The Stability of Systems with Command Saturation, Command Delay, and State Delay
Abstract
:1. Introduction
- Equating the stability of a system with state delay and input delay (point or distributed) with the stability of a system without delay by using the Arstein transform (and its generalization in case of delay in the state);
- Obtaining the command (input) of the initial system with delay by applying the inverse transform to the equivalent system without delay, in case of input saturation;
- Obtaining theorems on the stability of mono- and multiple-input systems, theorems on instability and the estimation of the stability region for systems with state delay and input delay (point or distributed) and input saturation. The main results are synthesized in twelve new theorems;
- A numerical solution to the transcendental matrix equation using the computational-intelligence PSO algorithm.
2. Systems with Saturation in Command
- 1.
- The open-loop system A is exponentially stable and diagonalizable;
- 2.
- The matrix is exponentially stable and diagonalizable;
- 3.
- The matrices A and BK commute under multiplication.
- 1.
- A and are exponentially stable;
- 2.
- is diagonalizable;
- 3.
- commutes with P, where is the diagonal form of and solve: .
- 1.
- Matrix A is unstable;
- 2.
- Matrix is exponentially stable.
3. Systems with Command Saturation, Saturation Delay, and State Delay
3.1. Systems with Point Delays
3.2. Systems with Distributed Delay
4. Main Results
4.1. Systems with Point Delays and Command Saturation
- 1.
- The matrix A is exponentially stable and diagonalizable;
- 2.
- The matrix is exponentially stable and diagonalizable;
- 3.
- The matrices A and commute under multiplication.
- 1.
- A and are exponentially stable;
- 2.
- is diagonalizable;
- 3.
- commutes with P, where is the diagonal form of , and solves: .
- 1.
- Matrix A is unstable, and all unstable eigenvalues of system (7) are contained in the spectrum of the matrix A;
- 2.
- Matrix is exponentially stable.
4.2. Systems with Distributed Delays and Command Saturation
- 1.
- The matrix A is exponentially stable and diagonalizable;
- 2.
- The matrix is exponentially stable and diagonalizable;
- 3.
- The matrices A and commute under multiplication.
- 1.
- A and are exponentially stable;
- 2.
- is diagonalizable;
- 3.
- commutes with P, where is the diagonal form of , and solves: .
- 1.
- Matrix A is unstable, and all the unstable eigenvalues of the system (27) are contained in the spectrum of the matrix A;
- 2.
- Matrix is exponentially stable.
5. Examples and Discussions
5.1. Example 1
5.2. Example 2
5.3. Example 3
5.4. Example 4
6. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
Correction Statement
Appendix A
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Nicola, M. The Stability of Systems with Command Saturation, Command Delay, and State Delay. Automation 2022, 3, 47-83. https://doi.org/10.3390/automation3010003
Nicola M. The Stability of Systems with Command Saturation, Command Delay, and State Delay. Automation. 2022; 3(1):47-83. https://doi.org/10.3390/automation3010003
Chicago/Turabian StyleNicola, Marcel. 2022. "The Stability of Systems with Command Saturation, Command Delay, and State Delay" Automation 3, no. 1: 47-83. https://doi.org/10.3390/automation3010003
APA StyleNicola, M. (2022). The Stability of Systems with Command Saturation, Command Delay, and State Delay. Automation, 3(1), 47-83. https://doi.org/10.3390/automation3010003