The aim of this work is to develop an effective numerical tool oriented to optimization of stabilizer parameters according to different criteria that appear in engineering practice. We formulate a special optimization problem that allows us to determine optimal parameters of a stabilizer. The obtained results are applied to choose parameters of a spacecraft stabilization system. We discuss the
choice of optimization criteria comparing the degree of stability objective function, the H _inf norm of the system transfer matrix, and the minimal “peak” criterion.Portuguese National Strategic Reference Framework (QREN)Fundação para a Ciência e a Tecnologia (FCT)European Regional Development Fund (FEDER)Portuguese Operational Programme for Competitiveness Factors (COMPETE

Guerman, Anna

Seabra, Ana M.

Smirnov, Georgi

Universidade do Minho: RepositoriUM

639INTRODUCTIONThe increase dynamic complexity of space missions motivates research on engineeringtools and methodologies capable to improve the performance of space systems. Develop-ment of effective high-precision control systems for large variety of applications, includingﬂexible spacecraft and large space structures, requires parameter optimization of the systemtransient processes.Assume that the differential equation describing the behavior of the system has zero asits asymptotically stable equilibrium position, and depends on a vector parameter. Duringthe system stabilization to the zero equilibrium position, the properties of transition processwould depend on the choice of this parameter. That choice should be done to optimize, insome sense, the behavior of the trajectories. In practice, this parameter can be selectedfrom a number of alternatives, based on various criteria, and it is impossible, of course, toconstruct a stabilizer optimal in all aspects.For example, for a linear controllable system, the pole assignment theorem guaranteesthe existence of a linear feedback yielding a linear differential equation with any given setof eigenvalues. One can choose a stabilizer with a very high damping speed. However,such a stabilizer is practically useless because of so-called peak-effect.2, 6, 9 Namely, thereexists a large deviation of the solutions from the equilibrium position at the beginning ofthe stabilization process, whenever the module of an eigenvalues real part is big enough.∗Professor, Department of Electromechanical Engineering, University of Beira Interior, Calc¸ada Fonte doLameiro, 6201− 001 Covilha˜, Portugal.†Professor, Scientiﬁc Area of Mathematics, ESTGV, Polytechnic Institute of Viseu, Campus Polite´cnico,3504− 510 Viseu, Portugal.‡Professor, Centre of Physics, Department of Mathematics and Applications, School of Sciences, Universityof Minho, Campus de Gualtar, 4710− 057 Braga, Portugal.IAA-AAS-DyCoSS1-07-01AAS 12-343PARAMETER OPTIMIZATION FOR STABILIZERSAnna D. Guerman,* Ana M. Seabra† and Georgi V. Smirnov‡The aim of this work is to develop an effective numerical tool oriented to opti-mization of stabilizer parameters according to different criteria that appear inengineering practice. We formulate a special optimization problem that allowsus to determine optimal parameters of a stabilizer. The obtained results are ap-plied to choose parameters of a spacecraft stabilization system. We discuss thechoice of optimization criteria comparing the degree of stability objective func-tion, the Hnorm of the system transfer matrix, and the minimal “peak” crite-rion.640The same phenomenon is observed if one chooses a robust stabilizer constructed using thedegree of stability concept.10Recently, the𝐻∞ approach has been developed as a tool for analyzing the robustness andperformance of stabilization systems.3 Usually an available model describes the plant onlyapproximately. 𝐻∞- optimal control is a frequency-domain synthesis method developedto solve the problem of inﬂuence of modeling errors. The main idea is to treat the worstcase scenario: it is necessary to design so-called robust controller capable to stabilize allsystems from some class. A robust controller should stabilize the system even in the caseof the largest possible modeling error. Due to the Nyquist criterion this can be formalizedin terms of the minimization of 𝐻∞-norm of the transfer matrix. If a linear system isasymptotically stable, then the maximum of the modulus of its transfer function over theclosed right-half of the complex plane will always occur on the imaginary axis. On theother hand, the smaller 𝐻∞- norm, the bigger the distance between the eigenvalues of theasymptotically stable matrix and the imaginary axis. Therefore, the result of 𝐻∞- normminimization is rather close to that of maximization of the degree of stability.In Reference 5 we develop a new tool to optimize the parameters of stabilizers, accord-ing to different criteria that appear in the engineering practice. We formulate a specialmathematical programming problem that allows us to determine optimal parameters of astabilizer. The objective function and the functions describing the constraints of the prob-lem are deﬁned by the maximum norm of solutions to the system. The functions havestrongly nonlinear nature and numerical methods have been developed to solve the respec-tive optimization problems. In this work we apply the above techniques to analyze theproperties of 𝐻∞-norm as an objective function and compare them with other optimalitycriteria for an example of parameter optimization of a spacecraft stabilization system.Throughout this paper, we denote the set of real numbers by 𝑅, and the usual n-dimen-sional space of vectors with components in𝑅 by𝑅𝑛. By ∣ ⋅ ∣ we denote the Euclidean norm.By 𝐵 we denote the set of vectors in 𝑅𝑛 satisfying ∣𝑥∣ ≤ 1. The conjugate transpose of amatrix 𝐴 is denoted by 𝐴∗.MATHEMATICAL BACKGROUNDConsider a linear sistem of differential equations?˙? = 𝐴(𝑢)𝑥, 𝑥 ∈ 𝑅𝑛, 𝑡 ∈ [0, 𝑇 ], (1)where 𝑢 is a parameter belonging to a compact set 𝑈 ⊂ 𝑅𝑘. Denote by 𝜆𝑖(𝑢), 𝑖 = 1, 𝑛 theeigenvalues of matrix 𝐴(𝑢). Assume thatRe𝜆𝑖(𝑢) < 0, 𝑖 = 1, 𝑛,i.e., all eigenvalues of matrix 𝐴(𝑢) have negative real parts, when 𝑢 ∈ 𝑈 . Let us brieﬂydescribe some well known criteria used to ﬁnd parameters of a linear system.641The Degree of StabilityOne of the most used optimality criteria is the degree of stability. It is deﬁned by𝛿(𝑢) = −max𝑖=1,𝑛Re𝜆𝑖(𝑢).The respective optimization problem takes the form𝛿(𝑢) → max,𝑢 ∈ 𝑈. (2)The problem can also include some additional constraints involving the trajectories of thesystem.The 𝐻∞-normAnother widely used objective function is the 𝐻∞-norm of the system transfer matrix.Denote by𝐺(𝑠, 𝑢) = (𝑠𝐼 − 𝐴(𝑢))−1,the transfer matrix of Equation (1). Here 𝑠 is a complex number and 𝐼 is the identitymatrix. The largest singular value of 𝐺(𝑠, 𝑢) is deﬁned as the square root of the maximumeigenvalue of the matrix 𝐺∗(𝑠, 𝑢)𝐺(𝑠, 𝑢) and is denoted by 𝜎(𝐺(𝑠, 𝑢)). The 𝐻∞-norm ofthe system transfer matrix is deﬁned by∥𝐺(⋅, 𝑢)∥∞ = sup𝜔∈𝑅𝜎(𝐺(𝑖 𝜔, 𝑢)).The parameters of System (1) can be choosen as solutions to the following optimizationproblem∥𝐺(⋅, 𝑢)∥∞ → min,𝑢 ∈ 𝑈. (3)As in the previous case the problem can also include some additional constraints involvingthe trajectories of the system.The above two criteria do not take into account a very important property of solutionsto linear asymptotically stable systems: a possible large deviation of trajectories during thestabilization process. This phenomenon, known as peak-effect, is a ”pay-off” for a fastor robust stabilization.2,6, 9 The minimal overshooting can also be an objective functions.Below we show how to formulate and solve such problems.Problem of Optimal Parameters DeterminationLet 𝑥(𝑡, 𝑥0, 𝑢) be the solution to the Cauchy problem?˙? = 𝐴(𝑢)𝑥, 𝑥 ∈ 𝑅𝑛, 𝑡 ∈ [0, 𝑇 ],𝑥(0) = 𝑥0.642Deﬁne the functions𝜑𝑖(𝑢) = max𝑡∈Δ𝑖max𝑥0∈𝐵𝑖∣𝑥(𝑡, 𝑥0, 𝑢)∣𝑖, 𝑖 = 0,𝑚. (4)Here 𝐵𝑖 = {𝑥 ∈ 𝑅𝑛 ∣ ∣ 𝑥∣𝑖 ≤ 1}, ∣ ⋅∣𝑖 are some norms in 𝑅𝑛, and Δ𝑖 ⊆ [0, 𝑇 ] are a closedintervals. Consider the following mathematical programming problem𝜑0(𝑢) −→ min,𝜑𝑖(𝑢) ≤ 𝜑𝑖, 𝑖 = 1,𝑚,𝑢 ∈ 𝑈.(5)Many problems of stabilization systems parameters optimization can be written in thisform. The involved functions depend only on the parameters of the system. Consider twoexamples.Minimization of the ﬁnal deviation The problem is to determine optimal values for thesystem parameters guaranteeing minimal deviation of the system state from the zero equi-librium position at the ﬁnal moment of time. This problem can be formalized as follows:max𝑥0∈𝐵∣𝑥(𝑇, 𝑥0, 𝑢)∣ −→ min,𝑢 ∈ 𝑈.For 𝑇 ≫ 1, this is an approximation to the problem of stability degree maximization.Minimization of the maximal deviation This problem consists in determination the pa-rameters corresponding to minimal deviation of a family of trajectories satisfying certainrestrictions at the ﬁnal moment of time. This problem can be formalized as follows:max𝑡∈ [0,𝑇 ]max∣𝑥0∣=1∣𝑥(𝑡, 𝑥0, 𝑢)∣ → min,max∣𝑥0∣=1∣𝑥(𝑇, 𝑥0, 𝑢)∣ ≤ 𝛿,𝑢 ∈ 𝑈.The above problems are of interest for stabilization theory; they both have Form (5).The study of these type of problems can hardly be performed analytically for more or lesscomplex systems. For this reason, we focus on the numerical aspects of this problem.Discrete Optimization ProblemLet 𝑡 ∈ Δ𝑖 and 𝑥0 ∈ 𝐵𝑖, 𝑖 = 0,𝑚. Consider the Euler approximation to solutions𝑥(𝑡, 𝑥0, 𝑢),?˜?(𝑡𝑖𝑘+1, 𝑥𝑖𝑗, 𝑢) = ?˜?(𝑡𝑖𝑘, 𝑥𝑖𝑗, 𝑢) + 𝜏𝑖𝐴(𝑢)?˜?(𝑡𝑖𝑘, 𝑥𝑖𝑗, 𝑢), 𝑘 = 0, 𝐾𝑖, (6)where 𝜏𝑖 = 𝑡𝑖𝑘+1 − 𝑡𝑖𝑘 are small enough constants, 𝑡𝑖0 = 0, 𝑡𝑖𝑘 ∈ Δ𝑖, and ?˜?(0, 𝑥𝑖𝑗, 𝑢) = 𝑥𝑖𝑗 .Let 𝜀 > 0 and 𝛿 > 0 be small enough. Let 𝐾𝑖(𝛿) and 𝐽𝑖(𝛿) be sets of indices such that the643points 𝑡𝑖𝑘 ∈ Δ𝑖, 𝑘 ∈ 𝐾𝑖(𝛿), and 𝑥𝑖𝑗 ∈ 𝐵𝑖, 𝑗 ∈ 𝐽𝑖(𝛿) form a 𝛿-net in Δ𝑖 and 𝐵𝑖, 𝑖 = 1,𝑚,respectively. Deﬁne the functions𝜑𝛿𝑖 (𝑢) = max𝑘∈𝐾𝑖(𝛿)max𝑗∈𝐽𝑖(𝛿)∣∣?˜?(𝑡𝑖𝑘, 𝑥𝑖𝑗, 𝑢)∣∣ , 𝑖 = 0,𝑚and consider the problem𝜑𝛿0(𝑢) −→ min,𝜑𝛿𝑖 (𝑢) ≤ 𝜑𝑖 + 𝜀, 𝑖 = 1,𝑚,𝑢 ∈ 𝑈.(7)Denote by ?ˆ? and 𝑢𝛿 the optimal parameters for problems (5) and (7), respectively. Recallthe following result obtained in Reference 5.Theorem 1 For any 𝜖 > 0 there exists 𝛿 > 0 such that 𝑢𝛿 is an admissible solution to thefollowing problem𝜑0(𝑢) −→ min,𝜑𝑖(𝑢) ≤ 𝜑𝑖 + 2𝜀, 𝑖 = 1,𝑚,𝑢 ∈ 𝑈,and𝜑0(𝑢𝛿) ≤ 𝜑0(?ˆ?) + 2𝜀.This theorem allows one to choose the parameters of discretization in order to obtainoptimal stabilizer parameters with a necessary precision (see Reference 5 for details).Note that the use of techniques developed in Reference 5 allows one to obtain moreprecise estimates for the number of points in the meshes needed to achieve a given dis-cretization accuracy in the case of asymptotically stable systems. Optimization problem(7) is a hard nonlinear nonsmooth problem and can be solved only using numerical meth-ods. Our computational experience shows that the multi-start Nelder - Mead method (seeReference 4) is the most adequate one. Note that the problem of optimal choice of param-eters is solved only once, at the stage of the control system’s development. Therefore it isworthwhile to dedicate more resources to its solution. The methods developed in Refer-ence 5 allow one to solve the problem within a reasonable time.SIMULATION RESULTSConsider motion of a connected two-body system in a circular orbit around the Earth.Body 1 is a satellite with the center of mass 𝑂1 , body 2 is a stabilizer with the center ofmass 𝑂2 . These two bodies are linked to each other at the point 𝑃 through a dissipativehinge mechanism. Let 𝑂 be the center of mass of the system. We use three referenceframes: 𝑂𝑋𝑌 𝑍 is the orbital coordinate frame, its axis 𝑂𝑍 is directed along the radiusvector of the point 𝑂 with respect to the center of the Earth, 𝑂𝑋 is directed along thevelocity of the point 𝑂 , and 𝑂𝑌 is normal to the orbit plane. The axes of referential644frames 𝑂1𝑥1𝑦1𝑧1 and 𝑂2𝑥2𝑦2𝑧2 are the central principal axes of inertia for bodies 1 and 2respectively. Consider motion of the system in the orbit plane supposing that the bodies areconnected in their centres of mass, i.e., the points 𝑂1, 𝑂2, 𝑂, and 𝑃 coincide. Let 𝛼1 and𝛼2 be the angles between the axis 𝑂𝑋 and the axes 𝑂1𝑥1 and 𝑂2𝑥2 respectively. Theequations of motion for this system can be written as?¨?1 + 3𝑝1 sin𝛼1 cos𝛼1 + 𝑘1(?˙?1 − ?˙?2) = 0,?¨?2 + 3𝑝2 sin𝛼2 cos𝛼2 − 𝑘1𝜇 (?˙?1 − ?˙?2) = 0.(8)Here the dot stands for the derivative with respect to the orbital time 𝜏 = 𝜔0 𝑡, where 𝑡 isthe time, and 𝜔0 is is the angular velocity of the system center of mass motion along thecircular orbit. Parameters (𝑝1, 𝑝2, 𝑘1, 𝜇) depend on the principal moments of inertia of thebodies and on the damping coefﬁcient of the stabilizer.1, 7, 8Linearizing the equations at the equilibrium position 𝛼10 = 0, 𝛼20 = 0, we get thefollowing linear system?˙? = 𝐴(𝑢) 𝑥, (9)where 𝑢 = (𝑝1, 𝑝2, 𝑘1, 𝜇), and the matrix 𝐴(𝑢) has the form𝐴 =⎛⎜⎜⎝0 0 1 00 0 0 1−3𝑝1 0 −𝑘1 𝑘10 −3𝑝2 𝑘1𝜇 −𝑘1𝜇⎞⎟⎟⎠ . (10)The set of admissible parameters is given by𝑈 = {(𝑝1, 𝑝2, 𝑘1, 𝜇) : 𝑘1 > 0, 𝜇 > 0, 0 < 𝑝1 ≤ 1, 0 < 𝑝2 ≤ 1, 𝑝1 ∕= 𝑝2} (11)(see Reference 7).Below we formalize the optimal parameter determination problem for this system indifferent manners and compare the results.Consider the following problemmax∣𝑥0∣=1∣ 𝑥(𝑇, 𝑥0, 𝑢) ∣ → min, (12)𝑢 ∈ 𝑈.Set 𝑇 = 3 𝜋 . The are two quartets of optimal parameters𝑝1 = 0.06928, 𝑝2 = 1.00757, 𝑘1 = 0.59209, 𝜇 = 0.33161, (13)and𝑝1 = 1.00521, 𝑝2 = 0.06920, 𝑘1 = 1.78178, 𝜇 = 3.01152. (14)The value of the global minimum is 𝑚 = 0.00378.645Recall that the maximum degree of stability for this system is achieved if the parameterstake the values𝑝1 = 0.0294, 𝑝2 = 1, 𝑘1 = 0.4203, 𝜇 = 0.1716, (15)or𝑝1 = 1, 𝑝2 = 0.0294, 𝑘1 = 2.4495, 𝜇 = 5.8284, (16)(see Reference 8). The corresponding eigenvalues are −0.59, −0.72 ± 0.12𝑖, −0.83. Inthis case we havemax∣𝑥0∣=1∣ 𝑥(3𝜋, 𝑥0, 𝑢)∣ = 0.47. (17)The minimization of the𝐻∞-norm of the transfer matrix leads to two sets of parameters𝑝1 = 0.12254, 𝑝2 = 1.00014, 𝑘1 = 0.66944, 𝜇 = 0.60089, (18)and𝑝1 = 1.00011, 𝑝2 = 0.12279, 𝑘1 = 1.11284, 𝜇 = 1.66412. (19)These sets of parameters correspond to the eigenvalues −0.48± 0.55𝑖, −0.41± 1.39𝑖. Forthis choice of parameters we ﬁndmax∣𝑥0∣=1∣ 𝑥(3𝜋, 𝑥0, 𝑢)∣ = 0.056. (20)Therefore the value of the problem∥𝐺(⋅, 𝑢)∥∞ → min,max∣𝑥0∣=1∣𝑥(3𝜋, 𝑥0, 𝑢)∣ ≤ 𝛿,𝑢 ∈ 𝑈,(21)does not depend on 𝛿, whenever 𝛿 > 0.0559.A different behaviour we observe if the maximal deviations of trajectories from the equi-librium position is considered as the objective function. For example, consider the problemmax𝑡∈ [0,3𝜋]max∣𝑥0∣=1∣𝑥(𝑡, 𝑥0, 𝑢)∣ → min,max∣𝑥0∣=1∣𝑥(3𝜋, 𝑥0, 𝑢)∣ ≤ 𝛿,𝑢 ∈ 𝑈.(22)Numerical solution of this problem leads to the following results:1. 𝛿 = 0.47. The optimal parameters for Problem (22) are𝑝1 = 0.22359, 𝑝2 = 0.65771, 𝑘1 = 0.44747, 𝜇 = 0.61902, (23)and𝑝1 = 0.57767, 𝑝2 = 0.16774, 𝑘1 = 0.74318, 𝜇 = 2.10926. (24)646The corresponding eigenvalues of the linear system and the value of the problem are−0.34± 0.97𝑖, −0.25± 1.20𝑖and𝑚 = 1.20, (25)respectively.2. 𝛿 = 0.06. The optimal parameters for Problem (22) are𝑝1 = 0.14827, 𝑝2 = 0.82299, 𝑘1 = 0.52301, 𝜇 = 0.61786, (26)and𝑝1 = 0.82390, 𝑝2 = 0.14867, 𝑘1 = 0.85127, 𝜇 = 1.60847. (27)The corresponding eigenvalues of the linear system and the value of the problem are−0.34± 0.69𝑖, −0.35± 1.24𝑖and𝑚 = 1.34, (28)respectively.3. 𝛿 = 0.005. The optimal parameters for Problem (22) are𝑝1 = 0.07048, 𝑝2 = 1.00220, 𝑘1 = 0.59611, 𝜇 = 0.33961, (29)and𝑝1 = 1.00175, 𝑝2 = 0.0743, 𝑘1 = 1.75718, 𝜇 = 2.94874. (30)The corresponding eigenvalues of the linear system and the value of the problem are−0.59± 0.34𝑖, −0.59± 1.02𝑖𝑚 = 1.59, (31)respectively.It is interesting that the optimal parameters for Problem (21) with 𝛿 = 0.005 are also𝑝1 = 0.07048, 𝑝2 = 1.00220, 𝑘1 = 0.59611, 𝜇 = 0.33961,and𝑝1 = 1.00175, 𝑝2 = 0.0743, 𝑘1 = 1.75718, 𝜇 = 2.94874.Note that for Parameters (15) and (16) we havemax𝑡∈ [0,3𝜋]max∣𝑥0∣=1∣∣𝑥(𝑡, 𝑥0, 𝑢(15))∣∣ = 1.92,647i.e, the overshooting is rather signiﬁcant.The analysis of the obtained results shows that the𝐻∞-norm is a good choice for the ob-jective function. It combines the robustness guaranteed by the maximal degree of stabilitycriterion with a reasonably small overshooting. However, the 𝐻∞ tool can only be appliedto linear systems. The direct approach based on solution of Problem (5) can be used alsofor nonlinear systems with well succeed implementation (see Reference 5).ACKNOWLEDGMENTThis research is supported by the Portuguese Foundation for Science and Technologies(FCT), the Portuguese Operational Programme for Competitiveness Factors (COMPETE),the Portuguese National Strategic Reference Framework (QREN), and the European Re-gional Development Fund (FEDER).REFERENCES[1] R. L. Borrelli and I.P. Leliakov. ”An optimization technique for the transient response of passively stablesatellites”, Journal of Optimization Theory and Applications, Vol. 10, no. 6, 344− 361, 1972.[2] V. Bushenkov, F. Miranda, and G. Smirnov, ”Advances on the Transient Growth Quantiﬁcation in LinearControl Systems”,International Journal of Applied Mathematics and Statistic, Vol. 14, 82− 92, 2009.[3] J. C. Doyle, K. Glover, P.P. Khargonekar, and B. A. Francis. ”State-Space Solutions to Standard𝐻2 and𝐻∞ Control Problems”. 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Parameter optimization for stabilizers

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