Robotics Research Laboratory 15 Note: Alexander Mikhailovitch Lyapunov (1857 – 1918)
Robotics Research Laboratory 16 Controllability ( Reachability ) Def : The system is controllable (reachable) if it is possible to find a control sequence such that the origin (an arbitrary state) can be reached from any initial state in finite time. Remark: finite time / unbounded input
Robotics Research Laboratory 34 Deadbeat Control (Minimum Settling Time Control) i. e., It will drive all the states to zero in at most n steps. i. e., nT is the settling time (only design parameter).
Robotics Research Laboratory 55 position velocity Input Fig 1. Control of the double integrator using the control law in (*) when T=0.5 Fig 2. Control of the double integrator using the modified control law in (**) when T=0.5 position velocity Input
Robotics Research Laboratory 56 Observability known
Robotics Research Laboratory 68 Observed-State Feedback Control With Estimator
Robotics Research Laboratory 69 State Feedback Separation principle State Observer
Robotics Research Laboratory 70 Remarks: i)As a rule of thumb, an observer response must be at least 4 to 5 times faster than the system response. ii)Using the output feedback only, we can not place poles at arbitrary locations. Compensator (Estimator and Control Mechanism) For the prediction estimator For the current estimator
Robotics Research Laboratory 71 Satellite Attitude Control (Franklin’s )
Robotics Research Laboratory 76 Reference Inputs for Full-State Feedback - + Objective : To make all the states and the outputs of the system follow the desired trajectory.
Robotics Research Laboratory 77 if the system is Type 0 steady-state error if Type 1 or higher no steady-state error For complex systems, the designer has no sufficient knowledge of the plant. In these cases, it is useful to solve for the equilibrium condition that satisfies We need a steady-state control term in order for the solution to be valid for all system types. For a step reference input,
Robotics Research Laboratory 80 Reference Inputs with Estimator - + + +
Robotics Research Laboratory 81 Output Error Command ex) thermostats (output error command) Remark (Fig. 8.17 vs. Figs. 8.19-20 in Franklin’s) state space approach > transfer function (I/O) approach state command > output-error command + -
Robotics Research Laboratory 85 05101520253035404550 0 0.5 1 1.5 2 2.5 Disk head reader response Time (msec) output ------------ state command structure ooooooooo output error command classical design..............
Robotics Research Laboratory 86 Integral Control by State Augmentation Integral action is useful in eliminating the steady-state error due to constant disturbance or reference input commands. - + - + + + +-
Robotics Research Laboratory 87 state augmentation
Robotics Research Laboratory 88 The design of [K K I ] requires the augmented system matrices. For the full-state feedback, an estimator is used to provide. The addition of the extra pole for the integrator state leads to a deteriorated command input response. Elimination of the excitation of the extra root by command inputs can be done by using the zero. Refer p. 327(b) The system has a zero at. So the selection of the zero location corresponds to a particular selection of N x.
Robotics Research Laboratory 89 0123456 -3 -2 0 1 2 3 OUTPUTS Time (sec) --o--- X1 --x--- X2 ------ U/5... w/2 Response with no integral control A unit reference input at t = 0 A step disturbance at t = 2
Robotics Research Laboratory 90 0123456 -3 -2 0 1 2 3 OUTPUTS Time (sec) Response with integral control --o--- X1 --x--- X2 ------ U/5... w/2
Robotics Research Laboratory 91 0123456 -3 -2 0 1 2 3 OUTPUTS Time (sec) Integral control with added zero --o--- X1 --x--- X2 ------ U/5... w/2