Fractional sliding mode control of MIMO nonlinear noncommensurable plants

Abstract

This paper addresses the fractional sliding mode control of MIMO nonlinear noncommensurable plants, which does not seem to have been covered until this moment in the literature on fractional sliding mode control. It includes simulation results to confirm the feasibility of the solution presented.

Keywords

Sliding mode control fractional control MIMO noncommensurable plants nonlinear control systems uncertain dynamic systems three-tank system

1. Introduction

Sliding mode control is a control technique developed in the late 1950s for MIMO (multiple-input, multiple-output) plants that can be linear or nonlinear. It has become widely used; an introduction thereto can be found in many books, such as Slotine (1991). The basic idea of sliding mode control is to use the input to make the pseudo-state-vector converge to a subspace, called sliding surface, of the space where it may otherwise evolve, and stay there. The sliding surface is chosen so that, on that surface, the control objective is fulfilled. Deviations from the sliding surface are corrected because all trajectories converge back to the sliding surface. This is graphically shown in Figure 1, where $s$ is the sliding surface, and the arrows are the system's trajectories, all converging thereto.

Figure 1.

A schematic of sliding mode control.

Sliding mode control can be adapted to plants that make use of fractional derivatives, or to sliding surfaces defined using fractional derivatives. This has been done, for instance, by Monje et al. (2010), Yin et al. (2012), Si-Ammour et al. (2009) and Delavari et al. (2010); the two latter references address its application to specific plants.

In such references, it seems the MIMO nonlinear noncommensurable case was never addressed. This paper addresses this case; a preliminary version appeared as Valério and Sá da Costa (2012a) (theoretical results) and Valério and Sá da Costa (2012b) (experimental results). Some basic results of fractional calculus and fractional dynamic systems are summed up in Section 2. Then it is expedient to consider first SISO (single-input, single-output) plants, addressed in Section 3, and MIMO nonlinear commensurable plants, in Section 4. The results will then be merged, to cope with MIMO nonlinear noncommensurable plants, in Section 5. The paper concludes with simulation and experimental results to confirm the feasibility of the solution (in Section 6) and some final remarks (in Section 7).

2. Fractional calculus and fractional dynamic systems

Derivatives $\frac{d n}{d t n} \equiv D n$ and integrals $\overset{n integrations}{\overset{︷}{\int_{0}^{t} \int_{0}^{t} \dots \int_{0}^{t} 1 mu d t \dots 1 mu d t 1 mu d t}} \equiv {_{0} D}_{t}^{- n}$ , with $n \in ℤ_{0}^{+}$ , are defined in calculus. Fractional calculus generalizes these notions to account for integration orders that do not need to be integer. A differential operator ${_{0} D}_{t}^{α}$ can thus be defined for any order $α \in ℂ$ ; it will coincide with the derivatives and integrals of integer order if $α \in ℤ$ . Note that fractional derivatives are called “fractional” but the order α may be irrational or complex. In this paper only real orders will be employed.

Readers interested in Fractional Calculus should consult, for instance, one of the papers Valério and Sá da Costa (2011a), Lavoie et al. (1976), or one of the books Podlubny (1999), Kilbas et al. (2006), or Valério and Sá da Costa (2013), for definitions and properties of fractional derivatives. (There are several alternative possible definitions to extend operator D for noninteger orders, which are equivalent in some cases but not in others.) For the purposes of this paper, the following results suffice.

Theorem 1.

The Laplace transform of a fractional derivative of real negative order α is

L [{_{0} D}_{t}^{α} f (t)] = s α F (s)

(1)

where

F (s) = L [f (t)]

Remark 1.

Remember that, if $α \in ℤ_{0}^{-}$ , then ${_{0} D}_{t}^{α} f (t)$ is the α-order integral of function f(t).

Theorem 2.

The Laplace transform of a fractional derivative of real positive order α is

\begin{matrix} L [{_{0} D}_{t}^{α} f (t)] = s α F (s) \\ + terms related to initial conditions \end{matrix}

(2)

where the terms related to initial conditions depend on how operator D is extended to noninteger orders.

Definition 1.

Dynamic systems described by differential equations involving fractional derivatives give rise (initial conditions being zero) to fractional transfer functions, i.e. those that involve fractional powers of s. A fractional transfer function is said to be commensurable of order α if all differentiation orders in the polynomials of the numerator and the denominator are integer multiples of α; in other words, a commensurable transfer function is a ratio of two polynomials in $s α$ :

G (s) = \frac{\sum_{k = 0}^{m} b_{k} s k α}{\sum_{k = 0}^{n} a_{k} s k α}

(3)

Theorem 3.

(Matignon's theorem). Let $ς_{k}, k = 1 \dots n$ , be the roots of the polynomial $A (ς) = \sum_{k = 0}^{n} a_{k} ς k$ , built with the denominator coefficients of G(s), given by (3). Then G(s) is stable if and only if

| ∠ ς_{k} | > α \frac{π}{2}, \forall k

(4)

(restricting

∠ ς_{k}

[- π, + π]

rad).

Theorem 4.

The transfer function

G (s) = \frac{1}{(\frac{s}{a}) α + 1}, α, a \in ℝ +

(5)

is stable when

0 < α < 2

. Its frequency response is as shown in Figure 2.

Remark 2.

In this paper, fractional orders are assumed to remain constant with time; variable-order fractional derivatives (handled in Valério and Sá da Costa, 2010, 2011b; Sun et al., 2011; Valério and Sá da Costa 2012c) will not be addressed.

A treatment of fractional Lyapunov functions and the fractional Lyapunov stability theory can be found in Li et al. (2009, 2010). For the present purposes it suffices to state that the existence of a Lyapunov function is enough to ensure system stability, and that in most cases function $\frac{1}{2} s T (x (t)) s (x (t))$ can be and is chosen for that role. In the SISO case, it reduces to function $\frac{1}{2} s 2 (x (t))$ . (The sliding surface $s$ will be defined below in Sections 3.2 and 4.2.)

3. The SISO case

3.1. The problem

Figure 2.

Bode diagram of $\frac{1}{(\frac{s}{a}) α + 1}, 0 < α < 2$ .

Let the SISO plant to control be a generic, nonlinear, noncommensurable plant given by

{\begin{matrix} {_{0} D}_{t}^{α_{1}} x (t) & = & x_{2} (t) \\ {_{0} D}_{t}^{α_{2}} x_{2} (t) & = & x_{3} (t) \\ : \\ {_{0} D}_{t}^{α_{n - 1}} x_{n - 1} (t) & = & x_{n} (t) \\ {_{0} D}_{t}^{α_{n}} x_{n} (t) & = & f_{A} (x (t)) + f_{B} (x (t)) u (t) \\ y (t) & = & x (t) \end{matrix}

(6)

where:

y(t) is the output;

$α_{i} > 0, \forall i = 1, \dots, n$ ;

all initial conditions are assumed to be zero;

pseudo-state vector $x (t)$ is given by

\begin{matrix} x (t) = [\begin{matrix} x (t) \\ x_{2} (t) \\ x_{3} (t) \\ : \\ x_{n - 1} (t) \\ x_{n} (t) \end{matrix}] = [\begin{matrix} x (t) \\ {_{0} D}_{t}^{α_{1}} x (t) \\ {_{0} D}_{t}^{α_{2}} {_{0} D}_{t}^{α_{1}} x (t) \\ : \\ {_{0} D}_{t}^{α_{n - 1}} \dots {_{0} D}_{t}^{α_{2}} {_{0} D}_{t}^{α_{1}} x (t) \end{matrix}] \\ = [\begin{matrix} x (t) \\ {_{0} D}_{t}^{α_{1}} x (t) \\ {_{0} D}_{t}^{α_{1} + α_{2}} x (t) \\ : \\ {_{0} D}_{t}^{\sum_{k = 1}^{n - 1} α_{k}} x (t) \end{matrix}]} n elements \end{matrix}

(7)

Remark 3.

Note that this last equality is a consequence of all initial conditions being zero, otherwise the last equality would not necessarily be true.

We want the output $y (t) = x (t)$ to follow a reference r(t), and for this we will make $x (t)$ follow a reference vector $r (t)$ , given by

r (t) = [\begin{matrix} r (t) \\ {_{0} D}_{t}^{α_{1}} r (t) \\ {_{0} D}_{t}^{α_{2}} {_{0} D}_{t}^{α_{1}} r (t) \\ : \\ {_{0} D}_{t}^{α_{n - 1}} \dots {_{0} D}_{t}^{α_{2}} {_{0} D}_{t}^{α_{1}} r (t) \end{matrix}] = [\begin{matrix} r (t) \\ {_{0} D}_{t}^{α_{1}} r (t) \\ {_{0} D}_{t}^{α_{1} + α_{2}} r (t) \\ : \\ {_{0} D}_{t}^{\sum_{k = 1}^{n - 1} α_{k}} r (t) \end{matrix}]

(8)

Error

ɛ (t) = x (t) - r (t)

ɛ (t) = [\begin{matrix} ε (t) \\ {_{0} D}_{t}^{α_{1}} ε (t) \\ {_{0} D}_{t}^{α_{1} + α_{2}} ε (t) \\ : \\ {_{0} D}_{t}^{\sum_{k = 1}^{n - 1} α_{k}} ε (t) \end{matrix}] = [\begin{matrix} [x (t) - r (t)] \\ {_{0} D}_{t}^{α_{1}} [x (t) - r (t)] \\ {_{0} D}_{t}^{α_{1} + α_{2}} [x (t) - r (t)] \\ : \\ {_{0} D}_{t}^{\sum_{k = 1}^{n - 1} α_{k}} [x (t) - r (t)] \end{matrix}]

(9)

Remark 4.

If the plant is commensurable, $α_{1} = α_{2} = \dots = α_{n}$ , the pseudo-state vector $x (t)$ is given by

\begin{matrix} x (t) = [\begin{matrix} x (t) \\ x_{2} (t) \\ x_{3} (t) \\ : \\ x_{n - 1} (t) \\ x_{n} (t) \end{matrix}] = [\begin{matrix} x (t) \\ {_{0} D}_{t}^{α_{1}} x (t) \\ {_{0} D}_{t}^{α_{2}} {_{0} D}_{t}^{α_{1}} x (t) \\ : \\ {_{0} D}_{t}^{α_{n - 1}} \dots {_{0} D}_{t}^{α_{2}} {_{0} D}_{t}^{α_{1}} x (t) \end{matrix}] \\ = [\begin{matrix} x (t) \\ {_{0} D}_{t}^{α_{1}} x (t) \\ {_{0} D}_{t}^{α_{1} + α_{2}} x (t) \\ : \\ {_{0} D}_{t}^{\sum_{k = 1}^{n - 1} α_{k}} x (t) \end{matrix}]} n elements \end{matrix}

(10)

instead of (7),

r (t)

is given by

r (t) = [\begin{matrix} r (t) \\ {_{0} D}_{t}^{α} r (t) \\ {_{0} D}_{t}^{2 α} r (t) \\ : \\ {_{0} D}_{t}^{(n - 1) α} r (t) \end{matrix}]

(11)

instead of (8),

ɛ (t)

\begin{matrix} ɛ (t) = [\begin{matrix} ɛ (t) \\ {_{0} D}_{t}^{α} ɛ (t) \\ {_{0} D}_{t}^{2 α} ɛ (t) \\ : \\ {_{0} D}_{t}^{(n - 1) α} ɛ (t) \end{matrix}] = y (t) - r (t) \\ = [\begin{matrix} [x (t) - r (t)] \\ {_{0} D}_{t}^{α} [x (t) - r (t)] \\ {_{0} D}_{t}^{2 α} [x (t) - r (t)] \\ : \\ {_{0} D}_{t}^{(n - 1) α} [x (t) - r (t)] \end{matrix}] \end{matrix}

(12)

instead of (9).

3.2. Sliding surface

A most reasonable choice for the sliding surface $s (x (t))$ is a linear combination of fractional derivatives of the error, including all orders found in $x (t)$ (and in $r (t)$ , and in $ɛ (t)$ ), and eventually some others:

s (x (t)) = 0

(13)

s (x (t)) = \sum_{i = 1}^{m} μ_{i} {_{0} D}_{t}^{β_{i}} ɛ (t)

(14)

\forall k = 1, \dots, n - 1, \exists i : β_{i} = \sum_{p = 1}^{k} α_{p}

(15)

Applying the Laplace transformation to (14), we get

\frac{E (s)}{S (X (s))} = \frac{1}{\sum_{i = 1}^{m} μ_{i} {_{0} D}_{t}^{β_{i}}}

(16)

It is of course reasonable to choose coefficients $μ_{i}$ so that transfer function (16) is stable.

Forcing $s (x (t)) = 0$ , we will obtain $ɛ (t) = 0$ , and hence $x (t) = r (t)$ .

Remark 5.

If the plant is commensurable, the sliding surface is commonly chosen as

s (x (t)) = 0

(17)

s (x (t)) = ({_{0} D}_{t}^{β} + λ) m ε (t)

(18)

where

β \in ℝ +

, such that

\frac{α}{β} \in ℕ

, and λ is a parameter defined by the control designer. In other words, α must be an integer multiple of β. We want all differentiation orders in

x (t)

(and in

r (t)

, and in

ɛ (t)

) to be present in

s (x (t))

, and for this reason

β m = (n - 1) α \Leftrightarrow m = (n - 1) \frac{α}{β}

(19)

Hence,

s (x (t)) = ({_{0} D}_{t}^{β} + λ) (n - 1) \frac{α}{β} ɛ (t)

(20)

For instance, when

m = 2

, then either

n = 3 \land α = β

n = 2 \land α = 2 β

, and

\begin{matrix} s (x (t)) = ({_{0} D}_{t}^{β} + λ) 2 ɛ (t) =_{0} D_{t}^{(n - 1) α} ɛ (t) \\ + 2 λ {_{0} D}_{t}^{(n - 1) \frac{1}{2} α} ɛ (t) + λ 2 ɛ (t) \end{matrix}

(21)

This way, $s (x (t))$ is a linear combination of the outputs of the plant. Applying the Laplace transformation to (18), we get

\frac{E (s)}{S (X (s))} = \frac{1}{(s β + λ) m}

(22)

For transfer function (22) to be stable, we must have

λ > 0

While it is likely that $s (x (t))$ cannot be always made exactly equal to zero, it is possible to find an upper bound for the error.

Theorem 5.

If $s (x (t))$ is given by (18) with $λ > 0$ , and $| s (x (t)) | < Φ, \forall t$ , then, after the transient regime has passed away,

| ɛ (t) | \leq \frac{Φ}{λ m} = \frac{Φ}{λ (n - 1) \frac{α}{β}}

(23)

| {_{0} D}_{t}^{i β} ɛ (t) | \leq \frac{2 i Φ}{λ m - i}, \forall i = 1, \dots, m

(24)

0 < β \leq 1

, or

| ɛ (t) | \leq \frac{Φ}{λ m \sin m \frac{β π}{2}} = \frac{Φ}{λ (n - 1) \frac{α}{β} \sin (n - 1) \frac{α}{β} \frac{β π}{2}}

(25)

| {_{0} D}_{t}^{i β} \int (t) | \leq \frac{(1 + \sin \frac{β π}{2}) i Φ}{λ m - i \sin m \frac{β π}{2}}, \forall i = 1, \dots, m

(26)

1 < β < 2

Proof.

Equation (22) means that $ɛ (t)$ and $| {_{0} D}_{t}^{i β} ɛ (t) |$ may be obtained from $s$ as seen in the top and bottom block diagrams of Figure 3, respectively. Equations (23)–(26) result from applying the properties of the frequency response in Theorem 4 to this block diagram.□

Theorem 6.

When the plant is not commensurable, expressions similar to (23)–(26) may be obtained from a decomposition of (22) in factors. The actual result depends on how many of the $β_{i}$ are in the ]0,1] range and how many are in the ]1,2[ range, e.g.

| ɛ (t) | \leq \frac{Φ}{\underset{i when 0 < α_{i} \leq 1}{Π} λ_{i} \underset{i when 1 < α_{i} \leq 2}{Π} λ_{i} \sin \frac{α_{i} π}{2}}

(27)

3.3. Following the sliding surface

As mentioned above, when the system remains on surface $s (x (t)) = 0$ , the desired reference is followed. Choosing $\frac{1}{2} s 2 (x (t))$ as Lyapunov function candidate, we are led to a control law that ensures

\begin{matrix} \frac{1}{2} \frac{d s 2 (x (t))}{d t} \leq - η | s (x (t)) | \Leftrightarrow \\ \Leftrightarrow s (x (t)) \frac{d s (x (t))}{d t} \leq - η | s (x (t)) | \Leftrightarrow \\ \Leftrightarrow σ (s (x (t))) \frac{d s (x (t))}{d t} \leq - η \end{matrix}

(28)

where

η \in ℝ +

is a value chosen by the control designer, and

σ (x)

is the sign function. Roughly speaking, since the sign of the derivative of

s 2 (x (t))

is always negative, and since

s 2 (x (t))

is always positive,

s (x (t))

is always tending to zero.

Figure 3.

Block diagrams corresponding to (22) to demonstrate Theorem 5.

If the dynamics of the system are known exactly, (28) suffices to follow surface $s$ . The system's dynamics are usually known with an error, however; but it is possible to circumvent this fact introducing a nonlinearity, as seen in the two following examples.

3.4. Example: commensurable plant, uncertainty in $f_{A} (x (t))$

Let us consider a commensurable plant with $f_{B} (x (t)) = 1$ :

{_{0} D}_{t}^{n α} x (t) = f_{A} (x (t)) + u (t)

(29)

Suppose that $f_{A} (x (t))$ is not known exactly, but that an approximate value ${\overset{\land}{f}}_{A} (x (t))$ is known.

We will define the sliding surface as in (20):

\begin{matrix} s (x (t)) = ({_{0} D}_{t}^{β} + λ) (n - 1) \frac{α}{β} ɛ (t) \\ =_{0} D_{t}^{(n - 1) α} ɛ (t) + S (ɛ (t)) \\ =_{0} D_{t}^{(n - 1) α} [x (t) - r (t)] + S (ɛ (t)) \\ =_{0} D_{t}^{- α} [f_{A} (x (t)) + u (t)] - {_{0} D}_{t}^{(n - 1) α} r (t) + S (ɛ (t)) \end{matrix}

(30)

All terms except ${_{0} D}_{t}^{(n - 1) α} ɛ (t)$ are collected in $S (ɛ (t)) = s (x (t)) -_{0} D_{t}^{(n - 1) α} ɛ (t)$ , and from now on we will omit most dependencies on $t, x$ and $ɛ$ , as well as the terminals of operator D, to alleviate the notation. Hence,

D s = D 1 - α f_{A} + D 1 - α u - D 1 + n α - α r + D S

(31)

and so

D s = 0 \Leftrightarrow u = - f_{A} + D n α r - D α S

(32)

but since $f_{A}$ is unknown and only ${\overset{\land}{f}}_{A}$ is known, we will rather implement the control action as

u = - {\overset{\land}{f}}_{A} + D n α r - D α S - kD α - 1 σ (s)

(33)

where the reason for the choice of the nonlinearity

- kD α - 1 σ (s)

will be clear in a moment, when desirable values for constant k are found. Replacing (33) in (31) we obtain

\begin{matrix} D s = D 1 - α f_{A} - D 1 - α {\overset{\land}{f}}_{A} + D 1 + n α - α r - D S \\ - k σ (s) - D 1 + n α - α r + D S \\ = D 1 - α (f_{A} - {\overset{\land}{f}}_{A}) - k σ (s) \end{matrix}

(34)

and replacing this in (28) we obtain

\begin{matrix} D 1 - α (f_{A} - {\overset{\land}{f}}_{A}) σ (s) - k \leq - η \Leftrightarrow \\ \Leftrightarrow k \geq η + D 1 - α (f_{A} - {\overset{\land}{f}}_{A}) σ (s) \end{matrix}

(35)

The term $D 1 - α (f_{A} - {\overset{\land}{f}}_{A}) σ (s)$ may be either positive or negative. If we can find an upper bound $F$ for the error committed with approximation $D 1 - α f_{A} \approx D 1 - α {\overset{\land}{f}}_{A}$ , that is to say, if $| D 1 - α (f_{A} - {\overset{\land}{f}}_{A}) | < F$ , then we make $k \geq η + F$ in control law (33) and thereby guarantee that (28) is verified, even in the presence of uncertainty in $f_{A}$ .

3.5. Example: integer plant, uncertainty in $f_{B} (x (t))$

Consider plant

{\begin{matrix} D 2 x (t) = bu (t) \\ y (t) = x (t) \end{matrix}

(36)

where we only know a variation range for b. We define

s (x (t))

s (x (t)) =_{0} D_{t}^{1 + α} ɛ (t) + λ ɛ (t), α > 0

(37)

which means that, even though the plant is of integer order, when the sliding surface

s (x (t)) = 0

is being followed, it will behave with fractional order dynamics. We will have (alleviating again from now on the notation as done in the example above)

\begin{matrix} D s = D 2 + α ɛ + λ D ɛ \\ = λ D ɛ + bD α u - D 2 + β r \end{matrix}

(38)

and so

D s = 0 \Leftrightarrow bu = D 2 r - λ D 1 - α ɛ

(39)

but since b is unknown and only

\overset{\land}{b}

is known, we implement the control action as

\tilde{u} = D 2 r - λ D 1 - α ɛ

(40)

u = \frac{\tilde{u} - kD - α σ (s)}{\overset{\land}{b}} = \frac{1}{\overset{\land}{b}} (D 2 r - λ D 1 - α ɛ - kD - α σ (s))

(41)

and from there we get

\begin{matrix} D s = λ D ɛ + \frac{b}{\overset{\land}{b}} D 2 + α r - \frac{b}{\overset{\land}{b}} λ D ɛ - \frac{b}{\overset{\land}{b}} k σ (s) - D 2 + α r \\ = (1 - \frac{b}{\overset{\land}{b}}) (λ D ɛ - D 2 + α r) - \frac{b}{\overset{\land}{b}} k σ (s) \end{matrix}

(42)

which, replaced in (28), gives

\begin{matrix} (1 - \frac{b}{\overset{\land}{b}}) (λ D ɛ - D 2 + α r) σ (s) - \frac{b}{\overset{\land}{b}} k \leq - η \Leftrightarrow \\ \Leftrightarrow k \geq \frac{\overset{\land}{b}}{b} η + (\frac{\overset{\land}{b}}{b} - 1) \underset{- D α \tilde{u}}{\underset{︸}{(- D 2 + α r + λ D ɛ)}} σ (s) \end{matrix}

(43)

and, consequently, taking into account that

\frac{1}{β} \leq \frac{\overset{\land}{b}}{b} \leq β

, and making

k \geq β η + (β - 1) | D α \tilde{u} |

(44)

we guarantee that (28) is verified.

3.6. Avoiding chattering

Owing to the hard nonlinearity in the definition of the control law, the system will likely oscillate around $s (x (t)) = 0$ , causing control chattering, which is undesirable. This can be improved replacing the sign function $σ (ξ)$ in the definition of control action u(t) with some function $ς (ξ)$ that smooths the nonlinearity. Two possible choices for $ς (ξ)$ are, for instance,

ς_{1} (ξ) = {\begin{matrix} ξ, if | ξ | \leq 1 \\ σ (ξ), if | ξ | > 1 \end{matrix}

(45)

ς_{2} (ξ) = \frac{2}{π} \arctan ζ ξ

(46)

where

ζ \in ℝ +

is chosen to regulate how far

ς_{2} (ξ)

is from

σ (ξ)

(notice that, when

ζ \to + \infty, ς_{2} (ξ)

approaches

σ (ξ)

The price to pay for avoiding chattering with such replacements is a performance degradation, but limits to the errors incurred can be found using the expressions in Theorem 5.

4. The MIMO commensurable case

4.1. The problem

Let the plant to control be a square (i.e. having as many outputs as inputs) commensurable MIMO plant, given by

{\begin{matrix} {_{0} D}_{t}^{n_{1} α} x_{1} (t) & = f_{1} (x (t)) + \sum_{j = 1}^{q} b_{1 j} (x (t)) u_{j} (t) \\ {_{0} D}_{t}^{n_{2} α} x_{2} (t) & = f_{2} (x (t)) + \sum_{j = 1}^{q} b_{2 j} (x (t)) u_{j} (t) \\ : \\ {_{0} D}_{t}^{n_{i} α} x_{i} (t) & = f_{i} (x (t)) + \sum_{j = 1}^{q} b_{ij} (x (t)) u_{j} (t) \\ : \\ {_{0} D}_{t}^{n_{q} α} x_{q} (t) & = f_{q} (x (t)) + \sum_{j = 1}^{q} b_{qj} (x (t)) u_{j} (t) \\ y (t) & = [x_{1} (t) x_{2} (t) \dots x_{q} (t)] T \end{matrix}

(47)

where

there are q inputs, q outputs and q equations;

$n_{i} \in ℕ, \forall i = 1, \dots, q$

the pseudo-state vector is

x (t) = [\begin{matrix} x_{1} (t) \\ {_{0} D}_{t}^{α} x_{1} (t) \\ {_{0} D}_{t}^{2 α} x_{1} (t) \\ : \\ {_{0} D}_{t}^{n_{1} α} x_{1} (t) \\ x_{2} (t) \\ : \\ {_{0} D}_{t}^{n_{2} α} x_{2} (t) \\ : \\ x_{q} (t) \\ : \\ {_{0} D}_{t}^{n_{q} α} x_{q} (t) \end{matrix}]

(48)

(this is why we could not write $y (t) = x (t)$ );

we want to track a reference $r (t) = [r_{1} (t) r_{2} (t) \dots r_{q} (t)] T$ ;

we define, to simplify notation,

u (t) = [u_{1} (t) u_{2} (t) \dots u_{q} (t)] T

(49)

f (t) = [f_{1} (x (t)) f_{2} (x (t) \dots f_{q} (x (t))] T

(50)

B (t) = [\begin{matrix} b_{11} (x (t)) & \dots & b_{1 q} (x (t)) \\ : & : \\ b_{q 1} (x (t)) & \dots & b_{qq} (x (t)) \end{matrix}]

(51)

\begin{matrix} ɛ (t) = y (t) - r (t) = [ε_{1} (t) ε_{2} (t) \dots ε_{q} (t)] T \\ = [x_{1} (t) - r_{1} (t) x_{2} (t) - r_{2} (t) \dots x_{q} (t) - r_{q} (t)] T \end{matrix}

(52)

n = [n_{1} n_{2} \dots n_{q}] T

(53)

This last vector will be used with derivatives of vectorial order, defined by

{_{0} D}_{t}^{α} x (t) = [{_{0} D}_{t}^{α_{1}} x_{1} (t) {_{0} D}_{t}^{α_{2}} x_{2} (t) \dots {_{0} D}_{t}^{α_{n}} x_{n} (t)] T

(54)

4.2. Uncertainty in $f (x (t))$ and in $B (t)$

Let the sliding surface be defined by

s (x (t)) = 0

(55)

\begin{matrix} s (x (t)) = ({_{0} D}_{t}^{β} + λ) (m) ɛ (t) \Leftrightarrow \\ \Leftrightarrow s_{i} (x (t)) = ({_{0} D}_{t}^{β} + λ_{i}) m_{i} ε_{i} (t), i = 1, \dots, q \end{matrix}

(56)

λ = [λ_{1} λ_{2} \dots λ_{q}] T

(57)

m = [m_{1} m_{2} \dots m_{q}] T

(58)

\frac{α}{β} \in ℕ

(59)

β m_{i} = (n_{i} - 1) α i = 1, \dots, q

(60)

where

a (b)

denotes the Hadamard power (or Schur power, or entrywise power) or vectors a and b. As for the SISO case, we want

\frac{1}{2} d s_{i} 2 (x (t)) d t 2 \leq - η_{i} | s_{i} (x (t)) |, i = 1, \dots, q \Leftrightarrow \Leftrightarrow σ (s_{i} (x (t))) \frac{d s_{i} (x (t))}{d t} \leq - η_{i}

(61)

where

δ = [η_{1} η_{2} \dots η_{q}] T \in (ℝ +) q

is a vector chosen by the control designer.

Let us consider plant (47), where $f (x (t))$ is not known exactly, while an approximation $\overset{\land}{f} (x (t))$ is known; and where $B (t)$ is not known exactly, while an invertible approximation $\overset{\land}{B} (t)$ is known, such that

\begin{matrix} B = (I + Δ) \overset{\land}{B} \Rightarrow \\ \Rightarrow B \overset{\land}{B} - 1 = I + Δ \Rightarrow \\ \Rightarrow I - B \overset{\land}{B} - 1 = - Δ \end{matrix}

(62)

Let there also be a matrix $D$ of majorants of $Δ$ such that

| Δ_{ij} | \leq D_{ij}, i, j = 1, \dots, q

(63)

We will write (56) as

\begin{matrix} s_{i} (x (t)) =_{0} D_{t}^{(n_{i} - 1) α} ɛ_{i} (t) + S_{i} (ɛ_{i} (t)) \\ =_{0} D_{t}^{(n_{i} - 1) α} [x_{i} (t) - r_{i} (t)] + S_{i} (ɛ_{i} (t)) \\ =_{0} D_{t}^{- α} [f_{i} (x (t)) + \sum_{j = 1}^{q} b_{ij} (x (t)) u_{j} (t)] \\ - {_{0} D}_{t}^{(n_{i} - 1) α} r_{i} (t) + S_{i} (ɛ_{i} (t)), i = 1, \dots, q \end{matrix}

(64)

collecting all terms save

{_{0} D}_{t}^{(n_{i} - 1) α} ɛ_{i} (t), i = 1, \dots, q

S (ɛ (t)) = [S_{1} (ɛ_{1} (t)) S_{2} (ɛ_{2} (t)) \dots S_{q} (ɛ_{q} (t))] T

(65)

We will from now on alleviate the notation as in the previous cases. Hence,

\begin{matrix} D s_{i} = D 1 - α f_{i} + \sum_{j = 1}^{q} b_{ij} D 1 - α u_{j} - D 1 + n_{i} α - α r_{i} + D S_{i} \Leftrightarrow \\ \Leftrightarrow D s = D 1 - α f + B D 1 - α u - D 1 + n α - α r + D S \end{matrix}

(66)

and so

D s = 0 \Leftrightarrow Bu = - f + D n α r - D α S

(67)

Since f and B are unknown and only

\overset{\land}{f}

and

\overset{\land}{B}

are known, we will implement the control action as

\tilde{u} = - \overset{\land}{f} + D n α r - D α S

(68)

\begin{matrix} u = \overset{\land}{B} - 1 (\tilde{u} - k \circ D α - 1 σ (s)) \\ = \overset{\land}{B} - 1 (- \overset{\land}{f} + D n α r - D α S - k \circ D α - 1 σ (s)) \end{matrix}

(69)

where the gain vector

k = [k_{1} k_{2} \dots k_{q}] T \in (ℝ +) q

will be determined in a moment, and ^ denotes the Hadamard product (or Schur product, or entrywise product). Replacing (69) in (66), and then adding and subtracting

D 1 - α \overset{\land}{f}

, we obtain

\begin{matrix} D s = D 1 - α f - B \overset{\land}{B} - 1 D 1 - α \overset{\land}{f} + B \overset{\land}{B} - 1 D 1 + n α - α r \\ - B \overset{\land}{B} - 1 D S - B \overset{\land}{B} - 1 k \circ σ (s) - D 1 + n α - α r + D S \\ = D 1 - α (f - \overset{\land}{f}) + \underset{Δ}{\underset{︸}{(I - B \overset{\land}{B} - 1)}} (D 1 - α \overset{\land}{f} - D 1 + n α - α r + D S) \\ - \underset{I + Δ}{\underset{︸}{B \overset{\land}{B} - 1}} k \circ σ (s) = D 1 - α (f - \overset{\land}{f}) \\ + Δ (D 1 - α \overset{\land}{f} - D 1 + n α - α r + D S) - k \circ σ (s) - Δ k \circ σ (s) \end{matrix}

(70)

It is now expedient to separate this into its q equations

\begin{matrix} D s_{i} = D 1 - α (f_{i} - {\overset{\land}{f}}_{i}) + \sum_{j = 1}^{q} Δ_{ij} (D 1 - α {\overset{\land}{f}}_{j} - D 1 + n_{j} α - α r_{j} + D S_{j}) \\ - k_{i} σ (s_{i}) - \sum_{j = 1}^{q} Δ_{ij} k_{j} σ (s_{j}) \\ = D 1 - α (f_{i} - {\overset{\land}{f}}_{i}) + \sum_{j = 1}^{q} Δ_{ij} (D 1 - α {\overset{\land}{f}}_{j} - D 1 + n_{j} α - α r_{j} + D S_{j}) \\ - (1 + Δ_{ii}) k_{i} σ (s_{i}) - \begin{matrix} \sum_{j = 1}^{q} \\ j \neq i \end{matrix} Δ_{ij} k_{j} σ (s_{j}) \end{matrix}

(71)

so as to replace them into (61) and obtain

\begin{matrix} D 1 - α (f_{i} - {\overset{\land}{f}}_{i}) σ (s_{i}) + \sum_{j = 1}^{q} Δ_{ij} (D 1 - α {\overset{\land}{f}}_{j} - D 1 + n_{j} α - α r_{j} \\ + D S_{j}) σ (s_{i}) - (1 + Δ_{ii}) k_{i} - \begin{matrix} \sum_{j = 1}^{q} \\ j \neq i \end{matrix} Δ_{ij} k_{j} σ (s_{j}) σ (s_{i}) \leq - η_{i} \end{matrix}

(72)

If we can find a vector $F = [F_{1} F_{2} \dots F_{q}] T$ such that $| D 1 - α (f_{i} - {\overset{\land}{f}}_{i}) | < F_{i}, i = 1, \dots, q$ , then, taking into account (63), we can make

\begin{matrix} (1 - D_{ii}) k_{i} + \begin{matrix} \sum_{j = 1}^{q} \\ j \neq i \end{matrix} D_{ij} k_{j} σ (s_{j}) σ (s_{i}) \geq η_{i} + F_{i} \\ + \sum_{j = 1}^{q} D_{ij} | D 1 - α {\tilde{u}}_{j} | \end{matrix}

(73)

We can allow the most favourable values, which correspond to equalities in the q equations above, and so we have a system of q equations from which the q gains k_i can be found. When replaced in control law (69), they guarantee that (61) is verified, even in the presence of uncertainty in f and B.

5. The MIMO noncommensurable case

For the MIMO noncommensurable case, let the plant be

{\begin{matrix} {_{0} D}_{t}^{α_{n_{1}, 1}} \dots {_{0} D}_{t}^{α_{2, 1}} {_{0} D}_{t}^{α_{1, 1}} x_{1} (t) & = & f_{1} (x (t)) + \sum_{j = 1}^{q} b_{1 j} (x (t)) u_{j} (t) \\ {_{0} D}_{t}^{α_{n_{1}, 2}} \dots {_{0} D}_{t}^{α_{2, 2}} {_{0} D}_{t}^{α_{1, 2}} x_{2} (t) & = & f_{2} (x (t)) + \sum_{j = 1}^{q} b_{2 j} (x (t)) u_{j} (t) \\ : \\ {_{0} D}_{t}^{α_{n_{1}}, i} \dots {_{0} D}_{t}^{α_{2, i}} {_{0} D}_{t}^{α_{1, i}} x_{i} (t) & = & f_{i} (x (t)) + \sum_{j = 1}^{q} b_{ij} (x (t)) u_{j} (t) \\ : \\ {_{0} D}_{t}^{α_{n_{q}, 1}} \dots {_{0} D}_{t}^{α_{2, q}} {_{0} D}_{t}^{α_{1, q}} x_{q} (t) & = & f_{q} (x (t)) + \sum_{j = 1}^{q} b_{qj} (x (t)) u_{j} (t) \\ y (t) & = & [x_{1} (t) x_{2} (t) \dots x_{q} (t)] T \end{matrix}

(74)

with the pseudo-state vector given by

x (t) = [\begin{matrix} x_{1} (t) \\ {_{0} D}_{t}^{α_{1, 1}} x_{1} (t) \\ {_{0} D}_{t}^{α_{2, 1}} {_{0} D}_{t}^{α_{1, 1}} x_{1} (t) \\ : \\ {_{0} D}_{t}^{α_{n_{1}, 1}} \dots {_{0} D}_{t}^{α_{2, 1}} {_{0} D}_{t}^{α_{1, 1}} x_{1} (t) \\ x_{2} (t) \\ : \\ {_{0} D}_{t}^{α_{n_{2}, 2}} \dots {_{0} D}_{t}^{α_{2, 2}} {_{0} D}_{t}^{α_{1, 2}} x_{2} (t) \\ : \\ x_{q} (t) \\ : \\ {_{0} D}_{t}^{α_{n_{q}, q}} \dots {_{0} D}_{t}^{α_{2, q}} {_{0} D}_{t}^{α_{1, q}} x_{q} (t) \end{matrix}]

(75)

and

r (t), u (t), f (t), B (t)

and

ɛ (t)

given as in Section 4.1 above. The sliding surface will be given by

s (x (t)) = 0

(76)

s_{i} (x (t)) = \sum_{j = 1}^{m_{i}} μ_{j, i} {_{0} D}_{t}^{β_{j, i}} ɛ (t), i = 1 \dots q

(77)

\begin{matrix} \forall i = 1, \dots, q, \forall k = 1, \dots, n_{i} - 1, \exists j : \\ β_{j, i} = \sum_{p = 1}^{k} α_{p, i} \end{matrix}

(78)

and the calculations proceed just as above.

6. Examples

6.1. Commensurable SISO plant, in simulation

Consider the plant with uncertainty in $f_{A} (x (t))$ and in $f_{B} (x (t))$

{_{0} D}_{t}^{\frac{3}{4}} x (t) = \overset{f_{A} (x (t))}{\overset{︷}{π \sqrt{x (t)} {_{0} D}_{t}^{\frac{1}{4}} x (t)}} + \overset{b}{\overset{︷}{\sqrt{2}}} u (t)

(79)

with ${\overset{\land}{f}}_{A} (x (t)) = 3 \sqrt{x (t)} {_{0} D}_{t}^{\frac{1}{4}} x (t)$ and $\overset{\land}{b} = 3 2$ , where $α = \frac{1}{4}, x (t) = [x (t) {_{0} D}_{t}^{\frac{1}{4}} x (t) {_{0} D}_{t}^{\frac{1}{2}} x (t)] T$ and $n = 3$ . If we make $β = α, k = 1.8$ and $λ = 10$ , we get

s (x (t)) =_{0} D_{t}^{\frac{1}{2}} ɛ (t) + 20 {_{0} D}_{t}^{\frac{1}{4}} ɛ (t) + 100 ɛ (t)

(80)

S (ɛ (t)) = 20 {_{0} D}_{t}^{\frac{1}{4}} ɛ (t) + 100 ɛ (t)

(81)

\begin{matrix} u (t) = \frac{2}{3} [- 3 \sqrt{| x (t) |} {_{0} D}_{t}^{\frac{1}{4}} x (t) - 20 {_{0} D}_{t}^{\frac{1}{2}} x (t) - 100 {_{0} D}_{t}^{\frac{1}{4}} x (t) \\ + {_{0} D}_{t}^{\frac{3}{4}} r (t) + 20 {_{0} D}_{t}^{\frac{1}{2}} r (t) + 100 {_{0} D}_{t}^{\frac{1}{4}} r (t) \\ - 1.8 {_{0} D}_{t}^{- \frac{3}{4}} σ ({_{0} D}_{t}^{\frac{1}{2}} x (t) + 20 {_{0} D}_{t}^{\frac{1}{4}} x (t) + 100 x (t) \\ - {_{0} D}_{t}^{\frac{1}{2}} r (t) - 20 {_{0} D}_{t}^{\frac{1}{4}} r (t) - 100 r (t))] \end{matrix}

(82)

Figure 4 shows simulations results obtained with the Grünwald–Letnikoff definition, $T_{s} = 0.1$ and making $σ (ξ) \approx \frac{2}{π} \arctan 20 ξ$ to avoid chattering.

Figure 4.

Simulation results from Section 6.1 for a sinusoidal reference.

6.2. Integer MIMO plant, experimental results

The plant shown in Figure 5 consists in three equal constant section cylindrical tanks, connected by pipes with valves (Amira, 2002). Water flows in a closed circuit, fed by two pumps, as seen in the diagram of Figure 6. Each of the six valves (identified in Figure 6) is actuated by a motor. This system, produced by Amira (model DTS200), is used in the Control, Automation and Robotics Laboratory of IST for didactic purposes, and conveniently reproduces the behaviour of industrial plants where some fluid flows in pipes connecting tanks because of gravity.

Figure 5.

The three-tank plant.

Figure 6.

Graphical user interface of the three-tank plant, schematizing interconnections.

Let us define:

h_n height of water in tank $n = 1, 2, 3$ ,

$Q_{nm}$ volume flow from tank $n = 1, 2, 3$ to tank $m = 0, 1, 2, 3$ (see Figure 6);

Q_n volume flow of water provided by pump $n = 1, 2$ ;

$g = 9.8 m / s 2$ .

Water heights $h_{1}, h_{2}$ and h₃ and pump flows Q₁ and Q₂ are normalized: $h_{n} = 1$ means that tank T_n is full, and $Q_{n} = 1$ means that pump n is delivering its maximum flow. Conservation of mass means that

A {\overset{\cdot}{h}}_{1} = Q_{1} - Q_{13} - Q_{10}

(83)

A {\overset{\cdot}{h}}_{2} = Q_{2} - Q_{23} - Q_{20} - Q_{200}

(84)

A {\overset{\cdot}{h}}_{3} = Q_{13} + Q_{23} - Q_{30}

(85)

(where

\overset{\cdot}{x} \equiv \frac{d x (t)}{d t}

). Flows can be found using the generalized Torricelli rule:

Q_{nm} = a_{nm} S_{v} σ (h_{n} - h_{m}) \sqrt{2 g | h_{n} - h_{m} |}

(86)

where

a_{nm}

is a flux coefficient for the valve connecting tanks n and m, and

h_{0} = 0

. Defining

h = [h_{1} h_{2} h_{3}] T

(87)

Q = [Q_{1} Q_{2}] T

(88)

A (h) = \frac{[\begin{matrix} - Q_{13} - Q_{10} \\ - Q_{23} - Q_{20} - Q_{200} \\ Q_{13} + Q_{32} - Q_{30} \end{matrix}]}{A}

(89)

B = \frac{1}{A} [\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}]

(90)

we can sum up (83)–(85) as

h . = A (h) + BQ

(91)

y = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}] h

(92)

where y is the plant output vector. Of course, if a valve is closed its flow is equal to 0 (because so is its coefficient

a_{nm}

The pumps receive a signal in the [0,1] range: 0 means the pump is stopped, 1 means the pump must work at maximum flow. The behaviour of the pumps is practically linear: the flow changes with the control signal as shown in Figure 7. This data was obtained filling up tanks 1 and 2, all output valves being closed, with the pump control signals at several constant values.

Figure 7.

Nonlinear behaviour of the pumps (left) and of the valves (right).

The outflow coefficients depend on the difference of water heights on either side of the valve, that is to say, $a_{nm} = a_{nm} (| h_{n} - h_{m} |)$ . They were experimentally found by filling up each tank and emptying it into the next, with the pumps stopped and all of the valves closed save the one under consideration. Their evolutions, depending of the water height of the respective tanks, are shown in Figure 7. Differences caused by the direction of the flow are minimal, and thus can be neglected. The approximations $a_{13} \approx a_{23} \approx 0.5$ and $a_{10} \approx a_{20} \approx a_{200} \approx a_{30} \approx 0.7$ were used; note that this overestimates the ability of the valves to supply a flow when the water height difference is small.

The plant was used so that water flows through pump 1 into tank 1, through valve VL1 into tank 3, through valve VL2 into tank 2, and through valve VD4 into the lower reservoir. So valves VL1, VL2 and VD4 are always open. The controlled variables are the three water heights in the three tanks. The manipulated variables are the pump flow Q₁ and the opening of valves VD2 and VD3, henceforth denoted ad $φ_{2} \in [0, 1]$ and $φ_{3} \in [0, 1]$ (0 denoting a closed valve and 1 an open valve). Valve VD1 is always closed, and pump 2 is always stopped. (See Figure 6.) The plant is run with a sampling time of $T_{s} = 0.1$ s. Inputs are filtered by

F (s) = \frac{0.3}{s + 0.3} \approx \frac{0.02955}{z - 0.9704}

(93)

to avoid harsh control actions, that might damage the hardware. (This filter is that chosen in Vinagre et al. (2010).) Note that, with this configuration of the plant, water heights must decrease from left to right.

The model of the plant can thus be rewritten as

\begin{matrix} \underset{x .}{\underset{︸}{[\begin{matrix} {\overset{\cdot}{h}}_{1} \\ {\overset{\cdot}{h}}_{2} \\ {\overset{\cdot}{h}}_{3} \end{matrix}]}} = \underset{f (x)}{\underset{︸}{[\begin{matrix} - \frac{a_{13} S_{v}}{A} ρ (h_{1} - h_{3}) \\ - \frac{a_{23} S_{v}}{A} ρ (h_{2} - h_{3}) - \frac{a_{200} S_{v}}{A} ρ (h_{2}) \\ \frac{a_{13} S_{v}}{A} ρ (h_{1} - h_{3}) + \frac{a_{23} S_{v}}{A} ρ (h_{2} - h_{3}) \end{matrix}]}} \\ + \underset{B (x)}{\underset{︸}{[\begin{matrix} \frac{1}{A} & 0 & 0 \\ 0 & - \frac{a_{20} S_{v}}{A} ρ (h_{2}) & 0 \\ 0 & 0 & - \frac{a_{30} S_{v}}{A} ρ (h_{3}) \end{matrix}]}} \underset{u}{\underset{︸}{[\begin{matrix} Q_{1} \\ φ_{2} \\ φ_{3} \end{matrix}]}} \end{matrix}

(94)

where

ρ (x) = σ (x) \sqrt{2 gx}

(95)

Let r be the reference that we want x to track, and

ɛ = x - r

. Let the sliding surface be

\begin{matrix} s = (D 1 + α + λ) ɛ \\ = D 1 + α x - D 1 + α r + λ ɛ \\ = D α f (x) + D α B (x) u - D 1 + α r + λ ɛ \end{matrix}

(96)

So as to have $s = 0$ , the control action must be

u = B - 1 (x) [D r - λ D - α ɛ - f (x)]

(97)

and as in Section 4 it is implemented thus

u = B - 1 (x) [D r - λ D - α ɛ - f (x) - k σ (s)]

(98)

Controller parameters were tuned using a genetic algorithm, similar to that of Valério and Sá da Costa (2005), that minimizes the integral of the squared error as follows.

A population with 20 sets of parameters is randomly generated.

The algorithm stops after 100 iterations, or when no improvements are registered after 10 consecutive iterations.

Performance of the 20 individuals is simulated for a typical reference.

Only the six best performing individuals survive and reproduce (elitism).

Six new individuals are found by mutation. A survival is chosen randomly and one of its parameters is randomly altered.

Six new individuals are found by cross-over. Two survivals are chosen randomly and each parameter is chosen randomly either from one or the other.

Two new individuals are found by spontaneous generation. That is to say, they are new individuals randomly generated.

After 39 iterations, the following parameters were obtained: $α = 0.3596, λ = - 0.8667, k = 18.6617$ . To avoid chattering, approximation $σ (s) \approx \frac{2}{π} \arctan 17 s$ was used.

Figure 8 shows the performance of fractional sliding mode control (and compares it to the results obtained for the integer case, $α = 1$ ) when there are no faults, when valves VL1, VL2 and VD4 are clogged (half open, that is to say, affected by $φ = 0.5$ ), and when tank 1 has a leak (valve VD1 is half open, with a coefficient $φ = 0.5$ ). It can be seen that fractional sliding mode control recovers slightly faster than the integer case.

Figure 8.

Experimental results from section 6.2: in the absence of faults (top left); when valves are clogged (top right); when tank 1 has a leak (bottom).

6.3. Fractional MIMO plant, in simulation

Consider plant

\begin{matrix} [\begin{matrix} {_{0} D}_{t}^{\frac{3}{10}} x_{1} \\ {_{0} D}_{t}^{\frac{5}{11}} x_{2} \end{matrix}] = [\begin{matrix} x_{2} \\ x_{1} \end{matrix}] + [\begin{matrix} u_{1} \\ 2 u_{2} \end{matrix}] = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] \\ + [\begin{matrix} 1 & 0 \\ 0 & 2 \end{matrix}] [\begin{matrix} u_{1} \\ u_{2} \end{matrix}] \end{matrix}

(99)

Suppose that B is known with a 10% error:

\overset{\land}{B} = [\begin{matrix} 0.91 & 0 \\ 0 & 1.82 \end{matrix}]

(100)

D = [\begin{matrix} 0.1 & 0 \\ 0 & . 1 \end{matrix}]

(101)

Making

s = [\begin{matrix} {_{0} D}_{t}^{\frac{1}{3}} ɛ_{1} \\ {_{0} D}_{t}^{\frac{1}{2}} ɛ_{2} \end{matrix}]

(102)

u = [\begin{matrix} 1 & 0 \\ 0 & \frac{1}{2} \end{matrix}] ([\begin{matrix} {_{0} D}_{t}^{\frac{1}{3}} r_{1} \\ {_{0} D}_{t}^{\frac{1}{2}} r_{2} \end{matrix}] - [\begin{matrix} x_{2} \\ x_{1} \end{matrix}] - k \circ D - 1 σ ([\begin{matrix} {_{0} D}_{t}^{\frac{1}{3}} ɛ_{1} \\ {_{0} D}_{t}^{\frac{1}{2}} ɛ_{2} \end{matrix}]))

(103)

and using $σ (ξ) \approx ς_{2} (ξ)$ with $ζ = 20$ , we obtain, for a sinusoidal input and $δ = [0 . 7 0 . 7] T$ , that $k = [1 1] T$ , leading to the results in Figure 9.

Figure 9.

Simulation results from Section 6.3 for a sinusoidal reference; B is known with a 10% error.

Now suppose that both B and $α$ are known with a 10% error:

\hat{α} = [\frac{1}{3} \frac{1}{2}] T

(104)

Errors in orders were not contemplated in the design method above, but results in Figure 10 show that fractional sliding mode control, found without taking such errors into account, is robust enough to deal with it.

Figure 10.

Simulation results from Section 6.3 for a sinusoidal reference; B and $α$ are known with a 10% error.

7. Conclusion

Fractional sliding mode control can be successfully applied to SISO and MIMO plants, linear or nonlinear, integer, commensurable or noncommensurable. The examples given in this paper concern specific sliding surfaces and a specific Lyapunov function: the reasonings used apply, with the necessary changes, to all alternatives that may be desired. Future work includes applying these control strategies to other simulation and experimental plants, enlarging the cases where it has proven well.

Footnotes

Funding

This work was supported by Fundação para a Ciência e a Tecnologia, through IDMEC under LAETA.

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Zhong

Chen

(2012) Design of sliding mode controller for a class of fractional-order chaotic systems. Communications in Nonlinear Science and Numerical Simulation 17: 356–366.

Fractional sliding mode control of MIMO nonlinear noncommensurable plants

Abstract

Keywords

1. Introduction

2. Fractional calculus and fractional dynamic systems

Theorem 1.

Remark 1.

Theorem 2.

Definition 1.

Theorem 3.

Theorem 4.

Remark 2.

3. The SISO case

3.1. The problem

Remark 3.

Remark 4.

3.2. Sliding surface

Remark 5.

Theorem 5.

Proof.

Theorem 6.

3.3. Following the sliding surface

3.4. Example: commensurable plant, uncertainty in f A ( x ( t ) )

3.5. Example: integer plant, uncertainty in f B ( x ( t ) )

3.6. Avoiding chattering

4. The MIMO commensurable case

4.1. The problem

4.2. Uncertainty in f ( x ( t ) ) and in B ( t )

5. The MIMO noncommensurable case

6. Examples

6.1. Commensurable SISO plant, in simulation

6.2. Integer MIMO plant, experimental results

6.3. Fractional MIMO plant, in simulation

7. Conclusion

Footnotes

Funding

References

3.4. Example: commensurable plant, uncertainty in $f_{A} (x (t))$

3.5. Example: integer plant, uncertainty in $f_{B} (x (t))$

4.2. Uncertainty in $f (x (t))$ and in $B (t)$