Sliding mode control in a two-axis gimbal system

This paper considers the performance of a two-axis gimbal system, which is typical of surface-to-air, air-to-air, and air-to-surface tactical missiles, when Sliding Mode Control is utilized. A representative gimbal system consisting of a flat planar antenna, pitch/yaw gimbals, connecting push rods, and separate armature controlled DC drive motors for the pitch and yaw gimbals is utilized to develop the theory and to serve as a testbed for digital simulation. The equations describing the gimbal system under consideration are highly nonlinear and include terms which are not accurately known or can not be accurately measured during the missile's flight. It is shown that the use of variable structure control or sliding mode control can be used to provide robust seeker performance in the face of nonlinearities and uncertainties.


INTRODUCTION
The majority of contemporary tactical missile systems utilize a gimbaled seeker. The control technique used for the gimbal system on a tactical missile must provide rapid and accurate tracking of boresight error signals generated by the missile's signal processing unit. Demands on the control system are most severe during the endgame portion of the missile's flight. Poor performance during endgame will result in large miss distances and a corresponding low probability of mission success.
The equations describing the gimbal system under consideration are highly nonlinear and include terms which are not accurately known or can not be accurately measured during the missile's flight. It is shown that the use of variable structure control or sliding mode control can be used to provide robust seeker performance in the face of non-linearities and uncertainties [I 2 1 .

BACKGROUND
The "Antenna Servo" unit positions the antenna in response to control signals, which may be provided either by 1) search scan circuiny in the indicator, or by 2) the angle tracking system [3]. A separate servo channel is provided for each gimbal. The voltage obtained from a transducer on the gimbal is subtracted from the control signal, thereby producing an error signal proportional to the error in the antenna's position. Classical control theory has been used for antenna control with system response improved by the following basic categories using frquency domain techniques [4]: series (cascade) compensation, feedback (parallel) compensation, and load compensation. Contemporary control theory has been applied to stabilization of a three-axis inertial platform in [SI. processing by the missile's signal processing unit, a boresight error signal is produced. It is the job of the gimbal control unit to drive this error to zero. Depending on the guidance law used by the missile (pursuit, proportional navigation, etc.) the orientation of the missile in inertial space will also be changed. However, in this paper the closed loop perfonnance of the missile is not evaluated.

PROBLEM FORMULATION
The overall objective of the gimbal control system is to follow a desired trajectory developed in real-time as rapidly as possible with minimum steady state error. A side view of the gimbal system is shown in Figure l shows rotation about the Y, axis, which is referred to as the pitch channel. The antenna plate is rotated about the YG axis by an armature controlled DC motor through a push rod assembly. The length 1 in Figure 10     A block diagram for an armature controlled DC motor is given in Figure 2(a). For our investigation we will assume that the "elecnical time constant" L/R is much smaller than the "mechanical time constant" .IS,,,. This is a reasonable assumption for many electro-mechanical systems and leads to a reduced order model as shown in Figure 2 where 1 -is the length given in Figure 1 g,is the radius of the motor gear meshing with the push rod.
Since the pitch and yaw channels are uncoupled and identical, only the pitch channel wiU be considered in the remainder of this paper. Equations for the yaw channel can be derived in a similar manner as those of the pitch channel. Subscript symbols for x and y have been dropped in the following derivations where possible.
Substituting eq (4) in eq (5) gives To transform eq (7) so that it is only a function of Og, it is necessary to find the first and second derivatives of eq (6) in terms of Bp and use them in eq (7). The first and second derivatives of eq (6) are Using eq (8) for the first and second derivatives of e , , in terms of Bg, in eq (7) results in the following relationship for the combined antenna and motor dynamics in terms of @ The differential equation (9) can be transformed to a state variable description using the following relationships The use of eq (10) transforms eq (9) into the following state variable form where y is the output of interest. For the following computations let which results in the following formnlation for eq (1 1) The quantity rp0(xI.x2) is the nominal value of g'xl,xd given in eq (12). This is the value assuming all of the motor parameters, moments of inertia, and other values are known. The parameters F and #tj of eq (12)

S D I N G MANIFOLD AND CONTROL DESIGN
A sliding mode controller design consists of two steps. In the first step a sliding surface is chosen such that the closed loop system motion on this surface exhibits a desired behavior regardless of plant uncertainties and disturbances.
In the second step a control function is chosen to provide reaching of the sliding surface in k i t e time and to guarantee system motion in this surface thereafter.

Sliding Manifold Design
The system ( Assuming the sliding manifold is of the form o = 2, + c1Zl then the system on the sliding surface can be witten as The coefficient e, can be found by noting that the settling time of this first order differential equation is approximately equal to: The description of the system (13) with sliding mode control is given as is positive definite and its derivative is given by In order to provide asymptotic stability to the origin of the system in eq (18) the following format for the derivative of the candidate to the Lyapunov function is required It is well known that if the inequality in eq (21) holds m e then the sliding surface defined by eq (17) will be reached in a fmite amount of time.
The following control law can be utilized to asymptotically stabilize a=Oineq(18) ..
Where U,, is an estimate of the equivalent control function U , that makes the derivative of the candidate to the Lyapnnov function, eq (20) equal to zero. The equivalent control function U,, is obtained from eq (20) and is given by (26)

GIMBAL SYSTEM SIMULATION
The pitch axis of the gimbal system with sliding mode control is defined by eq (26) and has been simulated using an all-digital simulation.  Q ( x l , x 2 ) , b(x1,x2) and the maximum value for angular velocity xz (4 is useful for determining the maximum value of p(nl, x,) .
It is assumed that the position (0, ) and angular velocity (e, ) of the antenna plate are measured perfectly and are available for use by the sliding mode cnutroller for the simulation. A desired trajectory y used for evaluation of the developed sliding surface and control is given in Figure  3. The chosen trajectory is realistic of what a missile's seeker would encounter in nying to track a maneuvering target. At a long range from the target the boresight errors would be relatively small however, as the missile approaches the target the boresight errors will rapidly grow. The first set of simulation runs is for the nominal case, i.e., A? and Ab are zero. This is to establish a reference case to determine the degradation in system performance when disturbances are added. The state variables x, and x2 for this case are given in Figure 5. The antenna was given an initial displacement of 20 deg and no initial angnlar velocity. The switching surface (U) versus time is shown in Figure 6.
For the case under consideration, once the sliding surface ( a = 0) is reached the system stays in the sliding mode, a desirable result. The motor voltage, designated the control U in the previous section, is given in Figure 7. Once the system reaches the sliding surface at approximately 1 sec, the voltage begins high speed switching. The culmination of the simulation is shown in Figure 8, which compares the actual antenna trajectory to the desired antenna trajectory. Figure 8 shows that once the initial displacement of the antenna with respect to the desired trajectory is compensated, the actual and desired trajectories are identical when there are no disturbances present.

System with Disturbances
One of the primary features of sliding mode controllers is their ability to operate. in the presence of unknown disturbances. Figure 9 shows the A q disturbance that was introduced to the nominal case. In addition to the disturbance shown in Figure 9 a Ab disturbance was added by reducing the motor parameter K, by 10% five seconds into the simulation. Figure 10 shows that the disturbances did not effect the systems ability to reach the sliding surface or its ability to stay on the surface. Figure 11 shows the control U (motor voltage) for the system with disturbances.
A comparison of Figures 7 and 11 reveals that the disturbances do not greatly change the character of the control signal. It is clear from Figure 12 that the disturbances do not effect the gimbal systems ability to track the target The system is able to operate satisfactorily for reductions in K, up to 20%. Beyond a 20% reduction the system is unable to track the reference profile.

Smooth Control
There are situations where high speed switching of the control signal is undesirable. The chattering can be removed by introducing a boundary layer about the sliding surface [9]. In this case, the system will not stay on the sliding surface but in some neighborhood about the sliding surface. The price paid for a smooth control signal is some loss in control system robusmess. To eliminate chattering the sign(@ function in eq (26) is replaced with the sut(oJ function. The sar function is shown in Figure 13 for reference. The size of the boundary layer about the sliding surface is determined by the choice of E in Figure 13. It should be noted that lim sat, ( x ) 3 sign ( x ) s 4 For the gimbal system under consideration an E of 50msec was empirically found to produce a smooth control signal and to allow the system to perform properly. Udlizmg the sat function required an increase of /j from 15.0 d s to 30.0 m&s. Sigma versus time is shown in Figure 14 and V, versus time is shown in Figure 15 for the smoothed case with disturbances. The antenna trajectory is shown in Figure  16 and it is clear that the control system allows the reference trajectory to be accurately tracked.
It should be noted that integration step size and integration technique are important items to consider when simulating the high speed switching in a sliding mode controller. To ensure simulation stability in these examples an integration step of lmsec and a first order Euler integration technique was utilized.   Tme (sec) Figure 12 Antenna Trajectory for the Disturbance Case.

Figure 13
The suz(x) Function Used to Eliminate Chattering.

CONCLUSION
A control scheme for a two axis gimbal system based on the sliding mode conuol technique has been designed and validated through simulation. The derived sliding mode controller provides excellent tracking in the face of Uknown or ill-defined disturbances and system parameters. In addition, realization of the developed control system is relatively simple, a fundamental amibute of the sliding mode control technique. Addition of noise on the measured values Bg , Bg used for control will degrade antenna tracking performance. The extent of this degradation is dependent on the severity of the noise. Areas for future study include: refined models for the antenna moments of inertia including effects of the push-rods, incorporation of parameters for a desired motor, impact of limiting antenna acceleration, gear non-linearities, and the impact of the antenna himng the gimbal stops. This paper has explored the theoretical groundwork for implementation of a sliding mode controller in a missile gimbal system.