Finite-dimensional control of linear discrete-time
This paper addresses the design of finite-dimensional feedback control laws for linear discrete-time fractional-order systems with additive state disturbance. A set of sufficient conditions are provided to guarantee convergence of the state trajectories to an ultimate bound around the origin with size increasing with the magnitude of the disturbances. Performing a suitable change of coordinates, the latter result can be used to design a controller that is able to track reference trajectories that are solutions of the unperturbed fractional-order system. To overcome the challenges associated with the generation of such solutions, we address the practical case where the references to be tracked are generated as a solution of a specific finite-dimensional approximation of the original fractional-order system. In this case, the tracking error trajectory is driven to an asymptotic bound that is increasing with the magnitude of the disturbances and decreases with the increment in the accuracy of the approximation. The proposed controllers are finite-dimensional, in the sense that the computation of the control input only requires a finite number of previous state and input vectors of the system. Numerical simulations illustrate the proposed design methods in different scenarios.
IST]Andrea Alessandretti REN]Sérgio Pequito PEN]George J. Pappas IST]A. Pedro Aguiar
Faculty of Engineering, University of Porto, Porto, Portugal.
Department of Industrial and Systems Engineering, Rensselaer Polytechnic Institute,Troy, USA
Department of Electrical & Systems Engineering School, Engineering and Applied Science, University of Pennsylvania, USA
A fractional-order derivative is a generalization of an integer order derivative, where the order can take non-integer values. Its discretization results in a fractional-order difference  that rules the dynamics of the discrete time counterpart. In this paper, we address the design of control laws for dynamical systems described by fractional-order difference equations, i.e., discrete-time fractional-order systems (FOSs).
In contrast to integer order differences, which use local information around a given point, fractional-order differences are obtained using all the previous values of a given function up to the point where the difference is computed. Because of the global nature of the fractional-order operators, models described by fractional-order differences seem to be particularly suitable to capture the behavior of systems with infinite memory and hence have found applications in many fields. Many biological systems manifest fractional-order behaviors . Fractional-order derivatives are used to model the cardiac tissue-electrode interface and electrically stimulated or mechanically stressed tissues [14, 13], drugs relies and adsorption , and the behavior of neural systems . Applications on polymer science are highlighted in . In , transfer functions with a pole of fractional power are used to model analog electronic circuits. The work  explores the benefits of fractional models for modeling of fuel cells.
The many applications and the control challenges associated with fractional-order systems motivated a wide literature in the control community. For the continuous-time case,  shows the equivalence of stability with the presence of the eigenvalues of a given polynomial in a specific angular sector, generalizing the integer order case where such a sector becomes the left half plane. Building on this, the works in [1, 5, 16] focus on developing linear matrix inequality (LMI) conditions for stability and extensions to the case of uncertain polytonic FOSs. The work  provides analytical expressions for the unit-step and unit-impulse of FOSs, that is then used to compare a proposed fractional-order PID controller with the traditional integer-order counterpart. The work  presents the concept of Mittag-Leffler stability for nonlinear FOSs and proposes a Lyapunov-like condition to certify such stability, using both integer and fractional-order derivative of the Lyapunov function.
For the discrete-time FOSs, the control design problem is addressed either using the infinite-dimensional formulation of the FOS , or adopting a finite-dimensional approximation of the FOS obtained by truncating the infinite-dimensional fractional-order difference operator and using the associated stabilizing controller to control the original FOS [9, 20]. The main drawback of the first approach is that the stability conditions that arise from the analysis of the whole infinite-dimensional form of FOS are generally difficult to verify and to apply for (finite-dimensional) control design purposes. The second approach was initially introduced in , without providing stability guarantees, and then adopted in the work  for the design of a stabilizing Model Predictive Control scheme. In the latter approach, the authors derive sufficient conditions for practical asymptotic stability and, upon satisfaction of an additional condition, asymptotic stability of the origin. In addition, the latter work is able to explicitly handle constraints in both the system state and input vectors.
In this work, we retain the simplicity in the design of the widely used approach proposed in , while at the same time providing closed-loop guarantees. The analysis adopted in this work builds upon the reasoning proposed in . Notwithstanding, as the main differences, this work does not consider system constraints and shows that convergence can be guaranteed globally, i.e., for any initial conditions of the system. The proposed controller is easy to implement, without the need of optimization-based schemes. In addition, we address (i) possible additive disturbances acting on the system and (ii) the trajectory-tracking problem. For the latter case, we also consider the practical scenario where the reference trajectories are solutions of an exogenous finite-dimensional approximation of the original infinite-dimensional FOS. Specifically, we provide sufficient conditions for the tracking error to converge to an asymptotic bound that is increasing with the magnitude of the disturbances and decreases with the increment in the accuracy of the approximation.
The remainder of this paper is structured as follows. Section 2 introduces the linear discrete-time fractional-order system and formalizes the control problem addressed in this paper. Section 3 contains the main results that are illustrated in Section 4 with numerical simulations. All the proofs are relegated to the Appendix.
Notation: The terms , , , , denote the set of reals, non-negative reals, integers, non-negative integers, and positive integers numbers, respectively. The terms and represent the set of column vectors of size and -by- matrices with real entries. A continuous function is said to belong to class if it is strictly increasing and . In addition, a function is said to belong to class if, in addition, as . Moreover, a continuous function is said to belong to class if, for a constant , belongs to class with respect to , and for a constant it is decreasing on and satisfies as (see ). The term denotes the identity function, i.e., . The term is used to denoted a identity matrix. For a sequence , we use the notation and , if with , and if . Given a square matrix , the notation indicates that the matrix is positive definite, i.e., for any . Moreover, whenever clear from the context, we use and to refer to and , respectively.
2 Control problem formulation
This section introduces the discrete-time FOS and defines the control objectives addressed in this paper.
2.1 Discrete-time fractional-order systems
Consider a left-bounded sequence over , i.e., . Then, for any the Grünwald-Letnikov fractional-order difference  is defined as
for all . Notice that the summation in (2.1) is well defined from the uniform boundedness of the sequence and the fact that
which implies that the sequence is absolutely summable for any . A discrete-time fractional-order system with additive disturbance can be modeled as
for scalars , with , , with , and , with , and where , , and denote the state, input, and disturbance vectors at time , respectively. The latter vector is considered to be bounded as
for some positive scalar . We denote by the initial condition of the FOS at time . In the computation of the fractional-order difference, the state, input, and disturbances are considered to be zero before the initial time, i.e., , , and for all .
It is worth noticing that the infinite dimensional case where the state is defined for all the times, can be addressed in the presented theorems considering the limit of the initial time that goes to minus infinity (or equivalently, considering the case ). In addition, given the robustness properties of the proposed control law, it is possible to addressed the case where the controller is activated at a given time by considering the previous absence of control input (or zero input signal) as a (bounded) disturbance.
2.2 Control objectives
The first control objective concerns the regulation of the state of the FOS toward the origin.
Problem 1 (Convergence)
Design a finite-dimensional controller that renders the associated closed-loop system with (2.1) globally bounded and globally ultimately bounded with respect to the disturbance (2.1), i.e., for any initial condition the associated closed-loop state trajectory is bounded and satisfies
for a function , with as , and a class- function .
Definition 1 (Finite-dimensional controller)
We denote by finite-dimensional controller a control law that only requires a finite number of previous state and input vectors to decide the current input to apply to the system, i.e., there exist two values such that the control law can be expressed in the form .
As it will be made explicit in the remainder of the paper, performing a suitable change of coordinate, a controller that solves Problem 1 can be used to track any reference trajectory that is a solution of (2.1) (similar to integer-order linear systems). It is interesting to note here that since (2.1) is infinite-dimensional, it often presents a major challenge in cases where one wishes to generate such a reference trajectory. Specifically, for any desired input reference trajectory , the computation of the associated (nominal) fractional-order state trajectory (i.e., the solution of (2.1) considered with , and for all ) becomes intractable as due to the sum of an always growing number of terms. To avoid this situation, we consider the case where the reference trajectory is generated using a finite-dimensional exogenous system.
Problem 2 (Tracking exogenous solutions)
Let , , and , with , be the state, input, and output vectors, respectively, of the finite-dimensional exogenous system
defined as the difference between the state of the FOS and the reference signal, satisfies
for a function , with as , a class- function , and a constant .
3 Main contribution
Before stating the main results, different formulations of the FOSs (2.1) are introduced, as well as mild technical assumptions.
The matrix is invertible.
Formulation (3) highlights one of the main peculiarities of FOSs, i.e., that the state does not depend only on the state, input, and disturbance at time , but it is a function of the whole past trajectory.
For any given integer , the model (2.1) can be equivalently rewritten as
using the augmented state vector
and appropriate matrices , , and , where denotes the initial condition. Specifically, the terms and are used to form the matrices and , whereas the remaining terms and the states and input components not included in are collected in the term .
Definition 3 (v-approximation)
Assumption 4 (v-stabilizability)
The pair of the v-approximation of the FOS system (2.1) is stabilizable.
Assumption 4 implies the existence of a matrix such that for any positive definite matrix the equality
holds for a positive definite matrix . At this point, we are ready to state one of the main results of this paper.
and , , , , , for any . Then, for any positive definite matrix , there exists a pair of matrices and such that (12) holds and the controller
with , solves Problem 1 with , where and .
Theorem 5 provides a set of sufficient conditions that guarantee the state vector of the closed-loop system (2.1) with (15) to converge to an ultimate bound with size proportional to the maximum magnitude of the noise. The controller (15) is designed to be any linear controller that stabilizes the -approximation of (2.1), where satisfies condition (13). It is worth noticing that since the state of the -approximation (with ) stores the previous state and input signals of the original fractional-order system, the control (15) is not a standard state-feedback controller for the FOS, but rather uses feedback of a window of past states and inputs, where the window size increases with .
In the control design for integer-order linear system, a controller designed to stabilize the origin of the state space can be used to stabilize the system around any of its solutions, i.e., performing a suitable change of coordinates, it is possible to stabilize the origin of the error space defined as the difference between the current state of the system and a solution of the system that we wish to track.
Corollary 6 (Tracking FOS solution)
Let the assumptions of Theorem 5 hold and consider a generic state and input reference trajectory pair and , respectively, that are the solution of (2.1) with for all , and the associated augmented state vector . Then, the closed-loop (2.1) with
leads to an error trajectory that is globally bounded and globally ultimately bounded with respect to the disturbance (2.1). Specifically, for any initial condition the error trajectory is bounded over time and satisfies
for a function , with as , and a class- function .
Corollary 6 highlights the robustness of the proposed control strategy, where the origin of the error space is ultimately bounded with the size of the ultimate bound being increasing with the bound of the disturbance.
In what follows, we consider the case where the trajectory that we want to track is not a solution of (2.1), but rather it is a solution of a finite-dimensional approximation.
Theorem 7 (Tracking solution v-approximation)
It is worth noticing that the existence of bounded state and input trajectories and required in (18) is always guaranteed by the Assumption 4. Moreover, we recall that since the exponential function can be rewritten as the infinite sum the function defined in Theorem 5 can be equivalently reformulated as
that decreases to zero as goes to infinity. As a consequence, the asymptotic bound of Theorem 7 is decreasing with the increment in the accuracy of the approximation.
In practice, Theorem 7 shows that the state and input trajectories of the original FOS can track arbitrarily well (by increasing the value of ) any references and that are generated as a solution of (2), where is the state of the v-approximation. As a consequence, we can control the original infinite-dimensional FOS by designing a finite-dimensional controller (using standard techniques for finite dimensional systems) for the auxiliary system (2). The overall controller (that includes the v-approximation system) achieves in closed loop a tracking error whose ultimate bound decreases with the increment in the accuracy of the approximation. An illustrative example is presented in the following section with the design of a finite-dimensional controller that is able to steer the infinite-dimensional FOS around a desired trajectory.
4 Numerical simulation
For different values of , we display the behavior of the closed-loop (2.1) with (15) and (19) for both the cases of convergence to the origin and trajectory-tracking displayed in Fig. 2 and Fig. 3, respectively. Moreover, in the first case, we consider both the noise-free case with and the case with noise where and the noise is generated as a random signal uniformly distributed within the interval (i.e., ).
Different approaches can be used to generate solutions of (2), that are reference trajectories for the trajectory-tracking algorithm. In this example, we used a model predictive control scheme (detailed in Appendix B) to compute reference trajectories for the -approximation that drives toward a pre-defined sinusoidal trajectory. It is worth noticing that for each , we generally have a different state and input solution of the -approximation (2) that achieve our objective (i.e., that drives ). This behavior is shown in Fig. 3 where the reference trajectory for in the case (dashed-line blue) differs from the case (dashed-line orange). Despite this, in both cases the state is driven toward the desired pre-defined sinusoidal trajectory.
It is worth noticing that the system under analysis is open-loop unstable. Moreover, despite the -approximation with being stabilizable, it follows that the associated linear controller does not stabilize the original infinite-dimensional system. Specifically, for the case , Assumption 4 is satisfied but the condition (13) is violated. Fig. 2 and Fig. 3 show that the closed-loop system is unstable with the error trajectory oscillating with increasing magnitude. To avoid this undesired behavior, Theorem 5 provides a suitable selection of the value of that satisfies (13) and therefore guarantees global convergence to the origin and, for the case of trajectory-tracking, global convergence to an ultimate bound around the reference signal.
It is worth noticing that the function strongly decreases as the value of increases, as depicted in Fig. 1. This plays an important role in the design and analysis of the controller, since small values of render condition (13) easy to satisfy and lead to small values of the term in (20). This is reflected in the following table where the value of is close to zero since .
This work introduces a set of procedures to design finite-dimensional controllers for FOSs that guarantee closed-loop convergence to the origin, in the case of regulation, or to a reference trajectory, in the case of trajectory-tracking. This is achieved exploiting a linear controller designed for a suitable finite-dimensional approximation of the original FOS. For the case of trajectory-tracking, we consider both cases where the trajectories to be tracked are solutions of a FOS or solution of an exogenous finite-dimensional linear system. The latter case is desired in many practical cases where the reference trajectories are to be generated. All the proposed results are global, in the sense that the convergence is guaranteed for any initial condition of the system, and robust to additive disturbances.
The following lemma is used in the proofs of the main results.
Consider the system (3) and let Assumptions 2 and 4 hold. Then, for any positive definite matrix there exists a pair of matrices and , with being a symmetric matrix, such that (12) holds and, for the closed-loop system (3) with (15), the state trajectory satisfies
with , for a class- function , and from Theorem 5.
PROOF. Consider the function
for any scalar . Notice that for any the inequality
holds with , , . Then, we can bound the last part of the right hand side of (A.3) as follows
with and where in the first inequality we used the fact that for any symmetric matrix the inequality
Therefore, we can write
with , , , , and , i.e., , with . Noticing that , the function is an ISS-Lyapunov function (see, e.g., Definition 3.2 of ), which implies that there exist a class- function and a class- function such that
for all . Now, let be any class- lower bound of such that , e.g., with for any . Moreover, let , with . Then, following  (specifically, from (13) to (17)) it is possible to show that (A.5) holds with where , i.e., , and for a class- function , which concludes the proof.
a.1 Proof of Theorem 5
The proof is structured as follows. First, we show that the state trajectory of the closed-loop system (3) with (15) is uniformly bounded over time and converges to an ultimate bound proportional to the value of , which also includes the contributes of the disturbance vectors . Then, noticing that the magnitude of is increasing with the increase of the norm of the past state vectors, a contraction argument can be invoked to show convergence to an ultimate bound that is proportional only to , which concludes the proof.
a.1.1 Ultimate boundedness
Notice that we used the fact that since, as described in the problem formulation, the state of the system is considered to be the zero vector before the initial time . Combining the latter with (2.1) and the defining (21) we obtain
that under condition (13) leads to
for a class- function .
We start by noticing that for any the bound on the norm of in (11b) can be decomposed as
The remainder of the proof is carried out by recursion. Specifically, for a generic , consider a constant and a function such that
holds, where for any given , is bounded over and satisfies as . Moreover, consider a that satisfies