Exercise: State-space models

Transcription

1 University College of Southeast Norway Exercise: State-space models A state-space model is a structured form or representation of a set of differential equations. Statespace models are very useful in Control theory and design. The differential equations are converted in matrices and vectors, which is the basic elements in MathScript. Assume we have the following differential equations: = a ## + a '# + + a )# x ) + b ## u # + b '# u ' + + b )# u ) This gives on vector form: x ) = a #- + a '- + + a )- x ) + b #- u # + b '- u ' + + b )# u ) x ). = a ## a )# a #- a )- 0 x ). + b ## b )# b #- b )- 1 u # u ' u ) 2 y # y ' y ) 4 = c ## c )# c #- c )- 6 x ). + d ## d )# d #- d )- 8 u # u ' u ) 2 A general linear State-space model may be written on the following general form: MathScript has several functions for creating state-space models: Function Description Example ss Constructs a model in state-space form. You also can use this >A = [1 2; 3 4] >B = [0; 1] function to convert transfer function models to state-space >C = B' form. >ssmodel = ss(a, B, C) Sys_order1 Sys_order2 Constructs the components of a first-order system model based on a gain, time constant, and delay that you specify. You can use this function to create either a state-space model or a transfer function model, depending on the output parameters you specify. Constructs the components of a second-order system model based on a damping ratio and natural frequency you specify. >K = 1; >T = 1; >H = sys_order1(k, T) >dr = 0.5 >wn = 20

2 2 step lsim Example: You can use this function to create either a state-space model or a transfer function model, depending on the output parameters you specify. Creates a step response plot of the system model. You also can use this function to return the step response of the model outputs. If the model is in state-space form, you also can use this function to return the step response of the model states. This function assumes the initial model states are zero. If you do not specify an output, this function creates a plot. Creates the linear simulation plot of a system model. This function calculates the output of a system model when a set of inputs excite the model, using discrete simulation. If you do not specify an output, this function creates a plot. >[A, B, C, D] = sys_order2(wn, dr) >ssmodel = ss(a, B, C, D) >num=[1,1]; >den=[1,-1,3]; >H=tf(num,den); >t=[0:0.01:10]; >step(h,t); >t = [0:0.1:10] >u = sin(0.1*pi*t)' >lsim(sysin, u, t) Given the following state-space model: = u y = 1 0 The MathScript code for implementing the model is: % Creates a state-space model A = [1 2; 3 4]; B = [0; 1]; C = [1, 0]; D = [0]; model = ss(a, B, C, D) Task 1: State-space models Task 1.1 = = 2 + 5u y = Task 1.2

3 3 = + u = x D = x D y # = x D y ' = + 2 Task 1.3 = = 3 +2u # +4u ' y = +u ' Task 2: Mass-spring-damper system Given a mass-spring-damper system: Using Newtons 2. law: The model of the system can be described as: F = ma

4 4 x = 1 m F cx kx Where x position, x speed/velocity, x acceleration c - damping constant, m - mass, k - spring constant, F - force Task 2.1 Assuming the control signal u is equal to the force F and that we only measure the position. Task 2.2 Define the state-space model above using the ss function in MathScript. Apply a step in u and use the step function in MathScript to simulate the result. Start with c = 1, m = 1, k = 1, then explore with other values. Task 2.3 Convert the state-space model defined in the previous task to a transfer function H s = 4(L) 2(L) using MathScript code. Use c = 1, m = 1, k = 1 Do the same using pen and paper and Laplace. Do you get the same answer? Task 3: Block Diagram

5 5 Task 3.1 Find the state-space model from the block diagram. Note! x1 and x2. Task 3.2 Implement the state-space model in MathScript and simulate the system using the step function in MathScript. Set And b = 1, c = 1 a # = 5 a ' = 2 Task 4: State-space model to Transfer functions = 2 = u # +8u ' Task 4.1 y = u # Find the state-space model on the form (pen and paper):

6 6 Task 4.2 Define the state-space model in MathScript and find the step response for the system. Discuss the results. Task 4.3 Find the following transfer functions: H # (s) = y(s) u # (s) H ' (s) = y(s) u ' (s) Additional Resources Here you will find tutorials, additional exercises, etc.