Period Assignment for Real-Time Cascade Control Tasks Under Stability and Schedulability Constraints

Many studies have been dedicated to the control-scheduling co-design problem. An early result [21] considers the optimization of global system performance using control performance index and subject to computing resources constraints. In [18], authors present an approach to optimize end-to-end timing constraints subject to performance and schedulability constraints of a real-time control system. Here periods of control tasks are modified to satisfy the objective of minimizing systems utilization while respecting the performance requirements. In [20], authors follow the same approach while considering the real-time control system subject to communication delays. [24, 25] targeted the control scheduling co-design, where the assignment of periods and priorities of multiple control applications running on the same processor is done considering some control performance metrics. Authors of [2] consider the effect of delays and jitter on the system stability and provided an approach for period assignment and scheduling scheme to guarantee stability. In [1], authors consider the problem of synthesizing timing contracts that guarantee stability and schedulability. All mentioned results considered hard-deadlines, meaning each task is required to meet all deadlines. In reality, control tasks could allow some relaxation of this constraint by allowing some deadline misses, while periods are chosen, often, as multiples of sensor periods leading to harmonic rates [22].

The relaxation of hard deadline requirements for real-time control systems is, also, considered in the literature. It is shown that the stability of control systems is not only dependent on the system dynamics and choice of periods, but also the stability depends on the number of consecutive deadline misses that the control task encounters. In [23], authors analyze stability and performance of control systems subject to burst of deadline misses, and show that stability analysis alone is not sufficient to properly quantify robustness regarding deadline misses. In [16], authors propose an adaptive control design to handle the sporadic overloads to counteract the effect on stability and performance. In [10], authors propose a new periodic fault-tolerant CPS task model, which generalize the existing task models. But also the model captures the stability and efficiency of the CPS by capturing tolerable number of control update misses. In this scheduling scheme tasks periods are considered as given.
Authors in [5] propose a first method to combine both previously mentioned working paths. This is done by proposing a model to jointly determine tasks periods and manage deadline misses. In their paper, a stable region of the physical system is presented, each point in this region corresponds to a pair of period and number of allowed consecutive deadline misses. This means in their proposed model to know that a control task can tolerate a certain number of consecutive deadline misses, its period could be chosen between a minimum and a maximum value based on the stability region.
To the best of our knowledge, no other result has been proposed to the co-design problem when the system is composed of multiple dependent control tasks in a cascade structure. So, there is no result on the effect of the dependence between control tasks on the region of stable periods and how do we quantify the maximum number of allowable deadline misses based on the region of stability.

We consider a task system $\tau$ of $n$ synchronous implicit deadline periodic tasks ${\tau }_i,\forall 1\le i\le n$ scheduled according to a preemptive fixed-priority scheduling policy on one processor $\pi$. The tasks are ordered according to their priority. A task ${\tau }_i$ has a higher priority than a task ${\tau }_j$ when . The priority assignment is given and finding such assignment is beyond the purpose of this paper. A task ${\tau }_i$ is defined by $C_i$, its worst-case execution time, $T_i$ its minimal inter-arrival time or period and $D_i$, its deadline with $D_i=T_i$. We denote by $U_i=\frac{C_i}{T_i}$ its utilization and the total utilization of the task system $\tau$ is denoted by $U={\sum }_{i=1}^n\frac{C_i}{T_i}$. We assume that all tasks are released for the first time synchronously at $t=0$.

A control system regulates or controls the behaviour of other devices or systems using control loops [7, 3, 11, 15]. There are, mainly, two types of control systems, open-loop and closed-loop control. Closed-loop control systems (also known as feedback control systems) have been the first examples of hard real-time systems [23]. In this paper, we deal with closed-loop control systems.

Cascade closed-loop control is the use of multiple control loops to control the behaviour of a system. In Figure 1, we illustrate $p$ cascade loops each containing a controller $G_j$ and a plant $P_j,\forall 1\le j\le p$. In a cascade control system, the output of the outer loop controller $G_2$ is the setpoint to the inner loop controller $G_1$. In addition, $G_1$ is faster than $G_2$ allowing the inner loop to react rapidly to fast changing dynamics which reduce the error of the outer loop.

The co-design problem is defined as the problem where the stability and the schedulability meet with the objective of finding a compromise solution for each of them. From a stability and performance point of view, one wants the control tasks to have the smallest possible periods, while this might overload the system from a scheduling point of view. So, the co-design problem requires a solution optimizing the control performance represented by Equation 1, while respecting stability and schedulability guarantees. Then, we define the optimization problem as follows:

For the control performance we choose sampling periods that decrease the control cost $J$ as much as possible. These periods should be chosen within the interval $[T_i^{min},T_i^{max}]$ which ensures the stability of the physical system. Our solution to compute this interval is described in Section 4. Also, the period of the inner loop should be smaller than that of the outer loop to reduce overall system error, where in the third condition $i$ corresponds to the inner loop and $j$ to the outer loop. Moreover, from scheduling point of view we chose periods that guarantee there is no violation of the sufficient condition of schedulability according to a fixed priority scheduling algorithm with the total utilization lower than $n(2^{\frac{1}{n}}-1)$.
This bound is introduced in [13] as follows: For a set of $n$ tasks scheduled with fixed priority, and the restriction that the ratio between any two request periods is less than $2$, the least upper bound to the processor utilization factor is $U=n(2^{\frac{1}{n}}-1)$.
For evaluation purposes and when $n\to \mathrm{\infty }$, $n(2^{\frac{1}{n}}-1)\to 0.69$, since $n(2^{\frac{1}{n}}-1)$ is a decreasing function with respect to the number of tasks, then for any number of tasks $0.69$ is a sufficient bound [13].

To illustrate our definitions and results, we consider an example of tasks set ${\tau }_{ex}$ with $9$ tasks ordered according to their priorities associated to the use-case described in Section 6. A task ${\tau }_i$ has a higher priority than a task ${\tau }_j$ when . Tasks are scheduled on one processor according to a preemptive fixed priority scheduling algorithm. Their parameters are described in Table 1. The worst-case execution time $C$ is obtained as the largest observed execution time during $5$ minutes of functioning for the associated use-case. The task system contains $9$ tasks, $3$ of which are control tasks. The relation between control tasks is described by cascade control loops in Figure 2. As for non-control tasks, we do not present them in Figure 2 because we consider that the output of the plant is complete and no data could be missed. Whereas in real implementation we use sensors to capture the behaviour of the system, in this case data should be estimated, filtered and corrected and this is the difference between real implementation and simulations.

Plants of real applications are non-linear, but for control design purposes usually the plant dynamics are linearized around an operation point. For our general stability analysis we consider two plants $P_1$ and $P_2$ described by transfer functions of the general form:

where $a_i$, $b_i$ and $c_i$ are constants derived from the dynamics of the physical plant as described in Section 6.1 (see Equation 14).
The controllers $G_1$ and $G_2$ are assumed to be PID controller and its general transfer function is as follows:

Jury’s stability is a criterion for discrete-time systems that is applied to the characteristic equations that are a function of $z$ to check there stability. Characteristic equation is the denominator of the closed-loop transfer function. Assuming a general characteristic equation of the form:

Jury’s stability provides necessary and sufficient conditions to ensure that $P(z)$ has no roots outside the unit circle.

A first step to control design or stability analysis of any control system is to derive its dynamics representation. As seen in the use-case the targeted system is a drone, so we start with a general drone representation. The dynamics of the drone consists of two parts translational dynamics and rotational dynamics. The dynamics could be derived from the physical laws and has been exhaustively studied in control theory.
It follows from [12, 26], that the dynamics of the drone should be studied with respect to two coordinates, earth coordinate which is the general one and drone body coordinate which is fixed to the drone body and moves with it. The translational position of the drone is studied with respect to the body frame, then transformed into the earth frame using a transformation matrix. Then we get the translational acceleration of the drone as follows:

Similarly the rotational position is also first derived in the body frame of the drone and then transformed to the earth frame. Here we present $p,q,r,\dot{p},\dot{q},\dot{r}$ the angular velocity and acceleration of drone around $x,y,z$ axis under body frame. Where as the roll, pitch and yaw: $\varphi ,\theta ,\psi ,\dot{\varphi },\dot{\theta },\dot{\psi }$ are angular velocity and angular acceleration of drone in the earth frame:

Generally the movement of the drone is controlled by varying the speed of its propellers. Here we represent the control outputs $u_i$ in terms of $K_T$ and $K_Q$ the thrust and drag coefficients, and $\mathrm{\Omega }$ the propeller speed. The $u_i$ would then be computed by the control tasks, and the propeller speeds would have to be calculated from them for a real implementation.

As could be noticed from the model the system is a nonlinear one. So for the design of the controller and stability analysis, we linearize the equations presented above. From [26] the linearized equations are as follows:

In order to reproduce a similar behavior to the PX4 autopilot, we aim to analyse three controllers:

• $\mathrm{■}$

  Position Control which controls both position and velocity of the translational motion $x,y,z$ and $\dot{x},\dot{y},\dot{z}$ respectively.

• $\mathrm{■}$

  Attitude Control which controls the angular velocity of the drone $\varphi ,\theta ,\psi$.

• $\mathrm{■}$

  Rate Control which controls the angular acceleration $\dot{\varphi },\dot{\theta },\dot{\psi }$.

Starting from the inner-most loop, which is the Rate Control, this controller takes as input setpoint the targeted angular acceleration values provided by the Attitude Control. The inner-most loop transfer functions from the linearized equations 13 also called the open-loop transfer functions are:

These three angular rates have similar dynamics, so we will do the representation for one of them. So, for one of the dynamics of Equation 14, the closed-loop equation becomes:

Now discretizing this equation as seen in Section 4.1 we obtain the following discrete closed-loop transfer function:

As a reminder, we have seen that the outer loop depends on the parameters of the inner loop, so that the closed-loop transfer function of the Attitude Control would depend on its period $T_2$ and the period of the Rate Control $T_1$ and so on. Then the closed-loop transfer functions of Attitude Control and Position Control could be represented as follows:

We implement the dynamics represented in equations 16 and 17, and the procedure of extracting the stability regions for the control tasks periods depicted in Figure 3 by using Maple ²²2Maple [14] is a powerful computer algebra system used for symbolic computation and mathematical modeling.. We present the results of this analysis in Figure 6, where the blue regions are the period combinations that make the physical system stable. We see that the periods $T_1$ and $T_2$ of the Rate Control and Attitude Control reaches a maximum of around $80ms$, whereas $T_3$ of the Position Control could tolerate a period of $120ms$ for some period combinations. This could be due to the fact that rotational dynamics should be acted upon more frequently than the translational dynamics.

Evaluating the performance of the control system represented by Equation 1 should be done during the execution of the system. For this we use Simulink [9] to monitor the step response of the system and measure its performance. The performance is done over all the stable combinations of $T_1$, $T_2$ and $T_3$. For the ease of representation we fix the value of $T_1$ and plot the variation of the cost $J$ with respect to the periods $T_2$ and $T_3$, the results are depicted in Figure 7. We can notice from the plots that the cost exhibits a general increasing trend as periods increase, the presence of some oscillations could be due to the discretization effect where some periods align better with the system dynamics resulting in lower cost. For this reason we consider the optimization since it is not always guaranteed that the lowest stable periods will result in the best performance.

After obtaining the stable periods and their corresponding performances it is time to evaluate the co-design problem presented in Section 2.1. From Table 1 we can see that the total utilization of the PX4 tasks excluding the three control tasks is $U_{nc}=0.59$ (non-control utilization), the output of the co-design problem is depicted in Table 4 Case $1$ which shows the values of the three periods that keeps the total utilization $U_{total}$ less than $0.69$ while having a low control cost. In an attempt to see what is the effect of adding more load to the system, we increase the value of $U_{nc}$ to $0.68$ (Case $2$) and we re-evaluate the co-design problem. Increasing the utilization is equivalent to assuming a multi-objective optimization where we add $U$ in the minimization and give it higher weight than the control cost J, this triggers period relaxation while allowing some performance degradation. We see that the periods are more relaxed compared to Case $1$ to account for the increased utilization, while we notice that the increase in the cost is not critical this is due to the good choice of controller parameters³³3More discussion on this in Section 8.2..

If we implement the $3$ controllers into a single program (task), this is equivalent to say that we run the $3$ controllers with the same period. To validate the behavior of such case, we check the step response of the system and performance when the period is set to be:

1. (a)

   equal to the shortest possible period for the $3$ controllers to be stable,

2. (b)

   equal to the longest possible period for the $3$ controllers to be stable.

We show in Table 5 the performance and total utilization of the two cases. For Case $a$ we see that it provides a slightly better performance than the optimal assignment seen in Case $1$ of Table 4. But this case wasn’t considered because it violates the condition on the allowable total utilization ($U_{total}$). Case $b$ shows good total utilization but a slightly higher cost $J$. It could be noticed from the step response of the two cases in Figure 8, that the two responses coincide which ensures that the performance is not changing by much thanks to a good choice of system parameters.

A question that could be asked, why don’t we assign periods to each controller separately as if they were independent from one another? To answer this question we do stability analysis and check if cascaded loop sampling periods impact each other. We start to solve for the period of the inner-most loop, i.e. the Rate Control in our case, by solving the Jury’s conditions as discussed in Section 4. We obtain that the inner-most loop by itself is stable for the values of period $T_1$ in the range $1ms$ to $750ms$. When we add the second loop, i.e. Attitude Control, we see that these values of $T_1$ are stable for small values of $T_2$, while for small values of $T_1$ the period of the second loop $T_2$ could reach a value of more than $1s$, see Figure 9(a) (red region). After adding the last loop, i.e. Position Control, we notice that the region of stability becomes much more constrained as in Figure 9(a) (blue region). If we zoom on this region, then we obtain Figure 9(b). So, yes there is an important impact of the cascade structure on the sampling period, we’ve seen a drop from $750ms$ to around $80ms$ for $T_1$ and a drop from around $1s$ to $80ms$ for $T_2$.

In this section, we discuss the importance of the controller parameters choice during the first step of the analysis. For this we attempt to change the integral gain of the velocity controller $K_i=0.4$ and the proportional gain of the position controller $K_p=0.01$, and repeat the whole process of evaluation we have seen in Section 7. If we compare Case $1^{\prime }$ of Table 6 with Case $1$ of Table 4, we see that although here the set of optimal periods is lower than that of Case $1$ the cost of the control is much higher. This could be seen on the plot of the step response of the two cases Figure 10, where the response $1^{\prime }$ converges slowly to the required setpoint whereas response $1$ is much faster.
Another observation is that if the controller parameters are badly chosen as in this case, relaxing the periods will lead to much higher cost where we see for Case $2^{\prime }$ an increase of $45\%$ in the cost with respect to Case $1^{\prime }$. Whereas when we had good parameters in the previous evaluation relaxing the control tasks periods did not show this drastic impact (Table 4).

Similarly if we replicate what is done in Section 7.4 for the current parameters we obtain the results in Table 7. Since as we said, the parameters are not chosen correctly in this section the choice of periods impact a lot the performance of the system. This could be seen from Figure 11, more precisely the “Zoomed In” part where we see that the response of Case $a^{\prime }$ will eventually converge faster than that of Case $b^{\prime }$.