C. Lorenz-Stenflo
III. Second order sliding mode control
5.4 Summary
6
Conclusions and Scope for future work
Contents
6.1 Conclusions . . . . 137 6.2 Scope for future work . . . . 139
6.1 Conclusions
6.1 Conclusions
This thesis is an attempt to design a robust optimal control methodology to minimize the control effort required for controlling an uncertain system. Specifically, this thesis aims to develop chattering free optimal sliding mode controllers (OSMCs) for both linear and nonlinear systems which are affected by matched and mismatched types of uncertainty. The common methodology adopted in the research work is to use classical optimal control technique and utilize sliding mode control method to impart robustness to the optimal controller for its unabated performance in the face of disturbances. The thesis work yielded robust optimal controllers which are discussed briefly as follows.
An optimal adaptive sliding mode controller (OASMC) is designed to control the linear system affected by matched uncertainty with unknown upper bound. The optimal controller is designed for the nominal linear system based on the LQR technique and is combined with the SMC by designing an integral sliding surface. As the upper bound of the uncertainty is unknown, an adaptive method is used to design the switching control in the SMC. The proposed OASMC shows satisfactory performance in stabilizing and tracking problems. Compared to conventional SMCs, the proposed optimal sliding mode controller spends a lower control energy but maintaining similar performance standard. For controlling linear systems affected by the mismatched uncertainty using minimum control input, a disturbance observer based optimal sliding mode controller is proposed. The optimal controller is designed for the nominal linear system based on the linear quadratic regulator (LQR) method and an integral sliding surface is combined with the optimal control for making it immune to uncertainties.
The sliding surface is designed using estimated value of the mismatched uncertainty. A disturbance observer is used for the disturbance estimation.
The main disadvantage of the OSMC is the presence of undesired high frequency chattering in the control input which is detrimental for the controller. In order to overcome this inherent difficulty of the OSMC, an optimal second order sliding mode controller (OSOSMC) is proposed. The optimal controller is designed for the nominal linear system using the LQR technique and integrated with a SOSMC. The second order sliding mode methodology is realized by designing a non-singular terminal sliding surface based on an integral sliding variable. The proposed controller is applied for both stabilization and tracking of linear uncertain SISO systems affected by matched uncertainty. The proposed OSOSMC is also applied for stabilization of linear uncertain decoupled MIMO systems affected by the matched uncertainty. The proposed optimal second order sliding mode controller uses
a substantially lower control effort than some existing sliding mode controllers while offering the same performance level.
For controlling nonlinear uncertain systems affected by matched uncertainty, an optimal second order sliding mode controller (OSOSMC) is proposed using the state dependent Riccati equation (SDRE). To design the SDRE based optimal controller, the nonlinear system needs to be represented as a linear like structure. Using extended linearization, the nonlinear system is represented as a linear like structure having state dependent coefficient (SDC) matrices. After designing the optimal con- troller, it is integrated with the second order sliding mode controller (SOSMC) which is implemented by designing an integral sliding variable based non-singular terminal sliding mode controller. The pro- posed controller is applied for stabilization and tracking problems and it was found that the control energy used in the proposed SDRE based optimal second order sliding mode controller is significantly reduced compared to some existing sliding mode controllers but maintaining comparable performance level. The proposed SDRE based OSOSMC is also successfully applied for stabilization of the chaotic system which is a special case of highly unstable nonlinear systems. The proposed control strategy can successfully stabilize those chaotic systems which can be represented as linear like structures.
The proposed SDRE based OSOSMC is not applicable for those nonlinear systems which cannot be represented as linear like structures. For such nonlinear uncertain systems, a control Lyapunov function (CLF) based optimal second order sliding mode controller (OSOSMC) is proposed. The CLF is chosen for the open loop system and then a feedback controller is designed to optimize the desired performance index. After designing the optimal controller for the nominal nonlinear system, it is integrated with the SOSMC designed by using integral sliding variable based terminal sliding mode. The proposed CLF based OSOSMC is applied for stabilization and tracking and in accordance with earlier results, it was observed that the control energy used in the proposed CLF based OSOSMC is significantly lower in comparison to some existing sliding mode controllers but without compromising on the performance standard. The proposed CLF based OSOSMC cannot tackle mismatched uncertainty. In order to handle mismatched uncertainty in nonlinear systems, a disturbance observer based optimal second order sliding mode controller (DOB-OSOSMC) is proposed. The optimal controller is designed for the nominal system based on the CLF and a disturbance observer is used to estimate the mismatched uncertainty. Based on the estimated value of the uncertainty, an integral sliding variable is designed to combine the optimal controller with the SMC. To mitigate chattering in the control input, the SMC