Adaptive controller design for nonlinear uncertain systems using multiple model based two level adaptation technique

In this thesis, an adaptive control method for nonlinear uncertain systems using multi-model based bilevel adaptation (MMTLA) is proposed. Vinay Kumar Pandey, Indrani Kar and Chitralekha Mahanta, “Controller Design for a Class of Nonlinear Coupled MIMO System Using Multiple Models and Level Two Adaptation,” ISA Transactions, Elsevier, Vol.

Introduction

Literature Review

25] developed an adaptive nonlinear model after control for a 3-DOF helicopter that contained high nonlinearity, cross-coupling, and large uncertainty. 20] proposed an MMAC technique with a relatively less number of models that cooperated among themselves to yield faster system identification.

Research Motivation

The basic idea of this controller design was the linearization of the input-output relationship of the system. [29] demonstrated an interesting use of multiple models in model predictive control (MPC) by using the switching strategy between multiple models to improve tracking performance.

Contributions of the Thesis

Then, a multi-model based multi-level adaptive control (MMTLA) technique is proposed for nonlinearly coupled MIMO systems. Experimental results show improvement in transient and steady-state performance using the proposed multi-model-based bi-level adaptation (MMTLA) method compared to the existing single-model-based adaptive control method.

Organization of the Thesis

Introduction

Change in Operating Environment

At some point, if controller Ci is in use and the performance index Jj of Sj is the minimum in the set {Jj}Nj=1, modelSj is selected and the controller switches from Ci to Cj, as shown in Figure 2.1 [ 18]. Parameter values of that chosen model are then considered as the actual parameters of the system and accordingly the controller can be designed at that particular time.

Figure 2.1: Multiple models in a system environment

The Adaptive Control Problem

Multiple Model Adaptive Control (MMAC) with Switching

Feedback control: Now the feedback control is used to ensure the stability of the system and the bound of x(t). HereN identification models with the same structure as given for the single identification model in section 2.4.3 can be used to establish N estimates of the parameter vector.

Summary

Design of an adaptive controller for linearly parameterized nonlinear SISO systems using a multi-model based two-level adaptation technique.

Introduction

The proposed adaptive controller, which uses multiple model based two-level adaptation method, is described in section 3.3, including stability analysis for the overall system.

Adaptive Control of Nonlinear Systems with Linear Parameter- ization

A stable estimation model of the system is chosen such that the states and output converge to those of the system as time t. To evaluate the stability of the closed loop of the nonlinear system (3.1) with a relative degree γ and in a linearized form as (3.3), the following assumptions are made: i) The null dynamics ϕ(0,ξ) is asymptotically stable and the internal dynamics ϕ(τ,ξ) is globally Lipschitz in τ and ξ.

Multiple Model based Two Level Adaptation (MMTLA) method

Since the parameter bounds are assumed to be known, the initial values of ˆθj(t0) at time t0 can be suitably chosen so that the system parameter vector θ lies on their convex hull. The tuning laws for the adaptive weights wj(t) can be found using the following theorem. Finally, using the combination of the first-level estimates of the system parameter vector ˆθj with the adaptive weights W(t) at each instant, the virtual estimate of the system parameter values θs(t) can be obtained as.

From the discussions in the previous section and using (3.23), the second level control error equation can be found as.

General Work Flow Chart

Simulation Results

Figure 3.5 shows the convergence of parameters and control input profiles for both methods. The simulation results obtained using the proposed MMTLA control method are shown in Figure 3.8 - Figure 3.12. In addition, Figure 3.11 shows the parameter estimation profile in the proposed MMTLA approach for the first and second levels for both parameters.

Finally, Figure 3.12 shows the adaptive weight convergence in all four models in the proposed MMTLA approach.

Summary

This is an important benefit of the proposed MMTLA control method that makes it suitable for practical applications. Adaptive controller design for nonlinear parametrized SISO systems using multiple model-based two-level adaptation technique. Adaptive controller design for nonlinear parametrized SISO systems using multiple model-based two-level adaptation technique.

Introduction

Nonlinearly Parameterized Nonlinear System Description

Adaptive Feedback Controller and Estimation Model Design

To investigate the stability properties of the nonlinear system (4.1), a suitable Lyapunov function is considered. ωTdθ)ωˆ Tf −ωTd(ωTfθ)ˆ −ωTdgu. 4.11). Now the closed-loop stability of the first-level system, including the control error limitation, will be discussed. Taking the first differentiation of (4.22) with respect to time and using (4.19) yields. 4.23) where Pi is a symmetric positive definite matrix, found by solving the Lyapunov equation ATP+PA=−I. Adaptive controller design for nonlinearly parameterized SISO nonlinear systems using two-level multi-model based adaptation techniques.

If the magnitude of the second term in (4.26) is smaller than the magnitude of the first term, ˙V(e) will become negative definite.

Multiple Model Based Two Level Adaptation Technique

Adaptive controller design for nonlinear parametrized SISO systems using multiple model-based two-level adaptation technique. 4.26) If the magnitude of the second term in (4.26) is smaller than the magnitude of the first term, ˙V(e) will become negative definite. The first-level models ˆθj(t) are combined convexly, using weights wj(t) which are also adaptive in nature, resulting in the desired second-level model θs(t) given as. Finally, using the system parameter estimation, the control input to the second level can be derived as .

To achieve overall system stability and tracking convergence, the control error equation (4.19) is modified here by replacing ˆθ with a second level estimate θs as.

Simulation Results

From Table 4.1, it is evident that transient performance of the MMTLA control technique is far superior to Ge et al.'s [12] method. However, smoother control input and faster convergence of parameters are the advantages of the proposed MMTLA control method. Further, Table 4.3 compares the transient and steady state performance of the system using the proposed MMTLA controller with those of Ge et al.'s method [12].

The control input in the proposed MMTLA control method is far smoother than Ge et al.'s method [12].

The simulation results are plotted in Figure 4.1 - Figure 4.4. Table 4.1 compares the transient and steady state performances of the MMTLA control method with those of Ge et al.’s method [12]

Summary

Adaptive controller design for nonlinearly coupled MIMO systems using two-level multi-model based adaptation techniques. Adaptive controller design for nonlinearly coupled MIMO systems using two-level multi-model based adaptation techniques.

Introduction

System Description

Design of Estimation Model

Controller Design Using Feedback Linearization

Then, an m×1 matrix B(x,θf) and am×m matrix A(x,θf,θg) are defined as respectively. 5.9) Here the matrix is called the decoupling matrix. As stated earlier, the decoupling problem is solvable if and only if the system has a relative vector degree or the decoupling matrix A is nonsingular. In addition, for x = 0, the equilibrium point of system (5.1), the dynamics ˙ξ = ϕ(0,ξ) is referred to as zero dynamics, which is assumed to be asymptotically stable for this case.

The structure of the m-set of equations given in (5.19) shows that non-interaction between the loops is achieved.

Introduction of Multiple Models

Two Level Adaptation for Nonlinear MIMO Systems

Therefore, following Theorem 3.3 and using the second-level parameter vectorθs(t) with adaptive weights wj(t) at each instant, the new control input can be obtained as This section discussed the overall stability of the system with two-level adaptation as well as the convergence of control errors and parameters. Further, to evaluate the closed-loop stability of the nonlinear coupled MIMO system (5.1) with relative scale γi and normal form as given in (5.17), the following assumptions are made: i) Zero dynamics ϕ(0,ξ ) is asymptotically stable and the internal dynamics ϕ(τ,ξ) is globally Lipschitz in τ and ξ.

Adaptive controller design for nonlinear coupled MIMO systems using multiple model-based two-level adaptation technique. iv) Since xis is a local diffeomorphism of τ and ξ,.

Simulation and Experimental Results

Oscillation tracking error Figure 5.6: Tracking error for step input. a) Simulation: pitch response for one model. Tables 5.4 and 5.5 confirm the superiority of the MMTLA control method over the single model-based adaptive control method in transient operation. In addition, the smoothness of the control signal in the MMTLA controller is comparable to the adaptive control method of one model.

The figures compare the trajectory tracking results for the MMTLA controller with the adaptive controller based on a single model.

Summary

Adaptive controller design for nonlinear MIMO model following control systems using multiple model-based two-level adaptation technique. Adaptive controller design for nonlinear MIMO model following control systems using multiple model-based two-level adaptation technique.

Introduction

System Model

Estimation Model Architecture

Model Following Controller Architecture

An appropriate Lyapunov function is chosen as V(eI,θ˜f,θ˜gi) =eTIPeI + ˜θTfθ˜f + ˜θTgiθ˜gi where Pis is the positive definite matrix solution of the Lyapunov equation ATP+PA =−Q and Q = QT >0. However, in cases where the decoupling matrix A(x,θ) is singular, static mode feedback technique cannot be used. Consequently, such a system cannot be decoupled by static state feedback. The well-known nonlinear structure algorithm [25, 78] is used here to design the model after checking for the nonlinear coupled MIMO system (6.1).

Now using (6.1) and (6.10) we find the error between the output of the reference model and the output of the system as .

Controller Design Using MMTLA Method

Simulation Results

It is found that simulation results with 4 number of models do not vary for the number of models greater than 4. Therefore, the number of models is chosen as 4. Adaptive Controller Design for Nonlinear MIMO Model Following Control Systems Using Multiple Model-Based Two- level adaptation technique. for both single and multiple models. Simulation results obtained using the proposed method with N = 4 number of models are compared with those obtained using only a single model as shown in Figure 6.2 - Figure 6.4. Table 6.2 summarizes the transient and steady state performances for. From Table 6.2, it can be seen that the overruns in the case of height control are completely eliminated with the proposed method.

The reduction in control energy for both pitch and travel control is an additional advantage of the proposed method.

Figure 6.1: 3-DOF laboratory helicopter model

Summary

While there is a slight improvement in the smoothness of the steer angle control signal, it is quite significant for pitch control. The combination of several first-level models provides a model with one virtual parameter, called the second-level adaptive model. Better tracking response, improved settling time, and smoother and lower control effort ensure the effectiveness of the proposed method.

Conclusions

Superior tracking response, improved transients, and smoother control effort with reduction in control energy establish the effectiveness of the proposed method. Results of real-time experiments performed on TRMS support theoretical propositions of the proposed MMTLA control method. Simulation studies confirm that the proposed MMTLA method outperforms existing multi-model switching-based adaptive control methods, but uses less number of models.

Further, parameter convergence of the proposed MMTLA method is reasonably fast, which makes it suitable for practical applications that require online tuning.

Scope for Future Work

After designing the control inputs, the proposed MMTLA method is applied to control a 3-degree-of-freedom (DOF) laboratory helicopter model which is an appropriate example of MIMO coupled nonlinear systems with singular decoupling matrix. The proposed technique can find application in electronic stability program (ESP) and anti-lock braking system (ABS). The proposed multiple models with two-level adaptive (MMTLA) control would be a potential choice for the above applications.

The proposed MMTLA technique is a promising candidate for adaptive vehicle navigation to improve navigation performance.

Definitions

Cezayirli et al.’s method

Ge et al.’s method

Extended Kalman Filter (EKF)

Performance Specifications

Finally, the total variation (TV) characterizes the smoothness of the control signal and input utilization and is given as .

Momentum equation for TRMS

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