Feedback Linearization of Nonlinear Multivariable Systems: Theory and Implementation

Feedback linearization represents a fundamental methodology in control theory for handling nonlinear multivariable systems. The core concept involves transforming the original nonlinear system into an equivalent linear system through precise state feedback and nonlinear coordinate transformations, thereby enabling the application of well-established linear control theories for system analysis and design. For multivariable nonlinear systems, the key to feedback linearization lies in identifying appropriate nonlinear feedback control laws and coordinate transformations that render the closed-loop system's dynamic behavior linear. This process typically manifests in two primary forms: input-output linearization and input-state linearization. Input-output linearization focuses on linearizing the relationship between outputs and inputs, while input-state linearization aims to transform the entire system's state equations into linear form. The critical implementation steps for feedback linearization involve calculating the system's relative degree, constructing Lie derivatives, and verifying integrability conditions. In code implementations, this often requires symbolic computation packages like MATLAB's Symbolic Math Toolbox to handle complex derivative calculations. For instance, the relative degree calculation algorithm involves recursively computing Lie derivatives until the output's dependence on inputs becomes explicit. If the system satisfies exact linearization conditions, designers can develop feedback control laws to eliminate nonlinear terms, rendering the system controllable or observable in transformed coordinates. A typical implementation approach might involve: 1. Defining system dynamics using state-space equations 2. Computing Lie derivatives using symbolic differentiation 3. Checking involutivity conditions for coordinate transformations 4. Designing feedback control laws to cancel nonlinearities Feedback linearization finds extensive applications in complex nonlinear systems such as robotic control, aircraft autopilots, and power systems. However, the method strongly depends on accurate system modeling and exhibits high computational complexity in higher-order nonlinear systems. Therefore, practical implementations often combine robust control or adaptive methods to enhance system disturbance rejection capabilities. Modern implementations may incorporate numerical optimization techniques to handle model uncertainties and computational challenges.