system poles are located on the right-half and the noise poles on the left-half of the z-plane, the sampling rate s f must be larger than four times the maximum frequency max s f of the system, and less than four times the minimum frequency of the noise, min

Identification of a plant operating in closed loop
In practice, and for a variety of reasons (for e.g.analysis, design and control), it is often necessary to identify a system that must operate in a closed-loop fashion under some type of feedback control.These reasons could also include safety issues, the need to stabilize an unstable plant and /or improve its performance while avoiding the cost incurred through downtime if the plant were to be taken offline for test.In these cases, it is therefore necessary to perform closed-loop identification.There are three basic approaches to closed-loop identification, namely a direct, an indirect and a two-stage one.A direct approach to identifying a plant in a closed-loop identification scheme using the plant input and output data is fraught with difficulties due to the presence of unknown and generally inaccessible noise, the complexity of the model or a combination of both.Although computationally simple, this approach can lead to parameter estimates that may be biased due mainly to the correlation between the input and the noise, unless the noise model is accurately represented or the signal-to-noise ratio is high (Raol, Girija, & Singh, 2004).The conventional indirect approach is based on identifying the closed-loop system using the reference input and the system (plant) output.Given an estimate of the system open-loop transfer function, an estimate of the closed-loop transfer function can be obtained from the algebraic relationship between the system's open-loop and closed-loop transfer functions.
The desired plant transfer function can then be deduced from the estimated closed-loop transfer function.However, the derivation of the plant transfer function from the closedloop transfer function may itself be prone to errors due to inaccuracies in the model of the subsystem connected in cascade with the plant.The two-stage approach, itself a form of an indirect method, is based on first identifying the sensitivity and the complementary sensitivity functions using a subspace Multi-Input, Multi-Output (MIMO) identification scheme (Shahab & Doraiswami, 2009).In the second stage, the plant transfer function is obtained from the estimates of the plant input and output generated by the first stage.

Two-stage identification
In the first stage, the sensitivity function () Sz and the complementary sensitivity functions () Tz are estimated using all the three available measurements, namely the reference input, r, plant input, u and the plant output, y , to ensure that the estimates are reliable.In other words, a Multiple-Input, Multiple-Output (MIMO) identification scheme with one input (the reference input r), and two outputs (the plant input u and the plant output y) is used here rather than a Single-Input, Single-Output (SISO) scheme using one input u and one output y.The MIMO identification scheme is based on minimizing the performance measure, J, as: where

[ ]
ˆˆT zy u = , û is the estimated plant input and ŷ is the estimated plant output.The plant input u, and the plant output y are related to the reference input r and the disturbance w by: As pointed out earlier, the proposed MIMO identification scheme will ensure that the estimates of the sensitivity and the complementary sensitivity functions are consistent (i.e. they have identical denominators), and hence will also ensure that the estimates of the plant input u and the plant output y , which are both employed in the second stage, are reliable.Note here that the reference signal r is uncorrelated with the measurement noise w and the disturbance v, unlike in the case where the plant is identified using the direct approach.This is the main reason for using the MIMO scheme in the first stage.In the second stage, the plant () Gz is identified from the estimated plant input, û , and plant output, ŷ , obtained from the stage 1 identification scheme, i.e.: Note that here the input û and the output ŷ are not correlated with the noise w and disturbance term v. Treating û as the input and ŷ as the output of the plant, and ŷ as the estimate of the plant output estimate, ŷ , the identification scheme is based on minimizing the weighted frequency-domain performance measure Wj ω is the weighting function.Furthermore, it is shown that: Lemma: If the closed-loop system is stable, then • The unstable poles of the plant must be cancelled exactly by the zeros of the sensitivity function if the reference input is bounded.

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The zeros of the plants form a subset of the zeros of the complementary transfer function This provides a cross-checking of the estimates of the poles and the zeros of the plant estimated in the second stage with the zeros of the sensitivity and complementary functions in the first stage, respectively.

Evaluation on a physical system: magnetic levitation system (MAGLEV)
The physical system is a feedback magnetic levitation system (MAGLEV) (Galvao, Yoneyama, Marajo, & Machado, 2003).Identification and control of the magnetic levitation system has been a subject of research in recent times in view of its applications to transportation systems, magnetic bearings used to eliminate friction, magnetically-levitated micro robot systems, magnetic levitation-based automotive engine valves.It poses a challenge for both identification and controller design.The model of the MAGLEV system, shown in Fig. 7, is unstable, nonlinear and is modeled by: 2 () () where y is the position, and u the voltage input.The poles, p, of the plant are real and are symmetrically located about the imaginary axis, i.e.: p α =± .The linearized model of the system was identified in a closed-loop configuration using LABVIEW data captured through both A/D and D/A devices.Being unstable, the plant was identified in a closedloop configuration using a controller which was a lead compensator.The reference input was a rich persistently-exciting signal consisting of a random binary sequence.An appropriate sampling frequency was determined by analyzing the input-output data for different choices of the sampling frequencies.A sampling frequency of 5msec was found to be the best as it proved to be sufficiently small to capture the dynamics of the system but not the noise artifacts.The physical system was identified using the proposed two-stage MIMO identification scheme.First, the sensitivity and complementary sensitivity functions of the closed-loop system were identified.The estimated plant input and output were employed in the second stage to estimate the plant model.The model order for identification was selected to be second order using the proposed scheme.Figure 8 below gives the pole-zero maps of both the plant and the sensitivity function on the left-hand side, and, on the righthand side, the comparison between the frequency response of the identified model ˆ() Gj ω , obtained through non-parametric identification, i.e. estimated by injecting various sinusoidal inputs of different frequencies applied to the system, and the estimate of the transfer function obtained using the proposed scheme.

Model validation
The identified model was validated using the following criteria: • The proposed model-order selection was employed.The identifications in stages I and II were performed for orders ranging from 1 to 4. A second-order model was selected in both stages since all the poles of the identified model were located in the right-half of www.intechopen.comthe z-plane.Note here that the dynamics of the actuator (electrical subsystem) was not captured by the model as it is very fast compared to that of the mechanical subsystem.

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A 4 th order model was employed in stage I to estimate the plant input and the output for the subsequent stage II identification.

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The plant has one stable pole located at 0.7580 and one unstable pole at 1.1158.The reciprocity condition is not exactly satisfied as, theoretically, the stable pole should be at 0.8962 and not at 0.7580.

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The zeros of the sensitivity function contain the unstable pole of the plant, i.e. the unstable pole of the plant located at 1.1158 is a zero of the sensitivity function.

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The frequency responses of the plant, computed using two entirely different approaches, should be close to each other.In this case, a non-parametric approach was employed and compared to the frequency response obtained using the proposed model-based scheme, as shown on the right-hand side of Fig. 8.The non-parametric approach gives an inaccurate estimate at high frequencies due to correlation between the plant input and the noise.

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The residual is zero mean white noise with very small variance.

H ∞ Mixed sensitivity H ∞ controller design
The weighting functions are selected by giving more emphasis on robust stability and less on robust performance: ( ) 0.001 To improve the robustness of the closed-loop system, a feed-forward control of the reference input is used, instead of the inclusion of an integrator in the controller.The H ∞ controller is given by: It is interesting to note here that there is a pole-zero cancelation between the nominal plant and the controller since a plant pole and a controller zero are both equal to 0.7578.In this case, the H ∞ norm is 0.1513 γ = and hence the performance and stability measure is The step response and magnitude responses of the weighted sensitivity, complementary sensitivity and the control input sensitivity of the closed-loop control system are all shown above in Fig. 9.

Evaluation on a physical sensor network: a two-tank liquid level system
The physical system under evaluation here is formed of two tanks connected by a pipe.A dc motor-driven pump supplies fluid to the first tank and a PI controller is used to control the fluid level in the second tank by maintaining the liquid height at a specified level, as shown in Fig. 10.This system is a cascade connection of a dc motor and a pump relating the input to the motor, u , and the flow i Q .It is expressed by the following first-order time-delay system: () where m a and m b are the parameters of the motor-pump subsystem and () u φ is a dead-band and saturation-type of nonlinearity.The Proportional and Integral (PI) controller is given by: With the inclusion of the leakage, the liquid level system is now modeled by : ( ) ( ) ( ) ( )  .The dual-tank fluid system structure can be cast into that of an interconnected system with a sensor network, composed of 3 subsystems , eu G uq G , and qh G relating the measured signals, namely the error e, control input u, flow rate Q and the height h, respectively.The proposed two-stage identification scheme is employed to identify these subsystems.It consists of the following two stages:

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In Stage 1, the MIMO closed-loop system is identified using data formed of the reference input r, and the subsystems' outputs measured by the 3 available sensors.

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In Stage 2, the subsystems eu G uq G , and qh G are then identified using the subsystem's estimated input and output measurements obtained from the first stage.

Fig. 2 .
Fig. 2. Co-prime factor-based uncertainty model for a SISO plant

Fig. 4 .
Fig. 4. Nominal closed-loop system relating the reference input and the weighted outputs The weighting functions () s Wj ω , and mixed-sensitivity optimization problem for robust performance and stability in the framework H ∞ − is then reduced to finding the controller 0 C such that : Fig. 5.The set ˆi S is a ball of radius 1/ i γ centered at ˆi G

Fig. 6 ..
Fig. 6.Figures A and B on the left show the Step responses (top) and Magnitude responses of sensitivity (bottom) when the model order is varied from 2 to 10 when the noise standard deviation is 0.001 v σ = .Similarly figures C and D on the right-hand show when the noise standard deviation v σ is varied in the range

Fig. 8 .
Fig. 8.A and B show pole-zero maps of the plant and of the sensitivity function (left) while C and D (right) show the comparison of the frequency response of the identified model with the non-parametric model estimate, and the correlation of the residual, respectively The nominal closed-loop input sensitivity function was identified as: ( ) 11 0 12 1.7124z 1 1.116 () 1 -1.7076z +0.7533z z Sz −− − − − == (29)

Fig. 9 .
Fig. 9.The step and frequency responses of the closed-loop system with H ∞ controller Fig. 10.Two-tank liquid level system

a
coefficients of the inter-tank and output valves, respectively.The linearized model of the entire system formed by the motor, pump, and the tanks is given by: , α and β are parameters associated with the linearization process, α is the leakage Figure 11 shows the estimation of the 4 key signals e, u, Q and h in our two-tank experiment, that are involved in the MIMO transfer function in stage I identification.Stage I identification yields the following MIMO closed-loop transfer function given by: