Discrete-Time Adaptive Command Following and Disturbance Rejection with Unknown Exogenous Dynamics

Abstract

This paper considers parameter-monotonic direct adaptive command following and disturbance rejection for single-input single-output minimum-phase linear time-invariant systems with knowledge of the sign of the high-frequency gain (first non-zero Markov parameter) and an upper bound on the magnitude of the high-frequency gain. We assume that the command and disturbance signals are generated by a linear system with known characteristic polynomial. Furthermore, we assume that the command signal is measured, but the disturbance signal is unmeasured. The first part of the paper is devoted to a fixed-gain analysis of a high-gain-stabilizing dynamic compensator for command following and disturbance rejection. The compensator utilizes a Fibonacci series construction to control systems with unknown-but-bounded relative degree. We then introduce a parameter-monotonic adaptive law and guarantee asymptotic command following and disturbance rejection.

Output feedback adaptive command following and disturbance rejection for nonminimum phase uncertain dynamical systems

“…As the controllable nonminimal state space realization of (11) and (12) is defined by a state that consists entirely of filtered inputs and outputs and their derivatives of the original system, an output feedback stabilization problem for (1) and (2) can be converted into a full-state feedback control design problem by equivalently considering (11) and (12). Furthermore, for an output feedback control design of the form (1) and (2) we typically require that (A p , B p ) be controllable (or stabilizable) and (A p , C p ) be observable (or detectable).…”

“…Furthermore, for an output feedback control design of the form (1) and (2) we typically require that (A p , B p ) be controllable (or stabilizable) and (A p , C p ) be observable (or detectable). In contrast, for a feedback control design using the input-output equivalent nonminimal state space model (11) and (12) we only require controllability of the pair (A f , B f ), which is automatic. Finally, it is important to note that only the system matrix A f in (11) is unknown for full-state feedback control design, whereas the triple (A p , B p , C p ) is unknown in (1) and (2) for an output feedback control design.…”

“…Nonminimal state space realizations for discrete-time adaptive control have been extensively developed in the literature, see [11,20,21] and the references therein. The proposed nonminimal state space realization for continuous-time adaptive control developed in this section is new and novel.…”

“…Specifically, consider the controlled linear multivariable uncertain dynamical system given by (11), where the known state vector x f (t), t 0, is given by (13), the unknown matrix A f is given by (15), and the known input matrix B f is given by (16). Next, consider the reference model given bẏ…”

“…Now, filtering the signals in (30) and (31), and using (36) and (40), a nonminimal state space realization of (1) and (2) is given by (11) and (12), where x f (t) ∈ R (m+l)n , t 0, is the known filtered expanded state vector given by (13), and A f ∈ R (m+l)n×(m+l)n , B f ∈ R (m+l)n×m , and C f ∈ R l×(m+l)n are given by (15)- (17), respectively, with…”