In this chapter, we consider briefly some important issues concerning the performance requirements to closed-loop linear systems and the fundamental design limitations in achieving the control aims. The performance specifications are formulated in continuous-time due to the clear physical interpretation in this case. First, we present the relatively simple case of single-input-single-output (SISO) systems which are well studied in the classical control theory. The trade-offs in the design of such systems are shown in some details. Then, we discuss the more complicated case of multiple-input-multiple-output (MIMO) systems whose performance is investigated by using the singular value plots of certain closed-loop transfer function matrices. An important issue in controller design is the closed-loop system performance in presence of different uncertainties. At the end of the chapter, we present some elements of the contemporary approach to the robustness analysis of uncertain linear systems based on the small gain theorem and structured singular value (SSV).

Chapter Contents:

• 3.1 SISO closed-loop systems
• 3.2 Performance specifications of SISO systems
• 3.2.1 Time-domain specifications
• 3.2.2 Frequency-domain specifications
• 3.3 Trade-offs in the design of SISO systems
• 3.3.1 Limitations on S and T
• 3.3.2 Right half-plane poles and zeros
• 3.3.3 Limitations imposed by time delays
• 3.3.4 Limitations imposed by measurement noise
• 3.3.5 Limitations, imposed by disturbances
• 3.3.6 Limitations on control action
• 3.3.7 Limitations due to model errors
• 3.4 MIMO closed-loop systems
• 3.5 Performance specifications of MIMO systems
• 3.5.1 Using singular values for performance analysis
• 3.5.2 H∞ Norm of a system
• 3.5.3 Hankel norm
• 3.6 Trade-offs in the design of MIMO systems
• 3.6.1 Disturbance rejection
• 3.6.2 Noise suppression
• 3.6.3 Model errors
• 3.7 Uncertain systems
• 3.8 Robust-stability analysis
• 3.8.1 Unstructured uncertainty
• 3.8.2 Structured singular value
• 3.8.3 Robust-stability analysis with
• 3.9 Robust performance analysis
• 3.9.1 Using for robust performance analysis
• 3.9.2 Worst case gain
• 3.9.3 Worst case margin
• 3.10 Numerical issues in robustness analysis
• 3.11 Notes and references

Inspec keywords: control system synthesis; closed loop systems; uncertain systems; continuous time systems; MIMO systems; single-input single-output systems (control); transfer function matrices

Other keywords: single-input-single-output systems; continuous-time system; small gain theorem; MIMO system; structured singular value; SISO system; closed-loop transfer function matrices; closed-loop system performance; multiple-input-multiple-output systems; design limitations; uncertain linear systems; robustness analysis; performance requirements; closed-loop linear systems; performance specifications; classical control theory; controller design; singular value plots

Subjects: Linear control systems; Multivariable control systems; Control system analysis and synthesis methods; Algebra