Review articleRobust control under parametric uncertainty: An overview and recent results
Introduction
Robustness of a system, the subject of this article, is its ability to remain functional despite large changes. In control engineering, robustness has played a central and pivotal role, since its beginning in the 1860s. Thus Black’s feedback amplifier (Kline, 1993), the Nyquist criterion (Nyquist, 1932), and gain and phase margins Bode (1945) were concepts dealing directly with robustness in the classical period.
Starting in 1960, the focus of control engineers shifted to optimization. However, the adequacy of an optimal design was ultimately judged by its robustness. Kalman’s Linear Quadratic Optimal Regulator (Kalman, 1959) was found to be deficient when measured by its ability to deliver stability margins under output feedback (Doyle & Stein, 1979). The remedy proposed was high order H∞ control (Doyle, Glover, Khargonekar, & Francis, 1989). In 1997, (Keel & Bhattacharyya, 1997) it was shown that even these controllers, and indeed all high order controllers, were fragile. This led of a renewed interest in direct studies on robustness resulting in a body of knowledge known as the parametric theory (Ackermann, 2012, Barmish, Jury, 1994, Bhattacharyya, Chapellat, Keel, 1995, Bhattacharyya, 1987). This theory has two components: analysis and synthesis. The present paper gives an overview account of the analysis results, Kharitonov’s theorem and its generalization (Chapellat, Bhattacharyya, 1989, Kharitonov, 1978), the Edge theorem (Bartlett, Hollot, & Lin, 1988), and related results as well as recent results on the parametric theory of synthesis and design (Bhattacharyya et al., 1995) of Proportional-Integral-Derivative (PID) controllers, Datta, Ho, and Bhattacharyya (2013), Silva, Datta, and Bhattacharyya (2007), Diaz-Rodriguez, Oliveira, and Bhattacharyya (2015), Diaz-Rodriguez and Bhattacharyya (2015).
In Kalman et al. (1960) introduced the state-variable approach and quadratic optimal control in the time-domain as new design approaches. This phase in the theory of automatic control systems arose out of the important new technological problems that were encountered at that time: the launching, maneuvering, guidance and tracking of space vehicles. A lot of effort was expended and rapid developments in both theory and practice took place. Optimal control theory was developed under the influence of many great researchers such as Pontryagin, Bellman, Kalman and Bucy. In the 1960s, Kalman introduced a number of key state-variable concepts. Among these were controllability, observability, optimal linear-quadratic regulator (LQR), state-feedback and optimal state estimation (Kalman filtering).
The optimal state feedback control produced by the LQR problem was guaranteed to be stabilizing for any quadratic performance index subject to mild conditions.
In a 1964 paper by Kalman (1964) which demonstrated that for SISO (single input-single output) systems the optimal LQR state-feedback control laws had some very strong guaranteed robustness properties, namely an infinite upper gain margin and a 60 ° phase margin, which in addition were independent of the particular quadratic index chosen. This is illustrated in Fig. 1 where the state feedback system designed via LQR optimal control has the above guaranteed stability margins at the loop breaking point “m”.
For some time, control scientists were generally led to believe that the extraordinary robustness properties of the LQR state feedback design were preserved when the control was implemented as an output feedback system through an observer. We depict this in Fig. 2 where the stability margin at the point m continues to equal that obtained in the state feedback system. However it was shown by Doyle and Stein (1979) that the margin at the point m′, which is much more meaningful, could be drastically less. This observation ushered in a period of renewed interest in robustness of closed loop designs.
Here we first describe a group of results which may be considered to be the central results of analysis in the field parametric theory of robust control. They are characterized by the important feature that they facilitate robust stability calculations by identifying apriori a small subset of parameters where stability or performance will be lost. Proofs of most of the results described here can be found in (Bhattacharyya, Chapellat, Keel, 1995, Bhattacharyya, Datta, Keel, 2009). We begin with the most spectacular of these results, namely Kharitonov’s Theorem Kharitonov (1978), which gives a surprisingly simple necessary and sufficient condition for the robust stability of an interval family of polynomials.
In 1997 it was shown that high order controllers were acutely sensitive to controller parameter perturbations. This led to a resurgence of interest in 3 term controllers, in particular PID controllers. An extensive theory of synthesis and design of PID controllers has been developed over the last 20 years (Bhattacharyya, Chapellat, Keel, 1995, Datta, Ho, Bhattacharyya, 2013, Diaz-Rodriguez, Bhattacharyya, 2015, Diaz-Rodriguez, Oliveira, Bhattacharyya, 2015, Silva, Datta, Bhattacharyya, 2007). We give an account of these elegant and useful results in the last part of the paper.
Section snippets
Kharitonov’s theorem
Consider the set of polynomials of degree n with real coefficients of the form where the coefficients lie within given ranges, Write and identify a polynomial δ(s) with its coefficient vector δ . Introduce the box of coefficients We assume that the degree remains invariant over the family, so that 0 ∉ [xn, yn]. Such a set of polynomials is called a real interval family and
Extremal properties of edges and vertices
In this section we state some useful extremal properties of the Kharitonov polynomials. Suppose that we have proved the stability of the family of polynomials with coefficients in the box Each polynomial in the family is stable. A natural question that arises now is the following: What point in Δ is closest to instability? The stability margin of this point is in a sense the worst case stability margin of the interval system. It turns out that
Robust state feedback stabilization
In this section we give an application of Kharitonov’s Theorem to robust stabilization by state feedback. We consider the following problem: Suppose that you are given a set of n nominal parameters together with a set of prescribed uncertainty ranges: Δa0, Δa1, and that you consider the family of monic polynomials, where To avoid trivial cases assume that the family
The edge theorem
The interval family dealt with in Kharitonov’s Theorem is a very special type of polytopic family. Moreover Kharitonov’s Theorem does not indicate where the roots of the polynomial family lie. The Edge Theorem deals with a general convex polytopic family of polynomials and gives a complete, exact and constructive characterization of the root set of the family. Such a characterization is obviously of value in the robustness and performance analysis of control systems. We describe this remarkable
The generalized kharitonov theorem
Kharitonov’s Theorem applies to polynomial families where the coefficients vary independently. In a typical control system problem, the closed loop characteristic polynomial coefficients vary interdependently. For example the closed loop characteristic polynomial coefficients may vary only through the perturbation of the plant parameters while the controller parameters remain fixed. The Generalized Kharitonov Theorem described below, deals with this situation and develops results that retain
Computation of the parametric stability margin
In this section we give a useful characterization of the parametric stability margin in the general case. This can be done by finding the largest stability ball in parameter space, centered at a “stable” nominal parameter value p0. The results to be described here were developed in Soh, Berger, and Dabke (1985), Tesi and Vicino (1989), Vicino (1991), Tsypkin and Polyak (1991), and Biernacki, Hwang, Bhattacharyya (1986). Let denote as usual an open set which is symmetric with respect to the
Controller fragility of high order controllers
In this section we focus on the 1997 results of Keel and Bhattacharyya (1997), where it was shown that high order controllers, even those designed to be robust to plant uncertainty, namely plant-robust, could be very fragile with respect to controller parameters, that is controller fragile.
Robust parametric synthesis: modern PID control
The demonstration of fragility of high order controllers in 1997 led to a resurgence of interest in low order controllers and in particular PID controllers. This led to the period of modern PID control, which started in 1997. These results complemented and built upon the classical results of Ziegler and Nichols (1942) and those of Åström and Hägglund (2006).
In the next subsection, we introduce PID control as a wonderful application of high gain feedback to the robust tracking and disturbance
PID synthesis for delay free continuous-time systems
In this section, we consider the synthesis and design of PID controllers for a continuous-time LTI plant, with underlying transfer function P(s) with n(m) poles (zeros). (see Fig. 20). We assume that the only information available to the designer is:
- 1.
Knowledge of the frequency response magnitude and phase, equivalently, P(ȷω), ω ∈ [0, ∞) if the plant is stable.
- 2.
Knowledge of a known stabilizing controller and the corresponding closed-loop frequency response G(ȷω).
Such assumptions are reasonable
PID controller synthesis for systems with delay
In this section, we show how the previous results can be extended to systems with delay. Consider the finite dimensional LTI plant PL with a cascaded delay in Fig. 21. Here P0 represents an LTI delay free system with a proper transfer function. The transfer functions of P0 and PL are denoted P0(s) and PL(s), respectively. We assume that frequency response measurements can be made at terminals “a” and “b,” that is on the delay system PL. Thus, the data we have is:
Computer-aided design (Bhattacharyya et al., 2009)
In this section we show some possibilities for computer-aided design using the above theory. The algorithm for the design of a PID Controller from the frequency response data of the system has been programmed in LabVIEW due to its user-friendly graphical environment. The Virtual Instrument (VI) has a front panel that is displayed to the user and a block diagram, where the computations are performed. The inputs to the LabVIEW program are the frequency response data and the number of RHP poles of
Continuous-time controllers: constant gain and phase Loci
For continuous-time controllers, it is possible to parametrize the controller parameters in a geometric form. For the cases of PI and PID controllers, the constant gain and phase loci result in ellipses and straight lines.
PI controllers Diaz-Rodriguez and Bhattacharyya (2016)
Let P(s) and C(s) denote the plant and controller transfer functions. The frequency response of the plant and controller are P(jω), C(jω) respectively where ω ∈ [0, ∞]. For a PI controller where KP and KI are design parameters. Then with we have
Discrete-time controllers: constant gain and phase loci
For discrete-time controllers, it is possible to parametrize the controller parameters in a geometric form. For the cases of PI and PID digital controllers, the constant gain and phase loci result in ellipses and straight lines.
Achievable performance with PI and PID controllers
The Gain-Phase Margin design curves represent the achievable performances, that is specified phase and gain margin, that our system can accomplish with a PI or PID controller. The procedure to construct these design curves is the following:
- 1.
Set a test range for phase margins and gain crossover frequencies.
- 2.
For discrete-time PI/PID controllers and continuous-time PI controllers, fix a value of phase margin and gain crossover frequency, plot the corresponding ellipse and straight line following
Multi-input multi-output (MIMO) control using single-Input single-Output (SISO) methods
In this section we describe a new approach to multivariable control using single-input single-output methods. The details may be found in Mohsenizadeh, Keel, and Bhattacharyya (2015). This has the potential to extend the design capabilities of SISO systems to multivariable systems.
Concluding remarks
This relatively brief overview of the subject of robust control under parametric uncertainty is necessarily incomplete and the author apologizes in advance for omissions of content or authorship and any personal bias in choice of topics. It is hoped that this article may be helpful to the reader who wishes to go deeper into specific topics. We have avoided some areas altogether such as W. M. Wonham’s geometric theory (Wonham, 1974) and robust adaptive control.
We have compiled an extensive list
Acknowledgments
The author acknowledges the help of Iván D. Díaz-Rodríguez and Sangjin Han in the preparation of this paper.
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