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1 Gain Scheduled Missile Control Using Robust Loop Shaping Examensarbete utfört i Reglerteknik vid Tekniska Högskolan i Linköping av Henrik Johansson Reg nr: LiTH-ISY-EX

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3 Gain Scheduled Missile Control Using Robust Loop Shaping Examensarbete utfört i Reglerteknik vid Tekniska Högskolan i Linköping av Henrik Johansson Reg nr: LiTH-ISY-EX Supervisor: Martin Enqvist Henrik Jonson Examiner: Torkel Glad Linköping, 1th January 23.

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5 Avdelning, Institution Division, Department Datum Date Automatic Control, Dept. of Electrical Engineering 1th January 23 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport URL för elektronisk version ISBN μ ISRN μ Serietitel och serienummer Title of series, numbering ISSN μ LITH-ISY-EX Titel Title Parameterstyrd missilstyrning med hjälp av robust kretsformning Gain Scheduled Missile Control Using Robust Loop Shaping Författare Author Henrik Johansson Sammanfattning Abstract Robust control design has become a major research area during the last twentyyears and there are nowadays several robust design methods available. One example of such a method is the robust loop shaping method that was developed by Glover and MacFarlane in the late 198s. The idea of this method is to use decentralized controller design to give the singular values of the loop gain a desired shape. This step is called Loop Shaping and it is followed by a Robust Stabilization procedure, which aimstogive the closed loop system a maximum degree of stability margins. In this thesis, the robust loop shaping method is used to design a gain scheduled controller for a missile. The report consists of three parts, where the first part introduces the Robust Loop Shaping controller design theory and a Gain Scheduling approach. The second part discusses the missile and its characteristics. In the third part a controller is designed and a short analysis of the closed loop system is performed. A scheduled controller is implemented in a nonlinear environment, in which performance and robustness are tested. Robust Loop Shaping is easy to use and simulations show that the resulting controller is able to cope with model perturbations without considerable loss in performance. The missile should to be able to operate in a large speed interval. There, it is shown that a single controller does not stabilize the missile everywhere. The gain scheduled controller is however able to do so, which has been shown by means of simulations in the nonlinear environment. Nyckelord Keywords Missile Control, Robust Loop Shaping, Gain Scheduling

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7 Abstract Robust control design has become a major research area during the last twenty years and there are nowadays several robust design methods available. One example of such a method is the robust loop shaping method that was developed by Glover and MacFarlane in the late 198s. The idea of this method is to use decentralized controller design to give the singular values of the loop gain a desired shape. This step is called Loop Shaping and it is followed by a Robust Stabilization procedure, which aims to give the closed loop system a maximum degree of stability margins. In this thesis, the robust loop shaping method is used to design a gain scheduled controller for a missile. The report consists of three parts, where the first part introduces the Robust Loop Shaping controller design theory and a Gain Scheduling approach. The second part discusses the missile and its characteristics. In the third part a controller is designed and a short analysis of the closed loop system is performed. A scheduled controller is implemented in a nonlinear environment, in which performance and robustness are tested. Robust Loop Shaping is easy to use and simulations show that the resulting controller is able to cope with model perturbations without considerable loss in performance. The missile should to be able to operate in a large speed interval. There, it is shown that a single controller does not stabilize the missile everywhere. The gain scheduled controller is however able to do so, which has been shown by means of simulations in the nonlinear environment. Keywords: Missile Control, Robust Loop Shaping, Gain Scheduling i

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9 Acknowledgment This work has been carried out at Saab Bofors Dynamics AB in Linköping, Sweden. Iwould like to thank my to supervisors Henrik Jonson and Martin Enqvist for their help and support during this project. I would also like to thank my examiner Torkel Glad for showing interest in my work. Finally, I would like to thank all the people at Saab Bofors Dynamics AB for making me feel like a part of the group. Linköping, December 22 Henrik Johansson iii

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11 Notation Symbols u u(t) input vector δ a aileron (u 1 ) δ e elevator (u 2 ) δ r rudder (u 3 ) x x(t) state vector ẋ the time derivative ofx ˆx the estimate of x y y(t) output vector G(s) transfer function ω the angular velocity vector p angular velocity around the x axis q angular velocity around the y axis r angular velocity around the z axis α angle of attack β sideslip angle V the missile velocity vector u velocity in the direction of the x axis v velocity in the direction of the y axis w velocity in the direction of the z axis V the forward speed of the missil ( V ) I mass moment of inertia or the unitary matrix I x the x part of I I y the y part of I the z part of I I z Operators and functions [ ] A B shorthand for the state space C D realization of G(s) =C(sI A) 1 B + D v

12 transpose complex conjugate transpose inverse pseudo inverse shorthand for (A 1 ) A. B element wisemultiplication of A and B λ(a) eigenvalue of A β absolute value of β f( ) H -norm A T A A 1 A A 1 vi

13 Contents 1 Introduction Background Objectives Limitations Thesis Outline The H Loop Shaping Design Basic Requirements for the Closed Loop Loop Shaping The Scaling Procedure Pairing of the Inputs and Outputs Decentralized Controller Design Alignment of Singular Values Summary of the Loop Shaping Procedure Robust Stabilization Normalized Left Coprime Factorizations The Stabilizing Controller K (s) Gain Scheduling Scheduling on Either One or Two Parameters The Missile Model The Missile The Guidance System of the Missile Assumptions and Limitations The Main Dynamics of the Missile Rigid Body Dynamics The Translation of the Missile The Rotation of the Missile Summary of the Main Dynamics of the Missile The Plant Choosing the Design Parameters The Newton-Raphson Method The Linear Plant Description vii

14 viii Contents 4 Controller Design for the Basic Plant The Plant Used for the Controller Design Loop Shaping and Robust Stabilization for the Basic Plant The Scaling Procedure Pairing the Inputs and Outputs Decentralized Controller Design Alignment of Singular Values Robust Stabilization Linear Analysis The Closed Loop System The Performance of the Closed Loop System Robustness of the Closed Loop System Nonlinear Evaluation Simulations The Controller Design Based on the Basic Model Transformation of the Reference Signals Simulations with a Single Controller Scheduling on the Forward Speed The Controller Design Based on the Extended Model Scheduling on the Angle of Attack Basic vs. Extended Model Conclusions Results Future Work Appendices A The System Matrices of the Nominal Plant 53 B The System Matrices of the Controller 55 C The Matlab Script Used for the Controller Design 57 D Simulation Plots 59

15 1 Introduction This report is a presentation of a project performed at Saab Bofors Dynamics AB in Linköping, Sweden. For the last twenty years Saab Bofors Dynamics AB has successfully been developing missile systems, where the Swedish defense is the main customer. Advanced automatic control is an important part of the missile systems of today. Hence, it is very important for Saab to stay updated in the area of guidance and control. 1.1 Background Robust control design has become a major research area during the last twenty years and there are nowadays several robust design methods available. One example of such a method is the robust loop shaping method that was developed by K. Glover and D. C. MacFarlane in the late 198s. The idea of this method is to use decentralized controller design to give the singular values of the loop gain a desired shape. This step is called Loop Shaping and it is followed by a Robust Stabilization procedure, which aims to give the closed loop system a maximum degree of stability margins. This method was used in [8] by R. A. Hyde and K. Glover in an aircraft application. 1.2 Objectives The objectives of this project is to use the ideas of K. Glover and R. A. Hyde [8] to design a scheduled H Loop Shaping controller for a missile application. The controller was supposed to be implemented in a nonlinear simulation environment, supplied by Saab Bofors Dynamics. Furthermore, the controller capability to handle model perturbations should be tested as well. 1

16 2 Introduction 1.3 Limitations In order to keep this project within reasonable limits there was a need for some restrictions. The missile is supposed to work well in a large operating area, which can be divided into speed and altitude. A large area means that several controllers have to be designed. In order to reduce the number of design cases, this project is restricted to designing and evaluating controllers for a fixed altitude. 1.4 Thesis Outline This report can be divided into three different parts. The first part describes the H Loop Shaping controller design and is found in Chapter 2. The controller design consists of a loop shaping procedure which specifies performance for the closed loop system. Moreover, a robust stabilization procedure, which is applied in order to give the closed loop system sufficient stability margins, and an approach for gain scheduling is described. An accurate system model is the cornerstone of a successful controller. Moreover, since gain scheduling is included in the controller design it is important to be able to derive several linear plant descriptions. A method that makes this easy is described in Chapter 3. This chapter can be viewed as the second part of the report. The third part of the thesis contains an evaluation of the designed controller. In Chapter 4 a short design example is given and in Chapter 5 the performance and robustness of the closed loop system is discussed. The best way toevaluate a controller is of course to let it control the real system. Since it, for obvious reasons, is impractical to implement the controller in a real missile, the next best thing is to evaluate the scheduled linear H Loop Shaping controller in a nonlinear simulation environment. This is described in Chapter 6 whereas conclusions and suggested future work is found in Chapter 7.

17 2 The H Loop Shaping Design The control design method that is used in this master thesis is H Loop Shaping. This particular method was proposed by D. C. MacFarlane and K. Glover, and was used by R. A. Hyde and K. Glover in [8]. The design method consists of three parts, all described in this chapter. A Loop Shaping technique is used as a first step to specify performance of the closed loop system. The loop shaping procedure is followed by a Robust Stabilization procedure with the purpose of maximizing the stability margins of the closed loop system. Finally, an approach to Gain Scheduling suitable for H controllers is presented. 2.1 Basic Requirements for the Closed Loop For a multivariable (MIMO) system there are some basic requirements for the Closed Loop System G c (s). These can be summarized as follows. The transfer function from the reference to the output should be close to I, i.e. G c (s) I. The transfer function from reference to input G ru (s) should not be too large. The sensitivity function S(s) should be small so that system disturbances and model perturbations have little or no effect on the output. For measurement disturbances to have little or no effect on the output, the complementary sensitivity function T (s) should be small. Furthermore, it should be small so that model perturbations do not affect the system stability. However, because these design objectives are usually conflicting it is not possible to fulfill them all. Furthermore, the missile is a poorly decoupled nonlinear system with six degrees of freedom. The H Loop Shaping design has proven to produce 3

18 4 The H Loop Shaping Design controllers that give a good compromise between the desired transfer functions mentioned earlier and that handle poorly decoupled systems. 2.2 Loop Shaping In [8] R. A. Hyde and K. Glover propose a Loop Shaping procedure used to specify performance as a first step in the H Loop Shaping controller design. The purpose of this step is to give the singular values of the loop gain a desired shape. This will after the robust stabilization procedure lead to suitable behavior of the closed loop system. According to [1] the desired shape of the singular values of the loop gain is high gain for lower frequencies and low gain for higher frequencies. During the transition between high and low gain, the loop gain should have a decrease of 1. In this master thesis project the Loop Shaping procedure consists of four steps, which are carried out in the following order 1. The Scaling Procedure D u and D y. 2. Pairing of the Inputs and Outputs. 3. Decentralized Controller Design W p (s). 4. Alignment ofsingularvalues W a. The result from the Loop Shaping procedure is a shaped plant G s (s) according to Figure 2.1. The singular values of the shaped plant should have the desired shape. W a W p ( s) W f D u G( s) Dy 1 Figure 2.1. The result of the Loop Shaping procedure is a loop gain that has characteristics suitable for the robust stabilization procedure and for the behavior of the closed loop system. The use of the W f matrix will be explained in Chapter The Scaling Procedure The first step in the Loop Shaping procedure is to scale the inputs and the outputs. This can be achieved, according to [1], by scaling the inputs and the outputs such that they vary in the interval between -1 and 1. A common scaling is to choose diagonal matrices D y and D u so that the physical true variables y p and u p satisfies y p = D y y and u p = D u u,wherey and u vary between 1 and 1. Hence, the first step of the Loop Shaping procedure is to choose the matrices D y and D u in (2.1). D y y = G(s)D u u y = Dy 1 G(s)D uu (2.1)

19 2.2 Loop Shaping 5 Notice that the scalings can be altered during the design phase and that preliminary scalings may be rough Pairing of the Inputs and Outputs In the second step of the Loop Shaping procedure the inputs and the outputs are paired to achieve a plant that is as diagonal as possible. The interactions between inputs and outputs reflect controller capabilities in the sense that strong cross couplings often lead to poor control. Hence, it is important for the plant tobeas diagonal as possible to make it easier for the controller to achieve good performance and robustness. In[1]itisshown how the Relative Gain Array canbeusedasan indication of how the inputs and outputs should be arranged to make the plant as diagonal as possible. The relative gain array can be calculated according to Definition 2.1. Definition 2.1 The Relative Gain Array of an arbitrary complex-valued matrix, A, is defined as RGA(A) =A. (A ) T where. denotes element wise multiplication and denotes the pseudo inverse. In the definition above, A is a static matrix. Hence, for the plant G(s) the RGA can only be calculated for fixed frequencies. The two most common frequencies to consider are ω =and the desired bandwidth, ω = ω b, of the closed loop system. The RGA of a matrix A has some useful characteristics, e.g. If rows (columns) are rearranged in A then the corresponding rows (columns) of RGA(A) are rearranged. RGA of a matrix A is independent of scalings, e.g. RGA(A) =RGA(Dy 1AD u) The sum of the elements in a row (column) is always 1. For a plant evaluated in ω =and in ω = ω b there are two formal results on how to arrange the inputs and outputs of the plant, namely I. The inputs and outputs should be paired such that the diagonal elements of RGA(G(iω b )) are as close to one, in the complex plane, as possible. II. Avoid pairings that implies negative diagonal elements for RGA(G()). The second step is to compute the relative gain array for ω =and for ω = ω b and pair the inputs and outputs together such that the results I and II are satisfied. An additional approach istochoose the matrix W f in Figure 2.1. The purpose of this matrix and how to choose it is explained in Section

20 6 The H Loop Shaping Design Decentralized Controller Design As the third step of the Loop Shaping procedure a decentralized controller W p (s) is designed such that the singular values of the shaped plant (loop gain) receives the desired shape. The idea of the decentralized control is to design a controller as if the system was diagonal, i.e. to disregard the influence of cross couplings. The precompensator W p (s) chosen for the controller design used in this thesis is, like in [8], supposed to add integral action to the scaled and diagonalized plant. The transfer function matrix W p (s) can be defined according to (2.2). K 1 (1 + 1 T ) 1s W p (s) = K 2 (1 + 1 T ) 2s K 3 (1 + 1 T 3s ) (2.2) The coefficients, K 1,T 1...K 3,T 3 are design parameters and can be altered during the design phase. In the next section, the fourth (and optional) step of the Loop Shaping procedure will be described. This step uses a certain method to align the singular values of the scaled, diagonalized and precompensated plant at a certain frequency Alignment of Singular Values As the fourth and last step of the Loop Shaping procedure, the alignment ofthe singular values of the shaped plant is carried out. The goal of the alignment procedure is to choose a matrix W a such that the singular values of the the loop gain are gathered near one at the desired bandwidth of the closed loop system. The idea is to use the W a matrix from Figure 2.1 to minimize the frequency interval where both the sensitivity function S(s) and the complementary sensitivity function T (s) are large. The method is described in [1] and uses the singular value decomposition (SVD). Definition 2.2 The singular value decomposition of a n m matrix A is A = UΣV where denotes the complex conjugate transpose of a matrix. U is a n n unitary (UU = I) matrix, Σ is a n m matrix which has the singular values of A along the diagonal and zeros elsewhere andv is a unitary m m matrix. In the alignment step it is here only necessary to consider SVD:s for quadratic 3 3 matrices. Let the SVD for the matrix A be U = ( ) u 1 u 2 u 3 Σ= σ 1 σ 2 σ 3 V = ( ) v 1 v 2 v 3

21 2.3 Robust Stabilization 7 where u i =(u 1i,u 2i,u 3i ) T and v i =(v 1i,v 2i,v 3i ) T. Based on the definitions above it is possible to write A as A = UΣV = 3 u i σ i v i As the equality Av j v j = u j σ j v j holds, A(I + αv j v j ) has the same singular values as A apart from the j:th which has been altered to σ j (1 + α). It is possible to alter the singular values of the 3 3 matrix G(iω) at the frequency ω with the following matrix i=1 W a = I + α 1 v 1 v 1 + α 2 v 2 v 2 + α 3 v 3 v 3 where the coefficients, α 1 to α 3, can be chosen such that the singular values of G(iω)W a will be aligned near one Summary of the Loop Shaping Procedure The main steps of the Loop Shaping procedure are 1. Scale the inputs and outputs by means of matrices Dy 1 scaled plant Dy 1 G(s)D u is obtained. and D u so that a 2. Compute the RGA for the frequencies and the desired bandwidth. Pair the inputs and the outputs according to the main results in Section Design a decentralized controller W p (s) so that the singular values of G(s)D uw f W p (s) have the desired shape. D 1 y 4. Align the singular values at the desired bandwidth by means of the W a matrix. This step is optional. The result of the design steps in the Loop Shaping procedure is the shaped plant G s (s) according to G s (s) =Dy 1 G(s)D u W f W p (s)w a The shaped plant G s (s) can be found in Figure 2.1 and it will be the subject of the robust stabilization procedure described in the next section. 2.3 Robust Stabilization The second part of the controller design phase is to stabilize the shaped plant G s (s). This is done in order to achieve maximum stability margins for the closed loop system G c (s), i.e. making the resulting closed loop system as robust to model perturbations and disturbances as possible. The shape of the singular values of the loop gain is usually maintained after the Robust Stabilization procedure. This means that the specified performance is maintained.

22 8 The H Loop Shaping Design Normalized Left Coprime Factorizations The Robust Stabilization procedure produce a controller K (s) that will stabilize a certain class of systems. These systems are those who can be described by normalized left coprime factorizations. A normalized left coprime factorization of a system G(s) is given by 1 G(s) = M (s)ñ(s) (2.3) where M(s) M (s)+ñ(s)ñ (s) =I. The normalized coprime factorization plant description of the plant G(s) can be derived according to the following theorem. Theorem 2.1 (Normalized left coprime factorization) Let G(s) be given by [ ] A B G(s) = C D and define R = I + DD > Suppose (C, A) is detectable and (A, B) controllable. Then there is a normalized left coprime factorization G(s) = M(s) 1 Ñ(s), with [ ] A + LC L M(s) = R 1/2 C R 1/2 and where [ ] A + LC B + LD Ñ(s) = R 1/2 C R 1/2 D L = (BD + ZC 1 ) R Z is the positive semidefinite solution to the algebraic riccati equation (A BD R 1 C)Z + Z(A BD R 1 C) ZC R 1 CZ + B R 1 B =. Proof. See Theorem 13.37, in [1]. The class of perturbations that K (s) stabilizes is given by (2.4). G (s) =( M(s)+ M (s)) 1 (Ñ(s)+ N (s)) (2.4) where M (s), N (s) are stable transfer functions that represent the uncertainty in the nominal plant. Notice that the uncertainties can introduce both new poles and zeros into the plant. This means that the perturbed plant might have more unstable poles and zeros than the nominal plant. The controller K (s) resulting from the H Loop Shaping controller design stabilizes perturbed systems G (s) with [ N (s) M (s)] < 1 (2.5) γ

23 2.3 Robust Stabilization The Stabilizing Controller K (s) This section is a review of the almost fully automatic procedure that, given a shaped plant G s (s), produces a stabilizing controller K (s). Forashapedplant given by (2.6) [ ] As B G s (s) = s (2.6) C s the following steps summarizes the Robust Stabilization procedure. First and foremost it is necessary to find the symmetric and positive definite solutions X and Z to the control algebraic riccati equation and the filtering algebraic riccati equation A sx + XA s XB s B s X + C s C s = A s Z + ZA s ZC s C sz + B s B s = (2.7) The optimal (smallest) γ opt maximizing (2.5) is given by γ opt = 1+λ m (XZ) (2.8) where λ m (XZ) is the largest eigenvalue of XZ. However, it is, according to [8], proven by experience that better results are often achieved with a slightly larger γ. With this in mind it is usually better to set γ to approximately 1.1γ opt. Define two matrices F and H as F =γ 2 B s X((1 γ2 )I + XZ) 1 H = ZC s (2.9) Then the stabilizing controller K (s) is given by K (s) = [ As + HC s + B s F H F ] (2.1) The controller K (s) can, for example, be implemented according to the block diagram in Figure 2.2. The benefit of having the controller K (s) in the feedback compared to a unity feedback is that an abrupt change in the inputs does not excite the dynamics of the controller.

24 1 The H Loop Shaping Design r K ( ) Σ G s ( s) y s K ( s) Figure 2.2. The closed loop system for the shaped plant G s(s) and the controller K (s). Notice the static gain K (). The implementation in Figure 2.2 is however not suitable due to the gain scheduling approach in this controller design method. An alternative implementation will be described in the next section. There the controller is written as an exact plant observer plus state feedback and this structure turns out to be more suitable for gain scheduling. 2.4 Gain Scheduling Linear controller design is often used to stabilize a nonlinear system, especially in flight applications. Since a linear controller design only is based on a linear model one controller is usually insufficient. The third step of the H Loop Shaping controller design is the Gain Scheduling approach. The idea is to implement the controller as a plant observer plus state feedback and to use linear interpolation between controllers of adjacent design points. In [2] itisshown how the controller K given by (2.1) can be written as an exact plant observer plus state feedback. The result is { ˆx = As ˆx + B s u s + H(C s ˆx y s ) (2.11) u s = F ˆx + P r All matrices in (2.11) apart from P can be recognized from (2.1). The P matrix is introduced in order to give the transfer function from reference signals to output signals the static gain I. In Section 5.1 it will be shown that P should be chosen as P = (C s (A s + B s F ) 1 B s ) 1

25 2.4 Gain Scheduling 11 The observer implementation is shown in the block diagram in Figure 2.3. For this master thesis project the controller is scheduled on either one or two parameters, the speed of the missile V and the angle of attack α. r P Σ u s G s ( s) y s F Observer Figure 2.3. The observer implementation of the H Loop Shaping controller Scheduling on Either One or Two Parameters For this master thesis project it is only necessary to design a gain scheduled controller that uses either the forward speed V, or V and the angle of attack α, for the interpolation. Scheduling on the Forward Speed Assume, for example, that the forward speed varies in the interval [V 1,V 2 [ and that two controllers with H matrices H 1 and H 2 are available. These controllers have been designed using the linearizations around V = V 1 and V = V 2, respectively. Let η be a function of V η = V V 1 (2.12) V 2 V 1 An interpolated H matrix H(η) can then be defined as H(η) =(1 η)h 1 + ηh 2 (2.13) Scheduling on the Forward Speed and the Angle of Attack Assume, in analogy with the previous section, that the missile forward speed V varies in the interval [V 1,V 2 [, that the angle of attack α varies in the interval [α 1,α 2 [ and that four adjacentcontrollers are available. These controllers have been designed around four linearizations. For example, the H matrix for the linearization around V = V 1 and α = α 1 is called H 11. Let ε be a function of α ε = α α 1 α 2 α 1 (2.14)

26 12 The H Loop Shaping Design An interpolated H matrix H(η, ε) can be defined as H(η, ε) =(1 η)[(1 ε)h 11 + εh 12 ]+ η[(1 ε)h 21 + εh 22 ] (2.15) It is straightforward to extend the scheduling to three or more variables. The topic of the next chapter is the missile and its characteristics. The chapter will discuss the missile in general and it will also show how a linear description of the main dynamics of the missile is derived.

27 3 The Missile Model An accurate mathematical description of the control object is the cornerstone of a successful controller. The plant is here the air-to-air missile in Figure 3.1. Although this missile does not exist yet, it has several features in common with existing ones. The H Loop Shaping controller design method is based on linear control theory. Thus, the topic of this chapter is to describe how a linear plant G(s), which describes the main dynamics of the missile in Figure 3.1, can be derived. Figure 3.1. The Missile. 13

28 14 The Missile Model 3.1 The Missile This section is a summary of the missile characteristics and the guidance system of the missile. The missile studied in this master thesis project is a bank-to-turn missile, which means that when the missile is going into a turn it should first roll in the direction of its velocity vector V and then accelerate in the direction of the z axis The Guidance System of the Missile This section will giveanintroduction to the guidance system of a missile. In Figure 3.2 a simplified description of the guidance system is given. Figure 3.2. A simplified description of the guidance system of the Missile. According to Figure 3.2, the guidance system consists of three parts, where each part is crucial to the behavior of the missile. A short description of these parts is given below.

29 3.1 The Missile 15 Strap Down Navigation. When the missile is released the strap down navigation system receives initial position, attitude, rotation and speed from the aircraft navigation system. The strap down navigation system uses measurements from rate gyros and accelerometers in combination with the initial conditions to calculate position, attitude, rotation and speed. The results of these calculations can be used by both Guidance and Autopilot. Guidance. The core of the guidance system is the block Guidance, which by predefined laws decides what the missile should do. Guidance receives information concerning the target partly from the aircraft and partly from the seeker. This information, combined with the missile state variables calculated by the strap down navigation system and the predefined laws, produces demanded reference signals to the autopilot. Autopilot. The autopilot receives reference signals as demanded output signals in terms of desired angular velocity around the x axis, desired acceleration in the direction of the y axis and desired acceleration in the direction of the z axis. These reference signals are p d, A yd and A zd and the main task of the autopilot is to produce commanded fin deflections such that the desired references are achieved. The H Loop Shaping controller of this project is implemented in the shaded block Autopilot in Figure 3.2. The autopilot receives demands in p d, A yd and A zd from the guidance system. Due to the Bank-to-Turn steering principle the demands in p d, A yd and A zd are suitable as reference signals Assumptions and Limitations The purpose of this chapter is to produce linear models of the main dynamics of the missile that can be used for the H Loop Shaping controller design. The linear model of the missile is called a plant and it is desirable that the plant description has as low degree as possible. Hence, it is necessary to make some assumptions during the modeling of the main dynamics of the missile. First, there is a need for an explanation of the notation in Figure 3.1. (x, y, z) is the fixed body frame of the missile. ω =(p, q, r) T is the angular velocity of the missile [rad/s]. V =(u, v, w) T is the missile velocity expressed in the body frame of the missile and V = V [m/s]. α is the angle of attack [rad]. β is the sideslip angle [rad]. m is the mass of the missile [kg].

30 16 The Missile Model The missile also has a mass moment of inertia matrix I, which due to the symmetry of the missile contain no mixed inertia terms and is given by I = I x I y [kgm 2 ] (3.1) I z The next two sections contain descriptions of the limitations and the assumptions made in this project. The Limitations of the Modeling Procedure In operating conditions there are loads acting on the missile, e.g. gravity, engine thrust, engine torque aerodynamic contributions etc. Some of the loads are not going to be considered for the modeling of the main dynamics of the missile. The gravity is neglected due to its small contribution and to maintain model simplicity. The contribution from the engine is neglected because the rate of change of the engine thrust is much slower than the other dynamics. The only force and torque contribution that are going to be considered are the force and torque that descend from the aerodynamics of the missile, see [5]. The aerodynamic force acting on the missile is according to [6] given by F a = q d S C T C C (3.2) C N where C T =1 C C =C Cβ β + C Cδr δ r C N =C Nα α + C Nδe δ e and the aerodynamic torque acting on the missile is according to [6] given by M a = q d Sd C l C m (3.3) C n where d C l = C lβ β + C lp 2V p + C lδ a δ a d C m = C mα α + C m β β + C mq 2V q + C mδ e δ e d C n = C nβ β + C nαβ αβ + C nr 2V r + C nδ a δ a + C nδr δ r In Equations (3.2) and (3.3) q d is the dynamic pressure, d a reference length and S = πd 2 /4 is a reference area. The different C i parameters are functions of the current Mach number and they are given by Table 3.1.

31 3.1 The Missile 17 Mach Mach C Cβ C Cδr C Nα C Nδe C lβ C lp C lδa C mα C m β C mq C mδe C nβ C nαβ C nr C nδa C nδr Table 3.1. The aerodynamic coefficients. The missile flight envelope is rather large, with a large span in both altitude and missile forward speed. The altitude and speed intervals that the missile should be able to operate in are Altitude: h 2 m. Forward speed: 45 V 12 m/s. Due to the limitations mentioned in Chapter 1 the controllers studied in this thesis are derived for the forward speed interval mentioned earlier and for the fixed altitude of h = 1 m. The Assumptions of the Modeling Procedure For the model to be sufficiently simple it is necessary to make some approximations. Besides neglecting some force and torque contributions four additional assumptions are made, namely 1. Assume that the angles α and β in Figure 3.1 are small, which for example means that sin β β and tan α α. 2. Assume small variations in the missile forward speed, i.e. V. 3. Assume that the mass of the missile is constant, i.e. neglect decrease of the mass of the missile caused by the combustion of fuel. 4. Assume that the mass moment of inertia is constant.

32 18 The Missile Model 3.2 The Main Dynamics of the Missile The main dynamics of the missile is described with basic rigid body dynamics and suitable approximations Rigid Body Dynamics When a rigid body is influenced by forces it will go into translation and when influenced by torque it will start rotating. Hence, it is necessary to understand the dynamics of the rigid body. The tools used to create a mathematical description are old but powerful. Isaac Newton first postulated the relationship between the force acting on a rigid body and its acceleration and this relation is hence called Newton's law of motion. The fact that the applied torque equals the rate of change of angular momentum was first postulated by Leonard Euler and this equality is usually known as Euler's equation. They are two well known laws of nature but a reminder might be appropriate. Definition 3.1 (Newton's law of motion) The force on a rigid body equals the mass of the rigid body times its inertial acceleration: F = ma. Definition 3.2 (Euler's equation) The applied torque equals the inertial rate of change of the angular momentum of a rigid body: M = Ḣ The angular momentum can be expressed in terms of mass moment of inertia and the rotation of the rigid body, i.e. H = Iω. Notice the use of inertial in the two definitions above. In order calculate the inertial time derivative of a rigid body it is necessary to pay attention to its rotation. The theorem below shows how the inertial time derivative of a rigid body can be calculated. Theorem 3.1 (Inertial derivative) The time derivative of a vector v with respect to the inertial (i) frame is related to the time derivative with respect to the body frame (b) by i d v = d b v + ω bi v dt dt where ω bi is the angular velocity of the body frame and v is the vector in the body frame. Proof. The proof is given in [9]. The main purpose of deriving a mathematical description of the missile is to use it for the controller design. The earlier mentioned simplifications have been made in order to make the resulting model easier to work with, hopefully without changing the main dynamics of the missile too much. The modeling approach is to apply Newton law of motion to describe the translation of the missile and to use the Euler equation to describe the rotation of the missile.

33 3.2 The Main Dynamics of the Missile The Translation of the Missile For the missile translation it is sensible to express the acceleration a in Definition 3.1 as the time derivative of the missile velocity V. This time derivative is obtained by applying Theorem 3.1 on the missile velocity according to (3.4). a = V + ω V (3.4) Since only the aerodynamic force F a is considered, the translation of the missile is obtained by combining (3.4) and Definition 3.1. The result is F ax F ay = m F az u v ẇ + m p q u v (3.5) r w Equation (3.5) contains the time derivatives of the suitable state variables u, v and w. The mathematical description which is the purpose of this section should express the time derivatives of the states as functions of the states and the inputs and it is therefore sensible to rewrite (3.5) according to (3.6). u = F ax m v = F ay m ẇ = F az m + rv qw + pw ru + qu pv (3.6) It is possible to include the two angles α and β as states instead of u, v and w. To do that there is a need for some further assumptions. As seen in Figure 3.1 the following relation between the angles, α and β, and the velocities, u, v and w, is obvious. sin β = v V tan α = w u (3.7) The assumption mentioned earlier considering small angles is useful and the approximations sin β β and tan α α are in this application sufficient up to α 3 and β 3. Hence, the relationship between α and β and u and w is β = v V α = w u (3.8) For small angles it is possible to assume that V Vx,which means that u V, andifthevariations in the speed V are small then V and so is u. This leads,

34 2 The Missile Model combined with Equations (3.6) and (3.8), to two first order differential equations in α and β α = F az mv + q pβ β = F (3.9) ay mv + pα r The forces F az and F ay are obtained from F a in (3.2) and this results in the description of the α and β dynamics according to (3.1). α = pβ + q q ds mv (C Nαα + C Nδe δ e ) β =pα r q ds mv (C Cββ + C Cδr δ r ) (3.1) The Rotation of the Missile The total torque acting on the missile equals the rate of change of the angular momentum of the missile. This is, due to the fact that H = Iω, the description of the rotation of the missile. The inertial time derivative ofh is then given by the following equation. Ḣ = I ω + ω Iω (3.11) The Euler equation from Definition 3.2 combined with (3.11), leads to the following equation. M ax M ay = I xṗ I y q + p q I xp I y q (3.12) M az I z ṙ r I z r Suitable state variables in (3.12) are p, q and r. By extracting the time derivatives of these variables from (3.12) the following expression is obtained. ṗ = M x I x q = M y I y ṙ = M z I z + I y I z qr I x + I z I x pr I y + I x I y pq I z (3.13)

35 3.3 The Plant 21 Equation (3.13) in combination with (3.3) leads to three first order differential equations in p, q and r according to (3.14). ṗ =qr I y I z I x q =pr I z I x I y + C mδe δ e ) + q dsd I x (C lβ β + C lp d 2V p + C lδ a δ a ) + q dsd I y (C mα α + C m β β + C mq d 2V q+ ṙ =pq I x I y + q dsd d (C nβ β + C nαβ αβ + C nr I z I z 2V r+ + C nδa δ a + C nδr δ r ) Summary of the Main Dynamics of the Missile (3.14) The main dynamics of the missile, described by equations (3.1) and (3.14), are summarized here. ṗ =qr I y I z I x q =pr I z I x I y + C mδe δ e ) + q dsd I x (C lβ β + C lp d 2V p + C lδ a δ a ) + q dsd I y (C mα α + C m β β + C mq d 2V q+ ṙ =pq I x I y + q dsd d (C nβ β + C nαβ αβ + C nr I z I z 2V r+ + C nδa δ a + C nδr δ r ) α = pβ + q q ds mv (C Nαα + C Nδe δ e ) β =pα r q ds mv (C Cββ + C Cδr δ r ) (3.15) There is need to be able to use several linearized models for the H Loop Shaping controller design. These linearizations can be derived from (3.15). 3.3 The Plant The plant used for the H Loop Shaping controller design is a linear transfer function G(s) mapping the input u to the output y. This plant is a linear description of the main dynamics of the missile in Figure 3.1 and it is desirable that the plant is a simple but yet accurate description of the missile. The result of the previous section was Equation (3.15), which is a nonlinear description of the main dynamics of the missile, and from this equation a plant is derived. Since gain scheduling is included in the controller design it is necessary to be able to produce several linear plant descriptions of the missile. Thus, the feature of this section is to generalize the process of deriving linear plant descriptions.

36 22 The Missile Model The result from this section will be the plant G(s) mapping the input u to the output y and it can either be written as a state space realization according to { ẋ = Ax + Bu (3.16) y = Cx + Du or as a transfer function matrix according to In (3.16) x is the states defined as G(s) =C(sI A) 1 B + D (3.17) x = p q r α β (3.18) and the input u is the fin deflections of the aileron, the elevator and the rudder according to u = δ a δ e (3.19) δ r In both (3.16) and (3.17) the matrices A, B, C and D are included and these matrices are called system matrices. The main dynamics of the missile described by equation (3.15) will in the future be referred to as ẋ = f(x, u,h,v) (3.2) and the linear description of (3.2) includes the two system matrices A and B and has the following appearance ẋ = Ax + Bu (3.21) The following theorem, which can be found in [1], is used to calculate the A and B matrices. Theorem 3.2 (Linearization) If f(x, u, h, V) is differentiable in a neighborhood of the stationary point x, u, h and V it is possible to approximate equation (3.2) with ż = Az + Bv + g(z, v) (3.22) where z = x x, v = u u and g(z,v) z + v matrices A and B are given by and A = B = f(x, u) x when z + v. The two (3.23) x=x,u=u,h=h,v =V f(x, u) u (3.24) u=u,x=x,h=h,v =V

37 3.3 The Plant 23 Proof. See Theorem 11.1 in [1]. A plant description of the missile can be obtained by choosing the C and D matrices of (3.16) such that the desired outputs are obtained. According to Theorem 3.2 the linearization is only valid in a neighborhood of the stationary point x and u of (3.2) and this point satisfies f(x, u,h,v )= (3.25) The Newton-Raphson method is a technique that makes it possible to find a numerical solution x, u, h and V to (3.25). This method is restricted to systems with n equations and n unknowns. This is not the case for (3.25), which hasfive equations and ten unknowns. The unknowns are the five states, the three input signals, the speed of the missile V and the dynamic pressure which is a function of V and the altitude h. Before it is possible to calculate the stationary points of (3.25) using the Newton-Raphson method it is thus necessary to choose five of the ten unknowns as design parameters Choosing the Design Parameters The flight envelope is divided into the speed of the missile V and the altitude h according to Section This implies that V and h are suitable design parameters. The main focus for this master thesis is to evaluate controllers for fixed altitudes and variations in speed. The angle of attack α is a gain scheduling parameter and should also be chosen as a design parameter. There are two remaining design parameters to choose. The angular velocity p around the x axis is a sensible choice as the fourth design parameter due to the fact that p d is a reference signal. Another reference signal is the acceleration in the direction of the y axis, which makes the sideslip angle β a good choice as the fifth design parameter The Newton-Raphson Method As seen in the previous section the five design parameters are chosen as The angular velocity around the x axis, p. The angle of attack, α. The sideslip angle, β. The forward speed of the missile, V. The altitude, h. This leaves a system of five equations and five unknowns which according to [7] can be solved by using the Newton-Raphson method. With the design points as fixed values (3.2) can be rewritten as ẋ = f((p,q,r,α,β ) T, u, h,v ) (3.26)

38 24 The Missile Model Define the new variable q as and let h(q) be defined as q = q r δ a δ e δ r (3.27) h(q) =f((p,q,r,α,β ) T, u, h,v ) (3.28) This implies that the following equation should be solved in order to find the stationary points to (3.25). h(q) = (3.29) The solution q that satisfies (3.29) can, according to [7], be obtained by the following steps. Define the functional matrix of h(q) as h(q) q = h 1(q) h 1(q) q 5 q h 5(q) q 1... h 5(q) q 5 Choose a starting point q and solve (3.31) by iteration. (3.3) h(q n )+ h(q n) (q n+1 q n )= (3.31) q A sufficiently good solution q is achieved when the following criteria is fulfilled h(q) = h(q) T h(q) λ (3.32) where λ is a small positive realnumber The Linear Plant Description From the linearizations of (3.15) there are two possible choices to make. A linear plant description according to (3.16) can be obtained by choosing the C and D matrices such that the output signals are suitable combinations of the states and input signals. It also possible to extend (3.21) by including the dynamics of the control surfaces and then choose suitable C and D matrices. The Basic Model The H Loop Shaping controller design is first applied on a model that does not consider the dynamics of the control surfaces. The outputs chosen for this controller design are

39 3.3 The Plant 25 The rotation around the x axis, p. The angle of attack, α. The sideslip angle, β. This choice of output signals is suitable because they are well connected to the reference signals p d, A zd and A yd. Let A and B be the matrices that are the result of a linearization procedure of (3.15) at a certain linearization point. These matrices are found in Appendix A. The D matrix is set to a zero matrix and the C matrix s chosen as C 1 = 1 1 (3.33) 1 which leads to a linearized plant description of the main dynamics of the missile according to (3.34). { ẋ1 = Ax 1 + Bu y = C 1 x 1 (3.34) The Extended Model The desired outputs p, A y and A z, which corresponds directly to the references that the autopilot receives from Guidance, cannot be achieved with a D matrix set to zero. However, if the dynamics of the control surfaces are included it is possible to choose a C matrix such that p, A y and A z can be used as outputs from the plant. This extension is necessary because these accelerations are given by A y = q ds m (C Cββ + C Cδr δ r ) A z = q ds m (C Nαα + C Nδe δ e ) (3.35) Notice that both δ e and δ r are included in (3.35), which can easily obtained from (3.2) and Definition 3.1. The dynamics of the control surfaces can be approximated as a second order system according to ω 2 δ = p 2 +2ξ ω p + ω 2 δ d (3.36) where ω is the distance from the poles to the origin and ξ is known as the relative damping. A large ω corresponds to a fast system and a ξ of at least.7 means that the system is properly damped. It is suitable to express (3.36) as a state space realization of the dynamics of the control surfaces since a better comprehension of the extended model is achieved

40 26 The Missile Model when working with state space realizations. Thus, with some manipulation of (3.36) the following expression is obtained (3.37). p(pδ) = 2ξ ω (pδ) ω 2 δ + ω 2 δ d (3.37) With x 1 = δ and x 2 = pδ the following state space realization of the dynamics of one control surface is obtained [ ] [ ][ ] [ ] ẋ1 1 x1 = ẋ 2 ω 2 + 2ξ ω x 2 ω 2 δ d δ = [ 1 ] [ ] (3.38) x 1 x 2 An extension of (3.38) gives the complete state space description of the dynamics of the control surfaces. Define x r and u r as δ a δ a x r = δ e δ e (3.39) δ r δ r and u r = δ ad δ ed (3.4) δ rd The following state space realization with six states, three inputs and three outputs describes the dynamics of the control surfaces { ẋr = A r x r + B r u r u = C r x r (3.41) where A r = 1 ω 2 2ξ ω 1 ω 2 2ξ ω 1 ω 2 2ξ ω B r = ω 2 ω 2 ω 2

41 3.3 The Plant 27 and C r = The autopilot receives, as mentioned earlier, a reference signal r in terms of desired p, A y and A z. The purpose of including the dynamics of the control surfaces is to be able to choose the outputs as The angular velocity around the x axis, p. The acceleration in the direction of the y axis, A y. The acceleration in the direction of the z axis, A z. Let the A and B matrices be a linearization according to (3.21) of the main dynamics of the missile given by (3.15). Hence, the extended plant description G 2 (s) is given by the following expression [ ] [ ] A BCr ẋ 2 = x A 2 + u r r Br (3.42) y = C 2 x 2 where [ x x 2 = xr ] and where the C 2 matrix defined to produce the desired outputs is given by 1 C 2 = q dsc Cβ m q dsc Cδr m q dsc Nα m q dsc Nδe m