Session S1C
A PROJECT -BASED COURSE OF NONLINEAR CONTROL M. De la Sen1 Abstract. -The main objective in the project of a last year Master Engineering course on discrete control where students develop projects based on discrete control at this University has been to emphasize that there are several points of view when focusing on discretization and the associate mathematical developments depending on the particular analysis technique and problem at hand. In particular, a combined non-linearity consisting of resolution (quantifier) and mechanical backlash has been assumed to operate on a linear plant subject to input discretization under a sampling and zero-order hold device. It has been proved through the project that because of sampling the system possess nonlinear inertia with the frequency of the first harmonics of nonlinear steady-state sustained oscillations being dependent on the sampling period and the critical locus being depending on frequency
after the "a priori" switching points between portions of the state trajectories, namely, at the current switching instants. Features of special relevance from an educational point of view which are then commented in more detail are: 1.
2.
3.
Index Terms. Resolution, Backlash, Describing Function, Limit Cycles, Non-linear Inertia. 4.
INTRODUCTION AND PRELIMINARIES Traditional courses on nonlinear systems and discrete control can be extracted from the existing literature[1-3,4] for M. Eng. Courses. An interesting particular sub-field is that of analysis and synthesis of discrete control systems. Its interest arises from the fact that both non-linearities and discretization are inherent to many physical systems and controller implementations. The introduction of case studies involving those topics as parts of projects is very convenient from an educational point of view. Such strategies also require to supply a previous related theoretical background to the students of advanced engineering courses. In the last years, one of the control engineering courses at this University includes a project to be developed by students divided into reduced groups while reducing the number of traditional lectures within the alternative classical course. The project is usually developed in groups of 5-6 people [4]. The use of discrete transfer functions or difference equations is suitable when only the sampling points are relevant to the problem. When other time instants are relevant as, for instance, those associated with the intersections of trajectories with switching surfaces in nonlinear problems [6] where the non-linearities are located in the actuators, the calculation of the trajectory solution for all time can be required. Such a need is due to the feature that real switches occur at the sampling instants which are located immediately
5.
6.
The switching hypersurfaces between zones (in particular, the separation and recombination curves in the case of backlash or combined backlash with resolution for second-order plants) are not the theoretical one expected in the continuous-time case. The (continuous-time) plant zeros (if any) are considered explicitly within a differential equation which modifies the location of the separation and recombination curves which would be associated with the plant transfer function in the zero-free case. The sampling and hold device causes an effective delay from the theoretical switching curves to the real curves since the last ones are necessarily reached at sampling instants. The calculation of the switching instants at the switching hypersurfaces is made by considering the complete state-trajectory for all time including the intersample times. The reason is that the current switching instants have to be sampling instants while the intersection points of the trajectories with the switching hypersurfaces may occur inbetween sampling instants. Both the describing function and its associate critical locus are calculated by considering the sampled shape of a pure sinusoidal plant input what is, roughly speaking, a periodic signal built with a sequence of steps obtained from a steady-state sinusoidal loop error prior to the discretization. This philosophy modifies the analytical expression obtained for the describing function and critical locus related to other existing approaches [1]. The fact that the switching hypersurfaces or curves depend on the sampling period and exhibit delays related to the (continuous) theoretical ones make the complex describing function and critical locus plots associated with memory nonlinearities to exhibit nonlinear inertia, in the sense that they are dependent on frequency and lead to multiple branches on the Nyquist plane. This phenomenon does not appear if the sampling and hold device is removed. It implies that the fundamental frequencies of any potential limit cycle are dependent on the linear hodograph plot of the frequency
1
Instituto de Invest igación y Desarrollo de Procesos. IIDP, Facultad de Ciencias, Universidad del País Vasco, Leioa (Bizkaia). Aptdo. 644 de Bilbao, SPAIN,
[email protected]
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Session S1C response of the plant and, furthermore, on the critical locus and, therefore, on the sampling period. Motivating Features Consider the continuous- time linear difference single-input, single- output system:
x& ( t ) = Ax ( t ) + bu ( t ) ,
y ( t) = c T x( t)
(1)
with x (0)=x 0 . Subscripts are now used for notational abbreviation to denote values of the various signals at sampling points. If T is the sampling period, then its solution on [k T , (k + 1) T ) in the presence of a zero-order hold is ´ A τ − τ A τ x + β ≡ τ e u x ( k T + τ ) = α ≡ e b d τ ∫ k 0 k
τ ∈ [0 , T
)
with y ( k T+ τ ) = c T x ( k T+ τ ) ; k ≥ 0
and x ( ( k + 1)T ) = x
k +1
and y ( ( k + 1)T ) = y
(2) k +1 ;
where
x(.), y(.) and u(.) are the state n- vector, scalar output and control input.Eq. 2 and the output equation lead to the following input-output ARMA model at sampling instants: y
k +1
= A ( q −1) y k + B ( q −1 ) u k = θ T φ k
(3)
and n −1 A (q −1 ) = ∑ α i q − i ; B ( q i= 0
−1
m
)=
∑α i= 0
i
q
−i
( β 0≠ 0 )
(4.a) θ = − α 1 , − α 2 , ..... , − α n , β 0 , β 1º , ...., β m T ) (4.b) T φ k = y k , y k − 1 , .... , y k − n + 1 , u k , u k −1 , ...., u
(
)
(
k −m
)
(4.c) Note that while (2) and its associated subsequent output equation provide the state and output trajectories at any time, the ARMA description (3) provides the output sequence sampling instants only. Therefore, the first description was used to calculate the switching instants at the separation and recombination surfaces since it is first necessary to deduce the theoretical switching points while the real ones are the next available sampling instants. It was a pedagogical issue in the project to emphasize that a continuous-type treatment of the problem is necessary to obtain such point even although the input is obtained from a nonlinear device. The following generic nonlinear controller is used:
u (t) = u
k
= f ( x ( τ ) , u ( τ ) ), τ ∈ [ t , t − t ) , some t < t
with y ( k T+ τ ) = c T x ( k T+ τ ) , some t < t ) (5) for all t ∈ [k T , ( k + 1) T ) where f is an appropriate nonlinear function which may include analytic nonlinearities, parasitic nonlinearities (as, for instance, backlash), bang- bang type controllers, and functions related to the synthesis of adaptive controllers in the case of unknown parameter vector θ etc. Two options have been selected for the nonlinear devices eq. 5, namely: . Option 1: Relay with dead- zone and hysteresis subject to resolution (i. e. quantifier effects) . Option 2: Backlash being associated with rotation of coupled gears substitutes to the relay of Option 1. . It is obvious that while the description (3)-(4) is relevant for discrete parameter estimation, some information has to be maintained about time instants which are not sampling instants. This is reasoned as follows. Define the switching instants as sampling instants occurring immediately after times associated with switching between zones of the nonlinear device. Those switching instants cannot be intersample instants since the discrete control input (5) with a zero-order hold only switches at sampling instants. The current switching instants are calculated as the next sampling points occurring after the "a priori" switching instants (i.e., those associated with the intersection trajectory-continuous switching hypersurface). In fact, the a priori switching surfaces can be analytically defined in the continuous case [1] while the current switching surfaces and instants are calculated with the time intervals from the "a priori" values for each next sampling instant. .
t ´1 is an "a priori" switching
Switching Instants. If
point, in general, being located in-between sampling instants, then its corresponding current sampling instant, necessarily being a sampling instant, is defined by
(
)
t 1 = k 1 T = Integer Part t ´1 / T + T
if
t ´1 is not a
multiple of T, and t 1 = t ´1 , otherwise. The increment ( t 1 - t ´1 ) may be interpreted as a switching delay caused by the sampling and hold process and it is, in general, nonconstant for different parts of the state- trajectory so that the switching surface is, in general, time-varying. The calculation of the current switching instants (in the following denoted simply as switching instants for abbreviation) requires some information about the solution of (1) for all time. The reason is that the "a priori" switching instants have to be first calculated. The fact that difference equations (2) lead to the solutions of (1) at sampling points is sometimes forgotten by students who have a tendency to
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Session S1C consider "continuous" and "discrete" as two different Worlds. In the subsequent sections, Option 2 is considered for the nonlinearity. ANALYSIS TECHNIQUES INVOLVED IN THE PROJECT Heuristic Foundations Backlash (or mechanical hysteresis) is due to the difference in motion between an increasing and a decreasing output, usually caused by mechanical gearing [1]. The two switching hypersurfaces between the backlash and linear zones are the separation hypersurface (Linear Zone to Backlash Zone) and the "pock-up", or recombination hypersurface (Backlash Zone to Linear Zone). The system is subject to sampling under a zero-order hold. The trajectory switchings between the linear and backlash zone and vice versa do not occur at the theoretical separation and recombination surfaces but at the next time instants where sampling occurs (i.e. , those sampling points located after the continuous switching curves have been crossed). This potential delay in the switching instants between zones is the natural consequence of the presence of the zero-order hold, which reduces the information in the control. Figure 1 shows the composite non-linearity with resolution and backlash when the sampling and zero-order hold device are in operation to discretize the plant. The closed-loop scheme is shown in Figure 2 which extends to the discrete case results presented in [3] for continuous plants. The plant zeros and poles are separated to facilitate the calculation of the "a priori" switching instants which are calculated from the modified switching hypersurfaces obtained from those associated with the continuous- time plant free oz zeros as follows. Assume that H (s)= Z(s) / A(s) is the continuoustime plant transfer function while H´(s)= 1/A(s) is the zerofree modified one with deg(A) ≥ deg (Z) . Assume for discussion simplicity that H (s) and H´(s) are of secondorder. Let s (y, y& )=0 a switching curve in the phase plane (y, y& ) for H ´(s) corresponding to switches between two operation zones of a nonlinear device with y being its output of time-derivative y& . Thus, if v(t) = Z (D) y(t) with D = d/dt denoting the time- derivative operator (being formally equivalent to the Laplace operator s) then s (v, v& )=0 is still the switching curve for the current plant H(s) in the phase plane ( v , v& ). By combining the implicit equation s (v, v& )=0 with v(t) = Z (D) y(t) and v& (t) = D. Z (D) y(t) = Z (D) y& (t), one gets the implicit switching curve s 1 ( y , y& ) = s ( Z( D) y ( t ) , Z ( D ) y& ( t ) ) =0
or
its
explicit
version v = f 1 ( y , &y ) , v& = f 2 ( y , y& ) . The input nonlinearity under consideration in the continuous-time transfer function is the composition of a backlash and resolution. It combines three effects, namely, that of an hysteresis of mechanical type in two coupled gears of distinct radius (backlash) , the effect of using a quantifier
for the signals useful to describe the use of a computer to calculate and generate the plant control law combined with a discretization device involving a sampler and zero-order hold. The discretization causes a time-delay in the generation of the switching hypersurfaces between two different operation modes of the nonlinear device. Phase- Space Analysis It is a time domain approach. The switching surface and the "a priori" switching instants are very relevant for the analysis as emphasized before. The sampling and hold device has typically the effect of generating a delay for such instants leading to the current switching instants which always are sampling instants if the actuator is nonlinear. This also implies a control delay in the sense that the plant control is typically the corresponding one to the preceding operation mode of the nonlinear characteristics during a fraction of sampling period after each switching takes place. In that way, a short time (for small sampling periods) after entering the backlash zone, the plant is controlled by a linear control and vice versa. By the reasons pointed out in the introductory section including the relevance of the " a priori" switching instants, which only are sampling instants excepcionally, it is requested to solve (1) for all time subject to piecewise continuous controls changing)-(3) of the solution of (1). Describing Function Analysis It is especially useful to calculate rapidly a good approximation of amplitude and frequency of limit cycles as well as to elucidate the stability regions in the parameter space [1]. If the sampling and hold device is physically located prior to the plant then it is useful to define the describing function as the complex gain of first-order harmonics from the plant input to the plant output as the plant input is a pure sinusoidal signal. In the proposed approach, we consider a plant input as a periodic sequence of steps built with the samples of a pure (continuous) sinusoidal signal by the zero-order sampling and hold device. This would be the real input to the plant provided by the zero-order sampling and hold device in sinusoidal steady state. In a general setting, this was a major pedagogical issue inherent to the project since in most of the approaches the steady-state describing function approach is usually investigated under a continuous- type sinusoidal input. Note that other approaches to the problem would be possible like considering the sampled frequency response of the plant n= ∞
2 nπ G j ω + when the input is T n = −∞ sinusoidal or that arising from reversing the order of the sampling and hold and the nonlinear device. Those approaches are, respectively, either computationally very involved or of difficult applicability because of the hysteresis effect (i.e., "memory") being inherently associated 1 G ( jω ) = T ∗
∑
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Session S1C with backlash. The phenomenon of the dynamic nature of the switching surface associated with the delay effect between "a priori" and (current) switching instants makes the critical locus (namely, the minus inverse of the describing function) to possess an inertia in the sense that it depends on the frequency and possess multiple branches. This phenomenon is not observed if the discretization process is removed and it has, therefore, an special educational relevance since the frequencies of possible limit cycles have to be considering from both complex plots in the Nyquist plane, namely, the linear hodograph portrait and the critical locus one. On the other hand, note that sampling and hold devices generate higher-order harmonics what is reflected in the critical locus plot compared to that obtained if the discretization is removed. Work Organization The Project was divided in three parts, namely: - State- space description including the analytic or numerical calculation of the trajectory for all times (i.e. "at" and "inbetween" sampling instants). - Analytic calculation of the describing function and critical locus of the composite non-linearity of resolution and backlash under a sampling and zero-order hold device. The particular sampled sinusoid mentioned above was used as plant input to perform the above calculations. - Computer Programs . A program was developed to calculate and plot the state trajectories for all time as well as to detect the " a priori" and " a posteriori" (current) switching points. A second program had the objective of detecting the amplitude and frequency of the fundamental harmonic of all possible limit cycles by calculating the describing function and critical locus for the composite nonlinearity with inertia and plotting the last one on a Nyquist plane. - Numerical simulations and interpretation of results. Second-order and third-order continuous-time plants were tested either being zero-free or involving the presence of zeros. The corresponding state-trajectories were plotted and possible limit cycles were detected from the above mentioned program. The obtained results were interpreted by comparing the limit cycles obtained from the state- space trajectories with their first harmonics obtained from the describing function analysis. Each group of students developed the four parts of the project with the second and third-order realizable plants including from none to two zeros. One of the poles was optionally a simple integrator since the state- trajectory is highly dependent on the presence of integrators or not since the nature of the differential equations to be solved is essentially distinct. The students were asked to find and discuss comparatively examples by modifying the various parameters of the plant transfer function, non-linearities and sampling period. For instance, the sampling period, the plant static gain, the plant poles and zeros and the backlash and
resolution parameters were modified. The comparative study included transient ands steady- state performances, stability, equilibrium points, limit cycles (which was the essential part of the study) with their amplitudes and frequencies. CONTENTS AND PROJECT DEVELOPMENT The resolution and backlash are, respectively described by the pairs of parameters (h , d) and (σ , k b ) - 2h is the quantifier step with amplitude d, and σ and k b are, respectively, the cut of the positive increasing ramp of the backlash with the horizontal axis and the slope of the linear part of the backlash characteristics. Saturated horizontal and vertical values vs and M are assumed for the backlash having a physical nature due to rotation limits of the gears. Both non-linearities are assumed to be symmetrical with respect to the origin. The switching hyperplane in the phase is given by the following implicit equations: v = - v s + 2 σ , - v s + 2 σ + h , ... , σ ,... , v s - h, ... , v s if v& ≥ 0 (Rising ramp of the backlash) (6.a) v = - v, - v s + h , ... , - σ ,... , v s -2 σ - h, ... , v s - 2 σ if v& < 0 (Descendent ramp of the backlash) (6.b)
where v s is the saturated value in abscissas. The transitions between the operation modes: Rising Ramp (RR), Descendent Ramp (DR), Backlash to the Left (BL) and Backlash to the Right (BR) when backlash operates can be considered as a finite automaton of four states (shown in Fig. 3) and they operate are as follows: (a) The state-space trajectory enters the backlash zones from the rising and descendent ramps when v& ( t ) changes its sign at some time t = kT, k being a positive integer and T being a sampling period. For a sufficiently small T, that test can be performed with the sign of the increment δ k = v k +1 − v k (memory effect associated with backlash). If a sign change is detected then v k b stores
v k . If the system enters the backlash zone then r
k
= K b Min (v k , v s sign ( v k )) which
takes
into
account a possible saturation in the plant output. While the system is operating in the backlash zones, r k = r k − 1 . (b) The trajectory leaves the left backlash zone moving to the descendent ramp if δ 'k = v k + 1 − v k b ≤ −2 σ and it reverses its motion (backward motion) when v k + 1 > v k . (c) The right backlash zone is left if v k +1 < v k b (towards the descendent ramp) or if δ `k + 1 ≥ 2 σ (toward the rising ramp). All those transitions occur at sampling points. While
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Session S1C the system operates in the linear ramps of the backlash, the backlash output is calculated similarly as an entering point of the backlash zones. The above strategies lead to the calculation of the backlash output from its input according to a relation r k = f v k , δ k −1 . For non-sampling points, the
(
)
above formula is modified as r ( t ) = f ( v ( t ) , δ
k −1
).
Switching Instants at the Switching Surfaces As pointed out before, the switching surfaces are delayed with respect to the continuous problem because of sampling and because of the presence of resolution. Choose the plant state variables as follows: x 1 ( t ) = x ( t ) ; x 2 ( t ) = x& ( t ) ; and the triple
( σ, h , d ) , the saturated values v s
. Step 1: For each sample k ≥ 0, calculate the parametrical equations of the state-trajectory
(
)
T
for
all
time
t ∈ [k T , ( k + 1) T ) from the state-space description of the
continuous-time plant under the piecewise constant control obtained from the sampling and zero-order hold. . Step 2: Calculate the backlash output according to where f (.) is defined according r ( t ) = f ( v ( t ) , δ k ) for all to the rules for the transitions between the operation modes of the backlash as discussed in the above subsection. Then, calculate for t = (k+1)T the resolution output at sampling instants u k +1 = d. Integer Part [2 r k +1 / h] which generates the plant input at sampling instants through the feedback loop as e k +1 = − u k +1 with e(t) = e k +1 = - u k+1 = - d. Integer Part [ 2 r k +1 / h ]= -
(
N (A , ω ) = q ( A , ω ) + j q ' ( A , ω )
(7)
After taking into account that the sinusoidal plant input is sampled by the sampling and hold device and, since ϕ is piecewise continuous, the real and imaginary parts of the describing function in (7) become:
(
q A,
ω
)=
1 πA
l ∑−1 u k = 0
ω ( k + α +1 )T 2π + ω α T sinβd β + u l ∫ sin β d β k ∫ ω ( k + α )T ω 1+ α T
(
)
d. Integer Part [ 2 . f v k +1 , δ k +1 / h ]. . Step 3: Make k → k + 1 and Go to Step 1.
Describing Function Analysis The main philosophy involved in deriving the describing function is to consider a sinusoidal discretized input signal acting on the non-linear device including resolution and to
(
q ' A,
ω
)=
1 πA
l −1 ∑ u ∫ ω ( k + α +1 )T cos βdβ + u ∫ 2π + ω α T cos β d β k ω ( k + α )T l ω 1+ α T k =0
(
)
(8.b) with α ∈ [0 , 1 ] being a real parameter which is an indicator for the fraction of sampling period [12] associated with the first sample in steady-state of the sinusoidal plant input. Possible limit cycles are given by the solution in (A, ω ) of the equation C (A, ω ) = G (j ω ) in the complex plane provided that both values are real and positive. The stability of the limit cycles is investigated by the application of the graphical Loeb's criterion[2]. Only stable sustained oscillations can be physically detected in practice. It has been seen through the numerical experimentation that the nonlinear inertia is irrelevant for l > 500 and sampling period T of the order of 10 − 4 or less since the relevant dynamics is close to the continuous one. For small l , of the order of ten, the loop filtering properties are poor since the sampling rate is large and higher-order harmonics have large amplitudes. As the sampling period increases, the nonlinear inertia becomes apparent for larger ranges of frequency.
NUMERICAL EXAMPLES and backlash are parametrized by σ = 2 h = 2 d = 0.2 , K b = 1 and . The linear plant has the
The
resolution
transfer function G ( s ) = Remark: Note that the switching surfaces only affect to the change of operation mode of the backlash non-linearity at sampling instants since the control is generated by a sampling and zero-order hold as mentioned above. This implies that the switching effective (separation and recombination) surfaces for the composite non-linearity are delayed with respect to the nominal theoretical ones obtained from a purely continuous- time description.
)
(8.a)
and M,
and the linear gains (K, K b ) of the plant and backlash characterize the composite non-linearity, The current switching times are calculated as follows for the given sampling period T. The subsequent discussion is limited for brevity to the third-order plant proposed.
x ( t ) = x 1 ( t ) = v ( t ), x 2 ( t ) , x 3 ( t )
calculate the gain corresponding to the first input-output harmonics. The describing function is defined as follows:
(
K s+z1
(s + p )(s + p 1
2
)
)( s + p 3 )
with
p 1 = 0 .1 , p 2 = 0 .5 , p 3 = 2 , z 1 = 1.6, K =1 and T (sampling period) = 0.01 sec. The α − parameter is zero in the critical locus unless otherwise stated. A stable limit cycle is detected with A = 0.258, ω = 0.530 rad /sec. . The nonlinear inertia for this frequency and the given sampling period is irrelevant . The real values obtained from the statespace analysis are A = 0.257 and ω = 0.529 rad. /sec. Thus, the absolute values of the relative errors in the results are 0.39% for the amplitude and 0.21% for the frequency. Modification of the plant zero and the plant gain. If the zero is move to unity then extensive simulations have shown
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Session S1C that the limit cycles disappears. Once a limit cycle has been eliminated by an appropriate choice of the zeros then a possible sampling period increase do not create new limit cycles. If the zero is increased up to 3 then four stable limit cycles appear. On the other hand, a gain increase generates a loss in the relative stability degree. A maximum of four limit cycles appear if the high frequency plant gain is increased sufficiently. Resolution modification. Limit cycles can disappear when h and d decrease sufficiently while keeping the values of all the remaining parameters fi the critical locus of the nonlinearity without resolution and the hodograph of the linear plant do not have any intersections. This is due to the negligible influence of small quantifier effects in the system as pointed out in [6] for the continuous case. Sampling period and α - parameter. The calculations via the describing function approach deteriorate as the sampling period increases as mentioned. Related simulation details are omitted by space reasons. It is found that for a sampling period T= 0.1 secs., and ( A , ω )= ( 0.262, 0.528), the parameters of a stable limit cycle of real values are 0.259 and 0.528, respectively, with respective relative errors from the method application of 1.2 % and 0 %. For T = 0.5 sec. , a stable limit cycle with ( A , ω )= ( 0.270, 0.520) is detected while the real values are 0.269 and 0.524, respectively. In summary, it has been found that if the sampling period is large then the plant filtering capability to high-order harmonics becomes poor and the obtained results from the method application are not good. Furthermore, the nonlinear inertia becomes significant. Figs. 4 display some simulations: ACKNOWLEDGMENTS The author is very grateful to MEC and UPV by its partial support of this work through Projects DPI 2000-0244 and 1/UPV/EHU I06. I06 EB 8235/2000.
[5] De la Sen, M., " Robust control of discrete critical systems with specifications in the frequency domain", Int. J. of Control, 67, No. 2, 1997, pp. 169-192. [6] De la Sen, M., Peña, A. and Esnaola, J.," Detection of limit cycles in discrete systems with backlash and resolution by using a discretization-oriented describing function", Int. J. of Control, 25, No. 2, 1997, pp. 48-55.
Figure 1. Composite non-linearity with resolution and backlash
Figure 2. Discrete system with composite non-linearity and linear plant
REFERENCES [1] Atherton, D.P., Nonlinear Control Engineering, Van Nostrand Reinhold Company (1975). [2] De la Sen, M., " The teaching of digital control design: A project approach", Int. J. Electrical Eng Educ., 28, No. 1, 1991 pp. 34-46. [3] Ramu, I. and Deekshatulu, B.L., " Analysis of systems with backlash and resolution", Int. J. of Control, 4, No. 4,1996, pp. 325-336. [4] De la Sen, M., " The design of a discrete robust linear
Figure 3. Automaton whose states are generated by composite non-linearity
feedback controller with nonlinear saturating actuator", Int. J. of Systems Science, 20, No. 3, 1989 pp. 495-521.
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Session S1C
Figure 4. Critical locus and plant hodograph in the Nyquist plane for the estimated frequencies of the stable limit cycles: (a) T=0.1 sec., ω=0.528 rad/sec. and T=0.5 sec., ω=0.52 rad/sec.; (b) T=1 sec., ω1 =0.478 rad/sec. and ω2 =0.598 rad/sec.
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