یک روش برنامه ریزی پویا برای کنترل پیش بین مدل صریح و روشن از سیستم های ترکیبی

This work presents an algorithm for explicit model predictive control of hybrid systems based on recent developments in constrained dynamic programming and multi-parametric programming. By using the proposed approach, suitable for problems with linear cost function, the original model predictive control formulation is disassembled into a set of smaller problems, which can be efficiently solved using multi-parametric mixed-integer programming algorithms. It is also shown how the methodology is applied in the context of explicit robust model predictive control of hybrid systems, where model uncertainty is taken into account. The proposed developments are demonstrated through a numerical example where the methodology is applied to the optimal control of a piece-wise affine system with linear cost function.

The potential of using multi-parametric programming in the context of model predictive control has been widely documented in the literature (Bemporad et al., 2002b, Pistikopoulos et al., 2002 and Pistikopoulos et al., 2007). The idea is motivated by the fact that, in closed-loop model predictive control implementations, an optimisation problem needs to be solved whenever a sample of the system state is made available. For systems with fast sampling rates, the computational time required to solve the optimisation problem may become prohibitive, therefore rendering the use of model predictive control impractical. By formulating the optimisation problem involved in model predictive control as a multi-parametric programming problem, with the state of the system being the vector of parameters (Bemporad et al., 2002b), it is possible to shift the computational effort involved in online optimisation to an offline step in which the optimal solutions for every possible realisation of the state vector are pre-computed as explicit functions. The regions in the state-space where these explicit functions are valid are referred to as critical regions. This method of designing controllers is called explicit model predictive control (Pistikopoulos et al., 2007). The use of an explicit model predictive controller as a control device consists of evaluating the state of the system, at every sampling instance, and looking-up the corresponding optimal control input in the pre-computed map of critical regions. This operation is usually significantly faster than repeatedly solving optimisation problems, and therefore the method may be used for systems with faster sampling rates. According to a recent survey paper (Alessio and Bemporad, 2009), explicit model predictive controllers are suitable for applications with sampling rates as fast as 50 ms. Currently available software tools for explicit model predictive control (ParOS, 2004 and Kvasnica et al., 2004) may be used to design controllers with 1–2 input variables and 5–10 parameters (Alessio and Bemporad, 2009). However, one potential limitation is the rapid increase in the computational burden involved in solving the offline problem when the prediction horizon increases (Bemporad et al., 2002b). The combination of multi-parametric programming and dynamic programming (Bellman, 1957) has been reported as a method suitable for reducing the complexity of the optimisation problem involved in multi-stage decision processes, such as explicit model predictive control (Borrelli et al., 2005, Faísca et al., 2008 and Kouramas et al., 2013). By using this method, the original problem is disassembled into a set of smaller sub-problems, with lower dimensionality, which are sequentially solved in a recursive manner. Recently, this approach has been extended to the problem of constrained dynamic programming of mixed-integer linear problems (Rivotti and Pistikopoulos, 2014). Hybrid systems, i.e. systems involving continuous and discrete elements, find relevance in most processes of practical interest (Pantelides et al., 1999 and Branicky et al., 1998). These often involve logical components and propositional logic statements that may be equivalently expressed through sets of linear constraints (Cavalier et al., 1990, Raman and Grossmann, 1991 and Bemporad and Morari, 1999a). This property is explored by the mixed logical dynamical framework (Bemporad and Morari, 1999a) that provides a systematic method of converting logical propositions into a mixed-integer linear formulation (Bemporad et al., 2002b). Due to this importance, including integer decision variables in an explicit model predictive control framework has been often identified as an important research direction (Morari and Lee, 1999 and Pistikopoulos, 2009). However, the modelling of hybrid systems results in models with integer variables (Raman and Grossmann, 1992 and Williams, 1999), and therefore in the need to use computationally complex multi-parametric mixed-integer programming algorithms to design the controllers. The problem of hybrid explicit model predictive control with a linear cost function has been addressed by Bemporad and Morari (1999a) and Baotic et al. (2006). Sakizlis et al. (2002) presented a method based on a mixed-integer quadratic programming algorithm (Dua et al., 2002) that handles quadratic cost functions. In this publication, recent developments in constrained dynamic programming for mixed-integer linear systems (Rivotti and Pistikopoulos, 2014) are used as the foundations for a novel algorithm for hybrid explicit model predictive control with linear cost function. Additionally, it is shown how a closed-loop robust control policy may be obtained by considering modelling uncertainty within this framework. The remaining sections are organised as follows. Fundamental concepts related to the modelling of hybrid systems are introduced in Section 2.1, with particular emphasis on piece-wise affine systems. The current literature and state of the art in hybrid explicit model predictive control is presented in Section 2.2. In Section 2.3, the main developments proposed are explained in detail and summarised in the form of an algorithm for hybrid explicit model predictive control with linear cost function. Different types of modelling uncertainty are introduced in Section 3.1 and Section 3.2 shows how the proposed algorithm is applied to the problem of robust explicit model predictive control of hybrid systems. Two numeric examples are presented in Section 4 to demonstrate the proposed developments. Concluding remarks and ongoing research topics are summarised in Section 5.

This work introduced an algorithm for explicit model predictive control of hybrid systems with linear cost function using multi-parametric programming and constrained dynamic programming. It was also shown how the method may be extended to address the problem of robust explicit model predictive control for hybrid systems. The main advantage of the proposed approach is that it provides a dynamic programming based solution in which non-convexities are avoided and, therefore, has the potential to be extended to the problem of hybrid explicit model predictive control with quadratic objective function. The numerical examples presented illustrate how the method can be applied in the context of hybrid explicit model predictive control of piece-wise affine systems. The results show that the algorithm involves significant computational times and, therefore, ongoing research is focused on developing more efficient solution techniques.