Disturbance observer–based control for missile non-strapdown seeker disturbance rejection

Abstract

Non-strapdown seeker disturbances which involve counter-electromotive force, spring restoring torque, and friction torque can corrupt the seeker tracking accuracy and degrade the missile guidance performance. In this article, the effects of guidance parameters and disturbances on the guidance stability and the miss distance are analyzed. It is shown that these disturbances have a pronounced deterioration in the guidance performance for the anti-air missile under high altitude and head-on interception. A new approach for disturbance rejection using disturbance observer–based control is presented. Disturbances in seeker stabilizing loop are estimated by the observer which has synthetic inputs based on the available seeker measurement signals. The estimates are fed back into the seeker inner loop to compensate the disturbances, and an additional loop is applied to improve the effect of disturbance compensation. The performance of the approach under non-nominal condition is analyzed. Simulation studies are performed to examine the effect of this control scheme. The result shows that this approach can effectively reduce the influences of corrupting disturbances and significantly improve the miss distance performance.

Introduction

In the missile homing guidance, the line-of-sight (LOS) rate is required to achieve proportional navigation. This information is often extracted by the seeker, which is mounted in the front of the missile. The optical sensors (such as infrared (IR), television, and laser) have heretofore been used to detect target. To achieve a high tracking accuracy, the sensor axis should be stabilized to point toward the target regardless of the body motion of missile. In a single-plane analysis, the relevant angles are shown in Figure 1.

Figure 1.

Angles in a single plane.

In Figure 1, $\epsilon$ is the bore-sight error (BSE) angle that is measured by the optical sensor, ${\phi }_r$ is the gimbal angle obtained by the angle sensor, $\vartheta$ is the missile body angle, $q_s$ is the angle between the optical sensor axis $x_s$ and inertial reference axis x, and $q_t$ is the line of sight angle, respectively.

With different target tracking stabilization architectures, seekers are mainly divided into three types: non-strapdown seeker, semi-strapdown seeker, and fully strapdown seeker. In a non-strapdown seeker, the optical sensor is suspended in a two-axis gimbal system, the measurement ${\stackrel{·}{q}}_s$ of the rate gyro, which is mounted on the gimbals and is used for tracking loop stabilization. The semi-strapdown seeker removes the rate gyro from the gimbals to satisfy the limits of the shape and volume. The output $\stackrel{·}{\vartheta }$ of the gyro located on the missile body is combined with the measured gimbal angle ${\phi }_r$ to stabilize the seeker tracking loop. The fully strapdown seeker straps the optical sensor on the missile body directly without using the gimbals, and the LOS rate ${\stackrel{·}{q}}_t$ is estimated based on measurements of the body angular rate $\stackrel{·}{\vartheta }$ and the BSE angle $\epsilon$ .

This article focuses on the disturbance rejection technique based on the non-strapdown seeker. The non-strapdown seeker contains two feedback loops: tracking loop and stabilizing loop. Major functions of the tracking loop are to maintain the BSE angle $\epsilon$ near zero and provide the desired LOS rate command to the stabilizing loop. The stabilizing loop is used to reduce the disturbances to stabilize the seeker despite base motion with high bandwidth. Traditionally, the LOS rate ${\stackrel{·}{q}}_t$ is directly approximated by the gyro measurement ${\stackrel{·}{q}}_s$ . With the advanced signal processing technique, an optimal estimator combines measurements of the BSE angle $\epsilon$ , rate gyro ${\stackrel{·}{q}}_s$ , and gimbal angle ${\phi }_r$ to obtain a much more accurate estimate of LOS rate. With the estimator, the guidance sensitivity to seeker disturbances can be greatly reduced depending on the quality of the measurements. To simplify the analysis, we assume that the LOS rate estimate is taken directly from the seeker gyro measurement in this article.

The seeker output is corrupted by several disturbances during the flight, such as random error, noise, counter-electromotive force (CEMF), spring restoring torque (SRT), and friction torque (FT). Among these factors, CEMF, SRT, and FT introduce base motion into the seeker control system, corrupting seeker output accuracy and degrading missile precision.

The disturbance effect has been focused on for several years. The influence of radar radome slope error was first proposed by Nesline and Zarchan.^1,2 Furthermore, the effect of the parasitic loop of semi-strapdown seeker induced by the CEMF, the friction coefficient, and the sensor dynamic mismatch on the stability and miss distance was analyzed by Song Jianmei. ³ In the study by Taylor and Price, ⁴ it was shown that the gimbal seeker tracking accuracy was quite sensitive to disturbance torque inputs. The miss distance was also given with the existence of seeker disturbance torque under nominal values of guidance parameters. ⁵ In guidance system design, guidance parameters are normally determined without seeker disturbances. However, the seeker disturbance effect may be amplified with unsuitable values of guidance parameters. Therefore, the relationship between the missile guidance performance and main guidance parameters should be provided with the existence of gimbal seeker disturbances.

In practice, there are two ways to improve tracking accuracy in the face of seeker disturbances: the first is to increase the stabilizing loop gain ⁶ and the other is to insert a compensator in the stabilizing loop. However, the stabilizing loop bandwidth is limited by hardware such as servo motor and rate gyro and also the seeker would become sensitive to sensor noise with these two methods. A seeker error-based loop decoupling filter was discussed by RN Bhattacharjee et al., ⁷ and this filter could significantly reduce seeker disturbances, but the seeker performance was decreased more or less because of high-frequency glitches. Chun-Liang Lin ⁵ developed a feedforward control employing a multilayer neural network to eliminate the seeker disturbance torque.

Another approach is provided to handle this disturbance problem using disturbance observer–based control (DOBC). DOBC has been developed and applied in engineering in the recent years.^8,9 In the DOBC approach, the disturbance is estimated by the disturbance observer and then is used to compensate the disturbance effects by feedback or feedforward control. It can greatly reduce the disturbances without significantly changing the nominal performance of the existing system. Smita Sadhu ¹⁰ extended this method for LOS rate estimation by treating the input signal of the seeker as a disturbance. Although this extension simplified the DOBC structure and did not require any additional sensor, the estimated output is degraded by mismatch dynamics of the mathematic model with the physical system, which is evaluated in simulation.

This article focuses on the effect of seeker disturbances (i.e. CEMF, SRT and FT) on missile guidance performance and then demonstrates a new control architecture that employs DOBC compensation to eliminate these disturbances. The disturbance estimates are extracted from available seeker measurement signals and then applied to compensate the disturbance effects. The different architecture from the nominal DOBC is that an additional loop (AL) is added inside of the seeker stabilizing loop to improve the compensation effect. The compensation effect is evaluated by the stability and miss distance analyses. The sensitivity of the performance of DOBC to seeker dynamic uncertainties is analyzed. Simulation results show that the seeker tracking quality and the guidance accuracy are significantly improved by the DOBC compensation approach.

Mathematical model of missile guidance system with gimbal seeker

The simplified missile lateral guided model is shown in Figure 1. Definitions and nominal values of the model parameters are listed in Table 1. In Figure 2, the BSE $\epsilon$ is detected by the tracking sensor. The desired stabilizing loop command is obtained by multiplying the sensor output with the tracking loop gain $K_1$ . The motor control torque $T_M$ is generated proportional to the stabilizing loop command minus the gyro output, and this torque drives the motor to derive the BSE to error. The gyro output ${\stackrel{·}{q}}_s$ is used both as the stabilizing loop feedback and the seeker output. The missile guided model is established under the following assumption.

Table 1.

Nominal values of guidance parameters.

Definition	Nominal value
Tracking loop gain, $K_1$	12
Stabilizing loop gain, $K_2$	12
Motor resistance, R	8Ω
Motor torque constant, $K_T$	0.2344 N m/A
Motor moment of inertia, J	0.0020995 kg m2
Guidance gain, N	4
Guidance time constant, $T_g$	0.3 s
Closing velocity, $V_c$	600 m/s
Missile velocity, $V_m$	300 m/s
Incidence lag, $T_{\alpha }$	0.9 s
Guidance time duration, $T_F$	5 s
Glint noise time constant, $T_{\mathrm{GN}}$	0.15 s

Figure 2.

Missile guidance and control system.

Assumption 1

The guidance filter is a simple first-order dynamic system with the time constant $T_g/4$ ; the autopilot is a first-order cubed transfer function with the same time constant $T_g/4$ .

Seeker glint noise $u_{\mathrm{GN}}$ of bandwidth $1/T_{\mathrm{GN}}$ and standard deviation ${\sigma }_{\mathrm{GN}}$ is approximated by passing white noise $u_w$ of power spectral density (PSD) ${\Phi }_{\mathrm{GN}}$ through a first-order lag of time constant $T_{\mathrm{GN}}$ where

$${\Phi }_{\mathrm{GN}}=2T_{\mathrm{GN}}{\sigma }_{\mathrm{GN}}^2$$

(1)

In Figure 2, $\widehat{\stackrel{·}{q}}$ is the seeker output, $E_a$ is the counter-electromotive, $T_M$ is the drive torque, $T_D$ is the disturbance torque, $a_c$ is the acceleration command, $a_m$ is the missile acceleration, and other variables are defined in Table 1.

CEMF

CEMF $E_a$ is the voltage caused by the magnetic induction line cutting motion of the servo motor. It has the opposite polarity with the motor drive voltage, and its value is proportional to the gimbal rotation rate. The simplified CEMF model is

$$E_a=K_E{\stackrel{·}{\phi }}_r$$

(2)

where $K_E$ is the CEMF coefficient with unit $\mathrm{V}\mathrm{s}/\mathrm{rad}$ and ${\stackrel{·}{\phi }}_r$ is the gimbal rotation rate with unit $\mathrm{rad}/\mathrm{s}$ .

SRT

SRT $T_s$ is generated by the wire constraint during the gimbal motion, and it is described as follows ⁵

$$T_s=K_{s,N}{\left(\frac{{\phi }_r}{{\phi }_{r\lim }}\right)}^{2n+1}$$

(3)

where $K_{s,N}$ is the nonlinear spring coefficient with unit $\mathrm{N\; m}$ , ${\phi }_r$ is the gimbal angle with unit rad, ${\phi }_{r\lim }$ is the threshold value of gimbal angle with the nominal value 0.698 rad, n is a nonnegative integer, and different values represent different types of SRT.

FT

The slip-stick FT $T_f$ is approximated as ¹¹

$$T_f=K_{f,N}f\left({\stackrel{·}{\phi }}_r\right)$$

(4)

where $K_{f,N}$ is the nonlinear friction coefficient with unit $\mathrm{N\; m\; s}/\mathrm{rad}$ . Different selections of ${\stackrel{·}{\phi }}_{r1}$ and ${\stackrel{·}{\phi }}_{r2}$ represent various types of FTs, with unit $\mathrm{rad}/\mathrm{s}$ (Figure 3(b)).

Figure 3.

Nonlinear disturbance torque model: (a) spring restoring torque and (b) friction torque.

Effect of seeker disturbances on missile performance

The SRT and FT are physically nonlinear, and for studying the effect of these disturbances on guidance performance, the disturbance torque can be simplified to the linear model, as shown in Figure 4.

Figure 4.

Linearized disturbance torque model.

In Figure 4, $K_s$ and $K_f$ are linearized coefficients of SRT and FT, with units $\mathrm{N\; m}/\mathrm{rad}$ and $\mathrm{N\; m\; s}/\mathrm{rad}$ , respectively.

The superposition principle is satisfied with the linearized disturbance models; therefore, the seeker output ${\stackrel{·}{q}}_s$ can be separated into two parts: the ideal seeker output $\stackrel{·}{q}$ according to the LOS rate ${\stackrel{·}{q}}_t$ and the bias part $\Delta {\stackrel{·}{q}}_s$ due to those disturbances. The equivalent missile guidance model without noise is shown in Figure 5. With the existence of disturbances, a parasitic loop is generated inside missile guided system.

Figure 5.

Equivalent guidance model.

Stability analysis of parasitic loop

The transfer function of the parasitic loop with LOS rate ${\stackrel{·}{q}}_t$ input and missile acceleration $a_m$ output is

$$\frac{a_m}{{\stackrel{·}{q}}_t}=\frac{N{V}_c}{{(\frac{T_g}{4}s+1)}^4-\frac{N{V}_c}{V_m}(T_{\alpha }s+1)\left(\frac{\Delta {\stackrel{·}{q}}_s}{\stackrel{·}{\vartheta }}\right)}$$

(5)

where $\Delta {\stackrel{·}{q}}_s/\stackrel{·}{\vartheta }$ is the transfer function of missile body rotational rate $\stackrel{·}{\vartheta }$ input and the bias of seeker output $\Delta {\stackrel{·}{q}}_s$ induced by CEMF, SRT, and FT

$$\frac{\Delta {\stackrel{·}{q}}_s}{\stackrel{·}{\vartheta }}=\frac{(K_E{K}_T+R{K}_f)s+R{K}_s}{\mathrm{RJ}{s}^2+(K_T{K}_2+K_{f}R+K_E{K}_T)s+R{K}_s+K_T{K}_1{K}_2}$$

(6)

The denominator of the equation is a sixth-order polynomial

$$\begin{array}{l}{D}_{PL}=\left[RJ{s}^2+\left(K_T{K}_2+K_{f}R+K_E{K}_T\right)s+R{K}_s+K_T{K}_1{K}_2\right]\\ {\left(\frac{T_g}{4}s+1\right)}^4-\frac{N{V}_c}{V_m}\left(T_{\alpha }s+1\right)\left[\left(K_E{K}_T+R{K}_f\right)s+R{K}_s\right]\end{array}$$

(7)

The stability regions of the parasitic loop are analyzed under assumption 2.

Assumption 2

The distance range between the missile and target is infinite, such that the time-to-go $(T_F-t)$ is big enough to treat the outer guidance loop open.

For $T_g=0.3$ , the stability regions of the parasitic loop can be obtained via Routh Criterion, shown in Figure 6. The lines in Figure 6 represent boundaries of the stability regions. It can be seen that the stability regions will become smaller by (1) larger disturbance coefficients ( $K_E$ , $K_s$ , and $K_f$ ) and (2) increasing $T_{\alpha }$ , N, and $V_c/V_m$ . The results also show that the missile guidance system may even become unstable in the cases of high altitude (large $T_{\alpha }$ ), head-on interception (large $V_c/V_m$ ), and large guidance gain (large N).

Figure 6.

Stability regions of the parasitic loop with disturbances: (a) CEMF, (b) SRT, and (c) FT.

Miss distance analysis

The miss distance is analyzed using adjoint method, ¹² and the adjoint model of guidance system is shown in Figure 7.

Figure 7.

Adjoint model of missile guidance system.

The ratio of miss distance $y_{\mathrm{GN}}$ and the square root of the PSD ${\Phi }_{\mathrm{GN}}$ (RMDTSRPSD) is

$$\mathrm{RMDTSRPSD}=\frac{y_{\mathrm{GN}}}{{\Phi }_{\mathrm{GN}}^{1/2}}={\left[\underset{0}{\overset{T}{\int }}{\hat{y}}_{\mathrm{GN}}^{2}d\tau \right]}^{1/2}$$

(8)

where $y_{\mathrm{GN}}$ is the miss distance with glint noise input and T is the rest guidance time.

Values of the parameters are shown in Table 1. Miss distances due to the glint noise are shown in Figure 8. The results show that the seeker disturbances would degrade the guidance accuracy. When the guidance system is stable, 5-s guidance time is sufficiently large to guarantee the miss distance reaching its asymptotic value.

Figure 8.

Miss distance varying with guidance time: (a) CEMF, (b) SRT, and (c) FT.

Assuming that T_g = 0.3 s and T_GN = 0.15 s, the effect of the guidance parameters and the disturbances on miss distances for 5 s guidance time T_F is given in Figure 11(a)–(c) (blue surface). It can be seen that the miss distance becomes larger with the increase in disturbance coefficients ( $K_E$ , $K_s$ , $K_f$ ), $V_c/V_m$ , and $T_{\alpha }$ . This result coincides with the conclusion of stability analysis that the guidance performance will seriously deteriorate especially in the case of head-on interception and large guidance gain.

Disturbance compensation with DOBC

As described previously, seeker disturbances seriously degrade the guidance performance. Compensation must be included to enhance the seeker tracking accuracy.

CEMF, SRT, and FT disturbances can be integrated into an equivalent disturbance torque d

$$d=\frac{K_T}{R}{E}_a+T_s+T_f$$

(9)

Typically, the DOBC generates an estimate of the disturbance $\hat{d}$ and then uses it to compensate the disturbance effects, shown in Figure 9(a). P is the actual physical system, $P_n$ is the nominal dynamic model of the physical system, G is a low-pass filter which is introduced to make the $P_n^{-1}G$ physical realizable, and r and y are input and output signals, respectively. In this article, G is chosen as

$$G=\frac{1}{T_{\mathrm{LF}}s+1}$$

(10)

Figure 9.

Disturbance compensation scheme: (a) standard configuration and (b) seeker disturbance compensation based on DOBC.

It is easy to deduce the transfer function from disturbance d input to y output

$$\frac{y}{d}=\frac{P(1-G)}{(1-G)+\mathrm{GP}/P_n}$$

(11)

The transfer function from input r to y output is

$$\frac{y}{r}=\frac{P}{(1-G)+\mathrm{GP}/P_n}$$

(12)

Two cases— $P/P_n=1$ and $P/P_n=0$ —are considered:

1. If $P/P_n=1$ , equations (11) and (12) are simplified to $y/d=P(1-G)$ , and $y/r=P$ . It is clear that the disturbance d is filtered by a high-pass filter; therefore, the effect of disturbance at low and moderate frequencies is reduced. Moreover, DOBC has no effect on seeker dynamics.

2. If $P/P_n=0$ , equations (11) and (12) are equal to $y/d=P$ and $y/r=P/(1-G)$ . The forward path gain of stabilizing loop is increased at low and moderate frequencies, which also reduces the effect of the disturbance on the seeker output.

The architecture of seeker disturbance compensation based on DOBC is presented in Figure 9(b). The observed value $\hat{d}$ is estimated and fed back to the inner stabilizing loop, and the disturbance is further reduced by applying an AL (red dash line). In Figure 9(b), $y_1$ is the stabilizing loop error signal and $y_2$ is the stabilizing loop error signal after DOBC. $G_{s1}$ agglomerates the transfer function from $y_2$ to the motor driving torque $T_m$ , and $G_{s2}$ is the motor loading transfer function. $G_{s1n}$ and $G_{s2n}$ are the nominal values of $G_{s1}$ and $G_{s2}$ , respectively

$$G_{s1n}=K_2{K}_T/R,G_{s2n}=1/\left(\mathrm{Js}\right)$$

(13)

Without AL (red line) in Figure 8(b), the transfer function $G_{d1}$ of the seeker output ${\stackrel{·}{q}}_s$ to the disturbance d input is given by

$$G_{d1}=\frac{G_{s2}(1-G)}{(1-G)+G_{s1}{G}_{s2}[(\frac{K_1}{s}+1)+\frac{G}{G_{s1n}{G}_{s2n}}]}$$

(14)

It can be seen that if the bandwidth of the low-pass filter is infinite $(G=1)$ , ${G_{d1}|}_{G=1}=0$ , the disturbance would be totally eliminated.

The transfer function $G_{d2}={\stackrel{·}{q}}_s/d$ with AL is deduced

$$G_{d2}=\frac{G_{s2}(1-G)}{(1-G)+G_{s1}{G}_{s2}\left[\begin{array}{c}(\frac{K_1}{s}+1)(G_{s1n}{G}_{s2n}+1)\hfill \\ \multicolumn{1}{c}{+\frac{G_{s1n}{G}_{s2n}+G}{G_{s1n}{G}_{s2n}}}\end{array}\right]}$$

(15)

Compare $G_{d2}$ to $G_{d1}$ , as $G=1$

$$G_{d2}=\frac{G_{d1}}{G_{s1n}{G}_{s2n}+1}$$

(16)

where $G_{s1n}{G}_{s2n}=K_2{K}_T/(\mathrm{JRs})$ is the nominal open-loop transfer function of seeker stabilizing loop. Conventionally, to guarantee an enough stabilizing loop bandwidth, the open-loop gain should be sufficiently large. Therefore, the disturbance effect can be greatly reduced by AL.

The transfer function $G_c={\stackrel{·}{q}}_s/{\stackrel{·}{q}}_t$ of the LOS rate ${\stackrel{·}{q}}_t$ input to seeker output ${\stackrel{·}{q}}_s$ with AL is

$$G_c=\frac{K_1{G}_{s1}{G}_{s2}(G_{s1n}{G}_{s2n}+1)}{[1-G+G\frac{G_{s1}{G}_{s2}}{G_{s1n}{G}_{s2n}}+G_{s1}{G}_{s2}+G_{s1}{G}_{s2}(G_{s1n}{G}_{s2n}+1)+K_1{G}_{s1}{G}_{s2}(G_{s1n}{G}_{s2n}+1)]s}$$

(17)

Under the nominal condition, that is, $G_{s1}=G_{s1n}$ , $G_{s2}=G_{s2n}$ , and $G=1$ , ${\stackrel{·}{q}}_s/{\stackrel{·}{q}}_t$ can be simplified to the nominal value

$$G_{c,n}=\frac{K_1{G}_{s1}{G}_{s2}}{[1+G_{s1}{G}_{s2}]s+K_1{G}_{s1}{G}_{s2}}$$

(18)

The above equation is equal to the ideal seeker dynamic function. It reflects that the disturbance compensation method does not change the seeker dynamics significantly. The only to-be-designed parameter is the time constant $T_{\mathrm{LF}}$ of the low-pass filter. The choice of the time constant $T_{\mathrm{LF}}$ should balance between the reduced disturbance and larger measurement noise effect. The higher open-loop gain of the stabilizing loop will also reduce the disturbance, but the gain is also limited by physical system (the servo motor and gyro bandwidth). By choosing $T_{\mathrm{LF}}=0.01s$ , T_g = 0.3 s, the comparison of the parasitic loop stability regions with DOBC and without DOBC under the head-on interception $(V_c/V_m=2)$ is presented in Figure 10. It is clear that the stability of the parasitic loop is significantly improved with the $\hat{d}$ compensation based on DOBC, and the stability region is further increased by applying A. The value of $T_{\alpha }$ generally falls within the range (0.1–5 s), and the stability analysis result shows that the compensation approach can guarantee the control system working within the stability region at a far target distance.

Figure 10.

Comparison of parasitic loop stability regions with and without DOBC compensation: (a) CEMF, (b) SRT, and (c) FT.

For T_g = 0.3 s, T_GN = 0.15 s, and T_F = 5 s, the improvement of missile guidance precisions with the DOBC compensation is presented in Figure 11. The guidance system is excited with glint noise $u_{\mathrm{GN}}$ . The guidance system with the DOBC compensation yields a remarkable decrease in the miss distance, and the guidance precision remains substantially unaltered with the increase in disturbance.

Figure 11.

Comparison of miss distances with and without DOBC: (a) CEMF, (b) SRT, and (c) FT.

Sensitivity analyses under non-nominal conditions

In practical applications, the seeker model may not be known exactly. Some unmodeled dynamics, such as fast dynamics of sensors and nonlinear dynamics of gimbal, perturb the seeker response from its nominal value and then derive estimation and compensation errors in DOBC. In this section, we discuss the sensitivity of the performance of DOBC to seeker dynamic uncertainties.

The previous section indicates that DOBC is designed based on nominal values $G_{s1n}$ and $G_{s2n}$ of the stabilizing loop; hence, the performance of DOBC is only sensitive to the deviations between real dynamics $G_{s1}$ , $G_{s2}$ and their nominal values $G_{s1n}$ , $G_{s2n}$ . Without loss of generality, the following assumptions of model uncertainties are satisfied in this study.

Assumption 3 (model uncertainties)

Seeker model uncertainties are investigated by satisfying the following assumptions:

• Model uncertainties are considered to be gain perturbation and phase perturbation.

• Gain perturbation is presented by the additive uncertainty, and real dynamics $G_{s1}$ and $G_{s2}$ can be expressed by

  $$G_{s1}=G_{s1n}+\Delta G_{s1}\mathrm{and}{G}_{s2}=G_{s2n}+\Delta G_{s2}$$

  (19)

  where $\Delta G_{s1}$ and $\Delta G_{s2}$ are dynamic perturbations of $G_{s1}$ and $G_{s2}$ , respectively, and $|\Delta G_{s1}/G_{s1n}|$ and $|\Delta G_{s2}/G_{s2n}|$ are small quantities.

• The phase perturbation is presented by the multiplicative uncertainty, which is approximated by an equivalent time delay, and $G_{s1}$ and $G_{s2}$ are rewritten as

  $$G_{s1}=G_{s1n}{e}^{-j\omega T_1}\mathrm{and}{G}_{s2}=G_{s2n}{e}^{-j\omega T_2}$$

  (20)

where $T_1$ and $T_2$ are equivalent time delays of $G_{s1}$ and $G_{s2}$ , respectively. Generally, $T_1$ and $T_2$ are small values, so that the phase perturbation $\omega (T_1+T_2)$ within the gain crossover frequency ${\omega }_s$ of stabilizing loop is less than 15° (0.25 rad).

• (d) Effects of model uncertainties are analyzed within the frequency range $0<\omega <{\omega }_g$ , with ${\omega }_g$ the gain crossover frequency of the seeker tracking loop.

Property 1 (seeker dynamics)

In an acceptable design, the seeker dynamics satisfies the following properties within the frequency range $(0,{\omega }_g)$ of interest:

• The open-loop transfer function of the stabilizing loop $\left|G_{\mathrm{SL}}\right|=\left|G_{s1}{G}_{s2}\right|$ would be a large number, and its corresponding closed-loop transfer function $|G_{\mathrm{SL}}/(G_{\mathrm{SL}}+1)|$ is nearly unity.

• The amplitude of the seeker (or tracking loop) closed-loop transfer function $\left|G_{c,n}\right|$ in equation (18) is close to unity, hence the amplitude of its open-loop transfer function $|(K_1/s)G_{\mathrm{SL}}/(G_{\mathrm{SL}}+1)|\ge 1$ .

• The gain crossover frequency ${\omega }_s$ of the stabilizing loop is much higher than ${\omega }_g$ , a reliable selection is ${\omega }_s~5{\omega }_g$ .

The sensitivity relations of the performance of DOBC to dynamic deviations are presented by

$${\stackrel{·}{q}}_s/{\stackrel{·}{q}}_t=G_{c,n}(1-M_{c,x})\mathrm{and}{\stackrel{·}{q}}_s/d=G_{d,n}(1-M_{d,x})$$

(21)

where $M_{c,x}$ and $M_{d,x}$ are sensitivity terms in the transfer functions ${\stackrel{·}{q}}_s/{\stackrel{·}{q}}_t$ and ${\stackrel{·}{q}}_s/d$ introduced by the specific perturbation x, respectively; for additive perturbation $x=\mathrm{add}$ and for multiplicative perturbation $x=\mathrm{multi}$ denotes the specific perturbation; $G_{c,n}$ is the nominal value of ${\stackrel{·}{q}}_s/{\stackrel{·}{q}}_t$ given in equation (18); and $G_{d,n}$ is nominal value of ${\stackrel{·}{q}}_s/d$ and has the following expression

$$G_{d,n}=\frac{G_{s2n}\left(1-G\right)}{\left(1-G\right)+G_{s1n}{G}_{s2n}\left[\left(\frac{K_1}{s}+1\right)\left(G_{s1n}{G}_{s2n}+1\right)+\frac{G_{s1n}{G}_{s2n}+G}{G_{s1n}{G}_{s2n}}\right]}$$

(22)

It is seen that smaller values of $\left|M_{c,x}\right|$ and $\left|M_{d,x}\right|$ denote the closer performance of DOBC under the non-nominal condition to the nominal. $\left|M_{c,x}\right|$ and $\left|M_{d,x}\right|$ for additive and multiplicative perturbations within the frequency range $0<\omega <{\omega }_g$ are evaluated.

Additive uncertainty

Effect of additive uncertainty on ${\stackrel{·}{q}}_s/{\stackrel{·}{q}}_t$

Using equation (19), the true open-loop dynamic $G_{\mathrm{SL}}$ of stabilizing loop is

$$G_{\mathrm{SL}}=G_{s1}{G}_{s2}=G_{s1n}{G}_{s2n}+\Delta G_{\mathrm{SL}}=G_{\mathrm{SLn}}+\Delta G_{\mathrm{SL}}$$

(23)

where $G_{\mathrm{SLn}}=G_{s1n}{G}_{s2n}$ is the nominal value of $G_{\mathrm{SL}}$ ; $\Delta G_{\mathrm{SL}}$ is the perturb dynamic of stabilizing loop with $\Delta G_{\mathrm{SL}}=G_{s1n}\Delta G_{s2}+G_{s2n}\Delta G_{s1}+\Delta G_{s1}\Delta G_{s2}$ .

Inserting equation (23) into equation (17) gives

$$\begin{array}{l}{\stackrel{.}{q}}_s/{\stackrel{.}{q}}_t\approx \frac{K_1{G}_{SL}}{\left[\frac{\Delta G_{SL}}{G_{SL}-\Delta G_{SL}}+1+G_{SL}\right]s+K_1{G}_{SL}}\\ =G_{c,n}\left(1-\frac{\Delta G_{SL}s}{\left[\left(1+G_{SL}\right)s+K_1{G}_{SL}\right]\left(G_{SL}-\Delta G_{SL}\right)}\right)\end{array}$$

(24)

$$M_{c,\mathrm{add}}\approx \frac{\Delta G_{\mathrm{SL}}s}{[(1+G_{\mathrm{SL}})s+K_1{G}_{\mathrm{SL}}](G_{\mathrm{SL}}-\Delta G_{\mathrm{SL}})}$$

(25)

Note that with Property 1(b), it is easy to prove that $|(1+G_{\mathrm{SL}})s+K_1{G}_{\mathrm{SL}}|\approx \left|K_1{G}_{\mathrm{SL}}\right|$ . Combining properties 1(a) and 1(b) gives $|K_1/\omega |\ge 1$ . Equation (25) becomes

$$\left|M_{c,\mathrm{add}}\right|\approx \left|\frac{\Delta G_{\mathrm{SL}}\omega }{K_1{G}_{\mathrm{SL}}^2}\right|\le \left|\frac{\Delta G_{\mathrm{SL}}}{G_{\mathrm{SL}}^2}\right|$$

(26)

$\left|M_{c,\mathrm{add}}\right|$ is expected to be a very small quantity, as $\left|G_{\mathrm{SL}}\right|$ would be a large number.

Effect of additive uncertainty on ${\stackrel{·}{q}}_s/d$

Inserting equation (19) into equation (15) gives

$$\begin{array}{l}{\stackrel{.}{q}}_s/d=\frac{\left(1-G\right)}{\frac{\left(1-G\right)}{G_{s2n}+\Delta G_{s2}}+\left(G_{s1}+\Delta G_{s1}\right)\left[\left(\frac{K_1}{s}+1\right)\left(G_{s1n}{G}_{s2n}+1\right)+1+\frac{G}{G_{s1n}{G}_{s2n}}\right]}\\ \approx G_{d,n}\left(1-\frac{\Delta G_{s1}}{G_{s1n}}\right)\end{array}$$

(27)

$$M_{d,\mathrm{add}}\approx \frac{\Delta G_{s1}}{G_{s1n}}$$

(28)

Under Assumption 3(b), $\left|M_{d,\mathrm{add}}\right|\approx |\Delta G_{s1}/G_{s1n}|$ is a small quantity. A practical simulation is shown in Figure 12(b).

Figure 12.

Seeker output with DOB compensation: (a) comparison of seeker outputs with and without DOBC, (b) comparison of DOB compensation and ELOSDOB with 20% mismatch in seekers dynamic, (c) comparison of DOB compensation and ELOSDOB with delay in seeker dynamics, (d) seeker output induced by disturbance input with different time constant of low-pass filter, and (e) seeker output with measurement noise.

Multiplicative uncertainty

From equation (20), the true dynamics $G_{s1}$ and $G_{s2}$ are

$$G_{s1}=G_{s1n}{e}^{-j\omega T_1}\approx G_{s1n}-G_{s1n}j\omega T_1$$

(29)

$$G_{s2}=G_{s2n}{e}^{-j\omega T_2}\approx G_{s2n}-G_{s2n}j\omega T_2$$

(30)

$$G_{\mathrm{SL}}=G_{\mathrm{SLn}}{e}^{-j\omega (T_1+T_2)}\approx G_{\mathrm{SLn}}-G_{\mathrm{SLn}}j\omega (T_1+T_2)$$

(31)

Effect of multiplicative uncertainty on ${\stackrel{·}{q}}_s/{\stackrel{·}{q}}_t$

Combining equations (31) and (26) gives

$$\left|M_{c,\mathrm{multi}}\right|\le \left|\frac{\omega (T_1+T_2)}{G_{\mathrm{SL}}}\right|$$

(32)

In equation (32), $\left|G_{\mathrm{SL}}\right|$ is large. Assumption 3(c) shows that $\omega (T_1+T_2)$ is a small number, typically less than 0.25 within ${\omega }_s$ . It would be much smaller than 0.25 within the frequency range of interest, as ${\omega }_s>>{\omega }_g$ . Consequently, $\left|M_{c,\mathrm{multi}}\right|$ would be small.

Effect of multiplicative uncertainty on ${\stackrel{·}{q}}_s/d$

From equation (28), we have

$$\left|M_{d,\mathrm{multi}}\right|\le \left|\omega T_1\right|$$

(33)

$\omega T_1$ is a small quantity which is much less than 0.25 within the frequency band $(0,{\omega }_g)$ . Therefore, $\left|M_{d,\mathrm{multi}}\right|$ is a small value. The analysis result is evaluated by the simulation, shown in Figure 12(c).

Case study

First, the seeker response is analyzed without considering both the outer parasitic loop and the guidance loop. Then, the parasitic loop is added into the discussion. Finally, the miss distance is simulated for the whole guidance system. Values of the parameters are shown in Table 1, and nonlinear disturbances are chosen as equations (3) and (4). Three sets of nonlinear disturbance coefficients are employed, shown in Table 2.

Table 2.

Parameters of nonlinear disturbance.

Disturbances	Parameters	First set	Second set	Third set
CEMF	$K_E(\mathrm{V\; s}/\mathrm{rad})$	$0.234$	0.468	0.468
SRT	$K_{s,N}(\mathrm{N\; m})$	0.15	0.30	0.50
	n	1	1	2
FT	$K_{f,N}(\mathrm{N\; m\; s}/\mathrm{rad})$	0.02	0.04	0.06
	${\stackrel{·}{\phi }}_{r1}(\mathrm{rad}/\mathrm{s})$	0.436	0.436	0.175
	${\stackrel{·}{\phi }}_{r2}(\mathrm{rad}/\mathrm{s})$	0.698	0.698	0.262

CEMF: counter-electromotive force; SRT: spring restoring torque; FT: friction torque.

Seeker response simulation

Without considering the outer parasitic loop and the guidance loop, the single seeker can be considered as a two-input one-output system with the LOS rate ${\stackrel{·}{q}}_t$ and base rotational rate $\stackrel{·}{\vartheta }$ input and the output ${\stackrel{·}{q}}_s$ .

Values of disturbances are chosen as the first set in Table 2, and the time constant of the low-pass filter is $T_{\mathrm{LF}}=0.01s$ . Now consider a platform rotational rate $\stackrel{·}{\vartheta }$ is 2 Hz sine wave of 30°/s amplitude. LOS rate ${\stackrel{·}{q}}_t$ is 0.2 Hz, square ware of 10°/s amplitude.

The seeker output ${\stackrel{·}{q}}_s$ with DOB compensation under nominal condition is shown in Figure 12(a). It is seen that this approach with DOB compensation can be effective in eliminating the disturbance.

The sensitivity of this approach to the deviation of nominal open-loop transfer function $G_{s1n}{G}_{s2n}$ is checked. It is assumed that $G_{s1n}{G}_{s2n}$ has ±20% deviation, which is

$$G_{s1n}{G}_{s2n}=1.2G_{s1}{G}_{s2}\mathrm{or}0.8G_{s1}{G}_{s2}$$

(34)

Smita Sadhu ¹⁰ used simplified DOB method to eliminate the effect of disturbances on the seeker outputs. In this approach, the true LOS rate ${\stackrel{·}{q}}_t$ is treated as the disturbance d, and it is estimated by the observed value $\hat{d}$ . The difference between the estimator and the standard DOBC is the observed value $\hat{d}$ and is not fed back to the control loop (red line of Figure 9(a)). Comparison of estimated LOS rate with DOB (ELOSDOB) is presented by Sadhu ¹⁰ and seeker output with DOBC is illustrated in Figure 12(b). With the deviation from the nominal case, ELOSDOB result significantly corrupts. It shows the output with DOBC is insensitive with the nominal parameters uncertainty.

The results of simulation with 20-ms time delay in stabilizing loop by two methods are shown in Figure 12(c). The similar conclusion shows an outperformance for DOBC.

According to the transfer function of the seeker output $\stackrel{·}{q}$ to the disturbance d, the bandwidth of the low-pass filter greatly influences the disturbance rejection effect. This phenomenon is shown in Figure 12(d) (without measurement noise). With the larger value of $T_{\mathrm{LF}}$ , the bigger seeker output induced by disturbance occurs. The choice of filter parameter should obtain a trade-off between the disturbance rejection and the measurement noise effect.

With zero mean, 0.3°/s standard deviation band-limited (10 Hz) white seeker gyro measurement noise, the seeker output is shown in Figure 12(e).

Parasitic loop response simulation

The analysis of the parasitic loop response is to check the seeker output and missile acceleration response to the LOS rate input with the closed disturbance feedback loop.

The guidance parameters are listed in Table 1. With 0.2 Hz, square ware of 10°/s amplitude LOS rate ${\stackrel{·}{q}}_t$ input, the seeker output ${\stackrel{·}{q}}_s$ is given in Figure 13(a), and the missile acceleration $a_m$ is shown in Figure 13(b). Without DOBC compensation, the seeker output ${\stackrel{·}{q}}_s$ diverges rapidly, resulting in a divergent acceleration response. This situation is greatly improved by DOBC compensation, and the seeker output error is effectively eliminated. The acceleration with DOBC compensation coincides with the guidance system without disturbances.

Figure 13.

Parasitic loop response with DOBC compensation: (a) comparison of seeker outputs with DOBC compensation and without compensation and (b) comparison of acceleration response with DOBC compensation and without compensation.

Miss distance simulation

The whole missile guidance system is analyzed to evaluate the effect of the DOBC compensation approach on miss distance. Three sets of nonlinear disturbances shown in Table 2 are employed.

The guidance time during $T_F$ is 5 s. The guidance system is simulated with the initial condition of 10° leading angle deviation and glint noise input. The glint noise is generated by a white noise entering a low-pass filter $1/(T_{\mathrm{GN}}s+1)$ , where $T_{\mathrm{GN}}$ is the glint noise time constant given in Table 1. The standard deviation of glint noise is 0.5 m. The PSD ${\Phi }_{\mathrm{GN}}=2T_{\mathrm{GN}}{\sigma }_{\mathrm{GN}}^2=0.075({\mathrm{m}}^2/\mathrm{Hz})$ . The seeker gyro measurement is corrupted by zero mean, 0.2°/s standard deviation band-limited (10 Hz) white seeker.

Figure 14(a) illustrates the result of the disturbance observer output $\hat{d}$ . It shows that in the case of DOB compensation control, the disturbance observer successfully generates a compensation signal $\hat{d}$ approximating the disturbance d for all disturbance sets. Figure 14(b) is the seeker output with DOB compensation. After the compensation, the most part of disturbances are reduced sufficiently, and their effect to the seeker output is eliminated. Seeker outputs with DOBC for different disturbance sets coincide with each other.

Figure 14.

Simulation result of guided system: (a) estimated disturbance by observer, (b) seeker output with DOB compensation, and (c) miss distance.

Monte-Carlo guidance simulation has been used to calculate miss distance as a function of engagement time with glint noise input. For each disturbance set shown in Table 2, the guidance time changes from 0 to 5 s with 0.1-s time step, and 1000 simulations are applied for each time step. The miss distance at each time step is obtained by averaging the 1000 simulation results. Miss distances are shown in Figure 14(c). It is obvious that miss distances are greatly reduced by the DOBC compensation approach.

Conclusion

The effect of seeker disturbances, which involve CEMF, SRT, and FT, on missile guidance stability and miss distance has been studied in this article, and the relationship between guidance performance and main guidance parameters is provided. The analysis shows that the guidance performance would be degraded by these disturbances, especially in situations of large guidance coefficient and head-on interception.

A new approach for disturbance rejection in seeker stabilizing loop using DOBC technique has been presented. This observer can estimate the disturbance accurately; with the compensation of DOBC, the disturbance is greatly reduced. Different from the normal DOBC architecture, an AL is added inside the seeker stabilizing loop to improve the compensation effect. The stability and miss distance analyses are repeated to evaluate the compensation effectivity. This scheme does not require any additional sensor. The sensitivity analyses are presented to investigate the effects of model uncertainties on the performance of proposed approach.

Simulation studies are performed to assess the performance of this compensation method. Compared to the case without compensation, the results show that this compensation controller can provide a significant improvement in seeker tracking accuracy and greatly reduce the missile miss distance.