Lee H. Keel

Robust, fragile, or optimal?

We show by examples that optimum and robust controllers, designed by using the H/sub 2/, H/sub /spl infin//, l/sup 1/, and /spl mu/ formulations, can produce extremely fragile controllers, in the sense that vanishingly small perturbations of the coefficients of the designed controller destabilize the closed-loop control system. The examples show that this fragility usually manifests itself as extremely poor gain and phase margins of the closed-loop system. The calculations given here should raise a cautionary note and draw attention to the larger issue of controller sensitivity which may be important in other nonoptimal design techniques as well.

Robust control with structure perturbations

The problem of making a given stabilizing controller robust so that the closed-loop system remains stable for prescribed ranges of variations of a set of physical parameters in the plant. The problem is treated in the state-space and transfer-function domains. In the state-space domain a stability hypersphere is determined in the parameter space using Lyapunov theory. The radius of this hypersphere is iteratively increased by adjusting the controller parameters until the prescribed perturbation ranges are contained in the stability hypersphere. In the transfer-function domain a corresponding stability margin is defined and optimized on the basis of the recently introduced concept of the largest stability hypersphere in the space of coefficients of the closed-loop characteristic polynomial. The design algorithms are illustrated by examples. >

Transient response control via characteristic ratio assignment

This note develops an approach to directly control the transient response of linear time-invariant control systems. We begin by considering all-pole transfer functions of order n for which we introduce a set of parameters /spl alpha//sub i/, i=1,...n called the characteristic ratios. We also introduce a generalized time constant /spl tau/. We prove that /spl alpha//sub 1/ and /spl tau/ can be used to characterize the system overshoot to a step input and the speed of response, respectively. By independently adjusting /spl alpha//sub 1/ and /spl tau/ in all-pole systems, arbitrarily small or no overshoot as well as arbitrarily fast speed of response can be achieved. These formulas are used to develop a procedure to design feedback controllers with feedforward or two parameter output feedback type for achieving time response specifications. For a minimum phase plant we show that arbitrary transient response specifications, namely one with independently specified overshoot and specified rise time or speed of response can be exactly attained.

Robust, fragile or optimal?

We show by examples that optimum and robust controllers, designed by using the H/sub 2/, H/sub /spl infin//, l/sup 1/ and /spl mu/ formulations, can produce extremely fragile controllers, in the sense that vanishingly small perturbations of the coefficients of the designed controller destabilize the closed loop control system. The examples show that this fragility also usually manifests itself as extremely poor gain and phase margins of the closed loop system.

Controller Synthesis Free of Analytical Models: Three Term Controllers

The main focus of this paper is on direct data driven synthesis and design of controllers. We show that the complete set of stabilizing proportional-integral-derivative (PID) and first-order controllers for a finite dimensional linear time-invariant plant, possibly cascaded with a delay, can be calculated directly from the frequency response (Nyquist/Bode) data P(jomega) for omega epsi [0, infin) without the need of producing an identified analytical model. It is also shown that complete sets achieving guaranteed levels of performance measures such as gain margin, phase margin, and Hinfin norms can likewise be calculated directly from only Nyquist/Bode data. The solutions have important new features. For example it is not necessary to know the order of the plant or even the number of left half plane or right half plane poles or zeros. The solution also identifies, in the case of PID controllers an exact low frequency band over which the plant data must be known with accuracy and beyond which the plant information may be rough or approximate. These constitute important new guidelines for identification when the latter is to be used for control design. The model free approach to control synthesis and design developed here is an attractive complement to modern and post modern model based design methods which require complete information on the plant and generally produce a single optimal controller. A discussion is included, with illustrative example, of the sharp differences between model-free and model based approaches when computing sets of stabilizing controllers. For example, it is shown, that the identified model of a high order system can be non-PID stabilizable whereas the original data indicates it is PID stabilizable. The results given here are also a significant improvement over classical control loop-shaping approaches since we obtain complete sets of controllers achieving the design specifications. It can enhance fuzzy and neural approaches which are model free but cannot guarantee stability and performance. Finally, these results open the door to adaptive, model free, fixed order designs of real world systems.

On the new method for the control of discrete nonlinear dynamic systems using neural networks

A new controller design method for nonaffine nonlinear dynamic systems is presented in this paper. An identified neural network model of the nonlinear plant is used in the proposed method. The method is based on a new control law that is developed for any discrete deterministic time-invariant nonlinear dynamic system in a subregion Phi(x) of an asymptotically stable equilibrium point of the plant. The performance of the control law is not necessarily dependent on the distance between the current state of the plant and the equilibrium state if the nonlinear dynamic system satisfies some mild requirements in Phi(x). The control law is simple to implement and is based on a novel linearization of the input-output model of the plant at each instant in time. It can be used to control both minimum phase and nonminimum phase nonaffine nonlinear plants. Extensive empirical studies have confirmed that the control law can be used to control a relatively general class of highly nonlinear multiinput-multioutput (MIMO) plants.

A new approach to digital PID controller design

In this note, we present a new approach to the problem of designing a digital proportional-integral-derivative (PID) controller for a given but arbitrary linear time invariant plant. By using the Tchebyshev representation of a discrete-time transfer function and some new results on root counting with respect to the unit circle, we show how the digital PID stabilizing gains can be determined by solving sets of linear inequalities in two unknowns for a fixed value of the third parameter. By sweeping or gridding over this parameter, the entire set of stabilizing gains can be recovered. The precise admissible range of this parameter can be predetermined. This solution is attractive because it answers the question of whether there exists a stabilizing solution or not and in case stabilization is possible the entire set of gains is determined constructively. Using this characterization of the stabilizing set we present solutions to two design problems: 1) maximally deadbeat design, where we determine for the given plant, the smallest circle within the unit circle wherein the closed loop system characteristic roots may be placed by PID control and 2) maximal delay tolerance, where we determine, for the given plant the maximal-loop delay that can be tolerated under PID control. In each case, the set of controllers attaining the specifications is calculated. Illustrative examples are included.

Linear control theory : structure, robustness, and optimization

Preface THREE TERM CONTROLLERS PID Controllers: An Overview of Classical Theory Introduction to Control The Magic of Integral Control PID Controllers Classical PID Controller Design Integrator Windup PID Controllers for Delay-Free LTI Systems Introduction Stabilizing Set Signature Formulas Computation of the PID Stabilizing Set PID Design with Performance Requirements PID Controllers for Systems with Time Delay Introduction Characteristic Equations for Delay Systems The Pade Approximation and Its Limitations The Hermite-Biehler Theorem for Quasipolynomials Stability of Systems with a Single Delay PID Stabilization of First-Order Systems with Time Delay PID Stabilization of Arbitrary LTI Systems with a Single Time Delay Proofs of Lemmas 3.3, 3.4, and 3.5 Proofs of Lemmas 3.7 and 3.9 An Example of Computing the Stabilizing Set Digital PID Controller Design Introduction Preliminaries Tchebyshev Representation and Root Clustering Root Counting Formulas Digital PI, PD, and PID Controllers Computation of the Stabilizing Set Stabilization with PID Controllers First-Order Controllers for LTI Systems Root Invariant Regions An Example Robust Stabilization by First-Order Controllers Hinfinity Design with First-Order Controllers First-Order Discrete-Time Controllers Controller Synthesis Free of Analytical Models Introduction Mathematical Preliminaries Phase, Signature, Poles, Zeros, and Bode Plots PID Synthesis for Delay-Free Continuous-Time Systems PID Synthesis for Systems with Delay PID Synthesis for Performance An Illustrative Example: PID Synthesis Model-Free Synthesis for First-Order Controllers Model-Free Synthesis of First-Order Controllers for Performance Data-Based Design vs. Model-Based Design Data-Robust Design via Interval Linear Programming Computer-Aided Design Data-Driven Synthesis of Three Term Digital Controllers Introduction Notation and Preliminaries PID Controllers for Discrete-Time Systems Data-Based Design: Impulse Response Data First-Order Controllers for Discrete-Time Systems Computer-Aided Design ROBUST PARAMETRIC CONTROL Stability Theory for Polynomials Introduction The Boundary Crossing Theorem The Hermite-Biehler Theorem Schur Stability Test Hurwitz Stability Test Stability of a Line Segment Introduction Bounded Phase Conditions Segment Lemma Schur Segment Lemma via Tchebyshev Representation Some Fundamental Phase Relations Convex Directions The Vertex Lemma Stability Margin Computation Introduction The Parametric Stability Margin Stability Margin Computation The Mapping Theorem Stability Margins of Multilinear Interval Systems Robust Stability of Interval Matrices Robustness Using a Lyapunov Approach Stability of a Polytope Introduction Stability of Polytopic Families The Edge Theorem Stability of Interval Polynomials Stability of Interval Systems Polynomic Interval Families Robust Control Design Introduction Interval Control Systems Frequency Domain Properties Nyquist, Bode, and Nichols Envelopes Extremal Stability Margins Robust Parametric Classical Design Robustness under Mixed Perturbations Robust Small Gain Theorem Robust Performance The Absolute Stability Problem Characterization of the SPR Property The Robust Absolute Stability Problem OPTIMAL AND ROBUST CONTROL The Linear Quadratic Regulator An Optimal Control Problem The Finite Time LQR Problem The Infinite Horizon LQR Problem Solution of the Algebraic Riccati Equation The LQR as an Output Zeroing Problem Return Difference Relations Guaranteed Stability Margins for the LQR Eigenvalues of the Optimal Closed Loop System Optimal Dynamic Compensators Servomechanisms and Regulators SISO Hinfinity AND l1 OPTIMAL CONTROL Introduction The Small Gain Theorem L Stability and Robustness via the Small Gain Theorem YJBK Parametrization of All Stabilizing Compensators (Scalar Case) Control Problems in the Hinfinity Framework Hinfinity Optimal Control: SISO Case l1 Optimal Control: SISO Case Hinfinity Optimal Multivariable Control Hinfinity Optimal Control Using Hankel Theory The State Space Solution of Hinfinity Optimal Control Appendix A: Signal Spaces Vector Spaces and Norms Metric Spaces Equivalent Norms and Convergence Relations between Normed Spaces Appendix B: Norms for Linear Systems Induced Norms for Linear Maps Properties of Fourier and Laplace Transforms Lp/lp Norms of Convolutions of Signals Induced Norms of Convolution Maps EPILOGUE Robustness and Fragility Feedback, Robustness, and Fragility Examples Discussion References Index Exercises, Notes, and References appear at the end of each chapter.

H/sub /spl infin// design with first order controllers

This paper considers the problem of determining the complete set of first order controller parameters for which the frequency weighted H/sub /spl infin// norm of some closed loop transfer function is less than a specified constant and the closed loop system is stable. The results apply to single-input single-output, linear, time invariant plants of arbitrary order. The problem of determining all first order controllers (C (s) = (x/sub 1/s+x/sub 2/)/(s+x/sub 3/)) which stabilize such a plant has been recently solved in [R.N. Rantaris et al., 2002]. In this paper, these results are extended to determine the subset of controllers which also satisfy various robustness and performance specifications which can be formulated as specific H/sub /spl infin// norm constraints. The problem is solved by converting the H/sub /spl infin// problem into the simultaneous stabilization of the closed loop characteristic polynomial and a family of related complex polynomials. The stability boundary of each of these polynomials can be computed explicitly for fixed x/sub 3/ by solving linear equations. The union of the resulting stability regions yields the set of all x/sub 1/ and x/sub 2/ which simultaneously satisfy the H/sub /spl infin// condition and closed loop stability for a fixed x/sub 3/. The entire three dimensional set meeting specifications is obtained by sweeping x/sub 3/ over the stabilizing range.

Robust adaptive control of nonaffine nonlinear plants with small input signal changes

Assuming small input signal magnitudes, ARMA models can approximate the NARMA model of nonaffine plants. Recently, NARMA-L1 and NARMA-L2 approximate models were introduced to relax such input magnitude restrictions. However, some applications require larger input signals than allowed by ARMA, NARMA-L1 and NARMA-L2 models. Under certain assumptions, we recently developed an affine approximate model that eliminates the small input magnitude restriction and replaces it with a requirement of small input changes. Such a model complements existing models. Using this model, we present an adaptive controller for discrete nonaffine plants with unknown system equations, accessible input-output signals, but inaccessible states. Our approximate model is realized by a neural network that learns the unknown input-output map online. A deadzone is used to make the weight update algorithm robust against modeling errors. A control law is developed for asymptotic tracking of slowly varying reference trajectories.