**4. The online optimal control law**

If a future output trajectory or an objective faction (usually a quadratic function of input and output) is given, in MPC, as mentioned above, the optimal control law can be solved.

At time *k*, a sequence of future input will be solved *u*(*k*), *u*(*k*+1), *u*(*k*+2), *u*(*k*+3), but only instant input *u*(*k*) will be carry out actually by the system. At the next sample time, time *k*+1, the whole process of prediction and optimization will be repeated and a new future input sequence is obtained. This is the essence of online optimization.

This operation can introduce information into the controller, such as the error between predictive output and real output, so model mismatch and other disturbances can be eliminated gradually. In some extent, the online optimization can be recognized as a kind of feedback control.

For linear systems, control law of MPC can often be obtained analytically, but for most nonlinear systems, we have to use numerical optimization algorithms to get the control solution. Nowadays modern numerical optimization methods, such as Genetic Algorithm (GA) (Yuzgec *et al*., 2006), ant colony optimization (ACO), Particle Swarm Optimization (PSO) *etc*. are the common solution tool for NMPC.

Compared to MPC for SISO system, for MIMO system or multi-objective problem, there is no special difference in optimization methods. While, constraints (on input, on output or on both of them) may cause big trouble in online optimization, for linear system, there is some method that can deal with simple constraints, but for complex constraints or for nonlinear systems, numerical methods are still the only usable means.

### **5. Application of MPC**

When MPC is invented, limited by modeling and optimization method and tools, it could only be used in process industry, with local linear model and large sample period. And the

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position of MPC in a whole process control project is shown in Fig. 2. We can see that, MPC is in a 'middle' level.

Now, the rapid development in computational science and technology leads to the second boom of MPC, especially on the applicative research of it. MPC's application can be found almost in every engineering field rather than process industry, such as MPC in motion control (Richalet, 1993), modern agriculture (Coelho *et al*., 2005), communication (Chisci *et al*., 2006) and even in decision making science (Kouvaritakis *et al*., 2006). In this book, there are also several recent successful applicative example of MPC for interesting plant for you.

It can be believed with much confidence, in the future, the great benefit of MPC could be shared by more and more practical domain for more and more people in the world.

Fig. 2. Position of MPC in a typical process control project
