Evaluation and Improvement of Control Vector Iteration Procedures for Optimal Control

An alternate graphical representation of linear, time-invariant, multi-input, multi-output (MIMO) system dynamics is proposed that is highly suited for exploring the influence of closedloop system parameters. The development is based on the adjustment of a scalar forward gain multiplying a cascaded multivariable controller/plant embedded in an output feedback configuration. By tracking the closed-loop eigenvalues as explicit functions of gain, it is possible to visualize the multivariable root loci in a set of "gain plots" consisting of two graphs: (i) magnitude of system eigenvalues versus gain and (ii) argument (angle) of system eigenvalues versus gain. The gain plots offer an alternative perspective of the standard MIMO root locus plot by depicting unambiguously the polar coordinates of each eigenvalue in the complex plane. Two example problems demonstrate the utility of gain plots for interpreting closed-loop multivariable system behavior. Introduction Since their introduction, classical control tools have been popular for analysis and design of single-input, single-output (SISO) systems. These tools may be viewed as specialized versions of more general methods that are applicable to multiinput, multi-output (MIMO) systems. Although modern "statespace" control techniques (relying on dynamic models of internal structure) are generally promoted as the predominant tools for multivariable system analysis, the classical control extensions offer several advantages, including requiring only an input-output map and providing direct insight into stability, performance, and robustness of MIMO systems. The understanding generated by these graphically based methods for the analysis and design of MIMO systems is a prime motivator of this research. An early graphical method for investigating the stability of linear, time-invariant (LTI) SISO systems was developed by ference transfer function matrix ([I + G(s)] where G(s) is the open-loop system transfer function matrix, rather than just 1 + g(s) for the SISO case where g{s) is the system transfer function). Despite the complication, significant research has supported the MIMO Nyquist extension for assessment of multivariable system stability and robustness The Bode plots Although promoted as an SISO tool, Evans root locus method To aid the controls engineer in extracting more information from the multivariable Evans root locus plot, we propose a set of "gain plots" that provide a direct and unique window into the stability, performance, and robustness of LTI MIMO systems. A conceptual framework motivating the gain plots and a discussion of their applicability to SISO systems has been presented previously Multivariable Eigenvalue Description Basic MIMO Concepts. A LTI MIMO plant can be represented in the standard state-space form as where state vector x p is length n, input vector u is length m, and output vector y is length m. Matrices A p , B p , C p , and D p are the system matrix, the control influence matrix, the output matrix, and the feed-forward matrix, respectively, of the plant with appropriate dimensions. The plant input-output dynamics are governed by the transfer function matrix, G p (s), GpW^CplsI-ApV'Bp + Vp (3) The system is embedded in the closed-loop configuration shown in Fig. 1 Ml MO closed-loop negative feedback configuration where A c , B c , C c , and D c are the controller matrices representing its internal structure, in similarity to Eqs. In the MIMO root locus plot, the migration of the eigenvalues of G*(5) in the complex plane is graphed for 0 < k < oo. (By equating the determinant of [I + kG p (s)G c (s)] to zero, the MIMO generalization of the SISO characteristic equation The presence of the determinant is the major challenge in generalizing the SISO root locus sketching rules to MIMO systems and complicates the root locus plot.) The closed-loop system dynamics can alternatively be cast in state-space form in terms of state vector r . The closed-loop system matrix then becomes where The eigenvalues of the closed-loop system,5 = X; = eig(A') (i = 1,2, . . . , «), may be computed numerically from Eq. (6). In the examples, the loci of the eigenvalues are calculated as k is monotonically increased from zero. High Gain Behavior. As the gain is swept from zero to infinity, the closed-loop eigenvalues trace out "root loci" in the complex plane. At zero gain, the poles of the closed-loop system are the open-loop eigenvalues. At infinite gain some of the eigenvalues approach finite transmission zeros, defined to be those values of s that satisfy the generalized eigenvalue problem. In the absence of pole/zero cancellation, the finite transmission zeros are the roots of the determinants of G p (s) and G c (s). Algorithms have been developed for efficient and accurate computation of transmission zeros The eigenvalues can be considered as always migrating from the open-loop poles to their matching transmission zeros MIMO Gain Plots. Just as the Bode plots embellish the information of the Nyquist diagram by exposing frequency explicitly in a set of magnitude versus frequency and angle (phase) versus frequency plots, it follows that a pair of gain plots (Kurfess and Nagurka, 1991) can enhance the standard root locus plot. As the gain-domain analog of the frequencydomain Bode plots, the gain plots explicitly depict the eigenvalue magnitude versus gain in a magnitude gain plot, and the eigenvalue angle versus gain in an angle gain plot. In similarity to the Bode plots, the magnitude gain plot employs a log-log scale whereas the angle gain plot uses a semi-log scale (with the logarithms being base 10). Although gain is selected as the variable of interest in the gain plots, it should be noted that any scalar parameter may be used in the geometric analysis, leading to the more generic idea of parametric plots. Gain plots can be drawn for both SISO and MIMO systems. In MIMO systems it is assumed that a single scalar gain amplifies all controller/plant inputs. For such systems, inspection of the magnitude and angle gain plots enables one to uniquely identify locus branches as a function of gain. As such, gain plots are a natural complement to multivariable root locus plots, where uncharacteristically confusing eigenvalue trajectories can result from being drawn in a single complex plane. Furthermore, it can be shown that the slopes of the lines in the gain plots are proportionally related to the root sensitivity function (Kurfess and Nagurka, 1992). MIMO Examples This section presents two multivariable examples. The first example introduces the concept of the gain plots and demonstrates the insight they offer by "unwrapping" the multivariable root locus and exposing unambiguous behavior. The second example highlights the power of the gain plots in revealing typical multivariable properties, such as high gain Butterworth patterns. Example 1: Coupled MIMO Example. The forward loop dynamics of this example are given by the transfer function matrix (Equation The gain plots presented in The gain plots highlight several other important features. For example, they show that the gains corresponding to the complex conjugate eigenvalue pairs break into the real axis and then proceed toward ± oo. Complex conjugate eigenvalues are shown as symmetric lines about either the 180 or 0 deg line with equal magnitudes. Purely real eigenvalues possess equal angles (180 or 0 deg) but distinct magnitudes. This behavior is demonstrated in The rates at which the eigenvalues increase toward infinite magnitude is seen in the magnitude gain plot of From Conclusions In typical MIMO root locus plots trajectories may be camouflaged as branches may overlap. Gain plots are promoted as a means to "untangle" MIMO eigenvalue trajectories. The major enhancement is the visualization of eigenvalue trajectories as an explicit function of gain, assumed here to be the same static gain applied to all error signals. The perspective presented in this note is intended to complement the many tools available to the controls engineer. In particular, for MIMO systems the gain plots provide: (/) a unique description of eigenvalues and their trajectories as a parameter, such as gain, is varied, (ii) a geometric depiction of the Riemann sheets at high gain, and (Hi) a rich educational tool for conducting parametric analyses of multivariable systems. Research efforts, currently underway, may shed additional light on gain plots for multivariable systems. In addition, work by MacFarlane and'Postlethwaite (1977 and In conclusion, gain plots enrich the multivarible root locus plot in much the same way that singular value frequency plots are an alternate and extended presentation of the multivariable Nyquist diagram. Their use in conjunction with the multivariable root locus provides a valuable geometric perspective on multivariable system behavior. Acknowledgment The authors wish to thank Mr. Ssu-Kuei Wang for his help, and for his earnest enthusiasm of gain plots for studying multivariable and optimal systems