Optimizing Nonlinear Adaptive Control Allocation

A control-Lyapunov approach is used to develop an adaptive optimizing control allocation algorithm for over-actuated mechanical systems where the actuator model is a¢ ne in the uncertain parameters. Uniform global (asymptotic) stability is guaranteed by the control allocation de…ned by the dynamic update laws in combination with an exponentially stable controller. Copyright c IFAC Keywords: Nonlinear systems. Adaptive control. Control allocation. 1. INTRODUCTION Consider the system _ x = f(t; x; ) = f1(t; x) + g(t; x) (1) = h(t; x; u; ) = (t; x; u) (2) where t 0; x 2 R; u 2 R; 2 R; 2 R; and m d r: The constant parameter vector contains parameters of the control allocation model (actuator and e¤ector model), that will be viewed as uncertain parameters to be adapted. Assume there exist a virtual control c = k(t; x) that uniformly exponentially stabilizes the equilibrium of (1). Introducing an instantaneous cost function J(t; x; u), the minimization problem min u J(t; x; u) s:t c (t; x; u)̂ = 0 (3) de…nes the nonlinear static control allocation problem. Since is an unknown parameter the idea is to use an indirect certainty equivalence adaptive control approach based on the estimate ̂: The cost function J incorporates objectives such as minimum power consumption and input constraints (implemented as barrier functions). Optimizing control allocation solutions have been derived for certain classes of over-actuated systems, such as for aircraft and marine vessels, (Enns, 1998), (Bu¢ ngton et al., 1998), (Sørdalen, 1997), (Bodson, 2002) and (Härkegård, 2002). The control allocation problem is generally viewed as 1 Supported by the Research Council of Norway, project 159556/I30 2 Also with SINTEF. Supported by the European Commission STREP project CEMACS. a static or quasi-dynamic problem that is solved independently of the dynamic control problem considering non-adaptive linear models = Gu. The main advantage of this is modularity and the ability to handle redundancy and constraints. In the present work we consider dynamic nonlinear adaptive optimal control allocation. Nonadaptive nonlinear control allocation has been recently studied using conventional nonlinear programming methods (Johansen et al., 2004). In (Johansen, 2004) a control Lyapunov function is used to derive an exponentially convergent dynamic update law for u (similar to a gradient/Newton-like optimization) such that the control allocation problem (3) is solved dynamically. It is shown that it is not necessary to solve the optimization problem (3) exactly at each time instant. It is shown that convergence and asymptotic optimality of the dynamic control allocation in combination with a uniform globally exponentially stable trajectory-tracking nonlinear controller guarantees uniform boundedness and uniform global exponential convergence of the system. One advantage of this approach is computational e¢ ciency, since the optimizing control allocation algorithm is implemented explicitly as a dynamic nonlinear controller. Solving (3) explicitly at each sampling instant requires a computationally more expensive numerical solution of a nonlinear program to guarantee optimality. In the present work we extend the results and ideas in (Johansen, 2004) with the introduction of setstability and adaptation in the control allocation model. 2. PARAMETER ADAPTATION IN THE ALLOCATION EQUATION The …rst order optimality conditions for the Lagrangian l(t; x; u; ; ̂) = J(t; x; u) + ( c (t; x; u)̂) de…nes local solutions to the optimizing control allocation problem (3). The design of the optimizing control allocation and adaptation laws are based on the following adaptive optimizing control Lyapunov function (aoclf) V1(t; x; ; ̂; u; ) = V0(t; x) + 1 2 ~ Q ~ + 1 2 Q + 1 2 @l @u @l @u + @l @ @l @ (4) where = x x̂ (5) _̂ x = f1(t; x) +A (x x̂) + g(t; x) (t; x; u)̂ (6) > 0 is an arbitrary constant, ~ = ̂ and the design matrices satis…es A = A > 0, Q = Q > 0 and Q = Q T > 0: The …rst term in (4) contains the Lyapunov function inherited from the exponential stable virtual controller: Assumption 1. There exists a di¤erentiable function V0 : [0;1) R ! R and positive constants c1; c2; c3; c4 such that 8t 0 c1 kxk V0(t; x) c2 kxk @V0 @t (t; x) + @V T 0 @x (t; x)f(t; x; k(t; x)) c3 kxk @V0 @x (t; x) c4 kxk The last term in (4) incorporates the …rst order optimality condition for the Lagrangian as in (Johansen, 2004). The second term is standard extension of the Lyapunov function for the certainty equivalence approach (Krstic et al., 1995), while the third term is introduced to make ̂ = ̂ ! ; such that ! c which will be shown to support the convergence of ~ ! 0 as t ! 1: The time derivative of V1 along the trajectories of the system is given by _ V1 = @V0 @t (t; x) + @V T 0 @x (t; x)f(t; x; k(t; x))