Robust Control with Classical Methods QT Per-Olof utman Review of the classical Bode-Nichols control problem QT in the basic Single nput Single Output (SSO) case undamental Design Limitations dentification of Uncertain Transfer functions QT for non-minimum phase and computer controlled systems QT for cascaded systems, and for a class of non-linear plants QT for Multi-nput Multi-Output (MMO) plants A comparison between QT and other robust and adaptive control the toolbox for robust control systems design P-O utman QT in the Multi-nput Multi- Output (MMO) case The 2x2 servo problem The structure of (s) The pre-compensator Equivalent SSO systems The equivalent plant /W jj Servo specifications The first design step - Servo bounds - orowitz-sidi bounds - Pre-filter design The second design step - true SSO Example the toolbox for robust control systems design P-O utman
The 2x2 MMO servo problem Closed loop block diagram of TTO servo system w/o disturbances the toolbox for robust control systems design P-O utman The 2x2 MMO servo problem, cont d the toolbox for robust control systems design P-O utman 2
The structure of (s) with the a priori user chosen Pre-compensator Let the full matrix feedback compensator and the to-be-designed Diagonal compensator be decomposed as the toolbox for robust control systems design P-O utman The pre-compensator P 0 (s) T s PsP sl s PsP sl ss = + 0 0 0 0 Choice of P 0 (s) connected to input-output pairing Example: invites the choice in order to make PP 0 - diagonal. Use RA or dynamic RA for pairing V(s)=P(s)P 0 - (s) = de facto plant Decentralized control P 0 (s) = Nominally decoupling control P 0 (s) = s d P nom (s) the toolbox for robust control systems design P-O utman P Ps = 0 P 0 2 2 0 P P s 02 0 = P 0 02 3
4 the toolbox for robust control systems design P-O utman Equivalent SSO systems Let V(s)=P(s)P 0 - (s) = de facto plant T s V s L s V s L s s = + 0 0 VsL sts VsL ss + = 0 0 V s L s T s L s s + = 0 0 W s W s L s T s L s s d b + + = 0 0 W s L sts L ss W sts d b + = 0 0 W s L s T s W L s s W st s d d b + = 0 0 T s W s L s W L s s W st s d d b = + 0 0 the toolbox for robust control systems design P-O utman Equivalent SSO systems, cont d orowitz: after the design, the specification would be satisfied. t would hold that T kj <b kj (ω) ence, -W ik T kj is replaced by a worst case disturbance D ij = max W ik b kj e jϕ T s W s L s W L s s W st s d d b = + 0 0
The equivalent plant /W jj Example: V=PP 0 - diagonal. W=V - =diag(/v, /V 22 ) fi /W jj = V jj V(s)=P(s)P 0 - (s) Decentralized control P 0 (s) = P P W s = 22 2 P P P P P P 22 2 2 2 W P P P = P 2 2 22 the toolbox for robust control systems design P-O utman Basically non-interacting specifications or the basically non-interacting servo problem, we postulate that T s V s L s = + V s L s s 0 0 the toolbox for robust control systems design P-O utman 5
Selection of the first design step Alternative : Design L 0, for {T, T 2 } and L 20, 22 for {T 2, T 22 } seperately, and independently. Disadvantage: Crosscoupling represented by worst-case disturbances only. Alternative : Design first the easiest of the two tasks. Then, the second design step becomes a true SSO problem with correct cross-coupling. Additional advantage: n a fully coupled system, the last loop to be closed in a sequential design, determines stability (Bode). the toolbox for robust control systems design P-O utman The first design step T 2 is replaced by b 2 Mixed servo & disturbance rejection problem fi Design of L 0 and is not separated, as in SSO T 22 is replaced by b 22 Modify the high frequency roll-off of b 22 or b 2, in order to ease the hf gain demand on L 0 or L 20. the toolbox for robust control systems design P-O utman 6
The first design step: Servo bounds or each ω and for each ϕ= arg(l 0 ), find b d (ω,ϕ): 0 b d (ω,ϕ) (b (ω)- a (ω))/2 such that the orowitz-sidi bounds become equal. the toolbox for robust control systems design P-O utman The first design step: Servo bounds, cont d or each ω and for each ϕ= arg(l 0 ), find b d (ω,ϕ): 0 b d (ω,ϕ) (b (ω)-a (ω))/2 s. t. the bounds become equal. fi conservativeness due to - inequality reduced. Modify the hf roll-off of b 2 (ω), see below. the toolbox for robust control systems design P-O utman 7
The first design step: orowitz-sidi bounds Servo bounds from Cross-coupling bounds from Sensitivity bounds from x ω + 0 L jω W jω Sensitivity bounds for Disturbance rejection Loop stability margins Avoiding nmp plant in the second design step, see below. and now, design L 0 = L * 0 ô recall, b ij hf roll-off the toolbox for robust control systems design P-O utman The first design step: a note on roll-off the toolbox for robust control systems design P-O utman 8
The first design step: Pre-filter design After the design of L * 0, b, * d (ω) is computed: Then design (s) to satisfy the toolbox for robust control systems design P-O utman The second design step With L 0 (s) and (s) designed, we get, with 2 = 2 =0 the toolbox for robust control systems design P-O utman 9
The second design step: true SSO Servo bounds from Cross-coupling bounds from Sensitivity bounds from + L jω W e jω 20 22 y ω the toolbox for robust control systems design P-O utman The second design step: true SSO Servo bounds from Cross-coupling bounds from Sensitivity bounds from y ω + L jω W e jω 20 22 As in SSO, design L 20 to satisfy the orowitz-sidi bounds, and such that the closed loop is stable, which will ensure MMO closed loop stability if the plant is fully connected. Design 22. Simulate in the frequency and time domains. Check closed loop stability Re-design, if necessary. the toolbox for robust control systems design P-O utman 0
Example the toolbox for robust control systems design P-O utman Example, cont d: templates and pre-compensator ere: nominally decoupling design with precompensator chosen as P 0 (s)=p nom (s). Exercise: carry out a decentralized design with P 0 (s)=. the toolbox for robust control systems design P-O utman
Example, cont d: specifications Servo specification Cross coupling specification 6 db sensitivity specification the toolbox for robust control systems design P-O utman Example, cont d: equivalent plants Note: k k 22 k 2 k 2 >0 P P s 0 = = 3 2 2 nom k k k k V = PP= 2 2 2 2 0 3 2k k 2k k 2 22 22 2 k k k k W = V = 2 2 22 2 2 k k k k k 2k 2k k 22 2 2 22 2 2 W = V = nom nom the toolbox for robust control systems design P-O utman 2
Example, cont d: bounds for L 0 /W nom mcbnd also computes the b d -vector for each servo bound into default matrix servobd the toolbox for robust control systems design P-O utman Example, cont d: design of L 0 /W nom the toolbox for robust control systems design P-O utman 3
Example, cont d: modified servo specification After the design of L * 0,, b* d (ω)is computed from Compute Compute the modified servo specification the toolbox for robust control systems design P-O utman Example, cont d: closing the first loop Compute the complementary sensitivity function the toolbox for robust control systems design P-O utman 4
Example, cont d: design of Design the prefilter, and display the first loop servo transfer function the toolbox for robust control systems design P-O utman Recall: The second design step Servo bounds from Cross-coupling bounds from Sensitivity bounds from y ω + L jω W e jω 20 22 As in SSO, design L 20 to satisfy the orowitz-sidi bounds, and such that the closed loop is stable, which will ensure MMO closed loop stability if the plant is fully connected. Design 22. Simulate in the frequency and time domains. Check closed loop stability Re-design, if necessary. the toolbox for robust control systems design P-O utman 5
Example, cont d: second design step templates Compute templates of x-coupling tf, wwc= and of equivalent plant iw22e = the toolbox for robust control systems design P-O utman Example, cont d: second design step bounds x-coupling ( couple2a ) servo ( servo2 ) sensitivity ( sens2 ) the toolbox for robust control systems design P-O utman 6
Example, cont d: second design step dominant bounds the toolbox for robust control systems design P-O utman Example, cont d: 2nd step feedback compensator the toolbox for robust control systems design P-O utman 7
8 the toolbox for robust control systems design P-O utman Example, cont d: 2nd step pre-filter design the toolbox for robust control systems design P-O utman Example, cont d: Design summary P P s 0 3 2 2 = = nom L s ss s ss 0 5 80 30 00 0 0 7 30 = + + + +
the toolbox for robust control systems design P-O utman Example end: Time domain simulation Example cont d: requency domain simulation the toolbox for robust control systems design P-O utman 9