A novel optimal approach for control law of multi-rate systems with different rate operations

It may be said that this was the first time in World War II that data sampling was used to control rotating radar systems (Zarchan, 2019).

One of the most serious studies and participations in the study of sampling methods, known as frequency analysis, is mentioned in reference (Jacquot, 2019). This article describes the use of this technique in developing a control system. In (Kim, 2020) published a book describing and explaining both methods, especially the vector decomposition method. Following this research, Friedland applied the technique of frequency analysis to periodic control structures, and this theory was developed by Salt et al. (2020). To systems with multiple feedback loops and different sampling rates. Since then, researchers have discovered the potential of multicast systems and beyond a single technique to analyze single-rate systems. At that time, the keyhole analysis method was used in a completely logical and effective way to design and expand such systems. In the same reference, Kranc came up with the idea of replacing multi-rate control instead of single-rate control in systems with variable dynamics (Li et al., 2002). Whitbeck presented a vectoring method for key analysis and generalized it to flight control issues (Harvey and Pope, 1982; Salt et al., 2020).

Methods on the stability of state-of-the-art space in the field of continuous time with the subject of analysis and design of multi-rate systems to achieve the stability of these systems were presented by (Dick, 2018) by publishing their article. This paper made a significant contribution to demonstrating the power of state space methods in describing a large number of single-rate sampling control systems with time changes.

After that, researchers at TASC developed a new approach to multi-rate control design based on optimal control and estimation formulation. This study included a mathematical formulation of the problem of designing, improving and upgrading design techniques, computational methods and the application of these methods in flight control examples (Perez-Montenegro et al., 2019; Ma et al., 2020).

Naturally, the guidance law system, autopilot, and search seeker of a guided missile have different sensors at different operating rates. Certainly, selecting these components in such a way that they all work at a single rate equal to the operating rate of the system will limit the choice and affect the accuracy and ultimately the performance of the system. In this paper, after introducing the operation of multi-rate systems, a guided missile, including the guidance law, automated pilot and searcher with different and different rates of input and output is simulated and its performance is examined in several different scenarios of target movement and optimal performance.

Consider a system with inputs and outputs at different rates according to Figure 1A.

For such system, it is assumed that the input and output are sampled in rate of 1 T m and 1 T n respectively, which T m and T n are related through relationship (1) to the lowest common sampling rate T b . (1) T n = l n , T m = l m T b , T b = T l

In these relationships, l m and l n are integer numbers and T is the highest common sampling rate is named main sampling interval. Input and output sampler are decomposed to m and n parallel sampling respectively at a slow rate, which is synchronized to 1 T for operation at a single rate (Figure 1A). Figure 1B and C presents decomposed form of switching and vectored decomposed form for system of Figure 1A, respectively.

Figure 1:

If G n + and G n − are defined as time vectors and delay vectors as equation (2), As a result, Figure 1B can be represented as a block diagram of Figure 1C. G n + and G n − are the necessary operators for vectored decomposition, also known as kranc operators: (2) G n + = [ 1 e s T n e 2 s T n . . . e ( n − 1 ) s T n ] T G n − = [ 1 e − s T n e − 2 s T n . . . e − ( n − 1 ) s T n ] T

The system descriptive equations of Figure 1C are written as equation (3): (3) G s 1 ( z ) = Z [ G m + R ( s ) ] G s 2 ( z ) = G s 1 ( z ) Z [ G n + G ( s ) G m − ] y ( z ) = G s 2 ( z ) Z [ G n − ]

The multi-rate operators Z [ G m + R ( s ) ] and Z [ G n − ] represent the rushed input signals and delayed output signals respectively, which can be shown as (4) (4) Z [ G m + R ( s ) ] T = [ R ( z ) Z [ R ( s ) e s T m ] Z [ R ( s ) e 2 s T m ] . . . Z [ R ( s ) e ( m − 1 ) s T m ] Z [ G n − ] = [ 1 z − k T n z − 2 k T n . . . z − ( n − 1 ) k T n ]

The term Z [ G n + G ( s ) G m − ] can also be defined in the same way. The multi-rate operator will be l m for input and l n for output: (5) Z [ G n + G ( s ) G m − ] = [ 1 e s T n e 2 s T n . . . e ( n − 1 ) s T n ] G ( s ) [ 1 e − s T m e − 2 s T m . . . e − ( m − 1 ) s T m ]

which can be expressed as (6) (6) Z [ G n + G ( s ) G m − ] = [ G T ( z ) G T ( z , Δ 1 ) . . . G T ( z , Δ m − 1 ) G T ( z , 1 + 1 n ) G T ( z , Δ 1 + 1 n ) . . . G T ( z , Δ m − 1 + 1 n ) . . . . . . . . . . . . . . . . . . G T ( z , 1 + ( n − 1 ) n ) G T ( z , Δ 1 + 1 + ( n − 1 ) n ) . . . G T ( z , Δ m − 1 + 1 + ( n − 1 ) n ) ]

which Δ k = 1 − k m and z defines as z = e − s T .

Re-expression of equation (6) in the form of vector, will include all the behavior of the internal samples of the multi-rate system, but they are not a desirable and appropriate form for use in classical analyzes. For example, it may not lead to an equation with analytical properties. Although we know that the Kranc method of operation ensures that the transmission of all routes is affected by multi-rate sampling by rushing and delaying blocks (Kellermann, 1988; van der Zee and Han, 2001; Han and Ding, 2010).

Therefore, by replacing the multi-rate system with different input and output rates with the equivalent one; the system can be expressed with the standard signal flow graph and simplified and analyzed with diagram block reduction techniques.

For this purpose, the general equations for a multi-rate system (integer and logical rates) and with multiple inputs–multiple outputs will be explained.

For a general model of multi-rate, multi-input, multi-output system with m subsystems, each of which may have different inputs and outputs at different rates or the number of different inputs and outputs, will be placed next to each other and combination of their responses will form general response of system. We must remember that input and output vectors are defined only at appropriate points with their sampling rates. Therefore, the control and output matrices of the general multi-rate, multi-input, multi-output model should reflect this fact. With this in mind, the state equations are defined as (7) (7) x [ ( k + 1 ) T ] = Φ x ( k T ) + Ψ u ( k T )

By defining n 0 as n 0 = T T b ( n 0 is the common fast rate, for example, if a system has two inputs with 2 T and 3 T rates, n 0 will be equal to 6) and n as the number of system inputs, matrix Φ has the dimensions of n n 0 × n n 0 and is defined in (8): (8) Φ = [ 0 0 . . . Φ ^ T b ]

Matrix Φ includes ( n 0 − 1 ) zero matrix with dimensions ( n n 0 × n ) .

Also, matrix Φ ^ T b is defined as (9) Φ ^ T b = [ Φ T b [ 1 , 0 ] T Φ T b [ 2 , 0 ] T . . . Φ T b [ n 0 , n ] T ] T

In this case, the Φ T b is discrete state transient matrix of in the [ K T , ( K + 1 ) T ] time interval, inputs are updated at a maximum of m different rates. Assigning each input to a specific state of the system, x i ( t ) at any time t that K T < t < ( K + 1 ) T should be calculated from different sampling points. This leads to the creation of a block control matrix with dimensions ( 1 × m ) shown in equation (10): (10) Ψ = [ Γ ^ 1 , 1 ... Γ ^ 1 , n 1 , ... , Γ ^ 1 , m 1 ... Γ ^ m , n m ]

In this regard, the Γ ^ k blocks are defined as (11) (11) Γ ^ k = { Γ ^ k , i } i = 1 , ... , n k

Index k in (11) indicates that all phrases of Γ are written according to kth column of the main control matrix Ψ .

Index i also represents the multi-rate control matrix column index. Each ( n 0 × 1 ) element of the block matrix Γ ^ k , i is defined by considering whether the kth input is updated in each T b , the result of which is the following statement: (12) Γ ^ k , i ( j ) = { 0 f o r j ≤ ( i − 1 ) ℓ k Γ [ j , ( i − 1 ) ℓ k ] f o r ( i − 1 ) ℓ k < j ≤ i ℓ k ϕ [ j , i ℓ k ] Γ [ i ℓ k , ( i + 1 ) ℓ k ] f o r j > i ℓ k

In relation (12), ℓ i represents the coefficient Tb in the range T i ( T i = ℓ i T b ).

The general output equation of a multi-rate system is similarly defined as (13) (13) y ( K T ) = [ ϒ [ n 1 , 1 , 0 ] , ... , ϒ [ n m , 1 , 0 ] x ( K T ) ] + [ D ^ 1 D ^ 2 ... D ^ m ] u ( K T )

which is defined in relation (14) ϒ [ k , j , i ] = [ C T b [ 1 , j − 1 ] C T b [ 2 , j ] Φ T b [ j , i ] C T b [ 3 , j ] Φ T b [ j + 1 , i ] . . . C T b [ k , j ] Φ T b [ j + k + 2 , i ] ] k > j > i

We also define D ^ k as a matrix ( n k × n k ) . (15) D ^ k = { D ^ k , i } i = 1 , ... , n k

The ith ( n k × 1 ) column vector is defined as follows: (16) D ^ k , i = I n 1 , i ϒ [ n k + 1 , i , i ] Γ T b [ i , i − 1 ]

All the equations for transferring a multi-rate system are calculated by considering how the x [ ( k + 1 ) T ] mode vector should be calculated from x ( K T ) and the series of control inputs in the [ K T , ( K + 1 ) T ] interval. These equations are able to calculate the states at all moments of baseline T b sampling.

It must be noted that for some internal sampling moments, there will be no any change in system input and output. At this point, multi-rate system matrices include phrases that represent the system’s natural response to the last input (output). Therefore, it can be said that these matrices include many terms that do not express any characteristics of input and output.

It is worth noting that the relationship (8) is non-minimal obviously, because it describes the multi-rate system with ( n − 1 ) n 0 poles at the origin and n non-zero poles, but in fact only n non-zero poles indicate the behavior of the system to their inputs and outputs. Therefore, we must obtain a method that is minimal to the stated method.

In the following, the minimal state space model for a multi-rate system with one input and one output will be discussed, noting that the input sampling rate includes several different rates and the output rate is constant.

To get the equations of a multi-input single output (MISO) system, we must first answer the question of how does x [ ( k + 1 ) T ] calculate from x ( K T ) and the control signals that generated by the u 1 input signal in the [ K T , ( K + 1 ) T ] interval.

For this system, the input signal is sampled at T 1 = T n 1 intervals and the output will be generated for each main sampling interval. Therefore, the related equation is written as (17) (17) x [ ( K + 1 ) T ] = Φ T b [ t , t 0 ] x [ t 0 ] + Γ T b [ t , t 0 ] u 1 [ t 0 ]

In which, t = ( K + 1 ) T , t 0 = ( K + 1 ) T − T b , Γ and Φ are only related to the single input single output subsystem (SISO).

Considering the T = n 1 T b sampling relationship, we will arrive at the following equation: (18) x [ ( K + 1 ) T ] = Φ [ n 1 T b , ( n 1 − 1 ) T b ] x [ t 0 ] + Γ [ n 1 T b , ( n 1 − 1 ) T b ] u 1 [ t 0 ]

If x [ ( K + 1 ) T − T b ] is written based on x [ ( K + 1 ) T − 2 T b ] , u 1 [ ( K + 1 ) − 2 T b ] and a reversal process is performed to get x [ K T ] , the state equation of the system with the T repetition interval will be obtained as follows: (19) x [ ( K + 1 ) T ] = Φ T b [ n 1 , 0 ] x ( K T ) + ∑ i = 0 n 1 − 1 Φ T b [ n 1 , ( i + 1 ) ℓ 1 ] Γ T b [ ( i + 1 ) ℓ 1 , i ℓ 1 ] u 1 ( K T , i T b )

In equation (19), ℓ 1 = n 0 n 1 and (20) Φ T b [ k , j ] = { ∏ i = j k − 1 Φ [ ( i + 1 ) T b , i T b ] k > j I n k = j 0 k < j

In (20), I n Is an Identity matrix of ( n × n ) .

It is worth noting that considering some conditions for Φ T b [ k , i ] k < l o r k < j so long as the discrete state transient can occur over time, both forward and backward, is not necessarily correct and can be also applied to the state equations of state defined in the K T < t < ( K + 1 ) T interval.

Equation (19) can also be rewritten in the following matrix form (21) x [ ( k + 1 ) T ] = Φ T b [ n 0 , 0 ] x ( k T ) + Γ ¯ ¯ T b , 1 ( n 0 ) u 1 ( k T )

In relation (21), u 1 ( k T ) is obtained from relation (22) and Γ ¯ ¯ T b , 1 ( n 0 ) Is defined as (23). (22) u i ( k T ) = [ u i [ k T ] u i [ k T + T i ] . . . u i [ k T + ( n i − 1 ) T i ] ] (23) Γ ¯ ¯ T b , 1 ( n 0 ) = [ Φ T b [ n 0 , ℓ 1 ] Γ T b [ ℓ 1 , 0 ] , Φ T b [ n 0 , 2 ℓ 1 ] Γ T b [ 2 ℓ 1 , ℓ 1 ] , ... , Γ T b [ n 0 , ( n 1 − 1 ) ℓ 1 ] ]

In relation (23), Γ ¯ ¯ T b , i ( n 0 ) expresses Γ T b that is formed by the ith column of the main control matrix of the system. For single input system i = 1 and for two inputs i = 2 .

Since the state and output vectors are defined only in the main sampling intervals, the output equation is defined as relation (24). (24) y ( K T ) = C T b [ n 0 , 0 ] x ( K T )

To achieve state equations for a system with multiple inputs at different rates, we can rewrite equation (21) as (25). (25) x [ ( K + 1 ) T ] = Φ x ( K T ) + Γ u ( K T )

Which in relation (25) (26) Φ = Φ T b [ n 0 , 1 ] & Γ = [ Γ ¯ ¯ T b , 1 ( n 0 ) ... Γ ¯ ¯ T b , m ( n 0 ) ]

Multi-rate system matrices in equation (25), use control and state matrices of multi-input multi-output.

The vector u ( K T ) is a multi-input multi-output system control vector.

The relation (26) shows that the multi-rate control matrix is a block related to the maximum value of m rates of updating of different inputs during the [ K T , ( K + 1 ) T ] interval. Therefore, we can say that relation (26) is a block description related to different update rates in the [ K T , ( K + 1 ) T ] interval.

A guided missile consists of three main parts (Farret et al., 2002): the seeker (Lee, 2019), autopilot (Lee and Tahk, 2019) and the guidance law (Sun et al., 2019) systems. In this simulation, it is assumed that the commands of the output lateral acceleration of these parts (finally the output of the auto-pilot) are sent to the canards at a constant rate of 120 ms. On the other hand, the output signal of seeker will be sampled at a rate of 40 ms and the final output will be sampled at a rate of 60 ms and sent to the guidance law subsystem. The output of the guidance law subsystem will also be sent to the autopilot at a rate of 30 ms. Block diagram of Figure 2 for the following reasons is an objective example of a multi-rate system:

Figure 2:

Existence of different input and output rates between different parts.

Existence of input signals with different rates for the base subsystem of guidance law and one output signal with a different rate of inputs.

In Figure 2, seeker, guidance law, and the autopilot are simulated according to the references. Also, after applying the explained multi-rate method and obtaining inputs and outputs, a scenario for target maneuvering and guided missile testing is considered, and after expressing the scenario, simulation results will be shown.

In the simulation results, Figure 3 shows the path of the missile and the target on the xyz coordinate and Figures 4–6 shows the missile–target engagement in x–y and x–z and y–z coordinates, respectively.

Figure 3:

Figure 4:

Figure 5:

Figure 6:

Acceleration commands for x and y and z screen over time are shown in Figures 7–9 respectively.

Figure 7:

Figure 8:

Figure 9:

This scenario is the linear movement of the target at constant altitude and speed without acceleration. The values of acceleration, velocity and target position in the three directions of x, y, and z are as follows: x t = 60.000 f t y t = 10.000 f t z t = 10.000 f t x ̇ t = − 2121 f t / s y ̇ t = − 2121 f t / s z ̇ t = 0 f t / s x ̈ t = 0 y ̈ t = 0 z ̈ t = 0 .

The simulation results are shown in Figures 3–9:

The lateral acceleration commands from the autopilot in x, y and z screen over time is shown in Figures 7–9.

As we can see, the simulation results show well that despite the use of different sampling rates in the subsystems, the missile guidance and control during the path to the target is done in the best possible optimal way with a smooth motion and it has not been affected by the existence of multi rate operation of the missile.

To validate the simulation results, it is enough to pay attention to the smooth and uniform movement of the missile along the flight path towards the target which shows the optimal performance of the proposed method. The results clearly show that along the flight path, the missile moved quite smoothly and no data was lost due to the difference in sampling rates in the system. Also, because the target maneuver is designed by the authors themselves in such a way that the maneuver is complete and similar to reality and on the other hand, the proposed method has been proposed and invented by the authors themselves and there is no similar case with these specifications for comparison and as it was stated, based on the smooth movement and no loss of data, despite the various sampling rates in the system, the optimality and acceptable performance of the proposed method can be concluded.

In process of digital system designing, choosing system performance rate, selecting of sensors and the components of processor are very important and the designer will be at a crossroads whether to act based on available performance rate or desired accuracy.

The results obtained in this study shows that by determining the rate of output and input of multi-rate systems, despite the existence of different rates at the input of different subsystems, without interfering with the performance of the whole system and losing data, the system output will be determined in one single rate. Simulation results also shows good performance and smooth movement without any data loss.