Ch.05 Reduction of Multiple Subsystems

2/3/2016 System Dynamics and Control 5.01 Reduction of Multiple Subsystems 05 Reduction of Multiple Subsystems HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.03 Nguyen Tan Tien Reduction of Multiple Subsystems System Dynamics and Control 5.02 Reduction of Multiple Subsystems Learning Outcome After completing this chapter, the student will be able to • Reduce a block diagram of multiple subsystems to a single block representing the transfer function from input to output • Analyze and design transient response for a system consisting of multiple subsystems • Convert block diagrams to signal-flow diagrams • Find the transfer function of multiple subsystems using Mason’s rule • Represent state equations as signal-flow graphs • Represent multiple subsystems in state space in cascade, parallel, controller canonical, and observer canonical forms • Perform transformations between similar systems using transformation matrices and diagonalize a system matrix HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.04 Nguyen Tan Tien Reduction of Multiple Subsystems §1.Introduction - Represent multiple subsystems in two ways • block diagrams: for frequency-domain analysis and design • signal-flow graphs: for state-space analysis - Develop techniques to reduce each representation to a single transfer function • Block diagram algebra will be used to reduce block diagrams ã Masons rule to reduce signal-flow graphs Đ2.Block Diagrams - The space shuttle consists of multiple subsystems Can you identify those that are control systems, or parts of control systems? HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.05 Nguyen Tan Tien Reduction of Multiple Subsystems §2.Block Diagrams - A subsystem is represented as a block with an input, an output, and a transfer function - Many systems are composed of multiple subsystems When multiple subsystems are interconnected, a few more schematic elements must be added to the block diagram HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien System Dynamics and Control 5.06 Nguyen Tan Tien Reduction of Multiple Subsystems §2.Block Diagrams Cascade Form Parallel Form HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 2/3/2016 System Dynamics and Control 5.07 Reduction of Multiple Subsystems §2.Block Diagrams Feedback Form System Dynamics and Control 5.08 Reduction of Multiple Subsystems §2.Block Diagrams Moving Blocks to Create Familiar Forms Block diagram algebra for summing junctions equivalent forms for moving a block to the left past a summing junction Block diagram algebra for summing junctions equivalent forms for moving a block to the right past a summing junction HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.09 Nguyen Tan Tien Reduction of Multiple Subsystems §2.Block Diagrams HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.10 Nguyen Tan Tien Reduction of Multiple Subsystems §2.Block Diagrams - Ex.5.1 Block Diagram Reduction via Familiar Forms Reduce the block diagram to a single transfer function Block diagram algebra for pickoff points equivalent forms for moving a block to the left past a pickoff point Block diagram algebra for pickoff points equivalent forms for moving a block to the right past a pickoff point HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.11 Nguyen Tan Tien Reduction of Multiple Subsystems §2.Block Diagrams Solution HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.12 Nguyen Tan Tien Reduction of Multiple Subsystems §2.Block Diagrams - Ex.5.2 Block Diagram Reduction via Familiar Forms Reduce the block diagram to a single transfer function Collapse summing junctions Form equivalent cascaded system in the forward path and equivalent parallel system in the feedback path Solution Form equivalent feedback system and multiply by cascaded 𝐺1 (𝑠) HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 2/3/2016 System Dynamics and Control 5.13 Reduction of Multiple Subsystems §2.Block Diagrams 5.14 Reduction of Multiple Subsystems §2.Block Diagrams HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control System Dynamics and Control 5.15 Nguyen Tan Tien Reduction of Multiple Subsystems §2.Block Diagrams Run ch5p1 in Appendix B Learn how to use MATLAB to • perform block diagram reduction HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.16 Nguyen Tan Tien Reduction of Multiple Subsystems §2.Block Diagrams Skill-Assessment Ex.5.1 Problem Find the equivalent TF, 𝑇 𝑠 = 𝐶(𝑠)/𝑅(𝑠), for the system Solution Combine the parallel blocks in the forward path Then, push 1/𝑠 to the left past the pickoff point HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.17 Nguyen Tan Tien Reduction of Multiple Subsystems §2.Block Diagrams System Dynamics and Control 5.18 Nguyen Tan Tien Reduction of Multiple Subsystems §2.Block Diagrams TryIt 5.1 Combine the parallel blocks in the forward path Then, push 1/𝑠 to the left past the pickoff point Combine the parallel feedback paths and get 2𝑠 Then, apply the feedback formula, simplify, and get 𝑠3 + 𝑇 𝑠 = 2𝑠 + 𝑠 + 2𝑠 HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Use the following MATLAB and Control System Toolbox statements to find the closed loop transfer function of the system in Ex.5.2 if all 𝐺𝑖 𝑠 = 1/(𝑠 + 1) and all 𝐻𝑖 𝑠 = 1/𝑠 G2=G1; G3=G1; H1=tf(1,[1 0]); H2=H1; H3=H1; System=append(G1,G2,G3,H1,H2,H3); input=1; output=3; Q= [1 -4 0 0; -5 0; -5 -6 0 0; 0 0; 0 0]; T=connect(System,Q,input,output); T=tf(T); T=minreal(T) HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 2/3/2016 System Dynamics and Control 5.19 Reduction of Multiple Subsystems §3.Analysis and Design of Feedback Systems - Consider the system which can model a control system such as the antenna azimuth position control system For example, the transfer function, 𝐾/𝑠(𝑠 + 𝑎), can model the amplifiers, motor, load, and gears The closed-loop transfer function, 𝑇(𝑠), for this system 𝐾 𝑇 𝑠 = 𝑠 + 𝑎𝑠 + 𝐾 𝐾 : models the amplifier gain, that is, the ratio of the output voltage to the input voltage HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.21 Nguyen Tan Tien Reduction of Multiple Subsystems §3.Analysis and Design of Feedback Systems - Ex.5.3 Finding Transient Response Given the system, find the peak time, percent overshoot, settling time Solution The closed-loop transfer function 25 52 𝑇 𝑠 = = 𝑠 + 5𝑠 + 25 𝑠 + × 0.5 × 5𝑠 + 52 and 𝜔𝑛 = 25 = 5, 𝜁 = 0.5 From these values of 𝜁 and 𝜔𝑛 𝜋 𝜋 𝑇𝑝 = = = 0.726𝑠 𝜔𝑛 − 𝜁 − 0.52 %𝑂𝑆 = 𝑒−𝜁𝜋/ 1−𝜁 × 100 = 𝑒 −0.5𝜋/ 4 𝑇𝑠 = = = 1.6𝑠 𝜁𝜔𝑛 0.5 × HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.23 1−0.52 System Dynamics and Control 5.20 Reduction of Multiple Subsystems §3.Analysis and Design of Feedback Systems 𝐾 𝑇 𝑠 = 𝑠 + 𝑎𝑠 + 𝐾 - As 𝐾 varies, the poles move through the three ranges of operation of a second-order system 𝑎 𝑎2 − 4𝐾 • overdamped: < 𝐾 < 𝑎2 /4 𝑠1,2 = − ± 2 As 𝐾 increases, the poles move along the real axis 𝑎 𝑠1,2 = − • critically damped: 𝐾 = 𝑎2 /4 𝑎 4𝐾 − 𝑎2 • underdamped: 𝐾 > 𝑎 /4 𝑠1,2 = − ± 𝑗 2 As 𝐾 increases, the real part remains constant and the imaginary part increases Thus, the peak time decreases and the percent overshoot increases, while the settling time remains constant HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control Nguyen Tan Tien 5.22 Reduction of Multiple Subsystems §3.Analysis and Design of Feedback Systems Run ch5p2 in Appendix B Learn how to use MATLAB to • perform block diagram reduction followed by an evaluation of the closed-loop system’s transient response by finding, 𝑇𝑝 , %𝑂𝑆, and 𝑇𝑠 • generate a closed-loop step response ã solve Ex.5.3 ì 100 = 16.303 Nguyen Tan Tien Reduction of Multiple Subsystems §3.Analysis and Design of Feedback Systems Learn how to use MATLAB’s Simulink to • explore the added capability of MATLAB’s Simulink using Appendix C • simulate feedback systems with nonlinearities through Ex.C.3 (p.842 Textbook) HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control Nguyen Tan Tien 5.24 Reduction of Multiple Subsystems §3.Analysis and Design of Feedback Systems - Ex.5.4 Gain Design for Transient Response Design the value of gain 𝐾 for the feedback control system so that the system will respond with a 10% overshoot Solution The closed-loop transfer function 𝐾 𝐾 𝑠(𝑠 + 5) 𝑇 𝑠 = = 𝐾 𝑠 + 5𝑠 + 𝐾 1+ 𝑠(𝑠 + 5) 𝐾 𝑠2 + × × 𝐾𝑠 + 𝐾 and 𝜔𝑛 = 𝐾, 𝜁 = 5/2 𝐾 = HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien HCM City Univ of Technology, Faculty of Mechanical Engineering 𝐾 Nguyen Tan Tien 2/3/2016 System Dynamics and Control 5.25 Reduction of Multiple Subsystems §3.Analysis and Design of Feedback Systems Percent overshoot is a function only of 𝜁 %𝑂𝑆 = 𝑒 −𝜁𝜋/ 1−𝜁 × 100 = 10% ⟹ 𝜁 = 0.591 From this damping ratio 𝜁= 𝐾 2 5 ⟹𝐾= = = 17.9 2𝜁 × 0.591 Although we are able to design for percent overshoot in this problem, we could not have selected settling time as a design criterion because, regardless of the value of 𝐾, the real parts, − 2.5, of the poles of 𝐾/(𝑠 + 5𝑠 + 𝐾) remain the same HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.27 Nguyen Tan Tien Reduction of Multiple Subsystems System Dynamics and Control 5.26 %𝑂𝑆 = 𝑒 −𝜁𝜋/ 1−𝜁 × 100 − ln %𝑂𝑆 − ln 0.05 ⟹𝜁= = = 0.69 𝜋 + ln2 %𝑂𝑆 𝜋 + ln2 0.05 ⟹ 𝑎 = 8𝜁 = × 0.69 = 5.52 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.28 §3.Analysis and Design of Feedback Systems 16 TryIt 5.2 𝐺𝑠 = Use the following MATLAB 𝑠(𝑠 + 𝑎) and Control System Toolbox statements to find 𝜁 , 𝜔𝑛 , a=2; numg=16; deng=poly([0 -a]); %𝑂𝑆, 𝑇𝑠 , 𝑇𝑝 , and 𝑇𝑟 for the G=tf(numg,deng); closed-loop unity feedback system described in Skill- T=feedback(G,1); Assessment Ex.5.2 Start [numt,dent]=tfdata(T,'v'); with 𝑎 = and try some other values A step wn=sqrt(dent(3)) response for the closed loop system will also be z=dent(2)/(2*wn) produced Ts=4/(z*wn) Tp=pi/(wn*sqrt(1-z^2)) pos=exp(-z*pi/sqrt(1-z^2))*100 Tr=(1.76*z^3-0.417*z^2+1.039*z+1)/wn step(T) §4.Signal-Flow Graphs - A signal-flow graph consists only of • branches: represent systems • Nodes: represent signals HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.29 Nguyen Tan Tien Reduction of Multiple Subsystems Nguyen Tan Tien Reduction of Multiple Subsystems - A system is represented by a line with an arrow showing the direction of signal flow through the system Adjacent to the line we write the transfer function A signal is a node with the signal’s name written adjacent to the node System Dynamics and Control 5.30 §4.Signal-Flow Graphs - Ex.5.5 Converting Common Block Diagrams to Signal-Flow Graphs Convert the cascaded, parallel, and feedback forms of the following block diagrams into signal-flow graphs Solution • Start by drawing the signal nodes for that system • Next interconnect the signal nodes with system branches a Cascaded form §4.Signal-Flow Graphs b Parallel form HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Reduction of Multiple Subsystems §3.Analysis and Design of Feedback Systems Skill-Assessment Ex.5.2 Problem For a unity feedback control system with a forward-path TF 𝐺 𝑠 = 16/𝑠(𝑠 + 𝑎), design the value of 𝑎 to yield a closed-loop step response that has 5% overshoot Solution The closed-loop transfer function 𝐺(𝑠) 16 42 𝑇𝑠 = = = + 𝐺(𝑠)𝐻(𝑠) 𝑠2 + 𝑎𝑠 + 16 𝑠2 + × 𝑎 × 4𝑠 + 42 and 𝜔𝑛 = 4, 𝜁 = 𝑎/8 Percent overshoot Nguyen Tan Tien Reduction of Multiple Subsystems Nguyen Tan Tien 2/3/2016 System Dynamics and Control 5.31 Reduction of Multiple Subsystems §4.Signal-Flow Graphs c Feedback form System Dynamics and Control 5.32 Reduction of Multiple Subsystems §4.Signal-Flow Graphs - Ex.5.6 Converting a Block Diagram to a Signal-Flow Graph Convert the block diagram to a signal-flow graph Solution Signal nodes HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.33 Nguyen Tan Tien Reduction of Multiple Subsystems §4.Signal-Flow Graphs System Dynamics and Control 5.34 Nguyen Tan Tien Reduction of Multiple Subsystems §4.Signal-Flow Graphs Signal-flow graph Simplified signal-flow graph HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control HCM City Univ of Technology, Faculty of Mechanical Engineering 5.35 Nguyen Tan Tien Reduction of Multiple Subsystems §4.Signal-Flow Graphs Skill-Assessment Ex.5.3 Problem Convert the block diagram to a signal-flow graph Solution Label nodes HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.36 Nguyen Tan Tien Reduction of Multiple Subsystems §4.Signal-Flow Graphs Draw nodes Nguyen Tan Tien HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 2/3/2016 System Dynamics and Control 5.37 Reduction of Multiple Subsystems §4.Signal-Flow Graphs System Dynamics and Control 5.38 Reduction of Multiple Subsystems §4.Signal-Flow Graphs Connect nodes and label subsystems Eliminate unnecessary nodes HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.39 Nguyen Tan Tien Reduction of Multiple Subsystems §5.Mason’s Rule System Dynamics and Control 5.40 Nguyen Tan Tien Reduction of Multiple Subsystems §5.Mason’s Rule - Loop gain: the product of branch gains found by traversing a path that starts at a node and ends at the same node, following the direction of the signal flow, without passing through any other node more than once Ex 𝐺2 (𝑠)𝐻1 (𝑠) 𝐺4 (𝑠)𝐻2 (𝑠) 𝐺4 (𝑠)𝐺5 (𝑠)𝐻3 (𝑠) 𝐺4 (𝑠)𝐺6 (𝑠)𝐻3 (𝑠) HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control HCM City Univ of Technology, Faculty of Mechanical Engineering 5.41 Nguyen Tan Tien Reduction of Multiple Subsystems §5.Mason’s Rule HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.42 Nguyen Tan Tien Reduction of Multiple Subsystems §5.Mason’s Rule - Nontouching loops: loops that not have any nodes in common Ex Loop 𝐺2 (𝑠)𝐻1 (𝑠) does not touch loops 𝐺4 (𝑠)𝐻2 (𝑠) , 𝐺4 (𝑠)𝐺5 (𝑠)𝐻3 (𝑠), and 𝐺4 (𝑠)𝐺6 (𝑠)𝐻3 (𝑠) HCM City Univ of Technology, Faculty of Mechanical Engineering - Forward-path gain: the product of gains found by traversing a path from the input node to the output node of the signal-flow graph in the direction of signal flow Ex 𝐺1 (𝑠)𝐺2 (𝑠)𝐺3 (𝑠)𝐺4 (𝑠)𝐺5 (𝑠)𝐺7 (𝑠) 𝐺1 (𝑠)𝐺2 (𝑠)𝐺3 (𝑠)𝐺4 (𝑠)𝐺6 (𝑠)𝐺7 (𝑠) Nguyen Tan Tien - Nontouching-loop gain: the product of loop gains from nontouching loops taken two, three, four, or more at a time Ex The product of loop gain 𝐺2 (𝑠)𝐻1 (𝑠) and loop gain 𝐺4 (𝑠)𝐻2 (𝑠) is a nontouching-loop gain taken two at a time In summary, all three of the nontouching-loop gains taken two at a time [𝐺2 𝑠 𝐻1 𝑠 ][𝐺4 𝑠 𝐻2 𝑠 ] [𝐺2 𝑠 𝐻1 𝑠 ][𝐺4 𝑠 𝐺5 𝑠 𝐻3 𝑠 ] [𝐺2 𝑠 𝐻1 𝑠 ][𝐺4 𝑠 𝐺6 𝑠 𝐻3 𝑠 ] HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 2/3/2016 System Dynamics and Control 5.43 Reduction of Multiple Subsystems System Dynamics and Control 5.44 Reduction of Multiple Subsystems §5.Mason’s Rule - Mason’s Rule The transfer function, 𝐶(𝑠)/𝑅(𝑠), of a system represented by a signal-flow graph is 𝐶(𝑠) 𝑘 𝑇𝑘 ∆𝑘 𝐺 𝑠 = = 𝑅(𝑠) ∆ 𝑘 : number of forward paths 𝑇𝑘 : the 𝑘th forward-path gain ∆ : − loop gains + nontouching-loop gains taken two at a time − nontouching-loop gains taken three at a time + nontouching-loop gains taken four at a time −⋯ ∆𝑘 : ∆ − loop gain terms in ∆ that touch the 𝑘th forward path In other words, ∆𝑘 is formed by eliminating from ∆ those loop gains that touch the 𝑘th forward path §5.Mason’s Rule - Ex.5.7 Transfer Function via Mason’s Rule Find the transfer function, 𝐶(𝑠)/𝑅(𝑠), for the signal-flow graph HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.45 Nguyen Tan Tien Reduction of Multiple Subsystems §5.Mason’s Rule System Dynamics and Control 5.46 Nguyen Tan Tien Reduction of Multiple Subsystems §5.Mason’s Rule Second, identify the loop gains 𝐺2 𝑠 𝐻1 𝑠 𝐺4 𝑠 𝐻2 𝑠 𝐺7 (𝑠)𝐻4 (𝑠) 𝐺2 (𝑠)𝐺3 (𝑠)𝐺4 (𝑠)𝐺5 (𝑠)𝐺6 (𝑠)𝐺7 (𝑠)𝐺8 (𝑠) HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control Solution First, identify the forward-path gains 𝐺1 (𝑠)𝐺2 (𝑠)𝐺3 (𝑠)𝐺4 (𝑠)𝐺5 (𝑠) 5.47 Third, identify the nontouching loops taken two at a time • loop does not touch loop 2: 𝐺2 𝑠 𝐻1 𝑠 𝐺4 𝑠 𝐻2 𝑠 • loop does not touch loop 3: 𝐺2 𝑠 𝐻1 𝑠 𝐺7 𝑠 𝐻4 𝑠 • loop does not touch loop 3: 𝐺4 𝑠 𝐻2 𝑠 𝐺7 𝑠 𝐻4 𝑠 Finally, the nontouching loops taken three at a time • loops 1,2 and 3: 𝐺2 𝑠 𝐻1 𝑠 𝐺4 𝑠 𝐻2 𝑠 𝐺7 𝑠 𝐻4 𝑠 Nguyen Tan Tien Reduction of Multiple Subsystems §5.Mason’s Rule System Dynamics and Control 5.48 Nguyen Tan Tien Reduction of Multiple Subsystems §5.Mason’s Rule Form ∆ ∆ = − [𝐺2 𝑠 𝐻1 𝑠 + 𝐺4 𝑠 𝐻2 𝑠 + 𝐺7 𝑠 𝐻4 𝑠 +𝐺2 𝑠 𝐺3 𝑠 𝐺4 𝑠 𝐺5 𝑠 𝐺6 𝑠 𝐺7 𝑠 𝐺8 𝑠 ] + [𝐺2 𝑠 𝐻1 𝑠 𝐺4 𝑠 𝐻2 𝑠 + 𝐺2 𝑠 𝐻1 𝑠 𝐺7 𝑠 𝐻4 𝑠 +𝐺4 𝑠 𝐻2 𝑠 𝐺7 𝑠 𝐻4 𝑠 ] − [𝐺2 𝑠 𝐻1 𝑠 𝐺4 𝑠 𝐻2 𝑠 𝐺7 𝑠 𝐻4 𝑠 ] HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Form ∆𝑘 by eliminating from ∆ the loop gains that touch the 𝑘th forward path ∆1 = − 𝐺7 𝑠 𝐻4 𝑠 The transfer function 𝑇1∆1 𝐺1 𝑠 𝐺2 𝑠 𝐺3 𝑠 𝐺4 𝑠 𝐺5 𝑠 [1 − 𝐺7 𝑠 𝐻4 𝑠 ] 𝐺 𝑠 = = ∆ ∆ HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 2/3/2016 System Dynamics and Control 5.49 Reduction of Multiple Subsystems §5.Mason’s Rule Skill-Assessment Ex.5.4 Problem Use Mason’s rule to find the transfer function of the signal-flow diagram System Dynamics and Control HCM City Univ of Technology, Faculty of Mechanical Engineering 5.51 Nguyen Tan Tien Reduction of Multiple Subsystems §5.Mason’s Rule Form ∆ ∆= + 𝐺1𝐺2𝐻1 + 𝐺2𝐻2 + 𝐺3𝐻3 + 𝐺1𝐺2𝐺3𝐻1𝐻3 + 𝐺2𝐺3𝐻2𝐻3 Form ∆𝑘 ∆1 = ∆2 = The transfer function 𝐶 𝑠 𝐺1𝐺3[1 + 𝐺2] 𝑘 𝑇𝑘∆𝑘 𝑇𝑠 = = = 𝑅𝑠 ∆ + 𝐺2𝐻2 + 𝐺1𝐺2𝐻1 [1 + 𝐺3𝐻3] HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control Reduction of Multiple Subsystems Loop gains • −𝐺1 𝐺2 𝐻1 • −𝐺2 𝐻2 • −𝐺3 𝐻3 Nontouching loops • −𝐺1 𝐺2 𝐻1 −𝐺3 𝐻3 = 𝐺1 𝐺2 𝐺3 𝐻1 𝐻3 • −𝐺2 𝐻2 −𝐺3 𝐻3 = 𝐺2 𝐺3 𝐻2 𝐻3 Solution Forward path gains • 𝐺1 𝐺2 𝐺3 • 𝐺1 𝐺3 System Dynamics and Control 5.50 §5.Mason’s Rule 5.53 Nguyen Tan Tien Reduction of Multiple Subsystems §6.Signal-Flow Graphs of State Equations - Then, feed to each node the indicated signals • 𝑠𝑋1 (𝑠) HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.52 Nguyen Tan Tien Reduction of Multiple Subsystems §6.Signal-Flow Graphs of State Equations - Consider the following state and output equations 𝑥1 = 2𝑥1 − 5𝑥2 + 3𝑥3 + 2𝑟 𝑥2 = −6𝑥1 − 2𝑥2 + 2𝑥3 + 5𝑟 𝑥3 = 𝑥1 − 3𝑥2 − 4𝑥3 + 7𝑟 𝑦 = −4𝑥1 + 6𝑥2 + 9𝑥3 - First, identify state variables, 𝑥1 , 𝑥2 , and 𝑥3 ; nodes, the input, 𝑟, and the output, 𝑦 - Next interconnect the state variables and their derivatives with the defining integration, 1/𝑠 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.54 Nguyen Tan Tien Reduction of Multiple Subsystems Đ6.Signal-Flow Graphs of State Equations ã 𝑠𝑋3 (𝑠) • 𝑠𝑋2 (𝑠) 𝑥3 = 𝑥1 − 3𝑥2 − 4𝑥3 + 7𝑟 𝑥1 = 2𝑥1 − 5𝑥2 + 3𝑥3 + 2𝑟, 𝑥2 = −6𝑥1 − 2𝑥2 + 2𝑥3 + 5𝑟 HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 2/3/2016 System Dynamics and Control 5.55 Reduction of Multiple Subsystems §6.Signal-Flow Graphs of State Equations - Finally, the output, 𝑦 5.56 Reduction of Multiple Subsystems §6.Signal-Flow Graphs of State Equations Skill-Assessment Ex.5.5 Problem Draw a signal-flow graph for the following state and output equations −2 0 𝒙 = −3 𝒙 + 𝑟 −3 −4 −5 𝑦= 0𝒙 Solution 𝑦 = −4𝑥1 + 6𝑥2 + 9𝑥3 𝑥1 = −2𝑥1 + 𝑥2 , 𝑥2 = −3𝑥2 + 𝑥3 , 𝑥3 = −3𝑥1 − 4𝑥2 − 5𝑥3 + 𝑟, 𝑦 = 𝑥2 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control System Dynamics and Control 5.57 Nguyen Tan Tien Reduction of Multiple Subsystems §7.Alternative Representations in State Space Cascade Form - Consider the system 𝐶(𝑠) 24 24 = = (5.37) 𝑅(𝑠) 𝑠 + 9𝑠 + 26𝑠 + 24 𝑠 + (𝑠 + 3)(𝑠 + 4) - A block diagram representation of this system formed as cascaded first-order systems HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.58 Nguyen Tan Tien Reduction of Multiple Subsystems §7.Alternative Representations in State Space - Solving for 𝑑𝑐𝑖 (𝑡)/𝑑𝑡 yields 𝑑𝑐𝑖 𝑡 𝑠 + 𝑎𝑖 𝐶𝑖 𝑠 = 𝑅𝑖 𝑠 ⟹ = −𝑎𝑖 𝑐𝑖 𝑡 + 𝑟𝑖 (𝑡) 𝑑𝑡 - Signal-flow graph Note: these state variables are not the phase variables - Transforming each block into an equivalent differential equation and cross-multiplying 𝐶𝑖 (𝑠) (5.39) = ⟹ 𝑠 + 𝑎𝑖 𝐶𝑖 𝑠 = 𝑅𝑖 (𝑠) 𝑅𝑖 (𝑠) 𝑠 + 𝑎𝑖 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.59 Nguyen Tan Tien Reduction of Multiple Subsystems §7.Alternative Representations in State Space System Dynamics and Control 5.60 Nguyen Tan Tien Reduction of Multiple Subsystems §7.Alternative Representations in State Space Parallel Form - Consider the system - The state equations for the new representation of the system 𝑥1 = −4𝑥1 + 𝑥2 𝑥2 = −3𝑥2 + 𝑥3 𝑥3 = −2𝑥3 + 24𝑟 with the system output 𝑦 = 𝑐 𝑡 = 𝑥1 - The state equations in vector-matrix form −4 0 𝒙 = −3 𝒙 + 𝑟 0 −2 24 𝑦= 0𝒙 HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 𝐶(𝑠) 24 12 24 12 = = − + (5.45) 𝑅(𝑠) 𝑠 + 9𝑠 + 26𝑠 + 24 𝑠 + 𝑠 + 𝑠 + - To arrive at a signal-flow graph, first solve for 𝐶(𝑠) 12 𝐶 𝑠 = +𝑅 𝑠 𝑠+2 24 −𝑅 𝑠 𝑠+3 12 +𝑅(𝑠) 𝑠+4 HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 10 2/3/2016 System Dynamics and Control 5.61 Reduction of Multiple Subsystems System Dynamics and Control 5.62 Reduction of Multiple Subsystems §7.Alternative Representations in State Space - The state equations for the new representation of the system 𝑥1 = −2𝑥1 + 12𝑟 𝑥2 = −3𝑥2 − 24𝑟 𝑥3 = −4𝑥3 + 12𝑟 - The output equation is found by summing the signals that give 𝑐(𝑡) 𝑦 = 𝑐 𝑡 = 𝑥1 + 𝑥2 + 𝑥3 - The state equations in vector-matrix form −2 0 12 (5.49) 𝒙 = −3 𝒙 + −24 𝑟 0 −4 12 𝑦= 1 1𝒙 §7.Alternative Representations in State Space Run ch5p3 in Appendix B Learn how to use MATLAB to • use MATLAB to convert a transfer function to state space in a specified form • solve the previous example by representing the transfer function in Eq.(5.45) by the state-space representation in parallel form of Eq.(5.49) HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.63 Nguyen Tan Tien Reduction of Multiple Subsystems System Dynamics and Control 5.64 Nguyen Tan Tien Reduction of Multiple Subsystems §7.Alternative Representations in State Space - If the denominator of the TF has repeated real roots 𝐶(𝑠) 𝑠+3 1 = = − + 𝑅(𝑠) 𝑠 + (𝑠 + 2) 𝑠+1 𝑠+1 𝑠+2 Proceeding as before, the signal-flow graph - The state equations 𝑥1 = −𝑥1 + 𝑥2 𝑥2 = −𝑥2 − 2𝑟 𝑥3 = −2𝑥3 + 12𝑟 𝑦 = 𝑐 𝑡 = 𝑥1 − 0.5𝑥2 + 𝑥3 or, in vector-matrix form −2 0 12 𝒙 = −3 𝒙 + −24 𝑟 0 −4 12 - Note: the system matrix will 𝑦= 1 1𝒙 not be diagonal §7.Alternative Representations in State Space Controller Canonical Form 𝐶(𝑠) 𝑠 + 7𝑠 + (5.55) - Consider the system = 𝑅(𝑠) 𝑠 + 9𝑠 + 26𝑠 + 24 - The phase-variable form 𝑥1 𝑥1 𝑥1 𝑥2 = 0 𝑥2 + 𝑟, 𝑦 = 𝑥2 (5.56) 𝑥3 𝑥3 −24 −26 −9 𝑥3 HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.65 Nguyen Tan Tien Reduction of Multiple Subsystems §7.Alternative Representations in State Space - Signal-flow graphs for obtaining forms for 𝐶(𝑠) 𝑠 + 7𝑠 + = 𝑅(𝑠) 𝑠 + 9𝑠 + 26𝑠 + 24 - Renumbering the phase variables in reverse order yields 𝑥3 𝑥3 𝑥3 𝑥2 = 0 𝑥2 + 𝑟, 𝑦 = 𝑥2 (5.57) 𝑥1 𝑥1 −24 −26 −9 𝑥1 - Finally, rearranging in the controller canonical form 𝑥1 𝑥1 −24 −26 −9 𝑥1 𝑥2 = 0 𝑥2 + 𝑟, 𝑦 = 𝑥2 (5.58) 𝑥3 𝑥3 𝑥3 System Dynamics and Control 5.66 Nguyen Tan Tien Reduction of Multiple Subsystems §7.Alternative Representations in State Space 𝐶(𝑠) 𝑠 + 7𝑠 + TryIt 5.3 (5.55) Use the following MATLAB 𝑅(𝑠) = 𝑠 + 9𝑠 + 26𝑠 + 24 and Control System Toolbox 𝑥1 −24 −26 −9 𝑥1 statements to convert the transfer function of Eq 𝑥2 = 0 𝑥2 + 𝑟 (5.58) (5.55) to the controller 𝑥3 𝑥3 canonical state-space 𝑥1 representation of Eqs (5.58) 𝑦 = 𝑥2 𝑥3 numg=[1 2]; deng=[1 26 24]; [Acc,Bcc,Ccc,Dcc]=tf2ss(numg,deng) HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 11 2/3/2016 System Dynamics and Control 5.67 Reduction of Multiple Subsystems §7.Alternative Representations in State Space Observer Canonical Form - Consider the system + + 𝐶(𝑠) 𝑠 + 7𝑠 + 𝑠 𝑠2 𝑠3 = = 𝑅(𝑠) 𝑠 + 9𝑠 + 26𝑠 + 24 + + 26 + 24 𝑠 𝑠2 𝑠3 - Cross-multiplying yields 26 24 + + 𝑅 𝑠 = + + + 𝐶(𝑠) 𝑠 𝑠2 𝑠3 𝑠 𝑠 𝑠 System Dynamics and Control 5.68 Reduction of Multiple Subsystems §7.Alternative Representations in State Space 1 𝐶 = (𝑅 − 9𝐶) + 7𝑅 − 26𝐶 + (2𝑅 − 24𝐶) 𝑠 𝑠 𝑠 (5.62) - Start with three integrations (5.59) - Signal-flow graph for observer canonical form variables (5.60) - Combining terms of like powers of integration gives 1 𝐶 = (𝑅 − 9𝐶) + 7𝑅 − 26𝐶 + (2𝑅 − 24𝐶) (5.62) 𝑠 𝑠 𝑠 This equation can be used to draw the signal-flow graph HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control Nguyen Tan Tien 5.69 Reduction of Multiple Subsystems §7.Alternative Representations in State Space - The state equation 𝑥1 = −9𝑥1 + 𝑥2 +𝑟 𝑥2 = −26𝑥1 + 𝑥3 + 7𝑟 𝑥3 = −24𝑥1 + 2𝑟 𝑦 = 𝑐 𝑡 = 𝑥1 - The state equations in vector-matrix form −9 1 𝒙 = −26 𝒙 + 𝑟 −24 0 𝑦= 0𝒙 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.71 System Dynamics and Control Nguyen Tan Tien Reduction of Multiple Subsystems Solution First, model the forward transfer function in cascade form • The gain of 100, the pole at −2, −3 → in cascaded form • The zero at −5 → obtained using the method for implementing zeros for a system represented in phasevariable form, as discussed in Section 3.5 Nguyen Tan Tien 5.70 Nguyen Tan Tien Reduction of Multiple Subsystems §7.Alternative Representations in State Space 𝐶(𝑠) 𝑠 + 7𝑠 + TryIt 5.4 Use the following MATLAB 𝑅(𝑠) = 𝑠 + 9𝑠 + 26𝑠 + 24 and Control System Toolbox −9 1 statements to convert the transfer function of Eq 𝒙 = −26 𝒙 + 𝑟 (5.55) to the observer −24 0 canonical state space 𝑦= 0𝒙 representation of Eqs (5.65) (5.55) (5.65) numg=[1 2]; deng=[1 26 24]; [Acc,Bcc,Ccc,Dcc]=tf2ss(numg,deng); Aoc=transpose(Acc) Boc=transpose(Ccc) Coc=transpose(Bcc) (5.65) §7.Alternative Representations in State Space - Ex.5.8 State-Space Representation of Feedback Systems Represent the feedback control system in state space Model the forward transfer function in cascade form HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.72 Nguyen Tan Tien Reduction of Multiple Subsystems §7.Alternative Representations in State Space Next add the feedback and input paths by inspection, write the state equations 𝑥1 = −3𝑥1 + 𝑥2 𝑥2 = −2𝑥2 + 100(𝑟 − 𝑐) The output 𝑐 = 5𝑥1 + 𝑥2 − 3𝑥1 = 2𝑥1 + 𝑥2 Then 𝑥1 = −3𝑥1 + 𝑥2 𝑥2 = −200𝑥1 − 102𝑥2 + 100 HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 12 2/3/2016 System Dynamics and Control 5.73 Reduction of Multiple Subsystems §7.Alternative Representations in State Space Then 𝑥1 = −3𝑥1 + 𝑥2 𝑥2 = −200𝑥1 − 102𝑥2 + 100 𝑐 = 2𝑥1 + 𝑥2 System Dynamics and Control 5.74 Reduction of Multiple Subsystems §7.Alternative Representations in State Space Skill-Assessment Ex.5.6 Problem Represent the feedback control system in state space Model the forward transfer function in controller canonical form Solution Draw the signal-flow graph in controller canonical form and add the feedback In vector-matrix form −3 𝒙= 𝒙+ 𝑟 −200 −102 100 𝑦= 1𝒙 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.75 Nguyen Tan Tien Reduction of Multiple Subsystems HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.76 Nguyen Tan Tien Reduction of Multiple Subsystems §7.Alternative Representations in State Space Writing the state equations from the signal-flow diagram −105 −506 𝒙= 𝒙+ 𝑟 𝑦 = 100 500 𝒙 §7.Alternative Representations in State Space - Writing the state equations from the signal-flow diagram 𝐶(𝑠) 𝑠+3 = 𝑅(𝑠) (𝑠 + 4)(𝑠 + 6) HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.77 Nguyen Tan Tien Reduction of Multiple Subsystems §7.Alternative Representations in State Space System Dynamics and Control 5.78 Nguyen Tan Tien Reduction of Multiple Subsystems §8.Similarity Transformations - A system represented in state space as 𝒙 = 𝑨𝒙 + 𝑩𝒖 𝒚 = 𝑪𝒙 + 𝑫𝒖 can be transformed to a similar system 𝒛 = 𝑷−1 𝑨𝑷𝒛 + 𝑷−1 𝑩𝒖 𝒚 = 𝑪𝑷𝑧 + 𝑫𝒖 where, for 2-shape 𝑝11 𝑝12 𝑷 = 𝑼𝒛1 𝑼𝒛1 = 𝑝 21 𝑝22 𝑝11 𝑝12 𝑧1 𝒙= 𝑝 = 𝑷𝒛 21 𝑝22 𝑧2 and 𝒛 = 𝑷−1 𝒙 HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 13 2/3/2016 System Dynamics and Control 5.79 Reduction of Multiple Subsystems §8.Similarity Transformations - Ex.5.9 Similarity Transformations on State Equations Given the system represented in state space 0 𝒙= 0 𝒙+ 𝑢 −2 −5 −7 𝑦= 0𝒙 transform the system to a new set of state variables, 𝑧, where the new state variables are related to the original state variables, 𝑥, as follows 𝑧1 = 2𝑥1 𝑧2 = 3𝑥1 + 2𝑥2 𝑧3 = 𝑥1 + 4𝑥2 + 5𝑥3 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.81 Nguyen Tan Tien Reduction of Multiple Subsystems §8.Similarity Transformations −1.5 𝑷−1 𝑨𝑷 = −1.25 0.7 0.4 −2.5 0.4 −6.2 𝑷−1 𝑩 = 𝑪𝑷 = 0.5 0 The transformed system is −1.5 𝒛 = −1.25 0.7 0.4 𝒛 + 𝑢 −2.5 0.4 −6.2 𝑦 = 0.5 0 𝒛 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.83 5.80 §8.Similarity Transformations Solution 𝑧1 = 2𝑥1 𝑧2 = 3𝑥1 + 2𝑥2 ⟹𝒛= 𝑧3 = 𝑥1 + 4𝑥2 + 5𝑥3 0 𝑷−1 𝑨𝑷 = 0 −2 −5 −1.5 = −1.25 0.7 0.4 −2.5 0.4 −6.2 0 0 −1 𝑷 𝑩= 0 = 5 0.5 𝑪𝑷 = 0 −0.75 0.5 0.5 −0.4 Reduction of Multiple Subsystems 0 𝒙 = 𝑷−1 𝒙 0.5 0 −0.75 0.5 −7 0.5 −0.4 0.2 0 = 0.5 0 0.2 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.82 Nguyen Tan Tien Reduction of Multiple Subsystems §8.Similarity Transformations Run ch5p4 in Appendix B Learn how to use MATLAB to • perform similarity transformations • Ex.5.9 Nguyen Tan Tien Reduction of Multiple Subsystems §8.Similarity Transformations Diagonalizing a System Matrix - The parallel form of a signal-flow graph can yield a diagonal system matrix - Advantage: each state equation is a function of only one state variable ⟹ each differential equation can be solved independently of the other equations (the equations are decoupled) Example −2 0 12 𝒙 = −3 𝒙 + −24 𝑟 0 −4 12 𝑦= 1 1𝒙 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control Nguyen Tan Tien HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.84 Nguyen Tan Tien Reduction of Multiple Subsystems §8.Similarity Transformations Diagonalizing a System Matrix - Eigenvector The eigenvectors of the matrix 𝐴 are all vectors, 𝒙𝑖 ≠ 𝟎, which under the transformation 𝐴 become multiples of themselves; that is, 𝑨𝒙𝑖 = 𝜆𝑖 𝒙𝑖 , 𝜆𝑖 : constant (5.80) • If 𝑨𝒙 is not collinear with 𝒙 after the transformation, 𝒙 is not an eigenvector • If 𝑨𝒙 is collinear with 𝒙 after the transformation, 𝒙 is an eigenvector HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 14 2/3/2016 System Dynamics and Control 5.85 Reduction of Multiple Subsystems System Dynamics and Control 5.86 Reduction of Multiple Subsystems §8.Similarity Transformations - Eigenvalue The eigenvalues of the matrix 𝑨 are the values of 𝜆𝑖 that satisfy 𝑨𝒙𝑖 = 𝜆𝑖 𝒙𝑖 , 𝜆𝑖 : constant (5.80) for 𝒙𝑖 ≠ 𝟎 - To find the eigenvectors, rearrange Eq (5.80) Eigenvectors, 𝜆𝑖 , satisfy 𝟎 = (𝜆𝑖 𝑰 − 𝑨)𝒙𝑖 (5.81) adj(𝜆𝑖 𝑰 − 𝑨) 𝒙𝑖 = (𝜆𝑖 𝑰 − 𝑨)−1 𝟎 = 𝟎 det(𝜆𝑖 𝑰 − 𝑨) Since 𝒙𝑖 ≠ 𝟎, a nonzero solution exists if det 𝜆𝑖 𝑰 − 𝑨 = 𝟎 (5.83) From which 𝜆𝑖 , the eigenvalues, can be found §8.Similarity Transformations - Ex.5.10 Finding Eigenvectors Find the eigenvectors of the matrix −3 𝑨= −3 Solution The eigenvectors, 𝒙𝑖 , satisfy Eq (5.81) First, use det(𝜆𝑖 𝑰 − HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering 𝟎 = (𝜆𝑖 𝑰 − 𝑨)𝒙𝑖 System Dynamics and Control 5.87 Nguyen Tan Tien Reduction of Multiple Subsystems §8.Similarity Transformations Using Eq (5.80) successively with each eigenvalue, we have 𝑨𝒙𝑖 = 𝜆𝑖 𝒙𝑖 Using eigenvalue 𝜆 = −2 𝑥1 −3 𝑥1 = −2 𝑥 −3 𝑥2 or −3𝑥1 + 𝑥2 = −2𝑥1 𝑥1 − 3𝑥2 = −2𝑥2 𝑐 From which 𝑥1 = 𝑥2 Thus 𝒙 = 𝑐 𝑐 Using eigenvalue 𝜆 = −4, 𝒙 = −𝑐 1 One choice of eigenvectors is 𝒙1 = and 𝒙2 = −1 𝑨𝒙𝑖 = 𝜆𝑖 𝒙𝑖 , 𝜆𝑖 : constant 5.88 Reduction of Multiple Subsystems §8.Similarity Transformations Run ch5p5 in Appendix B Learn how to use MATLAB to diagonalize a system, is similar (but not identical) to Ex.5.11 (5.80) HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control System Dynamics and Control (5.81) Nguyen Tan Tien 5.89 Nguyen Tan Tien Reduction of Multiple Subsystems HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.90 §8.Similarity Transformations Skill-Assessment Ex.5.7 Problem For the system represented in state space as follows 𝒙= 𝒙+ 𝑢, 𝑦 = 𝒙 −4 −6 convert the system to one where the new state vector −2 𝒛= 𝒙 −4 0.4 −0.2 −2 −1 Solution 𝑷 = ⟹𝑷= −4 0.1 −0.3 0.4 −0.2 −2 6.5 −8.5 𝑷−1𝑨𝑷 = = −4 −4 −6 0.1 −0.3 9.5 −11.5 −2 −3 𝑷−𝟏 𝑩 = = −4 −11 0.4 −0.2 𝑪𝑷 = = 0.8 −1.4 0.1 −0.3 §8.Similarity Transformations 6.5 −8.5 𝑷−1 𝑨𝑷 = 9.5 −11.5 −3 𝑷−𝟏 𝑩 = −11 𝑪𝑷 = 0.8 −1.4 The transformed system is 6.5 −8.5 −3 𝒛= 𝒛+ 𝑢 −11 9.5 −11.5 𝑦 = 0.8 −1.4 𝒛 HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Nguyen Tan Tien Reduction of Multiple Subsystems Nguyen Tan Tien 15 2/3/2016 System Dynamics and Control 5.91 Reduction of Multiple Subsystems System Dynamics and Control 5.92 Reduction of Multiple Subsystems §8.Similarity Transformations Skill-Assessment Ex.5.8 Problem For the system represented in state space as follows 𝒙= 𝒙+ 𝑢, 𝑦 = 𝒙 −4 −6 find the diagonal system that is similar Solution First find the eigenvalues 𝜆 𝜆 − −3 𝜆𝑖 𝑰 − 𝑨 = − = −4 −6 𝜆+6 𝜆 = 𝜆 + 5𝜆 + = (𝜆 + 2)(𝜆 + 3) From which the eigenvalues are −2 and −3 Now use 𝑨𝒙𝑖 = 𝜆𝒙𝑖 for each eigenvalue,𝜆 Thus, 𝑥1 𝑥1 =𝜆 𝑥 −4 −6 𝑥2 §8.Similarity Transformations 𝑥1 𝑥1 =𝜆 𝑥 −4 −6 𝑥2 For 𝜆 = −2 3𝑥1 + 3𝑥2 = −4𝑥1 − 4𝑥2 = ⟹ 𝑥1 = −𝑥2 For 𝜆 = −3 4𝑥1 + 3𝑥2 = −4𝑥1 − 3𝑥2 = ⟹ 𝑥1 = −0.75𝑥2 Let 0.707 −0.6 5.6577 4.2433 𝑷= ⟹ 𝑷−1 = −0.707 0.8 5 HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.93 Nguyen Tan Tien Reduction of Multiple Subsystems System Dynamics and Control 5.94 Nguyen Tan Tien Reduction of Multiple Subsystems §8.Similarity Transformations 0.707 −0.6 5.6577 4.2433 𝑷= ⟹ 𝑷−1 = −0.707 0.8 5 Hence −1 𝑫 = 𝑷 𝑨𝑷 0.707 −0.6 5.6577 4.2433 = −4 −6 −0.707 0.8 5 −2 = −3 18.39 5.6577 4.2433 𝑷−𝟏 𝑩 = = 20 5 0.707 −0.6 𝑪𝑷 = = −2.121 2.6 −0.707 0.8 The transformed system is −2 18.39 𝒛= 𝒛+ 𝑢 −3 20 𝑦 = −2.121 2.6 𝒛 §7.Alternative Representations in State Space TryIt 5.5 Use the following MATLAB and Control System Toolbox 𝒙 = 𝒙+ 𝑢, 𝑦 = 𝒙 −4 −6 statements to Skill- HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Assessment Ex.5.8 A=[1 3;-4 -6]; B=[1;3]; C=[1 4]; D=0;S=ss(A,B,C,D); Sd=canon(S, 'modal') Nguyen Tan Tien 16 ... Tien Reduction of Multiple Subsystems §2.Block Diagrams Solution HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics and Control 5.12 Nguyen Tan Tien Reduction of Multiple. .. Dynamics and Control 5.13 Reduction of Multiple Subsystems §2.Block Diagrams 5.14 Reduction of Multiple Subsystems §2.Block Diagrams HCM City Univ of Technology, Faculty of Mechanical Engineering System... 5.29 Nguyen Tan Tien Reduction of Multiple Subsystems Nguyen Tan Tien Reduction of Multiple Subsystems - A system is represented by a line with an arrow showing the direction of signal flow through