2014 American Control Conference (ACC) June 4-6, 2014. Portland, Oregon, USA
Adaptive Control of Uncertain Systems with Gain Scheduled Reference Models and Constrained Control Inputs Mehrdad Pakmehr and Tansel Yucelen
Abstract— This paper develops a new state feedback model reference adaptive control approach for uncertain systems with gain scheduled reference models in a multi-input multioutput (MIMO) setting with constrained control inputs. A single Lyapunov matrix is computed for multiple linearizations of the nonlinear closed-loop gain scheduled reference system, using convex optimization tools. This approach guarantees stability of the closed-loop gain scheduled reference model. Adaptive state feedback control architecture is then developed, and its stability is proven for the case with constrained control inputs. The resulting closed-loop system is shown to have bounded solutions with bounded tracking error, with the proposed stable gain scheduled reference model. Sufficient conditions for ultimate boundedness of the closed-loop system are derived. A semi-global stability result is proved with respect to the level of saturation for open-loop unstable plants while the stability result is shown to be global for open-loop stable plants. Simulation results show that the developed adaptive controller can be used effectively to control a degraded turboshaft engine for large thrust commands, with guaranteed stability and proper tracking performance.
I. I NTRODUCTION Adaptive control of systems which are operating in multiple operating points has been of interest to researchers recently; some of the related literature is briefly reviewed subsequently. Adaptive control of piecewise linear systems has been developed in [1]. In these kinds of adaptive control systems, multiple linear time invariant (LTI) systems are used and transitions between these models are modeled as switches. These switchings introduce discontinuities and jumps in the control inputs. Adaptive control of time varying systems with gain scheduling is done in [2], [3]. The stability analysis for these systems is also performed using a time varying quadratic Lyapunov function, with some conditions on time varying Lyapunov matrix P (t) and its rate P˙ (t). Various adaptive control approaches for systems with input saturation are described in [4], [3], [5]. The stability proofs in these works are shown for adaptive control systems with LTI reference models. To fulfill the need for a stable gain scheduled model reference model for systems with constrained control inputs, this paper develops a gain scheduled reference model without switching problem; which, in case of the gas turbine engine example its stability can be shown by finding a single Lyapunov function. Gain scheduled reference model design and Mehrdad Pakmehr is a Postdoctoral Fellow at the School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332,
[email protected] Tansel Yucelen is an Assistant Professor at the Mechanical and Aerospace Engineering Department, Missouri University of Science and Technology, Rolla, Mo 65409,
[email protected] 978-1-4799-3274-0/$31.00 ©2014 AACC 691
stability analysis is performed using the method presented in [6], [7]. The scheduling variable in our reference model design process is an endogenous parameter [8], which is a function of the measurable plant outputs. Then stability analysis is conducted for a state feedback adaptive control system with gain scheduled reference model and constrained control inputs. The analysis uses some results from [4], [3]. The constraints on the control inputs are implemented using a multi-dimensional rectangular saturation function. This controller, then, has been implemented on a high fidelity physics-based JetCat SPT5 turboshaft engine model for large throttle commands with constraints on the control inputs to keep the engine in its safe operating envelope. The contribution of this paper is the development of a stable state feedback model reference adaptive control algorithm for systems with gain scheduled reference models and constrained control inputs in a MIMO setting; the approach is applicable to systems, such as gas turbine engines, which the stability of their gain scheduled reference model is guaranteed by computing a single Lyapunov function [9], [10], [11]. A detailed stability analysis is performed for the proposed controller which can be used towards control software verification and certification [12]. The rest of this paper is organized as follows. In section II, a linear parameter dependent representation of the nonlinear system dynamics is presented, and stability analysis of the closed-loop system is shown. In section III, a model reference adaptive control with a gain scheduled reference model and constrained control inputs is designed with detailed stability proof. In section IV, simulations are performed for two different engine models including the nominal and degraded engine models. Simulation results show that the developed adaptive controller can be used effectively for the entire flight envelope of the turboshaft engine with guaranteed stability. Section V, concludes the paper. II. L INEAR PARAMETER D EPENDENT R EFERENCE M ODEL D ESIGN AND S TABILITY Consider the nonlinear dynamical system x˙ p (t) = f p (xp (t), u(t)), y(t) = g p (xp (t), u(t)),
(1)
where xp (t) ∈ 0∧Of T Γy>0, otherwise.
(12) Lemma 2: Given θ∗ ∈ Ω0 , then (θ − ∗ T θ ) (Γ−1 ProjΓ (θ, y) − y) ≤ 0. Lemma 3: Let ProjΓ (Θ, Y ) be defined like Definition 2, F = [f1 (θ1 )...fm (θm )]T ∈ Rm×1 be a convex vector function and Θ = [θ1 ...θm ], Θ∗ = ∗ ], Y = [y1 ...ym ], where Θ, Θ∗ , Y ∈ Rn×m then [θ1∗ ...θm trace (Θ − Θ∗ )T (Γ−1 ProjΓ (Θ, Y ) − Y ) ≤ 0. The constraints on the control inputs will be defined as a rectangular saturation function of v. The saturation function is given by Rs (v), where the elements of Rs are defined by
if |vi |≤vi,max , i=1,...,m,
(16)
Scheduling parameter is defined to be α(t) = ||xp (t)|| (i.e. scheduling parameter is the norm of the plant states). Note that, here δy(t) = δxp (t), i.e. C p (α(t)) = I, Dp (α(t)) = 0. For simplicity from now on we rename the variables δx(t), δy(t) and δr(t) as δx(t) := x(t), δy(t) := y(t) and δr(t) := r(t). Plant (16) can be written as
The definitions and lemmas presented here are mainly adopted from [17], [18], [3]. Definition 2: [17] A variant of the projection algorithm, Γ-projection, updates the parameter along a symmetric positive definite gain Γ as defined below
RSi =sat(vi )= v, i vi,max sgn(vi ),
}
∀α∈Ω.
A. Mathematical Preliminaries
ProjΓ (θ,y)= Γy−Γ Of (θ)(Of (θ))T Γyf (θ), (Of (θ))T ΓOf (θ) Γy,
{z
δx(t)
x˙ m (t) = Am (α(t))xm (t) + Br r(t), ∀α ∈ Ω.
(18)
In the previous section we showed the stability of this reference model. Note that r(t) ∈ vi,max .
This saturation function can be expressed as the sum of a direction preserving component and an error component, so 693
Assumption 3: Let K ∗ (α(t)) ∈ ΘK for all α ∈ Ω, where ΘK is a known convex compact set. We also assume that K ∗ (t) is continuously differentiable, and the derivative is uniformly bounded, ||K˙ ∗ (α(t))|| ≤ dk < ∞ for all α ∈ Ω. Remark 7: α(t) is defined to be α(t) = ||y(t)|| = ||xp (t)||; since it is a function of endogenous variables (i.e., the plant states), its boundedness is guaranteed by boundedness of the plant states. As a result, its derivative p (t)T x˙ p (t) (α(t) ˙ = x ||x ) is also bounded. More details can be p (t)|| found in [6], [7], [9], [10], [8].
Plant (25) with controller (20), for all α ∈ Ω, can be written as ˆ T (t)x(t)−B∆v(t)+Br r(t). x=A(α(t))x(t)+B ˙ K
Subtracting the reference model (18) and the plant (26), a closed-loop error dynamics equation is obtained as ˜T e(t)=A ˙ m (α(t))e(t)+B K (t)x(t)−B∆v(t),
e˙ ∆ (t)=Am (α(t))e∆ (t)−K∆ (t)∆v(t),
(20)
(21)
∆
where P = P > 0 is a solution of LMI (11). The gains in adaptive laws Γ ∈ 0. Theorem 2: The error ev (t) in q equation km ||Γ−1 || ≤ (29) is bounded, ||ev (t)||L∞ λmin (P ) , m P where km := 4 + max ||kj∗ (t)||2 j=1 kj ∈Θi
(P ) ||Γ−1 || 4 λmax q
m P
max ||kj∗ (t)||dkj .
∗
j=1 kj ∈Θkj
Proof: See [19] for detailed proof. Remark 9: The proof of Theorem 2 showed the boundedness of ev (t), however it can not guarantee the boundedness of the tracking error e(t). To prove the boundedness of e(t), we must prove that x(t) is bounded when the control inputs are constrained under rectangular saturation. ∗ ∗ We define Θ∗max h and Θmax to be Θmax i= sup||K (t)||, ˜ ˜ ∆ (t)|| . Since we asΘmax = max sup||K(t)||, sup||K sumed the control gains belong to a known compact set, then Θ∗max and Θmax are positive and finite, hence there exists a smallest n ∈ N such that Θ∗max ≤ nΘmax . For efficiency of notation we define γmax := max ||Γ−1 ||, ||Γ−1 ∆ || , vmin := min(vi,max ), vmax := max(vi,max ), and v0 := i i s q m P λmax (P ) 2 vi,max , ρ := λmin (P ) , where vi,max > 0 is the
In order to avoid the adaptive controller parameters to be adjusted improperly by the saturation error, we use the augmented error method in the adaptive control design developed in [4], [3] to provide the stability analysis for a gain scheduled model reference adaptive control system. The plant (17) with saturated control inputs can be written as
(24)
where v(t) is the adaptive control input which introduced in equation (20). the ultimate goal is to determine adaptive parameters such that all signals in the plant (24) are guaranteed to be bounded, and y(t) tracks r(t). The deficiency of v(t) is defined as ∆v(t) = v(t) − Rs (v(t)). Now, plant (24) can be written as ∀α∈Ω.
(29)
T
C. Stability Analysis
x=A(α(t))x(t)+Bv(t)−B∆v(t)+B ˙ r r(t),
Γ
∆
(22)
where Γ = ΓT > 0, P = P T > 0 is a solution of LMI (11), and ProjΓ (., .) is the Γ-projection operator defined in Definition 2.
∀α∈Ω,
(28)
˜ ∆ (t) = B − K∆ (t). Let K∆ (t) ∈ Θ∆ , where Θ∆ where K is a convex compact set. We now choose adaptive laws for adjusting the parameters ˆ˙ ˆ K(t) = ProjΓ K(t), −x(t)eT v (t)P B , (30) K˙ T (t) = Proj K T (t), ∆v(t)eT v (t)P ,
With the knowledge of lower and upper bounds of the parameters K ∗ (t), the parameter projection adaptive law is ˆ ˆ˙ (23) −x(t)eT (t)P B , K(t) = ProjΓ K(t),
x=A(α(t))x(t)+BR ˙ s (v(t))+Br r(t),
e∆ (t0 )=0
˜ T (t)x(t)−K ˜ ∆ (t)∆v(t), e˙ v (t)=Am (α(t))ev (t)+B K
˜ ˆ − K ∗ (t). Defining e(t) = x(t) − xm (t), where K(t) = K(t) the error dynamics are ˜ T (t)x(t), ∀α ∈ Ω. e(t) ˙ = Am (α(t))e(t) + B K
(27)
where K∆ (t) ∈ 0 if (i) ||x(0)|| < xmax ρ , p and (ii) V (0) < √Zγmax . Further ||x(t)|| < x , ∀t > 0, max max and error e(t) is in the order of ||e(t)|| = O[sup ||∆v(τ )||]. τ ≤t
Proof: See [19] for detailed proof. Remark 10: Theorem 3 implies that if the initial conditions of the state and the parameter error lie within certain bounds, then the adaptive system will have bounded solutions. The local nature of the result for unstable systems is because of the saturation limits on the control input. For open-loop stable systems the results are global.
Fig. 1.
the evolution of the infinity norm of the errors ||e(t)||∞ , ||ev (t)||∞ , ||e∆ (t)||∞ . The steady-state error in the Aged Engine (AgedEng) simulation case is because of the effect of the aging on the engine health parameters, and this causes a change in the equilibrium manifold of the aged engine in comparison to the nominal engine (NomEng). In other words, since we are using nominal engine equilibrium manifold to design a linear parameter dependant reference model, and the aged engine linear model has a different equilibrium manifold xpe,nom (α(t)) 6= xpe,aged (α(t)), and ue,nom (α(t)) 6= ue,aged (α(t)), then δxpaged (t) = xpaged (t) − xpe,aged (α(t)) 6= 0, and δuaged (t) = uaged (t) − ue,aged (α(t)) 6= 0, and this means ||δxaged (t)|| > δxmin 6= 0 for all t > 0, hence, there will be a steady-state error value greater than zero. Figure 3 shows
IV. T URBOSHAFT E NGINE E XAMPLE We apply the developed adaptive controller to a high fidelity physics-based model of JetCat SPT5 turboshaft engine driving a variable pitch propeller developed in [20]. The effect of engine degradation due to aging is modeled in the nonlinear simulation by modifying the efficiencies and flow capacities of key engine components such as: High Pressure Compressor (HPC), High Pressure Turbine (HPT) and Low Pressure Turbine (LPT). The values of these parameters used in this simulation are ηhpc = −1.470%, Wc,hpc = −2.455%, ηhpt = −1.315%, Wc,hpt = +0.880%, ηlpt = −0.269%, and Wc,lpt = +0.1294%, where their nominal values are zero. To show the stability of the closed-loop reference system, 40 different linearizations of the system have been used, to solve inequality (10), in Matlab with the aid of YALMIP [14] and SeDuMi [15] packages. The numerical value for the common matrix P is P =
0.491
0.079
0.102
−0.004
−0.072
−0.039
0.079
0.446
0.053
0.007
−0.097
−0.013
0.102
0.053
0.181
−0.041
−0.028
−0.022
−0.004
0.007
−0.041
0.130
0.023
0.013
−0.072
−0.097
−0.028
0.023
0.321
0.045
−0.039
−0.013
−0.022
0.013
0.045
0.332
High pressure spool speed and its reference signal
,
where its condition number is κ(P ) = 6.6303, and Q = 0.1 × I6 . Other controller parameters are c = 1, ηc = 3. The numerical values for the adaptive controller are set as Γ = diag([50, 50, 50, 50, 50, 50]), Γ∆ = diag([30, 30]), v1,max = 0.12, v2,max = 0.15, and the initial conditions and the compact sets are = [[−2, 0] [−2, 0]; [−2, 0] [−2, 0]], Θ∆ = ΘKi T [{0} {0} [0, 10] {0} {0} {0}; {0} {0} {0} [0, 10] {0} {0}] , ˆ i (0) = [−0.1950 − 0.1950; − 0.1970 − 0.1970], and K T K∆ (0) = [0 0 2.7 0 0 0; 0 0 0 2.7 0 0] . Simulations are conducted for two different cases including the control of the nominal model (NomEng), and the control of the deteriorated engine due to aging (AgedEng). These case studies, simulate the engine acceleration from the idle thrust to the cruise condition and then its deceleration back to the idle condition for a standard day at sea level condition. Simulation results are shown in figures 695
Fig. 2.
Norm of the error signals ||e(t)||∞ , ||ev (t)||∞ , ||e∆ (t)||∞
the evolution of the control inputs v(t) = [v1 (t), v2 (t)]T , which are inputs to the augmented system, each element is corresponding to one of the control inputs to the original system. For better performance and also to keep the engine in the safe range of operation, hard limits have been defined for both augmented control inputs, |vi | ≤ vi,max for i = 1, 2. Figure 4 shows the histories of fuel flow and propeller pitch angle as the control inputs to the plant. Figures 5 shows the evolution of the gain scheduling controller integral gain matrix (Ki (α)) and also adaptive controller ˆ i (t)). Figure 6 shows the evolution of the gain matrix (K nonzero elements of the augmented adaptive parameter for the saturated system (K∆ (t)).
control with constrained control inputs architecture by proving the ultimate boundedness of the error signal. Sufficient conditions for ultimate boundedness of the closed-loop system were derived. A semi-global stability result was proved with respect to the level of saturation for open-loop unstable plants while the stability result becomes global for openloop stable plants. Simulations result for adaptive control of JetCat SPT5 turboshaft engine physics-based model with some degradation due to aging, shows that the proposed adaptive controller tracks the reference model. Fig. 3.
Control inputs to the augmented system (v(t))
ACKNOWLEDGMENTS This material is based upon the work supported by the AFRL and also the NSF. R EFERENCES
Fig. 4.
[1] Q. Sang and G. Tao, “Adaptive Control of Piecewise Linear Systems: the State Tracking Case,” IEEE Transactions on Automatic Control, vol. 57, no. 2, pp. 522–528, Feb. 2012. [2] J. Jang, A. M. Annaswamy, and E. Lavretsky, “Adaptive Control of Time-Varying Systems with Gain-Scheduling,” in American Control Conference, Seattle, WA, 2008, pp. 3416–3421. [3] J. Jang, “Adaptive Control Design with Guaranteed Margins for Nonlinear Plants,” Ph.D. dissertation, MIT, 2009. [4] S. P. Karason and A. M. Annaswamy, “Adaptive Control in the Presence of Input Constraints,” IEEE Transactions on Automatic Control, vol. 39, no. 11, pp. 2325–2330, Nov. 1994. [5] R. V. Monopoli, “Adaptive Control for Systems with Hard Saturation,” in Proceedings of the IEEE Conference on Decision and Control, Piscataway, NJ, Dec. 1975, pp. 841–843. [6] J. S. Shamma, “Analysis and Design of Gain Scheduled Control Systems,” Ph.D. dissertation, MIT, 1988. [7] W. J. Rugh and J. S. Shamma, “Research on Gain Scheduling,” Automatica, vol. 36, no. 10, pp. 1401–1425, 2000. [8] J. S. Shamma, “Overview of LPV Systems,” in Control of Linear Parameter Varying Systems with Applications, J. Mohammadpour and C. W. Scherer, Eds. New York, NY: Springer, 2012, pp. 3–26. [9] M. Pakmehr, “Towards Verifiable Adaptive Control of Gas Turbine Engines,” Ph.D. dissertation, Georgia Institute of Technology, 2013. [10] M. Pakmehr, N. Fitzgerald, E. Feron, J. S. Shamma, and A. Behbahani, “Gain Scheduled Control of Gas Turbine Engines: Stability and Verification,” ASME Journal of Engineering for Gas Turbines and Power, vol. 136, no. 3, 2014. [11] M. Pakmehr, N. Fitzgerald, E. Feron, J. Shamma, and A. Behbahani, “Gain Scheduling Control of Gas Turbine Engines: Stability by Computing a Single Quadratic Lyapunov Function,” in Proceedings of the ASME Turbo Expo 2013, San Antonio, TX, June 2013. [12] E. Feron, “From Control Systems to Control Software,” IEEE Control Systems Magazine, vol. 30, no. 6, pp. 50–71, Dec. 2010. [13] D. J. Stilwell and W. J. Rugh, “Stability Preserving Interpolation Methods for the Synthesis of Gain Scheduled Controllers,” Automatica, vol. 36, no. 5, pp. 665–671, 2000. [14] J. L¨ofberg, “YALMIP: A Toolbox for Modeling and Optimization in MATLAB,” in Proceedings of the CACSD Conference, Taipei, Taiwan, 2004, uRL: http://users.isy.liu.se/johanl/yalmip. [15] J. F. Sturm, O. Romanko, and I. Plik, “SeDuMi (Self-DualMinimization): A MATLAB Toolbox for Optimization over Symmetric Cones,” 2001, uRL: http://sedumi.ie.lehigh.edu. [16] P. Miotto, “Fixed Structure Methods for Flight Control Analysis and Automated Gain Scheduling,” Ph.D. dissertation, MIT, 1997. [17] Lavretsky, E., and Gibson, T., and Annaswamy, A. M., “Projection Operator in Adaptive Systems,” arXiv:1112.4232v6, 16 Oct. 2012. [18] J. Pomet and L. Praly, “Adaptive Nonlinear Regulation: Estimation From the Lyapunov equation,” IEEE Transactions on Automatic Control, vol. 37, no. 6, pp. 729–740, 1992. [19] Pakmehr, M., and Yucelen, T., “Model Reference Adaptive Control of Systems with Gain Scheduled Reference Models,” arXiv:1403.3738v1, 15 Mar. 2014. [20] M. Pakmehr, N. Fitzgerald, E. Feron, J. Paduano, and A. Behbahani, “Physics-Based Dynamic Modeling of a Turboshaft Engine Driving a Variable Pitch Propeller,” submitted to the AIAA Journal of Propulsion and Power, 2013.
Fuel and prop pitch angle control inputs (u(t))
Fig. 5. Integral gain matrix elements for the gain scheduling controller ˆ i (t)) (Ki (α)), and for the adaptive controller (K
Fig. 6. Nonzero elements of the augmented adaptive parameter for the saturated system (K∆ (t))
V. C ONCLUSIONS Using convex optimization tools, a single quadratic Lyapunov function was computed, which guarantees the stability of the gain scheduled gas turbine engine reference model. Stability analysis was performed for the developed adaptive 696
