New Parametric Affine Modeling and Control for Skid-to-Turn Missiles

Transcription

1 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 9, NO. 2, MARCH New Parametric Affine Modeling and Control for Skid-to-Turn Missiles DongKyoung Chwa and Jin Young Choi, Member, IEEE Abstract This paper presents a new practical autopilot design approach to acceleration control for tail-controlled skid-to-turn (STT) missiles. The approach is novel in that the proposed parametric affine missile model adopts acceleration as the controlled output and considers the couplings between the forces as well as the moments and control fin deflections. The aerodynamic coefficients in the proposed model are expressed in a closed form with fittable parameters over the whole operating range. The parameters are fitted from aerodynamic coefficient lookup tables by the proposed function approximation technique, which is based on the combination of local parametric models through curve fitting using the corresponding influence functions. In addition, a feedback linearizing controller is designed by using the proposed parametric affine missile model. Stability analysis for the overall closed-loop system is provided, considering the uncertainties arising from approximation errors. The validity of the proposed modeling and control approach is demonstrated through simulations for an STT missile. Index Terms Aerodynamic coefficient in closed form, feedback linearizing controller, function approximation, parametric affine missile model, skid-to-turn (STT) missiles. NOMENCLATURE Missile body coordinate system. -, -, -components of the linear velocity vector of the missile. -, -, -components of the angular velocity vector of the missile. -, -, -components of the vector of aerodynamic forces. -, -, -components of the vector of aerodynamic moments. -, -, -components of the vector containing gravitational forces. -, -, -components of the moment of inertia of the missile. Products of inertia. Moment of inertia of the airframe represented in the body axis. Angle of attack, sideslip angle, and bank angle in radians [,, ]. Velocity of sound. Total velocity of missile. Manuscript received May 10, 1999; revised September 22, Recommended by Associate Editor N. Sundararajan. This work was supported by the Agency for Defense Development, the Automatic Control Research Center of Seoul National University, and the Brain Korea 21 Project. The authors are with the School of Electrical Engineering, Seoul National University, Seoul, Korea ( dkchwa@neuro.snu.ac.kr; jychoi@ee.snu.ac. kr). Publisher Item Identifier S (01) Mach number., Missile mass and air density. Dynamic pressure., Aerodynamic reference area and length of the missile. Deflection of yaw (pitch) control fin. Yaw (pitch) control fin command. Yaw (pitch) achieved acceleration. Yaw (pitch) commanded acceleration. Derivative of with respect to time. Euclidean norm of the vector.. I. INTRODUCTION RECENTLY, nonlinear control techniques have been applied to autopilot designs for missiles with highly nonlinear characteristics [1] [11]. Most articles [1] [9] have used minimum phase missile models with the controlled outputs of angular position or angle of attack to apply the well-known feedback linearization technique. In practice, however, it is desirable to control acceleration directly for high performance autopilot design. Unfortunately, the dynamics between the control fin deflection and missile acceleration in tail-controlled missiles has nonminimum phase characteristics [1]. To handle this nonminimum phase problem, a partially linearized system has been applied to autopilot design [10] and the singular perturbation technique has been used to obtain an approximate minimum phase model for feedback linearizing control [11]. The controllers described in those articles require lookup tables for the inverses of aerodynamic lookup tables. The use of a lookup table may give some limitations in applying various control techniques such as parametric adaptive control techniques. In practice, adaptive control may be required since uncertainties inevitably exist in aerodynamic coefficients obtained through wind-tunnel tests. Therefore, a parametric missile model will be useful for applications of various control techniques to autopilot designs. Furthermore, an affine missile model with respect to the first-order control input will simplify the autopilot design. Affine missile models have been proposed previously [1] [9]. The successful applications in [1] [8] arise from the fact that accelerations are not directly controlled. Instead, other variables such as angular position or angle-of-attack are chosen as the controlled outputs. Although modification of the desired output from the accelerations to angle-of-attack and sideslip angles can circumvent the nonminimum phase problem associated with zero dynamics, the system transient response cannot be precisely controlled. In particular, results in [1] are based on a linearized /01$ IEEE

2 336 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 9, NO. 2, MARCH 2001 model. On the other hand, parametric missile models (i.e., missile models with parameterized aerodynamic coefficients) were developed and utilized in [3], [4] for a fixed local operating range, and over the whole operating range in [5] [8]. In [5] [8], however, the coupling effects between the forces and control fin deflections were ignored. In [9], a pseudolinear model was constructed by chaining together the locally linearized modes at various flight conditions and the on-line affine model was obtained through the computational algorithms. These models suffered from performance degradation due to neglected nonlinearities and problems of implementation caused by the complex computational algorithm, respectively. In particular, the force components due to control fins of our missile system are not so small as to be neglected, as they were for several missile configurations [12], [13]. Thus, for practical applications, it becomes desirable to obtain the parametric affine missile model for acceleration control, the nonlinearities and also the coupling effects between the forces and control fin deflections are fully considered regardless of flight conditions. In this paper, we present a new parametric affine missile model over the whole operating range, accelerations are used as controlled outputs, and the couplings between the forces as well as moments and control fin deflections are fully considered. The proposed model uses a function approximation technique that is characterized by several local parametric models indexed by Mach number and the corresponding influence functions indicating the local influenced regions. The aerodynamic coefficients of the model can be expressed in closed form with fittable parameters. The parameters are fitted using the least-squares algorithm from aerodynamic coefficient lookup tables in off-line environments. In addition, a feedback linearizing controller [14] [16] is designed by using the proposed parametric affine missile model, which is simplified to a weak minimum phase model [14] by using the singular perturbation approach [11]. This weak minimum phase model is then used to design a controller that makes the overall closed-loop system follow a given reference model under the assumption that there are no approximation errors in the parametric model. The influences of the singular perturbation and the approximation errors are analyzed by deriving error dynamics and using Lyapunov stability theory. The validity of the proposed modeling and control approach is demonstrated through simulations for an STT missile. The present parametric affine model for the STT missile is described in Section II. Section III presents the controller design and provides a stability analysis for the overall closed-loop missile system. In Section IV, simulation results are included to verify the proposed modeling and control approach. Conclusions follow in Section V. A. STT Missile Dynamics Using the body axis equations of motion and assuming a rigid body, the linear and angular momentum vectors,,,, are differentiated with respect to time and are equated to the forces and moments acting on the body. Performing the differentiation relative to a nonrotating coordinate system, we can describe the complete six-degree-of-freedom missile dynamics as the following translational (1a) and rotational (1b) equations: (1a) (1b),,. For the derivation of missile dynamics, the following assumptions are made. Assumption 2.1: The variations of,, and are negligible ( ). Assumption 2.2: The missile has and symmetry (, ). Assumption 2.3: The missile is roll-stabilized ( ). Assumption 2.4: constant. Remark 2.1: Assumptions 2.1, 2.2, and 2.3 are usually made in modeling STT missiles. Despite Assumption 2.3, the generality of a missile model can be seen from the fact that even with Assumption 2.3, yaw and pitch dynamics are coupled through bank angle. Remark 2.2: The approximation in Assumption 2.4, which holds relatively well in realistic situations, was made in [10], fast maneuvering and high performance were achieved with this assumption. As for the equality in Assumption 2.4, the forward linear velocity is customarily assumed to be constant in the flight control area [6] and can be kept constant throughout maneuvers by throttle control [7], [8]. In fact, from the various simulation results in Section IV without making Assumption 2.4, we could see that sufficient performance can be obtained with the proposed model regardless of Assumption 2.4. The nonlinear differential equations for the missile model are given by Yaw dynamics: (2a) II. A PARAMETRIC AFFINE WEAK-MINIMUM-PHASE MODEL FOR STT MISSILES In this section, we first describe the missile dynamics together with parametric aerodynamics based on a function approximation technique, and then obtain a weak minimum phase parametric STT missile model by applying the singular perturbation technique. Pitch dynamics: (2b)

3 CHWA AND CHOI: NEW PARAMETRIC AFFINE MODELING AND CONTROL 337 Note that the terms related to can be included easily into the first rows in (2a) and (2b). These, however, only complicates the form of the resulting parametric affine model and were observed to make almost no difference in performance through simulation results. Thus, we derive the parametric affine model from the above dynamics. The aerodynamic coefficients, and are usually given in lookup tables and described in terms of, and. Since aerodynamic coefficients are described in this way, (2a) and (2b) are natural expressions for the derivation of a parametric missile model. The following relations between functions,,,,, and are used for (2a) and (2b) as in [11]: (3a) (3b) Note that and correspond to the dynamic damping terms for the moment coefficients. Substitution of (3) into (2) yields the following model: Yaw dynamics: Pitch dynamics: (4a) (4b) and denote the distances from the nose of the missile to the center of pressure of the control fins and the center of gravity, respectively. In the rest of this paper, we consider just the yaw dynamics as the same can be said of the pitch dynamics. B. Parametric Function Approximation for Aerodynamics In this subsection, a function approximation technique is introduced for a parametric affine model for acceleration control. The technique is characterized by several local parametric models indexed by Mach number and corresponding influence functions indicating the local influenced regions, The aerodynamic coefficients and are approximated using the proposed technique. The aerodynamic coefficient used in this paper is of the following form with scheduling variables,,, and : Here, and denote lateral aerodynamic coefficients with bank angle 0 and 45, respectively. In the following, and are approximated by local parametric models on local Mach regions, which is similar to the models in [3] [6]. The Mach operating range is divided into several local regions, which have their centers at,. Each local parametric model for and is proposed by the following form: (5) (6a) (6b) for, is the width of each local region. The,,, are all fitting parameters for,, which are obtained by a curve fitting technique from a lookup table of aerodynamic coefficients and can be tuned further in on-line environments. When using the above local parametric models, discontinuities occur at the borders of the local regions. To interpolate among the local models and maintain the continuity of the overall approximated function, we use locally defined influence functions, which form a partition of unity [17] over the whole Mach region. The partition of unity divides the whole operating region into nondisjoint regions, as defined in the following definition. Definition 2.1: Let a compact set exist. Then a collection of functions, defined in an open set containing, is called a partition of unity for, ifit satisfies the following properties: 1) for each, we have and 2) for each, wehave. Using the influence functions, the interpolated whole function is obtained by (7a) (7b) If we choose the influence functions as appropriate continuous functions, a smooth interpolation among local models results. Examples of generating functions for influence functions include radial basis functions and triangular functions

4 338 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 9, NO. 2, MARCH 2001 (i.e., first-order splines). Here, we use radial basis functions defined by is a nonzero constant representing the shape of the influence function and determining the controller performance via approximation accuracy. To satisfy the properties of Definition 2.1, the influence function is obtained by (8) (9),, and are fitting constants. Letting we have (14a) (14b) (15) is a time-varying coefficient depending on. The final forms of the aerodynamic coefficients in the yaw channel are given by The (5a), (6a) and (6b), (7a) and (7b) yield the following approximated function : (16) and are approximation errors. C. Weak Minimum Phase Affine STT Model Substituting (16) into (4a), we obtain the following form of yaw dynamics: (17) Defining the variables we finally get (10) (11a) (11b) (11c) (12) In order to make the above system almost linear, we employ the control input (18) is a new control input variable that will be defined later and is a nonzero parameter defined in (11). The resulting dynamics, controlled by (18), is,, and can be considered as slowly timevarying parameters depending on Mach number and bank angle. That is, can be described by a linear combination of,,. This approximated aerodynamic function gives a closed-form expression that can replace a three-dimensional lookup table of the aerodynamic coefficients. Aerodynamic coefficient does not depend on Mach number. Therefore, we do not need to use a local model. Similar to [5], [6], is parameterized by a curve fitting technique, the approximated functions are obtained by (13) (19) Feedback linearization techniques [14] [16] cannot be applied directly to the dynamics of tail-controlled missiles for acceleration, since a direct application of feedback linearization to

5 CHWA AND CHOI: NEW PARAMETRIC AFFINE MODELING AND CONTROL 339 a nonminimum phase system can leave the zero dynamics unstable, and thus it cannot guarantee the internal stability. Similar to [11], the dynamics described by (19) can be approximated to a weak minimum phase model [14] as follows. Since in the second row in (19) is physically a very large value, converges to a steady-state value relatively quickly. In other words, we can assume that can be equated to zero and is actually a steady-state value. Then, can be obtained as (20) Combining (19) and (20), we have the following simplified yaw dynamics: (21) as in (18), when the corresponding control input is chosen as and the compensator as (24) (25) then the behavior of the system (22) follows that of the reference model (26) and are design parameters for an appropriate reference model. Proof: Differentiating in the third row in (22) with respect to time, we obtain (27) and this in turn, together with (24), becomes The yaw dynamics in (21) is of weak minimum phase since the zero dynamics, which have bank angle as a solution, are physically stable [14]. (28) Combining the time derivative of (28) and compensator (25) gives III. CONTROL LAWS AND STABILITY ANALYSIS In this section, we present a procedure for controller design, and then analyze the stability of the original missile system. A weak minimum phase missile system with no approximation error ( ) can be given from (20) and (21) as (22) (29) This can be arranged to yield (26). (Q.E.D.) In the above equation (29), we can see that the yaw and pitch channels are now decoupled and behave as a second-order reference model when there is no approximation error. Remark 3.2: It is noted that since coupled parameters are considered in the design of a controller as in (23), the coupling effect can be reduced. In practice, the following control laws are applied to the full dynamics (19) instead of (23) (25): Here, we assume the following to simplify the control structure. Assumption 3.1: The time derivative of bank angle is zero, meaning that, i.e.,. Remark 3.1: This assumption has been commonly used in the STT missile control literature [12]. The time derivative of the coupled parameter due to bank angle is assumed here to be negligible in order to simplify the controller structure. However, the influence of the time varying bank angle is included in the stability analysis under Assumption 3.3. The controller design method using the feedback linearization technique for a weak minimum phase missile model (22) is summarized in the following proposition. Proposition 3.1 (Control Laws): Together with the control law for the system (22) with Assumption 3.1 described by (30) For the stability and performance analysis of the overall closed-loop system with approximation errors consisting of (19) and (30), error dynamics are derived in Proposition 3.2. Proposition 3.2 (Error Dynamics): The error dynamics between (19) and (22) is given by (31) (23)

6 340 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 9, NO. 2, MARCH 2001 (22) and (25) become (32) (41) and (33) Taking the time derivative of in (34), we have (34) as (35) (36) The output error (42) defined in (34) can be obtained easily as (43) (37) Proof: The overall closed-loop system consisting of (19) and (30), together with (15), can be given by Introducing the notation in (32) and (33) gives Meanwhile, the reduced system as follows. Together with (38) (39) in (22) can be expressed (40) Finally, the first and third rows in (31) can be directly obtained from (39), (41), and (43). This completes the proof. (Q.E.D.) Propositions 3.1 and 3.2 will be used in the following theorem on the stability and performance analysis for the overall closed-loop system with approximation errors. Here, we assume the following. Assumption 3.2: The approximation errors and are bounded and smooth. Assumption 3.3: The bank angle and its time derivative are bounded. Remark 3.3: Assumption 3.3, which is weaker than Assumption 3.1, is made here for the analysis of the influence of approximation errors on the stability in the next theorem as well as considering practical situations, as Assumption 3.1 is made for simplification of control laws. Theorem 3.1 (Stability Analysis of Error Dynamics): The error dynamics described by (31), with Assumptions 3.2 and 3.3, are stable in the sense that: 1) the tracking error is uniformly ultimately bounded, meaning that it remains in a small neighborhood of zero after some time, i.e.,. 2) when no modeling error exists, i.e., the approximation errors reduce to zero and also Assumption 3.1 is satisfied, also reduces to zero asymptotically, i.e., if Proof: See Appendix A. Remark 3.4: We analyzed the stability and performance of our approach under Assumptions 3.2 and 3.3, stating the boundedness of the modeling error, similar to robust or adaptive control schemes the uncertainty or modeling error is usu-

7 CHWA AND CHOI: NEW PARAMETRIC AFFINE MODELING AND CONTROL 341 (a) (a) (b) Fig. 1. C ( ;) from the lookup table (a) and its parametric model ^C ( ;)(b). ally assumed to be bounded. These assumptions imply that the approximation errors have the property that when the angle of attack and sideslip angle remain within the range of interest for the designed controller, approximation errors also remain bounded. The analysis described here has significance in that the validity of the weak minimum phase system with approximate parametric aerodynamics is discussed in a logical way. IV. SIMULATION RESULTS The proposed approach was applied to an STT missile. Simulations were performed in two directions: first, to show that the proposed parametric model based on function approximations is a valid representation of the actual model with aerodynamic lookup tables; second, to evaluate the performance of the designed controller using the parametric affine missile model. A. Parametric Expression of the Lookup Table for Aerodynamic Coefficients The fitting parameters,, and in the parametric model given by eqn. (13) have been obtained by curve fitting; these values are all positve constants. Fig. 1(a) and (b) show the values from a lookup table and the parametric model, respectively. The approximation error between them is less than 10% for every and.as shown in Fig. 1, is well fitted with respect to and. shows the same results. (b) Fig. 2. C from the lookup table (a) and its local parametric model ^C (b); M = M. Each and in (5) can be approximated also for each Mach number by the local parametric models of (6a) and (6b). The lookup tables for are given in six tables indexed by Mach number as. For the given, the comparison between the lookup table and the local parametric model is depicted in Fig. 2. For Mach numbers not provided in the lookup table, the aerodynamic coefficients are calculated by linear interpolation. In the proposed parametric model, the global expression form over the whole range is obtained by influence functions as shown in (12). Fig. 3 compares the aerodynamic coefficients for Mach number between and, (a) shows the result obtained by linear interpolation for lookup tables and (b) shows the result obtained by the proposed parametric expression of (12). As shown in Figs. 2 and 3, we can see that the proposed parametric expression shows a good approximation capability. For other Mach values, similar accuracy could be obtained in our experiments. Note that the shaping parameters,, in (8) are chosen as. The same results have been obtained for,, and but are omitted for brevity. The aerodynamic coefficients (or ) are given by 12 lookup tables, i.e., and for each Mach index. The proposed parametric model for includes 36 fitting parameters. However, these parameters are merged into only three parameters, which are dependent on the Mach number and bank angle. It is interesting to note that these time-varying parameters can be tuned further by on-line adaptive control. This issue will be pursued as a further study.

8 342 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 9, NO. 2, MARCH 2001 (a) (a) (b) (b) Fig. 3. C from a lookup table using a linear interpolation (a) and the proposed parametric model ^C (b); M <M <M. B. The Performance of the Nonlinear Controller The full six-degree-of-freedom nonlinear equations described by (1a) and (1b) have been simulated using the aerodynamic lookup tables. For tracking a square wave acceleration command and for missile-target interception engagement, we used only Assumptions 2.1 through 2.3, considering this to be a more practical situation. Note that Assumption 2.4 is used just for the derivation of the proposed model. Design parameters for a feedback linearizing controller in (25) are selected as (c) (44) (45) As an actuator model, we included the following low-pass filter: (46) for each yaw and pitch channel, the time constant s. In fact, we could see that time constants larger or smaller than this gave negligible influence on performance as in [10]. We evaluated the performance for tracking square wave commands with the initial velocity m/s. Fig. 4 shows the tracking performance of the designed controller and such Fig. 4. Tracking performance of a square wave command. (a) Commanded (dotted) and achieved (solid) output. (b) Control fin deflection. (c) Roll, pitch, and yaw rate. (d) Angle of attack and sideslip angle. (d)

9 CHWA AND CHOI: NEW PARAMETRIC AFFINE MODELING AND CONTROL 343 (a) (a) (b) Fig. 5. Missile-target intercept engagement. (a) Three-dimensional missile-target tracking trajectory (M: Missile, T: target). (b) Evolution of forward linear velocity U with time. parameters as the rise time, steady-state error, and overshoot are all satisfactory. In particular, the rise time can be made to be smaller by choosing other proper values of design parameters in (45) due to the feedback linearized system behavior. The results show the validity of the proposed parametric affine missile model. In addition, fin deflection angles do not reach the saturation limits and so do not require large control energy. Here, we can see coupling between the yaw and pitch dynamics, but it is effectively reduced. Furthermore, it shows that the functional approximation of aerodynamic coefficients is accurate enough for approximation errors to have little effect on the accuracy of acceleration outputs. Comparing the trajectories in Fig. 4(a) and (d), we can see the close relation between the sideslip-angle (or angle-of-attack) and yaw (or pitch) accelerations of the missile. Since the missile is assumed to be roll-stabilized, the roll rate is set identically to zero in Fig. 4(c). Simulations were also conducted for the scenario in Fig. 5(a), which shows the three-dimensional missile-target tracking trajectories. In this scenario, the target initially travels at constant velocity or maneuver, and two step-changes also occur in target accelerations. This simulation environment is specified in order to assess the performance characteristics in a closed-loop surface-to-air engagement scenario. Here, we assume that all measurements are noise-free. The proportional navigation (PN) law [18] is used in our simulation as a guidance law. In addition, the first-order lag, is the time constant of the (b) Fig. 6. (a) Y -, Z-axis accelerations of parametric affine missile system (dashed and dotted: guidance command, solid: actual missile output) and (b) their corresponding control fin deflection angles. target, is used as a target maneuver model. Although a control system is usually designed under the assumption that a missile has entered a burn-out state, and also that missile mass is constant, the changes of thrust, missile mass, and inertial moment of the missile are included for our simulation. The control start time, when a control action begins, is selected as one second after the missile is launched. It is assumed that the thrust of the missile changes with time and it enters a burn-out state after 3.2 s, until which the missile mass and the inertial moment of the missile are assumed to change linearly. A gravity effect is included in the missile dynamics and also in the guidance part. The acceleration command, which includes a gravity bias term, is generated considering this gravity effect. The evolution of forward velocity with time is depicted in Fig. 5(b), and the decrease of the velocity is due to external force such as drag force. Fig. 6 shows the result of the overall missile system under the scenario in Fig. 5(a), validating the practical effectiveness of the proposed scheme. Note that we conducted also several other engagement scenarios and could obtain the similar performance. In this section, we have shown that satisfactory tracking performance for square wave commands and for missile-target intercept engagement can be achieved by the nonlinear control of the proposed parametric affine missile model. V. CONCLUSION In this paper, we have proposed a parametric affine missile model for acceleration control and designed a nonlinear con-

10 344 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 9, NO. 2, MARCH 2001 troller together with stability analysis. The distinctive features of our model are summarized as follows. 1) It is effective for overall flight conditions contrary to those in [3], [4]. 2) It adopts accelerations as controlled outputs while existing parametric models [5], [6] use angle of attack and sideslip angle as controlled outputs in order to avoid nonminimum phase characteristics. 3) It fully considers the coupling terms among the forces or the moments and control fin deflections, unlike conventional parametric models [5] [8]. 4) A closed-form expression for the aerodynamic coefficients is given through a function approximation technique, based on several local parametric models obtained by curve fitting and locally defined influence functions. This has the strong advantage that versatile control techniques such as adaptive control can be applied easily to the autopilot design. 5) Unlike the model using linear velocities (i.e., and ) [10], [11], the whole dynamics are expressed in terms of angle of attack and sideslip angle (i.e.,, ), which is natural since and are obtained by either measurement or estimation and also aerodynamic coefficients are usually indexed by,,,, and. The parameters of the missile model in this paper have been obtained by a function approximation technique from aerodynamic lookup tables. Simulations showed there is a small amount of approximation error between the fitted functions and the lookup tables. In general, the lookup tables have inherent modeling errors since they are obtained through wind-tunnel tests. Hence, the actual approximation errors may be different from those presented in the simulations (they may be larger or smaller in practice). In any event, modeling uncertainties are inevitable in both the parametric model and the lookup table model. As a further study, an adaptive control with on-line tuning of the parameters will be considered to accommodate the remaining approximation errors. APPENDIX A PROOF OF THEOREM 3.1 First, and in (37) correspond to the modeling error that comes from the approximation errors and the controller, which is designed with Assumption 3.1. The first and second rows in (31), which are slowly-varying equations, can be rewritten in matrix form as (A1) Second, we derive the upper bound of the norms,,, and through the choice of Lyapunov functions and algebraic manipulations. Since is a Hurwitz matrix for, we can choose a positive definite matrix satisfying for some proper positive definite matrix function candidates The time derivative of yields (A2). Define Lyapunov (A3) and are the maximum and minimum eigenvalues of matrix, respectively. It is noted that (A1) is used in the first equality and (A2), (A3) are used in the first and second inequalities in the above relation for, respectively. Substituting into the above inequality, we can have Integrating the above inequality with respect to time gives (A4) (A5) ;. Therefore, from (A3) and (43), respectively, (A5) reduces to (A6a) Note that the third row in (31) shows fast-varying dynamic characteristics. From Assumptions 3.2 and 3.3, the time derivatives of fitting errors and bank angle are bounded. By Proposition 3.1, is stabilized by the control action (23). (A6b)

11 CHWA AND CHOI: NEW PARAMETRIC AFFINE MODELING AND CONTROL 345,.Now, we can take the time derivative of in (A3) and proceed as in to obtain (22a), this implies that use the inequalities and (A8) to obtain is bounded. Therefore, we can (A10) (A6c) From the above argument in (A6a) (A6c) we can see that the boundedness of,,, and depends on that of. Third, we therefore consider in more detail. The time derivative of (32) becomes (A11) Since and are bounded in the form of (A6a) and (A6b), by combining (A7), (A10), and (A11) we can have the following inequality for some proper constant, : (A12a) (A7) the second equality follows from the second row of (39). Taking the time derivative of the second row of (30a), we have (A12b) Now, we can have the inequality (for proof, see Appendix B) (A8) (A13) From Assumption 3.2, and are both bounded irrespective of the magnitude of. Since the reduced system can be stabilized by Proposition 3.1, together with Assumptions 3.2 and 3.3, we have (A9),, are positive constants. Note that when there is no approximation error, tends to zero asymptotically, as can be seen in (26a). Actually, the boundedness of can be seen more clearly in the following way. From (26a), is bounded. Together with (15a) and the last row of Finally, we complete the stability analysis for from the previous arguments. Substituting the inequality (A13) into (A6c) and using the Gronwall Bellman inequality [14], we have (A14)

12 346 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 9, NO. 2, MARCH 2001 and this, in turn, becomes (A15) is the base of the natural logarithm. In (A15), is bounded, but the boundedness of and depends on the modeling error. In addition, according to (A6a), (A6b), and (A15), we can see that the steady state tracking error depends on both the initial condition and the modeling error. Since, which is the inverse of forward velocity as in (33), is sufficiently small, can be expected to remain in a small neighborhood of zero after sufficient time under Assumptions 3.2 and 3.3. Accordingly, from (A6b) and (A10) the tracking error depends mainly on the modeling error and can be asymptotically reduced to zero when the modeling error is also reduced to zero. This point can be seen also from the simulation results presented later. This completes the theorem. (Q.E.D.) APPENDIX B PROOF OF EQN. (A13) Substituting (A12a) into the last term in (A6c), we have The second and third inequalities follow from the last row of (31) and (A6a), respectively, and the fact is used in the fourth inequality. (Q.E.D.) ACKNOWLEDGMENT The authors would like to thank J. Kim, C. Song, H. Cho, and I.-J. Ha for their valuable comments and assistance during this work. REFERENCES [1] M. Tahk, M. Briggs, and P. K. A. Menon, Applications of plant inversion via state feedback to missile autopilot design, in Conf. Decision Contr., Austin, TX, Dec. 1988, pp [2] K.-Y. Lian, L.-C. Fu, D.-M. Chuang, and T.-S. Kuo, Nonlinear autopilot and guidance for a highly maneuverable missile, in Proc. Amer. Contr. Conf., Baltimore, MD, Jun. 1994, pp [3] R. A. Hull, D. Schumacher, and Z. Qu, Design and evaluation of robust nonlinear missile autopilots from a performance perspective, in Proc. Amer Contr. Conf., Seattle, WA, Jun. 1995, pp [4] R. A. Hull and Z. Qu, Dynamic robust recursive control design and its application to a nonlinear missile autopilot, in Proc. Amer. Contr. Conf., Albuquerque, NM, Jun. 1997, pp [5] J. Huang and C. F. Lin, Sliding mode control of HAVE DASH II missile systems, in Proc. Amer. Contr. Conf., San Francisco, CA, Jun. 1993, pp [6] J. Huang, C. F. Lin, J. R. Cloutier, J. H. Evers, and C. D Souza, Robust feedback linearization approach to autopilot design, in Proc. IEEE Conf. Contr. Applicat., vol. 1, 1992, pp [7] J. J. Romano and S. N. Singh, I-O map inversion, zero dynamics and flight control, IEEE Trans. Aerospace Electron Syst., vol. 26, no. 6, pp , [8] S. N. Singh and M. Steinberg, Adaptive control of feedback linearizable nonlinear systems with application to flight control, J. Guidance, Contr., Dyn., vol. 19, no. 4, pp , [9] V. H. L. Cheng, C. E. Njaka, and P. K. Menon, Practical design methodologies for robust nonlinear flight control, in AIAA, Guidance Navigation Contr. Conf., San Diego, CA, Jul. 1996, [10] J. I. Lee, J. H. Oh, I. J. Ha, E. G. Kim, and H. J. Cho, A new approach to autopilot design for highly nonlinear missiles, in AIAA, Guidance Navigation Contr. Conf., San Diego, CA, July 1996, [11] J. H. Oh and I. J. Ha, Missile autopilot design via functional inversion and time-scaled transformation, IEEE Trans. Aerospace Electron Syst., vol. 33, no. 1, pp , [12] J. H. Blakelock, Automatic Control of Aircraft and Missiles: Wiley, [13] M. J. Hensch and J. N. Nielsen, Tactical Missile Aerodynamics, Progress in Astronautics and Aeronautics. New York: Amer. Inst. Aeronautics Astronautics, 1986, vol. 104.

13 CHWA AND CHOI: NEW PARAMETRIC AFFINE MODELING AND CONTROL 347 [14] H. K. Khalil, Nonlinear Systems, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, [15] A. Isidori, Nonlinear Control System, 3rd ed. New York: Springer- Verlag, [16] N. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems. New York: Springer-Verlag, [17] M. Spivak, Calculus on Manifold. New York: W. A. Benjamin, [18] C. F. Lin, Modern Navigation, Guidance, and Control Processing. Englewood Cliffs, NJ: Prentice-Hall, DongKyoung Chwa received the B.S. and M.S. degrees in control and instrumentation engineering from Seoul National University, Seoul, Korea, in 1995 and 1997, he is currently pursuing the Ph.D. degree from the School of Electrical Engineering. He is now joining the project of Automatic Control Research Center (ACRC) at Seoul National University. His fields of interests are nonlinear, robust, and adaptive control theory and their applications to the guidance and control of flight systems. Jin Young Choi (S 89 M 93) received the B.S., M.S., and Ph.D. degrees in control and instrumentation engineering from Seoul National University, Seoul, Korea, in 1982, 1984, and 1993, respectively. From 1984 to 1989, he joined the project of TDX switching system at the Electronics and Telecommunication Research Institute (ETRI). From 1992 to 1994, he was with the Basic Research Department of ETRI, he was a Senior Member of Technical Staff working on the neural information system. Since 1994, he has been with Seoul National University, he is currently an Assistant Professor in the School of Electrical Engineering. He is also affiliated with Automation and Systems Research Institute (ASRI), Engineering Research Center for Advanced Control and Instrumentation (ERC-ACI), and Automatic Control Research Center (ACRC) at Seoul National University. From 1998 to 1999, he was Visiting Professor at University of California, Riverside. His research interests are neuro computing and control, evolutionary computing, adaptive and learning control, and their applications to missile, nuclear power plant, rapid thermal processing systems, and motors.