\(L_{1}\) -Induced Output-Feedback Controller Synthesis for Interval Positive Systems
This chapter is concerned with the design of \(L_{1}\) -induced output-feedback controller for continuous-time positive systems with interval uncertainties. A necessary and sufficient condition for stability and \(L_{1}\) -induced performance of positive
- PDF / 1,837,760 Bytes
- 35 Pages / 439.37 x 666.142 pts Page_size
- 85 Downloads / 154 Views
L 1 -Induced Output-Feedback Controller Synthesis for Interval Positive Systems
This chapter is concerned with the design of L 1 -induced output-feedback controller for continuous-time positive systems with interval uncertainties. A necessary and sufficient condition for stability and L 1 -induced performance of positive linear systems is proposed in terms of linear inequalities. Based on this, conditions for the existence of robust static output-feedback controllers are established and an iterative convex optimization approach is developed to solve the conditions. The problem of controller synthesis is completely solved for single-input-multiple-output (SIMO) positive systems with the help of an analytical formula for L 1 -induced norm. The result is further extended to the multiple-input-multiple-output (MIMO) case by employing a structured controller. Moreover, the design of L 1 -induced sparse controller is investigated for continuous-time interval positive systems. In addition, a dynamic output-feedback controller is designed for interval positive systems under L 1 performance. Three illustrative examples are provided to show the effectiveness and applicability of the theoretical results. The organization of this chapter is as follows. In Sect. 3.1, the exact value of L 1 induced norm is computed for continuous-time positive system. Then, the L 1 -induced performance characterization is provided for interval positive system. In Sect. 3.2, a static output-feedback controller design method for interval positive systems is put forward based on the analysis conditions. Section 3.3 considers a special case of interval positive systems. Under the special case of SIMO interval positive systems, an analytical method for controller synthesis is proposed. Then, the results are further extended to the MIMO case. Moreover, the L 1 -induced sparse controller is designed for continuous-time positive systems with interval uncertainties. In Sect. 3.4, we turn to consider the dynamic output-feedback stabilization problem for interval positive systems. Section 3.5 presents several illustrative examples. Section 3.6 concludes the chapter.
© Springer Science+Business Media Singapore 2017 X. Chen, Analysis and Synthesis of Positive Systems Under 1 and L 1 Performance, Springer Theses, DOI 10.1007/978-981-10-2227-2_3
37
38
3 L 1 -Induced Output-Feedback Controller Synthesis …
3.1 Performance Analysis Consider a positive system:
x(t) ˙ = Ax(t) + Bw w(t), y(t) = C x(t) + Dw w(t),
(3.1)
where x(t) ∈ Rn , w(t) ∈ Rm and y(t) ∈ Rr are the system state, input and output, respectively. Now, we are in a position to give the definition of L 1 -induced norm. For a stable positive linear system given in (3.1), its L 1 -induced norm is defined as (L 1 ,L 1 )
sup
w=0, w∈L 1
y L 1 , w L 1
(3.2)
where : L 1 → L 1 denotes the convolution operator, that is, y(t) = ( ∗ w)(t). System (3.1) is said to have L 1 -induced performance at the level γ if, under zero initial conditions, (3.3) (L 1 ,L 1 ) < γ , where γ > 0 is
Data Loading...
