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1Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 4, pp. 585-597. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd231205
NONLINEAR ENGINEERING AND ROBOTICS
MSC 2010: 93C10, 93B35, 90C25
Optimization Driven Robust Control of Mechanical Systems with Parametric Uncertainties
C.A. Fam, S. Nedelchev
This paper presents a control algorithm designed to compensate for unknown parameters in mechanical systems, addressing parametric uncertainty in a comprehensive manner. The control optimization process involves two key stages. Firstly, it estimates the narrow uncertainty bounds that satisfy parameter constraints, providing a robust foundation. Subsequently, the algorithm identifies a control strategy that not only ensures uniform boundedness of tracking error but also adheres to drive constraints, effectively minimizing chattering. The proposed control scheme is demonstrated through the modeling of a single rigid body with parameter uncertainties. The algorithm possesses notable strengths such as maximal compensation for parametric uncertainty, chattering reduction, and consideration of control input constraints. However, it is applicable for continuous systems and does not explicitly account for uncertainty in the control input.
Keywords: optimization, sliding mode control, parametric uncertainty, stability
1. Introduction
Control of mechanical systems is an active research area that has attracted the attention of researchers for decades. During this time, many approaches and solution methods have been proposed and developed for systems of various types [1, 2].
One of the most common control approaches is to use proportional derivative (PD) controllers. These controllers are easy to implement and can provide satisfactory performance in many applications. However, their ability to deal with complex and nonlinear systems is limited.
To overcome the limitations of PD controllers, various nonlinear control methods have been developed. One such method is the use of the control Lyapunov functions (CLFs). The control
Received November 08, 2023 Accepted December 11, 2023
This work was supported by the Russian Science Foundation (Project number 22-41-02006).
Cham An Fam
c.fam@innopolis.university
Simeon Nedelchev
s.nedelchev@innopolis.university
Center for Technologies in Robotics and Mechatronics Components, Innopolis University ul. Universitetskaya 1, Innopolis, 420500 Russia
Lyapunov function is a powerful tool for developing stabilizing controllers for nonlinear systems. They are based on the concept of Lyapunov stability, which provides a measure of the stability of a system based on the behavior of its solutions. Applications of control Lyapunov functions can be found in [3, 4].
Another approach to controlling mechanical systems is based on the use of quadratic programming (QP) techniques to optimize control inputs subject to constraints. This method is widely used in controlling robots and other mechanical systems. For example, Spong [5] proposes an approach to optimizing torque for controlling manipulators. This method involves developing a nonlinear control law that solves the drive torque limitation problem using quadratic programming techniques. A number of studies have also examined control of quadruped robots based on quadratic programming and regressor based robust control [6-9].
Adaptive control is a technique in which controller parameters can be dynamically "adapted" in real time. This approach is especially useful in the case of mechanical systems characterized by parameter uncertainty or subject to external disturbances [10]. However, this method has a drawback: it is sensitive to sudden changes in the values of system parameters, which can lead to instability in control.
One of the methods for controlling nonlinear systems with uncertain parameters is sliding mode control [11, 12]. This method is robust to external disturbances and uncertainties in the system, but has the disadvantage of excessive power consumption since it requires constant control switching to maintain the system in the sliding surface. This frequent switch leads to the chattering in control. In addition, the user needs to customize control parameters to achieve system stability.
In practice, the system's property of linearity in terms of its parameters is often used, which leads to a new representation of the system through a regressor. In the current work, this property is to be utilized. The regressor provides a compact and unique representation of the equation of motion for a given set of parameters. This representation simplifies calculations and controller design of systems such as manipulators, drones and four-legged robots. Control over the systems represented in regressor form is considered in [9, 13-16]. The regressor form expresses the equation of motion through a linear combination of known system state functions and unknown parameters. This representation simplifies the calculation of the control law and makes it independent of specific parameter values. Using the regressor form, the control system designer can treat the parameters as unknown variables and design the controller based on the known functions. This approach reduces the complexity of the control design and simplifies its implementation. In addition, the regressor form provides a concise and comprehensive view of the system dynamics, which facilitates the analysis of system behavior and the development of control strategies. Overall, the use of regressors simplifies the control design of mechanical systems by providing a compact and independent representation of system dynamics, reducing complexity and facilitating control analysis and design.
In this field of mechanical systems research, various control approaches are actively used, each of which has its own strengths and weaknesses. The choice of a control method for a mechanical system must be carefully justified based on the specific application requirements and system characteristics, taking into account their advantages and limitations.
1.1. Dynamic models
This section reviews models of mechanical systems and highlights the representation of mechanical systems in regressor form.
The dynamics of the system are described by a system of second-order nonlinear ordinary differential equations (ODEs) of the form
M(q)q + C(q, q)q + g(q) + d(q, q) = B(q)u, (1.1)
where M(q) £ Rraxra is a positive definite inertia matrix, C(q, q) £ Rraxra is a skew-symmetric matrix, which usually describes Coriolis and centrifugal forces, g(q) represents gravitational force, and d(q, q) is a source of other forces (external forces, friction force etc.). Matrix B(q) £ £ Rraxm defines the input mapping. In this work, we consider fully-actuated and over-actuated systems (i.e., rank{B(q)} ^ n, Vq).
Expression (1.1) can be obtained in a standard way using the Euler - Lagrange approach:
d (dL\ dL &R ^ .
lU)-9i+«rQ = B(q)u' (L2)
where L = EK — En is a Lagrangian, EK(q, q), En(q) are kinetic and potential energy, respectively, Q represents generalized forces, and R(q, q) is a Rayleigh dissipation function which describes the half of power related to disturbance and dissipative forces.
1.2. Linearity to parameters
A notable property of most mechanical systems widely used in this work is the ability to linearly represent their energy through parameters. In particular, we consider mechanical systems with Lagrange and Rayleigh functions, which are described by the following equations:
L(q, q) = &(q, q)0, (1.3)
1 2(
where 0 £ Rp is a parameter vector (usually constant or slowly varying).
An immediate consequence of the linearity of L and R is the ability to linearly represent the differential equation of dynamics (1.2) in the following form:
Y(q, q, q)0 = Q = B(q)u, (1.5)
where Y(q, q, q) £ Rra is called regressor matrix. The advantage of this representation is that once the generalized coordinates q(i) are known as functions of time, the relationship between system inputs and uncertain parameters can immediately be linearly determined.
1.3. Parametric uncertainty
In practice, the exact values of the parameters 0 are unknown in advance; for example, there may be uncertainty in the payload and drive inertia or uncertainty may appear due to external disturbances.
However, in most cases, the actual system parameters can usually be factorized to their approximate estimates 00 and the uncertain part 0 as 0 = 00 + 0 such that the indefinite part 0 belongs to some set P. This set of parameters can be inherently complex, but in most practical cases it is usually possible to approximate P by one of the following relaxations:
||S(q, q, t)0\\2 < p(q, q, t) ellipsoid, (1.6)
A(q, q, t)0 ^ b(q, q, t) polytope. (1.7)
q) = ^qTQD(q, q) = ^(q, q)0, (1.4)
The advantage of ellipsoidal uncertainty (1.6) is that the extrema of ||S6||2 can be obtained analytically by examining the spectrum (possibly varying with time and system configuration) S, while the uncertainty expressed by a polytope is actually more suitable for accounting for most practical system uncertainties, such as a box constraint of the form 6min ^ 6 ^ 6max• Figure 1 shows the region of parametric uncertainty of three different types for a system with two parameters p1 and p2.
Vi A
Fig. 1. Illustration of uncertainty bounds for a system with two parameters. The blue area corresponds to a constraint defined as a polytope (rectangle), the orange area to an ellipoisoidal constraint, and the green area to a constraint defined as a circle
2. Conventional robust control
One of the most elegant controllers widely used in robotics is based on passivity and linearity of parameters, as proposed by Spong [9]. The idea of this controller comes from the famous property of passivity, linearity of parameters [17] and robust technique sometimes referred to as the Leitmann approach [18].
First we define the sliding variable s which is related to the tracking error q = q — qd (the difference between the actual and desired states) as follows:
s = q + Aq. (2.1)
Vector s conveys information about the boundedness and convergence of q and q, since s is a collection of stable (for A > 0) first-order differential equations with respect to q with s at the input. Thus, the convergence of s will in turn imply the exponential convergence of q, and the total synthesis can be considered as a stabilization in s. Further, it is necessary to introduce a "virtual velocity" vector qs = qd — Aq. Then the sliding variable can be redefined as s = q — qs. If the parameters of the system are precisely known, then the control law can be chosen as follows:
u = M(q)qs + C(q, q)qs + g(q) — Ks, (2.2)
which is equivalently redefined using the property of linearity of parameters (i. e., the control law can be rewritten in the form of a regressor):
u = Yc (q, q, qs, qs) 9 — Ks. (2.3)
Here, we refer to Yc (q, q, qs, qs) as the controller regressor, and K >- 0 is the gain matrix that regulates the speed of convergence to the sliding surface.
To illustrate the convergence of this law, it is convenient to define the following Lyapunov candidate:
V(t) = ±sTM(q)s. (2.4)
The time derivative will be equal to
V(t)= sT (u - Mqs - Cqs - g). (2.5)
Substituting the control law (2.2) gives the following derivative of the Lyapunov function:
V(t) = -sT Ks.
The resulting derivative is negative definite, which means convergence of the system to the surface s = 0 and proves the tendency of q and q to 0 as t ^-<x>.
However, in real applications, the robot's parameters are usually unknown. Actual values may differ from the nominal parameters and can be expressed by the difference 6 = 00 - 0, where 00 represent nominal parameters.
To compensate for parametric uncertainty, the following control law can be utilized [9]:
u = Yc (q, q, qr, qr)(0Q + 50) - Ks, (2.6)
where 50 is the new control input that needs to be found to compensate for parametric uncertainty.
In this case, the derivative of the Lyapunov candidate is defined as
F(t) = -sTKs - sTYc(6 + 50). (2.7)
Let us introduce an auxiliary variable z = Y^Ts. Then, the negativity of the derivative of the Lyapunov function (2.7) will correspond to the following restriction on the second term of the derivative of the Lyapunov function:
zT6 + zT50 ^ 0. (2.8)
To satisfy this condition, it is necessary to calculate the uncertainty bound, namely, the worst case for zT0. In classical robust control, it is assumed that the parameter uncertainty is norm-bounded, i.e.: ||6|| < p, and the Cauchy-Schwarz inequality can be used to obtain an analytical solution, assuming that the vector of new input data 50 points in the direction z, and its length is greater than the bound of parametric uncertainty p:
p-^-T if ||z|| > 0, 50 = { ||z| (2.9)
0 if ||z|| =0.
Substituting 50 into the expression (2.7) leads to V ^ 0 since zT6+p||z|| ^ 0 and ||6|| • ||z|| < < p||z||. Such a controller will ensure asymptotic convergence to the sliding surface and, consequently, convergence of the tracking error.
It should be noted that the value of p is constant and, unlike the uncertainty boundaries, in general does not depend on the desired and actual trajectory, as well as time [9]. This is in
contrast to the usual sliding mode controllers, when the sliding mode gain is usually depends on the trajectory and changes over time.
Due to the discontinuity of the above law (when ||z|| = 0), its direct implementation leads to chattering in the control input that can cause high-frequency unmodeled dynamic modes and may require insufficient drive bandwidth. Therefore, in practice, the smoothed control law is implemented through the following approximation:
In this case, asymptotic convergence is not guaranteed, but it is shown that the tracking error is ultimately uniformly bounded (UUB) [19] within an attractive and invariant boundary layer with a thickness determined by user-defined parameter n.
It is worth noting that the algorithm discussed above does not necessarily exactly compensate for the unknown parameters, but generates values that achieve the desired goal, namely, the convergence of the sliding variable s to zero.
Such a control design approach raises several issues:
• overly conservative uncertainty bounds;
• physical actuator constraints are not taken into account;
• careful selection of the thickness of the boundary layer is required to reduce control chat-
Therefore, the next section will propose an alternative method to solve the issues above.
3. Enhanced robust control
In this section, we propose several enhancements to control design for systems with parametric uncertainties:
1) tightening the parametric uncertainty bounds;
2) consideration of actuator constraints;
3) smoothing control and reducing chattering.
The control synthesis task will consist of two stages. First, to estimate the worst possible narrow uncertainty bounds that satisfy the system constraints. Second, to synthesize the control that: 1) provides ultimately uniformly bounded tracking error, 2) considers physical restrictions on the actuators, 3) takes into account input mapping, 4) eliminates control chattering.
3.1. Tightened uncertainty bounds
The contribution of parametric uncertainty to the control system can be analyzed using the corresponding part of the time derivative of the Lyapunov function (2.7), namely, the projection
(2.10)
tering.
of the uncertainty vector 6 to the auxiliary error variable z. Thus, we can find the extrema of the uncertainty contribution by solving the following optimization problem:
6* = argmax zT 6. (3-1)
0eP
Depending on the convex set P, the problem described above can be represented either by a linear (polytopic uncertainty) or a conic program (ellipsoidal uncertainty).
Then control synthesis is formulated as a set of convex optimization problems that can be solved in point-wise real time fashion:
minimize J(56, e)
¿0, £
s.t. - sup{zT0} - zT 56 < e (3-2)
h(56, e) < 0,
where J(56, e) is the loss function which is to be determined, sup{zT6} is a supremum of zT6, found in (3.1), e is a positive variable defining the thickness of the boundary layer (however, unlike the variable n in (2.10), e changes with time), and h(56, e) takes into account additional restrictions in the system.
The resulting linear constraint has the following geometric interpretation: the projection of the uncertainty vector 6 on the vector z must be less than the projection 56 onto the vector z.
3.2. Incorporating the actuator constraints
The convex optimization problem (3.2) can be modified by introducing new inputs to account for the constraint on the actuator. Then the control u must satisfy the condition umin ^ u ^ umax. Otherwise, this condition can be reformulated for the system represented in regressor form as
umin - Ym60 ^ Ym 56 ^ umax - Ym60.
The condition obtained is also linear in design variables, so it can be easily included in any convex optimization problem, which will allow us to take into account the practically important constraint, that is, the actuator constraint.
3.3. Smoothing the control
Instead of smoothing the control by manually tuning boundary layers, we propose to find smooth parametric control inputs by considering the trade-off between chattering and disturbance suppression and minimizing the following objective function:
J (56,e) = ||5'6|| + 7e2, (3.3)
where the constant 7 is used to adjust the trade-off between chattering and tracking accuracy. The larger the parameter 7, the smaller the value of e and thus the higher the tracking accuracy.
3.4. Implementation issues
Consolidation into one common optimization problem is possible, however, the current tendency is to embed controllers in control units equipped with multicore processors. Therefore, the division of the control design problem into several subtasks has practical benefits. The control calculation can be performed at a lower frequency in a separate thread (or process), increasing the overall computation speed. It is also possible to speed up the calculations by parallelizing the calculation of each row of the regressor matrix and separating the problem formulation and its solution.
4. Case study: actuated single rigid body
4.1. Dynamic model and control
In this work we use, as an example, the model of single a rigid body (SRB), which is described by the following system of ODEs:
f .
IT = v,
q = ^(q)^,
N
mV — mg = RT (q) ^ fi,
i=1 N
lbu + u x lbu = Y^ ri x fi,
(4.1)
i=1
where m and lb are the body mass and inertia, respectively, r is the position of the body's center of mass (CoM) in the world coordinate system, u is the angular velocity relative to the body's reference frame, fi is the external force relative to the body's reference frame, ri is the position of the point of application of force fi relative to the CM, expressed in the body's reference system, q is a unit quaternion describing the orientation of the body relative to the world coordinate system, g is a gravity vector, S is a quaternion rotation operator, RT(q) is a transposed rotation matrix, and N is the number of applied forces.
Regressor form
First, we determine the constant inertial parameters of the body as follows:
e
m i I I I I I
±xx *yy xy xz yz
T
where is a corresponding component in inertia matrix lb.
To apply the developed method, we rewrite the left side of the dynamics, namely, the last two equations of the system (4.1) in the regressor form (1.5) with the regressor specified as
Y(u, u, V)
0
3x6
V + g
03X1 L(u) + [u]xL(u)
(4.2)
The operator [-]x defines a skew-symmetric matrix used to describe the vector product [a]x b = a x b. Matrix L £ R3x6 is used to represent the scalar product lu linearly
to the vector of inertial parameters vech{I}, where vech{•} £ R6 is the semi-vectorization operator that transforms a symmetric 3 x 3 matrix into a column vector 6 x 1. For example, if we define the inertial parameters as the last 6 components of 0, vech {I} = [Ixx, Iyy, Izz, Ixy, Ixz, Iyz], then the scalar product Iu can be parameterized as L(^)vech{I} with L(w) defined as
LM
Now we consider the right side of the two last equations in (4.1) and rewrite it in the form Q = B(q)u. Generalized forces related to external forces and torques are defined as follows:
Ux 0 0 Uy 0
0 0 ^x 0 Uz
0 0 0 Uy
Q = B(q)u =
Given the equality assumption u = fi, we can define the input matrix B(q) as
N ■
RT(q) £ f
i=1
N
£ ri x fi . i=1
B(q)
RT (q) [riix
RT (q)'
[rN ]x
Algorithm 1 demonstrates a step-by-step software implementation for calculation of optimal control.
Algorithm 1: Simulation of an actuated single rigid body
1 for i ^ size(time array) do
2 get the user-defined desired state
3 solve numerically the optimization problem (3.2) and get S0k, S0k, e
4 calculate u from (2.6)
5 integrate the dynamical differential equation (4.1) from ti-1 to ti and get the actual state
6 update the state of the system with the actual state.
7 end for
4.2. Results
4.2.1. Implementation of an optimized robust control algorithm
During the simulation, the following dependencies of the position and orientation of the body were obtained. Figures 2 and 3 show transient processes for 7 = 500.
The control input graph is shown in Fig. 4. The maximum value of control input is 200 N • m and the minimum value is —200 N • m. The time dependence of the parameter e at 7 = 500 is shown in Fig. 5.
CO
O PL,
0.8 0.6 0.4 0.2 0.0
-— — X — V — z
_______ /
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Time (s)
Fig. 2. Position transient process
ö o
■ I—I ö in
<D
■s
o*
1.0 0.9 0.8 0.7
{ %
J
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Time (s)
(a) quaternion component q0
a
O ■ i—l
ö
!h ®
¡a
OV
0.0 -0.2 -0.4 -0.6
— h
—92" — Q3
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Time (s)
(b) quaternion component q0, qi, q2
Fig. 3. Orientation transient process
5 g,
CP
6
¡H
S
200 100 0 -100
-200
~u0 — u, -U2 -u3-
// - — tv
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Time (s)
Fig. 4. Control input
4.2.2. Implementation of the optimized robust control with different penalty parameter 7
As mentioned earlier, the parameter 7 plays the role of regularization between tracking accuracy and control smoothness. The larger 7, the more we penalize the second term in (3.3), which means that we provide higher accuracy in approximating the desired position or orientation. By reducing 7, tracking accuracy becomes the second priority, and system stabilization becomes the primary goal. Figure 6 illustrates this dependence.
xlO
-6
0.0 0.2 0.4 0.6 0.8 1.0 1.2 Time (s)
Fig. 5. Time dependence of the parameter s (design variable from the optimization problem (3.2))
0.1 0.0 -0.1 -0.2 -0.3 -0.4
g
y = 0.001 y = 0.005 y=l
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Time (s)
Fig. 6. Transient process of coordinate £ with different y
J, 0.0
h
0
(H ¡H Ol -0.2
N
-0.4
— slotine — proposed
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Time (s)
(a) error in z coordinate
0.05-^ 0.04 3 0.03 o 0.02
¡5 0.01 * 0.00 -0.01
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Time (s)
(b) error in z coordinate (zoomed)
Fig. 7. Transient process for the £ coordinate (red indicates the algorithm proposed by Slotine [12], blue indicates the optimized robust control)
4.2.3. Optimized robust control versus the conventional robust algorithm
Let us present the results of a comparative analysis of the classical robust algorithm and optimized robust control in the context of tracking the £ coordinate. Figure 7 shows the tracking error of the £ coordinate. The classical method exhibits a tracking error of approximately 0.01 m, while the optimized algorithm ensures accurate tracking without noticeable errors. From Fig. 8,
0.50 1.00 Time (s)
(a) Uo
0.50 1.00 Time (s)
(b) Ui
0.50 1.00 Time (s)
(c) U2
0.00 0.50 1.00 1.50 Time (s)
(d) U3
0.00 0.50 1.00 1.50 Time (s)
(e) U4
0.00 0.50 1.00 1.50 Time (s)
(f) U5
Fig. 8. Control input comparison of the classical approach with the optimized robust control (red indicates the algorithm proposed by Slotine [12], blue indicates the optimized robust control)
the conventional approach produces the control of high frequency, which leads to the chattering. In contrast, the proposed algorithm provides a smoothed control.
5. Conclusion
This work has demonstrated the effectiveness of the optimized robust algorithm in solving control problems. However, certain limitations of this method have been identified. First, more research needs to be done to optimize the gain coefficients, since in the current work they were considered as constant values, which may limit the applicability of the algorithm under different conditions. Secondly, it is necessary to take into account the uncertainty in the matrix B(q) in the differential dynamics equation (1.5), which reflects the relationship between the input forces and moments and the generalized forces. This aspect is important for control accuracy and stability in real systems. Finally, it is possible to study this control algorithm in discrete form, taking into account the features of discrete calculations in computer systems. Further research in these directions can help improve and expand the applicability of this method in various areas of systems control.
Conflict of interest
The authors declare that they have no conflict of interest.
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