Review article
Robust control under parametric uncertainty: An overview and recent results

https://doi.org/10.1016/j.arcontrol.2017.05.001Get rights and content

Abstract

Modern Robust Control has had two distinct lines of development: (a) Robustness through quadratic optimization and (b) Robustness under parametric uncertainty. The first approach consists of Kalman’s Linear Quadratic Regulator and H optimal control. The second approach is the focus of this overview paper. It provides an account of both analysis as well as synthesis based results. This line of results was sparked by the appearance of Kharitonov’s Theorem in the early1980s. This result was rapidly followed by further results on the stability of polytopes of polynomials such as the Edge Theorem and the Generalized Kharitonov Theorem, stability of systems under norm bounded perturbations and the computation of parametric stability margins. Many of these analysis results established extremal testing sets where stability or performance would breakdown. Starting in 1997, when it was established that high order controllers were fragile, attention turned to the synthesis and design of the parameters of low order controllers such as three term controllers and more particularly Proportional-Integral-Derivative (PID) controllers. An extensive theory of design of such systems has developed in the last twenty years. We provide a summary without proofs, of many of these 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 I(s) of polynomials of degree n with real coefficients of the form δ(s)=δo+δ1s+δ2s2+δ3s3+δ4s4++δnsnwhere the coefficients lie within given ranges, δ0[x0,y0],δ1[x1,y1],,δn[xn,yn].Write δ̲:=[δ0,δ1,,δn]and identify a polynomial δ(s) with its coefficient vector δ . Introduce the box of coefficients Δ:={δ̲:δ̲Rn+1,xiδiyi,i=0,1,,n}.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 δ(s)=δ0+δ1s+δ2s2++δnsn,with coefficients in the box Δ=[x0,y0]×[x1,y1]××[xn,yn].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 {a00,a10,,an10},together with a set of prescribed uncertainty ranges: Δa0, Δa1, ,Δan1, and that you consider the family I0̲(s) of monic polynomials, δ(s)=δ0+δ1s+δ2s2++δn1sn1+sn,where δ0[a00Δa02,a00+Δa02],,δn1[an10Δan12,an10+Δan12].To avoid trivial cases assume that the family I0̲(s)

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 SC 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, Pω), ω ∈ [0, ∞) if the plant is stable.

  • 2.

    Knowledge of a known stabilizing controller and the corresponding closed-loop frequency response Gω).

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: PL(jω)=ejωLP0(jω)=mL(ω)ejϕL(ω),

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(), C() respectively where ω ∈ [0, ∞]. For a PI controller C(s)=KPs+KIs,where KP and KI are design parameters. Then with s=jω, we have |C(jω)|2=KP2+

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.

References (36)

  • Y.Z. Tsypkin et al.

    Frequency domain criteria for l p-robust stability of continuous linear systems

    IEEE Transactions on Automatic Control

    (1991)
  • J. Ackermann

    Robust control: Systems with uncertain physical parameters

    (2012)
  • K.J. Åström et al.

    Advanced PID control

    (2006)
  • B.R. Barmish et al.

    New tools for robustness of linear systems

    IEEE Transactions on Automatic Control

    (1994)
  • A.C. Bartlett et al.

    Root locations of an entire polytope of polynomials: It suffices to check the edges

    Mathematics of Control, Signals and Systems

    (1988)
  • S. Bhattacharyya et al.

    Robust control: The parametric approach

    (1995)
  • S.P. Bhattacharyya

    Robust stabilization against structured perurbations

    (1987)
  • S.P. Bhattacharyya et al.

    Linear control theory: Structure, robustness, and optimization

    (2009)
  • R. Biernacki et al.

    Robust stabilization of plants subject to structured real parameter perturbations

    Technical Report TCSL Report

    (1986)
  • H.W. Bode

    Network analysis and feedback amplifier design

    (1945)
  • H. Chapellat et al.

    A generalization of kharitonov’s theorem; robust stability of interval plants

    IEEE Transactions on Automatic Control

    (1989)
  • A. Datta et al.

    Structure and synthesis of PID controllers

    (2013)
  • I.D. Diaz-Rodriguez et al.

    Modern design of classical controllers: Digital pi controllers

    Proceedings of the 2015 IEEE international conference on industrial technology (ICIT)

    (2015)
  • I.D. Diaz-Rodriguez et al.

    Pi controller design in the achievable gain-phase margin plane

    Proceedings of the 55th IEEE conference on decision and control (CDC)

    (2016)
  • I.D. Diaz-Rodriguez et al.

    Advanced tuning for Ziegler-Nichols plants

    Proceedings of the 2017 20th world congress of the international federation of automatic control (IFAC)

    (2017)
  • I.D. Diaz-Rodriguez et al.

    Modern design of classical controllers: Digital PID controllers

    Proceedings of the 2015 IEEE 24th international symposium on industrial electronics (ISIE)

    (2015)
  • J.C. Doyle et al.

    State-space solutions to standard h 2 and h control problems

    IEEE Transactions on Automatic Control

    (1989)
  • J.C. Doyle et al.

    Robustness with observers

    Technical Report

    (1979)
  • Cited by (84)

    • Calculation of robustly relatively stabilizing PID controllers for linear time-invariant systems with unstructured uncertainty

      2022, ISA Transactions
      Citation Excerpt :

      The uncertainty, which is incorporated into the model, can be considered as a cost for keeping the relative simplicity of an LTI model even for the systems with much more complicated (even nonlinear) behavior, the imprecise physical properties knowledge, or the parameters (“slowly”) depending on changing conditions. There are three principal approaches to modeling the uncertainty for the purpose of robust control, namely parametric uncertainty [8–11], unstructured uncertainty [9,12–16], and Linear Fractional Transformation (LFT) [13,17–20]. The first, and probably the most natural and comprehensible, approach uses so-called parametric uncertainty [8–11].

    • Towards stable milling: Principle and application of active contact robotic milling

      2022, International Journal of Machine Tools and Manufacture
    View all citing articles on Scopus
    View full text