Outer-Loop Sliding Mode Control Approach to Longitudinal Autopilot Missile Design. B. Kada*

Transcription

1 Outer-Loop Sliding Mode Control Approach to Longitudinal Autopilot Missile Design B. Kada* *Aeronautical Engineering Department, King Abdulaziz University, P O Box 80204, Jeddah, KSA (Tel: ext ; bkada@ kau.edu.sa). Abstract: This paper presents a new feedback control methodology to design robust Outer-Loop Sliding Mode (OLSM) controllers for systems with nonlinear rapidly time varying and highly uncertain dynamics such as tactical missiles. Within Sliding Mode Control (SMC) framework and based on relative degree and local diffeomorphic coordinate transformation concepts, OLSM controllers that feature short finite time convergence, heavy modeling uncertainties robustness, and chattering extinction and suppression are designed. Two OLSM pitch-axis autopilots for tail-controlled missile are designed and implemented in computer-simulation environment. Simulation results demonstrate the capabilities of OLSM approach in providing high performance levels while maintaining robustness against heavy uncertainties. 1. INTRODUCTION Due to large uncertainties of aerodynamics and modeling errors appearing in the modeling of missile s dynamics, robust integrated Flight Control System (FCS) is necessary to orchestrate the various missile subsystems to successfully track the desired guidance commands and to ensure that all performance requirements are met. The FCS is one element of the overall missile homing loop and its design depends on several factors such as system mission, performance requirements, operating range, control constraint, and cost. Longitudinal or pitch-axis autopilot is one of the common FCSs designed to track desired Angle-Of-Attack (AOA), pitching rate, and commanded normal acceleration using tailfin deflection control. As the overall performance of a guided missile system depends on the performance of its Guidance and Control Algorithms (GCA), during the few past decades considerable attention has been paid to the design and implementation of such algorithms. Using different control methodologies, researchers have sought to alleviate the problem of classical controllers by augmenting these controllers with modern robust algorithms. A standard method in designing missile autopilots has been to apply linear design techniques within gain scheduling framework (Robert et al. 1993). As this method is a tedious procedure, and leads to a sub-optimal controllers whose actual performance and robustness properties are unknown, various approaches have been developed to minimize the required gain scheduling. These approaches have focused on involving nonlinear control techniques and seeking to minimize the effect of system nonlinearities on performance. In (Nicholas 1993) a nonlinear pitch-axis autopilot was designed by scheduling linear controller designs at constant operating conditions bounding the operating range of the scheduling variables. Linear Parameter Varying (LPV) was also used to design quasi-lpv robust missile autopilots using H optimal control and µ-synthesis (Pellanda et al. 2002; Devant et al., 1999; Misgeld et al., 2011). Feedback linearization is also one of the most popular nonlinear control design methods that have been used for flight control applications (Zhou 2009), but the method shows instability when applied to minimum phase. For their online learning and functional approximation capabilities, neural networks have been also emerged as a means of explicitly accounting for uncertainties in the nonlinear plant dynamics (Lin and Hwang 2003). Recently, backstepping approach was used to design certain missile autopilots (Lin and Fan 2009; Fan and Su 2010), but the complexity of the designed autopilots and the lack of physical meaning restrict their application and integration within real time FCS. Due to its impressive performances and exceptional robustness, sliding mode control approach is becoming very popular in the field of design of GCA. Standard Sliding Modes (SSM) so-called 1-sliding modes provide the best achievable tracking performances in terms of asymptotic stability, finite time convergence, precise keeping of the desired constraints, and robustness against internal and external disturbances. Based upon this approach, certain longitudinal missile autopilot topologies have been recently designed (Fu-Kuang et al., 2004; Shima et al., 2006; Lee et al., 2009; Bahrami et al., 2010). Although their impressive features, 1-sliding modes suffer from certain drawbacks and restrictions regarding the design of sliding manifolds, convergence rate, and chattering effect. Also, it is found that 1-sliding modes only exist on the zero level set of the output function which means that they may be implemented only if the relative degree of the controlled system is one (Utkin 1992; Zinober 1994; Levant 1993, 2005). Due to these limitations, it appears that there is not much enthusiasm to use 1-sliding modes or inner-loop sliding modes for the design of high-performance FCS. Copyright by the International Federation of Automatic Control (IFAC) 11157

2 Higher relative degree based-design approach provides an alternate to overcome the aforementioned restrictions of 1- sliding mode controllers and yields feedback controllers that feature high-performance levels such as fast time convergence, heavy modeling uncertainties robustness, and chattering extinction and suppression. The relative degree appears as important knowledge about dynamical systems in many control issues such as local stabilization, disturbance decoupling, interaction decoupling, nonlinear adaptive control and Higher Order Sliding Mode (HOSM) control (Isidori 1989; Sira-Ramirez 1990, 1993; Levant 2001, 2003, 2005). Recently a certain missile autopilots have been designed based on HOSM strategy (Shang et al., 2004; Foreman et al., 2010). In this paper and within SMC framework, an outer loop control strategy is presented to design sliding regimes for a class of nonlinear smooth affine systems. Our primary concerns are to induce sliding regimes on systems with relative degree higher than one, and to examine the relevance of higher relative degree and local diffeomorphic coordinate transformation concepts to design enhanced sliding mode controllers that feature high-tracking performances. Within this strategy, two pitch-axis missile autopilot topologies are design and implemented in MATLAB environment. The agility and robustness of the controllers are analyzed via different computer-simulation scenarios. The remaining paper is organized as follows: In section 2, an extended nonlinear pitch-axis tail-controlled missile model is presented and mission performance requirements and control constraints are identified. Section 3 presents a new OLSM control strategy to design high-performance controllers. In section 4, two pitch-axis missile autopilot topologies are designed based upon the proposed strategy. Simulation results are shown is section 5 and comments on the proposed strategy are made in concluding section. 2. MISSILE DYNAMICS AND CONTROL OBJECTIVES The missile model used here for computer simulation and validation purposes is a hypothetical tailed-controlled pitchaxis missile so-called Reichert s missile model often used as a benchmark for investigation on GCA for supersonic vehicles (Reichert 1992; Mracek et al., 1996; Xin and Balakrishnan 2008). 2.1 Longitudinal Missile Dynamics The following set of nonlinear equations governs the longitudinal rigid body motion of a missile airframe in a motion space defined by Mach number, angle of attack, path angle, and pitch rate states where The aerodynamic coefficients and are estimated from wind-tunnel measurements as Actuator dynamics describing the tail-fin deflections are governed by the following second-order differential equation where and are, respectively, the actual and commanded tail-fin deflections, and are the actuator undamped natural frequency and actuator damping ratio, respectively. Description and numerical values of various aerodynamic coefficients and physical parameters of pitch axis missile model at an altitude of 6096 m are provided in the following table Table 1. Details of pitch-axis tail-controlled missile model Sym bol VALUE Quantity P N/m 2 static pressure S m 2 surface area d m diameter m kg mass v s speed of sound I y pitch moment of inertia C drag coefficient 0.7 actuator damping ratio 150 rad/s actuator undamped natural frequency a n deg -3 b n deg -2 c n deg -1 d n deg -1 a m deg -3 b m deg -2 c m deg -1 d m deg -1 The fin deflection has a negligible effect on the force aerodynamic coefficient (Devaud et al., 2000). Hence, the model (1)-(4) is reduced to the following Single-Input-Multi- Output (SIMO) state-space model 11158

3 with The output vector is defined by the measured variables available for feedback, and the actual normal acceleration is computed (in gs) from the following equation The field vector, control vector, and output vector are sufficiently smooth, bounded, and uncertain functions given as follows The vector summarizes all unknown internal and external disturbances such as matched and unmatched uncertainties, unmodeled dynamics, and measurement noises. is supposed to be bounded over the operating range of the system (8) with where and denotes the Euclidean norm of a given vector. For design purposes, we partition both vectors and into separate subvectors based on how the control input affects them with where the subscripts i and o are used to denote, respectively, the inner and outer variables (i.e. respectively, fast and slow dynamics with respect to ). The inner variable is the only variable directly controllable by. 2.2 Performance Goals The control objective is to design a robust controller which forces the system (8) to asymptotically truck a desired motion path in finite time and in spite of internal and external disturbances. The controller should maintain the desired closed-loop performance and robustness requirements over the following operating range The actuator constraints to the control input are specified in terms of position and speed where the tail deflection limits and maximum deflection rate for a 1 g step command in η are set as follows The closed-loop performance requirements over the operating range (24) are specified as follows Asymptotic stability. Finite time convergence. Fast responses to large commands. High robustness against internal and external disturbances. Recoverability from the effects of large commands. 3. RELATIVE DEGREE APPROACH FOR OLSM CONTROLLERS DESIGN Within SMC framework and based upon the local characteristics of dynamical systems, we propose here a new feedback control strategy to design OLSM controllers that ensure the above tracking requirements under the control constraints such as in (25) Local properties of smooth nonlinear systems By definition, the local relative degree of a dynamical system at an initial condition with respect to its output is interpreted as the minimum number of times one has to differentiate the output with respect to time until the control input explicitly appears. The importance of the local relative degree is that i-) it helps to understand how the input enters the system and how affects its outputs, ii-) it characterizes the dynamics smoothness degree in the vicinity of, iii-) it permits to define the zero dynamics manifold near equilibrium and the zero dynamics subsystems, iv-) it leads, via a local diffeomorphism, to a local normal form coordinates, which permits to apply an outer-loop feedback strategy, v-) it shows if a sliding regime locally exists on an open set surrounding

4 From now on, we consider the following tracking error vector to be the actual output of the system (8) that renders it globally minimum phase where the subscript d denotes the desired value. The internal behavior of the system, while being forced to locally exhibit desired output value, is asymptotically stable to an equilibrium point located on the manifold ). Based upon the investigations conducted in (Sira-Ramirez 1990; Slotine 1991; Utkin 1992; Isidori 1989; Levant 2001) about structure at infinity, local sliding regimes, and local feedback stabilization, we declare the following definitions for a SIMO system actuated by a scalar affine control input. Definition 1. The vector (p, output dimension) is said to be the relative degree vector of a SIMO system with respect to its new output vector, at a given initial condition, if the following conditions on Lie derivatives locally hold for any state in an open set containing, and for all, and. The consecutive derivatives that appear in (27) are computed as follows where n denotes the dimension of the state vector. Definition 2. If the system has a local relative degree with respect to an output, an asymptotically stable motion towards zero can be obtained for this output and its first time derivatives by a suitable design of a sliding manifold as follows where are the design parameters and. The state vector assures that the desired output is asymptotically reached. Definition 3. According to the standard form of sliding mode approach, the control variable u appears explicitly in the first-time derivative as follows with and. Definition 4. For each output function which locally exhibits relative degree on, there exists a locally (i.e. on the neighborhood ) unique solution of (30) such as where denotes the control input so-called equivalent control that corresponds to the output. We note that if all the design parameters are taken to be zero, then Under nominal conditions and in absence of external disturbances, the controller (31) locally constraints the system trajectories to the zero level set of. In other words, this controller asymptotically drives the state vector from towards the manifold as desired. The constrained dynamics resulting by substituting in (8) by (31) is known as the ideal sliding dynamics. 3.2 Design of OLSM Controllers We propose here a controller topology that is based upon the use of the smallest well-defined relative degree of a SIMO system. The main advantages of this design strategy are that i-) the smallest relative degree simplifies the design task and yields a practical controller, ii-) a well-defined relative degree guaranties the applicability of the controller over the operating range of the dynamical system. Definition 5. A relative degree is considered as welldefined relative degree in an open subset if the following condition holds where is a positive constant. In other words, is feedback invariant. Definition 6. If the definition 5 holds for all, then there exists a smallest relative degree (i.e. ) which is considered as the relative degree of the whole system with respect to the control input u. In the following, the sliding manifold and tracking error relative to are denoted by and, respectively. Theorem 1: Let definitions 1-6 hold. Then the following controller 11160

5 Preprints of the 18th IFAC World Congress produces robust OLSM dynamics provided that These dynamics feature the main characteristics of HOSM dynamics such as asymptotic stability, finite time convergence, and disturbance rejection. The directional derivative vector denotes the gradient vector of a scalar function and the operator is defined as follows The following block diagram depicts the implementation loop of the OLSM controller (34) Plant Dynamics Actuator Dynamics Fig. 1. Block diagram of the outer closed-loop sliding mode controller The signum function in (34) is selected as saturation function of a certain boundary layer as follows The gain is an adjustable gain introduced to compensate model uncertainties, unmodeled dynamics and external disturbances while the layer is introduced to soften the chattering. The parameters,, and constitute the controller design parameters specified by the control system designer. 3.3 Stability analysis Outer-Loop Controller To obtain sufficient conditions for the existence of sliding mode dynamics and to ensure the attraction of each trajectory towards the manifold, we define a Lyapunov candidate function as Sensor (Differentiator) The time derivative of (38) yields the existence condition of sliding regimes As consequence to the reaching condition (39), the tracking error and its successive time derivatives up to are asymptotically constrained to zero as long as. 4. OLSM CONTROL TO MISSILE AUTOPILOT DESIGN Now, we apply the methodology presented in section 3 to a two different missile autopilot topologies. The design task, in both cases, is achieved through a multi-step procedure under the following assumptions 1. All the outputs are bounded measureable or observable signals. 2. The reference signals are exogenous signals such that. 4.1 State Feedback Autopilot Topology Considering a full state system with, the tracking problem is focused on the system s state that shows as relative degree. First Step: The derivatives (28) for different outer states in (22) are computed as follows and with. As the function is different from zero over the operating range (24), the relative degree of the system is. This result shows clearly that the local relative degree of the triple ( ) is well-defined output-input assignment Ο (here Ο covers the entire operating range (24)). Hence, for any included in (24), a non-singular 11161

6 Preprints of the 18th IFAC World Congress controller can be implemented to asymptotically drive the state to its final value as desired. Second Step: According to the value of the sliding manifold is given from (28) as follows with. Third Step: To asymptotically achieve the closed-loop performance and robustness, we construct from (34) the following controller Using (13)-(17), the final feature of the controller (43) is as follows The autopilot above is designed upon a full state system consideration where the sliding dynamics could be selected as system s states which results in state-dependent controller topology. As The AOA is an unmeasured variable, the tracking mission cannot be completely achieved unless an observer for AOA reconstruction from measurements is designed. 4.2 Output Feedback Autopilot Design Assuming a full output availability and using (12) it is easy to show that the relative degree of the system is. Hence, an autopilot can be designed such that a commanded normal acceleration could be successfully tracked under input control generated by (34). In the following we show how to accommodate normal acceleration as the controlled output. (8) become functions of the measurable outputs. Using (45), the first time derivative of is given as Second, following the previous multi-step procedure developed for AOA autopilot design the control effort needed to track a commanded normal acceleration is generated from (34) as follows with where. 5. NUMERICAL SIMULATIONS AND ANALYSIS In absence of external disturbances, the control task is to use the two autopilots (44) and (47) to enforce the tail-controlled missile described in table 1 to track, in desired patterns, the extreme limits of its operating range (24). All the following simulation scenarios were run in MATLAB environment using Euler method at sampling step of second (i.e samples per second) and starting from the initial conditions. First scenario: The tracking performance of the AOAautopilot (44) was checked at the limits (24) introduced as step commands of. The resulting OLSM dynamics are shown in the following figures First and in order to avoid the construction of an estimator for AOA from measured quantities, an inverse transformation to (12) can be developed using an explicit inverse polynomial technique as follows with In this transformation, all the function vectors appearing in 11162

7 Fig. 2. Time history response and control effort corresponding to an extreme AOA tracking scenario Second scenario: In order to evaluate the robustness of the controller (44) against modeling uncertainties, the nominal values of the aerodynamic coefficients in (6) were simultaneously underestimated and overestimated by 50 per cent. The tracking responses under modeling uncertainties and the needed control efforts are shown in Fig. (3). Fig.4. Time history response and control effort corresponding to a normal acceleration pattern tracking The outcomes of the simulation scenarios described above reveal the high-tracking performances achieved by the proposed OLSM control strategy. The closed-loop performance enhancement is manifested by a fast asymptotic response (time repose less than 0.5 sec), strong robustness against modeling uncertainties (up to 50% of nominal values), and chattering extinction. In addition to this high tracking accuracy, it is worth noting that the tracking missions were successfully achieved within the operating range (24) and without violation of the control constraints (25). 6. CONCLUSIONS Fig. 3. Parameter uncertainty robustness at the extreme operating conditions Third scenario: The efficiency and robustness of the output feedback controller (47) are tested against a fast and highmaneuver normal acceleration tracking pattern. Here, the commanded accelerations are also selected at the extreme limits of the operating range (24) as shown in Fig. (4). In this paper, we have explored the relevance of high relative degree and local properties concepts to design GCA s for highly uncertain dynamics as tactical missiles. In order to overcome design limitations resulting from application of standard sliding modes and ovoid real-time implementation restrictions such as high order differentiation, the design was focused around the first well-known relative degree more than one. Two pitch-axis tail-controlled missile autopilot topologies were preliminary designed to successfully accomplish fast and high maneuverability tracking missions. Simulation results indicate that the design of GCA using OLSM strategy impressively enhances the system performance and ensures strong robustness properties even the system is operated at the limits of its flight envelope. REFERENCES Bahrami, M., B. Ebrahimi and G.R. Ansarifar (2010). Sliding mode observer and control design with adaptive parameter estimation for a supersonic flight vehicle. International Journal of Aerospace Engineering. Volume 2010, article ID ,

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