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The design of optimal strictly positive real (SPR) controllers using numerical optimization is considered. We focus on how to parameterize the SPR controllers being optimized and the effect of parameterization. Minimization of the closed-loop H2-norm is the optimization objective function. Various single-input single-output and multi-input multi-output controller parameterizations using transfer functions/matrices and state–space equations are considered. Depending on the controller form, constraints are enforced (i) using simple inequalities guaranteeing SPRness, (ii) in the frequency domain or, (iii) by implementing the Kalman–Yakubovich–Popov Lemma. None of the parameterizations we consider foster an observer-based controller structure. Simulated control of a single-link and a two-link flexible manipulators demonstrates the effectiveness of our proposed controller optimization formulations.

The passivity theorem is one of the most celebrated results in input–output systems theory. In general, a passive system is one that does not generate energy and a very strictly passive system is one that dissipates energy. The passivity theorem states that a passive system and a very strictly passive system connected in a negative-feedback loop are input–output stable [1]. This is an extremely powerful statement in the context of nonlinear control; stability of a nonlinear yet passive plant is guaranteed via control in the form of a very strictly passive operator. In the context of passive mechanical systems, inputs are forces and outputs are rates such as velocity. Flexible structures possess a large number of vibration modes, and in robotics applications have dynamics that are nonlinear. Flexible robotic manipulators are known to be passive via collocation of the joint torques and the angular velocity sensors, that is when the input–output map is between the joint torques and joint angular velocities. The passivity property for these collocated systems is independent of the system mass, stiffness, and modeled vibration modes. The noncollocated map between joint torques and end-tip velocity of a manipulator is not passive, but the map between a modified set of joint torques and a modified output, known as the μ-tip rate, has been shown to be passive, thus facilitating passivity-based control of the μ-tip rate [2]. Passive and very strictly passive systems that are linear and time-invariant (LTI) are closely related to positive real (PR) and strictly positive real (SPR) transfer functions or matrices [3]. The robust stability of nonlinear flexible robotic manipulators is assured via the passivity theorem when the controllers employed are SPR. In particular, spillover instabilities are avoided. In light of this important stability result, many authors have attempted to formulate rate controllers such that they are SPR. Benhabib et al. [4] suggested the use of SPR rate controllers to control large space structures where the controllers considered were not observer-based. Similarly, McLaren and Slater [5] investigated implementing positive real LQG controllers for the control of large space structures. Lozano-Leal and Joshi [6] investigated the design of LQG controllers, constraining the LQG weight matrices such that the resultant optimal controllers remain SPR. Haddad et al. [7] extend the work of Lozano-Leal and Joshi [6] to include an H∞ performance bound on the closed-loop, again by constraining the appropriate weighting matrices. The use of numerical optimization algorithms to find optimal SPR controllers has been considered in various papers. In Germoel and Gapsik [8] the design of observer-based SPR compensators using convex numerical optimization was considered. Using linear matrix inequality (LMI) constraints, the H2-optimal control problem was retooled to yield SPR controllers. The controllers were full order, meaning that the controllers and the plant model to be controlled have the same number of system states. Shimomura and Pullen [9] extended the work of Germoel and Gapsik [8], considering the use of iterative algorithms that overcome bilinear matrix inequality issues within the optimal SPR optimization formulation. Again, the resultant controllers were observer-based and full order. In both Germoel and Gapsik [8] and Shimomura and Pullen [9] the observer gains were those found via the solution to the unconstrained H2-optimal control problem. In Damaren [10], the optimization of single-input single-output SPR controllers of varying order was considered. The SPR controllers were not full order, nor observer-based compensators. Simple inequality constraints in the frequency domain via a transformation from the s-domain to the z-domain guaranteed SPRness. In Damaren et al. [11], optimal SPR controllers that approximate a given observer-based, full order controller were found by solving a quadratic programming problem with linear inequality constraints. In Henrion [12] a method using LMIs is presented whereby a transfer function can be designed to be robustly rendered SPR given a Hurwitz denominator polynomial. Other than the work of Benhabib et al. [4] and Damaren [10], the existing SPR design schemes (that is, optimal design schemes) yield controllers that are observer-based, and thus have the same order as the plant. It remains an open question as to whether or not the optimal SPR controller that solves the H2 control problem is observer-based, or even should be the same order as the plant being controlled. Because unconstrained H2-optimal controllers are observer-based, it does not mean that optimal SPR controllers must be observer-based. Also, if the dimension of the plant is large (as is the case with flexible robotic manipulators and structures), having a controller that is lower order yet still optimal is desirable. With this in mind, we will explore various controller parameterizations of various orders that are not observer-based, constrain the controllers to be SPR (in different ways, depending on the parameterization at hand), and optimize the controllers numerically by minimizing the closed-loop H2-norm of the system. One of the questions we hope to shed light on is that of controller order. Additionally, the existing literature often considers the control of a pinned–pinned Euler–Bernoulli beam [7], [8], [9] and [10]. In our work, we consider the tip control of single- and two-link flexible robotic manipulators. Recall that the SPR controllers we parameterize and optimize will be rate (velocity) controllers. To realize position control, the manipulators must also be compensated by position (i.e., proportional) control. We will include the proportional control gain as a design variable, which is equivalent to optimally designing the rigid-body mode of the plant in conjunction with the rate controller. The outline of the paper is as follows: in the next section we will briefly review the conditions which ensure that a system is SPR. We then consider flexible manipulator modeling and tip-based control. Both two- and single-link flexible manipulator models will be considered. In Section 4 we will state our numerical optimization objective function. Although we will consider various controller parameterizations and constraint methods in this paper, we will always be minimizing the closed-loop H2-norm of the system. We then move onto the main contributions of the paper. In Section 5 we consider three different SISO controller parameterizations, as well as ways to constrain the controllers based on the parameterizations. In Section 6 we then modify the parameterizations (and constraints) for MIMO controller design. Simulated control of the single- and two-link flexible manipulators is included. We close with a discussion and final remarks.

The objective function was minimized to a value of J2=80.9141. The convergence tolerance was set to 1 ×10−8. The frequency response of the controller G(s) created via the state–space parameterization and optimization is shown in Fig. 13. Shown in Fig. 14 is the system response of the two-link manipulator.The frequency response of the full-order SPR controller is quite interesting. Notice that the gain is high, but not as high as, for example, the variable-order SPR controller gain in Fig. 11. Also, note that the full-order SPR controller begins to roll off above 104 rad/s, unlike both the diagonal-decoupled (Section 6.1) and variable-order SPR controllers. The performance of the closed-loop system as controlled by the full-order SPR controller is quite good. There is only a moderate amount of overshoot at the end of the manipulator maneuver, with no residual vibration. The full-order SPR controller outperforms the diagonal-decoupled SPR controller significantly. However, the performance of the variable-order and full-order SPR controller is essentially the same. Recall that each controller parameterization is used within an optimization scheme that minimized the closed-loop H2-norm given a particular set of weights (i.e., the B1, C1, D12, D21 matrices). Interestingly, the present parameterization is able to attain a smaller closed-loop H2-norm as compared to the diagonal-decoupled and variable-order parameterizations. This is due to the fact that the full-order parameterization allows for controllers that are of greater order (compared to the other parameterizations).