Incremental SMC-based CNF control strategy considering magnetic ball suspension and inverted pendulum systems through cuckoo search-genetic optimization algorithm
- 75 Downloads
Abstract
A kind of incremental sliding mode control (SMC) approach in connection with the well-known composite nonlinear feedback (CNF) control strategy is newly considered in this research to deal with the nonlinear magnetic ball suspension and inverted pendulum systems, as well. The incremental SMC approach is in fact proposed to handle the aforementioned underactuated systems under control, which have a lower number of actuators than degrees of freedom. Based on the outcomes of the investigation presented here, the small overshoot and short settling time of the system response are fulfilled. In fact, the proposed CNF control strategy comprises two parts: the first term assures the stability of the closed-loop nonlinear system and provides a fast convergence response. The second term reduces its overshoot. The genetic-cuckoo hybrid algorithm is designed to minimize tracking errors for the purpose of finding the most suitable sliding surface coefficients. Finally, the finite time stability for the closed-loop system is proved, theoretically.
Keywords
Incremental sliding mode control approach Composite nonlinear feedback control approach Cuckoo search-genetic optimization algorithm Magnetic ball suspension system Inverted pendulum system Finite time stabilityIntroduction
The system uncertainty or mismatch is considered as one of the most important challenges in the area of nonlinear systems by now. It is to note that the uncertainty can be observed in the system parameters or the external disturbances that apply to the system. One of the popular approaches to deal with the uncertainties is known as the SMC strategy [1]. The SMC has indicated acceptable results since 1970, and comprises two parts: in the first part stable surfaces (sliding surfaces) are designed and in the second part, the control law for the trajectory of the closed-loop system is designed to converge the sliding surfaces in a finite time. The obvious feature of the SMC is the rapid response of the system, which leads to high overshoot. There exist contradictions between these characteristics; therefore, a tradeoff should be considered. The CNF is an efficient and simple approach which is employed to improve transient performance (small overshoot and acceptable settling time) and overcome the contradiction of simultaneous achievement of the mentioned transient performances. The CNF strategy is a relatively new approach that consists of a linear and a nonlinear section. The linear section plays an acceptable role in the closed-loop system stabilization and fast response. The nonlinear part attempts to change the damping ratio and decrease the steady state error according to the definition of the nonlinear function and the settling time response.
Recently, several types of research are established based on the CNF approach for the purpose of improving the performance of the closed-loop system [2, 3, 4, 5]. In [6], the CNF method is applied to synchronize the master/slave nonlinear systems with time-varying delays in chaotic systems with nonlinearities. In [7], for a particular type of vehicle suspension, a CNF with a band and a layer is used to reduce the chattering phenomenon. Then, the proportional-integral controller and intelligent algorithm have been used to improve the error situation and optimization. Combination of the CNF strategy with intelligent algorithms has been illustrated with acceptable results in recent years. In [8, 9, 10], the nonlinear system of level tank and electromagnetism suspension system has been described by Takagi–Sugeno (T–S) model then the stability of closed-loop system has been proved by the CNF strategy with the parallel distributed compensation and the LMI. In [11], the combination of the CNF with the SMC has been applied to a class of nonlinear systems.
To the best knowledge of recent considerations, a few investigations are applied to the underactuated and the nonlinear systems through the CNF approach. Tracking and regulation problem for practical systems has experienced a sweeping change over 1 decade. This paper proposes the SMC based on CNF approach for tracking control of a nonlinear magnetic ball suspension system and stabilization of an inverted pendulum system. The final object in a magnetic ball suspension system is to move a mass in a space without physical contact by magnetic characteristics. It is widely used in magnetic trains, accelerometers, etc. [12]. These systems have high nonlinearity and instability in the open-loop situation. Therefore, stabilization and tracking of the system are one of the engineering challenges. Several methods have been proposed to design a suitable control for linear and nonlinear types of the magnetic ball suspension system. In addition, investigation of underactuated systems has rapidly expanded in recent years. The underactuated systems are characterized by the fact that they have fewer actuators than the degrees of freedom to be controlled. The inverted pendulum is an example of an underactuated system with two degrees of freedom [13]. In these systems, the pendulum should be kept upright, meanwhile the cart must even be at the center of the line. It should be possible to control the position of the cart and the pendulum angle only with one control signal input. In fact, this model is a single input and two output (SIMO) system. In this paper, the idea of the CNF controller to the inverted pendulum system and nonlinear magnetic ball suspension system has been extended by the SMC and GC algorithm [14, 15, 16, 17]. The cuckoo search (CS) is a global random interactive search algorithm inspired by nature. The basis of this algorithm is the combination of the behavior of a particular species of cuckoo birds with the behavior of flying levy birds [18, 19, 20, 21, 22]. The Cuckoo search is applied owing to the fact that it is a simple, fast and efficient algorithm, which uses only a single parameter for search. The elimination of the genetic algorithm difficulty and providing global results are the main advantages of the cuckoo search algorithm; also, it does not trap in local optima and represents the proper coefficients for the sliding surfaces. Finally, it is theoretically proved that the trajectory of the closed-loop system converges to the sliding surface in a finite time manner in these cases.
The rest of the paper is organized as follows: in the next section, the formulation and preliminary concerning the incremental SMC-based CNF strategy is first studied and subsequently the genetic-cuckoo (GC) algorithm has been introduced to minimize tracking errors for the purpose of finding the suitable sliding surface coefficients. In following section, the main results regarding this research including the stability of the closed-loop system for magnetic ball suspension and inverted pendulum systems are proposed. In the section before the conclusion, the simulation results are carried out and finally, in last section, concluding remarks are provided.
The formulation and preliminary
The SMC-based CNF approach for the magnetic ball suspension system
where \( x \) and \( u \) are the state vector and the control input vector, respectively. d(x_{1}) obtained in the laboratory is a polynomial in x_{1} which illustrates the ratio between the amount of flow and the position of the ball. m is the mass of the ball and g is the gravitational force, L shows induction. x_{1} and x_{2} are the ball position and the velocity of ball, respectively.
It should be noted that the \( \psi (s) \) function increases the degree of freedom of the control rule [23]. Therefore, in this case, the CNF-based SMC approach is realized.
The incremental SMC-based CNF strategy for the inverted pendulum
The nonlinear \( \psi \,(s) \) function in the CNF
The optimization
- 1.
Each cuckoo bird collects an egg at a time and randomly places it in a selected nest.
- 2.
High-quality nests are selected for re-laying.
- 3.
The number of host nests is constant and a host with a certain probability identifies a foreign egg.
- 1.
Setting: production number is selected \( t = 1 \). Based on the cuckoo algorithm, the primary population is produced.
- 2.
Population update: as long as the conditions for the moratorium are not established, the new population is being implemented.
The cost function is calculated on the basis of the levy’s flight for each population.
The main results
In this section, the finite time stability for magnetic ball suspension and inverted pendulum system has been proved.
The magnetic ball suspension system stability
Equation (17) will be established, which means that \( \dot{V} \) has a negative value and ensures that the system is stable for a finite time.
The stability analysis
As a result, the closed-loop system will have finite time stability.
The simulation results
In this section, the examples illustrate the advantages of the proposed control strategy. In the first example, the SMC based on CNF is applied to the magnetic ball suspension system. In the second example, the inverted pendulum is given and the proposed controller designed in Eq. (10) is employed to stabilize the closed-loop system.
The constant parameters in the magnetic ball suspension system
Parameters | Magnitudes |
---|---|
Coil resistance (R) | 52 Ω |
Coil inductance (L) | 1.227 H |
Ball mass | 16.5 g |
The initial distance from the core | 50 mm |
Relative displacement and required current
x_{1}, ball position (mm) | i, coil current (amp) |
---|---|
30 | 0.114 |
40 | 0.236 |
50 | 0.376 |
60 | 0.523 |
70 | 0.746 |
Figure 6 shows that the SMC-CNF control is not sensitive to the system parameters changing, because there is no significant change in the system response even with a tenfold mass.
Figure 7 shows the control signal input, which is smooth.
The constant parameters in the inverted pendulum system
Pendulum mass (m) | 1 kg |
---|---|
Cart mass (M) | \( 1\,\,{\text{kg}} \) |
Friction of the cart | \( 0.1\,{\text{N/m/s}} \) |
Length of the pendulum (l) | \( 0.1\,\,{\text{m}} \) |
Inertia of the pendulum (i) | \( 0.006\,\,{\text{kg}} . {\text{m}}^{ 2} \) |
Gravity (g) | 9.8 m/s^{2} |
As can be seen, the sliding surfaces converge to the zero very fast.
By applying U_{T} controller in Eq. (1) to the model of the inverted pendulum, Eqs. (6) and (7) the cart and the pendulum position are obtained. As it can be seen, the SMC strategy stabilizes the closed-loop system and provides the high overshoot and long settling time; meanwhile, using the CNF-SMC the settling time has been reduced and the overshoot has been eliminated.
Transient response performance
CNF-SMC | SMC | |||
---|---|---|---|---|
Settling time | Over/undershoot | Settling time | Over/undershoot | |
Cart position | 19 | 2.1 | 18.2 | 2.43 |
Pendulum position | 2.2 | 0.24 | 5.3 | 0.41 |
Conclusion
In the investigation presented here, a kind of incremental SMC-based CNF strategy is newly designed considering the magnetic ball suspension and the inverted pendulum systems to be handled. The selection of all the tuning parameters regarding the aforementioned SMC-based CNF strategy is turned into a minimization problem and solved automatically by the GC algorithm. It should be noted that the Lyapunov stability theory is used to prove the finite time closed-loop stability of the magnetic ball suspension system and also the inverted pendulum system. By the proposed control approach, the convergence of the state variables to the sliding surfaces and the equilibrium points in the finite time is guaranteed. The main advantage of the proposed approach is that the controller does not show any sensitivity to the system parameters changing, such as ball mass and the sensors inaccuracy in determining the ball position for the tracking. The simulation results illustrate that adding the CNF approach improves the transient performance of the closed-loop system. Also, by applying the incremental SMC-based CNF strategy to the inverted pendulum system, the states variables converge to their equilibrium point with acceptable overshoot and its settling time. Using other control techniques such as the fuzzy-based solutions or in general, the intelligent control approaches instead of the SMC can be a new approach to the nonlinear systems via the CNF. Applying the CNF strategy to the singular systems and also the hybrid systems is the other suggestion in this area for the future researches.
Notes
References
- 1.Liu L, Pu J, Song X, Fu Z, Wang X (2014) Adaptive sliding mode control of uncertain chaotic systems with input nonlinearity. Nonlinear Dyn 76(4):1857–1865. https://doi.org/10.1007/s11071-013-1163-6 MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Zheng Z, Sun W, Chen H, Yeow JTW (2014) Integral sliding mode based optimal composite nonlinear feedback control for a class of systems. Control Theory Technol 12(2):139–146. https://doi.org/10.1007/s11768-014-0022-4 MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Wang J, Zhao J (2016) On improving transient performance in tracking control for switched systems with input saturation via composite nonlinear feedback. Int J Robust Nonlinear Control 26(3):509–518. https://doi.org/10.1002/rnc.3322 MathSciNetCrossRefzbMATHGoogle Scholar
- 4.Mobayen S, Majd VJ, Sojoodi M (2012) An LMI-based composite nonlinear feedback terminal sliding-mode controller design for disturbed MIMO systems. Math Comput Simul 85:1–10. https://doi.org/10.1016/j.matcom.2012.09.006 MathSciNetCrossRefzbMATHGoogle Scholar
- 5.Huang Y, Cheng G (2015) A robust composite nonlinear control scheme for servomotor speed regulation. Int J Control 88(1):104–112. https://doi.org/10.1080/00207179.2014.941408 MathSciNetCrossRefzbMATHGoogle Scholar
- 6.Mobayen S, Tchier F (2017) Composite nonlinear feedback control technique for master/slave synchronization of nonlinear systems. J Nonlinear Dyn 87(3):1731–1747. https://doi.org/10.1007/s11071-016-3148-8 CrossRefzbMATHGoogle Scholar
- 7.Yahaya M, Shahdan Sudin S, Ramli L, Khairi M, Ghazali R (2015) A reduce chattering problem using composite nonlinear feedback and proportional integral sliding mode control. In: IEEE international control conference Asian 10th (ASCC), pp. 1–6. https://doi.org/10.1109/ascc.2015.7244566
- 8.Ebrahimi Mollabashi H, Mazinan AH, Hamidi H (2018) Takagi–Sugeno fuzzy-based CNF control approach considering a class of constrained nonlinear systems. IETE J Res (TIJR). https://doi.org/10.1080/03772063.2018.1464969 CrossRefGoogle Scholar
- 9.Vrkalovic S, Teban T-A, Borlea I-D (2017) Stable Takagi–Sugeno fuzzy control designed by optimization. Int J Artif Intell 15(2):17–29Google Scholar
- 10.Sanchez MA, Castillo O, Castro JR (2015) Information granule formation via the concept of uncertainty-based information with Interval Type-2 fuzzy sets representation and Takagi Sugeno–Kang consequents optimized with cuckoo search. J Appl Soft Comput 27(C):602–609. https://doi.org/10.1016/j.asoc.2014.05.036 CrossRefGoogle Scholar
- 11.Ebrahimi Mollabashi H, Mazinan AH (2018) Adaptive composite non-linear feedback-based sliding mode control for non-linear systems. Inst Eng Technol (IET) 54(16):973–974. https://doi.org/10.1049/el.2018.0619 CrossRefGoogle Scholar
- 12.Ebrahimi Mollabashi H, Rajabpoor M, Rastegarpour S (2013) Inverted pendulum control with pole assignment, LQR and multiple layers sliding mode control. J Basic Appl Sci Res 3(1):363–368Google Scholar
- 13.Ebrahimi H, Shahmansoorian A, Rastegarpour S, Mazinan AH (2013) New approach to control of ball and beam system and optimization with a genetic algorithm. Life Sci J 10(5s):415–421Google Scholar
- 14.Gonzalez CI, Melin P, Castro JR, Castillo O, Mendoza O (2015) Optimization of interval type-2 fuzzy systems for image edge detection. J Appl Soft Comput 47:631–643. https://doi.org/10.1016/j.asoc.2014.12.010 CrossRefGoogle Scholar
- 15.Olivas F, Amador L, Perez J, Caraveo C, Valdez F, Castillo O (2017) Comparative study of type-2 fuzzy particle swarm, bee colony and bat algorithms in optimization of fuzzy controllers. Algorithms 10(3):101–109. https://doi.org/10.3390/a10030101 MathSciNetCrossRefzbMATHGoogle Scholar
- 16.Beatriz G, Fevrier V, Patricia M, German P (2015) Fuzzy logic in the gravitational search algorithm for the optimization of modular neural networks in pattern recognition. Expert Syst Appl 42(14):5839–5847. https://doi.org/10.1016/j.eswa.2015.03.034 CrossRefGoogle Scholar
- 17.Rodríguez L, Castillo O, Soria J, Melin P, Valdez F, Gonzalez CI, Martinez GE, Soto J (2017) A fuzzy hierarchical operator in the grey wolf optimizer algorithm. Appl Soft Comput 57:315–328. https://doi.org/10.1016/j.asoc.2017.03.048 CrossRefGoogle Scholar
- 18.Yang X-S, Deb S (2014) Cuckoo search: recent advances and applications. Neural Comput Appl 24(1):169–174. https://doi.org/10.1007/s00521-013-1367-1 CrossRefGoogle Scholar
- 19.Kanagaraj G, Ponnambalam SG, Jawahar N (2013) A hybrid cuckoo search and genetic algorithm for reliability–redundancy allocation problems. Comput Ind Eng 66:1115–1124. https://doi.org/10.1016/j.cie.2013.08.003 CrossRefGoogle Scholar
- 20.Olivas F, Valdez F, Castillo O, Gonzalez CI, Martinez G, Melin P (2017) Ant colony optimization with dynamic parameter adaptation based on interval type-2 fuzzy logic systems. Appl Soft Comput 53:74–87. https://doi.org/10.1016/j.asoc.2016.12.015 CrossRefGoogle Scholar
- 21.Saadat J, Moallem P, Koofigar H (2017) Training echo state neural network using harmony search algorithm. Int J Artif Intell 15(1):163–179. https://doi.org/10.1016/j.ins.2014.02.091 CrossRefGoogle Scholar
- 22.Valdez F, Melin P, Castillo O (2014) Modular neural networks architecture optimization with a new nature-inspired method using a fuzzy combination of particle swarm optimization and genetic algorithms. Inf Sci 270:143–153. https://doi.org/10.1016/j.ins.2014.02.091 CrossRefGoogle Scholar
- 23.Lin D, Lan W (2015) Output feedback composite nonlinear feedback control for singular systems with input saturation. J Frankl Inst 352(1):384–398. https://doi.org/10.1016/j.jfranklin.2014.10.018 MathSciNetCrossRefzbMATHGoogle Scholar
- 24.Precup R-E, David R-C, Petriu EM (2017) Grey wolf optimizer algorithm-based tuning of fuzzy control systems with reduced parametric sensitivity. IEEE Trans Ind Electron 64(1):527–534. https://doi.org/10.1109/TIE.2016.2607698 CrossRefGoogle Scholar
- 25.Cervantes L, Castillo O, Hidalgo D, Martinez R (2018) Fuzzy dynamic adaptation of gap generation and mutation in genetic optimization of type 2 fuzzy controllers. Adv Oper Res. https://doi.org/10.1155/2018/9570410 CrossRefzbMATHGoogle Scholar
- 26.Pazooki M, Mazinan AH (2018) Hybrid fuzzy-based sliding-mode control approach, optimized by genetic algorithm for quadrotor unmanned aerial vehicles. Complex Intell Syst 4(2):79–93. https://doi.org/10.1007/s40747-017-0051-y CrossRefGoogle Scholar
- 27.Guerrero M, Castillo O, García M (2015) Fuzzy dynamic parameters adaptation in the Cuckoo Search Algorithm using fuzzy logic. IEEE Congr Evol Comput (CEC). https://doi.org/10.1109/cec.2015.7256923 CrossRefGoogle Scholar
- 28.Sanchez MA, Castillo O, Castro JR (2015) Generalized type-2 fuzzy Systems for controlling a mobile robot and a performance comparison with interval type-2 and type-1 fuzzy systems. Int J Expert Syst Appl 42(14):5904–5914. https://doi.org/10.1016/j.eswa.2015.03.024 CrossRefGoogle Scholar
- 29.Marcek D (2018) Forecasting of financial data: a novel fuzzy logic neural network based on error-correction concept and statistics. Complex Intell Syst 2(2):95–104. https://doi.org/10.1007/s40747-017-0056-6 CrossRefGoogle Scholar
- 30.He Y, Chen BM, Wu C (2005) Composite nonlinear control with state and measurement feedback for general multivariable systems with input saturation. Syst Control Lett 54:455–469. https://doi.org/10.1016/j.sysconle.2004.09.010 MathSciNetCrossRefzbMATHGoogle Scholar
- 31.Naz N, Malik MB, Salman M (2013) Real-time implementation of feedback linearizing controllers for magnetic levitation system. In: IEEE conference on systems, process and control (ICSPC), pp 52–55Google Scholar
- 32.Mobayen S (2014) Design of CNF-based nonlinear integral sliding surface for matched uncertain linear systems with multiple state-delays. Nonlinear Dyn 77(3):1047–1054. https://doi.org/10.1007/s12555-015-0477-1 MathSciNetCrossRefzbMATHGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.