State Space

What is the most effective way of representing and solving complex systems in a way that is software programming friendly?

State-space models describe the behaviour of dynamic systems as a set of first order differential equations (ODEs). They solve many of the limitations of classical control theory that uses transfer functions: they replace an nth order differential equation with a single first order matrix differential equation. This is very useful for computer simulation. Hence the number of state variables is equal with the order of ODE describing the system; and the number of state variables is also equal with the number of conditions needed to completely solve the system.

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The first equation (containing A & B) is the state equation

The second equation (containing C & D) is the output equation

X is the state vector

U is the input vector

A is the state matrix

B is the input matrix

Y is the output vector

C is the output matrix

D is the direct transmission

Example 1: translational mass with spring and damper (zero friction) [1]

To obtain the State Space equations we follow this process:

Example 2: translational mass with two springs (zero friction) [2]

The choice of state variables is not unique. We can choose.

There are 3 energy storage elements so we expect 3 state equations.

For convenience we choose:

x: because spring2 energy is 1/2*k2*x^2

x_dot: because mass energy is 1/2*m*v^2

z: because spring1 energy is 1/2*k1*(z-x)^2

Example 3: mass with spring and damper

m*z_dot_dot + c*z_dot + k*z = F

z_dot_dot + c/m*z_dot + k/m*z = F

x1 = z

x2 = z_dot

x1_dot = z_dot = x2

x2_dot = z_dot_dot = 1/m*f - k/m*x1 - c/m*x2

therefore:

x1_dot = x2

x2_dot = 1/m*f - k/m*x1 - c/m*x2

Matrix: X_dot = Ax + Bu

| x1_dot | = | 0 1 | | x1 | + | 0 |*F

| x2_dot | = | -k/m -c/m | | x2 | | 1/m |

y = C*X

where C*X is the output we want

For displacement output:

y = | 1 0 | * | x1 |

| x2 |

For velocity output:

y = | 0 1 | * | x1 |

| x2 |

Example 4: two rotating masses [2]

The choice of state variables is not unique. We can choose.

There are 3 energy storage elements so we expect 3 state equations.

For convenience we choose:

theta_1: because spring2 energy is 1/2*kr*theta_1^2

theta_1_dot: because mass1 energy is 1/2*J1*alpha_1^2

theta_2_dot: because mass2 energy is 1/2*J2*alpha_2^2

References:

[1] x-engineer.org

[2] lpsa.swarthmore.edu

[3]

A useful video