3.1. Fleet description

We consider a fleet of vehicles characterised as in Lafferriere, Williams, Caughman, and Veerman (2005), where p vehicles with the same dynamics¹ are considered: (4) ${\dot{x}}_i(t)=A_{veh}^c{x}_i(t)+B_{veh}^c{u}_i(t),i=1\dots p,$(4) where the entries of $x_i\in {\mathbb{R}}^{2r}$ represent r configuration variables for vehicle i and their derivatives, and $u_i\in {\mathbb{R}}^r$ represents control inputs. As in Lafferriere et al. (2005), it is further assumed that matrices $A_{veh}^c$ and $B_{veh}^c$ have the form: (5) $\begin{array}{rl}{A}_{veh}^c& =\mathrm{diag}\{\left[\begin{array}{cc}0& 1\\ a_{21}^1& a_{22}^1\end{array}\right],\dots ,\left[\begin{array}{cc}0& 1\\ a_{21}^r& a_{22}^r\end{array}\right]\},\\ B_{veh}^c& =I_r\otimes \left[\begin{array}{c}0\\ 1\end{array}\right],\end{array}$(5) where ⊗ stands for the Kronecker product of two matrices. This model corresponds to the situation in which the individual configuration variables are decoupled and the acceleration is controlled separately. The odd-numbered entries of $x_i$ represent position-like variables, whereas the even-numbered entries represent velocity-like variables.

The previous dynamics are discretised with sampling time $T_s$ and additive disturbances are included: (6) $\begin{array}{rl}{x}_i(k+1)& =A_{veh}{x}_i(k)+B_{veh}{u}_i(k)+w_i(k),\\ & i=1\dots p,x_i\in {\mathbb{R}}^{2r},\end{array}$(6) where $A_{veh},B_{veh}$ correspond to the discrete-time version of $A_{veh}^c,B_{veh}^c$. The disturbances $w_i(k)\in {\mathbb{R}}^{2r}$ can be caused by various reasons, such as non-modelled motor dynamics, lack of accuracy in certain actuators, wind (aerial or ground vehicles), currents (water vehicles) and so on. Throughout the paper, we consider that these disturbances are unknown, but they always belong to known bounded sets, that is, $w_i(k)\in {\mathcal{W}}_i$. These sets are modelled as (7) ${\mathcal{W}}_i=⟨0,Q_i⟩.$(7) Let $x(k),u(k),w(k)$ be defined by stacking all the state vectors, control actions and disturbances of each vehicle, that is, $x(k)=[x_1(k{)}^T{x}_2(k{)}^T\dots x_p(k{)}^T{]}^{\mathrm{T}},u(k)=[u_1(k{)}^T{u}_2(k{)}^T\dots u_p(k{)}^T{]}^{\mathrm{T}},w(k)=[w_1(k{)}^T{w}_2(k{)}^T\dots w_p(k{)}^T{]}^{\mathrm{T}}$. Then, the dynamics of the complete fleet can be written as follows: (8) $x(k+1)=Ax(k)+Bu(k)+w(k),$(8) where $A\triangleq I_p\otimes A_{veh}$ and $B\triangleq I_p\otimes B_{veh}$.

3.1.1. Unicycle model for the vehicles

The presented formulation of the vehicles as linear systems in (4), although simple, can be used for more complicated models. In particular, this section summarises the results in Lawton, Beard, and Young (2003), in which they present an output feedback linearising control for the control of the hand position of a vehicle, whose dynamics can be written as a double integrator.

The hand position of the vehicle is a point $h_i=[h_i^x,h_i^y]$ that lies a distance $L_i$ along the line that is normal to the wheel axis and intersects the wheel axis at the centre point $b_i=[b_i^x,b_i^y]$, as shown in Figure 1. As it is well known, the dynamical equations of the unicycle model are: (9) $\left[\begin{array}{c}{\dot{b}}_i^x\\ {\dot{b}}_i^y\\ {\dot{\theta }}_i\\ {\dot{s}}_i\\ {\dot{\omega }}_i\end{array}\right]=\left[\begin{array}{c}{s}_i\cos ({\theta }_i)\\ s_i\sin ({\theta }_i)\\ {\omega }_i\\ 0\\ 0\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ \frac{1}{m_i}& 0\\ 0& \frac{1}{I_i}\end{array}\right]\left[\begin{array}{c}{F}_i\\ {\tau }_i\end{array}\right],$(9) where $b_i=[b_i^x,b_i^y{]}^{\mathrm{T}}$, ${\theta }_i$, $s_i$, ${\omega }_i$ are, respectively, the inertial position, orientation, linear speed and angular speed of vehicle i. The mass and moment of inertia of the vehicle are $m_i$ and $I_i$. The input signals are the torque ${\tau }_i$ and the force $F_i$.

Guaranteed estimation and distributed control of vehicle formations

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R. A. García, L. Orihuela, P. Millán, F. R. Rubio & M. G. Ortega

https://doi.org/10.1080/00207179.2020.1714074

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Figure 1. Unicycle model for the vehicle used for controlling the hand position.

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Figure 1. Unicycle model for the vehicle used for controlling the hand position.

Letting $z_i=[b_i^x,b_i^y,{\theta }_i,s_i,{\omega }_i{]}^{\mathrm{T}}$ and ${\nu }_i=[F_i,{\tau }_i{]}^{\mathrm{T}}$, the dynamical equations can be easily written as (10) ${\dot{z}}_i=f(z_i)+g_i{\nu }_i.$(10) Considering the hand position of each vehicle as output, it is shown in Lawton et al. (2003) that system (10) has constant relative degree equal to 2 and can, therefore, be output feedback linearised about the hand position. Now, by using the next output feedback linearising control: (11) $\begin{array}{rl}{\nu }_i& ={\left[\begin{array}{cc}\frac{1}{m_i}\cos ({\theta }_i)& -\frac{L_i}{I_i}\sin ({\theta }_i)\\ \frac{1}{m_i}\sin ({\theta }_i)& \frac{L_i}{I_i}\cos ({\theta }_i)\end{array}\right]}^{-1}\\ & \times (u_i-\left[\begin{array}{c}-s_i{\omega }_i\sin ({\theta }_i)-L_i{\omega }_i^2\cos ({\theta }_i)\\ s_i{\omega }_i\cos ({\theta }_i)-L_i{\omega }_i^2\sin ({\theta }_i)\end{array}\right]),\end{array}$(11) the input–output dynamics of each vehicle can be represented by the double integrator system: (12) ${\ddot{h}}_i=u_i.$(12) It should be mentioned that the internal dynamics, related to the orientation of the vehicle itself, are unobservable and uncontrollable. However, it is also shown in Lawton et al. (2003) that this internal dynamics is stable, but not asymptotically stable. Therefore, we will assume that the orientation of each vehicle is locally measured.

3.2. Agent description

It is assumed that each vehicle is equipped with an agent that has sensing, control, computational and communication capacities. In what respect to the sensing ability, each agent i is assumed to measure the following output: (13) $y_i(k)=C_{i}x(k)+v_i(k),$(13) where $y_i\in {\mathbb{R}}^{m_i}$ is the output vector, $C_i$ is the output matrix and $v_i$ represents the measurement noise. It is assumed that the noises are unknown, but they always belong to known bounded sets, that is, $v_i(k)\in {\mathcal{V}}_i$. These sets are modelled as (14) ${\mathcal{V}}_i=⟨0,R_i⟩.$(14) We distinguish two kinds of agents: (a) those who are able to measure the absolute position-like variables of its local vehicle (by using GPS-like sensors or similar) and (b) those who measure relative position-like variables (distances, with optical or acoustic sensors). Both kind of agents are assumed to get information concerning its local velocity-like variables. Matrix $C_i$, as supposed, is structurally different for each kind of agent.

The agents are endowed with communication devices that let them send/receive information to/from their neighbours. The communication topology is mathematically described by a graph G. It is assumed that two possible topologies are valid, chain or star. This assumption will be required later to solve the problem of distributed guaranteed estimation. For instance, for a four vehicle fleet, the two possible schemes are depicted in Figure 2.² The arrows represent the communication between the agents.

Guaranteed estimation and distributed control of vehicle formations

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R. A. García, L. Orihuela, P. Millán, F. R. Rubio & M. G. Ortega

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Figure 2. Scheme of the formation of the vehicles with the communication between agents: (a) Chain and (b) Star.

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It is worth mentioning that those topologies only apply for the communication, not for the very formation of the vehicles. It will be shown in the examples that both topologies can be used to keep different formations.

By using the available information, that is, the measured outputs and the information received from the neighbourhood, the agents must compute estimation sets ${\hat{\mathcal{X}}}_i(k|k)$ within which the state must be contained. The construction of this estimation set is described in Section 4.

Regarding the control of the vehicle, it is considered that each agent computes and applies the control signals to its local vehicle. The control signal applied by agent i to the vehicle is chosen as (15) $u_i(k)=K_i{c}_i(k|k),$(15) where the design of matrix $K_i$ will be tackled in Section 5.

3.3. Problem statement

The goal of control formation problem is to keep a prescribed position for the vehicles. The global reference to be tracked $z_{ref}(k)\in {\mathbb{R}}^{r\times p}$ is assumed to be known by each vehicle. Now, the problem can be formally stated.

Compared to Orihuela et al. (2017), the estimation objective is here more complex, since in that paper the agents' goal was to estimate the state of its associated subsystem $x_i(k)$. The ignorance of the control actions applied by the rest of agents, that is, $u_j(k)$ for $j\ne i$, does not lead to a difficulty in the estimation problem formulated in Orihuela et al. (2017). However, in Problem 1, this fact appears as the main complication.

4.1. Main obstacles and proposed solutions for the joint distributed control and estimation problem

When facing the joint problem of distributed estimation and control, the control engineer finds two main drawbacks:

1. Estimating the whole state with partial information: This difficulty is common in the literature of distributed estimation, no matter the sort of estimator used. The output measured by an agent does not contain enough information to estimate the whole state vector. Although the mathematical details vary between the different formulations, the solution typically consists in introducing the information received from neighbouring agents in the formulation of a centralised observer. For instance, distributed Kalman filter formulations modify the classical Kalman filter by adding the neighbouring information using consensus, as in Olfati-Saber (2007), Das and Moura (2015), Battistelli and Chisci (2016), del Nozal et al. (2017) or with diffusion strategies, see Cattivelli and Sayed (2010), Wang et al. (2017). Consensus algorithms have been also used to extend the Luenberger observer to a distributed formulation in Matei and Baras (2012), del Nozal et al. (2019), Mitra and Sundaram (2018), or for the distributed the Bayesian filter, see Forti et al. (2018).

2. The literature of distributed set-membership estimation is not that rich. However, some interesting solutions for this problem can be found. The information received from the neighbourhood can be incorporated in the estimated set by intersection, see Keiffer (2009), García et al. (2017), or by means of linear combinations of the sets, as in Orihuela et al. (2017), Combastel and Zolghadri (2018). Previous papers described the sets using zonotopes. Recently, it has been proposed the use of consensus techniques for distributed estimation with ellipsoidal sets, see Ma et al. (2017).

3. Predicting with an unavailable control signal: This problem only appears when the joint distributed and estimation scheme is tackled at the same time. When a local observer has to make predictions, it must incorporate, at least, two terms: the dynamics of the system and the control actions applied to it. Although a perfect knowledge of the dynamics is assumed, how can an individual agent know the actual control actions if they might be applied by agents that might be geographically distributed?

4. The solutions adopted in Millán et al. (2013), Rubio et al. (2014), Orihuela, Millán, Vivas, and Rubio (2016), Zhang et al. (2017) present different formulations of a cascade structure, where the estimation errors must converge to zero to achieve the control objectives. In all these papers, the authors assume that the control actions applied to the system at any moment are the same that the local agent would apply if it has access to all control channels. This incurs in an error that disappears when the estimated states of every agent converge to the system state.

5. This idea cannot be applied directly to set-membership paradigms, since the described assumption will lead to non-guaranteed estimations in the transient. The literature of distributed control and set-membership estimation is rather scarce, and it is limited to formulations of the plant divided into subsystems for which the estimation objectives are just local, see Orihuela et al. (2017), Orihuela et al. (2018), Combastel and Zolghadri (2018).

4.2. Proposed guaranteed estimation algorithm

This section presents the distributed estimation algorithm that will be executed by every agent deployed in the vehicles. In order to save the obstacle described before for the joint control and estimation problem, the proposed solution makes use of two key observations.

Recall that ${\hat{\mathcal{X}}}_i(k|k)$ stands for the estimation set (described by a zonotope) computed by agent i at instant k with all the information available at instant k. Let us denote by ${\mathcal{Z}}_i(k)$ to the zonotope sent by agent i to all its neighbours. This zonotope differs, in general, with the estimation set computed by agent i. Then, if agent j is going to use the zonotope received from agent i at instant k to obtain an estimation set ${\hat{\mathcal{X}}}_j(k|k)$ by intersection with ${\mathcal{Z}}_i(k)$, then agent i knows that the estimation set of agent j at instant k, at least, belong to ${\mathcal{Z}}_i(k)$. Hence, when agent j computes and applies its local control action to the plant based on (15), agent i will know a region into which the actual control action applied by j is guaranteed to belong. This region will be used to construct the guaranteed prediction sets.

Nonetheless, observe that the previous mechanism is only valid for direct neighbours. To be applicable in general, an additional observation is required. When agent i sends a zonotope ${\mathcal{Z}}_i(k)$ at instant k, the information carried in this set spreads through the network in direct paths (without loops) and will be received by every agent, at most, at instant $k+\overline{d}-1$. This is true due to the particular topologies considered. Since every agent $j\ne i$ will incorporate this information to construct their estimation set ${\hat{\mathcal{X}}}_j(k+\alpha |k+\alpha )$ by intersection, for some $\alpha \le \overline{d}-1$, then agent i will be able to compute a possible control region for the control action applied by agent j at instant $k+\alpha$ by means of open-loop predictions.

Before introducing the mathematical details, an illustrative example is presented that will surely help to clarify these ideas.

Example 4.1

Consider p = 3 vehicles with associated agents connected with a chain topology $1\leftrightarrow 2\leftrightarrow 3$. Consider that agent 3 applies a control action at instant k + 1. Agent 1 requires to compute a region that contains for sure this control action $u_3(k+1)$.

Figure 3 depicts, by using illustrative colours and shapes, the way the information spreads through the network. The zonotope sent by agent 1 at instant k (blue circle) affects the estimation set computed by agent 2 (brown triangle). Then, after an adequate evolution of this set (big brown triangle), it will affect the estimation set of agent 3 at instant k + 1. Since the information of each set is used by means of intersections, agent 1 will be able to evolve its estimation set at instant k (green rectangle) in open loop to obtain a predicted set (big green rectangle) that contains the actual control action $u_3(k+1)$.

Guaranteed estimation and distributed control of vehicle formations

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R. A. García, L. Orihuela, P. Millán, F. R. Rubio & M. G. Ortega

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Figure 3. Illustrative example of the transmission of information through the network. Intersections and evolutions of the sets are not pretended to be exact.

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Figure 3. Illustrative example of the transmission of information through the network. Intersections and evolutions of the sets are not pretended to be exact.

The complete control and estimation algorithm performed by every agent has five main steps. Next, we provide a description of this algorithm, which has been summarised in Table 1.

1. Measurement: Using the measured outputs, $y_i(k)$ in (13), the agents compute a strip of consistent states with each component of this vector, $S_i^l(k)$, for $l=1\dots ,m_i$. These strips are defined as in Section 2. In particular, considering for example a unidimensional output, the strip will be given by $S_i(k)=\{x(k):|C_{i}x(k)-y_i(k)|<R_i\}$, where $R_i$ is related with the output noise, see (14). Note that the width of this strip directly grows with the energy of the measurement noise. Then, the prediction zonotope available from the last time step ${\hat{\mathcal{X}}}_i(k|k-1)$ is intersected with every $S_i^l(k)$, obtaining ${\mathcal{Z}}_i(k)$ as a final result. The intersection between a strip and a zonotope can be written as a zonotope, as was explained in Section 2. Finally, an order reduction operation is executed to obtain zonotope $red({\mathcal{Z}}_i(k))$.

2. Communication: At this step of the algorithm the agents send the reduced-order zonotopes obtained before to all its neighbours. After that communication, each pair of connected agents computes the intersection zonotope ${\mathcal{I}}_{ij}(k)\triangleq red({\mathcal{Z}}_i(k))\cap red({\mathcal{Z}}_j(k))$, which allows them to reduce the estimation uncertainty. Observe that ${\mathcal{I}}_{ij}(k)={\mathcal{I}}_{ji}(k)$, as they are related to the same edge in an undirected graph. As explained in Section 2, the intersection between zonotopes can be written as a zonotope.

3. Estimation: By including the information of every edge (in other words, of every neighbour), the agents obtain the best estimation set with the information available at instant k, that is, ${\hat{\mathcal{X}}}_i(k|k)$.

4. Control: The control action applied by agent i to its vehicle is obtained using the centre of the estimation set according to (15).

5. Prediction: Finally, the prediction sets ${\hat{\mathcal{X}}}_i(k+1|k)$ are computed as (16) ${\hat{\mathcal{X}}}_i(k+1|k)=A{\hat{\mathcal{X}}}_i(k|k)\oplus B{\hat{\mathcal{U}}}_i(k)\oplus \mathcal{W}.$(16) Three terms appear in the previous equation, one for each term in (8): (1) the open-loop dynamics of the system, with the term $A{\hat{\mathcal{X}}}_i(k|k)$; (2) the set of possible control signals applied to the plant, $B{\hat{\mathcal{U}}}_i(k)$; and (3) the set of all possible disturbances, given by $\mathcal{W}\triangleq [{\mathcal{W}}_1^T{\mathcal{W}}_2^T\cdots {\mathcal{W}}_p^T{]}^{\mathrm{T}}$. The set ${\hat{\mathcal{U}}}_i(k)$ should contain every control action applied to the plant at instant k. Next, we present a method to compute this set.

Note that $Bu(k)$ can be written as $\sum _{m=1}^p{B}_i{u}_i(k)$, where $B_i$ is the submatrix of B for the control channels of agent i. With a little abuse of notation, let us denote ${\hat{\mathcal{U}}}_i(k)=[{\hat{\mathcal{U}}}_i^m(k)]$, where ${\hat{\mathcal{U}}}_i^m(k)$ is a set computed by agent i that must contain the vector $u_m(k)$, and it is given by (17) ${\hat{\mathcal{U}}}_i^m(k)\triangleq \{\begin{array}{ll}(a):& \text{if }m=i\\ & K_i{c}_i(k|k)\\ (b):& \text{if }m\ne i,m\in {\mathcal{N}}_i\\ & K_m{\mathcal{I}}_{im}(k)\\ (c):& \text{otherwise}\\ & K_m{\hat{\mathcal{I}}}_{im}(k|k-d_{im}+1),\end{array}$(17) being ${\hat{\mathcal{I}}}_{im}(k|k-d_{im}+1)$ the prediction of the intersection zonotope computed as (18) $\begin{array}{rl}{\hat{\mathcal{I}}}_{im}(k|k-d_{im}+1)& =A^{d_{im}-1}{\mathcal{I}}_{ij}(k-d_{im}+1)\\ & \oplus \underset{n=1}{\overset{d_{im}-1}{⨁}}{A}^{n-1}B{\hat{\mathcal{U}}}_i(k-n)\\ & \oplus \left[\sum _{n=1}^{d_{im}-1}{A}^{n-1}\right]\mathcal{W},\end{array}$(18) where $j\in {\mathcal{N}}_i$ is a neighbour of agent i such that there exists a path from m to j. Observer that the zonotope ${\hat{\mathcal{I}}}_{im}(k|k-d_{im}+1)$ represents the open-loop evolution of the zonotope ${\hat{\mathcal{I}}}_{ij}$ for $d_{im}-1$ time instants, which is the number of instants required for the zonotope ${\hat{\mathcal{I}}}_{ij}$ to reach agent m.

Please note that for $m\in {\mathcal{N}}_i$ expression (17)(b) matches (17)(c), for $d_{im}=1$. Nonetheless, three cases have been defined to distinguish between three important situations: $(a)$ agent i applies its local control signal $u_i$; $(b)$ agent i is a neighbour of the agent with access to the control channel and $(c)$ agent i is not neighbour of the agent applying $u_m$.

Next theorem states that, with the proposed algorithm, objective (1) of Problem 1 is met, that is, every agent is able to compute estimation sets containing the actual state vector of the whole fleet of vehicles.

The hypothesis of Theorem 1 is standard for induction-based proofs. Moreover, they can be easily fulfilled since the initial sets, namely ${\hat{\mathcal{X}}}_i(k_0|k_0-1),{\hat{\mathcal{X}}}_i(k_0-t|k_0-t),{\hat{\mathcal{U}}}_i(k_0-t|k_0-t)$ for $t=1,2,\dots ,d_{max}$, can be trivially enlarged so that they contain for sure the required state vector. Another initialisation scheme consists in a preliminary execution of the algorithm for $d_{max}$ time instants with $u_i(k)\equiv 0$ (open-loop dynamics). Then, it will be easily satisfied that $u(k_0-t)\in {\hat{\mathcal{U}}}_i(k_0-t)$ for $t=1,2,\dots ,d_{max}$. After these initial steps, the algorithm can run in its proposed manner.

Theorem 4.1 proves that every agent is able to find sets that contain for sure the actual state for the plant. However, it did not directly mention that these sets are found in an optimal way. In fact, the reader might think that the sets could grow until infinity, in such a way that the theorem would be fulfilled but the estimation would be inadequate.

Nevertheless, two main facts prevent the uncontrolled growing of the estimation sets. On the one hand, the mathematical operations involving zonotopes, such as intersection with strip, intersection between zonotopes and order reduction, are computed using published methods that ensure the optimality of those operations (see Section 2 and the references therein). On the other hand, under standard observability assumption, all the outputs collected by the whole fleet contain enough information to estimate the complete state. Since the algorithm is based in the inclusion of this information by means of intersections, the estimation sets are always kept under some bounds.

Finally, it is worth mentioning that the prediction step (Step 5 in Table 1) must be deeply analysed in the future, since its performance can be improved if it takes into consideration the particular topology. It is left for future research.

6.1. Example 1

We consider a fleet of four vehicles, being the vehicle 1 the one that is able to measure its absolute position, and the rest measure relative positions. We will compare both topologies considered in the paper. The communication topology, then, will be different in each case:

1. ${\mathrm{veh}}_1\leftrightarrow {\mathrm{veh}}_2\leftrightarrow {\mathrm{veh}}_3\leftrightarrow {\mathrm{veh}}_4$.

2. ${\mathrm{veh}}_1\leftrightarrow {\mathrm{veh}}_2$, ${\mathrm{veh}}_1\leftrightarrow {\mathrm{veh}}_3$, ${\mathrm{veh}}_1\leftrightarrow {\mathrm{veh}}_4$.

This example will show that the fleet is able to keep a given formation (which indeed is different to a linear or star formation) when the proposed algorithm is used. In addition, all the agents will be able to compute adequate estimation sets for the whole fleet. We assume, as in Lafferriere et al. (2005), that the disturbances affecting the model are equal to zero. Measurement noises are, however, taken into consideration. It is also assumed that the centre of the initial zonotopes is located exactly where the initial state of the vehicles is.

Without loss of generality, the reference trajectory is divided into four waypoints (see Section 5). Figures 4(a,b) depict, for both situations, the actual trajectories of the vehicles on the plane xy using solid lines. The initial position is marked with a circle, and the final position with a triangle. The references are represented in dashed lines. It can be seen that the proposed control formation obtains very nice results. No remarkable differences can be found between both scenarios in what respect to control formation.

Guaranteed estimation and distributed control of vehicle formations

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Figure 4. Trajectory of the four vehicles considering both communication topologies: (a) Chain and (b) Star.

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On the other hand, the estimation performance can be checked in Figure 5(a,b). The figures depict the horizontal position of the vehicle 1 (black solid line), its reference (blue solid line) and the estimation that every agent computes. Note that the zonotopes represent an interval when considering a unique variable (in this case, the horizontal position), and so, minimum and maximum values for the state are depicted by each agent (dashed lines of different colours).

Guaranteed estimation and distributed control of vehicle formations

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R. A. García, L. Orihuela, P. Millán, F. R. Rubio & M. G. Ortega

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Figure 5. Vehicle 1, actual horizontal position, reference and estimation of the position x by all agents: (a) Chain and (b) Star.

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It is easy to check that the state is always contained within the sets that all agents compute, this showing that the first requirement of a guaranteed estimator is fulfilled. Figure 5 also shows that the settling time of the tracking is around 40 sampling instants (4 s) with the chosen controller for both cases.

In addition, the example illustrates the effect of the distance in the estimation when the chain topology is considered, in the sense that the farther an agent is to the agent measuring the particular state (in this case, agent 1), the wider its bounds are. The centre of the zonotopes (the best estimation) does not suffer from this effect. On the contrary, for the star topology, all the agents can be considered symmetric and, therefore, they obtain a similar performance.

6.2. Example 2

This simulation example shows that the proposed distributed estimation and control method can be used for the nonlinear model of the unicycle, as long as the control of the hand is pursued.

We will assume, for the sake of simplicity, that the masses, moments of inertia and hand positions are the same for every vehicle, this is, $m_i=1,I_i=1,L_i=1,\mathrm{\forall }i$. Furthermore, the bounds of noises and disturbances are also the same, ${\mathcal{W}}_i=⟨0,0.05⟩,{\mathcal{V}}_i=⟨0,0.1⟩$. A chain topology is considered for the communication between the vehicles.

In order to compute the output feedback linearising control (11), it is assumed that each vehicle is equipped with an additional sensor that measures its local orientation ${\theta }_i$. The rest of variables, namely, $s_i,{\omega }_i$ can be obtained with the information of the estimation sets. The local cost functions in (23) are defined with ${\mathrm{\Phi }}_i=0.1I,{\mathrm{\Theta }}_i=I,\mathrm{\forall }i$.

Figure 6(a,b) depicts, respectively, the position of the hand and the position of the vehicle. Starting for different initial positions, the fleet is trying to join their hands at the origin. These figures effectively show how the proposed method is still valid for the control of the hand position of the fleet. The effect of the internal dynamics is clear in Figure 6(b).

Guaranteed estimation and distributed control of vehicle formations

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R. A. García, L. Orihuela, P. Millán, F. R. Rubio & M. G. Ortega

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Figure 6. Position of the hand and vehicle considering the unicycle model: (a) hand position and (b) vehicle position.

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Figure 6(b) also shows the bounds for the coordinate y obtained with the set-membership estimator. Similarly that in the previous example, wider bounds are obtained for the vehicles located far from the leader. Although it is possible to obtain bounds for the coordinate x as well, we have not included them in the figure to make it clearer for the reader. Note that the bounds could be used for collision avoidance. In particular, vehicles 2 and 3 might have a collision after some time when their bounds intersect each other.

6.3. Example 3

In this new example, we study the influence that the disturbances have in the estimation performance. The analysis is made considering again a chain communication topology, which increases the uncertainty. Since we are interested in the estimation bounds, we will assume that initial zonotopes are centred in the actual initial state of the vehicles.

Two scenarios are compared in Figure 7. In both cases, the observers consider that the disturbances might affect the vehicles, so zonotopes ${\mathcal{W}}_i=⟨0,0.2I⟩,\mathrm{\forall }i,$ in (7) are included in the prediction step. Whereas Figure 7(a) corresponds to the case in which no actual disturbances are applied, Figure 7(b) shows the case in which the disturbances are actually affecting the dynamics.

Guaranteed estimation and distributed control of vehicle formations

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R. A. García, L. Orihuela, P. Millán, F. R. Rubio & M. G. Ortega

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Figure 7. Comparative in the chain communication scenario between the simulation without disturbances and with them: (a) without disturbances and (b) with disturbances.

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It can be seen that the proposed control and estimation algorithm is able to track the prescribed reference in spite the fact that the estimation is degraded. Although the control signal applied by the rest of the fleet is ignored and the disturbances are affecting the vehicles, the guaranteed estimation is achieved.

The reader might think, from the previous simulations, that the reference is not tracked in the presence of disturbances. Then, we have made a longer simulation, depicted in Figure 8, that shows that the actual position of the vehicles evolves around the equilibrium point, within a confined region. The size of this region is related, as expected, to the energy of the disturbances.

Guaranteed estimation and distributed control of vehicle formations

All authors

R. A. García, L. Orihuela, P. Millán, F. R. Rubio & M. G. Ortega

https://doi.org/10.1080/00207179.2020.1714074

Published online:

23 January 2020

Figure 8. Vehicle 1, actual position, reference and bounds of the estimation of the position x by all agents.

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6.4. Example 4

This example analyses the relation between the control matrix $K_i$ designed and the estimation performance. We will consider two identical settings (same initial condition, no noises, no disturbances) to perform a fair comparison. Although no noises and disturbances actually affect the system, the agents will take into account the corresponding zonotopes. In particular, ${\mathcal{W}}_i=⟨0,0.1I⟩,{\mathcal{V}}_i=⟨0,0.1I⟩,\mathrm{\forall }i$. In addition, we will consider in this example that the initial position of the vehicles is unknown, so all the observers' initial condition is set to zero and the initial zonotope ${\hat{\mathcal{X}}}_i(0|-1)=⟨0,2.5I⟩$ is built big enough to encompass the initial state of the vehicles. This fact will let us compare the speed of convergence of the estimation with respect to the control.

Two families of LQR controller have been synthesised, with different aggressiveness. In particular, the local cost functions in (23) are defined with ${\mathrm{\Phi }}_i=10I,{\mathrm{\Theta }}_i=I,\mathrm{\forall }i,$ in the first scenario, and ${\mathrm{\Phi }}_i=1I,{\mathrm{\Theta }}_i=I,\mathrm{\forall }i,$ in the second one. A chain communication topology has been considered here.

The results are illustrated in Figure 9 using black colour lines for the first scenario and red colour lines for the second one. The figures show the actual state in solid lines, the centre of the estimation with circles, and the bounds with dashed lines. They all show the estimation of the vertical position of the first vehicle.

Guaranteed estimation and distributed control of vehicle formations

All authors

R. A. García, L. Orihuela, P. Millán, F. R. Rubio & M. G. Ortega

https://doi.org/10.1080/00207179.2020.1714074

Published online:

23 January 2020

Figure 9. Comparative of the gain $K_i$ based on a more aggressive control and on a less aggressive control, estimation of state 1 by all agents: (a) Agent 1, (b) Agent 2, (c) Agent 3 and (d) Agent 4.

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Several conclusions can be drawn from this set of figures:

• As in the previous results, the closer an agent is respect to the agent measuring the variable under consideration, the narrower the estimation is.

• The very first instants, in which the dynamics are frozen, represent the initialisation procedure described after Theorem 4.1.

• The speed of convergence of the estimators is much faster than that of the controllers, this implementing the standard cascade structure shown by other combined algorithms, such as LQG.

• The values of the weighting matrices in the cost functions have a double impact: first of all, they let us tune the speed of convergence to the desired reference. However, the more aggressive the controllers are, the wider the estimation bounds become. This is due to the calculations made in the prediction step (16) that uses the controller matrices of the neighbours to compute the predicted zonotope. More aggressive controller means bigger values for the control matrices $K_i$ and, then, bigger prediction zonotopes following Equation (17). As expected, this effect does not appear in the first agent, because this is the one measuring the vertical position.