Output Feedback Nonlinear 𝐻_∞-Tracking Control of a Nonminimum-Phase 2-DOF Underactuated Mechanical System

Abstract

Nonlinear 𝐻_∞ synthesis is developed to solve the tracking control problem restricted to a two degrees-of-freedom (DOF) underactuated mechanical manipulator where position measurements are the only available information for feedback. A local 𝐻_∞ controller is derived by means of a certain perturbation of the differential Riccati equations, appearing in solving the 𝐻_∞ control problem for the linearized system. Stabilizability and detectability properties of the control system are thus ensured by the existence of the proper solutions of the unperturbed differential Riccati equations, and hence the proposed synthesis procedure obviates an extra verification work of these properties. Due to the nature of the approach, the resulting controller additionally yields the desired robustness properties against unknown but bounded external disturbances. The desired trajectory is centered at the upright position where the manipulator becomes a nonminimum-phase system. Simulation results made for a double pendulum show the effectiveness of the proposed controller.

1. Introduction

The focus of this paper is to solve the tracking control problem for a 2-DOF underactuated mechanism via nonlinear 𝐻_∞-control for time-varying systems [1] where joint position measurements are the only available information for feedback. Further research applications in the control of underactuated systems have gone in many directions, for example, fully actuated robots where it is required that motion continues in spite of a failure of any of its actuators. Other typical examples are the systems where its desired operation mode is oscillatory such as biped walking robots where a periodic trajectory is required to produce a coordinated motion (see, e.g., [2]); hopping robots where thrust, decompression, flight, and compression phases are also governed by a periodic motion (see, e.g., [3]); tracking control in drive systems with backlash where usually the position sensor is placed on the side of the motor instead of the side of the load (see, e.g., [4] and [5, page 456]); juggling systems [6]; among others.

Objective
In the present paper, we address the output tracking control problem in nonminimum-phase underactuated mechanical system. Representative works in this topic include orbital stabilization of underactuated systems by means of reference models as generator of limit cycles (see, e.g., [7–10]). In particular, this paper is devoted to the solution of a periodic balancing problem for a two-link underactuated mechanical manipulator introduced in [7], whose first link is not actuated whereas the second joint is actuated.

Contribution
For nonlinear mechanical systems, tracking control problem is known to be more difficult than stabilization mainly for underactuated systems whose initial conditions are close to an unstable equilibrium point. The central problem in nonminimum-phase underactuated systems, solved here, is the specification and design of output feedback inner-tracking controllers to drive the output (joint position) to a nontrivial reference trajectory in spite of external disturbances.
The prior work on the tracking control of nonminiminum-phase systems includes, among others, the results of Consolini and Tosques [11] and Berkemeier and Fearing [7] who developed an exact tracking control via state-feedback. Wang [12] partially addressed the above problem by considering the regulation problem in linear systems. A unified treatment of the control of such systems via output feedback can be found in [13]. In the present paper, the nonlinear 𝐻_∞ control approach is extended for the time-varying nonlinear nonminimum-phase systems applied to tracking control problems for underactuated mechanical systems.

Methodology
The method we use for defining a desired trajectory for underactuated system is based on the work of Berkemeier and Fearing [7]. The method was successfully applied to derive a set of exact trajectories for the nonlinear equation which involve inverted periodic motion. This method was selected because the desired trajectories are at least twice-differentiable satisfying the smoothness assumption imposed on the system for the development of 𝐻_∞ control theory (see, e.g., [14]).
The above problem is locally resolved within the framework of nonlinear 𝐻_∞-control methods from [1, 14–16]. Those methods do not admit a straightforward application to the problem in question because in contrast to the standard case, a partial state stabilization (i.e., asymptotic stabilization of the output of the system) is only required provided that the complementary variables remain bounded. Their modification developed in the present paper is of the same level of simplicity, and it follows the common practice of proper solution to corresponding differential Riccati equations which is performed numerically.
The aforementioned 𝐻_∞ synthesis took its origins from game-theoretic approach from Basar and Bernhard [15], and the ℒ₂-gain analysis from Isidori and Astolfi [14]. It followed the line of reasoning, used in Orlov et al. [17], where the corresponding Hamilton-Jacobi-Isaacs expressions were required to be negative definite rather than semidefinite.
In contrast to the standard ℒ₂-gain analysis from Isidori and Astolfi [14] and Van der Shaft [18] the resulting 𝐻_∞ design procedure imposed the nonstabilizability-detectability conditions on the control systems. Under appropriate assumptions the existence of suitable solutions of Riccati differential equations, appearing in solving the 𝐻_∞ control problem for the linearized system, was shown to be necessary and sufficient condition for a local solution of the 𝐻_∞ control problem to exist. This mean that the verification of stabilizability and detectability conditions will be not required. A local solution was then derived by means of a certain perturbation of the Riccati equations when these unperturbed equations had bounded positive-semidefinite solutions. Thus, the local stabilizability and detectability properties of the control system were ensured by the existence of the proper solutions of the unperturbed Riccati equations, and hence the 𝐻_∞ synthesis obviated any extra work on verification of these properties.

Organization of the Paper
The paper is organized as follows. Background materials on time-varying 𝐻_∞-control synthesis are presented in Section 2. The tracking control problem of a 2-DOF underactuated system and its state equations are introduced in Section 3 while desired trajectory synthesis procedure is also discussed. A nonlinear 𝐻_∞-output control for time varying systems is also constructed. Performance issues of this controller are illustrated in a simulation study in Section 4. Finally, Section 5 presents conclusions.

2. Background Material on Nonlinear 𝐻_∞-Control of Time-Varying Systems

2.1. Basic Assumptions and Problem Statement

Consider a nonlinear system of the form 𝑥̇=𝑓(𝑥,𝑡)+𝑔₁(𝑥,𝑡)𝑤+𝑔₂(𝑥,𝑡)𝑢,𝑧=ℎ₁(𝑥,𝑡)+𝑘₁₂(𝑥,𝑡)𝑢,𝑦=ℎ₂(𝑥,𝑡)+𝑘₂₁(𝑥,𝑡)𝑤, (1) where 𝑥∈ℝ^𝑛 is the state space vector, 𝑡∈ℝ is the time, 𝑢∈ℝ^𝑚 is the control input, 𝑤∈ℝ^𝑟 is the unknown disturbance, 𝑧∈ℝ^𝑙 is the unknown output to be controlled, and 𝑦∈ℝ^𝑝 is the only available measurement on the system. The following assumptions are assumed to hold.

(A1)The functions 𝑓(𝑥,𝑡), 𝑔₁(𝑥,𝑡), 𝑔₂(𝑥,𝑡), ℎ₁(𝑥,𝑡), ℎ₂(𝑥,𝑡), 𝑘₁₂(𝑥,𝑡), and 𝑘₂₁(𝑥,𝑡) are piecewise continuous in 𝑡 for all 𝑥 and locally Lipschitz continuous in 𝑥 for all 𝑡. (A2)𝑓(0,𝑡)=0, ℎ₁(0,𝑡), and ℎ₂(0,𝑡)=0 for all 𝑡. (A3)ℎ^𝑇₁(𝑥,𝑡)𝑘₁₂(𝑥,𝑡)=0, 𝑘₁₂(𝑥,𝑡)^𝑇𝑘₁₂(𝑥,𝑡)=𝐼, 𝑘₂₁(𝑥,𝑡)𝑔₁(𝑥,𝑡)=0, 𝑘₂₁(𝑥,𝑡)𝑘^𝑇₂₁(𝑥,𝑡)=𝐼.

These assumptions are made for technical reasons. Assumption (A1) guarantees the well-posedness of the above dynamic system, while being enforced by integrable exogenous inputs. Assumption (A2) ensures that the origin is an equilibrium point of the nondriven (𝑢=0) disturbance-free (𝑤=0) dynamic system (1). Assumption (A3) is a simplifying assumption inherited from the standard 𝐻_∞-control problem.

A causal dynamic feedback compensator 𝑢=𝒦(𝜉,𝑡),𝜉̇=𝜂(𝜉,𝑦,𝑡) (2)(3) with internal state 𝜉∈ℝ^𝑠, is said to be globally (locally) admissible controller if the closed-loop systems (1)-(2) are globally (uniformly) asymptotically stable when 𝑤=0.

Given a real number 𝛾>0, it is said that systems (1), (2) haveℒ₂-gain less than 𝛾 if the response 𝑧, resulting from 𝑤 for initial state 𝑥(𝑡₀)=0, 𝜉(𝑡₀)=0, satisfies ^𝑡1_𝑡0‖𝑧(𝑡)‖²𝑑𝑡<𝛾^2𝑡1_𝑡0‖𝑤(𝑡)‖²𝑑𝑡 (4) for all 𝑡₁>𝑡₀ and all piecewise continuous functions 𝑤(𝑡).

The time-varying 𝐻_∞-control problem is to find a globally admissible controller (2)-(3) such that ℒ₂-gain of the closed-loop systems (1), (2), (3) is less than 𝛾. In turn, a locally admissible controller (2), (3) is said to be a local solution of the 𝐻_∞-control problem if there exists a neighborhood 𝑈 of the equilibrium such that inequality (4) is satisfied for all 𝑡₁>𝑡₀ and all piecewise continuous functions 𝑤(𝑡) for which the state trajectory of the closed-loop system starting from the initial point (𝑥(𝑡₀),𝜉(𝑡₀))=(0,0) remains in 𝑈 for all 𝑡∈[𝑡₀,𝑡₁].

2.2. Local State-Space Solution

Assumptions (A1)–(A3) allow one to linearize the corresponding Hamilton-Jacobi-Isaacs inequalities from [1] that arise in the state feedback and output-injection design thereby yielding a local solution of the time-varying 𝐻_∞-control problem. The subsequent local analysis involves the linear time-varying 𝐻_∞-control problem for the system 𝑥̇=𝐴(𝑡)𝑥+𝐵₁(𝑡)𝑤+𝐵₂(𝑡)𝑢,𝑧=𝐶₁(𝑡)𝑥+𝐷₁₂(𝑡)𝑢,𝑦=𝐶₂(𝑡)𝑥+𝐷₂₁(𝑡)𝑤, (5) where 𝐴(𝑡)=𝜕𝑓𝜕𝑥(0,𝑡),𝐵₁(𝑡)=𝑔₁(0,𝑡),𝐵₂(𝑡)=𝑔₂𝐶(0,𝑡),₁(𝑡)=𝜕ℎ₁𝜕𝑥(0,𝑡),𝐷₁₂(𝑡)=𝑘₁₂𝐶(0,𝑡),₂(𝑡)=𝜕ℎ₂𝜕𝑥(0,𝑡),𝐷₂₁(𝑡)=𝑘₂₁(0,𝑡). (6) Such a problem is now well understood if the linear system (A.1) is stabilizable and detectable from 𝑢 and 𝑦, respectively. Under these assumptions, the following conditions are necessary and sufficient for a solution to exist (see, e.g., [16]).

(C1)There exists a bounded positive semidefinite symmetric solution of the equation −𝑃̇(𝑡)=𝑃(𝑡)𝐴(𝑡)+𝐴^𝑇(𝑡)𝑃(𝑡)+𝐶^𝑇₁(𝑡)𝐶₁1(𝑡)+𝑃(𝑡)[𝛾²𝐵₁(𝑡)𝐵^𝑇₁(𝑡)−𝐵₂(𝑡)𝐵^𝑇₂(𝑡)]𝑃(𝑡), (7) such that the system 𝑥̇=[𝐴−(𝐵₂𝐵^𝑇₂−𝛾⁻²𝐵₁𝐵^𝑇₁)𝑃](𝑡)𝑥(𝑡) (8) is exponentially stable. (Throughout, a time-dependent 𝑛×𝑛-matrix 𝑃(𝑡) is positive semidefinite if and only if 𝑥^𝑇𝑃(𝑡)𝑥≥0 for all 𝑛-vectors 𝑥 and all time instants 𝑡 whereas 𝑃(𝑡) is positive definite if and only if 𝑥^𝑇𝑃(𝑡)𝑥≥𝑚𝑥^𝑇𝑥 for all 𝑥 and 𝑡, and some constant 𝑚>0. Respectively, 𝑃(𝑡) is bounded if and only if ‖𝑃(𝑡)‖≤𝑚₀ for all 𝑡 and some constant 𝑚₀>0. )(C2)There exists a bounded positive semidefinite symmetric solution to the equation 𝑍̇(𝑡)=𝐴(𝑡)𝑍(𝑡)+𝑍(𝑡)𝐴^𝑇(𝑡)+𝐵₁(𝑡)𝐵^𝑇₁1(𝑡)+𝑍(𝑡)×[𝛾²𝑃(𝑡)𝐵₂(𝑡)𝐵^𝑇₂(𝑡)𝑃(𝑡)−𝐶^𝑇₂(𝑡)𝐶₂(𝑡)]𝑍(𝑡) (9) specified with 𝐴(𝑡)=𝐴(𝑡)+𝛾⁻²𝐵₁(𝑡)𝐵^𝑇₁(𝑡)𝑃(𝑡), such that the system 𝑥̇=[𝐴−𝑍(𝐶^𝑇₂𝐶₂−𝛾⁻²𝑃𝐵₂𝐵^𝑇₂𝑃)](𝑡)𝑥(𝑡) (10) is exponentially stable.

According to the time-varying bounded real lemma [17], conditions (C1) and (C2) ensure that there exists a positive constant 𝜀₀ such that the system of the perturbed differential Riccati equations −𝑃̇_𝜀=𝑃_𝜀(𝑡)𝐴(𝑡)+𝐴^𝑇(𝑡)𝑃_𝜀(𝑡)+𝐶^𝑇₁(𝑡)𝐶₁(𝑡)+𝑃_𝜀1(𝑡)[𝛾²𝐵₁(𝑡)𝐵^𝑇₁(𝑡)−𝐵₂(𝑡)𝐵^𝑇₂(𝑡)]𝑃_𝜀(𝑡)+𝜀𝐼,𝑍̇_𝜀=𝐴(𝑡)𝑍_𝜀(𝑡)+𝑍𝐴^𝑇(𝑡)+𝐵₁(𝑡)𝐵^𝑇₁(𝑡)+𝑍_𝜀1(𝑡)×[𝛾²𝑃(𝑡)𝐵₂(𝑡)𝐵^𝑇₂(𝑡)𝑃(𝑡)−𝐶^𝑇₂(𝑡)𝐶₂(𝑡)]𝑍_𝜀(𝑡)+𝜀𝐼, (11)(12) has a unique positive definite symmetric solution (𝑃_𝜀(𝑡),𝑍_𝜀(𝑡)) for each 𝜀∈(0,𝜀₀) where 𝐴(𝑡)=𝐴(𝑡)+𝛾⁻²𝐵₁(𝑡)𝐵^𝑇₁(𝑡)𝑃_𝜀(𝑡).

Differential equations (11) and (12) are subsequently utilized to derive a local solution of the nonlinear 𝐻_∞-control problem for (1). The following resuls is extracted from [1].

Theorem 1. Let conditions (C1) and (C2) be satisfied, and let (𝑃_𝜀(𝑡),𝑍_𝜀(𝑡)) be the corresponding positive solution of (11), (12) under some 𝜀>0. Then the output feedback 𝜉̇=𝑓(𝜉,𝑡)+[1𝛾²𝑔₁(𝜉,𝑡)𝑔^𝑇₁(𝜉,𝑡)−𝑔₂(𝜉,𝑡)𝑔^𝑇₂(𝜉,𝑡)]×𝑃_𝜀(𝑡)𝜉+𝑍_𝜀(𝑡)𝐶^𝑇₂(𝑡)[𝑦−ℎ₂(𝜉,𝑡)],𝑢=−𝑔^𝑇₂(𝜉,𝑡)𝑃_𝜀(𝑡)𝜉 (13) is a local solution of the 𝐻_∞-control problem.

In what follows, Theorem 1 is used to design an 𝐻_∞ tracking controller for the underactuated system.

3. 𝐻_∞-Control of Underactuated System

3.1. Problem Statement

Consider the equation of motion of an underactuated mechanical system given by the Lagrange equation 𝑀(𝑞)𝑞̈+𝑁(𝑞,𝑞)̇=𝐵𝜏+𝑤_𝑥(𝑡), (14) where 𝑞=[𝑞₁,𝑞₂]∈ℝ² is a vector of generalized coordinates where 𝑞₁ and 𝑞₂ are the unactuated and actuated joints, respectively; 𝜏∈ℝ is the vector of applied joint torques; 𝐵=[0,1]^𝑇 is the input matrix that maps the torque input 𝜏 to the joint of coordinates space; 𝑤_𝑥(𝑡)∈ℝ² is the unknown disturbance vector to account for destabilizing model discrepancies due to hard-to-model nonlinear phenomena such as friction and backlash, 𝑡∈ℝ is the time; 𝑀(𝑞)∈ℝ^2×2 is the symmetric positive-definite inertia matrix; 𝑁(𝑞,𝑞)̇=[𝑁₁(𝑞,𝑞)̇,𝑁₂(𝑞,𝑞)̇]^𝑇∈ℝ² is the vector that contains the Coriolis, centrifugal, and gravity torques. Appendix A presents the dynamic model of the double pendulum.

The control objective is to design a nonlinear 𝐻_∞ tracking controller that ensures lim_𝑡→∞‖𝑞(𝑡)−𝑞_𝑑(𝑡)‖=0 (15) to be achieved asymptotically, while also attenuates the influence of external disturbances. Here, 𝑞_𝑑(𝑡)∈ℝ² is a continuously differentiable desired trajectory.

3.2. The Desired Trajectory

We point out that the present formulation is different from typical formulation of output tracking and regulation [1, 19], where the set point or the reference trajectory is a priori given because underactuated systems are not feedback or input-state linearizables due to its complexity. Therefore, special attention is required in the selection of the desired trajectory for the system under study. There are a few procedures to find desired trajectories for underactuated systems in literature [7, 8, 10, 20, 21], and under reasonable hypotheses all of them can be used to obtain a desired trajectory. The methodology from [7] is used here, where a set of exact trajectories is derived for the nonlinear equation of motion which involves inverted periodic motion. To this end, let us consider the desired trajectory which is solution of 𝑑𝑞𝑑𝑡𝑞̇^𝑑_𝑑=𝑞̇_𝑑𝑀⁻¹(𝑞_𝑑)[𝐵𝜏(𝑞_𝑑,𝑞̇_𝑑)−𝑁(𝑞_𝑑,𝑞̇_𝑑,)] (16) where 𝑞_𝑑(𝑡)∈ℝ²,𝑞̇_𝑑(𝑡)∈ℝ² are the desired joint positions and velocities, respectively, and 𝜏=𝑁₂(𝑞_𝑑,𝑞̇_𝑑)−2𝑀₂₂(𝑞_𝑑)−𝑀₁₂(𝑞_𝑑)2𝑀₁₂(𝑞_𝑑)−𝑀₁₁(𝑞_𝑑)𝑁₁(𝑞_𝑑,𝑞̇_𝑑) (17) is the control input that makes the desired virtual output 𝑦_𝑑(𝑡)=2𝑞_𝑑1(𝑡)+𝑞_𝑑2(𝑡)−𝜙 (18) remains at zero for all 𝑡≥0 when 𝑦_𝑑(𝑡) starts at 𝑦_𝑑(0)=𝑦̇_𝑑(0)=0,𝜙 is a constant parameter that parameterizes the equilibrium manifold of the pendulum, and the oscillations given by (16)–(18) are around this manifold. Throughout, we confine our research interest in desired oscillations around the upright position of the pendulum which correspond to the more difficult case due that the open-loop system has an unstable zero dynamics. Toward this end, we choose 𝜙=𝜋 for all 𝑡≥0 in (18). It was shown in [7] that (16) and (17) generate a set of exact periodic trajectories given by 𝑞̈_𝑑1=𝑐₄sin(𝑞_𝑑1)+𝑐₅sin(𝜙−𝑞_𝑑1)𝑐₁−𝑐₂, (19) that can be interpreted as the zero dynamics of the system (16) with respect to the output 𝑦_𝑑(𝑡). Time evolution of the desired trajectory is illustrated in Figure 1 where the value of 𝜙 is modified along the time: 𝜙=0,if𝑡<5,0.025,if5≤𝑡<12,0.1,if12≤𝑡≤20, (20) where 𝑡∈ℝ is given in seconds. Notice that frequency and amplitude of oscillations change according to variations in 𝜙. Figure 2 shows the profile of the frequency and amplitude of oscillations for several values of 𝜙.

(a)

(b)

Figure 1 

Plot of desired trajectories for Acrobot by selecting several values of 𝜙.

Figure 2 

Profile of the frequency and amplitude of oscillations for several values of 𝜙.

3.3. The Task

Our objective is to design a controller of the form 𝜏=𝜏(𝑞_𝑑,𝑞̇_𝑑)+𝑢 (21) with internal state 𝜉(𝑡)∈ℝ⁴, that ensures (15). Thus, the controller to be constructed consists of the trajectory compensator (17) and a disturbance attenuator 𝑢 given in (2), (3) internally stabilizing the closed-loop system around the desired trajectory. In the sequel, we confine our investigation to the 𝐻_∞ tracking problem, where

(1)the output to be controlled is given by 0𝑧=𝜌2𝑞₁+𝑞₂+1−𝜋0𝑢(𝑡) (22) with a positive weight coefficient 𝜌;(2)the joint position vector 𝑞(𝑡)∈ℝ² is the only available measurement, and this measurement is corrupted by the error vector 𝑤_𝑜(𝑡)∈ℝ², that is,

𝑦=𝑞(𝑡)+𝑤_𝑜(𝑡). (23) The 𝐻_∞ control problem in question is thus stated as follows. Given the system representation (14)–(23), the desired trajectory 𝑞_𝑑(𝑡)∈ℝ², and a real number 𝛾>0, it is required to find (if any) a causal dynamic feedback controller (2), (3) such that the undisturbed closed-loop system is uniformly asymptotically stable around the origin, and its ℒ₂-gain is locally less than 𝛾, that is, inequality (4) is satisfied for all 𝑡₁>𝑡₀ and all piecewise continuous functions 𝑤(𝑡)=[𝑤_𝑥(𝑡),𝑤_𝑜(𝑡)]^𝑇 for which the corresponding state trajectory of the closed-loop system, initialized at the origin, remains in some neighborhood of this point.

3.4. 𝐇_∞ Synthesis

To begin with, let us introduce the state deviation vector 𝑥=(𝑥₁,𝑥₂)^𝑇∈ℝ⁴ where 𝑥₁=(𝑞₁−𝑞_𝑑1,𝑞₂−𝑞_𝑑2)^𝑇 and 𝑥₂=(𝑞̇₁−𝑞̇_𝑑1,𝑞̇₂−𝑞̇_𝑑2)^𝑇. After that let us rewrite the system (14), the output to be controlled (22), and the output (23) in terms of the state vector 𝑥: 𝑑𝑥𝑑𝑡𝑥¹₂=𝑥₂𝑀⁻¹(𝑥₁+𝑞_𝑑)𝑁(𝑥₁+𝑞_𝑑,𝑥₂+𝑞̇_𝑑)−𝑞̈_𝑑+0𝑀⁻¹(𝑥₁+𝑞_𝑑)𝑤_𝑥+0𝑀⁻¹(𝑥₁+𝑞_𝑑)𝐵𝜏,0𝑧=𝜌2𝑥₁₁+𝑥₁₂+2𝑞_𝑑1+𝑞_𝑑2+1−𝜋0𝑢(𝑡),𝑦=𝑥₁+𝑞_𝑑(𝑡)+𝑤_𝑜(𝑡). (24)(25) Clearly, the above 𝐻_∞ tracking control problem is nothing else than a standard nonlinear 𝐻_∞ control problem from [1] stated for a time-varying nonlinear system (1) specified as 𝑥𝑓(𝑥,𝑡)=²𝑀⁻¹(𝑥₁+𝑞_𝑑)𝑁(𝑥₁+𝑞_𝑑,𝑥₂+𝑞̇_𝑑)−𝑞̈_𝑑,𝑔₁0(𝑥,𝑡)=^2×20_2×2𝑀⁻¹(𝑥₁+𝑞_𝑑)0_2×2,𝑔₂0(𝑥,𝑡)=^2×1𝑀⁻¹(𝑥₁+𝑞_𝑑,ℎ)𝐵₁0(𝑥,𝑡)=𝜌2𝑥₁₁+𝑥₁₂+2𝑞_𝑑1+𝑞_𝑑2,ℎ−𝜋₂(𝑥,𝑡)=𝑥₁+𝑞_𝑑,𝑘₁₂10,𝑘(𝑥,𝑡)=₂₁(𝑥,𝑡)=0_2×2𝐼₂. (26) Now by applying Theorem 1 to system (1) thus specified, we derive a local solution of the 𝐻_∞ tracking control problem.

Thus, the output feedback controller (13), specified according to (26), locally solves the 𝐻_∞ tracking control problem (4)–(24). Stabilizability and detectability properties of the control systems are ensured by the existence of the proper solutions of the unperturbed differential Riccati equations, and hence the corresponding synthesis procedure obviates an extra work (formidable in the nonlinear case) on verification of these properties.

4. Simulation Results

The controller performance was studied in simulation by applying the exposed ideas to the Acrobot, depicted in Figure 3, which is a two-link planar robot with no actuator at the shoulder (link 1) and actuator at the elbow (link 2). In the simulation, performed with MATLAB, the Acrobot was required to move from [𝑞₁(0),𝑞₂(0)]=[−0.07,3.3] to the desired trajectory 𝑞_𝑑(𝑡)∈ℝ² and 𝜙=𝜋. The initial velocity 𝑞(̇0)∈ℝ² and the initial compensator state 𝜉(0)∈ℝ⁴ were set to zero for all the simulations. The matrices 𝑀(𝑞) and 𝑁(𝑞,𝑞)̇ for the Acrobot are given in Appendix A. We seek for orbital stabilization of the unactuated link 𝑞₂ around the equilibrium point 𝑞^∗=(0,𝜋).

Figure 3 

Schematic diagram of the acrobot where 𝑙1 and 𝑙2 denote the length of the nonactuated and actuated links, respectively; and 𝑚1 and 𝑚2 are the masses of each link.

The control goal was achieved by implementing the nonlinear 𝐻_∞ controller with a weight parameter 𝜌=1 on the Acrobot. By iterating on 𝛾, we found the infimal achievable level 𝛾^⋆≃250. However, in the subsequent simulations 𝛾=2000 was selected to avoid an undesirable high-gain controller design that would appear for a value of 𝛾 close to the optimum. With 𝛾=2000 we obtained that for 𝜀=0.1 the corresponding differential Riccati equations (11)-(12) with 𝐴𝐴(𝑡)=00100001₃₁(𝑡)𝐴₃₂(𝑡)𝐴₃₃(𝑡)𝐴₃₄𝐴(𝑡)₄₁(𝑡)𝐴₄₂(𝑡)𝐴₄₃(𝑡)𝐴₄₄,𝐵(𝑡)₁0(𝑡)=^2×20_2×2𝑀⁻¹(𝑞_𝑑)0_2×2,𝐵₂0(𝑡)=^2×1𝑀⁻¹(𝑞_𝑑),𝐶₁(𝑡)=𝜌00002100,𝐶₂,𝐷(𝑡)=10000100₁₂10(𝑡)=,𝐷₂₂(𝑡)=0_2×2𝐼₂ (27) have positive-definite solutions. These solutions can be numerically found with MATLAB. Matrix 𝐴(𝑡) is given in Appendix B. It should be pointed out that the constant 𝜙=𝜋 does not appear in 𝐶₁(𝑡) due to straightforward calculation of (A.2), but it is definitely required in (22) to improve the selection of 𝛾 which affects inequality (4) thus avoiding the synthesis of a high-gain controllers. Resulting trajectories is depicted in Figure 4. This figure demonstrates that the 𝐻_∞ controller does asymptotically stabilize the system motion around the desired trajectory. In addition, the 𝐻_∞ controller was successfully applied to the Acrobot under external disturbances 𝑤_𝑥𝑖(𝑡)=𝑏_𝑖cos(0.1𝑡)exp(−2𝑡),𝑖=1,2, (28) where 𝑏₁=1×10⁻³ [N⋅m] and 𝑏₂=2×10⁻³ [N⋅m] are the disturbance levels at first and second joints, respectively. Resulting trajectories are depicted in Figure 5. Figures 6 and 7 show the time evolution of the determinant of minors of matrices 𝑃(𝑡)∈ℝ^4×4 and 𝑍(𝑡)∈ℝ^4×4 denoted as 𝑃_𝑚1(𝑡)=𝑃₁₁(𝑡),𝑃_𝑚2𝑃(𝑡)=¹¹(𝑡)𝑃₁₂𝑃(𝑡)₁₂(𝑡)𝑃₂₂,𝑃(𝑡)_𝑚3𝑃(𝑡)=¹¹(𝑡)𝑃₁₂(𝑡)𝑃₁₃𝑃(𝑡)₁₂(𝑡)𝑃₂₂(𝑡)𝑃₂₃𝑃(𝑡)₃₁(𝑡)𝑃₃₂(𝑡)𝑃₃₃,𝑍(𝑡)_𝑚1(𝑡)=𝑍₁₁(𝑡),𝑍_𝑚2𝑍(𝑡)=¹¹(𝑡)𝑍₁₂𝑍(𝑡)₁₂(𝑡)𝑍₂₂,𝑍(𝑡)_𝑚3𝑍(𝑡)=¹¹(𝑡)𝑍₁₂(𝑡)𝑍₁₃𝑍(𝑡)₁₂(𝑡)𝑍₂₂(𝑡)𝑍₂₃𝑍(𝑡)₃₁(𝑡)𝑍₃₂(𝑡)𝑍₃₃.(𝑡) (29) These Figures highlight that matrices 𝑃(𝑡) and 𝑍(𝑡), which are solution of (11) and (12), respectively, are bounded and positive definite for all 𝑡≥0.

(a)

(b)

Figure 4 

Phase portrait of the first joint trajectory and desired trajectory (+) for the unperturbed case (a) and perturbed case (b).

(a)

(b)

Figure 5 

Time evolution of the output (continuous line) following the desired trajectory (dashed line) under perturbed torques.

(a)

(b)

(c)

(d)

Figure 6 

Time evolution of the determinants of the principal minors of the matrix 𝑃∈ℝ4×4 𝑃𝑚1, 𝑃𝑚2, 𝑃𝑚3, 𝑃𝑚4.

(a)

(b)

(c)

(d)

Figure 7 

Time evolution of the determinants of the principal minors of the matrix 𝑍∈ℝ4×4 𝑍𝑚1, 𝑍𝑚2, 𝑍𝑚3, 𝑍𝑚4.

5. Conclusions

The output feedback Nonlinear 𝐻_∞ tracking control problem is locally solved for an underactuaded mechanical system. The desired periodic orbit is centered at the upright position where the open-loop plant becomes a nonminimum-phase system. The developed controller drives the trajectories of the robot into a set of inverted exact desired trajectories governed by its zero dynamics. Simulation studies, made for the Acrobot, showed the effectiveness of the controller. The design of methods to generate reference trajectories evolving more frequencies and amplitudes in the upright position is in progress, and few results have been published for double-pendulums in [20, 22]. In future work there are two extensions of the result of the paper. First, one would like to synthesize the 𝐻_∞ control taking into account reference trajectories derived from alternative methods. The other would also like to extend the result of the paper for the nonsmooth case.

Appendices

A. Dynamic Model of Acrobot

The equation motion of Acrobot, described by (14), was specified by applying the Euler-Lagrange formulation [23] where 𝑀𝑀(𝑞)=¹¹(𝑞)𝑀₁₂𝑀(𝑞)₁₂(𝑞)𝑀₂₂̇𝑁(𝑞),𝑁(𝑞,𝑞)=¹̇𝑁(𝑞,𝑞)₂(𝑞,𝑞)̇ (A.1) with 𝑀₁₁(𝑞)=𝑐₁+𝑐₂−2𝑐₃cos(𝑞₂𝑀),₁₂(𝑞)=𝑐₂−𝑐₃cos(𝑞₂𝑀),₂₂(𝑞)=𝑐₂,𝑁₁(𝑞,𝑞)̇=𝑐₃sin(𝑞₂)𝑞̇₁𝑞̇₂+𝑐₃𝑞̇₂(𝑞̇₁+𝑞̇₂)sin(𝑞₂)−𝑐₄sin(𝑞₁)+𝑐₅sin(𝑞₁+𝑞₂𝑁),₂(𝑞,𝑞)̇=−𝑐₃sin(𝑞₂)𝑞̇²₁+𝑐₅sin(𝑞₁+𝑞₂), (A.2) where the values of 𝑐_𝑖 (𝑖=1,…,5), given in Table 1, were taken from the experimental Acrobot provided in [7].

Table 1 

Parameter values for the Acrobot.

B. Matrix 𝐴(𝑡)

In this appendix we provide the computed matrix 𝐴(𝑡) for the Acrobot which was used in the solution of differential Riccati equations (11) and (12): 𝐴𝐴(𝑡)=00100001₃₁(𝑡)𝐴₃₂(𝑡)𝐴₃₃(𝑡)𝐴₃₄𝐴(𝑡)₄₁(𝑡)𝐴₄₂(𝑡)𝐴₄₃(𝑡)𝐴₄₄,(𝑡) (B.3) where 𝐴₃₁(𝑡)=𝑐₂Δ𝑀(𝑞_𝑑)⁻¹[𝑐₅cos(𝑞_𝑑1+𝑞_𝑑2)−𝑐₄cos(𝑞_𝑑1)]−Δ𝑀(𝑞_𝑑)⁻¹[𝑐₅cos(𝑞_𝑑1+𝑞_𝑑2)]𝑀₁₂(𝑞_𝑑𝐴),₃₂(𝑡)=𝑐₂Δ𝑀(𝑞_𝑑)⁻¹[𝑐₃cos(𝑞_𝑑2)𝑞̇_𝑑1𝑞̇_𝑑2+𝑐₃𝑞̇_𝑑2(𝑞̇_𝑑1+𝑞̇_𝑑2)cos(𝑞_𝑑2)+𝑐₅cos(𝑞_𝑑1+𝑞_𝑑2)]+−2𝑐₂𝑐₃Δ𝑀(𝑞_𝑑)⁻²sin(𝑞_𝑑2)cos(𝑞_𝑑2)𝑁₁(𝑞_𝑑,𝑞̇_𝑑)−𝑐₃Δ𝑀(𝑞_𝑑)⁻¹𝑁₂(𝑞_𝑑,𝑞̇_𝑑)sin(𝑞_𝑑2)+2𝑐₃Δ𝑀(𝑞_𝑑)⁻²sin(𝑞_𝑑2)cos(𝑞_𝑑2)𝑀₁₂(𝑞_𝑑)𝑁₂(𝑞_𝑑,𝑞̇_𝑑)−Δ𝑀(𝑞_𝑑)⁻¹[−𝑐₃cos(𝑞_𝑑2)𝑞̇_𝑑1+𝑐₅cos(𝑞_𝑑1+𝑞_𝑑2)]×𝑀₁₂(𝑞_𝑑𝐴),₃₃(𝑡)=2𝑐₂𝑐₃Δ𝑀(𝑞_𝑑)⁻¹sin(𝑞_𝑑2)𝑞̇_𝑑2+2𝑐₃Δ𝑀(𝑞_𝑑)⁻¹𝑀₁₂(𝑞_𝑑)sin(𝑞_𝑑2)𝑞̇_𝑑1,𝐴₃₄(𝑡)=2𝑐₂𝑐₃Δ𝑀(𝑞_𝑑)⁻¹[sin(𝑞_𝑑2)𝑞̇_𝑑1+sin(𝑞_𝑑2)𝑞̇_𝑑2𝐴],₄₁(𝑡)=Δ𝑀(𝑞_𝑑)⁻¹[𝑐₅cos(𝑞_𝑑1+𝑞_𝑑2)]𝑀₁₁(𝑞_𝑑)−Δ𝑀(𝑞_𝑑)⁻¹[𝑐₅cos(𝑞_𝑑1+𝑞_𝑑2)−𝑐₄cos(𝑞_𝑑1)]×𝑀₁₂(𝑞_𝑑𝐴),₄₂(𝑡)=2𝑐₃Δ𝑀(𝑞_𝑑)⁻¹𝑁₂(𝑞_𝑑,𝑞̇_𝑑)sin(𝑞_𝑑2)+Δ𝑀(𝑞_𝑑)⁻¹×[𝑐₅cos(𝑞_𝑑1+𝑞_𝑑2)−𝑐₃cos(𝑞_𝑑2)𝑞̇²_𝑑1]𝑀₁₁(𝑞_𝑑)−2𝑐₃Δ𝑀(𝑞_𝑑)⁻²𝑀₁₁(𝑞_𝑑)𝑁₂(𝑞_𝑑,𝑞̇_𝑑)sin(𝑞_𝑑2)cos(𝑞_𝑑2)−𝑐₃Δ𝑀(𝑞_𝑑)⁻¹𝑁₁(𝑞_𝑑,𝑞̇_𝑑)sin(𝑞_𝑑2)−Δ𝑀(𝑞_𝑑)⁻¹×[2𝑐₃cos(𝑞_𝑑2)𝑞̇_𝑑1𝑞̇_𝑑2+𝑐₃cos(𝑞_𝑑2)𝑞̇²_𝑑2+𝑐₅cos(𝑞_𝑑1+𝑞_𝑑2)]𝑀₁₂(𝑞_𝑑)+2𝑐₃Δ𝑀(𝑞_𝑑)⁻²𝑀₁₂(𝑞_𝑑)𝑁₁(𝑞_𝑑,𝑞̇_𝑑)sin(𝑞_𝑑2)cos(𝑞_𝑑2𝐴),₄₃(𝑡)=−2𝑐₃Δ𝑀(𝑞_𝑑)⁻¹[𝑀₁₁(𝑞_𝑑)+𝑀₁₂(𝑞_𝑑)]sin(𝑞_𝑑2)𝑞̇_𝑑1,𝐴₄₄(𝑡)=−2𝑐₃Δ𝑀(𝑞_𝑑)⁻¹[𝑞̇_𝑑1+𝑞̇_𝑑2]sin(𝑞_𝑑2)𝑀₁₂(𝑞_𝑑), (B.4) where Δ𝑀(𝑞_𝑑)=𝑀₁₁(𝑞_𝑑)𝑀₂₂(𝑞_𝑑)−𝑀²₁₂(𝑞_𝑑).