STATE SPACE MODELSWhy State Space ModelsBasicsPARTS OF A STATE SPACE REPRESENTATIONEXAMPLECont…PUTTING INTO VECTOR-MATRIX FORMExplanationEXACT REPRESENTATIONHOW TO INPUT THE STATE SPACE MODEL INTO MATLABExampleOutputSlide 13Slide 14EXTRACTING A, B, C, D MATRICES FROM A STATE SPACE MODELSlide 16Slide 17STEP RESPONSE USING THE STATE SPACE MODELSlide 19STATE SPACE MODELSMATLAB TutorialWhy State Space ModelsThe state space model represents a physical system as n first order differential equations. This form is better suited for computer simulation than an nth order input-output differential equation.BasicsVector matrix format generally is given by: where y is the output equation, and x is the state vectorPARTS OF A STATE SPACE REPRESENTATIONState Variables: a subset of system variables which if known at an initial time t0 along with subsequent inputs are determined for all time t>t0+State Equations: n linearly independent first order differential equations relating the first derivatives of the state variables to functions of the state variables and the inputs.Output equations: algebraic equations relating the state variables to the system outputs.EXAMPLEThe equation gathered from the free body diagram is: mx" + bx' + kx - f(t) = 0Substituting the definitions of the states into the equation results in: mv' + bv + kx - f(t) = 0Solving for v' gives the state equation: v' = (-b/m) v + (-k/m) x + f(t)/mThe desired output is for the position, x, so: y = xCont…Now the derivatives of the state variables are in terms of the state variables, the inputs, and constants.x' = vv' = (-k/m) x + (-b/m) v + f(t)/my = xPUTTING INTO VECTOR-MATRIX FORMOur state vector consists of two variables, x and v so our vector-matrix will be in the form:ExplanationThe first row of A and the first row of B are the coefficients of the first state equation for x'. Likewise the second row of A and the second row of B are the coefficients of the second state equation for v'. C and D are the coefficients of the output equation for y.EXACT REPRESENTATIONHOW TO INPUT THE STATE SPACE MODEL INTO MATLABIn order to enter a state space model into MATLAB, enter the coefficient matrices A, B, C, and D into MATLAB. The syntax for defining a state space model in MATLAB is: statespace = ss(A, B, C, D) where A, B, C, and D are from the standard vector-matrix form of a state space model.ExampleFor the sake of example, lets take m = 2, b = 5, and k = 3.>> m = 2;>> b = 5;>> k = 3;>> A = [ 0 1 ; -k/m -b/m ];>> B = [ 0 ; 1/m ];>> C = [ 1 0 ];>> D = 0;>> statespace_ss = ss(A, B, C, D)OutputThis assigns the state space model under the name statespace_ss and output the following:a = x1 x2 x1 0 1 x2 -1.5 -2.5Cont…b = u1 x1 0 x2 0.5c = x1 x2 y1 1 0Cont…d = u1 y1 0 Continuous-time model.EXTRACTING A, B, C, D MATRICES FROM A STATE SPACE MODELIn order to extract the A, B, C, and D matrices from a previously defined state space model, use MATLAB's ssdata command.[A, B, C, D] = ssdata(statespace) where statespace is the name of the state space system.Example>> [A, B, C, D] = ssdata(statespace_ss)The MATLAB output will be:A =   -2.5000 -0.3750 4.0000 0Cont…B = 0.2500 0 C = 0 0.5000 D = 0STEP RESPONSE USING THE STATE SPACE MODELOnce the state space model is entered into MATLAB it is easy to calculate the response to a step input. To calculate the response to a unit step input, use:step(statespace)where statespace is the name of the state space system.For steps with magnitude other than one, calculate the step response using:step(u * statespace)where u is the magnitude of the step and statespace is the name of the state space