Robust and Randomized Control Design of Mini- UAVs: The MH1000 Platform

Transcription

1 Robust and Randoized Control Design of Mini- UAVs: The MH Platfor L. Lorefice*,. Pralio* and R. Tepo** * Politecnico di Torino/Dipartiento di Ingegneria Aeronautica e Spaziale, Torino, Italy ** IEIIT-CNR, Torino, Italy Abstract Mini-UAVs (Unanned Aerial Vehicles) have been the subject of a large nuber of successful designs aied to research, coercial and ilitary purposes. A ini-uav platfor can be considered as a iniaturized aircraft so that classical design ethodologies can be extended to this aerial vehicles category. However, due the presence of plant uncertainties, flight dynaic analysis and control syste design are critical, and suitable robustness techniques need to be developed. In this paper, we present an innovative robust control strategy based on powerful tools provided by the theory of randoized algoriths. Within the context of the National Project Cofin 4, the proposed ethodology has been ipleented for guidance/trajectory tracking and platfor stabilization feedback loops of the autopilot syste. The nuerical results obtained are presented. I. INTRODUCTION Unanned Aerial Vehicles (UAVs) have becoe increasingly attractive for industries as well as for researchers in aeronautics for various econoic and safety reasons. These platfors can be reotely controlled or they can autonoously fly without pilot on-board, for surveillance and reconnaissance issions in dangerous areas. Clearly, they encounter a great nuber of useful applications both in civil and ilitary fields. Aong the ilitary applications, surveillance, targeting and counications can be counted. Civil issions also have several objectives, which in soe cases coincide with ilitary ones. In particular, UAV platfors enable localization of issing people, reconnaissance and surveillance of territories such as steep slopes, traffic onitoring, fire and rescue operations, biocheical sensing and sensor placeent. These platfors can be divided into different categories according to size, endurance or ission range, but one of the ost coon classifications of UAV typologies available in the literature is related to the platfor size. In the last decade, the acrony MAVs (Micro Aerial Vehicles) has been used to define flying objects characterized by physical size approxiately saller than 6 inches, in length, width or height. Recalling a classical definition [], these flying objects can be considered as ``aerial robots, as six-degree-of freedo achines whose obility can deploy a useful icropayload to a reote, or otherwise hazardous location, where it ay perfor any of a variety of issions.'' The concept of uninhabited aerial vehicles of reduced diensions, able to perfor ission profiles not copatible with any existing piloted platfors, was subsequently extended to bigger systes and it now incorporates the so-called Mini-UAVs, having axiu diension up to 6 ft. These platfors have been the subject of considerable interest and developent in recent years. A large nuber of successful designs have been perfored for either research, coercial or ilitary purposes by several universities, industries and governent-funded agencies both in the US as well as in Europe. As a consequence of different applications and, hence, a different range-payload perforance, several configuration concepts can be encountered within the Mini-UAV category. In fact, a fixed wing Mini-UAV can be considered as a iniaturized aircraft so that preliinary design issues and procedures applied to conventional platfors can be extended to this faily of aerial vehicles. In spite of that, aircraft dynaics and flight control syste definitions represent critical issues, influencing the approach for the design and flight testing phases. The reduced diensions of these vehicles lead to highly nonlinear syste behavior and unconventional dynaics in ters of natural frequencies and daping ratios. The inertial characteristics of the platfor yield to unconventional ode characterization thus resulting in undesirable abrupt responses to piloted coands. Furtherore, the sensitivity to changes in flight conditions (concerning velocity ore than altitude), the assuptions in aerodynaic database definition (e.g., stability and control derivatives), the inaccuracies in geoetric and inertial data represent a set of uncertainties in plant and environent odeling. The result is that conventional control design ethods are often not effective []. Therefore, the design of a flight control syste which guarantees a suitable level of tolerance to environental changes and platfor anufacturing/odelling inaccuracies plays a key role whenever stability and perforance requireents have to be fulfilled, see e.g. [3]. Micro and Mini-UAVs represent a ajor challenge for control design, and robust control in particular. Robust control algoriths ay be viewed as copleentary to gain scheduling and fault tolerant control ethods. In fact, the proble of searching the appropriate gains can be reforulated in ters of operating conditions representing the critical paraeters by eans of dynaic pressure and/or Mach nuber. On the other hand, the use of a fault tolerant approach strongly depends on the failure effects odeling. The failure types to be dealt with by the proposed ethodology are related to finding an explicit RTO-EN-SCI-95 4A -

2 Robust and Randoized Control Design of Mini-UAVs: The MH Platfor relationship between the failure entity and the plant paraeters. Whenever this relationship is available, the critical paraeters leading to instability or loss of perforance ay be easily detected. The applicability of various classical robustness techniques such as, for exaple, Linear Matrix Inequalities and related relaxation ethods, and µ-theory [4], has been explored for the MH platfor. However, the atheatical odel is affected by nonlinear uncertainty of paraetric type. In addition, this odel is obtained by nuerical linearization of the full order nonlinear syste representing the aircraft dynaics, so that explicit relationships between the state-space atrices and the uncertain paraeters are not available. For these reasons, in this paper we follow a different innovative approach for gain synthesis. This approach akes use of uncertainty randoization and is based on the theory of randoized algoriths [5]. These algoriths are easy to ipleent, have low coputational coplexity and are associated with robustness bounds that are generally less conservative than the classical ones, obviously at the expense of a probabilistic risk of failure. The theory of randoized algoriths follows the pioneering line of research that was initiated by Stengel [6] in the early eighties. In several subsequent papers [7][8][9] and references therein, various techniques, ainly based on Monte Carlo siulations, have been explored for the coputation of the probability of instability, and related perforance concepts. The application area providing otivations for the developent of these ethods is indeed aerospace control, and the results obtained are very successful. In this line of research, specific attention is devoted to reduced order odels of the aircraft dynaics by analyzing the effects of aerodynaic uncertainties. We now describe the organization of the paper and the ain results obtained. In Section II, we briefly introduce the MH platfor, representing the reference platfor for the control algoriths testing. In Section III, we present the atheatical forulation of the plant dynaics. This forulation is based upon a full six degrees-of-freedo nonlinear atheatical odel, see [] for details. Uncertainty characterization is the focus of Section IV. We consider the vector of l real uncertain paraeters δ i, restricted within upper and lower bounds. In addition, each paraeter δ i is a rando variable with an assigned probability density function, either unifor or truncated noral, within this interval. Uncertainties for the MH platfor include paraeters related to flight conditions, aerodynaic data, geoetric and inertial data, see Section IV for a precise description. In this section, we also briefly introduce the concept of specification property S which is utilized subsequently in the randoized algoriths. In Section V, we present three randoized algoriths which should be used sequentially. The first algorith is based upon the selection of a subset of critical uncertain paraeters, and has the objective to provide an initial set of randoly generated controller gains K rand. The specification property is explicitly defined and a stopping criterion in ters of the probabilistic confidence and accuracy is given. This stopping criterion is based upon the so-called Logover-log ound []. Algorith uses a set of gains K rand previously obtained and coputes the epirical probability that given perforance specifications are satisfied. To this end, a bound on the required saple size, the Chernoff ound [] is utilized. Finally, Algorith 3 has a structure siilar to Algorith, but different specification properties, derived fro standard flying qualities of piloted aircraft [3], are used. In Section VI we present the nuerical results related to the case study. In particular, in this section, classical graphical tools, such as the root locus plot and the standard bandwidth levels, highlighting the region of interest defined by the specification property, are used to copare various solutions and to single out the best fitting one. The conclusions are given in Section VII. II THE MH PLATFORM The aerial platfor MH is based on the MicroHawk configuration, developed at the Aerospace Engineering Departent, Politecnico di Torino (national patent no. TO3A7, holder Politecnico di Torino, international request PCT/I4/94). The reference platfor is characterized by a conventional layout consisting of fixed wing, tailless integrated wing-body configuration, tractor propeller driven. A scaled version of the MicroHawk configuration, naed MH (Fig. ), characterized by wingspan and a total weight of approxiately.5 kg, was designed and tuned to eet the ission specifications required by the project. The flight envelope of the platfor ranges fro 4 kph to 7 kph; an average flight speed of about 55 kph allows to achieve a flight endurance of at least 4 inutes. The platfor flight perforances, the autopilot syste effectiveness and the copliance of the integrated syste to the ission requireents have been tested by extensive on-site flight tests, also in cooperation with the Environent Protection and Daage Preeption Agency of Sicily. Fig. : The MH platfor during a flight test. III PLANT DYNAMICS The aerial platfor is described by a full six degrees-offreedo nonlinear atheatical odel, consisting of twelve, coupled, nonlinear, ordinary differential equations []. The odel is based on three point ass equations, three attitude dynaics equations, three kineatical relationships and three navigation equations for trajectory evaluation. The equations of otion, both for point ass center of gravity dynaics and for attitude dynaics, are written with reference to the body-axes reference frae, i.e. a vehicle body-fixed syste, having origin at the vehicle center of gravity and axes aligned to vehicle reference directions. Structural flexibility is neglected so that the rigid body assuption is ade. This assuption is coonly used in general flight siulations since the 4A - RTO-EN-SCI-95

3 Robust and Randoized Control Design of Mini-UAVs: The MH Platfor attention is focused on trajectory analysis and overall aircraft perforance. Furtherore, this assuption akes sense because we are dealing with a Mini-UAV characterized by sall diensions and weight. The flat and non-rotating Earth assuption is also used since it does not affect odel accuracy of low speed flight over a sall region of the Earth. Finally, the aerodynaic loads acting on the aerial platfor are obtained utilizing an aerodynaic odel, based on experiental wind tunnel testing and nuerical coputations. The point ass center of gravity dynaics and the attitude dynaics equations, written in atrix notation, are given by: { V& E} = { FA,T } [ Ωω ] { VE } [ TV] {} g { ω& } = [] I { M } [ Ω ] ([] I {} ω )} A,T ω where the body-axes coponents of the velocity vector {V E } and of the angular velocity vector {ω} are { V E} [ ] T = U V W {} ω = [ P Q R] T The aerodynaic and propulsive forces and oents vectors are represented by the vectors {F A,T } and {M A,T }, respectively. The atrix [I] and are the inertia atrix and the ass of the rigid body, respectively, and [Ω ω ] is the cross-product atrix of the angular velocity vector {ω}. Finally, {g} V is the gravity acceleration expressed in the local navigation reference (NED axis) syste and the atrix [T V ] represents the rotation atrix fro body to NED reference syste. The vehicle attitude is odeled by the Euler kineatical equations: Φ & = T VA ω { } [ ]{} where the Euler angles vector {Φ}={φ θ ψ} consists of roll, pitch and yaw, respectively. The atrix [T VA ] denotes the kineatical relationship between the Euler angles and the angular velocities. The navigation equations for trajectory evaluation is expressed by {} p & = [ T V ]{ V E } where the vector {p}={x V Y V Z V } represents the center of gravity coordinates in a geographic syste having origin at the vehicle center of gravity. The atheatical forulation also includes the odelling of the ain subsystes (engine, propeller, actuators). As to the propulsive syste, linear relationships are applied to odel the voltage supply and current drain trend of variation for a DC otor-based propulsion. The propeller perforance is estiated by ipleenting the blade eleent theory to copute propulsive forces and oents at a given regie of rotation [4]. V A state-space forulation is obtained by nuerical linearization of the full order syste around an equilibriu. condition (x, x, u ). The nuerical linearization algorith is based on the iplicit forulation of the ordinary differential equations representing the aircraft dynaics: ( x, & x,u) h = where x represents the vector of state variables and u is the vector of control variables. IV UNCERTAINTY CHARACTERIZATION To characterize paraetric uncertainty, the standard state-space forulation is used to re-write the linearized syste of the plant dynaics: () t = A( ) x() t ( ) u() t x & + where A R n,n, R n,p, and consists of l uncertain paraeters, i.e., = δ,δ, L, [ ] T δl The controller structure we use is a state-feedback of the for u = -K x, so that the proble is to find a gain atrix K satisfying given specifications k k = K M k p k L k L M O M k p L kn k n k pn Next, we assue that each uncertain paraeter δ i is a rando variable distributed according to a given probability density function (p.d.f.) p δi (δ i ), in the interval [δ - i, δ + i ]. We define the set = { : δ i [δ - i, δ + i ], i =,,, l} In particular, the paraeter δ i ay be uniforly distributed or distributed according to a truncated Gaussian density. Then, we consider N independent identically distributed (i.i.d.) saples j [ j j j ] T = δ,δ, L,δl of the rando vector. In particular, this eans that the j saple δ i of the rando paraeter δ i is drawn within the interval [δ - i, δ + i ] according to the distribution p δi (δ i ). The next step is to introduce a suitable specification property S, which is the set of all controller gains or uncertain paraeters satisfying suitably defined closed loop specifications. In particular, S refers to controller gains K, while S and S 3 are related to uncertain paraeters. Concrete instances of specification properties for the MH platfor are provided in Section VI and include, for exaple, liits on the step or frequency response, or restrict the natural frequency and daping ratio to a given range. RTO-EN-SCI-95 4A - 3

4 Robust and Randoized Control Design of Mini-UAVs: The MH Platfor In the first (synthesis) phase, we also assue that the gain atrix K is rando. Then, we take N K i.i.d. atrix saples {K, K,, K NK }. In particular, the saple k j i of the rando gain k i is drawn within the given interval [k - i, k + i] according to a unifor distribution. We define the set K = {K : k i [k - i, k + i], i =,,, p, =,,, n} The objective is to find a gain atrix K rand = K j which satisfies the specification property S. Clearly, it is iportant to deterine the saple size N K which provides a stopping rule in the randoized algoriths described later in Section V. To this end, we use an explicit bound often denoted Log-over-log which is based on two probabilistic quantities, restricted within the intervals (,), denoted as accuracy ε and confidence η, see [] and [5] for details. Let {K, K,, K s } be the rando atrices satisfying S, where s denotes the nuber of successes, i.e., the nuber of atrix gains satisfying S. The second (robustness) phase utilizes K rand =K j, j=,,,s, previously coputed and has the objective to estiate the true probability p true that the specification property S is satisfied. Forally, we define an initial set of controller gains, while Algoriths and 3 are then used to iprove robustness. The perforance requireents specified in Algorith 3 are different than those of Algorith, so that refineents on the choice of gains are provided. Algorith (Fig. ) is based upon the selection of a subset of l critical uncertain paraeters. That is, we consider a vector c containing only coponents of : c = [ δ,δ, L, δ ] T ptrue = { } p S ( ) d Then, we introduce the indicator function j ( ) j if S I = otherwise The estiated probability that the specification property S is satisfied is iediately given by pˆ N = N N I j= j ( ) The estiate p N is usually referred to as epirical probability. The saple size needed to obtain a reliable probabilistic estiate p N is given by the Chernoff ound, see [] and [5] for further discussions on this topic. V RANDOMIZED ALGORITHMS FOR GAIN SYNTHESIS AND ROUSTNESS ANALYSIS Three randoized algoriths for gain synthesis and robustness analysis are shown in this section. As previously discussed, the controller structure is a state-feedback of the for u = -K x, so that the proble is to find a suitable gain atrix K. Three algoriths are presented for gain synthesis, stability analysis and perforance analysis, respectively. The algoriths are strongly coupled and closely related to the operating flight conditions and to the flying quality standards. In particular, the algoriths are used sequentially: Algorith is used for synthesis to provide Fig. : Algorith Rando Gain Selection. Then, setting the critical uncertain paraeters to all cobinations of their upper or lower values δ i + or δ i -, suitable controller gains are randoly selected. This selection akes use of a given p.d.f. of controller gains and their a priori bounds. The reaining non-critical paraeters are set to their corresponding noinal value, i.e., δ i, i = +,, l. Clearly, since the state-space atrices A( ) and ( ) depend on, this requires to copute M = critical atrices of the for: A c = A δ Ac = A δ M M Ac = A δ and M atrices of the for c = δ c = δ M M c = δ (,δ, L,δ,δ +, L, δl ) + + (,δ, L,δ,δ, L, δ ) + (,δ, L,δ,δ, L, δ ) (,δ, L,δ,δ +, L, δl ) + + (,δ, L,δ,δ, L, δ ) + (,δ, L,δ,δ, L, δ ) + We observe that in our context, the nuber of critical paraeters is very liited (two or three at ost) and l l l l 4A - 4 RTO-EN-SCI-95

5 Robust and Randoized Control Design of Mini-UAVs: The MH Platfor therefore the construction of the critical atrices is not a serious coputational proble. Next, we consider a specification property S for Algorith which depends on the controller gains K. The property of satisfying S is verified only for the upper and lower values of the critical paraeters. We reark that the algorith terinates after a nuber of iterations dictated by the Log-over-log ound providing a set of gains {K, K,,K s }. The starting point of Algorith (Fig. 3) is the use of these gains. For each gain K rand = K j, j =,,,s, the algorith is based on randoization of all uncertain paraeters (and not only the critical ones) within their given intervals, according to a specified distribution p( ). Given accuracy ε (,) and confidence η (,), the Chernoff ound is used for coputing the required saple size for the specification property S. The epirical probability of satisfaction is coputed. Algorith 3 (Fig. 4) is finally utilized. This algorith has a structure siilar to Algorith, but a different specification property S 3 based on flying qualities [3] is introduced. The corresponding epirical probability is coputed using the sae saples size utilized in Algorith. Fig. 4: Algorith 3 Probabilistic Robust Perforance. VI NUMERICAL RESULTS In this section, nuerical results are provided. The proposed control design ethodology has been applied to the design of an ebedded real-tie syste for autonoous flight control. The autopilot syste includes guidance, navigation and loop stabilization. The obtained gain set is used for guidance/trajectory tracking and platfor stabilization feedback loops. The guidance laws include altitude/velocity and heading hold loops, while the stabilization issues are related to hold attitude angles and to dap attitude rates by coanding aerodynaic control surface deflections. The proposed gain synthesis ethodology can be applied to full state feedback as well as to output feedback control design. The case study reported here is based on the assuption of decoupled dynaics and it is focused on the longitudinal plane dynaics stabilization. In particular, we restrict our attention to a full state feedback longitudinal control, just deanded to the elevon control surfaces. The state vector x R 4 is defined as: Fig. 3: Algorith Probabilistic Robustness Analysis. x = [ V α q θ] T where V is the flight speed, α is the angle of attack, q is the pitch rate and θ is the pitch angle. The control vector u R is defined as: u = [ δ e ] where δ e consists of the syetrical elevon deflection. State and control atrices characterizing the noinal reference condition (V = 3 /s; h = 5 ) are obtained by nuerical linearization: RTO-EN-SCI-95 4A - 5

6 Robust and Randoized Control Design of Mini-UAVs: The MH Platfor.93.3 A = = Next, we observe that structured paraeter uncertainties are taken into account, including those related to flight conditions (dynaic pressure), aerodynaic data (stability and control derivatives), geoetric and inertial data. Uncertainties related to the flight conditions can be ascribed to the real flight in a non-ideally-cal air and to the need to cover a portion of the flight envelope as large as possible. Uncertainties concerning the aerodynaic data can be related to experiental easureent errors or coputational approxiations due to nuerical evaluation. Finally, uncertainties in ters of geoetric and inertial data ay take into account anufacturing inaccuracies. Therefore, the vector R 6 consisting of uncertain δ i is described in Table I. For siplicity, only approxiate values of upper and lower bounds are shown. TALE I. PARAMETER UNCERTAINTIES Plant and flight condition uncertainties # paraeter p.d.f. δ i % - δ i + δ i flight speed [/s] U 3. ±5. 5. altitude [] U 5. ±.. 3 ass [kg] U.5 ± wingspan [] U. ± ean aero chord [] U.536 ± wing surface [ ] U.5 ± oent of inertia [kg ] U.566 ± Aerodynaic database uncertainties # paraeter p.d.f. ean % σ i 8 C X coefficient [-] G -.5 ±.4 9 C Z coefficient [-] G ±5.5 C coefficient [-] G -.4 ±5.4 C Xq derivative [rad - ] G.435 ±.65 C Zq derivative [rad - ] G ±.5 3 C q derivative [rad - ] G ±5. 4 C Xδ derivative [rad - ] G.77 ±.54 5 C Zδ derivative [rad - ] G ±5. 6 C δ derivative [rad - ] G ±5.5 In the reaining of the Section the obtained nuerical results characterizing the three phases previously described are presented. Phase : Rando Gain Synthesis As critical uncertain paraeters for rando gain selection flight speed and ass are chosen, i.e. the vector c is defined as: c = [ δ, δ ] T The flight speed has been chosen as critical paraeter in order to optiize gain scheduling issues, while the take-off ass represents a key paraeter in ission profile definition and flight perforance evaluation. The lower and upper bounds of the critical uncertain paraeters are reported in Table I. The specification property S is defined as follows: S = {K K : A cl (K) = A c - c K satisfies the properties listed below} I( λ i ) Re( λ i ) < 4. rad/s < ωsp < 6. rad/s.5 < ζsp <.9. rad/s < ωph <.5 rad/s. < ζ PH <.3 ωsp = ω ωsp < ± 45% SP upper lower ωph = ωph ωph < ± % upper lower where λ i represents the eigenvalues and ω and ζ are the undaped natural frequency and daping ratio of the syste characteristic odes, respectively. The subscript SP and PH refer to the short period and the phugoid ode, representing the two dynaic odes characterizing the aircraft otion in the longitudinal plane. The specification property definition is strictly related to the user needs in ters of ission profile and to the reference platfor. Fro the properties specified above, it can be observed that a copletely decoupled dynaics with classical odal characterization (two periodic stable odes) are required for the MH platfor. The nuber of rando saples for the K atrix, coputed according to the Log-over-log ound with ε = 4-5 and η = 3-4, is N K =,. The obtained set of rando gains satisfying the specification property S are reported in Table II. TALE II. RANDOM GAIN SET FROM GAIN SYNTHESIS Gain set K V K α K q K θ K K K K K Phase : Probabilistic Robustness Analysis The uncertain paraeters considered in the case study, their noinal values, probability density function and tolerances are given in Table I. As can be observed fro Table I, unifor and gaussian probability density functions have been used to characterize paraeter uncertainties: geoetric, inertial and operational uncertainties are characterized by unifor p.d.f. while the set of uncertainties related to the aerodynaic database are characterized by a gaussian p.d.f. This criterion has been adopted due to the nature of the paraeters: the uncertain 3 4A - 6 RTO-EN-SCI-95

7 Robust and Randoized Control Design of Mini-UAVs: The MH Platfor value of the aerodynaic derivative has to be intended with higher probability close to the noinal value experientally or nuerically obtained. Since the aerodynaic data vary as a function of the velocity, the ean value as well as the variance of the uncertainties corresponding to these paraeters are approxiate values referring to a fixed operating condition. The specification property S is defined as follows: S = { : A cl ( ) = A c ( )- c ( )K rand satisfies the related properties} The user defined properties related to S are the sae as for S. The nuber of saples for the specification property S, is obtained by the Chernoff bound with ε =.45 and η =.45, and it is equal to N = 5,. The results in ters of estiated probability of stability are reported in Table III. TALE III. ESTIMATED PROAILITY OF STAILITY RELATED TO PHASE (SSRA) AND PHASE 3 (SPRA) Gain set SSRA SPRA K K K K K Phase 3: Probabilistic Robust Perforance The uncertain paraeters and the rando saples N considered for the Phase 3 of the proposed ethodology are the sae as for the Phase (Table I). The specification property S 3 is defined as follows: S 3 = { : A cl ( ) = A c ( )- c ( )K rand satisfies the properties listed below}.5 rad/s < ωw < 5. rad/s. < τp <.5 where ω W and τ P are the bandwidth and the phase delay of the frequency response, respectively. The specification property S 3 is defined accordingly to the bandwidth criterion reported in the aeronautical standards [3]. A graphical approach to the perforance robustness analysis, based on the bandwidth criterion plot, has been added to the evaluation of the epirical probability of satisfaction of the specification property. The case study discussed here does not include properties related to tie doain response. The estiated probability of stability characterizes the level of atching the desired dynaics and it represents a key feature to choose the best robust solution. To this end, graphical tools have been also utilized to perfor robustness analysis of the designed closed loop syste. As an exaple, Fig. 5 reports the root locus plot for two different solutions, characterized by different probability levels p N. It can be observed that the closed loop syste obtained by the rando gain set K fulfilling specification property S could give raise to soe unstable conditions when all uncertain paraeters are considered. Therefore, even if the associated probability of stability is high, it is not considered as a good gain and it is consequently discarded. Fig. 5: Root locus plot for K (black) and K 4 (blue) highlighting the regions of interest for stability robustness analysis. Fig. 6: andwidth criterion plot for K (black) and K 3 (blue) highlighting the regions of interest for perforance robustness analysis. The results in ters of estiated probability of stability are reported in Table III. Differences could be shown for the probability values between SSRA and SPRA results: they can be due to the dependence of the stability criteria on the user s experience and knowledge about the plant dynaics despite of the relationships of the perforance etrics with the standard requireents. For the present case study, the K solution has been chosen as the best fit solution according to a coproise between the stability and the perforance etrics. VII CONCLUSION AND REMARKS In this paper, we developed a ethodology, based on the theory of randoized algoriths, for gain selection and robustness analysis of the Mini-UAV platfor MH. The proposed ethodology deonstrates to resolve successfully soe critical issues arising when classical/odern control techniques are used. In particular, the nuerical results given in Section VI enable the aerospace engineer to copute the estiated probability that specific perforance requireents are et. Subsequent research will be perfored utilizing different control structures, thus showing the flexibility of the approach in different contexts. RTO-EN-SCI-95 4A - 7

8 Robust and Randoized Control Design of Mini-UAVs: The MH Platfor An extended version of this paper has been subitted to an international journal for possible publication. ACKNOWLEDGMENT This project is supported by the Italian Ministry for Instruction, University, and Research (MIUR) within the frae of the project PRIN 4 (Project Nuber ): Study and developent of a real-tie land control and onitoring syste for fire prevention. The aerial platfor is based on the MicroHawk configuration, developed at the Aerospace Engineering Dept. of Politecnico di Torino (national patent no. TO3A7, holder Politecnico di Torino, international request PCT/I4/94). REFERENCES [] W.R. Davis, MicroUAVs, 3rd Annual AUVSI Syposiu, 996. [] F. Quagliotti, G. Guglieri G.,. Pralio and L. Lorefice, Siulation Tools for Flight Dynaics Analysis in the Design of MAVs, 4th ICAS Congress, Yokohaa, Japan, 4. [3] M. Elgersa, S. Gangli, V. Ha and T. Saad, Statistical Perforance Verification for an Autonoous Rotorcraft, Proc. of IEEE International Syposiu on Intelligent Control, Taipei, Taiwan, 4. [6] R.F. Stengel, Soe Effects of Paraeter Variations on the Lateral- Directional Stability of Aircraft, AIAA Journal of Guidance and Control, Vol. 3, pp. 4-3, 98. [7] L.R. Ray and R.F. Stengel, A Monte Carlo Approach to the Analysis of Control Syste Robustness, Autoatica, Vol. 9, pp. 9-36, 993. [8] Q. Wang and R.F. Stengel, Robust Nonlinear Flight Control of a High Perforance Aircraft, IEEE Transactions on Control Systes Technology, Vol. 3, pp. 5-6, 5. [9] Q. Wang and R.F. Stengel, Probabilistic Control of Nonlinear Uncertain Dynaic Systes, in Probabilistic and Randoized Methods for Design under Uncertainty, Springer-Verlag, London, pp , 6. [].L. Stevens and F.L. Lewis, Aircraft Control and Siulation, John Wiley & Sons Inc., 3. [] R. Tepo, E.W. ai and F. Dabbene, Probabilistic Robustness Analysis: Explicit ounds for the Miniu Nuber of Saples,' Systes and Control Letters, Vol. 3, pp. 37-4, 997. [] H. Chernoff, A Measure of Asyptotic Efficiency for Tests of a Hypothesis ased on the Su of Observations, Annals of Matheatical Statistics, Vol. 3, pp , 95. [3] Anonyous, Flying Qualities of Piloted Aircraft, MIL-HDK- 797, Departent of Defense, USA, 997. [4]. McCorick, Aerodynaics, aeronautics and flight echanics, John Wiley & Sons Inc., 995. [4] K. Zhou and J.C. Doyle and K. Glover, Robust and Optial Control, Prentice-Hall, Upper Saddle River, 996. [5] R. Tepo, G. Calafiore and F. Dabbene, Randoized Algoriths for Analysis and Control of Uncertain Systes, Springer-Verlag, London, 5. 4A - 8 RTO-EN-SCI-95