The Remote Control of the Artillery Rocket Set as a Strongly Nonlinear System Subject to Random Loads

Abstract

On the modern battlefield, fighting capabilities, such as speed, target detection range, target identification capabilities, and shooting effectiveness, of short-range artillery rocket sets (ARSs) are constantly being improved. Problems arise when attempting to successfully fire such kits in the face of disruption from both the cannon and the moving platform on which the cannon is mounted. Furthermore, the set is a variable mass system since it can fire anywhere from a few to dozens or even hundreds of missiles in a brief period of time, implying that the ARS is a highly nonlinear system of variable parameters (non-stationary). This work shows how to control such a system. If the ARS is placed on a moving basis where there is both a system and measurement noise, the state variables must be restored, and the ARS data must be filtered. Therefore, in addition to the LQR regulator, an extended Kalman filter was used. As a consequence of this synthesis, an LQG (linear quadratic Gaussian) regulator of ARS was obtained, which was used to follow the target along the line of sight. The key goal of this paper is to develop control algorithms that will increase the performance of ARS control in elevation and azimuth, as well as the accuracy of achieving and eliminating maneuverable air targets. Moreover, through the quality criterion adopted, we hope to affect control energy costs while maintaining control precision. Graphical representations of certain computational simulation results are provided.

1. Introduction

The aim of modern artillery rocket sets (ARSs), is to capture low-flying, maneuvering air targets, not only in all weather conditions but also during the motion of the carrier on the unevenness of the surface on which such a set is mounted; this is applicable to both land and water surfaces [1,2,3,4].

The ARS described in this paper is a very short-range anti-aircraft system dedicated to the defense of important military and civilian objects, both fixed and mobile, from air attacks from up to 5 km. It has an integrated computerized system for detecting, identifying, and managing targets, which ensures high efficiency with high mobility and low cost of exploitation [5,6].

The set is equipped with a stabilized optoelectronic day–night head, which can work independently of the armament in the scope of observation, detection, and target identification. This optoelectronic head is not only an element of the guidance system but also a source of information for the entire system since the data on the detected and observed objects are exchanged throughout the chain of command. Each set is also equipped with a laser radiation warning system.

A double 23 mm caliber gun with a theoretical rate of 2000 shots per minute and an effective range of fire of up to 3 km, as well as two anti-aircraft GROM/PIORUN missiles with a range of 5.5 km, are included in the set’s armament. The set is capable of destroying not only planes and combat helicopters but also unmanned aerial vehicles and even maneuvering missiles, thanks to its increased tracking speed and precision. Lightly armored targets, on both land and water surfaces, can also be destroyed. A single battery can defend a 350-square kilometer area from an aerial attack coming from any direction.

This paper analyses the issue of the selection of such a stabilization and control system with this type of set (Figure 1) so that the search and tracking process can reliably take place in the disadvantageous conditions mentioned before. This process is required to develop a suitable mathematical model of the set in question because, in the systems of automatic self-propelled rocket control, PD or PID controllers are most commonly used. This control fails in cases in which a change in the structure or parameters of the set occurs (e.g., during ammunition storage firing or damage in combat conditions) [7,8,9]. Inevitably, during the operation of the set, there are both process and measurement noises, so it is necessary to restore state variables and filter the measurement data. Therefore, an extended Kalman filter was used to control the set, along with a modified LQG (linear quadratic Gaussian) regulator in which Jacobian was used instead of a state matrix [10,11]. It is worth noting that new, robust control methods for nonlinear systems are currently being developed and studied. Examples of works in this field include [12,13,14]. However, these require learning to be performed and are mostly quite computationally demanding. In the case of the ARS, there is a need to simultaneously filter the signals and develop real-time controls with sampling rates of 0.001 s or, often, even more frequent rates.

Although scientific research is currently being conducted into the LQR regulator in defense applications [16], it should be emphasized that, in the available literature, excluding the works of the authors of this article, the authors did not find a description of the nonlinear mathematical model or a study on a remote-controlled artillery rocket set, especially with LQR or LQG control. Therefore, in their opinion, there is a need to conduct a theoretical simulation and experimental research into such sets, which are now becoming increasingly common short-range anti-aircraft weapons on the modern battlefield. It is also important to consider the real conditions affecting the set as comprehensively as possible. The development of an LQG controller for the aforementioned set operating under random disturbances is a new topic discussed in the scientific literature.

Section 2 presents a mathematical model of the dynamics of a remotely controlled artillery rocket set under the influence of random disturbances and linearized using Jacobians.

Section 3 contains an algorithm for controlling the set using a modified LQG controller. The kinematic equations of motion of the target observation line are provided, from which the software trajectory (set) of the ARS motion is determined.

Selected results of simulation studies are provided in graphical form in Section 4. A comparative analysis of the set control using PID, LQR, and LQG regulators was performed under the conditions of the occurrence of interference from the base on which the set is placed (ARS) and during a short series of shots.

A summary of the test results and conclusions are presented in the last Section 5.

2. Mathematical Model of the Set Movement

A 3D model was designed to obtain physical parameters such as mass and moments of inertia before specifying the set’s equations of motion. Figure 2 shows the model, along with a list of the most significant elements.

A mechanical model is depicted in Figure 3. The generalized moment Q₁ is assumed to rotate the body (1) by an azimuth angle θ₁ around the axis z₁. The generalized moment Q₂ rotates the body (2) by the elevation angle θ₂ around the axis y₂ that travels with the body (1).

The nonlinear state equations describing the ARS dynamics model are based on those presented in paper [9]. Some changes have been adopted in this paper: a viscotic friction model has been assumed, and kinematic disturbances (from the ARS base movement) will be part of the disturbances of generalized torques Q₁ and Q₂.

$$\frac{d{x}_1}{dt}=x_2=g_1$$

(1)

$$\frac{d{x}_2}{dt}=\stackrel{g_2}{\overbrace{\frac{-\left(3a{x}_3^2+2b{x}_3+c\right)x_2{x}_4-{\eta }_1{x}_2}{I_1+I_a+I_s}}}+\frac{Q_1}{I_1+I_a+I_s}$$

(2)

$$\frac{d{x}_3}{dt}=x_4=g_3$$

(3)

$$\frac{d{x}_4}{dt}=\stackrel{g_4}{\overbrace{\frac{0.5\left(a{x}_3^2+2b{x}_3+c\right)x_2^2-mgr \cos \left(x_3+\gamma \right)-{\eta }_2{x}_4}{I_2}}}+\frac{Q_2}{I_2}$$

(4)

Equations (1)–(4) have been divided into two parts to facilitate the presentation of the model in linearized form (5). The first part $g_i=g_i\left(x_{\mathrm{ARS}}\right), i=1,2,3,4$ is related to the dynamics of the system. The second part, which is present in (2) and (4), is related to the control of the system. The system state vector is assumed to be $x_{\mathrm{ARS}}\left(t\right)={\left[\begin{array}{cccc}{x}_1\left(t\right)& x_2\left(t\right)& x_3\left(t\right)& x_4\left(t\right)\end{array}\right]}^{\mathrm{T}}$. The initial conditions $x_1\left(0\right), x_2\left(0\right), x_3\left(0\right), x_4\left(0\right)$ are known (estimated) after calibration of the system before the firing.

With further consideration, the time variable for state variables and matrices is omitted. The linearized time-variant ARS motion equations are represented in the following form [9,11]:

$$\left\{\begin{array}{c}\Delta {\dot{x}}_{\mathrm{ARS}}=J_{\mathrm{ARS}}\Delta x_{\mathrm{ARS}}+B_{\mathrm{ARS}}{u}_{\mathrm{ARS}}+w_{\mathrm{ARS}}\hfill \\ \mathsf{\Delta }{z}_{\mathrm{ARS}}=H_{\mathrm{ARS}}\Delta x_{\mathrm{ARS}}+v_{\mathrm{ARS} }\hfill \end{array}\right.$$

(5)

where:

$$\mathsf{\Delta }{x}_{\mathrm{ARS}}=x_{\mathrm{ARS}}-x_{\mathrm{ARS}}^{*}=\left[\begin{array}{c}{x}_1\\ x_2\\ \begin{array}{c}{x}_3\\ x_4\end{array}\end{array}\right]-\left[\begin{array}{c}{x}_1^{*}\\ x_2^{*}\\ \begin{array}{c}{x}_3^{*}\\ x_4^{*}\end{array}\end{array}\right]$$

$$B_{\mathrm{ARS}}\left(x_{\mathrm{ARS}}^{*}\right)=\left[\begin{array}{cc}0& 0\\ \frac{1}{I_1+I_a\left(x_{\mathrm{ARS}}^{*}\right)+I_s}& 0\\ 0& 0\\ 0& \frac{1}{I_2}\end{array}\right]; u_{\mathrm{ARS}}=\left[\begin{array}{c}{Q}_1\\ Q_2\end{array}\right]$$

while: $x_1={\theta }_1$ is the ARS azimuth angle. $x_2={\dot{\theta }}_1$ is the ARS azimuth angle speed. $x_3={\theta }_2$ is the ARS elevation angle. $x_4={\dot{\theta }}_2$ is the ARS elevation angle speed. $\mathsf{\Delta }{x}_{\mathrm{ARS}}$ is the vector of state variables’ deviation from desired values. $x_{\mathrm{ARS}}={\left[\begin{array}{cccc}{x}_1& x_2& x_3& x_4\end{array}\right]}^{\mathrm{T}}$ is the vector of real variables of the ARS state. $x_{\mathrm{ARS}}^{*}=\left[\begin{array}{cccc}{x}_1^{*}& x_2^{*}& x_3^{*}& x_4^{*}\end{array}\right]$ is the vector of ARS state variables at work point, i.e., the vector of the desired values of state variables. $H_{\mathrm{ARS}}=I^{4x4}$ is the measurement (output) matrix. $\mathsf{\Delta }{z}_{\mathrm{ARS}}$ is the output vector of deviation from the desired values. $w_{\mathrm{ARS}}={\left[w_1, w_2, w_3, w_4\right]}^{\mathrm{T}}$ is the ARS process noise (white Gaussian). $v_{\mathrm{ARS}}$ is the ARS measurement noise (white Gaussian). $Q_i=M_{ci}+M_{bi}+M_{si}$, $M_{ci}$ is the control moment. $M_{bi}$ is the base movement disturbance moment. $M_{si}$ is the shooting disturbance moment. $T_i={\eta }_i{\dot{\theta }}_i$ is the friction moment. ${\eta }_i$ are the coefficients of moments of friction forces acting in the ARS (${\eta }_1$ is the azimuth and ${\eta }_2$ is the elevation). $I_a=a{x}_3^3+b{x}_3^2+c{x}_3+d;$ $I_s=pn+q;$ a, b, c, d, p, q, n are the parameters of the set described in detail in the paper [8]. It should be emphasized that the Taylor series method was used to linearize state Equations (1)–(4), and this method requires a Jacobian calculation.

The matrix $J_{\mathrm{ARS}}$ (in Equation (5)) is a Jacobian of the following general form:

$$J_{\mathrm{ARS}}\left(x_{\mathrm{ARS}}\right)=\left[\begin{array}{cccc}{\left.\frac{\partial g_1}{\partial x_1}\right|}_{x_{\mathrm{ARS}}^{*}}& {\left.\frac{\partial g_1}{\partial x_2}\right|}_{x_{\mathrm{ARS}}^{*}}& {\left.\frac{\partial g_1}{\partial x_3}\right|}_{x_{\mathrm{ARS}}^{*}}& {\left.\frac{\partial g_1}{\partial x_4}\right|}_{x_{\mathrm{ARS}}^{*}}\\ {\left.\frac{\partial g_2}{\partial x_1}\right|}_{x_{\mathrm{ARS}}^{*}}& {\left.\frac{\partial g_2}{\partial x_2}\right|}_{x_{\mathrm{ARS}}^{*}}& {\left.\frac{\partial g_2}{\partial x_3}\right|}_{x_{\mathrm{ARS}}^{*}}& {\left.\frac{\partial g_2}{\partial x_4}\right|}_{x_{\mathrm{ARS}}^{*}}\\ {\left.\frac{\partial g_3}{\partial x_1}\right|}_{x_{\mathrm{ARS}}^{*}}& {\left.\frac{\partial g_3}{\partial x_2}\right|}_{x_{\mathrm{ARS}}^{*}}& {\left.\frac{\partial g_3}{\partial x_3}\right|}_{x_{\mathrm{ARS}}^{*}}& {\left.\frac{\partial g_3}{\partial x_4}\right|}_{x_{\mathrm{ARS}}^{*}}\\ {\left.\frac{\partial g_4}{\partial x_1}\right|}_{x_{\mathrm{ARS}}^{*}}& {\left.\frac{\partial g_4}{\partial x_2}\right|}_{x_{\mathrm{ARS}}^{*}}& {\left.\frac{\partial g_4}{\partial x_3}\right|}_{x_{\mathrm{ARS}}^{*}}& {\left.\frac{\partial g_4}{\partial x_4}\right|}_{x_{\mathrm{ARS}}^{*}}\end{array}\right]$$

(6)

The full form of I_a and I_s was substituted before calculating the derivatives, and the derivatives are:

$$\begin{gathered} {\left.\frac{\partial g_1}{\partial x_1}\right|}_{x_{\mathrm{ARS}}^{*}}=0; {\left.\frac{\partial g_1}{\partial x_2}\right|}_{x_{\mathrm{ARS}}^{*}}=1; {\left.\frac{\partial g_1}{\partial x_3}\right|}_{x_{\mathrm{ARS}}^{*}}=0; {\left.\frac{\partial g_1}{\partial x_4}\right|}_{x_{\mathrm{ARS}}^{*}}=0; {\left.\frac{\partial g_2}{\partial x_1}\right|}_{x_{\mathrm{ARS}}^{*}}=0; \\ \begin{array}{c}{\left.\frac{\partial g_2}{\partial x_2}\right|}_{x_{\mathrm{ARS}}^{*}}=\end{array}-\frac{\left(3a{\left(x_3\right)}^2+2b{x}_3+c\right)x_4-{\eta }_1}{h_{\mathrm{ARS}}}; \\ {\left.\frac{\partial g_2}{\partial x_3}\right|}_{x_{\mathrm{ARS}}^{*}}=-\frac{6a{x}_3+2b}{h_{\mathrm{ARS}}}{x}_2{x}_4+\frac{x_2\left(3a{\left(x_3\right)}^2+2b{x}_3+c\right)\left[x_4\left(3a{\left(x_3\right)}^2+2b{x}_3+c\right)+{\eta }_1\right]}{h_{\mathrm{ARS}}{}^2}; \\ {\left.\frac{\partial g_2}{\partial x_4}\right|}_{x_{\mathrm{ARS}}^{*}}=\frac{3a{\left(x_3\right)}^2+2b{x}_3+c}{h_{\mathrm{ARS}}}{x}_2; {\left.\frac{\partial g_3}{\partial x_1}\right|}_{x_{\mathrm{ARS}}^{*}}=0; {\left.\frac{\partial g_3}{\partial x_2}\right|}_{x_{\mathrm{ARS}}^{*}}=0; {\left.\frac{\partial g_3}{\partial x_3}\right|}_{x_{\mathrm{ARS}}^{*}} \\ =0; \\ {\left.\frac{\partial g_3}{\partial x_4}\right|}_{x_{\mathrm{ARS}}^{*}}=1; {\left.\frac{\partial g_4}{\partial x_1}\right|}_{x_{\mathrm{ARS}}^{*}}=0; {\left.\frac{\partial g_4}{\partial x_2}\right|}_{x_{\mathrm{ARS}}^{*}}=-\frac{3a{\left(x_3\right)}^2+2b{x}_3+c}{I_2}{x}_2; \\ {\left.\frac{\partial g_4}{\partial x_3}\right|}_{x_{\mathrm{ARS}}^{*}}=-\frac{a{\left(x_3\right)}^2+b}{I_2}{\left(x_2\right)}^2+\frac{mgr}{I_2}\sin \left(x_3+\gamma \right); {\left.\frac{\partial g_4}{\partial x_4}\right|}_{x_{\mathrm{ARS}}^{*}}=-\frac{{\eta }_2}{I_2}; \end{gathered}$$

where:

$$h_{\mathrm{ARS}}=I_1+np+q+a{\left(x_3\right)}^3+b{\left(x_3\right)}^2+c{x}_3+d$$

3. ARS Control Algorithm with Modified LQG Controller

Control law ${\mathbf{u}}_{\mathrm{ARS}}$ for the ARS will be defined by means of linear quadratic control method [17,18,19], with the goal function I_ARS in the form (7). The first part of the sum under the integral of the goal function relates to control accuracy. The second part relates to optimizing the energy required to control. Thus, adequate control accuracy while reducing the power required can be obtained. This is also important in order to ensure that the controller does not force the drives, i.e., hydraulic or electric motors, to operate at saturation for an extended period of time. The goal function is written for a continuous-time system with stochastic noises. However, for a discretized system discussed later in this paper, these criteria will be equivalent [20]. Moreover, there is no third term connected with state value at the terminal time in (7) because it is much more important to achieve accuracy during the entire control.

$$I_{\mathrm{ARS}}\left(\Delta x_{\mathrm{ARS}},u_{\mathrm{ARS}}\right) =E\left[\underset{0}{\overset{t_t}{{\displaystyle \int }}}\left(\Delta x_{\mathrm{ARS}}^{\mathrm{T}}{Q}_{\mathrm{ARS}}\Delta x_{\mathrm{ARS}}+u_{\mathrm{ARS}}^{\mathrm{T}}{R}_{\mathrm{ARS}}{u}_{\mathrm{ARS}}\right)dt\right]$$

(7)

where: t_t is the terminal time. E[·] is the expected value. $Q_{\mathrm{ARS}}$ and $R_{\mathrm{ARS}}$ are the weight matrices for state variables deviation and control, respectively.

We can present this law using the formula

$$u_{\mathrm{ARS}}=-K_{\mathrm{ARS}}·\left(x_{\mathrm{ARS}}-x_{\mathrm{ARS}}^{*}\right)$$

(8)

where: $K_{\mathrm{ARS}}$ is the ARS control gain matrix. $x_{\mathrm{ARS}}^{*}=\left[\begin{array}{cccc}{x}_1^{*}& x_2^{*}& x_3^{*}& x_4^{*}\end{array}\right]=\left[\begin{array}{cccc}\sigma & \dot{\sigma }& \epsilon & \dot{\epsilon }\end{array}\right]$ are the preset (desired) variables that determine the location of the line of sight (LOS). They are obtained from the following equations [14]:

$$\left\{\begin{array}{c}\frac{d{R}_{\mathrm{LOS}}}{dt}=V_{\mathrm{c}}\left[{\cos \mathsf{\chi }}_{\mathrm{c}}\cos \mathsf{\sigma }\cos \left(\epsilon -{\gamma }_c\right)+\sin {\chi }_c\sin \sigma \right]=f_1\\ \frac{d\epsilon }{dt}=-\frac{V_c\cos {\chi }_c\sin \sigma \sin \left(\epsilon -{\gamma }_c\right)}{R_{LOS}\cos \sigma }=f_2\\ \frac{d\mathsf{\sigma }}{\mathrm{dt}}=-\frac{{\mathrm{V}}_{\mathrm{c}}\left[{\cos \mathsf{\chi }}_{\mathrm{c}}\sin \mathsf{\sigma }\cos \left(\mathsf{\epsilon }-{\mathsf{\gamma }}_{\mathrm{c}}\right)-{\sin \mathsf{\chi }}_{\mathrm{c}}\cos \mathsf{\sigma }\right]}{R_{LOS}}=f_3\end{array}\right.$$

(9)

where: $\epsilon , \sigma$ are the pitch and yaw angles of the line of sight, respectively. $R_{\mathrm{LOS}}$ is the distance between the ARS and aerial target.$V_c$ are the velocities of the target. ${\chi }_c, {\gamma }_c$ are the pitch and yaw angles of the target velocity vector, respectively (Figure 4).

Gain matrix $K_{\mathrm{ARS}}$ occurring in Equation (8) is derived from the following relationship:

$$K_{\mathrm{ars}}=R_{\mathrm{ARS}}^{-1}·B_{\mathrm{ARS}}^{\mathrm{T}}·P_{\mathrm{ARS}}$$

(10)

Matrix $P_{\mathrm{ARS}}$ is a solution to the Riccati algebraic equation

$$J_{\mathrm{ARS}}^{\mathrm{T}}{P}_{\mathrm{ARS}}+P_{\mathrm{ARS}}{J}_{\mathrm{ARS}}-P_{\mathrm{ARS}}{B}_{\mathrm{ARS}}{R}_{\mathrm{ARS}}^{-1}{B}_{\mathrm{ARS}}^{\mathrm{T}}{P}_{\mathrm{ARS}}+Q_{\mathrm{ARS}}=0$$

(11)

The weight matrices $R_{\mathrm{ARS}}$ and $Q_{\mathrm{ARS}}$, occurring in Equations (10) and (11), are selected as diagonal matrices and fine-tuned experimentally while the search is starting with values equal to [17]:

$$q_{ii}=\frac{1}{2x_{i\max }},r_{jj}=\frac{1}{2u_{j\max }}$$

(12)

where:$x_{imax}$ is the maximum scope of changes of i-th value of the state variable (i = 1, 2, 3, 4). $u_{jmax}$ is the maximum scope of changes of j-th control variable value (j = 1, 2).

In the case of ARS interference in the form of process noise and measurement noise (disturbances), we will use the extended discrete Kalman filter. Therefore, the system (5) was subjected to discretization and written in the form of the difference Equations (13) and (14). Moreover, the system is now written in terms of the output variables x_ARS and z_ARS rather than the increments Δx_ARS and Δz_ARS.

$$x_{\mathrm{ARS} k}=J_{\mathrm{ARS} k-1}{x}_{{\mathrm{ARS}}_{ }k-1}+B_{\mathrm{ARS} k-1}{u}_{\mathrm{ARS} k-1}^{\mathrm{LQG}}+w_{\mathrm{ARS} k-1}$$

(13)

$$z_{\mathrm{ARS} k}=H_{\mathrm{ARS} k}{x}_{\mathrm{ARS} k}+v_{\mathrm{ARS} k}$$

(14)

where: k = 1, 2, 3, … is the discrete-time index and initial values, where k–1 = 0 are estimated from the ARS calibration. $z_{\mathrm{ARS} k}$ is the vector containing the measurement of the output of state variables ARS. $u_{\mathrm{ARS} k-1}^{\mathrm{LQG}}$ are the previous control values. $H_{\mathrm{ARS} k}=H_{\mathrm{ARS}}$ is the measurement matrix. $w_{\mathrm{ARS} k-1}$ is the discrete process noise vector (white Gaussian) of zero expected value and known covariance matrix $Q_{\mathrm{ARS}}^{\mathrm{FK}}$. $v_{\mathrm{ARS} k}$ is the discrete measurement noise vector (white Gaussian) of zero expected value and known covariance (independent of $w_{\mathrm{ARS} k-1}$), $J_{\mathrm{ARS} k-1}$ is the discrete ARS transition matrix (15) and $B_{\mathrm{ARS} k-1}$ is the discrete input matrix (16) given by [18,21]:

$$J_{\mathrm{ARS} k-1}={\mathrm{e}}^{J_{\mathrm{ARS}}\left({\widehat{x}}_{\mathrm{ARS} k-1/k-1}\right)\mathsf{\Delta }t}\approx I+J_{\mathrm{ARS}}\left({\widehat{x}}_{\mathrm{ARS} k-1/k-1}\right)\mathsf{\Delta }t$$

(15)

$$\begin{gathered} B_{\mathrm{ARS} k-1}=J_{\mathrm{ARS}}^{-1}\left({\widehat{x}}_{k-1/k-1}\right)\left(J_{\mathrm{ARS} k-1}-I\right) B_{\mathrm{ARS}}\left({\widehat{x}}_{\mathrm{ARS} k-1/k-1}\right) \\ \approx B_{\mathrm{ARS}}\left({\widehat{x}}_{\mathrm{ARS} k-1/k-1}\right)\mathsf{\Delta }t \end{gathered}$$

(16)

where: $\mathsf{\Delta }t$ is the discretization time step. For the additional numerical simulation, an approximation of the above matrices was used because, for the selected small $\mathsf{\Delta }t$, the accuracy is highly satisfactory and the calculations are much faster, as $J_{\mathrm{ARS} k-1}$ and $B_{\mathrm{ARS} k-1}$ are updated in every discrete time instance. Vector ${\widehat{x}}_{\mathrm{ARS} k-1/k-1}$ is the most recent ARS state estimate known from the Kalman filter.

The predicted ARS state at time instant k on the basis of the estimate of state and control from the previous time instant [10,22] is governed by:

$${\widehat{x}}_{\mathrm{ARS} k/k-1}=J_{\mathrm{ARS} k-1}{\widehat{x}}_{\mathrm{ARS} k-1/k-1}+B_{\mathrm{ARS} k-1}{u}_{\mathrm{ARS} k-1}^{\mathrm{LQG}}$$

(17)

$$P_{\mathrm{ARS} k/k-1}^{FK}=J_{\mathrm{ARS} k-1}{P}_{\mathrm{ARS} k-1/k-1}^{FK}{J}_{\mathrm{ARS} k-1}^{\mathrm{T}}+Q_{\mathrm{ARS}}^{FK}$$

(18)

where: ${\widehat{x}}_{\mathrm{ARS} \mathrm{k}-1/\mathrm{k}-1}$ is the previous ARS state estimate. ${\widehat{x}}_{\mathrm{ARS} \mathrm{k}/\mathrm{k}-1}$ is the assessment of ARS variables a priori (before measurement). $P_{\mathrm{ARS} k-1/k-1}^{FK}$ is the previous covariance matrix of predictive error. $P_{\mathrm{ARS} k/k-1 }^{FK}$ is the covariance matrix of predictive error before the measurement for ARS. $Q_{\mathrm{ARS}}^{FK}=\mathrm{E}\left[w_{\mathrm{ARS} k-1}{w}_{\mathrm{ARS} k-1}^{\mathrm{T}}\right]$ is the covariance matrix of process noise for the ARS.

An update (i.e., correction) to the state estimate and covariance error matrix based on input measurement at present:

$$K_{\mathrm{ARS} k}^{FK}=P_{\mathrm{ARS} k/k-1}^{FK}{H}_{\mathrm{ARS}}^{\mathrm{T}}{\left(H_{\mathrm{ARS}}{P}_{\mathrm{ARS} k/k-1}^{FK}{H}_{\mathrm{ARS}}^{\mathrm{T}}+R_{\mathrm{ARS}}^{FK}\right)}^{-1}$$

(19)

$${\widehat{x}}_{\mathrm{ARS} k/k}={\widehat{x}}_{\mathrm{ARS} k/k-1}+K_{\mathrm{ARS} k} \left(z_{\mathrm{ARS} k}-H_{\mathrm{ARS}}^{FK}{\widehat{x}}_{\mathrm{ARS} k/k-1}\right)$$

(20)

$$P_{\mathrm{ARS} k/k}^{FK}=\left(I-K_{\mathrm{ARS} k}^{FK}{H}_{\mathrm{ARS}}^{FK}\right)P_{\mathrm{ARS} k/k-1}^{FK}$$

(21)

where: $K_{\mathrm{ARS} k}^{FK}$ is the Kalman filter gain matrix for the ARS. $R_{\mathrm{ARS}}^{FK}=\mathrm{E}\left(v_{\mathrm{ARS} k-1}{v}_{\mathrm{ARS} k-1}^{\mathrm{T}}\right)$ is the covariance matrix of measurement noise for the ARS. ${\widehat{x}}_{\mathrm{ARS} k/k}$ is the current ARS state estimate (a posteriori). $P_{\mathrm{ARS} k/k}^{\mathrm{FK}}$ is the covariance matrix of filtration error for the ARS.

As a result of the control synthesis, we obtain an LQG regulator in the form of

$$u_{\mathrm{ARS} k}^{\mathrm{LQG}}=-K_{\mathrm{ARS} k}\cdot \left({\widehat{x}}_{\mathrm{ARS} k/k}-x_{\mathrm{ARS} k/k}^{*}\right)$$

(22)

In the case of random interactions on the target tracking system, a Jacobian should be created for the LOS movement model, as described by Equation (9). It will be as follows [10,17]:

$$J_{\mathrm{LOS}}\left(x_{\mathrm{LOS}}\right)=\left[\begin{array}{ccc}\frac{\partial f_1}{\partial R_{\mathrm{LOS}}}& \frac{\partial f_1}{\partial \epsilon }& \frac{\partial f_1}{\partial \sigma }\\ \frac{\partial f_2}{\partial R_{\mathrm{LOS}}}& \frac{\partial f_2}{\partial \epsilon }& \frac{\partial f_2}{\partial \sigma }\\ \frac{\partial f_3}{\partial R_{\mathrm{LOS}}}& \frac{\partial f_3}{\partial \epsilon }& \frac{\partial f_3}{\partial \sigma }\end{array}\right]$$

(23)

where:

$$\begin{gathered} \frac{\partial f_1}{\partial R_{\mathrm{LOS}}}=0; \frac{\partial f_1}{\partial \epsilon }=-V_c\cos {\chi }_c\cos \sigma \sin \left(\epsilon -{\gamma }_c\right); \\ \frac{\partial f_1}{\partial \sigma }=V_c\left[\sin {\chi }_c\cos \sigma -\cos {\chi }_c\sin \sigma cos\left(\epsilon -{\gamma }_c\right)\right]; \\ \frac{\partial f_2}{\partial R_{LOS}}=\frac{V_c\cos {\chi }_c\sin \sigma \sin \left(\epsilon -{\gamma }_c\right)}{ R_{LOS}{}^2 \cos \sigma }; \frac{\partial f_2}{\partial \epsilon }=-\frac{V_c\cos {\chi }_c\sin \sigma \cos \left(\epsilon -{\gamma }_c\right)}{ R_{LOS}\cos \sigma }; \\ \frac{\partial f_2}{\partial \sigma }=-\frac{V_c\cos {\chi }_c\sin \left(\epsilon -{\gamma }_c\right)}{R_{LOS}{\left(\cos \sigma \right)}^2}; \\ \frac{\partial f_3}{\partial R_{LOS}}=V_c\frac{\cos {\chi }_c\cos \left({\gamma }_c-\epsilon \right)\sin \sigma -\cos \sigma \sin {\chi }_c}{R_{LOS}{}^2}; \\ \frac{\partial f_3}{\partial \epsilon }=\frac{V_c\cos {\chi }_c\sin \sigma \sin \left(\epsilon -{\gamma }_c\right)}{R_{LOS}}; \\ \frac{\partial f_3}{\partial \sigma }=-\frac{V_c\left(\cos {\chi }_c\cos \sigma \cos \left({\gamma }_c-\epsilon \right)+\sin {\chi }_c\sin \sigma \right)}{R_{LOS}}. \end{gathered}$$

The formulas for the discrete-time system of LOS and the extended Kalman filter will then be written as:

$$x_{\mathrm{LOS} k}=J_{\mathrm{LOS} k-1}{x}_{\mathrm{LOS} k-1}+w_{\mathrm{LOS} k-1}$$

(24)

$$z_{\mathrm{LOS} k}=H_{\mathrm{LOS}}{x}_{\mathrm{LOS} k}+v_{\mathrm{LOS} k}$$

(25)

$${\widehat{x}}_{\mathrm{LOS} k/k-1}=J_{\mathrm{LOS} k-1}{\widehat{x}}_{\mathrm{LOS} k-1/k-1}$$

(26)

$$P_{\mathrm{LOS} k/k-1}^{FK}=J_{\mathrm{LOS} k-1}{P}_{\mathrm{LOS}\frac{k}{k}-1}^{FK}{J}_{LOS k-1}^{\mathrm{T}}+Q_{LOS}^{FK}$$

(27)

$$K_{\mathrm{LOS} k}^{FK}=P_{\mathrm{LOS} k/k-1}^{FK}{H}_{\mathrm{LOS}}^{\mathrm{T}}{\left(H_{\mathrm{LOS}}{P}_{\mathrm{LOS}\frac{k}{k}-1}^{FK}{H}_{\mathrm{LOS}}^{\mathrm{T}}+R_{\mathrm{LOS}}^{FK}\right)}^{-1}$$

(28)

$${\widehat{x}}_{\mathrm{LOS} k/k}={\widehat{x}}_{\mathrm{LOS} k/k-1}+K_{\mathrm{ARS} k}\left(z_{\mathrm{LOS} k}-H_{\mathrm{LOS}}^{FK}{\widehat{x}}_{\mathrm{LOS} k/k-1 }\right)$$

(29)

$$P_{\mathrm{LOS} k/k}^{FK}=\left(I-K_{\mathrm{LOS} k}^{FK}{H}_{\mathrm{LOS}}^{FK}\right)P_{\mathrm{LOS} k/k-1}^{FK}$$

(30)

where: $x_{\mathrm{LOS} k}$ is the LOS output variable vector. ${\widehat{x}}_{\mathrm{LOS} k-1/k-1}$ is the previous LOS state estimation. ${\widehat{x}}_{\mathrm{LOS} k/k-1}$ is the assessment of the LOS state variables a priori (before measurement). $J_{\mathrm{LOS} k-1}=I+J_{\mathrm{LOS}}\left({\widehat{x}}_{\mathrm{LOS} k-1/k-1}\right)\mathsf{\Delta }t$ is the LOS state matrix in discrete form. $P_{\mathrm{LOS} k-1/k-1}^{FK}$ is the previous covariance matrix of predictive error for LOS. $z_{\mathrm{LOS} k}$ is the current measurements from the scan-track head at time k. $P_{\mathrm{LOS} k/k-1}^{FK}$ is the covariance matrix of predictive error after measurement for LOS. ${\widehat{x}}_{\mathrm{LOS} k/k}$ is the current LOS state estimate (a posteriori),$w_{\mathrm{LOS} k}={\left[w_{R \mathrm{LOS} \mathrm{k}}, w_{\epsilon \mathrm{LOS} \mathrm{k}}, w_{\mathsf{\sigma } \mathrm{LOS}}\right]}^{\mathrm{T}}$, $Q_{\mathrm{LOS}}^{FK}=\mathrm{E}\left[w_{\mathrm{LOS} k-1}{w}_{\mathrm{LOS} k-1}^T\right]$ is the covariance matrix of process noise for LOS. $R_{\mathrm{LOS}}^{FK}=\mathrm{E}\left(v_{\mathrm{LOS} k-1}{v}_{\mathrm{LOS} k-1}^{\mathrm{T}}\right)$ is the covariance matrix of measurement noise for LOS. $w_{\mathrm{LOS} k}$, $v_{\mathrm{LOS} k}$ are the process and measurement noises (white Gaussian), i.e., inaccuracy of the assumed flight path of the aggressor and scan-track head measurement errors. The angular positions of LOS are determined using a passive gyroscopic target coordinator (scan-track head). Laser is used to measuring the distance of the target.

Taking into account the Kalman filtering of the target sight lines, the LQG regulator for control of the artillery rocket set in terms of random interferences influence will be as follows [9,11,17]:

$$\begin{gathered} \begin{array}{c}{u}_{\mathrm{ARS} k}^{\mathrm{LQG}}=-K_{\mathrm{ARS} k}\cdot \left({\widehat{x}}_{\mathrm{ARS} k/k}-{\widehat{x}}_{\mathrm{ARS} k/k}^{*}\right)\hfill \end{array} \\ =-K_{\mathrm{ARS} k}\cdot \left({\widehat{x}}_{\mathrm{ARS} k/k} \\ -{\left[\begin{array}{c}\begin{array}{cccc}{\widehat{\sigma }}_{\mathrm{ARS} k/k}& {\widehat{\dot{\sigma }}}_{\mathrm{ARS} k/k}& {\widehat{\epsilon }}_{\mathrm{ARS} k/k}& {\widehat{\dot{\epsilon }}}_{\mathrm{ARS} k/k}\end{array}\end{array}\right]}^{\mathrm{T}}\right) \end{gathered}$$

(31)

where ${\widehat{\dot{\sigma }}}_{\mathrm{ARS} k/k}$ and ${\widehat{\dot{\epsilon }}}_{\mathrm{ARS} k/k}$ can be approximated using the finite difference method.

Therefore, the optimal control moments that we will use to control the ARS while tracking and shooting to the maneuvering air target will take the form of

$$M_{c1}=u_{\mathrm{ARS} k}^{\mathrm{LQG}}\left(1\right)$$

(32a)

$$M_{\mathrm{c}2}=u_{\mathrm{ARS} k}^{\mathrm{LQG}}\left(2\right)$$

(32b)

Moments (32a) and (32b) are entered at the input of nonlinear system (1)–(4) and include the following limitations:

$$\left|M_{c1}\right|\le M_{c1max}; \left|M_{c2}\right|\le M_{c2max}$$

(33)

where: $M_{c1max}$ is the maximum allowable control moment in azimuth and $M_{c2max}$ is the maximum allowable control moment in elevation. It models the saturation of the ARS’s driving torques.

The formulation and application of the LQR and LQG regulators significantly improve the stability of a remotely controlled artillery rocket set when firing at a maneuvering air target, particularly relative to PID-type controllers. This will be presented in the next section.

4. Numerical Example and Results

Consider a hypothetical artillery rocket set that detects and tracks a maneuvering low-flying target. The basic parameters of the considered set are taken from [8]. Numerical simulations were performed for nonlinear model (1)–(4) in a MATLAB environment with integration step $\mathsf{\Delta }t$ = 0.001 s [23,24]. The duration of the process is $t_t=10 \mathrm{s}$. Given that the initial conditions for the system under consideration are very important, taking into account [25], the following initial conditions for the ARS state variables and their estimations were supposed: $x_{\mathrm{ARS} 0}=\left[\begin{array}{cccc}0& 1.5& 0& 2.5\end{array}\right]$, ${\widehat{x}}_{\mathrm{ARS} 0}=\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]$. The weight matrices for optimal controls LQR are:

$$\begin{gathered} Q_{\mathrm{ARS}}=\left[\begin{array}{cccc}40000& 0& 0& 0\\ 0& 0.1& 0& 0\\ 0& 0& 40000& 0\\ 0& 0& 0& 0.1\end{array}\right]; \\ {R}_{\mathrm{ARS}}=\left[\begin{array}{cc}0.0001& 0\\ 0& 0.0001\end{array}\right]. \end{gathered}$$

The selection of the matrices Q_ARS and R_ARS was made by using the knowledge and importance of controlled state variables and then fine-tuned.

The effectiveness of the modified LQR and LQG regulators has been examined by comparing the optimum settings of the PID controller described in paper [26]. These parameters were optimized using the Nelder–Mead algorithm without constraints. The criterion function was integral of absolute error (IAE), where the decision variables were PID gains.

It was considered that the ARS state variables measurement matrix is as follows:

$$H_{\mathrm{ARS}}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],$$

thus, all state variables are measured: angular positions with absolute encoders and angular velocity with an angular Hall effect velocity sensor.

The covariance matrix of the ARS process noise is $Q_{\mathrm{ARS}}^{FK}={\alpha }_{proc}^2{H}_{\mathrm{ARS}}{H}_{\mathrm{ARS}}^{\mathrm{T}}$, where ${\alpha }_{proc}=0.1$ is the amplitude of process noise. The covariance matrix of the ARS measurement noise is $R_{\mathrm{ARS}}^{FK}={\alpha }_{mes}^2{H}_{\mathrm{ARS}}{H}_{\mathrm{ARS}}^{\mathrm{T}}$, where ${\alpha }_{mes}=0.1$ is the measurement noise amplitude, which is mainly related to the quantization noise and the noise introduced by the angular speed calculation. The initial covariance matrix of error of estimation of the ARS state variables is $P_{\mathrm{ARS}}=Q_{\mathrm{ARS}}^{FK}$. On the other hand, air targets data have the following values: Location of the target relative to the ARS when detected by the observing-tracking head is $x_{c0}=1000 \mathrm{m},$ $y_{c0}=1000 \mathrm{m},$ and $z_{c0}=500 \mathrm{m}.$ The speed of the moving target is $V_c=100 \mathrm{m}/\mathrm{s}=\mathrm{const}$. The target observation initial conditions, $R_{\mathrm{LOS}0}=\sqrt{x_{c0}^2+y_{c0}^2+z_{c0}^2}$, ${\epsilon }_0=\mathrm{arctg}\frac{y_{c0}}{x_{c0}}$, ${\sigma }_0=\arcsin \frac{z_{c0}}{R_{\mathrm{LOS}0}}$, ${\gamma }_{c0}=0; {\chi }_{c0}=0$, and ${\omega }_{c0}=0.75 \frac{rad}{s}$ are the angular velocity of the target maneuver.

It has been assumed that the angles of a target flight change according to the law (the target is maneuverable):

$${\gamma }_c={\omega }_{c0}t; {\chi }_c={\omega }_{c0}t.$$

Taking into consideration the fact that the distance $R_{LOS}$ is measured with a laser distance meter, it is assumed that the target observation system measurement matrix has the form of:

$$H_{\mathrm{LOS}}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

The initial conditions for the individual state vectors are as follows:

$$\begin{gathered} x_{\mathrm{LOS} 0}={\left[\begin{array}{ccc}{R}_{\mathrm{LOS} 0}& {\epsilon }_0& {\sigma }_0\end{array}\right]}^{\mathrm{T}} \\ {\widehat{x}}_{\mathrm{LOS} 0}={\left[\begin{array}{ccc}0.8R_{\mathrm{LOS} 0}& 0.9{\mathsf{\epsilon }}_0& 0.9{\sigma }_0\end{array}\right]}^{\mathrm{T}} \\ {\widehat{x}}_{\mathrm{ARS} 0}^{*}={\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]}^{\mathrm{T}} \end{gathered}$$

The gain matrix of the LOS process noise is $Q_{\mathrm{LOS}}^{FK}={\alpha }_{proc}^2{H}_{\mathrm{LOS}}{H}_{\mathrm{LOS}}^{\mathrm{T}}$. The covariance matrix of the LOS measurement noise is $R_{\mathrm{LOS}}^{FK}={\alpha }_{proc}^2{H}_{\mathrm{LOS}}{H}_{\mathrm{LOS}}^{\mathrm{T}}$. The initial covariance matrix of LOS state variables estimation error is $P_{\mathrm{LOS}}=Q_{\mathrm{LOS}}^{FK}$. The measurement noise is related to the principle of operation of the optoelectronic scan-track head. Moreover, the long-range laser distance measurement is subject to a certain error.

Interferences were introduced in the form of three shots simulated as rectangular pulses in the form of:

$$\begin{gathered} M_{s1}=\mathsf{\Pi }\left(t_{z11},t_{z12}\right)M_{01}+\mathsf{\Pi }\left(t_{z21},t_{z22}\right)M_{01}+\mathsf{\Pi }\left(t_{z31},t_{z32}\right)M_{01}, \\ {M}_{s2}=\mathsf{\Pi }\left(t_{z11},t_{z12}\right)M_{02}+\mathsf{\Pi }\left(t_{z21},t_{z22}\right)M_{02}+\mathsf{\Pi }\left(t_{z31},t_{z32}\right)M_{02}; \end{gathered}$$

where:

$$\begin{gathered} t_{z11}=3.00 \mathrm{s}, t_{z12}=3.02 \mathrm{s}, t_{z21}=3.22 \mathrm{s}, \\ {t}_{z22}=3.24 \mathrm{s}, t_{z31}=3.44 \mathrm{s}, t_{z32}=3.46 \mathrm{s} \end{gathered}$$

are the moments of firing individual shots. $M_{01}=1000 \left[\mathrm{Nm}\right], M_{02}=12000 \left[\mathrm{Nm}\right]$ are the moments of forces acting from the shot in azimuth and elevation, respectively. $\mathsf{\Pi }\left(\dots \right)$ is the rectangular stroke function in the time interval $\left[t_{zi1}; t_{zi2}\right], i=1, 2, 3$.

It was assumed that the base on which the set is mounted is affected by both azimuth and elevation in the form of the following moments:

$$\begin{gathered} M_{b1}=100·\left[\sin \left(0.9t-\frac{\pi }{2}\right)+1\right], \\ {M}_{b2}=200·\left[\sin \left(0.5t-\frac{\pi }{2}\right)+1\right], \\ {M}_{c1max}=1000 \mathrm{Nm},M_{c2max}=12000 \mathrm{Nm}. \end{gathered}$$

Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17 present the results of the conducted simulation tests. Graphs of the assumed disturbance moments from the base for simulation are shown in Figure 5. Figure 6 presents the sum of the disturbing moments from the base and shots $M_{ei}=M_{bi}+M_{si}$. Figure 7 shows the angular displacement of the ARS in azimuth and elevation as a function of time during interference of the base on which the set is placed (moments $M_{bi}$) and during the firing of three shots (moments $M_{si}$). In Figure 8, we can observe the realized (simulated) and desired trajectories for the ARS movement with the abovementioned interferences. To control and stabilize the set, a PID controller was used with optimally selected parameters due to the minimum integral absolute error. While Figure 9 and Figure 10 show the same relationships with the same interactions when using an optimal LQR control, it can be observed that the modified LQR more effectively reduces the impact of three shots being fired from the base on which ARS is mounted.

With additional measurement and process random interference on the ARS base, both PID and LQR controllers fail—the dynamic effects are very large and unacceptable. However, it should be noted that the modified LQR controller works better than the optimal PID controller. This is particularly evident in Figure 11 and Figure 13. However, in the case of random interferences, the LQR regulator is also insufficient to ensure LOS tracking accuracy (see Figure 13).

In this case, the most effective in action is the modified LQG regulator. This is shown in Figure 14, Figure 15, Figure 16 and Figure 17. They show that random disturbances that affect the ARS during movement on uneven terrain can be effectively neutralized. It should be emphasized that, in order to destroy an air target, such sets currently fire bullets while being in a stationary position. Using the modified LQG method presented in this paper, it becomes possible to shoot a maneuverable air target during the set movement. This increases the effectiveness and mobility of the ARS.

5. Conclusions

The algorithm presented in this paper allows for the precise control of an ARS system in case of disturbances. The example presented in this article shows that ARS tracking of the maneuvering air target using a Jacobian in a closed control loop is more effective than using a classical PID or LQR control. As the preliminary results show, improving the precision of an ARS control can be crucial in reaching a target in such artillery rocket systems.

As a result, the algorithm allows for a degree of control so that it is possible to minimize the impact of kinematic effects on the side of a moving carrier (off-road vehicle or ship) and random external interferences. This increases effectiveness and mobility of ARS and allows it to attack air targets during its movement on uneven surfaces. In the case of a land vehicle, the shooting can take place without the necessity to stop it.

Theoretical considerations and simulation studies have shown that, in conditions of ARS interference, it is preferable to use Jacobians in both extended Kalman filtering and optimal LQR control. Therefore, the effectiveness of the modified LQG regulator has been demonstrated. The presented control algorithm updates the model of the set at each time step, thus making the model linearized at a preset operating point. In contrast, however, if the real system deviated significantly from the preset operating point for an extended period of time (due to, e.g., failure or inaccurate calibration), then the control actually developed would not be optimal. In this case, a different type of controller, e.g., a sliding mode controller, should be used for the transition period until the real system and set operating point converge sufficiently, after which a switchover to the LQG controller could take place.

The mentioned issue should be investigated in additional studies. Moreover, in further research, the effectiveness of this regulator should be tested in field conditions when firing an artillery rocket set to a low-flying maneuvering air target.