Fragility analysis for control systems

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1 DOI: /CTA Control Theory & Applications Vol. 3 No. 1 Jan , 1, 2, 1, 1 (1., 151; 2., 158) :. ( 1, j), ( 1, j)., Bode...,,,, Bode. : ; Bode ; ; ; : TP273 : A Fragility analysis for control systems HE Zhen 1, WANG Guang-xiong 1, ZHANG Jing 2, QI Jian-cheng 1, ZHOU Di 1 (1. Department of Control Science and Engineering, Harbin Institute of Technology, Harbin Heilongjiang 151, China; 2. Department of Automation, Harbin University of Science and Technology, Harbin Heilongjiang 158, China) Abstract: Fragility means that the stability of a closed-loop system is extremely sensitive to parameter perturbations. For these systems, the open-loop frequency response is closely passing by the critical stability point ( 1, j), and the fragile relationship with the point ( 1, j) can easily be violated by any possible perturbation. It is analyzed and pointed out that the fragility of a control system can be quantitatively analyzed by using the Bode integrals. For non-minimumphase unstable plants, the unstable pole of the controller in addition with the unstable pole of the plant may greatly increase the peak of the sensitivity function. So the resulting design for such systems is unavoidably fragile. For multivariable unstable plants, there may be a rather large unstable pole in an equivalent output feedback loop, which can also cause a large peak of the sensitivity function and results in a fragile design. The Bode integral can also be used to avoid the fragility of the design. Key words: fragility; Bode integral; output feedback; non-minimum-phase system; unstable plant 1 (Introduction) [1], [1], [2 3].,. [4],.,,,.,..,,. ( ),,.,,,.,.. [1], [1], [5 7].,,,. [1] : ; : Tel.: : (61341); ( ).

2 96 3, [8]. [8],,.,.,,.. Bode. 2 Bode (Bode integrals),, [9], -- G(s) = 5.715[1 (.1334s)2 ] = s 2 [(.12168s) 2 1] ( s 2 ) s 2 s (1) G(s) ( ) ( ). (1)., 3,, 18.,. Bode [1], Bode. 1 Bode [1 11]. L(s) p 1, p 2,, p N, ν = n m., S(jω) : ln S(jω) dω = π N Re(p i ), ν > 1, (2) i=1 ln S(jω) dω = γ π 2 + π N Re(p i ), ν = 1. i=1 (3) n L(s), m, γ = lim s sl(s). S(jω) = L(jω). (4) [11]. Bode,., ( v > 1), ln S(jω) dω =. (5), S = 1, 1, 1. (5),.,., L, S = 1/(1 + L), S 1,. (2),.,, ( 2).,, (2),. 1 Nyquist, ρ L(jω) 1. ρ = min 1 + L. (6) M S, M S = max S(jω) = 1/ρ. (7) M S, 1,,, M S ( ). M S [12], M S = 3 [1]. 1 Nyquist Fig. 1 Nyquist plot of the system, --,.,, (2)., ( ), M S,,., M S 2, 1.5.,,,..

3 1 : 97 3 (Fragility analysis).,. H [13]. H,,, H. 1. G(s) = 1 s s(s 2). (8) [13] K(s) = 1 41s s 2 + 6s 23. (9) (8) (9) s 4 + 4s 3 + 6s 2 + 4s + 1 = (s + 1) 4 =. (1) (1), 4 1,. K(s) : , , G(s) 2, 2. Bode (2) ln S(jω) dω = π ( ) = rad/s. (4), ( ) 1 rad/s, ln S max = , S max = (8) (9). 1,, ω =.22 rad/s, ln S. 2, ω = ln S >. ln S, ln S,. ln S 2.77, M S = M S = 16, Nyquist [14] 1.6( 3),., G(s), G (s), G (s) = 1 s 1.5 s(s 2) (.5s + 1). 5%,. G (s) K(s)( (9)) 22.43,.7145 ± j2.8821,.852 ± j ( (1)), 2.,,,. M S = 16, M S > 2,, (.5 s ),. 3 1 Nyquist Fig. 3 Nyquist plot of example ,, x. (1) [9]. Bode [9]. [9] H PS/T [15], H 2 1 Fig. 2 Sensitivity function of example 1 : K(s) = q 3s 3 + q 2 s 2 + q 1 s + q s 4 + p 3 s 3 + p 2 s 2 + p 1 s + p, (12) q 3 = , p 3 = ,

4 98 3 q 2 = , p 2 = , q 1 = , p 1 = , q = , p = , 57.39, , , (1) , 2, Bode (2) ln S(jω) dω = π ( ) = rad/s. (5), 8 rad/s,,, ln S max 5, S max Fig. 4 Sensitivity function of the inverted pendulum system S max, Nyquist (<.1) 1,. [9] 392.9, 8.195, 6.421, ± j13.52, ,.226 ± j θ() =.1 rad. 5 : x, θ(t) Fig. 5 Transient responses of the inverted pendulum system θ. θ -- 6 [16], (1) u x G(s)., x K(s)( (12)), u. 5, (θ = ), Fig. 6 Dynamic model of the inverted pendulum system 5, S max,, Nyquist 1..,.., (12) 1.1 ( 1%), 393, 8.198, 6.931, 3.2 ± j15.136,.7184,.667 ± j1.693,.226 ± j1.666., 1%,. H,, 2 Bode, --. Isidori [17],, Bode, -- [18]. Ishida [19], Ishida H,.., [19]. 4 (Discussions about the fragility and the control design methods) 1, [1] ( 1 2)

5 1 : 99. 1, G(s) = s 1 s 2 s 2, H, 6, Bode ( (13) )., Nyquist [1].,..,., H, synthesis( ), Bode,.,., [1],. 2,. [14] : -- H., -- H PS/T,, [14] ,,, ( ), Bode (2), ln S max > π( : ), S max > 23.,., Bode, S max [14]. [2],,.. 3 ν = 1( (3)), [21]..,,, ( ),,, ν 1., ν > 1. 5 (Conclusions),,. 1, 1..,.. Bode M S., M S 2 2,, M S,. M S > 2,, M S > 2,,, ( ).., M S > 2. Bode., --. (References): [1] KEEL L H, BHATTACHARYYA S P. Robust, fragile, or optimal? [J]. IEEE Transactions on Automatic Control, 1997, 42(8): [2] MÄKILÄ P M. Comments on robust, fragile, or optimal? [J]. IEEE Transactions on Automatic Control, 1998, 43(9): [3] PAATTILAMMI J, MÄKILÄ P M. Fragility and robustness: a case study on paper machine headbox control [J]. IEEE Control Systems Magazine, 2, 2(1): [4] EL-SAKKARY A K. The gap metric: robustness of stabilization of feedback systems [J]. IEEE Transactions on Automatic Control, 1985, 3(3): [5] FAMULARO D, DORATO P, ABDALLAH C T. Robust non-fragile LQ controller: the static state feedback case [J]. International Journal of Control, 2, 73(2): [6] WHIDBORNE J F, ISTEPANIAN R, WU J. Reduction of controller fragility by pole sensitivity minimization [J]. IEEE Transactions on Automatic Control, 21, 46(2): [7],,,. H [J]., 21, 14(2): (SUN Wen an, SUN Fengjie, PEI Bingnan, et al. Non-fragile H guaranteed cost control for a class of networked control systems with uncertain time delay [J]. Electric Machines and Control II, 21, 14(2): ) [8] KEEL L H, BHATTACHARYYA S P. Structural instability and minimal realizations [J]. IEEE Transactions on Automatic Control, 21, 55(4): [9],,. : [J]., 22, 6(3): (WANG Guangxiong, ZHANG Jing, ZHU Jingquan. Output feedback control of inverted pendulum: fragility and robustness [J]. Electric Machines and Control, 22, 6(3): )

6 1 3 [1] STEIN G. Respect the unstable [J]. IEEE Control Systems Magazine, 23, 23(4): [11],. [M]. :, 28. (WANG Guangxiong, HE Zhen. Control System Design [M]. Beijing: Tsinghua University Press, 28.) [12] ASTRÖM K J, PANAGOPOULOS H, HäGGLUND T. Design of PI controllers based on non-convex optimization [J]. Automatica, 1998, 34(5): [13] FRANCIS B A. A Coure in H control theory [R] //Lecture Notes in Control and Information Sciences. New York: Springer-Verlag, 1987, 88. [14],,,. [J]., 211, 15(4): (HE Zhen, RAO Dan, WANG Guangxiong, et al. Fragility and robustness of the control system [J]. Electric Machines and Control, 211, 15(4): 8 84.) [15],. H [M]. :, 21. (WANG Guangxiong, HE Zhen. Applied H Control [M]. Harbin: HIT Press, 21.) [16],,. [J]., 25, 22(1): (ZHANG Jing, WANG Guangxiong, HE Zhen. Robust control of systems with unstable zero-dynamics [J]. Electric Machines and Control, 25, 22(1): ) [17] ISIDORI A. A tool for semiglobal stabilization of uncertain nonminimum-phase nonlinear systems via output feedback [J]. IEEE Transactions on Automatic Control, 2, 45(1): [18] WANG G X, ZHANG J. Robustness of the design by unlocking the unstable zero-dynamics [J]. Acta Automatica Sinica, 24, 3(1): [19] ISHIDA T, GANEKO S. Consideration of H control of an inverted pendulum(in Japanese) [C] //The 19th Symposium on Control Theory. Tokyo: Hobonsya Press, 199: [2] BALINI H, SCHERER C, WITTE J. Performance enhancement for AMB systems using unstable H controllers [J]. IEEE Control Systems Technology, 211, 19(6): [21] KOKOTOVIC P, ARCAK M. Constructive nonlinear control: a historical perspective [J]. Automatica, 21, 37(5): : (1972 ),,, H, hezhen@hit.edu.cn; (1933 ),,, H, gxwang@hit.edu.cn; (1968 ),,,, zhangjing@hrbust.edu.cn; (1987 ),,,, zhangjing@hrbust.edu.cn; (1969 ),,,, zhoud@hope.hit.edu.cn.

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