State-Space Models Date: 24 th July 2008 Prepared by: Megat Syahirul Amin bin Megat Ali

 Introduction  State-Space Model  Signal Flow Graph

 The basic questions that will be addressed in state-space approach include: i. What are state-space models? ii. Why should we use them? iii. How are they related to the transfer function used in classical control system? iv. How do we develop a space-state model?

A representation of the dynamics of N th -order system as a first-order equation in an N-vector, which is called the state. Convert the Nth-order differential equation that governs the dynamics of the system into N first- order differential equation.

 The state of a system is described by a set of first-order differential equations written in terms of the state variable.

 In a matrix form, we have:  State vector:

Input equation Output equation

 Example: Springer-mass-damper system  The 2 first-order equations are: Therefore, we define variable x 1 and x 2. Dynamic equation of the system:

 Example: Springer-mass-damper system If the measured output of the system is position, then we have: In matrix form: General State-Space Model:

 Example: Simple mechanical system The we will obtain: Dynamic equation of the system: Let us define variable x 1 and x 2.

 Example: Simple mechanical system In vector form: The output equation:

 Problem: Find the space-state for the following mechanical system.

 Problem: Find the space-state for the following RLC circuit.

 A signal flow graph is a graphical representation of the relationships between the variables of a set of linear algebraic equations.  The basic element of a signal flow graph is a unidirectional path segments called branch.  The input and output points or junctions are called nodes.  A path is a branch or continuous sequence or branches that can be traversed from one signal node to another signal node.  A loop is a closed path that originates and terminates on the same node, and along the path no node is met twice.  Two loops are said to be non-touching if they do not have a same common node.

 Signal flow graph of control systems

 Mason’s Gain Formula for Signal Flow Graph Where, P ijk : k th path from variable x i to x j ∆: Determinant of the graph ∆ ijk : Cofactor of the path P ijk

 Example: Transfer function of interacting system

The paths connecting input R(s) to output Y(s) are: P 1 = G 1 G 2 G 3 G 4 P 2 = G 5 G 6 G 7 G 8 There are four individual loops: L 1 = G 1 H 1 L 2 = G 2 H 2 L 3 = G 3 H 3 L 4 = G 4 H 4

 Example: Transfer function of interacting system Loops L 1 and L 2 does not touch loops L 3 and L 4. Therefore, the determinant is: The cofactor of the determinant along path 1 is evaluated by removing the loops that touch path 1 from ∆. Therefore have: and, Similarly, the cofactor for path 2 is:

 Example: Transfer function of interacting system Therefore, the transfer function of the system is:

 Problem: Obtain the closed-loop transfer function by use of Mason’s Gain Formula

 Chapter 3 i. Dorf R.C., Bishop R.H. (2001). Modern Control Systems (9th Ed), Prentice Hall. ii. Nise N.S. (2004). Control System Engineering (4th Ed), John Wiley & Sons.  Chapter 5 i. Nise N.S. (2004). Control System Engineering (4th Ed), John Wiley & Sons.

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