The complete results can be found in .2. Design of the ControllersSince we have derived several models for the plant, also we must design controllers for www.selleckchem.com/products/Vorinostat-saha.html each of the models that will be optimized in a specific way. Besides the standard MPC design and switch MPC design, here we introduce a hybrid MPC and a hybrid multiple-model predictive controller in order to improve the control performance of the industrial furnace. The idea for hybrid control and some results regarding constraints and stability have been explored in details in [10, 11].The results presented here show a multiple-model MPC of piecewise affine (PWA) system  and the design of a complete hybrid MPC for the temperature control of the furnace . The results obtained here clearly justify the use of the hybrid control algorithms over the conventional methods.
For this simulation, we have used one linear MPC, one linear multiple-model MPC, one hybrid MPC, and one hybrid multiple-model MPC. These controllers are tested in equal simulation conditions, and the results are compared.2.1. Controller SynthesisThe optimization problem of linear MPC is known for a long time, and it is not a subject of this paper. For the design of the controller, standard design methods are used. Regarding the hybrid optimization, the problem in control science is relatively new. In this case, k��[0,N?1],Sxx(N?�O?t)��Tx,(2)where?k��[0,N],ymin?��y(t+k)��ymax?,?k��[0,N?1],xmin?��x(t+k?�O?t)��xmax?,?x(0?�O?t)=x(t),x(k+1?�O?t)=Ax(k?�O?t)+B1u(k?�O?t)+B2��(k?�O?t)+B3z(k?�O?t),y(k?�O?t)=Cx(k?�O?t)+D1u(k?�O?t)+D2��(k?�O?t)+D3z(k?�O?t),E2��(k?�O?t)+E3z(k?�O?t)��E1u(k?�O?t)+E4x(k?�O?t)+E5,umin?��u(t+k)��umax?,?+||Qy(y(k?�O?t)?yr)||p,(1)?s.
t.?+��k=0N?1||Qu(u(k)?ur)||p+||Qz(z(k?�O?t)?zr)||p?we have designed a cost function in the form given in (1) and (2).Consider?min?u,��,z0N?1J(u,��,z0N?1,x(t))=��||QxN(x(N?�O?t)?xr)||p+��k=1N?1||Qx(x(k)?xr)||p N is the optimal control interval and x(k | t) represents the state predicted at moment t + k resulting from the input u(t + k). The initial value of the system at time t is x(0 | t) = x(t); umin , umax , ymin , ymin , and xmin , xmin are hard bound on the inputs, outputs, and states, respectively; and x : Sxx �� Tx is a final target polyhedral subset of the state-space n. In (1), ||Qx||p = x��Qx for p = 2 and ||Qx||p = ||Qx|| for p = ��.
In (1) and (2), with x(t) we represent the continuous states of the system and with z(t) the discrete AV-951 states of the system; the inputs are denoted by u(t) and the outputs by y(t).We use the Hybrid Toolbox for Matlab  as a design tool for the controller for the high consumption industrial furnace. This toolbox can work with several different types of hybrid system models (e.g., Mixed Logical Dynamical Systems, Piecewise Affine Systems, and Discrete-Time Hybrid Automata) and presents a formal mathematical equivalence between these models. We use HYSDEL to represent the model of the furnace [14, 15].