Real-Time Optimization (RTO) is a process automation technology whose objective is to predict the economically optimal process operating policy in the near term. The application of RTO typically requires a considerable expenditure for manufacturers. It involves the development of a process model and the implementation of a suitable optimization routine that can solve the optimization problem in real-time. To circumvent these requirements, a number of alternative techniques have been proposed to solve steady-state optimization problems. One such RTO technique is extremum-seeking control (ESC). This approach, which dates back to the 1920s provides a mechanism by which a system can be driven to the optimum of a measured variable of interest. ESC can be viewed as an empirical RTO implementation in which no exact model description is required, but where the objective function of the RTO problem is available from process measurements. ESC provides an effective control system design technique that can be used to steer an unknown dynamical system to an equilibrium that optimizes a cost function.
When dealing with complex dynamical systems, it is generally recognized that overall process objectives are difficult to achieve due to the computational complexity associated with centralized approaches. Thus, a decentralized or a distributed optimization approach is usually favoured in large-scale RTO systems design. In this approach, global process objectives are achieved by solving several local RTO subproblems. The distributed optimization task is said to be non-cooperative when each local RTO achieves its local optimization objectives. Non-cooperative RTO problems have been tackled using ESC by several researchers. The distributed optimization task can also be cooperative when the local RTOs coordinate actions to optimize the sum of their assigned costs. A particular class of distributed cooperative optimization has been the subject several studies. For a class of unconstrained optimization problems, it is shown that it is possible to achieve overall system objectives by solving local problems and communicating the optimization results via the network. Few ESC techniques have been proposed to solve decentralized and distributed optimization problems. For constrained optimization problems, the Alternating Direct Method of Multipliers (ADMM) can be used to solve distributed and coordinated optimization problems. A constrained distributed optimization approach was presented where a projection operator approach is used. The approach can effectively solve distributed optimization problems subject to a known computable projection to a known convex set. Extensions of cooperative optimization techniques to more general network architectures have been proposed.
The control of networks of multi-agent systems with unknown dynamics has been treated in a number of studies have presented a consensus approach for a class of time-varying dynamical networks with unknown stable, decoupled system dynamics and first order agent dynamics with dynamic coupling. In general, existing design approaches for multi-agent systems with unknown agent dynamics are limited to multi-agent dynamics where the only dynamic coupling arises from consensus or communication protocols. This study proposes the design of a method of distributed optimization over networks of dynamic agents with unknown coupled unstable dynamics. The network dynamics are described by a large-scale unknown unstable nonlinear dynamical system operating over a local actuator and sensor communication network. Each agent has access to a local sensor measurement and a certain number of actuators. It is also able to communicate its sensor information with neighbouring agents. The local input-output dynamics of each agent are assumed to be unknown and can be affected by actuator variables and state variables from the network dynamics. To the best of our knowledge this class of multi-agent dynamical systems has not been treated in the literature. A distributed extremum-seeking controller is proposed to solve the optimization problem. The main feature of existing ESC approaches is the use of some variation of a gradient descent algorithm. This approach requires the integration of an estimate of the gradient of the unknown cost function subject to some form of filter and a dither signal. ESC problems cannot be systematically solved in the absence of time-scale separations. As a result, the stabilization of unknown nonlinear systems with the help of ESC cannot be tackled. This paper proposes a novel proportional-integral ESC (PIESC) design technique that is implemented in a distributed environment to design cooperative systems to solve a distributed optimization problem over networks of unknown unstable dynamic agents. The agent dynamics implement a PIESC controller that can solve the real-time optimization without the explicit need for a two time-scale approach. The main contribution of this paper is to show that the distributed PIESC can be effectively applied to the design of real-time optimization control systems that can stabilize the network of unstable dynamics to the unknown optimum of the total plant cost. The result offers a novel mechanism to design distributed control systems with unknown dynamic interactions. Application to large-scale economic model-predictive controller design is addressed.
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