07B Pole-Zero Placement - SYS635 Adaptive Control Ka C....

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SYS635 Adaptive Control Ka C. Cheok 07B Pole-Zero Placement 1 POLE AND ZERO PLACEMENT CONTROLLER DESIGN TECHNIQUE Objective: The controller design technique allows the poles and to some extent zeros of a process or plant to be placed at suitable pole and zero location. Example: Let’s consider a pole-zero placement design for a second order system where the plant/process polynomials are given by 2 12 01 () A zza z a Bz bz b = ++ =+ and the controller polynomials are Rz rz r Sz sz s Tz tz t = + = + = + Analysis: Show that, when w = 0 and v = 0, the closed-loop transfer function from r to y is Expand the terms and observe ( )( ) ( ) ( ) 2 0 1 ()() () () bz b tz t BzTz AzRz BzSz za zar z r b z bs z s = + + + + + Requirement: We specify the desire that the closed-loop system is to behave according to a new model ( ) 2 mm m m bzb Az + = Design: We set + (denominator of the controller) to equal to ( ) + so that a pole-zero cancellation takes place in the closed-loop transfer function. That is ( )( ) 0 1 22 2 10 21 zab z b b z z s za s z s s z a s + == + + + +++ + + = So we set 0 1 b z b += + and 1 2 s z a sz a z a ++=+ + so that 2 BzSz za + = + Rzuz Tzu z Szyz c =− ( ) Azxz Bz uz wz y u r x v w Controller Plant y u c +
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SYS635 Adaptive Control Ka C. Cheok 07B Pole-Zero Placement 2 General Pole & Zero Placement Control Design where the plant/process polynomials are given by 1 1 0 () aa a b b nn n n nb a Az z az a Bz bz b n n =+ + + + L L and the controller polynomials are 1 01 1 1 , , rr r ss s tt t n ns r nt r Rz rz r Sz sz s n n Tz tz t n n + + + + + + L L L Note that A(z) is monic, i.e., the coefficient of the highest power term is one. ab is the number of pole excess Show that, when w = 0 and v = 0, the pole-zero placement closed-loop transfer function from r to y is Notice that we can now modify the closed-loop denominator polynomial with R(z) and S(z), and some closed-loop zeros with T(z). _________________________________________________________________ Note: Compare this to the conventional pole placement controller where T(z) = S(z). Then we have the familiar ()() cc Rzuz Sz u z yz o r uz u z =− = Tzu z Szyz c () () ( ) Azxz Bz uz wz y u r x y u c y u c BzTz AzRz BzSz + + v w
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07B Pole-Zero Placement - SYS635 Adaptive Control Ka C....

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