Robotics Research Laboratory 1 Chapter 6 Design Using State-Space Methods

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3 Solution of LTI Discrete-Time State Equation

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Robotics Research Laboratory 5 Lyapunov Stability Analysis Def : Equilibrium State

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Robotics Research Laboratory 11 ex) In 2 –dim stable i.s. L asymptotically stable

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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

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Robotics Research Laboratory 20 Is it controllable ?

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Robotics Research Laboratory 23 Output Controllability

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Robotics Research Laboratory 26 Regulation via Pole Placement arbitrary pole placement completely state controllable Nec. & Suff.

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Robotics Research Laboratory 28 Controllable canonical form

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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).

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Robotics Research Laboratory 49 State Feedback Design by Emulation (Redesign)

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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 57 given known

Robotics Research Laboratory 58 State Observer (Estimator) state observer open-loop vs. closed-loop full-order vs. reduced-order prediction vs. current

Robotics Research Laboratory 59 Prediction Estimator correcting term or innovation term

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Robotics Research Laboratory 61 Current Estimator

Robotics Research Laboratory 62 Reduced-Order Estimator (Minimum-Order Estimator)

Robotics Research Laboratory 63 very inconvenient because of same time instant

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Robotics Research Laboratory 65   

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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 )

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Robotics Research Laboratory OUTPUTS Time (sec) Time histories with prediction estimator --o-- X1 --*-- X2 TILDE --x-- X U/4

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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 78  - +  + +

Robotics Research Laboratory 79  - + Remark:

Robotics Research Laboratory 80 Reference Inputs with Estimator  - +  + +

Robotics Research Laboratory 81 Output Error Command ex) thermostats (output error command) Remark (Fig vs. Figs in Franklin’s) state space approach > transfer function (I/O) approach state command > output-error command + - 

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Robotics Research Laboratory 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 OUTPUTS Time (sec) --o--- X1 --x--- X 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 OUTPUTS Time (sec) Response with integral control --o--- X1 --x--- X U/5... w/2

Robotics Research Laboratory OUTPUTS Time (sec) Integral control with added zero --o--- X1 --x--- X U/5... w/2

Robotics Research Laboratory 92 Disturbance Estimation-Input Disturbance  - +  - +  + +

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Robotics Research Laboratory 94 augmented state

Robotics Research Laboratory 95 A unit reference input at t=0 and a step disturbance at t=2 sec with bias estimation

Robotics Research Laboratory 96 A sinusoidal input disturbance with disturbance rejection

Robotics Research Laboratory 97 Disturbance Estimation-Output Disturbance  - +  + +

Robotics Research Laboratory OUTPUTS Time (sec) Sinusoidal output disturbance rejection ---o--- X1, output y w, disturbance w bar, disturbance estimate X1+w, measured output, y`..... A sinusoidal sensor disturbance (measured error) with disturbance rejection

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