B. Xian , N. Jalili , D. M. Dawson , and Y. Fang

1. Adaptive Tracking Control of Linear Uncertain Mechanical Systems in Presence of Unknown Sinusoidal Disturbances American Control Conference, Alaska, 2002 B. Xian, N. Jalili, D. M. Dawson, and Y. Fang Department of Electrical and Computer Engineering Department of Mechanical Engineering CLEMSON U N I V E R S I T Y

2. Presentation Overview • Introduction • System Model • Control Objective • State Estimation and Filters Design • Controller Design • Stability Analysis • Simulation and Experiment Results • Conclusions

3. Introduction - Motivation • Many mechanical systems are subject to periodic disturbances. • Traditionally, the controller design is influenced by a-prior knowledge of the disturbance signal to a large extent. • In most cases, the disturbance signals are completely unknown or not directly measurable, or the system parameters are not known precisely; hence, the need for a adaptive controller is apparent.

4. Introduction – Past Research • When the disturbance is partially known or directly measurable, it is often possible to directly cancel the disturbance based on measurement. • Adaptive feedforward (AFC) algorithms (W. Messner, 1995) • Linear quadratic control (LQG) method (G. Celantano, 1996) • When the disturbance signals are unknown, adaptive techniques can be used. • Internal model principle (G. Feng,1992 and M. Bodson,1992) • External model principle (W. C. Wang,1994 and C. K. Chak,1994) • Disturbance observer (V. O. Nikiforov,1996 and R. Marino, 2000)

5. Introduction – Current Research • The subject of this paper is the design of an output feedback tracking controller for a SISO system with unknown plant parameters. • The technique proposed here is based on constructing a set of stabilizing controllers using a state estimate observer which does not require disturbance model estimation • The control design utilizes only the system output (displacement) since the rest of states are assumed to be unmeasurable.

6. K M u(t) C y(t) Fd(t) System Model • A SDOF linear mass-spring-damper trio subjected to a simple sinusoidal disturbance Fd(t) is considered. u(t) • Its dynamic representation can be rewritten as • In order to separate measurable states from unknown parameters, the system dynamics is rewritten into the observer-based canonical form.

7. System Model system output , where , , .

8. Control Objective • The control objective is to asymptotically force y(t) to track a reference signal yr(t) in the presence of an unknown sinusoidal disturbance and parametric uncertainties in the plant. • Assumption1: The sign of the high-frequency gain is known. • Assumption2: The reference signal and its first two time derivatives are known and bounded, and in addition, its second derivative is piecewise continuous.

9. State Estimation and Filters Design • The inclusion of the disturbance model into the plant dynamics will increase the order of the combined plant-disturbance system. As a result, some additional states are introduced which do not necessarily have physical meanings and are not measurable. • To address this issue, we employ the following filters gain vector regression matrix where and

10. State Estimation and Filters Design • With the proposed filter, the state estimate is defined as • And the state estimation error is defined as • A further step is to lower the dynamic order of the Ω-filter. Denote Ω(t) as • The first three columns of Ω(t) can be obtained through the expression • and is generated through the filter

11. y u B ( s ) A ( S ) - - 1 - - 1 ( sI A ) e ( sI A ) e 0 4 4 0 l h + - ( ) ( ) B A A A 0 0 State Estimation and Filters Design • The remaining four columns of Ω(t) are obtained through the expression and the filter • can be obtained through the equation • An equivalent expression for the virtual state estimation is

12. Control Design • In a recursive backstepping fashion, we design a controller for the system by considering some of the state variables as ``virtual controls'' and designing for the intermediate control laws. • The state variable can be viewed as a control input to the subsystem consisting of • To begin the tuning function design, we start with the following equations available for measurement • We use the proposed state estimate to rewrite the first equation of above expression as

13. Control Design • Therefore, the system for which the control is now designed for is given as follows • To quantify the control objective, we define the error signals as estimate of • with the stabilization function being designed as and • We define the parameter estimation error as

14. Control Design • After taking the time derivative of and performing some manipulations, we obtain the expression for as follows . • After taking the time derivative of and performing some manipulations, we obtain the expression for as follows • After designing the control input as we can manipulate as follows stabilizing function

15. Control Design • To obtain the update law for , we define the following non-negative function • The auxiliary control input and update law for are designed as then we can obtain time derivative of as • To obtain the update law for and expression for , we define the following non-negative function

16. Control Design • After designing the update law for , auxiliary control input and stabilization function as we can obtain time derivative of as

17. Control Design t 2 t 1 t n ~ – q a a 1 z 2 z 1 2 a = u n z n G s • To summarize the controller design procedure, the following block diagram of successive step of design procedure is given, where for a n-th order plant, the update laws and tuning functions are designed in a similar way.

18. Stability Analysis Theorem 1:All the signals in the proposed closed-loop adaptive system consisting of the plant, the adaptive control laws, the update laws, the filters are globally bounded and asymptotic tracking is achieved in the sense that Proof :We start our proof by defining the non-negative function then we can show

19. Stability Analysis • After taking the Laplace transform for , we have where It can now be proved that Now we can show all the remaining signals are bounded and Consequently, the Barbalat’s theorem can be employed to conclude that

20. Numerical Simulations • The nominal parameters for the SDOF system are given in the following table • Below is the experiment setup

21. Numerical Simulations control motor encoder disturbance motor damper

22. Numerical Simulations • For regulation, we consider the set-point problem • The filter gains are selected as • The initial condition for estimated parameters are set as • For tracking, the desired trajectory is selected as yr( )

23. Numerical Simulations • The disturbance is set as • The control parameters were tuned for the best performance and are recorded in the following table

24. Numerical Simulations-Regulation 15 5 Error z1, mm Control u, N 10 0 5 -5 0 -10 0 10 20 30 40 0 10 20 30 40 1.8 ^ 0.8 q 1, 1/kg ^ r 1.6 , kg 0.6 1.4 0.4 0.2 1.2 0 1 -0.2 0 10 20 30 40 0 10 20 30 40 -5 -5 x 10 x 10 4 1.5 ^ 3 2 ^ q 5, rad /sec q 6, N/(m.kg) 1 2 0.5 0 0 0 10 20 30 40 0 10 20 30 40

25. Numerical Simulation-Tracking 15 20 Control u, N error z1, mm 10 10 5 0 0 -5 -10 -10 0 10 20 30 40 0 10 20 30 40 5 0.4 ^ 0.3 ^ 4 r , kg q 1, 1/kg 0.2 3 0.1 2 0 1 -0.1 0 10 20 30 40 0 10 20 30 40 -4 -4 x 10 x 10 5 4 ^ ^ q 3 6, N/(m.kg) 2 q 0 3 5, rad /sec 2 -5 1 -10 0 0 10 20 30 40 0 10 20 30 40 Time (sec) Time (sec)

26. Experiment Results-Regulation 20 error z1, mm Control u, N 0 10 -5 0 -10 -10 0 10 20 30 40 50 60 0 10 20 30 40 50 60 1.15 2 ^ ^ q 1, 1/kg r , kg 1.1 1.5 1.05 1 1 0.5 0.95 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 2.5 ^ 2 3 1 q 5, rad /sec ^ 2 q 6, N/(m.kg) 0.8 1.5 0.6 1 0.4 0.5 0.2 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 5 Time (sec) Time (sec)

27. Experiment Results-Tracking 0.4 6 Control u, N error z1, mm 4 0.2 2 0 0 -2 -0.2 -4 0 20 40 60 80 0 20 40 60 80 0.4 3 ^ ^ r , kg q 1, 1/kg 0.3 2 0.2 0.1 1 0 0 -0.1 0 20 40 60 80 0 20 40 60 80 0.6 0.8 ^ ^ 0.6 2 3 q 6, N/(m.kg) 0.4 q 5, rad /sec 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 0 20 40 60 80 0 20 40 60 80 Time (sec) Time (sec)

28. Comparison Between Simulation and Experiment-Regulation 15 5 5 20 error z1, mm Control u, N Error z1, mm Control u, N 10 0 0 10 5 -5 -5 0 -10 -10 0 -10 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 0 10 20 30 40 1.15 2 1.8 ^ ^ q 1, 1/kg ^ q 1, 1/kg 0.8 r , kg ^ r 1.1 1.5 1.6 , kg 0.6 1.05 1 1.4 0.4 0.2 1 0.5 1.2 0 0.95 0 1 -0.2 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 0 10 20 30 40 -5 -5 x 10 x 10 2.5 4 1.5 ^ ^ 2 3 1 2 q ^ ^ 3 2 q /sec 5, rad 5, rad /sec q q 6, N/(m.kg) 6, N/(m.kg) 0.8 1 1.5 0.6 2 1 0.4 0.5 0.5 0.2 0 0 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 0 10 20 30 40 Time (sec) Time (sec) Simulation Experiment

29. Comparison Between Simulation and Experiment-Tracking 15 20 0.4 6 Control u, N error z1, mm Control u, N 10 error z1, mm 4 10 0.2 5 2 0 0 0 0 -2 -5 -10 -0.2 -4 -10 0 10 20 30 40 0 10 20 30 40 0 20 40 60 80 0 20 40 60 80 0.4 5 0.4 3 ^ 0.3 ^ ^ ^ 4 r r , kg q , kg 1, 1/kg q 1, 1/kg 0.3 0.2 2 0.2 3 0.1 0.1 1 2 0 0 -0.1 1 0 -0.1 0 10 20 30 40 0 10 20 30 40 20 40 60 80 20 40 60 80 0 0 -4 -4 x 10 x 10 5 4 0.6 0.8 ^ ^ ^ 0.6 2 q 3 ^ 3 6, N/(m.kg) q 6, N/(m.kg) 2 0.4 q 3 5, rad /sec q 0 /sec 5, rad 0.4 2 0.2 0.2 -5 0 1 0 -0.2 -10 0 -0.2 -0.4 0 10 20 30 40 0 10 20 30 40 0 20 40 60 80 0 20 40 60 80 Time (sec) Time (sec) Time (sec) Time (sec) Simulation Experiment

30. Conclusions • We have presented an output feedback adaptive tracking controller for SISO LTI mechanical system with unknown parameters and subjected to an unknown sinusoidual disturbance. • By constructing a set of stabilizing tuning functions with a state estimate observer, a non-linear adaptive controller was designed to achieve global asymptotic tracking. • Simulation and experimental results for a SDOF mass-spring-damper setup were presented to demonstrate the performance of the control law.