A new robust LMI-based model predictive control for continuous-time uncertain nonlinear systems
This paper presents a new robust predictive controller for a special class of continuous-time non-linear systems with uncertainty. These systems have bounded disturbances with unknown upper bound as well as constraints on input states. The controller is designed in the form of an optimization problem of the 'worst-case' objective function over an infinite moving horizon. Through this objective function, constraints and uncertainties can be applied explicitly on the controller design, which guarantees the system stability. Next, LMI tool is used to improve the calculation time and complexity. To do this, in order to find the optimum gain for state-feedback, the optimization problem is solved using LMI method in each time step. Finally, to show the efficiency and effectiveness of the proposed algorithm, a surge phenomenon avoidance problem in centrifugal compressors is solved.
Keywords: Robust model predictive control; LMI; surge; centrifugal compressors
1. Introduction
In recent years, attention to robust approaches to controller design has increased strictly [[1]]. Meanwhile, model predictive controller is widely used to control industrial systems [[4]] due to its unique features. Given the fact that most operational systems are non-linear and bounded systems, the predictive controllers are used on the basis of linear or non-linear models with uncertainty [[9]]. One reason for this is that non-linear predictive control algorithms lead to non-convex, non-linear optimization problems, the solution requires reiterative methods along with extended calculation times [[8]]. In addition, the convergence region of these kinds of algorithms is qualitative which increases online calculation time. On the other hand, using linear model and square cost function lead to a convex square optimization problem that can be solved easily for predictive control algorithms. However, in many systems, the non-linear effects cannot be ignored. In these conditions, the system can be approximated by a linear model and considered on the scope approximation error [[12]]. Predictive control has a strong advantage: it can consider constraints explicitly in the problem, but it cannot calculate the model uncertainty explicitly in the formulation. Using robust predictive control methods, uncertainty in the process model can be combined explicitly with the problem [[15]].
Modelling the system is a major prerequisite in predictive control design. Also, the model accuracy plays an important role in controller good performance. Practically, systems have uncertainty in models, which should be considered in designing robust controller in order to guarantee the stability of closed-loop system throughout the uncertainty scope. Various algorithms have been presented for robust predictive controller design. These algorithms should have three important characteristics. First, the time required for calculation should be functional and appropriate. Second, it should provide a vast stability region for the system. Third, the controller performance given the system uncertainties should be in an appropriate condition. There are different methods that cope with these conditions.
In first method, the non-linear system is presented along with a linear model and a non-linear term which is Lipchitz. Then, in robust stability, sentences including this non-linear term are substituted by its upper bound and turn into a linear matrix inequality. In this method, the robust predictive controller problem turns into a state physic design, obtained by solving a linear matrix inequality (LMI). Therefore, in each moment an optimization problem is solved in the format of LMI which compared to an online, non-linear, convex optimization problem, can have a smaller calculation time and complexity [[12]],[19]].
In another algorithm for robust predictive controller, an offline predictive control method has been used to reduce the calculation time and complexity. In this method, state-feedback has been calculated for pre-specified regions of the state space in an offline manner and the results are stored in a table. Then, for online implementation, the calculation time is only assigned to search for corresponding physic with the existing state [[20]]. In this method, if the working area of a non-linear system is big, stating it with only one single linearized model around the desired working point is not an exact specification for the non-linear system. Therefore, in [[21]] a tabulated predictive controller or a multi-point predictive controller has been used for limited, non-linear systems with extensive working range. In this method, the working area is divided into smaller parts according to the algorithm which include a set of approximate stability regions around different equilibrium points on the surface of the equilibrium system. Then, for each region, a local controller is calculated. These regions are chosen in a way that have overlap and the system stability is guaranteed by this algorithm. However, since this algorithm states the non-linear model in each region with a time-variable linear model along with multi-dimensional structural uncertainty, it needs a huge calculation time.
In order to calculate the predictive controller problems in an online manner, one of the appropriate tools is LMIs that have small calculation time and complexity. In this paper, for a special class of non-linear systems, some LMI-based robust predictive controllers are presented. As it was mentioned, in first controlling method, the non-linear system is presented along with a linear model and a non-linear term which is Lipchitz. In the design of an LMI-based robust MPC, the Lipchitz condition is an essential requirement in the design process [[23]] and somehow leads to the linearization of the controller design while in the paper, it has been tried to present a method that removes the Lipchitz condition and includes a greater class of non-linear system in the model. The most important innovation of this paper is to provide an LMI-based predictive control method for a class of continuous non-linear systems in the presence of disturbance and uncertainty, while ensuring simplicity and less time for computation, confirms the optimal control signal and in addition to compliance with constraints of variables and states, it is not necessary to have the Lipchitz condition on the non-linear part.
This paper is composed of the following sections: Section two presents the new robust MPC. Section three designs a robust MPC controller to stabilize the compressor system. Section four, by presenting a simulation, proves the efficiency of the controller. Finally, section five concludes the paper.
2. Robust LMI-based MPC
Consider the following continuous-time non-linear system
Graph
(1)
where
Graph
shows the system states,
Graph
the control input,
Graph
continuous non-linear uncertainty function. The
Graph
is considered in the following set
Graph
(2)
The system has the following limitations
Graph
. Where
Graph
is bounded and
Graph
is compact.
Lemma 1:
([[26]]). Let
Graph
be a continuously differentiable function and
Graph
, where
Graph
are
Graph
class functions. Suppose
Graph
is chosen, and there exit
Graph
and
Graph
such that
(3)
Graph
with
Graph
. Then, the system trajectory starting from
Graph
, will remain in the set
Graph
, where
(4)
Graph
Lemma 2:
([[12]]). Let
Graph
be real constant matrices and
Graph
be a positive matrix of compatible dimensions. Then
(5)
Graph
Holds for any
Graph
.
Lemma 3:
(Schur complements [[27]]). The LMI
(6)
Graph
In which,
Graph
,
Graph
and
Graph
are affine functions of
Graph
, and are equivalent to
Graph
(7)
The state-feedback control law for system (1) in
Graph
time is chosen as
Graph
(8)
This control signal is true for the following constraint
Graph
(9)
Finally, the chosen infinite horizon quadratic cost function is specified as
Graph
(10)
where
Graph
and
Graph
are positive definite weight matrices. In the objective function (4), the uncertain but negative effect with weight
Graph
is introduced, where
Graph
is positive constant [[28]].
Theorem 1:
Consider system (1),
Graph
is the measure value in sampling time of
Graph
. There is a state-feedback control law (8) that is true in stability condition and in input constraint (9) in every moment. If the optimization problem with LMI constraints can be feasible.
(11)
Graph
where
Graph
and
Graph
are matrixes obtained from the above-mentioned optimization problem. As such, state-feedback matrix in every moment is obtained as
Graph
.
Proof.:
Considering a quadratic Lyapunov function, we have
(12)
Graph
In sampling time for
Graph
assume that
Graph
is true in the following condition
(13)
Graph
(14)
Graph
In order to obtain the robust efficiency, we should have
Graph
which results in
Graph
. By integrating both sides of the Equation (14), we have
(15)
Graph
where
Graph
is a positive scalar (the upper bound of the objective (10)).
In order to obtain an MPC robust algorithm, the Lyapunov function should be minimized considering the upper bound [[29]]. So
Graph
(16)
By defining
Graph
and using Schur Complements, we have
Graph
(17)
In the following, according to Lemma 1, for system (1) we have
Graph
(18)
Then, according to (12) we have
Graph
(19)
Graph
(20)
According to Lemma 2, we have
Graph
(21)
By substituting (21) in (20), it is obtained that
Graph
(22)
Consider
Graph
(23)
where
Graph
is the maximum eigenvalue of
Graph
and
Graph
is the corresponding upper bound [[12]], then
Graph
(24)
By choosing
Graph
(25)
Equation (24) is reduced to
Graph
(26)
Substituting
Graph
and
Graph
,
Graph
(27)
Pre and post multiplying by
Graph
,
Graph
(28)
Given (23), we have
Graph
(29)
Substituting
Graph
and pre multiplying by
Graph
, we have
Graph
(30)
Finally, the input constraint is investigated [[12]]. According to (9), we have
Graph
(31)
Given (13) and (15), it is known that the states
Graph
determine and ellipsoid invariant set
Graph
(32)
Therefore
Graph
(33)
From (31) and (32), we can rewrite the input two-norm constraint in (31) as
Graph
(34)
By applying Schur complement. (34) is equivalent to
Graph
(35)
So, the proof is completed.
3. Robust model predictive control on surge
Surge is a condition that occurs on compressors when the amount of gas is insufficient to compress and the turbine blades lose their forward thrust, causing a reverse movement in the shaft. It can cause extensive structural damage in the machine because of the violent vibration and high thermal loads that generally accompany the instability. For this reason, compressor system control as one of the most practical systems is considered in this section.
3.1. Compressor model
Pure surge model of Moore and Greitzer [[30]] for the centrifugal compressor are as the followings
Graph
(36)
where
Graph
is the coefficient of increase in compressor pressure,
Graph
is the coefficient of compressor's mass flow,
Graph
and
Graph
are the disturbances of flow and pressure. Also,
Graph
is the characteristic of throttle valve and
Graph
is the characteristic of the compressor.
Graph
is the Greitzer's parameter and
Graph
shows the length of channels (ducts). Moor and Greitzer's [[30]] compressor characteristic is defined as
Graph
(37)
where
Graph
is the value of characteristic curve in 0db,
Graph
is half of the height of the characteristic curve, and
Graph
is the half of the width of the characteristic curve. The equation for throttle valve characteristic is also derived from [[31]] as follows
Graph
(38)
Figure 1 is the diagram of compression system with Close Couple Valve (CCV).
Graph: Figure 1. The compressor system with CCV [[32]].
The system model equations, considering a CCV, are
Graph
(39)
Considering
Graph
as the input for system control and
Graph
, the equations of compressor state space are
Graph
Graph
(40)
3.2. Controller design
In designing a surge controller in the compressor system (40), it is assumed that the values of throttle valve, as well as the compressor characteristic are unknown. So, rewriting the compressor equation in the form of Equation (1) yields:
Graph
(41)
Graph
(42)
According to the compressor model,
Graph
,
Graph
and
Graph
are 2, 1 and 2, respectively. Since the controlling signal has a CCV output, so we have
Graph
(43)
The next constraint and limitation is that the flow has some maximum and minimum values. This constraint should also be considered.
Graph
(44)
The LMI parameters are selected
Graph
(45)
4. Simulation
This section includes a simulation to prove the robustness and effectiveness of the presented controller. Several studies have recently been conducted on optimal and robust control for surge instability in the compressor system [[33]]. In this paper, reference [[33]] is used to perform a comparison as well as to examine the capability of the proposed method. To do this, the system is simulated in three different modes. In all three modes, the scenario is that until the time
Graph
the value of throttle valve is
Graph
, which is the compressor working point on the right side of the surge line. After
Graph
, the value of throttle valve is reduced to
Graph
which causes the compressor working point to go to the left side of the surge line and the system suffers from limit cycling. Values of compressor parameters used in simulation are according to [[39]].
Graph
(46)
The initial points of process were
Graph
.
In the first mode, it is assumed that there is no external disturbance on the compressor system.
Graph
(47)
The states and control signal for compression system using proposed method are shown in Figures 2–4. Also, it is compared with robust adaptive tube MPC [[33]] to demonstrate the effectiveness of the proposed controller. As it can be seen, simulation results illustrate the robustness of the proposed controller to steer the centrifugal compressor against change in throttle valve opening percentage. As shown in Figures 2 and 3, using the proposed controller, the compressor operates at high pressure away from the surge condition. Figure 4 presents the controlling signal. Since CCV is considered as the controller operator, its output must be positive. According to Figure 4, the control signal has lower amplitude than the robust adaptive tube MPC which indicates better optimization in the proposed controller.
PHOTO (COLOR): Figure 2. Pressure of compressor.
PHOTO (COLOR): Figure 3. Flow of compressor.
PHOTO (COLOR): Figure 4. Control signal.
Figure 5 shows the trajectory of compressor in a performance curve. It shows how the controller prevents the compressor from entering to the surge area.
PHOTO (COLOR): Figure 5. Compression system trajectories.
In the second scenario, it is assumed that some transitory disturbances are applied to the system. These disturbances are modeled as
Graph
(48)
Figures 6 and 7 present compressor pressure and flow, respectively. As it can be seen, simulation results illustrate the robustness of the proposed controller to steer the centrifugal compressor against disturbance and change in throttle valve opening percentage. It is clear from Figure 6, the proposed method yields higher pressure rate. It is also seen from Figure 7 that the compressor operates at steady state oscillations less than robust adaptive tube MPC [[33]]. Limited states oscillations and higher pressure rate away from the surge condition should be considered as advantages of the proposed controller.
PHOTO (COLOR): Figure 6. Pressure of compressor.
PHOTO (COLOR): Figure 7. Flow of compressor.
The control signal is also presented in Figure 8. According to Figure 8, the control signal from proposed method has lower amplitude and changes in comparison with the robust adaptive tube MPC.
PHOTO (COLOR): Figure 8. Control signal.
The trajectory of the compressor is also shown in Figure 9 and it conveys the ability of the controller for disturbance rejection.
PHOTO (COLOR): Figure 9. Compression system trajectories.
In the third scenario, it is assumed that some stable disturbances are applied to the system. These disturbances are modeled as follows
Graph
(49)
As it can be seen in Figures 10 and 11, simulation results illustrate the robustness of the proposed controller to steer the centrifugal compressor against stable disturbance and change in throttle valve opening percentage. By using the proposed controller, the compressor operates at higher pressure rate away from the surge condition, however, the flow rate is higher than the robust adaptive tube MPC.
PHOTO (COLOR): Figure 10. Pressure of compressor.
PHOTO (COLOR): Figure 11. Flow of compressor.
PHOTO (COLOR): Figure 12. Control signal.
The CCV output is also presented in Figure 12 that satisfies the (43) and (44) constraints.
The performance curve for the compressor is also shown in Figure 13. It can be seen that, in spite of the existence of disturbance, the controller has been able to prevent the surge.
PHOTO (COLOR): Figure 13. Compression system trajectories.
From the simulation results for these scenarios, it can be concluded the proposed controller provides higher pressure rate than robust adaptive tube MPC [[33]]. Also, the flow has less fluctuating than the reference [[33]], of course, the control signals obtained from the proposed method have less amplitude and smoother behaviour. In addition, the computational time required for these two methods is given in Table 1.
Table 1. The computational time required for proposed method (RMPC) and RAMPC [33].
RMPC | RAMPC |
Self-time of cost function | 5.711 s | Self-time of cost function | 11.592 s |
Self-time of LMI | 3.428 s | Self-time of estimator | 43.896 s |
Total time | 161.789 s | Total time | 1189.919 s |
In the RAMPC method [[33]], for solving the objective function and obtaining the control signal, an estimation is made on each prediction horizon which makes the problem complexity and time-consuming. But in the proposed LMI-based method, in spite of the infinite prediction horizon, this complexity is reduced and less computation time is achieved because of nonexistence of an estimator.
5. Conclusion
A new robust predictive controller for a special class of continuous-time non-linear systems with uncertainty and unknown bounded disturbances has been presented in this paper. The controller system is trying to designate a state-feedback control law in order to minimize the upper bound of LMI-based infinite horizon cost function in the framework of linear matrix inequalities. The proposed method offers advantages such as robustness to uncertainty and boundary disturbance, lower computational complexity and thus lower computation time. Finally, the proposed controller is used to solve a surge problem in centrifugal compressors. The results obtained from simulation show the efficiency and resistance of this controller.
Disclosure statement
No potential conflict of interest was reported by the author(s).
[
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By Luo Huofa and Hashem Imani Marrani
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