Guidance law implementation with performance recovery using an extended high-gain observer☆
Introduction
In the terminal guidance phase of a homing missile with bearing only measurement (BOM) system, as in the case of a homing missile with a passive infrared seeker, only line-of-sight (LOS) angle signals are available for guidance. The widely used Proportional Navigation Guidance (PNG) law, as well as many other improved guidance laws based on PNG, feeds back the LOS rate signals to stabilize the LOS rate. Thus to implement the guidance laws, it is necessary to estimate the differentials of LOS angle signals. Beside the LOS angle and LOS rate signals, modern guidance laws utilize the relative range, relative range rate, and target maneuver signals to cancel out the nonlinearities and compensate for the target maneuvers. Conventional implementation methods include observers and digital filters, both of which require the observability of the signals to be estimated [1], [2], [3].
Difficulties arise from the lack of observability of the signals to be estimated. Simple analysis shows that if no additional measurement is available, different ranges and target maneuvers are indistinguishable from the LOS angles [2]. A common approach to tackle this difficulty is to incorporate the target motion dynamics to the guidance model [1]. The guidance performance will be seriously deteriorated if there exists significant difference between the real target maneuver and model prediction.
In this paper, a different idea is proposed. All the signals necessary to the guidance law implementation are estimated using a high gain observer. In the relative motion equations of the missile and its target, the nonlinear terms, consisting of the signals of the LOS rates, relative range and its rate, are treated as model uncertainties, and the target maneuvers are treated as external disturbances. The model uncertainties and external disturbances can be estimated by extending the order of the observer. Using the estimates of the extended observer, the cancellation of nonlinear terms and compensation for target maneuvers in the guidance laws can be implemented. In doing so, the observability assumptions of target maneuver, relative range and relative range rate, required by other methods, are removed, and no target maneuver model is needed for the guidance law implementation.
The use of high-gain observers is a practical approach to implementation of state feedback control of nonlinear systems [4]. A high-gain observer can robustly estimate the derivatives of the output, which are used to replace the true states in state feedback control. The observer state will converge very fast, but with a peaking phenomenon. To protect the state of the system from peaking, the state feedback control is required to be globally bounded. This requirement is usually met by saturating a continuous state feedback function outside a compact region of interest. The theoretical base for the use of high-gain observers in output feedback control is the so-called separation principle. For the closed-loop system of a class of nonlinear systems with output feedback control using high-gain observers, a fairly general separation principle can be guaranteed in the sense that the performance of a globally bounded partial state feedback control can be recovered, and the performance recovery includes asymptotic stability of the origin, the region of attraction and trajectories [5]. Theoretical investigations and practical applications of high-gain observers include stabilization, nonlinear servomechanisms, adaptive control, sliding mode control, robustness to fast un-modeled dynamics, discrete-time implementation and application to induction motors [6].
Although widely discussed, literature concerning high-gain observer technique includes only the systems whose stability is with respect to all state variables. In [5], asymptotic stability of the origin is discussed. In [7], bounded-input–bounded-state stability is assumed on the zero dynamics. In other cases, similar stability assumptions, such as input-to-state stability, on zero dynamics are required. However, for the guidance problem of a homing missile against a target, only the LOS rate signals, which are only a part of the whole systemʼs state variables, are to be stabilized [8], [9], [10], [11]. In many engineering applications, consideration of stability with respect to part of the systemʼs state variables, the so-called partial stability, is also necessary [12], [13]. Other examples of partial stability can be found in many fields such as electro-magnetics, inertial navigation systems, spacecraft stabilization via flywheels, biocenology [14].
The rest of the paper is organized as follows. In Section 2, the guidance dynamics of a homing missile are discussed. Section 3 designs an extended observer to recover the performance of a state feedback control law. In Section 4, the extended high-gain observer is used to implement the guidance law of a missile in a terminal guidance phase, and simulation is conducted. We concludes the paper with Section 5.
Section snippets
Guidance dynamics of a homing missile
For a homing missile in the terminal guidance phase, the task of guidance laws is to steer the missile to intercept its designated target with a tolerable miss distance. The general three-dimensional attack geometry, in an inertial coordinates frame, is shown in Fig. 1. M and T represent the missile and the target, respectively, both of which are considered as mass points in designing guidance laws. r is the relative range between the missile and the target, and and are velocity vectors
High-gain observer design and performance recovery
The three-dimensional relative motion dynamics developed in the above section can be formulated as a general multi-input–multi-output nonlinear system in the following normal form where and are the state variables, is the control input, is a vector of unknown external disturbances, and is the measured output. The matrix A, the matrix B, and the matrix C, given by
High-gain observer design and guidance law implementation
Take and the relative motion, shown by Eqs. (1), (2), (3), writes in the form of (7), (8), (9), (10), where , , , , and Due to the existence of the effective operation range of the seeker , in the guided flight phase of the missile,
Conclusions
For a class of multi-input–multi-output nonlinear systems, extended high-gain observers are used in the output feedback control design. Motivated by the guidance of a homing missile, partial practical stabilization is considered for the class of systems. Since only a part of state variables are practically stabilized, no non-minimum phase conditions are required, unlike the common cases where stabilization of full state is considered. So systems with unstable zero dynamics can also be included
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This work was supported in part by the National Natural Science Foundation of China under grant number 61174001. The work of the second author was supported in part by the National Science Foundation under grant number ECCS-0725165.