IMPROVING THE PERFORMANCE OF AN UNDER-DAMPED MASS SPRING DAMPER SYSTEM THROUGH SWITCHED PARAMETERS By Amer Lafi Allafi A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mechanical Engineering - Master of Science 2014 ABSTRACT IMPROVING THE PERFORMANCE OF AN UNDER-DAMPED MASS SPRING DAMPER SYSTEM THROUGH SWITCHED PARAMETERS By Amer Lafi Allafi In this thesis we propose to improve the performance of the standard single degree-offreedom under-damped mass-spring-damper (MSD) system using variable structure control. Two controllers are proposed: both of them switch the parameters of the system between their nominal values and their negative values. This approach results in a hybrid system comprised of the nominal system, which is asymptotically stable, and an unstable system. The first controller is based on switched stiffness whereas the second controllers is based on switched stiffness and damping. For both controllers, the parameters are switched based on the location of the system in its configuration space. A phase portrait analysis indicates that the resulting hybrid systems are asymptotically stable although they switch between an asymptotically stable and an unstable system. A comparison of the step response of the hybrid systems with that of the original underdamped system indicates that the switched systems have better performance in terms of rise time, settling time, and reduced or no overshoot. Different designs of the switching logic have been investigated and they provide clues on how the switching logic can be designed to achieve the maximum improvement in performance. ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisor Dr. Ranjan Mukherjee for his patience, motivation, vast knowledge, and guidance throughout this work. I would like to thank the other members of my committee, Dr. Hassan K. Khalil, and Dr. Jongeun Choi for the assistance they provided. I would also like to thank my family: my parents Lafi Allafi and Hessa Almuzaini, and my wife Najla Aleid, for supporting me spiritually throughout my life. Finally, I would like to thanks the Saudi Arabia Cultural Mission for the financial support during my Master program. iii TABLE OF CONTENTS LIST OF FIGURES…………………………...…………………………………………………. vi Chapter 1 Introduction......................................................................................................................1 Chapter 2 Mass-Spring-Damper System with Switched Stiffness and Damping..............................4 2.1. Mass-Spring-Damper System with Switched Stiffness…………………………………...4 2.1.1. Stability Characteristics of the Individual Systems………………………………..5 2.1.2. Behavior of the First System (𝛼 = +1)……………………………………………5 2.1.3. Behavior of the Second System (α = −1)………………………………………...7 2.1.4. Different Cases of the VSS with Switched Stiffness………………………………7 2.2. Mass-Spring-Damper System with Switched Stiffness and Damping…………………...8 2.2.1. Stability Characteristics of the Individual Systems………………………………..9 2.2.2. Behavior of the First System (𝛼 = +1)……………………………………………9 2.2.3. Behavior of the Second System (α = −1)…………………………………..…...10 2.2.4. Different Cases of the VSS with Switched Stiffness………………………..……11 Chapter 3 Variable Structure System with Switched Stiffness……………………….…………..13 3.1. Phase Portrait……………………………………………………………………………13 3.2. Case A: 𝜇1 < λ < 0….…………………………………………………………………..13 3.2.1. Stability…………………………………………………………………………..15 3.2.2. Performance – Response to Step Input…………………………………………...20 3.2.3. Performance – Speed of Convergence……………………………………………26 3.3. Case B: λ ≤ 𝜇1 < 0 ……………………………...……………………………………...30 3.3.1. Stability…………………………………………………………………………..30 3.3.2. Performance – Response to Step Input…………………………………………...34 3.3.3. Performance – Speed of Convergence……………………………………………43 3.4. Conclusion………………………………………………………………………………49 Chapter 4 Variable Structure System with Switched Stiffness and Damping…………….……..51 4.1. Phase Portrait……………………………………………………………………………51 4.2. Case A: 𝜇1 < λ < 0….…………………………………………………………………..53 4.2.1. Stability…………………………………………………………………………..53 4.2.2. Performance – Response to Step Input…………………………………………...58 4.2.3. Performance – Speed of Convergence……………………………………………63 4.3. Case B: λ ≤ 𝜇1 < 0 ……………………………...……………………………………...67 4.3.1. Stability…………………………………………………………………………..67 4.3.2. Performance – Response to Step Input…………………………………………...74 4.3.3. Performance – Speed of Convergence……………………………………………83 4.4. Conclusion………………………………………………………………………………89 Chapter 5 Conclusion…..………………………………………………………………………...90 iv REFERENCES……………………..……………………………………………………………92 v LIST OF FIGURES Figure 2.1. Phase portrait for 𝛼 = +1 and 𝜁 < 1…….…………………..……………... 6 Figure 2.2. Phase portrait for 𝛼 = −1….…..…………………………………..……….. 8 Figure 2.3. Phase portrait for 𝛼 = +1 and 𝜁 < 1……………………………………….. 11 Figure 2.4. Phase portrait for 𝛼 = −1………………...………...………………………. 12 Figure 3.1. Phase portrait of the Variable Structure System (VSS) for 𝛼 = +1…..………. 14 Figure 3.2. Phase portrait of the Variable Structure System (VSS) for 𝛼 = −1...…..…….. 14 Figure 3.3. Phase portrait of the Variable Structure System (VSS) for λ > μ 1 is the union of the phase portrait in Figs.3.1 and 3.2..…………………………….. 15 Figure 3.4. Convergence of VSS trajectories to the origin for ζ < 1 and 𝜇1 < λ < 0…. Figure 3.5. Comparison of the settling times of the VSS and the original system............ 24 Figure 3.6. Comparison of the settling times of the VSS and the original system...……. 25 Figure 3.7. Comparison of the settling times of the VSS and the original system….…... 27 Figure 3.8. Comparison between the settling time of the VSS and the original system... 29 Figure 3.9. Phase portrait of the Variable Structure System (VSS) for 𝛼 = +1……...…… 31 Figure 3.10. Phase portrait of the Variable Structure System (VSS) for 𝛼 = −1……...…… 31 Figure 3.11. Phase portrait of the Variable Structure System (VSS) for λ > μ 1 ……...…… 32 Figure 3.12. Step response in the phase plane of the Variable Structure System (VSS) for λ < μ 1 ………..…………………………………………………………….. 36 Figure 3.13. Step response of the Variable Structure System (VSS) for λ < μ 1 ……...…… 37 20 Figure 3.14. Comparison of the rise times of the VSS and original system…..………….. 41 Figure 3.15. Comparison of the percentage of overshoots of the VSS and original system……………………………………………………………………….. 42 vi Figure 3.16. Comparison of the settling time of the VSS and original system…...……… 43 Figure 3.17. Step response in the Phase plane for the Variable Structure System (VSS) for λ < μ 1 ………………………………………………………………………. 45 Figure 3.18. Step response for the Variable Structure System (VSS) for λ < μ 1 ………….. 45 Figure 3.19. The rise time for VSS and original system …………………….…………... 47 Figure 3.20. The percentage of overshoot for VSS and original system……..…………... 48 Figure 3.21. The settling times of VSS and original system………………..…….……… 49 Figure 4.1. Phase portrait of the Variable Structure System (VSS) for 𝛼 = +1…..…...….. 52 Figure 4.2. Phase portrait of the Variable Structure System (VSS) for 𝛼 = −1……..……. 52 Figure 4.3. Phase portrait of the Variable Structure System (VSS) for λ > μ 1 is the union of the phase portrait in Figs.4.1 and 4.2………..……….……………. 53 Figure 4.4. Convergence of VSS trajectories to the origin for ζ < 1 and 𝜇1 < λ < 0…. Figure 4.5. Comparison of the settling times of the VSS and the original system.....…... 62 Figure 4.6. Comparison of the settling times of the VSS and the original system….…... 64 Figure 4.7. Comparison of the settling times of the VSS and the original system…..….. 66 Figure 4.8. Phase portrait of the Variable Structure System (VSS) for 𝛼 = +1……...…… 68 Figure 4.9. Phase portrait of the Variable Structure System (VSS) for 𝛼 = −1……...…… 68 Figure 4.10. Phase portrait of the Variable Structure System (VSS) for λ > μ 1 ……...…… 69 Figure 4.11. Phase portrait of the Variable Structure System…...……………………….. 70 Figure 4.12. The percentage of the contraction in energy of the VSS between points A and C…………………………………………………...…………………… 74 58 Figure 4.13. Step response in the phase plane of the Variable Structure System (VSS) for λ < μ 1 …………………………………………………...………………...... 76 Figure 4.14. Step response of the Variable Structure System (VSS) for λ < μ 1 …………... 77 Figure 4.15. Comparison of the rise times of the VSS and original system…...…………. 80 vii Figure 4.16. Comparison of the percentage of overshoots of the VSS and original system……………………………………………………………………….. 81 Figure 4.17. Comparison of the settling time of the VSS and original system...………… 82 Figure 4.18. Step response in the Phase plane for the Variable Structure System (VSS) for λ < μ 1 ………………………………………………………………………. 84 Figure 4.19. Step response for the Variable Structure System (VSS) for λ < μ 1 ………….. 84 Figure 4.20. The rise time for VSS and original system ……………...…….…………… 87 Figure 4.21. The percentage of overshoot for VSS and original system……..……...…… 87 Figure 4.22. The settling times of VSS and original system…………...………………… viii 88 Chapter 1 Introduction The single degree-of-freedom mass-spring-damper (MSD) system describes the dynamics of many real physical systems. The behavior of the MSD is well understood and therefore controllers are also designed to make the closed-loop dynamics of many physical systems emulate the dynamics of the MSD. The behavior of the MSD is largely dependent on the damping ratio ζ: a small damping ratio results in fast response (small rise time) but a large overshoot and a large settling time; on the other hand, a large damping ratio results in sluggish response (large rise and settling times) but no overshoot. While emulating the dynamics of the MSD system, it is common to choose the damping ratio to lie in the range 0.5 < ζ < 0.8 since smaller values of ζ result in large overshoots and larger values of ζ result in sluggish response [1]. To improve the performance of the MSD and have fast response with no overshoot, we investigate the stability and the performance of two hybrid MSD systems. The first system uses switched stiffness; the stiffness of this system is switched between its nominal value and its negative value. The second system uses switched stiffness and damping coefficients; the stiffness and damping coefficients are switched between their nominal values and their negative values. In both hybrid systems, switching is based on the location of the system in its configuration space. 1 Each hybrid system, namely, the MSD with switched stiffness and the MSD with switched stiffness and damping, can be described by a differential equation with a discontinues forcing function, where the forcing function can be viewed as a sliding mode controller [2-5]. Sliding mode control (SMC) is typically used for robust stabilization of dynamical systems in the presence of uncertainties but here it is used for a completely different purpose, namely, to improve the performance of the system. The improvement in performance is achieved by switching between the original system, which is asymptotically stable, and the system with negative stiffness, which is unstable. The idea of using switched stiffness for performance improvement has been investigated by several researchers [6-9]. In all of these studies, the stiffness of a mass-spring system is switched between two positive values; this approach results in a hybrid system comprised of two stable subsystems. In this work, the stiffness of the MSD system is switched between a positive value and a negative value; the resulting hybrid system is comprised of two sub-systems that are asymptotically stable and unstable, respectively. The improvement in performance using negative stiffness has also been investigated [10-12]; in most of these studies, negative stiffness has been incorporated in the design for improved vibration isolation. These studies are different from our study since the stiffness is not switched. The idea of varying the damping coefficient for performance improvement has also been investigated [14-15]. In these studies, the stiffness and the damping coefficients are varied with respect to time. The results have been applied to different application problems such as improved vibration isolation in high-speed rotors. In our work, switching of the stiffness and the damping parameters is based on the location of the system in its configuration space. Also, the stiffness and damping coefficients are switched between a constant negative value and a positive value. 2 This thesis is organized as follows. The dynamics of the hybrid MSD system and the stability characteristics of the individual subsystems are presented in Chapter 2. The stability characteristics and performance of the MSD with switched stiffness is analyzed in Chapter 3. The time response of the hybrid system is then compared to that of the original system. Chapter 4 repeats the analysis for the MSD with switched stiffness and damping. Chapter 5 contains concluding remarks. 3 Chapter 2 Mass-Spring-Damper System with Switched Stiffness and Damping 2.1. Mass-Spring-Damper System with Switched Stiffness We consider the following mass-spring-damper system with switched stiffness: 𝑚𝑥̈ + 𝑐𝑥̇ + 𝛼𝑘𝑥 = 0 (2.1) where m, c, and k are the mass damping coefficient, and stiffness, respectively, α switches between ±1 according to the logic: 𝛼={ +1 −1 𝑖𝑓 𝑖𝑓 (𝑆 > 0 and 𝑥 ≥ 0 ) 𝑜𝑟 (𝑆 < 0 and 𝑥 ≤ 0 ) 𝑆𝑥<0 (2.2) and S is defined by the relation: 𝑆 = 𝑥2 − 𝜆 𝑥1 , λ< 0 (2.3) The unforced system in (2.1) can be viewed as the following forced system: 𝑚𝑥̈ + 𝑐𝑥̇ + 𝑘𝑥 = 𝐹 (2.4) where, the forcing function F is defined as: 𝐹={ 0 2𝑘𝑥 𝑖𝑓 (𝑆 > 0 and 𝑥 ≥ 0 ) 𝑜𝑟 (𝑆 < 0 and 𝑥 ≤ 0 ) 𝑖𝑓 𝑆𝑥<0 4 (2.5) Equation (2.1) can be rewritten as: 𝑥̈ = − 𝑐 𝑘 𝑥̇ − 𝛼 𝑥 𝑚 𝑚 (2.6) The variable structure system (VSS) in Eq.(2.6) is comprised of two linear systems: one where 𝛼 = 1, and the other where 𝛼 = −1. We will study the behavior of these two individual systems with the objective of investigating the stability and performance of the variable structure system for different values of m, c, and, k. 2.1.1. Stability Characteristics of the Individual Systems To investigate the stability characteristics of the VSS in Eq.(2.6) we need to first investigate the stability properties of the individual systems. To this end we define the states 𝑥1 = 𝑥, and 𝑥2 = 𝑥̇ . Then, the state space equations are: 𝑥̇ 1 = 𝑥2 𝑥̇ 2 = − (2.7) 𝑐 𝑘 𝑥2 − 𝛼 𝑥1 𝑚 𝑚 (2.8) If we define 𝜔𝑛 2 = 𝑘⁄𝑚 and 𝜁 = 𝑐⁄2𝑚𝜔 the state space equation can be rewritten as 𝑛 𝑥̇ 1 = 𝑥2 (2.9) 2 𝑥̇ 2 = −𝛼𝜔𝑛 𝑥1 − 2𝜁𝜔𝑛 𝑥2 2.1.2. Behavior of the First System (𝛂 = +𝟏) For the first system, where 𝛼 = +1, we have 𝑥1 𝑥̇ [ 1 ] = [𝐴1 ] [𝑥 ] , 𝑥̇ 2 2 𝐴1 ≜ [ 0 −𝜔𝑛 2 1 ] −2𝜁𝜔𝑛 The eigenvalues of A1 are: 5 (2.10) 𝜎1 , 𝜎2 = 𝜔𝑛 [−𝜁 ± √𝜁 2 − 1 ] < 0 The original system is under-damped, i.e. 𝜁 < 1. Therefore, the eigenvalues are complex and are given by: 𝜎1 = 𝜔𝑛 [−𝜁 + 𝑖√1 − 𝜁 2 ] (2.11) 𝜎2 = 𝜔𝑛 [−𝜁 − 𝑖√1 − 𝜁 2 ] (2.12) and the origin is a stable focus. [13]. The eigenvectors corresponding to the eigenvalues in Eqs.(2.11) and (2.12) are described by the equations: 𝑥2 = 𝜎1 𝑥1 (2.13) 𝑥2 = 𝜎2 𝑥1 (2.14) Figure 2.1. Phase portrait for 𝛼 = +1 and 𝜁 < 1 6 and the phase portrait of the system is shown in Fig.2.1. 2.1.3. Behavior of the Second System (𝛂 = −𝟏) For the second system, when 𝛼 = −1, we have 𝑥1 𝑥̇ [ 1 ] = [𝐴2 ] [𝑥 ] , 𝑥̇ 2 2 𝐴2 ≜ [ 0 𝜔𝑛 2 1 ] −2𝜁𝜔𝑛 (2.15) The eigenvalues of A2 are: 𝜇1 = 𝜔𝑛 [− 𝜁 − √𝜁 2 + 1 ] < 0 (2.16) 𝜇2 = 𝜔𝑛 [− 𝜁 + √𝜁 2 + 1 ] > 0 (2.17) Therefore, the origin is a saddle [13]. The stable and unstable eigenvectors corresponding to the eigenvalues in Eqs.(2.16) and (2.17) are described by the equations: 𝑥2 = 𝜇1 𝑥1 (2.18) 𝑥2 = 𝜇2 𝑥1 (2.19) and the phase portrait of the system is shown in Fig.2.2. 2.1.4. Different Cases of the VSS with Switched Stiffness The VSS with switched stiffness has the same behavior as that of the first system when (𝑆 > 0 and 𝑥 ≥ 0) or (𝑆 < 0 and 𝑥 ≤ 0), and that of the second system when 𝑆𝑥1 < 0 – see Eqs.(2.2) and (2.6). Since the first system (where 𝛼 = +1) can be only under-damped (𝜁 < 1), we study of the VSS Switched Stiffness and we carry out our investigating separately for 𝜆 > 𝜇1 and 𝜆 ≤ 𝜇1 in Chapter 3. 7 Figure 2.2. Phase portrait for 𝛼 = −1 2.2. Mass-Spring-Damper System with Switched Stiffness and Damping Consider the following mass-spring-damper system with switched stiffness: 𝑚𝑥̈ + 𝛼𝑐𝑥̇ + 𝛼𝑘𝑥 = 0 (2.20) where m, c, and k are the mass damping coefficient, and stiffness, respectively, α switches between ±1 according to the logic: 𝛼={ +1 −1 𝑖𝑓 𝑖𝑓 (𝑆 > 0 and 𝑥 ≥ 0 ) 𝑜𝑟 (𝑆 < 0 and 𝑥 ≤ 0 ) 𝑆𝑥<0 (2.21) and S is defined by the relation: 𝑆 = 𝑥2 − 𝜆 𝑥1 , λ< 0 (2.22) The unforced system in (2.20) can be viewed as the following forced system: 8 𝑚𝑥̈ + 𝑐𝑥̇ + 𝑘𝑥 = 𝐹 (2.23) where, the forcing function F is defined as: 𝐹={ 0 2𝑘𝑥 + 2𝑐𝑥̇ 𝑖𝑓 (𝑆 > 0 and 𝑥 ≥ 0 ) 𝑜𝑟 (𝑆 < 0 and 𝑥 ≤ 0 ) 𝑖𝑓 𝑆𝑥<0 (2.24) Equation (2.20) can be rewritten as: 𝑥̈ = −𝛼 𝑐 𝑘 𝑥̇ − 𝛼 𝑥 𝑚 𝑚 (2.25) The variable structure system (VSS) in Eq.(2.25) is comprised of two linear systems: one where 𝛼 = 1, and the other where 𝛼 = −1. We will study the behavior of these two individual systems with the objective of investigating the stability and performance of the variable structure system for different values of m, c, and, k. 2.2.1. Stability Characteristics of the Individual Systems To investigate the stability characteristics of the VSS in Eq.(2.25) we need to first investigate the stability properties of the individual systems. To this end we define the states 𝑥1 = 𝑥, and 𝑥2 = 𝑥̇ . Then, the state space equations are: 𝑥̇ 1 = 𝑥2 𝑥̇ 2 = −𝛼 (2.26) 𝑐 𝑘 𝑥2 − 𝛼 𝑥1 𝑚 𝑚 (2.27) If we define 𝜔𝑛 2 = 𝑘⁄𝑚 and 𝜁 = 𝑐⁄2𝑚𝜔 the state space equation can be rewritten as 𝑛 𝑥̇ 1 = 𝑥2 𝑥̇ 2 = −𝛼𝜔𝑛 2 𝑥1 − 2𝛼𝜁𝜔𝑛 𝑥2 (2.28) 2.2.2. Behavior of the First System (𝛂 = +𝟏) 9 For the first system, where 𝛼 = +1, we have 𝑥1 𝑥̇ [ 1 ] = [𝐴1 ] [𝑥 ] , 𝑥̇ 2 2 𝐴1 ≜ [ 0 −𝜔𝑛 2 1 ] −2𝜁𝜔𝑛 (2.29) The eigenvalues of A1 are: 𝜎1 , 𝜎2 = 𝜔𝑛 [−𝜁 ± √𝜁 2 − 1 ] < 0 The original system is under-damped, i.e. 𝜁 < 1. Therefore, The eigenvalues are complex and are given by: 𝜎1 = 𝜔𝑛 [−𝜁 + 𝑖√1 − 𝜁 2 ] (2.30) 𝜎2 = 𝜔𝑛 [−𝜁 − 𝑖√1 − 𝜁 2 ] (2.31) and the origin is a stable focus [13]. The eigenvectors corresponding to the eigenvalues in Eqs.(2.30) and (2.31) are described by the equations 𝑥2 = 𝜎1 𝑥1 (2.32) 𝑥2 = 𝜎2 𝑥1 (2.33) and the phase portrait of the system is shown in Fig.2.3. 2.2.3. Behavior of the Second System (𝛂 = −𝟏) For the second system, when 𝛼 = −1, we have 𝑥1 𝑥̇ [ 1 ] = [𝐴2 ] [𝑥 ] , 𝑥̇ 2 2 𝐴2 ≜ [ 0 𝜔𝑛 2 1 ] 2𝜁𝜔𝑛 10 (2.34) Figure 2.3. Phase portrait for 𝛼 = +1 and 𝜁 < 1 The eigenvalues of A2 are: 𝜇1 = 𝜔𝑛 [𝜁 − √𝜁 2 + 1 ] < 0 (2.35) 𝜇2 = 𝜔𝑛 [𝜁 + √𝜁 2 + 1 ] > 0 (2.36) Therefore, the origin is a saddle [13]. The stable and unstable eigenvectors, corresponding to the eigenvalues in Eqs.(2.35) and (2.36), are described by the equations 𝑥2 = 𝜇1 𝑥1 (2.37) 𝑥2 = 𝜇2 𝑥1 (2.38) and the phase portrait of the system is shown in Fig.2.4. 2.2.4. Different Cases of the VSS with Switched Stiffness and Damping 11 Figure 2.4. Phase portrait for 𝛼 = −1 The VSS with switched stiffness and damping has the same behavior as that of the first system when (𝑆 > 0 and 𝑥 ≥ 0) or (𝑆 < 0 and 𝑥 ≤ 0), and that of the second system when 𝑆𝑥1 < 0 – see Eq.(2.21). Since the first system (where 𝛼 = +1) can be only under-damped (𝜁 < 1), we study of the VSS Switched Stiffness and Damping and we carry out our investigating separately for 𝜆 > 𝜇1 and 𝜆 ≤ 𝜇1 in Chapter 4. 12 Chapter 3 Variable Structure System with Switched Stiffness 3.1. Phase Portrait In this chapter, we investigate the behavior of the VSS with switched stiffness. The phase portrait of the VSS is the union of the phase portrait of the first system (𝛼 = +1) in the region (𝑆 > 0 and 𝑥 ≥ 0) or (𝑆 < 0 and 𝑥 ≤ 0), and the phase portrait of the second system (𝛼 = −1) in the region 𝑆𝑥1 < 0. The phase portrait of the first system (𝛼 = +1) in the region (𝑆 > 0 and 𝑥 ≥ 0) or (𝑆 < 0 and 𝑥 ≤ 0) is shown in Fig.3.1 and the phase portrait of the second system (𝛼 = −1) in the region 𝑆𝑥1 < 0 is shown in Fig.3.2. The union of these phase portraits is shown in Fig.3.3. For the phase portrait in Fig.3.3, the slope of the line S = 0 or 𝑥2 = 𝜆 𝑥1 is shown to be greater than the slope of the line 𝑥2 = 𝜇1 𝑥1, i.e., λ > 𝜇1 . This is not necessarily true always, and therefore we need to additionally investigate the case where λ ≤ 𝜇1. These two cases, namely, (𝜇1 < λ < 0), and ( λ ≤ 𝜇1 < 0), are investigated in sections 3.2 and 3.3. 3.2. Case A: 𝝁𝟏 < 𝛌 < 𝟎 13 Figure 3.1. Phase portrait of the Variable Structure System (VSS) for 𝛼 = +1 Figure 3.2. Phase portrait of the Variable Structure System (VSS) for 𝛼 = −1 14 Figure 3.3. Phase portrait of the Variable Structure System (VSS) for λ > μ 1 is the union of the phase portrait in Figs.3.1 and 3.2 3.2.1. Stability The VSS is a hybrid system that switches between an asymptotically stable system, the phase portrait of which is shown in Fig.3.1, and an unstable system whose phase portrait is shown in Fig.3.2. The union of these phase portraits is shown in Fig.3.3. To investigate the stability of the equilibrium of the VSS, the phase plane is divided into three zones, namely: 𝑍1 = {𝑥 ∈ 𝑅 2 |𝑆𝑥1 > 0} 𝑍2 = {𝑥 ∈ 𝑅 2|(𝑥2 − 𝜇1 𝑥1 )(𝑥2 − 𝜆 𝑥1 ) < 0} 𝑍3 = {𝑥 ∈ 𝑅 2|𝑥1 (𝑥2 − 𝜇1 𝑥1 ) < 0} The three zones are separated by the following three lines: L1: 𝑥1 = 0 15 L2: 𝑥2 = 𝜇1 𝑥1 L3: 𝑥2 = 𝜆 𝑥1 We investigate the system trajectories in these zones and on these lines next. To study the behavior of the trajectories on L1, we investigate the direction of the vector field on L1. For 𝑥2 > 0 (and 𝑥1 = 0),we have the following equations and vector field: 𝑥̇ 1 = 𝑥2 > 0 𝑥̇ 2 = − 𝑐 𝑥 <0 𝑚 2 Therefore, the trajectories enter 𝑍1 . For 𝑥2 < 0 (and 𝑥1 = 0), we have the following equations and vector field: 𝑥̇ 1 = 𝑥2 < 0 𝑥̇ 2 = − 𝑐 𝑥 𝑚 2 Therefore, all trajectories on L1 will enter 𝑍1 . The line L2 is the stable eigenvector of the second system where 𝛼 = −1 – see Eqs.(2.16) and (2.18). On L2 we have 𝑥2 = 𝑥̇ 1 = 𝜇1 𝑥1 → 𝑥1 (𝑡) = 𝑥1 (0) 𝑒 𝜇1 𝑡 , 𝜇1 < 0. Therefore, all trajectories on L2 converge to the origin. The boundaries of 𝑍1 are L1 and L3. We have already shown that trajectories on L1 enter 𝑍1 . We now show all trajectories in 𝑍1 reach L3 in finite time. To show this we consider the equation of motion of the system for 𝛼 = 1, given by Eq.(2.9): 𝑥̇ 2 = 𝑥̈ 1 = −2𝜁𝜔𝑛 𝑥̇ 1 − 𝜔𝑛 2 𝑥1 The general form of the solution of this differential equation is 𝑥1 (𝑡) = 𝑒 −𝜁𝜔𝑛𝑡 [ 𝐴 cos 𝜔𝑑 𝑡 + 𝐵 sin 𝜔𝑑 𝑡] 16 (3.1) and 𝑥2 (𝑡) = −𝜁𝜔𝑛 𝑒 −𝜁𝜔𝑛𝑡 [ 𝐴 cos 𝜔𝑑 𝑡 + 𝐵 sin 𝜔𝑑 𝑡] +𝑒 −𝜁𝜔𝑛𝑡 [ −𝐴 𝜔𝑑 sin 𝜔𝑑 𝑡 + 𝐵 𝜔𝑑 cos 𝜔𝑑 𝑡] (3.2) where 𝜔𝑑 = 𝜔𝑛 √1 − 𝜁 2 Substitution of the initial conditions 𝑥1 (0) = 𝑥10 , and 𝑥2 (0) = 𝑥20 into Eqs.(3.1) and (3.2), where (𝑥10 , 𝑥20 ) lies in 𝑍1 , we get: 𝐴 = 𝑥10 , 𝐵= 𝑥20 + 𝜁𝜔𝑛 𝑥10 (3.3) 𝜔𝑛 √1 − 𝜁 2 To find the time when trajectory in 𝑍1 reaches L3, we substitute Eqs.(3.1) and (3.2) into the equation of L3, namely: 𝑥2 (𝑡) = 𝜆 𝑥1 (𝑡) This gives: −𝜁𝜔𝑛 𝑒 −𝜁𝜔𝑛𝑡 [𝐴 cos 𝜔𝑑 𝑡 + 𝐵 sin 𝜔𝑑 𝑡] + 𝜔𝑑 𝑒 −𝜁𝜔𝑡 [− 𝐴 sin 𝜔𝑑 𝑡 + 𝐵 cos 𝜔𝑑 𝑡] = 𝜆 𝑒 −𝜁𝜔𝑛𝑡1 [A cos 𝜔𝑑 𝑡 + 𝐵 sin 𝜔𝑑 𝑡] ⇒ tan 𝜔𝑑 𝑡 = 𝜆𝐴 − 𝜔𝑑 𝐵 + 𝜁𝜔𝑛 𝐴 −𝜁𝜔𝑛 𝐵 − 𝜔𝑑 𝐴 − 𝜆𝐵 (3.4) Substitution of the values of A and B from Eq.(3.3) into Eq.(3.4) gives 𝜆𝑥10 − 𝑥20 − 𝜁𝜔𝑛 𝑥10 + 𝜁𝜔𝑛 𝑥10 tan 𝜔𝑑 𝑡 = − ⇒𝑡= 𝜁𝑥20 + 𝜁 2 𝜔𝑛 𝑥10 𝜔𝑛 (1 − 𝜁 2 )𝑥10 𝑥 + 𝜁𝜔𝑛 𝑥10 − − 𝜆 ( 20 ) 2 2 √1 − 𝜁 √1 − 𝜁 𝜔𝑛 √1 − 𝜁 2 (𝜆𝑥10 − 𝑥20 )√1 − 𝜁 2 ] 𝑥20 + 𝜁𝜔𝑛 𝑥10 ) 20 − 𝜔𝑛 𝑥10 − 𝜆 ( 𝜔𝑛 1 tan−1 [ 𝜔𝑑 −𝜁𝑥 (3.5) This expression implies that all trajectories in 𝑍1 reaches L3 at finite time. The boundaries of 𝑍2 are L2 and L3. Furthermore, the trajectories in 𝑍2 cannot cross L2 since L2 represents a trajectory itself, and trajectories cannot cross each other. We now show all 17 trajectories in 𝑍2 reach L3 in finite time. To show this we consider the equation of motion of the system for 𝛼 = −1, given by Eq.(2.9) : 𝑥̇ 2 = 𝑥̈ 1 = −2𝜁𝜔𝑛 𝑥̇ 1 + 𝜔𝑛 2 𝑥1 The general form of the solution of this differential equation is 2 𝑥1 (𝑡) = 𝐷 𝑒 𝜔𝑛𝑡(−𝜁−√1+𝜁 ) + 𝐸 𝑒 𝜔𝑛𝑡(−𝜁+√1+𝜁 and (3.6) 2) 𝑥2 (𝑡) = 𝜔𝑛 (−𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 𝜔𝑛𝑡(−𝜁−√1+𝜁 +𝜔𝑛 (−𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜔𝑛𝑡(−𝜁+√1+𝜁 2) 2) (3.7) Substitution of the initial conditions 𝑥1 (0) = 𝑥10 , and 𝑥2 (0) = 𝑥20 into Eqs.(3.6) and (3.7) , where (𝑥10 , 𝑥20 ) lies in 𝑍2 we get: 𝐷 = 𝑥10 − 𝑥20 + 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 , 𝐸= 𝑥20 + 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 (3.8) To find the time when trajectory in 𝑍2 reaches L3, we substitute Eqs.(3.6), and (3.7) into the equation of L3, namely: 𝑥2 (𝑡) = 𝜆 𝑥1 (𝑡) This gives: 𝜔𝑛 (−𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 𝜔𝑛𝑡(−𝜁−√1+𝜁 = 𝜆 (𝐷 𝑒 𝜔𝑛𝑡(−𝜁−√1+𝜁 𝑒 𝜔𝑛𝑡(2√1+𝜁 2) = 2) 2) + 𝜔𝑛 (−𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜔𝑛𝑡(−𝜁+√1+𝜁 2 + 𝐸 𝑒 𝜔𝑛𝑡(−𝜁+√1+𝜁 ) ) 𝐷[𝜆 − 𝜔𝑛 (−𝜁 − √1 + 𝜁 2 ) ] 𝐸[𝜔𝑛 (−𝜁 + √1 + 𝜁2 ) −𝜆] Substitution of the values of D and E from Eq.(48)into Eq.(3.9) gives: 𝑒 𝜔𝑛 𝑡(2√1+𝜁 2 ) = 2) − (𝑥20 + 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 )) [𝜆 − 𝜔𝑛 (−𝜁 − √1 + 𝜁 2 ) ] (𝑥20 + 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 )) [𝜔𝑛 (−𝜁 + √1 + 𝜁 2 ) − 𝜆 ] 18 (3.9) ⇒𝑡= 1 2𝜔𝑛 √1 + 𝜁 2 ln [ 𝑄1 ] 𝑄2 (3.10) where 𝑄1 and 𝑄2 , which necessarily have the same sign, are 𝑄1 = − (𝑥20 + 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 )) [𝜆 − 𝜔𝑛 (−𝜁 − √1 + 𝜁 2 ) ] 𝑄2 = (𝑥20 + 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 )) [𝜔𝑛 (−𝜁 + √1 + 𝜁 2 ) − 𝜆 ] This expression implies that all trajectories in 𝑍2 reaches L3 at finite time. The line L3 is a sliding surface since all trajectories in 𝑍1 and 𝑍2 approach L3 – see Fig.3.3. On L3 we have 𝑥2 = 𝑥̇ 1 = 𝜆𝑥 1 → 𝑥1 (𝑡) = 𝑥1 (0) 𝑒 𝜆𝑡 . Therefore, all trajectories on L3 converge to the origin. The boundaries of 𝑍3 are L1 and L2. Furthermore, the trajectories in 𝑍3 cannot cross L2 since L2 represent a trajectory itself and trajectories cannot cross each other. Moreover, the trajectories on L1 enter 𝑍1 . We now show all trajectories in 𝑍3 reach L1 in finite time. The general form of the solution of the differential equation is described by Eqs.(3.6), and (3.7). To find the time when trajectory in 𝑍3 reaches L1, we substitute Eq. (3.6) into the equation of L1, namely: 𝑥1 = 0 This gives: 2 𝐷 𝑒 𝜔𝑛𝑡(−𝜁−√1+𝜁 ) + 𝐸 𝑒 𝜔𝑛𝑡(−𝜁+√1+𝜁 𝑒 𝜔𝑛𝑡(2√1+𝜁 2) =− 2) =0 𝐷 𝐸 (3.11) Substitution of the values of D and E from Eq.(3.8) into Eq.(3.11) gives 𝑒 𝜔𝑛𝑡(2√1+𝜁 2) = 𝑄3 𝑄4 19 Figure 3.4. Convergence of VSS trajectories to the origin for ζ < 1 and 𝜇1 < λ < 0 ⇒𝑡= 1 2𝜔𝑛 √1 + 𝜁 2 ln [ 𝑄3 ] 𝑄4 (3.12) where 𝑄3 and 𝑄4 , which necessarily have the same sign, are 𝑄3 = [𝑥20 + 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 )] 𝑄4 = [𝑥20 + 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 )] This expression implies that the trajectories in 𝑍3 reaches L1 at finite time. The VSS is a hybrid system that switches between an asymptotically stable system and unstable system. However, from the above discussion it is clear that the origin of the VSS is asymptotically stable. The trajectories of the VSS converge to the origin through L2 or L3 – see Fig.3.4. 3.2.2. Performance – Response to Step Input To investigate the performance of the VSS, we will study the response of the VSS to a step input and compare its performance with that of the original system for the same step input. 20 For the original system, the response to a step input would be described by the following equation and initial conditions: 𝑚𝑥̈ + 𝑐 𝑥̇ + 𝑘 𝑥 = 𝑢 = 𝑘, 𝑥(0) = 0, 𝑥̇ (0) = 0 (3.13) We choose a step input of variable k such that the steady state value of x is unity. On change of variable 𝑦 = (𝑥 − 1) and 𝑦̇ = 𝑥̇ , we get the unforced system: 𝑦̈ = − 𝑐 𝑘 𝑦̇ − 𝑦, 𝑚 𝑚 𝑦(0) = −1, 𝑦̇ (0) = 0 (3.14) To obtain the same VSS as in Eqs.(2.2) and (2.6), we redefine F in Eq.(2.5) as follows: 0 𝐹 =𝑘+{ 2𝑘(𝑥 − 1) 𝑖𝑓 (𝑆 > 0 and 𝑥 ≥ 1) 𝑜𝑟 (𝑆 < 0 and 𝑥 ≤ 1) 𝑖𝑓 𝑆 (𝑥 − 1) < 0 (3.15) Substitution of the values of Eq.(55) into Eq.(1) and change of variables 𝑦 = (𝑥 − 1) and 𝑦̇ = 𝑥̇ , we get: 𝑦̈ = − where 𝛼={ 𝑐 𝑘 𝑦̇ − 𝛼 𝑦, 𝑚 𝑚 +1 −1 𝑖𝑓 𝑖𝑓 𝑦̇ (0) = 0 𝑦(0) = −1, (3.16) (𝑆 > 0 and 𝑦 ≥ 0 ) 𝑜𝑟 (𝑆 < 0 and 𝑦 ≤ 0 ) 𝑆𝑦<0 Note that Eqs.(3.14) and (3.16) are very similar and have the same initial conditions. Also, note that Eq.(3.16) is identical to Eq.(2.6); therefore, the analysis presented in the section 3.2.1 is applicable. The phase portrait of the VSS in Eq.(3.16) has been studied in section 3.2.1. From this study we know that no trajectories of the VSS will undergo a phase change of more than 2π, and therefore, the VSS will not exhibit oscillations for a step input. Consequently, the rise time and percentage overshoot of the system are not relevant. In the absence of oscillations, we investigate the performance of the VSS using the metric of settling time (the time required for the response curve to reach and stay within 2% of the final value) [1]. To this end, we first consider the time 21 needed by the VSS to reach the sliding surface L3, given by Eq.(3.5), which is now rewritten using variables 𝑦 and 𝑦̇ . If 𝑦0 and 𝑦̇ 0 donate the values of 𝑦(0) and 𝑦̇ (0) respectively, the time needed to reach L3 is given by : 𝑡= (𝜆𝑦0 − 𝑦̇ 0 )√1 − 𝜁 2 1 tan−1 [ ] 𝑦̇ + 𝜁𝜔 𝑦 𝜔𝑑 −𝜁𝑦̇ 0 − 𝜔𝑛 𝑦0 − 𝜆 ( 0 𝜔 𝑛 0 ) 𝑛 Substitution of the values of 𝑦0 = −1 and 𝑦̇ 0 = 0 , gives: 𝑡1 = 1 −𝜆√1 − 𝜁 2 tan−1 [ ] 𝜔𝑑 𝜔𝑛 + 𝜆 𝜁 (3.17) we substitute the values of 𝑦0 = −1, 𝑦̇ 0 = 0, and 𝑡1 described by Eq.(3.16) into Eqs.(3.1) and (3.2), with variables 𝑥1 and 𝑥2 replaced by 𝑦 and 𝑦̇ respectively. This gives the values of 𝑦 and 𝑦̇ when the VSS reaches the sliding surface L3: 𝑦𝑠 = 𝑒 −𝜁𝜔𝑛𝑡1 [− cos 𝜔𝑑 𝑡1 − 𝜁 √1 − 𝜁 2 𝑦̇𝑠 = −𝜁𝜔𝑛 𝑒 −𝜁𝜔𝑛𝑡1 [− cos 𝜔𝑑 𝑡1 − +𝑒 −𝜁𝜔𝑛𝑡1 [ 𝜔𝑑 sin 𝜔𝑑 𝑡1 − where sin 𝜔𝑑 𝑡1 ] 𝜁 √1 − 𝜁 2 𝜁 √1 − 𝜁 2 (3.18) sin 𝜔𝑑 𝑡1 ] 𝜔𝑑 cos 𝜔𝑑 𝑡1 ] (3.19) 𝜔𝑑 = 𝜔𝑛 √1 − 𝜁 2 To find the time needed for the trajectories on L3 to enter the region |𝑦| ≤ 0.02, we consider the dynamics on L3, namely: 𝑦(𝑡) = 𝑦𝑠 𝑒 𝜆𝑡 , (3.20) 𝜆<0 Substitution of 𝑦(𝑡) = −0.02 yields 𝑒 𝜆𝑡 = −0.02 1 −0.02 ⇒ 𝑡2 = ln ( ) 𝑦𝑠 𝜆 𝑦𝑠 (3.21) 22 The settling time of the VSS is given by: 1 −𝜆√1 − 𝜁 2 1 −0.02 tan−1 [ ] + ln ( ) 𝜔𝑑 𝜔𝑛 + 𝜆 𝜁 𝜆 𝑦𝑠 𝑡𝑠 = 𝑡1 + 𝑡2 = (3.22) where 𝑦𝑠 is defined by Eq.(3.18). For proper comparison of the performance, we nondimensionalize time using the variable 𝜏 = 𝜔𝑛 𝑡. Concurrently, we define 𝜆 = 𝜅 𝜇1 where 𝜅, 0 < 𝜅 < 1, is a constant. The nondimensional settling time is now given by the following equation: 𝜏𝑠 = 𝜏1 + 𝜏2 where 𝜏1 = 𝜏2 = 1 √1 − 𝜁 2 (3.23) tan−1 𝜅 [ 𝜁 + √𝜁 2 + 1 ]√1 − 𝜁 2 1 − 𝜁 𝜅 [ 𝜁 + √𝜁 2 + 1 ] −1 𝜅[ 𝜁 + √𝜁 2 + 1 ] ln ( −0.02 ) 𝑦𝑠 (3.24) (3.25) and where 𝑦𝑠 = 𝑒 −𝜁𝜏1 [− cos 𝜏1 √1 − 𝜁 2 − 𝜁 √1 − 𝜁2 sin 𝜏1 √1 − 𝜁 2 ] (3.26) Figure 3.5 shows a comparison of the settling time of the VSS for different value of 𝜅 (𝜅 = 0.1, 𝜅 = 0.3, 𝜅 = 0.5, 𝜅 = 0.9) and the original system. The settling time for the original system shows several discontinuities in the settling time; this is due to the sensitivity of the settling time to changes in 𝜁 [Ogata, 2009]. For 𝜅 = 0.1, the VSS has better performance than that of the original system only when 𝜁 < 0.11. For 𝜅 = 0.3, the performance is better for a larger range of 𝜁, namely 𝜁 < 0.39. As we increase 𝜅, the performance is better for a larger range of 𝜁: for 𝜅 = 0.9 the performance is better for 𝜁 < 0.79. The settling time of the original system can be reduced by increasing the value of 𝜁 but for most 𝜁 values, a VSS can be found (𝜅 can be chosen) that has better performance in terms of the settling time. For 23 Figure 3.5. Comparison of the settling times of the VSS and the original system. example, for 𝜁 = 0.2, the settling time for the original system is 𝜏𝑠 = 19.55, whereas it is equal to 32.13, 10.87, 6.72, and 4.0 for VSS with 𝜅 = 0.1, 0.3, 0.5, and 0.9 respectively. It is clear that the VSS improves the settling time for 𝜁 = 0.2 by 44.4% when 𝜅 = 0.3, by 65,6% when 𝜅 = 0.5, and by 79.1% when 𝜅 = 0.9. The best performance is achieved by the VSS when 𝜅 → 1 or 𝜆 → 𝜇1 . However, the maximum value of κ can be chosen is less than unity (0.95, for example) because of the finite time needed to switching of α from +1 to −1. If 𝜅 ≈ 1 the area of Z2 in Fig.3.3 is almost zero and there will be no switching in this zone and the switching will be in Z3. Figure 3.5 also indicates that there is no difference in the settling time of the original system and the VSS with 𝜅 = 0.9 for 𝜁 ≥ 0.9. This is because the trajectories of the VSS enters the region |𝑦| ≤ 0.02 before it reaches the sliding surface L3. This implies that the 24 VSS has no advantage over the original system when the system is under-damped and has values of 𝜁 close to unity. It is important to note that 𝜇1 = 𝜇1 (𝜁) and 𝜇1 decreases as ζ increases. If λ = κ 𝜇1 but λ is constant, we have a situation where κ decreases as ζ increases. Since the performance is better (settling time is smaller) for larger values of κ, a constant value of λ results in poorer performance for large values of ζ. This is shown with the help of Fig.3.6. Note ζ and µ1 are plotted along x-axis in the figure and it can be seen that the settling time 𝜏𝑠 for VSS increases as ζ increases. Figure 3.6. Comparison of the settling times of the VSS and the original system. 25 In this section, we investigated the performance of the VSS with switched stiffness for a step input for different values of sliding surface slope 𝜆. The VSS has smaller settling time than that of the original mass-spring-damper system for a range of 𝜁 values. For higher values of 𝜆 the settling time of the VSS decreases and the best performance is achieved when 𝜆 → 𝜇1 . 3.2.3. Performance – Speed of Convergence In last section, we investigated the performance of the VSS for a step input. This problem was recast as an initial value problem with the initial conditions at (−1, 0) in the phase plane. Since (−1, 0) lies in Z1, we now compare the performance of the VSS with switched stiffness and the original system for arbitrary initial conditions in Z2 and Z3. For an initial conditions in Z2 (see Fig.3.3), the time needed by the VSS to reaches the sliding surface L3 (𝑥2 = 𝜆 𝑥1 ) has been studied in section 3.2.1, and was described by Eq.(3.10). The values of 𝑥1 and 𝑥2 when the VSS reaches the sliding surface L3 has been studied in section 3.2.1, and was described by Eq.(3.6), and (3.7). The time needed for the trajectories on L3 to enter the region |𝑥1 | ≤ 0.02 has been studied in section 3.2.1, and was described by Eq.(3.21). Using these relations, the settling time of the VSS can be shown to be: 𝜏𝑠 = 𝜏1 + 𝜏2 where 𝜏1 = 𝜔𝑛 𝑡1 = ln [ (3.27) 1 2√1 − 𝜁 2 [−𝑥20 − 𝑥10 (𝜁 − √1 + 𝜁 2 )][(𝜁 + √1 + 𝜁 2 ) − 𝜅(𝜁 + √1 + 𝜁 2 ) ] [𝑥20 + 𝑥10 (𝜁 + √1 + 𝜁 2 )][(−𝜁 + √1 + 𝜁 2 ) + 𝜅(𝜁 + √1 + 𝜁 2 ) ] 𝜏2 = 𝜔𝑛 𝑡2 = −1 −0.02 ln ( ) 𝑥1𝑠 [ 𝜁 + √𝜁 2 + 1 ] 26 ] (3.28) (3.29) where 𝑥 = 𝐷 𝑒 𝜏1 (−𝜁−√1+𝜁 2) + 𝐸 𝑒 𝜏1 (−𝜁+√1+𝜁 2) 1𝑠 and 𝐷 = 𝑥10 − 𝑥20 + 𝑥10 (𝜁 + √1 + 𝜁 2 ) 2√1 + 𝜁 2 , (3.30) 𝐸= 𝑥20 + 𝑥10 (𝜁 + √1 + 𝜁 2 ) 2√1 + 𝜁 2 where 𝑥10 and 𝑥20 are the initials conditions in Z2. Figure 3.7 shows a comparison of the settling time of the VSS for different value of 𝜅 (𝜅 = 0.1, 𝜅 = 0.3, 𝜅 = 0.5, 𝜅 = 0.9) and the original system for initial conditions in Z2. The initial conditions were arbitrarily chosen as 𝑥10 = −1, and 𝑥20 = −0.95 𝜇1 . For 𝜅 = 0.1, the VSS has better performance than that of the original system only when 𝜁 < 0.18. For 𝜅 = 0.3, the performance is better for a larger range of 𝜁, namely 𝜁 < 0.68. As we increase 𝜅, the performance is better for a larger range of ζ: for 𝜅 = 0.5 and 0.9, the performance is better for all value of 𝜁. The settling time of the original Figure 3.7. Comparison of the settling times of the VSS and the original system 27 system can be reduced by increasing the value of 𝜁 but for most ζ values, a VSS can be found (𝜅 can be chosen) that has better performance in terms of the settling time. For example, for 𝜁 = 0.2, the settling time for the original system is 𝜏𝑠 = 19.29, whereas it is equal to 22.13, 8.48, 5.59, and 3.55 for VSS with 𝜅 = 0.1, 0.3, 0.5, and 0.9 respectively. It is clear that the VSS improves the settling time for 𝜁 = 0.2 by 56% when 𝜅 = 0.3, by 70.66% when 𝜅 = 0.5, and by 81.58% when 𝜅 = 0.9. The best performance is achieved by the VSS when 𝜅 → 1 or 𝜆 → 𝜇1 . For an initial conditions in Z3 (see Fig.3.3), the time needed by the VSS to reaches L1 has been studied in section 3.2.1, and was described by Eq.(3.12). The values of 𝑥1 and 𝑥2 when the VSS reach the sliding surface L1 (𝑥1 = 0) has been studied in section 3.2.1, and was described by Eq.(3.6), and (3.7). The time needed by the VSS to reaches the sliding surface L3 (𝑥2 = 𝜆 𝑥1 ) has been studied in section 3.2.1, and was described by Eq.(3.5). The values of 𝑥1 and 𝑥2 when the VSS reach the sliding surface L3 has been studied in section 3.2.1, and was described by Eqs.(3.1) and (3.2). The time needed for the trajectories on L3 to enter the region |𝑥1 | ≤ 0.02 has been studied in section 3.2.1, and was described by Eq.(3.21). Using these relations, the settling time of the VSS can be shown to be: 𝜏𝑠 = 𝜏1 + 𝜏2 + 𝜏3 where 𝜏1 = 𝜔𝑛 𝑡1 = 𝜏2 = 𝜔𝑛 𝑡2 = 𝜏3 = 𝜔𝑛 𝑡3 = (3.31) 1 2√1 + 𝜁 2 1 √1 − 𝜁 2 ln [ [𝑥20 + 𝑥10 (𝜁 − √1 + 𝜁 2 )] ] (3.32) [𝑥20 + 𝑥10 (𝜁 + √1 + 𝜁 2 )] tan−1 [ √1 − 𝜁 2 𝜁 − 𝜅(𝜁 + √1 + 𝜁 2 ) −1 0.02 ln ( ) 𝑥1𝑠 𝜅[ 𝜁 + √𝜁 2 + 1 ] 28 ] (3.33) (3.34) and 𝑥1𝑠 = 𝑒 −𝜁𝜔𝑛𝜏2 [ where 𝑥2𝑙1 √1 − 𝜁 2 sin √1 − 𝜁 2 𝜏2 ] (3.35) 2) (3.36) 𝑥2𝑙1 = (−𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 𝜏1 (−𝜁−√1+𝜁 + (−𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜏1 (−𝜁+√1+𝜁 where 𝐷 = 𝑥10 − 𝑥20 + 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 , 2) 𝐸= 𝑥20 + 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 where 𝑥10 and 𝑥20 are the initials conditions in Z3. Figure 3.8 shows a comparison of the settling time of the VSS for different value of 𝜅 (𝜅 = 0.1, 𝜅 = 0.3, 𝜅 = 0.5, 𝜅 = 0.9) and the original system for initial conditions in Z3. The initial conditions were arbitrarily chosen as 𝑥10 = −1, and 𝑥20 = −1.1 𝜇1. For 𝜅 = 0.1, the VSS has better performance than that of the original Figure 3.8. Comparison between the settling time of the VSS and the original system 29 system only when 𝜁 < 0.16. For 𝜅 = 0.3, the performance is better for a larger range of 𝜁, namely 𝜁 < 0.48. For 𝜅 = 0.5, the performance is better for a larger range of 𝜁, namely 𝜁 < 0.71. As we increase 𝜅, the performance is better for a larger range of 𝜁: for 𝜅 = 0.9, the performance is better for all value of 𝜁. The settling time of the original system can be reduced by increasing the value of 𝜁 but for most 𝜁 values, a VSS can be found (𝜅 can be chosen) that has better performance in terms of the settling time. For example, for 𝜁 = 0.2, the settling time for the original system is 𝜏𝑠 = 21.67, whereas it is equal to 24.87, 10.32, 7.51, and 5.78 for VSS with 𝜅 = 0.1, 0.3, 0.5, and 0.9 respectively. It is clear that the VSS improves the settling time for 𝜁 = 0.2 by 52.34% when 𝜅 = 0.3, by 65.31% when 𝜅 = 0.5, and by 73.33% when 𝜅 = 0.9. The best performance is achieved by the VSS when 𝜅 → 1 or 𝜆 → 𝜇1 . In this section, we investigated the performance of the VSS with switched stiffness for arbitrary initial conditions in Z2 and Z3, and for different values of sliding surface slope 𝜆. The VSS has a smaller settling time than that of the original mass-spring-damper system for a range of 𝜁 values. For higher value of 𝜆 the settling time of the VSS decreases and with the best performance is achieved by the VSS when 𝜆 → 𝜇1 . 3.3. Case B: 𝛌 ≤ 𝝁𝟏 < 𝟎 3.3.1. Stability The VSS is a hybrid system that switches between an asymptotically stable system, the phase portrait of which is shown in Fig.3.9, and an unstable system whose phase portrait is shown in Fig.3.10. The union of these phase portraits is shown in Fig.3.11. To investigate the stability of the equilibrium of the VSS, the phase plane is divided into two zones, namely: 30 Figure 3.9. Phase portrait of the Variable Structure System (VSS) for 𝛼 = +1 Figure 3.10. Phase portrait of the Variable Structure System (VSS) for 𝛼 = −1 31 Figure 3.11. Phase portrait of the Variable Structure System (VSS) for λ > μ 1 𝑍1 = {𝑥 ∈ 𝑅 2 |𝑆𝑥1 > 0} 𝑍2 = {𝑥 ∈ 𝑅 2|𝑥1 (𝑥2 − 𝜆 𝑥1 ) < 0} The two zones are separated by the following two lines: L1: 𝑥1 = 0 L2: 𝑥2 = 𝜆 𝑥1 We investigate the system trajectories in these zones and on these lines next. To study the behavior of the trajectories on L1, we investigate the direction of the vector field on L1. For 𝑥2 > 0 (and 𝑥1 = 0), we have the following equations and vector field: 𝑥̇ 1 = 𝑥2 > 0 𝑥̇ 2 = − 𝑐 𝑥 <0 𝑚 2 32 Therefore, the trajectories enter 𝑍1 . For 𝑥2 < 0 (and 𝑥1 = 0), we have the following equations and vector field: 𝑥̇ 1 = 𝑥2 < 0 𝑥̇ 2 = − 𝑐 𝑥 >0 𝑚 2 Therefore, all trajectories on L1 will enter 𝑍1 . The boundaries of 𝑍1 are L1 and L2. We have already shown that trajectories on L1 enter 𝑍1 . We now show all trajectories in 𝑍1 reaches L2 in finite time. The time when the trajectory in 𝑍1 reaches L2 has been studied in section 3.2.1, and was described by Eq.(3.5). For convenience, it is provided below: 𝑡= (𝜆𝑥10 − 𝑥20 )√1 − 𝜁 2 ] 𝑥20 + 𝜁𝜔𝑛 𝑥10 − 𝜔 𝑥 − 𝜆 ( ) 20 𝑛 10 𝜔𝑛 1 tan−1 [ 𝜔𝑑 −𝜁𝑥 This expression implies that all trajectories in 𝑍1 reaches L2 at finite time. The boundaries of 𝑍2 are L2 and L1. We have already shown that the trajectories on L1 enter 𝑍1 . We now show that all trajectories in 𝑍2 reaches L1 in finite time. The time when a trajectory in 𝑍2 reaches L1 has been studied in section 3.2.1, and was described by Eq.(3.12). For convenience, it is provided below: 𝑡= 1 2𝜔𝑛 √1 + 𝜁 2 ln [ [𝑥20 + 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 )] [𝑥20 + 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 )] ] This expression implies that all trajectories in 𝑍2 reaches L1 at finite time. From the above discussion it is clear that the trajectories of the VSS move alternately between zones 𝑍1 and 𝑍2 by crossing lines L2 and L1. We now show that all these trajectories 33 converge to the origin. We consider the Lyapunov Function candidate given below, which is equal to the total energy of the mass-spring-damper: 𝑉= 1 1 𝑘 𝑥12 + 𝑚 𝑥22 2 2 (3.37) The derivative of V with respect to time gives: 𝑉̇ = 𝑘 𝑥1 𝑥̇ 1 + 𝑚 𝑥2 𝑥̇ 2 (3.38) In 𝑍1 , where 𝛼 = 1, we have 𝑉̇ = 𝑘 𝑥1 𝑥2 + 𝑚 𝑥2 𝑥̇ 2 = 𝑘 𝑥1 𝑥2 − 𝑐 𝑥22 − 𝑘 𝑥1 𝑥2 = −𝑐 𝑥22 ≤ 0 (3.39) In 𝑍2 , where 𝛼 = −1, we have 𝑉̇ = 𝑘 𝑥1 𝑥2 + 𝑚 𝑥2 𝑥̇ 2 = 𝑘 𝑥1 𝑥2 − 𝑐 𝑥22 + 𝑘 𝑥1 𝑥2 = 2 𝑘 𝑥1 𝑥2 − 𝑐 𝑥22 In 𝑍2 , 𝑥2 < 𝜆 𝑥1 when 𝑥1 > 0, and 𝑥2 > 𝜆 𝑥1 when 𝑥1 < 0. Therefore where 𝑉̇ < 2 𝑘 𝜆 𝑥12 − 𝑐 𝑥22 where 𝜆 < 0 𝑉̇ < 0 for all (𝑥1 , 𝑥2 ) ≠ (0, 0) (3.40) From the above discussion, we have 𝑉̇ ≤ 0 in the entire domain. It is negative definite in 𝑍2 but negative semidefinite in 𝑍1 . Using LaSalle’s theorem [Khalil, 2009], we can show that in 𝑍1 𝑉̇ ≡ 0 ⇒ 𝑥2 ≡ 0 ⇒ 𝑥̇ 2 ≡ 0 ⇒ 𝑥1 ≡ 0 Furthermore, 𝑉 → ∞ as 𝑥 → ∞. Therefore, the origin of the VSS is globally asymptotically stable. 3.3.2. Performance – Response to Step Input 34 To investigate the performance of the VSS, we will study the response of the VSS to a step input and compare its performance with that of the original system for the same step input. For the original system, the response to a step input has been studied in section 3.2.1, and was described in Eq.(3.14). For convenience, it is provided below: 𝑦̈ = − 𝑐 𝑘 𝑦̇ − 𝑦, 𝑚 𝑚 𝑦(0) = −1, 𝑦̇ (0) = 0 (3.41) The response of the VSS to the same step input was described in Eq.(3.16). For convenience, it is provided below: 𝑦̈ = − where 𝛼={ 𝑐 𝑘 𝑦̇ − 𝛼 𝑦, 𝑚 𝑚 +1 −1 𝑖𝑓 𝑖𝑓 𝑦̇ (0) = 0 𝑦(0) = −1, (3.42) (𝑆 > 0 and 𝑦 ≥ 0 ) 𝑜𝑟 (𝑆 < 0 and 𝑦 ≤ 0 ) 𝑆𝑦<0 Note that Eqs.(3.41) and (3.42) are very similar and have the same initial conditions. Also, note that Eq.(3.42) is identical to Eq.(2.6); therefore, all the analysis presented in section 3.3.1 will be applicable. The phase portrait of the VSS in Eq.(3.42) has been studied in section 3.3.1. From this study we know that the trajectories of the system will spiral to the origin. Unlike the previous case, it is not possible to claim that the phase change of the trajectories will be less than 2π. In other words, a step input will result in oscillation, similar to the original system. Therefore, the performance of the original system and the VSS can be compared by on the basis of: (a) the rise time 𝑡𝑟 (the time required for the response to reach the final value for the first time) [Ogata, 2009], (b) the maximum percentage of overshoot 𝑀𝑝 (the maximum peak value of the response of the VSS measured from the final value) [Ogata, 2009], and 35 (c) the settling time. The rise time and the maximum percentage of overshoot can be calculated analytically but the same is not true for the settling time. To calculate the rise time and the maximum percentage of overshoot, we define three points A, B, and C on the phase portrait of the VSS, shown in Fig 3.12. These points are also shown in Fig.3.13, which is a plot of the time response of the VSS to the step input. Consider the time needed by the VSS to reach point A; this has been studied in section 3.2.1, and was described by Eq.(3.5). Using variables 𝑦 and 𝑦̇ we can write it as follows: (𝜆𝑦0 − 𝑦̇ 0 )√1 − 𝜁 2 1 1 −𝜆√1 − 𝜁 2 −1 −1 𝑡𝐴 = tan [ ]= tan [ ] (3.43) 𝑦̇ 0 + 𝜁𝜔𝑛 𝑦0 𝜔𝑑 𝜔 𝜔 + 𝜆 𝜁 𝑑 𝑛 −𝜁𝑦̇ 0 − 𝜔𝑛 𝑦0 − 𝜆 ( ) 𝜔 𝑛 where 𝑦0 = −1 and 𝑦̇ 0 = 0 are the initial conditions. The values of 𝑦 and 𝑦̇ at A has been Figure 3.12. Step response in the phase plane of the Variable Structure System (VSS) for λ < μ 1 36 studied in section 3.2.1, and was described by Eq.(3.1), and (3.2). For convenience, it is provided below: 𝑦𝐴 = 𝑒 −𝜁𝜔𝑛𝑡𝐴 [ 𝐴 cos 𝜔𝑑 𝑡𝐴 + 𝐵 sin 𝜔𝑑 𝑡𝐴 ] (3.44) 𝑦̇𝐴 = −𝜁𝜔𝑛 𝑒 −𝜁𝜔𝑛𝑡𝐴 [ 𝐴 cos 𝜔𝑑 𝑡𝐴 + 𝐵 sin 𝜔𝑑 𝑡𝐴 ] + 𝑒 −𝜁𝜔𝑛𝑡𝐴 [ −𝐴 𝜔𝑑 sin 𝜔𝑑 𝑡𝐴 + 𝐵 𝜔𝑑 cos 𝜔𝑑 𝑡𝐴 ] where 𝜔𝑑 = 𝜔𝑛 √1 − 𝜁 2 , 𝐴 = −1 , 𝐵 = − (3.45) 𝜁 √1−𝜁 2 The time needed for the VSS to move from point A to point B has been studied in section 3.2.1, and was described by Eq.(3.12), For convenience, it is provided below: 𝑡𝐴𝐵 = 1 2𝜔𝑛 √1 + 𝜁 2 ln [ [𝑦̇𝐴 + 𝜔𝑛 𝑦𝐴 (𝜁 − √1 + 𝜁 2 )] [𝑦̇𝐴 + 𝜔𝑛 𝑦𝐴 (𝜁 + √1 + 𝜁 2 )] ] (3.46) where 𝑦𝐴 , and 𝑦̇𝐴 are the initial conditions and given by Eqs.(3.44) and (3.45). The values of 𝑦 Figure 3.13. Step response of the Variable Structure System (VSS) for λ < μ 1 37 and 𝑦̇ at B has been studied in section 3.2.1, and was described by Eq.(3.4) and (3.7). For convenience, it is provided below: 𝑦𝐵 = 0 (3.47) 𝑦̇ 𝐵 = 𝜔𝑛 (−𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 𝜔𝑛𝑡𝐴𝐵 (−𝜁−√1+𝜁 2) + 𝜔𝑛 (−𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜔𝑛𝑡𝐴𝐵 (−𝜁+√1+𝜁 where 𝐷 = 𝑦𝐴 − 𝑦̇𝐴 + 𝜔𝑛 𝑦𝐴 (𝜁 + √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 , 𝐸= 2) (3.48) 𝑦̇𝐴 + 𝜔𝑛 𝑦𝐴 (𝜁 + √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 To find the time needed for the VSS to move from point B to point C, we substitute Eq.(3.2) into the equation of y-axis where point C lie on this axis, namely: 𝑦̇ = 0 This gives: −𝜁𝜔𝑛 𝑒 −𝜁𝜔𝑛𝑡 [ 𝐴 cos 𝜔𝑑 𝑡 + 𝐵 sin 𝜔𝑑 𝑡] + 𝑒 −𝜁𝜔𝑛𝑡 [ −𝐴 𝜔𝑑 sin 𝜔𝑑 𝑡 + 𝐵 𝜔𝑑 cos 𝜔𝑑 𝑡] = 0 ⟹ tan 𝜔𝑑 𝑡 = −𝜁𝜔𝑛 𝐴 + 𝐵𝜔𝑑 𝜁𝜔𝑛 𝐵 + 𝜔𝑑 𝐴 (3.49) Substitution of the values of A and B from Eq.(3.3) into Eq.(3.49) gives: ⟹ tan 𝜔𝑑 𝑡 = ⟹ 𝑡𝐵𝐶 = √1 − 𝜁 2 𝜁 1 √1 − 𝜁 2 tan−1 [ ] 𝜔𝑑 𝜁 (3.50) Substitution of the value of 𝑡𝐵𝐶 from Eq.(3.50) into Eqs.(3.1) and (3.2) gives the values of 𝑦 and 𝑦̇ at C, namely: 𝑦̇ 𝐵 𝑦𝐶 = 𝑒 −𝜁𝜔𝑛𝑡𝐵𝐶 [ sin 𝜔𝑛 √1 − 𝜁 2 𝑡𝐵𝐶 ] 𝜔𝑛 √1 − 𝜁 2 38 (3.51) 𝑦̇ 𝐶 = 0 (3.52) The rise time of the VSS can now be calculated as follows: 𝑡𝑟 = 𝑡𝐴 + 𝑡𝐴𝐵 = 1 2𝜔𝑛 √1 + 𝜁 2 ln [ [𝑦̇𝐴 + 𝜔𝑛 𝑦𝐴 (𝜁 − √1 + 𝜁 2 )] [𝑦̇𝐴 + 𝜔𝑛 𝑦𝐴 (𝜁 + √1 + 𝜁 2 )] ]+ 1 −𝜆√1 − 𝜁 2 tan−1 [ ] (3.53) 𝜔𝑑 𝜔𝑛 + 𝜆 𝜁 where 𝑦𝐴 , and 𝑦̇𝐴 are defined by Eqs.(3.44) and (3.45) respectively. The maximum percent overshoot of the VSS depend on 𝑡𝐵𝐶 and is given by: 𝑦̇ 𝐵 𝑀𝑝 = 𝑒 −𝜁𝜔𝑛𝑡𝐵𝐶 [ sin 𝜔𝑛 √1 − 𝜁 2 𝑡𝐵𝐶 ] × 100 𝜔𝑛 √1 − 𝜁 2 = 𝑒 ( √1−𝜁 2 −𝜁 tan−1 [ ]) 𝑦̇ 𝐵 𝜁 √1−𝜁 2 [ 𝜔𝑛 ] × 100 (3.54) where 𝑦̇ 𝐵 is defined by Eq.(3.48). For proper comparison of the performance, we nondimensionalize the time using the variable 𝜏 = 𝜔𝑛 𝑡. Concurrently, we define 𝜆 = 𝜌 𝜇1 where 𝜌 ≥ 1 is a constant. The nondimensional rise time and the maximum percentage overshoot are given by the following equations respectively: 𝜏𝑟 = 𝜔𝑛 𝑡𝑟 = 1 √1 − 𝜁 2 + 𝑀𝑝 = tan−1 [ 1 2√1 + 𝜁 2 ln [ √1−𝜁 2 −𝜁 tan−1 𝜁 2 𝑒 √1−𝜁 𝑦̇ 𝐵 𝜌 [ 𝜁 + √𝜁 2 + 1 ]√1 − 𝜁 2 1 − 𝜁 𝜌 [ 𝜁 + √𝜁 2 + 1 ] [𝑦̇𝐴 + 𝑦𝐴 (𝜁 − √1 + 𝜁 2 )] [𝑦̇𝐴 + 𝑦𝐴 (𝜁 + √1 + 𝜁 2 )] ] ] (3.55) (3.56) × 100 where 𝑦 = 𝑒 −𝜁𝜏𝐴 [ 𝐴 cos √1 − 𝜁 2 𝜏 + 𝐵 sin √1 − 𝜁 2 𝜏 ] 𝐴 𝐴 𝐴 𝑦̇𝐴 = −𝜁𝑒 −𝜁𝜏𝐴 [ 𝐴 cos √1 − 𝜁 2 𝜏𝐴 + 𝐵 sin √1 − 𝜁 2 𝑑 𝜏𝐴 ] +√1 − 𝜁 2 𝑒 −𝜁𝜏𝐴 [ −𝐴 sin √1 − 𝜁 2 𝜏𝐴 + 𝐵 cos √1 − 𝜁 2 𝜏𝐴 ] 39 (3.57) (3.58) 𝑦̇ 𝐵 = (−𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 𝜏𝐴𝐵 (−𝜁−√1+𝜁 + (−𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜏𝐴𝐵 (−𝜁+√1+𝜁 𝜏𝐴 = 1 √1 − 𝜁 2 𝜏𝐴𝐵 = tan−1 [ 1 2√1 + 𝜁 2 𝐴 = −1 , 𝐷 = 𝑦𝐴 − ln [ 2) (3.59) 2) 𝜌 [ 𝜁 + √𝜁 2 + 1 ]√1 − 𝜁 2 1 − 𝜁 𝜌 [ 𝜁 + √𝜁 2 + 1 ] [𝑦̇𝐴 + 𝑦𝐴 (𝜁 − √1 + 𝜁 2 )] [𝑦̇𝐴 + 𝑦𝐴 (𝜁 + √1 + 𝜁 2 )] 𝐵=− ] (3.60) ] (3.61) 𝜁 √1 − 𝜁 2 𝑦̇𝐴 + 𝑦𝐴 (𝜁 + √1 + 𝜁 2 ) 2√1 + 𝜁 2 , 𝐸= 𝑦̇𝐴 + 𝑦𝐴 (𝜁 + √1 + 𝜁 2 ) 2√1 + 𝜁 2 The settling time for the VSS is the time needed for the VSS trajectory to enter the region |𝑦| ≤ 0.02. This cannot be calculate analytically because the VSS may enter the region |𝑦| ≤ 0.02 in 𝑍1 or in 𝑍2 . Therefore, we calculate the settling time numerically. Figure 3.14 shows a comparison of the rise times of the VSS for different values of 𝜌 (𝜌 = 1.1, 𝜌 = 1.5, 𝜌 = 2, 𝜌 = 5) and the original system. It is clear that the original system has smaller rise time than the VSS. For example, for 𝜁 = 0.2, the rise time for the original system is 𝜏𝑟 = 1.2, whereas it is equal to 2.5, 1.97, 1.87, and 1.81 for the VSS with 𝜌 = 1.1, 1.5, 2, and 5 respectively. It is clear that the rise time for the VSS increases for 𝜁 = 0.2 by 106.9% when 𝜌 = 1.1, by 63.9% when 𝜌 = 1.5, by 55% when 𝜌 = 2, and by 50.2% when 𝜌 = 5. As 𝜌 → ∞ or 𝜆 → ∞ the rise time of the VSS tend to be equal to that of the original system. 40 Figure 3.14. Comparison of the rise times of the VSS and original system Figure 3.15 shows a comparison of the maximum percentage of overshoots of the VSS for different values of (𝜌 = 1.1, 𝜌 = 1.5, 𝜌 = 2, 𝜌 = 5) and the original system. It is clear that the VSS has lower percentage of overshoot than the original system. For example, for 𝜁 = 0.2, the percentage of overshoot for the original system is 𝑀𝑝 = 52.6%, whereas it is equal to 15.67%, 34.04%, 42.43%, and 51.13% for VSS with 𝜌 = 1.1, 1.5, 2, and 5 respectively. It is clear that the VSS reduces the percentage of overshoot for 𝜁 = 0.2 by 70.23% when 𝜌 = 1.1, by 35.34% when 𝜌 = 1.5, by 19.4% when 𝜌 = 2, and by 2.88% when 𝜌 = 5. The best percentage of overshoot is achieved by the VSS when 𝜌 → 1 or 𝜆 → 𝜇1 . Figure 3.16 shows a comparison of the settling time of the VSS for different value of 𝜌 (𝜌 = 1.1, 𝜌 = 1.5, 𝜌 = 2, 𝜌 = 5) and the original system. It is clear that the VSS has better settling 41 Figure 3.15. Comparison of the percentage of overshoots of the VSS and original system time than the original system. For example, for 𝜁 = 0.2, the settling time for the original system is 𝜏𝑠 = 19.55, whereas it is equal to 8.43, 11.24, 14.03 , and 17.08 for VSS with 𝜌=1.1, 1.5, 2, and 5 respectively. It is clear that the VSS reduces the settling time for 𝜁 = 0.2 by 56.97% when 𝜌 = 1.1, by 42.61% when 𝜌 = 1.5, by 28.36% when 𝜌 = 2, and by 12.83% when 𝜌 = 5. The best performance is achieved by the VSS when 𝜌 → 1 or 𝜆 → 𝜇1 . In this section, we investigated the performance of the VSS with switched stiffness for a step input for different values of sliding surface slope 𝜆. The original system has a smaller rise time than the VSS. on the other hand, the VSS has a smaller maximum percentage of overshoot and a smaller settling time than that of the original system. Therefore, the VSS improves the performance of the mass-spring-damper system, the best performance achieved when 𝜆 → 𝜇1 . 42 Figure 3.16. Comparison of the settling time of the VSS and original system 3.3.3. Performance – Speed of Convergence In last section, we investigated the performance of the VSS for a step input. This problem was recast as an initial value problem with the initial conditions at (−1, 0) in the phase plane. Since (−1, 0) lies in Z1, we now compare the performance of the VSS with switched stiffness and the original system for arbitrary initial conditions in Z2. For initial conditions in Z2 (see Fig.3.17), the performance of the original system and the VSS can be compared by computing: (a) the rise time 𝑡𝑟 , (b) the maximum percentage of overshoot 𝑀𝑝 , (c) the settling time. 43 The rise time and the maximum percentage of overshoot can be calculated analytically but the same is not true for the settling time. To calculate the rise time and maximum percentage of overshoot, we define two points A and B on the phase portrait of the VSS, shown in Fig.3.17. These points are also shown in Fig.3.18, which is a plot of the time response of the VSS to the step input. Consider the time needed by the VSS to reach point B; this B has been studied in section 3.2.1, and was described by Eq.(3.12). The values of 𝑥1 and 𝑥2 at A has been studied in section 3.2.1, and was described by Eq.(3.6) and (3.7). To find the time needed for the VSS to move from point A to point B, we substitute Eq.(3.2) into the equation of y-axis where point B lies on this axis, namely: 𝑥2 = 0 This gives −𝜁𝜔𝑛 𝑒 −𝜁𝜔𝑛𝑡 [ 𝐴 cos 𝜔𝑑 𝑡 + 𝐵 sin 𝜔𝑑 𝑡] + 𝑒 −𝜁𝜔𝑛𝑡 [ −𝐴 𝜔𝑑 sin 𝜔𝑑 𝑡 + 𝐵 𝜔𝑑 cos 𝜔𝑑 𝑡] = 0 ⟹ tan 𝜔𝑑 𝑡 = −𝜁𝜔𝑛 𝐴 + 𝐵𝜔𝑑 𝜁𝜔𝑛 𝐵 + 𝜔𝑑 𝐴 (3.62) Substitution of the values of A and B from Eq.(3.3) into Eq.(3.62) gives: √1 − 𝜁 2 ⟹ tan 𝜔𝑑 𝑡 = 𝜁 ⟹ 𝑡𝐴𝐵 1 √1 − 𝜁 2 −1 = tan [ ] 𝜔𝑑 𝜁 (3.63) Substitution of the value of 𝑡𝐴𝐵 into Eqs.(3.1), and (3.2) gives the values of 𝑥1 and 𝑥2 at B, namely: 𝑥1𝐵 = 𝑒 −𝜁𝜔𝑛𝑡𝐴𝐵 [ 𝑥2𝐴 𝜔𝑛 √1 − 𝜁 2 sin 𝜔𝑛 √1 − 𝜁 2 𝑡𝐴𝐵 ] 44 (3.64) Figure 3.17. Step response in the Phase plane for the Variable Structure System (VSS) for λ < μ 1 Figure 3.18. Step response for the Variable Structure System (VSS) for λ < μ 1 45 𝑥2𝐵 = 0 (3.65) The rise time of the VSS can now be calculated as follows: 𝑡𝑟 = 𝑡𝐴 = 1 2𝜔𝑛 √1 + 𝜁 2 ln [ [𝑥20 + 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 )] ] (3.66) [𝑥20 + 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 )] where 𝑥10 , and 𝑥20 are the initial conditions. The maximum percent overshoot of the VSS depend on 𝑡𝐵𝐶 and is given by: 𝑥2𝐴 𝑀𝑝 = 𝑒 −𝜁𝜔𝑛𝑡𝐴𝐵 [ sin 𝜔𝑛 √1 − 𝜁 2 𝑡𝐴𝐵 ] × 100 𝜔𝑛 √1 − 𝜁 2 = 𝑒 ( √1−𝜁 2 −𝜁 tan−1 [ ]) 𝑥2𝐴 𝜁 √1−𝜁 2 [ 𝜔𝑛 ] × 100 (3.67) where 𝑥2𝐴 is defined by Eq.(3.7). For proper comparison of the performance, we nondimensionalize the time using the variable 𝜏 = 𝜔𝑛 𝑡. Concurrently, we define 𝜆 = 𝜌 𝜇1 where 𝜌 ≥ 1 is a constant. The nondimensional rise time and the maximum percent overshoot are given by the following equations respectively: 𝜏𝑟 = 𝜔𝑛 𝑡𝑟 = 𝑀𝑝 = 1 2√1 + 𝜁 2 ln [ [𝑥20 + 𝑥10 (𝜁 − √1 + 𝜁 2 )] ] (3.68) [𝑥20 + 𝑥10 (𝜁 + √1 + 𝜁 2 )] √1−𝜁 2 −𝜁 tan−1 𝜁 2 𝑒 √1−𝜁 𝑥2𝐴 (3.69) × 100 where 𝑥2𝐴 = (−𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 𝜏𝐴 = 1 2√1 + 𝜁 2 𝐷 = 𝑥10 − ln [ 𝜏𝐴 (−𝜁−√1+𝜁 2 ) + (−𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜏𝐴(−𝜁+√1+𝜁 [𝑥20 + 𝑥10 (𝜁 − √1 + 𝜁 2 )] [𝑥20 + 𝑥10 (𝜁 + √1 + 𝜁 2 )] 𝑥20 + 𝑥10 (𝜁 + √1 + 𝜁 2 ) 2√1 + 𝜁 2 , 46 ] 𝐸= 2) (3.70) (3.71) 𝑥20 + 𝑥10 (𝜁 + √1 + 𝜁 2 ) 2√1 + 𝜁 2 where 𝑥10 and 𝑥20 are the initials conditions in Z2. The settling time of the VSS is the time needed for the VSS to enter the region |𝑥1 | ≤ 0.02. There is no way to calculate the settling time analytically, because the VSS may enter the region |𝑥1 | ≤ 0.02 in 𝑍1 or 𝑍2 . Therefore we calculate the settling time numerically. Figure 3.19 shows a comparison between the rise time of the VSS with 𝜌 = 1.1 and the original system for initial conditions in Z2. The initial conditions were arbitrarily chosen as 𝑥10 = −1, and 𝑥20 = 2.88. It is clear that the original system has better rise time than the VSS. For example, for 𝜁 = 0.2, the rise time for the original system is 𝜏𝑟 = 0.35, whereas it is equal to 0.39 for the VSS with 𝜌 = 1.1. It is clear that the VSS increases the rise time for 𝜁 = 0.2 by 9.8%. As 𝜌 → ∞ or 𝜆 → ∞, the rise time of the VSS tend to be equal to that of the original system. Figure 3.19. The rise time for VSS and original system 47 Figure 3.20 shows a comparison between the percentage of overshoot of the VSS with 𝜌 = 1.1 and the original system for initial conditions in Z2. The initial conditions were arbitrarily chosen as 𝑥10 = −1, and 𝑥20 = 2.88. It is clear that the VSS has lower percentage of overshoot than the original system. For example, for 𝜁 = 0.2, the percentage of overshoot for the original system is 𝑀𝑝 = 200.8%, whereas it is equal to 173.2% for VSS with 𝜌 = 1.1. It is clear that the VSS reduces the percentage of overshoot for 𝜁 = 0.2 by 13.75% when 𝜌 = 1.1. The best percentage of overshoot is achieved by the VSS when 𝜌 → 1 or 𝜆 → 𝜇1 . Figure 3.21 shows a comparison between the settling time of the VSS with 𝜌 = 1.1 and the original system for initial conditions in Z2. The initial conditions were arbitrarily chosen as 𝑥10 = −1, and 𝑥20 = 2.88. It is clear that the VSS has lower settling time than the original system. For example, for 𝜁 = 0.2, the settling time for the original system is 𝜏𝑠 = 24.69, Figure 3.20. The percentage of overshoot for VSS and original system 48 whereas it is equal to 10.75, for the VSS. It is clear that the VSS reduces the settling time for 𝜁 = 0.2 by 56.45%. The best performance is achieved by the VSS when 𝜌 → 1 or 𝜆 → 𝜇1 . In this section, we investigated the performance of the VSS with switched stiffness for arbitrary initial conditions in Z2. The original system has a smaller rise time than the VSS. on the other hand, the VSS has a smaller maximum percentage of overshoot and a smaller settling time than that of the original system. Therefore, the VSS improves the performance of the mass-spring-damper system, with the best performance achieved when 𝜆 → 𝜇1 . 3.4. Conclusion In this Chapter, we investigated the behavior of the VSS with switched stiffness. For Case A (𝜇1 < λ < 0), the VSS is asymptotically stable and no trajectories of the VSS undergo a Figure 3.21. The settling times of VSS and original system 49 phase change of more than 2π; therefore, the VSS does not exhibit oscillations. Also, the VSS has a smaller settling time than that of the original system for a range of 𝜁 values. As we increase 𝜆, the settling time of the VSS decreases for a larger range of 𝜁 values. Therefore, the VSS improve the performance for the mass-spring-damper system. For Case B (λ ≤ 𝜇1 < 0), the VSS is asymptotically stable and the VSS exhibits oscillations. The original system has a smaller rise time than the VSS. on the other hand, the VSS has a smaller maximum percentage of overshoot and a smaller settling time than that of the original system. Therefore, the VSS will improve the performance of the mass-spring-damper system. For both cases, the best performance is achieved by the VSS when 𝜆 → 𝜇1 . 50 Chapter 4 Variable Structure System with Switched Stiffness and Damping 4.1. Phase Portrait In this chapter, we investigate the behavior of the VSS with switched stiffness and damping. The phase portrait of the VSS is the union of the phase portrait of the first system (𝛼 = +1) in the region (𝑆 > 0 and 𝑥 ≥ 0) or (𝑆 < 0 and 𝑥 ≤ 0), and the phase portrait of the second system (𝛼 = −1) in the region 𝑆𝑥1 < 0. The phase portrait of the first system (𝛼 = +1) in the region (𝑆 > 0 and 𝑥 ≥ 0) or (𝑆 < 0 and 𝑥 ≤ 0) is shown in Fig.4.1 and the phase portrait of the second system (𝛼 = −1) in the region 𝑆𝑥1 < 0 is shown in Fig.4.2. The union of these phase portraits is shown in Fig.4.3. For the phase portrait in Fig.4.3, the slope of the line S = 0 or 𝑥2 = 𝜆 𝑥1 is shown to be greater than the slope of the line 𝑥2 = 𝜇1 𝑥1, i.e., λ > 𝜇1 . This is not necessarily true always, and therefore we need to additionally investigate the case where λ ≤ 𝜇1. These two cases, namely, ( 𝜇1 < λ < 0), and (λ ≤ 𝜇1 < 0), are investigated in sections 4.2 and 4.3. 51 Figure 4.1. Phase portrait of the Variable Structure System (VSS) for 𝛼 = +1 Figure 4.2. Phase portrait of the Variable Structure System (VSS) for 𝛼 = −1 52 Figure 4.3. Phase portrait of the Variable Structure System (VSS) for λ > μ 1 is the union of the phase portrait in Figs.4.1 and 4.2 4.2. Case A: 𝝁𝟏 < 𝛌 < 𝟎 4.2.1. Stability The VSS is a hybrid system that switches between an asymptotically stable system, the phase portrait of which is shown in Fig.4.1, and an unstable system whose phase portrait is shown in Fig.4.2. The union of these phase portraits is shown in Fig.4.3. To investigate the stability of the equilibrium of the VSS, the phase plane is divided into three zones, namely: 𝑍1 = {𝑥 ∈ 𝑅 2 |𝑆𝑥1 > 0} 𝑍2 = {𝑥 ∈ 𝑅 2|(𝑥2 − 𝜇1 𝑥1 )(𝑥2 − 𝜆 𝑥1 ) < 0} 𝑍3 = {𝑥 ∈ 𝑅 2|𝑥1 (𝑥2 − 𝜇1 𝑥1 ) < 0} The three zones are separated by the following three lines: 53 L1: 𝑥1 = 0 L2: 𝑥2 = 𝜇1 𝑥1 L3: 𝑥2 = 𝜆 𝑥1 We investigate the system trajectories in these zones and on these lines next. To study the behavior of the trajectories on L1, we investigate the direction of the vector field on L1. For 𝑥2 > 0 (and 𝑥1 = 0),we have the following equations and vector field: 𝑥̇ 1 = 𝑥2 > 0 𝑥̇ 2 = − 𝑐 𝑥 <0 𝑚 2 Therefore, the trajectories enter 𝑍1 . For 𝑥2 < 0 (and 𝑥1 = 0), we have the following equations and vector field: 𝑥̇ 1 = 𝑥2 < 0 𝑥̇ 2 = − 𝑐 𝑥 𝑚 2 Therefore, all trajectories on L1 will enter 𝑍1 . The line L2 is the stable eigenvector of the second system where 𝛼 = −1 – see Eqs.(2.35) and (2.37). On L2 we have 𝑥2 = 𝑥̇ 1 = 𝜇1 𝑥1 → 𝑥1 (𝑡) = 𝑥1 (0) 𝑒 𝜇1 𝑡 , 𝜇1 < 0. Therefore, all trajectories on L2 converge to the origin. The boundaries of 𝑍1 are L1 and L3. We have already shown that trajectories on L1 enter 𝑍1 . We now show all trajectories in 𝑍1 reach L3 in finite time. To show this we consider the equation of motion of the system for 𝛼 = 1, given by Eq.(2.28): 𝑥̇ 2 = 𝑥̈ 1 = −2𝜁𝜔𝑛 𝑥̇ 1 − 𝜔𝑛 2 𝑥1 The general form of the solution of this differential equation is: 54 𝑥1 (𝑡) = 𝑒 −𝜁𝜔𝑛𝑡 [ 𝐴 cos 𝜔𝑑 𝑡 + 𝐵 sin 𝜔𝑑 𝑡] and (4.1) 𝑥2 (𝑡) = −𝜁𝜔𝑛 𝑒 −𝜁𝜔𝑛𝑡 [ 𝐴 cos 𝜔𝑑 𝑡 + 𝐵 sin 𝜔𝑑 𝑡] +𝑒 −𝜁𝜔𝑛 𝑡 [ −𝐴 𝜔𝑑 sin 𝜔𝑑 𝑡 + 𝐵 𝜔𝑑 cos 𝜔𝑑 𝑡] (4.2) where 𝜔𝑑 = 𝜔𝑛 √1 − 𝜁 2 Substitution of the initial conditions 𝑥1 (0) = 𝑥10 , and 𝑥2 (0) = 𝑥20 into Eqs.(4.1) and (4.2), where (𝑥10 , 𝑥20 ) lies in 𝑍1 , we get: 𝐴 = 𝑥10 , 𝐵= 𝑥20 + 𝜁𝜔𝑛 𝑥10 (4.3) 𝜔𝑛 √1 − 𝜁 2 To find the time when trajectory in 𝑍1 reaches L3, we substitute Eqs.(4.1) and (4.2) into the equation of L3, namely: 𝑥2 (𝑡) = 𝜆 𝑥1 (𝑡) This gives: −𝜁𝜔𝑛 𝑒 −𝜁𝜔𝑛𝑡 [𝐴 cos 𝜔𝑑 𝑡 + 𝐵 sin 𝜔𝑑 𝑡] + 𝜔𝑑 𝑒 −𝜁𝜔𝑡 [− 𝐴 sin 𝜔𝑑 𝑡 + 𝐵 cos 𝜔𝑑 𝑡] = 𝜆 𝑒 −𝜁𝜔𝑛𝑡1 [A cos 𝜔𝑑 𝑡 + 𝐵 sin 𝜔𝑑 𝑡] ⇒ tan 𝜔𝑑 𝑡 = 𝜆𝐴 − 𝜔𝑑 𝐵 + 𝜁𝜔𝑛 𝐴 −𝜁𝜔𝑛 𝐵 − 𝜔𝑑 𝐴 − 𝜆𝐵 (4.4) Substitution of the values of A and B from Eq.(4.3) into Eq.(4.4) gives: 𝜆𝑥10 − 𝑥20 − 𝜁𝜔𝑛 𝑥10 + 𝜁𝜔𝑛 𝑥10 tan 𝜔𝑑 𝑡 = − 𝜁𝑥20 + 𝜁 2 𝜔𝑛 𝑥10 𝜔𝑛 (1 − 𝜁 2 )𝑥10 𝑥 + 𝜁𝜔𝑛 𝑥10 − − 𝜆 ( 20 ) √1 − 𝜁 2 √1 − 𝜁 2 𝜔𝑛 √1 − 𝜁 2 (𝜆𝑥10 − 𝑥20 )√1 − 𝜁 2 ] 𝑥20 + 𝜁𝜔𝑛 𝑥10 − 𝜔 𝑥 − 𝜆 ( ) 20 𝑛 10 𝜔𝑛 1 ⇒𝑡= tan−1 [ 𝜔𝑑 −𝜁𝑥 This expression implies that all trajectories in 𝑍1 reach L3 at finite time. 55 (4.5) The boundaries of 𝑍2 are L2 and L3. Furthermore, the trajectories in 𝑍2 cannot cross L2 since L2 represents a trajectory itself, and trajectories cannot cross each other. We now show all trajectories in 𝑍2 reach L3 in finite time. To show this we consider the equation of motion of the system for 𝛼 = −1, given by Eq.(2.28) 𝑥̇ 2 = 𝑥̈ 1 = 2𝜁𝜔𝑛 𝑥̇ 1 + 𝜔𝑛 2 𝑥1 The general form of the solution of this differential equation is 2 𝑥1 (𝑡) = 𝐷 𝑒 𝜔𝑛𝑡(𝜁−√1+𝜁 ) + 𝐸 𝑒 𝜔𝑛𝑡(𝜁+√1+𝜁 and 𝑥2 (𝑡) = 𝜔𝑛 (𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 𝜔𝑛𝑡(𝜁−√1+𝜁 +𝜔𝑛 (𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜔𝑛𝑡(𝜁+√1+𝜁 (4.6) 2) 2) 2) (4.7) Substitution of the initial conditions 𝑥1 (0) = 𝑥10 , and 𝑥2 (0) = 𝑥20 into Eqs.(4.6) and (4.7) , where (𝑥10 , 𝑥20 ) lies in 𝑍2 we get: 𝐷 = 𝑥10 − 𝑥20 − 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 , 𝐸= 𝑥20 − 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 (4.8) To find the time when trajectory in 𝑍2 reaches L3, we substitute Eqs.(4.6), and (4.7) into the equation of L3, namely: 𝑥2 (𝑡) = 𝜆 𝑥1 (𝑡) This gives: 𝜔𝑛 (𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 = 𝜆 (𝐷 𝑒 𝜔𝑛𝑡(𝜁−√1+𝜁 𝑒 𝜔𝑛𝑡(2√1+𝜁 2) = 2) 𝜔𝑛 𝑡(𝜁−√1+𝜁 2 ) + 𝜔𝑛 (𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜔𝑛 𝑡(𝜁+√1+𝜁 2 ) 2 + 𝐸 𝑒 𝜔𝑛𝑡(𝜁+√1+𝜁 ) ) 𝐷[𝜆 − 𝜔𝑛 (𝜁 − √1 + 𝜁 2 ) ] 𝐸[𝜔𝑛 (𝜁 + √1 + 𝜁2 ) −𝜆] Substitution of the values of D and E from Eq.(4.8) into Eq.(4.9) gives: 56 (4.9) 𝑒 𝜔𝑛𝑡(2√1+𝜁 ⇒𝑡= 2) = 𝑄1 𝑄2 1 2𝜔𝑛 √1 + 𝜁 2 ln [ 𝑄1 ] 𝑄2 (4.10) where 𝑄1 and 𝑄2 , which necessarily have the same sign, are: 𝑄1 = − (𝑥20 − 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 )) [𝜆 − 𝜔𝑛 (𝜁 − √1 + 𝜁 2 ) ] 𝑄2 = (𝑥20 − 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 )) [𝜔𝑛 (𝜁 + √1 + 𝜁 2 ) − 𝜆 ] This expression implies that all trajectories in 𝑍2 reach L3 at finite time. The line L3 is a sliding surface since all trajectories in 𝑍1 and 𝑍2 approach L3 – see Fig.4.3. On L3 we have 𝑥2 = 𝑥̇ 1 = 𝜆𝑥 1 → 𝑥1 (𝑡) = 𝑥1 (0) 𝑒 𝜆𝑡 . Therefore, all trajectories on L3 converge to the origin. The boundaries of 𝑍3 are L1 and L2. Furthermore, the trajectories in 𝑍3 cannot cross L2 since L2 represent a trajectory itself and trajectories cannot cross each other. Moreover, the trajectories on L1 enter 𝑍1 . We now show all trajectories in 𝑍3 reach L1 in finite time. The general form of the solution of the differential equation is described by Eqs.(4.6), and (4.7). To find the time when trajectory in 𝑍3 reaches L1, we substitute Eq. (4.6) into the equation of L1, namely: 𝑥1 = 0 This gives: 2 𝐷 𝑒 𝜔𝑛𝑡(𝜁−√1+𝜁 ) + 𝐸 𝑒 𝜔𝑛𝑡(𝜁+√1+𝜁 𝑒 𝜔𝑛𝑡(2√1+𝜁 2) =− 2) =0 𝐷 𝐸 (4.11) Substitution of the values of D and E from Eq.(4.8) into Eq.(4.11) gives: 57 𝑒 𝜔𝑛𝑡(2√1+𝜁 ⇒𝑡= 2) = 𝑄3 𝑄4 1 2𝜔𝑛 √1 + 𝜁 2 ln [ 𝑄3 ] 𝑄4 (4.12) where 𝑄3 and 𝑄4 , which necessarily have the same sign, are: 𝑄3 = [𝑥20 − 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 )] 𝑄4 = [𝑥20 − 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 )] This expression implies that the trajectories in 𝑍3 reach L1 at finite time. The VSS is a hybrid system that switches between an asymptotically stable system and unstable system. However, from the above discussion it is clear that the origin of the VSS is asymptotically stable. The trajectories of the VSS converge to the origin through L2 or L3 – see Fig.4.4. 4.2.2. Performance – Response to Step Input To investigate the performance of the VSS, we will study the response of the VSS to a step input and compare its performance with that of the original system for the same step input. Figure 4.4. Convergence of VSS trajectories to the origin for ζ < 1 and 𝜇1 < λ < 0 58 For the original system, the response to a step input has been studied in section 3.2.1, and was described in Eq.(3.14). For convenience, it is provided below: 𝑦̈ = − 𝑐 𝑘 𝑦̇ − 𝑦, 𝑚 𝑚 𝑦̇ (0) = 0 𝑦(0) = −1, (4.13) To obtain the same VSS as in Eqs.(2.21) and (2.25), we redefine F in Eq.(2.24) as follows: 𝐹 =𝑘+{ 0 2𝑘(𝑥 − 1) + 2𝑐𝑥̇ 𝑖𝑓 (𝑆 > 0 and 𝑥 ≥ 1) 𝑜𝑟 (𝑆 < 0 and 𝑥 ≤ 1) 𝑖𝑓 𝑆 (𝑥 − 1) < 0 (4.14) Substitution of the values of Eq.(4.14) into Eq.(2.23) and change of variables 𝑦 = (𝑥 − 1) and 𝑦̇ = 𝑥̇ , we get: 𝑦̈ = −𝛼 where 𝛼={ 𝑐 𝑘 𝑦̇ − 𝛼 𝑦, 𝑚 𝑚 +1 −1 𝑖𝑓 𝑖𝑓 𝑦(0) = −1, 𝑦̇ (0) = 0 (4.15) (𝑆 > 0 and 𝑦 ≥ 0 ) 𝑜𝑟 (𝑆 < 0 and 𝑦 ≤ 0 ) 𝑆𝑦<0 Note that Eqs.(4.13) and (4.15) are very similar and have the same initial conditions. Also, note that Eq.(4.15) is identical to Eq.(2.25); therefore, the analysis presented in the section 4.2.1 is applicable. The phase portrait of the VSS in Eq.(4.15) has already been studied in section 4.2.1. From this study we know that no trajectories of the system will undergo a phase change of more than 2π, and therefore, the VSS will not exhibit oscillations for a step input. Consequently, the rise time and percentage overshoot of the system are not relevant. In the absence of oscillations, we investigate the performance of the VSS using the metric of settling time. To this end, we first consider the time needed by the VSS to reach the sliding surface L3, given by Eq.(4.5), which is now rewritten using variables 𝑦 and 𝑦̇ . If 𝑦0 and 𝑦̇ 0 donate the values of 𝑦(0) and 𝑦̇ (0) respectively, the time needed to reach L3 is given by: 59 𝑡= (𝜆𝑦0 − 𝑦̇ 0 )√1 − 𝜁 2 1 tan−1 [ ] 𝑦̇ 0 + 𝜁𝜔𝑛 𝑦0 𝜔𝑑 −𝜁𝑦̇ 0 − 𝜔𝑛 𝑦0 − 𝜆 ( ) 𝜔 𝑛 Substitution of the values of 𝑦0 = −1 and 𝑦̇ 0 = 0 , gives 𝑡1 = 1 −𝜆√1 − 𝜁 2 tan−1 [ ] 𝜔𝑑 𝜔𝑛 + 𝜆 𝜁 (4.16) we substitute the values of 𝑦0 = −1, 𝑦̇ 0 = 0, and 𝑡1 described by Eq.(4.16) into Eqs.(4.1) and (4.2), with variables 𝑥1 and 𝑥2 replaced by 𝑦 and 𝑦̇ respectively. This gives the values of 𝑦 and 𝑦̇ when the VSS reaches the sliding surface L3: 𝑦𝑠 = 𝑒 −𝜁𝜔𝑛𝑡1 [− cos 𝜔𝑑 𝑡1 − 𝜁 √1 − 𝜁 2 𝑦̇𝑠 = −𝜁𝜔𝑛 𝑒 −𝜁𝜔𝑛𝑡1 [− cos 𝜔𝑑 𝑡1 − +𝑒 −𝜁𝜔𝑛𝑡1 [ 𝜔𝑑 sin 𝜔𝑑 𝑡1 − where sin 𝜔𝑑 𝑡1 ] 𝜁 √1 − 𝜁 2 𝜁 √1 − 𝜁 2 (4.17) sin 𝜔𝑑 𝑡1 ] 𝜔𝑑 cos 𝜔𝑑 𝑡1 ] (4.18) 𝜔𝑑 = 𝜔𝑛 √1 − 𝜁 2 To find the time needed for the trajectories on L3 to enter the region |𝑦| ≤ 0.02, we consider the dynamics on L3, namely: 𝑦(𝑡) = 𝑦𝑠 𝑒 𝜆𝑡 , (4.19) 𝜆<0 Substitution of 𝑦(𝑡) = −0.02 yields: 𝑒 𝜆𝑡 = −0.02 1 −0.02 ⇒ 𝑡2 = ln ( ) 𝑦𝑠 𝜆 𝑦𝑠 (4.20) The settling time of the VSS is given by: 𝑡𝑠 = 𝑡1 + 𝑡2 = 1 −𝜆√1 − 𝜁 2 1 −0.02 tan−1 [ ] + ln ( ) 𝜔𝑑 𝜔𝑛 + 𝜆 𝜁 𝜆 𝑦𝑠 60 (4.21) where 𝑦𝑠 is defined by Eq.(4.17). For proper comparison of the performance, we nondimensionalize time using the variable 𝜏 = 𝜔𝑛 𝑡. Concurrently, we define 𝜆 = 𝜅 𝜇1 where 𝜅, 0 < 𝜅 < 1, is a constant. The nondimensional settling time is now given by the following equation: 𝜏𝑠 = 𝜏1 + 𝜏2 where 𝜏1 = 𝜏2 = (4.22) 1 √1 − 𝜁 2 tan−1 𝜅 [− 𝜁 + √𝜁 2 + 1 ]√1 − 𝜁 2 1 + 𝜁 𝜅 [ 𝜁 − √𝜁 2 + 1 ] 1 𝜅[𝜁 − √𝜁 2 + 1 ] ln ( −0.02 ) 𝑦𝑠 (4.23) (4.24) and where 𝑦𝑠 = 𝑒 −𝜁𝜏1 [− cos 𝜏1 √1 − 𝜁 2 − 𝜁 √1 − 𝜁2 sin 𝜏1 √1 − 𝜁 2 ] (4.25) Figure 4.5 shows a comparison of the settling time of the VSS for different value of 𝜅 (𝜅 = 0.3, 𝜅 = 0.5, 𝜅 = 0.7, 𝜅 = 0.9) and the original system. The settling time for the original system shows several discontinuities in the settling time. For 𝜅 = 0.3, the VSS has better performance than that of the original system only when 𝜁 < 0.24. For 𝜅 = 0.5, the performance is better for a larger range of 𝜁, namely 𝜁 < 0.34. As we increase κ, the performance is better for a larger range of 𝜁: for 𝜅 = 0.7 and 0.9 the performance is better for 𝜁 < 0.39 and 𝜁 < 0.52 respectively. The settling time of the original system can be reduced by increasing the value of 𝜁 but for most 𝜁 values, a VSS can be found (𝜅 can be chosen) that has better performance in terms of the settling time. For example, for 𝜁 = 0.2, the settling time for the original system is 𝜏𝑠 = 19.55, whereas it is equal to 16.03, 9.75, 7.11, and 5.67 for VSS with 𝜅 = 0.3, 0.5, 0.7, and 0.9 respectively. It is clear that the VSS improves the settling time for 𝜁 = 0.2 by 18.18% when 𝜅 = 0.3, by 50,2% when 61 Figure 4.5. Comparison of the settling times of the VSS and the original system. 𝜅 = 0.5, by 63.71% when 𝜅 = 0.7, and by 71.04% when 𝜅 = 0.9. The best performance is achieved by the VSS when 𝜅 → 1 or 𝜆 → 𝜇1 . However, the maximum value of κ can be chosen is less than unity (0.95, for example) because of the finite time needed to switching of α from +1 to −1. If 𝜅 ≈ 1 the area of Z2 in Fig.4.3 is almost zero and there will be no switching in this zone and the switching will be in Z3. In this section, we investigated the performance of the VSS with switched stiffness and damping for a step input for different values of 𝜆. The VSS has a smaller settling time than that of the original mass-spring-damper system for a range of 𝜁 values. As we increase 62 𝜆, the settling time of the VSS is smaller for a larger range of 𝜁 values, and the best performance achieved when 𝜆 → 𝜇1 . 4.2.3. Performance – Speed of Convergence In last section, we investigated the performance of the VSS for a step input. This problem was recast as an initial value problem with the initial conditions at (−1, 0) in the phase plane. Since (−1, 0) lies in Z1, we now compare the performance of the VSS with switched stiffness and the original system for arbitrary initial conditions in Z2 and Z3. For an initial conditions in Z2 (see Fig.4.3), the time needed by the VSS to reaches the sliding surface L3 (𝑥2 = 𝜆 𝑥1 ) has been studied in section 4.2.1, and was described by Eq.(4.10). The values of 𝑥1 and 𝑥2 when the VSS reaches the sliding surface L3 has been studied in section 4.2.1, and was described by Eq.(4.6), and (4.7). The time needed for the trajectories on L3 to enter the region |𝑥1 | ≤ 0.02 has been studied in section 4.2.1, and was described by Eq.(4.20). Using these relations, the settling time of the VSS can be shown to be: 𝜏𝑠 = 𝜏1 + 𝜏2 where (4.26) 𝜏1 = 𝜔𝑛 𝑡1 = ln [ 1 2√1 − 𝜁 2 [−𝑥20 + 𝑥10 (𝜁 + √1 + 𝜁 2 )][𝜅(𝜁 − √1 + 𝜁 2 ) − (𝜁 − √1 + 𝜁 2 ) ] [𝑥20 − 𝑥10 (𝜁 − √1 + 𝜁 2 )][(𝜁 + √1 + 𝜁 2 ) − 𝜅(𝜁 − √1 + 𝜁 2 ) ] 𝜏2 = 𝜔𝑛 𝑡2 = ] −1 −0.02 ln ( ) 𝑥1𝑠 [ 𝜁 + √𝜁 2 + 1 ] (4.28) where 𝑥 = 𝐷 𝑒 𝜏1 (𝜁−√1+𝜁2 ) + 𝐸 𝑒 𝜏1 (𝜁+√1+𝜁2 ) 1𝑠 and 𝐷 = 𝑥10 − 𝑥20 − 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 63 (4.27) (4.29) , 𝐸= 𝑥20 − 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 where 𝑥10 and 𝑥20 are the initials conditions in Z2. Figure 4.6 shows a comparison of the settling time of the VSS for different value of 𝜅 (𝜅 = 0.3, 𝜅 = 0.5, 𝜅 = 0.9) and the original system for initial conditions in Z2. The initial conditions were arbitrarily chosen as 𝑥10 = −1, and 𝑥20 = −0.95 𝜇1 . For 𝜅 = 0.3, the VSS has better performance than that of the original system only when 𝜁 < 0.25. For 𝜅 = 0.5, the performance is better for a larger range of 𝜁, namely 𝜁 < 0.36. As we increase 𝜅, the performance is better for a larger range of 𝜁: for 𝜅 = 0.9, the performance is better for 𝜁 < 0.54. The settling time of the original system can be reduced by increasing the value of 𝜁 but for most ζ values, a VSS can be found (𝜅 can be chosen) that has better performance in terms of the settling time. For example, for 𝜁 = 0.2, the settling time for the original system is 𝜏𝑠 = 19.31, whereas it is equal to 13.8, 8.81, and 5.29 for VSS Figure 4.6. Comparison of the settling times of the VSS and the original system 64 with 𝜅 = 0.3, 0.5, and 0.9 respectively. It is clear that the VSS improves the settling time for 𝜁 = 0.2 by 28.58% when 𝜅 = 0.3, by 54.37% when 𝜅 = 0.5, and by 72.6% when 𝜅 = 0.9. The best performance is achieved by the VSS when 𝜅 → 1 or 𝜆 → 𝜇1 . For an initial conditions in Z3 (see Fig.4.3), the time needed by the VSS to reaches L1 has been studied in section 4.2.1, and was described by Eq.(4.12). The values of 𝑥1 and 𝑥2 when the VSS reach the sliding surface L1 (𝑥1 = 0) has been studied in section 4.2.1, and was described by Eq.(4.6), and (4.7). The time needed by the VSS to reaches the sliding surface L3 (𝑥2 = 𝜆 𝑥1 ) has been studied in section 4.2.1, and was described by Eq.(4.5). The values of 𝑥1 and 𝑥2 when the VSS reach the sliding surface L3 has been studied in section 4.2.1, and was described by Eqs.(4.1) and (4.2). The time needed for the trajectories on L3 to enter the region |𝑥1 | ≤ 0.02 has been studied in section 4.2.1, and was described by Eq.(4.20). Using these relations, the settling time of the VSS can be shown to be: 𝜏𝑠 = 𝜏1 + 𝜏2 + 𝜏3 where 𝜏1 = 𝜔𝑛 𝑡1 = 𝜏2 = 𝜔𝑛 𝑡2 = 𝜏3 = 𝜔𝑛 𝑡3 = and where (4.30) 1 2√1 + 𝜁2 1 √1 − 𝜁2 ln [ [𝑥20 − 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 )] [𝑥20 − 𝜔𝑛 𝑥10 (𝜁 − √1 + tan−1 [ √1 − 𝜁 2 𝜁 + 𝜅(𝜁 − √1 + 1 𝑥2𝑙1 √1 − 𝜁2 (4.33) sin √1 − 𝜁 2 𝜏2 ] 𝑥2𝑙1 = (𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 𝜏1 (𝜁−√1+𝜁 𝐷 = 𝑥10 − (4.32) ] 0.02 ln ( ) 𝑥1𝑠 𝜅[ 𝜁 − √𝜁 2 + 1 ] 𝑥1𝑠 = 𝑒 −𝜁𝜔𝑛𝜏2 [ where 𝜁2) (4.31) ] 𝜁 2 )] 2) (4.34) + (𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜏1 (𝜁+√1+𝜁 𝑥20 − 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 65 , 𝐸= 2) 𝑥20 − 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 (4.35) Figure 4.7. Comparison between the settling time of the VSS and the original system where 𝑥10 and 𝑥20 are the initials conditions in Z3. Fig.4.7 shows a comparison of the settling time of the VSS for different value of 𝜅 ( 𝜅 = 0.3, 𝜅 = 0.5, 𝜅 = 0.9) and the original system for initial conditions in Z3. The initial conditions were arbitrarily chosen as 𝑥10 = −1, and 𝑥20 = −1.1 𝜇1 . For 𝜅 = 0.3, the VSS has better performance than that of the original system only when 𝜁 < 0.24. For 𝜅 = 0.5, the performance is better for a larger range of 𝜁, namely 𝜁 < 0.32. As we increase 𝜅, the performance is better for a larger range of 𝜁: for 𝜅 = 0.9, the performance is better for 𝜁 < 0.41. The settling time of the original system can be reduced by increasing the value of 𝜁 but for most 𝜁 values, a VSS can be found (𝜅 can be chosen) that has better performance in terms of the settling time. For example, for 𝜁 = 0.2, the settling time for the original system is 𝜏𝑠 = 19.3, whereas it is equal to 15.62, 10.71, and 7.53 for 66 VSS with 𝜅 = 0.3, 0.5, and 0.9 respectively. It is clear that the VSS improves the settling time for 𝜁 = 0.2 by 19.04% when 𝜅 = 0.3, by 44.5% when 𝜅 = 0.5, and by 60.9% when 𝜅 = 0.9. The best performance is achieved by the VSS when 𝜅 → 1 or 𝜆 → 𝜇1 . In this section, we investigated the performance of the VSS with switched stiffness and damping for arbitrary initial conditions in Z2 and Z3, and for different values of 𝜆. The VSS has a smaller settling time than that of the original mass-spring-damper system. As we increase 𝜆 the settling time of the VSS is smaller for a larger range of 𝜁 values, and the best performance achieved when 𝜆 → 𝜇1 . 4.3. Case B: 𝛌 < 𝝁𝟏 < 𝟎 4.3.1. Stability The VSS is a hybrid system that switches between an asymptotically stable system, the phase portrait of which is shown in Fig.4.8, and an unstable system whose phase portrait is shown in Fig.4.9. The union of these phase portraits is shown in Fig.4.10. To investigate the stability of the equilibrium of the VSS, the phase plane is divided into two zones, namely: 𝑍1 = {𝑥 ∈ 𝑅 2 |𝑆𝑥1 > 0} 𝑍2 = {𝑥 ∈ 𝑅 2|𝑥1 (𝑥2 − 𝜆 𝑥1 ) < 0} The two zones are separated by the following two lines: L1: 𝑥1 = 0 L2: 𝑥2 = 𝜆 𝑥1 We investigate the system trajectories in these zones and on these lines next. 67 Figure 4.8. Phase portrait of the Variable Structure System (VSS) for 𝛼 = +1 Figure 4.9. Phase portrait of the Variable Structure System (VSS) for 𝛼 = −1 68 Figure 4.10. Phase portrait of the Variable Structure System (VSS) for λ > μ 1 To study the behavior of the trajectories on L1, we investigate the direction of the vector field on L1. For 𝑥2 > 0 (and 𝑥1 = 0), we have the following equations and vector field: 𝑥̇ 1 = 𝑥2 > 0 𝑥̇ 2 = − 𝑐 𝑥 <0 𝑚 2 Therefore, the trajectories enter 𝑍1 . For 𝑥2 < 0 (and 𝑥1 = 0), we have the following equations and vector field: 𝑥̇ 1 = 𝑥2 < 0 𝑥̇ 2 = − 𝑐 𝑥 >0 𝑚 2 Therefore, all trajectories on L1 will enter 𝑍1 . The boundaries of 𝑍1 are L1 and L2. We have already shown that trajectories on L1 enter 𝑍1 . We now show all trajectories in 𝑍1 reaches L2 in finite time. The time when the 69 trajectory in 𝑍1 reaches L2 has been studied in section 4.2.1, and was described by Eq.(4.5). For convenience, it is provided below: 𝑡= (𝜆𝑥10 − 𝑥20 )√1 − 𝜁 2 ] 𝑥20 + 𝜁𝜔𝑛 𝑥10 ) 20 − 𝜔𝑛 𝑥10 − 𝜆 ( 𝜔𝑛 1 tan−1 [ 𝜔𝑑 −𝜁𝑥 This expression implies that all trajectories in 𝑍1 reaches L2 at finite time. The boundaries of 𝑍2 are L2 and L1. We have already shown that the trajectories on L1 enter 𝑍1 . We now show that all trajectories in 𝑍2 reaches L1 in finite time. The time when a trajectory in 𝑍2 reaches L1 has been studied in section 4.2.1, and was described by Eq.(4.12). For convenience, it is provided below: 𝑡= 1 2𝜔𝑛 √1 + 𝜁 2 ln [ [𝑥20 − 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 )] ] [𝑥20 − 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 )] This expression implies that all trajectories in 𝑍2 reaches L1 at finite time. Figure 4.11. Phase portrait of the Variable Structure System 70 From the above discussion it is clear that the trajectories of the VSS move alternately between zones 𝑍1 and 𝑍2 by crossing lines L2 and L1. Since the phase plane is symmetric about the 𝑥2 axis (see Fig. 4.10), we now investigate if there will be a contraction or an expansion in the total energy of the system when the VSS undergoes a phase change equal to 𝜋. Fig.4.11 shows the phase portrait of the VSS undergoing a phase change of 𝜋 between points A and C. The values of 𝑥1 and 𝑥2 when the VSS is at point A are: 𝑥1𝐴 = 0 (4.36) 𝑥2𝐴 = 𝑥20 (4.37) where 𝑥20 is some positive number. The time needed by the VSS to move from point A to point C been studied in section 4.2.1, and was described by Eq.(4.5). For convenience, it is provided below: 𝑡𝐴𝐵 = √1 − 𝜁 2 tan−1 [ ] 1 𝜔𝑛 √1 − 𝜁 2 𝜁 + 𝜆 (𝜔 ) 1 (4.38) 𝑛 The values of 𝑥1 and 𝑥2 when the VSS reach point B has been studied in section 4.2.1, and was described by Eq.(4.1), and (4.2). For convenience, it is provided below: 𝑥2𝐴 𝑥1𝐵 = 𝑒 −𝜁𝜔𝑛𝑡 ( ) sin 𝜔𝑛 √1 − 𝜁 2 𝑡𝐴𝐵 𝜔𝑛 (4.39) 𝑥2𝐴 𝑥2𝐵 = 𝜆 𝑥1𝐵 = 𝜆 𝑒 −𝜁𝜔𝑛𝑡 ( ) sin 𝜔𝑛 √1 − 𝜁 2 𝑡𝐴𝐵 𝜔𝑛 (4.40) The time needed for the VSS to move from point B to point C was studied in section 4.2.1, and was described by Eq.(4.12). For convenience, it is provided below: 𝑡𝐵𝐶 (𝜆 − 𝜔𝑛 (𝜁 + √1 + 𝜁 2 )) = ln [ ] 2𝜔𝑛 √1 + 𝜁 2 (𝜆 − 𝜔𝑛 (𝜁 − √1 + 𝜁 2 )) 1 71 (4.41) The values of 𝑥1 and 𝑥2 when the VSS reaches point C has been studied in section 4.2.1, and was described by Eq.(4.6), and (4.7). For convenience, it is provided below: 𝑥1𝐶 = 0 (4.42) 𝑥2𝐶 = 𝜔𝑛 (𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 𝜔𝑛𝑡𝐵𝐶 (𝜁−√1+𝜁 2) +𝜔𝑛 (𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜔𝑛𝑡𝐵𝐶 (𝜁+√1+𝜁 (4.43) 2) where 𝐷 = 𝑥1𝐵 [ −𝜆 + 𝜔𝑛 (𝜁 + √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 ], 𝐸 = 𝑥1𝐵 [ 𝜆 − 𝜔𝑛 (𝜁 − √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 ] (4.44) The total energy of the mass-spring-damper at point A is: 𝑇𝐴 = 1 𝑚 𝑥2𝐴 2 2 (4.45) The total energy of the mass-spring-damper at point C is: 𝑇𝐴 = 1 𝑚 𝑥2𝐶 2 2 (4.46) The contraction in the total energy of the VSS between points A and C is: 𝑇𝐴 − 𝑇𝐶 = 1 1 𝑚 𝑥2𝐴 2 − 𝑚 𝑥2𝐶 2 2 2 (4.47) The percentage of the contraction in the total energy of the VSS between points A and C can be shown to be: 1 1 2 𝑚 𝑥 − 𝑚 𝑥2𝐶 2 𝑇𝐴 − 𝑇𝐶 𝑥2𝐴 2 − 𝑥2𝐶 2 2𝐴 2 2 × 100 = × 100 = × 100 1 𝑇𝐴 𝑥2𝐴 2 2 𝑚 𝑥 2𝐴 2 (4.48) For proper comparison of the contraction in the total energy, we nondimensionalize the time using the variable 𝜏 = 𝜔𝑛 𝑡. Concurrently, we define 𝜆 = 𝜌 𝜇1 where 𝜌 ≥ 1 is a constant. The 72 percentage of the contraction in the total energy of the VSS between points A and C given by the following equation: 𝐸𝑝 = (1 − 𝑥𝑐 2 ) × 100 (4.49) where 𝑥𝑐 = 𝑥2𝐶 2 2 = (𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 𝜏𝐵𝐶 (𝜁−√1+𝜁 ) + (𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜏𝐵𝐶 (𝜁+√1+𝜁 ) (4.50) 𝑥2𝐴 where 𝜏𝐵𝐶 = 𝑡𝐵𝐶 𝜔𝑛 = 𝐷 = 𝑥𝐵 ( 𝐸 = 𝑥𝐵 ( 𝑥𝐵 = 1 2√1 + 𝜁 2 ln [ (𝜌[𝜁 − √𝜁 2 + 1 ] − [𝜁 + √1 + 𝜁 2 ]) (𝜌[𝜁 − √𝜁 2 + 1 ] − [𝜁 − √1 + 𝜁 2 ]) −𝜌[𝜁 − √𝜁 2 + 1 ] + (𝜁 + √1 + 𝜁 2 ) 2√1 + 𝜁 2 𝜌[𝜁 − √𝜁 2 + 1 ] − (𝜁 − √1 + 𝜁 2 ) 2√1 + 𝜁 2 ) 1 √1 − 𝜁 2 tan−1 [ (4.51) (4.52) ) (4.53) 𝑥1𝐵 = 𝑒 −𝜁𝜔𝑛𝑡 sin √1 − 𝜁 2 𝜏𝐴𝐵 𝑥2𝐴 𝜏𝐴𝐵 = 𝑡𝐴𝐵 𝜔𝑛 = ] (4.54) √1 − 𝜁 2 𝜁 + 𝜌[𝜁 − √𝜁 2 + 1 ] ] (4.55) The VSS will be an asymptotically stable if there is a contraction in total energy between point A and point C (𝐸𝑝 > 0), stable if the total energy remains constant between point A and point C (𝐸𝑝 = 0), and an unstable if there is an increasing in total energy between point A and point C (𝐸𝑝 < 0). Fig.4.12 shows the percentage of the contraction in total energy of the VSS between point A and point C for different value of 𝜌 (𝜌 = 1.1, 𝜌 = 1.2, 𝜌 = 1.3, 𝜌 = 1.4, 𝜌 = 1.5). For 𝜌 = 1.1, the VSS is asymptotically stable only when 𝜁 < 0.7, stable for 𝜁 = 0.7, and unstable for 𝜁 > 0.7. For 𝜌 = 1.2, the VSS is asymptotically stable only when 𝜁 < 0.74, a stable for 𝜁 = 0.74, and unstable for 𝜁 > 0.74. For 𝜌 = 1.3, the VSS is asymptotically stable 73 Figure 4.12. The percentage of the contraction in energy of the VSS between points A and C when 𝜁 < 0.79 or 𝜁 > 0.98 , stable for 𝜁 = 0.79 or 𝜁 = 0.98, and unstable for 0.78 < 𝜁 < 0.98. As we increase 𝜌, the VSS is unstable for a smaller range of 𝜁 values: for 𝜌 = 1.4, the VSS is an asymptotically stable when 𝜁 < 0.85 or 𝜁 > 0.96 , stable when 𝜁 = 0.85 or 𝜁 < 0.96, and unstable when 0.85 < 𝜁 < 0.96. For 𝜌 = 1.5, the VSS is an asymptotically stable for all values of 𝜁. By trial and error, it was determined that the VSS is an asymptotically stable for all values of 𝜁 for 𝜌 > 1.44. 4.3.2. Performance – Response to Step Input To investigate the performance of the VSS, we will study the response of the VSS to a step input and compare its performance with that of the original system for the same step input. 74 For the original system, the response to a step input has been studied in section 4.2.1, and was described in Eq.(3.14). For convenience, it is provided below: 𝑦̈ = − 𝑐 𝑘 𝑦̇ − 𝑦, 𝑚 𝑚 𝑦̇ (0) = 0 𝑦(0) = −1, (4.56) The response of the VSS to the same step input was described in Eq.(130). For convenience, it is provided below: 𝑦̈ = −𝛼 where 𝛼={ 𝑐 𝑘 𝑦̇ − 𝛼 𝑦, 𝑚 𝑚 +1 −1 𝑖𝑓 𝑖𝑓 𝑦(0) = −1, 𝑦̇ (0) = 0 (4.57) (𝑆 > 0 and 𝑦 ≥ 0 ) 𝑜𝑟 (𝑆 < 0 and 𝑦 ≤ 0 ) 𝑆𝑦<0 Note that Eqs.(4.56) and (4.57) are very similar and have the same initial conditions. Also, note that Eq.(4.57) is identical to Eq.(2.25); therefore, all the analysis presented in section 4.3.1 will be applicable. The phase portrait of the VSS in Eq.(4.57) has already been studied in section 4.3.1. From this study we know that the trajectories of the system will spiral to the origin and the system will be asymptotically stable for 𝜌 > 1.44. Unlike the previous case, it is not possible to claim that the phase change of the trajectories will be less than 2π. In other words, a step input will result in oscillations, similar to the original system. Therefore, the performance of the original system and the VSS can be compared by on the basis of (a) the rise time 𝑡𝑟 , (b) the maximum percentage of overshoot 𝑀𝑝 , and (c) the settling time. The rise time and the maximum percentage of overshoot can be calculated analytically but the same is not true for the settling time. To calculate the rise time and the maximum percentage of overshoot, we define three points A, B, and C on the phase portrait of the VSS, shown in Fig 75 Figure 4.13. Step response in the phase plane of the Variable Structure System (VSS) for λ < μ 1 4.13. These points are also shown in Fig.4.14, which is a plot of the time response of the VSS to the step input. Consider the time needed by the VSS to reach point A; this has been studied in section 4.2.1, and was described by Eq.(120). Using variables 𝑦 and 𝑦̇ we can write it as follows: 𝑡𝐴 = (𝜆𝑦0 − 𝑦̇ 0 )√1 − 𝜁 2 1 1 −𝜆√1 − 𝜁 2 tan−1 [ ]= tan−1 [ ] (4.58) 𝑦̇ 0 + 𝜁𝜔𝑛 𝑦0 𝜔𝑑 𝜔 𝜔 + 𝜆 𝜁 𝑑 𝑛 −𝜁𝑦̇ 0 − 𝜔𝑛 𝑦0 − 𝜆 ( ) 𝜔 𝑛 where 𝑦0 = −1 and 𝑦̇ 0 = 0 are the initial conditions. The values of 𝑦 and 𝑦̇ at A has been studied in section 4.2.1, and was described by Eq.(116), and (117). For convenience, it is provided below: 𝑦𝐴 = 𝑒 −𝜁𝜔𝑛𝑡𝐴 [ 𝐴 cos 𝜔𝑑 𝑡𝐴 + 𝐵 sin 𝜔𝑑 𝑡𝐴 ] (4.59) 𝑦̇𝐴 = −𝜁𝜔𝑛 𝑒 −𝜁𝜔𝑛𝑡𝐴 [ 𝐴 cos 𝜔𝑑 𝑡𝐴 + 𝐵 sin 𝜔𝑑 𝑡𝐴 ] +𝑒 −𝜁𝜔𝑛𝑡𝐴 [ −𝐴 𝜔𝑑 sin 𝜔𝑑 𝑡𝐴 + 𝐵 𝜔𝑑 cos 𝜔𝑑 𝑡𝐴 ] 76 (4.60) Figure 4.14. Step response of the Variable Structure System (VSS) for λ < μ 1 where 𝜔𝑑 = 𝜔𝑛 √1 − 𝜁 2 , 𝐴 = −1 , 𝐵 = − 𝜁 √1−𝜁 2 The time needed for the VSS to move from point A to point B has been studied in section 4.2.1, and was described by Eq.(4.12), For convenience, it is provided below: 𝑡𝐴𝐵 = 1 2𝜔𝑛 √1 + 𝜁 2 ln [ [𝑦̇𝐴 − 𝜔𝑛 𝑦𝐴 (𝜁 + √1 + 𝜁 2 )] [𝑦̇𝐴 − 𝜔𝑛 𝑦𝐴 (𝜁 − √1 + 𝜁 2 )] ] (4.61) where 𝑦𝐴 , and 𝑦̇𝐴 are the initial conditions and given by Eqs.(4.58) and (4.60). The values of 𝑦 and 𝑦̇ at B has been studied in section 4.2.1, and was described by Eq.(121) and (122). For convenience, it is provided below: 𝑦𝐵 = 0 (4.62) 𝑦̇ 𝐵 = 𝜔𝑛 (𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 𝜔𝑛𝑡𝐴𝐵 (𝜁−√1+𝜁 +𝜔𝑛 (𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜔𝑛𝑡𝐴𝐵 (𝜁+√1+𝜁 77 2) 2) (4.63) where 𝐷 = 𝑦𝐴 − 𝑦̇𝐴 − 𝜔𝑛 𝑦𝐴 (𝜁 − √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 , 𝐸= 𝑦̇𝐴 − 𝜔𝑛 𝑦𝐴 (𝜁 − √1 + 𝜁 2 ) 2𝜔𝑛 √1 + 𝜁 2 To find the time needed for the VSS to move from point B to point C, we substitute Eq.(4.2) into the equation of y-axis where point C lie on this axis, namely: 𝑦̇ = 0 This gives −𝜁𝜔𝑛 𝑒 −𝜁𝜔𝑛𝑡 [ 𝐴 cos 𝜔𝑑 𝑡 + 𝐵 sin 𝜔𝑑 𝑡] + 𝑒 −𝜁𝜔𝑛𝑡 [ −𝐴 𝜔𝑑 sin 𝜔𝑑 𝑡 + 𝐵 𝜔𝑑 cos 𝜔𝑑 𝑡] = 0 ⟹ tan 𝜔𝑑 𝑡 = −𝜁𝜔𝑛 𝐴 + 𝐵𝜔𝑑 𝜁𝜔𝑛 𝐵 + 𝜔𝑑 𝐴 (4.64) Substitution of the values of A and B from Eq.(4.3) into Eq.(4.64) gives ⟹ tan 𝜔𝑑 𝑡 = ⟹ 𝑡𝐵𝐶 = √1 − 𝜁 2 𝜁 1 √1 − 𝜁 2 tan−1 [ ] 𝜔𝑑 𝜁 (4.65) Substitution of the value of 𝑡𝐵𝐶 from Eq.(4.64) into Eqs.(4.1) and (4.2) gives the values of 𝑦 and 𝑦̇ at C, namely: 𝑦̇ 𝐵 𝑦𝐶 = 𝑒 −𝜁𝜔𝑛𝑡𝐵𝐶 [ sin 𝜔𝑛 √1 − 𝜁 2 𝑡𝐵𝐶 ] 2 𝜔𝑛 √1 − 𝜁 (4.66) 𝑦̇ 𝐶 = 0 (4.67) The rise time of the VSS can now be calculated as follows: 𝑡𝑟 = 𝑡𝐴 + 𝑡𝐴𝐵 = 1 −𝜆√1 − 𝜁 2 tan−1 [ ] 𝜔𝑑 𝜔𝑛 + 𝜆 𝜁 78 + 1 2𝜔𝑛 √1 + 𝜁 2 ln [ [𝑦̇𝐴 − 𝜔𝑛 𝑦𝐴 (𝜁 + √1 + 𝜁 2 )] [𝑦̇𝐴 − 𝜔𝑛 𝑦𝐴 (𝜁 − √1 + 𝜁 2 )] ] (4.68) where 𝑦𝐴 , and 𝑦̇𝐴 are defined by Eqs.(4.59) and (4.60) respectively. The maximum percent overshoot of the VSS depend on 𝑡𝐵𝐶 and is given by: 𝑦̇ 𝐵 𝑀𝑝 = 𝑒 −𝜁𝜔𝑛𝑡𝐵𝐶 [ sin 𝜔𝑛 √1 − 𝜁 2 𝑡𝐵𝐶 ] × 100 𝜔𝑛 √1 − 𝜁 2 = 𝑒 ( √1−𝜁 2 −𝜁 tan−1 [ ]) 𝑦̇ 𝐵 𝜁 √1−𝜁 2 [ 𝜔𝑛 ] × 100 (4.69) where 𝑦̇ 𝐵 is defined by Eq.(178). For proper comparison of the performance, we nondimensionalize the time using the variable 𝜏 = 𝜔𝑛 𝑡. Concurrently, we define 𝜆 = 𝜌 𝜇1 where 𝜌 ≥ 1 is a constant. The nondimensional rise time and the maximum percentage overshoot are given by the following equations respectively: 𝜏𝑟 = 𝜔𝑛 𝑡𝑟 = + 𝑀𝑝 = 1 √1 − 𝜁 2 1 2√1 + 𝜁 2 ln [ tan−1 [ 𝜌 [−𝜁 + √𝜁 2 + 1 ]√1 − 𝜁 2 1 + 𝜁 𝜌 [ 𝜁 − √𝜁 2 + 1 ] [𝑦̇𝐴 − 𝑦𝐴 (𝜁 + √1 + 𝜁 2 )] [𝑦̇𝐴 − 𝑦𝐴 (𝜁 − √1 + 𝜁 2 )] √1−𝜁 2 −𝜁 tan−1 𝜁 2 √1−𝜁 𝑒 𝑦̇ 𝐵 ] ] (4.70) × 100 (4.71) where 𝑦 = 𝑒 −𝜁𝜏𝐴 [ 𝐴 cos √1 − 𝜁 2 𝜏 + 𝐵 sin √1 − 𝜁 2 𝜏 ] 𝐴 𝐴 𝐴 (4.72) 𝑦̇𝐴 = −𝜁𝑒 −𝜁𝜏𝐴 [ 𝐴 cos √1 − 𝜁 2 𝜏𝐴 + 𝐵 sin √1 − 𝜁 2 𝑑 𝜏𝐴 ] +√1 − 𝜁 2 𝑒 −𝜁𝜏𝐴 [ −𝐴 sin √1 − 𝜁 2 𝜏𝐴 + 𝐵 cos √1 − 𝜁 2 𝜏𝐴 ] 𝑦̇ 𝐵 = (𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 𝜏𝐴𝐵 (𝜁−√1+𝜁 2) + (𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜏𝐴𝐵 (𝜁+√1+𝜁 79 (4.73) 2) (4.74) 𝜏𝐴 = 1 √1 − 𝜁 2 𝜏𝐴𝐵 = tan−1 [ 1 2√1 + 𝜁 2 𝐴 = −1 , 𝐷 = 𝑦𝐴 − ln [ 𝜌 [−𝜁 + √𝜁 2 + 1 ]√1 − 𝜁 2 1 + 𝜁 𝜌 [ 𝜁 − √𝜁 2 + 1 ] [𝑦̇𝐴 − 𝑦𝐴 (𝜁 + √1 + 𝜁 2 )] [𝑦̇𝐴 − 𝑦𝐴 (𝜁 − √1 + 𝜁 2 )] 𝐵=− ] (4.75) ] (4.76) 𝜁 √1 − 𝜁 2 𝑦̇𝐴 − 𝑦𝐴 (𝜁 − √1 + 𝜁 2 ) 2√1 + 𝜁 2 , 𝐸= 𝑦̇𝐴 − 𝑦𝐴 (𝜁 − √1 + 𝜁 2 ) 2√1 + 𝜁 2 The settling time for the VSS is the time needed for the VSS trajectory to enter the region |𝑦| ≤ 0.02. This cannot be calculate analytically because the VSS may enter the region |𝑦| ≤ 0.02 in 𝑍1 or in 𝑍2 . Therefore, we calculate the settling time numerically. Figure 4.15 shows a comparison of the rise times of the VSS for different values of 𝜌 Figure 4.15. Comparison of the rise times of the VSS and original system 80 (𝜌 = 1.5, 𝜌 = 2, 𝜌 = 5) and the original system. 𝜌 has been chosen to be greater than 1.44 to make sure that the VSS will be asymptotically stable. For 𝜌 = 1.5, the VSS has better performance than that the original system only when 𝜁 > 0.72. It is clear that the original system has better rise time than the VSS for 𝜌 = 2 and 5. Figure 4.16 shows a comparison of the maximum percentage of overshoots of the VSS for different values of (𝜌 = 1.5, 𝜌 = 2, 𝜌 = 5) and the original system. Once again, 𝜌 has been chosen to be greater than 1.44 to make sure that the VSS will be asymptotically stable. For 𝜌 = 5, the VSS has lower maximum percentage of overshoot than that of the original system only when 𝜁 < 0.06. For 𝜌 = 2, the maximum percentage of overshoot is lower for a larger range of 𝜁, namely 𝜁 < 0.15. As we decreases 𝜌, the maximum percentage of overshoot is lower for a larger range of 𝜁: for 𝜌 = 1.5, the maximum percentage of overshoot is lower for 𝜁 < 0.21. Figure 4.16. Comparison of the percentage of overshoots of the VSS and original system 81 Figure 4.17 shows a comparison of the settling time of the VSS for different value of 𝜌 (𝜌 = 1.5, 𝜌 = 2, 𝜌 = 5) and the original system. 𝜌 has been chosen to be greater than 1.44 to make sure that the VSS will be asymptotically stable. For 𝜌 = 5, the VSS has larger settling time that that of the original system for all values of ζ. For 𝜌 = 2, the VSS has lower settling time than that of the original system only when 𝜁 < 0.15 As we decrease 𝜌, the settling time is lower for a larger range of 𝜁: for 𝜌 = 1.5, the settling time is lower for 𝜁 < 0.18. In this section, we investigated the performance of the VSS with switched stiffness and damping for a step input for different values of 𝜌. The value of 𝜌 was chosen to be greater than 1.44 to make sure that the VSS will be asymptotically stable. The VSS has a smaller settling time than that of the original mass-spring-damper system for a small range of 𝜁 values. As we decrease 𝜆, the settling time of the VSS is smaller for a larger range of 𝜁 values, Figure 4.17. Comparison of the settling time of the VSS and original system 82 and the best performance is achieved by the VSS when 𝜌 → 1.44. 4.3.3. Performance – Speed of Convergence In last section, we investigated the performance of the VSS for a step input. This problem was recast as an initial value problem with the initial conditions at (−1, 0) in the phase plane. Since (−1, 0) lies in Z1, we now compare the performance of the VSS with switched stiffness and the original system for arbitrary initial conditions in Z2. For initial conditions in Z2 (see Fig.4.10), the performance of the original system and the VSS can be compared by computing: (a) the rise time 𝑡𝑟 , (b) the maximum percentage of overshoot 𝑀𝑝 , and (c) the settling time. The rise time and the maximum percentage of overshoot can be calculated analytically but the same is not true for the settling time. To calculate the rise time and maximum percentage of overshoot, we define two points A and B on the phase portrait of the VSS, shown in Fig.4.18. These points are also shown in Fig.4.19, which is a plot of the time response of the VSS to the step input. Consider the time needed by the VSS to reach point B; this has been studied in section 4.2.1, and was described by Eq.(4.12). The values of 𝑥1 and 𝑥2 at A has been studied in section 4.2.1, and was described by Eq.(4.6) and (4.7). To find the time needed for the VSS to move from point A to point B, we substitute Eq.(4.2) into the equation of y-axis where point B lies on this axis, namely: 𝑥2 = 0 This gives: 83 Figure 4.18. Step response in the Phase plane for the Variable Structure System (VSS) for λ < μ 1 Figure 4.19. Step response for the Variable Structure System (VSS) for λ < μ 1 84 −𝜁𝜔𝑛 𝑒 −𝜁𝜔𝑛𝑡 [ 𝐴 cos 𝜔𝑑 𝑡 + 𝐵 sin 𝜔𝑑 𝑡] + 𝑒 −𝜁𝜔𝑛𝑡 [ −𝐴 𝜔𝑑 sin 𝜔𝑑 𝑡 + 𝐵 𝜔𝑑 cos 𝜔𝑑 𝑡] = 0 ⟹ tan 𝜔𝑑 𝑡 = −𝜁𝜔𝑛 𝐴 + 𝐵𝜔𝑑 𝜁𝜔𝑛 𝐵 + 𝜔𝑑 𝐴 (4.77) Substitution of the values of A and B from Eq.(4.3) into Eq.(4.77) gives: ⟹ tan 𝜔𝑑 𝑡 = ⟹ 𝑡𝐴𝐵 = √1 − 𝜁 2 𝜁 1 √1 − 𝜁 2 tan−1 [ ] 𝜔𝑑 𝜁 (4.78) Substitution of the value of 𝑡𝐴𝐵 into Eqs.(4.2), and (4.3) gives the values of 𝑥1 and 𝑥2 at B, namely: 𝑥1𝐵 = 𝑒 −𝜁𝜔𝑛𝑡𝐴𝐵 [ 𝑥2𝐴 𝜔𝑛 √1 − 𝜁 2 sin 𝜔𝑛 √1 − 𝜁 2 𝑡𝐴𝐵 ] (4.79) 𝑥2𝐵 = 0 (4.80) The rise time of the VSS can now be calculated as follows: 𝑡𝑟 = 𝑡𝐴 = 1 2𝜔𝑛 √1 + 𝜁 2 ln [ [𝑥20 − 𝜔𝑛 𝑥10 (𝜁 + √1 + 𝜁 2 )] ] [𝑥20 − 𝜔𝑛 𝑥10 (𝜁 − √1 + 𝜁 2 )] (4.81) where 𝑥10 , and 𝑥20 are the initial conditions. The maximum percent overshoot of the VSS depend on 𝑡𝐵𝐶 and is given by: 𝑥2𝐴 𝑀𝑝 = 𝑒 −𝜁𝜔𝑛𝑡𝐴𝐵 [ sin 𝜔𝑛 √1 − 𝜁 2 𝑡𝐴𝐵 ] × 100 𝜔𝑛 √1 − 𝜁 2 = 𝑒 ( √1−𝜁 2 −𝜁 tan−1 [ ]) 𝑥2𝐴 𝜁 √1−𝜁 2 [ 𝜔𝑛 ] × 100 (4.82) where 𝑥2𝐴 is defined by Eq.(4.7). For proper comparison of the performance, we nondimensionalize the time using the variable 𝜏 = 𝜔𝑛 𝑡. Concurrently, we define 𝜆 = 𝜌 𝜇1 where 85 𝜌 ≥ 1 is a constant. The nondimensional rise time and the maximum percent overshoot are given by the following equations respectively: 𝜏𝑟 = 𝜔𝑛 𝑡𝑟 = 𝑀𝑝 = where 1 2√1 + 𝜁 2 ln [ [𝑥20 − 𝑥10 (𝜁 + √1 + 𝜁 2 )] [𝑥20 − 𝑥10 (𝜁 − √1 + 𝜁 2 )] √1−𝜁 2 −𝜁 tan−1 𝜁 2 𝑒 √1−𝜁 𝑥2𝐴 1 2√1 + 𝜁 2 𝐷 = 𝑥10 − ln [ (4.84) × 100 𝑥2𝐴 = (𝜁 − √1 + 𝜁 2 ) 𝐷 𝑒 𝜏𝐴(𝜁−√1+𝜁 𝜏𝐴 = (4.83) ] 2) + (𝜁 + √1 + 𝜁 2 ) 𝐸 𝑒 𝜏𝐴(𝜁+√1+𝜁 [𝑥20 − 𝑥10 (𝜁 + √1 + 𝜁 2 )] [𝑥20 − 𝑥10 (𝜁 − √1 + 𝜁 2 )] 𝑥20 − 𝑥10 (𝜁 − √1 + 𝜁 2 ) 2√1 + 𝜁 2 , 2) ] 𝐸= (4.85) (4.86) 𝑥20 − 𝑥10 (𝜁 − √1 + 𝜁 2 ) 2√1 + 𝜁 2 where 𝑥10 and 𝑥20 are the initials conditions in Z2. The settling time of the VSS is the time needed for the VSS to enter the region |𝑥1 | ≤ 0.02. There is no way to calculate the settling time analytically, because the VSS may enter the region |𝑥1 | ≤ 0.02 in 𝑍1 or 𝑍2 . Therefore we calculate the settling time numerically. Figure 4.20 shows a comparison between the rise time of the VSS with 𝜌 = 1.5 and the original system for initial conditions in Z2. 𝜌 has been chosen to be greater than 1.44 to make sure that the VSS will be asymptotically stable. The initial conditions were arbitrarily chosen as 𝑥10 = −1, and 𝑥20 = 2.88. The VSS has lower rise time than that of the original system only when 𝜁 < 0.12. Therefore, the VSS may have a lower rise time than that of the original system if the initial conditions are in Z2. 86 Figure 4.20. The rise time for VSS and original system Figure 4.21. The percentage of overshoot for VSS and original system 87 Figure 4.21 shows a comparison between the percentage of overshoot of the VSS with 𝜌 = 1.5 and the original system for initial conditions in Z2. The initial conditions were arbitrarily chosen as 𝑥10 = −1, and 𝑥20 = 2.88. The VSS has lower maximum percentage of overshoot than that of the original system only when 𝜁 < 0.09. Therefore, the VSS may have a lower maximum percentage of overshoot than that of the original system if the initial conditions are in Z2. Figure 4.22 shows a comparison between the settling time of the VSS with 𝜌 = 1.5 and the original system for initial conditions in Z2. 𝜌 has been chosen to be greater than 1.44 to make sure that the VSS will be asymptotically stable. The initial conditions were arbitrarily chosen as 𝑥10 = −1, and 𝑥20 = 2.88. The VSS has lower settling time than that of the original system only when 𝜁 < 0.19. Therefore, the VSS may have a lower rise time than that of the original Figure 4.22. The settling times of VSS and original system 88 system if the initial conditions are in Z2. In this section, we investigated the performance of the VSS with switched stiffness and damping for a step input. The value of 𝜌 was chosen to be greater than 1.44 to make sure that the VSS will be asymptotically stable. The VSS has a smaller settling time than that of the original mass-spring-damper system for a small range of 𝜁. As we decrease 𝜆, the settling time of the VSS is smaller for a larger range of 𝜁 values, and the best performance is achieved by the VSS when 𝜌 → 1.44. 4.4. Conclusion In this Chapter, we investigated the behavior of the VSS with switched stiffness and damping. For Case A (𝜇1 < λ < 0), the VSS is asymptotically stable and no trajectories of the VSS undergo a phase change of more than 2π; therefore, the VSS does not exhibit oscillations. Also, the VSS has a smaller settling time than that of the original system for a range of 𝜁 values. As we increase 𝜆, the settling time of the VSS is smaller for a larger range of 𝜁 values. The best performance is achieved by the VSS when 𝜆 → 𝜇1 . Therefore, the VSS improves the performance for the mass-spring-damper system. For Case B (λ ≤ 𝜇1 < 0), the VSS may be asymptotically stable, stable, or unstable. However, if we chose 𝜌 > 1.44, the VSS will be asymptotically stable. For 𝜌 > 1.44 the trajectories of the VSS will undergo a phase change of more than 2π; therefore, the VSS will exhibit oscillations. The VSS improves the performance of the mass-spring-damper system for a small range of 𝜁 values. As we decrease 𝜆 the settling time of the VSS is smaller for a larger range of 𝜁 values, and the best performance is achieved by the VSS when 𝜌 → 1.44. 89 Chapter 5 Conclusion We investigated the behavior of two hybrid mass-spring-damper (MSD) systems. The first system uses switched stiffness, where the stiffness in the model switches between its nominal value and its negative value. The second system uses switched stiffness and damping, where both the stiffness and damping coefficients in the model are switched between their nominal values and their negative values. Both hybrid MSD systems are switched based on the location of the system in its configuration space. Each hybrid system is asymptotically stable even though they are individually comprised of an asymptotically stable and an unstable sub-system. Through proper design of the switching logic, the hybrid systems can have significantly better performance than the nominal system in terms of rise time, settling time, and overshoot. Based on the design of the switching logic, the MSD system with switched stiffness has two behaviors. In the first case, all trajectories in the phase plane asymptotically converge to the origin, but more importantly, no trajectory is capable of undergoing a phase change of more than 2π rad. This establishes the fact that the hybrid system will not exhibit oscillations, even if the original system is underdamped. Additionally, the hybrid system has a rise time comparable to that of the original system but a significantly lower settling time. In the second case, all trajectories in the phase plane asymptotically converge to the origin, but the hybrid system exhibits oscillations. 90 The original system has a smaller rise time than the hybrid system but the hybrid system has a smaller maximum percentage of overshoot and a smaller settling time. The hybrid system therefore improves the performance of the nominal system if the nominal system is underdamped. The analysis presented in this work provides clues on how the switching logic can be designed to achieve the best performance improvement for a given value of damping ratio. Similar to the hybrid system with switched stiffness, the hybrid system with switched stiffness and damping has two behaviors depending on the design of the switching logic. In the first case, the origin is asymptotically stable since and all trajectories in the phase plane converge to origin. Also, the hybrid system does not exhibit oscillations due to the fact that no trajectory is capable of undergoing a phase change of more than 2π rad. Additionally, the hybrid system has a rise time comparable to that of the original system but a lower settling time for a small range of damping ratios (𝜁 < 0.5, for example). For the second case, the hybrid system can be asymptotically stable, stable, or unstable based on the design of the switching logic. For the asymptotically stable case, the hybrid system does exhibit oscillations but it has better performance than the nominal system (in terms of what) for a small range of damping ratios (𝜁 < 0.2, for example). The analysis presented in this work provides clues on how the switching logic can be designed to guarantee asymptotic stability for the hybrid system and achieve the best performance improvement for a given value of damping ratio. 91 REFERENCES 92 REFERENCES [1] Ogata, Katsuhiko. 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