AIAA 2002-4759 LPV Controller Interpolation for Improved Gain-Scheduling Control Performance Fen Wu Dept. of Mechanical and Aerospace Engineering North Carolina State University Raleigh, NC 27695 SungWan Kim Dynamics and Control Branch NASA Langley Research Center Hampton, VA 23681-2199 AIAA Guidance, Navigation, and Control Conference August 5-8, 2002/Monterey, CA For permission to copy or republish, contact the American Institute ofAeronautics andAstronautics 1801Alexander Bell Drive, Suite 500, Reston, VA 20191-4344 LPV Controller Interpolation for Improved Gain-Scheduling Control Performance Fen W_* Dept. of Mechanical and Aerospace Engineering North Carolina State University Raleigh, NC 27695 SungWan Kim* Dynamics and Control Branch NASA Langley Research Center Hampton, VA 23681-2199 In this paper, a new gain-scheduling control design guaranteed stability and performance properties, and approach is proposed by combining LPV control the- simplifies the interpolation and realization problems ory with interpolation techniques. The improvement associated with conventional gain-scheduling meth- of gain-scheduled controllers can be achieved from lo- ods. However, one potential problem associated with cal synthesis of Lyapunov functions and continuous parameter-dependent Lyapunov function approach is construction of a global Lyapunov function by interpo- the lack of guidance for choosing "right" basis func- lation. It has been shown that this combined LPV con- tions to parameterize infinite dimensional functional trol design scheme is capable of improving closed-loop space. performance derived from local performance improve- Interpolation is an important step toward synthesis ment. The gain of the LPV controller will also change of gain-scheduled controllers and has not been ade- continuously across parameter space. The advantages quately addressed in a systematic way. Some ad-hoc of the newly proposed LPV control is demonstrated interpolation techniques have been proposed in the through a detailed AMB controller design example. past: (1) linear interpolation of poles, zeros, and gains of local controllers; (2) linear interpolation of solu- Introduction tions of Riccati equations; and (3) linear interpolation The gain-scheduling approach is perhaps one of the of state-space matrices of balanced controller real- most popular nonlinear control design techniques that izations. These approaches are intuitively appealing has been widely used in fields ranging from aerospace but could generate destabilizing controllers. Stilwell to process control. Although it seems to work well in and Rugh 9 proposed a gain-scheduled control design practice, this heuristic design procedure does not take based on interpolation techniques. The interpolated the parameter variations into account, x'2 In its early parameter-varying controller preserves point-wise sta- practice, the control design came with virtually no bility of local LTI controllers for all frozen parameter guarantee on performance, robustness, or even nom- values. However, the global control does not provide a inal stability. Recently, a systematic gain-scheduling priori stability and performance guarantee fl'om locally design technique was developed in the form of linear designed controllers for fast time-varying parameters. parameter-varying (LPV) control theory. 3 8 This class In this research, we propose an interpolating LPV of systems is different from the standard linear time- control approach by combining LPV control theory varying counterpart due to the causal dependence of with interpolation techniques. The interpolated LPV its controller gains on the variations of the plant dy- control design is capable of improving controlled per- namics. The implications of parameter-dependent sys- fornmnce by finding the most appropriate Lyapunov tems theory for gain scheduling is obvious because gain functions in a local sense. Moreover, the local stabil- scheduling conceptually involves a linear, parameter- ity property will be extended to the entire parameter dependent plant. The LPV design technique provides range by a globally constructed Lyapunov function through interpolation. The proposed interpolating *Assistant Professor, AIAA Member. E-mail: fwu©eos.ncsu.edu, Phone: (919) 515-5268, Fax: (919) LPV control technique has great potential to many 515-7968. This research was supported by Grant from NASA industrial applications including active magnetic bear- Langley Research Center (NAG-I-01119). ings (AMBs), for which the controller adjusts its gain lResearch Engineer, AIAA Senior Member. based on changing rotor speed and provides accurate Copyright @ 2002 by Fen Wu. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. control of flexible mode variation in terms of natural 1OF9 AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2002-4759 fl'equency, damping, and mode shape. In this research, matrix function Qx(p) >_ O, if there exists a group we use a simple rigid magnetic bearing example to of continuously diffcrentiablc positive-definite matrix demonstrate advantages over existing gain-scheduling functions Xi(p), i = 1, 2 such that for any p E 7)i, approaches. Improved LPV Analysis Condition + ] Consider an LPV system -vd DT(p)| C(R) D(p) -'_iI J '(t)] [A(p(t)) B(p(t))] [x(t)] (4) e(t)] = LC(p(t)) D(p(t))] [d(t)J (1) and for" all p ¢ T)12 where x,a_' ¢ R '_, d ¢ R '_d, e ¢ R _. All X2(p) - Xs (p) + (b - a)Qx(p) >_0 (5) the matrices have compatible dimensions. It is as- X2(p) - Xs(p) - (b- a)Qx(p) <_0 (6) sumed that the vector-valued parameter p evolves continuously over time and its range is limited to then the LPV system (1) is exponentially stable and a compact subset P C R*. Its time derivative is its induced fl--2norm with x(O) = 0 is bounded by 7 = bounded and satisfies the constraint -ui <_ fli <_ rnax{Ts, 79}. ui,i = 1,2,.-. ,s. For notational purposes, F := {v : -ui <_vi _<ui, i = 1,2,.-. ,s}, where Y is a given Given the above conditions over each parameter sub- convex polytope in R* that contains the origin. Given set, a continuous Lyapunov function in the form of the sets P and V, we define the parameter u-variation set as b-p p-a (7) 5_;={p•dS(R+,R*): p(t) • P, fl(t) •1;, Vt > O} (2) can be constructed and used to verify the desired sta- Therefore the dynamics of the LPV system are char- bility and performance properties of the system for any acterized by the parameter value p and its variation parameter trajectories within 3_. Then a non-smooth along time. Previous research on LPV control the- dissipative system theory is applied to address possible ory mainly focused on a single Lyapunov function discontinuity of the Lyapunov function derivative at (quadratic or parameter-dependent) over the entire the boundary of the parameter subsets, l° The perfor- parameter set. For the given LPV system, it is clear rnance bound derived in Theorem 1 states the "worst- that the achievable performance relies on the choice of case" performance, which is not often achievable. For the Lyapunov function. However, it would be bene- example, if the parameter trajectory is constrained in ficial if one can analyze the performance of the LPV the subset pi, then the system's performance is over- system over different parameter ranges using different bounded by "7i, which could be less than max {%, %}. Lyapunov functions with a stability guarantee. Theoretically, this theorem simply states the fact To simplify the presentation, let us assume that the that the performance level of the LPV system relates parameter set has dimension one. That is, 7) C R s. to the existence of a continuous Lyapunov function. Suppose Ps,P2 is an overlapped partition of the pa- However, in practical situations, it is usually hard rameter set P, and define the intersection of 7)s and to identify a suitable parameter-dependent Lyapunov T)2 aS function without resorting to a global search. In this _12 : 77)1 _ T)2 (3) sense, the improved analysis condition will be help- ful to find a sharper performance bound through local Then the parameter space 5° is the union of two subsets studies. Then the global Lyapunov function will be _1, p2. Through linear interpolation of the Lyapunov constructed by interpolation. functions Xi(p),Xs(p) over subsets _ol and 5°2, one would obtain a continuous Lyapunov function over Interpolating LPV Control Synthesis the entire parameter space. The continuity property Given a generalized open-loop LPV system of the Lyapunov function is important to guarantee monotonic decrease of its value over any allowable tra- 1 [A(p(t)) BI(P(t)) Bz(P(t))I [x(t)l jectories dictated by the LPV system dynamics. A 4t)I = Dll(p(t)) Ds'e(p(t))| |d(t)| new LPV stability and performance analysis condition y(t)J L&@(t)) D21(p(t)) D22(p(t))J kn(t)J based on the partitioned parameter subsets is then pro- (8) posed. where all of the state-space data are continuous func- Theorem 1 For the parameter set P = [p_,p] with tions of the scheduling parameter p, it is assumed that its overlapped partition 7)s = [p,b] and p2 = [a,p] the parameter trajectory resides in the set 5_. For (a < b), the LPV system (1), and a given symmetric simplicity, we assume that 2OF9 AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2002-4759 (A1) (A(p), B2(p), 6'2(p)) is parameter-dependently Theorem 2 For a partition of parameter space 7) = stabilizable and detectable for all p, __, p] as subsets _:)1 = JR, b] and 7)2 = [a, p] (a < b), and given QR, Os >_0 matrix functions, if one of the (A2) Dx2(p) and D2x(p) have full column and row following equivalent conditions are satisfied: rank respectively for all p, 1. there exist continuously differentiable matrix fanc- (A3) Dis(p) = 0 and D22(p) = 0. tions Ri(p),Si(p), i = 1,2 such that for p E 79i We consider the class of interpolating LPV con- troller in the form of A_ +AR_ + R_A YR <0 C1]_i --7i I 0 = LCk(p(t),/_(t)) Dk(p(t),l)(t))] Ly(t) J B T 0 --7i I L_\_/ j (9) (12) Illgeneral,the controllergain isacontinuous function of parameter p and its derivative. The control de- A_T +A_Si + S_/ A;s < o sign objective is to minimize the energy (E2 norm) of BT si --7i I the output signal e(t) of the closed-loop system in the [ { _[_/] ((_, _}_0SI )} _i1_1 CoT] presence of bounded energy disturbance d(t). Specif- C1 0 -7ilJ (13) ically for the magnetic bearing control problem, we would like to synthesize a gain-scheduled control with combined disturbance rejection, gyroscopic compensa- [R_p) S_{p)] _>0 (14) tion, and automatic balancing capability. This can be with_(p) = KerIBm(p) Dl_(p) 0],a:_(p)= formulated as an optimization problem with minimiza- tion of displacement of the rotor fi'om its centerfine Ker [C2(p)D21 (p)O],andforpeT _12 at selected points subject to unknown torque distur- /_2(P) -- /_I(P) -- (b - a)QR > 0 (15) bances. J_2(P) -- ]_I(P) -- (b - a)0 R < 0 (16) Define xTd := [xg xTk], then the closed-loop system can be written as S2(p) -- Sl(p) + (b -- a)QS > 0 (17") S2(p) -- Sl(p) -- (b -- a)Os < O. (18) LCc_(p,p) Dc,(p, fl)J 2. there exist continuously differentiable matrix func- tions Ri(p),Si(p) > 0, i = 1,2 and continuous where matrix functions _(i) j_(i) _(i) == 1, 2 such that ]:o°o k, k ,_k ,_ for p E 7)i, LG, D_lJ C 1 0 Dll +b_)c_ + (.) oB2 o] + I 0 Ck Dk C2 0 D21 0 D12 +AR_+B_0_/+ (,)J (11) BT& + Dr2/_1(or BT n ,_(i) Next, we propose a synthesis condition for an Cs ClJ_i + L"12'-_ k improved LPV controller using multiple parameter- c: SiB1 -- 1_i)D21 dependent Lyapunov functions and an interpolation f%(i)T I3T B1 RicT1 + "_k _12 <0 scheme. For clarity, only one parameter is considered --7i I 0 and the parameter space P is covered by two over- 0 --7i I lapped subsets 77)1 and p2. For each parameter subset, (19) we seek to design one LPV controller as stated in the form of eqn. (9). The overall gain-scheduling con- (20) troller is then constructed by interpolating local LPV controllers. It is clear that the global controller is ca- and for p E 5D12 pable of achieving tighter performance due to smaller R_(p)- Rl(p)+ (b- a)QR>0 parameter range. However, a critical issue associated (21) with the proposed controller interpolation scheme is R_(p)- Rl(p) - (b- a)QR<0 (22) the stability of the global LPV controller. This will (23) S2(p) -- Sl(fl) + (b - a)Qs >_0 be guaranteed by constructing a globally continuous Lyapunov function over the entire parameter set. S2(p) -- SI(p) -- (b - a)Qs <_O, (24) 3OF9 AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2002-4759 then the closed-loop LPV system can be stabilized by is necessary and the amount of allowable change de- a continuous LPV controller with induced £2 pcrfor- pends on the maximal parameter variation. When mancc less than 7 = max{71,72 }- u + 0 (slowly varying case), the conditions (16)-(18) Furthermore, let will have no effect on the synthesis result. Thus the synthesis condition over each subset is essentially de- p<a coupled and can be solved independently. 7(p)= b_-p 71+ _p-a 72 a<p<b The solvability conditions in eqns. (16)-(18) are clearly infinite-dimensional, as is the solution 72 p_b space. To approximate, we restrict the search of the parameter-dependent Lyapunov function to a span of finite numbers of basis functions. That is, let ]_1 (P), $1 (P) p<a (b-b-a--)p {/_l(fl),_l(P)} NI Ng a<p<b re(p) : Z k(P)tgk,, s,(p) : Za k(p)k& (35) n(p),s(p) = { + {a(p),s2(p)} k 1 k 1 n2(p),&(p) p_b (26) where {f_ (p) }N*1 and {gik(P)}kN_1 are user-specified scalar basis functions. Note that different basis func- and M(p)NT(p) := I - R(p)S(p), calculate tions may be used over different parameter subsets. R_,,S) are new optimization variables to be deter- T dMT mined. After such a parameterization, the LPV Ak(p, fi) = S(P)_t + _ (p)_ AT(p) synthesis conditions can be solved using a gridding 1 method over each parameter subset. 7(P) [S(p)Bl(p) + Bk(p)D21(p) CT(p)] V BI (p) ] Advanced AMB Gain-Scheduling × (27) LC1 (p)i_(p) -FD21 (p)Ck (p)] Control /_k(P) = - [7(P)C T(P) + S(p)B1 (p)DT1 (p)] Active magnetic bearings (AMBs) use an electro- magnetic force to provide noncontact support for to- x [D2s (p)DT1 (p)] -s (28) tors in high-speed rotating machinery. AMBs have Ok(p) = - [DT12(p)D12 (p)] -1 several unique characteristics that make them well suited for aerospace applications: they provide non- x [7(p)BT(p) + DT2(p)C] (p)R(p)] (29) contacting support of the flywheel rotor, virtually &(p) : o (30) eliminating bearing friction and wear, and they elim- inate concerns with lubrication, heating, and power Then the interpolated LPV controller Kp will be con- consumption typical of standard bearing systems. Ad- structed as ditionally, AMBs can effectively eliminate synchronous vibrations associated with mass imbalance and shaft Ak(p,/5) = N-'(p) {Jk(p, ,5)- S(p)B2(p)Ok(p) run out, making them highly desirable for spacecraft applications where pointing accuracy is critical. Mag- -J_k (P)C2 (p)R(p) - S(p)A(p)R(p)} M-T(p) netic bearings are well suited as flywheels in replace- (31) ment of chemical batteries to store energy on a space- Bk(p)= :v (p) (32) craft, ]1 and they provide integrated attitude control and momentum management functionality. 1214 Al- Ck (p) = C,_(p)M- T(p) (33) though this technique has tremendous potential for a Dk(p) = 0. (34) variety of industrial applications, AMBs are open-loop unstable, thus making the controller design problem a It is noted that we ask for different performance challenge. Moreover, the flexibility of high-speed ro- levels over each parameter subregion. The proposed tors adds to the complexity of control design task. interpolation scheme could provide a global stabiliz- Most controllers in use today for magnetic bear- ing LPV controller with the potential to improve its ings were designed using PID strategies. However, local performance. The global Lyapunov function for it is difficult to satisfy the stringent performance re- the closed-loop LPV system is derived from the matrix quirernents with PID control. Mohamed and Busch- functions R_(p) and S_(p). Because of the interpola- Vishmiac la and Matsumura et al.16 designed gain- tion scheme used, it relaxes continuity requirement of scheduled 7-/o_ controllers for rigid magnetic bear- individual Lyapunov functions over the intersections ings utilizing the stabilizing controller parameteriza- of parameter subsets. In particular, only finite vari- tion with the free parameter Q playing the schedul- ation of Lyapunov functions over intersected regions ing mechanisms. Unfortunately, this ad-hoc gain- 4OF9 AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2002-4759 scheduling design technique suffers from slow vary- Parameter Value ing parameter requirements. The resulting gain- A area of each pole 1531.79mm 2 scheduling controller could render the closed-loop sys- h pole width 40.OOmm tem unstable when the rotor speed changes rapidly, s6 Go nominal gap 0.55ram Rotor flexibility was considered using the #-synthesis Jr radial moment of inertia 0.333kg •rn2 technique, sr It was shown that the # controller exhib- J_ axial moment of inertia O.O136hg •rn2 ited much greater stability robustness to variations in half the length of the shaft O.13m rotor mass. However, the controller caused significant k 4.6755576 x 108 performance degradation when the actual rotor speed N number of coil turns 400 differs from the pre-specified design condition. One R coil resistance 14.6Ohm way to address this problem is to design a series of con- _0 nominal airgap 2.09 x lO-4Wb trollers for each operating speed and then interpolate Table 1 Magnetic bearing parameters. between these controllers. 16 However, this approach does not provide any stability and performance guar- antee. In Tsiotras and Mason 18 and Tsiotras and In the absence of disturbances and modeling errors, Knospe, s9 the LPV control theory was applied to ad- the above equation specifies an equilibrium. Lineariz- dress variable rotor operating speeds using a rigid rotor ing the nonlinear equations at the equilibrium, we obtain model and conservatism of LPV control design was recognized. Similar controllers were also developed by Sivrioglu and Nonami. 2° = - _ + T(-4c_gO + 2c_0o + fdo) (4o) Rotor Dynamics Modeling ;} = P&O + g (-4c_g_ + 2c1_ + Ida) (41) Owing to the linear dependence of the rotor speed in the plant dynamics, the nonlinear gyroscopic equa- NOo = co + 2d2gO - dlOo (42) tions of AMB can be simplified to a set of linear, N_ = e_ + 2d2g_q)- ds0g, (43) time-varying differential equations. However, the ro- tor dynamics are inherently unstable; that is, even where 00, q_ are the differential magnetic flux fi'om small unbalanced masses can create large synchronous electromagnetic pairs, and e0, ce are the correspond- disturbances with the same frequency as the rotor's. ing differences of electric voltage. The constants Therefore it is necessary to develop a gain-scheduled cs,c2,dl,d2 depend on _o, Go, R, A, N, uo and the ge- controller capable of rejecting the periodic disturbance ometry of the bearings as follows as the rotor speed changes. The rotational motion of a magnetic bearing can be 2Co) (44) derived from its rigid body dynamics, which issa'18 Cl = 2k_o 1+ _-j , c2 - rch ' 2RGo 2R¢Po Jr dl - uoAN' d2- uoAN' m= 7-. (45) 0= -- _;'q- _7(frl -- fr2 q- re2 -- fgl q-fdo) (36) The imbalance forces fao and fd_ are typically mod- _;= PJ"ojr+ f7 (re3 -- fr4 + fg4 -- ft3 + fd_) (37) eled as sinusoidal disturbances, and are given by where 0,_ are the Euler angles denoting the orienta- tion of the rotor centerline. J,_,J,. are the moment of fdo -- (Jr -g J_,)p2Tcos(pt) ' (46) inertia of the rotor in axial and radial directions, re- (Jr - J¢,)p2Tsin(pt)" (4r) spectively. The parameter p denotes the rotor speed. fdW - g The magnetic forces generated by four pairs of elec- In automatic balancing design, the imbalance forces tromagnets are denoted by f,.i,fei for i = 1,2,3,4. will be treated as a sinusoidal sensor noise nr = f_,o,fdv, are disturbance forces caused by gravity, mod- [eTsin(v* + a) eTCosIpt + a)] on the measured rotor eling errors, imbalances, etc. displacement, which can be written in a state-space The electromagnetic force fj is related to the voltage form of ej across the jth coil through the magnetic flux _¢jby the equations (38) fj =/_j 1+ _h/ 2R (39) where _ = -0.001. The following system parameters are chosen for the active magnetic bearing example (Table 1). and ur = [e0 ee]. Combining rotor dynamics and the 5OF9 AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2002-4759 disturbance equation, the linearized equation can be written as :b = A(p)x + Blw ÷ B2u (50) d Z = Clx ÷ Dllw ÷ D12u (51) G y = C2x ÷ D2sw ÷ D22u (52) I with 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 _ 4(-'2 0 0 _v& 2c_ 0 0 0 7n J_ m 0 4c2 _ 0 0 2c_ 0 0 A= ?I_ Je 2d2 0 0 0 N 0 0 0 dNl 0 2d_ 0 0 0 N dNl 0 0 Fig. 1 Weighted open-loop interconnection for the 0 0 0 0 0 0 _ -p magnetic bearing system. 0 0 0 0 0 0 p rr (53) range. The LPV synthesis problem can be solved us- ing either a single or parameter-dependent quadratic Fo4x2] B1 _ F/L°_o1×1/J1, B2 x11i I (54) Ldiymapenusnioovnal funpcatriaomneter overspaalcle)'griddsinTghe ppoerifnotsrmanicne a onoeb-- tained through a single quadratic Lyapunov function D12= [02x2] (SQLF) approach is 7 = 320.78, which is quite con- C1= [&02×8%×_lj'Dl1=04×1, L_2J' servative. Also, multiple LPV control syntheses are (55) conducted with the whole parameter space divided into two subsets ([315,720] U[700, 1100]) and four sub- 6'2= [/2 02×4 /2], D21 =02×1, D22=02x2 (56) sets ([315,520] U [500,720] U [700,920] U [900, 1100]), respectively. Then five points are used to grid each Note that the linearized rotor dynamic equation is in parameter subspace uniformly. The free parameters the form of a linear paralneter-varying system with QR and Qs are chosen as 0.111.) for p E pi NPJ or the rotor speed serving as the parameter. Fox" gain- zero otherwise. In each partitioned parameter space scheduled control, the rotational speed is assumed to case, two sets of identical basis functions are used to be available in real-time fox" controller use. parameterize the functional space, Interpolating LPV Control Design f/1(p) = 1, (60) The design objectives of the LPV controller are g_,(p) = 1, g_(p) = p (61) to asymptotically stabilize the system over the whole range of rotor speeds and to minimize an error signal for each i. Note that the basis function for Ri(p) representing a weighted sum of the forces at the bear- is chosen as a constant over each parameter subset. ings, the gap displacement at the bearing locations, This will only cause small performance degradation and the control input used. compared with more complicated basis function se- lections. Since both R and S become functions of Although the scheduling variable is time-varying in the LPV dynamics, it is simply treated as a fixed pa- scheduling parameters after interpolation, the inter- rameter in the design stage. The design objective is polated LPV controller gain will depend on both the quantified fl'om a frozen parameter design viewpoint parameter and its derivative. The performance level by weighting functions, and the weighted open-loop of the interpolating LPV controller versus number of interconnection is given in Figure 1. parameter subsets under different parameter variation In Figure 1 the weighting functions are chosen as rates is shown in Table 2. Note that the "7 value represents the "worst-ease" interpolating LPV control 200@ ÷ 100)/2, (57) performance over different parameter subsets. The ac- _(_)- _7O.Ol tual performance over each parameter subset could be O.O01s less than the "worst-ease" performance. w_(s) - 0.055+ 1&' (58) For comparison, a single parameter-dependent Lya- W,,(s) = 0.001 (59) punov function is also considered to demonstrate the sharpness of the newly proposed LPV synthesis condi- The rotor speed is assumed to change freely be- tion. The calculated performance bound is about the tween 315rad/s to llOOrad/s. The rotor dynam- same as the "worst-ease" performance of interpolated ics exhibit significant gyroscopic effects in this speed LPV cases. 6OF9 AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2002-4759 Variation rate PDLF Interpolating LPV xl0 3 2 subsets 4 subsets 1 1 9.863 9.936 9.954 _o5 100 10.064 10.107 10.149 10000 141.24 144.43 143.50 1f L/ 15 Table 2 Achievable induced-£2 performance us- 20 00I5 01I 0'15 0!2 025' ing single and multiple parameter-dependent Lya- Time (s) punov functions (PDLFs). xlO 3 Simulation Results 5 Next, the LPV controller synthesized using two sub- sets is used for comparison study and simulation work. To compare the LPV controller with the optimal 01 015 02 025 7-to_controller, eight points are chosen fl'om the param- Time (s) eter set P and 7-to_ controller is synthesized for each (a) rotor displacement Xl,.Z'2 operating condition. Then their 7-/0, norm values are compared with the sub-optimal control performance that is achieved by LPV controllers evaluated at fixed ° parameter values. These are provided in the Table 3. 5o p(rad/s) ?-/o_ Interpolating LPV 315 9.778 9.833 01 015 02 025 03 Time (s) 427.1 7.487 8.147 539.3 6.149 6.719 651.4 5.289 5.772 200 3oo 763.6 4.696 4.988 100 875.7 4.264 4.831 2 0 987.9 3.936 4.600 100 1100 3.678 4.383 200 01 015 02 025 03 Time (s) Table 3 Frozen optimal/LPV closed-loop 7-{_ (b) control force (Ul, ?t2 norm. Fig. 2 Fixed parameter unit-step response with 7-to_optimal controller. Since the LPV controller is designed for a range of parameters, it is not surprising that they are only sub- tions for AMB with time-varying rotational speed. A optimal for each fixed parameter value. However, it time-varying rotor speed profile is chosen as is observed that the optimal performance level using 7-to_ is very close to the achievable performance de- d(t) = sin(wt), w=655+50t 0.5s_<t< 1.5s rived from switching the LPV controller at each frozen sin(680t) 0 < t < 0.5s parameter. However, the optimal controller is highly sin(730t) 1.5s _<t < 2s (62) tuned to its designated rotor speed. When a magnetic bearing operates at a different rotational speed, signif- Note that the rotor speed trajectory is deliber- icant performance degradation or even loss of stability ately chosen to cross the intersection of two subsets is expected. For example, the optimal 7-to_controller [315,720] and [700, 1100] of the scheduling parame- designed for p = 700r'ad/s results in a performance ter to illustrate LPV controller interpolating effect. Disturbance cl = 1.3 × 10-_ is used for the simula- level 7 = 1.16 × 106 when the rotor is actually rotat- ing with the speed of llOOrad/s. tion purpose. Except for small glitches during the controller interpolation period, the simulation results Finally, we compare the closed-loop step response demonstrate good performance of interpolating LPV for both optimal and LPV controllers. The time re- sponses for the optimal controller and the LPV con- control as shown in Figure 4. troller are shown in Figures 2 and Figure 3, respec- tively. It can be seen that the step responses of rotor Concluding Remarks displacement from both controllers are quite similar, In this paper, an LPV control interpolation algo- whereas the control actions are slightly different due rithm was proposed to achieve high-performance of to sub-optimality of LPV controllers. gain-scheduled control. The proposed LPV control After analyzing the frozen point LPV controller approach unified the systematic LPV control theory property, we are ready to do some nonlinear simula- with interpolation technique. The stability of the 7OF9 AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2002-4759 xl0 3 750 1 740 _o5 7S0 720 V 15 710 20 005I 01I 01I5 O2 O25i O3 J700 Time (s) xl0 3 69O 68O 5 67O 66O 02 04 06 08 Timel(s) 12 14 16 18 (a) Rotor speed profile 01 015 O2 O25 :1o' Time (s) (a) rotor displacemen_ 921,X2 ° 5°o_ ¢ 015 02 025 03 Time (s) 3 200 4 02 04 06 08 Timel(s) 12 14 16 18 100 (b) Displacement: xs solid, x._ dash 2 o Fig. 4 Performance of interpolating LPV control 100 for time-varying rotor speed. 200 01 015 02 025 03 Time (s) of providing guaranteed stability and performance for (b) control force us, u2 large flight envelope. Fig. 3 Fixed parameter unit-step response with LPV controller. References 1W.J. Rugh, "Analytical framework for gain scheduling," interpolated LPV control was achieved by enforcing IEEE Contr. Sys. Mag., 11(1):74 84, 1991. continuity of Lyapunov function over entire parameter 2J.S. Shamma and M. Athans, "Analysis of Gain Scheduled space. With modest increase of computational effort, Control Ibr Nonlinear Plants," IEEE Trans. Automat. Contr., the proposed LPV interpolation scheme improved the AC-35:898 907, 1990. LPV control performance considerably. The newly de- ap. Apkarian and R.J. Adams, "Advanced Gain-scheduling Techniques for Uncertain Systems," IEEE Trans. Contr. Syst. veloped LPV control synthesis method was applied to Tech., 6(1):21-32, 1997. the magnetic bearing control problem to reject un- 4p. Apkarian and P. Gahinet, "A Convex Characteriza- balancing sinusoidal disturbances and accommodate tion of Gain-scheduled 7/o0 Controllers," IEEE T_'ans. Automat. changing rotor speed. 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