REENTRY TERMINAL GUIDANCE THROUGH SLIDING MODE CONTROL
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REENTRY TERMINAL GUIDANCE THROUGH SLIDING MODE CONTROL
SEMINAR REPORT submitted in partial fulfillment of the requirements for the award of M.Tech Degree in Electrical and Electronics Engineering (Guidance and Navigation Control) of the University of Kerala Submitted by DEVIKA K.B. First Semester M.Tech, Guidance and Navigation Control Guided By Smt. R.Lethakumari Lecturer, Dept. of EEE DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING COLLEGE OF ENGINEERING TRIVANDRUM 2010 ABSTRACT This paper presents a terminal guidance method for a reusable launch vehicle dur ing approach and landing phase. This guidance technology do not require any predeter mined reference trajectories, instead would be capable of obtaining feasible trajectories online. In scenarios in which the reentry vehicle is significantly deviated from its nor mal trajectories upon entry into the landing phase, the usefulness of such an online method can be clearly realised. To solve the approach and landing guidance problem, a concept called the sliding mode terminal guidance is presented. This approach takes the advantage of the finite time reaching phase of a sliding mode technique to ensure that any desired state constraints can be fulfilled within a finite time. Through a set of sim ulations, it was made clear that some robustness to variations in the initial downrange and velocity is possible. REENTRY TERMINAL GUIDANCE THROUGH SLIDING MODE CONTROL.pdf (Size: 518.25 KB / Downloads: 83) TABLE OF CONTENTS 1 Introduction 1 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Sliding mode control 3 3 System Model and Equations of Motions 4 4 Reentry SMTG Design 6 4.1 Sliding Mode Control for Systems with Relative Degree of 2 . . . . 6 4.2 Application of SMTG to Reentry Terminal Guidance . . . . . . . . 8 5 Results and Analysis 12 6 Conclusions 18 i LIST OF FIGURES 3.1 . . . . . . . . . . . . . . . . 4 Diagram of the Approach and Landing Phase 5.1 . . . . . . . 12 Attitude vs downrange for varying value of final vertical velocity 5.2 . . . . . . . . . . 13 Vertical velocity vs time for varying final vertical velocity 5.3 . . . . . . . . . . . 13 Lift coefficient vs time for varying final vertical velocity 5.4 . . . . . . . . . . . . . . 14 Attitude vs downrange for varying lift coefficient 5.5 . . . . . . . . . . . . . 14 Vertical velocity vs time for varying lift coefficients 5.6 . . . . . . . . . . . . . . 15 Lift coefficient vs time for varying lift coefficients. 5.7 . . . . . . . . . . . . . . . . . . . . 15 Attitude vs downrange for varying n 5.8 . . . . . . . . . . . . . . . . . . . 16 Vertical velocity vs time for varying n 5.9 . . . . . . . . . . . . . . . . . . . . 16 Lift coefficient vs time for varying n ii CHAPTER 1 Introduction 1.1 Overview In recent years, it has become apparent that there is a need for developing advanced reentry guidance technologies that can improve the safety and reliability of reusable launch vehicles (RLVs). In particular, guidance technologies are desired that can accommodate for aerosurface failures, poor vehicle performance, or dispersions from the desired trajectory . These types of technologies are particularly critical during the approach and landing (A and L) reentry phase. Of these sources of uncertainty, this method focuses on accounting for dispersions from the desired trajectory. However, the method presented here is not intended to compensate for the aerosurface failures. For the space shuttle, the A and L phase is the final phase of the reentry pro cess, beginning at the end of the terminal area energy management phase at an altitude of roughly 10,000 ft and ending with touchdown on the runway. The goal of this flight phase is for the RLV to land at a desired runway with a nearzero vertical velocity. The vertical velocity at touchdown is desired to be below 5ft/s , but velocities up to 9 ft/s are generally still considered acceptable for the space shuttle. One of the concerns with guidance during the A and L phase is that there is the possibility that crucial control sur faces on the RLV may be damaged during the previous reentry phases, leading to a loss of controllability. Another concern is that upon entering the A and L phase there may be a significant deviation of the RLV from its desired trajectory. This concern is very significant because current space shuttle guidance methods during the A and L phase rely on the shuttle to follow a predetermined trajectory. There is thus a need for A and L guidance technologies that do not require predetermined reference trajectories to be implemented, but instead would be capable of obtaining feasible trajectories online. There have been many previous papers that solve the A and L guidance prob lem while attempting to minimize the amount of offline information required. Schier man [2] presented an approach for obtaining feasible A and L trajectories in the pres ence of major control surface errors. He solved the problem using an optimumpath togo (OPTG) approach. In this approach, a large database of neighboring optimal trajectories is first generated offline. Then, when integrating the trajectory online, the states are observed at each point and the trajectory is reconfigured to follow the partic ular offline trajectory that leads to the best performance. This method presents a novel method for generating online A and L trajectories by taking advantage of the finitetimereaching phase of the sliding mode control tech nique. The only information the control law requires in addition to the instantaneous state of the system is the desired runway location and the A and L initial conditions. Thus, a certain level of robustness to initial conditions is obtained. To solve this prob lem, a concept of sliding mode terminal guidance (SMTG) is used. In SMTG the sliding surface is chosen as a terminal constraint, which is the altitude at the landing site in this case. Then, by making use of the finitetimereaching phase of sliding mode, it can be guaranteed that the terminal constraint will be reached in a finite time. A key principle in these types of problems is that once the sliding surface is reached, the problem is complete, and so no movement along the sliding surface is required. It should be noted that while there have been previous approaches to reentry guidance with the sliding mode technique, these approaches involved tracking a trajectory generated offline and not creation of new trajectories to adapt to current conditions. 2 CHAPTER 2 Sliding mode control In control theory, sliding mode control, or SMC, is a form of variable structure control (VSC). It is a nonlinear control method that alters the dynamics of a nonlinear system by application of a highfrequency switching control. The statefeedback control law is not a continuous function of time. Instead, it switches from one continuous structure to another based on the current position in the state space. Hence, sliding mode control is a variable structure control method. The multiple control structures are designed so that trajectories always move toward a switching condition, and so the ultimate trajectory will not exist entirely within one control structure. Instead, the ultimate trajectory will slide along the boundaries of the control structures. The motion of the system as it slides along these boundaries is called a sliding mode and the geometrical locus consisting of the boundaries is called the sliding (hyper)surface. CHAPTER 3 System Model and Equations of Motions Figure 3.1: Diagram of the Approach and Landing Phase This section details the system model of an RLV during the approach and landing phase. The equations of motion of an unpowered RLV during the A and L phase are, D ̇ V = − − gsinγ (3.1) m L gcosγ − γ= ̇ (3.2) mV V ̇ h = V sinγ (3.3) x = V cosγ ̇ (3.4) In Eqs. (3.1) and (3.2), the lift and drag forces are defined as, L = q SCL ̄ (3.5) D = q SCD ̄ (3.6) q is the dynamic pressure. q = 1 ρV 2 2 ρ = ρ0 e( − h/H) (3.7) CL = CL0 sin2 αcosα (3.8) 2 CD = CD0 + KCL (3.9) H = Scale height, 8.5 km CD = Drag Coeffcient CD0 = Zero lift drag coeffcient CL = Lift Coeffcient CL0 = Zero angle of attack lift coeffcient D = Drag force g = Earth’s gravitational acceleration h = Altitude K = Liftinduced drag coefficient parameter L = Lift force m = Reusable launch vehicle mass S = Surface area V = Velocity magnitude α = Angle of attack γ = Flightpath angle ρ = Air density 5 CHAPTER 4 Reentry SMTG Design This section covers the development of SMTG and its application to the A and L reentry phase and consists of two parts. First, a general approach for solving sliding mode problems with a relative degree of 2 is derived. Afterwards, the steps in the use of this approach for solving the A and L guidance problem are given. 4.1 Sliding Mode Control for Systems with Relative Degree of 2 To handle problems with larger relative degrees, a higherorder sliding mode (HOSM) approach must be used. In HOSM, both the sliding surface and its succes sive derivatives are driven to zero. Most HOSM algorithms are of the second order and can be used to either handle problems with a relative degree of 2 or to reduce chattering in problems with a relative degree of 1. One of the more popular secondorder sliding mode approaches is the twisting algorithm, in which the sliding surface and its deriva tive make an infinite number of rotations (twists) about the origin and reach zero in a finite time. Although the twisting algorithm is a very useful approach that has been used for a variety of applications, since it forces the sliding surface to oscillate between positive and negative values before converging to zero, there are certain applications for which it cannot be used. In particular, the twisting algorithm cannot be used for cer tain terminal sliding mode problems, in which once the sliding surface crosses zero, the problem is generally considered to be over. Thus, there is a need for a new secondorder sliding mode approach that can be used to solve these types of problems. In this section, a novel secondorder sliding mode method based on an adaptive backstepping approach is derived. The usefulness of this method is that it guarantees the sliding surface and its derivative will go to zero in a finite time, while also ensuring that the sliding surface will not cross zero until the final time. First assume that is desired to reach a sliding surface of the form, s1 = f (x) = 0 (4.1) s ̇1 = h(x) (4.2) s1 = l(x) + g(x)u ̈ (4.3) functions h(x),l(x) and g(x) are assumed to be known functions of x, the state vector. It can be seen that the control input u appears in the second derivative of the sliding surface, so the surface s1 has a relative degree of 2. The goal is to find an expression for u such that the sliding surface s1 and its first derivative will both go to zero at some desired finite time tr. This goal is achieved through the use of a backstepping approach, where first derivative of s1 is taken as a virtual control input. To find the value of first derivative of s1 that would be needed to drive s1 to zero, a candidate Lyapunov functionVl1 is chosen as, 1 Vl1 = s2 (4.4) 2 1 ̇ Vl1 = s1 s ̇1 (4.5) To guarantee that the surface s1 will go to zero in a finite time, first derivative of s1 must be chosen such that Vl1 is negativedefinite. This negativedefinite property can be achieved by choosing first derivative of s1 as, ns2 1 s ̇1 = − ,n > 1 (4.6) tr − t so that ns2 Vl = − tr −t < 0 1 One way to guarantee that the desired trajectory will be reached in a finite time is to consider Eq. (4.6) as a new sliding surface: ns1 =0 s2 = s ̇1 + (4.7) tr − t 7 ns ̇1 (tr − t) + ns1 s ̇2 = l(x) + + g(x)u (4.8) (tr − t)2 It can be seen from Eq. (4.8) that the new sliding surface s2 has a relative degree of one with respect to the control input u. To find an expression for u that will drive s2 to zero in some finitetime t∗ , a candidate Lyapunov function is chosen as, r 1 Vl2 = s2 (4.9) 2 2 ns ̇1 (tr − t) + ns1 V ̇l2 = s2 s ̇2 = s2 l(x) + + g(x)u (4.10) tr − t)2 make Eq. (4.10) negativedefinite, u is selected as, ns ̇1 (tr − t) + ns1 ns1 1 u=− + ηsgn s ̇1 + l(x) + (4.11) 2 (tr − t) tr − t) g(x) s2 (0) η= (4.12) t∗r necessary condition for the convergence, t∗ < tr (4.13) r 4.2 Application of SMTG to Reentry Terminal Guidance This section details the steps in the application of the SMTG method to the reen try terminal guidance problem. For the A and L reentry phase, the two primary terminal constraints are as follows: 1) The RLV must land on a desired runway located at some distance downrange of the entry point of the A and L phase. 2) The RLV must land with a sufficiently small vertical velocity. The SMTGbased guidance law developed in this section will lead to both of the above objectives being 8 satisfied for a desired runway location relative to the A and L entry point. It should be noted that only longitudinal twodimensional motion is analyzed in this work. Thus, the vehicle attitude/orientation and any crossrange errors are not consid ered. A sliding surface is first chosen to make the final altitude zero as, s=h=0 (4.14) ̇ ̇ s = h = V sinγ (4.15) s=A ̈ (4.16) ̇ s = B + C CL ̈ (4.17) terms A, B, and C are highly nonlinear functions with the expressions, 1 S 1 S A = −g + ρV 2 cosγCL − KsinγCL − ρV 2 2 sinγCD0 (4.18) 2 m 2 m S ̇ 2 cosγCL − KsinγCL − sinγCD0 V B = ρV m (4.19) 1 S + ρV 2 2 −sinγCL − − cosγCD0 γ KcosγCL ̇ 2 m S 1 C = ρV 2 [cosγ − 2KsinγCL ] (4.20) 2 m By inspecting Eqs. (4.14) and (4.15), it can be seen that the RLV will be at a zero altitude with a zero vertical velocity. However, a zero vertical velocity at touchdown is unnecessarily restrictive, as the maximum vertical velocity at touchdown for the space shuttle is typically between 7 and 10 ft/s. A more reasonable approach would thus involve constraining the vertical velocity at touchdown to be some specific finite value 9 rather than zero. To allow for this constraint, a new variable is defined as, z = s − sdes = 0 ̇ ̇ ̇ (4.21) where, ̇ sdes = hdes ̇ (4.22) Integrating Eq.(4.21) leads to the expression, z = s − sdes t + c ̇ (4.23) where c is a constant. Now define a new term, called the timetogo, as, r tgo = (4.24) V where r is the instantaneous range from the RLV to the runway given by, (xr − x)2 + h2 r= (4.25) The timetogo gives an approximation of the time it will take the RLV to reach the runway if it follows its present course. Using this timetogo expression, the term c in Eq. (4.21) is now chosen as , c = sdes (tgo − t) ̇ (4.26) Substituting Eq. (4.26) into Eq. (4.21) leads to, z = s − sdes t − sdes (tgo − t) ̇ ̇ (4.27) Finally, simplifying Eq.(4.27) and substituting Eqs. (4.14) and (4.22), expression for z ̇ z = h + hdes tgo (4.28) Now, suppose that z is driven to zero at the runway. Since the timetogo, tg o, be comes, goes to zero as the distance to the runway (i.e. the range r) goes to zero, the 10 second term in Eq. (4.28) will go to zero at the runway. Thus, in order for Eq. (4.28) to be zero at the runway, the first term in the equation (h) must also be zero. This im plies that the altitude will be zero at the runway, satisfying the touchdown requirement. Equation (4.28) can be considered as a new sliding surface, which satisfies the same property as the original sliding surface in Eq. (4.14). As was done before, Eq. (4.28) is now differentiated until the control input appears, leading to, z = y − ydes ̇ ̇ ̇ (4.29) z=A ̈ (4.30) ̇ z = B + C CL ̈ (4.31) where the terms A, B, and C were defined previously. Note that the condition z =0 and first derivative of z= 0 implies that the RLV will be at a zero altitude with the desired amount of vertical velocity. Thus, the reentry guidance problem can be solved by finding an expression for the derivative of the lift coefficient that will guarantee that the sliding surface z and its derivative will go to zero at some desired downrange position xr corresponding to the runway. 11 CHAPTER 5 Results and Analysis In this section, simulations of the proposed reentry guidance approach is presented. For each of these simulations, the initial values for the altitude, velocity, and flight path angle are taken as 10,000 ft, 570 ft/s, and 25 degree respectively. First, it was desired to see the effect of varying the desired vertical velocity at touchdown, initial lift coefficient, and value for n. To analyze these effects, a set of simulations were performed for an initial downrange of 28,000 ft. Following figures contain the results of a simulation of varying the desired vertical velocity at touchdown for a specific initial lift coefficient and n of 0.6 and 2, respectively. Figure 5.1: Attitude vs downrange for varying value of final vertical velocity Following are the results of the simulation of varying the initial lift coefficient for a set desired vertical velocity at touchdown and n of 5 ft/s and 2 respectively . Following are the results of a simulation of varying n for specific initial lift coeffi cient and desired vertical velocity at touchdown velocities of 0.6 and 5 ft/s. Inspect ing the results of varying the desired vertical velocity at touchdown , it is interesting to observe that the final vertical velocity has a fairly negligible effect on the resulting trajectories. However, the values of the initial lift coefficient and n have a large im Figure 5.2: Vertical velocity vs time for varying final vertical velocity Figure 5.3: Lift coefficient vs time for varying final vertical velocity 13 Figure 5.4: Attitude vs downrange for varying lift coefficient Figure 5.5: Vertical velocity vs time for varying lift coefficients 14 Figure 5.6: Lift coefficient vs time for varying lift coefficients. Figure 5.7: Attitude vs downrange for varying n 15 Figure 5.8: Vertical velocity vs time for varying n Figure 5.9: Lift coefficient vs time for varying n 16 pact on the transient performance and nature of the trajectories. By increasing the lift coefficient, the RLV begins to perform an increasingly large pullup maneuver at the beginning of the trajectory. Also, each initial lift coefficient satisfies the desired vertical velocity condition of 5 ft/s at touchdown, the horizontal velocity at touchdown increases as the initial lift coefficient increases. 17 CHAPTER 6 Conclusions A method for A and L reentry phase guidance has been introduced through a concept of sliding mode terminal guidance (SMTG). A control law is obtained that allows for the rapid construction of feasible trajectories that depend only on the instantaneous states of the system and the initial and final conditions of the A and L phase. Thus, the availability of a predetermined reference trajectory is not required. Through a set of simulations, it was shown that the approach shows some robustness with respect to variations in the initial downrange and velocity. From various simulations, it was found that for a downrange of 28000 ft, a high initial lift coefficient and a low value of n lead to acceptable transient performance. REFERENCES [1] Nathan Harl and S.N.Balakrishnan,“Reentry Terminal Guidance Through Sliding Mode Control,” Journal Of Guidance, Control, And Dynamics,Vol. 33, no. 1, JanuaryFebruary pp.186199.2010. [2] Schierman, J., Hull, J., and Ward, D, “Online Trajectory Command Reshaping for Reusable Launch Vehicles,”AIAA Paper 20035439, August. 2003. [3] Shtessel, Y., and Krupp, D,“Reusable Launch Vehicle Trajectory Control in Slid ing Modes,”Proceedings of the American Control ConferenceInst. of Electrical and Electronics Engineers, Piscataway,NJ, June 1997, pp. 25572561. [4] Schierman, J., Hull, J., and Ward, D,“Adaptive Guidance with Trajectory Reshaping for Reusable Launch Vehicles,”AIAA Paper 20024458, August. 2002. [5] Bollino, K., Ross, M., and Doman, D, “Optimal Nonlinear Feedback Guidance for Reentry Vehicles,” AIAA Paper 20066074, August. 2006, pp.376387. [6] Schierman, J., Ward, D., Hull, J., Gahndi, N., Oppenheimer, M., and Doman, D,“Integrated Adaptive Guidance and Control for ReEntry Vehicles with Flight Test Results,”Journal Of Guidance, Control, And Dynamics, Vol. 27, No. 6, NovemberDecember. 2004, pp. 975988. 


