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Automated Controller Design for a Missile Using Convex Optimization

ABSTRACT

The focus of the present master thesis is the automation of an existing controller design for a missile using two aerodynamic actuating systems. The motivation is to evaluate more missile concepts in a shorter period of time.

The option used is trimming and linearization of a highly nonlinear missile at specfic conditions. According to these conditions, either a two-dimensional operating point grid defined by Mach number and height or three-dimensional operating point grid defined by Mach number, height and angle of attack is generated for the whole operating range of the missile. The controllers are designed at these points using convex optimization.

The convex set defines the pole placement area which is constrained by linear matrix inequalities according to the dynamic behavior of the missile at the operating point conditions. These controllers describe a validity area where the missile can be stabilized. This area consists all neighboring operating points and defines therefore the grid density which can differ at specfic regions of the operating range.

Controlling the missile to the target make sit necessary to apply gain-scheduling in order to get the manipulated variable by interpolation of adjacent operating points. During this blending of the controllers a problem called windup can occur when an actuator is saturated. This might lead to instability in worst case but can be counteracted by a model-recovery anti-windup network which guarantees stability in the presence of saturation. Thisanti-windup design is automated by an ane linear parameter dependency of the grid parameters and has the same validity area like the controllers.

The whole design was successfully developed and tested in MATLAB/Simulink on missiles using one or two aerodynamic actuating systems. The controllers have a good performance at small and high acceleration steps and the anti-windup keeps the missile stable even though the actuators are saturated. Stability and robustness of the controllers and anti-windup networks was verfied as well as an air defense maneuver where the missile starts at the ground and intercepts at high altitude was successfully simulated for different grids and missiles.

BACKGROUND THEORY FOR THE CONTROL OF MISSILES

Figure 2.1: Illustration of the body-xed frame of a missile

Figure 2.1: Illustration of the body-fixed frame of a missile

Figure 2.2: Figure 2.1 enhanced by the aerodynamic angles

Figure 2.2: Figure 2.1 enhanced by the aerodynamic angles

The control of a technical system requires a detailed mathematical copy of it. This can be realized by the related differential equations which describe its motion. Therefore, all exterior forces and moments which act on the missile are needed in order to get all equations of motion, representing the six Degrees of Freedom (DOF) behavior of the missile. The forces and moments are caused by gravity, aerodynamics and the missile’s thrust.

Figure 2.9: Transformation of a model recovery anti-windup control loop in general representation to the mismatch representation

Figure 2.9: Transformation of a model recovery anti-windup control loop in general representation to the mismatch representation

The advantage of this transformation or rather the mismatch representation is the decomposition of the anti-windup network in two well arranged subsystems. The first linear subsystem describes the unlimited control loop by equations with the output ys,ub

p-00911--automated-controller-1

AUTOMATION

Figure 3.1: Flow chart of the whole automation process

Figure 3.1: Flow chart of the whole automation process

In order to provide a better understanding of the automated controller design procedure, a step by step description of the procedure is presented below. The main tasks are also illustrated in ow chart 3.1.

Figure 3.13: Example of the meaning of the validity areas of the controllers

Figure 3.13: Example of the meaning of the validity areas of the controllers

Consider figure 3.13 where M represents the Mach number and H the height. The blue illustrated areas around controller one and two are defined by the Δ-values where only this controller provides the control input. We can see a pink area between them where the input will be interpolated by controller one and two. The affiliation parameter dM weights then how much of the control input is provided by controller one and how much by controller 2. First, the input will be interpolated along M by keeping H constant and afterwards the just determined values will be interpolated along H. For clarification see figure 3.14 where a = M and b = H.

Figure 3.17: Realization of a model recovery anti-windup network in Simulink

Figure 3.17: Realization of a model recovery anti-windup network in Simulink

The decoupled anti-windup network introduces an anti-windup compensator which would not replace existing controllers but extend them by an anti-windup network without a new design. This method ensures local stability and robust control even if the system contains poles in the open-right-half complex plane. Therefore, only local anti-windup compensation is covered.

RESULTS

Figure 4.1: Step responses of an arbitrary system

Figure 4.1: Step responses of an arbitrary system

A missile should be simulated by applying the designed controllers and anti-windup networks and by using gain-scheduling to blend the manipulated variable in order to stabilize the missile on its way to the target. It was also required to hold specific performance criteria, i.e. the magnitude of the overshoot (±15 %), settling time (where the deviation from the set value is smaller or equal 5%) and rise time (maximal 520mt) according to figure 4.1.

Figure 4.2: Derivatives as a function of Mach number. The light red marked patch illustrated the critical region

Figure 4.2: Derivatives as a function of Mach number. The light red marked patch illustrated the critical region

During the controller design and automated grid adaption, problems occurred at Mach numbers in a range of 0:6 ≤ M ≤ 2. As the EoMs have been described in section 2.1.1, the derivatives for the representation of the missile’s moments and forces have been presented. These derivatives differ a lot in this range which is due to the huge differences between the subsonic and supersonic physics of the missile. These changes are shown in figure 4.2 for some derivatives to provide an idea of the problem.

CONCLUSION AND OUTLOOK

One concept for an automated controller design was worked out and together with the background theory for the control of missiles in the course of the present master thesis presented. Applying this design to a missile which uses aerodynamic actuating systems, a good working irregular operating point grid will be determined. Either two-dimensional of Mach number and height or three-dimensional of Mach number, height and angle of attack. The missile will be trimmed and linearized over a local area at the operating points. Afterwards, dynamic output feedback controllers for the execution of lateral acceleration steps as well as state feedback controllers to keep the roll angle zero are designed.

The controllers stabilize the missile in a validity area which is big enough to stabilize the missile at adjacent grid points. Since the aerodynamics differ a lot between subsonic and supersonic, controllers to keep a defined roll rate in this “critical region” are automatically designed as well. For the stabilization of the missile in the presence of saturation of the actuators, a model-recovery anti-windup network was used and due to affine linear parameter dependency among the grid parameters automatically designed at the grid points. The whole design can then be simulated in Simulink which is linked via S-functions to the missile simulation program to get almost real conditions and feedback. The method of gain-scheduling allows the interpolation of the manipulated variable computed by neighboring operating points to keep the missile stable on its mission.

Figure 5.1: Example of a sparse grid

Figure 5.1: Example of a sparse grid

As you have seen in section 4, the controllers have a good working performance but are not optimized. To reach an optimal controller performance was not part of this master thesis but can be achieved by improving the determination of the design limits for the pole placement area. Additionally, the weighting factors on the control error and actuator dynamics for the output energy optimization can be enhanced. To improve computing time you should consider recursive grid generation and using sparse grids (see Figure 5.1).

Source: Lulea University of Technology
Authors: Christoph Auenmuller

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