Linear Control Systems Project session 3: Design in state-space 6 th October 2017 Kathleen Coutisse kathleen.coutisse@student.ulg.ac.be 1
Content 1. Closed loop system 2. State feedback 3. Observer 4. Simulations 2
Closed loop system General control structure: reminder 3
Closed loop system General control structure 4
Content 1. Closed loop system 2. State feedback 3. Observer 4. Simulations 5
State feedback Structure A proportional state feedback controller provides a control signal u which is proportional to the values of the states and the reference r: u = -Kx+k r r Assumption: we consider that we can measure the value of all states. 6
State feedback Proportional controller u = -Kx+k r r dx/dt = (A-BK)x+Bk r r Determine the feedback gain K thanks to pole placement problem = eigenvalue assignment Determine k r that affects the steady-state solution Dynamic performance Static performance Stabilization by state feedback for the TJA ( Development done in class ) 7
Content 1. Closed loop system 2. State feedback 3. Observer 4. Simulations 8
Observer Reminder Assumption: we consider that we can measure the value of all states. we consider the case where all the states are not directly measured. If the system is observable, we can estimate the state by measuring the input(s) and output(s) of the system. construct an observer whose dynamics are given by 9
Observer Reminder The observer is a dynamical system characterized by this equation: We now want to make the observer converge faster. To do so, we use the estimated output of the system and compare it to the measured output: The convergence of the error is described by the dynamical system: Equations for the TJA ( Development done in class ) Determine the observer gain L thanks to the pole placement problem = eigenvalue assignment 10
Content 1. Closed loop system 2. State feedback 3. Observer 4. Simulations 11
Simulations General constraints Design constraints Maximum acceleration: 4m/s 2 Maximum derivation: <2cm Delays: 50ms Noise: rejection as much as possible Load disturbance: need to follow as close as possible the car in front good load disturbance reaction Simulations specifications Control signal a= F r m Choice of a real/plausible load disturbance signal = a real displacement of the car in front 12
Simulations General constraints Simulations specifications Choice of a real/plausible load disturbance signal = a real displacement of the car in front (car 1 ) 0.5 Position of the car 1 0.49 0.48 0.47 Position (m) 0.46 0.45 0.44 0.43 0.42 0.41 0.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time(s) 13
Simulations Control using estimated state: general structure 14
Simulations Control using estimated state: general structure General Simulink block diagram Open-Loop System F r u 1 Perturbation input y Output d 1 u 2 u 1 u 2 y Observer x 1 x 2 r u 2 Reference 0.4 0.2 x 1 F r x 2 State feedback 15
Simulations Block diagram Moving from theory to the project: Using the theoretical equations with the computed matrix to obtain the block diagram ( Development done in class ) 16
Simulations Choice of the state feedback parameters 17
Simulations Choice of the state feedback parameters Tests w c = 1000 w o = 1000 ξ c = 1 ξ o = 1 Impossible! Find realistic solutions Relative distance Acceleration 0.4005 80 0.4004 60 0.4003 40 0.4002 Distance(m) 0.4001 0.4 0.3999 Acceleration (m/s 2 ) 20 0-20 0.3998-40 0.3997 0.3996-60 0.3995 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time(s) -80 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time(s) 18
Simulations Choice of the state feedback parameters Simulations by varying one of the parameters Demonstration in class for ξ c as a parameter w c = param w o = 450 ξ c = 0.75 ξ o = 0.75 Relative distance Acceleration 0.42 8 0.415 c = 20 c = 45 c = 75 6 0.41 4 Distance(m) 0.405 0.4 0.395 0.39 Acceleration(m/s 2 ) 2 0-2 -4 c = 10 c = 45 c = 75 0.385-6 0.38 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time(s) -8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time(s) 19
Simulations Choice of the observer parameters How to simulate? Useless to compare the output of the observer and the states ensured convergence Compare the states when different initial conditions are applied to the system and to the observer Common practice: w o =10w c ξ o = ξ c
Simulations Choice of the observer parameters Simulations by testing the convergence of one state State x 1 with w o varying State x 1 with ξ o varying 0.025 0.025 o = 45 o = 100 o = 450 Real state o = 0.25 o = 0.75 o = 1 Real state 0.02 0.02 x 1 (m) 0.015 x 1 (m) 0.015 0.01 0.01 0.005 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 Time(s) 0.005 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 Time(s) Check with the second state
Simulations Summary Step 1: realisation of the state feedback (by assuming that all states are directly measurable) Step 2: adding an observer Step 3: fixing the real constraints and simulations specifications Step 4: tuning the different parameters of the state feedback and the observer to obtain a good design thanks to simulations
0.408 Simulations Summary Results in the time domain Matlab and Simulink demonstrations performed in class Relative distance 4 Acceleration 0.406 3 0.404 2 Distance(m) 0.402 0.4 0.398 Acceleration(m/s 2 ) 1 0-1 0.396-2 0.394-3 0.392 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time(s) Conclusions? Good results (nice tracking, small deviations, ) But static error -4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time(s) Move to the FREQUENCY DOMAIN