Positive systems are rapidly gaining more attention and popularity due to their appearance in numerous applications. The response of these systems to positive initial conditions and inputs remains in the positive orthant of state-space. They offer nice robust stability properties which can be employed to solve several control and estimation problems. Due to their specific structural as well as stability properties, it is of particular interest to solve constrained stabilization and control problems for general dynamical systems such that the closed-loop system admits the same desirable properties. However, positive systems are not the only special class of systems with lucrative features. The class of symmetric systems with eminent stability properties is another important example of structurally constrained systems. It has been recognized that they are appearing combined with the class of positive systems. The positive symmetric systems have found application in diverse area ranging from electromechanical systems, industrial processes, and robotics to financial, biological, and compartmental systems. This dissertation is devoted to separately analyzing positivity and symmetry properties of two classes of positive and symmetric systems. Based on this analysis, several critical problems concerning the constrained stabilization, estimation and control have been formulated and solved. First, positive stabilization problem with maximum stability radius is tackled and the solution is provided for general dynamical systems in terms of both LP and LMI. Second, the symmetric positive stabilization is considered for general systems with state-space parameters in regular and block controllable canonical forms.
Next, the positive unknown input observer (PUIO) is introduced and a design procedure is provided to estimate the state of positive systems with unknown disturbance and/or faults. Then, the PI observer is merged with UIO to exploit their benefits in robust fault detection. Finally, the unsolved problems of positive eigenvalue assignment (which ties to inverse eigenvalue problem) and symmetric positive control are addressed.
- Professor Bahram Shafai (Advisor)
- Professor Mario Sznaier
- Professor Rifat Sipahi
- Professor Mikhail Malioutov