Analytical Design of a Closed Control Loop Controller, Based on a Suboptimal Kalman-Busy Filter

Abstract

The Analytical Design (AD) of a closed, negative feedback control loop, when only single design criteria (potentially achievable accuracy) can be considered at the first stage of development for the synthesis of desired system dynamics. Such an approach, based on a modified suboptimal Kalman-Busy Filter (KBF) with Bounded Grows of Memory (FBGM), was presented in several previous author’s papers. In some cases, the required optimal controller should work, mainly, in the stationary stabilization mode in stationary conditions and, actually, is a regulator. In these cases, FBGM can be essentially simplified to a stationary Kalman’s state estimator, with a switched matrix weight coefficient (transient/stationary). The coefficient can, practically, be found rather from the conventional conditions for providing the system sufficient dynamics, than from the solution of KBF Riccati eq. A successful tuning makes the steady state accuracy be close to the optimal, provided by the KBF. The estimator is used for the estimation/filtering and control/regulation purposes simultaneously. This approach is considered in the below chapter to draw developer’s attention. A simple example of the 2nd order unit, assuming regulation of system angular position and angular velocity is presented.

1. Introduction

As far as the author knows, the term “Analytical design (AD)” appeared in the seventieth-century in the former USSR. There were published [1, 2], where authors applied optimal filtering [3, 4] (KBF) and optimal control theory [5, 6] to find an optimal dynamic for the typical aerospace applications. The fundamental base for this approach was presented by R. Kalman in Ref. [3], as a “dual (separation) principle”. Further general consideration of above-mentioned optimal estimation/filtering and control theories and applications can be found in the following publications [7, 8, 9, 10, 11]. It must be mentioned that despite the term AD was not directly used in the Western scientific literature, in fact, a similar approach was presented in some publications [7, 10]. This approach can be called a synthesis of optimal closed control loop dynamic with minimum of the quadratic quality criterion. To be short, in the below consideration, the term AD is used. What is essential for the engineering practice is that the AD is very consistent with developing generic International methodology and tools (MATLAB/Simulink), developed by MathWorks under the concept of Model-Based Design [12].

Talking about the direct using of KBF in AD, we need to take into account that stochastic system model, assumed in KBF theory [3, 4], is usually very idealistic. In reality takes place more complex than centered Gaussian white noises (the mathematical abstraction) disturbances and measured errors. We can clearly see this when comparing experimental data with KBF idealistic model, especially for a long-time observed intervals. Often, exact tuning the KBF in accordance with its idealistic model can lead to an unstable filter that cannot be practically used. An additional problem is the limited calculation capability of used real-time computers. Even today, for essentially developed sophisticated Aerospace on-board computers [13, 14] computational and software debugging difficulties for the direct KBF implementation have not been completely overcome. However, if implemented, then often it can result in useless wasting of computational power and idle processing where simpler and more reliable results can be achieved. All above says that almost always is worth sacrificing theoretical minimum, provided by KBF, for the robust and simple implementation of a suboptimal KBF. Such suboptimal versions of KBF the author began developing in the early seventieth-century. At this time he was urged by the necessity to simplify and implement it in the first generation of USSR aviation on-board digital computers (OBC) that had then very modest computational capabilities. Then was developed a sufficient suboptimal KBF form- the FBGM, where the filter weight coefficient matrix $K$ was presented as consisting from two components; not stationary $\tilde{K}\left(t\right)$ and stationary $K^{\ast }$(constant). Both parts were pre-calculated and the not stationary one ($\tilde{K}\left(t\right)$) was approximated by simple polynomial functions. A special analytical procedure, based on using of the ratio signal/noise (estimation indicators), was developed. Working in the steady state with the stationary coefficient $K^{\ast }$, filter periodically can be restarted to upgrade the estimates. Detailed information about this can be found in [15, 16, 17]. In this final analytical form, the filter coefficients were “stitched“into the OBC memory. Some examples of using the filter as the estimator and the controller simultaneously were published in [18] (pair of satellites, flying in formation (FF)) and [19] (navigation accelerometer with the electric spring). The general approach of using FBGM for the AD was published in [20].

The following approach is presented below. The filter is suggested to be used as the filter and controller at the same time. It is in the scope of the linear, time-invariant systems under the assumptions of the linear KBF theory. It is considered a special, however, a wide class of control systems: the stabilization systems, when the controller mainly acts as a regulator, stabilizing regulated parameters with respect to the desired set level. Two modes are typical for such a system: the initial activation (transient process) and the stabilization (stationary/steady filtering process). For such a case the FBGM can be represented (degenerats) as a simple form: a stationary filter with switching in time bandwidth: wide at the transient stage and narrow at the filtering stage. With proper tuning, the matrix of the filter coefficients is robust and can provide estimation (control) accuracy close to the theoretically achievable maximum (minimum of estimation errors), as with KBF.

As with regards to the AD and Model-based design, the filter synthesis can be considered as the first step of the design. Preliminary (conceptual model – CM or Low Fidelity Model - LFM) design model. It can be used to synthesize the system state estimator/observer and the controller and to evaluate system feasibility and potentially achievable performance. Further steps of the design take into account some realistic constraints and restrictions, and develop this LFM, by adding more detail elements, to complex, non-linear, and not-stationary High Fidelity Model (HFM).

More recent work about AD taking into account some additional constraints can be found in [21].

It must be added that recent developments in the Control Theory introduced new effective methods for syntheses controllers that can work in more complex than stabilization operational modes and conditions, demonstrating better than LTI - KBF dynamics. At least, two new approaches should be mentioned here: the Fuzzy Logic (FL) and the Artificial Intelligence (AI).

Works, dedicated to using FL for synthesis of PID controller and for Fault detection and diagnostic, were published by Prof. C. Volosenku in Refs. [22, 23, 24].

Broad survey of using AI in various engineering applications for technical diagnostic readers can be found in Ref. [25].

2. Suboptimal Kalman-Busy filter

Let us assume that we have had acceptable for further analytical design system conceptual model, which is LFM; linear time invariant (LTI), fully observable and controllable model of a stochastic system, presented by the following state equation [4].

$$\begin{array}{l}\dot{x}=\mathit{Fx}+\mathit{Gw},\\ z=\mathit{Hx}+v,\end{array}$$E1

where: $x$ is an $n$ state vector, $w$ is an $m$ -external disturbances vector, $z$ is a $p$ measurement vector, $v$ is a $p$ is measurement noise vector, $F$ is $\left(n\times n\right)$ matrix of system, $G$ is an $\left(n\times m\right)$ matrix of disturbances, $H$ is $\left(p\times n\right)$ matrix of measurements.

Let us assume about (1) that $F,G,H$ are given matrixes and

$$\begin{array}{l}E\left[x\left(t_0\right)\right]=0,E\left[w\left(t\right)\right]=0,E\left[v\left(t\right)\right]=0,\\ E\left[x\left(t_0\right)x^T\left(t_0\right)\right]=P_0,\\ E\left[w\left(t\right)v^T\left(\tau \right)\right]=E\left[v\left(t\right)w^T\left(\tau \right)\right]=E\left[w\left(t\right)x^T\left(t_0\right)\right]=0,\\ E\left[w\left(t\right)w^T\left(\tau \right)\right]=Q\left(t\right)\delta \left(t-\tau \right),\\ E\left[v\left(t\right)v^T\left(\tau \right)\right]=R\left(t\right)\delta \left(t-\tau \right),\end{array}$$E2

where: $P_0$ is the covariance matrix of the initial state, $R\left(t\right)$ is the measurement noise covariance matrix, $Q\left(t\right)$ is the disturbance noise covariance matrix, $\delta \left(t-\tau \right)$ is the Dirac’s standard pulse. In other words, $w\left(t\right)$ and $v\left(t\right)$ are normal Gaussian white noise processes. Often, $Q$ and $R$ are diagonal and constant matrixes of the white noises $w\left(t\right)$ and $v\left(t\right)$ accordingly. Instead of the idealistic white noises, in a wide frequency band, they can be interpreted as correlated processes with the following matrixes spectral densities

$$\begin{array}{l}Q={\left[2{\sigma }_{\mathit{wi}}^2{T}_{\mathit{wi}}\right]}_{\mathit{diag}},{}_{i=1,2,\mathrm{..}m}\\ R={\left[2{\sigma }_{\mathit{vj}}^2{T}_{\mathit{vj}}\right]}_{\mathit{diag}},{}_{j=1,2,\mathrm{..}m}\end{array}$$E3

where ${\sigma }_{\mathit{wi}}\text{and}{\sigma }_{\mathit{vj}}$ are the standard deviations, $T_{\mathit{wi}}\text{and}{T}_{\mathit{vj}}$ are the correlation times of these stochastic processes $w$ and $v$ .

We want to find an optimal $\widehat{x}$ estimate of the system state vector $x$, using measurements $z$.

The estimation error $\tilde{x}=\widehat{x}-x$ covariance matrix is as follows $P=E\left[\tilde{x}\left(t\right)\cdot {\tilde{x}}^T\left(t\right)\right]$.

Let us use the following optimization criteria

$$J_{\min }=min\left(\mathit{\text{traceP}}\right)$$E4

Then, the Kalman-Busy filter (KBF) is to be considered as the optimal estimator [4, 5, 6].

Let us consider KBF in its continuous form [6].

$$\begin{array}{l}\widehat{\dot{x}}=F\widehat{x}+K\left(z-H\widehat{x}\right),{\widehat{x}}_0,\\ K={\mathit{PH}}^T{R}^{-1},\\ \dot{P}=\mathit{FP}+{\mathit{PF}}^T-{\mathit{PH}}^T{R}^{-1}\mathit{HP}+{\mathit{GQG}}^T,P_0\end{array}$$E5

where: $K=K\left(t\right)$ is matrix weight coefficient, $P=P\left(t\right)$ is estimated state vector errors covariance matrix.

Matrix $P$ is determined by the 3-rd equation of (5), aka as the differential equation Riccati of the KBF filter theory).

Eq. (5) provides an optimal solution with minimum (4), when measurements $z$ are available $z\ne 0$, then the weight coefficient $K\left(t\right)$ in (5) is not equal to zero and takes place continuous update of the current estimates. This mode of working (5) is called the “filtering mode”. If, for some reason, the measurement vector $z$ is not available ($z\equiv 0$) then, for this time interval the filter (5) can also be switched to another mode of operation, called the “prediction mode”. In this mode weight coefficient in (5) should be set to zero $K\left(t\right)\equiv 0$ and initial conditions should be taken as the last available past estimates: ${\widehat{x}}_p\left(t_0\right)={\widehat{x}}_f\left(t_f\right)$. Then original KBF equations degenerate to the following:

$$\begin{array}{l}\widehat{\dot{x}}=F\widehat{x},{\widehat{x}}_0={\widehat{x}}_p,\\ K\equiv 0,\\ \dot{P}=\mathit{FP}+{\mathit{PF}}^T,P_0=P_p\end{array}$$E6

The KBF, providing optimal system state estimation, was presented by R. Kalman in the early sixtieth-century, in the recursive, (convenient for direct digital computer implementation) [5] and analogue [6] forms. Right away, its presentation attracted attention of many developers to apply it for diverse technical applications. However, low computing power of electronic analog and digital computers at that time hardly allowed implementing the KBF practically, specifically for real-time aerospace applications in on-board computers. And even today, despite significant progress, achieved with on-board computer characteristics [7, 8], implementation of KBF still is quite a challenging problem for a programmer, requiring from him a certain experience and skill. It is worth noting that for many practical applications, direct KBF implementation would follow just grinding the steady state data in idle and/or creating the computational divergence trend for the filter, followed by the system instability. Therefore, the author’s position, related to the considered problem (using KBF for analytical design) is that is almost always worth analyzing and implementing a suboptimal KBF (sacrificing by the theoretical maximum estimation accuracy, but making the filter robust and simple for its implementation and debugging) than direct optimal KBF (5) implementation (Quid pro quo).

Such sub-optimal form of KBF - the Filter with Bounded Growth of Memory (FBGM) is presented below. The simplest, but practically common and important case is when the model (1) and the filter (5) are linear stationary (time-invariant) systems (LTI).

The filter development stages and it’s not-stationary form for suboptimal estimation in transient process readers can be found in Refs. [1, 2, 9].

The filter operation in time is consecutively divided, by switching its weight coefficient $K$, in two periods: estimation of the transient process $t<t_{\ast }$ (initial filter-IF) and estimation of the steady process $t>t_{\ast }$ (steady filter-SF). It can be presented by the following equations

$$\begin{array}{l}\widehat{\dot{x}}=F\widehat{x}+K\left(z-H\widehat{x}\right)\\ K=\left\{\begin{array}{l}{K}_{\mathit{\text{IF}}}^{\ast }\text{if}{t}_0\le t\le t_{\ast },\\ K_{\mathit{SF}}^{\ast }\text{if}t>t_{\ast },\\ 0\mathit{\text{if}}z\equiv 0\end{array}\right.\end{array}$$E7

In (7)$K_{\mathit{\text{IF}}}^{\ast }$ and $K_{\mathit{SF}}^{\ast }$ are filter matrix weight coefficients.

$K_{\mathit{\text{IF}}}^{\ast }$ for IF and $K_{\mathit{SF}}^{\ast }$ for SF filtering mode, and $K=0$ for the prediction mode. These coefficients and boundary time $t_{\ast }$ can be found empirically or from the KF sub-optimization procedure [1, 2, 9]. The suboptimal coefficient for the transient process $K_{\mathit{\text{IF}}}^{\ast }=\tilde{K}\left(t\right)$ can be found as the polynomial approximation of KBF solution (3-rd differential equations, Riccati type) in (5). And for the stationary (steady) state ($t\to \infty ,\dot{P}\left(t\to \infty \right)=0$) estimation $K_{\mathit{SF}}^{\ast }$ is, in fact, optimal and can be found from the equations below

$$K_{\mathit{SF}}^{\ast }=P^{\ast }{H}^T{R}^{-1}$$E8

$$F^{\ast }{P}^{\ast }+P^{\ast }{F}^{\ast T}-P^{\ast }{H}^T{R}^{-1}{\mathit{HP}}^{\ast }+{\mathit{GQG}}^T=0$$E9

where $F^{\ast }=F-K_{\mathit{SF}}^{\ast }H$.

In this steady state case, the differential non-linear Riccati equation in [5] degenerates to the algebraic eq. (9) that can be solved numerically and required $P^{\ast }$ and $K^{\ast }=K_{\mathit{SF}}^{\ast }$ can be found.

3. Closed loop, negative feedback control, and estimation

The estimation error $\tilde{x}$ is presented by the following formula

$$\tilde{x}=x-\widehat{x}$$E10

Let us assume that all state vector $x$ components are directly measured (in some cases this assumption is not directly applicable).

Then, using eqs. (1) and (5), we can get the following matrix equation

$$\dot{\tilde{x}}=\tilde{F}\tilde{x}+\mathit{Gw}-\mathit{Kv}$$E11

where $\tilde{F}=F-\mathit{KH}$ and $K$ can be determined as in (9).

Closed negative feedback control loop with the state estimator performing both: estimation and control tasks is shown in Figure 1.

Figure 1.

Sate estimator (filter) as system closed loop controller.

Let us talk about the selection of the appropriate control-estimation matrix coefficient $K$ for (7), (11).

As it can be seen from (7), the KBF optimal matrix weight coefficient is entirely dependent on the matrixes $Q$ and $R$ that in many cases not known. System disturbance $w\left(t\right)$ (matrix $Q$) usually, hardly can be approximated by white noise and is rather almost a deterministic signal (for example, close to the bias).

However, the measuring error $V\left(t\right)$ (matrix $R$) often can be indeed considered as a Gaussian wide bandwidth white noise with a small correlation time ($\tau$).

The following simple idea for choosing suboptimal coefficients $K_{\mathit{\text{IF}}}^{\ast }$ and $K_{\mathit{SF}}^{\ast }$ in (7) is used. Both are constant and are chosen by rather conventional engineering practices than optimal KBF formulas. Two filter studies are considered: a-Transient; IF mode with high band-with and b-Stationary; SF mode with low bandwidth Physically, a closed control loop (11) can be considered as an LTI filter having a big cut frequency ${\omega }_{\mathit{\text{IF}}}^{\ast }$ (wide bandwidth) during the transient process $t_0\le t\le t_{\ast },$ and a low cut frequency ${\omega }_{\mathit{SF}}^{\ast }<{\omega }_{\mathit{\text{IF}}}^{\ast }$.

Its weight matrix coefficient $K^{\ast }$ is switched by a step at the time $t=t_{\ast }$ accordingly to provide desired ${\omega }_{\mathit{SF}}^{\ast }\text{and}{\omega }_{\mathit{\text{IF}}}^{\ast }$. By choosing an appropriate ${\omega }_{\mathit{\text{IF}}}^{\ast }$ (wide filter bandwidth) we can achieve a high performance and a short duration of the filter transient process $t_{\ast }$. After the bandwidth is narrowed, a steady state filtering process will take place, which has the purpose of passing through the filter slowly changing in time (almost deterministic) input signal and cutting high frequency wide range measured noise.

In [19] author proposes to apply a rather conventional engineering than pure mathematical optimization approach: splitting system operation time into two typical parts transient process ($t_0<t<t_{\ast }$) and steady operation (filtering) process ($t_{\ast }<t$).

• At the first (transient) stage original system model (1) is considered as “quasi-deterministic”

$$\begin{array}{l}\dot{x}=\mathit{Fx},x_0\\ z=\mathit{Hx}\end{array}$$E12

and neglecting by noises $w$ and $v$ choosing the gate $K_{\mathit{\text{IF}}}^{\ast }$ as being aimed by the single criterion: fast decaying of the transfer process, caused by the initial conditions $x_0$.

• At the second (steady) stage, the original system model [1] is considered as “purely statistic”, when after decaying of the transferring process the steady state filtering takes place: passing through the low-frequency bandwidth almost deterministic signal measuring with an additive random white noise.

$$\begin{array}{l}\dot{x}=\mathit{Fx},x_0=0\\ z=\mathit{Hx}+v\end{array}$$E13

Then, $K_{\mathit{SF}}^{\ast }$ can be found from the equation

$$F^{\ast }P+F^{\ast }P+K_{\mathit{SF}}^{\ast T}{\mathit{RK}}_{\mathit{SF}}^{\ast }=0$$E14

where $F^{\ast }=F-K_{\mathit{SF}}^{\ast }H$,

This equation is covariance equation for the filter estimation errors caused by the measuring noise $v\left(t\right)$

$$\dot{\tilde{x}}=F^{\ast }\tilde{x}-K_{\mathit{SF}}^{\ast }v$$E15

In fact, in practice the real physical processes $w\left(t\right)$ and $v\left(t\right)$ have more complex than Gaussian stationary white noise structure and hardly can be expressed by the matrixes $Q$ and $R$, assumed in KBF theory. However, this abstraction could be helpful for practical needs if some “appropriate” levels of $Q$ and $R$ are taken for the AD that would result in a solution that is compatible with conventional engineering practice.

It is presented that both matrixes control/filtering gates $K_{\mathit{\text{IF}}}^{\ast }$ and $K_{\mathit{SF}}^{\ast }$ can be chosen from usual for the Automatic Control system theory [4, 10] engineering criteria, such as stability, overshooting, and filtering bandwidth, rather than from solving KBF Riccati equation.

Let us consider the filter SF characteristic polynomial

$$\mathrm{\Delta }\left(s\right)=\mathit{sI}-\left(F+K^{\ast }H\right)=s^n+b_1{s}^{n-1}+..\dots +b_{n-1}s+b_0$$E16

and choosing its coefficients $b_i$ to have in (3.6) desired roots. For example, for the multiple real number roots $\lambda$

$$\mathrm{\Delta }\left(s\right)={\left(s+\lambda \right)}^n$$E17

where $\lambda$ is the physical meaning of system bandwidth.

The matrix coefficient $K_{\mathit{\text{IF}}}^{\ast }$ can be found, considering desired bandwidth $\lambda$ and standard polynomial coefficients in (17).

It also should be noted that if the filter realization (estimation process only) is digital, then the following trick to eliminate transient process can be applied

$${\widehat{x}}_1=H^{-1}{z}_1$$E18

The estimate vector at the first step is equal to the first measurement.

4. Example

Let us consider simple single-axis $X$ flat rotation of a spherical rigid body, with angular dynamics equation

$$\ddot{x}=\frac{M}{J}$$E19

where $\ddot{x}$ is point acceleration, $J$ is its axial inertia $J=J_x$, $M$ is external disturbing torque.

This equation can be rewritten in the state equation form as follows

$$\left\{\begin{array}{l}{\dot{x}}_1=w_1\\ {\dot{x}}_2=x_1\end{array}\right.$$E20

where $x_1$ is point velocity $x_1=\dot{x}=V$, $x_2$ is point position $x_2=x$, $w_1$ is the specific external disturbing force, applied to the point $w_1=\frac{M}{J}$.

Let us also assume that system for (10) (plant) states vector $x$ both components are measured by some position and angular velocity sensors

$$z=\mathit{Hx}+v$$E21

where.

$H=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\text{is measurement matrix},v=\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]\text{is measuremnt error vector}$

Gaussian white noises having covariance matrixes $Q=\left[\begin{array}{cc}{q}_1& 0\\ 0& 0\end{array}\right]$ and $R=\left[\begin{array}{cc}{r}_1& 0\\ 0& r_2\end{array}\right]$.

In this example, we assume that both state vector components $x_1$ and $x_2$ are measured and estimation filter will be used for the control purposes in the closed control loop as well. In other words, (4) is a common quality criterion for the estimation and the control.

If minimum diagonal $P^{\ast }$ is a common quality criterion for estimation and control of (19), then.

Applying to the system KBF and subtracting from the system vector its estimation, we can get equation of estimated errors

$$\left\{\begin{array}{l}{\dot{\tilde{x}}}_1=-k_{12}{\tilde{x}}_2-k_{12}{v}_2+w_1\\ {\dot{\tilde{x}}}_2={\tilde{x}}_1-k_{22}\delta {\tilde{x}}_2-k_{22}{v}_2\end{array}\right.$$E22

that can be interpreted as the negative feedback closed control loop. Converting (22) to the second-order differential equation, we will get the following 2-nd order differential equation unit

$${\ddot{\tilde{x}}}_2=-\frac{1}{T^2}{\dot{\tilde{x}}}_2-2\frac{d}{T}{\tilde{x}}_2-\frac{1}{T^2}{\dot{v}}_2-2\frac{d}{T}{v}_2+w_1$$E23

where $T=\frac{1}{\sqrt{k_{12}}}$ is the unit time constant, $d=\frac{k_{22}}{2\sqrt{k_{12}}}$ is specific damping coefficient.

The above example was simulated with Simulink simulator.

The following data were used:

$J=50{\mathrm{kgm}}^2$ is satellite inertia, ${\sigma }_M={10}^{-6}\mathrm{Nm}$ is satellite disturbing torque (noise) standard deviation (STD), $T_M=10\mathrm{s}$ is disturbing torque correlation time,

${\sigma }_{v1}=0.01deg/\mathrm{s}$is satellite angular rate measured noise STD and correlation time $T_{v1}=0.1\mathrm{s}$.

${\sigma }_{v2}=0.1deg$is satellite attitude measured noise STD and $T_{v2}=0.1\mathrm{s}$ is satellite attitude error correlation time.

For this example, steady state algebraic Riccati eq. (9) cannot be solved analytically. The numerical solution, which was found, gives the following parameters:

${\sigma }_1=0.0008deg/s{\sigma }_2=0.05deg$ are STD of estimation errors and $k_{11}=0.0054,k_{12}=0.000272,k_{21}=0.272,k_{22}=0.0236$ are the filter/controller coefficients.

These coefficients can be converted into the time constant $T=\frac{1}{\sqrt{k_{11}{k}_{22}-k_{12}{k}_{21}+k_{12}}}=55.3\mathrm{s}$ and the specific damping coefficient $d=\frac{k_{11}+k_{22}}{2T}=2.62\cdot {10}^{-4}$. So, the calculated optimal damping coefficient for the KBF cannot practically be used as not providing sufficient damping.

Therefore, to demonstrate the applicability from the practical point of view a different (suboptimal) approach.

The coefficient $d$ was set as $d=0.707$, but chosen filter time constant - is optimal $T=55.3\mathrm{s}$.

There were added some additional factors to demonstrate essential differences with the idealistic model for KBF; constant disturbing torque ${\sigma }_{M_0}={10}^{-5}\mathrm{Nm}$ and constant angular velocity measurement error (bias) ${\mathrm{\Delta }}_{x1}=10deg/h=0.0028deg/s$.

For IF mode was taken that: $T_{\mathit{\text{IF}}}=3\mathrm{s}$, $d=0.707$. Switching IF/SF time was $t^{\ast }=9\mathrm{s}$.

The simulation results are presented in Figure 2.

Figure 2.

Attitude and attitude velocity are measured. Control with optimal estimator, acting simultaneously also as the controller.

1-Zero initial conditions and additional disturbing torque: $x_{10}=0,x_{20}=0,M_0=0$.

The time of FBGM switch from IF to SF mode is $t^{\ast }=9\mathrm{s}$.

5. Conclusion

AD, based on using the sub-optimal KBF in closed control loop for the filtering and control-stabilization purposes simultaneously, can be considered as a helpful synthesis method for a broad spectrum of engineering applications. Simple implementation of the two stages stationary filter with switched: wide/narrow bandwidth can guarantee a satisfactory system dynamic with close to optimal steady state KBF accuracy.

Acknowledgments

Author acknowledges the Academic Editor Prof. C. Volosenku and the Publisher IntechOpen contributions to the chapter editorial improvement and publishing.