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Engineering LibreTexts

9.0: Prelude to Controllers for State Variable Models

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  • The state variable models of dynamic systems comprises first-order differential equations that express the time derivatives of a set of state variables. For linear time-invariant (LTI) systems, these equations are commonly expressed in vector-matrix form.

    The controller design for the state variable models involves feeding back all the state variables using appropriate weights to generate the error signal. State feedback allows arbitrary placement of roots of the closed-loop characteristic polynomial. It is more powerful and offers greater flexibility than the output feedback that allows only selective placement of closed-loop roots. State feedback assumes that the complete set of state variables are availabe for feedback.

    The pole placement design refers to the selection of feedback gains for placing the roots of the closed-loop characteristic polynomial at the desired locations in the complex plane. The pole placement design is performed with ease when the state variable model is in the controller form. Alternately, Ackermann’s and Bass-Gura formulas, or the Sylvester’s equation can be used for this purpose.

    Output regulation refers to finding a control law to asymptotically track prescribed refernce signals and/or asymptotically reject undesired disturbance inputs. It includes imparting desired degree of dynamic stability to the system through arbitrary selection of the closed-loop characteristic polynomial.

    The tracking system design involves reducing the steady-state error to a given reference input to zero. Though reference signal can be used to cancel the tracking error, the design is not robust against parameter variations. A more robust design involves integrating the error signal inside the feeback loop to form an augmented system model, which can used for pole placement. The augmented model includes the differential equation describing the out put of the integrator.

    The state-space design methods primarily cater to the design of multi-input multi-output (MIMO) systems. This chapter, however, introduces the state-space design methods using examples from single-input single-output (SISO) systems.