The root locus (RL) is a systematic technique for investigating,

Transcription

1 LECTURE NOTES Complete Root Contours for Circle Criteria and Relay Autotune Implementation By Aldo Balestrino, Alberto Landi, and Luca Sani The root locus (RL) is a systematic technique for investigating, in a simple way, the effect of feedback on the closed-loop system poles It provides an easy method for designing controllers with a varying parameter if the open-loop transfer function is rational and accurately known [] Experience gained during several undergraduate control system courses confirms the RL as an effective technique in controller design Moreover, the RL method is extremely easy to use: in educational terms, the method is user-friendly for students, judging from their excellent results in solving exercises The basic properties of the RL method were first defined by Evans [], [3] in the 95s; since then, the RL has been included in all the leading textbooks in the field of automatic control Today, computer software (eg, MATLAB) provides substantial support for the teaching of automatic control; moreover, computer-aided plotting of root loci reduces the difficulties in applying the method and its extensions Multiple-parameter root loci are easily implementable, and extensions to standard RL applications can be treated with low-cost software (eg, the so-called Robust Root Locus (RRL) provides a visual method for studying variations in the root loci under multiparameter perturbation of the plant transfer functions [4], [5]) Unfortunately, the RL method hides important information from the designer with respect to standard frequency domain techniques For instance, in Bode, Nyquist, or Nichols plots, the primary measures of relative stability (ie, the gain margin (GM) and the phase margin (PM)) can be directly visualized, but when using the RL, the same information is hidden on first observation The gain margin can be computed by looking for the complete root loci intersection with the imaginary axis of the s-plane [6], so that the itical gain, and hence the GM, can be determined from the condition of magnitude The phase margin cannot be easily evaluated, and, to our knowledge, basic textbooks do not cover this topic In the 99s, this problem was considered and solved using the so-called phase RL (practically a constant phase chart) [7], [8] Such solutions, although ingenious, are too complicated to be adopted in teaching undergraduate students Besides the phase margin, evaluated at the frequency where the Nyquist plot of the process intersects the unitary circle centered at the origin, different and more interesting circle iteria in the Nyquist plot provide information on the sensitivity, on the closed-loop bandwidth, and on the absolute stability (see Fig ) This wealth of information in the Nyquist plane is apparently not available in the RL analysis The main objective of the first part of this article is to provide a general and simple framework for exploring how the information supplied by circles in the Nyquist plane can be recovered in the RL plane All information visualized from the circles of Fig can be extended to the RL plane by considering the so-called complete root contours defined as a root contour, for both positive and negative varying parameters (in this article, we use the terminology proposed in [6], briefly recalled in the next section) In such a framework, even the phase margin evaluation can be included in a natural manner Applications to phase margins, sensitivity, bandwidth computations, and absolute stability are reported in the next sections With respect to previous results, presented first in [9], the class of off-axis circles is also included, along with an example of constant N-contours for determination of the closed-loop phase response using the open-loop transfer function Unfortunately, the class of off-axis circles leads to equations with complex coefficients Therefore, the RL loses its symmetry with respect to the real axis of the plot, and the root loci can be easily constructed only via a software approach and not manually In the second part of this article, a procedure based on the application of a relay feedback technique is proposed for a laboratory implementation of the previous framework An extension of the relay feedback method originally proposed in [] is considered A modified method, called sinusoidal autotune variation (SATV), is based on a phaselocked loop module [] added to a standard relay test for eliminating the low-pass filtering hypothesis due to the use of the desibing function analysis in the relay feedback In the final section, illustrative examples are solved to demonstrate the ease of the proposed approach for experimental evaluation of phase margin, 3-dB bandwidth, sensitivity, and resonance frequency in closed-loop processes Circle Criteria and Complete Root Contours: The General Case The RL plot is a graphical method for determining the location of the closed-loop poles in the complex plane, as a changing parameter (usually the proportional control gain) varies in the open loop system Following the definitions proposed in [6, p 399], the general RL problem refers to the following equation in the complex variable s: Landi (landi@dseaunipiit), Balestrino, and Sani are with the Department of Electrical Systems and Automation (DSEA), University of Pisa, Via Diotisalvi,, I-566 Pisa, Italy 7-78//$7 IEEE 8 IEEE Control Systems Magazine October Authorized licensed use limited to: University of Michigan Library Downloaded on May 9, at ::58 UTC from IEEE Xplore Restrictions apply

2 ds ( ) + K ns ( ) =, () where ds ( )and ns ( )are pth and mth order polynomials, with p and m positive integers, and K is a real constant (the RL gain) that can vary in the range (, + ) The plot displaying the migration of the roots of () in the complex plane as K varies from to + is known as root loci (RL) The loci of negative gain are known as complementary root loci (CRL) A complete root loci is the combination of the root loci and the CRL Root loci of multiple variable parameters can be displayed by varying one parameter at a time The plot of the migration of the roots of () in the case of multiparameter variations is known as root contours (RCs) In this section we consider an extension of the above definitions to the case of complete root contours (CRCs), defined as an RC, for both positive and negative varying parameters The main objective is to explore how the information supplied by circles in the Nyquist plane can be recovered in the RL plane by considering CRCs A circle in the Nyquist plane can be desibed in parametric form by s τ cfr( s) = c+ r, + s τ () where r = radius and c = center are known parameters; if c is a real number, the diameter of the circle lies on the real axis The fraction in () has unitary modulus and phase (for s = jω) jω τ = tan ( ω τ) + jω τ Now consider a rational transfer function G( jω) = n( jω) /( d jω ) In all circle-based iteria, the knowledge of the intersecting points between the transfer function and the circle, if any, is relevant information The intersecting condition in the Nyquist plane is (3) where, if c is real, G ( s)represents a c-shift of Gs ( )along the real axis and H( s, τ) is a transfer function equivalent to a phase shift Note that the c-shift of Gs ( )along the real axis corresponds to analysis of the intersection between G ( s) and a circle centered at the origin Equation (6) can be interpreted as a characteristic equation and can be reconsidered as an RC by varying the parameter τ for different rs From (5), after a straightforward computation, the parameter τ for a fixed r can be evidenced as ds ( ) α ns ( ) =, α τ s ( d( s) + α n( s)) where α = ( r c)/( r + c), α = /( r + c), α = /( r c) The root loci of (7) may be drawn by varying the parameter τ for fixed parameters α and α It emanates from the root loci of Gs ( ) at fixed α (zeroes) and α (poles); moreover, a pole is added at the origin These root loci are called complete RCs : more than one parameter varies, and in the hypothesis of α = α, ie, c =, it originates from a complete root loci A simple correlation between the Nyquist plot and the root-locus plot was illustrated in [6] The most intuitive correlation between the two planes is related to the points where the loci oss the imaginary axis This condition corresponds to a migration of at least one closed-loop root into the right-half plane (ie, a loss of stability visualized in the Nyquist plot of the open-loop transfer function), as the ossing of the( + j) point In the case of circle-based iteria, (4) through (7) hold and the root loci of (7) oss the imaginary axis if and only if the Nyquist plot of Gs ( )intersects the circle, so that (4) is satisfied Sensitivity Circle (7) n( jω) jω τ = c+ r d( jω) + jω τ (4) The purpose of the following computations is to reduce (4) to the Evans form, where τ is the varying parameter A simple algebraic computation leads to or n( jω) + jω τ c = r d( jω) jω τ (5) Constant M - Circle 3-dB Closed-Loop Bandwidth + Critical Circle Nonlinearity in the Sector [ / k / k] /k /k = r G ( s) H( s, τ ), (6) Figure Circle iteria in the Nyquist plane October IEEE Control Systems Magazine 83 Authorized licensed use limited to: University of Michigan Library Downloaded on May 9, at ::58 UTC from IEEE Xplore Restrictions apply

3 Examples of Complete Root Contours Phase Margin Computation Now consider an open-loop transfer function Gs ( ) The gain ossover frequencyω is the frequency at which the magnitude of Gs ( )is equal to one ( db); ie, 3 3 PM G( j ) ω = (8) Unitary Circle G ( jω) 3 4 Figure Phase margin (PM) in the Nyquist plot for a generic Gs ( ) The phase margin is measured at the gain ossover frequency ω as PM = 8 + G( jω ) (9) In the case of minimum-phase systems, a necessary condition for the closed-loop stability is that the PM must be positive A graphical interpretation of PM in the Nyquist plane is given in Fig It should be emphasized that the gain ossover frequency is the frequency at which the Nyquist plot of G( jω) intersects the unitary circle centered at the origin Now consider the root loci plane with c = and r =in (), so that the complete RCs (7) are given by ds ( ) ns ( ) = τ s ( d( s) + n( s)) () The gain ossover frequency ω is the frequency at which the complete RCs () intersect the imaginary axis of the complex plane Corresponding to this frequency, the itical gain τ may be easily computed by means of the condition on the magnitude of the root loci From (5), the phase shift term ( + jω τ /( jω τ )) can be considered as the additive phase delay leading the transfer function Gs ( ) to pass through the point ( + j) of the Nyquist plane Therefore, the PM is computed as PM = + jω jω τ τ = tan ( ω τ ) () poles K = 8 ω = 3 rad/s Figure 3 Root loci plot of Gs ( ) = k/( s+) 3, showing the itical gain k Example Problem: Consider the open-loop transfer function Gs ( ) = k/( s+) 3 and evaluate the phase margin PM, selecting k so that the gain margin is GM = Solution: The root loci for positive ks are shown in Fig 3 The RL begins at the open-loop poles, located at s =, denoted as X in the s-plane and conventionally representing the closed-loop poles when k = The third-order pole at s = has three branches asymptotically migrating toward three infinite transmission zeroes, as the gain approaches + From the Evans rules, the pole-zero configuration is symmetrical with respect to the real axis, and the asymptotes are subdividing the s-plane in three equi-angle sectors Therefore, the angles of the asymptotes intersecting the imaginary axis are ±6 (see Fig 3) and from simple geometrical considerations ω = 3 3 The characteristic equation ( s+ ) + k =, solved for s = jω, has a solution k = k = 8 The condition on the gain margin is verified if k pos = k /= 4(note that a dual analysis may be performed for negative ks, leading to k neg = / ) 84 IEEE Control Systems Magazine October Authorized licensed use limited to: University of Michigan Library Downloaded on May 9, at ::58 UTC from IEEE Xplore Restrictions apply

4 Now consider the phase margin (PM) evaluation for k = k pos It emanates from the complete root loci of Gs ( )at fixed gain: the zeroes of () are given by the roots of 3 ( s+ ) kpos =, whereas the poles are given by the roots of 3 ( s+ ) + kpos =, plus a pole at the origin of the s-plane As shown in Fig 4, where poles are denoted as X and finite transmission zeroes as O, the CRC plot (red lines) emanates from the complete root loci of Gs ( ) (dashed black lines) The closed-loop roots are intersecting the imaginary axis for τ = 96 and ω =33 rad/s, ie, the frequency at which the Nyquist plot ofg( jω) intersects the unitary circle centered at the origin The phase margin is computed from () as 74 It must be noted that a different choice of GM leads to a different value of PM Therefore, a systematic way to recover phase information of the open-loop transfer function from the root loci plane could be established, varying the GM Sensitivity Analysis For single-input, single-output systems, the sensitivity function is defined as S( s) = /( + G( s)), where Gs ( )is a given open-loop transfer function Assuming W( s) as the closed-loop transfer function of Gs ( )with unitary feedback, the sensitivity function S( s) characterizes the behavior of the closed-loop system with respect to the open-loop system in the presence of small deviations in the nominal parameters of the open-loop plant In the Nyquist plane, the sensitivity function can be visualized by drawing a unitary circle centered at( + j ) : sensitivity is less than one at frequencies for which G( jω) is outside the unitary circle The itical frequency ω is evaluated at the frequency where the Nyquist plot of the process intersects the unitary circle centered at ( + j ) Expression (7) cannot be directly applied, since α= ; from (5), the complete RCs for sensitivity analysis (c = and r = ) become ns ( ) + = sτ ( ns ( ) + ds ( )) () The itical frequency ω is the frequency at which the complete RCs of () intersect the imaginary axis of the complex plane Example Problem: Evaluate the itical frequency for the sensitivity function in the case of Example Solution: As in Example, the problem may be solved by a complete RC analysis (red lines), emanating from the complete root loci ofg( s)(dashed black lines), as shown in Fig 5 The closed-loop roots of () intersect the imaginary axis for the itical frequency ω = rad/s Constant M-Circles A graphical method for determination of the closed-loop frequency response using the open-loop transfer function is based on the so-called Hall circles, or constant M N contours The constant M-contours yield a quick tool for evaluating the peak resonance and/or the bandwidth of the closed-loop plant in the Nyquist plane In the complete RC framework, all constant M-circles can be easily considered (apart from the case of M =, where the M locus degenerates into a straight line perpendicular to the real axis at 5) The constant M-circles have their center at ( xc = M /( M ), y c = ) and radius r = M /( M ), where M is the magnitude of the closed-loop plant ω = 33 rad/s τ = 96 Figure 4 CRC plot for Gs ( ) = k/( s+) 3 in the case of k = k pos ω = rad/s Figure 5 Sensitivity analysis for Gs ( ) = 4/( s+ ) 3 October IEEE Control Systems Magazine 85 Authorized licensed use limited to: University of Michigan Library Downloaded on May 9, at ::58 UTC from IEEE Xplore Restrictions apply

5 Expression (7) can be directly applied for a given M Example 3 Problem: Evaluate the 3-dB closed-loop bandwidth for the system Gs ( ) = / ss ( + ) Solution: The M-circle to be considered has its center at x c = and radius r = The problem may be solved by a complete RC analysis (red lines), emanating from the complete root loci ofg( s)(dashed black lines), as shown in Fig 6 The application of the complete RCs (7), where α= ( )/( + ),α = /( + ),α = /( ), leads to ω = 3 db 94 rad/s ω 3dB = 94 rad/s Figure 6 Closed-loop 3-dB bandwidth in the case of Gs ( ) = / ss ( + ) Absolute Stability As an application of the above theory, consider the circle theorem [] for absolute stability analysis By considering a closed control loop with a nonlinearity bounded in the sector [ k, k] and a transfer function G( jω ), if the Nyquist plot of G( jω) neither intersects nor encircles the itical circle shown in Fig, then the system is absolutely stable As in the previous case, the problem can be directly solved by using the complete RCs (7), where the circle is centered at ( xc = 5 ( / k + / k); yc = ) with radius r = 5 ( / k / k ) Example 4 Problem: Study the absolute stability for an open-loop transfer function Gs ( ) = K/( s+) 3 and a nonlinearity sector [/ ] Solution: By varying the gain K, different intersections with the imaginary axis can be drawn In Fig 7, three examples of complete RCs for K ( 6, 6, 95) are illustrated For a better understanding of the figure, the complete root loci ofg( s)and arrows showing the migration of the closed-loop poles toward the zeroes are not plotted In the case of K = 6, there are no intersections with the y-axis and the system is absolutely stable In the case of K = 6, there are two intersections with the imaginary axis, revealing two intersections with the itical circle, and the circle theorem cannot predict stability In the case of K = 95, the root loci do not intersect the imaginary axis, but a branch is totally drawn in the right-half plane, revealing instability In Fig 8, the standard Nyquist plots are shown for a visual comparison of the results obtained via the complete root loci 6 4 K = 95 K = 95 4 K = 6 K = 6 4 Critical Circle K = 6 K = Figure 7 Complete RCs for Gs ( ) = K/( s+) 3 for different values of K Green continuous line: K = 6 ; blue dashed line: K = 6; red dashed-dotted line: K = 95 Figure 8 Nyquist plots forg( s) = K/( s+) 3 for different values of K Green continuous line: K = 6 ; blue dashed line: K = 6;red dashed-dotted line: K = IEEE Control Systems Magazine October Authorized licensed use limited to: University of Michigan Library Downloaded on May 9, at ::58 UTC from IEEE Xplore Restrictions apply

6 Off-Axis Circles: Examples of Complete Root Contours We now consider the class of off-axis circles The coefficient c in (7) is a complex number (ie, the class of off-axis circles leads to equations with complex coefficients) Therefore, the RL loses its symmetry with respect to the real axis of the plot, and its construction is less intuitive and must be based on a software approach Nevertheless, commercial software that includes root loci routines can automatically solve such complex cases In the following example, the case of constant N-contours is shown, but the case of the off-axis circle iterion for evaluating absolute stability can be treated in a similar way Constant N-Contours Now consider the Hall circles for determination of the closed-loop phase response using the open-loop transfer function, the so-called N-contours The constant N-circles have their center at ( xc = 5, yc = 5 / N) and radius r = 5 ( N + )/ N, where N = tan( φ)and φ is the phase of the closed-loop transfer function Example 5 Problem: Evaluate the frequency corresponding to a closed-loop phase of φ= 6 for the system Gs ( ) = /( s+ ) Solution: The N-circle to be considered has N = 3, with its center at ( 5, 5 / 3) and radius r =/ 3 From (7), the application of the complete RCs with complex coefficients α,α, andα (Fig 9) leads toω = 95 rad/s Note that since the center c is a complex number, the root loci lose their symmetry property and the loci are no longer emanating from the complete root loci of Gs ( ),asinthe above examples The itical frequency evaluated from the points where the root loci oss the imaginary axis corresponds to point A in the Nyquist plot of Fig, where the open-loop transfer function Gs ( ) (red line) is intersecting the circle (blue line) corresponding to a closed-loop phase of φ= 6 Complete Root Contours and Relay Feedback Tests Now we investigate how the previous framework, based on the CRC, can be tested experimentally The problem at hand may be reconsidered as: Is there a simple way to measure the itical frequencies typical of the intersections between circles and a real process in the s-plane? It should be clear that answering this question would significantly benefit students of control theory courses by giving them useful insight into system properties Besides the educational advantages, an experimental validation of design parameters, such as phase margin, closed-loop bandwidth, etc, may be useful for testing the effectiveness of controllers applied to real processes (the test is more valuable for processes with uncertainties in their models) Relay feedback is a simple, powerful, and commonly used method of finding system parameters useful for designing and tuning standard proportional-integral-derivative (PID) controllers Since the pioneering studies of Aström and Hagglund in [], much research has been directed at extending and improving the relay tests, often denoted with the aonym ATV (automated tuning variation) The logic behind such an approach is that the existing controller is replaced by a nonlinearity (eg, of relay type, as shown in Fig ), so that most processes are brought into a condition of permanent oscillation In this way, the output signal (y) is a periodic signal, lagging behind the input (u)by 4 4 ω = 95 rad/s 5 5 Figure 9 CRC plot for Gs ( ) = /( s+ ) Case of off-axis circle intersection 5 6 N Circle 6 6 A 5 Figure Nyquist plot of G( s) and its intersection with the N circle of φ= 6 October IEEE Control Systems Magazine 87 Authorized licensed use limited to: University of Michigan Library Downloaded on May 9, at ::58 UTC from IEEE Xplore Restrictions apply

7 π radians and oscillating as a limit cycle with a pulsation ω C, usually termed ultimate frequency The desibing function [] provides a useful tool for a simple first-order harmonic balance in a limit cycle analysis From the standard theory, the desibing function is defined as the amplitude ratio of the fundamental harmonic component of the nonlinearity output (u ) to the input of the nonlinearity In the case shown in Fig, N( E)is the desibing function of an ideal relay with amplitude A, and E represents the amplitude of the sinusoidal signal at the input of the nonlinear block, so that A N( E) = 4 π E (3) A sustained oscillation is generated from a relay feedback when the following relation in the Nyquist plane holds: + G( jω ) N( E) =, (4) where G( jω) is the linear transfer function of the process The value of the itical (ultimate) gain K C is r = + e K C NE ( ) A A PID A = 4, πe C u Gs () Figure Block diagram of a relay feedback loop r = + A u B A Sinusoidal Oscillator ω u A Figure SATV scheme δ A B Switch Control u G( s) y Phase-Detector (5) y ϕ PI + ϕ* = π where E C is the signal amplitude at the input of the nonlinear block for the itical frequency ω C The basic advantages of this approach are that the peak-to-peak amplitude and the frequency of the limit cycle are normally measured, the limit cycle amplitude can be controlled by the relay output, and a mathematical model of the process is not required The controller settings can be easily determined from the knowledge of the ultimate gain and frequency, eg, using Ziegler-Nichols rules, and the PID controller of Fig can be automatically or manually switched in with the new parameters ATV is a successful technique for industrial applications, and several commercial autotuners have been built In the case of a relay nonlinearity, the real signal output of relay (u) is a square wave Therefore, relation (5) constitutes an acceptable approximation if and only if the linear process Gs ( ) behaves as a low-pass filter, cutting off any nonlinearity-generated higher harmonic components Otherwise, the higher-order harmonics cannot be neglected Unfortunately, the ATV technique provides only an approximation of the itical parameters of the oscillation Thus, significant errors occur (the errors in the estimated values are up to % compared to the exact values); in any case, the obtainable results vary greatly with the process at hand The estimation inaccuracy is due mainly to a itical point estimation based on the desibing function analysis As stated previously, the objective of our work is to utilize the relay feedback method for extending the analysis to estimation of the different itical frequencies due to the intersections with circles in the s-plane Unfortunately, the intersecting condition (4) leads to a linear part in the scheme of Fig for which the low-pass filter hypothesis becomes false and the desibing function analysis is inadequate Therefore, a modified method, the sinusoidal autotune variation (SATV), presented in [], can be considered; the SATV is based on a phase-locked loop (PLL) module added to a standard relay test for eliminating the low-pass filtering hypothesis due to use of the desibing function analysis in the relay feedback The SATV differs from a conceptually similar method proposed in [3] in that a shorter estimation time is obtained, due to the application of mixed relay plus PLL instead of a pure PLL technique SATV Method The simplest solution is to modify the nonlinearity output such that there is no production of harmonics other than the fundamental one (ie, the output of the nonlinear block is forced to be sinusoidal and the first harmonic linearization is correctly applicable) The signal output u of the nonlinear block is therefore locked to be a sinusoidal wave with a frequency such that the phase shift between the input e and output of the nonlinear block is π rad (SATV method) The question is how to achieve such an objective automatically 88 IEEE Control Systems Magazine October Authorized licensed use limited to: University of Michigan Library Downloaded on May 9, at ::58 UTC from IEEE Xplore Restrictions apply

8 Consider the block diagram shown in Fig First, the switch is in position A The plant Gs ( ) is included in a closed-loop system with a relay-type nonlinearity, as in the traditional ATV The limit cycle occurs, and a sustainable oscillation with a frequency ω near the itical frequency ω c of Gs ( )can be detected The input signal u A is a square wave The switch is commuted at the level B when the transient is exhausted,u A =, and the relay output osses zero in a positive direction The plantg( s)is now connected with a sinusoidal oscillator, generating the input signal u B as a sinusoidal wave with frequency and amplitude equal, respectively, toω and to the fundamental harmonic of u A In this way the switching between the two states is smooth A PLL adjusts the phase between the input and output signals toϕ = ϕ = π radians by changing the frequency of the oscillator The phase detector is a device able to detect the phase difference between two periodic signals at the same frequency; unfortunately, the device is very sensitive to disturbances Its action becomes robust even in the presence of disturbance by a suitable modification of the algorithm, based on an integration of the periodic input signals over m periods The sinusoidal oscillator is a nonlinear system desibed by the following equations: where & x x x& = ω x u B = ω x, δ= t < t t t + δ (6) (7) and t is the starting time With a null initial condition on the integrators, the output u B is u = 4 A B sin ( ω t ω t) π (8) The block diagram of the sinusoidal oscillator is shown in Fig 3, and its output for a lock-in time of t = 6 s is shown in Fig 4 An important feature of SATV is the ease with which it obtains different information from the ultimate frequency The reference angle in the feedback loop can be changed, and various points of the linear process on the Nyquist plane can be easily identified The effectiveness of the solution proposed was tested extensively in [] in typical cases of chemical processes where the desibing function analysis often loses its validity SATV Experimental Tests Consider () with the parameter τ=, leading to s cfr( s) = c+ r + s (9) The intersecting condition (4) in the Nyquist plane can be rewritten as Gs ( ) / α s Gs ( ) + / α By defining the transfer function = () Gs ( ) / α P( s) = s Gs ( ) + / α () from (), the intersecting condition is reduced to detecting the ultimate frequency ω C of P( jω) in a relay feedback loop + s δ + s u Figure 3 Sinusoidal oscillator scheme δ u B 5 5 X Lock-In Time Time [s] Output of the Sinusoidal Oscillator Time [s] Figure 4 Sinusoidal oscillator output for t = 6 s ω X u B October IEEE Control Systems Magazine 89 Authorized licensed use limited to: University of Michigan Library Downloaded on May 9, at ::58 UTC from IEEE Xplore Restrictions apply

9 u s + α α α Gs () Figure 5 Block diagram implementing the P(s) transfer function R R R 3 + y Unfortunately, the low-pass filter hypothesis in the standard relay feedback method often becomes false in the case of () Therefore, the ATV method is inadequate for estimating the itical frequencies representing the intersections of a transfer function with circles in the s-plane On the other hand, the SATV method can be successfully applied to the transfer function P( jω ), so that the process Gs ( )under test has to be rearranged as in Fig 5 and then inserted into the SATV scheme of Fig An experimental test was performed for evaluating the effectiveness of the proposed method in the case of a resistive capacitive (RC) filter with a third-order transfer function The RC filter, shown in Fig 6, has a sinusoidal ac input voltage Vin and can be amplified by a factor K so that different phase margins can be evaluated The input signal is amplified by a factor of The nominal filter transfer function is K V in C C C 3 V OUT F( s) = ( RCs) + 5( RCs) + 6RCs + 3 () Figure 6 RC filter: R = R =R =R 3 = C=C =C =C 3 = F RT Workshop PC SATV Output Board In Out Out In Input Board RC Filter Figure 7 Real-Time Workshop implementation Conclusions The objective of this article was to provide a general and simple framework to explore how the information supplied by circles in the Nyquist plane can be recovered in the RL plane In this article, all circle iteria (including off-axis circles) in the frequency domain can be reformulated as complete root contours In this way, some difficulties of analysis in the frequency domain are overcome by means of the Evans technique Applications to phase margins, sensitivity, bandwidth computations, constant N-contours, and absolute sta- Two- Channels Scope CH = mv CH = mv 5 ms/div ks/s Figure 8 Lock-in phase of the RC filter: experimental results The hardware acquisition equipment is based on a data acquisition board (National Instruments Lab-PC- AI) installed on a PC; the SATV method is then implemented via executable files eated by the Real-Time Workshop Toolbox of MATLAB (see Fig 7) In Fig 8, the input and output signals of the RC filter are shown during the lock-in phase of the SATV, as revealed from the two-channel scope Table lists experimental results for the hardware implementation of the method proposed By varying the factor K, different tests can be easily performed In Fig 9, the two-channel scope window is shown; the phase margin is measured by comparing the phase shift between the input and output signals As a final remark, note that in Table, the nominal values are the nominal R and C values of the filter, and the simulated SATV is the software implementation of the technique with the nominal filter The real-time SATV considers the real implementation of the RC filter, and the estimated implementation is the software implementation of the technique in the case of an RC filter, whose parameters are identified by using for example, a modulating function approach [4] 9 IEEE Control Systems Magazine October Authorized licensed use limited to: University of Michigan Library Downloaded on May 9, at ::58 UTC from IEEE Xplore Restrictions apply

10 yt ( ) CH = 5 mv CH = mv ut ( ) Sensitivity Values (Hz) Phase Margin (deg) (K = /) Nominal values ( f = 39 Hz) Estimated values ( f = 5 Hz) Simulated SATV ( f = 39 Hz) Real Time SATV ( f = 8 Hz) 5 ms/div ks/s [5] Y Tong and NK Sinha, A computational technique for the robust root locus, IEEE Trans Indust Electron, vol 4, pp 79-85, 994 [6] BC Kuo, Automatic Control System Englewood Cliffs, NJ: Prentice-Hall, 99 [7] ML Nagurka and TR Kurfess, Gain and phase margins of SISO systems from modified root locus plots, IEEE Contr Syst Mag, vol, pp3-7, 99 [8] TJ Cavicchi, Phase root locus and relative stability, IEEE Contr Syst Mag, vol 69, pp 69-77, 996 [9] A Balestrino and A Landi, Circle iteria and complete root contour, in Proc 4th IFAC Int Triennial World Congress, Beijing, China, 999, pp [] KJ Aström and T Hagglund, Automatic tuning of simple regulators with specifications on phase and amplitude margins, Automatica, vol, pp , 984 [] A Balestrino, A Landi, and L Sani, ATV techniques: Troubles and remedies, in Proc Int Symp ADCHEM, Pisa, Italy,, pp [] DP Atherton, Nonlinear Control Engineering London: Van Nostrand Reinhold, 98 [3] J Crowe and MA Johnson, Process identifier and its application to industrial control, IEE Proc Contr Theory Appl, vol 47, pp 96-4, [4] HA Preising and DWT Rippin, Theory and application of the modulating function method: I Review and theory of the method and theory of the spline-type modulating functions, Comput Chem Eng, vol 7, pp -6, 993 Figure 9 Experimental scope window of the phase margin evaluation bility were reported, along with a practical implementation in laboratory tests via a modified relay autotune method, called SATV The SATV technique was successfully used for a practical evaluation of the itical frequencies of the plant intersecting circles Its robustness was tested, even in the presence of measurement noise, based on an integration of the periodic input signal over m periods This approach offers engineers and students an opportunity to verify in practice fundamental concepts of control system design, such as phase margin, bandwidth, sensitivity, and absolute stability, for controlled plants with uncertain or unknown transfer functions References [] WS Levine, The root locus plot, in The Control Handbook, WS Levine, Ed CRC: Boca Raton, FL, and IEEE: Piscataway, NJ, 996, pp 9-98 [] WR Evans, Graphical analysis of control systems, Trans AIEE, vol 67, pp , 948 [3] WR Evans, Control system synthesis by root locus method, Trans AIEE, vol 69, pp 66-69, 95 [4] BR Barmish and R Tempo, The robust root locus, Automatica, vol 6, pp 83-9, 99 Aldo Balestrino graduated in 965 in electronics engineering From 98 to 983, he was a full professor of systems theory at the University of Naples Since 984, he has been a professor of automatic control at the University of Pisa From 989 to 993, he was the Head of the Department of Electrical Systems and Automation, and from 989 to he was the Coordinator of the PhD course in automation and industrial robotics His research interests include modeling and simulation of nonlinear systems, design of advanced regulators, neural networks, and innovative control for high-speed railway systems Alberto Landi received the electrical engineering degree in 986 from the University of Genova and the PhD degree in electrical engineering in 99 from the University of Pisa He is currently at the Department of Electrical Systems and Automation, University of Pisa, as an Associate Professor, where he chairs courses on automatic control His research interests include motion control, nonlinear identification and control, and electrical traction high-speed railway systems Luca Sani received the electrical engineering degree and the PhD degree in automation and industrial robotics from the University of Pisa, Italy, in 996 and, respectively He is currently with the Department of Electrical Systems and Automation, University of Pisa His current research interests include converters, identification in the continuous domain, and drives for high-speed railway systems October IEEE Control Systems Magazine 9 Authorized licensed use limited to: University of Michigan Library Downloaded on May 9, at ::58 UTC from IEEE Xplore Restrictions apply