Multi-Target Tracking Control Using Continuous ... - Semantic Scholar
Multi-Target Tracking Control Using Continuous Double Auction Parameter Selection Alexander Charlish
Karl Woodbridge and Hugh Griffiths
Fraunhofer FKIE Wachtberg, Germany Email: [email protected]
University College London London, U.K. Email: {k.woodbridge, h.griffiths}@ee.ucl.ac.uk
Abstract—Multifunction sensor resource management can be formulated as a distributed parameter selection problem, whereby parameter selections are sought for each task which maximises the global performance of all tasks, subject to a resource constraint. This paper describes the application of the Continuous Double Auction Parameter Selection (CDAPS) algorithm to the sensor resource management problem, and demonstrates the algorithm on multi-target tracking control scenarios for a multi-function radar. CDAPS is shown to give a significant improvement in performance over conventional rule based methods. The continuous nature of CDAPS allows for the resource allocation to evolve over time instead of in fixed allocation frames, and the algorithm is computationally efficient as only partial participant preferences are computed.
I. I NTRODUCTION Sensor systems are increasingly being deployed with complex requirements for information fusion, which can necessitate the system to fulfil a variety of differing functions. Also, sensor systems are increasingly operating in complex environments, where it can be required to support numerous tasks for each function, such as the surveillance of multiple regions, or the tracking of multiple targets. An example of such a system is the multifunction radar, whose multifunction capability is primarily attributed to the use of an electronically steered array, which allows the dynamic allocation of the finite time-energy resource between numerous heterogeneous tasks. However, the multifunction capability is more generally derived from the ability to dynamically select an array of operational parameters which can be matched to each task. Yet, the selection of operational parameters, given the finite resource constraint, is beyond the response time and processing capability of the human operator. So, novel sensor resource management techniques are required, to automate this process. The sensor resource management problem can be thought of as a branch of sensor management, specifically addressing how the sensors’ finite resource should be allocated between the numerous tasks which support the varied functions. This can be approached as a constrained optimisation problem, whereby operational parameters must be selected for the tasks, such that the combined sensor loading of all tasks does not exceed the capability of the sensor. Existing approaches to the problem loosely fit into two categories, rule based methods [1], [2], [3] or optimisation methods [4], [5]. Generally, rule based methods generate sub-optimal solutions but benefit from
light computational demand, whereas optimisation methods can generate optimal solutions but are hindered by excessive computation. Hence, it is desired to mix the desirable characteristics of both methods to produce computationally light algorithms capable of producing quality solutions. This paper presents the Continuous Double Auction Parameter Selection (CDAPS) algorithm [6] which can be used to tackle the sensor resource management problem. The CDAPS algorithm uses a market mechanism, known as the continuous double auction [7], to select the global optimum parameters for each individual task given the global finite resource constraint. In previous works CDAPS has been demonstrated on surveillance control [8] problems. However, as CDAPS is inherently multifunction, it is described in this paper in the context of multi-target tracking control. Section II gives a general formulation of the sensor resource management problem, which is followed by a description of CDAPS in Sec. III and specifics of the models used for multi-target tracking in Sec. IV. The algorithm and the models are demonstrated on example multi-target tracking control scenarios in Sec. V and finally conclusions are drawn in Sec. VI II. S ENSOR R ESOURCE M ANAGEMENT P ROBLEM F ORMULATION The sensor resource management problem requires that the finite sensor resource is allocated between numerous, potentially conflicting tasks. To this end, operatal parameters should be selected for each of the individual tasks, to globally maximise the combined performance of all tasks without exceeding the resource capability of the sensor. This constrained optimisation problem can be fulfilled through two aspects, the development of a resource allocation model for each individual task, and the development of global allocation mechanism which allocates the resource between tasks and imposes the global finite resource constraint. 1) Task Allocation Model: Following the sensor resource management formulation in [4], task allocation models are sought, which map operational and environmental parameters to quality and utility. Operational parameters are parameters which are under the sensors’ control, such as the frequency at which the task is performed. Environmental parameters are parameters which are outside of the sensors’ control but still
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impact on the performance and loading of the task, such as the target location. •
Quality function - Relates operational parameters Xi to task quality space Qi , given some uncontrollable environmental parameters Ei : qi : Xi × Ei → Qi
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Resource function - Relates operational parameters to resource loading, given some uncontrollable environmental parameters: gi : Xi × Ei → Ri
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Utility function - Describes the satisfaction associated with each point in quality space: ui : Qi → R
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Therefore each operational parameter selection for a task achieves some quality, given some loading it exerts on the system. This quality is to some extent satisfactory, depending on the requirements of the task, which is quantified in terms of utility. As the utility functions capture the aim or requirements of all the tasks, global maximisation of utility is considered complementary to the goals or requirements of the system. These allocation functions are steady state, and so describe the continuous, non-myopic loading, quality and utility. 2) Global Allocation Mechanism: Given the task allocation model described, the global constrained optimisation problem can be formulated as the maximisation of the sum of the utility across all tasks:
III. C ONTINUOUS D OUBLE AUCTION PARAMETER S ELECTION Continuous Double Auction Parameter Selection (CDAPS) is an agent based auction mechanism which selects operational parameters for numerous tasks given a finite resource constraint. An agent is an adaptive, proactive, autonomous element which is directed to achieving a goal. Additionally, agents possess a social capability, which allow them to engage in multi-agent systems (MAS). In MAS, the synergy of the interactions between agents generates globally desirable behaviour. Mechanisms, which define the protocol for interaction between agents, are required to translate the individual actions of the agents into a globally suitable outcome. Of particular relevance to sensor resource management are auction and market mechanisms, which have been successfully applied in human societies for generations. Specifically, the continuous double auction is a mechanism where agents assuming the role of buyer or seller can trade a finite resource at continuous points in time. The CDAPS algorithm utilises the continuous double auction mechanism to tackle the parameter selection problem for sensor resource management. A. CDAPS Mechanism The CDAPS algorithm hosts a market mechanism where agents representing the numerous sensor tasks can trade resource assuming the role of both buyer and seller. The mechanism adheres to the following protocol: •
f (x1 , x2 , ...xk ) = ΣTk=1 uk (qk (xk ))
(4)
where T is the number of tasks, xk is an operational parameter selection for task k. The sum of the resource loading across all tasks must not exceed the loading available: g(x1 , x2 , ..., xk ) = ΣTk=1 rk ≤ rm
•
•
(5)
where rm is the total resource available. So, the sensor resource management problem is optimised by maximising the utility across the numerous, heterogeneous tasks maintained. Given this formulation of the sensor resource management problem, it is possible to distinguish between the optimisation and rule based methods for sensor resource management. Optimisation methods attempt to tackle the global optimisation problem, i.e. maximisation of some objective or utility functions. However, due to the number of parameter dimensions in the problem, optimisation methods are severely hindered by the curse of dimensionality and so suffer from excessive computational demand. Rule based methods specify rules for each individual task, such as a single point in quality space. These are fast to calculate, but can only at best produce the summation of local optimums, which can produce poor and unpredictable performance, especially when the sensor is heavily loaded.
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The resource held by task agent k, which is denoted rk , represents the allowed sensor loading of its represented task. The total resource held by all task agents, cannot exceed the P sensor resource loading available rm for all tasks, i.e. k rk ≤ rm . Each agent may publicly announce an offer to trade comprising of a quantity s, unit price p, and an identifier. At any time each task agent may announce a bid to buy (bn (sn , pn , n)), an ask to sell (am (sm , pm , m)) or both. Each offer remains active until it is cleared or updated by the agent, giving a set of active asks A = {a1 , ...., am } and a set of active bids B = {b1 , ...., bn } at P any time. A subset of asksP I ⊆ A, with value VI = i pi si and quantity SI = Pi si , and a subset of bidsPJ ⊆ B, with value VJ = j pj sj and quantity SJ = j sj can generate a transaction if VI < VJ and UI > UJ . The subsequent transaction price pˆ is a weighted average of the lowest price in the ask set imin and the bid set jmin , i.e pˆ = 0.5imin + 0.5jmin .
Trading in the mechanism is driven by each agent having differing and potentially dynamic valuations of the resource in terms of the system currency, which is known as utility. These utility valuations depend on the models of the task, outlined in Sec. II-1 which is represented by the agent.
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Fig. 1. Example operational parameter selections in resource-utility space, with gradient between two possible parameter selections marked.
Fig. 2. Percentage of target tracks maintained against resource availability for the static scenario.
B. Task Agents
marginal utilities as the emergent competitive equilibrium from the market mechanism. However, as the possible parameter selections are discrete, the solution is optimal for the given discrete parameter set, but only near optimal in contrast to a continuous parameter set [9].
Each agent represents a sensor task and aims to maximise its utility production by acquiring as much resource as possible given the competition from the other task agents. To ensure the validity of the allocation it is essential that each agent accurately evaluates the utility associated with changes in resource. This is determined by the functions in Eq. (1) - Eq. (3), which inherently allow for different tasks, that may be quantified by differing performance metrics. Each agent is able to evaluate the quality, utility and loading of every parameter selection, an example of which is shown in resource-utility space in Fig. 1. Potential changes from the current parameter selection can be evaluated as the difference in utility, ∆u, given the change in resource, ∆r, which is the gradient between resource-utility points. This gradient is the agents true price valuation, p∗ = ∆u ∆r , of the potential change in operational parameters. Each agent uses a hill-climbing search to find the potential change in parameters which gives the best ask and bid offers, with the lowest price and largest price respectively. The best bid and ask prices are local due to the monotonic nature of each parameter dimension, which reduces the search space. As the CDAPS mechanism is continuous new bids and asks are generated over a time scale of seconds, as new data is received or the environment changes. This spreads the search over time and reduces the computation in comparison to fixed resource allocation frames. C. Optimality For a non-linear programming solution to be optimal, it is required to satisfy the Karush-Kuhn-Tucker (KKT) conditions. Hence it is required that the marginal utilities, or gradients in resource-utility space, are equal. This concept was used to develop the Q-RAM [4] algorithm which produces solutions which maximise resource utilisation whilst satisfying the KKT conditions. The CDAPS algorithm also relies on the KKT conditions by producing an optimal solution with equal
IV. T RACKING R ESOURCE A LLOCATION M ODELS To apply CDAPS to a multi-target tracking control scenario it is necessary to use a tracking resource allocation model, to form the functions defined in Sec. II-1. Such a model is described by Van Keuk [2] for active tracking using a radar. The strategy adopted is to schedule a track update such that the track angular estimation error, along the major axis of the uncertainty ellipse, is maintained within a fraction of the beamwidth. This strategy allows for individual track loading to be minimised by balancing the trade between using long revisit intervals whilst minimising the number of looks per update. An increasing number of looks per update occurs as the uncertainty of the target position increases during the revisit interval so that the beam can not be accurately centred on the target. Assuming Singer target dynamics [10], empirical formulas are presented, which can be used to relate the coherent dwell length and target revisit interval to angular estimation error and resource loading. So this model, which is briefly described in the following subsection, can completely describe the required functions in Sec. II-1. In this case, the operational parameters are the coherent dwell length and the target revisit interval, the environmental parameters are the Singer target dynamics and the target location. A. Quality Function In the Van Keuk model, the track revisit interval fT is related to the steady state angular estimation error according to: Rσ √Θ 0.4 υ 2.4 fT = 0.4 (6) Σ 1 + 21 υ 2
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Mean Track Utility Against Resource Available 0.9 0.8 0.7 Mean Track Utility
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Mean Angular Estimation Error Standard Deviation Against Resource Available 3.5 CDAPS RBPS1 RBPS2 3
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Fig. 4. Average track utility against resource availability for the static scenario.
where υ is the variance reduction ratio: Bν0 (7) υ= σ and R is the range, ν0 is the angular estimation error in units of half beamwidth, B is the half beamwidth, Θ is the Singer manoeuvre variance and Σ is the manoeuvre time constant. The measurement accuracy σ can be calculated as:
and
B2 SNRT where SNRT is the instantaneous signal to noise ratio: σ2 =
SNRT =
SNR0 − ln PF 1 + 2ν02
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(9)
where PF is the probability of false alarm and SNR0 is the boresight signal to noise ratio. The boresight signal to noise ratio is dependent on the dwell length d, the target position and the radar cross section, according to the standard radar equations [11]. B. Resource Function In the Van Keuk model, track updates are requested at the track revisit interval. A detection may not occur on the track update, due to beam positioning loss. Therefore the resource loading, which is the steady state percentage of available time the task uses, can be approximated from the expected number of looks at each update and the revisit interval: r=
E[n] E[fT ]
(10)
where the expected number of looks n can be approximated using: 1/2 1 E[n] = 1 + (αν02 )2 (11) PD0 where PD0 is the boresight probability of detection which for a Swerling 1 target is equal to: PD0 = PF
1 1+SNR0
(12)
α ' 1 + 14(| ln Pf |/SNR0 )1/2
(13)
This gives an approximation of the resource loading of an active tracking task. C. Utility Function The utility function describes the satisfaction associated with each point in quality space. A low angular estimation error, which relates to high tracking accuracy is desirable. In this work the utility function is taken as: −k u(ν0 ) = 1 − exp (14) ν0 where k is a sensitivity parameter, which determines the sensitivity to tracking accuracy. V. S IMULATIONS The CDAPS algorithm was implemented in Java for verification and also to gauge the scale of performance improvement over rule based techniques. The agent functionality was enabled by the JADE Framework [12] which includes agent-orientated capabilities such as messaging passing and independent threads of control. It was required for the resource manager to select operational parameters to perform active tracking for a number of targets assuming an electronically steered phased array antenna. For all simulations target environmental parameters are randomly chosen at ranges between 10km − 120km, at bearings between ±45o and with radar cross sections between 0 − 10m2 . The targets were randomly generated to be of the three types listed in Table I, and so Singer manoeuvre parameters were randomly generated from the respective ranges listed in Table I. The operational parameters for each active tracking task is the dwell length d and the revisit interval f . In the simulations the following algorithms are compared: • CDAPS (Continuous Double Auction Parameter Selection) - Applied using an exponential utility function as described in Sec. IV-C.
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Location of Active and Inactive Targets for CDAPS
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TABLE I S INGER MANOEUVRE PARAMETERS FOR
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THREE TARGET TYPES
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RBPS1 (Rule Based Parameter Selection 1) - Coherent dwell length is chosen to maintain a 13dB SNR, and a revisit interval chosen to maintain the angular accuracy to 0.11 beamwidths. • RBPS2 (Rule Based Parameter Selection 2) - Coherent dwell length is chosen to maintain a 13dB SNR, and a revisit interval chosen to maintain the angular accuracy to 0.11 beamwidths. When resource available is unable to meet the resource required, the scheduler drops the tracks exerting the highest load first. RBPS1 and RBPS2 are based on the methodology applied by Van Keuk and so aim to minimise the radar loading per track so as to maximise the number of targets in track. In contrast, CDAPS aims to maximise the global utility of the resource allocation. The timing constraints which result from these parameter selections are passed to the scheduler, which forms a timeline •
based on an earliest deadline first approach [1]. The timeline which is produced is used for performance assessment, by again using the Van Keuk model on the realised parameter values in the radar schedule. So, real tracks are not maintained in the simulations and the same model is used for performance assessment as for resource allocation. Although this is not valid in reality, and the realised performance would differ from the allocation model, this is a suitable framework for comparative analysis of the respective allocation mechanisms. Consequently, for RBPS1 and RBPS2, the choice of a 13dB SNR is unrealistically low, however, this is chosen to match with the Van Keuk model, so as not to disadvantage the rule based methods in the performance assessment. A. Static Scenario The first simulation generated a static scenario, whereby the number of targets and the environmental parameters of each target did not change over the duration of the simulation. In the simulation it was required to select operational parameters for 300 targets given the finite resource available. The simulation ran for 60s and the CDAPS algorithm used a tracking sensitivity parameter of k = 0.003. Fig. 2 plots the percentage of the targets which were maintained, over a range of resource availabilities. The resource availability was synthesised in the scheduler by only allowing
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a percentage of the total time available to be used for the tracking tasks. It can be seen that the RBPS2 is able to track the highest percentage of targets, and is equivalent to RBPS1 when excess resource is available (> 13%). However, below 13% resource availability there is not enough resource to track all targets at the quality specified by RBPS1 and RBPS2. When this is the case, RBPS1 is unable to track a large number of targets, as its specified rules do not consider the global finite resource constraint. Both RBPS2 and CDAPS are able to track a high percentage of the targets as they both consider the global finite resource constraint. However, RBPS2 is able to track the most, as tracking the greatest number of targets is the aim of the rule specification for RBPS2. The average angular estimation error, according to Van Keuk’s model, is plotted in Fig. 3 for CDAPS, RBPS1 and RBPS2 against varying resource availabilities. It can be seen that both RBPS1 and RBPS2 maintain an average angular accuracy around 2.88mrad, as this is the 0.11 beamwidths quality specified in their rules, until 13% resource availability.
After 13% the angular accuracy improves, as the tracking tasks are scheduled early due to the excess resource available. However, it can be seen that CDAPS significantly outperforms both the RBPS approaches with a much lower average angular estimation error. Fig. 2 and Fig. 3 highlight the difficultly of performance assessment for multifunction sensor systems, as RBPS2 outperforms in the percentage of targets tracked, whereas CDAPS outperforms in the angular estimation error. From these figures alone it cannot be clear which is the superior approach, because the mission dependent relative importance of the two measures is not known. However, the utility function has been defined as the satisfaction associated with each point in quality space and so, the total global utility ultimately represents the satisfaction of the complete resource allocation. The average track utility against resource availability is shown in Fig. 4 for CDAPS, RBPS1 and RBPS2 under varying resource availabilities. In the figure, it can be seen that CDAPS significantly outperforms the rule based approaches, as CDAPS is designed
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Fig. 7. Percentage of target tracks maintained against varying resource availability for the dynamic scenario.
to maximise the global utility. In fact, by producing the global optimum through satisfying the KKT conditions, CDAPS must always outperform the locally optimised rules. When the resource available is insufficient, it is necessary for the resource manager to make decisions about which subset of the tracks should be maintained. The locations of the active and inactive targets are shown in Fig. 5 for CDAPS, RBPS1 and RBPS2 with 3% resource availability. It can be seen that RBPS1, which does not consider the global finite resource constraint, randomly chooses the targets to maintain. This leads to poor performance as a few distant targets, which have long dwell lengths, are maintained at the expense of a greater number of targets at close range, which have short dwell lengths. However, RBPS2 maintains more targets closer to the platform due to the additional rule addressing the global finite resource constraint. This additional rule manifests itself as an improvement in performance over RBPS1. The maintenance of targets closer to the platform emerges naturally from CDAPS as a consequence of the formulation as a constrained optimisation problem, which converges on the global optimuml solution. The amount of utility each active target produces, for 3% resource availability, is shown in Fig. 6 for CDAPS, RBPS1 and RBPS2. It can be seen that the utility produced by every target with RBPS1 and RBPS2 is the same, and this is a natural consequence of the rule which specifies an angular estimation error of 0.11 beamwidths. CDAPS however, produces a greater utility from targets which are closer to the platform, as these targets are easier to track accurately and so can produce a greater utility per unit resource. From this simulation it can be concluded that the simulations are in agreement with the assertion in Sec. III-C that CDAPS must always outperform the conventional, locally optimised rule based approaches. Rule based methods can be improved, as shown by RBPS2 outperforming RBPS1, however they always perform worse than CDAPS which achieves the global optimum.
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B. Dynamic Scenario The continuous nature of the CDAPS mechanism allows the resource allocation to evolve over time, and so it is especially suited to dynamic problems. Consequently, a second dynamic scenario was generated where the number of targets increased from 200 to 300 in the first 30s and decreased from 300 to 200 in the last 30s. This is a highly contrived scenario, but is useful for highlighting the differences between CDAPS and the rule based approaches. The average number of targets maintained, average angular uncertainty and global utility against resource availability is shown for CDAPS, RBPS1 and RBPS2 in Fig 7, 8 and 9 respectively. As with the static scenario, RBPS2 is able to track the greatest number of targets but CDAPS achieves the lowest, and so the best, average angular estimation error. CDAPS significantly outperforms the rule based methods in the average utility per track, which is a representative measure of the goal of the sensor system. The competitive market equilibrium prices for varying resource availabilities is shown in Fig. 10 over the duration of the simulation. It can be seen that the lower the resource availability, the greater the price, and the greater the resource availability, the lower the price, which is intuitively expected for a market mechanism. It can also be seen that over the first 30s when targets are arriving, the market price is increasing, whereas over the last 30s when the targets are leaving, the market price is decreasing. This competitive market equilibrium in the algorithm is the value of the marginal utility, which must be equal across all tasks. It is this equal marginal utility, resulting from the competitive market equilibrium, which ensures that the KKT conditions are satisfied and the solution converges to the global optimum. Consequently, resource for arriving targets is taken away from targets which lose the least amount of utility per resource, and resource from departing targets is given to the targets which gain the most utility per resource.
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VI. C ONCLUSION
R EFERENCES
The multifunction sensor resource management problem is such that operational parameters are sought for numerous heterogeneous tasks such that performance is optimised and a global finite resource constraint is satisfied. The CDAPS algorithm is a distributed resource allocation mechanism which addresses this problem by selecting task operational parameters using a continuous double auction. The competitive market equilibrium present in the continuous double auction, facilitates the Karush-Kuhn-Tucker conditions to be satisfied which ensures that the algorithm converges on the optimal solution. The CDAPS algorithm has been demonstrated in this paper for multi-target tracking control using Van Keuk’s active tracking models. It is shown to give a significant improvement in performance over conventional rule based methods as it converges to the global optimum allocation, in contrast to the rule based methods which generate a summation of local optimums. It has also been shown that the quality of rule based methods can be improved, but they do not exceed the performance of CDAPS. The CDAPS algorithm is continuous, which allows the resource allocation to evolve over time as necessary instead of in fixed allocation frames, and so it is suited to dynamic problems. As each agent in the algorithm calculates only partial preferences the algorithm is computationally efficient and subsequently the simulations in the paper all ran in real time. Although demonstrated in this paper on multi-target tracking control, the CDAPS algorithm is general and can be applied to any distributed, constrained, parameter selection problem. Future work will demonstrate the algorithm in more realistic sensor environments, and extend the algorithm to sensor suites and networks.
[1] S. Blackman and R. Popoli, Design and Analysis of Modern Tracking Systems. Artech House, 1999. [2] G. van Keuk and S. Blackman, “On phased-array radar tracking and parameter control,” IEEE Transactions on Aerospace and Electronic Systems, vol. 29, no. 1, pp. 186 –194, Jan 1993. [3] T. Kirubarajan, Y. Bar-Shalom, W. Blair, and G. Watson, “IMMPDAF for radar management and tracking benchmark with ECM,” IEEE Transactions on Aerospace and Electronic Systems, vol. 34, no. 4, pp. 1115 –1134, Oct. 1998. [4] J. Hansen, R. Rajkumar, J. Lehoczky, and S. Ghosh, “Resource management for radar tracking,” in IEEE Conference on Radar, 2006, p. 8. [5] A. Hero, D. Castanon, D. Cochran, and K. Kastella, Eds., Foundations and Applications of Sensor Management. Springer, 2007. [6] A. Charlish, “Autonomous agents for multifunction radar resource management,” Ph.D. dissertation, University College London, 2011. [7] D. Friedman and J. Rust, The Double Auction Market: Institutions, Theories and Evidence. Cambridge, MA: Perseus Publishing, 1993. [8] A. Charlish, K. Woodbridge, and H. Griffiths, “Agent based multifunction radar surveillance control,” in 2011 IEEE Radar Conference, May 2011, pp. 824 –829. [9] A. Irci, A. Saranli, and B. Baykal, “Study on Q-RAM and feasible directions based methods for resource management in phased array radar systems,” IEEE Transactions on Aerospace and Electronic Systems, vol. 46, no. 4, pp. 1848 –1864, Oct. 2010. [10] R. A. Singer, “Estimating optimal tracking filter performance for manned maneuvering targets,” IEEE Transactions on Aerospace and Electronic Systems, vol. AES-5, pp. 473–483, 1970. [11] M. I. Skolnik, Ed., Radar Handbook, 3rd ed. New York: McGraw-Hill, 2008. [12] P. A. Bellifemine, F. and G. Rimassa, “JADE a FIPA-compliant agent framework,” in Proc. Practical Applications of Intelligent Agents and Multi-Agent Technology, London, April 1999, pp. 97–108.
ACKNOWLEDGMENT The authors would also like to acknowledge the contribution of Professor Chris Baker at Ohio State University and Ulrich Nickel at Fraunhofer FKIE.
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Karl Woodbridge and Hugh Griffiths
Fraunhofer FKIE Wachtberg, Germany Email: [email protected]
University College London London, U.K. Email: {k.woodbridge, h.griffiths}@ee.ucl.ac.uk
Abstract—Multifunction sensor resource management can be formulated as a distributed parameter selection problem, whereby parameter selections are sought for each task which maximises the global performance of all tasks, subject to a resource constraint. This paper describes the application of the Continuous Double Auction Parameter Selection (CDAPS) algorithm to the sensor resource management problem, and demonstrates the algorithm on multi-target tracking control scenarios for a multi-function radar. CDAPS is shown to give a significant improvement in performance over conventional rule based methods. The continuous nature of CDAPS allows for the resource allocation to evolve over time instead of in fixed allocation frames, and the algorithm is computationally efficient as only partial participant preferences are computed.
I. I NTRODUCTION Sensor systems are increasingly being deployed with complex requirements for information fusion, which can necessitate the system to fulfil a variety of differing functions. Also, sensor systems are increasingly operating in complex environments, where it can be required to support numerous tasks for each function, such as the surveillance of multiple regions, or the tracking of multiple targets. An example of such a system is the multifunction radar, whose multifunction capability is primarily attributed to the use of an electronically steered array, which allows the dynamic allocation of the finite time-energy resource between numerous heterogeneous tasks. However, the multifunction capability is more generally derived from the ability to dynamically select an array of operational parameters which can be matched to each task. Yet, the selection of operational parameters, given the finite resource constraint, is beyond the response time and processing capability of the human operator. So, novel sensor resource management techniques are required, to automate this process. The sensor resource management problem can be thought of as a branch of sensor management, specifically addressing how the sensors’ finite resource should be allocated between the numerous tasks which support the varied functions. This can be approached as a constrained optimisation problem, whereby operational parameters must be selected for the tasks, such that the combined sensor loading of all tasks does not exceed the capability of the sensor. Existing approaches to the problem loosely fit into two categories, rule based methods [1], [2], [3] or optimisation methods [4], [5]. Generally, rule based methods generate sub-optimal solutions but benefit from
light computational demand, whereas optimisation methods can generate optimal solutions but are hindered by excessive computation. Hence, it is desired to mix the desirable characteristics of both methods to produce computationally light algorithms capable of producing quality solutions. This paper presents the Continuous Double Auction Parameter Selection (CDAPS) algorithm [6] which can be used to tackle the sensor resource management problem. The CDAPS algorithm uses a market mechanism, known as the continuous double auction [7], to select the global optimum parameters for each individual task given the global finite resource constraint. In previous works CDAPS has been demonstrated on surveillance control [8] problems. However, as CDAPS is inherently multifunction, it is described in this paper in the context of multi-target tracking control. Section II gives a general formulation of the sensor resource management problem, which is followed by a description of CDAPS in Sec. III and specifics of the models used for multi-target tracking in Sec. IV. The algorithm and the models are demonstrated on example multi-target tracking control scenarios in Sec. V and finally conclusions are drawn in Sec. VI II. S ENSOR R ESOURCE M ANAGEMENT P ROBLEM F ORMULATION The sensor resource management problem requires that the finite sensor resource is allocated between numerous, potentially conflicting tasks. To this end, operatal parameters should be selected for each of the individual tasks, to globally maximise the combined performance of all tasks without exceeding the resource capability of the sensor. This constrained optimisation problem can be fulfilled through two aspects, the development of a resource allocation model for each individual task, and the development of global allocation mechanism which allocates the resource between tasks and imposes the global finite resource constraint. 1) Task Allocation Model: Following the sensor resource management formulation in [4], task allocation models are sought, which map operational and environmental parameters to quality and utility. Operational parameters are parameters which are under the sensors’ control, such as the frequency at which the task is performed. Environmental parameters are parameters which are outside of the sensors’ control but still
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impact on the performance and loading of the task, such as the target location. •
Quality function - Relates operational parameters Xi to task quality space Qi , given some uncontrollable environmental parameters Ei : qi : Xi × Ei → Qi
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Resource function - Relates operational parameters to resource loading, given some uncontrollable environmental parameters: gi : Xi × Ei → Ri
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Utility function - Describes the satisfaction associated with each point in quality space: ui : Qi → R
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Therefore each operational parameter selection for a task achieves some quality, given some loading it exerts on the system. This quality is to some extent satisfactory, depending on the requirements of the task, which is quantified in terms of utility. As the utility functions capture the aim or requirements of all the tasks, global maximisation of utility is considered complementary to the goals or requirements of the system. These allocation functions are steady state, and so describe the continuous, non-myopic loading, quality and utility. 2) Global Allocation Mechanism: Given the task allocation model described, the global constrained optimisation problem can be formulated as the maximisation of the sum of the utility across all tasks:
III. C ONTINUOUS D OUBLE AUCTION PARAMETER S ELECTION Continuous Double Auction Parameter Selection (CDAPS) is an agent based auction mechanism which selects operational parameters for numerous tasks given a finite resource constraint. An agent is an adaptive, proactive, autonomous element which is directed to achieving a goal. Additionally, agents possess a social capability, which allow them to engage in multi-agent systems (MAS). In MAS, the synergy of the interactions between agents generates globally desirable behaviour. Mechanisms, which define the protocol for interaction between agents, are required to translate the individual actions of the agents into a globally suitable outcome. Of particular relevance to sensor resource management are auction and market mechanisms, which have been successfully applied in human societies for generations. Specifically, the continuous double auction is a mechanism where agents assuming the role of buyer or seller can trade a finite resource at continuous points in time. The CDAPS algorithm utilises the continuous double auction mechanism to tackle the parameter selection problem for sensor resource management. A. CDAPS Mechanism The CDAPS algorithm hosts a market mechanism where agents representing the numerous sensor tasks can trade resource assuming the role of both buyer and seller. The mechanism adheres to the following protocol: •
f (x1 , x2 , ...xk ) = ΣTk=1 uk (qk (xk ))
(4)
where T is the number of tasks, xk is an operational parameter selection for task k. The sum of the resource loading across all tasks must not exceed the loading available: g(x1 , x2 , ..., xk ) = ΣTk=1 rk ≤ rm
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•
(5)
where rm is the total resource available. So, the sensor resource management problem is optimised by maximising the utility across the numerous, heterogeneous tasks maintained. Given this formulation of the sensor resource management problem, it is possible to distinguish between the optimisation and rule based methods for sensor resource management. Optimisation methods attempt to tackle the global optimisation problem, i.e. maximisation of some objective or utility functions. However, due to the number of parameter dimensions in the problem, optimisation methods are severely hindered by the curse of dimensionality and so suffer from excessive computational demand. Rule based methods specify rules for each individual task, such as a single point in quality space. These are fast to calculate, but can only at best produce the summation of local optimums, which can produce poor and unpredictable performance, especially when the sensor is heavily loaded.
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The resource held by task agent k, which is denoted rk , represents the allowed sensor loading of its represented task. The total resource held by all task agents, cannot exceed the P sensor resource loading available rm for all tasks, i.e. k rk ≤ rm . Each agent may publicly announce an offer to trade comprising of a quantity s, unit price p, and an identifier. At any time each task agent may announce a bid to buy (bn (sn , pn , n)), an ask to sell (am (sm , pm , m)) or both. Each offer remains active until it is cleared or updated by the agent, giving a set of active asks A = {a1 , ...., am } and a set of active bids B = {b1 , ...., bn } at P any time. A subset of asksP I ⊆ A, with value VI = i pi si and quantity SI = Pi si , and a subset of bidsPJ ⊆ B, with value VJ = j pj sj and quantity SJ = j sj can generate a transaction if VI < VJ and UI > UJ . The subsequent transaction price pˆ is a weighted average of the lowest price in the ask set imin and the bid set jmin , i.e pˆ = 0.5imin + 0.5jmin .
Trading in the mechanism is driven by each agent having differing and potentially dynamic valuations of the resource in terms of the system currency, which is known as utility. These utility valuations depend on the models of the task, outlined in Sec. II-1 which is represented by the agent.
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Fig. 2. Percentage of target tracks maintained against resource availability for the static scenario.
B. Task Agents
marginal utilities as the emergent competitive equilibrium from the market mechanism. However, as the possible parameter selections are discrete, the solution is optimal for the given discrete parameter set, but only near optimal in contrast to a continuous parameter set [9].
Each agent represents a sensor task and aims to maximise its utility production by acquiring as much resource as possible given the competition from the other task agents. To ensure the validity of the allocation it is essential that each agent accurately evaluates the utility associated with changes in resource. This is determined by the functions in Eq. (1) - Eq. (3), which inherently allow for different tasks, that may be quantified by differing performance metrics. Each agent is able to evaluate the quality, utility and loading of every parameter selection, an example of which is shown in resource-utility space in Fig. 1. Potential changes from the current parameter selection can be evaluated as the difference in utility, ∆u, given the change in resource, ∆r, which is the gradient between resource-utility points. This gradient is the agents true price valuation, p∗ = ∆u ∆r , of the potential change in operational parameters. Each agent uses a hill-climbing search to find the potential change in parameters which gives the best ask and bid offers, with the lowest price and largest price respectively. The best bid and ask prices are local due to the monotonic nature of each parameter dimension, which reduces the search space. As the CDAPS mechanism is continuous new bids and asks are generated over a time scale of seconds, as new data is received or the environment changes. This spreads the search over time and reduces the computation in comparison to fixed resource allocation frames. C. Optimality For a non-linear programming solution to be optimal, it is required to satisfy the Karush-Kuhn-Tucker (KKT) conditions. Hence it is required that the marginal utilities, or gradients in resource-utility space, are equal. This concept was used to develop the Q-RAM [4] algorithm which produces solutions which maximise resource utilisation whilst satisfying the KKT conditions. The CDAPS algorithm also relies on the KKT conditions by producing an optimal solution with equal
IV. T RACKING R ESOURCE A LLOCATION M ODELS To apply CDAPS to a multi-target tracking control scenario it is necessary to use a tracking resource allocation model, to form the functions defined in Sec. II-1. Such a model is described by Van Keuk [2] for active tracking using a radar. The strategy adopted is to schedule a track update such that the track angular estimation error, along the major axis of the uncertainty ellipse, is maintained within a fraction of the beamwidth. This strategy allows for individual track loading to be minimised by balancing the trade between using long revisit intervals whilst minimising the number of looks per update. An increasing number of looks per update occurs as the uncertainty of the target position increases during the revisit interval so that the beam can not be accurately centred on the target. Assuming Singer target dynamics [10], empirical formulas are presented, which can be used to relate the coherent dwell length and target revisit interval to angular estimation error and resource loading. So this model, which is briefly described in the following subsection, can completely describe the required functions in Sec. II-1. In this case, the operational parameters are the coherent dwell length and the target revisit interval, the environmental parameters are the Singer target dynamics and the target location. A. Quality Function In the Van Keuk model, the track revisit interval fT is related to the steady state angular estimation error according to: Rσ √Θ 0.4 υ 2.4 fT = 0.4 (6) Σ 1 + 21 υ 2
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Fig. 4. Average track utility against resource availability for the static scenario.
where υ is the variance reduction ratio: Bν0 (7) υ= σ and R is the range, ν0 is the angular estimation error in units of half beamwidth, B is the half beamwidth, Θ is the Singer manoeuvre variance and Σ is the manoeuvre time constant. The measurement accuracy σ can be calculated as:
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B2 SNRT where SNRT is the instantaneous signal to noise ratio: σ2 =
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SNR0 − ln PF 1 + 2ν02
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where PF is the probability of false alarm and SNR0 is the boresight signal to noise ratio. The boresight signal to noise ratio is dependent on the dwell length d, the target position and the radar cross section, according to the standard radar equations [11]. B. Resource Function In the Van Keuk model, track updates are requested at the track revisit interval. A detection may not occur on the track update, due to beam positioning loss. Therefore the resource loading, which is the steady state percentage of available time the task uses, can be approximated from the expected number of looks at each update and the revisit interval: r=
E[n] E[fT ]
(10)
where the expected number of looks n can be approximated using: 1/2 1 E[n] = 1 + (αν02 )2 (11) PD0 where PD0 is the boresight probability of detection which for a Swerling 1 target is equal to: PD0 = PF
1 1+SNR0
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α ' 1 + 14(| ln Pf |/SNR0 )1/2
(13)
This gives an approximation of the resource loading of an active tracking task. C. Utility Function The utility function describes the satisfaction associated with each point in quality space. A low angular estimation error, which relates to high tracking accuracy is desirable. In this work the utility function is taken as: −k u(ν0 ) = 1 − exp (14) ν0 where k is a sensitivity parameter, which determines the sensitivity to tracking accuracy. V. S IMULATIONS The CDAPS algorithm was implemented in Java for verification and also to gauge the scale of performance improvement over rule based techniques. The agent functionality was enabled by the JADE Framework [12] which includes agent-orientated capabilities such as messaging passing and independent threads of control. It was required for the resource manager to select operational parameters to perform active tracking for a number of targets assuming an electronically steered phased array antenna. For all simulations target environmental parameters are randomly chosen at ranges between 10km − 120km, at bearings between ±45o and with radar cross sections between 0 − 10m2 . The targets were randomly generated to be of the three types listed in Table I, and so Singer manoeuvre parameters were randomly generated from the respective ranges listed in Table I. The operational parameters for each active tracking task is the dwell length d and the revisit interval f . In the simulations the following algorithms are compared: • CDAPS (Continuous Double Auction Parameter Selection) - Applied using an exponential utility function as described in Sec. IV-C.
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TABLE I S INGER MANOEUVRE PARAMETERS FOR
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THREE TARGET TYPES
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RBPS1 (Rule Based Parameter Selection 1) - Coherent dwell length is chosen to maintain a 13dB SNR, and a revisit interval chosen to maintain the angular accuracy to 0.11 beamwidths. • RBPS2 (Rule Based Parameter Selection 2) - Coherent dwell length is chosen to maintain a 13dB SNR, and a revisit interval chosen to maintain the angular accuracy to 0.11 beamwidths. When resource available is unable to meet the resource required, the scheduler drops the tracks exerting the highest load first. RBPS1 and RBPS2 are based on the methodology applied by Van Keuk and so aim to minimise the radar loading per track so as to maximise the number of targets in track. In contrast, CDAPS aims to maximise the global utility of the resource allocation. The timing constraints which result from these parameter selections are passed to the scheduler, which forms a timeline •
based on an earliest deadline first approach [1]. The timeline which is produced is used for performance assessment, by again using the Van Keuk model on the realised parameter values in the radar schedule. So, real tracks are not maintained in the simulations and the same model is used for performance assessment as for resource allocation. Although this is not valid in reality, and the realised performance would differ from the allocation model, this is a suitable framework for comparative analysis of the respective allocation mechanisms. Consequently, for RBPS1 and RBPS2, the choice of a 13dB SNR is unrealistically low, however, this is chosen to match with the Van Keuk model, so as not to disadvantage the rule based methods in the performance assessment. A. Static Scenario The first simulation generated a static scenario, whereby the number of targets and the environmental parameters of each target did not change over the duration of the simulation. In the simulation it was required to select operational parameters for 300 targets given the finite resource available. The simulation ran for 60s and the CDAPS algorithm used a tracking sensitivity parameter of k = 0.003. Fig. 2 plots the percentage of the targets which were maintained, over a range of resource availabilities. The resource availability was synthesised in the scheduler by only allowing
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a percentage of the total time available to be used for the tracking tasks. It can be seen that the RBPS2 is able to track the highest percentage of targets, and is equivalent to RBPS1 when excess resource is available (> 13%). However, below 13% resource availability there is not enough resource to track all targets at the quality specified by RBPS1 and RBPS2. When this is the case, RBPS1 is unable to track a large number of targets, as its specified rules do not consider the global finite resource constraint. Both RBPS2 and CDAPS are able to track a high percentage of the targets as they both consider the global finite resource constraint. However, RBPS2 is able to track the most, as tracking the greatest number of targets is the aim of the rule specification for RBPS2. The average angular estimation error, according to Van Keuk’s model, is plotted in Fig. 3 for CDAPS, RBPS1 and RBPS2 against varying resource availabilities. It can be seen that both RBPS1 and RBPS2 maintain an average angular accuracy around 2.88mrad, as this is the 0.11 beamwidths quality specified in their rules, until 13% resource availability.
After 13% the angular accuracy improves, as the tracking tasks are scheduled early due to the excess resource available. However, it can be seen that CDAPS significantly outperforms both the RBPS approaches with a much lower average angular estimation error. Fig. 2 and Fig. 3 highlight the difficultly of performance assessment for multifunction sensor systems, as RBPS2 outperforms in the percentage of targets tracked, whereas CDAPS outperforms in the angular estimation error. From these figures alone it cannot be clear which is the superior approach, because the mission dependent relative importance of the two measures is not known. However, the utility function has been defined as the satisfaction associated with each point in quality space and so, the total global utility ultimately represents the satisfaction of the complete resource allocation. The average track utility against resource availability is shown in Fig. 4 for CDAPS, RBPS1 and RBPS2 under varying resource availabilities. In the figure, it can be seen that CDAPS significantly outperforms the rule based approaches, as CDAPS is designed
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Fig. 7. Percentage of target tracks maintained against varying resource availability for the dynamic scenario.
to maximise the global utility. In fact, by producing the global optimum through satisfying the KKT conditions, CDAPS must always outperform the locally optimised rules. When the resource available is insufficient, it is necessary for the resource manager to make decisions about which subset of the tracks should be maintained. The locations of the active and inactive targets are shown in Fig. 5 for CDAPS, RBPS1 and RBPS2 with 3% resource availability. It can be seen that RBPS1, which does not consider the global finite resource constraint, randomly chooses the targets to maintain. This leads to poor performance as a few distant targets, which have long dwell lengths, are maintained at the expense of a greater number of targets at close range, which have short dwell lengths. However, RBPS2 maintains more targets closer to the platform due to the additional rule addressing the global finite resource constraint. This additional rule manifests itself as an improvement in performance over RBPS1. The maintenance of targets closer to the platform emerges naturally from CDAPS as a consequence of the formulation as a constrained optimisation problem, which converges on the global optimuml solution. The amount of utility each active target produces, for 3% resource availability, is shown in Fig. 6 for CDAPS, RBPS1 and RBPS2. It can be seen that the utility produced by every target with RBPS1 and RBPS2 is the same, and this is a natural consequence of the rule which specifies an angular estimation error of 0.11 beamwidths. CDAPS however, produces a greater utility from targets which are closer to the platform, as these targets are easier to track accurately and so can produce a greater utility per unit resource. From this simulation it can be concluded that the simulations are in agreement with the assertion in Sec. III-C that CDAPS must always outperform the conventional, locally optimised rule based approaches. Rule based methods can be improved, as shown by RBPS2 outperforming RBPS1, however they always perform worse than CDAPS which achieves the global optimum.
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B. Dynamic Scenario The continuous nature of the CDAPS mechanism allows the resource allocation to evolve over time, and so it is especially suited to dynamic problems. Consequently, a second dynamic scenario was generated where the number of targets increased from 200 to 300 in the first 30s and decreased from 300 to 200 in the last 30s. This is a highly contrived scenario, but is useful for highlighting the differences between CDAPS and the rule based approaches. The average number of targets maintained, average angular uncertainty and global utility against resource availability is shown for CDAPS, RBPS1 and RBPS2 in Fig 7, 8 and 9 respectively. As with the static scenario, RBPS2 is able to track the greatest number of targets but CDAPS achieves the lowest, and so the best, average angular estimation error. CDAPS significantly outperforms the rule based methods in the average utility per track, which is a representative measure of the goal of the sensor system. The competitive market equilibrium prices for varying resource availabilities is shown in Fig. 10 over the duration of the simulation. It can be seen that the lower the resource availability, the greater the price, and the greater the resource availability, the lower the price, which is intuitively expected for a market mechanism. It can also be seen that over the first 30s when targets are arriving, the market price is increasing, whereas over the last 30s when the targets are leaving, the market price is decreasing. This competitive market equilibrium in the algorithm is the value of the marginal utility, which must be equal across all tasks. It is this equal marginal utility, resulting from the competitive market equilibrium, which ensures that the KKT conditions are satisfied and the solution converges to the global optimum. Consequently, resource for arriving targets is taken away from targets which lose the least amount of utility per resource, and resource from departing targets is given to the targets which gain the most utility per resource.
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VI. C ONCLUSION
R EFERENCES
The multifunction sensor resource management problem is such that operational parameters are sought for numerous heterogeneous tasks such that performance is optimised and a global finite resource constraint is satisfied. The CDAPS algorithm is a distributed resource allocation mechanism which addresses this problem by selecting task operational parameters using a continuous double auction. The competitive market equilibrium present in the continuous double auction, facilitates the Karush-Kuhn-Tucker conditions to be satisfied which ensures that the algorithm converges on the optimal solution. The CDAPS algorithm has been demonstrated in this paper for multi-target tracking control using Van Keuk’s active tracking models. It is shown to give a significant improvement in performance over conventional rule based methods as it converges to the global optimum allocation, in contrast to the rule based methods which generate a summation of local optimums. It has also been shown that the quality of rule based methods can be improved, but they do not exceed the performance of CDAPS. The CDAPS algorithm is continuous, which allows the resource allocation to evolve over time as necessary instead of in fixed allocation frames, and so it is suited to dynamic problems. As each agent in the algorithm calculates only partial preferences the algorithm is computationally efficient and subsequently the simulations in the paper all ran in real time. Although demonstrated in this paper on multi-target tracking control, the CDAPS algorithm is general and can be applied to any distributed, constrained, parameter selection problem. Future work will demonstrate the algorithm in more realistic sensor environments, and extend the algorithm to sensor suites and networks.
[1] S. Blackman and R. Popoli, Design and Analysis of Modern Tracking Systems. Artech House, 1999. [2] G. van Keuk and S. Blackman, “On phased-array radar tracking and parameter control,” IEEE Transactions on Aerospace and Electronic Systems, vol. 29, no. 1, pp. 186 –194, Jan 1993. [3] T. Kirubarajan, Y. Bar-Shalom, W. Blair, and G. Watson, “IMMPDAF for radar management and tracking benchmark with ECM,” IEEE Transactions on Aerospace and Electronic Systems, vol. 34, no. 4, pp. 1115 –1134, Oct. 1998. [4] J. Hansen, R. Rajkumar, J. Lehoczky, and S. Ghosh, “Resource management for radar tracking,” in IEEE Conference on Radar, 2006, p. 8. [5] A. Hero, D. Castanon, D. Cochran, and K. Kastella, Eds., Foundations and Applications of Sensor Management. Springer, 2007. [6] A. Charlish, “Autonomous agents for multifunction radar resource management,” Ph.D. dissertation, University College London, 2011. [7] D. Friedman and J. Rust, The Double Auction Market: Institutions, Theories and Evidence. Cambridge, MA: Perseus Publishing, 1993. [8] A. Charlish, K. Woodbridge, and H. Griffiths, “Agent based multifunction radar surveillance control,” in 2011 IEEE Radar Conference, May 2011, pp. 824 –829. [9] A. Irci, A. Saranli, and B. Baykal, “Study on Q-RAM and feasible directions based methods for resource management in phased array radar systems,” IEEE Transactions on Aerospace and Electronic Systems, vol. 46, no. 4, pp. 1848 –1864, Oct. 2010. [10] R. A. Singer, “Estimating optimal tracking filter performance for manned maneuvering targets,” IEEE Transactions on Aerospace and Electronic Systems, vol. AES-5, pp. 473–483, 1970. [11] M. I. Skolnik, Ed., Radar Handbook, 3rd ed. New York: McGraw-Hill, 2008. [12] P. A. Bellifemine, F. and G. Rimassa, “JADE a FIPA-compliant agent framework,” in Proc. Practical Applications of Intelligent Agents and Multi-Agent Technology, London, April 1999, pp. 97–108.
ACKNOWLEDGMENT The authors would also like to acknowledge the contribution of Professor Chris Baker at Ohio State University and Ulrich Nickel at Fraunhofer FKIE.
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