General Integral Control Design via Singular Perturbation Technique

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1 International Journal o Modern Nonlinear heor and pplication, 214, 3, Published Online September 214 in SciRes. General Integral Control Design ia Singular Perturbation echnique Baishun Liu, Xiangqian Luo, Jianhui Li cadem o Naal Submarine, Qingdao, China baishunliu@163.com, qdqtlq@sina.com, jianhui_li@163.com Receied 1 ugust 214; reised 8 September 214; accepted 15 September 214 Copright 214 b authors and Scientiic Research Publishing Inc. his work is licensed under the Creatie Commons ttribution International License (CC BY). bstract his paper proposes a sstematic method to design general integral control with the generic integrator and integral control action. No longer resorting to an ordinar control along with a known Lapuno unction, but snthesiing singular perturbation technique, mean alue theorem, stabilit theorem o interal matri and Lapuno method, a uniersal theorem to ensure regionall as well as semi-globall asmptotic stabilit is established in terms o some bounded inormation. Its highlight point is that the error o integrator output can be used to stabilie the sstem, just like the sstem state, such that it does not need to take an etra and special eort to deal with the integral dnamic. heoretical analsis and simulation results demonstrated that: general integral controller, which is tuned b this design method, has super strong robustness and can deal with nonlinearit and uncertainties o dnamics more orceull. Kewords General Integral Control, Nonlinear Control, Robust Control, General Integrator, General Integral ction, Singular Perturbation Method, Output Regulation 1. Introduction Integral control [1] plas an important role in practice because it ensures asmptotic tracking and disturbance rejection when eogenous signals are constants or planting parametric uncertainties appear. Howeer, integral control design is not triial matter because it depends on uncertain parameters and disturbances. hereore, it is o important signiicance to deelop the design method on the integral control. In 29, or oercoming the restriction o traditional integral control, the idea o general integral control irstl was proposed b [1], which presented some general integrators and controllers. Howeer, their justiication was not eriied b mathematical analsis. General integral control designs based on linear sstem theor, How to cite this paper: Liu, B.S., Luo, X.Q. and Li, J.H. (214) General Integral Control Design ia Singular Perturbation echnique. International Journal o Modern Nonlinear heor and pplication, 3,

2 sliding mode technique and eedback lineariation technique were presented b [2]-[4], respectiel. he main shortage o these design methods proposed b literature [2]-[4] is that the were all achieed b using a kind o particular integrator and linear integral action, which are a serious obstruction to design a high perormance integral controller. In addition, general concae integral control [5], general cone integral control [6], constructie general bounded integral control [7] and the generaliation o the integrator and integral control action [8] were all deeloped b resorting to an ordinar control along with a known Lapuno unction. his results in that design methods presented b [5]-[8] are all suspended in midair. hus, it is a er aluable and challenging problem to establish a solid oundation or designing general integral control with the generic integrator and integral control action. Motiated b the cognition aboe, this paper proposes a sstematic method to design general integral control with the generic integrator and integral control action. he main contributions are that: 1) B mean alue theorem, the nonlinear actions in the subsstem and integral dnamics are all reormulated as the linear orms on the interal matri such that stabilit theorem o interal matri can be used to deal with them; 2) he error o integrator output can be used to stabilie the sstem, just like the sstem state, such that it does not need to take an etra and special eort to deal with the integral dnamic; 3) No longer resorting to an ordinar control along with a known Lapuno unction, but snthesiing singular perturbation technique, mean alue theorem, stabilit theorem o interal matri and Lapuno method, a uniersal theorem to ensure regionall as well as semigloball asmptotic stabilit is established in terms o some bounded inormation. Consequentl, this uniersal theorem is not suspended in midair but is deeloped with a solid oundation. Moreoer, simulation results showed that general integral controller, which is tuned b this design method, has superstrong robustness and can deal with nonlinearit and uncertainties o dnamics more orceull. hroughout this paper, we use the notation λ m ( ) and λ ( ) M to indicate the smallest and largest eigen- n, or an R. he norm o ector alues, respectiel, o a smmetric positie deined bounded matri is deined as =, and that o matri is deined as the corresponding induced norm λm ( ) =. For two n m matrices and B, B denotes element-b-element inequalit. amil o interal matrices is deined as, n m Λ (, ) = R : where = a ij and = a ij are ied matrices. he amil Λ is described geometricall as hperrectangle in the space R n n n m o the coeicients a ij. We sa that a R amil matri Λ is Hurwit stable i eer Λ is Hurwit stable. he remainder o the paper is organied as ollows: Section 2 describes the sstem under consideration, assumption and deinition. Section 3 addresses the design method. Eample and simulation are proided in Section 4. Conclusions are presented in Section Problem Formulation Consider the ollowing controllable nonlinear sstem, (, ) (,, ) (,, ) = = w + g wu (1) n m m l where R and R are the states; u R is the control input; w R is a ector o unknown constant parameters and disturbances. he partial deriatie o unction on (, ) is bounded in the control domain n m D D R R, and (,) =. he unctions, and g are continuous in ( w,, ) on the control don m l main D D Dw R R R. We want to design a control law, u such that t and ( t) as t.,,u that satisies the equations, ssumption 1: here is a unique pair =,, w+ g,, wu (2) so that = = is the desired equilibrium point and u is the stead-state control that is needed to maintain equilibrium at = =. 174

3 ssumption 2: No loss o generalit, suppose that the unctions ( w,, ) and (,, ) ollowing inequalities, (,, ) (,, ) B. S. Liu et al. g w satisies the w w l + l (3) g g w,, g> (4) (,, ) (,, ) g w g w l + l (5) or all D, D and w Dw. where l, l, l g and l g are all positie constants. n Deinition 1: Fφ ( aφ, bφ, cφ, ) with a φ >, b φ > and R denotes the set o all continuousl dierential increasing unctions [8], ( ) ( ) ( ) g g = n n φ φ φ φ such that φ ( ) =, : φ b R > a i i φ i i c dφ d > R i = 1, 2,, n. φ i i i i where stands or the absolute alue. Figure 1 depicts the eample cures o one component o the unctions belonging to the unction set F φ. For instance, or all R, the unctions, a ( a > ), tanh ( ), arcsinh ( ) and so on, all belong to unction set F φ. Deinition 2: (,, n m F c ) with c >, D R and D R denotes the set o all integrable unctions [8], φ such that (, ) = (, ) (, ) (, ) 1 2 n i, i, i (, ) = +,, = ςi, ςi, = ςi, ςi i i c, >, = ςi, ςi, = ςi, ςi Figure 1. Eample cures o one component o the unctions belonging to the unction set F φ. 175

4 hold or all i 1, 2,, n i, i, to the origin. Figure 2 depicts the eample cures o one component o the unctions belonging to the unction set F. For instance, or all ( 1,1), and ( 1,1), the unctions, + 3, tanh + sinh, sinh ( + 2), and so on, all belong to the unction set F. 3. Control Design =, and ς ς is a point on the line segment connecting In general, integral controller comprises three components: the stabiliing controller, the integral control action and the integrator, and then the general integral controller can be gien as, ( σφ( σ) ) (, ) ; belong to the unc- where K, c β βi( σi) > ( i 1, 2,, m) tion sets F φ and F, respectiel. hus, substituting (6) into (1), obtain the augmented sstem, u = ε K+ K+ K (6) σ = β σ K and K σ are the m n, m m and m m gain matrices, respectiel; σi = βi( σi) i(, ) = ; ε is a positie constant; the unctions φ ( ) and (, ) (,, ) (,, )( σ ) (, ) = ε = ε w g w K+ K+ Kφ σ σ = β σ B ssumption 1 and choosing ε K σ to be nonsingular and large enough, and then setting = and = = o the sstem (7), we obtain, (,, ) ε (,, ) g w Kσφ σ = w (8) hereore, we ensure that there is a unique solution,,σ is a unique equilibrium point o the closed-loop sstem (7) in the domain o interest. t the equilibrium point, = =, irrespectie o the alue o w.,, we hae, σ, and then Now, b Mean Value heorem or each component o the ector unction i (, ) i (, ) i (, ) = + (, ) = ( ςi, ςi ) (, ) = ( ςi, ςi ) where i 1, 2,, n = and ( i, i ) ς ς is a point on the line segment connecting (, ) For conenience, the unction, (, ) (, ) = θ + θ to the origin. can be written as a compact ormulation, that is, (7) Figure 2. Eample cures o one component o the unctions belonging to the unction set F. 176

5 where, =, 1 2 n = 1 2 n θ θ θ θ θ θ θ θ. hus, b the bound o partial deriatie o unction,, we can ensure that the matrices θ belong to the amilies o interal matrices, respectiel, that is, on θ,,, θ θ θ θ θ. In the same wa, we obtain, (, ) = +, = ( ) ϕ β σ θ θ φ σ φ σ θ σ σ θ and hereore, b c β β ( σ ) >, and Deinitions 1 and 2, we hae, i i θ,,, θ θ θ θ θ, θ φ θφ, θ φ, and then substituting them and (8) into (7), obtain, where Now, deining [ ] = θ + θ ε = gk gk gk θ ( σ σ ) + ε( ) ε( g g ) g σ = θ + θ σ φ (,, ), (,, ), (,, ), and (,, ) = w = w g= g w g = g w. = σ σ, = h, (9) h = K K K Kσθ φ( σ σ ), and then the closed-loop sstem (9) can be rewritten as, = + δ d = Λ + εδ dτ (, ) where θ θ K K θ K Kσθ φ =, θ θk K θk Kσθφ (1) Λ= gk, τ = t ε, and In the absence o δ and (, ) δ θ = θ, 1 (, ) ( ) ( g g ) g h δ =. δ, the asmptotic stabilit o the closed-loop sstem (1) can be achieed b designing the interal matrices and Λ are all Hurwit stable [9]. hus, b linear sstem theor, two quadratic Lapuno unctions, V = P (11) V = P (12) Λ 177

6 can be obtained, respectiel. Where P and P Λ are the solutions o Lapuno equations P + P = I and P Λ+Λ P = I, respectiel. It is obious that P and P Λ are all interal matrices, that is, Λ Λ P P, P and P P, P. Λ Λ Λ Based on the Lapuno Functions (11) and (12), a composite Lapuno unction candidate [1] or the closed-loop sstem (1) can be written as, ( ) V, = 1 d V + dv < d < 1 (13) Obiousl, Lapuno unction candidate (13) is positie deine. hereore, our task is to show that its time deriatie along the trajectories o the closed-loop sstem (1) is negatie deine, which is gien b, (, ) = ( 1 ) + = ( ) + ( ) δ + ( ) δ + δ ( ) + δ ( ) V d V dv 1 d 1 d P 1 d P d d P, d, P. B deinitions o h( ) and φ( σ ), we hae, Λ = φ( σ) h K K K K σ φ( σ) φ σ φ( σ) = σ = β σ σ σ and then substituting φ( σ),, and = + h( ) into h( ) and 2, we hae, (, ) In addition, b θ θ, θ, θ θ, θ where γ δ, γ δ, and γ δ are all positie constants. Substituting (15) and (16) into (14), obtain, where δ (, ) (14), and using ssumptions 2, and Deinitions 1 δ γ + γ (15) δ and deinition o δ δ γ δ, obtain, 2 2 (, ) ( 1 ) ( ) 2( ( 1 ) 1 2) (16) ε γ + β + β = ξ ξ (17) 1 V d d d d Q 1 δ P P, P 2 δ PΛ PΛ, PΛ ( 1 ) ( 1) ( ) ( Λ ) β = γ P, P = Ma P, β = γ P, P = Ma P, γ = 2 γδ P,, ξ = d d β1 dβ2 Q =. ( d ) β1 dβ2 d( ε γ ) he right-hand side o the inequalit (17) is a quadratic orm, which is negatie deine when, ( 1 d) d( ε γ) (( 1 d) β ) 2 1 dβ2 > + (18) which is equialent to, ε < ε = d ( 1 d) d ( 1 d) dγ + (( 1 d) β ) 2 1+ dβ2 (19) 178

7 B the dependence o [1] and is gien b, ε d on d, it is obious that the maimum alue o 1 2 d B. S. Liu et al. ε occurs at d = β ( β + β ) ε < ε d = (2) γ + 4ββ It ollows that the origin o closed-loop sstem (1) is asmptoticall stable or all ε < ε. Consequentl, b = and =, we hae =, σ = σ and = h( ) =. his means that the closed-loop sstem (7) is asmptoticall stable, too. his established the ollowing heorem. heorem 1: Under ssumptions 1 and 2, i there eist gain matrices K, K and K σ such that the interal matrices and Λ are all Hurwit stable and the ollowing inequalit, m ( gmkσ ( aφ) ) (,, w) λ φ > ε (21) holds to ensure that there eist positie constants d,,σ is an eponentiall stable equilibrium point o the closed-loop sstem (7) or all ε < ε. Moreoer, i all assumptions hold globall, and then it is globall eponentiall stable. Discussion 1: It is not hard to see that: Just using singular perturbation technique, two ke points in stabilit analsis are soled, that is, one is that it decomposes the whole sstem into two interconnection subsstems such that it is er eas to obtain two quadratic Lapuno unctions; another is that it deries the condition on the controller gains to ensure the asmptotic stabilit. hereore, although mean alue theorem, stabilit theorem o interal matri, singular perturbation technique and Lapuno method are all indispensable components, singular perturbation technique plas a decisie role. his is wh our design method is called as singular perturbation one. Discussion 2: B special concerns o the equation and its matri, it is not hard to ind that the error o integrator output σ σ appears in not onl the sstem dnamic but also the integral dnamic σ. his results in that integral dnamic σ has the same ormulation as the sstem dnamic. hus, the error o integrator output can be used to stabilie the sstem, just like the sstem state. his means that in stabilit analsis, the integral dnamic plas a positie role and it does not need to take an etra and special eort to deal with it. s a result, this is a highlight point o this paper. Discussion 3: Compared with general integral control proposed b [2]-[8], the main dierences are that: 1) he integrator and integral action here are all generalied b two unction sets, respectiel. Howeer, the are all particular in [2]-[4]; 2) s like reerence [7], the integrator here increases a positie deine ector unction β( σ ) on base o the integrator presented b [8], which can be used as an additional reedom o degree to improe the integrator perormance. Howeer, it is not completel reedom and mainl used to construct the bound condition in [7]; 3) Control design here is achieed b snthesiing singular perturbation technique, mean alue theorem, stabilit theorem o interal matri and Lapuno method. Howeer, in reerence [2]-[8], the designs are achieed b linear sstem theor, sliding mode technique, eedback lineariation technique and Lapuno method, respectiel; 4) For the stabilit analsis, the integral dnamic here not onl plas a positie role but also its negatie eects can be eectiel attenuated b decreasing ε. Howeer, the integral dnamics not onl almost hae not positie actions in [5]-[8] but also there no eectie method was proposed to deal with its negatie eects; 5) heorem 1 is not suspended in midair but is established on a solid oundation. Howeer, stabilit theorems [5]-[8] were all deeloped b resorting to an ordinar control along with a known Lapuno unction. Discussion 4: From the design procedure aboe, it is obious that: First, the equation in (1) is transormed into the singular perturbation orm such that singular perturbation technique can be used to attenuate the nonlinearities and/or uncertainties, δ (, ). Second, b mean alue theorem, the nonlinear terms, (, ), φ( σ ) and β( σ ) (, ) are all reormulated as the linear orms on the interal matri such that stabilit theorem o interal matri can be used to deal with them. Finall, b Lapuno method, a uniersal theorem to ensure regionall as well as semi-globall asmptotic stabilit is established in terms o some bounded inormation. ll o them snthesie a sstematic method to design general integral control with the generic integrator and integral control action. Speciall, the error o integrator output can be used to stabilie the sstem, just like the sstem state. ll those mean that the design method here can more eectiel deal with nonlinearit and uncertaint o dnamics, and then makes the engineers more easil design a stable controller. ε and ε, and then 179

8 4. Eample and Simulation Consider the pendulum sstem [1] described b, θ = asinθ b θ + c where a, b, c >, θ is the angle subtended b the rod and the ertical ais, and is the torque applied to the pendulum. View as the control input and suppose we want to regulate θ to r. Now, taking = θ r, = θ, k = 3 and k = = 9, general integral controller can be written as, k σ ( ( )) u = ε σ + tanh σ, σ = 5 + sinh + + tanh and then the pendulum sstem with the normal parameters a = c = 1 and b = 1 can be written as, = + δ d = 3 + εδ, dτ ( ) where,, 3 6 = 2, 3 3] = ], (22) δ 1 = 2, (, ) = ( sin ( + ) sin ) δ φ σ r r. B stabilit theorem o interal matri [9], it is eas to eri that the interal matri is Hurwit stable. hus, b soling Lapuno equation P + P = I and PΛΛ+Λ PΛ = I, the maimum alues o P and P Λ can be obtained, respectiel, that is, and then using δ, P.177 and P and 5 δ, we hae, β 1 =.4, β 2 = 1.74 and γ =.47. ε < and [ ] hereore, the stabilit o the closed-loop sstem (22) can be ensured or all.3 r π, π. For illustrating the perormance o controller aboe, the simulations are achieed under normal and perturbed parameters, respectiel. he normal parameters are a = c = 1 and b = 1. In the perturbed case, b and c are reduced to.25 and 2.5, respectiel, corresponding to our times the mass. Figure 3 showed the simulation results under normal (solid line) and perturbed (dashed line) cases. he ollowing obserations can be made: the optimum response in the whole domain o interest can all be achieed b a set o the same control gains, een under the case that our times paload changes. his demonstrates that although the design method here is too conseratie, general integral controller, which is tuned b onl the normal parameters, has superstrong robustness, ast conergence, and good leibilit and can deal with nonlinearit and uncertainties o dnamics more orceull. 5. Conclusions his paper proposes a sstematic method to design general integral control with the generic integrator and integral control action. he main contributions are that: 1) B mean alue theorem, the nonlinear actions in the subsstem and integral dnamics are all reormulated as the linear orms on the interal matri such that stabilit theorem o interal matri can be used to deal with them; 2) he error o integrator output can be used to 18

9 Figure 3. Sstem output under normal (solid line) and perturbed case (dashed line). stabilie the sstem, just like the sstem state, such that it does not need to take an etra and special eort to deal with the integral dnamic; 3) No longer resorting to an ordinar control along with a known Lapuno unction, but snthesiing singular perturbation technique, mean alue theorem, stabilit theorem o interal matri and Lapuno method, a uniersal theorem to ensure regionall as well as semi-globall asmptotic stabilit is established in terms o some bounded inormation. Consequentl, this uniersal theorem is not suspended in midair but is deeloped with a solid oundation. Simulation results showed that general integral controller, which is tuned b this design method, has superstrong robustness and can deal with nonlinearit and uncertainties o dnamics more orceull. Reerences [1] Liu, B.S. and ian, B.L. (29) General Integral Control. Proceedings o the International Conerence on danced Computer Control, Singapore, Januar 29, [2] Liu, B.S. and ian, B.L. (212) General Integral Control Design Based on Linear Sstem heor. Proceedings o the 3rd International Conerence on Mechanic utomation and Control Engineering, Baotou, Jul 212, Vol. 5, [3] Liu, B.S. and ian, B.L. (212) General Integral Control Design Based on Sliding Mode echnique. Proceedings o the 3rd International Conerence on Mechanic utomation and Control Engineering, Baotou, Jul 212, Vol. 5, [4] Liu, B.S., Li, J.H. and Luo, X.Q. (214) General Integral Control Design ia Feedback Lineariation. Intelligent Control and utomation, 5, [5] Liu, B.S., Luo, X.Q. and Li, J.H. (213) General Concae Integral Control. Intelligent Control and utomation, 4, [6] Liu, B.S., Luo, X.Q. and Li, J.H. General Cone Integral Control. International Journal o utomation and Computing. [7] Liu, B.S. (214) Constructie General Bounded Integral Control. Intelligent Control and utomation, 5, [8] Liu, B.S. (214) On the Generaliation o Integrator and Integral Control ction. International Journal o Modern Nonlinear heor and pplication, 3, [9] Krans, F.J. and Mansour, M. (1991) Suicient Conditions or Hurwit and Schar Stabilit o Interal Matrices. Proceeding o the 3th conerence on decision and control, Brighton, December 1991, [1] Khalil, H.K. (27) Nonlinear Sstems. 3rd Edition, Electronics Industr Publishing, Beijing,

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