IMECE NEW APPROACH OF TRACKING CONTROL FOR A CLASS OF NON-MINIMUM PHASE LINEAR SYSTEMS

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1 Proceedings of IMECE 27 ASME International Mechanical Engineering Congress and Exposition November -5, 27, Seattle, Washington,USA, USA IMECE NEW APPROACH OF TRACKING CONTROL FOR A CLASS OF NON-MINIMUM PHASE LINEAR SYSTEMS Bo Xie Intelligent and Precision Control Laboratory School of Mechanical Engineering Purdue University W.Lafayette, Indiana 4797 Tel: (37) , FAX: (765) bxie@purdue.edu Bin Yao Intelligent and Precision Control Laboratory School of Mechanical Engineering Purdue University W.Lafayette, Indiana 4797 Tel: (765) , FAX: (765) byao@purdue.edu ABSTRACT The paper presents a new tracking control approach for a class of non-minimum phase linear systems. The proposed approach consists of two parts: trajectory planning and tracking controller design. The trajectory planning is solved as an optimization problem to improve the achievable transient performance under the fundamental constraints associated with perfect tracking of non-minimum phase systems. The recently proposed adaptive robust tracking controller for a class of non-minimum phase systems is then applied to guarantee that the tracking error dynamics can be stabilized with bounded internal states. The effectiveness of the proposed approach is illustrated through simulation on tracking control of a second order non-minimum phase linear system. Further works are underway to extend the proposed control strategy and trajectory design to a class of nonminimum phase nonlinear systems. Introduction In the past two decades, a great deal of effort has been devoted to the development of global or semi-global analysis and design of nonlinear feedback controller. Recent results include the notion and properties of input-to-state stability [4], the methods of adaptive output feedback control [6], the approaches for THE WORK IS SUPPORTED IN PART BY THE US NATIONAL SCI- ENCE FOUNDATION GRANT NO. CMS-656. Address all correspondence to this author. nonlinear adaptive control with backstepping [7], the adaptive robust control [7], and nonlinear adaptive robust observer design []. A good survey paper [2] reviewed the constructive nonlinear control development from the historical point of view and predicted the trend of the constructive procedure in the nonlinear control design. Most of the current control designs have been focusing on minimum phase nonlinear systems, in which the internal dynamics are stable. In [9], the original nonlinear system is transformed into two subsystems via coordination transformations and the controller design then focuses on the linear subsystem via input-output linearization. Tracking control of nonlinear systems has been one of the most challenging problems in the control community due to its practical importance and the theoretical difficulties in solving the problem, especially when dealing with non-minimum phase systems whose internal dynamics are unstable. An interesting dynamic inversion method [2] and its modified version [2] have been proposed for tracking problems of non-minimum phase nonlinear systems. The main drawback is that it is non-causal as the future information is needed to compute the desired trajectories. Another approach proposed in [22] [23] needs to solve the partial differential equation to find the center manifold of the composite system including the socalled exo-system. How to solve the partial differential equation is still a challenging problem in the computation field. In [24], a robust output feedback stabilization controller is developed for a class of non-minimum phase nonlinear systems. Copyright c 27 by ASME

2 Motivated by the fundamental limitation on the feedback control of non-minimum phase systems [], the unstable internal dynamics is stabilized first when the output is viewed as the virtual control input. The subsequent controller design is then to use the backstepping procedure to ensure the error between the desired output and the virtual control input and its derivatives converge to zeros asymptotically. With the input-to-state stability concept, the resulting overall control law guarantees that the control input and the internal signals are bounded and the system is stabilized even with un-structural uncertainties. In [2], the state feedback adaptive robust tracking controller design is proposed for a class of non-minimum phase nonlinear systems. One strong assumption there is that there exists an analytical solution to the internal dynamics. This paper is to investigate possible ways to relax this assumption and to demonstrate how to achieve that on the tracking control of a class of nonminimum phase linear systems. The paper is organized as follows. Section 2 gives the motivation from the cheap control strategy for the non-minimum phase linear systems. Section 3 applies the cascade control approach to design the tracking controller. Section 4 proposes an optimal trajectory design approach to minimize the maximum undershoot when the control effort is limited. The proposed control scheme are illustrated through the simulation of a second order non-minimum phase linear system in Section 5. 2 Stabilization of Second Order Non-minimum Phase Linear Systems For a non-minimum phase linear system with relative degree equal to r and transfer function give by G(s) = b + b s + + b n r s n r a + a s + + a n s n r + s n = n(s) d(s) in which b n r, and n(s) and d(s) are coprime, it is well known that the system can be transformed into the following normal form via coordinate transformation ζ i = ζ i+, i n r ζ n r = b n r ( b ζ b n r ζ n r ) + b n r x ẋ i = x i+, i r n r ẋ r = i= y = x c i ζ i + r i= d i x i + b n r u where c i and d i depend on a i and b i. () In this section, tracking control of a simple second order non-minimum phase linear system is used to illustrate the motivation to the proposed approach. Specifically, consider a second order linear system transformable into the cascade form ż = a z + b y ẏ = a z + b y + bu where z,y,u R and a,a,b,b R. b, and b, and a >. As the zero of the transfer function from control input u to output y is at a, which is unstable, the above system is of non-minimum phase. The objective of the controller design is to find the state feedback stabilization controller for the dynamical system (2) which minimizes the functional J = 2 Z (2) [y 2 + ε 2 u 2 ]dt (3) where < ε <<. It is well known from the view of the cheap optimal control [] that the minimum energy to stabilize the system (2) is equivalent to the minimum of the functional J = 2 Z y 2 dt (4) while the output y is viewed as the control input to stabilize the zero dynamics It is easy to know that ż = a z + b y (5) J = a b 2 z2 (6) with control input y = 2a b z where z is the initial condition of state z. In other words, the smallest achievable 2-norm of the output y is equal to the least amount control energy needed to stabilize the unstable internal dynamics []. Thus, the original stabilization problem is decomposed into two steps, one that stabilizes the unstable internal dynamics with the virtual control α(z) and the other that regulates y α(z) to zero quickly. 2 Copyright c 27 by ASME

3 3 Regulation and Tracking Control for Nonminimum Phase Linear Systems In order to illustrate the issues on the feedback control design for non-minimum phase linear systems, let us consider the following second order system, where the control input is designed as u =u f + u s =ẏ d K ζ ( (K ζ ) ζ + z) K z z (4) ζ =ζ + x ẋ =u y =x where u is the control input, y is the output, ζ and x are measurable states. The control objective is to design a tracking controller such that the system is stable and the output y = x tracks the desired trajectory y d (t) as close as possible. In the regulation problem, the desired trajectory y d (t) is constant, i.e., y d (t) = y d, t > where y d is a known constant. For the above simple linear dynamics, it is easy to verify that ζ d (t) = y d, t > is a bounded solution for the internal dynamics. Define Then the error dynamics of ζ becomes (7) ζ = ζ ζ d (8) ζ =ζ + x ζ d y d = ζ + x y d (9) Design virtual controller of α for x y d as where K ζ >. Define the error variable z as α = K ζ ζ () z = x y d α () The error dynamics of the internal states becomes ζ = (K ζ ) ζ + z (2) Then the control input u is designed to stabilize the dynamics of z. The error dynamics of z is ż =ẋ ẏ d α ( ζ) =u ẏ d + K ζ ( (K ζ ) ζ + z) = K z z (3) From (), x y d =z + α =z (K ζ ) ζ (5) From (2), the ζ dynamics satisfies the input-to-state stability with respect to z. And from (3), z = is globally uniformly asymptotically stable. So ( ζ,z) = (,) is globally uniformly asymptotically stable. What is the limitation of the 2-norm of the error signal e = y y d when the control effort is free? In [], it is shown that this limitation can be simply and completely characterized by the number and locations of the unstable zeros. Z e 2 (t)dt = (6) 2 since only one open-loop zeros c = is located in the open right half plane. The initial condition is (ζ(),x()) = (,). In the frequency domain, the feedback design limitations are well known through the Bode Integral on the complementary sensitivity [28]. Let T ( jω) be the complementary sensitivity function for the closed-loop system. Then, Z ln T ( jω) dω + = (7) π ω2 2k v where T () =, k v is the velocity constant of the open loop transfer function which is defined [28] as dt (s) = lim k v s ds From (6) and (7), it is interesting to note that the integral value of the T in the Bode integral constraint is equal to the least amount of energy to stabilize the unstable internal dynamics [], i.e., Z ln T ( jω) dω + = Z e 2 (t)dt π ω2 2k v 2 3 Copyright c 27 by ASME

4 4 Optimal Trajectory Design Approach with Input Constraint In the tracking control design approach, the controller is designed first and then motion path is computed through dynamical programming approach. The proposed controller can guarantee the system will converge to zero globally asymptotically. The transient performance, however, may not be acceptable in practical situation. In industry application, the maximum allowable undershoot is specified and the control effort is limited. In this section, the optimal trajectory is designed to minimize the maximum undershoot with control input limitation. 4. Motivation for motion planning So In the above controller design, the error dynamics is: z(t) =z()e K zt ζ = (K ζ ) ζ + z ż = K z z ζ(t) = ζ()e (K ζ )t + Z t x(t) =z(t) + y d (t) (K ζ ) ζ(t) The initial condition: ζ() =ζ() ζ d () e (K ζ )(t τ) z()e K zτ dτ z() =x() y d () + K ζ ζ() In practice, x() = ;y d () =. The undershoot may be too large to be applied in the industrial application such as gantry system. 4.2 Optimal Trajectory Design In order to minimize the maximum allowable undershoot for point-to-point motion, the problem is formulated as follows: min(y u ) = min( max t T ( y d(t))) (8) such that ζ d = ζ d + y d ẏ d = u y d () =,y d (T ) = ζ d (T ) = y u y d (t) + y o ẏ d (t) v max u(t) u max where y u is the maximum allowable undershoot amplitude, and y o is the maximum allowable overshoot amplitude. Let us suppose y u and y o are the same. It makes sense when the physical system follows point-to-point trajectory for most industrial system. The initial feasible solution for the optimal problem is determined through the traditional approach. Given the maximal velocity and acceleration, y d (t) can be computed through B-spline approaches. For illustration purpose, let me assume that y d (t) and ζ d (t) is computed as y d (t) = { (( T t )3 5( T t )4 + 6( T t )5 )y d t T y d = const t > T (9) The internal state ζ d (t) is derived via the backward integration with the initial condition ζ d (T ) = y d (T ). ζ d (t) = { ζ d (T )e t T R T t e t τ y d (τ)dτ t T ζ d = const t > T (2) 5 Simulation In this section, two cases are simulated to illustrate the issues associated with trajectory planning for regulation and tracking control for nonminimum phase linear systems. Case : In this case, both the desired output y d (t) and internal state ζ d (t) are smooth functions without optimization. t < 2 y d (t) = ( t 2 2 )3 5( t 2 2 )4 + 6( t 2 2 )5 2 t 4 t > 4 t < 2 ζ d (t) = e t 4 R T t e t τ y d (τ)dτ 2 t 4 t > 4 (2) (22) 4 Copyright c 27 by ASME

5 Table. Undershoot and Control Effort Comparison.5 desired trajectory: yd approach Optimal approach y u.25.9 u max Case 2: In this case, both the desired output y d (t) and internal state ζ d (t) are optimized to reduced the maximum undershoot while the control effort is limited. From above table, the maximum undershoot amplitude and control input is reduced 28% and 72%. It is claimed that the transient performance is improved through the proposed optimal trajectory design for desired output trajectory. From Figure, the desired trajectory in the optimized case has small undershoot. Then it is possible for the desired internal state ζ to be zero at the beginning. From error dynamics, the error becomes small and then the control effort is also small. When the control effort is reduced, it is also possible to reduce the settling time while the control effort and the motion velocity are still within the specification Figure. Desired trajectory actual trajectory: x 6 Conclusion This paper presents a new approach to design track control for nonminimum phase linear system with input constraint. With a new desired path, the control effort and the maximum undershoot amplitude is reduced significantly. Simulation results demonstrated favorably the effectiveness of proposed approach. Further work will be focused to extend the proposed control scheme for a class of nonminimum phase nonlinear systems such as classical inverted pendulum. REFERENCES [] M.Seron, J.Braslavsky, P.Kokotovic, Feedback limitation in nonlinear systems: from bode integrals to cheap control, IEEE Trans.Automat.Contr, vol.44, no.4, pp , 999. [2] P.Kokotovic, and M.Arcak, Constructive nonlinear control: a historical perspective, Automatica,vol.37, pp , 2. [3] M.Krstic, I.Kanellakopoulos, and P.Kokotovic, Nonlinear and adaptive control design, John Wiely and Sons Inc., New York, 995. [4] E.Sontag, and Y.Wang, New characteristics of input-tostate stability,ieee Trans.Automat.Contr, vol.4, no.9, pp , Figure 2. Actual trajectory [5] A.Isidori, A tool for semiglobal stabilization of uncertain nonminimum phase nonlinear systems via output feedback, IEEE Trans.Automat.Contr, vol.45, no., pp , 2. [6] R.Marino, and P.Tomei, A class of globally output feedback stabilizable nonlinear nonminimum phase systems, in IEEE Proc. of Conference on Decision and Control, pp , 24. [7] B.Yao, High performance adaptive robust control of nonlinear systems: a general framework and new schemes in Proc. of IEEE Conf. Decision and Control, San Diego, pp , 997. [8] B.Yao, Integrated direct/indirect adaptive robust control of siso nonlinear systems in semi-strict form, in Proc. of American Control Conference, pp , 23. [9] B.Yao, and L.Xu, Output feedback adaptive robust con- 5 Copyright c 27 by ASME

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