ISA Transactions 57 (2015) 295–300
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Novel controller design for plants with relay nonlinearity to reduce amplitude of sustained oscillations: Illustration with a fractional controller Ameya Anil Kesarkar, N. Selvaganesan n, H. Priyadarshan Department of Avionics, Indian Institute of Space Science and Technology (IIST), Thiruvananthapuram-695547, India
art ic l e i nf o
a b s t r a c t
Article history: Received 4 September 2013 Received in revised form 18 November 2014 Accepted 7 January 2015 Available online 28 January 2015 This paper was recommended for publication by Y. Chen
This paper proposes a novel constrained optimization problem to design a controller for plants containing relay nonlinearity to reduce the amplitude of sustained oscillations. The controller is additionally constrained to satisfy desirable loop specifications. The proposed formulation is validated by designing a fractional PI controller for a plant with relay. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Limit cycles Relay nonlinearity Describing function PI α
1. Introduction For systems containing separable nonlinearity in cascade, designed controllers usually produce undesirable sustained periodic oscillations in the plant output response due to the existence of stable limit cycles [1]. For the analysis of such limit cycles, Describing Function (DF) is a commonly used tool [2]. The studies in [3–5] show also the usefulness of DF for synthesis of controllers to satisfy the given limit cycle details. The work in [3] focuses on designing a robust limit cycle controller for a plant with relay nonlinearity by maintaining Nyquist plot of loop Transfer Function (TF) orthogonal to negative inverse of DF of relay nonlinearity. This is subsequently generalized for other nonlinearities in [4,5]. Motivated from the above works, we consider the problem of shaping the loop involving relay nonlinearity to reduce the amplitude of sustained oscillations. Additionally, we constrain the loop to meet certain performance specifications. For meeting such a stringent control requirement, we explore the potential of Fractional-Order Controllers (FOCs). FOCs are the controllers whose dynamics are governed by fractional-order differential equations [6–8]. FOCs such as PI α , ½PIα , PDβ , ½PDβ , PI α Dβ are superclass of their integer counterparts. Therefore, one expects them to perform better [9]. For instance, PI α has the n
Corresponding author. Tel.: þ 91 471 256 8456. E-mail address: [email protected] (N. Selvaganesan).
http://dx.doi.org/10.1016/j.isatra.2015.01.005 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
capacity to outperform integer-order PI controller. In the literature, the design of FOCs for linear plants has been widely studied [10–16]. In the limit cycle context, however, only a few works are seen for FOCs [17– 19]. For the design problem in the current paper, we investigate the applicability of FOCs. The contributions of this paper are: (i) An optimization problem is proposed to design a controller for plants containing relay nonlinearity in order to get: Reduced sustained oscillation amplitudes. Desirable loop performance. (ii) Demonstration using fractional PI controller design.
2. Basics of relay nonlinearity and stable limit cycles Let us consider a plant consisting of a relay nonlinearity in cascade with the TF G(s). The closed loop control schematics with controller C(s) is shown in Fig. 1. Mathematically, the relay nonlinearity in Fig. 1 is given by ( y2 ðtÞ ¼
M
if y1 ðtÞ Z 0
M
if y1 ðtÞ o 0
ð1Þ
We concentrate on a case where the Nyquist plot of the designed loop is assumed to intersect with negative inverse of
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Reference input +−
C (s)
y1( t)
y2 (t )
M −M
G( s)
controlling the ‘transient’ behavior of a closed loop system in the presence of nonlinearity as follows:
y3 ( t )
For the closed loop schematics shown in Fig. 1, one usually
Relay
neglects the plant nonlinearity and considers only the TF G(s) while designing the controller C(s). This means that one assumes the nonlinearity to behave as a unity gain during the transient time. At a particular amplitude A of step reference, if the signal occurring at input of nonlinearity roughly takes shape of a sine-wave with peak amplitude P such that NðPÞ ¼ 1, then the transient performance of the designed closed loop control system with and without nonlinearity is ‘same’ for such A.
Fig. 1. Closed loop control schematics.
DF in the complex plane only once1 as shown in Fig. 2. Fig. 2 contains superposition of: (i) 1=NðXÞ plot of relay nonlinearity. Where, NðXÞ ¼ 4M=πX is its DF [1] and X is the peak amplitude of sinusoidal signal at the input of nonlinearity. (ii) Nyquist plot of loop TF LðsÞ ¼ CðsÞGðsÞ. The arrow in the Nyquist plot indicates increasing ω direction ðωA ½0; 1ÞÞ. The arrow in 1=NðXÞ plot shows increasing X direction. The limit cycle point ① is intersection point between Nyquist plot and 1=NðXÞ curve. For sustained oscillations to be produced at the output in the presence of nonlinearity, the nature of limit cycle point must be ‘stable’. For the relay nonlinearity case, the occurrence and stability of limit cycles is ensured if the conditions explained below are met by the loop TF L(s) (Refer Fig. 2). 1. Nyquist Condition for Limit Cycle Existence [1]: 1 ¼ ½LðjωÞω ¼ ω0 NðXÞ X ¼ X 0
ð2Þ
3. A novel optimization problem for controller design Many times, the presence of nonlinearity in the control loop shown in Fig. 1 produces undesirable sustained oscillations at the steady state due to the presence of stable limit cycles. It is usually desirable to design a controller which reduces the amplitude of sustained oscillations. Also, it will be advantageous if the controller can additionally meet certain loop specifications. For the plant with relay nonlinearity, we intend the closed loop system to meet specified gain crossover frequency, phase margin and closed loop stability. In the following subsections, such design aspects are discussed to subsequently propose a constrained optimization problem. 3.1. Incorporating describing function to control transient behavior In general, DF is used for computing amplitude and frequency of limit cycles, which sustain at the input of the nonlinearity. In the present subsection, we propose an additional application of DF for Ref. [3] also focuses on such a specific case.
We extend the above concept for a general P ðNðPÞ a 1Þ and
where, ω0 and X0 are limit cycle frequency and amplitude at point ①. 2. Tsypkin's Condition [20] for Stability of Limit Cycle: For stability of limit cycle, it is essential that for the given Nyquist curve seen in its arrow direction, the 1=NðXÞ curve in its arrow direction crosses from right to left (refer [1,2] for details.). For relay nonlinearity case, this leads to the following Tsypkin's condition [20]: d ðImðLðjωÞÞ 40 ð3Þ dω ω ¼ ω0
1
Remark 1. Limit cycles are the ‘sustained’ sine-waves that occur at the input of nonlinearity in the steady state. Whereas, we currently focus on a few cycles of sine-wave that occurs at the input of nonlinearity during the transient time. Therefore, P is different from limit cycle amplitude X0.
design C(s) for NðPÞGðsÞ. For a particular step reference amplitude A, such designed control system meets the transient performance in the presence of plant nonlinearity. This is because, at such amplitude the input to the nonlinearity takes the form of sine-wave with amplitude P. It must be noted that the assumption of a sine-wave shaped input during the transient time is ideal, since input to the relay is not freely assigned. Therefore, it must be true only for a certain sub-class of loop TFs containing relay. Determining such a sub-class is a possible future direction to this work. Interestingly, we observe that a loop containing type-1 motion control plant and fractional PI controller satisfies such a sine-wave assumption as will be explained later in Section 4.2.
3.2. Desired loop specifications Based on the discussion in the previous subsection, we consider the loop CðsÞNðPÞGðsÞ to meet certain performance specifications. The specifications are as presented below (refer Fig. 3):
Gain Crossover Frequency (ωgc): j Cðjωgc ÞNðPÞGðjωgc Þj ¼ 1: Phase Margin (ϕm): ∠½Cðjωgc ÞNðPÞGðjωgc Þ ¼ π þ ϕm .
3.3. Proposed conditions for closed loop stability To ensure the closed loop stability, necessary conditions need to be evaluated. One knows that the gain margin in decibels is GM dB ¼ 20 log 10 1=a (refer Fig. 3). Therefore, for one requires a o 1 for positive gain margin. It is seen from Fig. 3 that ∠½Cðjωpc ÞNðPÞGðjωpc Þ ¼ π. The relay nonlinearity does not introduce any phase shift, which results into its describing function N(P) being a real quantity NðPÞ ¼ 4M=πP , i.e. ∠NðPÞ ¼ 0. It is also noticed from Fig. 2 that ∠½Cðjω0 ÞGðjω0 Þ ¼ π. Therefore, one can conclude that the limit cycle frequency ω0 and phase crossover frequency ωpc are equal for the case of relay nonlinearity. i.e. ω0 ¼ ωpc
ð4Þ
From Fig. 3, we have, a ¼ Cðjωpc ÞNðPÞGðjωpc Þ ¼ NðPÞCðjωpc ÞGðjωpc Þ
ð5Þ
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297
Remark 3. Using (4), (5), and (6), we get a ¼ NðPÞ=NðX 0 Þ. Therefore,
because N(P) is a positive real quantity. Recalling (2), 1 ¼ ½LðjωÞω ¼ ω0 ¼ ½CðjωÞGðjωÞω ¼ ω0 NðXÞ X ¼ X 0
GM dB ¼ 20 log 10
On taking modulus on both the sides and simplifying, one gets: ð6Þ 1 ¼ NðX 0 ÞCðjω0 ÞGðjω0 Þ Using (4), (5), and (6), the condition a o1 implies that: NðPÞ oNðX 0 Þ
1 NðX 0 Þ 4M ¼ 20 log 10 ¼ 20 log 10 a NðPÞ NðPÞπX 0
ð9Þ
In (9), as X0 decreases, the GMdB increases. Therefore, controller which minimizes X0 is useful in maximizing GMdB for plants with relay. 3.4. Proposal for maximization of limit cycle frequency
Therefore, X0 o P
ð7Þ
Thus, we get an interesting condition for positive GMdB in terms of: (i) Limit cycle amplitude X0 which ‘sustains’ in the steady state at the input of the nonlinearity. (ii) Amplitude P of the sine-wave that occurs ‘only during’ the transient time at the input of the nonlinearity. For the closed loop stability, it is clear that the positive phase and gain margins are necessary along with the following relation between phase crossover frequency (ωpc) and gain crossover frequency (ωgc): ωpc 4 ωgc Since, ω0 ¼ ωpc , we get, ω0 4 ωgc
ð8Þ
Remark 2. Since a ¼ j Cðjωpc ÞNðPÞGðjωpc Þj o 1,the system is stable. Therefore, the sine-wave of amplitude P occurs at the input of nonlinearity ‘only’ in the transient time and does not sustain thereafter.
Imaginary Axis
1 − N(X ) Real Axis
L( jω )
In Fig. 1, if we consider the signal y1 ðtÞ of the form X 0 sin ðω0 tÞ, then y2 ðtÞ is a square-wave signal having frequency ω0 and amplitudes M and M during ON and OFF-time respectively. For such y2 ðtÞ, the plant TF G(s) usually produces a response y3 ðtÞ which increases and decreases monotonically during the ON and OFF-time of y2 ðtÞ respectively. This is the case with many plants such as type-0 first order, type-1 second order, and type-1 third order. For such plants, the peak amplitude of y3 ðtÞ is decided by the frequency of y2 ðtÞ, i.e. ω0. For instance, if the frequency ω0 is high (i.e. time period is less), ON and OFF times are less. Therefore, y3 ðtÞ reaches lesser peak value. Thus, if the limit cycle frequency ω0 is maximized, it minimizes the peak amplitude of y3 ðtÞ, which is the amplitude of sustained oscillations. Therefore, for reducing amplitude of sustained oscillations, one can consider maximization of ω0 (or minimization of 1=ω0 ) as an objective function. For practical applications, there is an upper limit on ω0 as one cannot allow relay to toggle beyond a certain rate. Therefore, maximization of ω0 needs to be considered within such bound. 3.5. Proposed constrained optimization problem Based on the above discussions, we frame minimization of X 0 þ ð1=ω0 Þ as the objective function and construct the following constrained optimization problem for the controller design: 1 Minimize subject to: X0 þ ω0 ðController Parameters; ω0 ;X 0 Þ
1. Occurrence and Stability of Limit Cycles: Nyquist Condition for Limit Cycle Existence: 1 ¼ ½LðjωÞω ¼ ω0 ð10Þ NðXÞ X ¼ X 0
Fig. 2. Stability of limit cycles for relay nonlinearity.
Tsypkin's Stability of Limit Cycle Condition: d ðImðLðjωÞÞ 40 ð11Þ dω ω ¼ ω0
Imaginary Axis
Crossing at ω pc Unit Radius Circle
a
−
Real Axis
φm at ωgc C ( jω ) N ( P )G ( jω ) Fig. 3. Nyquist plot of CðsÞNðPÞGðsÞ with performance specifications.
Fig. 4. Limit cycle and loop performance together when NðPÞ ¼ 1.
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Fig. 5. Performance analysis using graph.
Fig. 6. Zoomed view of Fig. 5.
4. Illustration with PI α controller design
2. Performance Specifications: Gain Crossover Frequency (ωgc): j Cðjωgc ÞNðPÞGðjωgc Þj ¼ 1
ð12Þ
Phase Margin (ϕm): ∠½Cðjωgc ÞNðPÞGðjωgc Þ ¼ π þϕm
ð13Þ
Condition for Positive Gain Margin: X0 o P
ð14Þ
Condition on ω0 and ωgc: ω0 4 ωgc
ð15Þ
Let us consider a plant consisting of relay nonlinearity in cascade with a TF GðsÞ ¼ K=sðs þ bÞ. The TF G(s) satisfies the required monotonicity property as discussed in Section 3.4. For the design purpose, a PI α controller is considered. Since the PI α controller takes the form of integer PI when α ¼ 1, on considering PI α controller, one also takes into account the possibility of integer PI as a solution. Tsypkin's Condition of limit cycle stability (3) for loop TF LðsÞ ¼ CðsÞGðsÞ leads to the following inequality: h π
π
2 ðω3 þ b ωÞ K i ð1 αÞω α sin α þ bK i αω α cos α 2 2
Ki π K π 2 ð3ω2 þ b Þ sin α b 1 þ αi cos α 40 α 1 2 2 ω ω ω ¼ ω0 ð16Þ
The constraints (10) and (11) ensure the occurrence and stability of limit cycles. Eqs. (12)–(15) shape the loop to meet desirable performance.
Remark 4. When NðPÞ ¼ 1, CðSÞNðPÞGðsÞ ¼ CðsÞGðsÞ. Therefore, one can observe limit cycle and performance shaping together as in Fig. 4.
For demonstration, let the numerical values be: K ¼5, b¼ 0.7, M¼1, ωgc ¼ 0:5 rad=s, ϕm ¼ 501 ¼ 5π=18 rad. The bounds for α are selected as ð0; 1 so as to include PI controller (for which α¼ 1) as a possible candidate for solution. Also, for convenience, P is taken as 4M=π so that NðPÞ ¼ 1. Therefore, one may refer Fig. 4 to visualize the controller design problem graphically. The selected intervals for optimization parameters are as follows:
K p A ½0:01; 1, K i A ½0:01; 1, α A ½0:01; 1.
A.A. Kesarkar et al. / ISA Transactions 57 (2015) 295–300
ω0 A ð0:5; 11:7769. Recalling (15), ω0 4 ωgc . Since, ωgc ¼ 0:5, lower
299
frequency range ½0:001; 1000 rad=s. As seen from Fig. 6, there is a ‘single’ crossing point ( 0:0026173 þ j0) between 1=NðXÞ, which leads to X 0 ¼ 0:0033. This value matches with X0 ¼0.0034 obtained after solving optimization. Furthermore, from the expression (9), we get GM dB ¼ 51:7279, which closely matches to 51.6 dB as obtained from Fig. 6. Also, from Fig. 6, ω0 ¼ ωpc ¼ 11:9 rad=s, which is very near to the ω0 ¼ 11:7728 rad=s obtained as a result of optimization. Additionally, Fig. 6 shows that the closed loop system is stable.
bound is taken as 0.5. The upper bound 11.7769 is arbitrarily chosen. X 0 A ½0; 1:15 (Recalling (14), X 0 oP. Since, P ¼ 4M=π ¼ 1:2732, upper bound for X0 is chosen as 1.15.)
For solving the optimization problem, fmincon() solver available in MATLAB [21] is used. For each controller case, 20 random initial guesses are taken and the corresponding converged values are preserved. Best among the 20 (i.e. the one with least X 0 þð1=ω0 Þ ) is selected for the performance analysis. At the end of the simulation, we obtain: Kp ¼0.0532, Ki ¼0.5711, α¼ 0.1291, ω0 ¼ 11:7728, X0 ¼0.0034. It shows that the resultant controller is a non-integer order (fractional) PI with α¼ 0.1291.
4.2. Verification of limit cycles with closed loop simulation A simulink patch-up is constructed for the schematics shown in Fig. 1 to verify the limit cycle details using closed loop simulation. A step reference input of magnitude 16 is given to the patch-up and consequently the limit cycles are observed ðy1 ðtÞÞ as shown in Fig. 7. As seen from Fig. 7, the limit cycles have the following details: X0 ¼ 0.00346, ω0 ¼ 11:5224. The values are very close to those obtained with DF approach. Corresponding to such limit cycles, closed loop response ðy3 ðtÞÞ shows sustained oscillations in the steady state as shown in Fig. 8. Fig. 8 also shows the closed loop response obtained without relay nonlinearity. It is observed in Fig. 7 that P¼ 1.2834 for which
4.1. Graphical performance analysis of designed controller Fig. 5 shows Nyquist plot of L(s) over which 1=NðXÞ curve is superimposed. It is observed in Fig. 5 that the required ωgc and ϕm are met by the designed loop. The zoomed view of the selected portion of Fig. 5 is shown in Fig. 6. For analyzing the performance of designed PI α controller, its Oustaloup approximation [22] is considered with order 9 and
1.4 1.2
Transient Roughly Takes the Shape of a Sinusoidal with Amplitude, P=1.2834
1
0.02
0.6
Limit Cycles: Amplitude= 0.00346 Frequency=11.5224 rad/s y (t) 1
y1(t)
0.8
0.4
0 −0.02
0.2
32
0
t
34 Zoomed View
−0.2 −0.4
0
5
10
15
20
25
30
35
40
45
50
t Fig. 7. Limit cycles.
20 18 16 14
Zoomed View
y3(t)
12 16.5
10
Sustained Oscillations: Amplitude= 0.0480 Frequency=11.5224 rad/s
8 16
6 4 2 0
Nonlinear Simulation
15.5 25
0
5
10
15
30
35
t
20
25
Linear Simulation 30
t Fig. 8. Closed loop response.
35
40
45
50
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Table 1 Limit cycle details. Source
X0
ω0
Optimization results Graphical analysis Closed loop simulation
0.0034 0.0033 0.00346
11.7728 11.9 11.5224
NðPÞ ¼ 0:9921 1. Since we designed the controller to meet loop performance for NðPÞ ¼ 1, the transient responses for this reference with linear and nonlinear simulation match closely as can be seen from Fig. 8. Table 1 summarizes the limit cycle details obtained from optimization results, graphical analysis, and closed loop simulation. It is seen in Table 1 that the results are close to each other, which validates our design. 5. Conclusion In this paper, we proposed a novel constrained optimization problem for controller design so as to reduce the amplitude of sustained oscillations for the plant with relay nonlinearity. Additionally, desirable loop specifications were also imposed in the design. The formulation incorporated DF of the nonlinearity for shaping the limit cycle details as well as controlling the transient behavior. Under this proposed construction, tuning of fractional PI controller was demonstrated in detail. The formulation in this paper focused on a specific case where the Nyquist plot of designed loop was assumed to intersect negative inverse of the DF only once in the complex plane. Future investigations are needed in this aspect to deal with multiple crossing points. Subsequent extension of this work for other kind of nonlinearities is also an interesting future direction. References [1] Gopal M. Digital control and state variable methods. third edition. New Delhi: Tata McGraw-Hill Publishing Company Limited; 2009. [2] Khalil HK. Nonlinear systems. third edition. Upper Saddle River, New Jersey: Prentice Hall; 2002. [3] Oliveira NMF, Kienitz KH, Misawa EA. An algebraic approach to the design of robust limit cycle controllers. In: Proceedings of American control conference. Denver, Colorado; 2003. p. 2419–23.
[4] Oliveira NMF, Kienitz KH, Misawa EA. A describing function approach to limit cycle controller design. In: Proceedings of American control conference. Minneapolis, Minnesota, USA; 2006. p. 1511-6. [5] Oliveira NMF, Kienitz KH, Misawa EA. A describing function approach to the design of robust limit-cycle controllers. Nonlinear Dyn 2012;67(1):357–63. [6] Oldham KB, Spanier J. The fractional calculus. New York: Academic Press; 1974. [7] Podlubny I. Fractional-order systems and PI λ Dμ -controllers. IEEE Trans Autom Control 1999;44(1):208–14. [8] Xue D, Chen YQ, Atherton D. Linear feedback control-analysis and design with Matlab. Chapter 8: fractional-order controller-an introduction. SIAM Press; 2007. [9] Chen YQ, Petras I, Xue Du. Fractional order control—a tutorial. In: Proceedings of American control conference, WeC02 Tutorial Session, Grand Ballroom B (east); 2009. p. 1397–411. [10] Padula F, Visioli A. Tuning rules for optimal PID and fractional-order PID controllers. J Process Control 2011;21(1):69–81. [11] Monje CA, Vinagre BM, Chen YQ, Feliu V, Lanusse P, Sabatier J. Proposals for fractional PI λ Dμ tuning. In: The first IFAC symposium on fractional differentiation and its applications. Bordeaux, France; 2004. [12] Li HS, Luo Y, Chen YQ. A fractional order proportional and derivative (FOPD) motion controller: tuning rule and experiments. IEEE Trans Control Syst Technol 2010;18(2):516–20. [13] Li HS, Chen YQ. A fractional order proportional and derivative (FOPD) controller tuning algorithm. In: Proceedings of Chinese control and decision conference; 2008. p. 4059–63. [14] Luo Y, Chen YQ. Fractional-order [proportional derivative] controller for robust motion control: tuning procedure and validation. In: Proceedings of American control conference. St. Louis, Missouri; 2009. p. 1412–7. [15] Wang CY, Luo Y, Chen YQ. Fractional order proportional integral (FOPI) and [Proportional Integral] (FO[PI]) controller designs for first order plus time delay (FOPTD) systems. In: Proceedings of Chinese control and decision conference; 2009. p. 329–34. [16] Wang CY, Luo Y, Chen YQ. An analytical design of fractional order proportional integral and [proportional integral] controllers for robust velocity servo. In: Proceedings of 4th IEEE conference on industrial electronics and applications. Xian, China; 2009. p. 3448–53. [17] Luo Y, Chen YQ, Pi Y. Fractional order ultra low speed position servo: improved performance via describing function analysis. ISA Trans 2011;50(1):53–60. [18] Barbosa RS, Machado JAT, Galhano AM. Performance of fractional PID algorithms controlling nonlinear systems with saturation and backlash phenomena. J Vib Control 2007;13:1407–18. [19] Kesarkar AA, Selvaganesan N. Fractional control of precision modular servo setup for better limit cycle suppression. In: Proceedings of IEEE multiconference on systems and control (MSC). India; 2013. p. 467–71. [20] Tsypkin Ya 2.: Teorija relejnykh sistem avtomaticheskovo regulirovanija, Gostekhizdat, Moscow, 1955, chaps. 8 and 9. Translation: sampling systems theory and its application, vols. 1 and 2. New York: Pergamon Press and The Macmillan Company; 1964. [21] MATLAB version 7.10.0 (R2010a). Natick, Massachusetts: The MathWorks Inc.; 2010. [22] Oustaloup A, Levron F, Mathieu B, Nanot FM. Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans Circuits Syst I: Fundam Theory Appl 2000;47(1):25–40.
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Novel controller design for plants with relay nonlinearity to reduce amplitude of sustained oscillations: Illustration with a fractional controller Ameya Anil Kesarkar, N. Selvaganesan n, H. Priyadarshan Department of Avionics, Indian Institute of Space Science and Technology (IIST), Thiruvananthapuram-695547, India
art ic l e i nf o
a b s t r a c t
Article history: Received 4 September 2013 Received in revised form 18 November 2014 Accepted 7 January 2015 Available online 28 January 2015 This paper was recommended for publication by Y. Chen
This paper proposes a novel constrained optimization problem to design a controller for plants containing relay nonlinearity to reduce the amplitude of sustained oscillations. The controller is additionally constrained to satisfy desirable loop specifications. The proposed formulation is validated by designing a fractional PI controller for a plant with relay. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Limit cycles Relay nonlinearity Describing function PI α
1. Introduction For systems containing separable nonlinearity in cascade, designed controllers usually produce undesirable sustained periodic oscillations in the plant output response due to the existence of stable limit cycles [1]. For the analysis of such limit cycles, Describing Function (DF) is a commonly used tool [2]. The studies in [3–5] show also the usefulness of DF for synthesis of controllers to satisfy the given limit cycle details. The work in [3] focuses on designing a robust limit cycle controller for a plant with relay nonlinearity by maintaining Nyquist plot of loop Transfer Function (TF) orthogonal to negative inverse of DF of relay nonlinearity. This is subsequently generalized for other nonlinearities in [4,5]. Motivated from the above works, we consider the problem of shaping the loop involving relay nonlinearity to reduce the amplitude of sustained oscillations. Additionally, we constrain the loop to meet certain performance specifications. For meeting such a stringent control requirement, we explore the potential of Fractional-Order Controllers (FOCs). FOCs are the controllers whose dynamics are governed by fractional-order differential equations [6–8]. FOCs such as PI α , ½PIα , PDβ , ½PDβ , PI α Dβ are superclass of their integer counterparts. Therefore, one expects them to perform better [9]. For instance, PI α has the n
Corresponding author. Tel.: þ 91 471 256 8456. E-mail address: [email protected] (N. Selvaganesan).
http://dx.doi.org/10.1016/j.isatra.2015.01.005 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
capacity to outperform integer-order PI controller. In the literature, the design of FOCs for linear plants has been widely studied [10–16]. In the limit cycle context, however, only a few works are seen for FOCs [17– 19]. For the design problem in the current paper, we investigate the applicability of FOCs. The contributions of this paper are: (i) An optimization problem is proposed to design a controller for plants containing relay nonlinearity in order to get: Reduced sustained oscillation amplitudes. Desirable loop performance. (ii) Demonstration using fractional PI controller design.
2. Basics of relay nonlinearity and stable limit cycles Let us consider a plant consisting of a relay nonlinearity in cascade with the TF G(s). The closed loop control schematics with controller C(s) is shown in Fig. 1. Mathematically, the relay nonlinearity in Fig. 1 is given by ( y2 ðtÞ ¼
M
if y1 ðtÞ Z 0
M
if y1 ðtÞ o 0
ð1Þ
We concentrate on a case where the Nyquist plot of the designed loop is assumed to intersect with negative inverse of
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A.A. Kesarkar et al. / ISA Transactions 57 (2015) 295–300
Reference input +−
C (s)
y1( t)
y2 (t )
M −M
G( s)
controlling the ‘transient’ behavior of a closed loop system in the presence of nonlinearity as follows:
y3 ( t )
For the closed loop schematics shown in Fig. 1, one usually
Relay
neglects the plant nonlinearity and considers only the TF G(s) while designing the controller C(s). This means that one assumes the nonlinearity to behave as a unity gain during the transient time. At a particular amplitude A of step reference, if the signal occurring at input of nonlinearity roughly takes shape of a sine-wave with peak amplitude P such that NðPÞ ¼ 1, then the transient performance of the designed closed loop control system with and without nonlinearity is ‘same’ for such A.
Fig. 1. Closed loop control schematics.
DF in the complex plane only once1 as shown in Fig. 2. Fig. 2 contains superposition of: (i) 1=NðXÞ plot of relay nonlinearity. Where, NðXÞ ¼ 4M=πX is its DF [1] and X is the peak amplitude of sinusoidal signal at the input of nonlinearity. (ii) Nyquist plot of loop TF LðsÞ ¼ CðsÞGðsÞ. The arrow in the Nyquist plot indicates increasing ω direction ðωA ½0; 1ÞÞ. The arrow in 1=NðXÞ plot shows increasing X direction. The limit cycle point ① is intersection point between Nyquist plot and 1=NðXÞ curve. For sustained oscillations to be produced at the output in the presence of nonlinearity, the nature of limit cycle point must be ‘stable’. For the relay nonlinearity case, the occurrence and stability of limit cycles is ensured if the conditions explained below are met by the loop TF L(s) (Refer Fig. 2). 1. Nyquist Condition for Limit Cycle Existence [1]: 1 ¼ ½LðjωÞω ¼ ω0 NðXÞ X ¼ X 0
ð2Þ
3. A novel optimization problem for controller design Many times, the presence of nonlinearity in the control loop shown in Fig. 1 produces undesirable sustained oscillations at the steady state due to the presence of stable limit cycles. It is usually desirable to design a controller which reduces the amplitude of sustained oscillations. Also, it will be advantageous if the controller can additionally meet certain loop specifications. For the plant with relay nonlinearity, we intend the closed loop system to meet specified gain crossover frequency, phase margin and closed loop stability. In the following subsections, such design aspects are discussed to subsequently propose a constrained optimization problem. 3.1. Incorporating describing function to control transient behavior In general, DF is used for computing amplitude and frequency of limit cycles, which sustain at the input of the nonlinearity. In the present subsection, we propose an additional application of DF for Ref. [3] also focuses on such a specific case.
We extend the above concept for a general P ðNðPÞ a 1Þ and
where, ω0 and X0 are limit cycle frequency and amplitude at point ①. 2. Tsypkin's Condition [20] for Stability of Limit Cycle: For stability of limit cycle, it is essential that for the given Nyquist curve seen in its arrow direction, the 1=NðXÞ curve in its arrow direction crosses from right to left (refer [1,2] for details.). For relay nonlinearity case, this leads to the following Tsypkin's condition [20]: d ðImðLðjωÞÞ 40 ð3Þ dω ω ¼ ω0
1
Remark 1. Limit cycles are the ‘sustained’ sine-waves that occur at the input of nonlinearity in the steady state. Whereas, we currently focus on a few cycles of sine-wave that occurs at the input of nonlinearity during the transient time. Therefore, P is different from limit cycle amplitude X0.
design C(s) for NðPÞGðsÞ. For a particular step reference amplitude A, such designed control system meets the transient performance in the presence of plant nonlinearity. This is because, at such amplitude the input to the nonlinearity takes the form of sine-wave with amplitude P. It must be noted that the assumption of a sine-wave shaped input during the transient time is ideal, since input to the relay is not freely assigned. Therefore, it must be true only for a certain sub-class of loop TFs containing relay. Determining such a sub-class is a possible future direction to this work. Interestingly, we observe that a loop containing type-1 motion control plant and fractional PI controller satisfies such a sine-wave assumption as will be explained later in Section 4.2.
3.2. Desired loop specifications Based on the discussion in the previous subsection, we consider the loop CðsÞNðPÞGðsÞ to meet certain performance specifications. The specifications are as presented below (refer Fig. 3):
Gain Crossover Frequency (ωgc): j Cðjωgc ÞNðPÞGðjωgc Þj ¼ 1: Phase Margin (ϕm): ∠½Cðjωgc ÞNðPÞGðjωgc Þ ¼ π þ ϕm .
3.3. Proposed conditions for closed loop stability To ensure the closed loop stability, necessary conditions need to be evaluated. One knows that the gain margin in decibels is GM dB ¼ 20 log 10 1=a (refer Fig. 3). Therefore, for one requires a o 1 for positive gain margin. It is seen from Fig. 3 that ∠½Cðjωpc ÞNðPÞGðjωpc Þ ¼ π. The relay nonlinearity does not introduce any phase shift, which results into its describing function N(P) being a real quantity NðPÞ ¼ 4M=πP , i.e. ∠NðPÞ ¼ 0. It is also noticed from Fig. 2 that ∠½Cðjω0 ÞGðjω0 Þ ¼ π. Therefore, one can conclude that the limit cycle frequency ω0 and phase crossover frequency ωpc are equal for the case of relay nonlinearity. i.e. ω0 ¼ ωpc
ð4Þ
From Fig. 3, we have, a ¼ Cðjωpc ÞNðPÞGðjωpc Þ ¼ NðPÞCðjωpc ÞGðjωpc Þ
ð5Þ
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Remark 3. Using (4), (5), and (6), we get a ¼ NðPÞ=NðX 0 Þ. Therefore,
because N(P) is a positive real quantity. Recalling (2), 1 ¼ ½LðjωÞω ¼ ω0 ¼ ½CðjωÞGðjωÞω ¼ ω0 NðXÞ X ¼ X 0
GM dB ¼ 20 log 10
On taking modulus on both the sides and simplifying, one gets: ð6Þ 1 ¼ NðX 0 ÞCðjω0 ÞGðjω0 Þ Using (4), (5), and (6), the condition a o1 implies that: NðPÞ oNðX 0 Þ
1 NðX 0 Þ 4M ¼ 20 log 10 ¼ 20 log 10 a NðPÞ NðPÞπX 0
ð9Þ
In (9), as X0 decreases, the GMdB increases. Therefore, controller which minimizes X0 is useful in maximizing GMdB for plants with relay. 3.4. Proposal for maximization of limit cycle frequency
Therefore, X0 o P
ð7Þ
Thus, we get an interesting condition for positive GMdB in terms of: (i) Limit cycle amplitude X0 which ‘sustains’ in the steady state at the input of the nonlinearity. (ii) Amplitude P of the sine-wave that occurs ‘only during’ the transient time at the input of the nonlinearity. For the closed loop stability, it is clear that the positive phase and gain margins are necessary along with the following relation between phase crossover frequency (ωpc) and gain crossover frequency (ωgc): ωpc 4 ωgc Since, ω0 ¼ ωpc , we get, ω0 4 ωgc
ð8Þ
Remark 2. Since a ¼ j Cðjωpc ÞNðPÞGðjωpc Þj o 1,the system is stable. Therefore, the sine-wave of amplitude P occurs at the input of nonlinearity ‘only’ in the transient time and does not sustain thereafter.
Imaginary Axis
1 − N(X ) Real Axis
L( jω )
In Fig. 1, if we consider the signal y1 ðtÞ of the form X 0 sin ðω0 tÞ, then y2 ðtÞ is a square-wave signal having frequency ω0 and amplitudes M and M during ON and OFF-time respectively. For such y2 ðtÞ, the plant TF G(s) usually produces a response y3 ðtÞ which increases and decreases monotonically during the ON and OFF-time of y2 ðtÞ respectively. This is the case with many plants such as type-0 first order, type-1 second order, and type-1 third order. For such plants, the peak amplitude of y3 ðtÞ is decided by the frequency of y2 ðtÞ, i.e. ω0. For instance, if the frequency ω0 is high (i.e. time period is less), ON and OFF times are less. Therefore, y3 ðtÞ reaches lesser peak value. Thus, if the limit cycle frequency ω0 is maximized, it minimizes the peak amplitude of y3 ðtÞ, which is the amplitude of sustained oscillations. Therefore, for reducing amplitude of sustained oscillations, one can consider maximization of ω0 (or minimization of 1=ω0 ) as an objective function. For practical applications, there is an upper limit on ω0 as one cannot allow relay to toggle beyond a certain rate. Therefore, maximization of ω0 needs to be considered within such bound. 3.5. Proposed constrained optimization problem Based on the above discussions, we frame minimization of X 0 þ ð1=ω0 Þ as the objective function and construct the following constrained optimization problem for the controller design: 1 Minimize subject to: X0 þ ω0 ðController Parameters; ω0 ;X 0 Þ
1. Occurrence and Stability of Limit Cycles: Nyquist Condition for Limit Cycle Existence: 1 ¼ ½LðjωÞω ¼ ω0 ð10Þ NðXÞ X ¼ X 0
Fig. 2. Stability of limit cycles for relay nonlinearity.
Tsypkin's Stability of Limit Cycle Condition: d ðImðLðjωÞÞ 40 ð11Þ dω ω ¼ ω0
Imaginary Axis
Crossing at ω pc Unit Radius Circle
a
−
Real Axis
φm at ωgc C ( jω ) N ( P )G ( jω ) Fig. 3. Nyquist plot of CðsÞNðPÞGðsÞ with performance specifications.
Fig. 4. Limit cycle and loop performance together when NðPÞ ¼ 1.
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Fig. 5. Performance analysis using graph.
Fig. 6. Zoomed view of Fig. 5.
4. Illustration with PI α controller design
2. Performance Specifications: Gain Crossover Frequency (ωgc): j Cðjωgc ÞNðPÞGðjωgc Þj ¼ 1
ð12Þ
Phase Margin (ϕm): ∠½Cðjωgc ÞNðPÞGðjωgc Þ ¼ π þϕm
ð13Þ
Condition for Positive Gain Margin: X0 o P
ð14Þ
Condition on ω0 and ωgc: ω0 4 ωgc
ð15Þ
Let us consider a plant consisting of relay nonlinearity in cascade with a TF GðsÞ ¼ K=sðs þ bÞ. The TF G(s) satisfies the required monotonicity property as discussed in Section 3.4. For the design purpose, a PI α controller is considered. Since the PI α controller takes the form of integer PI when α ¼ 1, on considering PI α controller, one also takes into account the possibility of integer PI as a solution. Tsypkin's Condition of limit cycle stability (3) for loop TF LðsÞ ¼ CðsÞGðsÞ leads to the following inequality: h π
π
2 ðω3 þ b ωÞ K i ð1 αÞω α sin α þ bK i αω α cos α 2 2
Ki π K π 2 ð3ω2 þ b Þ sin α b 1 þ αi cos α 40 α 1 2 2 ω ω ω ¼ ω0 ð16Þ
The constraints (10) and (11) ensure the occurrence and stability of limit cycles. Eqs. (12)–(15) shape the loop to meet desirable performance.
Remark 4. When NðPÞ ¼ 1, CðSÞNðPÞGðsÞ ¼ CðsÞGðsÞ. Therefore, one can observe limit cycle and performance shaping together as in Fig. 4.
For demonstration, let the numerical values be: K ¼5, b¼ 0.7, M¼1, ωgc ¼ 0:5 rad=s, ϕm ¼ 501 ¼ 5π=18 rad. The bounds for α are selected as ð0; 1 so as to include PI controller (for which α¼ 1) as a possible candidate for solution. Also, for convenience, P is taken as 4M=π so that NðPÞ ¼ 1. Therefore, one may refer Fig. 4 to visualize the controller design problem graphically. The selected intervals for optimization parameters are as follows:
K p A ½0:01; 1, K i A ½0:01; 1, α A ½0:01; 1.
A.A. Kesarkar et al. / ISA Transactions 57 (2015) 295–300
ω0 A ð0:5; 11:7769. Recalling (15), ω0 4 ωgc . Since, ωgc ¼ 0:5, lower
299
frequency range ½0:001; 1000 rad=s. As seen from Fig. 6, there is a ‘single’ crossing point ( 0:0026173 þ j0) between 1=NðXÞ, which leads to X 0 ¼ 0:0033. This value matches with X0 ¼0.0034 obtained after solving optimization. Furthermore, from the expression (9), we get GM dB ¼ 51:7279, which closely matches to 51.6 dB as obtained from Fig. 6. Also, from Fig. 6, ω0 ¼ ωpc ¼ 11:9 rad=s, which is very near to the ω0 ¼ 11:7728 rad=s obtained as a result of optimization. Additionally, Fig. 6 shows that the closed loop system is stable.
bound is taken as 0.5. The upper bound 11.7769 is arbitrarily chosen. X 0 A ½0; 1:15 (Recalling (14), X 0 oP. Since, P ¼ 4M=π ¼ 1:2732, upper bound for X0 is chosen as 1.15.)
For solving the optimization problem, fmincon() solver available in MATLAB [21] is used. For each controller case, 20 random initial guesses are taken and the corresponding converged values are preserved. Best among the 20 (i.e. the one with least X 0 þð1=ω0 Þ ) is selected for the performance analysis. At the end of the simulation, we obtain: Kp ¼0.0532, Ki ¼0.5711, α¼ 0.1291, ω0 ¼ 11:7728, X0 ¼0.0034. It shows that the resultant controller is a non-integer order (fractional) PI with α¼ 0.1291.
4.2. Verification of limit cycles with closed loop simulation A simulink patch-up is constructed for the schematics shown in Fig. 1 to verify the limit cycle details using closed loop simulation. A step reference input of magnitude 16 is given to the patch-up and consequently the limit cycles are observed ðy1 ðtÞÞ as shown in Fig. 7. As seen from Fig. 7, the limit cycles have the following details: X0 ¼ 0.00346, ω0 ¼ 11:5224. The values are very close to those obtained with DF approach. Corresponding to such limit cycles, closed loop response ðy3 ðtÞÞ shows sustained oscillations in the steady state as shown in Fig. 8. Fig. 8 also shows the closed loop response obtained without relay nonlinearity. It is observed in Fig. 7 that P¼ 1.2834 for which
4.1. Graphical performance analysis of designed controller Fig. 5 shows Nyquist plot of L(s) over which 1=NðXÞ curve is superimposed. It is observed in Fig. 5 that the required ωgc and ϕm are met by the designed loop. The zoomed view of the selected portion of Fig. 5 is shown in Fig. 6. For analyzing the performance of designed PI α controller, its Oustaloup approximation [22] is considered with order 9 and
1.4 1.2
Transient Roughly Takes the Shape of a Sinusoidal with Amplitude, P=1.2834
1
0.02
0.6
Limit Cycles: Amplitude= 0.00346 Frequency=11.5224 rad/s y (t) 1
y1(t)
0.8
0.4
0 −0.02
0.2
32
0
t
34 Zoomed View
−0.2 −0.4
0
5
10
15
20
25
30
35
40
45
50
t Fig. 7. Limit cycles.
20 18 16 14
Zoomed View
y3(t)
12 16.5
10
Sustained Oscillations: Amplitude= 0.0480 Frequency=11.5224 rad/s
8 16
6 4 2 0
Nonlinear Simulation
15.5 25
0
5
10
15
30
35
t
20
25
Linear Simulation 30
t Fig. 8. Closed loop response.
35
40
45
50
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Table 1 Limit cycle details. Source
X0
ω0
Optimization results Graphical analysis Closed loop simulation
0.0034 0.0033 0.00346
11.7728 11.9 11.5224
NðPÞ ¼ 0:9921 1. Since we designed the controller to meet loop performance for NðPÞ ¼ 1, the transient responses for this reference with linear and nonlinear simulation match closely as can be seen from Fig. 8. Table 1 summarizes the limit cycle details obtained from optimization results, graphical analysis, and closed loop simulation. It is seen in Table 1 that the results are close to each other, which validates our design. 5. Conclusion In this paper, we proposed a novel constrained optimization problem for controller design so as to reduce the amplitude of sustained oscillations for the plant with relay nonlinearity. Additionally, desirable loop specifications were also imposed in the design. The formulation incorporated DF of the nonlinearity for shaping the limit cycle details as well as controlling the transient behavior. Under this proposed construction, tuning of fractional PI controller was demonstrated in detail. The formulation in this paper focused on a specific case where the Nyquist plot of designed loop was assumed to intersect negative inverse of the DF only once in the complex plane. Future investigations are needed in this aspect to deal with multiple crossing points. Subsequent extension of this work for other kind of nonlinearities is also an interesting future direction. References [1] Gopal M. Digital control and state variable methods. third edition. New Delhi: Tata McGraw-Hill Publishing Company Limited; 2009. [2] Khalil HK. Nonlinear systems. third edition. Upper Saddle River, New Jersey: Prentice Hall; 2002. [3] Oliveira NMF, Kienitz KH, Misawa EA. An algebraic approach to the design of robust limit cycle controllers. In: Proceedings of American control conference. Denver, Colorado; 2003. p. 2419–23.
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