Robust Control Design with Matlab

Boubaker Krim

Robust Control Design with Matlab

Boubaker Krim

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Abstract

Robustness is of crucial importance in control systems design, because real engineering systems are vulnerable to external disturbance and measurement noise, and there are always discrepancies between mathematical models used for design and the actual system in practice. Typically a control engineer is required to design a controller which will make the closed loop system stable and achieve certain performance levels in the presence of disturbance signals, noise interference, unmodeled plant dynamics and plant parameter variations. The purpose of this book is to help post-graduate students and control engineers learn how to use well-developed, advanced robust control system design methods and the state-of-the-art MATLAB® tools in practical design cases.

Figures (389)

and noises in a real work environment, and the effect due to these signals would ad- versely affect the expected, normal system output in an unstable system. Feedback control techniques may reduce the influence generated by uncertainties and achieve desirable performance. However, an inadequate feedback controller may lead to an unstable closed-loop system though the original open-loop system is stable. In this section, control-system stabilities and stabilizing controllers for a given control sys- tem will be discussed.

Fig. 2.10 Analog block diagram of Example 2.2

Fig. 2.11 Structured uncertainties block diagram of Example 2.2

The block diagram in Fig. 2.12 can be generalized to be a standard configuration to represent how the uncertainty affects the input/output relationship of the control system under study. This kind of representation first appeared in the circuit analysis back in the 1950s [140, 141]. It was later adopted in the robust control study [145] for uncertainty modeling. The general framework is depicted in Fig. 2.13. The interconnection transfer function matrix M in Fig. 2.13 is partitioned as

3.2 Performance Considerations Figure 3.3 depicts a typical closed-loop system configuration, where G is the plant and K the controller to be designed. r, y, u, e, d, n are, respectively, the reference input, output, control signal, error signal, disturbance, and measurement noise. With a little abuse of notations, we do not distinguish the notations of signals in the time or frequency domains. The following relationships are immediately available:

LFT technique introduced in Chap. 2 and by specifying/grouping signals into sets of external inputs, outputs, input to the controller and output from the controller, which of course is the control signal. Note that in Fig. 4.2 all the external inputs are denoted by w; z denotes the output signals to be minimized/penalized that includes both performance and robustness measures, y is the vector of measurements available to the controller K, and u the vector of control signals. P(s) is called the generalized plant or interconnected system. The objective is to find a stabilizing controller K to minimize the output z, in the sense of energy, over all w with energy less than or equal to 1. Thus, it is equivalent to minimizing the H.o-norm of the transfer function from w to z.

4.3.4 Normalization Transformations In general, the system data given would not be in the normalization form as dis- cussed in Sect. 4.3.1, though D1 will be of full column rank and D>, of full row rank, respectively, for any realistic control systems. In order to apply the results in Theorem 4.1, certain transformations must be used first. Using the singular value decomposition (SVD) or otherwise, we may find orthonormal matrices Uj2, Vj2, U1, and V1, such that

Fig. 5.1 Robust stabilization with regard to coprime factor uncertainty

The closed-loop transfer function between the reference r and the plant output y then becomes

$.3 A Few More Properties Concerning LMI

Example 9.1 Let us consider a two-input, two-output system described in the state- space by the equations Creation of a state-space model of time-invariant continuous-time system is il- lustrated by the following example. produces the following result:

Fig. 9.9 Transient responses due to input disturbances

Fig. 9.10 Transient responses due to output disturbances Mass—damper-spring system

Nyqudit diagram of uncertain system Fig. 9.15 Nyquist diagram of uncertain system

Step response for a grid of 50 values of uncertain parameters Fig. 9.16 Step response for a grid of 50 values of uncertain parameters 9.2.4 Other Functions to Build Uncertain Models

Approximation of uncertain transfer function by additive unceratinty use 20 data points of the upper bound, positioning the cursor using a mouse. Dat points are entered by pressing the left mouse button. As aresult we obtain

Approximation of uncertain transfer function by multiplicative perturbation

Approximation of uncertain time delay by multiplicative uncertainty

obtain the closed-loop transient responses due to step reference r and step input and output disturbances dj, d. Exercise 9.2 For the system with plant and controller transfer matrices

Fig. 10.3. Closed-loop system with input multiplicative uncertainty As a result we obtain an eighth order uncertain object,

Example 10.2 Consider the control system described in Example 10.1 and let G de- notes the plant transfer function matrix and K —the controller transfer function ma- trix. Let the requirements to system performance are to ensure reference (r) tracking with admissible error (e) and limited control (uw) magnitude in the presence of dis- turbance d. According to the first requirement, the output sensitivity function S; connecting e with r and d should be sufficiently small in the low-frequency range (r and d are low-frequency signals). According to the second requirement the trans- fer function matrix K S, connecting u with r and d should be sufficiently small. Tc take into account both requirements it is appropriate to use as a system performance index the quantity

Fig. 10.8 Upper and lower bounds of ju The maxima of upper and lower bounds of structured singular values with respect to frequency are found by the lines

Fig. 10.10 Singular values of output complementary sensitivity

Fig. 10.13 Transient responses of the uncertain system

Fig. 10.14 Closed-loop singular values for nominal and worst-case gain

Fig. 10.15 Step responses of the nominal and worst-case gain systems

Fig. 10.16 Random and worst-case singular value plots

Fig. 10.17 Random and worst-case gain step responses

If the unstructured uncertainty in the plant model G is represented by additive perturbation, i.e., Gp(s) = G(s) + Aq(s) then the size of the smallest destabilizing additive perturbation is

Fig. 11.2. Specifying the desired singular values of L, S, and T

Fig. 11.7 Sensitivity of the uncertain system

Fig. 11.8 Complementary sensitivity of the uncertain system

Fig. 11.11 Design results obtained by mixsyn

Fig. 11.15 Closed-loop system with model

Fig. 11.17 Open-loop system structure Fig. 11.18 Generalized

The number of measurements and the number of controls are equal to 1. The range for y -iteration is chosen between the numbers 0.1 and 10 with tolerance tol = 0.001. At each iteration, the program shows the current value of y and the results from five tests for existence of suboptimal controller. At the end of each iteration is shown the symbol p or £ which indicates whether the current value of y is accepted or dismissed. The symbol # is used to show which of the five conditions for existence of Hoo suboptimal controller is violated for the y used. When the iteration procedure ends, the minimal achievable value of y is given. The transfer function matrix of the closed-loop system from dist to e is saved in the variable clp. The minimal achievable value of y is 0.586 with the suboptimal controller ob-

Fig. 11.22 Transient response of the nominal system with respect to reference

Fig. 11.24 Closed-loop system with 2-degree-of-freedom controller

The open-loop system has three inputs and four outputs (Fig. 11.26). In this case the Hoo design is done by the commands

Fig. 11.27 Frequency responses of the Hoo 2-degree-of-freedom controller

Fig. 11.28 Transient responses of the nominal system with respect to reference

Fig. 12.4 Block-diagram of the closed-loop system

Since the closed-loop system achieves robust performance, the condition

Fig. 12.9 Sensitivity functions of the closed-loop system

Fig. 12.8 Frequency responses of the closed-loop system

yeing equal to 0.578. In this way the jz-synthesis produces better results compared O Hoo design in which the controller does not ensure even robust stability of the -losed-loop system. When the j-synthesis does not lead to robust performance, as n the case under consideration, it is necessary either to change the performance ‘equirements to the closed-loop system (i.e., the weighting functions) or to try 2-degree-of-freedom controller design.

Fig. 12.14 Frequency responses of the controller

‘ig. 12.17 Worst-case gain magnitude frequency response Worst case closed-loop system Bode plot

lhe string ‘pol’ indicates that the parameter range is defined as a polytope.

Fig. 13.3 Parameter trajectory for system simulation

Fig. 13.4 Time response of the parameter-dependent system

This interpolation yields a smooth scheduling of the controller matrices by the pa- rameter measurements p(t).

Fig. 13.12 Transient response of the closed-loop system

are mounted at the bottom of sliders. Today’s hard disks read data using giant- magneto-resistive (GMR) heads and write data with thin-film inductive heads. The sliders with read/write heads are mounted onto head arms. The arms are lightweight rigid constructions allowing them to be moved rapidly on the platter surface. There is one arm per read/write head and all arms are mounted in a stack assembly that moves over multiple disk surfaces simultaneously. The heads are suspended less than 1 microinch above the disk surface. The appropriate flying height of the heads is achieved thanks to the air flow generated by the spinning disk. The data recorded on the platters are in concentric circles called tracks. Modern

Fig. 14.2 Block-diagram of the hard disk drive servo system

Table 14.1 Rigid body model parameters and tolerances

The next step is to take into account the high-frequency resonant modes of the head disk assembly represented by the transfer function Hg(s). In the given case H,(s) consists of four resonant modes and is obtained as

The hard disk model has two inputs and one output (Fig. 14.8).

Fig. 14.6 Block-diagram of a resonant mode Fig. 14.7 Parallel connection of the four resonant modes Fig. 14.7 Parallel connection Fig. 14.8 Block-diagram of the hard disk considered as a control plant

Fig. 14.10 Block-diagram of the closed-loop system with performance specifications The system has a reference input (7), an input disturbance (d), a noise (m) an two output costs (ey and e,). The system M is an ideal model of performance to which the designed closed-loop system tries to match. The transfer function o the uncertain plant is denoted by G. The performance objective requires the transfe matrix from r, d, and n to ey and e, to be small in the sense of || - ||oo, for all possibl values of the uncertain parameters. The position noise signal is obtained by passin; the unit-bounded signal n through the weighting transfer matrix W,. The transfe matrices W, and W, are used to reflect the relative significance of the differen frequency ranges for which the performance is required. Hence, the performanc: objective can be recast, with possible slight conservativeness, as that the || - loo 0 the transfer function matrix from r, d, and n to ey and e, is less than 1. It is straightforward to show that

Fig. 14.11 Block-diagram of the open-loop system with performance specifications

The noise shaping function W,, is determined on the basis of the spectral density of the position noise signal. In the given case it is taken as the high-pass filter In the jz-synthesis we shall take into account only the uncertainty in the rigid-body model (i.e. only the uncertainty in the parameters k,;, J, and ky) will be consid- ered. The inclusion of the uncertainties of resonant modes would make the D—K iterations difficult to converge which is the reason to ignore these uncertainties. This confirms that the resonant modes may create difficulties in the controller de- sign. However, these resonant modes (with nominal values) are included in the plant dynamics and will be considered, with parametric variations, in the assessment of designed controllers in Sect. 14.6. The block- diagram of the closed- -loop system used i in the p- synthesis 1 is s shown —y qaQn4 > ™ wm %< «4 ras

Inverse Performance Weighting Functions the aim is to achieve a small difference between the system and model outputs, and a small effect of the disturbance on the system outputs. This will ensure good tracking of the reference input and small error due to low-frequency disturbances. Changing the performance weighting function from Wp; to W,3 moves the inverse weighting frequency response to the right (to higher frequencies) which forces the system to match the model in over a larger frequency range. The control weighting function is usually chosen as high-pass filters in order to

‘ig. 14.17 Transient responses for three jz-controllers

Fig. 14.20 Closed-loop magnitude plots of three j-controllers

Fig. 14.19 Transient responses to disturbance of three j1-controllers

The results obtained from different weighting functions show that moving the frequency response of the inverse performance weighting function to the right would lead to larger closed-loop system bandwidth, and consequently, faster time- responses of the closed-loop system, though may introduce larger over(under)shoot. However, at the same time, this may reduce the robustness of the closed-loop sys- tem. Hence, one has to compromise between the different objectives in the design. In the present design case, it seems that the second controller leads to a good trade- off between the requirements in terms of transient response, disturbance rejection, and robustness. Hence, we will use the weighting functions W,2 and W,2 in both the jz-synthesis and Ho design.

Frequency responses of the plant and shaped plant

Fig. 14.24 Nominal performance of the closed-loop systems

Fig. 14.25 Robust performance of the closed-loop systems

Closed-loop Bode plots for three controllers

Fig. 14.30 Transient responses with three controllers

Fig. 14.34 Output disturbance sensitivity for random plant parameters

Fig. 14.33 Magnitude closed-loop plots for random plant parameters

Fig. 14.35 Output sensitivity to noise for random plant parameters

Fig. 14.36 Bode plots of full- and reduced-order jz-controllers

Fig. 14.38 Control action of the j.-controller for random plant parameters

Fig. 14.39 Transient disturbance responses with j-controller for random plant parameters

Bode plots of continuous-time and discrete-time plants

Fig. 14.41 Robust stability of f; = 36 kHz design

QAUIUILY dildlysis dll LODUSLE PECIIOTIMANCe dallalyosis, LOspCCuvely, i VOUT PlOls UI orst results are seen around the second resonant frequency of 2.2 kHz. The transient responses of the closed-loop system for 30 random combination: f the uncertain parameters are obtained by the file ds1_hdd.m, which imple ents the function sdlsim. The function sdlsim also computes the control ac on signal obtained at the output of the digital-to-analog converter. The closed-loo] ansient response is shown in Fig. 14.43 and the corresponding control action i ig. 14.44. The undershoot for all parameter values is less than 50 % and the maxi lum magnitude of the control signal is 1 V. The transient response to disturbance is shown in Fig. 14.45. Overall, the result: ptained are almost as good as the results obtained with the continuous-time, ju wa

Fig. 14.43 Transient response of f,; = 36 kHz design

Fig. 14.44 Control action of f, = 36 kHz design

Fig. 14.47 Transient responses of the nonlinear systems

After linearization of (15.1) under the assumptions of small deviations of the pen- dulum from the vertical position and of small velocities, one obtains the following equation: The outputs are measured by linear potentiometers, whose voltages are given by we obtain

Fig. 15.3 Block-diagram of the triple inverted pendulum system

The whole, perturbed, triple inverted pendulum system can be described by the equation

Fig. 15.15 Open-loop interconnection of the triple inverted pendulum system

Fig. 15.16 Schematic diagram of the open-loop system

Fig. 15.18 Schematic diagram of the closed-loop system Fig. 15.19 Block diagram for Hoo design

The singular value plot of the Hoo controller transfer matrix is shown in Fig. 15.20.

Fig. 15.20 Singular values of the Ho controller

Fig. 15.24 Robust stability test of Ho. controller

Fig. 15.26 Singular values of the jz-controller

Fig. 15.27 Robust stability of jz controller

Fig. 15.28 Robust performance of jz controller

Fig. 15.29 Closed-loop transient response of j-controller

Closed-—loop disturbance response The closed-loop frequency responses of the uncertain closed-loop system are ob- tained by the M-file £rs_pend.m. The singular values of the closed-loop transfer matrix with the j-controller are plotted in Fig. 15.33. The comparison with the plot shown in Fig. 15.23 reveals that the closed-loop bandwidth in the case of a u- controller is slightly smaller, which leads to a slower response and worse disturbance attenuation. In contrast to the case of H controller, the closed-loop frequency re- sponses do not have peaks. — gh wal’ as r r oe -— rs ak ulCUM

Fig. 15.34 Singular values of the disturbance rejection transfer function matrix

Singular Value Plot of the Closed—loop Transfer Function Matrix Fig. 15.33 Closed-loop singular value plot

Fig. 15.35 Singular values of the noise-attenuation transfer function matrix

Fig. 15.36 Frequency responses of the full- and reduced-order controllers

Fig. 15.38 Transient response of the nonlinear system—small reference

Approximation of uncertain time delay by multiplicative perturbation

sensitivity function for G. The block diagrams of the closed-loop systems with 1-degree-of-freedom and 2- degree-of-freedom controllers, incorporating the design requirements consideration represented by weights, are shown in Figs. 16.5 and 16.6, respectively. The plant G, enclosed by the dashed rectangle, consists of the nominal scaled model G plus the input multiplicative uncertainty. The controller K implements a feedback from outputs yp and xg and a feedforward from the reference signal r. The measurement of the distillate and bottom products composition is corrupted by the noise n. The desired dynamics of the closed-loop system is sought by implementation of a suit- ably chosen model M. The model M represents the desired dynamic behavior of the closed-loop system from the reference signal to the outputs. The usage of a model of the desired dynamics allows us to take easily into account the design specifications. The transfer function matrix of the model M is selected as

Inverse Performance Weighting Function Fig. 16.8 Inverse of performance weighting function

16.5 Open-Loop and Closed-Loop System Interconnections The open-loop system interconnections for both types of controller are obtained by the M-file olp_col.

ig. 16.14 Schematic diagram of the open-loop interconnection for 2DOF controller

Fig. 16.16 Schematic diagram of the closed-loop interconnection for 1DOF controller

Fig. 16.18 Schematic diagram of the closed-loop interconnection for 2DOF controller

Frequency responses of the original plant and shaped plant

Fig. 16.20 Robust stability for loop-shaping controller

Singular Value Plot of the Closed—loop Transfer Function Matrix

Singular Value Plot of the Sensitivity Function

Fig. 16.26 Perturbed transient response y,; for loop-shaping controller

Fig. 16.30 Perturbed control action u1; for loop-shaping controller

Fig. 16.32 Perturbed control action v2; for loop-shaping controller

Fig. 16.34 Robust stability for jz-controller

Fig. 16.35 Robust performance for jz-controller

Fig. 16.38 Frequency responses with respect to noises Singular Value Plot of the Noise Transfer Function

Fig. 16.42 Perturbed transient response y1; for sz-controller

Fig. 16.46 Perturbed control action 1; for jz-controller

Fig. 16.47 Perturbed control action w12 for jz-controller

Fig. 16.48 Perturbed control action u2, for jz-controller

Fig. 16.49 Perturbed control action u22 for -controller

Singular values of the controller transfer matrices

stability and robust performance of the closed-loop system. Finally, the advantages of the jz-controller over a conventional, collocated PD controller are demonstrated.

Fig. 17.3, Bode plot of the flexible-link manipulator with input t and output a

my =0.15 kg. In the expressions (17.9) and (17.10) the uncertain parameter m; appears once. Since in (17.8) the coefficients a; and a2 are repeated twice, and b; and b2 three times, it appears that as a whole the parameter my is repeated in (17.8) 13 times. However, in the manipulator model, built on the basis of the state space represen- tation of (17.8), the uncertain parameter m, is repeated 32 times, which makes the robust analysis and design of the system difficult. This number may be reduced significantly, building the manipulator model by using the function sysic repre- senting appropriately (17.8). Se SE, ee ee es (ee ee eT ee Ms fe, ee ke eee 2. eee

Bode plots of exact and approximated manipulator models Fig. 17.9 Bode plots of exact and approximated manipulator models with input t and output © for my = 0.125 kg

Bode plots of exact and approximated manipulator models

system to be close to that model. The use of such a model to represent the desired dynamics allows us to take into account the requirements on system performance more easily and directly. In other words, such a model (named M in Fig. 17.11) pre- scribes the desired dynamic behavior of the closed-loop system from the reference signal to the tip position. The uncertainty plant is denoted by G in Fig. 17.11. The internal, current control loop of the servo drive W, is modeled as a first order lag with the time constant 0.004 s and gain equal to 1. Let the 3 x 1 transfer matrix G be partitioned as

The model transfer function to be matched is taken in this design as

Inverse of performance weighting function

Fig. 17.14 Open-loop interconnection structure of the flexible-link manipulator system Fig. 17.15 Schematic ee Sn III NE diagram of the open-loop interconnection

Closed-loop transient response ‘ig. 17.22 Closed-loop transient response for ;z-controller

Fig. 17.24 Control action for jz-controller

Singular value plots of full-order and reduced-order controllers

Closed-loop transient response for the PD controller ing the link flexibility the closed-loop transfer function coincides with the transfer function of the model. The results by using the j-analysis method in this case (i.e. with the PD controller) are 3.4212 and 0.1398 for the stability margin and perfor- mance margin, respectively. Therefore, the PD controller leads to poor robust perfor- mance in comparison to the jz-controller designed. This can be seen by comparing Figs. 17.30 and 17.31 with Figs. 17.22 and 17.23, respectively. It has to be noticed that good results in the design may also be obtained by using a collocated controller on the feedback from the joint angle 6 and the velocity 0. The use of the tip acceleration @, however, allows better results with respect to the

Fig. 17.31 Tip-deflection transient response for the PD controller

Nonlinear transient response Fig. 17.33 Transient response of the nonlinear system

Manipulator elastic deflation Fig. 17.34 Elastic deflection w(x, t) for the case of jz-controller

Fig. 17.35 Elastic deflection w(x, t) for the case of PD-controller

Fig. 18.5 Tail rotor velocity characteristic

Main rotor velocity Fig. 18.4 Main rotor velocity characteristic

The uncertain TRAS system is described as a control plant by the equation

Fig. 18.9 Frequency responses of the uncertain plant

Fig. 18.10 Closed-loop system with 2-degree-of-freedom controller

Fig. 18.13 Schematic input—output diagram of the open-loop system structure

Fig. 18.15 Schematic input-output diagram of the closed-loop system structure

Inverse Performance Weighting Function The frequency responses of the inverse performance weighting function are given 1 Fig. 18.18.

Control Action Inverse Weighting Function ‘ig. 18.19 Inverse control action weighting functions

Robust stability Fig. 18.22 Robust stability of the closed-loop system

Fig. 18.23 Robust performance of the closed-loop system

Singular Value Plot of the Closed—-loop Transfer Function Matrix Fig. 18.24 Frequency responses of the closed-loop system (—) and model (- -, -.)

Singular Value Plot of the Noise Transfer Function Matrix

Singular Value Plot of the Sensitivity Transfer Function Matrix

Singular Value Plot of the Input Sensitivity Transfer Function Matrix

Singular Value Plot of the Noise to Input Transfer Function Matrix Fig. 18.28 Frequency response of the noise-to-control loop (—) and inverse control weighting , a: me

Azimuth and pitch control actions Fig. 18.31 Control actions of the uncertain linearized closed-loop system

Fig. 18.30 Transient responses of the uncertain linearized closed-loop system

Nonlinear system transient responses Fig. 18.33 Transient responses of the nonlinear system

Nonlinear system control actions Fig. 18.34 Control actions in the nonlinear system

Fig. 18.37 Experimental control actions (noisy data) Fig. 18.36 Experimental transient responses

In this chapter we present the design of a robust controller for NXTway-GS robot with the aim to implement in maximum degree the available software for real-time control presented in [188]. Since the robust controllers are of higher order the basic problem is to check the possibility to implement such controllers on the available microcontroller working with sampling frequency jf; = 250 Hz in the stabilization loop. The results obtained show that the microcontroller under consideration im- plements without difficulties a robust discrete controller of 12th order that allows to improve the closed-loop system performance. Results from the simulation of the closed-loop system as well as experimental results obtained during the real imple- mentation of the controller designed are given.

Table 19.1 Nominal parameters of the robot model

Singular value plot of the uncertain model Fig. 19.3 Magnitude responses of the nominal and uncertain plant

allows to design easily a robust controller that ensures robust stability and robust performance of the closed-loop system with respect to the given uncertainty. In par- ticular, we assume that the coefficient fj, is known with 20 % uncertainty and the coefficient f,, with 100 % uncertainty. The analysis performed later on shows that the coefficient f,, has more significant influence on the system dynamics. The two real uncertain parameters f,, and f,, are defined in the program by using the following command lines:

Fig. 19.4 Block-diagram of the closed-loop system with performance requirements

Fig. 19.6 Schematic diagram of the open-loop system interconnection

In the given case the following weighting functions are found as appropriate in the controller design. De! pe a <n 2

Fig. 19.8 Singular value plot of jz-controller

Singular value plot of the closed—loop system ©(z)/r(z)

Singular value plot of the output sensitivity e,(2)/r(z)

Sensitivity of control to references and noises Fig. 19.12 Sensitivity of control action to references and noises

‘ig. 19.13 Robust stability of the closed-loop system

Singular value plot of the uncertain closed-loop system Fig. 19.15 Worst-case magnitude response of the closed-loop system

Closed-loop transient response Fig. 19.17 Transient response of the closed-loop system: body tilt angle

Closed-loop transient response Fig. 19.16 Transient response of the closed-loop system

Fig. 19.21 Control action to the right wheel motor

Richard Chiang

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Robust control system design for multivariable plants with lightly damped modes

Dominique Gruel

A robust controller design is proposed for the active suspension system bench-mark problem. The CRONE control system design used is extended to unstable multivariable plants with lightly damped modes and RHP zeros. Decoupling and stabilizing controller K, is achieved for the open-loop transfer matrix. Fractional order transfer functions are used to define all the components of the diagonal open-loop transfer matrix, β. In defining the fractional open-loop transfer function β0i some elements of the plants, G0 and its inverse must be considered to achieve the stable controller. Optimisation provides the best fractional open-loop βopt. Finally, frequency domain system identification is used to find controller K=G0-1βopt.

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Robust controller design based on model reference approach

Manuel Armando Duarte Mermoud

Kybernetes, 2002

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