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Outline • • • • • State Variables M. Sami Fadali Professor of Electrical Engineering UNR State variables. State-space representation. Linear state-space equations. Nonlinear state-space equations. Linearization of state-space equations. 1 2 Linear Systems Input-output Description • Single-input-single-output (SISO) The description is valid for a) time-varying systems: parameters are explicit functions of time. b) multi-input-multi-output (MIMO) systems: l input-output differential equations, l = # of outputs. c) nonlinear systems: differential equations include nonlinear terms. ିଵ ିଵ ିଵ ିଵ ିଵ ିଵ ଵ ଵ • Time-dependent coefficients for timevarying system. 3 4 State Variables Definitions To solve the differential equation we need for the period of interest. (1) The system input (2) A set of constant initial conditions. ଶ ିଵ ଶ ିଵ • Minimal set of initial conditions: incomplete knowledge of the set prevents complete solution but additional initial conditions are not needed to obtain the solution. • Initial conditions provide a summary of the history of the system up to the initial time. System State: minimal set of numbers , needed together with the input to uniquely determine the behavior of the system in the interval . = order of the system. State Variables: As increases, the state of the system evolves and each of the becomes a time variable. numbers State Vector: vector of state variables 5 6 Notation Definitions • Column vector bolded • Row vector bolded and transposed -dimensional vector space where represent the coordinate axes State plane: state space for a 2nd order system Phase plane: special case where the state variables are proportional to the derivatives of . ଵ Phase variables: state variables in phase plane. State trajectories: Curves in state space State portrait: plot of state trajectories in the plane (phase portrait for the phase plane). State Space: ் . ଵ ଵ ଶ ் ଶ 7 8 Example 7.1 Example 7.1 • State for equation of motion of a point mass driven by a force = displacement of the point mass. ௧ ௧ Write state-space equations for the springmass-damper system driven by a force = mass = spring constant = damper constant = displacement of the point mass. ௧బ ௧బ 2 system is second order 9 Solution (cont.) Solution • Obtain a complete solution given the force, if 2 initial conditions are known : state of the system at time . • 2 I.C.s system is second order. • State (phase)Variables: ଵ ଶ • State Vector: 10 ଵ ଶ ் ଵ • Two first order differential equations ଵ ଶ ଶ ଵ ଶ ଵ 1.First equation: from definitions of state variables. 2.Second equation: from equation of motion ଶ 11 12 State-space Representation State Equations • Representation for the system described by a differential equation in terms of state and output equations. • Linear Systems: More convenient to write state (output) equations as a single matrix equation. • Set of first order equations governing the state variables obtained from the input-output differential equation and the definitions of the state variables. • In general, n state equations for a nth order system. • The form of the state equations depends on the nature of the system (equations are time-varying for time-varying systems, nonlinear for nonlinear systems, etc.) • State equations for linear time-invariant systems can also be obtained from their transfer functions. 13 Output Equation 14 Solution of State Equations For the spring-mass-damper system • Solve the 1st order differential equations then substitute in ଵ • 2 differential equations + algebraic expression are equivalent to the 2nd order differential equation. • Feedback Control Law 2nd order underdamped system • Algebraic equation expressing the output in terms of the state variables and the input. • Multi-output systems: a scalar output equation is needed to define each output. • Substitute from solution of state equation to obtain output. ଶ 15 ଵ 16 More on the Solution Phase Portrait 1.Solution depends only on initial conditions. 2.Obtain phase portrait using MATLAB command lsim or initial 3.Time is an implicit parameter. 4.Arrows indicate the direction of increasing time. 5.Choice of state variables is not unique. x2 3 2 1 0 -1 -2 -3 -2 x1 -1.5 -1 -0.5 0 0.5 1 1.5 17 ଵ Matrix Form ଶ ଶ ଵ ଵ 18 General Form for Linear Systems Example 7.2: Matrix Form Scalar Eqns. 2 ଵ ଵ ଶ ଶ ଶ ଵ ଶ 19 real vectors. Matrices: state or system matrix input or control matrix output matrix direct transmission or feedforward matrix 20 State Space in MATLAB Linear Vs. Nonlinear State-Space » a = [0, 1; 5, 4];b=[0;1];c=[1,1];d=0; » p = ss(a,b,c,d) % State-space quadruple Example 7.3: The following are examples of state-space equations for linear systems a) 3rd order 2-input-2-output (MIMO) LTI a= x1 x2 x1 0 -5 x1 x2 u1 0 1 y1 x1 1 x2 1 -4 b= . 0.3 15 . x1 01 . 0 x1 11 u1 x 01 . 35 . . 2.2 x 2 0 11 2 u 2 . x 3 10 . 10 . x 3 0.4 2.4 11 c= x2 1 d= u1 y1 0 Continuous-time system. 21 Example 7.3 (b) 2nd order 2-output-1-input (SIMO) linear time-varying ଵ ଶ ଵ ିଷ௧ ଶ ଵ ଵ ଶ ଶ 1. Zero D, constant + and . 2. Time-varying system: are functions of . has entries that 23 x1 y1 1 2 0 1 2 u1 y 0 0 1 x 2 0 1 u 2 2 x 3 22