LINEAR SYSTEMS WITH POLYNOMIAL UNCERTAINTY STRUCTURE: STABILITY MARGINS AND CONTROL

LINEAR SYSTEMS WITH POLYNOMIAL UNCERTAINTY STRUCTURE: STABILITY MARGINS AND CONTROL Mohammad Bozorg Deatment of Mechanical Engineering University of Yazd P. O. Box 89195-741 Yazd Iran Fax: +98-351-750110 E-mail: bozorg@yazduni.ac.ir Keywords: Parameter Uncertainty - Polynomial Uncertainty Structure - Controller Design - Pole Placement Robust Control Abstract In this aer the stability and the control of linear systems with arameter uncertainty are considered. The characteristic euations of such systems are derived as functions of uncertain arameters. The cases in which the coefficients are olynomial functions of the uncertain arameters are studied. An algorithm is resented for the calculation of stability margins in the arameter sace in general l -norms. The stability is defined with resect to arbitrary regions in the comlex lane socalled D-stability. The algorithm contains a search on the contour of the region freuency search and solving a system of euations at each ste. Using the above algorithm a method is also resented for the design of robust controllers for systems with uncertain arameters. Robust ole lacement inside the regions of interest is addressed. 1. Introduction For linear systems the locations of the roots of the characteristic euations determine many erformance secifications such as stability daming and the seed of the time resonse. These secifications can be assured by the lacement of the roots of the characteristic euations in an aroriate region D in the roots lane so-called D-stabilization. Locating the roots inside the lefthalf lane LHP guarantees the stability of the system lacing the roots inside the left-sectors in the LHP guarantees a minimum daming ratio for the roots and clustering the roots inside the shifted LHP guarantees a maximum settling-time for the time resonse of the system [1 ] Fig. 1. To have the combined effects of the left-sector and the shifted LHP one may choose a hyerbolic region in the LHP as the root assignment region. 1

A fundamental tye of roblem encountered in robust control of uncertain systems is the calculation of maximum allowable erturbations in arameters of a D-stable system without losing D- stability. In [3 4 5] general l -norm erturbations are considered and several algorithms are resented for the case of olynomials with linearly-deendent coefficients. However in most alications the coefficients of the characteristic euation of the system are multilinear or olynomial functions of uncertain arameters. One of the instances where the multilinear uncertainty structure arises is where there exist several uncertain transfer functions in a forward ath See Examle of this aer. In [] the control system of a sar ignition engine is studied and it is observed that the characteristic euation of the system has a multilinear uncertainty structure. Also several alications in flight control and the steering of vehicles are examined in [] and it is shown that the characteristic euations aear as olynomial functions of uncertain arameters. For olynomials with multilinear uncertainty structure the coefficients are multilinear functions of uncertain arameters the well-nown Maing Theorem [7] is one of the most owerful available results for checing the robust stability. The Maing Theorem has been used in [8] and [9] to develo some algorithms for the calculation of stability margins. In the analysis context in [10] it was shown that to chec the stability of interval multilinear olynomials it is sufficient to chec only a set of manifolds in the arameter sace. The results of [10] and [7] were used in [11] to resent a numerical algorithm for the calculation of Hurwitz-stability margin of multilinear systems. Since the Maing Theorem rovides the sufficient condition for stability the method of [11] leads to conservative results unless multile decomosition of the uncertainty region is erformed and a considerable comutational effort is invested. Although the multilinear uncertainty structure is a subclass of olynomial uncertainty structure it has been shown that a olynomial with olynomial uncertainty structure and with a olytoic arameter uncertainty domain can be transformed to a olynomial with a multilinear uncertainty structure and a new olytoic arameter uncertainty domain [9]. However this is not alicable to non-olytoic shaes of uncertainty domains such as sheres and l -balls. For olynomials with olynomial uncertainty structure the coefficients are olynomial functions of uncertain arameters very few results are available. Among them are the wors of [1] where the stability domains of olynomials are visualized in the arameter sace. The roosed method allow for the grahical evaluation of D-stability margin in the arameter sace. As mentioned in the conclusions of [1] the method best suits the systems with a small number of uncertain arameters. The wor of [13] resents two algorithms for Hurwitz-stability checing of olynomials with olynomial uncertainty structure using the Bernstein olynomials. The interesting roerties of the Bernstein olynomials are used to reduce the comutational reuirements although the algorithms still reuire considerable amount of comutations for general cases. The calculation of general l -norm D-stability margins for olynomials with multilinear or olynomial uncertainty structure still remains an oen roblem.

In this aer a new rocedure is resented to calculate l -norm D-stability margins for olynomials with olynomial uncertainty structure. The rocedure reuires sweeing the contour of the region and at each oint on the contour solution of a system of euations. The methods of [14] and [15] on the calculation of stability margins is generalized here for the case of olynomial uncertainty structure. Another imortant roblem in robust control is the design of robust controllers. The robustness roerties of olynomials has less been used in the synthesis context. Pole lacement is one of the techniues widely used for controller design. An imortant issue in the ole lacement method is the sensitivity of the closed-loo systems to the variation of the arameters of the system. These sensitivities were taen into account in [1] by comuting the variation of the oles with resect to the variation of the arameters. In this aer the design of robust controllers for olynomial systems is osed as an otimization roblem. A method is resented for the robust lacement of closed-loo oles of uncertain systems inside the regions of interest. The algorithm develoed for the calculation of D-stability margin is used in the formulation and the solution of the synthesis roblem. Two examles are also resented to demonstrate the results. It is to be mentioned that some reliminary results of this wor were resented in [17].. Preliminaries Consider the olynomial n s an s + + a1 s + a0 ; an 0 L 1 where the coefficients a n 1 m [ ] R L. 1 m a a are olynomial functions of the system arameters: 0 The results of the well-nown Zero Exclusion Theorem is used in this aer for the calculation of D- stability margins. Theorem 1. Zero Exclusion: Suose the family of invariant-degree olynomials 1 with where is an uncertainty set which is ath-wise connected. Let D be an oen subset of the comlex lane and C D be its contour. Furthermore assume that the family of olynomials has at least one D- stable member. Then the family of olynomials is D-stable if and only if 0 u; u C D. Proof. See [1]. To chec the necessary and sufficient conditions of Theorem 1 for D-stability the contour of the region must be swet. The oint u on the contour C D can be exressed as a function of a sweeing arameter i.e. u u ; Ω j u e ; [ 0 π. The substitution of u u in 1 results where. For instance the contour of the unit circle can be reresented by u R + j I 3

Re[ u ] 3 R Im[ u ]. 4 I Define the nominal arameter vector [ L ] 5 1 m at which s is D-stable. The calculation of the minimum distance of to the instability region in the system arameter sace will be addressed in the next section. The weighted l -distance norm of two arbitrary oints m R is defined by where δ 1/ m / w 0 w 0; 1 K m > are weights and 1< < is a constant. If ~ reresents a erturbed value of the ~ arameter vector δ gives the l -norm magnitude of erturbation from to ~. 3. D-Stability Margins In the wae of the Minimum Distance aroach resented in [15] for the calculation of D- stability margins for olynomials with linearly-deendent coefficients a relevant aroach is taen in this aer for the case of olynomial uncertainty structure. From Theorem 1 it is conferred that the contour of D-stability region must be swet. At an arbitrary value of the sweeing arameter which corresonds to the oint u on the contour the minimum distance of the nominal arameter vector to the set of arameter vectors at which D- stability conditions of Theorem 1 fail can be found by solving the following otimization roblem: subject to: ρ & min 7.a δ 0 7.b R 0. 7.c I ~ At each the erturbations δ must be et smaller than the minimum distance function to ensure the zero exclusion of the erturbed olynomials. If the constraints 7 are satisfied D- stability of the erturbed olynomials is concluded. Then D-stability margin of the nominal olynomial s is obtained sweeing in its entire range i.e. min ρ Ω ρ. Several relevant otimization methods have been used in [14] and [18] for the case of olynomials with linearlydeendent coefficients. Here the more general case of olynomials with olynomial uncertainty structures is considered. A necessary condition of Theorem 1 for D-stability is the invariance of the degree of the erturbed olynomials. This means that the erturbation of the arameters must not result in the nullification of a n. To satisfy the condition the erturbations δ ~ must be less than the otimal value 4

5 ˆ min ˆ δ η & 8.a constrained by 0 ˆ n a. 8.b If the erturbations ~ δ in the arameters of the system are smaller than both minima of 7 and 8 considering that s is D-stable all conditions of Theorem 1 are satisfied and the family of erturbed olynomials are D-stable. The above discussion leads to the following theorem. Theorem. The family of olynomials { } B b s Ρ where { } b b B < ~ : ~ δ 9 is the l -ball of arameter uncertainty is D-stable if and only if b γ < Ω ρ η γ : min min. 10 In 9 b is the size of the l -ball and is the nominal value of the arameter vector and the family is assumed to be D-stable at. The l -ball becomes a hyershere for a box for and a diamond for 1. Fig. shows some samle l -balls in the two-dimensional case for different values of. In the rest of this section we roceed to erform the otimizations 7 and 8. The otimization 7 with two constraints will be discussed. Similar stes can be taen to erform the otimization 8. The method of Lagrange multiliers is used to erform the otimization 7. The functional [ ] + µ + λ δ I R F is defined. At the critical oints one has. 1 0; sgn 1 m w F I R K + + µ λ Then with some maniulation of the euations it is obtained 1 1/ + I R w µ λ & + sgn sgn I R S µ λ &. At a corresonding to a oint on the contour the following system of euations must be solved to obtain the closest distance to the instability border: 0 0 1 ; I R m S K 11

where the m + unnowns are ; 1 K m and the Lagrange multiliers λ and µ. The critical oints obtained solving the system of euations 11 can be bac-substituted in the definition ; 1 K m to obtain the minimum distance of the nominal arameter vector to the instability border i.e. ρ min δ. The deviations of the otimization 8 from the above rocedure are: 1 the objective and the constraint of 8 is indeendent of sweeing is not reuired and only one system of euations is formed for each roblem there exists only one constraint and only one Lagrange multilier must be used. 4. Synthesis of Robust Controllers Consider a standard feedbac control system Fig. 3. The lant transfer function is defined by where N G s NG s DG s G s and D G s arameter vector m [ ] R 1 m 1 L. are olynomials whose coefficients are olynomial functions of the The arameters of the lant are uncertain and are erturbed inside the l -ball 9 where is the nominal value of the arameter vector. In most alications the bounds of erturbations of the arameters of the system can be estimated a riori and the erturbation domain ball is redefined for the design roblem. The controller is reresented by the ratio of two fixed olynomials i.e. NC x s C x s DC x s l NC s cs l x DC x s ds 0 0 13 and the controller vector is defined by x & [ c L c c d L d ]. 14 l 1 0 l 1 d0 The characteristic olynomial of the closed-loo control system is given by CL x s NG s NC x s + DG s DC x s To comly with the definition of the uncertain characteristic olynomial given in 1 noting that the controller x is fixed one may write n CL x s s an s + L + a1 s + a0 where the coefficients are olynomial functions of the lant arameters. Then the results of Section 3 can be used for this feedbac control system. For the robust synthesis of the system the Robust Pole Assignment RPA can be defined as [19]:

Given the uncertain lant model of 1 and a region D in the comlex lane find a controller C x s such that all closed-loo oles lie in D for every B ρ. As discussed in the introduction various erformance secifications for a control system can be achieved by the lacement of the roots of its characteristic olynomial in aroriate regions. The results of Section 3 on D-stability of olynomials with olynomial uncertainty structure can be used directly for the solution of the RPA roblem for the feedbac configuration described in this section. The main aim in RPA is to ensure that the roots of the family of characteristic olynomials lie inside D. However a set of controllers rather than a uniue controller usually satisfies this reuirement. Then to select a controller among all D-stabilizing controllers an additional objective can be introduced [1]. In this aer the controller design is osed as the following otimization roblem: subject to: min f x D x CL 15.a x s is D - stable 15.b γ b 15.c where γ is obtained from 10. The function f x can be any convex function whose otimal value is desired. Several examle D functions are resented in [1] and [19]. One of the common objectives for 15.a is the minimization of the l -distance of the designed controller from the nominal controller i.e. f x D f x δ x x where x is the nominal controller vector. The constraint 15.b is to ensure that the closed-loo system is D-stable at the nominal values of the arameters. When the ineuality 15.c is satisfied it is ensured that the roots of the characteristic olynomials corresonding to all lants with B b lie inside D. However it might be referred to relace the ineuality sign in 15.c by the euality sign to revent unnecessary conservatism. 5. Examles In this section two examles are resented to illustrate the methods of this aer. In Examle 1 the results of Section 3 is used to calculate D-stability margin of a olynomial with olynomial uncertainty structure. Examle imlements the method of Section 4 for the synthesis of a robust controller for an uncertain lant with multilinear uncertainty structure. Examle 1. The model of the steering system of Daimler Benz 0305 electric bus is given as [1] 109.81 s + 38800s + 48801 s 3 s 1 s + 10771 s + 1.81 + 70000 G 1 where 1 and the uncertain arameters of the lant are the mass tons and the velocitym/s of the bus. The variation range of these arameters are obtained from the bus oerating secifications as min [ 9.95 3] 17 [ 3 7.5]. 18 max 7

The controller is given as 344s + 10938s + 9375 s 3 s + 50s + 150s + 155 C x. 19 It is desirable to investigate the location of the oles of the closed-loo control system of Fig. 3 as the mass and the velocity of the bus vary in the ranges 17 and 18. The transfer functions 1 and 19 in the closed-loo configuration of Fig. 3 form a characteristic euation whose coefficients are olynomial functions of the uncertain arameters 1 and. The coefficients are resented in the Aendix. Due to the symmetry of the erturbations around the nominal values of the arameters the nominal characteristic olynomial is considered at the middle of the uncertainty domain i.e. 8 i s a i s i 0 [ ] [ 15.5 0.975]. 0 1 From 17 and 18 it is noticed that the ranges of the erturbations in 1 and are not eual. Then the erturbation of each arameter is weighted according to its range to normalize the erturbations: [ 4.5.05] max min.; w 1 1 4. 5 ; w. 05. In [1] a hyerbola in the left half lane Fig. 1d is considered as the root assignment region D and the contour of the region is defined as 0.35] u + j 5 1.75 ;. At each the system of euations 11 is formed and solved to comute the minimum distance function. In Fig. 4 the minimum distance functions are lotted for the nominal arameter vector ρ in some selected ranges for 1. Since the erturbations are in the intervals between min and max this case corresonds to. The lowest value of the minimum distance function ρ is obtained as 0.504 at 0. 490. Also otimization 8 leads to η 0.3 as the degree droing bound. Finally the D-stability margin is calculated from Theorem as min { 0.30.504} 0. 504 γ. The size of the erturbation box in the arameter sace in 1 -direction is comuted as γ w 1 4.9 and in -direction is obtained as γ w.3. The calculated D-stability margin which corresonds to the radius of the box of erturbations corresonds the results of [1]. In contrast to the most of the current methods which consider only the interval erturbations the method of this aer can be used for any l -norm erturbations. The size of the diamond and the hyershere of stability which resectively corresond to l 1 and l -norm erturbations are obtained as γ 1 and 979 1 1. γ 0.. Examle. The following feedbac control system introduced in [11] is examined here. The lant transfer function is given by 3 s + s + 1.s + 13.5s + 15.5s + 0.4 s 3 3 s + s + 4s + 1 s + 3s + 3.5s +.4 G 1 with the nominal arameter vector [ 3 3.5] and the weights 1 ; i 1 3. The numerator of the lant transfer function is multilinear in. w i 8

a Calculation of stability margin: The fixed nominal controller is given by s + s s + 1 C x ; x [ 1 1 1]. The characteristic euation of the closed-loo system formed by 1 and in the feedbac configuration of Fig. 3 has a multilinear uncertainty structure. The multilinear case is a secial case of the olynomial uncertainty structure. It is desired to calculate the Hurwitz-stability margin of the nominal arameters of the lant. The contour of the region is the imaginary axis and a oint on the contour can be described by the sweeing function u j; [0. At each the system of euations 11 is formed and solved to comute the minimum distance function. In Fig. 5 the minimum distance function is lotted for the nominal arameter vector in a selected range for ρ. The minimum of ρ is obtained as 0.40 at 1. 9. For this system considering that the coefficient of the highest order coefficient of the characteristic euation is unerturbed 1 is obtained. Therefore the Hurwitz-stability margin is calculated from Theorem as { 0.40} 0. 40 γ min. a n η Investing more comutational effort a close value was obtained in [11] by dividing the uncertainty domain to smaller regions and alying the results of the Maing Theorem. The size of the diamond and the hyershere of stability are also calculated resectively as γ 77 and γ 94. For this 1. 0. system the otimization roblem 15 is defined as follows. The objective 15.a is given by f x D f x δ xx. The controller arameters c 1 and d 1 are et unchanged by assigning small numbers as their weights. The weights in δ. 9 9 xx are then defined as [ 10 1 10 1] b Design of a robust controller: For the samle lant it is assumed that the arameters are uncertain in the ranges between [ 1 4.5] and [ 3 4.5]. Therefore the uncertainty set is the box B b with the size b min max 1. The nominal controller does not stabilize the uncertain lant. The locations of the roots of the characteristic euation using this controller is shown in Fig. for five evenly-saced oints in the ranges of the uncertain arameters between min and max. It can be seen that some roots lie in the right hand lane. Then the nominal controller does not stabilize the entire family of lants. Solving the otimization roblem 15 for this examle leads to the otimal controller x ot [ 1.878 1 0.10]. The closed-loo oles of the control system with the controller x ot and the uncertain arameters evenly saced in ten intervals are lotted in Fig. 7. It can be seen that the controller laces all the oles inside the left-hand lane and D-stabilizes the family of lants.. Conclusions In this aer the roblem of D-stability margins in the arameter sace for linear systems is investigated. A new algorithm is develoed for the calculation of these margins. Comared with 9

existing methods such as that of [] the method resented here is more straightforward and demands less comutations. It reuires sweeing of the contour and solving a system of euations at each oint on the contour. The number of euations to be solved deends uon the number of uncertain arameters of the system. This number is usually small in real alications. Then the comutational burden of the method is contained. For the articular case of multilinear uncertainty structure the derivatives / and / will be indeendent of. This roerty can be used to R I simlify the numerical solution of the system of euations 11. The comutational feasibility of the resented method rovides the grounds for its use in iterative design algorithms. In this aer a method has also been resented for the design of robust controllers for lants with olynomial uncertainty structures. Robust ole lacement in the regions of interest has been addressed. The method can be used to obtain otimal controllers that D-stabilize the uncertain lants. Considering that very few tools are available for the design of robust controllers for systems with olynomial uncertainty structure the design rocedure resented in this aer is as a ste forward in this direction. Further wor on the otimization roblems aearing in 7 and 15 can lead to fruitful results. In articular olynomial otimization techniues can be used in this direction. Also develoing efficient numerical methods for solving the system of euations 11 will facilitate the use of the method resented here for the cases with a higher number of uncertain arameters. Alication of the results to the control systems with uncertain arameters aearing in the steering of vehicles robotic and rocess control is also romising as very few results are available to deal with such systems in a non-conservative and comutationally feasible manner secially for the cases with multilinear or olynomial uncertainty structure [13]. Aendix In this aendix the coefficients of the characteristic euation of the closed-loo system of Examle 1 is resented: a 8 1 a 7 501 + 10801 3 3 3 a 1.5 10 1 + 1.81 + 53.9 10 1 + 70 10 3 a 5 15. 10 1 + 8401 + 1.35 10 1 + 13.5 10 a 4 1.45 10 1 + 1.8 10 1 + 338 10 a 3.93 10 1 + 911 10 1 + 40 10 a 5.7 10 1 + 113 10 1 + 450 10 1 a 1 58 10 1 + 340 10 1 a 0 453 10 1. 10

References [1] R. B. Barmish New Tools for Robustness of Linear Systems Macmillan New Yor 1994. [] J. Acermann Robust Control: The Parameter Sace Aroach nd edition Sringer-Verlag London 00. [3] D. Hinrichsen and A. J. Pritchard An alication of state sace methods to obtain exlicit formulae for robustness measures of olynomials in Robustness in Identification and Control eds. M. Milanese R. Temo and A. Vicino New Yor: Plenum Press 1989. [4] L. iu and E. J. Davison A simle rocedure for the exact stability robustness comutation of olynomials with affine coefficient erturbations Systems and Control Letters vol. 13. 413-40 1989. [5] M. Teboulle and J. Kogan Alications of otimization methods to robust stability of linear systems Journal of Otimization Theory and Alications vol. 81. 19-19 1994. [] N. Safari-Shad and M. Taabe Refined robust stability analysis of a sar ignition engine model IEEE Transactions on Control Systems Technology vol. 5. 1-5 1997. [7] L.A. Zadeh and C. A. Desoer Linear Systems Theory McGraw-Hill New Yor 193. [8] R. R. De Gaston. and M. G. Safonov Exact calculation of the multiloo stability margin IEEE Transactions on Automatic Control vol. 33. 15-171 1988. [9] S. Sideris and R. S. Sánchez Peña Fast comutation of multivariable stability margin for real interrelated uncertain arameters IEEE Transactions on Automatic Control vol. 34. 17-17 1989. [10] H. Chaellat L. H. Keel and S. P. Bhattacharyya Robust stability manifolds for multilinear interval systems IEEE Transactions on Automatic Control vol. 34. 314-318 1993. [11] L.H. Keel and S. P. Bhattacharyya Parametric stability margin for multilinear interval control systems Proceedings of American Control Conference San Francisco California USA. - 1993. [1] J. Acermann D. Kaesbauer and R. Muench Robust Gamma-stability analysis in a lant arameter Sace Automatica vol. 7. 75-85 1990. [13] M. Zettler and J. Garloff Robustness analysis of olynomials with olynomial arameter deendency using Bernstein exansion IEEE Transactions on Automatic Control vol. 43. 45-431 1998. [14] Yu. I. Neimar Robust stability of linear systems Soviet Physics Dolady vol. 3. 578-580 1991. [15] M. Bozorg and E. M. Nebot l arameter erturbation and design of robust controllers for linear systems International Journal of Control vol. 7. 7-75 1999. [1] Y.C. Soh and R. J. Evans Characterization of robust controllers Automatica vol. 5. 115-117 1989. [17] M. Bozorg Polynomial uncertainty structure: stability margins and control Proceedings of Euroean Control Conference Porto Portugal. 351-351 001. [18] A. Tesi and A. Vicino Robustness analysis for linear dynamical systems with linearly correlated arametric uncertainties IEEE Transactions on Automatic Control vol. 35. 18-191 1990. [19] H. Rotstein R. Sanchez Peña J. Bandoni A. Dessages and J. Romagnoli Robust characteristic olynomial assignment Automatica vol. 7. 711-715 1991. 11

Im Im O Re O Re a b Im Im φ O Re O Re c d Fig. 1 Regions of interest in the comlex lane: a left hand lane b shifted left hand lane c left-sector and d left-hyerbola 3.5 11.1..... inf Nominal Value 1.5 1 0.5 0 0.5 1 1.5.5 3 1 Fig.. l -balls in the arameter sace. 1

u + C x s G s y - Fig.3 Standard feedbac control configuration. 3.5 3.5 x : + :.. : 1 M D 1.5 1 0.5 - -1.8-1. -1.4-1. -1-0.8-0. -0.4 Fig. 4 The minimum distance function for Examle 1. 13

70 0 M D 50 40 30 0 10 0 1 1. 1.4 1. 1.8..4..8 3 Fig. 5 The minimum distance function for Examle. 4 3 Im 1 0-1 - -3-4 -7 - -5-4 -3 - -1 0 1 Re Fig. Closed-loo oles with the nominal controller. 14

5 4 3 Im 1 0-1 - -3-4 -5-5 -4.5-4 -3.5-3 -.5 - -1.5-1 -0.5 0 Re Fig. 7 Closed-loo oles with the otimal controller. 15