A Continuous Asymptotic Tracking C ontrol Strategy for Uncertain Multi-Input Nonlinear Systems

1. A Continuous Asymptotic Tracking Control Strategy for Uncertain Multi-Input Nonlinear Systems Submitted to ACC2003 by B. Xian, D. M. Dawson, M. S. de Queiroz, and J. Chen Mechatronics Lab CLEMSON U N I V E R S I T Y

2. Presentation Overview • Introduction • First-Order Single-Input System • - System Model and Assumptions • - Control Objective • - Open Loop Error System • - Control Formulation • - Stability Analysis • Extension to Higher-Order Multi-Input System • Simulation Results • Conclusion Remarks

3. Introduction - Motivation • The control of uncertain nonlinear dynamic systems is a challenging topic. • The obstacles associated with dealing with high-order, multi-input nonlinear systems represent additional control design challenges. • The choice of a specific control design method is influenced by the type of uncertainty associated with the system model.

4. Introduction – Past Research feedforward control term • If is linearly parameterized, adaptive control is often considered to be a choice. (S. Sastry, 1989 and M. Krstic, 1995 ) • If can be upper bounded by a norm-based inequality, sliding mode controller can be developed. (J.J. Slotine 1991 ) • If is periodic and the period is known, learning controller can be designed. (W. Messener 1991) • Much progress has been made in the areas of adaptive control for time-varying systems(P. A. Ioannou, 1996), robust control (Z. Qu, 1998), adaptive control for classes of nonlinearly parameterized system (A. Fradkov, 2001) and neural network-based control (F. L. Lewis, 1999 ).

5. Introduction – Current Research • In this presentation, a new continuous control mechanism which is inspired by a simple example ( Z. Qu, 2002) is utilized for a class of uncertain, multi-input nonlinear systems • Based on Lyapunov-based approach, a full-state feedback controller is developed to produce semi-global asymptotic tracking under some assumptions. • The proposed controller leads to an interesting fact that the control law learns the unknown nonlinearity.

6. SISO System-Model and Assumptions • We first consider the following first-order, single-input system Assumption 1: The function is positive and bounded as Assumption 2: The functions and are second-order differentiable with respect to as

7. SISO System-Control Objective desired trajectory • To quantify our control objective, we define the tracking error as follows • where the desired trajectory and its first third time derivative are assumed to be bounded. • The control goal is to obtain asymptotic tracking with a continuous control law using norm-based, inequality bounds on the functions and

8. SISO System-Open Loop Error System • To facilitate the subsequent control development and stability analysis, we define a filtered tracking error variable as follows • After some manipulations, the following open-loop dynamics for the filtered tracking error can be obtained where

9. SISO System-Control Formulation • Given the previous open-loop error system for the filtered tracking error variable, the following continuous control input is designed • The following closed-loop system for the filtered tracking error variable can be obtained where and can be upper bounded as

10. SISO System-Stability Analysis • Lemma 1: Define the scalar auxiliary function if the control gain is selected to large enough then • Lemma 2: Region is defined as Let be a continuous differentiable function such that and and If , we have where and

11. SISO System-Stability Analysis • Theorem 1: The continuous control law ensures that all systems signals remain bounded under closed-loop operation and that provided the control gain satisfy the previous requirement and the control gain is selected according to Proof: To prove Theorem 1, we define the following non-negative function as where is defined as

12. SISO System-Stability Analysis • can be bounded as • Taking time derivative of and after some manipulations yields for • Define region as , then we can show , and , , , .

13. SISO System-Stability Analysis • Define region as • We can now use Lemma 2 to show • The above region can be made arbitrarily large to include any initial conditions by increasing . • Remark 1:The control tends to behave like an exact-model-knowledge, nonlinear feedforward controller as time approaches infinity.

14. MIMO System-Model and Assumptions • We consider a system of the form Assumption 1: is symmetric and positive-definite, and is bounded by Assumption 2: and are second-order differentiable

15. MIMO System- Open Loop Error System • To quantify our control objective, the tracking errorvector is defined as • where the desired trajectory and is continuously differentiable up to its (n+2)th derivative. • We begin by defining the following auxiliary error signals • A general form of can be expressed as follows

16. MIMO System- Open Loop Error Systems • The filtered tracking error as • where is a positive-definite, diagonal, control gain matrix. • The following open-loop error system for can be obtained where and can be upper bounded as follows

17. MIMO System- Control Formulation • Based on the previous open-loop error system and the subsequent stability analysis, the control law is designed as follows where are positive-definite, diagonal, control matrices, and is defined as • Taking the time derivative of and substituting the result into the previous open-loop error system yields

18. MIMO System- Stability Analysis • Theorem 2: The continuous control law ensures that all systems signals remain bounded under closed-loop operation and that • provided that the elements of the control gain matrices and are selected sufficiently large. Proof:The same procedure in SISO section can be followed to prove Theorem 2. • Remark 2:Similar to SISO system, we can obtain the interesting fact that

19. unmodeled dynamic effects Simulation Results • The following dynamic model of the two-link IMI robot is employed for the numerical simulation • The reference trajectory is selected as follows

20. Simulation Results • The control gains were tuned by trial-and-error until the best tracking performance was obtained, this results in following gains : • For comparison purposes, a standard variable structure controller with the following form is also simulated • The control gains that resulted in the best tracking performance were recorded as .

21. 5 0 -5 Link 1 (deg) -10 -15 -20 -25 0 5 10 15 20 25 30 Time (sec) 5 0 -5 -10 Link 2 (deg) -15 -20 -25 -30 -35 0 5 10 15 20 25 30 Time (sec) Simulation Results-Proposed Controller tracking error

22. 10 u1 ud1 5 0 Link 1 (N.m) -5 -10 0 5 10 15 20 25 30 Time (sec) 4 u2 3 ud2 2 1 Link 2 (N.m) 0 -1 -2 -3 0 5 10 15 20 25 30 Time (sec) Simulation Results-Proposed Controller control inputs and desired dynamics

23. 5 5 0 0 -5 -5 Link 1 (deg) -10 Link 1 (deg) -10 -15 -15 -20 -25 -20 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time (sec) Time (sec) 5 5 0 0 -5 -5 -10 -10 Link 2 (deg) -15 Link 2 (deg) -15 -20 -20 -25 -30 -25 -35 -30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time (sec) Simulation Results-Variable Structure Controller Time (sec) proposed controller variable structure controller

24. 8 6 4 2 0 Link 1 (N.m) -2 -4 -6 -8 0 5 10 15 20 25 30 Time (sec) 4 3 2 1 0 Link 2 (N.m) -1 -2 -3 -4 0 5 10 15 20 25 30 Time (sec) Simulation Results-Variable Structure Controller control inputs

25. Conclusions • A continuous control method was demonstrated to achieve asymptotic tracking for a class of uncertain, multi-input nonlinear systems under very limited restrictions on the uncertain system nonlinearities. • The proposed controller also has an interesting capability of learning the unknown system model. • Simulation results illustrated that the performance of the proposed controller outperforms a standard variable structure controller. • Future research work will examine extensions of the proposed controller.

26. Thank you!