Miroslav Krstic

Stabilization of stochastic nonlinear systems driven by noise of unknown covariance

This paper poses and solves a new problem of stochastic (nonlinear) disturbance attenuation where the task is to make the system solution bounded by a monotone function of the supremum of the covariance of the noise. This is a natural stochastic counterpart of the problem of input-to-state stabilization in the sense of Sontag (1989). Our development starts with a set of new global stochastic Lyapunov theorems. For an exemplary class of stochastic strict-feedback systems with vanishing nonlinearities, where the equilibrium is preserved in the presence of noise, we develop an adaptive stabilization scheme (based on tuning functions) that requires no a priori knowledge of a bound on the covariance. Next, we introduce a control Lyapunov function formula for stochastic disturbance attenuation. Finally, we address optimality and solve a differential game problem with the control and the noise covariance as opposing players; for strict-feedback systems the resulting Isaacs equation has a closed-form solution.

Delay Compensation for Nonlinear, Adaptive, and PDE Systems

Preface 1. Introduction Part I. Linear Delay-ODE Cascades 2. Basic Predictor Feedback 3. Predictor Observers 4. Inverse Optimal Redesign 5. Robustness to Delay Mismatch 6. Time-Varying Delay Part II. Adaptive Control 7. Delay-Adaptive Full-State Predictor Feedback 8. Delay-Adaptive Predictor with Estimation of Actuator State 9. Trajectory Tracking Under Unknown Delay and ODE Parameters Part III. Nonlinear Systems 10. Nonlinear Predictor Feedback 11. Forward-Complete Systems 12. Strict- Feedforward Systems 13. Linearizable Strict-Feedforward Systems Part IV. PDE-ODE Cascades 14. ODEs with General Transport-Like Actuator Dynamics 15. ODEs with Heat PDE Actuator Dynamics 16. ODEs with Wave PDE Actuator Dynamics 17. Observers for ODEs Involving PDE Sensor and Actuator Dynamics Part V. Delay-PDE and PDE-PDE Cascades 18. Unstable Reaction-Diffusion PDE with Input Delay 19. Antistable Wave PDE with Input Delay 20. Other PDE-PDE Cascades Appendices A. Poincare, Agmon, and Other Basic Inequalities B. Input-Output Lemmas for LTI and LTV Systems C. Lyapunov Stability and ISS for Nonlinear ODEs D. Bessel Functions E. Parameter Projection F. Strict-Feedforward Systems: A General Design G. Strict-Feedforward Systems: A Linearizable Class H. Strict-Feedforward Systems: Not Linearizable References Index

Closed-form boundary State feedbacks for a class of 1-D partial integro-differential equations

In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equations well posedness and the kernels smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.

Performance improvement and limitations in extremum seeking control (

We propose the inclusion of a dynamic compensator in the extremum seeking algorithm which improves the stability and performance properties of the method. This compensator is added to the integrator used for adaptation to improve the overall relative degree and phase response of the extremum seeking loop. The compensator is potentially more effective in accounting for the plant dynamics than the often used phase shifting of the demodulation signal. We present a detailed analysis of the extremum seeking system based on averaging. This analysis provides two linear models, one for tracking reference changes and the other for sensitivity to noise, which offer insight into how different parameters influence the performance. This analysis is less conservative than in previous cases and allows the use of faster adaptation for improved transients. We extend the extremum seeking method to problems of tracking changes in the set point which are more general than step functions.

Stochastic nonlinear stabilization—I: a backstepping design

While the current robust nonlinear control toolbox includes a number of methods for systems affine in deterministic bounded disturbances, the problem when the disturbance is unbounded stochastic noise has hardly been considered. We present a control design which achieves global asymptotic (Lyapunov) stability in probability for a class of strict-feedback nonlinear continuous-time systems driven by white noise. In a companion paper, we develop inverse optimal control laws for general stochastic systems affine in the noise input, and for strict-feedback systems. A reader of this paper needs no prior familiarity with techniques of stochastic control.

Adaptive nonlinear design with controller-identifier separation and swapping

We present a new adaptive nonlinear control design which achieves a complete controller-identifier separation. This modularity is made possible by a strong input-to-state stability property of the new controller with respect to the parameter estimation error and its derivative as inputs. These inputs are independently guaranteed to be bounded by the identifier. The new design is more flexible than the Lyapunov-based design because the identifier can employ any standard update law gradient and least-squares, normalized and unnormalized. A key ingredient in the identifier design and convergence analysis is a nonlinear extension of the well-known linear swapping lemma. >

Backstepping observers for a class of parabolic PDEs

In this paper we design exponentially convergent observers for a class of parabolic partial integro-differential equations (P(I)DEs) with only boundary sensing available. The problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system. Observer gain (output injection function) is shown to satisfy a well-posed hyperbolic PDE that is closely related to the hyperbolic PDE governing backstepping control gain for the state-feedback problem. For several physically relevant problems the observer gains are obtained in closed form. The observer gains are then used for an output-feedback design in both collocated and anti-collocated setting of sensor and actuator. The order of the resulting compensator can be substantially lowered without affecting stability. Explicit solutions of a closed loop system are found in particular cases.

Inverse optimal stabilization of a rigid spacecraft

The authors present an approach for constructing optimal feedback control laws for regulation of a rotating rigid spacecraft. They employ the inverse optimal control approach which circumvents the task of solving a Hamilton-Jacobi equation and results in a controller optimal with respect to a meaningful cost functional. The inverse optimality approach requires the knowledge of a control Lyapunov function and a stabilizing control law of a particular form. For the spacecraft problem, they are both constructed using the method of integrator backstepping. The authors give a characterization of (nonlinear) stability margins achieved with the inverse optimal control law.

Input Delay Compensation for Forward Complete and Strict-Feedforward Nonlinear Systems

We present an approach for compensating input delay of arbitrary length in nonlinear control systems. This approach, which due to the infinite dimensionality of the actuator dynamics and due to the nonlinear character of the plant results in a nonlinear feedback operator, is essentially a nonlinear version of the Smith predictor and its various predictor-based modifications for linear plants. Global stabilization in the presence of arbitrarily long delay is achieved for all nonlinear plants that are globally stabilizable in the absence of delay and that satisfy the property of forward completeness (which is satisfied by most mechanical systems, electromechanical systems, vehicles, and other physical systems). For strict-feedforward systems, one obtains the predictor-based feedback law explicitly. For the linearizable subclass of strict-feedforward systems, closed-loop solutions are also obtained explicitly. The feedback designs are illustrated through two detailed examples.

Nonlinear design of adaptive controllers for linear systems

A new approach to adaptive control of linear systems abandons the traditional certainty-equivalence concept and treats the control of linear plants with unknown parameters as a nonlinear problem. A recursive design procedure introduces at each step new design parameters and incorporates them in a novel Lyapunov function. This function encompasses all the states of the adaptive system and forces them to converge to a manifold of smallest possible dimension. Only as many controller parameters are updated as there are unknown plant parameters, and the dynamic order of the resulting controllers is no higher (and in most cases is lower) than that of traditional adaptive schemes. A simulation comparison with a standard indirect linear scheme shows that the new nonlinear scheme significantly improves transient performance without an increase in control effort. >