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The paper deals with an online optimization control method for dynamical processes called Model Predictive Control (MPC). It is a popular control method in industry and frequently treated in academic areas as well. The standard predictive controllers usually do not guarantee stability especially for the case of short horizons and large control error penalization. Terminal state is one way to ensure stability or at least increase the controller robustness. In the paper, deviation of the predicted terminal state from the desired terminal state is considered as one term of the cost function. Effect of the stability and control quality is demonstrated in the simulated experiments. The application area for online optimization methods is very broad including various logistics and transport problems. If the dynamics of the controlled processes cannot be neglected, the optimization problem must be solved not only for steady state but also for transient behaviour, e.g., by MPC.

Minimization of a quadratic cost function is a common method for solving many engineering problems. In the control area, this method is fundamental not only for standard control design methods like optimal control, e.g., [

The paper is structured as follows: standard controller design is described in Section

Under the assumptions of linear controlled system and quadratic cost function it is possible to formulate the task of the optimal controller design as a standard mathematical problem, extreme finding with an analytic solution. A unique solution exists also in the case of constrains existence in a form of linear inequalities.

The key part of the controller design is to incorporate maximum of the known information and demands into the properly formulated cost function. It is possible to involve various (even conflicting) control demands. Then the controller tuning consists in weightings of the particular demands.

From practical point of view, it is appropriate to formulate the task in discrete-time domain with receding (finite) control horizon [

The cost function always contains the fundamental control requirement, the term _{x} in the cost function can be used only in the case of finite control horizon and state space description. It introduces into the cost function a dependence on the system state at the end of the control horizon called terminal state. The predictive controller design based on input-output description does not use it in a basic formulation of the cost function. The terminal state is obviously introduced in the extensions concerning the stability and robustness; see, e.g., [

In some cases the terminal state is important from the mathematical point of view. In case of LQ control design on finite horizon, the mathematical importance of the terminal state is that the matrix

In literature the terminal state is obviously mentioned only in the context of the controller stability. The use of the terminal state has also an implication to the controller performance. The standard formulation of the terminal state in the form of (

Clear and unique additional requirements can be formulated because the state vector contains complete information about the state of the system. The predictive controller can ensure, e.g., demand of minimum energy cost of a system with more inputs than outputs (nonsquare, overactuated system). The problem of how to determine an optimal steady state for such systems is discussed, e.g., in [

Application area of predictive control methods is not limited to refinery, chemical, pulp, and paper industries, but it is becoming very broad. It can be advantageously applied also in transport industry, as demonstrated in [

The controller design starts from a discrete-time state space model of the controlled MIMO (Multi-Input Multi-Output) system with_{u} inputs,_{x} state variables, and_{y} outputs. The model is in a standard form (

_{u},1],

_{x},1] and

_{y},1].

Matrix equations (

Matrices

With respect to a terminal state in the cost function (

The cost function in matrix form (

_{y},1],

_{u},1],

_{u},1],

_{x},_{x}],

_{y},_{y}] and

_{u}, (_{u}].

First item of the vector

Computation of the desired terminal state is trivial in case of the system with identical number of inputs and outputs and if we consider steady state. The controlled system steady state behaviour is given by

The solution for the desired output_{0}=_{0} is

The aim of the following control simulations is to demonstrate the effect of the terminal state in predictive controller design to the control quality and stability. The simulations are supposed as an ideal case; controlled system is identical with the process model used for the controller design and neither noises nor disturbances are considered. The controller is designed for the set-point tracking task.

Two different controlled systems are treated in the simulations. The first system is a standard system of a higher order (

Characteristics of controlled systems.

Standard predictive controller described in Section

_{0} is steady state vector,

The set-point shape consists from tree parts. The first part takes the same time as a control horizon plus 5 sampling periods and the set-point is constant. The second part lasts as a system settling time (50 s) and the set-point linearly increases from the first to third part. The third part is as long as the second one and the set-point is constant again. The control quality measure is calculated as an integral of the absolute control error

The effect of the terminal state can be observed from IAE measure values for simulated control experiments summarized in Table _{x}.

IAE quality measure for system (

T = 1 s | ||||
---|---|---|---|---|

N | | | | |

| ||||

15 | 0.3507 | 4.0338 | 27.240 | 65.718 |

| ||||

20 | 0.4231 | 4.6390 | 28.661 | 62.632 |

| ||||

35 | 0.4032 | 3.0894 | 16.149 | 30.107 |

| ||||

50 | 0.4034 | 1.3764 | 5.685 | 9.295 |

IAE quality measure for system (

T = 1 s | ||||
---|---|---|---|---|

N | | | | |

| ||||

15 | unstable | unstable | 13.878 | 10.911 |

| ||||

20 | unstable | 10.842 | 7.697 | 8.017 |

| ||||

35 | 4.0766 | 1.769 | 2.117 | 3.069 |

| ||||

50 | 1.2727 | 0.658 | 0.807 | 1.239 |

The control responses of two selected control experiments are plotted in Figure

Control response of system (

Control response of system (

Effect of the terminal state to the stability of the control is definitely positive. Even in the case of wrong choice of the controller parameters (control horizon is too short) the terminal state increases dramatically the controller stability.

Nevertheless the control quality is obviously worse if the terminal state is used. The control deterioration is evident even if the terminal state was considered in a form of the deviation from the desired terminal state calculated from the steady state. The deviation form solves the main problem, permanent control error in the steady state. Control quality decrease is caused by stronger effect of the terminal state in the cost function then the other two terms (although in one point of the control horizon). This statement is not true generally because the effect depends on the controlled system and on the controller parameters, firs of all on the length of the control horizon.

This effect was strong especially by the control of system (

Prediction along control horizon.

The desired terminal state in our case is a steady state corresponding to the set-point at the end of the control horizon. The question remains whether to use more accurate desired terminal state. This should increase the control quality, but it will make the whole controller more complicated and we will lose interesting feature for the steady state optimization especially for nonsquare systems.

The process models and controller parameters data used to support the findings of this study are included within the article. We can provide also MATLAB scripts with the code for the simulations if suitable.

The authors declare that they have no conflicts of interest.

The work was supported by ERDF/ESF “Cooperation in Applied Research between the University of Pardubice and companies, in the Field of Positioning, Detection and Simulation Technology for Transport Systems (PosiTrans)” (No. CZ.02.1.01/0.0/0.0/17_049/0008394).