Chapter 7 State-Variable TechniqueHomeworkWhy numerical calculation possible? Convergence ofExample 7-1 Easy exampleExample 7-2 Not easy example. I can not do it!Step 1: Select each iL and vc as state variablesStep 2: For each iL , write a KVL ( will be included)For each vc , write a KCL ( will be included)Step 1: Label vc as x1Step 2: For vc , write KCL :For iL , write KVLFrom transfer Function => state-equationEE 422G Notes: Chapter 7 Instructor: ZhangChapter 7 State-Variable Technique7-1 Introduction What is it about? Another method for system description and analysis. What we already have? (Why introduced Laplace Transform? to avoid the complexity of solving high-order differential equations!)Is Laplace Transform method good enough or do we need more or other techniques?(1) How effective Laplace Transform is in solving high-order differential equations with non-zero initial conditions?(2) How effective Laplace Transform is in handing multivariable (multi-input multi-output) system?(3) Classical control theory: Laplace Transform based Modern control theory: state equation based Many modern control system design methods require state equation description of system to apply!Characteristics of state-variable technique (state equation description based):(1) Uniform structure (form) for all linear-systems: despite the order, the numbers of inputs and outputs, and forms of the input functions.Always : BuAxx state equation DuCxy output equation x : state vector, x = (x1, … xn)T xjs : state variable u = (u1, …, um)T : input vector ujs : inputs y = (y1,…yp)T : output vector yjs : outputsPage 7-1EE 422G Notes: Chapter 7 Instructor: Zhang A : nxn B: nxm C: pxn D: pxmDifference: dimensions of the vectors and matrices. (2) Uniform method of solution: Laplace: different equations, different input function different solution! State: Different solution method for different inputs? No difference ! Different solution method for different systems (with different orders)? No! The same! uniform form, uniform methods for analysis and design!Fundamental CharacteristicLaplace Transform, differential equation, …, convolution: external inputs  external outputs!State equation (state-variable technique):External inputs  internal state variables (as a bridge)  external outputs help to understand the system better because of use of “internal state”! What’s the state of a system?7-2 State-Variable Concepts(1) Example: University’s rank (simplified and idealized)What’s the fundamental measurement of university system (dynamic system)?Money? Very important, not a direct measurement of university with great inertia! What are the direct measurement which need long time to build and change?Faculty qualityStudent qualityGood faculty => Research , Teaching  Reputation , Good student Good students => student Reputation Graduate quality Good Faculty Faculty quality and student quality:Page 7-2EE 422G Notes: Chapter 7 Instructor: ZhangState of university (dynamic system)State equations: UK’s ranking: output sfxcxc21 ------- output equation (xf Faculty quality , xs: student quality) State equation: Change of faculty quality: salaryaxaxaxsff131211 Change of student quality: BashetballbpScholorshibxbxbxsfs24232221At 2000: )2000(),2000(sfxx: initial stateAt and after 2000: (state appreciation, private donations) => (salary, scholarship)Administration, basketball: future inputs => UK’s ranking (determined by initial state and future inputs)(2) State, state-variables, state-space, trajectory:State of a system at 0t: includes the minimum information necessary to specifycompletely the condition of the system at t0 and allow determination of all system outputs at t>t0 when inputs up to time t are specified. State : a set of state variables State:sfxxx (state vector) State Space: Set of all possible sfxxx (All possible paired values ));...)2(),2(());1(),1((sfsfxxxx) Trajectory of the state )2000()2000(sfxxx)2001()2001(sfxxx… … a curve in two-dimensional space. Page 7-3EE 422G Notes: Chapter 7 Instructor: Zhang In general nxxx 17-3 Form of the state equations1. FormExample :BuAxxububxaxaxububxaxax22212122212122121112121111DuCxyududxcxcyududxcxcy22212122212122121112121111 21xxx, 22211211aaaaA, 22211211bbbbB 21yyy, 22211211ccccC, 22211211ddddD 21uuuPage 7-4EE 422G Notes: Chapter 7 Instructor: Zhang In general : nxxx 1, muuu 1, pyyy 1, equationoutputducxyequationstateBuAxxnnA :mnB :npC :mcD : (1: nx, 1: mu , 1: py)(2) Simulation example 2211221122212122212122121112121111ududxcxcyububxaxaxububxaxaxPage 7-5EE 422G Notes: Chapter 7 Instructor: Zhang (3) Block Diagram of state equationducxyBuAxxHomework1. Given )()()( tButAxtx  with initial condition 00)( xtx  where x0 is the given initial state. Construct an