A Robust State Feedback Adaptive Controller with Improved Transient Tracking Error Bounds for Plants with Unknown Varying Control Gain

and design such as Feedback Linearization, Lyapunov Based Control, Adaptive Control, Optimal Control and Robust Control. different applications that are basically independent of each other. The will provide the with information on modern control techniques and results which cover a very wide application area. Each chapter attempts to demonstrate how one would apply these techniques to real-world systems through both simulations and experimental settings.

The state adaptive backstepping (SAB) of (Kanellakopoulos et al., 1991) is a common framework for the design of adaptive controllers for plants in controllable form. As is well known, a major difficulty in introducing robustness techniques to SAB based schemes is that the states z i and the stabilizing functions must be differentiable to certain extent (see (Yao & Tomizuka, 1997), (Yao, 1997), (Ge & Wang, 2003)).
The robust SAB scheme of (Zhou et al., 2004), (Su et al., 2009), (Feng, Hong, Chen & Su, 2008) has the advantage that the knowledge on the upper or lower bounds of the plant coefficients can be relaxed if the controller is properly designed and the control gain is constant or known. The approach is based on the truncation method of (Slotine & Li, 1991), pp. 309. The stabilizing functions are smoothed at each i-th step in order to render it differentiable enough. The following benefits are obtained: i) the scheme is robust with respect to unknown varying but bounded coefficients, ii) upper or lower bounds of the plant coefficients are not required to be known, and iii) the tracking error converges to a residual set whose size is user-defined.
The specific case of unknown varying control gain is an important issue, more difficult to handle than other unknown varying coefficients. The varying control gain is usually handled by means of robustness techniques (cf. (Wang et al., 2004), (Huang & Kuo, 2001), (Bechlioulis (Li et al., 2004)) or the Nussbaum gain technique (cf. (Su et al., 2009), (Feng, Hong, Chen & Su, 2008), (Feng et al., 2006), (Ge & Wang, 2003)). The above methods are applicable to plants in parametric-pure feedback or controllable form, and with controllers that use the SAB or the MRAC as the control framework.
In (Wang et al., 2004), a system with dead zone in the actuator is considered, assuming that both dead zone slopes have the same value. The input term is rewritten as the sum of an input term with constant control gain plus a bounded disturbance-like term. The disturbance term is rejected by means of a robust technique, based on (Slotine & Li, 1991) pp. 309. Nevertheless, this strategy is not valid for different values of the slopes. Other robustness techniques comprise a control law with compensating terms and either a projection modification of the update law, as in cf. (Huang & Kuo, 2001), or a σ modification as in (Bechlioulis & Rovithakis, 2009), (Li et al., 2004). Nevertheless, some lower or upper bounds of the plant coefficients are required to be known.
The Nussbaum gain technique can relax this requirement, as can be noticed from (Su et al., 2009), (Feng, Hong, Chen & Su, 2008), . The main drawbacks of the Nussbaum gain method are (see (Su et al., 2009), (Feng, Hong, Chen & Su, 2008), (Feng et al., 2006), (Ge & Wang, 2003), (Feng et al., 2007), , (Ren et al., 2008), (Zhang & Ge, 2009), (Du et al., 2010)): i) the upper bound of the transient behavior of the tracking error is significantly modified in comparison with that of the disturbance-free case: the value of this bound depends on the time integral of terms that comprise Nussbaum terms, and ii) the controller involves an additional state, which is necessary to compute the Nussbaum function.
iii) The control gain is assumed upper bounded by some known function as in (Bechlioulis & Rovithakis, 2009). iv) Upper or lower bounds of plant parameters are required to be known to achieve asymptotic convergence of the tracking error to a residual set of user-defined size, as in (Bashash & Jalili, 2009), (Chen, 2009), (Ho et al., 2009), (Park et al., 2009) and (Bechlioulis & Rovithakis, 2009).
In this chapter, we develop a controller that overcomes the above drawbacks, so that: Bi) The upper bound of tracking error transient value does not depend on time integral terms.
Bii) Additional states are not used in the controller.
Biii) The control gain is not required to be upper bounded by a constant.
Biv) The control gain is not required to be bounded by a known function.
Bv) Upper or lower bounds of the plant parameters are not required to be known.
Bvi) The control and update laws do not involve signum type signals.
Bvii) The tracking error converges to a residual set whose size is user-defined.
We consider systems described by the controllable form model with arbitrary relative degree, unknown varying but bounded coefficients and varying control gain. We use the SAB of (Kanellakopoulos et al., 1991) as a basic framework for the control design, preserving a simple definition of the states resulting from the backstepping procedure. We use the Lyapunov-like function method to handle the unknown time varying behavior of the plant parameters. All closed loop signals remain bounded so that parameter drifting is prevented.
The key elements to handle the varying behavior of the control gain are: i) introduce the control gain in the term involving the adjusted parameter vector, by means of the inequality that relates the control gain and its lower bound, and ii) apply the Young's inequality.
In current works that deal with plants in controllable form and time varying parameters and use the state transformation based on the backstepping procedure, they modify the defined states at each step of the state transformation in order to tackle the unknown time varying behavior of the plant parameters. Instead of altering the state transformation, we formulate a Lyapunov-like function, such that its magnitude and time derivative vanish when the states resulting from the state transformation reach a target region.
The control design and proof of boundedness and convergence properties are simpler in comparison to current works that use the Nussbaum gain method. The controller is also simpler as it does not introduce additional states that would be necessary to handle the unknown time varying control gain.
The chapter is organized as follows. In section 2 we detail the plant model. In section 3 we present the goal of the control design. In section 4 we carry out a state transformation, based on the state backstepping procedure. In section 5 we derive the control and update laws. In section 6 we prove the boundedness of the closed loop signals. In section 7 we prove the convergence of the tracking error e, finally, in section 8 we present an example.

Problem statement
In this section we detail the plant and the reference model. Consider the following plant in controllable form: where y(t) ∈ R is the system output, u(t) ∈ R the input, a a vector of varying entries, γ n a known vector, b the control gain, and d a disturbance-like term. We make the following assumptions: Ai) The vector a involves unknown, time varying, bounded entries a 1 , ···, a j , which satisfy: |a 1 |≤μ 1 , ···, |a j |≤μ j , whereμ 1 , ···,μ j are unknown, positive constants.
Aii) The entries of the vector γ n are known linear or nonlinear functions of y, ···, y (n−1) .
Aiv) The term d represents either external disturbances or unknown model terms that satisfy: where μ d is an unknown positive constant, and f d is a known function that depends on y, ···, y (n−1) . In the case that d is bounded, we have f d = 1. The term d may come from the product of a known function g d with an unknown varying but bounded coefficient Av) The control gain b satisfies: where b m is an unknown lower bound, and the value of the signum of b is constant and known.
Remark. We recall that μ d , b m ,μ 1 , ···,μ j are unknown constants. In contrast, the values of y, ···, y (n−1) , γ n , f d , sgn(b) are required to be known. Notice in assumption Av that we do not require the control gain b to be upper bounded by any constant. That is a major contribution with respect to current works that use the Nussbaum gain method, e.g (Su et al., 2009), (Feng, Hong, Chen & Su, 2008), , (Feng et al., 2007), (Ge & Wang, 2003). The requirement about the value of the signum of b is a common and acceptable requirement.

Control goal
Let where e(t) is the tracking error, y d (t) is the desired output, Ω e is a residual set, r is the reference signal. Moreover, a m,n−1 , ···, a m,o are constant coefficients defined by the user, such that the polynomial K(p) is Hurwitz, being K(p) defined as K(p)=p (n) + a m,n−1 p (n−1) + ···+ a m,o . The reference signal r(t) is bounded and user-defined. The constant C be is positive and user-defined. The objective of the MRAC design is to formulate a controller, provided by the plant model (1) subject to assumptions Ai to Av, such that: i) The tracking error e converges asymptotically to the residual set Ω e .
ii) The control signals are bounded and do not involve discontinuous signals.

State transformation based on the state backstepping
In this section we carry out a state transformation by following the steps 0, ···, n of the backstepping procedure. The plant model (1) can be rewritten as follows: The model (7, 8) can be obtained by making γ 1 = ··· = γ n−1 = 0 in the parametric -pure feedback form of (Kanellakopoulos et al., 1991). We use the SAB of (Kanellakopoulos et al., 1991) as the basic framework for the formulation of the control and update laws.
We develop the SAB for the plant model (7, 8), and introduce a new robustness technique.
Since the order of the plant is n, the procedure comprises the steps 0, ···, n, to be carried out in a sequential manner.
Step 0. We begin by defining the state z 1 as the tracking error: Step i (1 ≤ i ≤ n − 1). At each i-th step, we obtain the dynamics of the state z i by deriving it with respect to time, and using the definitions ofẋ i+1 provided by (7). For the sake of clarity, we develop the step 1 and then we state a generalization for (1 ≤ i ≤ n − 1).
For the case i = 1, we differentiate z 1 defined in (9) and use the definition ofẋ 1 provided by (7) with i = 1:ż where ϕ 1 is a known function ofẏ d . Equation (10) can be rewritten as: where c 1 ≥ 2 is a positive constant of the user choice. The dynamic equation of z 2 is obtained by differentiating it with respect to time. The same procedure must be followed until the step i = n − 1. To state a generalization, we expressż i as: where ϕ i is a known scalar term, that is function of Equation (13) can be rewritten as:ż where c i ≥ 2 is a positive constant of the user choice. At the step i = n − 1 we obtain the dynamic equation for z n−1 and the expression for z n as indicated by (15, 16).

Remark.
Notice that the definition of the states z i is similar to that of the disturbance free case, so that a simple design is preserved. This is due to the following facts: i) Disturbance like terms are absent in the dynamicsẋ 1 , ···,ẋ n−1 given by (7), so that they are also absent in the dynamicsż 1 , ···,ż n−1 , as can be noticed in (13).
ii) Dead zone functions of the states z i are not used.
Step n. We obtain the dynamics of z n by differentiating it with respect to time and using the expression ofẋ n provided by (8): ϕ n = ϕ n (z 1 , ··· , z n , y d , ··· , y where ϕ n is a known scalar that is function of z 1 , ···, z n , y d , ···, y d . Notice that the disturbance like term d and the control input u appear explicitly at the dynamics of z n ,a t the step n of the procedure. Thus, we have completed the state transformation, which allows us to develop the controller.

Control and update laws
In this section we develop the control and update laws, taking into account the assumptions stated in section 2 and the goals of section 3. The key elements of the procedure are: i) Incorporate the assumptions Ai and Aiv, concerning the unknown time varying parameter a 1 , ···, a j and the disturbance like term d.
ii) Carry out a linear parameterization.
iii) Express the parameterization in terms of adjustment error and adjusted parameter vector. iv) Introduce the control gain b within the adjusted parameter vector. v) Formulate the control law.
vi) Formulate a Lyapunov-like function and find its time derivative. vii) Formulate the update law.
We begin by rewriting (17) as follows: z n = −c n z n + bu + a ⊤ γ n + ϕ n + c n z n + d, where c n ≥ 2 is a positive constant of the user choice. Multiplying (19) by z n , we obtain: z nżn = −c n z 2 n + bz n u + z n a ⊤ γ n + z n (ϕ n + c n z n )+z n d The term z n a ⊤ γ n + z n (ϕ n + c n z n )+z n d can be rewritten as follows: z n a ⊤ γ n + z n (ϕ n + c n z n )+z n d = z n (a ⊤ γ n + d + ϕ n + c n z n ) using assumptions Ai and Aiv of section 2 and parameterizing, we obtain: z n a ⊤ γ n + z n (ϕ n + c n z n )+z n d ≤|z n | μ 1 |γ n Notice that the entries of the vector θ are unknown, positive, constant, because bounds of the time varying parameters a i and d have been introduced, according to the properties in assumptions Ai and Aiv of section 2. Now, we express (22) in terms of adjustment error and adjusted parameter vector: z n a ⊤ γ n + z n (ϕ n + c n z n )+z n d ≤− b mn |z n |φ ⊤θ + b mn |z n |φ ⊤θ beingθ an adjusted parameter vector andθ an adjustment error. Using the property (3) in the term √ b mn |z n |φ ⊤θ of (25), we obtain: b mn |z n ||φ ⊤θ |≤ 3C bvz 2 2 3C bvz |b||z n ||φ ⊤θ | where C bvz =(1/2)C 2 be (28) using the Young's inequality (cf. (Royden, 1988) pp. 123), we obtain: Substituting (29) into (25), we obtain: z n a ⊤ γ n + z n (ϕ n + c n z n )+z n d ≤− √ b mn |z n |φ ⊤θ + 3 4 C bvz + 1 3C bvz |b|z 2 n (φ ⊤θ ) 2 (30)

Remark.
We have proposed a new method to handle the unknown varying behavior of the control gain b, alternative to the current Nussbaum gain method. We parameterized the model term z n a ⊤ γ n + z n (ϕ n + c n z n )+z n d in terms of adjustment errorθ and adjusted parameter vectorθ, and developed the following steps: i) Introduce the constant √ b mn in the parameterization, see (22).
iii) Apply the Young's inequality to obtain b, see (29).
Recall that the value of b mn is not required to be known.
Substituting (30) into (20), we obtain: we choose the following control law: whereφ, z n are defined in (23), (16), respectively. Substituting (32) into (31), we obtain: z nżn ≤−c n z 2 n +(3/4)C bvz − b mn |z n |φ ⊤θ To handle the effect of the constant (3/4)C bvz , we formulate the following Lyapunov-like function: where C bvz is defined in (28). We need the following properties: Proposition 5.1. The functionV z defined in (34) has the following properties: Proof. From (34) it follows thatV z ≥ 0∀t ≥ t o , the property i of proposition 5.1. In addition, from (34) it follows that Applying the Young's inequality (cf. (Royden, 1988) pp. 123), we obtain: This completes the proof of property ii. From (42) it follows that ∂V z /∂V z = 0i fV z = C bvz and that ∂V z /∂V z is continuous. From (34) it follows thatV z is continuous. This completes the proof of property iii of proposition 5.1.

Remark. The control and update laws stated in
i) The control law uses the adjusted parameter vectorθ, so that it does not rely on upper or lower bounds of the plant coefficients, i.e.μ 1 , ···,μ j , μ d ,b mn , and excessive control effort is also avoided.
ii) Additional states are not required to handle the unknown varying behavior of the control gain, what is an important benefit with respect to closely related schemes that use the Nussbaum gain method.
iii) The control and update laws do not involve discontinuous signals. In fact, the vectorial field of the closed loop system is Lipschitz continuous, so that trajectory unicity is preserved.

Boundedness analysis
In this section we prove that the closed loop signals z 1 , ···, z n ,θ, u are bounded if the developed controller is applied.

Convergence analysis
In this section we prove that if the developed controller is applied, then the signal z 1 converges asymptotically to Ω z , where Ω z = {z 1 : |z 1 |≤C be }.
Proof. In view of (42), equation (58) can be rewritten as: The derivative ∂ f d /∂V z is not continuous, as it involves an abrupt change at V = C bvz . Thus, the Barbalat's Lemma can not be applied on f d . To remedy that, we shall express (73) in terms of a function with continuous derivative: Arranging and integrating (75), we obtain: Thus, f g ∈ L 1 . We have to prove that f g ∈ L ∞ ,ḟ g ∈ L ∞ to apply the Barbalat's Lemma. Since V z ∈ L ∞ , it follows from (76) that f g ∈ L ∞ . Differentiating (76) with respect to time, we obtain: Notice that ∂ f g /∂V z is continuous. Since V z ∈ L ∞ , then ∂ f g /∂V z ∈ L ∞ . Since z 1 ∈ L ∞ , ···, z n ∈ L ∞ , it follows from (11), (15) thatż 1 ∈ L ∞ , ···,ż n−1 ∈ L ∞ . Since u ∈ L ∞ , it follows from (17) thatż n ∈ L ∞ . Therefore, from (43) it follows thatV z ∈ L ∞ .
The procedure of section 4 is followed in order to establish the terms involved in the control and update laws, mentioned in remark 5. Eq. (80) can be rewritten as: x 1 = x 2 (88) x 1 = y, x 2 =ẏ, n = 2 (90) since n = 2, the state transformation based on the backstepping procedure involves the steps 0, 1, 2.
The results are shown in figures 2 and 3. We have choosen y d (t 0 ) ≈ y(t 0 ) in order to obtain a rapid convergence of y towards y d . Figure 2 shows that. i) the tracking error e converges asymptotically towards Ω e = {e : |e|≤0.1}. ii) The output y converges towards y d with threshold 0.1 without large transient differences. Figure 3 shows thatθ 1 , ...,θ 4 are not decreasing with respect to time. This occurs because˙θ is non-negative. The procedure for the sample plant (80) is simpler in comparison with adapive controllers that use the Nussbaum gain method.