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Passivity Based Controller Design Based on EL and PCHD Model
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Procedia Engineering
Procedia Procedia Engineering 00 (2011) Engineering 15000–000 (2011) 33 – 37 www.elsevier.com/locate/procedia
Advanced in Control Engineering and Information Science
Passivity Based Controller Design Based on EL and PCHD Model Jiuhe Wanga, Hongren Yinb a
School of Automation, Beijing Information Science & Technology University ,Beijing and 100192， China b Library, Beijing Information Science & Technology University ,Beijing and 100110， China
Abstract In this paper, the method of passivity based controller design, is introduced according to passivity, EL (EulerLagrange) and PCHD(Port Controlled Hamiltonian with Dissipation) model of nonlinear system. In order to improve the property of system, energy shaping and damping injection are adopted in passivity based controller design. Passivity based controller is used in PWM rectifier, the properties of PWM rectifier is improved. The paper points out deficiencies of passivity based controller design based on EL and PCHD model, and further research.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [CEIS 2011] Keywords: Passivity based controller design; Storage function; Damping injection; EL model; PCHD model;PWM rectifier.
1. Introduction The system designed by passivity based control theory can achieve global stabile, non-singular point, and strong robustness to parameter variation of system and disturbance from outside. Passivity-based control theory is a essence control one of energy and is widely used in engineering[1]. In practice, many system such as mechanical, electrical and electromechanical, can be modeled by EL or PCHD equations. Passivity based controller can be designed based on passivity, EL or PCHD model of system. In order to improve the property of system, energy shaping and damping injection are adopted in passivity based
1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.08.008
342
Jiuhe andal/ Hongren YinEngineering / Procedia Engineering 15 (2011) 33 – 37 JiuheWang Wang,et Procedia 00 (2011) 000–000
controller design. The paper points out deficiencies of passivity based controller design based on EL and PCHD model, and further research. 2. Systems model 2.1. System EL model An n degrees of freedom dynamical system with generalized coordinates q and external forces τ is described by the EL equations[2] M (q )q&& + C (q, q& )q& + g (q ) = τ (1) where q = [q1 L qn ]T ; M (q ) is the generalized inertia matrix that satisfies M (q ) = M T (q ) > 0 ; & (q ) − 2C (q, q& ) is skew-symmetric matrix that satisfies ξ T ( M & (q) − 2C (q, q& ) ) ξ = 0 ; g (q ) is derivative M
of potential function relative to q . Many system, such as converters, can be modelled by EL equations in the following form && + Jx& + ℜ x& = u (2) Mx where M is a positive matrix, M = M T > 0 ; J is skew-symmetric matrix, J = − J T ; ℜ is a positive matrix, ℜ = ℜ T > 0 , which express the dissipation of system; u is external input. For equation(1), let H denotes the storage function in the following form 1 H ( q, q& ) = q& T M (q)q& + P(q) (3) 2 Then 1 & (q )q& + q& T ∂P (q ) = q& Tτ + 1 q& T ( M & (q ) − 2C (q, q& ) ) q& = q& Tτ H& ( q, q& ) = q& T M (q)q&& + q& T M (4) 2 2 ∂q From equation (4), if output y = q& T ,the system described by equation (1) is passive. For equation(2), let V denotes the storage function in the following form 1 V = x& T Mx& 2 Then V& = x& T Mx&& = x& T (u − Jx& − ℜ x& ) = x& T u − x& Tℜ x&
(5) (6)
From equation (6), if output y = x& ,the system described by equation (2) is strict passive. T
2.2. System PCHD model
System PCHD model can be described[3] as ∂H ( x ) ⎧& ⎪⎪ x = ( J ( x ) − R( x ) ) ∂x + g ( x )u (7) ⎨ ⎪ y = g T ( x ) ∂H ( x ) ∂x ⎩⎪ n where x ∈ R are the energy variables; the smooth function H ( x ) : R n → R represents the total stored energy and u, y ∈ R m are the port power variables. The port variables u and y are conjugated variables, in the sense that their duality product defines the power flows exchanged with the environment of the system. The interconnection structure is captured in the n×n skew-symmetric matrix J ( x ) = − J T ( x ) and
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Jiuhe Wang Hongren Yin / Procedia Engineering 15 (2011) 33 – 37 Jiuhe and Wang,et al / Procedia Engineering 00 (2011) 000–000
the n×m matrix g ( x ) , while R( x ) = RT ( x ) > 0 represents the dissipation. According to equation(7), the differential of H ( x ) is given as dH ( x (t )) ∂T H ( x (t )) ∂H ( x (t )) = uT (t ) y (t ) − R( x (t )) ≤ uT (t ) y (t ) dt ∂x ∂x From equation (8), the system described by equation (10) is strict passive.
(8)
3. Passivity based controller design
3.1. Passivity based controller design based on EL model Passivity based controller design based on equation(2) is stated as follow: Let He denotes the error storage function in the following form 1 H e = x& eT Mx& e xe = x − x ∗ 2
(9)
∗
where x is desired equilibrium points of system. In order to accelerate H e to zero and energy dissipation of system, damping injection is used in passivity based controller design. Method Ⅰ Damping injection dissipation is set as ℜ d xe = (ℜ + Ra ) xe where Ra is a positive dissipative matrix. Equation(2) can be expressed as Mx&&e + Jx& e + ℜ x& e = u − ( Mx& * + Jx& * + ℜ x& * − Ra x& e ) Based on equation(11), passivity based control law is obtained by u = Mx&&* + Jx& * + ℜ x& * − Ra x& e From equation(2), Mx&&e + Jx& e + ℜ x& e = 0 ,then H& = − x& T (ℜ + R ) x& < 0 e
e
a
e
From equation(13), the convergence rate of system is determined by Ra . Method Ⅱ Damping injection dissipation is same as equation(10). Equation(2) can be expressed as &&e + ℜ d x& e = u − ( Mx &&* + J ( x& * + x& e ) + ℜ x& * − Ra x& e ) Mx Passivity based control law can be written[4] as &&* + Jx& + ℜ x& * − Ra x& e u = Mx Passivity based control law（15） ensures H& = − x T (ℜ + R ) x < 0 . e
a
e
(10)
(11) (12) (13)
(14) (15)
3.2 Passivity based controller design based on PCHD model Passivity based controller based on PCHD model is designed by standard feedback interconnection[2], interconnection based on cyclo-passivity[5] and passivity of system. Passivity based controller based on passivity of system[1,3] is widely used in practice, and is introduced as follow. The IDA–PBC (interconnection and damping assignment-passivity based control, IDA–PBC) design methodology is used. IDA–PBC can realize energy shaping and damping injection.
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Jiuhe andal/ Hongren YinEngineering / Procedia Engineering 15 (2011) 33 – 37 JiuheWang Wang,et Procedia 00 (2011) 000–000
The central idea of IDA-PBC is to assign to the closed-loop a desired storage function via the modification of the interconnection and dissipation matrices. The desired dynamics is a Hamiltonian system of the form ∂H ( x ) x& = ( J d ( x ) − Rd ( x ) ) d (16) ∂x where J d and Rd are the new interconnection matrix and dissipation matrix respectively, H d ( x ) = H ( x ) + H a ( x ) is the total storage function with a minimum when x = x ∗ . The problem how to get the control law is transformed into the search for the functions J d ( x ) , Rd ( x ) ∂H a ( x ) satisfying the PDE(Partial Derivative Equation, PDE)[3] ∂x ∂H ( x ) ∂Η ( J ( x , u) − ℜ ) + gu = ( J d ( x ) − Rd ( x ) ) d ∂x ∂x and such that 1) (Structure preservation) ⎧⎪ J d ( x ) = J ( x , u) + J a ( x ) = 0 − J dT ( x ) ⎨ T ⎪⎩ Rd ( x ) = R( x ) + Ra ( x ) = Rd ( x ) ≥ 0
and a vector function K ( x ) =
(17)
(18)
2) (Integrability) K ( x ) is the gradient of a scalar function T
∂K ( x ) ⎛ ∂K ( x ) ⎞ =⎜ ⎟ ∂x ⎝ ∂x ⎠ 3) (Equilibrium assignment) when ∂H ( x ) K ( x* ) = − ∗ ∂x x = x 4) (Lyapunov stability)The Jacobian of K ( x ) at x ∗
(19)
(20)
∂K ( x ) ∂ 2 H ( x) (21) ∗ > − ∗ ∂x x = x ∂x 2 x = x Substituting J d ( x ) , Rd ( x ) , K ( x ) and H a ( x ) into (20), the control can be obtained. by solving the PDE(17).
3.2. Passivity based controller design application in PWM rectifier In order to unity power factor and constant control of DC voltage, desired points of PWM rectifier are id = I m ( I m is amplitude of AC phase current in stable state), iq = 0 and uDC = uDCR ; id, iq are current in synchronous rotating dq coordinate, respectively; uDC is the DC output voltage. According to Method Ⅱ, passivity based control law based on EL model can be obtained. Under conditions: source phase voltage 220V, f=50Hz, L=16mH,R=0.3Ω, C=2200μF, rating load RL=100Ω, Ra=10Ω, modulation frequency 5kHz and uDCR = 600V ; R and L mean resistance and inductance of filter reactor, respectively; C is the DC side capacitance; RL is the DC side load. Simulation results of PWM rectifier is shown in Fig.1. Passivity based controller design based on PCHD model is obtained according to 3.2. Simulation parameters are same as above. Simulation results of PWM rectifier is shown in Fig.2. Simulation results in Fig.1 and Fig.2 show that PWM rectifiers with passivity based controller designed based on EL model and PCHD model has good static and dynamic performances.
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Jiuhe Wang Hongren Yin / Procedia Engineering 15 (2011) 33 – 37 Jiuhe and Wang,et al / Procedia Engineering 00 (2011) 000–000 800
800
u DCR
600
600
u DC
400
400
200 0
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0
id /A
iq /A
Fig.1 Simulation results of PWM rectifier based on EL model
Fig.2 Simulation results of PWM rectifier based on PCHD model
4. Conclusions
The system controlled by passivity based controller based on EL and PCHD model has good properties. But passivity based controller is designed according to constant desired points in a general way popularly. For passivity based controller design based on EL, damping injection can be carried out and energy shaping don’t. But passivity based control law based on EL is simple. Passivity based controller design based on PCHD, which damping injection and energy shaping can be carried out, has better properties . But passivity based control law based on PCHD is complicated. Passivity based controller under variation desired points is researched further. Acknowledgements
This work is supported by national natural science foundation of China(51077005)/ by funding project for academic human resources development in institutions of higher learning under the jurisdiction of beijing municipality（PHR201007130）. References [1] Wang Jiuhe. Passivity-Based Control Theory and Its Applications. Publishing House of Electronics Industry, Beijing, china, 2010. [2] Arjan van der Schaft.L2 Gain and Passivity Techniques in Nonlinear Control, Springer, Berlin , Germany ,2000. [3] Romeo Ortega, Arjan van der Schaft, Bernhard Maschke, Gerardo Escobar. Interconnection and damping assignment passivitybased control of port-controlled Hamiltonian systems. Automatica ,38 (2002), pp.585 – 596. [4] Wang Jiuhe，Huang Lipei，Zhang Jinlong，Xia Peirong. Passive controller for three -phase voltage-source PWM rectifier. Electric Power Automation Equipment, 2008，28（10）：38-41(in Chinese)． [5] Romeo Ortega, Arjan van der Schaft, Fernando Casta n% os, and Alessandro Astolfi. Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian System. IEEE Transactions on automatic control, 2008, 53(11):2527-2542.
Procedia Engineering
Procedia Procedia Engineering 00 (2011) Engineering 15000–000 (2011) 33 – 37 www.elsevier.com/locate/procedia
Advanced in Control Engineering and Information Science
Passivity Based Controller Design Based on EL and PCHD Model Jiuhe Wanga, Hongren Yinb a
School of Automation, Beijing Information Science & Technology University ,Beijing and 100192， China b Library, Beijing Information Science & Technology University ,Beijing and 100110， China
Abstract In this paper, the method of passivity based controller design, is introduced according to passivity, EL (EulerLagrange) and PCHD(Port Controlled Hamiltonian with Dissipation) model of nonlinear system. In order to improve the property of system, energy shaping and damping injection are adopted in passivity based controller design. Passivity based controller is used in PWM rectifier, the properties of PWM rectifier is improved. The paper points out deficiencies of passivity based controller design based on EL and PCHD model, and further research.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [CEIS 2011] Keywords: Passivity based controller design; Storage function; Damping injection; EL model; PCHD model;PWM rectifier.
1. Introduction The system designed by passivity based control theory can achieve global stabile, non-singular point, and strong robustness to parameter variation of system and disturbance from outside. Passivity-based control theory is a essence control one of energy and is widely used in engineering[1]. In practice, many system such as mechanical, electrical and electromechanical, can be modeled by EL or PCHD equations. Passivity based controller can be designed based on passivity, EL or PCHD model of system. In order to improve the property of system, energy shaping and damping injection are adopted in passivity based
1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.08.008
342
Jiuhe andal/ Hongren YinEngineering / Procedia Engineering 15 (2011) 33 – 37 JiuheWang Wang,et Procedia 00 (2011) 000–000
controller design. The paper points out deficiencies of passivity based controller design based on EL and PCHD model, and further research. 2. Systems model 2.1. System EL model An n degrees of freedom dynamical system with generalized coordinates q and external forces τ is described by the EL equations[2] M (q )q&& + C (q, q& )q& + g (q ) = τ (1) where q = [q1 L qn ]T ; M (q ) is the generalized inertia matrix that satisfies M (q ) = M T (q ) > 0 ; & (q ) − 2C (q, q& ) is skew-symmetric matrix that satisfies ξ T ( M & (q) − 2C (q, q& ) ) ξ = 0 ; g (q ) is derivative M
of potential function relative to q . Many system, such as converters, can be modelled by EL equations in the following form && + Jx& + ℜ x& = u (2) Mx where M is a positive matrix, M = M T > 0 ; J is skew-symmetric matrix, J = − J T ; ℜ is a positive matrix, ℜ = ℜ T > 0 , which express the dissipation of system; u is external input. For equation(1), let H denotes the storage function in the following form 1 H ( q, q& ) = q& T M (q)q& + P(q) (3) 2 Then 1 & (q )q& + q& T ∂P (q ) = q& Tτ + 1 q& T ( M & (q ) − 2C (q, q& ) ) q& = q& Tτ H& ( q, q& ) = q& T M (q)q&& + q& T M (4) 2 2 ∂q From equation (4), if output y = q& T ,the system described by equation (1) is passive. For equation(2), let V denotes the storage function in the following form 1 V = x& T Mx& 2 Then V& = x& T Mx&& = x& T (u − Jx& − ℜ x& ) = x& T u − x& Tℜ x&
(5) (6)
From equation (6), if output y = x& ,the system described by equation (2) is strict passive. T
2.2. System PCHD model
System PCHD model can be described[3] as ∂H ( x ) ⎧& ⎪⎪ x = ( J ( x ) − R( x ) ) ∂x + g ( x )u (7) ⎨ ⎪ y = g T ( x ) ∂H ( x ) ∂x ⎩⎪ n where x ∈ R are the energy variables; the smooth function H ( x ) : R n → R represents the total stored energy and u, y ∈ R m are the port power variables. The port variables u and y are conjugated variables, in the sense that their duality product defines the power flows exchanged with the environment of the system. The interconnection structure is captured in the n×n skew-symmetric matrix J ( x ) = − J T ( x ) and
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Jiuhe Wang Hongren Yin / Procedia Engineering 15 (2011) 33 – 37 Jiuhe and Wang,et al / Procedia Engineering 00 (2011) 000–000
the n×m matrix g ( x ) , while R( x ) = RT ( x ) > 0 represents the dissipation. According to equation(7), the differential of H ( x ) is given as dH ( x (t )) ∂T H ( x (t )) ∂H ( x (t )) = uT (t ) y (t ) − R( x (t )) ≤ uT (t ) y (t ) dt ∂x ∂x From equation (8), the system described by equation (10) is strict passive.
(8)
3. Passivity based controller design
3.1. Passivity based controller design based on EL model Passivity based controller design based on equation(2) is stated as follow: Let He denotes the error storage function in the following form 1 H e = x& eT Mx& e xe = x − x ∗ 2
(9)
∗
where x is desired equilibrium points of system. In order to accelerate H e to zero and energy dissipation of system, damping injection is used in passivity based controller design. Method Ⅰ Damping injection dissipation is set as ℜ d xe = (ℜ + Ra ) xe where Ra is a positive dissipative matrix. Equation(2) can be expressed as Mx&&e + Jx& e + ℜ x& e = u − ( Mx& * + Jx& * + ℜ x& * − Ra x& e ) Based on equation(11), passivity based control law is obtained by u = Mx&&* + Jx& * + ℜ x& * − Ra x& e From equation(2), Mx&&e + Jx& e + ℜ x& e = 0 ,then H& = − x& T (ℜ + R ) x& < 0 e
e
a
e
From equation(13), the convergence rate of system is determined by Ra . Method Ⅱ Damping injection dissipation is same as equation(10). Equation(2) can be expressed as &&e + ℜ d x& e = u − ( Mx &&* + J ( x& * + x& e ) + ℜ x& * − Ra x& e ) Mx Passivity based control law can be written[4] as &&* + Jx& + ℜ x& * − Ra x& e u = Mx Passivity based control law（15） ensures H& = − x T (ℜ + R ) x < 0 . e
a
e
(10)
(11) (12) (13)
(14) (15)
3.2 Passivity based controller design based on PCHD model Passivity based controller based on PCHD model is designed by standard feedback interconnection[2], interconnection based on cyclo-passivity[5] and passivity of system. Passivity based controller based on passivity of system[1,3] is widely used in practice, and is introduced as follow. The IDA–PBC (interconnection and damping assignment-passivity based control, IDA–PBC) design methodology is used. IDA–PBC can realize energy shaping and damping injection.
364
Jiuhe andal/ Hongren YinEngineering / Procedia Engineering 15 (2011) 33 – 37 JiuheWang Wang,et Procedia 00 (2011) 000–000
The central idea of IDA-PBC is to assign to the closed-loop a desired storage function via the modification of the interconnection and dissipation matrices. The desired dynamics is a Hamiltonian system of the form ∂H ( x ) x& = ( J d ( x ) − Rd ( x ) ) d (16) ∂x where J d and Rd are the new interconnection matrix and dissipation matrix respectively, H d ( x ) = H ( x ) + H a ( x ) is the total storage function with a minimum when x = x ∗ . The problem how to get the control law is transformed into the search for the functions J d ( x ) , Rd ( x ) ∂H a ( x ) satisfying the PDE(Partial Derivative Equation, PDE)[3] ∂x ∂H ( x ) ∂Η ( J ( x , u) − ℜ ) + gu = ( J d ( x ) − Rd ( x ) ) d ∂x ∂x and such that 1) (Structure preservation) ⎧⎪ J d ( x ) = J ( x , u) + J a ( x ) = 0 − J dT ( x ) ⎨ T ⎪⎩ Rd ( x ) = R( x ) + Ra ( x ) = Rd ( x ) ≥ 0
and a vector function K ( x ) =
(17)
(18)
2) (Integrability) K ( x ) is the gradient of a scalar function T
∂K ( x ) ⎛ ∂K ( x ) ⎞ =⎜ ⎟ ∂x ⎝ ∂x ⎠ 3) (Equilibrium assignment) when ∂H ( x ) K ( x* ) = − ∗ ∂x x = x 4) (Lyapunov stability)The Jacobian of K ( x ) at x ∗
(19)
(20)
∂K ( x ) ∂ 2 H ( x) (21) ∗ > − ∗ ∂x x = x ∂x 2 x = x Substituting J d ( x ) , Rd ( x ) , K ( x ) and H a ( x ) into (20), the control can be obtained. by solving the PDE(17).
3.2. Passivity based controller design application in PWM rectifier In order to unity power factor and constant control of DC voltage, desired points of PWM rectifier are id = I m ( I m is amplitude of AC phase current in stable state), iq = 0 and uDC = uDCR ; id, iq are current in synchronous rotating dq coordinate, respectively; uDC is the DC output voltage. According to Method Ⅱ, passivity based control law based on EL model can be obtained. Under conditions: source phase voltage 220V, f=50Hz, L=16mH,R=0.3Ω, C=2200μF, rating load RL=100Ω, Ra=10Ω, modulation frequency 5kHz and uDCR = 600V ; R and L mean resistance and inductance of filter reactor, respectively; C is the DC side capacitance; RL is the DC side load. Simulation results of PWM rectifier is shown in Fig.1. Passivity based controller design based on PCHD model is obtained according to 3.2. Simulation parameters are same as above. Simulation results of PWM rectifier is shown in Fig.2. Simulation results in Fig.1 and Fig.2 show that PWM rectifiers with passivity based controller designed based on EL model and PCHD model has good static and dynamic performances.
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Jiuhe Wang Hongren Yin / Procedia Engineering 15 (2011) 33 – 37 Jiuhe and Wang,et al / Procedia Engineering 00 (2011) 000–000 800
800
u DCR
600
600
u DC
400
400
200 0
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0
id /A
iq /A
Fig.1 Simulation results of PWM rectifier based on EL model
Fig.2 Simulation results of PWM rectifier based on PCHD model
4. Conclusions
The system controlled by passivity based controller based on EL and PCHD model has good properties. But passivity based controller is designed according to constant desired points in a general way popularly. For passivity based controller design based on EL, damping injection can be carried out and energy shaping don’t. But passivity based control law based on EL is simple. Passivity based controller design based on PCHD, which damping injection and energy shaping can be carried out, has better properties . But passivity based control law based on PCHD is complicated. Passivity based controller under variation desired points is researched further. Acknowledgements
This work is supported by national natural science foundation of China(51077005)/ by funding project for academic human resources development in institutions of higher learning under the jurisdiction of beijing municipality（PHR201007130）. References [1] Wang Jiuhe. Passivity-Based Control Theory and Its Applications. Publishing House of Electronics Industry, Beijing, china, 2010. [2] Arjan van der Schaft.L2 Gain and Passivity Techniques in Nonlinear Control, Springer, Berlin , Germany ,2000. [3] Romeo Ortega, Arjan van der Schaft, Bernhard Maschke, Gerardo Escobar. Interconnection and damping assignment passivitybased control of port-controlled Hamiltonian systems. Automatica ,38 (2002), pp.585 – 596. [4] Wang Jiuhe，Huang Lipei，Zhang Jinlong，Xia Peirong. Passive controller for three -phase voltage-source PWM rectifier. Electric Power Automation Equipment, 2008，28（10）：38-41(in Chinese)． [5] Romeo Ortega, Arjan van der Schaft, Fernando Casta n% os, and Alessandro Astolfi. Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian System. IEEE Transactions on automatic control, 2008, 53(11):2527-2542.