Runge-Kutta MIMO NARMA-L2 Controller

Abstract: This paper introduces a novel control architecture which combines nonlinear autoregressive moving average (NARMA)-L2 control method with Runge Kutta based system modeling. Control law is developed by employing NARMA-L2 model via Taylor expansion. Jacobian information required for the controller is constituted by making use of RK model of plant to be controlled. To evaluate performance, simulations are carried out on the Three Tank System. Runge-Kutta based Generalized Predictive Controller (GPC) is deployed to compare the performance of the introduced controller. Acquired results illustrate that introduced controller has substantially good performance on multiple input multiple output (MIMO) nonlinear systems.


Introduction
Controller design which is as simple as possible is an intended goal in terms of implementation and analysis of the control architectures. NARMA-L2, which is an affine linear structure derived from nonlinear dynamics using Taylor series expansion, is one of the most effective techniques to design control law for nonlinear systems. Since control signal is derived via subdynamics of the NARMA-L2 model, any additional control architecture is not needed. Literature contains numerous implementations of NARMA-L2 based controllers. Majstorovic et al [1] deployed a NARMA-L2 controller derived from offline NARMA model for a two-tank system. Pedro et al. realized NARMA-L2 to control slip in an anti-lock braking system (ABS) [2]. Hagan et al. utilized a NARMA-L2 control method using a neural network for a continuously stirred tank reactor (CSTR) system [3,4]. To diminish oscillations in NARMA-L2 controller, Pukritttayakamee et al. [5] introduced to employ an additional term that includes a combination of reference and output signals. Wahyudi, Mokri and Shafie [6] utilized a smoothed NARMA-L2 controller to overcome the appearing nonlinearities and parametric uncertainties in a robotic link. The control of underactuated systems is difficult, and static feedback law can be deployed for asymptotically stabilization of the system, therefore Akbarimajd and Kia [7] applied NARMA-L2 control method for a 2-dof underactuated robot. Vesselenyi et al [8] used NARMA-L2 position controller for a pneumatic actuator. The common point of the studies in literature is that NARMA-L2 controller is derived from the offline system model [1][2][3][4][5][6][7][8][9]. Presented work acquaints an original Runge-Kutta (RK) based NARMA-L2 control technique for nonlinear MIMO systems. Parameters are identified via a RK system model. Then, using the future system outputs, a NARMA-L2 control signal is derived. To assess the performance, three tank system (TTS) dynamics are used and controller has been compelled to work under various conditions vital for control. The performance of the controller has been compared with Runge-Kutta based generalized predictive control (GPC) introduced in [10]. The obtained results indicate that the RK-NARMA-L2 controller can be conveniently applied for nonlinear MIMO systems. The paper is organised as: NARMA model is overviewed in section 2. The details related to the introduced controller and Runge-Kutta identification technique are summarized in section 3. Results of the control performance assessment for nonlinear three tank system are illustrated in Section 4. The paper is finalized with detailed discussion in conclusion part.

NARMA-L2 Model and Controller
Nonlinear dynamical systems can be expressed via NARMA model around the equilibrium state [11]. NARMA model can be given in companion form: (1) Here, d denotes relative degree [12] [13]. For , if the NARMA model is rewritten as (2) and if "n-1" is substituted in place of "n" time index, the model is obtained as (3) NARMA-L2 model is a very effective system identification structure derived via Taylor expansion. For system model given in (3), NARMA-L2 model where control signal appears linearly [4], [7], [11]- [15] is written as (4) where (5) It is possible to express Taylor expansion for MIMO system as: In (7), q and m denote output and input dimensions and: (9) reference signal can be substituted in place of in order to attain control signal so that system output equals the reference signal. Required control input that forces the controlled outputs to reference signals is computed as follows: (10) As observed in (8) and (9), the entries of the matrix are system Jacobian which is obtained using RK system model. Figure 1 demonstrates mechanism of RK NARMA-L2 controller where number of the system inputs-outputs are represented by R and Q respectively. The adjustment mechanism is composed of two crucial parts to be examined: RK NARMA-L2 controller to compute required control input and RK system model to estimate system dynamics required to constitute the matrix. To enhance the understandability of the proposed control architecture, RK NARMA-L2 is used as an abbreviation to Runge-Kutta NARMA-L2 controller and RK model is utilized to denote Runge-Kutta system model. The adjustment mechanism given in Figure 1 consists of two main phases which are executed online consecutively: prediction and control phases.

Prediction Phase (RK model )
The candidate signal ( ) is calculated as given in (10) by using the system Jacobian derived via RK model . Then, to observe the system response to the acquired , it is applied to the RK model . RK model detailed in Figure 2 incorporates three subblocks to capture the MIMO nonlinear system dynamics: raw Runge-Kutta system model, Runge-Kutta Model based EKF(RK EKF ) and Runge-Kutta based Model parameter estimation (RK estimator ) sub-blocks [10], [16]- [18].

Runge-Kutta System Model
Let a nonlinear continuous-time MIMO system be represented by: (12) In (12) denotes state vector and stands for input vector, denotes system outputs, and are continuously differentiable functions with respect to system states and control inputs [10] [18]. The following constraints are valid for x and u [10,[16][17][18] The continuous nonlinear system given in (12) and (13) can be discretized using 4 th order RK integration method, and system states are approximated with: (14) and (15) where (16) The discrete system model in (14)- (16) can be rewritten as follows: Using approximated states as entry in (17) repetitively, a predictive model can be derived as: (18) where K indicates prediction horizon. The model in (18) necessitates state information ( ) as well as accurate system parameters ( ) if there is deterioration in parameters. Therefore, RK EKF is utilized to obtain current system states and RK estimator is required to identify the uncertain parameters via optimization theory. Using the current system states and model parameters in recursive predictive model in (18), the system behavior can be accurately identified.

An overview of RK EKF
EKF is one of the effective filters to attain system states via observed system input-output. Let a discrete-time nonlinear MIMO system be represented with: is system noise vector with covariance matrix , expresses measurement noise vector with covariance matrix [10], [16]. In prediction phase of EKF, states and covariance matrix can be written with state transition matrix as [10], [16], [19]: (20) Then, Kalman gain matrix is derived, and in prediction step can be straightened as [10], [16]: (21) and in (21) are derived as [10], [16]: (22) Here, for i=1,..,N and j=1,…,N. (24)

An overview of RK estimator
The identification performance of RK model is directly dependent on parametric accuracy of the model( ). If system model parameters diverge from their actual values or system model parameters cannot be measured accurately, success of the identification carried out by RK model gets worse [16]. Thus, in order to improve the identification performance of RK model , it is required to utilize online adjustment mechanism [16]. For this purpose, the following adjustment rule is deployed to optimize parameter vector of RK model ( ) [10]: where (30) and (31) For the computation of Jacobian matrix in (30), is required. The term is derived as [10]: n n n n n n n n n n n n n n n n x n x n n n x n x n

Control Phase (RK NARMA-L2 )
In order to constitute control law in (10), it is required to compute matrix. The Jacobian information can be approximated via RK model as follows [10]: term is calculated as [10]: Thus, all derivations required to attain matrix for the computation of by RK NARMA-L2 controller are given in (7)-(10) and (34)-(36). When NARMA controller is derived, some high order terms are neglected in Taylor expansion. Therefore, in order to minimize the tracking error resulting from linearization, a correction term for the control input ( ) is summed up with RK NARMA-L2 controller output. To compute , objective function given in (37) is minimized. Here, . By considering a small deviation as from control signal, the objective function in (37) can be expanded via Taylor expansion: Using first order optimality conditions, it is possible to acquire correction term as follows: By approximating the Hessian as [10] (40) is calculated as [10] (41) where is ,ˆq   Fig. 3. Three Tank System [10,16,17,22,23]. (42) In a nutshell, the suboptimal correction term ( ) is summed up to the control signal computed with RK NARMA-L2 controller, optimal control vector ( ) is attained as [10,16].

Symbol
Parameter description Value coefficient of outflow (among tank 1 and tank 3) Two pumps above the tanks are deployed to pump the water, and valves restrict the flow between tanks [10,16,17,20,21]. The differential equations given in (43) represent the dynamics of TTS: where; (44) shows height of the liquid, indicates the control signal, is rate of flow between tanks [10] [16], [17], [22]. The description of the symbols in (43),(44) and Figure 2 are given in Table 1. In this context, and represent outputs to be controlled, and are control input signals. is an uncontrollable system dynamic [21]. The control signal limitations of the TTS are given as and and Ts=1s in the simulations [10], [16] [23]. The RK NARMA-L2 controller performance is compared with RK GPC for nominal, measurement and parametric uncertainty cases.
) n n n n n Q t az S y t y t g y t y t Q t az S y t y t g y t y t Q t az S g y t Q t az S g y t Q t az S g y t

Nominal Case
The reference tracking performance of RK NARMA-L2 is given in Figure 4 (a,d). Control inputs and correction terms computed by RK NARMA-L2 are also depicted in Figure 4 (b,c,e,f). It is clearly seen from Figure 4 that RK NARMA-L2 controller provides small tracking errors for controlled outputs. The impact percentage of the RK NARMA-L2 controller and correction terms on construction of control signal is illustrated in Figure 5. It can be seen that almost all control task is carried out by RK NARMA-L2 controller.

Measurement Noise Case
The RK NARMA-L2 controller performance has been assessed for the case where measurement devices are contaminated by measurement noise. Therefore, , are incurred measurement noise having standard deviation values ( ) = ( ) = 0.03 . Closed -loop performance is depictured in Figure  6. Also, Figure 7 illustrates percentage of control task.

Parametric Uncertainty
By considering the outflow parameter as the parameter with uncertainty given as ( ) = 0.52 + 0.28 sin(0.0133 ), prediction achievement of RK estimator subblock in RK model ,and tracking ability of RK NARMA-L2 controller can be examined as given in Figure 8. The convergence of the uncertain parameter to its actual value is illustrated in Figure 8(g). The control task share is given in Figure 9.

Computation Time
In this section, the computational load of the algorithms has been examined. For this purpose, the total computation times of the algorithm have been illustrated for the mentioned three different cases in Figure 10. In Figure 10   nominal case, and for the cases where measurement noise and parametric uncertainty are implemented are illustrated respectively. Maximum computational time is less than 5 ms. Thus, controller is convenient for real time applications.

Comparison with Runge-Kutta based GPC
The performance of the controller has been compared with Runge-Kutta GPC(RK GPC ) proposed in [10] to provide a basis for evaluation. Since RK NARMA-L2 controller is constructed via one-step ahead model behavior, and to carry out a fair evaluation, the prediction horizon for predictive mechanism is chosen as "1".  The tracking performance of RK GPC is depicted in Figure 11 for nominal condition. The performance of the RK GPC for measurement noise and parametric uncertainty cases are illustrated in Figure 12 and 13, respectively. The magnitude of chattering control signals is larger in RK GPC compared to RK NARMA-L2 This chattering can be suppressed by increasing the prediction horizon. The following performance index function is deployed to constitute Table 2 for performance comparison.   As can be seen from Table 2, RK NARMA-L2 has better performance than RK GPC . It is observed that RK NARMA-L2 is 1.4841 %, 11.1430 % and 2.9724 % better than RK GPC for nominal, measurement noise and parametric uncertainty cases, respectively. In order to evaluate the applicability of the control algorithms, the computation times are illustrated in Figure 10 and Figure 14 for RK NARMA-L2 and RK GPC respectively.   prediction horizon was chosen as "1" to avoid injustice to RK NARMA-L2 controller and to provide a fair comparison.

Conclusion
In this study, a new control architecture with a RK NARMA-L2 controller is proposed for MIMO and nonlinear systems. Control performance is evaluated for three different operating conditions on TTS. RK GPC is utilized to evaluate the control performance of the introduced controller. The computational loads of the control algorithms have also been evaluated to justify the real time applicability of the methods. The attained results and computational loads indicate that it is possible to employ the proposed RK NARMA-L2 control architecture successfully.
It is intended to work on stable and adaptive control methods derived via Runge-Kutta identification method in future studies.