Robust nonlinear control design for a missile using backstepping

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(1)Robust nonlinear control design for a missile using backstepping Examensarbete utf¨ ort i Reglerteknik vid Tekniska H¨ ogskolan i Link¨ oping av Johan Dahlgren Reg nr: LiTH-ISY-EX-3300-2002.

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(3) Robust nonlinear control design for a missile using backstepping Examensarbete utf¨ ort i Reglerteknik vid Tekniska H¨ ogskolan i Link¨ oping av Johan Dahlgren Reg nr: LiTH-ISY-EX-3300-2002. Supervisor: Ola H¨ arkeg˚ ard Henrik Jonson Examiner: Torkel Glad Link¨ oping, 27th January 2003..

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(5) Avdelning, Institution Division, Department. Datum Date 2002-12-19. Institutionen för Systemteknik 581 83 LINKÖPING Språk Language Svenska/Swedish X Engelska/English. Rapporttyp Report category Licentiatavhandling X Examensarbete C-uppsats D-uppsats. ISBN ISRN LITH-ISY-EX-3300-2002 Serietitel och serienummer Title of series, numbering. ISSN. Övrig rapport ____. URL för elektronisk version http://www.ep.liu.se/exjobb/isy/2002/3300/ Titel Title. Robust olinjär missilstyrning med hjälp av backstepping Robust nonlinear control design for a missile using backstepping. Författare Author. Johan Dahlgren. Sammanfattning Abstract This thesis has been performed at SAAB Bofors Dynamics. The purpose was to derive a robust control design for a nonlinear missile using backstepping. A particularly interesting matter was to see how different design choices affect the robustness. Backstepping is a relatively new design method for nonlinear systems which leads to globally stabilizing control laws. By making wise decisions in the design the resulting closed loop can receive significant robustness. The method also makes it possible to benefit from naturally stabilizing aerodynamic forces and momentums. It is based on Lyapunov theory and the control laws and a Lyapunov function are derived simultaneously. This Lyapunov function is used to guarantee stability. In this thesis the control laws for the missile are first derived by using backstepping. The missile dynamics are described with aerodynamic coeffcients with corresponding uncertainties. The robustness of the design w.r.t. the aerodynamic uncertainties is then studied further in detail. One way to analyze how the stability is affected by the errors in the coeffcients is presented. To improve the robustness and remove static errors, dynamics are introduced in the control laws by adding an integrator. One conclusion that has been reached is that it is hard to immediately determine how a certain design choice affects the robustness. Instead it is at the point when algebraic expressions for the closed loop system have been obtained, that it is possible to analyze the affects of a certain design choice. The designed control laws are evaluated by simulations which shows satisfactory results.. Nyckelord Keyword backstepping, nonlinear, robust, missile, integral action.

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(7) Abstract This thesis has been performed at SAAB Bofors Dynamics. The purpose was to derive a robust control design for a nonlinear missile using backstepping. A particularly interesting matter was to see how different design choices affect the robustness. Backstepping is a relatively new design method for nonlinear systems which leads to globally stabilizing control laws. By making wise decisions in the design the resulting closed loop can receive significant robustness. The method also makes it possible to benefit from naturally stabilizing aerodynamic forces and momentums. It is based on Lyapunov theory and the control laws and a Lyapunov function are derived simultaneously. This Lyapunov function is used to guarantee stability. In this thesis the control laws for the missile are first derived by using backstepping. The missile dynamics are described with aerodynamic coefficients with corresponding uncertainties. The robustness of the design w.r.t. the aerodynamic uncertainties is then studied further in detail. One way to analyze how the stability is affected by the errors in the coefficients is presented. To improve the robustness and remove static errors, dynamics are introduced in the control laws by adding an integrator. One conclusion that has been reached is that it is hard to immediately determine how a certain design choice affects the robustness. Instead it is at the point when algebraic expressions for the closed loop system have been obtained, that it is possible to analyze the affects of a certain design choice. The designed control laws are evaluated by simulations which shows satisfactory results. Keywords:. backstepping, nonlinear, robust, missile, integral action. i.

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(9) Acknowledgment First of all a would like to thank my two supervisors, Henrik Jonsson at SAAB Bofors Dynamics and Ola H¨ arkeg˚ ard at Link¨ opings University. The have given me good support and advises. I would also like to thank Professor Torkel Glad, my examiner. Besides these key persons i also would like to thank: all the people at floor 5 at SAAB Bofors Dynamics in Link¨ oping for their kind reception. Rebecca, Linda and Torbj¨ orn Crona for proof-reading the report and providing valuable comments.. iii.

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(11) Notation Symbols x, X R u, U x1 . . . xn V (x) xref 1 xdes 1 a ˆ. boldface letters are used for vectors, matrices and sets. the set of real numbers control input state variables Lyapunov function reference value of x1 desired value of x1 estimated value of the variable a that is used in the control laws. Operators and functions ||x|| V˙ =. dV dt. ∇V (x, y) = ∂V∂x(x) , ∂V∂y(x) (x) ∂V (x) Vx (x) = ∂V , . . . , ∂x1 ∂xn . F (z1 ) =. dF (z1 ) dz1. Euclidian norm time derivative of V gradient of V (x, y) gradient of V (x) derivative of F w.r.t. its only argument z1. Abbreviations clf. Control Lyapunov Function v.

(12) Missile nomenclature State variables Symbol α β p q r V = (u, v, w) u v w az ay. Unit rad,degrees rad,degrees rad/s rad/s rad/s m/s m/s m/s m s2 m s2. Definition angle of attack sideslip angle roll rate pitch rate yaw rate body-axes velocity longitudinal velocity lateral velocity normal velocity accleration in the normal direction acceleration in the lateral direction. Fin deflections Symbol δa δe δr. Unit rad,degrees rad,degrees rad/s. Definition aileron deflection elevator deflection rudder deflection. Missile data Symbol m S d  Ixx I= 0 0. 0 Iyy 0.  0 0  Izz. Unit kg m2 m. Definition mass aerodynamic reference area diameter. kgm2. inertial matrix. vi.

(13) Contents 1 Introduction 1.1 Background . 1.2 Purpose . . . 1.3 Method . . . 1.4 Limitations . 1.5 Outline of the. . . . . . . . . . . . . . . . . thesis. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 2 Basic theory 2.1 General non-linear theory . . . . . . . . . . . . . . . . . 2.2 Lyapunov theory . . . . . . . . . . . . . . . . . . . . . . 2.2.1 A geometrical interpretation of the direct method 2.3 Lyapunov theory and control design . . . . . . . . . . .. . . . . . . . . . . . . . . of Lyapunov . . . . . . .. 3 Backstepping 3.1 Backstepping design procedure . . . 3.2 Aspects on backstepping . . . . . . . 3.2.1 Design choices . . . . . . . . 3.2.2 Which system can be handled. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 4 The missile 4.1 Introduction to the missile . . . . . . . . . . . . . . . . . . . 4.1.1 Guidance, Navigation and Control system (GN&C) . 4.1.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Maneuvering . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Actuator dynamics . . . . . . . . . . . . . . . . . . . 4.2 Deriving a model of the missile . . . . . . . . . . . . . . . . 4.2.1 Basic rigid body dynamics . . . . . . . . . . . . . . . 4.2.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Equations . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 State-space representation . . . . . . . . . . . . . . . 4.2.5 Comments about the aerodynamic coefficients . . . . 4.2.6 The simulation model . . . . . . . . . . . . . . . . . vii. . . . .. . . . . . . . . . . . .. . . . .. . . . . . . . . . . . .. . . . .. . . . . . . . . . . . .. 1 1 1 1 2 2 3 3 4 5 7. . . . .. 9 . 9 . 10 . 10 . 11. . . . . . . . . . . . .. 13 13 14 15 16 17 17 17 18 19 21 21 22. . . . . . . . . . . . ..

(14) viii 5 Backstepping design for the missile 5.1 Control objectives . . . . . . . . . . . . . 5.1.1 Conversion of the accelerations . . 5.2 Backstepping design . . . . . . . . . . . . 5.2.1 Controlling the pitch-dynamics . . 5.2.2 Controlling the yaw -dynamics . . . 5.2.3 Controlling the roll-dynamics . . . 5.2.4 Combining the control inputs . . . 5.3 Aspects on the design . . . . . . . . . . . 5.3.1 Cascaded control structure . . . . 5.3.2 Tuning of the design parameters . 5.3.3 Robustness of the design . . . . . . 5.3.4 Exploring some design flexibilities 5.4 Simulation . . . . . . . . . . . . . . . . . . 5.4.1 Conditions . . . . . . . . . . . . . 5.4.2 Controller parameters . . . . . . . 5.4.3 Simulation results . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . 5.6 Simulation plots . . . . . . . . . . . . . .. Contents. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. 23 23 23 25 25 28 30 30 31 31 31 33 35 37 37 37 37 38 39. 6 Dealing with uncertainties 6.1 Extended robustness discussion . . . . . . . . . . . . . 6.1.1 Problem review . . . . . . . . . . . . . . . . . . 6.1.2 Using level curves for stability analysis . . . . . 6.1.3 Identifying the ∆Ez2 -terms and ∆Ez1 z2 -terms . 6.1.4 Simulations . . . . . . . . . . . . . . . . . . . . 6.2 Adding dynamics to the controller . . . . . . . . . . . 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . 6.2.2 Theoretical background . . . . . . . . . . . . . 6.2.3 Adding the integrator to the previous design . 6.2.4 Simulation results . . . . . . . . . . . . . . . . 6.3 Increasing the gain in the backstepping loop . . . . . . 6.3.1 A level curve discussion . . . . . . . . . . . . . 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Simulation plots . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. 41 41 41 42 45 46 49 49 49 50 51 54 55 56 57. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. 7 Conclusions and future work 63 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 A Missile data. 67. B Aerodynamic coefficients. 69.

(15) Chapter 1. Introduction 1.1. Background. Over the last two decades there has been a development in design methods for control of nonlinear dynamic systems. Several new methods have been invented and one of the more recent method is backstepping. At SAAB Bofors Dynamics there is an interest to follow how this development is proceeding.. 1.2. Purpose. The main purpose of this thesis is to present a controller for a missile designed with backstepping for SAAB. From the results reached in this report, SAAB can decide whether backstepping is a design method of interest for their applications. The missile dynamics are described with aerodynamic coefficients with corresponding uncertainties. Focus is therefore drawn to make the design robust against these uncertainties in the aerodynamic coefficients. One goal is to determine how different design choices affect the robustness. Since backstepping leads to static feedback, another goal is to investigate how dynamics can be added to the controller.. 1.3. Method. Several books and papers on backstepping, control theory and other relevant areas have been studied. A model of the missile is derived and the control laws are designed using backstepping. Then the robustness of the design is studied and the dynamic part (the integrator) is added. To verify the controller, simulations are performed. 1.

(16) 2. 1.4. Introduction. Limitations. This thesis does not discuss how the derived controller handles disturbances, i.e sensitivity. In the simulations the angular velocities of the fins are not limited.. 1.5. Outline of the thesis. Chapter 2: Presents general nonlinear theory and Lyapunov theory. This chapter gives the theoretical background needed in order to understand the backstepping design. Chapter 3: Here the backstepping design procedure is presented. Some aspects on the design such as different design choices and which systems that can be handled are also covered. Chapter 4: Describes some basic facts about the missile and a model describing the missile dynamics is derived. Chapter 5: In this chapter the backstepping control laws are derived. Some aspects of the design, including a robustness discussion, are outlined. Simulation results are also presented. Chapter 6: This chapter includes a robustness analysis using phase diagrams. The dynamic part is added to the backstepping control laws and simulations are performed. Chapter 7: Concludes the thesis by compiling the main results that have been reached. Also includes some proposals for future work..

(17) Chapter 2. Basic theory In this chapter a short general discussion on nonlinear theory is given, see [3]. Then the theory which backstepping is based upon is presented, including some important definitions and theorems.. 2.1. General non-linear theory. A standard description of a nonlinear system is x˙ =f (x) y =h(x). (2.1). When all the state variables are constant the system is said to be in equilibrium or at an equilibrium point. Nonlinear systems can have many different equilibrium points. The following definition defines the stability of a certain equilibrium point. Definition 2.1 Given the system (2.1), assume that xe is an equilibrium point, x(0) represents the initial state. Then the equilibrium point is said to be • stable, if there for each > 0 exists δ() > 0 such that ||x(0) − xe || < δ ⇒ ||x(t) − xe || < . f or. all. t≥0. (2.2). • unstable, if not stable • asymptotically stable, if stable and there exists a r > 0 such that ||x(0) − xe || < r ⇒ x(t) → xe. as. t→∞. (2.3). • globally asymptotically stable if it is asymptotically stable for all initial states This definition is often referred to as Lyapunov stability. A global asymptotically stable equilibrium point means that all solutions, regardless of starting point, will converge to the point. Clearly, this is in most cases a desirable property of a control system. 3.

(18) 4. Basic theory. 2.2. Lyapunov theory. In order to show which type of stability a certain equilibrium point correspond to, equation (2.1) must be solved to find x(t). This is not, in general, possible to do analytically. Stability can however be proved by using the direct method of Lyapunov. This method determines the stability properties from the properties of f (x(t)) and its relation to a so called Lyapunov function V (x). The Lyapunov function can be interpreted as a generalized measurement of how far from the equilibrium the system is. If this measurement decreases then the system moves towards the equilibrium point. Before this is concluded in a theorem some definitions are made. Definition 2.2 A function V (x) is said to be • positive definite if V (0) = 0 and V (x) > 0,. x = 0. • positive semidefinite if V (0) = 0 and V (x) ≥ 0,. x = 0. • negative (semi-)definite if −V (x) is positive (semi-)definite • radially unbounded if V (x) → ∞. as. x→∞. Theorem 2.1 (LaSalle-Yoshizawa) Let x=0 be an equilibrium point for (2.1). Let V(x) be a scalar, continuously differentiable function of the state x such that • V(x) is positive definite • V(x) is radially unbounded • V˙ (x) = Vx (x)f (x) ≤ −W (x) where W(x) is positive semidefinite and Vx (x) ∂V ∂V represents the row vector ∂x1 , . . . , ∂xn Then all solutions satisfy limt→∞ W (x(t)) = 0. In addition if W(x) is positive definite then the equilibrium x=0 is globally asymptotically stable. Proof. See [7].. 2. When V˙ (x) is negative semidefinite the following theorem can be used to prove stability. Theorem 2.2 Let x=0 be an equilibrium point for (2.1). Let V(x) be a scalar, continuously differentiable function of the state x such that • V(x) is positive definite • V(x) is radially unbounded • V˙ (x) is negative semidefinite Let E = {x : V˙ (x) = 0} and suppose that no other solution than x(t) ≡ 0 can stay forever in E. Then x=0 is globally asymptotically stable..

(19) 2.2 Lyapunov theory. 5. Proof. See [7]. 2. Using Lyapunov functions is a very powerful tool when determine the stability or instability of an equilibrium point. No knowledge of the solution to the system of differential equations is necessary. Many other methods to determine the stability are local theories, whereas the Lyapunov theory presents a more global result. For example, it is possible to get an estimate of the extent of the basin of attraction of an equilibrium point. The basin of attraction is known as the domain such that all solutions starting within the domain approach the equilibrium point, see [1]. A theorem for “deciding” this domain is presented below. It is taken from [3]. Theorem 2.3 (Basin of attraction) Assume that for the system (2.1) there exists a function V and a number d > 0 that satisfies the conditions for Theorem 2.2 in the set (2.4) Md = {x : V (x) < d} Then all solutions starting in the interior of Md remains there. If, in addition, no other solutions but the equilibrium point, x0 , remain in the subset of Md where V˙ (x) = 0, then all solutions starting in the interior of Md will converge to x0 . 2. Proof. see [3].. This theorem is useful in cases when the properties in Theorem 2.2 are not valid in the whole state-space.. 2.2.1. A geometrical interpretation of the direct method of Lyapunov. The following section is taken from [1]. Assume that V (x, y) is a valid Lyapunov function of two state variables, x and y, for the system (x, ˙ y) ˙ = f (x, y). Then V˙ (x, y) can be written as V˙ (x, y) = ∇V (x, y) ◦ (x, ˙ y) ˙ = ∇V (x, y) ◦ f (x, y). (2.5). where ∇V (x, y) is the gradient of V (x, y) and f (x, y) can be thought of as the tangent of the trajectory for the system. In the following discussion keep in mind that ∇V (x, y) and f (x, y) are vectors. In order for V˙ = 0, one of the following conditions must be met • ∇V (x, y) = 0 • f (x, y) = 0 • ∇V (x, y) and f (x, y) are orthogonal The second condition corresponds to that the system is in equilibrium. The third meaning that there are situations when V˙ = 0 but the system is not in equilibrium..

(20) 6. Basic theory. The function V (x, y) can be plotted as curves in the xy-plane. These curves are known as level curves. Figure 2.1 shows two level curves for the Lyapunov function. One for V (x, y) = c1 and one for V (x, y) = c2 where c2 > c1 > 0. The gradient ∇V (x, y), f (x, y) and the angle, θ, between them are also shown together with the trajectory of the system. The following discussion is based on this figure. Assume the origin to be the goal state, i.e. the equilibrium point. A correctly designed Lyapunov function will then have a gradient which is pointing away from the origin as shown in figure 2.1. Equation (2.5) is a scalar product and it follows from the definition of scalar products that in order for V˙ (x, y) ≤ 0, the angle θ between f (x, y) and the gradient must be in the range [π/2, 3π/2]. This means that f (x, y) must be pointing inwards with respect to V (x, y) = c1 or at worst tangent to this curve. Since f (x, y) represents the direction of motion of the system, it moves inwards. It will then meet another level curve and if V˙ (x, y) < 0 the system continues to move inwards. This is repeated until it reaches the origin. From this discussion it follows that a trajectory starting inside c2 will never cross the curve V (x, y) = c2 and thus c2 makes up the outer limit for the basin of attraction for this trajectory.. y. V ( x,y). = c2. ∇V ( x,y). θ. f ( x,y). x. V ( x,y) = c1. Figure 2.1. Geometrical interpretation of the Lyapunov functions. The figure shows the gradient, f (x, y) and the angle θ between them. The shaded curve is the trajectory of the system..

(21) 2.3 Lyapunov theory and control design. 2.3. 7. Lyapunov theory and control design. In the previous section it was shown that if a Lyapunov function is found so that its time derivative is negative definite within an area, all trajectories starting within that area will converge to same equilibrium point. How can closed loop systems be designed so that they have this property? Consider the system x˙ = f (x, u). (2.6). We would like to find a control law u = k(x) which makes some desired state of the closed loop system asymptotically stable. By picking a Lyapunov function, V (x), and choosing k(x) so that V˙ = Vx f (x, k(x)) = −W (x). (2.7). (where W (x) is positive definite) closed loop stability is given by Theorem 2.1. It is not obvious how V (x) and W (x) should be chosen. To make it easier the following definition is of interest. Definition 2.3 (Control Lyapunov Function) A smooth, positive definite, radially unbounded function V (x) is called a control Lyapunov function(clf ) for (2.6) if for all x = 0, V˙ = Vx f (x, u) < 0 for some u (2.8) A theorem known as Artstein’s theorem has been developed to make the clf meaningful. The theorem says that the existence of a clf is equivalent to the existence of a control law which will make the desired state global asymptotically stable, see [4]..

(22) 8. Basic theory.

(23) Chapter 3. Backstepping In the previous chapter the theory which backstepping is based upon was presented. There, the need of a control lyapunov function (clf) was shown. It was stated that if a clf exists, a control law which make the system globally asymptotically stable can be found. However, no clue to how to find a clf or the control law was given. Backstepping is a procedure which finds both a clf and a control law simultaneously and is the topic of this chapter. First, a description of the backstepping design procedure is given. Then different aspects of the design, such as different design choices and which type of system that can be handled, is considered.. 3.1. Backstepping design procedure. To show how to find a clf and a control law, a short design example is considered. It is a variant of a design example presented in [7]. The system that is to be controlled is given below x˙ =f (x) + g(x)ξ ξ˙ =a(x, ξ) + b(x, ξ)u. (3.1). where x ∈ Rn and ξ ∈ R are state variables and u ∈ R is the control input. First ξ is regarded as a control input for the x-subsystem. ξ can be chosen in any way to make the x-subsystem globally asymptotically stable. The choice is denoted ξ des (x) and is called a virtual control law. For the x-subsystem a clf, V1 (x), can be chosen so that with the virtual control law inserted its time derivative becomes negative definite. V˙ 1 (x) = V1x x˙ = V1x (x) f (x) + g(x)ξ des (x) < 0, 9. x = 0. (3.2).

(24) 10. Backstepping. A new state is introduced which represents the residual ξ˜ = ξ −ξ des (x). The system (3.1) is then written in terms of these new variables, resulting in x˙ =f (x) + g(x)(ξ˜ + ξ des (x)) ∂ξ des (x) ˙ f (x) + g(x)(ξ˜ + ξ des (x)) ξ˜ =a(x, ξ˜ + ξ des (x)) + b(x, ξ˜ + ξ des (x))u − ∂x (3.3) For the system (3.3) a clf is constructed from V1 (x) by adding a quadratic term ˜ = V1 (x) + 1 ξ˜2 . Differentiating V2 (x, ξ) ˜ ˜ V2 (x, ξ) which penalizes the residual ξ, 2 w.r.t. time yields ˜ =V1x (x) f (x) + g(x)ξ des (x) + g(x)ξ˜ + ξ˜ a(x, ξ˜ + ξ des (x)) + V˙ 2 (x, ξ). (3.4) ∂ξ des (x) ξ˜ b(x, ξ˜ + ξ des (x))u − f (x) + g(x)(ξ˜ + ξ des (x)) ∂x Equation (3.4) can be rewritten in the following way if the variables that the functions depend on are omitted. V˙ 2 =V1x (f + gξ des )+. . ∂ξ des f + g(ξ˜ + ξ des ) ξ˜ V1x g + a + bu − ∂x. (3.5). To guarantee stability V˙ 2 has to be negative definite. This can be achieved by choosing the control input, u in (3.5) as. 1 ∂ξ des des ˜ ˜ u= f + g(ξ + ξ ) − a − V1 g − k ξ , k > 0 (3.6) b ∂x Then V˙ 2 becomes. V˙ 2 = V (f + gξ des ) − k ξ˜2 ≤ 0. (3.7). If u is not the actual control input but a virtual control law consisting of state variables, then the system can be further expand by starting over again. Hence the backstepping design procedure is recursive.. 3.2 3.2.1. Aspects on backstepping Design choices. When deriving a control law using backstepping many variations can be done. Among other opportunities this enables the designer to benefit from useful nonlinearities. With useful means that the nonlinear terms naturally stabilizes the system. This is done by choosing the virtual control laws properly. For examples see [4]..

(25) 3.2 Aspects on backstepping. 11. Another design choice is using a non-quadratic clf. To render equation (3.4) negative definite, u was chosen so that it canceled some dynamics, ∂ξ des ∂x. . f + g(ξ˜ + ξ des ) , a, V1x g. ˜ Some of these terms may instead and replaced them with linear dynamics (−k ξ). be canceled by choosing V1x (x) properly. This is done by not deciding what V1x (x) should look like until the expression corresponding to V˙ 2 has been derived. Which dynamics that V1x (x) can cancel and still be a valid clf for the x-subsystem is then investigated. This design choice usually results in improved robustness and is used later in Chapter 5.. 3.2.2. Which system can be handled. To be able to apply backstepping to a system, it must have a so called lower triangular form. Pure-feedback form systems (3.8) is one example of this form. x˙ =f (x, ξ1 ) ξ˙1 =g1 (x, ξ1 , ξ2 ) .. . ξ˙i =gi (x, ξ1 , . . . , ξi , ξi+1 ) .. . ξ˙m =gm (x, ξ1 , . . . , ξm , u). (3.8). Also systems which can be written on strict-feedback form (3.9) can be handled. x˙ =f (x) + g(x)ξ1 ξ˙1 =f1 (x, ξ1 ) + g1 (x, ξ1 )ξ2 .. . ξ˙i =fi (x, ξ1 , . . . , ξi ) + gi (x, ξ1 , . . . , ξi )ξi+1 .. . ˙ξm =fm (x, ξ1 , . . . , ξm ) + gm (x, ξ1 , . . . , ξm )u. (3.9). Many physical systems can not be written on a lower triangular form. However, by neglecting some physical properties when modeling the system, it can be written on this form. Then it is possible to apply the backstepping technique. Of course some form of analysis or simulation have to be carried out to verify that the neglected physical property does not affect the stability of the closed loop system..

(26) 12. Backstepping.

(27) Chapter 4. The missile To be able to derive a controller, it is necessary to have a model of the system which is to be controlled. The better the model is, the bigger the chances are that the closed loop system will have some desired properties. On the other hand, a very detailed model will be complex and it can be hard to find a controller. In this chapter a model describing the missile dynamics is derived in section 4.2. A short presentation of the missile is first given in section 4.1, including some important definitions.. 4.1. Introduction to the missile. The missile that is considered in this thesis is an air to air missile and is pictured in Figure 4.1. Notice the air inlet on the missile’s underside. It is there to provide the engine with air.. Figure 4.1. The missile.. 13.

(28) 14. 4.1.1. The missile. Guidance, Navigation and Control system (GN&C). The guidance, navigation and control system of the missile consist of three major blocks, see Figure 4.2. A strap down navigation block, a guidance block and finally an autopilot block. Below follows a short description of each block. The strap down navigation block When the missile is launched its position, attitude, speed, acceleration and rotation is known. This block updates these variables during the flight. This is done by using sensor data and strap down navigation algorithms. The variables are supplied to the guidance block and the autopilot block. The guidance block The guidance block receives information from principally three sources. The strap down navigation block, the weapon carrier (the fighter) and the missile’s seeker. From this information calculations are made on where the missile should fly and are passed on to the autopilot.. GN&C Strap down navigation. sensors. position,attitude,rotation,speed, altidue. seeker head. Guidance. weapon carrier accelerations,angular velocity. Autopilot. Fin angles. Figure 4.2. Illustration of the three block that make up the missile’s guidance system..

(29) 4.1 Introduction to the missile. 15. The autopilot The autopilot receives information from the guidance block on where the missile should fly. This information is expressed in terms of demanded directional accelerations and angular velocities . The task of the autopilot is to fulfill that the missile receives the wanted accelerations and angular velocities. It also receives measured accelerations and angular velocities from the strap down navigation block that are used for feed-back when calculating the fins deflections. This thesis treats how this is done.. 4.1.2. Definitions. A fixed body frame coordinate system (x,y,z) is introduced and shown i Figure 4.3. Its origin is placed at the center of gravity. The movement and position of the missile is described in this coordinate system, using the variables in Table 4.1.. Figure 4.3. Fixed body frame (x,y,z) and definitions..

(30) 16. The missile x, y, z V p, q, r α β m. fixed body frame coordinates speed vector angular velocities round the (x,y,z)-axis angle of attack side slip angle mass of the missile. Table 4.1. Explanation of the variables given in Figure 4.3.. The velocity vector is divided into three components as explained below. u = speed component of the center of gravity in the x-direction v = speed component of the center of gravity in the y-direction w = speed component of the center of gravity in the z-direction In the thesis the following terms are also used. roll-channel, is used to describe actions that give rise to movement round the x-axis. pitch-channel, to describe movement round the y-axis. yaw-channel, movement round the z-axis.. 4.1.3. Maneuvering. To steer the missile, movement in the roll, pitch and yaw-channels are generated. In the pitch-channel the variables that are to be controlled are the acceleration in the z-direction and the pitch rate q. The angle of attack α is closely related to these variables and appears naturally in the equations which describes the pitchdynamics. In the yaw-channel the typical goal is to keep β small. The reason for this is, a large value can lead to that the engine does not get enough air and goes out. For a very short period β is allowed to have large values. When the missile is very close to its target the overall goal is to hit that target. In order to do that it may require a β that is so large that the engine dies. The variables to control here are the acceleration in the y-direction or the yaw rate r. These have a strong connection to β. It is possible to rotate the missile either round the x-axis or the velocity vector V. When rotating round V, α and β are unchanged. It is possible to write the controller so that a rotation always is done round the velocity vector. In this thesis this possibility is not used. Instead it always rotates round the x-axis but by simultaneously create a movement in the yaw-channel rolling round V can be achieved. This missile is a bank to turn missile. It means that it turns by first rolling followed.

(31) 4.2 Deriving a model of the missile. 17. by an acceleration in the negative z-direction, pitch-channel. The reason for this is the placement of the air inlet. By performing a turn this way the side slip angle, β, is kept small so that the engine can be provided with air. The placement of the air inlet also restricts the maneuvering in the pitch channel. The acceleration in the positive z-direction must not be larger than 50 sm2 . So if the missile is flying straight forward and a strong dive is desired the missile must first roll over on its back followed by an acceleration in the negative z-direction.. 4.1.4. Actuator dynamics. A servo sets the desired fin deflections. This has the following dynamics [5]. δ=. s2. ω02 δa,e,r + 2ζω0 s + ω02. (4.1). Here ω0 = 250 and ζ = 0.7. This gives poles in −175 ± 178.54i.. 4.2. Deriving a model of the missile. In this section a model of the missile is derived. First a theoretical background describing some basic rigid body dynamics, see [8], is given followed by a section describing the assumptions. Then the actual model is derived.. 4.2.1. Basic rigid body dynamics. The relationship between the sum of all forces, F acting on a body and its inertial acceleration, a, is given by the well known Newton’s second law. Definition 4.1 (Newton’s second law)

(32). F = ma = m. d v dt. (4.2). Here v represents the velocity vector. The next definition is also very useful. Definition 4.2 (Euler’s equation)

(33). ˙ = d Iω M=H dt. (4.3). Euler’s equation states that the moment of all forces acting on a body equals the inertial time rate of change of angular momentum of the body. The two definitions above involves a time derivate of a vector in the inertial frame. The inertial frame is in this application the earth and is considered fixed. When a body frame rotates relatively a fixed frame the following theorem is used to calculate the derivate of a vector expressed in the fixed frame..

(34) 18. The missile. Theorem 4.1 (Time derivate of a vector in a rotating frame) Let XYZ represent the fixed frame, xyz the body frame and ωxyz the angular velocity of the body. Then the time derivate of a vector V is . V dV = + ωxyz × V (4.4) dt XY Z dt xyz 2. Proof. see [8].. The moment of inertia matrix, I is a part of Euler’s equation. In this case when the origin of the body’s frame is placed in the body’s center of gravity the matrix I equals the principal axis frame showed below.   0 Ixx 0 I =  0 Iyy 0  (4.5) 0 0 Izz. 4.2.2. Assumptions. The following assumptions are made when the model is derived. Some consequences of these assumptions are also outlined. • Assumption 1: The thrust and torque produced by the engine are neglected. The air resistance is also neglected. • Assumption 2: The effect of gravity is neglected. • Assumption 3: Small values of α and β are assumed. • Assumption 4: The velocity in the x-direction, u is constant. • Assumption 5. The fin servo dynamics are much faster than the dynamics in the roll, pitch and yaw-channels. This dynamics are therefore not included in the model. Assumption 3 means that sinα ≈ α, tanα ≈ α and cosβ ≈ β. Also u ≈ V holds due to this assumption. Together with the geometric relations given in Figure 4.3 the following can be stated. α=. w , u. β=. v v = |V | u. From assumption 4 it follows that u˙ = 0. The assumptions are made to simplify the model..

(35) 4.2 Deriving a model of the missile. 4.2.3. 19. Equations. Force equations Since the effect of gravity and the thrust are neglected the only forces acting on the missile are aerodynamic forces Fa .   CT (4.6) Fa = −qd S  CC  CN Here qd is the dynamic pressure S is a reference area. CT , CC and CN are short notations for the following expressions. CT =1 CC =CCβ β + CCδr δr CN =CN α α + CN δe δe. (4.7). δe is the elevator deflection and δr is the rudder deflection. The CC -coefficients and the CN -coefficients are explained in section 4.2.5. From definition 4.1 and theorem 4.1 it follows that       u u˙ p (4.8) F¯ = m(V¯˙ + ω × V¯ ) = m  v˙  +  q  ×  v  w˙ r w Replacing F in (4.8) with (4.6) together with (4.7) yields the following −qd S =m(u˙ + qw − vr) −qd S(CCβ β + CCδr δr ) =m(v˙ − pw + ur). (x-direction) (y-direction). −qd S(CN α α + CN δr δr ) =m(w˙ + pv − qu). (z-direction). (4.9). Using assumption 4, u˙ = 0, the equation in the x-direction becomes a static relationship which means there are no dynamics that have to be modeled in this direction. Using assumption 3 and assumption 4 again, the equation in the ydirection can be written.. d (uβ) − pαu + ur −qd S(CCβ β + CCδr δr ) =m dt (4.10) =um β˙ − pα + r =V m β˙ − pα + r The equation in z-direction can also be rewritten using assumption 3 and assumption 4.. d (uα) + pβu − qu −qd S(CN α α + CN δr δr ) =m dt (4.11) =um (α˙ + pβ − q) =V m (α˙ + pβ − q).

(36) 20. The missile. Finally equation (4.10) and equation (4.11) are rewritten. qd S (CCβ β + CCδr δr ) β˙ =pα − r − (y-direction) Vm qd S α˙ = − pβ + q − (CN α α + CN δr δr ) (z-direction) Vm. (4.12). Momentum equations The aerodynamic momentums, Ma , that affect the missile are written   Cl Ma = qd Sd  Cm  Cn. (4.13). where d is a reference length, here the diameter of the missile. Cl , Cm and Cn are short notations for d p + Clδa δa Cl =Clβ β + Clp 2V d (4.14) q + Cmδe δe Cm =Cmα α + Cm|β| |β| + Cmq 2V d r + Cnδa δa + Cnδr δr Cn =Cnβ β + Cnαβ αβ + Cnr 2V Euler’s equation together with equation (4.1), (4.5) and following momentum equations for the missile       Ixx 0 0 p˙ p Ixx ¯ =  0 Iyy 0   q˙  +  q  ×  0 M 0 0 Izz 0 r˙ r   Ixx p˙ + qr (Izz − Iyy ) =  Iyy q˙ + pr (Ixx − Izz )  Izz r˙ + pq (Iyy − Ixx ). Theorem 4.1 forms the 0 Iyy 0.   0 p 0  q  Izz r. (4.15) Inserting equation (4.13) and (4.14) into (4.15) yields  . 1  d p˙ = p + Clδa δa  qr (Iyy − Izz ) +qd Sd Clβ β + Clp Ixx 2V =0. 1 d q˙ = q + Cmδe δe pr (Izz − Ixx ) + qd Sd Cmα α + Cm|β| |β| + Cmq Iyy 2V. d 1 r + Cnδa δa + Cnδr δr r˙ = pq (Ixx − Iyy ) + qd Sd Cnβ β + Cnαβ αβ + Cnr Izz 2V (4.16) Since the missile is symmetric, Iyy = Izz , the qr (Iyy − Izz )-term in the p˙ equation equals zero..

(37) 4.2 Deriving a model of the missile. 4.2.4. 21. State-space representation. Equation (4.12) and equation (4.16) make up a set of equations that describes the missile dynamics. The system can be written on the state-space representation x˙ =f (x) + g(x)u y =h(x) where.    x˙ =    .    f (x) =   . . p˙ q˙ r˙ α˙ β˙.      . (4.18). . qd Sd d Ixx Clβ β + Clp 2V p 1 d q Iyy pr (Izz − Ixx ) + qd Sd Cmα α + Cm|β| |β| + Cmq 2V 1 d Izz pq (Ixx − Iyy ) + qd Sd Cnβ β + Cnαβ αβ + Cnr 2V r qd S −pβ + q − V m (CN α α) S pα − r − Vqdm (CCβ β). .    g(x)u =   . qd Sd Ixx Clδa δa qd Sd Iyy Cmδe δe qd SdCnδr qd Sd δr Izz Cnδa δa + Izz qd S C δ N δ e e Vm qd S V m CCδr δr. .   h(x) =   . 4.2.5. (4.17). p q r α β.       . (4.19).       . (4.20).      . (4.21). Comments about the aerodynamic coefficients. Here follows a short explanation of the aerodynamic coefficients CT , CC , CN , Cl , Cm and Cn . The side force coefficient CC = CCβ β + CCδr δr is taken as an example. It is build up by a term depending on the side slip angle, β, and a term depending on fin deflection δr . CCβ and CCδr are known as aerodynamic derivatives and defines ∂Cc c the side force curve slope in relation to β ( ∂C ∂β ) and δ ( ∂δr ), also see [6]. The value of this derivative changes with the speed according to Table B.1 in appendix B. Usually these values are marred with uncertainties. This is, among other reasons, because it is hard to model aerodynamics correctly. In this thesis, when referred to the aerodynamic coefficients it is the aerodynamic derivatives that is though of if nothing else is said..

(38) 22. 4.2.6. The missile. The simulation model. The model that has been derived in this chapter is only used to find the controller. When simulations are performed a more complex model of the missile is used to calculate the missile dynamics. This model does not use the assumptions stated in section 4.2.2 to the same extend and is a more realistic model of the true missile. Data on the missile used in the simulation is found in Appendix A. The simulation environment is implemented in ADA..

(39) Chapter 5. Backstepping design for the missile In this chapter control laws for the missile are derived using backstepping. First the control objectives are described, then the control laws are derived. This is followed by a section which considers aspects of the design, including a short robustness discussion. In section 5.3.4 some design flexibilities are explored and finally simulation results are presented.. 5.1. Control objectives. The control objectives are that the missile should follow reference signals given in the roll, pitch and yaw-channel. The reference signals in the pitch and yaw channel are expressed as directional accelerations while in the roll channel it is expressed as an angular velocity. If the notation az is used for directional acceleration along the z-axis and ay for directional acceleration along the y-axis, the control objectives can be expressed as p =pref (roll) az =aref z (pitch) ay =aref y (yaw). 5.1.1. Conversion of the accelerations. Since ay and az are not states in the state-space model, described by equations (4.18)-(4.21), they can not be used directly in the design. First they need to be converted into α and β. To find this conversion the force equation in y-direction and aref and z-direction are used, see equation (4.9). The reference signals, aref z y , are expressed in terms of the missile’s own coordinate system. Therefore the part 23.

(40) 24. Backstepping design for the missile. which arises from the rotation of the velocity vector, the (ω × V )-term, is neglected and equation (4.9) can be written (with v˙ = ay and u˙ = az ) qd S (CCβ β + CCδr ) m qd S az = − (CN α α + CN δe ) m. ay = −. (5.1). Here δe and δr are the elevator and rudder deflection, respectively. From equation (5.1) β and α can be separated. az m − C δ −Q N δ e e S d (5.2) α= CN α a m − Qyd S − CCδr δr (5.3) β= CCβ This conversion form az to α and ay to β involve the fin deflections and can be regarded as a form of feedback. Simulations that has been performed showed that the missile can become instable due to this feedback. Therefore the δr and δe terms in equation (5.1) is replaced with an approximation. This approximation is derived below. Start with the expressions for the momentum equation in the y and z directions, see equation (4.16). Iz q˙ =Qd Sd(Cmα α + Cmδe δe + f (β, q, p, r)) Iy r˙ =Qd Sd(Cnβ β + Cnαβ αβ + Cnδr δr + g(α, q, p, r)). (5.4). Equilibrium is assumed and the contribution from f and g are neglected to make the conversion independent the other states variables. The equations may then be written as Cmα α Cmδe Cnβ β + Cnαβ αβ δr = − Cnδr δe = −. (5.5). Inserting the expression for δe and δr in equation (5.2) respectively (5.3) finally yields the following expression for converting the accelerations into angles α=− β =−. . maz. Qd S CN α − . Qd S CCβ −. CN δe Cmα Cmδe. . may CCδr (Cnβ +Cnαβ α) Cnδr. . (5.6).

(41) 5.2 Backstepping design. 5.2. 25. Backstepping design. In this section backstepping design is used to derive control laws for the missile. The design follows the ideas given in Chapter 7.2 in [7]. The first thing that has to be done is to make the state space model applicable to backstepping. In the statespace model, described by equations (4.17)-(4.21), the control inputs are present in all states and the system is not written on lower triangular form. Therefore it is not possible perform backstepping design. However, if the force contributions of the fins are neglected the system is on the correct form for backstepping design. This assumption is valid since the coefficients CN δe and CCδr are quite small in comparison with CN α respectively CCβ . Therefore their contribution to the force equation can be neglected during the backstepping design. Thus, using the notation in Table 5.1 the system which will be considered for the design of the controller can be written x˙1 =p1 x5 + p2 x1 + p3 U1 x˙2 =q1 x1 x3 + q2 x4 + q3 |x5 | + q4 x2 + q5 U2 x˙3 =r1 x1 x2 + r2 x5 + r3 x4 x5 + r4 x3 + r5 U1 + r6 U3 x˙4 = − x1 x5 + x2 − a1 x4. (5.7). x˙5 =x1 x4 − x3 − b1 x5. x1 = p. p1 =. x2 = q. p2 =. x3 = r. p3 =. qd dS Ix Clβ qd d2 S 2Ix V Clp qd dS 2Ix V Clδa. x4 = α. q1 = q2 = q3 = q4 =. x5 = β. a1 = b1 =. qd S V m CN α qd S V m CCβ. q5 =. (Iz −Ix ) Iy qd dS Iy Cmα qd dS Iy Cm|β| qd d2 S 2Iy V Cmq qd dS Iy Cmδe. r1 = r2 = r3 = r4 = r5 = r6 =. Ix −Iy Iz qd dS Iz Cnβ qd dS Iz Cnαβ qd d2 S 2Iz V Cnr qd dS Iz Cnδa qd dS Iz Cnδr. Table 5.1. Notation used in equation (5.7).. 5.2.1. Controlling the pitch-dynamics. Step 1: Introducing new variables First, a new variable, z1 , is introduced which moves the equilibrium point to the origin. z1 = α − αref = x4 − αref (5.8) Its time derivative is z˙1 = x˙ 4 − α˙ ref = −x1 x5 + x2 − a1 x4 − α˙ ref. (5.9).

(42) 26. Backstepping design for the missile. Rewriting equation (5.8) using x4 = z1 + αref and inserting into equation (5.9) yields (5.10) z˙1 = x˙ 4 − α˙ ref = −x1 x5 + x2 − a1 z1 − a1 αref − α˙ ref From equation (5.10) we can see that the state x2 can be regarded as a control input for the z1 -dynamics. In physical properties this means that the pitch-dynamics is controlled with the pitch rate, which make sense. The desired value of x2 is the virtual control law and is denoted, xdes 2 . It is chosen such that it will give the z1 -dynamics some desired properties. = x1 x5 + a1 αref + α˙ ref − k1 z1 xdes 2. (5.11). Here k1 is a design parameter which is determined later. The next step is to introduce the residual x2 -xdes 2 , which is denoted z2 . = x2 − x1 x5 − a1 αref − α˙ ref + k1 z1 z2 = x2 − xdes 2. (5.12). The pitch-dynamics are written in terms of z1 and z2 . z˙1 = − (a1 + k1 )z1 + z2 z˙2 =(−k12 − k1 a1 − k1 q4 + q2 − x21 )z1 + (k1 + q4 )z2 + (q1 + 1)x1 x3 + q3 |x5 | + (q4 + b1 − p2 )x1 x5 − αref x21 − p1 x25 + (q2 + q4 a1 )α. ref. + (q4 − a1 )α˙. ref. −α ¨. (5.13). ref. + q5 U2 − p3 x5 U1 The first equation in (5.13) leads to a constraint on k1 , k1 > −a1. (5.14). in order for the z1 -dynamics to be stable. Using the following notations, ϕ1 (x) =(q1 + 1)x1 x3 + q3 |x5 | + (q4 + b1 − p2 )x1 x5 − αref x21 − p1 x25 A =(q2 + q4 a1 )αref + (q4 − a1 )α˙ ref − α ¨ ref equation (5.13) can be rewritten in a more compact form z˙1 = − (a1 + k1 )z1 + z2 z˙2 =(−k12 − k1 a1 − k1 q4 + q2 − x21 )z1 + (k1 + q4 )z2 + ϕ1 (x) + A + q5 U2 − p3 x5 U1. (5.15). Step 2: Finding the clf A non-quadratic clf for the system (5.15) is selected as 1 V (z1 , z2 ) = F (z1 ) + z22 2. (5.16).

(43) 5.2 Backstepping design. 27. where F (z1 ) is any valid clf for the z1 -dynamics which is determined below. Differentiating the clf w.r.t. time yields V˙ (z1 , z2 ) = − F (z1 )(a1 + k1 )z1 + z2 [F (z1 ) + z1 (−k12 − k1 (a1 + q4 ) + q2 − x21 ). (5.17). + z2 (q4 + k1 ) + ϕ1 (x) + A + q5 U2 − p3 x5 U1 ] The complexity of the z2 -dynamics may be reduced by selecting F (z1 ) so that it cancels the z1 -dynamics inside the []-brackets. By picking F (z1 ) as F (z1 ) = −z1 (−k12 − k1 (a1 + q4 ) + q2 − x21 ). (5.18). this is achieved. In order for the first term in (5.17) to be negative definite, F (z1 ) must lie in the 1st or 3rd quadrant. This constraint means that F (z1 ) = −z1 (−k12 − k1 (a1 + q4 ) + q2 − x21 ) > 0 which is the same as −k12 − k1 (a1 + q4 ) + q2 − x21 < 0. (5.19). ⇔ k12 + k1 (a1 + q4 ) − q2 + x21 > 0. (5.20). Since k1 is a design parameter it can be adjusted so that inequality (5.20) always holds. To be able to determine the value of k1 some knowledge about the coefficients a1 , q2 and q4 are necessary. It is known that: • q2 < 0, always • (a1 + q4 ) > 0, always This, and the fact that the term x21 always is greater or equal to zero, gives the following constraints on k1 k1 > 0, k1 ∈ R (5.21) Earlier it was also stated that k1 > −a1 , which still must hold. Step 3: Determine the control input Inserting the selected F (z1 ) , equation (5.18), into (5.17) yields V˙ (z1 , z2 ) = − (−k12 − k1 (a1 + q4 ) + q2 − x21 )(a1 + k1 )z12 + z2 [z2 (q4 + k1 ) + ϕ1 (x) + A + q5 U2 − p3 x5 U1 ]. (5.22). in which the first term is negative definite as long as (5.14) and (5.21) holds. To make the second term negative definite as well, the control input is used. Choosing q5 U2 − p3 x5 U1 = −z2 k2 − ϕ1 (x) − A. (5.23). where k2 is a new design parameter, leads to V˙ (z1 , z2 ) = (−k12 − k1 (a1 + q4 ) + q2 − x21 )z12 + (−k2 + k1 + q4 )z22. (5.24). For (5.24) to be negative definite, the following constraint on k2 must hold k2 > (k1 + q4 ). (5.25).

(44) 28. Backstepping design for the missile. 5.2.2. Controlling the yaw -dynamics. Step 1: Introducing new variables First a new variable, z3 , is introduced. z3 = β − β ref = x5 − β ref. (5.26). It makes the origin the equilibrium point. Differentiating z3 w.r.t. time and rewriting it in terms of z3 results in z˙3 = x1 x4 − x3 − b1 z3 − b1 β ref − β˙ ref. (5.27). In equation (5.27) x3 can be regarded as a control input and the virtual control is chosen as law xdes 3 xdes = x1 x4 − b1 β ref − β˙ ref + k21 z3 3. (5.28). Let z4 represent the residual z4 = x3 − xdes = x3 − x1 x4 + b1 β ref + β˙ ref − k21 z3 3. (5.29). Writing the yaw -dynamics in terms of z3 and z4 yields z˙3 = − (b1 + k21 )z3 − z4 2 z˙4 =(r2 + r3 x4 + r4 k21 + x21 − x4 p1 + k21 b1 + k21 )z3 + (r4 + k21 )z4. + r1 x1 x2 + r3 x4 β ref + r4 x1 x4 + x21 β ref − x1 x2 + a1 x1 x4 − p2 x1 x4 + r2 β ref − r4 b1 β ref − p1 x4 β ref − r4 β˙ ref + br β˙ ref + β¨ref. (5.30). + (r5 − p3 x4 )U1 + r6 U3 As with the pitch-dynamics the first equation in (5.30) gives a constraint on k21 , k21 > −b1. (5.31). to make the z3 -dynamics stable. The second equation may be written in more compact manner. z˙3 = − (b1 + k21 ) − z4 2 z˙4 =(r2 + r3 x4 + r4 k21 + x21 − x4 p1 + k21 b1 + k21 )z3 + (r4 + k21 )z4. (5.32). + ϕ2 (x) + B + (r5 − p3 x4 )U1 + r6 U3 Here the following notations were used. ϕ2 (x) =r1 x1 x2 + r3 x4 β ref + r4 x1 x4 + x21 β ref − x1 x2 + a1 x1 x4 − p2 x1 x4 (5.33) B = + r2 β ref − r4 b1 β ref − p1 x4 β ref − r4 β˙ ref + br β˙ ref + β¨ref.

(45) 5.2 Backstepping design. 29. Step 2: Finding the clf Now a clf for (5.32) can be found. The benefit of picking a non-quadratic clf is once again used. 1 V22 (z3 , z4 ) = G(z3 ) + z42 (5.34) 2 where G(z3 ) is any valid clf for the z3 -dynamics. Differentiating V22 (z3 , z4 ) w.r.t. time yields V˙ 22 (z3 , z4 ) = − G (z3 )(k21 + b1 )z3 2 + z4 [−G (z3 ) + z3 (r2 + r3 x4 + r4 k21 + x21 − x4 p1 + k21 b1 + k21 ) + z4 (r4 + k21 ) + ϕ2 (x) + B + (r5 − p3 x4 )U1 + r6 U3 ] (5.35). The first term in (5.35) tells that G (z3 ) must lie in the 1st or 3rd quadrant to render it negative definite. This corresponds to G (z3 ) > 0. In the second term, describing the z4 -dynamics, one would like to cancel the z3 -dynamics by choosing G (z3 ) as 2 G (z3 ) = (r2 + r3 x4 + r4 k21 + x21 − x4 p1 + k21 b1 + k21 )z3. (5.36). Since G (z3 ) > 0, the following inequality must hold 2 (r2 + r3 x4 + r4 k21 + x21 − x4 p1 + k21 b1 + k21 )>0. ⇔ 2 k21 + k21 (r4 + b1 ) + r2 + x21 + (r3 − p1 )x4 > 0. (5.37). k21 ∈ R, but when x4 < 0 k21 becomes a complex number. Therefore the x4 -term can not be canceled by G (z3 ). A new G (z3 ) is chosen by simply omitting the x4 -terms in (5.36). 2 G (z3 ) = (r2 + r4 k21 + x21 + k21 b1 + k21 )z3. (5.38). This leads to a new constraint on k21 2 k21 + k21 (r4 + b1 ) + r2 + x21 > 0. (5.39). When examine what values of k21 which will satisfy (5.39) the following knowledge is used • x21 > 0, always • r2 ∈]124, 317[ • (r4 + b1 ) ∈] − 0.5, 0.5[ This indicates that equation (5.39) hold for any k21 > 0, but equation (5.31), k21 > −b1 , must be fulfilled as well..

(46) 30. Backstepping design for the missile. Step 3: Determine the control input Inserting (5.38) in (5.35) yields 2 V˙ 22 = − (r2 + r4 k21 + x21 + k21 b1 + k21 )z32 (k21 + b1 ) + z4 [z3 (r3 − p1 )x4 + z4 (r4 + k21 ) + ϕ2 (x) + B + (r5 − p3 x4 )U1 + r6 U3 ] (5.40). By choosing the control inputs U1 and U3 as (r5 − p3 x4 )U1 + r6 U3 = −ϕ2 − B − z3 (r3 − p1 )x4 − z4 k22. (5.41). where k22 is a design parameter, equation (5.40) now becomes 2 V˙ 22 = − (r2 + r4 k21 + x21 + k21 b1 + k21 )z32 (k21 + b1 ). + (k21 − k22 + r4 )z42. (5.42). It is made positive definite if k22 > (k21 + r4 ). 5.2.3. (5.43). Controlling the roll-dynamics. To control the roll-dynamics a simple PI-controller with a term which cancels the p1 x5 -term is used. t p1 x5 U1 = −Kp e(t) − Ki e(τ )dτ − (5.44) p3 0 where e(t) is the control error. In the roll-channel the loop-gain is rather large, around 60 dB, so therefore reasonable values for Kp and Ki are 0.001-0.01.. 5.2.4. Combining the control inputs. From the equations (5.23), (5.41) and (5.44) a fairly complicated equation system can be derived, which solution provides the control inputs for the whole missile. q5 U2 − p3 x5 U1 = −z2 k2 − ϕ1 (x) − A (r5 − p3 x4 )U1 + r6 U3 = −ϕ2 − B − z3 (r3 − p1 )x4 − z4 k22 t p1 x5 U1 = −Kp e(t) − Ki e(τ )dτ − p3 0. (5.45). The solution to the equation system is U1 = −Kp e(t) − Ki 0. t. e(τ )dτ −. p1 x5 p3. p3 x5 U1 − z2 k2 − ϕ1 (x) − A U2 = q5 −(r5 − p3 x4 )U1 − ϕ2 − B − z3 (r3 − p1 )x4 − z4 k22 U3 = r6. (5.46).

(47) 5.3 Aspects on the design. 5.3. 31. Aspects on the design. This section considers some aspects on the backstepping design from the previous section.. 5.3.1. Cascaded control structure. To realize how the control laws are build up U2 and U3 are examined closer. Substituting z1 − z4 in equation (5.46) by their correspondents in terms of x1 , x2 , . . . , x5 yields, U2 = [−x2 k2 − (x4 − αref )(k1 (q4 + k2 )) − x1 x3 (q1 + 1) − q3 |x5 | + p1 x25 + x1 x5 (k2 − b1 + p2 ) 1 + αref (a1 k2 − q2 + x21 ) + p3 x5 U1 ] q5 U3 = [−x3 k22 + (x5 − β ref )(k21 (r4 + k22 )). (5.47). + x1 x5 (r3 − p1 ) − x1 x4 (−k22 + a1 − p2 ) − x1 x2 (r1 − 1) 1 − β ref (x21 + b1 k22 + b1 k22 ) + (x4 p3 − r5 )U1 ] r6 Notice the similarities between the two expressions. Especially the first row in the two expressions are almost identical. Consider U2 . It is the control signal for the fin which is used to control the pitch-dynamics given a reference signal. The main objective is to control α, i.e. x4 . To do that q = x2 , is used. A closer look at U2 shows that it is in fact a cascaded control structure, with feedback from x2 as the inner loop and the feedback from x4 as the outer. See Figure 5.1. In cascaded control structures the inner loop must normally have faster dynamics then the outer loop. In most cases this is equivalent to that the inner loop has higher gain then the outer loop. In this case, the inner loop has the gain k2 and the outer loop k1 . Thus k2 > k1 which is almost the same as the constraint that equation (5.17) led to. The same holds for the U3 -expression. Here, feedback from x3 represents the inner loop and feedback from x5 the outer loop.. 5.3.2. Tuning of the design parameters. In 5.3.1 one way to gain more understanding of the backstepping design was shown. Here another way is presented which also suggests how the design parameters should be chosen. In (5.15) and (5.32) insert U2 and U3 respectively. This results in the following closed loop dynamics for the pitch and yaw-channels. z˙1 = −(a1 + k1 )z1 + z2 z˙2 = [−k12 − k1 (q4 + a1 ) − q2 − x21 ]z1 + (−k2 + k1 + q4 )z2. (5.48).

(48) 32. Backstepping design for the missile. − x1 x3 ( q1 + 1) − q3x5 + p1x25. αre f. − ∑. k1. + x4. +. ∑ x2. (. − q 4 − k2 ). +. + ∑. +. 1 q5. Missile. +. x1 x5 ( k2 − b1 + p2) + αre f (a1 k2 − q2 + x21 ) + p3 x5U1. Figure 5.1. Cascaded control structure. z˙3 = −(b1 + k21 )z3 + z4 2 z˙4 = [k21 + k21 (r4 + b1 ) − r2 + x21 ]z3 + (−k22 + k21 + r4 )z4. (5.49). Note that (5.48) is linear in terms of z1 and z2 . Equation (5.49) is also linear but in terms of z3 and z4 . This means that a linear tool, such as pole-placement, can be used when the values of k1 − k22 are to be decided. The poles should be placed in the left half plane inside the bisectors in the 3rd and the 4th quadrant, see [2]. Since the aerodynamic coefficients depend on the speed of the missile, the poles will move if the speed is changed. Table 5.2 shows how. The gain in the outer loops (k1 and k21 ) are quite small. A higher value would have given a greater imaginary part and thereby quicker response in the case when the speed is Mach 1.5. At Mach 3.0 the same value of k1 and k2 would have placed the poles outside the bisectors and the response might oscillate. In section 5.3.4 this problem is further discussed.. Mach 1.5 Mach 1.5, x1 = 2rad/s Mach 3.0. pith-channel −12.7719 ± 3.6112i −12.7719 ± 4.1280i −13.4356 ± 12.0182i. yaw-channel −15.4287 ± 2.2919i −15.4287 ± 3.0419i −16.1047 ± 14.5999i. Table 5.2. The table shows where the poles ends up for different speeds when the design parameter are k1 = 2.3,k2 = 22,k21 = 3 and k22 = 28. It also shows that when there is movement in the roll-channel(x1 ) the imaginary part increases..

(49) 5.3 Aspects on the design. 5.3.3. 33. Robustness of the design. All models of physical systems suffer from model errors and uncertainties. This means that the model on which the design is based differs from the actual physical system. For the missile case the aerodynamic coefficients are an example of such model uncertainties. Sometimes one also has to simplify the model and make assumptions. In the design, the force contribution from the fins were neglected. Despite that the closed loop system must be stable. In this section the robustness of the design is therefore studied. Uncertainties in the aerodynamic coefficients In equation (5.7) all coefficients p1 , p2 , . . . , b1 are more or less marred by uncertainties. The design led to constraints on k1 ,k2 ,k21 and k22 that have a direct connection to some of these coefficients. Equation (5.14) gave the constraint k1 > −a1 , equation (5.20) showed that k12 + k1 (a1 − q4 ) − q2 + x21 > 0 and finally (5.25) said that k2 > k1 . For k21 and k22 similar constraints were found. This gives a direct possibility to design the system so that it is robust against some model uncertainties by simply picking the design parameters with a good margin. However, the values of k1 − k22 cannot be chosen arbitrary large. The higher the value the more sensitive the system is to disturbances. Above it was shown that the design could be made robust against some uncertainties in the aerodynamic coefficients, namely the ones that appear directly in the expressions. There are however others that do not appear in the expressions, or they do appear but they enter the design in other ways. These also have to be examined. Consider equation (5.22) V˙ (z1 , z2 ) =(−k12 − k1 (a1 + q4 ) + q2 − x21 )z12 + z2 [z2 (q4 + k1 ) + ϕ1 (x) + A + q5 U2 − p3 x5 U1 ]. (5.50). In the design, q5 U2 −p3 x5 U1 = −z2 k2 −ϕ1 (x)−A, was chosen so that it cancels the ϕ(x)-term and the A-term. Note that both ϕ(x) and A are made up by aerodynamic coefficients. What happens if the true values of these coefficients are uncertain. Then V˙ (z1 , z2 ) would not look like the desired equation (5.24). Instead it will look like V˙ (z1 , z2 ) =(−k12 − k1 (a1 + q4 ) + q2 − x21 )z12 + [(−k2 + k1 + r4 )z2 + ∆ϕ1 (x) + ∆A] z2. (5.51). Here ∆ϕ1 (x) and ∆A represent the difference between the true values of the coefficients and the values that are used in the model. With these extra terms in the z2 -term it is much harder to make any comment on the stability properties. For the system to be globally asymptotically stable V˙ (z1 , z2 ) must be negative definite. The first term in equation (5.51) is still negative definite, provided that one have taken good margins against model uncertainties according to what is said in the.

(50) 34. Backstepping design for the missile. beginning of this section. To make the z2 -term negative definite the following must be achieved (5.52) (−k2 + k1 )z22 + ∆ϕ1 (x)z2 + ∆Az2 < 0 This is not easy since neither the sign or magnitudes of ∆ϕ1 (x) and ∆A are known. The solution to this problem is postpone to the next chapter. The fact that a non-quadratic clf, F (z1 ) and G(z3 ), are used gives us a possibility to improve robustness against uncertainties in the aerodynamic coefficients. By the choices of the clf:s the number of terms that includes the coefficients which the control inputs have to cancel, were reduced. Instead the coefficients turn up in the constraints for k1 , k2 , k3 and k4 . Here less knowledge about them is needed compared to what one need in order to cancel them. Using non-quadratic clf:s is one great advantage of Backstepping. This becomes even more clear when dealing with nonlinearities. In the same manner, as with the uncertainties, you need less knowledge about the nonlinearities. For examples see [4] and [10]. Force contribution of the fins In the backstepping design the force contribution of the fins had to be neglected. In the true system the force contributions exists. It is therefore interesting to see if the design still is globally asymptotically stable. The true system for the pitch-dynamics is x˙ 2 =q1 x1 x3 + q2 x4 + q3 x5 + q4 x2 + q5 U2 x˙ 4 = − x1 x5 + x2 − a1 x4 − a2 U2. (5.53). where a2 is an aerodynamic coefficient. In terms of z1 and z2 equation (5.53) can be written z˙1 = − (a1 + k1 )z1 + z2 − a2 U2 z˙2 =(−k12 − k1 a1 − k1 q4 + q2 − x21 )z1 + (k1 + q4 )z2 + ϕ1 (x) + A + q5 U2 − p3 x5 U1. (5.54). and the control input U2 is as earlier U2 =. p3 x5 U1 − z2 k2 − ϕ1 (x) − A q5. Inserting equation (5.55) into equation (5.54) yields. k2 a2 a2 z˙1 = − (a1 + k1 )z1 + z2 1 + (p3 x5 U1 + ϕ1 (x) + A) − q5 q5. (5.55). (5.56). z˙2 =(−k12 − k1 (a1 q4 ) + −x21 )z1 + (−k2 + k1 + q4 )z2 The same non-quadratic clf as in the design is used 1 V = F (z1 ) + z22 2. (5.57).

(51) 5.3 Aspects on the design. 35. Differentiating equation (5.57) w.r.t. time and inserting equation (5.56) and yields a2 V˙ = − F (z1 )((a1 + k1 ) z1 − (p3 x5 U1 + ϕ1 (x) + A)) q5 k a 2 2 + z2 [F (z1 ) + (−k2 + k1 + q4 )z2 ] q5. (5.58). If F (z1 ) is chosen as earlier the problem of ensuring that V˙ in equation (5.58) is negative definite is similar to the problem that was discussed in the previous section.. 5.3.4. Exploring some design flexibilities. When the control laws were derived, some design choices were made. In this section additional design flexibilities are explored. Often, as we shall see, a choice which is good from a performance point of view, might be bad from a robustness point of view. One design choice has already been pointed out, namely the benefit of using a non-quadratic clf. A consequence of this choice is that a design flexibility is introduced. This is explored in the following text. To make V˙ (z1 , z2 ) and V˙ 22 (z3 , z4 ) negative definite, some dynamics had to be canceled. This can be archived in two ways. Either by the non-quadratic clf, F’ and G’, or by the control input. In section 5.3.3 it was shown that it is appealing, in order to achieve good robustness, to let the non-quadratic terms cancel them. However, the dynamics will then affect the closed loop dynamics and thereby the performance of the system. So from a performance point of view it is better to let the control input cancel all dynamics in the clf. Then replace them so that the desired performance of the closed loop system is achieved. This method is in fact the same as exact linearization. It requires very good knowledge about the system that is to be controlled for the closed loop system to be robust. Since there are uncertainties in the aerodynamic coefficients this solution is not good from a robustness point of view. Of course it is possible to design for something in between these two situations, i.e. letting the non-quadratic clf cancel some dynamics and the control input cancel some others. This could lead to improved performance without affecting the robustness too much. Bellow follows an example. Recall the closed loop dynamics for the pitch-channel that was derived in section 5.3.2, z˙1 = −(a1 + k1 )z1 + z2 z˙2 = [−k12 − k1 (q4 + a1 ) − q2 − x21 ]z1 − (k2 − k1 + q4 )z2. (5.59). In that section it was shown that the poles for the closed loop system varied with the speed of the missile. At low speed the imaginary part is small compared to the real part and therefore the system response is quite slow. But at high speed.

(52) 36. Backstepping design for the missile. the imaginary part is bigger and the system response is quicker and less damped. The reason for this behavior is that the value of q2 varies between -80 at minimum speed and -200 at maximum speed. So, it would be good if q2 does not affect the closed loop dynamics. This can be achieved by letting the control input, U2 , cancel the z1 q2 -dynamics in equation (5.17) instead of F (z1 ). In the yaw-channel, r2 has the same effect on the closed loop system and therefore let U3 cancel the z3 r2 -dynamics in equation (5.35). Now the control laws look like . t. p1 x5 p3 0 p3 x5 U1 − z2 k2 − ϕ1 (x) − A − z1 q2 U2 = q5 −(r5 − p3 x4 )U1 − ϕ2 − B − z3 ((r3 − p1 )x4 + r2 ) − z4 k22 U3 = r6. U1 = −Kp e(t) − Ki. e(τ )dτ −. (5.60). and the closed loop dynamics in the pitch and yaw-channels are described by the following equations z˙1 = −(a1 + k1 )z1 + z2 z˙2 = [−k12 − k1 (q4 + a1 ) − x21 ]z1 − (k2 − k1 + q4 )z2 z˙3 = −(b1 + k21 )z3 + z4 2 z˙4 = [k21 + k21 (r4 + b1 ) + x21 ]z3 + (−k22 + k21 + r4 )z4. (5.61). (5.62). Since the real and the imaginary part of the poles now changes little with the speed, it enables an increase in the value of k1 and k21 . This makes the closed loop dynamics faster and thereby the performance is improved. On the other hand if the value of q2 and r2 are not known exactly the robustness is affected. Let ∆q2 and ∆r2 represent the difference between the true values of the coefficient and the value that is used in the model. Then V˙ (z1 , z2 ) and V˙ 22 (z3 , z4 ) have the following appearance V˙ (z1 , z2 ) =(−k12 − k1 (a1 + q4 ) − x21 )z12 + [(−k2 + k1 + r4 )z2 + ∆ϕ1 (x) + ∆A + ∆q2 z1 ] z2 2 V˙ 22 (z3 , z4 ) = − (r4 k21 + x21 + k21 b1 + k21 )z32 (k21 + b1 ) + [(k21 − k22 + r4 )z4 + ∆ϕ2 (x) + ∆B + ∆r2 z3 ] z4. (5.63). (5.64). ∆q2 and enters V˙ (z1 , z2 ) in the same way as ∆ϕ1 (x) and ∆A and this is dealt with in the next chapter..

(53) 5.4 Simulation. 37. 5.4. Simulation. 5.4.1. Conditions. The following conditions are valid during the simulations. • In the simulations the missile dynamics are calculated using the simulation model described in section 4.2.6, i.e. not the one that the design was built upon. • The aerodynamics coefficients are assumed to be known exactly. This means that the coefficients used in the control laws are the same as the coefficients which are used to calculate the dynamics of the missile. • The initial speed of the missile is Mach 1.5 in simulation 5.1 and Mach 3.0 in simulation 5.2. • No noise or disturbances such as wind gusts are added. • The sensors are ideal. The actuator dynamics are described in section 4.1.4 and the fins have zero friction. • In the controller, the derivatives of the reference signals are not used. • The controller used in the simulations are the one that was derived in equation (5.60) in Section 5.3.4.. 5.4.2. Controller parameters. The controller parameters are chosen according to Table 5.3. Table 5.4 shows where the poles in the pitch-channel and yaw-channel ends up.. roll-channel pitch-channel yaw-channel. Kp Ki k1 k2 k21 k22. 0.0025 0.060 15.0 32.0 13.0 27.0. Table 5.3. Controller parameters. 5.4.3. Simulation results. The maneuver performed is a bank to turn maneuver. First a roll is performed followed by an acceleration in the pitch-channel. Also an acceleration in the yawchannel is added to see how the derived controller handles the cross coupling effects..

(54) 38. Backstepping design for the missile. α-channel β-channel. Mach Mach Mach Mach. 1.5 3.0 1.5 3.0. −17.7719 ± 15.8055i −18.4356 ± 16.3654i −14.9287 ± 12.8879i −15.6047 ± 13.4098i. Table 5.4. Placement of the poles. The simulation results where the initial speed is Mach 1.5 (called simulation 5.1) are shown in Figure 5.2 and 5.3. The result when the speed is Mach 3.0 (called simulation 5.2) is shown in Figure 5.4 and 5.5. The simulation plots are found at the end of the chapter in section 5.6. Comments on the simulation The poor reference tracking in the ay -channel depends on that decoupling from α and p is not fully accomplished. In simulation 2, the reference tracking in ay is improved. This is because the angle of attack, α, is smaller and thus the cross coupling effect on β. In both simulations all variables such as the incidence angles, α and β, and the fin angles are within reasonable levels.. 5.5. Summary. In this chapter the control laws were derived and some aspects on the design were considered. During the design some design choices were made. Since there is no existing design tool for backstepping the consequences of the choices could not be investigated until the closed loop expressions for the Lyapunov function, its time derivative or the closed loop dynamics were derived. However these expressions are straightforward and easy to understand. For example the closed loop dynamics in the pitch-channel (5.48) and yaw-channel (5.49) are linear differential equations which are easy to analyze. Some other aspects of the design were also outlined. In section 5.3.1 it was shown that the design led to a cascaded control structure and short discussion on the robustness was covered in section 5.3.3..

(55) 5.6 Simulation plots. 5.6. 39. Simulation plots a response (m/s2). Roll response (rad/s). a response (m/s2). y. 2.5. z. 40 p p (rad/s). 35. 2. 1.5. 1. 0.5. 0. 50. aref y ay. ref. 0. 30. −50. 25. −100. 20. −150. 15. −200. 10. −250. 5. −300. 0. −350. −5 −0.5. 0. 2. 4 6 Time (s). 8. 10. −10. aref z az. −400 0. 2. Fin angles (deg). 4 6 Time (s). 8. −450. 10. 0. 2. Rate response (rad/s). 4. 2. 8. 10. Incidence angles (deg). 2.5. δa δe δr. 4 6 Time (s). 30 p q r. 2. α β. 25. 0. 20 1.5. −2. 15 1. −4. 10 0.5. −6. 5 0. −8. −10. 0. 2. 4 6 Time (s). 8. 10. −0.5. 0. 0. 2. 4 6 Time (s). 8. −5. 10. 0. 2. 4 6 Time (s). 8. 10. Figure 5.2. Simulation 5.1. Simulation of the bank to turn maneuver with the initial speed of Mach 1.5.. 0.5 0 −0.5 −1 −1.5 −2 −2.5 −3 βref β. −3.5 −4. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Figure 5.3. Simulation 5.1.β ref and β in simulation 1. Notice the step in the reference signal at T ≈ 4s this is due to cross coupling effects..

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