4.3.1 Sliding Mode Dynamics Analysis

Design the following integral switching function:

(4.5)

where for each , G_i ∈ R^{m × n} and K_i ∈ R^{m × n} are real matrices to be designed later. The matrix G_i is designed to satisfy that G_iB_i is nonsingular and G_iD_i = 0.

The solution of Ex(t) is given as

(4.6)

It follows from (4.5) and (4.6) that

According to SMC theory, when the system state trajectories reach onto the sliding surface, it follows that s(t) = 0 and . Then, by , we obtain the equivalent control law as

(4.7)

Thus, by substituting (4.7) into (4.3), the sliding mode dynamics can be obtained as

(4.8)

Now, we will analyze the stability of the sliding mode dynamics in (4.8) based on Lemma 4.2.2.

Proposition 4.3.1 The sliding mode dynamics in (4.8) is stochastically stable if there exist nonsingular matrices X_i such that the following conditions hold for ,

(4.9a)

(4.9b)

Remark 4.1 Notice that the conditions in Proposition 4.3.1 are not all of strict LMI form owing to the matrix equality constraint of (4.9a). This may cause problems in checking the conditions numerically, since a matrix equality constraint is fragile and usually not satisfied perfectly. Therefore, the strict LMI conditions are more desirable than non-strict ones from the numerical point of view.

In the following, we present a new condition in terms of strict LMIs for the stochastic stability of the sliding mode dynamics in (4.8).

Proposition 4.3.2 The sliding mode dynamics in (4.8) is stochastically stable if there exist matrices P_i > 0 and nonsingular matrices Q_i such that for ,

(4.10)

where R ∈ R^{(n − r) × n} and S ∈ R^{n × (n − r)} are matrices satisfying RE = 0 and ES = 0.

Proof. Letting X_i≜P_iE + R^TQ_iS^T in (4.10), we can obtain (4.9a)–(4.9b). Thus, by Proposition 4.3.1 we know that the sliding mode dynamics in (4.8) is stochastically stable. This completes the proof.

In the following, we present a strict LMI condition for solving parameter K_i in the switching function of (4.5). Before proceeding, we use the following lemma (i.e. Lemma 2.4.3 in Chapter 2) which will play a key role in the sequel.

Lemma 4.3.3 Let P_i be symmetric matrices such that E^T_LP_iE_L > 0 and suppose Q_i is nonsingular. Then, P_iE + R^TQ_iS^T is nonsingular and

where are symmetric matrices and are nonsingular matrices with

Theorem 4.3.4 The sliding mode dynamics in (4.8) is stochastically stable if there exist symmetric matrices , nonsingular matrices , and matrices , such that for ,

(4.12)

where and

Moreover, the parametric matrices K_i in the switching function of (4.5) can be computed by

(4.11)

Proof By Proposition 4.3.2 we know that the sliding mode dynamics in (4.8) is stochastically stable if there exist matrices P_i > 0 and nonsingular matrices Q_i, such that the conditions of (4.10) hold for . Moreover, according to Lemma 4.3.3, we know that P_iE + R^TQ_iS^T are nonsingular and . Now, performing a congruence transformation on (4.10) by matrices , we have

Letting and , we have

(4.13)

But the following fact is true:

Considering π_ii < 0, thus we have

Therefore, (4.13) holds if the following inequalities hold for :

(4.14)

By Schur complement, (4.14) is equivalent to (4.11). This completes the proof. ▀

4.3.2 SMC Law Design

In the following, we shall design an SMC law, by which the state trajectories of the Markovian jump singular stochastic system in (4.1) can be driven onto the designed sliding surface s(t) = 0 in a finite time and maintained there for all subsequent time.

Theorem 4.3.5 Consider the Markovian jump singular stochastic system in (4.1). Suppose that the switching function is given as (4.5) with K_i being solved by (4.12), and G_i is chosen to satisfy that G_iB_i is nonsingular and G_iD_i = 0. Then, the state trajectories of system (4.1) can be driven onto the sliding surface s(t) = 0 by the following SMC law:

(4.15)

where λ > 0 is an adjustable scalar.

Proof We choose G_i as G_i = B^T_iY_i, where Y_i > 0 are matrices to be designed such that G_iB_i = B^T_iY_iB_i > 0 for . Choose the following Lyapunov function:

According to (4.5), we have

(4.16)

Substituting (4.15) into (4.16) yields

(4.17)

Thus, taking the derivation of V(t) and considering (4.17), we have

which implies that the state trajectories of the system in (4.1) will be driven onto the sliding surface s(t) = 0 in a finite time. This completes the proof.

In the implementation of the SMC law in (4.15), the upper bound scalar ϵ of f_i(t) in (4.2) is required to be known a priori. If the value of ϵ is not available, we have to estimate it. In the following theorem, we shall consider this case, and first design an adaptive law to estimate ϵ, thus an adaptive SMC law will be presented for system (4.1).

Theorem 4.3.6 Consider the Markovian jump singular stochastic system in (4.1), and assume that the exact value of the upper bound scalar ϵ is not available. Suppose that the switching function is given as (4.5) with K_i being solved by (4.12), and G_i is chosen to satisfy that G_iB_i are nonsingular and G_iD_i = 0. Then, the state trajectories of system (4.1) can be driven onto the sliding surface s(t) = 0 by the following adaptive SMC law:

where represents the estimate of ϵ, and the adaptive law is given as

with ϵ(0) = 0, where δ > 0 is an adjustable scalar.

Proof Select the following Lyapunov function:

The rest of the proof can be followed along the same lines as the proof of Theorem 4.3.5. ▀

4.4.1 Performance Analysis and SMC Law Design

Consider the following singular stochastic systems with Markovian jump parameters and an external disturbance:

(4.18a)

(4.18b)

where ω(t) ∈ R^p is the disturbance input which belongs to ; z(t) ∈ R^q is the controlled output; C_i, F_i and H_i are real constant matrices. Unless other specified, the notations in (4.18a)–(4.18b) have the same meanings as those in (4.1).

Designing the same switching function as in (4.5) and employing the methods used in Section 4.3, we can obtain the following sliding mode dynamics:

(4.19)

Remark 4.2 Notice from (4.19) that if matrix F_i in (4.18a)–(4.18b) satisfies the so-called matching condition – that is, there exist matrices satisfying – it follows that the sliding mode dynamics in (4.19) becomes (4.8). This implies that the sliding mode dynamics is adaptive to the disturbance ω(t). In this case, the methods proposed in the previous section of this chapter can be applied directly. In the following, we assume that matrix F_i does not satisfy the matching condition, thus there will exist a disturbance ω(t) in the sliding mode dynamics, and the results will be sharply different from the matching case.

Definition 4.4.1 Given a scalar γ > 0, the sliding mode dynamics in (4.19) is said to be stochastically stable with an performance level γ, if it is stochastically stable with ω(t) = 0, and under zero condition, for nonzero , it holds that

(4.20)

Now, we will analyze the stability and the performance of the sliding mode dynamics in (4.19).

Theorem 4.4.2 Given a scalar γ > 0, the sliding mode dynamics in (4.19) is stochastically stable with an performance level γ, if there exist nonsingular matrices such that for ,

(4.21a)

(4.21b)

(4.21c)

where

Proof Choose the following Lyapunov function:

where and (denoted by when r_t = i) are nonsingular matrices to be specified such that are positive definite for .

Let be the infinitesimal generator of the Markov process {(x(t), r_t), t ≥ 0}. Then, the average derivative emanating from point (x, i) at time t is given by the following expression:

(4.22)

where

Here, we choose in the above derivation, which guarantees that is nonsingular since . In addition, are introduced due to G_iD_i = 0. Therefore, when ω(t) = 0 in (4.22), it follows that . By Schur complement, (4.21b) implies ϒ_i < 0 for , thus,

Therefore, we know that the the sliding mode dynamics in (4.19) with ω(t) = 0 is stochastically stable.

Now, we will establish the performance. To this end, assume zero initial condition (that is, x(0) = 0, thus W(0, r₀) = 0) and consider index:

Dynkin’s formula gives

and together with (4.22), we have

(4.23)

where Ξ_12i and Ξ_22i are defined in (4.21b). By Schur complement, (4.21b) implies

Then from (4.23), thus (4.20) holds. This completes the proof. ▀

The following theorem will give a sufficient condition by which the sliding mode dynamics in (4.19) is guaranteed to be stochastically stable with an performance level γ, and the switching function in (4.5) can be solved.

Theorem 4.4.3 Given a scalar γ > 0, the sliding mode dynamics in (4.19) is stochastically stable with an performance level γ, if there exist symmetric matrices , positive definite matrices , , nonsingular matrices , and matrices , such that for ,

(4.24a)

(4.24b)

(4.24c)

(4.24d)

where

Moreover, the parametric matrices K_i in the switching function of (4.5) can be computed by

(4.25)

Proof From Proposition 4.3.2, it can be seen that the sliding mode dynamics in (4.19) is stochastically stable with an performance level γ, if there exist matrices P_i > 0 and nonsingular matrices Q_i such that for ,

(4.26)

(4.27)

where

By Lemma 4.3.3, we know that Z_i≜P_iE + R^TQ_iS^T are nonsingular and , where are symmetric matrices and are nonsingular matrices with

Now, performing a congruence transformation on (4.26) by , we have

(4.28)

where are defined in (4.24a) and

Notice that

Letting and , we know that (4.28) holds if for ,

(4.29)

where

where , and the other notations are defined in (4.24a). Furthermore, considering (4.24b) and (4.24d), inequality (4.24a) yields (4.29), and (4.24c) yields (4.27). This completes the proof. ▀

Now, by applying the same procedures as in Section 4.3, we design a discontinuous SMC law to drive the system state trajectories onto the predefined sliding surface in a finite time and maintain it there for all the subsequent time.

Theorem 4.4.4 Consider the Markovian jump singular stochastic system in (4.1). Suppose that the switching function is given as (4.5) with K_i being solved by (4.25), and G_i are chosen as , where is the solution of (4.24a)–(4.24d). Then, the state trajectories of system (4.1) can be driven onto the sliding surface s(t) = 0 by the SMC law u(t) designed in (4.15).

4.4.2 Computational Algorithm

Notice that there exist three matrix equalities of (4.24b)–(4.24d) in Theorem 4.4.3, which can not be solved directly by applying the LMI procedures. In the following, we will propose some algorithms to solve them. Firstly, to solve (4.24b)–(4.24c), we consider the following matrix inequalities for scalars α > 0 and β > 0,

(4.30)

(4.31)

By Schur complement, (4.30) and (4.31) are respectively equivalent to

(4.32)

(4.33)

Therefore, when α > 0 and β > 0 are chosen as two sufficiently small scalars, matrix equalities (4.24b) and (4.24c) can be solved through LMIs (4.32) and (4.33), respectively.

We use the CCL method [66] to solve (4.24d) by formulating it into a sequential optimization problem subject to LMI constraints. We suggest the following minimization problem involving LMI conditions instead of the original nonconvex feasibility problem in Theorem 4.4.3.

Problem SMDA (Sliding mode dynamics analysis):

(4.34)

If the solution of the aforesaid minimization problem is Nr, then the conditions in Theorem 4.4.3 are solvable. Although it is still not possible to always find the global optimal solution, the proposed minimization problem is easier to solve than the original nonconvex feasibility problem. We suggest the following algorithm to solve Problem SMDA.

Algorithm SMDA

• Step 1. Choose α > 0 and β > 0 as sufficiently small scalars.
• Step 2. Find a feasible set satisfying (4.24a), (4.32)–(4.33), and (4.34). Set κ = 0.
• Step 3. Solve the following optimization problem
  
  and denote f* as the optimized value.
• Step 4. Substitute the obtained matrices into (4.29). If (4.29) is satisfied, with
  
  for a sufficiently small scalar ϵ > 0, then output the feasible solutions , so EXIT.
• Step 5. If where is the maximum number of iterations allowed, so EXIT.
• Step 6. Set κ = κ + 1, , and go to Step 3.