MindMap Gallery modern control theory
This is a mind map about modern control theory, including the state space description of the control system, the state space response analysis of the linear system, the controllability and observability of the linear control system, etc.
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modern control theory
Chapter 1 State Space Description of Control System
1. Establish a state space expression based on the system mechanism
Kirchhoff's voltage and current laws
2. Simulation structure diagram of state space expression
3. Establish state space expressions from the system block diagram
4. Establish state space expressions from differential equations
(1) The input of the differential equation does not contain derivatives
(2) When the input of the differential equation contains derivatives
5. Build state space expressions from transfer functions
(1) No duplicate roots
(2) There are duplicate roots
Transfer function matrix, converting state space expressions into transfer function form
6. Convert the state space expression into a Jordan matrix
1. Array A is in any form
(1) The eigenvalues of array A have no duplicate roots.
(2) The eigenvalues of array A have multiple roots
2. Array A is the standard type
Must be able to control and observe
(1) When there is no double root
(2) When there are double roots
Chapter 2 State Space Response Analysis of Linear Systems
1. Solutions to linear steady homogeneous state equations (free solutions)
2.State transition matrix
definition
nature
Special matrix exponential functions
calculate
3. Solutions to non-homogeneous equations of linear steady systems
Chapter 3 Controllability and Observability of Linear Control Systems
1. Controllability
definition
Determination of controllability
single input system
(1) Judging the controllability of the system from the Jordan standard type
(2) Determine the controllability of the system from A and B
Wux(s) is the transfer function matrix
2. Observability
definition
Determination of observability
(1) Judging the controllability of the system from the Jordan standard type
When the system matrix A has multiple roots, if and only if the first column elements in the output matrix C are not all 0, y(t) always contains all the free components of the system and is completely observable.
(2) Directly judge the observability of the system from the A and C arrays
3. The dual relationship between controllability and observability
system duality
Duality principle
4. Controllable standard form and observable standard form of state space expressions
(1) Controllable standard type I
Convert the linear steady single input system on the left to the form on the right
(2) Controllable standard type II
(3) Energy observation standard type I
(4) Energy observation standard type II
subtopic
5. Structural decomposition of linear systems
(1) Decompose according to controllability
(2) Decompose according to observability
Chapter 5 Synthesis of Linear Stationary Systems
1. Basic structure and characteristics of linear feedback control system
status feedback
State space expression of closed loop system
Output feedback
First derivative feedback from output to state vector x
Controllability and observability of closed-loop systems
2. Pole configuration problem
(1) Use state feedback to configure poles
3. Status Observer
Definition of state observer
Chapter 4 Stability and Lyapunov Method
1. Lyapunov’s first method (indirect method)
Determining the stability of linear systems
system state stability
System output stability
①② must be satisfied at the same time
Zero-pole cancellation: that is, the same zeros and poles are eliminated at the same time
Transfer function matrix, converting state space expressions into transfer function form
Determining the stability of nonlinear systems
Find equilibrium state
linearization
Put x1, 2=0 into the equation on the left
2. Lyapunov’s second method (direct method)
x≠[0,0,0]
3. Application of Lyapunov method in linear systems
Asymptotic Stability Criterion for Linear Steady Continuous Systems
Matrix <0 means negative definite, not less than 0 in the scalar sense.
Non-singular matrix, i.e. determinant ≠0
Non-singular transformation is coordinate transformation