The paper deals with the multimodel and robust control system design and their combination based on
When hydraulic turbine power control is considered, the water inertia effect is a factor that makes it difficult to maintain stability under isolated operation or to have fast response in case of load change in the whole operational range. Nonlinear models of hydraulic turbine control systems are needed in those cases where the turbine power varies in the whole range. The turbine model parameters vary significantly with the unpredictable load variations. Such nonlinearities make the controller design a nontrivial task due to the fact that a controller designed for specific operating conditions may not perform well under different loads [
There are several approaches to model hybrid systems [
In this paper, hybrid control, often denoted by multimodel control, will be used. Several switching algorithms will be compared to show how the switching speed affects performance and stability. Well-known robust control method will be presented as a control solution which ensures stability on whole operating range, but with lower performance. Combination of multimodel control and robust control which ensure stability and good performance will be presented on numerical example of nonlinear hydraulic turbine model.
Consider nonlinear system as uncertain system described by set of transfer functions
To deal with uncertainties instead of a single model, the behavior of a class of models
Multimodel control is based on several linear models identified in operating points. For each linear model, local controller is designed which ensures stability and perhaps even performance only for this model. As nonlinear system moves from one operating point to another, controller switches from one local controller to another one. Performance of this control approach can be much better as using only one controller on whole range, but this method does not yield stability when nonlinear system is between operating points. Fast switching between local controllers can decrease performance and also lead to instability.
In robust control, several linear models identified in operating points are used for uncertainty model calculation. Then, for system control, one controller is used, which ensures stability on whole operating range, between all operating points. Performance (of robust control) is not so good as performance of multimodel control; however, there is no problem with stability by controllers switching. When nonlinearity is increased, uncertainty is increased too and performance of robust control decreases. If the system is strongly nonlinear, uncertainty could be too big to be able to control the system using robust control.
In this paper, designing a set of local controllers similar to those in multimodel control design will be done, but each local controller will be designed as robust to ensure stability between operating points. By robust design of local controller, instead of all operating points, only given and neighbourhood operating points will be considered. This type of control will be called robust multimodel control and will ensure stability and higher performance than robust control.
Consider a system with nonlinear static characteristic (Figure
Static characteristic is nonlinear and could be approximate using
These models are simple to obtain using input-output identification. For each model (operating point (OP)), local PID controller can be tuned using any well-known and tested method. Depending on system output, multimodel controller switches between local PID controllers. Local controllers ensure stability only in operating point.
Characteristic equation of closed-loop feedback system (Figure
Standard feedback control configuration.
Static characteristic of nonlinear system.
Static characteristic with linear models.
Equation (
Local controllers ensure only stability (negative roots) of (
A simple uncertainty model is obtained using unstructured model approach. In the sequel (see Figure Additive uncertainty model for operating points Input multiplicative uncertainty model for operating points
where
where
Standard feedback configuration with unstructured uncertainty of any type can be rearranged to obtain the general
Illustration of how the full operating area of nonlinear system is covered by stability area of single robust controller.
Robust stability condition for unstructured perturbations is formulated in terms of stability of the for additive uncertainty:
for input multiplicative uncertainty:
In view of (
Robust control approach based on unstructured uncertainty and with robust stability ensured using Nominal model calculation: nominal model For nominal model standard feedback controller is designed: good choice could be controller design method with desired phase margin. Robust stability verification according to (
In this case, for each operating point of switched system, with arbitrary switch algorithm, robust controller
Let derive robust stability condition for operating point
Robust controller design procedure for each operating point
Robust multimodel control approach based on unstructured uncertainty and with robust stability ensured using For each operation point, nominal model is calculated using not all but only neighborhood operating points. Nominal model For nominal model Robust stability verification according to (
These three steps have to be done
Stability areas of local controllers (Figure
Illustration of how the full operating area of nonlinear system is covered by stability area of multimodel robust controller.
Consider hydraulic turbine with power operating range (0–100)%. In our case, nonlinear model described in [
Verification of linear models identified in operating points can be found in appendix (Figures
In first step, multimodel control approach will be used for hydraulic turbine control. For each transfer function, PI controller will be designed using D-partition method [
Figure
Step responses of linear models with PI controllers.
First switching algorithm will be based on system output. Since system output will be lower than 30%, first controller will be active. If output will be more than 30% but less than 50%, second controller will be active and so on (Table
Switching algorithm based on system output.
Output [%] | <0, 30) | <30, 50) | <50, 70) | <70, 85) | <85, 100) |
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Active contr. |
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Second algorithm will be based on comparison of system output
In each sample time comparison will be done and
where
Nonlinear turbine model has strong nonminimal phase behavior. Performance comparison (Figure
Control performance comparison by different switching algorithms.
Switching between controllers
When high control performance is needed, it is recommended to avoid fast switching. Possible solutions could be increasing switching period (not at each sample time), weighting of controller outputs, or, in this case, using first switching algorithm.
In second step by robust control design instructions from previous chapter will be used.
From transfer functions (
For nominal model, PI controller was calculated using design method ensuring desired phase margin. Controller ensuring phase margin
Robust stability condition for controllers ensuring phase margins
Transfer function of controller ensuring phase margin
Comparison of multimodel and robust control (Figure
IAE comparison of multimodel control (two switching algorithms) and robust control.
Control | Multimodel 1 | Multimodel 2 | Robust |
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IAE | 4,82 | 7,06 | 5,94 |
Control performance comparison multimodel and robust control.
In third step, robust multimodel control will be designed, which is a combination of robust and multimodel control (see Table
By controller design for first operating point, nominal model
For this nominal model controller ensuring phase margin
Robust stability condition (Figure
Robust stability condition, first operating point.
By controller design for second operating point, nominal model
For this nominal model controller ensuring phase margin
Robust stability condition (Figure
Robust stability condition, second operating point.
Controller design for the rest of operating points goes the same way. Controller parameters for operating points 3–5 are as follows:
Comparison of three control approaches (Figure
IAE comparison of multimodel control, robust control, and robust multimodel control.
Control | Multimodel 1 | Robust | Multimodel robust |
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IAE | 4,82 | 5,94 | 4,92 |
IAE comparison of multimodel and robust multimodel control algorithm 2.
Control algorithm 2 | Multimodel | Robust multimodel |
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IAE | 7,06 | 5,7 |
Comparison of multimodel, robust, and robust multimodel control.
Performance of multimodel and robust multimodel approaches will be compared also using second switching algorithm (Figure
Comparison of multimodel and robust multimodel control using second switching algorithm.
Comparison of data measured on nonlinear system and on identified linear model, OP 20%.
Comparison of data measured on nonlinear system and on identified linear model, OP 40%.
Comparison of data measured on nonlinear system and on identified linear model, OP 60%.
Comparison of data measured on nonlinear system and on identified linear model, OP 80%.
Comparison of data measured on nonlinear system and on identified linear model, OP 90%.
In this paper, hybrid control, often denoted by multimodel control, was used. Two switching algorithms were compared to show how the switching speed affects performance and stability. Robust control method based on
For identification results, in form of model verification, in all operating points , see Figures
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Scientific Grant APVV 0772-12.