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Lecture Multivariable Control Sytem Ali Karimpour Aociate Profeor Ferdowi Univerity of Mahhad Lecture Reference are appeared in the lat lide. Dr. Ali Karimpour May 6

Uncertainty in Multivariable Sytem and Quantitative feedback theory Lecture Topic to be covered include: Introduction Type of Uncertainty in Multivariable Sytem Robut Stability of Uncertain Sytem. Quantitative Feedback Theory QFT Deign Procedure. Remark: Evaluation of thi lecture can be done before end of cla under pecial circumtance. Dr. Ali Karimpour May 6

Introduction Lecture Reaon of uncertainty: Linearization or ignoring ome non-linear part of ytem. Ignoring enor or actuator dynamic in modeling. Ignoring high frequency behavior or model order reduction. Changing operating point. Fault in ome part of ytem or fatigue. Why do one need to examine uncertainty in control ytem? 3 Dr. Ali Karimpour May 6

Introduction Lecture LQ Control: Optimal tate feedback J r z T Qz u T Ru dt where z Mx, Q Q T and R R T The optimal olution for any initial tate i where u t K x t K r R r B T X Where X=X T i the unique poitive-emidefinite olution of the algebraic Riccati equation A T X XA XBR B T X M T QM 4 Dr. Ali Karimpour May 6

Introduction Lecture Robutne Propertie For an LQR-controlled ytem if the weight R i choen to be diagonal, then S I K I A B atifie S j, r Nyquit plot in MIMO cae i and.5 ki, i,,..., m ki and i 6, i,,..., Thi wa brought tarkly to the attention of the control community by Doyle 978 in a paper entitled uaranteed Margin for LQR Regulator with a very compact abtract which imply tate There are none. m - 5 Dr. Ali Karimpour May 6

Introduction Lecture Example -: LQR deign of a firt order proce. 3 3 The cot function to be minimized i J r y Ru dt Let R. x x u y x K r T R B X [86.78-75.386] A BK r 7.596 6. 988i Stable for and. 5 k A BK k table for all k. 5 r Let uncertainty in b a b For.9ytemi untable. Exercie -: Derive for R=.. 6 Exercie -: Derive curve of veru /R Dr. Ali Karimpour May 6

Dr. Ali Karimpour May 6 Lecture 7 Example -: LQR deign of a firt order proce. 3 3 x y u x x Introduction

Dr. Ali Karimpour May 6 Lecture 8 Example -: Decoupling controller 5 4 56 47 3 We have good tability margin in both channel. 7 6 8 7 7 6 8 7 d The pre compenator approach may be extended by introducing a pot compenator W W p The overall controller i then W K W K p k k k k K 7 6 8 7 7 6 8 7 Exercie -3: Derive tability margin for different value of δ if k k K For k= o find the mallet δ that lead to intability. Repeat for k=. Introduction

Uncertainty in Multivariable Sytem and Quantitative feedback theory Lecture Introduction Type of Uncertainty in Multivariable Sytem Robut Stability of Uncertain Sytem. Quantitative Feedback Theory QFT Deign Procedure. 9 Dr. Ali Karimpour May 6

Lecture Type of Uncertainty in Multivariable Sytem Type of uncertainty Parametric real uncertainty. tructured uncertainty Model tructure and order are known, but ome parameter value are uncertain. k a Dynamic frequency-dependent uncertainty or nonparametric uncertainty. untructured uncertainty There exit ome erroneou or miing dynamic. Uually unmodeled dynamic i in high frequencie. Dr. Ali Karimpour May 6

Dr. Ali Karimpour May 6 Lecture Type of untructured uncertainty Type of Uncertainty in Multivariable Sytem w w p w w I p w w I p

Dr. Ali Karimpour May 6 Lecture Type of untructured uncertainty Type of Uncertainty in Multivariable Sytem w w I p w w I p w w I p

Dr. Ali Karimpour May 6 Lecture 3 Parametric uncertainty Nonparametric uncertainty Example -3: Conider a plant with parametric uncertainty max min p p p Now let, / max min max min r r p, w r w w w r p w w p Nonparametric uncertainty ha more conervativene. Type of Uncertainty in Multivariable Sytem

Type of Uncertainty in Multivariable Sytem Lecture Parametric uncertainty Nonparametric uncertainty Example -4: Conider a plant with two parametric uncertainty p k e k,, 3 4 Dr. Ali Karimpour May 6

Dr. Ali Karimpour May 6 Lecture 5 3,, k e k p, ; j w A p Conider additive uncertainty a: Additive uncertainty can be repreent By multiplicative one:, ; j w j j w j w A M M p Type of Uncertainty in Multivariable Sytem

Type of Uncertainty in Multivariable Sytem Lecture Sytem without uncertainty z P P P P w u u K z P P K I P K P w Nw N F P, K l Sytem with uncertainty N tructure Pull out uncertainty Suitable for robut performance analyi 6 Dr. Ali Karimpour May 6

Dr. Ali Karimpour May 6 Lecture 7 Sytem without uncertainty Nw w P K P I K P P z, K P F N l Sytem with uncertainty NΔ tructure u w P P P P z u K w u N N N N z y y u Fw w N N I N N z, N F F u Type of Uncertainty in Multivariable Sytem

Robut Stability of Uncertain Sytem Lecture Sytem without uncertainty Sytem with uncertainty N tructure Suitable for nominal performance analyi z P P K I P K P w Nw Suitable for robut performance analyi z N N I N N w Fw eneral Control Configuration Checking robut tability? Suitable for controller deign 9 Dr. Ali Karimpour May 6

Robut Stability of Uncertain Sytem Lecture Sytem without uncertainty Sytem with uncertainty N tructure Suitable for nominal performance analyi eneral Control Configuration z N M tructure If there i no uncertainty we have nominal tability o: Suitable for robut performance analyi N I N N w Fw N, N, N and N are table M N Suitable for robut tability analyi Dr. Ali Karimpour May 6

Robut Stability of Uncertain Sytem Lecture Sytem without uncertainty Sytem with uncertainty N tructure Suitable for nominal performance analyi eneral Control Configuration Suitable for controller deign Suitable for robut performance analyi z N N I N N w Fw M tructure If there i no uncertainty we have nominal tability o: N, N, N and N are table M N Suitable for robut tability analyi Dr. Ali Karimpour May 6

Robut Stability of Uncertain Sytem Lecture M tructure NS: N i internally table RS: NS and F=F u N,Δ i table for any Δ Suitable for robut tability analyi Theorem -: RS for untructured full perturbation. Aume that the nominal ytem M i table NS and that the perturbation are table. Then The M -tructure i table for all atifying γ M M / j / The M -tructure i table M j / M j Dr. Ali Karimpour May 6

Robut Stability of Uncertain Sytem M tructure Lecture Sytem without uncertainty Suitable for robut tability analyi Sytem with additive uncertainty p w w y Mu M w K I K w Robut tability condition: In the cae of M w K I K w In the cae of free M 3 Dr. Ali Karimpour May 6

Robut Stability of Uncertain Sytem M tructure Lecture Sytem without uncertainty Suitable for robut tability analyi p I w w Sytem with multiplicative input uncertainty y Mu M w K I K w Robut tability condition: In the cae of M w K I K w In the cae of free M 4 Dr. Ali Karimpour May 6

Robut Stability of Uncertain Sytem M tructure Lecture Sytem without uncertainty Suitable for robut tability analyi p I ww Sytem with multiplicative output uncertainty M y Mu wk I K w Robut tability condition: In the cae of M wk I K w In the cae of free M 5 Dr. Ali Karimpour May 6

Robut Stability of Uncertain Sytem M tructure Lecture Sytem without uncertainty Suitable for robut tability analyi p I w w Sytem with invere additive uncertainty y Mu M w I K w Robut tability condition: In the cae of M w I K w In the cae of free M 6 Dr. Ali Karimpour May 6

Robut Stability of Uncertain Sytem M tructure Lecture Sytem without uncertainty Suitable for robut tability analyi p I w w w w Sytem with invere multiplicative input uncertainty M y Mu w I K w Robut tability condition: In the cae of M w I K w In the cae of free M 7 Dr. Ali Karimpour May 6

Robut Stability of Uncertain Sytem M tructure Lecture Sytem without uncertainty Suitable for robut tability analyi p I w w w w Sytem with invere multiplicative output uncertainty M w I K w Robut tability condition: In the cae of M w I K w In the cae of free M 8 Dr. Ali Karimpour May 6

Robut Stability of Uncertain Sytem M tructure Lecture Uncertainty Perturbed Plant Suitable for robut tability analyi M in MΔ-tructure Additive uncertainty p w w M w K I K w Multiplicative input uncertainty p I w w M w K I K w Multiplicative output uncertainty p I ww M wk I K w Invere additive uncertainty p I w w M w I K w Invere multiplicative input uncertainty p I w w M w I K w Invere multiplicative output uncertainty I w w p M w w I K 9 Dr. Ali Karimpour May 6

Robut Stability of Uncertain Sytem M tructure Lecture Sytem with coprime factor uncertainty Suitable for robut tability analyi M l N l p M l M N l N N M M K I I K M l Since there i no weight for uncertainty o the theorem i RS : M / N M 3 Dr. Ali Karimpour May 6

Dr. Ali Karimpour May 6 Lecture 3 Remind Example -: Decoupling controller 5 4 56 47 3 k k k k K 7 6 8 7 7 6 8 7 I p Conider ytem with multiplicative input uncertainty K I K K I K / Robut Stability of Uncertain Sytem

Robut Stability of Uncertain Sytem M tructure Lecture / K I K Suitable for robut tability analyi K I K 4. db 5. 85 /5.85.63 3 Dr. Ali Karimpour May 6

Introduction Lecture Many deign method require deign objective to be tated in term which are familiar from claical SISO deign method. Some of thee pecification, may be derived from an accurate knowledge of noie and diturbance tatitic, or of poible perturbation to the nominal plant model, but they are more often obtained in a le quantitative manner. Typically they are initially obtained from previou experience with imilar plant, and then refined in a pecify-deign-analyze cycle until derived acceptable C.L. behavior. Thi approach ha been forcefully criticized a inadequate by Horowitz 98. The baic reaon for uing feedback i: Unpredictable noie or diturbance. To combat uncertainty Unpredictable variation in the behavior of the plant. If we have a quantitative decription of the amount of uncertainty which may be preent, and a precie pecification of the range of behavior which may be tolerated in the face of uch uncertainty, then we hould aim to develop a deign technique 34 Dr. Ali Karimpour May 6

Uncertainty Model and Plant Template Lecture The variou origin of model uncertainty: I Parametric uncertainty Parametric uncertainty implie pecific knowledge of variation in parameter of the tranfer function. II Non-parametric uncertainty The main ource of non-parametric uncertainty i error in the model. j 35 Dr. Ali Karimpour May 6

Lecture Uncertainty Model and Plant Template I Parametric uncertainty Nichol Chart for ab a f a [ 4] b[.... 5].5 5 Nichol Chart Template for ab/+a. 4 db. 3.5 db.5 db...5. Open-Loop ain db - db 3 db 6 db a [ 4] b 5 a[ 4] b a b[ a 4 b[ 5] 5] - -3-4 -5-8 -35-9 -45 Open-Loop Phae deg 36 Dr. Ali Karimpour May 6

Uncertainty Model and Plant Template Lecture I Parametric uncertainty Sometime edge are not ok! Nichol Chart for a b -5 - Nichol Chart Template Edge for /a+b+ a [ 3] b a[ 3] b[ 5] Open-Loop aindb -5 b[ 5] a 3 b[ 5] a rad /ec - a [ 3] b 5-5 -6-5 -4-3 - - - -9 Open-Loop Phaedeg 37 Dr. Ali Karimpour May 6

Uncertainty Model and Plant Template Lecture I Parametric uncertainty Be careful to ue the edge. Nichol Chart for a b -5 - Nichol Chart Template Edge and Complete for /a+b+ a [ 3] b a[ 3] b[ 5] a[ 3] b[ rad /ec 5] Open-Loop aindb -5 - b[ 5] a 3 b[ 5] a a [ 3] b 5-5 -6-5 -4-3 - - - -9 Open-Loop Phaedeg 38 Dr. Ali Karimpour May 6

Amplitude db Amplitude db Uncertainty Model and Plant Template How a QFT controller work? Lecture k Nichol Chart for a k [ ] a [ ] Nichol Chart template for k/-a 4 db Nichol 4 Chart for K 3 Nichol Chart template With Controller k a db 3.5 db.5 db db - db.5 3.5 db db.5 db - db Open-Loop ain db - - -3 3 db 6 db -3 db -6 db - db - db 5 Open-Loop ain db - - -3 3 db 6 db -3 db -6 db - db - db -4-4 db -36-35 -7-5 -8-35 -9-45 Open-Loop Phae deg Frequency rad/ec -4-4 db -36-35 -7-5 -8-35 -9-45 Open-Loop Phae deg Frequency rad/ec 39 Dr. Ali Karimpour May 6

Amplitude db Uncertainty Model and Plant Template How a QFT controller work? Lecture Sytem with and without Controller Nichol Chart for.5 k a 5 K 3 4 Nichol Chart template With and Withoput Controller db Open-Loop ain db 3 -.5 db.5 db db 3 db 6 db - db -3 db -6 db - db.5 5 - - db -3-4 -4 db -36-35 -7-5 -8-35 -9-45 Open-Loop Phae deg Frequency rad/ec 4 Dr. Ali Karimpour May 6

Uncertainty Model and Plant Template How a QFT controller work? Lecture Step Repone of Sytem with and without Controller K= k a K 3 5 Step Repone of All Plant Without Controller.8 Step Repone of All Plant With Controller.6.4 5. Amplitude Amplitude.8-5.6.4 -. -5.5.5.5 3 3.5 4 4.5 5 Time ec.5.5.5 3 3.5 4 4.5 5 Time ec 4 Dr. Ali Karimpour May 6

QFT Deign Procedure Lecture Choice of Frequency Array An appropriate frequency band for a computing template and bound ha to be elected. Choice of Nominal Plant In order to compute bound, it i neceary to chooe a plant from the uncertainty et a the nominal plant. It i common practice to elect a nominal plant which we think i mot convenient for deign. QFT Bound Computation iven the plant template, QFT convert cloed-loop magnitude pecification into magnitude and phae contraint on a nominal openloop function. QFT Loop-haping The final tep in a QFT deign involve the deign loop haping of a nominal loop function that meet it bound. The controller deign then proceed uing the Nichol chart and claical loop-haping idea. 4 Dr. Ali Karimpour May 6

QFT Deign Procedure Lecture he QFT approach aume that the plant uncertainty i repreented by a et of template n the complex plane at ome frequency ω k. It alo aume that the deign pecification i in the form of bound on the magnitude of the frequency-repone tranfer function S j M a T j b The QFT technique lead to a deign which atifie thee pecification for all permiible plant variation. Horowitz and Sidi 97 have obtained ufficient condition on frequency-domain bound which imply the atifaction of time-domain bound, the frequency-domain bound obtained in thi way do not appear to be unduly conervative. 44 Dr. Ali Karimpour May 6

QFT Deign Procedure Bound for S Lecture Let we need <.5 o: L mut be outide of black curve. Now conider the template at one frequency with nominal plant a blue curve. 5 Nichol Chart nominal ab a. a [ b [ a b 4] 5] Bound for template i red curve. Open-Loop ain db 4 3 - - -3.5 db.5 db db 3 db 6 db db - db -3 db -6 db - db - db Effect of nominal plant? -4 db -4 45-7 -5-8 -35-9 -45 Open-Loop Phae deg Dr. Ali Karimpour May 6

QFT Deign Procedure Bound for S Lecture Let we need <.5 o: L mut be outide of black curve. Changing Nominal plant ab a [ 4] a b [ 5] nominal nominal..5.5 a b a.5 b.5 Open-Loop ain db 5 4 3 - - -3 db 3 db 6 db Nichol Chart.5 db.5 db -4 db -4 46-7 -5-8 -35-9 -45 Open-Loop Phae deg Dr. Ali Karimpour May 6 db New bound Perviou bound - db -3 db -6 db - db - db

QFT Deign Procedure Bound for T Lecture A feedback configuration with two degree of freedom. We need: Since: a T j b T L L P If L would not interect any pair of M-circle whoe value differed by more than b / a M / M L M M L Nominal plant j M 47 M Dr. Ali Karimpour May 6

Dr. Ali Karimpour May 6 Lecture 48 P L L T If L would not interect any pair of M-circle whoe value differed by more than / / M M a b QFT Deign Procedure j Nominal plant Bound for T M M M L L M c b L L c a b L L c a P

QFT Deign Procedure Bound for T Lecture Suppoe that we know the template of all poible value of at ω : But we need: Since: T a T j b L A feedback configuration with two degree of freedom. L P If L would not interect any pair of M-circle whoe value differed by more than Nominal plant j L j B b / a 49 Dr. Ali Karimpour May 6

QFT Deign Procedure Bound for T Lecture A feedback configuration with two degree of freedom. Changing Nominal plant Lead Different Bound Nominal plant j L j New bound Perviou bound B B 5 Dr. Ali Karimpour May 6

Dr. Ali Karimpour May 6 Lecture 5 i i i i i i c b j L j L c a In order to meet the deign pecification i i i b j T a i i c j P the pre-filter i choen to have the gain QFT Deign Procedure Similarly one can found bound for S Bound for T

QFT Algorithm: QFT Deign Procedure Lecture. Formulating of the cloed-loop control performance pecification, i.e., tability margin, tracking and diturbance rejection.. enerating template. For a given uncertain plant PP, elect a erie of frequency point i, i =,,...,m according to the plant characteritic and the pecification. 3. Computation of QFT bound. Find the interection of bound. An arbitrary member in the plant et i choen a the nominal cae. 4. Loop haping for QFT controller. The deign of the QFT controller, K i accomplihed on the Nichol Chart. The QFT bound at all frequencie mut be atified and the cloed-loop nominal ytem i table; 5. Deign of prefilter P. The Final tep in QFT i to deign the prefilter, P, uch that the performance pecification are atified. 5 Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture Example -5: Suppoe that the et of our plant i: ab a [ 4] b[ 5] a Performance pecification are: a S <.5. for robut tability b Zero teady tate error to tep input. c Le than 5% overhoot to tep input. Contrain on S Contrain on T d Output mut be above 9% after econd when tep input applied. 53 Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture a S <.5. for robut tability b Zero teady tate error to tep input. c Le than 5% overhoot to tep input. Contrain on S Contrain on T d Output mut be above 9% after econd when tep input applied..5 T n n n Acceptable tep repone.9.5 54 3 Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture T n n n.6 Step Repone Bound.4 n min max min max Amplitude..8.6.4..5.5.5 3 3.5 4 4.5 5 Timeec 55 Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture T n n min max min max n n a i T ji b i f.....5. b / a..44.4.583 3.55 7.77 Amplitudedb - - -3-4 -5 Frequency Repone Bound b i a i - - i FrequencyHz 56 Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture Template are: 5 f ab a a [ 4] b[ 5]... Nichol Chart Template for ab/+a..5 4 Open-Loop ain db 3 -.5 db.5 db db 3 db 6 db db.....5 -. -3-4 -5-8 -35-9 -45 Open-Loop Phae deg 57 Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture Nominal Plant i: ab a a= and b= 5 Nichol Chart Template for ab/+a 4 3.5 db db. Open-Loop ain db -.5 db db 3 db 6 db....5 -. -3-4 -5-8 -35-9 -45 Open-Loop Phae deg 58 Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture Bound for S.5 5 Nichol Chart Contraint <.5 4 db. Open-Loop ain db 3 -.5 db.5 db db 3 db 6 db - db -3 db -6 db - db....5. - - db -3-4 db -4-7 -5-8 -35-9 -45 Open-Loop Phae deg 59 Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture Bound for a T b f.....5. b / a..44.4.583 3.55 7.77 Open-Loop ain db 5 4 3 - - Nichol Chart for Contraint aw < Tw <bw db.5 db.5 db db 3 db 6 db - db -3 db -6 db - db - db.....5. -3-4 db -4-7 -5-8 -35-9 -45 Open-Loop Phae deg 6 Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture Interection of Bound for a T b and S.5 5 Nichol Chart for Contraint <.5 & aw < Tw <bw Open-Loop ain db 4 3 -.5 db.5 db db 3 db 6 db db - db -3 db -6 db - db.....5. - - db -3-4 db -4-7 -5-8 -35-9 -45 Open-Loop Phae deg 6 Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture Three different controller applied K 4 K 5 5 K 3 5 3.6 4 3 5 Open-Loop ain db Nichol Chart for Contraint <.5 & aw < Tw <bw and Some Different Controller 5 4 db 3.5 db.5 db db 3 db 6 db - - db -3 db -6 db - db.....5. - - db -3-4 db -4-7 -5-8 -35-9 -45 Open-Loop Phae deg 6 Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture ab a.6.4 K 3 Step Repone Bound + Some Different Controller a [ 4] b[ 5]. Three Different Controller K 3 K P K 4 5 5 5 3.6 4 3 5 Amplitude.8.6.4. K K.5.5.5 3 3.5 4 63 4.5 5 Timeec Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture Frequency Repone Bound + Some Different Controller Three Different Controller K 3 K K ab a a [ 4] b[ P 4 5 5 5 3.6 4 3 5 5] Amplitudedb - -4-6 -8 - - FrequencyHz K K 3 K 64 Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture ab a.6.4 Step Repone of All Poible Plant with Controller a [ 4] b[ 5]. With Controller P K 3 5 3.6 4 3 5 Amplitude.8.6.4..5.5.5 3 3.5 4 4.5 65 5 Timeec Dr. Ali Karimpour May 6

QFT Deign Procedure Example Frequency Repone of All Poible Plant with Controller Lecture ab a a [ 4] b[ 5] With Controller 5 3.6 4 K3 P 3 5 Let Amplitude db db.9 P.45-5 - -5 Frequency rad/ec? Amplitudedb - - -3-4 -5 - - FrequencyHz 66 Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture ab a.6.4 Step Repone of All Poible Plant with Controller + Prefilter a [ 4] b[ 5]. With Controller.9 P.45 K 3 5 3.6 4 3 5 Amplitude.8.6.4..5.5.5 3 3.5 4 4.5 67 5 Timeec Dr. Ali Karimpour May 6

QFT Deign Procedure Example Lecture K 3 ab a a [ 4] b[ 5] With Controller.9 P.45 5 3.6 4 3 5 Amplitudedb - - -3-4 Frequency Repone of All Poible Plant with Controller + Prefilter -5-6 68 - - FrequencyHz Dr. Ali Karimpour May 6

Extenion to MIMO ytem Lecture Thi deign method i extended to multivariable problem a follow. We are going to find t uv vth input to uth output y t l lv r v k t ˆ Ky g u l ul ul lv r v I K y KP r ˆ Ky KP r Let k ij If r j KP r KP u for uv r v j v k l ul p lv r v r v lu gˆ ul t lv r v gˆ uu kuu tuv kuu puvrv Let h / ˆ ij g ij t uv h k p h d uu uu uv uu uv d huukuu huuk uv uu lu t h lv ul 69 Dr. Ali Karimpour May 6

Dr. Ali Karimpour May 6 Lecture 7 uu uu uv uu uu uu uv uu uu uv k h d h k h p k h t uv uu uu uu uu uu uv uu uu uv d k h h k h p k h t d K r K KP y u l ul lv uv h t d Extenion to MIMO ytem

Exercie Lecture Exercie -: Mentioned in the lecture. Exercie -: Mentioned in the lecture. Exercie -3: Mentioned in the lecture. Exercie -4: Conider following block diagram. We have both input and output uncertainty. a Find the et of poible plant p, and b Find M and derive robut tability condition. o i Exercie -5: Aume we have derived the following detailed model: Suppoe we choe =3/+ with multiplicative uncertainty. Derive uitable caling 7 Matrix. Dr. Ali Karimpour May 6

Reference Lecture Reference Web Reference Multivariable Feedback Deign, J M Maciejowki, Weley,989. Multivariable Feedback Control, S.Skogetad, I. Potlethwaite, Wiley,5. Control Configuration Selection in Multivariable Plant, A. Khaki-Sedigh, B. Moaveni, Springer Verlag, 9. Control Sytem Deign QFT. Bo Bernhardon, K,J. Atrom, Department of Automatic Control LTH, Lund Univerity تحليل و طراحی سيستم های چند متغيره دکتر علی خاکی صديق http://www.um.ac.ir/~karimpor http://aba.kntu.ac.ir/eecd/khakiedigh/coure/mv/ 7 Dr. Ali Karimpour May 6