Guaranteed robustness properties of multivariable, nonlinear, stochastic optimal regulators

John Tsitsiklis, Michael Athans
1983 The 22nd IEEE Conference on Decision and Control  
We study the robustness of optimal regulators for nonlinear, deterministic and stochastic, multi-input dynamical systems, under the assumption that all state variables can be measured. We show that, under mild assumptions, such nonlinear regulators have a guaranteed infinite gain margin; moreover, they have a guaranteed 50 percent gain reduction margin and a 60 degree phase margin, in each feedback channel, provided that the system is linear in the control and the penalty to the control is
more » ... the control is quadratic, thus extending the well-known properties of LQ regulators to nonlinear optimal designs. These results are also valid for infinite horizon, average cost, stochastic optimal control problems. This paper has been submitted to the 22nd IEEE Conference on Decision and Control, and to the IEEE Transactions on Automatic Control. I. INTRODUCTION. Regulator design for dynamical systems is usually performed on the basis of a nominal model of the plant to be controlled. Modelling errors are unavoidable and, in fact, often desirable because they may result in simpler designs. It is therefore essential that the regulator based on the nominal model is robust; that is, it preserves its qualitative properties (namely, the stability of the closedloop system) in the face of modelling errors. The robustness and sensitivity to modelling errors of controlled linear systems has been extensively studied in the past [2, 6]. The robustness (stability margins) of regulators has been traditionally described in terms of gain and phase margins, although more recent approaches [3, 9, 12] focus on the singular values of the return difference or of the inverse return difference matrix. One of the most appealing features of optimal linear quadratic (LQ) regulators are their guaranteed stability margins. Namely, LQ regulators remain stable when the control gains are multiplied by any number greater than 1/2. They also have guaranteed phase margins of sixty degrees 11, 13, 14, 16]. These results can be obtained directly by appropriately manipulating the associated Riccati equation [13]. A recent paper by Glad [5] has shown that gain margins of optimal regulators for nonlinear systems can be derived from the associated Hamilton-Jacobi-Bellman (HJB) equation, under suitable assumptions. This result ties nicely with the results on LQ regulators, because the Riccati equation is a direct consequence of the IHJB equation associated with LQ problems. However, the results of [5] are only applicable to single-input, deterministic systems, perturbed by memoryless nonlinearities thus allowing only derivation of gain margin results; no phase margin results were derived in [5] . In this paper we derive general robustness margins of optimal regulators for multi-input nonlinear systems. Our results are valid for both deterministic and stochastic systems (controlled diffusion processes). In contrast to [5], we allow dynamical, (i.e. not just memoryless) perturbations inside the loop and obtain, as a corollary, a generalization of the phase margin results of [13]. In particular, we show (Theorem 3) that the robustness margins of LQ regulators (including the 60 degree phase margin) hold for optimal regulators of any nonlinear plant which is linear in the control, provided that the cost functional contains a quadratic control penalty. In the stochastic case, we consider two distinct classes of controlled processes: a) Those for which the state can be steered to an equilibrium point (assumed to be the origin). Such is the case for
doi:10.1109/cdc.1983.269867 fatcat:qbipylvai5dvroo4npid4vlhmq