**2. The predictive model**

Any model that could be used to predict the future behaviour can be the system model in MPC, and it is usually called predictive model.

MPC itself has no special request on the choice of model, the only need is that the model could predict the future behaviour of the system, no matter how we get the system model and how we obtain the future output by the model. But many researchers still classify MPC into different types by their models, since different model usually lead to quite different optimization method in solution of control law. Because all MPC have the same basic structure, the optimization method may be the most important part of a novel MPC algorithm indeed, and it also can determine the algorithm's practical applicability in industry. In Certain Meaning, the develop history of MPC is mainly the develop history of the predictive model of MPC.

When MPC was invented in 1970s, limited by the modelling and computational method, the scientist and engineers often use simple models, such as discrete time linear model (Richalet *et al*., 1978, Culter *et al*., 1980, Rouhani *et al*., 1982 and Clarke *et al*., 1987), to build MPC, while using this kind of models could already satisfy the requirement on control performance in process industry of that days. Later, based on modern control theory, a lot of MPC based on linear state-space system model is proposed (Ordys *et al*., 1993, Lee *et al*., 1994). These mentioned references can also help the readers of this book to understand the basic characters thoroughly if they still have problems after reading this short guidance, because these references were work of the precursors, who paid special attention to explain what MPC's essential properties are.

But, nonlinearity, constraints, stochastic characters and other complex factors exist naturally in the physical world, especially in control engineering.

Model Predictive Control: Basic Characters 3

next future output *y*(*k*+1|*k*) can be predicted. This operation is usually called as one-step

With similar process, if we know the sequence of future control input *u*(*k*), *u*(*k*+1), *u*(*k*+2), *u*(*k*+3)…, we can predict the sequence of future output *y*(*k*+1|*k*), *y*(*k*+2|*k*), *y*(*k*+3|*k*) …, here, the length of prediction or the number of predictive steps is called predictive horizon in MPC. In MPC, though we cannot know he sequence of future control input *u*(*k*), *u*(*k*+1), *u*(*k*+2), *u*(*k*+3)…, we can still predict *y*(*k*+1|*k*), *y*(*k*+2|*k*), *y*(*k*+3|*k*), with he sequence of future control input *u*(*k*), *u*(*k*+1), *u*(*k*+2), *u*(*k*+3)… remaining in these predictive values as unknown

If certain expectation future output is given, such as the future trajectory shown in Fig. 1. (the expect way of output how it reaches the setpoint in certain time), to the contrary of prediction mentioned in the second and the third paragraph of this section, the sequence of future control input *u*(*k*), *u*(*k*+1), *u*(*k*+2), *u*(*k*+3)… can be solved by the given *y*(*k*+1|*k*), *y*(*k*+2|*k*), *y*(*k*+3|*k*) …, and this is exactly the way how MPC can get a optimal control law

If a future output trajectory or an objective faction (usually a quadratic function of input and

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

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

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

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

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

output) is given, in MPC, as mentioned above, the optimal control law can be solved.

sequence is obtained. This is the essence of online optimization.

(PSO) *etc*. are the common solution tool for NMPC.

systems, numerical methods are still the only usable means.

prediction.

variables that need to be solved.

**4. The online optimal control law** 

from model prediction.

feedback control.

**5. Application of MPC** 

For highly nonlinear processes, and for some moderately nonlinear processes, which have large operating regions, MPC based on local linear model is often inefficient. Since the nonlinearity is the most important essential nature, and the increasing demand on the control performances, controller designers and operators have to face it directly. In 1990s, nonlinear model predictive control (NMPC) became one of the focuses of MPC research and it is still difficult to handle today as Prof. Qin mentioned in his survey (Qin *et al*., 2003). The direct incorporation of a nonlinear process into the MPC formulation will result in a nonconvex nonlinear programming problem, which needs to be solved under strict sampling time constraints. In general, there is still no analytical solution to this kind of nonlinear programming problem. To solve this difficulty, many kinds of simplified model is chosen to present nonlinear systems, such as nonlinear affine model (Cannon, 2004), bilinear model (Yang *et al*., 2007), block-oriented model (including Hammerstein model, Wiener model, *etc*.)(Harnischmacher *et al*., 2007, Arefi *et al*., 2008).

Stochastic characters and other complex factors also special expression models, such as Markov chain description and other method. Limited by the length, we won't introduce them in detail here, readers who are interested in these models can read more surveys on MPC and then find clue to research on them.
