Analysis of Implicit Uncertain Systems. Part II: Constant Matrix Problems and Application to Robust H2 Analysis

This paper introduces an implicit framework for the analysis of uncertain systems, of which the general properties were described in Part I. In Part II, the theory is specialized to problems which admit a finite dimensional formulation. A constant matrix version of implicit analysis is presented, leading to a generalization of the structured singular value μ as the stability measure; upper bounds are developed and analyzed in detail. An application of this framework results in a practical method for robust H2 analysis: computing robust performance in the presence of norm-bounded perturbations and white-noise disturbances

Validation of Neural Network Controllers for Uncertain Systems Through Keep-Close Approach: Robustness Analysis and Safety Verification

Among the major challenges in neural control system technology is the validation and certification of the safety and robustness of neural network (NN) controllers against various uncertainties including unmodelled dynamics, non-linearities, and time delays. One way in providing such validation guarantees is to maintain the closed-loop system output with a NN controller when its input changes within a bounded set, close to the output of a robustly performing closed-loop reference model. This paper presents a novel approach to analysing the performance and robustness of uncertain feedback systems with NN controllers. Due to the complexity of analysing such systems, the problem is reformulated as the problem of dynamical tracking errors between the closed-loop system with a neural controller and an ideal closed-loop reference model. Then, the approximation of the controller error is characterised by adopting the differential mean value theorem (DMV) and the Integral Quadratic Constraints (IQCs) technique. Moreover, the Relative Integral Square Error (RISE) and the Supreme Square Error (SSE) bounded set are derived for the output of the error dynamical system. The analysis is then performed by integrating Lyapunov theory with the IQCs-based technique. The resulting worst-case analysis provides the user a prior knowledge about the worst case of RISE and SSE between the reference closed-loop model and the uncertain system controlled by the neural controller. The suitability of the proposed technique is demonstrated by the results obtained on a nonlinear single-link robot system with a NN trained to control the movement of this mechanical system while keeping close to an ideal closed-loop reference model.Comment: 19 pages, 10 figures, Journal Paper submitted to IEEE Transactions on Control Systems Technolog

An Overview of Integral Quadratic Constraints for Delayed Nonlinear and Parameter-Varying Systems

A general framework is presented for analyzing the stability and performance of nonlinear and linear parameter varying (LPV) time delayed systems. First, the input/output behavior of the time delay operator is bounded in the frequency domain by integral quadratic constraints (IQCs). A constant delay is a linear, time-invariant system and this leads to a simple, intuitive interpretation for these frequency domain constraints. This simple interpretation is used to derive new IQCs for both constant and varying delays. Second, the performance of nonlinear and LPV delayed systems is bounded using dissipation inequalities that incorporate IQCs. This step makes use of recent results that show, under mild technical conditions, that an IQC has an equivalent representation as a finite-horizon time-domain constraint. Numerical examples are provided to demonstrate the effectiveness of the method for both class of systems

Stability Analysis of Piecewise Affine Systems with Multi-model Model Predictive Control

Constrained model predictive control (MPC) is a widely used control strategy, which employs moving horizon-based on-line optimisation to compute the optimum path of the manipulated variables. Nonlinear MPC can utilize detailed models but it is computationally expensive; on the other hand linear MPC may not be adequate. Piecewise affine (PWA) models can describe the underlying nonlinear dynamics more accurately, therefore they can provide a viable trade-off through their use in multi-model linear MPC configurations, which avoid integer programming. However, such schemes may introduce uncertainty affecting the closed loop stability. In this work, we propose an input to output stability analysis for closed loop systems, consisting of PWA models, where an observer and multi-model linear MPC are applied together, under unstructured uncertainty. Integral quadratic constraints (IQCs) are employed to assess the robustness of MPC under uncertainty. We create a model pool, by performing linearisation on selected transient points. All the possible uncertainties and nonlinearities (including the controller) can be introduced in the framework, assuming that they admit the appropriate IQCs, whilst the dissipation inequality can provide necessary conditions incorporating IQCs. We demonstrate the existence of static multipliers, which can reduce the conservatism of the stability analysis significantly. The proposed methodology is demonstrated through two engineering case studies.Comment: 28 pages 9 figure

A Unified Analysis of Stochastic Optimization Methods Using Jump System Theory and Quadratic Constraints

We develop a simple routine unifying the analysis of several important recently-developed stochastic optimization methods including SAGA, Finito, and stochastic dual coordinate ascent (SDCA). First, we show an intrinsic connection between stochastic optimization methods and dynamic jump systems, and propose a general jump system model for stochastic optimization methods. Our proposed model recovers SAGA, SDCA, Finito, and SAG as special cases. Then we combine jump system theory with several simple quadratic inequalities to derive sufficient conditions for convergence rate certifications of the proposed jump system model under various assumptions (with or without individual convexity, etc). The derived conditions are linear matrix inequalities (LMIs) whose sizes roughly scale with the size of the training set. We make use of the symmetry in the stochastic optimization methods and reduce these LMIs to some equivalent small LMIs whose sizes are at most 3 by 3. We solve these small LMIs to provide analytical proofs of new convergence rates for SAGA, Finito and SDCA (with or without individual convexity). We also explain why our proposed LMI fails in analyzing SAG. We reveal a key difference between SAG and other methods, and briefly discuss how to extend our LMI analysis for SAG. An advantage of our approach is that the proposed analysis can be automated for a large class of stochastic methods under various assumptions (with or without individual convexity, etc).Comment: To Appear in Proceedings of the Annual Conference on Learning Theory (COLT) 201

A Framework for Worst-Case and Stochastic Safety Verification Using Barrier Certificates

This paper presents a methodology for safety verification of continuous and hybrid systems in the worst-case and stochastic settings. In the worst-case setting, a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region. No explicit computation of reachable sets is required in the construction of barrier certificates, which makes it possible to handle nonlinearity, uncertainty, and constraints directly within this framework. In the stochastic setting, our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set, a bound whose validity is proven by the existence of a barrier certificate. For polynomial systems, barrier certificates can be constructed using convex optimization, and hence the method is computationally tractable. Some examples are provided to illustrate the use of the method

Sets and Constraints in the Analysis Of Uncertain Systems

This thesis is concerned with the analysis of dynamical systems in the presence of model uncertainty. The approach of robust control theory has been to describe uncertainty in terms of a structured set of models, and has proven successful for questions, like stability, which call for a worst-case evaluation over this set. In this respect, a first contribution of this thesis is to provide robust stability tests for the situation of combined time varying, time invariant and parametric uncertainties. The worst-case setting has not been so attractive for questions of disturbance rejection, since the resulting performance criteria (e.g., ℋ∞,) treat the disturbance as an adversary and ignore important spectral structure, usually better characterized by the theory of stochastic processes. The main contribution of this thesis is to show that the set-based methodology can indeed be extended to the modeling of white noise, by employing standard statistical tests in order to identify a typical set, and performing subsequent analysis in a worst-case setting. Particularly attractive sets are those described by quadratic signal constraints, which have proven to be very powerful for the characterization of unmodeled dynamics. The combination of white noise and unmodeled dynamics constitutes the Robust ℋ2 performance problem, which is rooted in the origins of robust control theory. By extending the scope of the quadratic constraint methodology we obtain a solution to this problem in terms of a convex condition for robustness analysis, which for the first time places it on an equal footing with the ℋ∞ performance measure. A separate contribution of this thesis is the development of a framework for analysis of uncertain systems in implicit form, in terms of equations rather than input-output maps. This formulation is motivated from first principles modeling, and provides an extension of the standard input-output robustness theory. In particular, we obtain in this way a standard form for robustness analysis problems with constraints, which also provides a common setting for robustness analysis and questions of model validation and system identification