05A IO Pole Place Controller

# 05A IO Pole Place Controller - SYS635 Adaptive Control...

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SYS635 Adaptive Control Systems Ka C. Cheok 05A IO Pole Place Controller 110/1/2005 1 POLE PLACEMENT CONTROLLER DESIGN In this chapter, we investigate a pole placement control design technique for altering the dynamic characteristics of systems that can be described by simple transfer functions. The controller design calculations turn out to be mathematically simple and numerically easy to implement. A 1st Order System A.1 Constant Gain Compensator for a 1-pole system Analysis : The closed-loop transfer function system, ignoring the noise v and disturbance w, is given by (show this) We see that the pole of the closed-loop system is 1 () o ab s λ = −+ . For the closed-loop system to be stable, this pole must be lie within the unit circle of the z-plane. That is 1 1 < . (WHY?) Design : In the pole placement technique, we first specify the value of 1 , and proceed to calculate the gain as 1 o a s b −− = . For example, if we choose the closed-loop system pole to be 1 0.9 = , then we set the controller gain to 0.9 o a s b = , which depends on the value of parameters a & b . Self-tuning or adaptive: If the parameters a & b are unknown, a parameter estimation scheme can used to generate estimates ˆ ˆ & . The controller gain can then be computed using the estimated parameters as ˆ 0.9 ˆ ˆ o a s b = . This is an example of the so-called self-tuning controller. Gz Bz Az b za == + Gz s c = 0 y v w u r e - yz bs zab s rz bs z o = ++ = 00 1

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SYS635 Adaptive Control Systems Ka C. Cheok 05A IO Pole Place Controller 210/1/2005 2 A2. Proportional+Integral (PI) Controller PI action : 01 () 1 11 ( 1 ) pi p cp i Kz K K s zs Gz K K zz z + + =+ = = −− Analysis: The characteristic polynomial for the closed-loop control system is given by ( ) ( )( ) ( ) ( ) 2 0 1 z a z b s z s z a bs z bs a +− + + = + + + We see how the coefficients and s s of the controller (1 ) c s z + = affect the closed-loop system polynomial. Design . We specify 12 & λ as the desired closed-loop poles and this require that the desired characteristic polynomial to be 2 1 2 2 2 1 2 ( ) z c z c c c λλ − =++ = −+ = Matching the closed-loop control polynomial to the desired characteristic polynomial yields ( ) 1 ab s c bs a c += = To place the closed-loop poles as desired, we set the controller gains to 1 0 2 1 1 ca s b s b + = + = Self-tuning or adaptive : An adaptive version of the controller gain based on parameter estimates is given by 1 0 2 1 ˆ 1 ˆ ˆ ˆ ˆ ˆ s b s b + = + = _ b za = + u(z) ) c s z + = y(z) r(z) e(z) ( )( ) ( ) ( ) ( ) () () 1( ) ( ) 1 c c bs GzG z z a z b sz s + = ++ + + y(z) r(z)
SYS635 Adaptive Control Systems Ka C. Cheok 05A IO Pole Place Controller 310/1/2005 3 A.3 Compensator + Integrator Action for Accurate Tracking & Performance Equating the controlled polynomial (AR+BS) to the desired polynomial A cl yields Well, you know how to solve for the controller parameters. Bz Az b za () = + Sz Rz sz s rz r z () ( ) ( ) = + +− 01 1 y v w u r e - Closed - loop transfer function: ( ) = = , ignoring w & v The closed - loop characteristic polynomial yields: We would like assign the poles of the closed - loop system to locations, Or equivalently, assign th 1 yz BzSz AzRz uz z a z rz r bsz s zaz a r z r b s z s and cl cl () () () () ( ) ( ) ( ) ( ) [()] ( ) ( ) ,, .

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## 05A IO Pole Place Controller - SYS635 Adaptive Control...

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