State-Space Models DNT 354 - CONTROL PRINCIPLE Date: 24th July 2008 Prepared by: Megat Syahirul Amin bin Megat Ali Email: [email protected] CONTENTS Introduction State-Space Model Signal Flow Graph INTRODUCTION 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? STATE-SPACE MODEL A representation of the dynamics of Nth-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 firstorder differential equation. STATE-SPACE MODEL The state of a system is described by a set of first-order differential equations written in terms of the state variable. STATE-SPACE MODEL In a matrix form, we have: State vector: STATE SPACE MODEL Input equation Output equation REPRESENTATION OF DYNAMIC SYSTEM Example: Springer-mass-damper system Dynamic equation of the system: d 2 y (t ) dy (t ) M b ky(t ) r (t ) 2 dt dt Therefore, we define variable x1 and x2. x1 y x2 y The 2 first-order equations are: REPRESENTATION OF DYNAMIC SYSTEM Example: Springer-mass-damper system In matrix form: If the measured output of the system is position, then we have: General State-Space Model: REPRESENTATION OF DYNAMIC SYSTEM Example: Simple mechanical system Dynamic equation of the system: d 2 y (t ) dy (t ) M b ky(t ) r (t ) 2 dt dt Let us define variable x1 and x2. x1 y The we will obtain: x2 y REPRESENTATION OF DYNAMIC SYSTEM Example: Simple mechanical system In vector form: The output equation: y x1 REPRESENTATION OF DYNAMIC SYSTEM Problem: Find the space-state for the following mechanical system. REPRESENTATION OF DYNAMIC SYSTEM Problem: Find the space-state for the following RLC circuit. t di(t ) 1 L Ri (t ) i( )d v(t ) dt C0 SIGNAL FLOW GRAPH 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 Signal flow graph of control systems SIGNAL FLOW GRAPH Signal flow graph of control systems SIGNAL FLOW GRAPH Mason’s Gain Formula for Signal Flow Graph Tij P ijk ijk k Where, Pijk ∆ ∆ijk : kth path from variable xi to xj : Determinant of the graph : Cofactor of the path Pijk Δ 1 - (sum of all different loop gains) (sum of the gain products of all combinatio ns of 2 nontouchin g loops) - (sum of the gain products of all combinatio ns of 3 nontouchin g loops) ... SIGNAL FLOW GRAPH MODEL Example: Transfer function of interacting system SIGNAL FLOW GRAPH MODEL Example: Transfer function of interacting system The paths connecting input R(s) to output Y(s) are: P1 = G1G2G3G4 P2 = G5G6G7G8 There are four individual loops: L1 = G1H1 L2 = G2H2 L3 = G3H3 L4 = G4H4 SIGNAL FLOW GRAPH MODEL Example: Transfer function of interacting system Loops L1 and L2 does not touch loops L3 and L4. Therefore, the determinant is: 1 ( L1 L2 L3 L4 ) ( L1L3 L1L4 L2 L3 L2 L4 ) The cofactor of the determinant along path 1 is evaluated by removing the loops that touch path 1 from ∆. Therefore have: L1 L2 0 and, 1 1 ( L3 L4 ) Similarly, the cofactor for path 2 is: 2 1 ( L1 L2 ) SIGNAL FLOW GRAPH MODEL Example: Transfer function of interacting system Therefore, the transfer function of the system is: Y ( s) P P T ( s) 1 1 2 2 R( s ) G1G2G3G4 (1 L3 L4 ) G5G6G7G8 (1 L1 L2 ) 1 L1 L2 L3 L4 L1 L3 L1 L4 L2 L3 L2 L4 SIGNAL FLOW GRAPH MODEL Problem: Obtain the closed-loop transfer function by use of Mason’s Gain Formula FURTHER READING… Chapter 3 i. ii. Dorf R.C., Bishop R.H. (2001). Modern Control Systems (9th Ed), Prentice Hall. 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. “A mystic is someone who wants to understand the universe, but are too lazy to study physics…” THE END…
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