Dynamic modeling and optimal robust approximate constraint-following control of constrained mechanical systems under uncertainty: A fuzzy approach

Abstract

A novel and systematic approach for dynamic modeling and approximate constraint-following servo control of constrained mechanical systems under uncertainty is proposed. Fundamental equation of constrained mechanical systems is first obtained based on Udwadia-Kalaba approach which is applicable to holonomic and nonholonomic constrained systems no matter whether they satisfy the D’Alember’s principle. The uncertainty (possibly fast time-varying) in mechanical systems is bounded and the bound is unknown. Fuzzy set theory is used to describe the unknown bound. The performance requirement is modeled as servo constraints in second-order form. A model-based robust control is presented to approximately follow the servo constraints. The proposed control is deterministic and is not IF-THEN rules-based. The uniform boundedness and uniform ultimate boundedness of the tracking error are guaranteed, regardless of the uncertainty. A performance index (the combined cost, which includes average fuzzy system performance and control effort) is proposed based on the fuzzy information. The optimal design problem associated with the control can then be solved (explicitely or numerically) by minimizing the performance index. The resulting control design is systematic and is able to guarantee the deterministic performance as well as minimizing the cost.

Keywords

Mechanical system fundamental equation constraint robust control coordinated robot

1 Introduction

In order to control the constrained discrete mechanical system, the dynamic model should first be obtained. Obtaining equations of motion for constrained discrete mechanical systems is one of the central issues in multi-body dynamics. The problem has been energetically and continuously worked on by many scientists, engineers and mathematicians since constrained motion was initially described by Lagrange (1787). He invented the special Lagrange multiplier method to deal with constrained motion. However, the Lagrange multiplier method relies on problem-specific approaches to the determination of the multipliers and it is often very difficult to find the multipliers to obtain the explicit equations of motion for systems which have large numbers of degrees of freedom and a mass of non-integrable constraints. Gauss (1829) introduced a general, new principle of mechanics for handling constrained motion. Gauss’s Principle gives a clear description of the general nature of constrained motion in terms of the minimization of a function of the accelerations of the particles of a system. Formulations of the equations of constrained motion, when the constraints satisfy d’Alember’s principle, were independently offered by Gibbs (1879) and Appell (1899). Pars (1979) in his treatise on the analytical dynamics refers to the Gibbs-Appell equations as ‘probably the most comprehensive equations of motion so far discovered’. But Gibbs-Appell equations require a felicitous choice of problem-specific quasi-coordinates and suffer from similar problems in dealing with systems with a large number of degrees of freedom and many non-integrable constraints. Dirac (1964) developed, using Poisson brackets, a recursive scheme for determining the Lagrange multipliers for singular, Hamiltonian systems where the constraints do not exactly depend on time.

Udwadia & Kalaba (1992, 1996) obtained a concise, explicit set of equations of motion for constrained discrete dynamical systems which lead to a simple and new fundamental view of Lagrangian mechanics. They derived the fundamental equation of motion that describes the dynamics of constrained systems from Gauss’s principle which seems somewhat less popular than the principles of Lagrange, Hamilton, Gibbs and Appell. The equations can deal with holonomic and also non-holonomic constraints. Udwadia & Kalaba (2001, 2005) observed that all the above researches have used D’Alember’s principle as their starting point which indicates that the forces of constraints are considered to be ideal and the total work done by the forces of constraints under virtual displacement is always zero. This assumption works well in many situations and is regarded as the core of classical analytical dynamics, but it is not applicable when the constraints are nonideal. Thus, Udwadia & Kalaba generalized their previous equations to constrained mechanical systems that may not satisfy D’Alember’s principle.

In this paper, we use fundamental equation of constrained system proposed by Udwadia and Kalaba to get the dynamic equations of motion of the constrained mechanical system. According to this approach, the unconstrained system is first considered whose equations of motion can be obtained by Lagrangian mechanics in terms of the generalized coordinates. Then constraint equations is written in the form of second order. The dynamic equations can be obtained by imposing the additional generalized forces of constraint obtained from the second order constraint equations upon the unconstrained system. Based on the obtained dynamic model, robust approximate constraint-following servo control is used to realize the performance requirement modeled as servo constraints (also called active constraints, program constraints, prescribed constraints). That is to use a set of servo controls to generate the appropriate constraint force for the system to approximately obey the constraints.

For a mechanical system confined to a set of constraints, constraint forces are needed for the system to obey the constraints. In Lagrangean mechanics, the constraint forces are governed by d’Alembert’s principle. This is what Lagrange asserted the Nature would do (Papastavridis 2002). Over the past century and a half, key developments in Lagrangian mechanics include the Maggi equation (Maggi, 1901; Neimark & Fufaev, 1972), the Boltzmann and Hamel equation (Hamel, 1904, 1949; Neimark and Fufaev, 1972), the Gibbs and Appell equation (Appell, 1899; Gibbs, 1879; Hamel, 1949; Pars, 1979), and the Udwadia and Kalaba equation (Udwadia and Kalaba, 1996). However, the emphasis of all these developments mainly falls into the passive constraint (also called material constraint) problem in which the constraint is followed in a passive manner. That is, as the constraint is determined, the environment (such as the structure of the machine) can generate the required constraint force automatically. Taking a particle constrained to move on a table surface as an example, the table surface can automatically generate the constraint force to support the particle. One is not necessary to consider generating the constraint force. So, the generation of constraint force is often considered to be a design task: design the machine structure to generate the constraint force automatically.

We consider the issue of generation of constraint forces as a control task (that is servo constraint problem), rather than a design task. The servo constraint problem has rarely been studied in analytical mechanics. Perhaps because the concept of servos is relatively new in analytical mechanics. In the servo constraint problem, a set of servo controls are equipped to provide the required constraint force for the system to obey a set of required constraints. The servo constraint problem focuses on what the engineer should do, so that the constraints are followed. Earlier work of the servo constraint problem are mainly on precise model-based control design (Kirgetov 1967, Chen 2008). The engineer, however, is always limited to a rather confined domain of knowledge. As a result, uncertainty tends to be inevitable. Prominent efforts dealing with uncertainty can be found in (Song, et al . 2005, Tseng & Chen 2003, Wang, et al . 2004, Wang, et al . 2006).

Optimal robust approximate constraint-following servo control of constrained mechanical systems under uncertainty is proposed. The uncertainty (possibly fast time-varying) in mechanical systems is assumed to be bounded, but the bound is unknown. Fuzzy set theory is used to describe the unknown bound. The performance requirement is modeled as servo constraints in second-order form. With the uncertainty in presence and no restrictions on the initial condition, it is only reasonable to expect the system to follow the prescribed constraints approximately. The proposed control is deterministic and is not IF-THEN rules-based. The uniform boundedness and uniform ultimate boundedness of the tracking error are guaranteed, regardless of the uncertainty. From a different angle, we employ the fuzzy theory to describe the uncertainty in the mechanical system and then propose optimal robust control design of fuzzy mechanical systems.

The main contributions of the article are fourfold. First, Udwadia-Kalaba approach is described in detail. The procedure of obtaining fundamental equation of constrained motion is shown. Second, robust approximate constraint-following servo control of constrained mechanical systems under uncertainty is proposed. Third, a performance index (the combined cost, which includes average fuzzy system performance and control effort) is proposed based on the fuzzy information. The optimal design problem associated with the control can then be solved by minimizing the performance index. In the end, the resulting control design is systematic and is able to guarantee the deterministic performance as well as minimizing the cost.

2 Fundamental equation of constrained systems

According to Udwadia-Kalaba approach, first we should consider an unconstrained mechanical system whose configuration is described by the n generalized coordinates q : = [q ₁, q ₂, …, q _n] ^T. Its equation of motion can be obtained, using Newtonian or Lagrangian mechanics, by the relation $M (q, t) \ddot{q} = Q (q, \dot{q}, t),$ (1) where M (q, t) ∈ R ^n×n is symmetric and positive definite inertia matrix, $\dot{q} \in R^{n}$ is the velocity and $\ddot{q} \in R^{n}$ is the acceleration, $Q (q, \dot{q}, t) \in R^{n}$ is the force imposed on the system whose constraints are released. The imposed forces could include centrifugal force, gravitational force and control input. The generalized acceleration of the unconstrained system, which we denote by $a (q, \dot{q}, t)$ , is thus given by $\ddot{q} = M^{- 1} (q, t) Q (q, \dot{q}, t) = a (q, \dot{q}, t),$ (2)

Second, constraints present in the system should be considered. We shall assume that the system is subjected to h holonomic constraints of the form $φ_{i} (q, t) = 0, i = 1, 2, \dots, h,$ (3) and m - h nonholonomic constraints of the form $φ_{i} (q, \dot{q}, t) = 0, i = h + 1, h + 2, \dots, m .$ (4) We can get the differentiation of the usual constraint equations in Lagrangian mechanics which are usually pfaffian form, that is to say, under the assumption of sufficient smoothness, take time derivative once on non-holonomic constraints and twice on holonomic constraints, then it will derive the constraint equations of the matrix form: $\hat{A} (q, \dot{q}, t) \ddot{q} = \hat{b} (q, \dot{q}, t),$ (5) where $\hat{A} (q, \dot{q}, t) \in R^{m \times n}$ is referred to as constraint matrix and $\hat{b} (q, \dot{q}, t)$ is a m-vector.

Remark. The constraint (5) has to be met strictly in the whole process of the system’s moving regardless of the initial condition and the uncertainty. We call it fixed constraint or structure constraint.

The final step is to form the explicit equations of motion with constraints. Due to the presence of constraints, additional ‘generalized forces of constraints’ should be imposed on the system. So, the actual explicit equation of motion of the constrained system could be assumed to take the form: $M (q, t) \ddot{q} = Q (q, \dot{q}, t) + Q^{c} (q, \dot{q}, t),$ (6) where $Q^{c} (q, \dot{q}, t) \in R^{n}$ , arising by virtue of the holonomic and non-holonomic constraints, is the additional term of forces imposed on the system. In Lagrangian mechanics, $Q^{c} (q, \dot{q}, t)$ is considered to be ideal and it is governed by the usual D’Alember’s Principle which indicates that forces of constraint do zero work under virtual displacements. However, the constraints can also be nonideal. Ideal constraints generate ideal constraint forces which subject to D’Alember’s Principle, while nonideal constraints generate nonideal constraint forces such as friction force, electro-magnetic force and so on. If there exist both ideal and nonideal constraints in the system, Udwadia and Kalaba put $Q^{c} (q, \dot{q}, t)$ in the form of $Q^{c} (q, \dot{q}, t) = Q_{id}^{c} (q, \dot{q}, t) + Q_{nid}^{c} (q, \dot{q}, t),$ (7) where $Q_{id}^{c} (q, \dot{q}, t)$ is the ideal constraint force and $Q_{nid}^{c} (q, \dot{q}, t)$ is the nonideal constraint force.

Udwadia and Kalaba generalize D’Alember’s Principle to include forces of constraint that may do positive, negative, or zero work under virtual displacement at any instant time during the motion of the constrained system, that is to say, they extend the Lagrange’s form of D’Alember’s Principle to include nonideal constraints. Now, we denote the specified constraint force as $c (\dot{q}, q, t) \in R^{n}$ (Udwadia & Kalaba 2001,2005). The constraint force does the work W = v ^T c (the same as the work done by $Q^{c} (q, \dot{q}, t)$ ) in any displacement v which subjects to $\hat{A} (q, \dot{q}, t) v = 0$ . So we write it down in the form of (we will omit arguments of functions where no confusions may arise from now on) $W = v^{T} Q^{c} = v^{T} c,$ (8) which is the extended Lagrange’s form of D’Alembert’s Principle. The work done by the ideal constraint force $Q_{id}^{c}$ under virtual displacements is: $v^{T} Q_{id}^{c} = 0,$ (9) while the work done by nonideal constraint force is: $v^{T} Q_{nid}^{c} \neq 0 .$ (10)

Udwadia and Kalaba have proved that the ideal constraint force takes the form $Q_{id}^{c} = M^{\frac{1}{2}} {\hat{B}}^{+} (\hat{b} - \hat{A} M^{- 1} Q),$ (11) and non-ideal constraint force can be written in the form $Q_{nid}^{c} = M^{\frac{1}{2}} (I - {\hat{B}}^{+} \hat{B}) M^{- \frac{1}{2}} c,$ (12) where $\hat{B} = \hat{A} M^{- \frac{1}{2}}$ , and the superscript ‘+’ denotes the Moore-Penrose generalized inverse (Moore 1920; Penrose 1955).

From (6), (7), (11) and (12), the explicit equation of motion that governs the evolution of the constrained system (including both ideal and nonideal constraints) is

$\begin{matrix} M \ddot{q} & = & Q + M^{\frac{1}{2}} {\hat{B}}^{+} (\hat{b} - \hat{A} M^{- 1} Q) \\ + & M^{\frac{1}{2}} (I - {\hat{B}}^{+} \hat{B}) M^{- \frac{1}{2}} c, \end{matrix}$ (13) where the vector c is determined by the mechanician and could be obtained by experimentation and/orobservation.

When c is always zero, (13) reduces to the D’Alember’s Principle, which means the total work done under virtual displacement is zero, the constraints are ideal, the constraint force is $Q^{c} = Q_{id}^{c} = M^{\frac{1}{2}} {\hat{B}}^{+} (\hat{b} - \hat{A} M^{- 1} Q),$ (14) and the explicit equation of motion of the constrained system (only including ideal constraints) is $M \ddot{q} = Q + M^{\frac{1}{2}} {\hat{B}}^{+} (\hat{b} - \hat{A} M^{- 1} Q),$ (15) which will be applied in the coordinated robot problem for there is no nonideal constraints in the system.

Remark. Udwadia and Kalaba have provided explicit general equations of motion for constrained mechanical systems with their new initiated approach which is applicable to all holonomic and nonholonomic constrained systems no matter whether they satisfy the D’Alember’s Principle. They generalize Lagrangian mechanics to include both ideal and nonideal constraint forces by using a new fundamental principle governing the motion of constrained systems. The equation of motion obtained by using Udwadia and Kalaba’s approach is general, simple and understandable.

Remark. The advantage of determining the explicit equations of motion for this constrained coordinated robot system through Udwadia-Kalaba equation is obvious. Compared to the Lagrange multiplier method, there is no need for this new approach to determine the multiplier which is often very difficult to obtain for systems having a large number of degrees of freedom and many non-integrable constraints. Also, if using the formulations offered by Gibbs, Volterra, Appell and Boltzmann, we will be required a proper quasi-coordinates, but we don’t have this problem using the new initiated approach.

3 Robust approximate constraint-following control approach

Consider the following constrained mechanical system under uncertainty whose fundamental equation can be obtained by Udwadia-Kalaba approach:

$\begin{matrix} M (q (t), σ (t), t) \ddot{q} (t) - Q (q (t), \dot{q} (t), σ (t), t) \\ - Q^{c} (q (t), \dot{q} (t), σ (t), t) = τ (t) . \end{matrix}$ (16)

Here t ∈ R is the independent variable (time), q ∈ R ⁿ is the coordinate, $\dot{q} \in R^{n}$ is the velocity, $\ddot{q} \in R^{n}$ is the acceleration, σ ∈ ∑ ⊂ R ^p is the uncertain parameter (possibly fast time-varying), and τ ∈ R ⁿ is the control input. Furthermore, M (q, σ, t) is the inertia matrix, $Q (q, \dot{q}, σ, t)$ is the imposed unconstraint force, and $Q^{c} (q, \dot{q}, σ, t)$ is the constraint force. The matrices/vector are of appropriate dimensions. The functions M (·) , Q (·) , and Q ^c (·) are assumed to be Lebesgue measurable in t.

Remark. The coordinate q is not necessarily the generalized coordinate. It can be selected based on the specific problem. The bounding set ∑ is compact and prescribed.

3.1 Second-order performance servo constraint

The required performance is modeled as servo constraints (i.e. the control input serves as the constraint force to follow the constraint). We model the required performance as the following constraints: $\sum_{i = 1}^{n} A_{li} (q, t) {\dot{q}}_{i} + A_{l} (q, t) = 0, l = 1, \dots, m$ (17) where 1 ≤ m ≤ n, A _li (·) and A _l are both C ¹ in q and t. These constraints imply restrictions on the velocities as well as the displacement, and are the first order form of the constraints. The matrix form is $A (q, t) \dot{q} = c (q, t),$ (18) where A = [A _li] _m×n, c = [c ₁, c ₂, ⋯ , c _m] ^T.

Now the constraints are converted into second order form. Differentiating constraint equation (17) with respect to t yields $\sum_{i = 1}^{n} \frac{d}{dt} A_{li} (q, t) {\dot{q}}_{i} + \sum_{i = 1}^{n} A_{li} (q, t) {\ddot{q}}_{i} + \frac{d}{dt} A_{l} (q, t) = 0$ (19) where $\frac{d}{dt} A_{li} (q, t) = \sum_{k = 1}^{n} \frac{\partial A_{li} (q, t)}{\partial q_{k}} {\dot{q}}_{k} + \frac{\partial A_{li} (q, t)}{\partial t}$ (20) and $\frac{d}{dt} A_{l} (q, t) = \sum_{k = 1}^{n} \frac{\partial A_{l} (q, t)}{\partial q_{k}} {\dot{q}}_{k} + \frac{\partial A_{l} (q, t)}{\partial t} .$ (21) Equation (19), the second order form of the constraints, can be rewritten as $\begin{matrix} \sum_{i = 1}^{n} A_{li} (q, t) {\ddot{q}}_{i} & = - \sum_{i = 1}^{n} \frac{d}{dt} A_{li} (q, t) {\dot{q}}_{i} - \frac{d}{dt} A_{l} (q, t) \\ = : b_{i} (\dot{q}, q, t) (l = 1, \dots, m) \end{matrix}$ (22) or, in matrix form, $A (q, t) \ddot{q} = b (\dot{q}, q, t),$ (23) where A = [A _li] _m×n and b = [b ₁ b ₂ ⋯ b _m] ^T. Equation (23), the second order form of the constraints (Chen 1998, 1999), is linear with respect to $\ddot{q}$ , and so contains the acceleration ${\ddot{q}}_{j}$ explicitly and linearly. The possible acceleration component $\ddot{q}$ is governed by equation (23) when $\dot{q}$ is the possible velocity.

Remark. In fact, the second order form constraint equation (23) is a very general form. It includes typical constraints as illustrated by Rosenberg (1977) and Papastavridis (2002). It also includes a number of standard control problems such as stabilization, trajectory following and optimality (Chen 1998, 1999). All one needs to do is to prescribe a desirable system performance and then convert it to the second order form.

Remark. (23) is the second order form servo constraint (or required performance). It is not like the structure (or fixed) constraint (5) which should be strictly met. This performance constraint only needs to be approximately followed, because in real life people do not need too strict performance requirement. Furthermore, with the uncertainty in presence and no restrictions on the initial condition, it is only reasonable to expect approximate constraint following.

The servo constraint problem can now be stated as follows: Determine the servo control τ such that the resulting controlled system observes the constraint (18) approximately.

3.2 Constraint force if there is no uncertainty

We first show the constraint force when there is no uncertainty in the mechanical system (16). Thus, σ is known.

Assumption 1. For each (q, t) ∈ R ⁿ × R, σ ∈ ∑, the inertia matrix is positive definite. That is M (q, σ, t) >0.

Remark. This assumption is vital in later development. In the past, the positive definiteness was often believed to be a fact rather than an assumption. However, there exist counter examples (Chen, et al . 1998), when q is not selected to be the generalized coordinate.

Definition 1. For given A and b, the constraint equation (23) is said to be consistent if there exists at least one solution $\ddot{q}$ .

The consistency is a necessary but not a sufficient condition to the servo constraint problem. That is, if the constraint equation (23) is not consistent, then there is no solution τ to the servo constraint problem.

Assumption 2. Considering constraint equation (23), for each q, t, rank A (q, t) ≥1. Also, the constraint is consistent. (This assumption ensures the existence of the Moore-Penrose inverse of A.)

Theorem 1. (Udwadia & Kalaba 1996 p. 233) Consider the mechanical system (16) and the constraint (23). Subject to Assumptions 1 and 2, the constraint force $\begin{matrix} Q_{s}^{c} & = M^{1 / 2} (q, σ, t) (A (q, t) M^{- 1 / 2} (q, σ, t))^{+} [b (q, \dot{q}, t) \\ - A (q, t) M^{- 1} (q, σ, t) (Q + Q^{c})] \end{matrix}$ (24) complies with Lagrange’s form of d’Alembert’s principle (23) and renders the system to meet the constraint.

Remark. Lagrange’s form of d’Alembert’s principle renders the constraint force (24) to be the one with minimum norm, out of all possible alternative forces which can also meet (23) (Udwadia & Kalaba 1996). Theorem 1 suggests the Nature will undertake the strategy to meet the constraint. The constraint force is based on the exact model information. If the uncertainty was known and the initial condition was met, one could apply the control input $τ = Q_{s}^{c}$ to drive the system to meet (23). In reality, the uncertainty is unknown. The robust approximate constraint-following control dealing with uncertainty will be investigated in the next section.

3.3 Robust servo control design

We now consider the uncertainty of the mechanical system while designing the control τ. Suppose the matrices/vector M, Q, Q ^c can be decomposed asfollows:

$\begin{matrix} M (q, σ, t) & = \bar{M} (q, t) + ▵ M (q, σ, t), \\ Q (q, \dot{q}, σ, t) & = \bar{Q} (q, \dot{q}, t) + ▵ Q (q, \dot{q}, σ, t), \\ Q^{c} (q, \dot{q}, σ, t) & = \bar{Q^{c}} (q, \dot{q}, t) + ▵ Q^{c} (q, \dot{q}, σ, t) . \end{matrix}$ (25) Here $\bar{M}$ , $\bar{Q}$ and $\bar{Q^{c}}$ are the nominal terms of corresponding matrix/vector with $\bar{M} > 0$ (this is always feasible because it is the designer’s discretion). while ▵M, ▵Q and ▵Q ^c are the uncertain terms which depend on σ. The functions $\bar{M} (\cdot)$ , $\bar{Q} (\cdot)$ , $\bar{Q^{c}} (\cdot)$ ,▵M (·),▵Q (·), and ▵Q ^c (·) are all continuous. Let $D (q, t) : = {\bar{M}}^{- 1} (q, t), ▵ D (q, t) : = M^{- 1} (q, σ, t) - {\bar{M}}^{- 1} (q, t), E (q, σ, t) : = \bar{M} (q, t) M^{- 1} (q, σ, t) - I$ (thus ▵D (q, t) = D (q, t) E (q, σ, t)).

Assumption 3. For each (q, t) ∈ R ⁿ × R, A (q, t) is of full rank. This means A (q, t) A ^T (q, t) is invertible.

Assumption 4. There exists ${\hat{ρ}}_{E} (\cdot) : R^{n} \times R \to (- 1, \infty)$ such that for all (q, t) ∈ R ⁿ × R, $\frac{1}{2} min_{σ \in \sum} λ_{m} (E (q, σ, t) + E^{T} (q, σ, t)) \geq {\hat{ρ}}_{E} (q, t)$ (26) where λ _m (·) represents the minimum eigenvalue and min(·) means the smallest number.

Remark. If there is no uncertainty in M (i.e. $\bar{M} = M, E = 0$ ), one can choose ${\hat{ρ}}_{E} = 0$ to meet the assumption. By continuity, there is a unidirectional threshold for the allowable uncertainty in E. For the current setting, it is less restrictive than the standard assumption in this area (Chen 1986) that $max_{σ \in \sum} ∥ E (q, σ, t) ∥ < 1$ .

Assumption 5. For given P ∈ R ^n×n, P > 0, let $Ψ (q, t) : = PA (q, t) D (q, t) D (q, t) A^{T} (q, t) P .$ (27) There exists a scalar constant $\underline{λ} > 0$ such that $inf_{(q, t) \in R^{n} \times R} λ_{m} (Ψ (q, t)) \geq \underline{λ} .$ (28)

Remark. Under Assumptions 1,2 and 3, Ψ (q, t) is always positive definite. Thus the Assumption 5 is to assure λ _m (Ψ (q, t)) positively bounded from below.

We now consider the approximate constraint following problem. That is, it is possible that $A \dot{q} \neq c$ or $A \ddot{q} \neq b$ (let $β (q, \dot{q}, t) : = A (q, t) \dot{q} - c (q, t)$ , hence β ≠ 0). This may be due to modeling uncertainty. In addition, the system may not start with the constraint manifold in the beginning (i.e., β ≠ 0 as t = t ₀). Consider the following robust control design:

$\begin{matrix} τ (t) & = p_{1} (q (t), \dot{q} (t), t) + p_{2} (q (t), \dot{q} (t), t) \\ + p_{3} (q (t), \dot{q} (t), t), \end{matrix}$ (29) with

$\begin{matrix} p_{1} (q (t), \dot{q} (t), t) & = Q_{s}^{c}, \\ p_{2} (q (t), \dot{q} (t), t) & = - κ {\bar{M}}^{- 1} (q, t) A^{T} (q, t) P β, \\ p_{3} (q (t), \dot{q} (t), t) & = - γ (q, \dot{q}, t) μ (q, \dot{q}, t) ρ (q, \dot{q}, t), \end{matrix}$ (30) where ɛ, κ ∈ R, ɛ, κ > 0, $γ (q, \dot{q}, t) = {\begin{matrix} \frac{(1 + {\hat{ρ}}_{E} (q, t))^{- 1}}{∥ \bar{μ} (q, \dot{q}, t) ∥ ∥ μ (q, \dot{q}, t) ∥} if ∥ μ (q, \dot{q}, t) ∥ > ɛ, \\ \frac{(1 + {\hat{ρ}}_{E} (q, t))^{- 1}}{{∥ \bar{μ} (q, \dot{q}, t) ∥}^{2} ɛ} if ∥ μ (q, \dot{q}, t) ∥ \leq ɛ, \end{matrix}$ (31)

$\begin{matrix} μ (q, \dot{q}, t) & = η (q, \dot{q}, t) ρ (q, \dot{q}, t), \\ η (q, \dot{q}, t) & = \bar{μ} (q, \dot{q}, t) β (q, \dot{q}, t), \\ \bar{μ} (q, \dot{q}, t) & = {\bar{M}}^{- 1} (q, t) A^{T} (q, t) P . \end{matrix}$ (32) The function ρ (·) : R ⁿ × R ⁿ × R → R ₊ is chosen such that

$\begin{matrix} ρ (q, \dot{q}, t) & \geq max_{σ \in \sum} ∥ PA Δ D (Q + Q^{c} + p_{1} + p_{2}) \\ + PAD (Δ Q + Δ Q^{c}) ∥ \end{matrix}$ (33) which actually defines the upper bound of the term of the right-hand side.

Theorem 2. Subject to Assumptions 1,2,3,4 and 5, consider the mechanical system under uncertainty (16). The control (29) renders the errorβ

uniformly bounded: For any r > 0, there is a d (r)< ∞ such that if $∥ β (q (t_{0}), \dot{q} (t_{0}), t_{0}) ∥ \leq r$ , then $∥ β (q (t), \dot{q} (t), t) ∥ \leq d (r)$ for all t ≥ t ₀.

uniformly ultimately bounded: For any r > 0 with $∥ β (q (t_{0}), \dot{q} (t_{0}), t_{0}) ∥ \leq r$ , there exists a $\underline{d} > 0$ such that $∥ β (q (t), \dot{q} (t), t) ∥ \leq \bar{d}$ for any $\bar{d} > \underline{d}$ as $t \geq t_{0} + T (\bar{d}, r)$ , where $T (\bar{d}, r) < \infty$ . Furthermore, $\bar{d} \to 0$ as ɛ → 0.

Proof. Consider the Lyapunov function candidate V (β) = β ^T Pβ. For any given uncertainty σ (·), the derivative of V along a trajectory is evaluated as (henceforth, for simplicity, arguments of functions are sometimes omitted when no confusions are likely to arise): $\begin{matrix} \dot{V} & = 2 β^{T} P (A \ddot{q} - b) \\ = 2 β^{T} P {A [M^{- 1} (Q + Q^{c}) \\ + M^{- 1} (p_{1} + p_{2} + p_{3})] - b} . \end{matrix}$ (34) After decomposing M ^-1, Q and Q ^c, we have

$\begin{matrix} A [M^{- 1} (Q + Q^{c}) + M^{- 1} (p_{1} + p_{2} + p_{3})] - b \\ = & A [(D + Δ D) (\bar{Q} + \bar{Q^{c}} + Δ Q + Δ Q^{c}) \\ + (D + Δ D) (p_{1} + p_{2} + p_{3})] - b \\ = & A [D (\bar{Q} + \bar{Q^{c}} + p_{1} + p_{2}) + D (Δ Q + Δ Q^{c}) \\ + Δ D (Q + Q^{c} + p_{1} + p_{2}) \\ + (D + Δ D) p_{3}] - b . \end{matrix}$ (35) First, we know, when there is no uncertainty (recall Theorem 1), $A \ddot{q} - b = 0$ and $\bar{M} \ddot{q} - \bar{Q} - \bar{Q^{c}} = p_{1} \Rightarrow \ddot{q} = {\bar{M}}^{- 1} (\bar{Q} + \bar{Q^{c}} + p_{1}) .$ (36) Thus

$\begin{matrix} A [{\bar{M}}^{- 1} (\bar{Q} + \bar{Q^{c}} + p_{1})] - b = 0 \\ \Rightarrow A [D (\bar{Q} + \bar{Q^{c}}) + {Dp}_{1})] - b = 0 . \end{matrix}$ (37) Next, by (33),

$\begin{matrix} 2 β^{T} PA [D (Δ Q + Δ Q^{c}) + Δ D (Q + Q^{c} + p_{1} + p_{2})] \\ \leq & 2 ∥ β ∥ ∥ PA [D (Δ Q + Δ Q^{c}) \\ + Δ D (Q + Q^{c} + p_{1} + p_{2})] ∥ \\ \leq & 2 ∥ β ∥ ρ . \end{matrix}$ (38) Based on (30) and (33),

$\begin{matrix} 2 β^{T} {PADp}_{2} & = 2 β^{T} PAD (- κ {\bar{M}}^{- 1} A^{T} P β) \\ = - 2 κ η^{T} η = - 2 κ {∥ η ∥}^{2} . \end{matrix}$ (39) By $Δ D = DE, {\bar{M}}^{- 1} = D$ , (30), (32) and (26),

$\begin{matrix} 2 β^{T} PA (D + Δ D) p_{3} = 2 β^{T} PA (D + DE) (- γ μ ρ) \\ = & 2 ({DA}^{T} P β ρ)^{T} (I + E) (- γ μ) \\ = & 2 μ^{T} (I + E) (- γ μ) \\ = & - 2 γ μ^{T} μ - 2 γ μ^{T} E μ \leq \\ - 2 γ {∥ μ ∥}^{2} - γ λ_{m} (E + E^{T}) {∥ μ ∥}^{2} \\ \leq & - 2 γ (1 + {\hat{ρ}}_{E}) {∥ μ ∥}^{2} . \end{matrix}$ (40) As ∥μ ∥ > ɛ, by(31) and (32),

$\begin{matrix} - 2 γ (1 + {\hat{ρ}}_{E}) {∥ μ ∥}^{2} & = & - 2 \frac{(1 + {\hat{ρ}}_{E})^{- 1}}{∥ \bar{μ} ∥ ∥ μ ∥} (1 + {\hat{ρ}}_{E}) {∥ μ ∥}^{2} \\ = & - 2 ∥ β ∥ ρ . \end{matrix}$ (41) As ∥μ ∥ ≤ ɛ, by(31),

$\begin{matrix} - 2 γ (1 + {\hat{ρ}}_{E}) {∥ μ ∥}^{2} & = & - 2 \frac{(1 + {\hat{ρ}}_{E})^{- 1}}{{∥ \bar{μ} ∥}^{2} ɛ} (1 + {\hat{ρ}}_{E}) {∥ μ ∥}^{2} \\ = & - 2 {∥ β ∥}^{2} ρ^{2} / ɛ . \end{matrix}$ (42) With (37)-(42), we have for ∥μ ∥ > ɛ, $\dot{V} \leq - 2 κ {∥ η ∥}^{2} - 2 ∥ β ∥ ρ + 2 ∥ β ∥ ρ = - 2 κ {∥ η ∥}^{2},$ (43) and for ∥μ ∥ ≤ ɛ,

$\begin{matrix} \dot{V} & \leq - 2 κ {∥ η ∥}^{2} - 2 {∥ β ∥}^{2} ρ^{2} / ɛ + 2 ∥ β ∥ ρ \\ \leq - 2 κ {∥ η ∥}^{2} + ɛ / 2 . \end{matrix}$ (44) Finally we conclude that $\dot{V} \leq - 2 κ {∥ η ∥}^{2} + ɛ / 2 .$ (45) By Rayleigh’s principle (Chen et al . 1998) and Assumptoin 5,

$\begin{matrix} {∥ η ∥}^{2} & = η^{T} η = β^{T} {PADDA}^{T} P β \\ \geq λ_{m} ({PADDA}^{T} P) {∥ β ∥}^{2} \\ \geq \underline{λ} {∥ β ∥}^{2} . \end{matrix}$ (46) Therefore, $\dot{V} \leq - 2 κ \underline{λ} {∥ β ∥}^{2} + ɛ / 2 .$ (47) Upon invoking arguments as in Chen (1986), we conclude uniform boundedness with $d (r) = {\begin{matrix} r {[\frac{λ_{M} (P)}{λ_{m} (P)}]}^{\frac{1}{2}} if r > R, \\ R {[\frac{λ_{M} (P)}{λ_{m} (P)}]}^{\frac{1}{2}} if r \leq R . \end{matrix}$ (48) $R = {[\frac{ɛ}{4 κ \underline{λ}}]}^{\frac{1}{2}} .$ (49) Uniform ultimate boundedness also follows with $\bar{d} = R {[\frac{λ_{M} (P)}{λ_{m} (P)}]}^{\frac{1}{2}} = {[\frac{ɛ λ_{M} (P)}{4 κ \underline{λ} λ_{m} (P)}]}^{\frac{1}{2}},$ (50) $T (\bar{d}, r) = {\begin{matrix} 0 if r \leq \bar{d} {[\frac{λ_{m} (P)}{λ_{M} (P)}]}^{\frac{1}{2}}, \\ \frac{λ_{M} (P) r^{2} - (λ_{m}^{2} (P) / λ_{M} (P)) {\bar{d}}^{2}}{2 κ \underline{λ} {\bar{d}}^{2} (λ_{m} (P) / λ_{M} (P)) - (ɛ / 2)} otherwise . \end{matrix}$ (51) The ultimate boundedness ball size $\bar{d}$ can be made arbitrarily small by a suitable choice of ɛ.

Remark. With the uncertainty in presence and no restrictions on the initial condition, it is only reasonable to expect approximate constraint following (shown in the ultimate boundedness of β). In special case when there is no uncertainty (i.e., ΔD ≡ 0, ΔQ ≡ 0, ΔQ ^c ≡ 0), One can choose ρ = 0 and hence p ₃ = 0. This means τ = p ₁ + p ₂ and $\dot{V} \leq - 2 κ \underline{λ} {∥ β ∥}^{2}$ . Therefore β → 0 as t→ ∞. Furthermore, if we choose p ₂ = 0, then $\dot{V} = 0$ . This means if the constraint is met initially (i.e., β = 0), then β = 0 for all t ≥ t ₀. This special case falls into Theorem 1, the perfect constraint following case.

Remark. There are three theories to describe the uncertainty, which are deterministic theory, probability theory and fuzzy theory. However, only fuzzy theory can describe the uncertainty quantitatively via membership function of uncertainty and this provides a possibility for the optimization of robust control in this paper. In the practical engineering, there are numerous uncertain systems, such as the room air-condition control system, which can not apply the deterministic theory or the probability theory (Etik et al. 2009). The comfort level to the temperature for the human is vague in nature, which can not be described quantitatively by the deterministic theory or probability theory. But the fuzzy theory can solve for it via the use of membership function perfectly.

4 Optimal gain design

We have shown that a system performance can be guaranteed by a deterministic robust control scheme. By the analysis, the size of the uniform ultimate boundedness region decreases as κ increases. As κ approaches to infinity, the size approaches to 0. This rather strong performance is accompanied by a (possibly) large control effort, which is reflected by κ. From the practical design point of view, the designer may be interested in seeking an optimal choice of κ for a compromise among various conflicting criteria. This is associated with the minimization of a performance index.

We use the fuzzy theory to describe the uncertainty quantitatively via membership function of uncertainty and this provides a possibility for the optimization of robust control.

Assumption 6. (i) For each entry of q ₀ (i.e., q (t ₀)), namely q _0i, i = 1, 2, ⋯ , n, there exits a fuzzy set Q _0i in a universe of discourse Ξ _i ⊂ R characterized by a membership function μ _{Ξ
_i} : Ξ _i → [0, 1]. That is, $Q_{0 i} = {(q_{0 i}, μ_{Ξ_{i}} (q_{0 i})) | q_{0 i} \in Ξ_{i}} .$ (52) Here Ξ _i is known and compact. (ii) For each entry of the vector σ (t), namely σ _i (t), i = 1, 2, ⋯ , p, the function σ _i (·) is Lebesgue measurable. (iii) For each σ _i (t), there exists a fuzzy set S _i in a universe of discourse Σ _i ⊂ R characterized by a membership function μ _i : Σ _i → [0, 1]. That is, $S_{i} = {(σ_{i}, μ_{i} (σ_{i})) | σ_{i} \in Σ_{i}} .$ (53) Here Σ _i is known and compact.

Remark. Assumption 6 imposes fuzzy restriction on the uncertainty q ₀ and σ (t). We employ the fuzzy description on the uncertainties in the mechanical system. This fuzzy description earns much more advantage than the probability avenue which often requires a large number of repetitions to acquire the observed data (always limited by nature).

We first explore more on the deterministic performance of the uncertain mechanical system. We know $- 2 κ {∥ β ∥}^{2} \leq \frac{- 2 κ}{λ_{M} (P)} β^{T} P β .$ (54) Then, by (47), we get

$\begin{matrix} \dot{V} & \leq - 2 κ \underline{λ} {∥ β ∥}^{2} + ɛ / 2 = \underline{λ} (- 2 κ {∥ β ∥}^{2}) + ɛ / 2 \\ \leq \underline{λ} (\frac{- 2 κ}{λ_{M} (P)} β^{T} P β) + ɛ / 2 \\ \leq \frac{- 2 κ \underline{λ}}{λ_{M} (P)} V + ɛ / 2 = : - ψ V + θ . \end{matrix}$ (55) with $V_{0} = V (t_{0}) = V (\underline{e} (t_{0}))$ . This is a differential inequality (Hale 1969) whose analysis can be made according to Chen (1996) as follows.

Definition 2. If w (ψ, t) is a scalar function of the scalars ψ, t in some open connected set D, we say a function $ψ (t), t_{0} \leq t \leq \bar{t}, \bar{t} > t_{0}$ is a solution of the differential inequality (Hale 1969) $\dot{ψ} (t) \leq w (ψ (t), t)$ (56) on $[t_{0}, \bar{t})$ if ψ (t) is continuous on $[t_{0}, \bar{t})$ and its derivative on $[t_{0}, \bar{t})$ satisfies (56).

Theorem 3. Let w (φ (t) , t) be continuous on an open connected set D $\in R^{2}$ and such that the initial value problem for the scalar equation (Hale 1969) $\dot{φ} (t) = w (φ (t), t), φ (t_{0}) = φ_{0}$ (57) has a unique solution. If φ (t) is a solution of (57, 56) on $t_{0} \leq t \leq \bar{t}$ and ψ (t) is a solution of (57, 56) on $t_{0} \leq t \leq \bar{t}$ with ψ (t ₀) ≤ φ (t ₀), then ψ (t) ≤ φ (t) for $t_{0} \leq t \leq \bar{t}$ .

Instead of exploring the solution of the differential inequality, which is often non-unique and not available, the above theorems suggests that it may be feasible to study the upper bound of the solution. The reason is, however, based on that the solution of (57) is unique.

Theorem 4. Consider the differential inequality (56, 57) and the differential equation (56, 57). Suppose that for some constant L > 0, the function w (·) satisfies the Lipschitz condition (Chen 1996) $| w (v_{1}, t) - w (v_{2}, t) | \leq L | v_{1} - v_{2} |$ (58) for all points (v ₁, t) , (v ₂, t) ∈ D. Then any function ψ (t) that satisfies the differential inequality 956) for $t_{0} \leq t \leq \bar{t}$ satisfies also the inequality $ψ (t) \leq φ (t)$ (59) for $t_{0} \leq t \leq \bar{t}$ .

We consider the differential equation $\dot{r} (t) = - ψ r (t) + V + θ, r (t_{0}) = V_{0} .$ (60) The right-hand side satisfies the global Lipschitz condition with L = ψ. We proceed with solving the differential equation (60). This results in $r (t) = (V_{0} - \frac{θ}{ψ}) exp [- ψ (t - t_{0})] + \frac{θ}{ψ} .$ (61) Therefore $V (t) \leq r (t) .$ (62) or $V (t) \leq (V_{0} - \frac{θ}{ψ}) exp [- ψ (t - t_{0})] + \frac{θ}{ψ}$ (63) for all t ≥ t ₀. By the same argument, we also have, for any t _s and any τ ≥ t _s, $V (τ) \leq (V_{s} - \frac{θ}{ψ}) exp [- ψ (τ - t_{s})] + \frac{θ}{ψ}$ (64) where $V_{s} = V (t_{s}) = V (\underline{e} (t_{s}))$ . The time t _s is when the control scheme starts to be executed. It does not need to be t ₀.

We know, $V (\underline{e}) \geq λ_{m} (P) ∥ \underline{e} ∥^{2}$ , the right-hand side of (64) provides an upper bound of $λ_{m} (P) ∥ \underline{e} ∥^{2}$ . This in turn leads to an upper bound of $∥ \underline{e} ∥^{2}$ . For each τ ≥ t _s, let $η (κ, t, t_{s}) : = (V_{s} - \frac{θ}{ψ}) exp [- ψ (τ - t_{s})],$ (65) $η_{\infty} (κ) : = \frac{θ}{ψ} .$ (66) Notice that for each κ, t _s, η (κ, τ, t _s) →0 as τ→ ∞.

One may relate η (κ, t, t ₀) to the transient performance and η _∞ (κ) the steady state performance. Since there is no knowledge of the exact value of uncertainty, it is only realistic to refer to η (κ, t, t ₀) and η _∞ (κ) while analyzing the system performance. We also notice that both η (κ, t, t ₀) and η _∞ (κ) are dependent on θ. The value of θ is not known except that it is characterized by a membership function.

Definition 3. Consider a fuzzy set (Chen 2010) $N = {(v, μ_{N} (ν)) | v \in N} .$ (67) For any function $f : N \to R$ , the D-operation D [f (ν)] is given by $D [f (ν)] = \frac{\int_{N} f (ν) μ_{N} (ν) d ν}{\int_{N} μ_{N} (ν) d ν} .$ (68)

Remark. In a sense, the D-operation D [f (ν)] takes an average value of f (ν) over μ _N (ν). In the special case that f (ν) = ν, this is reduced to the well-known center-of-gravity defuzzification method (Klir & Yuan 1995). Particularly, if N is crisp (i.e., μ _N (ν) =1 for all ν ∈ N), D [f (ν)] = f (ν).

Lemma 1. For any crisp constant $a \in R$ ,

$\begin{matrix} D [af (ν)] & = \frac{\int_{N} af (ν) μ_{N} (ν) d ν}{\int_{N} μ_{N} (ν) d ν} \\ = a \frac{\int_{N} f (ν) μ_{N} (ν) d ν}{\int_{N} μ_{N} (ν) d ν} \\ = aD [f (ν)] . \end{matrix}$ (69)

We now propose the following performance index: For any t _s, let $\begin{matrix} J (κ, t_{s}) & : = D [\int_{t_{s}}^{\infty} η^{2} (κ, τ, t_{s}) d τ] + α D [η_{\infty}^{2} (κ)] + β κ^{2} \\ = : J_{1} (κ, t_{s}) + α J_{2} (κ) + β J_{3} (κ) \end{matrix}$ (70) where α, β > 0 are scalars. The performance index consists of three parts. The first part J ₁ (κ, t _s) may be interpreted as the average (via the D-operation) of the overall transient performance (via the integration) from time t _s. The second part J ₂ (κ) may be interpreted as the average (via the D-operation) of the steady state performance. The third part J ₃ (κ) is due to the control cost. Both α and β are weighting factors. The weighting of J ₁ is normalized to be unity. Our optimal design problem is to choose κ > 0 such that the performance index J (κ, t _s) is minimized.

Remark. A standard LQG (i.e., linear-quadratic-Gaussian) problem in stochastic control is to minimize a performance index, which is the average (via the expectation value operation in probability) of the overall state and control accumulation. The proposed new control design approach may be viewed, loosely speaking, as a parallel problem, though not equivalent, in fuzzy mechanical systems. However, one cannot be too careful in distinguishing the differences. For example, the Gaussian probability distribution implies that the uncertainty is unbounded (although a higher bound is predicted by a lower probability). In the current consideration, the uncertainty bound is always finite. Also, LQG does not take parameter uncertainty into account. One can show that

$\begin{matrix} \int_{t_{s}}^{\infty} η^{2} (κ, τ, t_{s}) d τ \\ = {(V_{s} - \frac{θ}{ψ})}^{2} \int_{t_{s}}^{\infty} exp [- 2 ψ (τ - t_{s})] d τ \\ = \frac{V_{s}^{2}}{2 ψ} - V_{s} \frac{θ}{ψ^{2}} + \frac{θ^{2}}{2 ψ^{3}} . \end{matrix}$ (71) Taking the D-operation yields

$\begin{matrix} D [\int_{t_{s}}^{\infty} η^{2} (κ, τ, t_{s}) d τ] \\ = \frac{1}{2 ψ} D [V_{s}^{2}] - \frac{1}{ψ^{2}} D [V_{s} θ] + \frac{1}{2 ψ^{3}} D [θ^{2}] . \end{matrix}$ (72) The last equality is due to Lemma 1. Next, we analyze the cost J ₂ (κ). Again by Lemma 1, we have

$\begin{matrix} D [η_{\infty}^{2} (κ)] \\ = D [{(\frac{θ}{ψ})}^{2}] \\ = \frac{1}{ψ^{2}} D [θ^{2}] . \end{matrix}$ (73) With (72) and (73) into (70), we also have

$\begin{matrix} J (κ, t_{s}) : & = \frac{1}{2 ψ} D [V_{s}^{2}] - \frac{1}{ψ^{2}} D [V_{s} θ] \\ + \frac{1}{2 ψ^{3}} D [θ^{2}] + α \frac{1}{ψ^{2}} D [θ^{2}] + β κ^{2} . \end{matrix}$ (74) Recalling that $ψ = 2 \underline{λ} κ / λ_{M} (P)$ , we have

$\begin{matrix} J (κ, t_{s}) : \\ = & \frac{λ_{M} (P)}{4 \underline{λ} κ} D [V_{s}^{2}] - \frac{λ_{M}^{2} (P)}{4 {\underline{λ}}^{2} κ^{2}} D [V_{s} θ] \\ + \frac{λ_{M}^{3} (P)}{16 {\underline{λ}}^{3} κ^{3}} D [θ^{2}] + α \frac{λ_{M}^{2} (P)}{4 {\underline{λ}}^{2} κ^{2}} D [θ^{2}] \\ + β κ^{2} \\ = : & \frac{κ_{1}}{κ} - \frac{κ_{2}}{κ^{2}} + \frac{κ_{3}}{κ^{3}} + α \frac{κ_{4}}{κ^{2}} + β κ^{2}, \end{matrix}$ (75) where $κ_{1} = \frac{λ_{M} (P)}{4 \underline{λ}} D [V_{s}^{2}]$ , $κ_{2} = \frac{λ_{M}^{2} (P)}{4 {\underline{λ}}^{2}} D [V_{s} θ]$ , $κ_{3} = \frac{λ_{M}^{3} (P)}{16 {\underline{λ}}^{3}} D [θ^{2}]$ , $κ_{4} = \frac{λ_{M}^{2} (P)}{4 {\underline{λ}}^{2}} D [θ^{2}]$ .

The optimal design problem is then equivalent to the following constrained optimization problem: For any t _s, $min_{κ} J (κ, t_{s}) subject to κ > 0 .$ (76) By using the performance index in (75), we will then pursue the optimal solution in the next section.

For any t _s, taking the first order derivative of J with respect to κ

$\begin{matrix} \frac{\partial J}{\partial κ} & = - \frac{κ_{1}}{κ^{2}} + 2 \frac{κ_{2}}{κ^{3}} - 3 \frac{κ_{3}}{κ^{4}} - 2 α \frac{κ_{4}}{κ^{3}} + 2 β κ \\ = \frac{1}{κ^{4}} (- κ_{1} κ^{2} + 2 κ_{2} κ - 3 κ_{3} - 2 α κ_{4} κ \\ + 2 β κ^{5}) . \end{matrix}$ (77) That $\frac{\partial J}{\partial κ} = 0$ (78) leads to $- κ_{1} κ^{2} + 2 κ_{2} κ - 3 κ_{3} - 2 α κ_{4} κ + 2 β κ^{5} = 0$ (79) or $2 β κ^{5} + 2 κ_{2} κ = κ_{1} κ^{2} + 2 α κ_{4} κ + 3 κ_{3} .$ (80) Equation (80) is a quintic polynomial equation.

Theorem 5. Suppose D [θ] ¬ =0. For given κ₁, κ₂, κ₃, κ₄, the solution κ > 0 to (80) always exists and is unique, which globally minimizes the performance index (75).

Proof. Let Θ (κ) : =2βκ ⁵ + 2κ ₂ κ. Then Θ (0) =0 and Θ (·) is continuous in κ. In addition, since κ ₂ ≥ 0 and β > 0, Θ (·) is strictly increasing in κ. Since D [θ] ¬ =0, we have D [θ] >0, D [θ ²] >0, κ ₃, κ ₄ > 0, κ ₁ ≥ 0 and therefore κ ₁ κ ² + 2ακ ₄ κ + 3κ ₃ > 0 (notice that α, κ > 0). As a result, the solution κ > 0 to (80) always exists and is unique. Therefore the positive solution κ > 0 of the quintic equation (80) solves the constrained minimization problem (76) Q.E.D.

Remark. In the special case that the fuzzy sets are crisp, D [θ] = θ, D [θ ²] = θ ². The current setting still applies. The optimal design can be found by numerically solving (80).

Remark. The robust control scheme using the optimal design of κ > 0 renders the tracking error $\underline{e}$ of the closed-loop mechanical system uniformly ultimately bounded (with the initial state $\underline{e} (t_{s})$ ). In addition, the performance index J is globally minimized.

5 Conclusions

A mechanical system under uncertainty subject to a class of (possibly nonholonomic) constraints is considered. Fundamental equation of constrained mechanical systems is obtained based on Udwadia-Kalaba approach which is applicable to holonomic and nonholonomic constrained systems no matter whether they satisfy the D’Alembert’s principle. The proposed robust control is motivated by the perfect constraint following control design which is based on the D’Alembert’s principle (the Nature’s action). It is proved that the robust control can render the system to follow the performance constraint sufficiently close (or approximately), even in the presence of uncertainty. Fuzzy description of uncertainties in mechanical systems is employed. The control is deterministic and is not if-then rules-based. The uniform boundedness and uniform ultimate boundedness of the tracking error β are assured. A performance index is proposed and by minimizing the performance index, the optimal design problem associated with the control can be solved. The resulting control design is systematic and is able to guarantee the deterministic performance as well as minimizing the cost.

Acknowledgments

The research is supported by “the Fundamental Research Funds for the Central Universities” (No. JZ2014HGBZ0336).

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