Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 476367, 9 pages

http://dx.doi.org/10.1155/2013/476367

Research Article

Convergence Guaranteed Nonlinear Constraint Model Predictive Control via I/O Linearization

Xiaobing Kong,¹ Xiangjie Liu,¹ and Xiuming Yao²

¹State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China

²Department of Automation, North China Electric Power University, Baoding 071003, China

Received 14 July 2013; Revised 23 October 2013; Accepted 28 October 2013

Academic Editor: Baoyong Zhang

Copyright © 2013 Xiaobing Kong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Constituting reliable optimal solution is a key issue for the nonlinear constrained model predictive control. Input-output feedback linearization is a popular method in nonlinear control. By using an input-output feedback linearizing controller, the original linear input constraints will change to nonlinear constraints and sometimes the constraints are state dependent. This paper presents an iterative quadratic program (IQP) routine on the continuous-time system. To guarantee its convergence, another iterative approach is incorporated. The proposed algorithm can reach a feasible solution over the entire prediction horizon. Simulation results on both a numerical example and the continuous stirred tank reactors (CSTR) demonstrate the effectiveness of the proposed method.

1. Introduction

Model predictive control (MPC) is a popular algorithm in process control which online solves an optimization problem at each time step [1]. MPC considers process input, output, and state constraints directly in the control variable calculation. Under the linear model, with a quadratic objective function, MPC utilizes the convex quadratic program (QP), which can easily find the optimal solution [2].

Industrial processes are generally nonlinear, due to the frequent changes of the operating point right across the whole operation range. In general, the nonlinear model predictive control (NMPC) also online solves an optimization problem, by using the sequential quadratic program (SQP). The resulting nonlinear programming problems are usually nonconvex, and the online computational burden is generally large for most complex systems. A general way to solve this nonlinear optimal problem is to use approximate approach. For example, the first control move is calculated exactly, since it is actually implemented. The rest control moves can be approximated, since they are not implemented [3]. Thus the number of decision variables in the online optimization problem is equal to the number of inputs, instead of the number of inputs multiplied by the control horizon for a conventional NMPC algorithm. Paper [4] extended the first prediction of the linear constraints to the whole control horizon. Paper [5] studies the stability and region of attraction properties of a family of nonlinear MPC systems. Paper [6] presents nonlinear multivariable predictive control using neurofuzzy networks.

“Jacobian linearization” and “the input-output feedback linearization (IOFL)” are the two popular approaches in nonlinear control area. While the former can reflect the nonlinear model only at some certain point, the later represents the nonlinear system over a much wider operating range. Thus, IOFL is utilized in this paper for constituting nonlinear MPC, since the IOFL can offer a linear dynamic system so that the total optimal problem can be solved using the QP routine. Nevertheless, this can make the constraints to be nonlinear and state dependent. The approximate method developed in paper [7] can guarantee a feasible solution over the entire prediction horizon. The neural network is utilized to model the nonlinear discrete-time system. Paper [8] also introduces a technique with an affine transformation of the feasible region using feedback linearization scheme for handling input constraints. The neural network is also utilized to model the nonlinear system.

For digital MPC controller design, since the real-time control systems are in discrete-time forms, most NMPC research adopts the discrete-time feedback linearization. It should be noticed that the IOFL in differential geometry has been well developed for continuous-time system. Some famous methods in continuous-time system [9], for example, the extended system method, are not suitable to feedback linearise a general discrete-time system. Direct application of feedback linearization for a general discrete-time system may either be impossible or involve the time-consuming search algorithm. In [10], the authors proposed a linear model predictive control strategy via input/output linearization to the nonlinear process, by using one-step constraint algorithm. This paper aims to make full use of the advantage of continuous-time system IOFL and then try to reach the convergence to a feasible solution over the entire prediction horizon within the available time. Since the input constraints are transferred into nonlinear constraints from initial linear constraints, the convergent algorithm is constituted to guarantee a feasible solution without constraints violation. Simulation results on both a numerical example and the continuous stirred tank reactors (CSTR) demonstrate the effectiveness of the NMPC method.

2. The Linear Control Structure via Input-Output Feedback Linearization

The IOFL is to transform the original nonlinear system into linear input-output relationship, generally by using a static state feedback control law [11]. Consider the SISO affine state-space model as follows:

where is state variables, is the manipulated input variable, is the controlled output variable, and , , and are smooth functions in a domain .

Definition 1 (relative degree). The nonlinear system (1) is said to have relative degree , , in a region if

If the relative degree , then for every , a neighborhood of exists such that the map restricted to is a diffeomorphism on .

The mapping transfers the system (1) to a new system as follows:

where , , which can also be expressed as , in the new coordinates.

The feedback law is

where is the transformed input variable. The new linear system is

where

The resulting linear state-space system (5) could be used for constituting “standard” linear MPC [1]. Discretization (5) can result in

The matrices , , and are expressed as , , , where is sampling time.

Define ; ; . Choose a new group of state variables ; the augmented system is

where . Based on the state-space model , the future outputs along the predictive horizon are calculated sequentially as

Define vectors

where is the control horizon.

Collect (9) and (10) together in a compact matrix form as

where

Similarly, we can have the future state variables along the predictive horizon based on the model of

where

The object function to be minimised is a quadratic criterion on as

By using (12), the optimization problem can be attributed to

where , , and indicates the minimal and maximum value of .

3. Convergence Algorithm for Constraint Optimal MPC

3.1. The Nonlinear Constraint Handling

In practice, the process inputs are frequently subjected to the following level inequality constraints:

where represents the control inputs, and and indicate the minimal and maximum value of .

After the realization of the IOFL, the input vector is transformed into a new one . With this nonlinear and state dependent input constraint, traditional QP cannot be used to calculate the optimal control sequence. Though the sequential quadratic programming (SQP) technique is generally used on nonlinear constraint MPC, it is often nonconvex and can cause large computation burden. This property will be demonstrated later in the simulation example of CSTR.

In order to solve this nonlinear constraints problem, an iterative QP is first used to try to get the optimal solution. The key issue is to establish the nonlinear relationship between vectors and and then get the explicit expression of from .

To establish the nonlinear relationship between vectors and , expand the feedback law (4) over the control horizon ,

Considering the expression (14), (19) can be rearranged in the matrix form

The above inequality can be rewritten in the following form:

due to

In general, can be rewritten as

with the constraints

where , indicate the minimal and maximum value of , which are state dependent.

Thus

The above inequality can be rewritten over the entire horizon,

where

From (26), the above equation can be rewritten as

This iterative QP algorithm is presented below:

 Step  1: Initializing within the input constraint. Step  2: Solve QP for subject to  Step  3: Calculate  Step  4: Test if , end; otherwise, , go to Step  2.

The above iterative QP is effective for solving the problem if the initial condition is properly chosen. Nevertheless, it has no guaranteed convergence to a feasible solution. To overcome this problem, another iterative algorithm is needed to guarantee a feasible solution over the entire prediction horizon is incorporated. This idea is previously presented by paper [7] on a input/output model. Extending it to the state-space equation is straightforward.

In order to solve the convergence problem, linear approximation is necessary. Now consider the approximation by linearization through Taylor’s expansion about of the nonlinear constraints (21) as

where is the chosen initial operating trajectory, is obtained from (21) when , the dimensional matrix represents the Jacobian matrix in (21) at the operating trajectory, and corresponds to the higher order terms of the approximation given by

where , .

Neglect the higher order terms as

Equation (31) can lead to an explicitly linear function relationship between the control sequence and :

Now the optimization problem is defined as

subject to the convex set of approximate linear constraints

with , .

In order for the algorithm to reach the convergence to a feasible solution, the optimal problem can be rewritten as

where is decreasing parameter with initial value . Define and , respectively, by

Define to be the linear control sequence and to be the nonlinear control sequence obtained from . Then the major goal is to keep

Thus the optimization problem with the approximate constraints is a feasible solution, which can be solved iteratively using QP.

This approximate constraints algorithm is presented follows.

 Step  1: assure feasibility of the new operating trajectory: . Solve for ,  Step  2:  Step  3: actualize the squeezing factor :

 Step  4: new linearized constraints at : . Step  5: Solve QP for the new linearized constraints:

 Step  6: test if violate the original constraint. If: . Step  7: let , go to 3.

3.2. The Convergence Property

With the object function (35), assume the practical input sequence constraints

Assume a given initial vector representing the operating trajectory, , verifying the conditions

where is small positive scalar values, .

For any given scalar , the solution of the optimization problem (35) is subjected to

which can be further written as

In the limit, as , , , the solution of the optimization problem is given by

Due to the smooth function and definition (42), this last result leads to

signifying that the resulting practical nonlinear control sequence is always within the bounds, and so a feasible solution is always guaranteed.

3.3. The Algorithm for Constraint Optimal MPC

The MPC algorithm, which can guarantee convergence to a feasible solution, has to deal with the constraints as the following steps:

 Step  1: . Step  2: if , go to 8. Step  3: assure feasibility of the new operating trajectory: . Step  4: solve QP at :

 Step  5: test if violates the constraints: if

         where is nonlinear control sequence, obtained from .

 Step  6: . Go to 2 Step  7: , . Solve for , . Step  8: . Step  9: actualize the squeezing factor :

 Step  10: new linearized constraints at :  Step  11: Solve QP for the new linearized constraints:

 Step  12: test if violate the original constraint. If:  Step  13: let . Go to 9.

The total algorithm performance is quite related to the tuning of two parameters:

 : the number of iterations of the QP algorithm; : the squeezing parameter of the convergence guaranteed algorithm.

The convergence speed of this iterative algorithm can be set using the decreasing rate , defined here through parameter .

Choosing the initial value is a key issue in solving the optimal problem, since it is quite related to the resulting computation burden. Suppose that the final input sequence at instant can be expressed as . Then, the initial value at instant should be chosen as  ; for example, the first factors are taken directly from the prediction of .

4. Case Studies

Two examples are presented to illustrate the implementation and the performance of the proposed nonlinear MPC. In example 1, a numerical state-space equation is used, and in example 2, the control problem of CSTR is considered.

4.1. Numerical Example

Assume the state-space equations to be

The input constraint is .

Using (4), the static feedback law:

The resulting feedback linearization linear system is

Choose the predictive horizon , the control horizon , and the sampling time  s. Figure 1 shows the system response for a step signal using the proposed IOFL MPC method. The algorithm guarantee’s that the process input does not violate the constraints. The system reaches stability in 3 seconds without overshoot. Figure 2 shows the number of iterations at each optimization step. From Figure 2, the process needs 65 iterations at the beginning but soon reduces the iteration and finally reaches the convergence using only one iteration.

Figure 1: The closed-loop response for a step signal under the proposed IOFL MPC.

Figure 2: The number of iterations at each optimization step.

Simulations were then repeated under different predictive horizons. Table 1 lists the comparison of the control performance. From Table 1, the computing burden increases when the prediction horizon increases, meanwhile the tracking property improves with the increment of the prediction horizon.

Table 1: The comparison of control performance at different prediction horizon.

Prediction horizon
Computing burden

(CPU time)
Sum squared error

8
1.0469
11.4042

10
1.1875
11.4036

12
1.2813
11.4032

4.2. The Control of a First-Order CSTR

CSTRs are typical chemical processes representing a variety of complex industry systems. The CSTR problem discussed here represents a first-order, irreversible, exothermic kinetics reaction, which can be described by the following equation [12]:

where the two state variables and are the reactor temperature and the normalized reactant concentration. The control variable is the normalized cooling water temperature. , , and are the constants of the system.

The CSTR control objectives contain two aspects: set point tracking and regulation of reactant concentration under the perturbation of feed water temperature.

With the static feedback law,

where

The resulting feedback linearization linear system is

Suppose the constraint to be . Let , , the predictive horizon , the control horizon , and the sampling time  s. Figure 3 shows the system output and the control under step changes in the reference signal.

Figure 3: The system output and the control under step changes in the reference signal.

From the simulation result, the manipulated variable does not violate the defined constraint. The system reaches stability in 30 steps without obvious overshoot.

As mentioned before, there exist two general approaches for NMPC. One is to use the SQP method. Another is to approximate the first constraints as the constraints over the entire control horizon. The proposed method is then compared with these two methods, considering this CSTR plant. The simulation results using the three predictive control schemes for are given in Figure 4, adopting the best and for each in the proposed IOFL NMPC.

Figure 4: The comparison of the three responses.

Computation burden is one of the most concerning problems in CSTR real-time control. In using MPC, though choosing larger predictive horizon can improve system performance, it can, meantime, result in the increment of computation burden, especially in nonlinear system. For further comparison, the controller performance is analysed with respect to two variables, for example, sum squared output error (SSE) and the relative optimization time for the simulation. Table 2 shows the performance of each controller configuration for various prediction horizons. The three methods, namely, the one-step constraints, the MPC with IOFL and the nonlinear MPC, can all guarantee a feasible control solution over the complete prediction horizon. The comparison of the three shows a similar trend performance in terms of the system tracking error, although the one-step constraint is less computational time demanding, as shown in Figure 5. With the increase of the prediction horizon, the time increase in nonlinear MPC is obvious, while the time increase in one-step constraints and MPC with IOFL optimization is quite minor. This is due to the efficient iterative process of the proposed IOFL MPC. The reduced computing burden for the one-step constraints method is obvious, since only the first time-step-ahead inputs are constrained, leaving the other inputs unconstrained. The shortcoming is that the resulting optimum may not be feasible. From Figure 5, the total trade-off between the computational demands and the gain in optimality is obviously favourable to the proposed MPC with IOFL.

Table 2: Left: CPU time of the entire simulation. Right: sum squared output error (SSE) in the simulation.

Predictive horizon
CPU time
SSE

One-step constraints
IOFL MPC
SQP
One-step constraints
IOFL MPC
SQP

10
0.1563
0.2011
0.3054
11.4181
11.4045
11.4042

12
0.1719
0.2407
0.3602
11.4180
11.4034
11.4031

14
0.2031
0.2795
0.4415
11.4177
11.4026
11.4018

16
0.2614
0.3306
0.5728
11.4130
11.4019
11.4003

Figure 5: The comparison of control performance.

5. Conclusions

MPC has been widely used in industrial process. One limitation of MPC is that it is mostly based on a linear model. The performance will deteriorate when the current output is relatively far away from the operating point at which the linear control model was generated, especially in the presence of constrains. The approach presented in this paper aims at combining the nonlinear differential geometry technique with existing MPC technique to avoid nonlinear optimization when using a nonlinear model of the process. Considering the continuous-time system, the detailed routine for reaching a feasible solution through iterative QP optimization is presented in this paper. The initial value is properly chosen to greatly reduce the computation burden. Comparing results on the CSTR considering, both the SSE and the relative optimization time needed for completing the simulation have also been addressed in detail. The advantage is that it avoids a direct nonlinear optimization on a nonlinear model of the process and incorporates the linear optimization technique in the controller design.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 60974051, 61273144, and 61203041, the Natural Science Foundation of Beijing under Grant 4122071, the Chinese National Postdoctor Science Foundation under Grants 2011M500217 and 2012T50036, the Doctoral Fund of Ministry of Education of China under Grant 20120036120013, and the Fundamental Research Funds for the Central Universities 13XS09.

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