Improvement in Controller Performance for Distillation Column Using IMC Technique

Output Stabilization of Linear Systems in Given Set

“…It is evident that for a given c > 0 and any fixed σ, α, β, inequalities (25) always have finite solutions (K, τ 1 , τ 2 ). As a result, the control law (10) with K satisfying (12) guarantees the input-to-state stability in the closed-loop system (11), implying a boundedness of ε(t). According to Theorem 1, the goal (2) is achieved.…”

“…where K ∈ R m×m is the control gain matrix to be designed. Next, we will demonstrate that selecting the control law (10) enables us to ensure the negativity of the derivative of Lyapunov function (8) when analyzing the stability of the closed-loop system. By substituting the control law (10) into Equation ( 7), we obtain…”

“…In particular, if the system (1) is one-dimensional (i.e., m = 1), then the gain K of the control law (10) can be determined if the following inequalities are satisfied…”

“…The LMI technique and S-procedure allow us to design the controller for a MIMO system under the influence of unknown bounded disturbances based on analysis the input to state stability of the closed-loop system. Moreover, the problem of finding the control gain matrix of (10) can be reduced to the problem of finding the solutions to the feasible problems ( 12) and ( 13), which can be easily solved using popular solvers for semidefinite programming (such as SEDUMI [32], SDPT3 [33], CSDP [34] and others.) Remark 2.…”

“…In practice, there are many control problems associated with a guarantee of finding outputs in given sets. For example, it is required to maintain the frequency and output voltage of electric generators within specified bounds in the electric power network [1][2][3][4][5][6][7] or pressure and flow rate at the wellhead in a given band when controlling the formation pressure stabilization process [8][9][10], etc. Violating these constraints during operation may degrade the performance and lead to system instability.…”