Comparison on

## 1. Introduction

Model predictive control (MPC) refers to the class of computer control algorithms in which a dynamic process model is used to predict and optimize process performance. Since its lower request of modeling accuracy and robustness to complicated process plants, MPC for linear systems has been widely accepted in the process industry and many other fields. But for highly nonlinear processes, or for some moderately nonlinear processes with large operating regions, linear MPC is often inefficient. To solve these difficulties, nonlinear model predictive control (NMPC) attracted increasing attention over the past decade (Qin et al., 2003, Cannon, 2004). Nowadays, the research on NMPC mainly focuses on its theoretical characters, such as stability, robustness and so on, while the computational method of NMPC is ignored in some extent. The fact mentioned above is one of the most serious reasons that obstruct the practical implementations of NMPC.

Analyzing the computational problem of NMPC, the direct incorporation of a nonlinear process into the linear MPC formulation structure may result in a non-convex nonlinear programming problem, which needs to be solved under strict sampling time constraints and has been proved as an NP-hard problem (Zheng, 1997). In general, since there is no accurate analytical solution to most kinds of nonlinear programming problem, we usually have to use numerical methods such as Sequential Quadric Programming (SQP) (Ferreau et al., 2006) or Genetic Algorithm (GA) (Yuzgec et al., 2006). Moreover, the computational load of NMPC using numerical methods is also much heavier than that of linear MPC, and it would even increase exponentially when the predictive horizon length increases. All of these facts lead us to develop a novel NMPC with analytical solution and little computational load in this chapter.

Since affine nonlinear system can represents a lot of practical plants in industry control, including the water-tank system that we used to carry out the simulations and experiments, it has been chosen for propose our novel NMPC algorithm. Follow the steps of research work, the chapter is arranged as follows:

In Section 2, analytical one-step NMPC for affine nonlinear system will be introduced at first, then, after description of the control problem of a water-tank system, simulations will be carried out to verify the result of theoretical research. Error analysis and feedback compensation will be discussed with theoretical analysis, simulations and experiment at last.

Then, in Section 3, by substituting reference trajectory for predicted state with stair-like control strategy, and using sequential one-step predictions instead of the multi-step prediction, the analytical multi-step NMPC for affine nonlinear system will be proposed. Simulative and experimental control results will also indicate the efficiency of it. The feedback compensation mentioned in Section 2 is also used to guarantee the robustness to model mismatch.

Conclusion and further research direction will be given at last in Section 4.

## 2. One-step NMPC for affine system

### 2.1. Description of NMPC for affine system

Consider a time-invariant, discrete, affine nonlinear system with integer k representing the current discrete time event:

In the above,

Assume

The function F (.) and G (.,.) represent the terminal state penalty and the stage cost respectively, where p is the predictive horizon.

In general,

Corresponding to (1) and (3), the NMPC for affine system at each sampling time now is formulated as the minimization of

By the way, for simplicity, In (3), part of

And sometimes,

### 2.2. One-step NMPC for affine system

Except for some special model, such as Hammerstein model, analytic solution of multi-step NMPC could not be obtained for most nonlinear systems, including the NMPC for affine system mentioned above in Section 2.1. But if the analytic inverse of system function exists (could be either state-space model or input-state model), the one-step NMPC always has the analytic solution. So all the research in this chapter is not only suitable for affine nonlinear system, but also suitable for other nonlinear systems, that have analytic inverse system function.

Consider system described by (1a-1d) again, the one-step prediction can be deduced directly as follow with only one unknown data

In (5),

If there is no model mismatch, the predictive error of (5) will be

Then if the setpoint is

To minimize

Mark

But in practical control problem, limitations on input and output always exist, so the result of (8) is usually not efficient. To satisfy the constraints, we can just put logical limitation on amplitudes of

First, suppose every constraint in (1b) and (1c) could be rewritten in the form as

In which,

Choose Lagrange function as

In which,

### 2.3. Control problem of the water-tank system

Our plant of simulations and experiments in this chapter is a water-tank control system as that in Fig. 1. and Fig. 2. (We just used one water-tank of this three-tank system). Its affine nonlinear model is achieved by mechanism modeling (Chen et al., 2006), in which the variables are normalized, and the sample time is 1 second here:

s. t.

In (12), _{1} and valve V_{1}, while valve V_{2} is always open. In the control problem of the water-tank, for convenience, we choose the system state as the output, that means

To change the height of the water level, we can change the velocity of input flow, by adjusting control current of valve V_{1,} and the normalized relation between the control current and the velocity

## 2.4 One-step NMPC of the water-tank system and its feedback compensation

Choose objective function

Suppose there is no model mismatch, the simulative control result of one-step NMPC for water-tank system is obtained as Fig. 4. and it is surely meet the control objective.

To imitate the model mismatch, we change the simulative model of the plant from

Since

Update the process of one-step NMPC at time k, we have:

(13)-(14), and notice that

Proof end.

Because the soften factor

Table 1. is the comparison on

| e=x_{s}-x_{sp}Simulation(%) | e=x_{s}-x_{sp}Value of (15)(%) | |

0.975 | 0 | -8.3489 | -8.3489 |

0.001 | -8.3489 | -8.3489 | |

0.01 | -8.3489 | -8.3489 | |

0.95 | 0 | -4.5279 | -4.5279 |

0.001 | -4.5279 | -4.5279 | |

0.01 | -4.5279 | -4.5279 |

From (15) we know, we cannot eliminate this steady-state error by adjusting

In which,

Then add

Use this new predictive value to carry out one-step NMPC, the simulation result in Fig. 6. verify its robustness under model mismatch, since there is no steady-state error with this feedback compensation method.

The direct feedback compensation method above is easy to understand and carry out, but it is very sensitive to noise. Fig. 7. is the simulative result of it when there is noise add to the system state, we can see that the input vibrates so violently, that is not only harmful to the actuator in practical control system, but also harmful to system performance, because the actuator usually cannot always follow the input signal of this kind.

To develop the character of feedback compensation, simply, we can use the weighted average error

Choose

## 3. Efficient multi-step NMPC for affine system

Since reference trajectory and stair-like control strategy will be used to establish efficient multi-step NMPC for affine system in this chapter, we will introduce them in Section 3.1 and 3.2 at first, and then, the multi-step NMPC algorithm will be discussed with theoretical research, simulations and experiments.

### 3.1. Reference trajectory for future state

In process control, the state usually meets the objective in the form of setpoint along a softer trajectory, rather than reach the setpoint immediately in only one sample time. This may because of the limit on control input, but a softer change of state is often more beneficial to actuators, even the whole process in practice. This trajectory, usually called reference trajectory, often can be defined as a first order exponential curve:

In which,

### 3.2. Stair-like control strategy

To lighten the computational load of nonlinear optimization, which is one of the biggest obstacles in NMPC’s application, stair-like control strategy is introduced here. Suppose the first unknown control input’s increment

Instead of the full future sequence of control input’s increment:

Since the dynamic optimization process will be repeated at every sample time, and only instant input

### 3.3. Multi-step NMPC for affine system

The one-step NMPC in Section 2 is simple and fast, but it also has one fatal disadvantage. Its predictive horizon is only one step, while long predictive horizon is usually needed for better performance in MPC algorithms. One-step prediction may lead overshooting or other bad influence on system’s behaviour. So we will try to establish a novel efficient multi-step NMPC based on proposed one-step NMPC in this section.

In this multi-step NMPC algorithm, the first step prediction is the same as (5), then follows the prediction of

Since

For most nonlinear f(.) and g(.), the embedding form above makes it impossible to get an analytic solution of

Using the stair-like control strategy, mark

Here,

At last, the instant input

Though there are approximate predictions in the novel NMPC which may take in some inaccuracy, the feedback compensation mentioned above and the new optimization process at every sample time will eliminate the error before its accumulation, to keep the efficiency of the algorithm. The constraints also could be handled by methods mentioned Section 2 or by other numerical optimizing algorithm, thus we would not discuss about it here again.

### 3.4. Multi-step NMPC of the water-tank system

Choose

Choose

## 4. Conclusion

Using a series of approximate one-step predictions instead of the traditional multi-step prediction, the proposed multi-step NMPC leaded to an analytic result for nonlinear control of affine system. The use of stair-like control strategy caused a very little computational load and the feedback compensation brought robustness of model mismatch to it.

The simulations and experiments verify the practicability and efficiency of this multi-step NMPC for affine system, while the theoretical stability and other analysis will be the future work with considerable value.

## Acknowledgments

This work is supported by National Natural Science Foundation of China (Youth Foundation, No. 61004082) and Special Foundation for Ph. D. of Hefei University of Technology (No. 2010HGBZ0616, from the Fundamental Research Funds for the Central Universities).